[{"Name":"Lectures And Exercise","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Intro to Waves","Duration":"11m 25s","ChapterTopicVideoID":11988,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/11988.jpeg","UploadDate":"2018-05-12T10:15:43.3370000","DurationForVideoObject":"PT11M25S","Description":null,"MetaTitle":"Intro to Waves: Video + Workbook | Proprep","MetaDescription":"Waves - Lectures And Exercise. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/high-school-physics/waves/lectures-and-exercise/vid12456","VideoComments":[],"Subtitles":[],"ID":12456},{"Watched":false,"Name":"Transverse and Longitudinal Waves","Duration":"7m 29s","ChapterTopicVideoID":11989,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.715","Text":"Hello, in this lesson we\u0027re going to be focusing on learning different definitions,"},{"Start":"00:05.715 ","End":"00:07.995","Text":"such as the definition of"},{"Start":"00:07.995 ","End":"00:13.450","Text":"a 1-dimensional wave and also of transverse and longitudinal waves."},{"Start":"00:13.700 ","End":"00:18.240","Text":"First, let\u0027s speak about what a transverse wave is."},{"Start":"00:18.240 ","End":"00:20.220","Text":"In a transverse wave,"},{"Start":"00:20.220 ","End":"00:26.010","Text":"the disturbance is perpendicular to the wave\u0027s direction of travel."},{"Start":"00:26.010 ","End":"00:31.935","Text":"An example of the transverse wave is what we saw in the previous video,"},{"Start":"00:31.935 ","End":"00:36.210","Text":"where we had some string attached to a wall."},{"Start":"00:36.210 ","End":"00:38.580","Text":"It could be a whip, for instance,"},{"Start":"00:38.580 ","End":"00:42.380","Text":"and we could see that the particles in the string or in"},{"Start":"00:42.380 ","End":"00:47.405","Text":"the rope moved up and down in the y-direction."},{"Start":"00:47.405 ","End":"00:53.250","Text":"However, the direction of travel of the wave itself was in this direction,"},{"Start":"00:53.250 ","End":"00:54.835","Text":"in the x-direction,"},{"Start":"00:54.835 ","End":"00:58.625","Text":"so the disturbance is in the y-direction,"},{"Start":"00:58.625 ","End":"01:00.260","Text":"but the wave\u0027s direction of travel,"},{"Start":"01:00.260 ","End":"01:03.420","Text":"we can see, is in the x-direction."},{"Start":"01:04.430 ","End":"01:08.360","Text":"If this is the direction,"},{"Start":"01:08.360 ","End":"01:10.580","Text":"if we take a particle over here,"},{"Start":"01:10.580 ","End":"01:14.830","Text":"so this is the direction of the disturbance,"},{"Start":"01:14.830 ","End":"01:19.370","Text":"but the wave is moving in this direction."},{"Start":"01:19.370 ","End":"01:24.410","Text":"We can see that the angle between these 2 arrows is 90 degrees;"},{"Start":"01:24.410 ","End":"01:29.840","Text":"they\u0027re perpendicular, so that is why this wave is a transverse wave."},{"Start":"01:29.840 ","End":"01:33.005","Text":"Another example of the transverse wave"},{"Start":"01:33.005 ","End":"01:36.155","Text":"is the other example that we saw in the previous video,"},{"Start":"01:36.155 ","End":"01:44.164","Text":"which was where we had our water or a pond and then we dropped a pebble into the center,"},{"Start":"01:44.164 ","End":"01:51.905","Text":"which formed these circular waves around the pebble,"},{"Start":"01:51.905 ","End":"01:57.245","Text":"where each line that I\u0027m drawing is the peak of the wave."},{"Start":"01:57.245 ","End":"02:02.090","Text":"Here we can see that the water particles were moving up and"},{"Start":"02:02.090 ","End":"02:08.330","Text":"down in the direction above the water level and below the water level,"},{"Start":"02:08.330 ","End":"02:11.840","Text":"so perpendicular to the water\u0027s surface."},{"Start":"02:11.840 ","End":"02:20.150","Text":"However, the waves themselves were moving out radially in this direction,"},{"Start":"02:20.150 ","End":"02:27.185","Text":"so moving out in a perpendicular direction to the z-direction, if you will."},{"Start":"02:27.185 ","End":"02:30.755","Text":"That\u0027s why this example that we saw also in the previous lesson"},{"Start":"02:30.755 ","End":"02:34.140","Text":"is also a transverse wave."},{"Start":"02:34.140 ","End":"02:41.330","Text":"1 example of a longitudinal wave is if we take some spring or"},{"Start":"02:41.330 ","End":"02:44.930","Text":"slinky and we hold it on 1 end and"},{"Start":"02:44.930 ","End":"02:48.910","Text":"the other end is attached to a wall or someone else\u0027s holding the other end,"},{"Start":"02:48.910 ","End":"02:51.440","Text":"and if you move your hand backwards and then"},{"Start":"02:51.440 ","End":"02:55.685","Text":"forwards to stretch the slinky and then bring your hand back,"},{"Start":"02:55.685 ","End":"02:57.740","Text":"or in another way,"},{"Start":"02:57.740 ","End":"03:00.170","Text":"to your hand forward and then backwards,"},{"Start":"03:00.170 ","End":"03:04.010","Text":"in other words, to compress the slinky and then back,"},{"Start":"03:04.010 ","End":"03:11.495","Text":"what you\u0027ll see is that the coils of the slinky or of the spring will clump up together,"},{"Start":"03:11.495 ","End":"03:14.900","Text":"so there\u0027ll be an area of higher density,"},{"Start":"03:14.900 ","End":"03:17.555","Text":"and then they\u0027ll loosen up,"},{"Start":"03:17.555 ","End":"03:20.540","Text":"and then we\u0027ll get another compression again,"},{"Start":"03:20.540 ","End":"03:23.650","Text":"and so on and so forth."},{"Start":"03:24.350 ","End":"03:28.475","Text":"What we can see is that the disturbance,"},{"Start":"03:28.475 ","End":"03:30.859","Text":"which are these compressions,"},{"Start":"03:30.859 ","End":"03:32.790","Text":"so this is,"},{"Start":"03:32.790 ","End":"03:36.090","Text":"in actual fact, the disturbance."},{"Start":"03:36.090 ","End":"03:41.185","Text":"We can see that it\u0027s parallel to the wave\u0027s direction of travel."},{"Start":"03:41.185 ","End":"03:44.890","Text":"This is all working on 1 dimension and"},{"Start":"03:44.890 ","End":"03:48.200","Text":"there\u0027s lots of different compressions and they\u0027re happening in"},{"Start":"03:48.200 ","End":"03:51.860","Text":"the same plane or in the same direction as the direction of"},{"Start":"03:51.860 ","End":"03:56.740","Text":"travel of the disturbances themselves,"},{"Start":"03:57.080 ","End":"03:59.375","Text":"so that\u0027s 1 example."},{"Start":"03:59.375 ","End":"04:04.805","Text":"Another example is if we have a row of dominoes."},{"Start":"04:04.805 ","End":"04:07.895","Text":"Each of these lines represents a domino,"},{"Start":"04:07.895 ","End":"04:12.955","Text":"and then let\u0027s imagine that I tip 1 of the dominoes over."},{"Start":"04:12.955 ","End":"04:14.820","Text":"It will begin falling in"},{"Start":"04:14.820 ","End":"04:22.325","Text":"this direction and it will slowly knock down all of these dominoes."},{"Start":"04:22.325 ","End":"04:26.390","Text":"We can see that the disturbance is going in"},{"Start":"04:26.390 ","End":"04:33.665","Text":"this rightwards direction and also the transfer of energy,"},{"Start":"04:33.665 ","End":"04:35.570","Text":"the wave propagation itself,"},{"Start":"04:35.570 ","End":"04:39.515","Text":"is also going in that same direction."},{"Start":"04:39.515 ","End":"04:44.970","Text":"Pushing over dominoes is another example of a longitudinal wave."},{"Start":"04:44.990 ","End":"04:51.080","Text":"Another example to a longitudinal wave, are sound waves."},{"Start":"04:51.080 ","End":"04:55.205","Text":"For instance, if I\u0027m talking,"},{"Start":"04:55.205 ","End":"05:03.430","Text":"I\u0027m over here and you are over here and I\u0027m busy talking."},{"Start":"05:03.430 ","End":"05:07.875","Text":"There\u0027s lots and lots of different particles in the air."},{"Start":"05:07.875 ","End":"05:11.000","Text":"All of these messy dots are the air particles."},{"Start":"05:11.000 ","End":"05:15.350","Text":"When I talk, I\u0027m forming some kind of pulse,"},{"Start":"05:15.350 ","End":"05:21.615","Text":"which pushes these particles slightly."},{"Start":"05:21.615 ","End":"05:26.480","Text":"Then these particles move a little bit in this direction,"},{"Start":"05:26.480 ","End":"05:30.850","Text":"therefore, pushing these particles and so on and so forth."},{"Start":"05:30.850 ","End":"05:34.760","Text":"Then we get some kind of compression like we saw in the slinky."},{"Start":"05:34.760 ","End":"05:38.645","Text":"You can imagine this exact same thing happening,"},{"Start":"05:38.645 ","End":"05:40.954","Text":"just instead of coils and a slinky,"},{"Start":"05:40.954 ","End":"05:44.090","Text":"it\u0027s where the particles in the air and slowly, slowly,"},{"Start":"05:44.090 ","End":"05:49.250","Text":"the sound waves move across via compressions,"},{"Start":"05:49.250 ","End":"05:53.670","Text":"all the way until they reach your ear."},{"Start":"05:53.950 ","End":"05:58.070","Text":"Sound waves are also an example of longitudinal waves because"},{"Start":"05:58.070 ","End":"06:02.675","Text":"their disturbance is parallel to the wave\u0027s direction of travel."},{"Start":"06:02.675 ","End":"06:07.385","Text":"Another way that you can maybe see this and imagine this more is,"},{"Start":"06:07.385 ","End":"06:10.543","Text":"for example, when you hear the Doppler effect."},{"Start":"06:10.543 ","End":"06:13.939","Text":"If some race car drives past,"},{"Start":"06:13.939 ","End":"06:18.980","Text":"you\u0027ll notice that the sound that the race car makes changes in tone;"},{"Start":"06:18.980 ","End":"06:21.595","Text":"it becomes a slightly higher pitch."},{"Start":"06:21.595 ","End":"06:25.770","Text":"Why is that? That\u0027s exactly due to the longitudinal wave,"},{"Start":"06:25.770 ","End":"06:30.350","Text":"so the sound waves are moving from the race car as it travels towards you."},{"Start":"06:30.350 ","End":"06:34.190","Text":"The sound waves are traveling towards your ear and the race car is moving"},{"Start":"06:34.190 ","End":"06:38.270","Text":"so fast and still producing the sound waves as it travels"},{"Start":"06:38.270 ","End":"06:46.640","Text":"towards you that the new sound waves that it\u0027s producing by its continued motion is"},{"Start":"06:46.640 ","End":"06:50.900","Text":"creating this kind of pulse or compression on"},{"Start":"06:50.900 ","End":"06:56.343","Text":"the sound waves that have already left the race car from when it was further away."},{"Start":"06:56.343 ","End":"06:59.780","Text":"Then these compressions make you hear the sound of"},{"Start":"06:59.780 ","End":"07:04.830","Text":"the race car as a higher pitch or at a higher tone."},{"Start":"07:04.830 ","End":"07:08.585","Text":"You can really hear the effects of"},{"Start":"07:08.585 ","End":"07:15.210","Text":"this longitudinal wave and of the compression of the air particles."},{"Start":"07:16.130 ","End":"07:23.240","Text":"I hope that this was a good introduction to transverse and longitudinal waves,"},{"Start":"07:23.240 ","End":"07:27.020","Text":"and these 2 definitions are important to remember."},{"Start":"07:27.020 ","End":"07:29.880","Text":"That\u0027s the end of this lesson."}],"ID":12457},{"Watched":false,"Name":"Basic Definitions","Duration":"14m 55s","ChapterTopicVideoID":11990,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"Hello. In this lesson we\u0027re going to be going a little bit"},{"Start":"00:03.420 ","End":"00:07.215","Text":"more into detail about transverse waves."},{"Start":"00:07.215 ","End":"00:17.595","Text":"Let\u0027s imagine that we choose this point over here and this point is the origin 0."},{"Start":"00:17.595 ","End":"00:20.940","Text":"When our rope is flat,"},{"Start":"00:20.940 ","End":"00:27.915","Text":"this is our direction of the axis and this is the x-direction."},{"Start":"00:27.915 ","End":"00:33.240","Text":"If let\u0027s say I wanted to know the position of"},{"Start":"00:33.240 ","End":"00:38.550","Text":"this particle and let\u0027s say that this particle is located,"},{"Start":"00:38.550 ","End":"00:46.235","Text":"let\u0027s say that this is 1.5 meters away from where I define the origin."},{"Start":"00:46.235 ","End":"00:50.293","Text":"Then if I wanted to speak about the position,"},{"Start":"00:50.293 ","End":"00:53.180","Text":"this is something that they can ask"},{"Start":"00:53.180 ","End":"00:58.490","Text":"you about wave so the position of each particle and a wave."},{"Start":"00:58.490 ","End":"01:01.940","Text":"Then I could say if I was speaking about this particle,"},{"Start":"01:01.940 ","End":"01:08.045","Text":"that its position is at x is equal to 1.5 meters."},{"Start":"01:08.045 ","End":"01:13.255","Text":"If let\u0027s say I wanted to look at a particle 3 meters away."},{"Start":"01:13.255 ","End":"01:15.280","Text":"Let\u0027s say that\u0027s over here,"},{"Start":"01:15.280 ","End":"01:19.825","Text":"3 meters, that means I want to look at this particle."},{"Start":"01:19.825 ","End":"01:23.210","Text":"If I was asking for the position of this particle,"},{"Start":"01:23.210 ","End":"01:29.470","Text":"I\u0027d say that its position is at x is equal to 3 meters."},{"Start":"01:29.740 ","End":"01:34.565","Text":"The next thing that we can be asked when dealing with a transverse wave"},{"Start":"01:34.565 ","End":"01:40.200","Text":"is about particle displacement or displacement amplitude."},{"Start":"01:40.280 ","End":"01:43.460","Text":"What is this particle displacement?"},{"Start":"01:43.460 ","End":"01:49.170","Text":"It\u0027s the distance a particle has moved from its point of equilibrium."},{"Start":"01:50.390 ","End":"01:56.990","Text":"All right. This particle displacement is simply just the amplitude of the wave."},{"Start":"01:56.990 ","End":"02:00.650","Text":"Let\u0027s draw another axis going in this direction and let\u0027s call this"},{"Start":"02:00.650 ","End":"02:06.145","Text":"the y-axis and let\u0027s say that this height over here is 0."},{"Start":"02:06.145 ","End":"02:12.170","Text":"So then if I wanted to know the amplitude or"},{"Start":"02:12.170 ","End":"02:18.226","Text":"the displacement amplitude or the particle displacement of this point over here at 1.5,"},{"Start":"02:18.226 ","End":"02:28.320","Text":"what I would write is that y at this point x which is equal to 1.5 meters,"},{"Start":"02:28.320 ","End":"02:31.820","Text":"so that\u0027s this point over here, is equal to,"},{"Start":"02:31.820 ","End":"02:39.380","Text":"and then I could just draw a line and see what value lies here."},{"Start":"02:39.380 ","End":"02:46.376","Text":"Let\u0027s just say that this is equal to 0.4 meters,"},{"Start":"02:46.376 ","End":"02:48.695","Text":"this value over here."},{"Start":"02:48.695 ","End":"02:50.870","Text":"I can also ask,"},{"Start":"02:50.870 ","End":"02:53.015","Text":"what is my amplitude?"},{"Start":"02:53.015 ","End":"02:57.585","Text":"y at x is equal to 3 meters."},{"Start":"02:57.585 ","End":"03:02.765","Text":"So x is equal to 3 meters is talking about this particle over here."},{"Start":"03:02.765 ","End":"03:06.260","Text":"I can see that my amplitude over here,"},{"Start":"03:06.260 ","End":"03:09.625","Text":"this particle\u0027s amplitude, is 0."},{"Start":"03:09.625 ","End":"03:16.470","Text":"Then let\u0027s say I wanted to look at this particle right at the top over here."},{"Start":"03:16.960 ","End":"03:27.240","Text":"Let\u0027s say that it is located at position 1.1 meters. It doesn\u0027t really matter."},{"Start":"03:27.240 ","End":"03:30.670","Text":"Then I could see that the amplitude is its height,"},{"Start":"03:30.670 ","End":"03:33.805","Text":"its deviation away from the point of equilibrium."},{"Start":"03:33.805 ","End":"03:36.350","Text":"Which if I look over here,"},{"Start":"03:36.350 ","End":"03:44.035","Text":"I can say that the amplitude of my x when x is equal to 1.1 meters,"},{"Start":"03:44.035 ","End":"03:47.005","Text":"which is relating to this particle is,"},{"Start":"03:47.005 ","End":"03:53.795","Text":"we can say, 0.5 meters."},{"Start":"03:53.795 ","End":"03:55.465","Text":"It doesn\u0027t really matter."},{"Start":"03:55.465 ","End":"03:57.220","Text":"I\u0027m just making up these values,"},{"Start":"03:57.220 ","End":"03:58.660","Text":"but just so that you see what it is."},{"Start":"03:58.660 ","End":"04:02.195","Text":"Because we can see that the displacement of this point"},{"Start":"04:02.195 ","End":"04:07.320","Text":"over here is greater than the displacement of this point over here."},{"Start":"04:08.390 ","End":"04:13.700","Text":"What we learned now was the particle displacement or the displacement amplitude."},{"Start":"04:13.700 ","End":"04:19.100","Text":"The next term that we\u0027re going to learn is the term amplitude."},{"Start":"04:19.100 ","End":"04:22.820","Text":"The amplitude, not to be confused,"},{"Start":"04:22.820 ","End":"04:27.125","Text":"although it\u0027s similar with the particle displacement or the displacement amplitude,"},{"Start":"04:27.125 ","End":"04:31.093","Text":"is the maximum, let\u0027s underline that,"},{"Start":"04:31.093 ","End":"04:36.090","Text":"the maximum particle displacement of a wave."},{"Start":"04:36.100 ","End":"04:39.425","Text":"If the particle displacement was the distance"},{"Start":"04:39.425 ","End":"04:42.065","Text":"a particle move from its point of equilibrium,"},{"Start":"04:42.065 ","End":"04:45.965","Text":"the amplitude is that maximum distance."},{"Start":"04:45.965 ","End":"04:48.530","Text":"For instance, we can see that in this wave we have"},{"Start":"04:48.530 ","End":"04:52.355","Text":"a multitude of possible displacements that our particle can be at."},{"Start":"04:52.355 ","End":"04:53.677","Text":"It can be here,"},{"Start":"04:53.677 ","End":"04:55.659","Text":"slightly higher, higher, higher,"},{"Start":"04:55.659 ","End":"04:59.030","Text":"then it reaches this peak and then it can be over here, lower,"},{"Start":"04:59.030 ","End":"05:00.061","Text":"lower, lower,"},{"Start":"05:00.061 ","End":"05:04.230","Text":"lower until it returns to its point of equilibrium at 0."},{"Start":"05:04.490 ","End":"05:10.438","Text":"Those are the possible particle displacements or possible displacement amplitudes."},{"Start":"05:10.438 ","End":"05:12.910","Text":"All of these little black dots over here."},{"Start":"05:12.910 ","End":"05:18.200","Text":"The amplitude is the maximum particle displacement."},{"Start":"05:18.200 ","End":"05:24.575","Text":"Over here, the maximum particle displacement would be this red point over here."},{"Start":"05:24.575 ","End":"05:27.859","Text":"The point that is right at the peak, the highest,"},{"Start":"05:27.859 ","End":"05:34.080","Text":"the greatest possible displacement that it can be at from its point of equilibrium."},{"Start":"05:34.670 ","End":"05:37.980","Text":"This is the amplitude."},{"Start":"05:37.980 ","End":"05:41.390","Text":"The amplitude of a wave can only be 1 value"},{"Start":"05:41.390 ","End":"05:45.005","Text":"and it has the maximum value of the displacement."},{"Start":"05:45.005 ","End":"05:48.990","Text":"It\u0027s how high the peak of the wave is at."},{"Start":"05:49.850 ","End":"05:52.320","Text":"Like we saw in the previous diagrams,"},{"Start":"05:52.320 ","End":"05:59.435","Text":"we know that this wave is going to carry on moving along in this x direction."},{"Start":"05:59.435 ","End":"06:05.810","Text":"The peak of the wave is going to remain at this height, at this amplitude."},{"Start":"06:05.810 ","End":"06:07.526","Text":"The peak is always going to be here,"},{"Start":"06:07.526 ","End":"06:11.200","Text":"what we said, 0.5 meters."},{"Start":"06:11.200 ","End":"06:14.255","Text":"This is going to always be the peak."},{"Start":"06:14.255 ","End":"06:17.165","Text":"The peak, as the wave moves along,"},{"Start":"06:17.165 ","End":"06:22.160","Text":"the peak is going to travel along in the x direction,"},{"Start":"06:22.160 ","End":"06:25.400","Text":"maintaining this value of"},{"Start":"06:25.400 ","End":"06:30.235","Text":"0.5 meters which is just the random value that I gave here for this example."},{"Start":"06:30.235 ","End":"06:33.950","Text":"But the peak is going to stay at this height and it\u0027s just"},{"Start":"06:33.950 ","End":"06:37.550","Text":"going to travel along the x-axis."},{"Start":"06:37.550 ","End":"06:40.280","Text":"That is the amplitude,"},{"Start":"06:40.280 ","End":"06:44.320","Text":"the maximum displacement of the wave."},{"Start":"06:44.320 ","End":"06:47.450","Text":"Now a little note the amplitude,"},{"Start":"06:47.450 ","End":"06:51.125","Text":"which I\u0027ve squared in red so that you remember what this term means."},{"Start":"06:51.125 ","End":"06:56.630","Text":"The amplitude is always the absolute value of a maximum displacement."},{"Start":"06:56.630 ","End":"07:00.500","Text":"If we had again our axis,"},{"Start":"07:00.500 ","End":"07:03.416","Text":"here is our y and here is our x,"},{"Start":"07:03.416 ","End":"07:05.555","Text":"just scroll down a little bit,"},{"Start":"07:05.555 ","End":"07:10.820","Text":"and then let\u0027s say that I had a wave that instead of"},{"Start":"07:10.820 ","End":"07:16.505","Text":"traveling like in our example here in the positive y direction,"},{"Start":"07:16.505 ","End":"07:20.720","Text":"it went in the negative y direction like so."},{"Start":"07:20.720 ","End":"07:26.168","Text":"We can see that the peak of this wave is here."},{"Start":"07:26.168 ","End":"07:33.270","Text":"Let\u0027s say that this peak is at a value of negative 0.4"},{"Start":"07:33.270 ","End":"07:41.355","Text":"in the y direction and let\u0027s say that it corresponds to a value of 1 in the x-direction."},{"Start":"07:41.355 ","End":"07:44.840","Text":"Then if I wanted to write down its position,"},{"Start":"07:44.840 ","End":"07:50.345","Text":"so I would say that its position is at x is equal to 1."},{"Start":"07:50.345 ","End":"07:53.795","Text":"If I wanted to find its particle displacement,"},{"Start":"07:53.795 ","End":"07:58.115","Text":"so I would say that y is at"},{"Start":"07:58.115 ","End":"08:05.490","Text":"x is equal to 1 is equal to negative 0.4."},{"Start":"08:05.490 ","End":"08:07.475","Text":"That\u0027s its particle displacement."},{"Start":"08:07.475 ","End":"08:10.775","Text":"However, if I wanted to know its amplitude,"},{"Start":"08:10.775 ","End":"08:15.665","Text":"so the amplitude of x is equal to 1."},{"Start":"08:15.665 ","End":"08:19.155","Text":"The amplitude is always the absolute value,"},{"Start":"08:19.155 ","End":"08:22.320","Text":"which means that even if I have a negative number,"},{"Start":"08:22.320 ","End":"08:24.905","Text":"I convert it into a positive number."},{"Start":"08:24.905 ","End":"08:30.190","Text":"Instead of writing that its amplitude is negative 0.4 because it\u0027s an absolute value,"},{"Start":"08:30.190 ","End":"08:34.095","Text":"its amplitude is simply 0.4."},{"Start":"08:34.095 ","End":"08:39.950","Text":"The next term that we\u0027re going to be learning about is the phase velocity."},{"Start":"08:39.950 ","End":"08:45.455","Text":"The phase velocity is how fast the peak of the wave propagates."},{"Start":"08:45.455 ","End":"08:49.400","Text":"Let\u0027s imagine that we have the same diagram from before."},{"Start":"08:49.400 ","End":"08:53.710","Text":"A hand holding some kind of rope and they flick"},{"Start":"08:53.710 ","End":"08:59.015","Text":"the rope and then we have this peak over here."},{"Start":"08:59.015 ","End":"09:06.185","Text":"Here we have our y-axis and here we have our x-axis."},{"Start":"09:06.185 ","End":"09:09.525","Text":"Then we know that this wave,"},{"Start":"09:09.525 ","End":"09:14.620","Text":"this peak, is going to be moving along as we saw before."},{"Start":"09:14.900 ","End":"09:18.690","Text":"Let\u0027s draw the wave in red."},{"Start":"09:18.690 ","End":"09:24.040","Text":"A second later here we\u0027ll have a flat or sometime"},{"Start":"09:24.040 ","End":"09:30.520","Text":"later and the peak would have moved to this section over here and carry on this way."},{"Start":"09:30.520 ","End":"09:35.930","Text":"Then again, even later,"},{"Start":"09:35.930 ","End":"09:41.065","Text":"the peak of the wave will be here and like so."},{"Start":"09:41.065 ","End":"09:46.400","Text":"We can see that the peak of the wave is propagating, it\u0027s traveling forward."},{"Start":"09:46.400 ","End":"09:49.880","Text":"The phase velocity is to see how long it"},{"Start":"09:49.880 ","End":"09:54.070","Text":"takes for the peak to move from 1 place to another."},{"Start":"09:54.070 ","End":"09:55.825","Text":"Or in 1 second,"},{"Start":"09:55.825 ","End":"09:59.445","Text":"how far this peak is going to move."},{"Start":"09:59.445 ","End":"10:04.625","Text":"Let\u0027s say we start off from this point over here."},{"Start":"10:04.625 ","End":"10:06.970","Text":"This is point number 1,"},{"Start":"10:06.970 ","End":"10:09.550","Text":"let\u0027s say and our peak is right over here."},{"Start":"10:09.550 ","End":"10:12.845","Text":"Then we wait another second."},{"Start":"10:12.845 ","End":"10:18.105","Text":"Now our peak has moved to point number 3."},{"Start":"10:18.105 ","End":"10:22.275","Text":"Between moving from point 1 to point 3,"},{"Start":"10:22.275 ","End":"10:24.725","Text":"1 second has gone past."},{"Start":"10:24.725 ","End":"10:31.700","Text":"What we can do is we can see that the phase velocity of the wave, so v,"},{"Start":"10:31.700 ","End":"10:35.985","Text":"is simply going to be equal to the distance traveled,"},{"Start":"10:35.985 ","End":"10:38.170","Text":"so the change in position,"},{"Start":"10:38.170 ","End":"10:42.215","Text":"so Delta x divided by the change in time,"},{"Start":"10:42.215 ","End":"10:44.800","Text":"the time taken to travel this distance."},{"Start":"10:44.800 ","End":"10:46.600","Text":"Here, the Delta x,"},{"Start":"10:46.600 ","End":"10:53.190","Text":"so it\u0027s the final position minus the initial position divided by the time it\u0027s taken."},{"Start":"10:53.190 ","End":"10:59.500","Text":"We said that the time taken traveling from here to here was 1 second."},{"Start":"10:59.500 ","End":"11:04.940","Text":"Obviously, all of these values are just to give an example."},{"Start":"11:04.940 ","End":"11:07.935","Text":"Then we divide it by this 1 second."},{"Start":"11:07.935 ","End":"11:12.770","Text":"We can see that this is going to be equal to 2 divided by 1,"},{"Start":"11:12.770 ","End":"11:18.580","Text":"which is simply 2 meters per second,"},{"Start":"11:19.040 ","End":"11:22.045","Text":"if we\u0027re working here in meters."},{"Start":"11:22.045 ","End":"11:23.770","Text":"If we\u0027re working in centimeters,"},{"Start":"11:23.770 ","End":"11:26.460","Text":"then we\u0027ll be using centimeters here."},{"Start":"11:26.460 ","End":"11:28.510","Text":"That\u0027s the phase velocity."},{"Start":"11:28.510 ","End":"11:32.345","Text":"We\u0027re using the kinematic equation that we know and love,"},{"Start":"11:32.345 ","End":"11:36.570","Text":"which is that velocity is distance over time."},{"Start":"11:36.570 ","End":"11:39.065","Text":"We look at the distance traveled by"},{"Start":"11:39.065 ","End":"11:44.320","Text":"this peak and the time it took to travel that distance,"},{"Start":"11:44.320 ","End":"11:47.610","Text":"and then we have the phase velocity."},{"Start":"11:47.610 ","End":"11:49.795","Text":"Now, just a little point,"},{"Start":"11:49.795 ","End":"11:53.900","Text":"the phase velocity I said is how fast the peak of the wave propagates."},{"Start":"11:53.900 ","End":"11:55.785","Text":"That means that we were measuring"},{"Start":"11:55.785 ","End":"12:00.550","Text":"the different positions of where the peak was and how long it took to get there."},{"Start":"12:00.550 ","End":"12:03.700","Text":"That\u0027s usually how the phase velocity is measured."},{"Start":"12:03.700 ","End":"12:05.970","Text":"However, of course at the same time,"},{"Start":"12:05.970 ","End":"12:10.340","Text":"one could take any point on the wave."},{"Start":"12:10.340 ","End":"12:17.985","Text":"I could take this point over here and then I\u0027d have to find the same point in the wave,"},{"Start":"12:17.985 ","End":"12:19.870","Text":"but a few moments later."},{"Start":"12:19.870 ","End":"12:21.790","Text":"I would have to choose it exactly,"},{"Start":"12:21.790 ","End":"12:23.110","Text":"so let\u0027s say over here,"},{"Start":"12:23.110 ","End":"12:27.785","Text":"which has a lot of the time much harder to find than just finding the peak of the wave,"},{"Start":"12:27.785 ","End":"12:30.230","Text":"which is relatively easy to measure."},{"Start":"12:30.230 ","End":"12:35.320","Text":"But theoretically you could use even these 2 points and then you find"},{"Start":"12:35.320 ","End":"12:39.340","Text":"the velocity of this point moving from"},{"Start":"12:39.340 ","End":"12:43.670","Text":"this position over here to this position over here."},{"Start":"12:43.670 ","End":"12:46.330","Text":"Of course, you can use any point on the wave."},{"Start":"12:46.330 ","End":"12:48.910","Text":"You could also use the right at the bottom of"},{"Start":"12:48.910 ","End":"12:51.820","Text":"the wave and then find right at the bottom of this wave,"},{"Start":"12:51.820 ","End":"12:54.755","Text":"and then find the time frame between"},{"Start":"12:54.755 ","End":"12:59.780","Text":"this point being at this position and then it being at this position over here."},{"Start":"12:59.780 ","End":"13:02.555","Text":"You\u0027re just doing the same thing that we did before by"},{"Start":"13:02.555 ","End":"13:05.165","Text":"just using different points on the wave."},{"Start":"13:05.165 ","End":"13:06.725","Text":"You can also do that."},{"Start":"13:06.725 ","End":"13:09.860","Text":"Usually people use the peak,"},{"Start":"13:09.860 ","End":"13:13.285","Text":"but you can use every other point on the wave as long as"},{"Start":"13:13.285 ","End":"13:17.415","Text":"if you choose this point on the wave at this time,"},{"Start":"13:17.415 ","End":"13:22.030","Text":"then you have to use the exact same point on the wave a few moments later."},{"Start":"13:22.030 ","End":"13:26.350","Text":"You can get confused because then your velocity won\u0027t be accurate."},{"Start":"13:26.490 ","End":"13:29.050","Text":"That is the end of the lesson."},{"Start":"13:29.050 ","End":"13:30.180","Text":"A quick little recap."},{"Start":"13:30.180 ","End":"13:34.850","Text":"The position is just the x-coordinate of"},{"Start":"13:34.850 ","End":"13:41.235","Text":"a certain particle of the wave so that we can know which particle we\u0027re referencing."},{"Start":"13:41.235 ","End":"13:43.865","Text":"We also learned about the particle displacement or"},{"Start":"13:43.865 ","End":"13:47.700","Text":"another way of saying particle displacement is the displacement amplitude,"},{"Start":"13:47.700 ","End":"13:50.625","Text":"which is the distance the particle has moved."},{"Start":"13:50.625 ","End":"13:54.135","Text":"In the case that we\u0027re speaking of transverse waves,"},{"Start":"13:54.135 ","End":"13:55.470","Text":"which is this lesson,"},{"Start":"13:55.470 ","End":"14:01.280","Text":"so the distance the particles move perpendicularly from its point of equilibrium."},{"Start":"14:01.280 ","End":"14:05.290","Text":"The distance it\u0027s moved in this direction or"},{"Start":"14:05.290 ","End":"14:11.705","Text":"this direction if the wave is moving along the x-axis."},{"Start":"14:11.705 ","End":"14:14.940","Text":"Then we learned about the term amplitude,"},{"Start":"14:14.940 ","End":"14:17.935","Text":"which is the maximum particle displacement."},{"Start":"14:17.935 ","End":"14:20.670","Text":"We learned that it\u0027s always a positive value,"},{"Start":"14:20.670 ","End":"14:22.665","Text":"so we always take the absolute value."},{"Start":"14:22.665 ","End":"14:25.885","Text":"Even if our peak is in the negative y-direction,"},{"Start":"14:25.885 ","End":"14:32.479","Text":"our amplitude is going to be the absolute value of that position."},{"Start":"14:32.479 ","End":"14:35.600","Text":"We learned about the phase velocity."},{"Start":"14:35.600 ","End":"14:40.565","Text":"That is just how fast a specific point on the wave is traveling."},{"Start":"14:40.565 ","End":"14:42.785","Text":"Usually we measure it from the peak."},{"Start":"14:42.785 ","End":"14:45.940","Text":"It\u0027s important to remember this equation which is"},{"Start":"14:45.940 ","End":"14:52.955","Text":"the basic kinematic equation of velocity is the distance traveled in the time taken."},{"Start":"14:52.955 ","End":"14:55.850","Text":"That\u0027s the end of this lesson."}],"ID":12458},{"Watched":false,"Name":"Representing Waves","Duration":"11m 12s","ChapterTopicVideoID":11991,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.175","Text":"Hello. In this lesson,"},{"Start":"00:02.175 ","End":"00:07.185","Text":"we\u0027re going to learn how to mathematically represent a wave."},{"Start":"00:07.185 ","End":"00:12.585","Text":"Let\u0027s look at this picture over here,"},{"Start":"00:12.585 ","End":"00:17.909","Text":"where a hand is up and moving back down and will cause"},{"Start":"00:17.909 ","End":"00:24.480","Text":"a wave or a pulse to move through this string or rope over here."},{"Start":"00:24.480 ","End":"00:29.730","Text":"What we want to do is we want to find our size y."},{"Start":"00:29.730 ","End":"00:39.130","Text":"This is our y-axis and this is our x-axis."},{"Start":"00:39.200 ","End":"00:45.245","Text":"Now we\u0027re going to see that we need 2 variables in order to"},{"Start":"00:45.245 ","End":"00:51.970","Text":"define the y position of the rope or of the wave."},{"Start":"00:51.970 ","End":"00:58.910","Text":"Let\u0027s see, the first is the position on the x-axis."},{"Start":"00:58.910 ","End":"01:03.095","Text":"We can see that if we choose this x position,"},{"Start":"01:03.095 ","End":"01:09.050","Text":"our y value is going to be some positive number."},{"Start":"01:09.050 ","End":"01:11.585","Text":"If we choose this x position,"},{"Start":"01:11.585 ","End":"01:14.540","Text":"we can see that our y is now different."},{"Start":"01:14.540 ","End":"01:18.710","Text":"It\u0027s changed and here specifically it\u0027s 0."},{"Start":"01:18.710 ","End":"01:28.775","Text":"We can see that our value for y is dependent on our x values."},{"Start":"01:28.775 ","End":"01:33.755","Text":"However, that was specifically at this exact moment."},{"Start":"01:33.755 ","End":"01:36.035","Text":"But what if I move down the timeline?"},{"Start":"01:36.035 ","End":"01:39.485","Text":"I was looking at this point over here,"},{"Start":"01:39.485 ","End":"01:46.140","Text":"and I move down the timeline to a different time a few seconds later."},{"Start":"01:46.180 ","End":"01:49.850","Text":"Then, if I look over here and again,"},{"Start":"01:49.850 ","End":"01:54.690","Text":"I have my y-axis going in this direction and my x-axis going in this."},{"Start":"01:54.690 ","End":"02:01.595","Text":"Here I can see that at t is equal to 0 or this time over here,"},{"Start":"02:01.595 ","End":"02:05.460","Text":"I could see that I had some amplitude."},{"Start":"02:05.460 ","End":"02:09.740","Text":"It was some deviation away from the point of equilibrium."},{"Start":"02:09.740 ","End":"02:12.470","Text":"But over here, a few moments later,"},{"Start":"02:12.470 ","End":"02:22.535","Text":"I can see that at the exact same x position but at a different time,"},{"Start":"02:22.535 ","End":"02:29.885","Text":"my deviation away from the point of equilibrium is now 0."},{"Start":"02:29.885 ","End":"02:34.580","Text":"If I look at this point over here,"},{"Start":"02:34.580 ","End":"02:38.180","Text":"which originally I marked as having a deviation from the point of"},{"Start":"02:38.180 ","End":"02:43.800","Text":"equilibrium of 0 and I go down in time,"},{"Start":"02:43.800 ","End":"02:48.455","Text":"so I can see that now suddenly it does have"},{"Start":"02:48.455 ","End":"02:54.930","Text":"some deviation away from the equilibrium point."},{"Start":"02:55.610 ","End":"03:00.170","Text":"Because we can see that this deviation or"},{"Start":"03:00.170 ","End":"03:05.735","Text":"this displacement from the point of equilibrium is different at different times,"},{"Start":"03:05.735 ","End":"03:11.510","Text":"we can see that the amplitude or the particle displacement in"},{"Start":"03:11.510 ","End":"03:18.605","Text":"general is also dependent on the time-frame that we\u0027re looking at."},{"Start":"03:18.605 ","End":"03:22.850","Text":"Now we can see that our y value,"},{"Start":"03:22.850 ","End":"03:24.650","Text":"so the value that we\u0027re trying to measure,"},{"Start":"03:24.650 ","End":"03:25.925","Text":"this height over here,"},{"Start":"03:25.925 ","End":"03:28.355","Text":"is dependent on 2 variables."},{"Start":"03:28.355 ","End":"03:36.250","Text":"It\u0027s dependent on the position of the particle and on the time."},{"Start":"03:36.250 ","End":"03:42.425","Text":"Now what we have here is a function with 2 variables."},{"Start":"03:42.425 ","End":"03:45.950","Text":"This is a little bit difficult to work with."},{"Start":"03:45.950 ","End":"03:50.825","Text":"Again, it\u0027s difficult to solve questions with a function that has 2 variables."},{"Start":"03:50.825 ","End":"03:56.520","Text":"What we\u0027re going to do is we\u0027re going to split up this 1 function into"},{"Start":"03:56.520 ","End":"04:03.030","Text":"2 different functions where each 1 is dependent on 1 of the variables."},{"Start":"04:03.710 ","End":"04:09.200","Text":"The first function that we\u0027re going to create is the position displacement function."},{"Start":"04:09.200 ","End":"04:12.770","Text":"This describes the particle displacement for"},{"Start":"04:12.770 ","End":"04:19.985","Text":"every position along the x-axis for a given specific time."},{"Start":"04:19.985 ","End":"04:23.885","Text":"That means that our function over here,"},{"Start":"04:23.885 ","End":"04:27.170","Text":"let\u0027s say it\u0027s y because we\u0027re measuring the amplitude,"},{"Start":"04:27.170 ","End":"04:33.150","Text":"is going to be as a function of the position x."},{"Start":"04:33.220 ","End":"04:40.280","Text":"That means that if we take a snapshot of the wave traveling through the rope,"},{"Start":"04:40.280 ","End":"04:45.945","Text":"we choose a specific time for it."},{"Start":"04:45.945 ","End":"04:49.115","Text":"Then we look at every single x position"},{"Start":"04:49.115 ","End":"04:52.610","Text":"at that specific moment of time that\u0027s been frozen."},{"Start":"04:52.610 ","End":"04:56.085","Text":"We look at the particle displacements."},{"Start":"04:56.085 ","End":"04:59.355","Text":"We look at the y value."},{"Start":"04:59.355 ","End":"05:05.030","Text":"Let\u0027s imagine that we\u0027re looking at this snapshot over here."},{"Start":"05:05.160 ","End":"05:11.815","Text":"We\u0027re looking at this picture right over here."},{"Start":"05:11.815 ","End":"05:15.130","Text":"Here we have our time frozen,"},{"Start":"05:15.130 ","End":"05:22.105","Text":"so here we can say t is equal to 2, let\u0027s say."},{"Start":"05:22.105 ","End":"05:26.330","Text":"This is a specific moment in time."},{"Start":"05:26.330 ","End":"05:34.940","Text":"Now again we have a y-axis going like so and our x-axis going like so."},{"Start":"05:34.940 ","End":"05:38.200","Text":"Then for every single point along the x-axis,"},{"Start":"05:38.200 ","End":"05:42.820","Text":"we can define 1 y value."},{"Start":"05:42.820 ","End":"05:47.284","Text":"Here at this value of x,"},{"Start":"05:47.284 ","End":"05:51.575","Text":"we can see that our y displacement is 0,"},{"Start":"05:51.575 ","End":"05:52.970","Text":"at this value of x,"},{"Start":"05:52.970 ","End":"05:55.115","Text":"our y displacement is again 0."},{"Start":"05:55.115 ","End":"05:56.510","Text":"At this value of x,"},{"Start":"05:56.510 ","End":"05:59.105","Text":"we have some positive y displacement."},{"Start":"05:59.105 ","End":"06:01.489","Text":"At this value of x,"},{"Start":"06:01.489 ","End":"06:06.140","Text":"we have our peak y displacements, so our amplitude."},{"Start":"06:06.140 ","End":"06:07.670","Text":"At this value of x,"},{"Start":"06:07.670 ","End":"06:12.230","Text":"we still have a positive y value of positive particle displacement,"},{"Start":"06:12.230 ","End":"06:13.580","Text":"but it\u0027s slightly lower."},{"Start":"06:13.580 ","End":"06:24.690","Text":"Then back to every single point or position on the x-axis having a y displacement of 0."},{"Start":"06:26.870 ","End":"06:30.565","Text":"Now we\u0027re going to speak about the second split,"},{"Start":"06:30.565 ","End":"06:34.510","Text":"which is the time displacement function."},{"Start":"06:34.510 ","End":"06:38.785","Text":"Here we are speaking about our y values."},{"Start":"06:38.785 ","End":"06:46.030","Text":"Our particle displacement as a function of t this time and only t."},{"Start":"06:46.030 ","End":"06:50.365","Text":"The time displacement function describes the particle displacement of"},{"Start":"06:50.365 ","End":"06:56.835","Text":"a specific position for all time values."},{"Start":"06:56.835 ","End":"07:01.480","Text":"Here we\u0027re looking at 1 specific point,"},{"Start":"07:01.480 ","End":"07:04.990","Text":"we\u0027re isolating that point and we\u0027re looking at"},{"Start":"07:04.990 ","End":"07:12.780","Text":"that specific point in our medium\u0027s position as time goes by."},{"Start":"07:12.780 ","End":"07:18.215","Text":"Here we\u0027re just freezing our x value and just looking at time as a variable."},{"Start":"07:18.215 ","End":"07:22.025","Text":"Whereas in number 1, we froze our time variable and"},{"Start":"07:22.025 ","End":"07:26.265","Text":"we\u0027re only looking at our x variable."},{"Start":"07:26.265 ","End":"07:28.515","Text":"How do we do this?"},{"Start":"07:28.515 ","End":"07:29.960","Text":"Now we can say,"},{"Start":"07:29.960 ","End":"07:33.635","Text":"let\u0027s look at a snapshot of a particle."},{"Start":"07:33.635 ","End":"07:36.535","Text":"Now we\u0027re looking at this particle."},{"Start":"07:36.535 ","End":"07:42.295","Text":"Now we\u0027re looking as time goes by what happens to this particle."},{"Start":"07:42.295 ","End":"07:47.174","Text":"Originally, the y-value for this particle is equal to 0. You know what?"},{"Start":"07:47.174 ","End":"07:51.270","Text":"We can even draw a table."},{"Start":"07:51.270 ","End":"07:59.300","Text":"At what time and what particle displacement value y we have."},{"Start":"07:59.300 ","End":"08:03.865","Text":"I\u0027m just reminding you that this"},{"Start":"08:03.865 ","End":"08:10.020","Text":"is the y-direction and this is our x-direction."},{"Start":"08:10.020 ","End":"08:12.705","Text":"These are just different time frames."},{"Start":"08:12.705 ","End":"08:15.300","Text":"Originally at t is equal to 0,"},{"Start":"08:15.300 ","End":"08:16.426","Text":"let\u0027s say that here,"},{"Start":"08:16.426 ","End":"08:19.310","Text":"our y displacement is 0."},{"Start":"08:19.310 ","End":"08:21.020","Text":"Then we\u0027re going to look,"},{"Start":"08:21.020 ","End":"08:24.635","Text":"let\u0027s say the snapshot is for 2 seconds later,"},{"Start":"08:24.635 ","End":"08:26.720","Text":"so t is equal to 2."},{"Start":"08:26.720 ","End":"08:31.060","Text":"We can see that our y position is still 0."},{"Start":"08:31.060 ","End":"08:35.630","Text":"Now let\u0027s say that this is a t is equal to 4 seconds."},{"Start":"08:35.630 ","End":"08:38.795","Text":"We can see again our y displacement is still 0."},{"Start":"08:38.795 ","End":"08:41.780","Text":"Then we\u0027re following the particle down,"},{"Start":"08:41.780 ","End":"08:44.030","Text":"so it\u0027s the exact same particle."},{"Start":"08:44.030 ","End":"08:47.715","Text":"Then we get to this position over here."},{"Start":"08:47.715 ","End":"08:52.526","Text":"Now we can see that a t is equal to 6 seconds,"},{"Start":"08:52.526 ","End":"08:56.090","Text":"we have some positive displacement."},{"Start":"08:56.090 ","End":"08:58.055","Text":"It\u0027s even almost at the peak."},{"Start":"08:58.055 ","End":"09:05.000","Text":"Let\u0027s say that it\u0027s at a particle displacement of 0.9."},{"Start":"09:05.000 ","End":"09:10.010","Text":"Then again, the same particle just at a different time frame."},{"Start":"09:10.010 ","End":"09:12.130","Text":"We get to here, this is,"},{"Start":"09:12.130 ","End":"09:13.900","Text":"let\u0027s say after 8 seconds."},{"Start":"09:13.900 ","End":"09:19.350","Text":"Now we can see that our particles still has some positive displacement,"},{"Start":"09:19.350 ","End":"09:20.580","Text":"but it\u0027s slightly less."},{"Start":"09:20.580 ","End":"09:24.490","Text":"Let\u0027s say it\u0027s approximately 0.2."},{"Start":"09:25.290 ","End":"09:28.990","Text":"This time displacement function is looking at"},{"Start":"09:28.990 ","End":"09:33.295","Text":"the exact same position, specific x position."},{"Start":"09:33.295 ","End":"09:38.380","Text":"Seeing how as a variable of time, this x position,"},{"Start":"09:38.380 ","End":"09:44.275","Text":"that\u0027s still fixed, how the y position changes as time changes."},{"Start":"09:44.275 ","End":"09:52.025","Text":"Whereas before we took 1 picture of a specific time at 4 seconds."},{"Start":"09:52.025 ","End":"09:58.360","Text":"We saw at every single x position at t is equal to 4 seconds."},{"Start":"09:58.360 ","End":"10:03.950","Text":"Every single x position what its y component was."},{"Start":"10:03.950 ","End":"10:08.350","Text":"Now, I can even draw a graph of this information."},{"Start":"10:08.350 ","End":"10:12.310","Text":"This will be a graph of the time displacement function."},{"Start":"10:12.310 ","End":"10:16.915","Text":"Of course, we can also do the same with the position displacement function."},{"Start":"10:16.915 ","End":"10:22.450","Text":"Here I would have my particle displacement and here I would have my time variable."},{"Start":"10:22.450 ","End":"10:26.095","Text":"I will just plot where it is so a t is equal to 0,"},{"Start":"10:26.095 ","End":"10:28.480","Text":"I had y is 0, t is equal to 2,"},{"Start":"10:28.480 ","End":"10:30.910","Text":"y is 0, t is equal to 4,"},{"Start":"10:30.910 ","End":"10:33.295","Text":"y is 0 and then 6,"},{"Start":"10:33.295 ","End":"10:35.460","Text":"it reached some peak."},{"Start":"10:35.460 ","End":"10:38.475","Text":"Let\u0027s draw it here. Then at 8 seconds,"},{"Start":"10:38.475 ","End":"10:40.065","Text":"it was much lower."},{"Start":"10:40.065 ","End":"10:43.850","Text":"Then of course, if you take closer shorter time intervals,"},{"Start":"10:43.850 ","End":"10:46.250","Text":"then you\u0027ll get a much more accurate graph."},{"Start":"10:46.250 ","End":"10:50.000","Text":"Then you can connect the points."},{"Start":"10:50.000 ","End":"10:58.245","Text":"Then we can get a picture of what this specific x-position looks like as time goes by."},{"Start":"10:58.245 ","End":"11:03.620","Text":"Then of course, we can join these 2 equations to get back to"},{"Start":"11:03.620 ","End":"11:10.100","Text":"our original equation of y as a function of both of these variables."},{"Start":"11:10.100 ","End":"11:13.320","Text":"That\u0027s the end of this lesson."}],"ID":12459},{"Watched":false,"Name":"Interference","Duration":"26m 34s","ChapterTopicVideoID":11992,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.635","Text":"Hello. In this lesson we\u0027re going to be speaking about interference."},{"Start":"00:04.635 ","End":"00:09.480","Text":"Now, interference is a phenomenon that waves experience when"},{"Start":"00:09.480 ","End":"00:13.260","Text":"2 or more waves interact with one"},{"Start":"00:13.260 ","End":"00:18.330","Text":"another when they\u0027re traveling to meet at the same point."},{"Start":"00:18.330 ","End":"00:20.684","Text":"Let\u0027s give an example."},{"Start":"00:20.684 ","End":"00:27.890","Text":"Here we have some rope and 1 friend is standing"},{"Start":"00:27.890 ","End":"00:35.435","Text":"at one end and causes a pulse in this end of the rope."},{"Start":"00:35.435 ","End":"00:44.720","Text":"The friend created this pulse and this pulse is traveling in this direction and"},{"Start":"00:44.720 ","End":"00:48.920","Text":"at the exact same time another friend standing at"},{"Start":"00:48.920 ","End":"00:53.645","Text":"the other end of the rope created this pulse,"},{"Start":"00:53.645 ","End":"00:57.650","Text":"which is traveling in the opposite direction."},{"Start":"00:57.650 ","End":"01:03.115","Text":"At some stage these 2 waves are going to meet somewhere in the middle."},{"Start":"01:03.115 ","End":"01:07.430","Text":"The question is, what\u0027s going to happen when these 2 waves"},{"Start":"01:07.430 ","End":"01:12.930","Text":"meet and this is exactly what the topic of interference deals with."},{"Start":"01:13.060 ","End":"01:16.700","Text":"Here\u0027s a slightly better diagram."},{"Start":"01:16.700 ","End":"01:21.440","Text":"What we can see is here we have some wave which is traveling in"},{"Start":"01:21.440 ","End":"01:26.455","Text":"this direction and here we have another wave traveling in this direction."},{"Start":"01:26.455 ","End":"01:28.040","Text":"This is at point 1,"},{"Start":"01:28.040 ","End":"01:29.555","Text":"they\u0027re traveling towards each other."},{"Start":"01:29.555 ","End":"01:32.480","Text":"Here at point 2 we can see that they\u0027re getting"},{"Start":"01:32.480 ","End":"01:36.605","Text":"closer and here at point 3 the 2 waves have met."},{"Start":"01:36.605 ","End":"01:39.830","Text":"We\u0027re going to return to this point to see exactly what is"},{"Start":"01:39.830 ","End":"01:43.250","Text":"happening at this point so I\u0027m not going to explain it yet."},{"Start":"01:43.250 ","End":"01:47.740","Text":"Then we can see at point 4 as if nothing has happened,"},{"Start":"01:47.740 ","End":"01:50.785","Text":"the 2 waves move apart."},{"Start":"01:50.785 ","End":"01:55.135","Text":"We can see that the wave that was originally traveling in the left direction,"},{"Start":"01:55.135 ","End":"01:58.450","Text":"the exact same wave still traveling in"},{"Start":"01:58.450 ","End":"02:02.320","Text":"the left direction as if nothing happened to it and the original wave,"},{"Start":"02:02.320 ","End":"02:04.360","Text":"which was traveling in the right direction,"},{"Start":"02:04.360 ","End":"02:09.505","Text":"the exact same wave traveling still in the right direction, again,"},{"Start":"02:09.505 ","End":"02:15.310","Text":"as if nothing happened to it and at point 5 the waves have just carried on"},{"Start":"02:15.310 ","End":"02:22.340","Text":"moving in their respective directions as defined at point 1 and they just move away."},{"Start":"02:23.240 ","End":"02:28.320","Text":"This is a common or 1 of the properties of waves,"},{"Start":"02:28.320 ","End":"02:32.880","Text":"that after the waves have passed one another,"},{"Start":"02:33.130 ","End":"02:38.765","Text":"the waves continue their original motion undisturbed."},{"Start":"02:38.765 ","End":"02:41.465","Text":"Like we saw here, this original wave,"},{"Start":"02:41.465 ","End":"02:46.350","Text":"exactly the shape traveling in this leftwards direction"},{"Start":"02:46.350 ","End":"02:48.650","Text":"and once the waves pass one another it\u0027s"},{"Start":"02:48.650 ","End":"02:51.800","Text":"the exact same shape still traveling in the leftwards direction."},{"Start":"02:51.800 ","End":"02:56.240","Text":"This is very different to what you learned about let\u0027s say if"},{"Start":"02:56.240 ","End":"03:01.080","Text":"you have some basketball and a tennis ball,"},{"Start":"03:01.080 ","End":"03:07.541","Text":"which you throw together to meet at some point over here,"},{"Start":"03:07.541 ","End":"03:13.445","Text":"you know that after the collision or the meeting between this basketball and tennis ball,"},{"Start":"03:13.445 ","End":"03:17.120","Text":"you know that the tennis ball will bounce in"},{"Start":"03:17.120 ","End":"03:21.430","Text":"the opposite direction as well the basketball."},{"Start":"03:21.430 ","End":"03:26.165","Text":"They\u0027ll interact with one another and then bounce away from one another."},{"Start":"03:26.165 ","End":"03:30.365","Text":"This is very different to how waves work."},{"Start":"03:30.365 ","End":"03:36.750","Text":"Waves just meet and then carry on in their original motion undisturbed."},{"Start":"03:36.830 ","End":"03:40.640","Text":"An experiment that you can do at home to see"},{"Start":"03:40.640 ","End":"03:43.820","Text":"this property of the waves is that if you get"},{"Start":"03:43.820 ","End":"03:48.560","Text":"some green laser and you shine it in"},{"Start":"03:48.560 ","End":"03:54.615","Text":"this direction and a red laser and you shine it this direction,"},{"Start":"03:54.615 ","End":"03:58.460","Text":"even though they meet at this point of contact you\u0027ll see that on"},{"Start":"03:58.460 ","End":"04:03.020","Text":"the wall over here you\u0027ll see a red dot like as if"},{"Start":"04:03.020 ","End":"04:07.700","Text":"this laser wasn\u0027t shone through anything and on a wall over"},{"Start":"04:07.700 ","End":"04:13.120","Text":"here you\u0027ll see a green dot as if this laser wasn\u0027t shone through anything."},{"Start":"04:13.120 ","End":"04:16.865","Text":"This is a property that you have to remember."},{"Start":"04:16.865 ","End":"04:21.005","Text":"Now, let\u0027s speak about what is happening over here."},{"Start":"04:21.005 ","End":"04:27.020","Text":"Let\u0027s speak about what is going on at position 3."},{"Start":"04:27.020 ","End":"04:33.945","Text":"What we can see is that we have some resultant wave."},{"Start":"04:33.945 ","End":"04:38.840","Text":"What has actually happened is that we had our small wave, this one over here,"},{"Start":"04:38.840 ","End":"04:44.660","Text":"which looks approximately like this and then we had this wave over here,"},{"Start":"04:44.660 ","End":"04:48.870","Text":"which is a bit bigger, which looks approximately like this."},{"Start":"04:48.970 ","End":"04:53.315","Text":"What happens is that these 2 waves,"},{"Start":"04:53.315 ","End":"04:54.920","Text":"this one and this one,"},{"Start":"04:54.920 ","End":"05:02.345","Text":"when they meet, are superimposed on one another forming a resultant wave."},{"Start":"05:02.345 ","End":"05:04.980","Text":"What on earth does that mean,"},{"Start":"05:04.980 ","End":"05:06.465","Text":"what did I just say?"},{"Start":"05:06.465 ","End":"05:11.359","Text":"Superposition literally just means addition."},{"Start":"05:11.359 ","End":"05:17.960","Text":"All that we\u0027re doing here is we\u0027re taking the amplitude."},{"Start":"05:17.960 ","End":"05:22.940","Text":"Remember the amplitude is the maximum particle displacement of a wave."},{"Start":"05:22.940 ","End":"05:29.075","Text":"We\u0027re taking this amplitude of this wave over here, so that\u0027s this,"},{"Start":"05:29.075 ","End":"05:34.430","Text":"and then we\u0027re also taking the amplitude over here of the taller wave,"},{"Start":"05:34.430 ","End":"05:37.090","Text":"so that\u0027s this over here."},{"Start":"05:37.090 ","End":"05:40.215","Text":"Then we have 2 sizes."},{"Start":"05:40.215 ","End":"05:45.320","Text":"Let\u0027s say that this amplitude is 0.1 meters"},{"Start":"05:45.320 ","End":"05:51.730","Text":"and the magnitude of this amplitude is 0.5 meters."},{"Start":"05:51.730 ","End":"05:55.070","Text":"We\u0027re literally superimposing these waves,"},{"Start":"05:55.070 ","End":"05:59.515","Text":"which means that we\u0027re just adding up these 2 amplitudes."},{"Start":"05:59.515 ","End":"06:04.400","Text":"Then the total amplitude of this resultant wave,"},{"Start":"06:04.400 ","End":"06:09.305","Text":"so the wave that results from this meeting of these 2 waves,"},{"Start":"06:09.305 ","End":"06:13.346","Text":"is literally going to be equal to 0.1, this one,"},{"Start":"06:13.346 ","End":"06:16.520","Text":"plus 0.5, the amplitude of this one,"},{"Start":"06:16.520 ","End":"06:20.940","Text":"which is equal to 0.6 meters."},{"Start":"06:21.110 ","End":"06:24.829","Text":"We\u0027ve superimposed these 2 waves."},{"Start":"06:24.829 ","End":"06:29.704","Text":"That\u0027s what happens in interference when 2 waves meet, they super impose."},{"Start":"06:29.704 ","End":"06:33.910","Text":"That means we add up their amplitudes."},{"Start":"06:33.910 ","End":"06:37.610","Text":"Here I wrote that the resultant wave is the superposition,"},{"Start":"06:37.610 ","End":"06:43.580","Text":"which simply is a fancy word for the word sum of the interfering waves amplitudes."},{"Start":"06:43.580 ","End":"06:49.790","Text":"Now, notice that because the 2 waves have amplitudes in the same direction,"},{"Start":"06:49.790 ","End":"06:54.564","Text":"both are in this positive y direction,"},{"Start":"06:54.564 ","End":"06:58.490","Text":"we just add them up and we get another positive number."},{"Start":"06:58.490 ","End":"07:02.510","Text":"However, if we had 2 amplitudes in the opposite direction,"},{"Start":"07:02.510 ","End":"07:08.210","Text":"so let\u0027s say this one was 0.5 but this wave went in"},{"Start":"07:08.210 ","End":"07:15.666","Text":"this direction instead and it was just negative 0.1,"},{"Start":"07:15.666 ","End":"07:19.490","Text":"we would again sum as usual and so the sum would be negative"},{"Start":"07:19.490 ","End":"07:27.235","Text":"0.1 plus 0.5 and then we just get that the total amplitude is 0.4."},{"Start":"07:27.235 ","End":"07:35.180","Text":"That\u0027s what happens if the pulse is in the opposite direction, if the 2 pulses."},{"Start":"07:35.180 ","End":"07:39.455","Text":"Interference itself is simply just that."},{"Start":"07:39.455 ","End":"07:45.395","Text":"It\u0027s the resultant wave resulting from the superposition of 2 or more waves."},{"Start":"07:45.395 ","End":"07:47.330","Text":"This is just interference,"},{"Start":"07:47.330 ","End":"07:51.980","Text":"two waves meeting and a resultant or a new wave being"},{"Start":"07:51.980 ","End":"07:58.185","Text":"formed for that moment when they\u0027re meeting an interfering or pattern."},{"Start":"07:58.185 ","End":"07:59.790","Text":"That\u0027s interference."},{"Start":"07:59.790 ","End":"08:05.460","Text":"Now let\u0027s give an example and in this example we\u0027re going to use the square wave."},{"Start":"08:07.200 ","End":"08:15.655","Text":"Here is our line and this line rep represents a time of t=0."},{"Start":"08:15.655 ","End":"08:23.110","Text":"Let\u0027s say that we have a big square wave like this in red and"},{"Start":"08:23.110 ","End":"08:31.880","Text":"another smaller square wave in blue and they\u0027re both traveling towards one another."},{"Start":"08:32.160 ","End":"08:36.505","Text":"Now let\u0027s take the next time interval."},{"Start":"08:36.505 ","End":"08:40.135","Text":"Let\u0027s say some time t1 it doesn\u0027t really matter."},{"Start":"08:40.135 ","End":"08:42.670","Text":"This could be 5 seconds, 10 seconds,"},{"Start":"08:42.670 ","End":"08:47.905","Text":"depending on the velocity of these 2 waves."},{"Start":"08:47.905 ","End":"08:53.080","Text":"Now we can see that they\u0027re moving closer together."},{"Start":"08:53.080 ","End":"08:57.220","Text":"Now imagine that I\u0027m maintaining the same dimensions and"},{"Start":"08:57.220 ","End":"09:01.720","Text":"shape each time and that both of these waves are perfect squares."},{"Start":"09:01.720 ","End":"09:04.900","Text":"Here we can see the waves are still moving towards one"},{"Start":"09:04.900 ","End":"09:08.630","Text":"another but they still haven\u0027t met."},{"Start":"09:09.330 ","End":"09:18.820","Text":"Now let\u0027s take some other random time t=t2 and here the waves are meeting."},{"Start":"09:18.820 ","End":"09:23.500","Text":"Now I\u0027m going to draw this in dots because"},{"Start":"09:23.500 ","End":"09:28.825","Text":"afterwards I want to show you in solid line what the resulted wave looks like."},{"Start":"09:28.825 ","End":"09:37.090","Text":"Let\u0027s imagine that here we have our large red square wave that\u0027s over here."},{"Start":"09:37.090 ","End":"09:43.180","Text":"Then we have, over here they\u0027re literally just meeting now,"},{"Start":"09:43.180 ","End":"09:46.690","Text":"our blue wave and they\u0027ve just meet in here."},{"Start":"09:46.690 ","End":"09:49.075","Text":"Through the idea of superposition,"},{"Start":"09:49.075 ","End":"09:51.625","Text":"what will the resultant wave look like?"},{"Start":"09:51.625 ","End":"09:55.030","Text":"I\u0027m going to draw the resultant wave in green."},{"Start":"09:55.030 ","End":"09:57.955","Text":"The resultant wave is just going to look like this."},{"Start":"09:57.955 ","End":"10:02.380","Text":"We go along the red and then we get to"},{"Start":"10:02.380 ","End":"10:07.510","Text":"this area over here where the red has to be superimposed with the blue."},{"Start":"10:07.510 ","End":"10:10.855","Text":"That means that we have this whole length of the red,"},{"Start":"10:10.855 ","End":"10:15.655","Text":"the amplitude of the red plus this amplitude of the blue."},{"Start":"10:15.655 ","End":"10:20.380","Text":"We\u0027ll add the amplitude of the blue until we get to this place over here"},{"Start":"10:20.380 ","End":"10:24.985","Text":"where we can see that they haven\u0027t met,"},{"Start":"10:24.985 ","End":"10:31.045","Text":"they\u0027re not meeting here anymore and then we just follow this shape."},{"Start":"10:31.045 ","End":"10:34.570","Text":"That is what the superposition means and this is"},{"Start":"10:34.570 ","End":"10:42.175","Text":"our interference wave or our interference at time t2."},{"Start":"10:42.175 ","End":"10:47.990","Text":"Now let\u0027s take a look at time t3."},{"Start":"10:48.000 ","End":"10:52.270","Text":"Here the waves have still met but now"},{"Start":"10:52.270 ","End":"10:56.995","Text":"our blue wave is located right in the center of our red wave."},{"Start":"10:56.995 ","End":"10:58.945","Text":"Let\u0027s draw this."},{"Start":"10:58.945 ","End":"11:07.075","Text":"Here is our red wave and our blue wave is right over here."},{"Start":"11:07.075 ","End":"11:11.575","Text":"Now what is our resultant wave going to look like due to this interference."},{"Start":"11:11.575 ","End":"11:14.170","Text":"We go along the red wave and then we reach"},{"Start":"11:14.170 ","End":"11:17.335","Text":"this position over here where the blue wave is located."},{"Start":"11:17.335 ","End":"11:21.160","Text":"Then we\u0027re going to add up the 2 amplitudes."},{"Start":"11:21.160 ","End":"11:26.770","Text":"We have the amplitude of the red plus the amplitude of the blue which will look like"},{"Start":"11:26.770 ","End":"11:29.245","Text":"this up until we reach"},{"Start":"11:29.245 ","End":"11:34.480","Text":"the end of the blue wave and then we carry on following the red wave and that\u0027s it."},{"Start":"11:34.480 ","End":"11:38.020","Text":"That\u0027s our resultant wave over here."},{"Start":"11:38.020 ","End":"11:41.875","Text":"Of course, now we understand the idea of"},{"Start":"11:41.875 ","End":"11:45.753","Text":"whilst the blue wave is traveling through the red wave,"},{"Start":"11:45.753 ","End":"11:48.880","Text":"we\u0027re just going to carry on adding it up depending on where"},{"Start":"11:48.880 ","End":"11:52.600","Text":"the blue wave is located relative to the red wave."},{"Start":"11:52.600 ","End":"11:55.045","Text":"Then, now let\u0027s take this t4."},{"Start":"11:55.045 ","End":"11:58.045","Text":"Well, our waves have again separated."},{"Start":"11:58.045 ","End":"12:02.230","Text":"Then as if nothing happened we\u0027re going to have our red wave like"},{"Start":"12:02.230 ","End":"12:06.475","Text":"so and our blue wave like so and"},{"Start":"12:06.475 ","End":"12:09.880","Text":"just like before the red wave was moving in the right direction and"},{"Start":"12:09.880 ","End":"12:15.280","Text":"the blue wave was moving in the left direction and that is just how it\u0027s going to look."},{"Start":"12:15.280 ","End":"12:21.520","Text":"These are the 2 resulting waves that we\u0027ve given for these random times."},{"Start":"12:21.520 ","End":"12:25.420","Text":"Obviously, as the wave carriers are moving there are more resultant waves but"},{"Start":"12:25.420 ","End":"12:32.470","Text":"these 2 waves in green are examples of resultant waves and examples of interference."},{"Start":"12:32.470 ","End":"12:36.790","Text":"Interference can be split up into 2 sub-categories,"},{"Start":"12:36.790 ","End":"12:40.735","Text":"constructive interference and destructive interference."},{"Start":"12:40.735 ","End":"12:45.970","Text":"Here are some definitions for constructive and destructive interference."},{"Start":"12:45.970 ","End":"12:51.085","Text":"Definition Number 1 is for a constructive interference."},{"Start":"12:51.085 ","End":"12:56.500","Text":"It says that the displacement amplitude of each wave at a certain point in"},{"Start":"12:56.500 ","End":"12:58.960","Text":"absolute value is smaller than"},{"Start":"12:58.960 ","End":"13:03.595","Text":"the resultant wave\u0027s displacement value at that point and again,"},{"Start":"13:03.595 ","End":"13:08.380","Text":"where the resultant wave\u0027s displacement value is also an absolute value."},{"Start":"13:08.380 ","End":"13:13.525","Text":"What does that mean? That\u0027s just this previous example that we dealt with."},{"Start":"13:13.525 ","End":"13:17.230","Text":"Let\u0027s say that this displacement amplitude is"},{"Start":"13:17.230 ","End":"13:21.850","Text":"1 and let\u0027s say that this displacement amplitude is 2."},{"Start":"13:21.850 ","End":"13:26.710","Text":"We can see that over here the total is equal to,"},{"Start":"13:26.710 ","End":"13:28.120","Text":"we said it\u0027s the sum,"},{"Start":"13:28.120 ","End":"13:30.505","Text":"the superposition, so 1+2."},{"Start":"13:30.505 ","End":"13:34.770","Text":"Here the total displacement amplitude is 3 and we can"},{"Start":"13:34.770 ","End":"13:39.135","Text":"see that 3 is bigger than both 2 and 1."},{"Start":"13:39.135 ","End":"13:42.285","Text":"That means that we have constructive interference."},{"Start":"13:42.285 ","End":"13:46.330","Text":"Now, I alluded to this before but the same"},{"Start":"13:46.330 ","End":"13:51.940","Text":"applies if our waves are on the opposite side of the number line."},{"Start":"13:51.940 ","End":"14:01.645","Text":"Let\u0027s say that my red wave is like so and my blue wave is like so."},{"Start":"14:01.645 ","End":"14:07.330","Text":"Now we can see that the amplitude over here is let\u0027s say it\u0027s the same dimensions but"},{"Start":"14:07.330 ","End":"14:09.970","Text":"here it\u0027s in the negative side of the number line so"},{"Start":"14:09.970 ","End":"14:13.405","Text":"negative 2 and here we have negative 1."},{"Start":"14:13.405 ","End":"14:17.845","Text":"Absolute values of these become 1 and 2 respectively."},{"Start":"14:17.845 ","End":"14:22.075","Text":"Then if we look at the resultant wave,"},{"Start":"14:22.075 ","End":"14:26.725","Text":"it will go like this up until it reaches the line of the blue and then we"},{"Start":"14:26.725 ","End":"14:31.195","Text":"add this amplitude onto the red\u0027s amplitudes."},{"Start":"14:31.195 ","End":"14:34.840","Text":"We add an amplitude of 1 all the way to the end of the wave,"},{"Start":"14:34.840 ","End":"14:39.130","Text":"imagine that this is a square and we carry on."},{"Start":"14:39.130 ","End":"14:44.860","Text":"Now again, we can see that the total amplitude of this or displacement amplitude of"},{"Start":"14:44.860 ","End":"14:52.735","Text":"this green resultant wave is the absolute value of -1+-2."},{"Start":"14:52.735 ","End":"15:00.445","Text":"This is -3 or an absolute value, it\u0027s just 3."},{"Start":"15:00.445 ","End":"15:07.570","Text":"We can see that 3 is bigger than the absolute value of 1 and the absolute value of 2."},{"Start":"15:07.570 ","End":"15:11.305","Text":"Again, we\u0027re dealing with constructive interference."},{"Start":"15:11.305 ","End":"15:14.905","Text":"That\u0027s simply what constructive interference means."},{"Start":"15:14.905 ","End":"15:18.685","Text":"Now let\u0027s look at destructive interference."},{"Start":"15:18.685 ","End":"15:20.950","Text":"This is definition Number 2."},{"Start":"15:20.950 ","End":"15:25.240","Text":"Destructive interference, the displacement amplitude of"},{"Start":"15:25.240 ","End":"15:29.695","Text":"the resultant wave at a specific point,"},{"Start":"15:29.695 ","End":"15:32.080","Text":"and again we\u0027re talking in absolute value,"},{"Start":"15:32.080 ","End":"15:36.160","Text":"is smaller than the displacement amplitude of"},{"Start":"15:36.160 ","End":"15:41.150","Text":"at least 1 of the original waves in absolute value."},{"Start":"15:41.340 ","End":"15:45.530","Text":"Let\u0027s draw out an example."},{"Start":"15:46.230 ","End":"15:52.090","Text":"Let\u0027s say that we have our red wave over"},{"Start":"15:52.090 ","End":"15:59.425","Text":"here and it has the amplitude an absolute value of 2."},{"Start":"15:59.425 ","End":"16:04.510","Text":"Then let\u0027s say that we have the blue wave over"},{"Start":"16:04.510 ","End":"16:09.580","Text":"here and its amplitude in absolute value is 1."},{"Start":"16:09.580 ","End":"16:13.435","Text":"Its amplitude is actually -1 but in absolute value it\u0027s 1."},{"Start":"16:13.435 ","End":"16:16.045","Text":"What will the resultant wave look like?"},{"Start":"16:16.045 ","End":"16:18.590","Text":"Here, we\u0027re going to start."},{"Start":"16:18.590 ","End":"16:25.565","Text":"We have here our red wave until we get to this point over here."},{"Start":"16:25.565 ","End":"16:28.595","Text":"Now we see we have our blue wave."},{"Start":"16:28.595 ","End":"16:34.400","Text":"We can see that our blue wave is a negative so we have to minus,"},{"Start":"16:34.400 ","End":"16:38.195","Text":"we have to subtract this amplitude 1."},{"Start":"16:38.195 ","End":"16:44.740","Text":"We go minus 1 until the end of the blue wave and then we\u0027re back to positive 2."},{"Start":"16:44.740 ","End":"16:48.190","Text":"Here we can see that the amplitude of"},{"Start":"16:48.190 ","End":"16:51.850","Text":"the section that was experiencing interference so that\u0027s"},{"Start":"16:51.850 ","End":"16:59.125","Text":"this section is equal to 2-1 because this is a negative so it\u0027s equal to 1."},{"Start":"16:59.125 ","End":"17:01.555","Text":"We\u0027re adding the amplitude,"},{"Start":"17:01.555 ","End":"17:04.450","Text":"the total amplitude of the interference wave,"},{"Start":"17:04.450 ","End":"17:12.835","Text":"AI amplitude of the interference is equal to 2+-1 which is equal to 1."},{"Start":"17:12.835 ","End":"17:19.150","Text":"We can see that 1 is smaller than the displacement amplitude of"},{"Start":"17:19.150 ","End":"17:26.170","Text":"at least 1 of the original waves because 1 of the original waves has an amplitude of 2."},{"Start":"17:26.170 ","End":"17:30.625","Text":"Our interference amplitude is 1 which is smaller than 2."},{"Start":"17:30.625 ","End":"17:33.890","Text":"This is destructive interference."},{"Start":"17:33.890 ","End":"17:39.305","Text":"Now another example is if we have our red wave"},{"Start":"17:39.305 ","End":"17:47.805","Text":"again and it has again an amplitude of 2 and then we have our blue wave."},{"Start":"17:47.805 ","End":"17:53.090","Text":"Let\u0027s draw it over here and it\u0027s just in the opposite direction but it also has"},{"Start":"17:53.090 ","End":"18:01.770","Text":"an amplitude in absolute value of 2 but it\u0027s in the negative direction so it\u0027s -2."},{"Start":"18:01.770 ","End":"18:06.260","Text":"Now when we\u0027re trying to find the amplitude of"},{"Start":"18:06.260 ","End":"18:09.965","Text":"the interference wave, how are we going to draw that?"},{"Start":"18:09.965 ","End":"18:16.810","Text":"We start here but then we can see that we have +2 -2 so it\u0027s at 0."},{"Start":"18:16.810 ","End":"18:25.615","Text":"We\u0027re always going to be at a height of +2 and dealing with this -2 amplitude over here."},{"Start":"18:25.615 ","End":"18:30.770","Text":"We can see that our interference is going to be completely destructive."},{"Start":"18:30.770 ","End":"18:34.035","Text":"We\u0027re not going to have a wave over here."},{"Start":"18:34.035 ","End":"18:38.570","Text":"The amplitude is 0 of the interference wave."},{"Start":"18:38.570 ","End":"18:47.365","Text":"The amplitude of the interference wave is equal to 2+ -2 which is equal to 0."},{"Start":"18:47.365 ","End":"18:54.990","Text":"Zero is smaller than the displacement amplitudes in absolute value of both of the waves."},{"Start":"18:54.990 ","End":"18:59.850","Text":"That\u0027s constructive and destructive interference where here"},{"Start":"18:59.850 ","End":"19:05.120","Text":"when we get a destructive interference where the amplitude of the resultant wave is 0,"},{"Start":"19:05.120 ","End":"19:08.775","Text":"is called complete destructive interference."},{"Start":"19:08.775 ","End":"19:13.620","Text":"The next term that we\u0027re going to learn about is called the node."},{"Start":"19:13.620 ","End":"19:17.885","Text":"A node is a point where the amplitude of the wave is always"},{"Start":"19:17.885 ","End":"19:22.535","Text":"0 and it\u0027s an example of destructive interference."},{"Start":"19:22.535 ","End":"19:25.740","Text":"Let\u0027s see how this works."},{"Start":"19:26.730 ","End":"19:33.200","Text":"Let\u0027s imagine over here that this is our number line."},{"Start":"19:34.250 ","End":"19:40.540","Text":"Here we have 2 waves moving towards one another."},{"Start":"19:41.090 ","End":"19:45.390","Text":"Our red wave is moving in this direction and"},{"Start":"19:45.390 ","End":"19:50.580","Text":"our blue wave is moving in this direction and these are 2 triangle waves."},{"Start":"19:50.580 ","End":"19:58.140","Text":"Let\u0027s say that this is what the system looks like at t is equal to 0 seconds."},{"Start":"19:59.330 ","End":"20:09.160","Text":"Let\u0027s say that our waves are moving 1^2 every second."},{"Start":"20:09.770 ","End":"20:12.825","Text":"Here we can see as the midpoint,"},{"Start":"20:12.825 ","End":"20:17.460","Text":"which means that we have to wait 4.5 seconds until"},{"Start":"20:17.460 ","End":"20:23.220","Text":"the front of each wave will meet the front of the other wave."},{"Start":"20:23.220 ","End":"20:27.060","Text":"Now let\u0027s draw what happens at that exact moment,"},{"Start":"20:27.060 ","End":"20:31.840","Text":"so t is equal to 4.5 seconds."},{"Start":"20:32.990 ","End":"20:37.245","Text":"Now we can see that both of the waves are meeting at this point."},{"Start":"20:37.245 ","End":"20:42.270","Text":"We can see that here we have an amplitude and we still don\u0027t have any superposition,"},{"Start":"20:42.270 ","End":"20:44.310","Text":"because the waves aren\u0027t overlapping."},{"Start":"20:44.310 ","End":"20:50.955","Text":"The only place that the waves are overlapping is at this point over here at the center,"},{"Start":"20:50.955 ","End":"20:54.270","Text":"so we can see that we\u0027re going to have"},{"Start":"20:54.270 ","End":"20:58.275","Text":"this amplitude that reaches a maximum just before here."},{"Start":"20:58.275 ","End":"21:00.000","Text":"Then at that exact point,"},{"Start":"21:00.000 ","End":"21:05.010","Text":"we\u0027re going to have a 0 amplitude and the same with the red wave."},{"Start":"21:05.010 ","End":"21:07.830","Text":"We have an amplitude which in the absolute value"},{"Start":"21:07.830 ","End":"21:10.440","Text":"is increasing until it reaches this point which"},{"Start":"21:10.440 ","End":"21:16.500","Text":"is overlapping with this point of the blue wave and the amplitude is here going to be 0."},{"Start":"21:16.500 ","End":"21:20.175","Text":"What we\u0027re going to see is a node,"},{"Start":"21:20.175 ","End":"21:24.930","Text":"because this point is going to be 0 throughout for all time,"},{"Start":"21:24.930 ","End":"21:28.230","Text":"for the whole motion of these 2 waves."},{"Start":"21:28.230 ","End":"21:34.750","Text":"Now let\u0027s look at t is equal to 6 seconds."},{"Start":"21:35.480 ","End":"21:38.770","Text":"Let\u0027s see what happens."},{"Start":"21:45.620 ","End":"21:52.305","Text":"The red wave has carried on moving in the left direction and now let\u0027s see what we have."},{"Start":"21:52.305 ","End":"21:56.190","Text":"In this section of the wave,"},{"Start":"21:56.190 ","End":"21:58.605","Text":"up until this point,"},{"Start":"21:58.605 ","End":"22:02.400","Text":"we\u0027re going up and the same here up until we"},{"Start":"22:02.400 ","End":"22:06.554","Text":"reach this point where the blue wave starts overlapping."},{"Start":"22:06.554 ","End":"22:13.890","Text":"Then we can see that here we go up 1 and here we are peak as at negative 2,"},{"Start":"22:13.890 ","End":"22:19.875","Text":"so plus 1 negative 2 will make us at minus 1."},{"Start":"22:19.875 ","End":"22:22.560","Text":"Then we go here and the same over here."},{"Start":"22:22.560 ","End":"22:24.255","Text":"Here we\u0027re minus 1,"},{"Start":"22:24.255 ","End":"22:25.560","Text":"but plus 2,"},{"Start":"22:25.560 ","End":"22:30.135","Text":"so plus 2 minus 1 will take us to 1."},{"Start":"22:30.135 ","End":"22:34.665","Text":"Then our wave begins to look like so."},{"Start":"22:34.665 ","End":"22:39.209","Text":"Now we can see that exactly at this midpoint,"},{"Start":"22:39.209 ","End":"22:41.790","Text":"our original midpoint over here,"},{"Start":"22:41.790 ","End":"22:44.940","Text":"we follow it down that it\u0027s at this point."},{"Start":"22:44.940 ","End":"22:52.380","Text":"We can see that here we go up approximately 1.5 and here we go down approximately 1.5."},{"Start":"22:52.380 ","End":"22:56.505","Text":"We\u0027re assuming that I drew these waves in the same dimensions."},{"Start":"22:56.505 ","End":"23:01.890","Text":"That means plus 1.5 minus 1.5 is 0 again."},{"Start":"23:01.890 ","End":"23:04.260","Text":"We can see that this point,"},{"Start":"23:04.260 ","End":"23:09.150","Text":"even when we\u0027ve moved forward 1.5 seconds is still 0."},{"Start":"23:09.150 ","End":"23:12.090","Text":"Then we can see that here,"},{"Start":"23:12.090 ","End":"23:16.995","Text":"this point is a little bit in absolute value"},{"Start":"23:16.995 ","End":"23:20.010","Text":"larger than this point over here and this point over"},{"Start":"23:20.010 ","End":"23:23.760","Text":"here is a little bit larger in absolute value than this point over here,"},{"Start":"23:23.760 ","End":"23:29.070","Text":"so we get a shape like this when we join our lines."},{"Start":"23:29.070 ","End":"23:33.824","Text":"In black, we can see the shape of the resultant wave at 6 seconds,"},{"Start":"23:33.824 ","End":"23:43.230","Text":"and here we can see our resultant wave of 4.5 seconds."},{"Start":"23:43.230 ","End":"23:49.740","Text":"Now let\u0027s look at our wave at t is equal to 7 seconds."},{"Start":"23:49.740 ","End":"23:53.250","Text":"This is what the 2 waves will look like relative to"},{"Start":"23:53.250 ","End":"23:56.340","Text":"one another at t is equal to 7 seconds."},{"Start":"23:56.340 ","End":"24:00.120","Text":"Here we can see we have 0 minus 2,"},{"Start":"24:00.120 ","End":"24:03.015","Text":"so the first point will be over here."},{"Start":"24:03.015 ","End":"24:06.790","Text":"Then here we have 2 plus 0,"},{"Start":"24:07.640 ","End":"24:10.665","Text":"sorry, the point will be over here."},{"Start":"24:10.665 ","End":"24:15.110","Text":"Then we can draw in this section of the resultant wave."},{"Start":"24:15.110 ","End":"24:18.800","Text":"Then, obviously, there\u0027s inaccuracies in my drawing."},{"Start":"24:18.800 ","End":"24:20.420","Text":"These aren\u0027t perfect triangles,"},{"Start":"24:20.420 ","End":"24:24.170","Text":"but they\u0027re meant to represent the same dimension and be the exact same shape,"},{"Start":"24:24.170 ","End":"24:25.715","Text":"just in opposite form."},{"Start":"24:25.715 ","End":"24:27.980","Text":"What we can see, even in this diagram,"},{"Start":"24:27.980 ","End":"24:30.325","Text":"is that at the same time over here,"},{"Start":"24:30.325 ","End":"24:36.015","Text":"where up until now this point has had an amplitude of 0."},{"Start":"24:36.015 ","End":"24:38.370","Text":"We can see that over here it also cancels out."},{"Start":"24:38.370 ","End":"24:43.200","Text":"We go up approximately 1^2 and here we go down approximately 1^2,"},{"Start":"24:43.200 ","End":"24:47.160","Text":"so here is also 0."},{"Start":"24:47.160 ","End":"24:51.240","Text":"Then we can just connect the dots and then"},{"Start":"24:51.240 ","End":"24:54.975","Text":"we can see what our resultant wave will look like."},{"Start":"24:54.975 ","End":"24:57.135","Text":"Then obviously as time goes by,"},{"Start":"24:57.135 ","End":"24:59.835","Text":"our blue wave will carry on moving like this,"},{"Start":"24:59.835 ","End":"25:04.110","Text":"looking as its original fashion and the red wave will"},{"Start":"25:04.110 ","End":"25:09.400","Text":"carry on moving like so and will carry on looking exactly like this."},{"Start":"25:10.160 ","End":"25:14.550","Text":"What\u0027s important to see here is that at all of these points,"},{"Start":"25:14.550 ","End":"25:16.890","Text":"the same position, 0,"},{"Start":"25:16.890 ","End":"25:19.650","Text":"let\u0027s call it the origin 0,"},{"Start":"25:19.650 ","End":"25:24.960","Text":"it always has an amplitude of 0 as we can see."},{"Start":"25:24.960 ","End":"25:27.810","Text":"We can say that it\u0027s y amplitude,"},{"Start":"25:27.810 ","End":"25:34.560","Text":"so the y position of this coordinate 0 is always 0."},{"Start":"25:34.560 ","End":"25:39.270","Text":"Of course, also when the waves have moved past each other,"},{"Start":"25:39.270 ","End":"25:44.020","Text":"the amplitude of this point will still be 0."},{"Start":"25:44.570 ","End":"25:48.540","Text":"This is exactly what a node is."},{"Start":"25:48.540 ","End":"25:53.640","Text":"This point is always with an amplitude of 0 and it\u0027s caused by"},{"Start":"25:53.640 ","End":"26:01.350","Text":"this destructive interference as we saw, so this happens."},{"Start":"26:01.350 ","End":"26:04.260","Text":"This node is generated by"},{"Start":"26:04.260 ","End":"26:09.660","Text":"a very specific type of symmetry between the 2 waves that we won\u0027t go into now,"},{"Start":"26:09.660 ","End":"26:12.570","Text":"because it\u0027s not important for this course."},{"Start":"26:12.570 ","End":"26:14.640","Text":"Maybe in future courses,"},{"Start":"26:14.640 ","End":"26:17.220","Text":"this will be an important point for you to learn,"},{"Start":"26:17.220 ","End":"26:21.930","Text":"but just know that it\u0027s true to a symmetry between the 2 waves."},{"Start":"26:21.930 ","End":"26:25.845","Text":"The definition of a node or what a node exactly is,"},{"Start":"26:25.845 ","End":"26:27.705","Text":"is very important to remember,"},{"Start":"26:27.705 ","End":"26:31.530","Text":"so I\u0027ve highlighted this in red."},{"Start":"26:31.530 ","End":"26:34.600","Text":"That\u0027s the end of this lesson."}],"ID":12460},{"Watched":false,"Name":"Wave Characteristics","Duration":"30m 45s","ChapterTopicVideoID":11993,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.889","Text":"Hello. In this lesson,"},{"Start":"00:01.889 ","End":"00:05.280","Text":"we\u0027re going to be learning about the characteristics of waves."},{"Start":"00:05.280 ","End":"00:08.519","Text":"The first characteristic is that the wave speed,"},{"Start":"00:08.519 ","End":"00:12.044","Text":"which in a previous lesson we referred to as the phase velocity,"},{"Start":"00:12.044 ","End":"00:17.410","Text":"depends only on the properties of the transmission medium."},{"Start":"00:17.660 ","End":"00:22.620","Text":"If we have a pulse traveling through some transmission medium,"},{"Start":"00:22.620 ","End":"00:27.240","Text":"be it through a rope or water or whatever it will be,"},{"Start":"00:27.240 ","End":"00:30.735","Text":"irrespective of the frequency of the wave,"},{"Start":"00:30.735 ","End":"00:34.170","Text":"how many peaks it has or its size,"},{"Start":"00:34.170 ","End":"00:36.510","Text":"its width or its amplitude,"},{"Start":"00:36.510 ","End":"00:40.679","Text":"its speed, how fast or how quickly it propagates through"},{"Start":"00:40.679 ","End":"00:43.729","Text":"this transmission medium depends only on"},{"Start":"00:43.729 ","End":"00:47.649","Text":"the transmission medium and not on the wave itself."},{"Start":"00:47.649 ","End":"00:54.660","Text":"Let\u0027s imagine that we have a rope and we send in 2 different size pulses through it."},{"Start":"00:54.740 ","End":"00:58.020","Text":"Here we have a small wave."},{"Start":"00:58.020 ","End":"01:00.139","Text":"It\u0027s narrower and shorter,"},{"Start":"01:00.139 ","End":"01:01.969","Text":"it has a lower amplitude."},{"Start":"01:01.969 ","End":"01:03.590","Text":"Here we have a larger wave,"},{"Start":"01:03.590 ","End":"01:07.265","Text":"and we can see it\u0027s wider and it has a higher amplitude."},{"Start":"01:07.265 ","End":"01:10.850","Text":"These 2 waves are traveling through a rope."},{"Start":"01:10.850 ","End":"01:13.415","Text":"Let\u0027s say, that\u0027s the transmission medium."},{"Start":"01:13.415 ","End":"01:16.650","Text":"If we wait 1 second, the wave,"},{"Start":"01:16.900 ","End":"01:25.575","Text":"this small wave, would have traveled the same distance as the larger wave."},{"Start":"01:25.575 ","End":"01:29.330","Text":"The center of this small wave would now be here and"},{"Start":"01:29.330 ","End":"01:33.005","Text":"the center of the large wave would now be here."},{"Start":"01:33.005 ","End":"01:37.245","Text":"They travel the same distance."},{"Start":"01:37.245 ","End":"01:42.350","Text":"The equation for this wave speed is given like so."},{"Start":"01:42.350 ","End":"01:51.080","Text":"V_wave is equal to the square root of the ratio between Tau,"},{"Start":"01:51.080 ","End":"01:52.970","Text":"Greek letter Tau,"},{"Start":"01:52.970 ","End":"01:57.605","Text":"and Mu, which is also another Greek letter."},{"Start":"01:57.605 ","End":"02:00.045","Text":"What is Tau?"},{"Start":"02:00.045 ","End":"02:07.179","Text":"Tau is the tension in the transmission medium."},{"Start":"02:07.179 ","End":"02:12.249","Text":"The more taut the string or the rope is,"},{"Start":"02:12.249 ","End":"02:17.514","Text":"the faster the wave or the pulse will travel through it."},{"Start":"02:17.514 ","End":"02:23.790","Text":"Let\u0027s say that our rope is pulled to a tension of 4."},{"Start":"02:23.790 ","End":"02:27.785","Text":"We can see that the square root of 4 is 2."},{"Start":"02:27.785 ","End":"02:38.764","Text":"A rope with a tension of 4 will mean that the velocity of the wave will be twice as fast."},{"Start":"02:38.764 ","End":"02:40.929","Text":"That\u0027s how that works."},{"Start":"02:40.929 ","End":"02:42.899","Text":"Then what is Mu?"},{"Start":"02:42.899 ","End":"02:47.440","Text":"Mu is the mass per unit length."},{"Start":"02:47.440 ","End":"02:50.074","Text":"Mass per unit length,"},{"Start":"02:50.074 ","End":"02:52.910","Text":"which is the Greek letter Mu,"},{"Start":"02:52.910 ","End":"02:56.119","Text":"is defined as the total mass."},{"Start":"02:56.119 ","End":"02:58.715","Text":"Here it would be the total mass of the rope,"},{"Start":"02:58.715 ","End":"03:01.414","Text":"divided by the length,"},{"Start":"03:01.414 ","End":"03:05.269","Text":"the total length of the rope."},{"Start":"03:05.269 ","End":"03:07.159","Text":"Let\u0027s give an example."},{"Start":"03:07.159 ","End":"03:11.000","Text":"If we had a rope whose mass,"},{"Start":"03:11.000 ","End":"03:13.564","Text":"this is capital M, not a Mu,"},{"Start":"03:13.564 ","End":"03:22.860","Text":"whose mass was 5 kilograms and whose length was equal to 2 meters."},{"Start":"03:22.860 ","End":"03:26.535","Text":"Then its mass per unit length,"},{"Start":"03:26.535 ","End":"03:32.835","Text":"Mu would be equal to 5 divided by 2,"},{"Start":"03:32.835 ","End":"03:36.645","Text":"which is equal to 2.5."},{"Start":"03:36.645 ","End":"03:41.030","Text":"We can see that the larger value for Mu,"},{"Start":"03:41.030 ","End":"03:43.894","Text":"the larger the masses per unit length,"},{"Start":"03:43.894 ","End":"03:48.809","Text":"the slower the wave will propagate through this medium."},{"Start":"03:49.240 ","End":"03:53.644","Text":"If we have a very small mass per unit length,"},{"Start":"03:53.644 ","End":"03:57.114","Text":"the wave will travel very quickly."},{"Start":"03:57.114 ","End":"04:02.840","Text":"What\u0027s important to learn from this first characteristic is number 1,"},{"Start":"04:02.840 ","End":"04:07.520","Text":"this equation and what each value represents."},{"Start":"04:07.520 ","End":"04:14.329","Text":"To remember that if we\u0027re transferring 2 waves through the same medium,"},{"Start":"04:14.329 ","End":"04:16.579","Text":"if the medium remains the same,"},{"Start":"04:16.579 ","End":"04:20.869","Text":"they\u0027re going to be traveling at a constant speed through the medium."},{"Start":"04:20.869 ","End":"04:24.830","Text":"It\u0027s not going to change. They\u0027re not going to accelerate or slow down at"},{"Start":"04:24.830 ","End":"04:28.850","Text":"any point and the speed that they\u0027re going to be"},{"Start":"04:28.850 ","End":"04:32.899","Text":"traveling at is only dependent on the medium itself and"},{"Start":"04:32.899 ","End":"04:37.940","Text":"not on any other features that you might see of the wave."},{"Start":"04:37.940 ","End":"04:43.924","Text":"The second characteristic is that waves carry energy."},{"Start":"04:43.924 ","End":"04:50.735","Text":"I\u0027m sure if you\u0027ve ever been backs and hit by someone whipping a rope,"},{"Start":"04:50.735 ","End":"04:55.069","Text":"then you felt the pain because that comes from the energy,"},{"Start":"04:55.069 ","End":"04:59.310","Text":"or if you\u0027ve ever been to the beach and had a wave come crashing over your head,"},{"Start":"04:59.310 ","End":"05:03.430","Text":"you can feel that these waves are carrying energy."},{"Start":"05:03.430 ","End":"05:07.084","Text":"We\u0027re not going to give the exact equation"},{"Start":"05:07.084 ","End":"05:11.870","Text":"for the energy of a wave because it\u0027s complicated to calculate."},{"Start":"05:11.870 ","End":"05:16.895","Text":"However, we do know that the energy of a wave"},{"Start":"05:16.895 ","End":"05:23.989","Text":"is proportional to its amplitude squared."},{"Start":"05:23.989 ","End":"05:28.149","Text":"Now, this sign means proportional."},{"Start":"05:28.149 ","End":"05:34.459","Text":"We say proportional because it\u0027s mainly dependent on the amplitude squared."},{"Start":"05:34.459 ","End":"05:38.509","Text":"However, there might be some constant that appears before here,"},{"Start":"05:38.509 ","End":"05:41.419","Text":"which will change our number slightly."},{"Start":"05:41.419 ","End":"05:46.475","Text":"That means that we can put an equals sign over here but if it\u0027s proportional,"},{"Start":"05:46.475 ","End":"05:49.310","Text":"then we can say something like so."},{"Start":"05:49.310 ","End":"05:56.060","Text":"If we have our medium and we send in a pulse-like so through the medium."},{"Start":"05:56.060 ","End":"06:01.100","Text":"Let\u0027s say that this pulse has an amplitude of A."},{"Start":"06:01.100 ","End":"06:06.329","Text":"Then we send in another pulse, like so."},{"Start":"06:06.329 ","End":"06:11.965","Text":"This pulse has an amplitude double this, 2A."},{"Start":"06:11.965 ","End":"06:16.999","Text":"We can\u0027t say exactly what the energy of this wave is equal to because we don\u0027t know"},{"Start":"06:16.999 ","End":"06:21.740","Text":"the full equation or of this wave but we can say that the energy,"},{"Start":"06:21.740 ","End":"06:23.915","Text":"let\u0027s say this is wave 1 and wave 2,"},{"Start":"06:23.915 ","End":"06:32.029","Text":"the energy of wave 1 is proportional to A^2 from this equation over here."},{"Start":"06:32.029 ","End":"06:37.435","Text":"Then we can say that the energy of wave 2,"},{"Start":"06:37.435 ","End":"06:40.610","Text":"because it might be multiplied by another constant,"},{"Start":"06:40.610 ","End":"06:45.065","Text":"but we know that it\u0027s proportional to 2A,"},{"Start":"06:45.065 ","End":"06:48.395","Text":"because that\u0027s its amplitude, squared."},{"Start":"06:48.395 ","End":"06:52.010","Text":"That means that it\u0027s proportional to 4A^2."},{"Start":"06:52.010 ","End":"06:55.880","Text":"We can see that if we have a wave with double the amplitude,"},{"Start":"06:55.880 ","End":"07:02.760","Text":"that means that its energy will be 4 times greater."},{"Start":"07:02.760 ","End":"07:05.970","Text":"What\u0027s important to remember from point 2,"},{"Start":"07:05.970 ","End":"07:14.525","Text":"characteristic number 2 is this equation or formula that\u0027s written over here."},{"Start":"07:14.525 ","End":"07:20.900","Text":"The next characteristic is that waves passing one another."},{"Start":"07:20.900 ","End":"07:24.155","Text":"During the overlap, they experience"},{"Start":"07:24.155 ","End":"07:28.205","Text":"interference with one another or they interfere with one another,"},{"Start":"07:28.205 ","End":"07:30.500","Text":"which is what we learned in the previous lesson about"},{"Start":"07:30.500 ","End":"07:33.625","Text":"constructive and destructive interference."},{"Start":"07:33.625 ","End":"07:36.125","Text":"After passing one another,"},{"Start":"07:36.125 ","End":"07:40.325","Text":"the waves just continue their original motion undisturbed."},{"Start":"07:40.325 ","End":"07:42.665","Text":"This we learned in the previous lesson."},{"Start":"07:42.665 ","End":"07:44.449","Text":"If this isn\u0027t clear to you,"},{"Start":"07:44.449 ","End":"07:46.550","Text":"please watch the previous lesson."},{"Start":"07:46.550 ","End":"07:52.489","Text":"Of course, during overlap we have a certain instances where nodes are formed."},{"Start":"07:52.489 ","End":"07:55.174","Text":"Again, we speak about that in the previous lesson."},{"Start":"07:55.174 ","End":"08:00.035","Text":"The fourth characteristic is that waves are reflected."},{"Start":"08:00.035 ","End":"08:04.279","Text":"There are three types of reflection that we\u0027re going to speak about."},{"Start":"08:04.279 ","End":"08:11.139","Text":"In the 1, we\u0027re speaking about transmission medium that has no end."},{"Start":"08:11.139 ","End":"08:16.835","Text":"What does that mean? Let\u0027s say that here is our hand."},{"Start":"08:16.835 ","End":"08:19.189","Text":"We\u0027re holding the transmission medium,"},{"Start":"08:19.189 ","End":"08:20.525","Text":"which is a rope,"},{"Start":"08:20.525 ","End":"08:23.540","Text":"and it just continues endlessly along."},{"Start":"08:23.540 ","End":"08:30.935","Text":"If I send over some right angle triangular wave in this direction,"},{"Start":"08:30.935 ","End":"08:33.970","Text":"it\u0027s just going to as time goes by."},{"Start":"08:33.970 ","End":"08:36.465","Text":"This is the wave a few moments later."},{"Start":"08:36.465 ","End":"08:38.510","Text":"Imagine the dimensions are the same,"},{"Start":"08:38.510 ","End":"08:39.800","Text":"I\u0027m not drawing it accurately,"},{"Start":"08:39.800 ","End":"08:43.430","Text":"but this wave is meant to be the exact same size and shape as"},{"Start":"08:43.430 ","End":"08:48.984","Text":"this one and just will carry on traveling down like so."},{"Start":"08:48.984 ","End":"08:52.660","Text":"This is just i."},{"Start":"08:53.420 ","End":"08:56.300","Text":"The first one, there\u0027s no end to"},{"Start":"08:56.300 ","End":"08:59.510","Text":"the transmission medium and the wave just continues to travel."},{"Start":"08:59.510 ","End":"09:04.040","Text":"Now let\u0027s take a look at the second type ii."},{"Start":"09:04.040 ","End":"09:09.919","Text":"This is that our transmission medium is attached to something but it\u0027s end,"},{"Start":"09:09.919 ","End":"09:11.600","Text":"its attachment is loose."},{"Start":"09:11.600 ","End":"09:13.384","Text":"What does that mean?"},{"Start":"09:13.384 ","End":"09:17.705","Text":"If this is our hand over here and this is the transmission medium."},{"Start":"09:17.705 ","End":"09:20.239","Text":"We can say that it\u0027s attached to this pole,"},{"Start":"09:20.239 ","End":"09:21.965","Text":"but it\u0027s attached loosely."},{"Start":"09:21.965 ","End":"09:25.749","Text":"It\u0027s not fixed over here and it can move up and down."},{"Start":"09:25.749 ","End":"09:27.709","Text":"Let\u0027s see what this means."},{"Start":"09:27.709 ","End":"09:33.080","Text":"That means that the wave pattern returns with left-right reflection."},{"Start":"09:33.080 ","End":"09:34.850","Text":"What does that mean?"},{"Start":"09:34.850 ","End":"09:41.035","Text":"We can consider this pole over here or this wall over here as the y-axis,"},{"Start":"09:41.035 ","End":"09:45.024","Text":"and let\u0027s consider this direction as the x-direction."},{"Start":"09:45.024 ","End":"09:47.720","Text":"If we draw our waves, again,"},{"Start":"09:47.720 ","End":"09:50.659","Text":"let\u0027s deal with this right-angled wave,"},{"Start":"09:50.659 ","End":"09:52.879","Text":"and is traveling in this direction."},{"Start":"09:52.879 ","End":"09:59.385","Text":"A few moments later it\u0027s going to reach the end of the wall over here."},{"Start":"09:59.385 ","End":"10:02.045","Text":"Of course, the red represents the wave"},{"Start":"10:02.045 ","End":"10:05.929","Text":"and the black is the transmission medium but obviously,"},{"Start":"10:05.929 ","End":"10:10.229","Text":"over here the transmission medium will be in the shape of the wave."},{"Start":"10:10.450 ","End":"10:13.730","Text":"Because of course, we learned that the wave is just"},{"Start":"10:13.730 ","End":"10:17.584","Text":"displacing the particles in the transmission medium."},{"Start":"10:17.584 ","End":"10:20.900","Text":"We can see that our rope, our transmission medium,"},{"Start":"10:20.900 ","End":"10:23.900","Text":"it\u0027s going to take this shape when we have a loose end."},{"Start":"10:23.900 ","End":"10:26.540","Text":"What we can see is,"},{"Start":"10:26.540 ","End":"10:29.149","Text":"let\u0027s show it in gray,"},{"Start":"10:29.149 ","End":"10:35.165","Text":"that this end of the wave reaches the wall first."},{"Start":"10:35.165 ","End":"10:39.815","Text":"What we\u0027re going to see is if this end of the wave reaches the wall first,"},{"Start":"10:39.815 ","End":"10:44.595","Text":"it\u0027s also going to be the first to return back to the origin."},{"Start":"10:44.595 ","End":"10:46.710","Text":"This tail of the wave,"},{"Start":"10:46.710 ","End":"10:50.675","Text":"this edge over here is going to reach the wall last,"},{"Start":"10:50.675 ","End":"10:56.045","Text":"which means it\u0027s also going to be last when coming back to the origin."},{"Start":"10:56.045 ","End":"11:00.974","Text":"Now let\u0027s draw the wave after its hit the wall,"},{"Start":"11:00.974 ","End":"11:03.719","Text":"once it begins its turn."},{"Start":"11:03.719 ","End":"11:09.125","Text":"Now it\u0027s left the area of the wall and it\u0027s now traveling in this direction."},{"Start":"11:09.125 ","End":"11:12.109","Text":"We can see that the section that was"},{"Start":"11:12.109 ","End":"11:16.320","Text":"first in the original wave that was traveling towards the wall,"},{"Start":"11:16.320 ","End":"11:20.449","Text":"the section with the right angle is now also first."},{"Start":"11:20.449 ","End":"11:25.504","Text":"It\u0027s also right at the front in its journey back to the origin."},{"Start":"11:25.504 ","End":"11:29.605","Text":"What we get is this right-left reflection."},{"Start":"11:29.605 ","End":"11:33.919","Text":"We get a mirror image in the y-direction."},{"Start":"11:33.919 ","End":"11:37.115","Text":"What we can say is if we have a loose end,"},{"Start":"11:37.115 ","End":"11:44.047","Text":"we can say that we have y-axis reflection."},{"Start":"11:44.047 ","End":"11:49.735","Text":"The wave is just going to look as if we put a mirror over here on the y-axis."},{"Start":"11:49.735 ","End":"11:54.370","Text":"This is the waves reflection in the y-axis."},{"Start":"11:54.370 ","End":"11:57.160","Text":"This is in a loose end."},{"Start":"11:57.160 ","End":"12:03.189","Text":"Then the blue wave will just carry on traveling back to the origin, and that\u0027s it."},{"Start":"12:03.189 ","End":"12:06.160","Text":"Now let\u0027s look at our third option."},{"Start":"12:06.160 ","End":"12:10.435","Text":"The third option is a fixed end. What does that mean?"},{"Start":"12:10.435 ","End":"12:14.409","Text":"Here\u0027s our hand and it\u0027s holding our transmission medium."},{"Start":"12:14.409 ","End":"12:17.184","Text":"We can carry on this y-axis. It doesn\u0027t matter."},{"Start":"12:17.184 ","End":"12:23.005","Text":"Then when, where it\u0027s attached to the y-axis to this polar wall, it\u0027s fixed."},{"Start":"12:23.005 ","End":"12:26.575","Text":"It can\u0027t move up, down, left or right."},{"Start":"12:26.575 ","End":"12:33.790","Text":"In this case, the wave pattern returns with left-right reflection as well,"},{"Start":"12:33.790 ","End":"12:39.235","Text":"so y-axis reflection, but also it returns upside down."},{"Start":"12:39.235 ","End":"12:46.520","Text":"This is y and x-axis reflection."},{"Start":"12:46.740 ","End":"12:52.060","Text":"In this case, it says if we put a mirror on the y-axis and we put"},{"Start":"12:52.060 ","End":"12:57.670","Text":"a mirror on the x-axis and then that is how the wave will return."},{"Start":"12:57.670 ","End":"13:00.415","Text":"Again, let\u0027s see what this looks like."},{"Start":"13:00.415 ","End":"13:02.589","Text":"Here we\u0027re sending off this red wave,"},{"Start":"13:02.589 ","End":"13:05.245","Text":"our original wave, and it\u0027s traveling."},{"Start":"13:05.245 ","End":"13:12.310","Text":"Then our wave hits the end over here where we have a fixed end."},{"Start":"13:12.310 ","End":"13:15.760","Text":"What we can see is Number 1, again,"},{"Start":"13:15.760 ","End":"13:19.810","Text":"this section is the first to reach this area over here."},{"Start":"13:19.810 ","End":"13:24.505","Text":"It\u0027s also going to be the first section to return back to the origin"},{"Start":"13:24.505 ","End":"13:29.290","Text":"but also we have reflection in the x-axis,"},{"Start":"13:29.290 ","End":"13:32.650","Text":"which means that the wave has to return upside down."},{"Start":"13:32.650 ","End":"13:35.230","Text":"What does that mean? The wave,"},{"Start":"13:35.230 ","End":"13:37.360","Text":"this section is going to return first,"},{"Start":"13:37.360 ","End":"13:38.589","Text":"so it will be here,"},{"Start":"13:38.589 ","End":"13:41.799","Text":"but it\u0027s upside down, so it\u0027s here."},{"Start":"13:41.799 ","End":"13:46.030","Text":"Then the tail hits the y-axis last."},{"Start":"13:46.030 ","End":"13:51.100","Text":"The tail is traveling back to the origin on the reflected wave, also last."},{"Start":"13:51.100 ","End":"13:55.480","Text":"The tail is also going to be here and it\u0027s just upside down."},{"Start":"13:55.480 ","End":"14:01.180","Text":"All we do is we draw it like this and a travel is like so."},{"Start":"14:01.250 ","End":"14:05.220","Text":"Then it will just carry on traveling back to the origin."},{"Start":"14:05.220 ","End":"14:07.889","Text":"What we can see is that we reflected it in"},{"Start":"14:07.889 ","End":"14:13.510","Text":"the y-axis so that it looked like this but then we also,"},{"Start":"14:13.510 ","End":"14:15.160","Text":"because we have a fixed end,"},{"Start":"14:15.160 ","End":"14:17.350","Text":"we reflected it in the x-axis."},{"Start":"14:17.350 ","End":"14:19.675","Text":"We put a mirror over here."},{"Start":"14:19.675 ","End":"14:27.930","Text":"What the mirror image of the blue wave would be in this side of our graph,"},{"Start":"14:27.930 ","End":"14:30.840","Text":"so on the negative y scale."},{"Start":"14:30.840 ","End":"14:34.260","Text":"Then it will just return back to the origin."},{"Start":"14:34.260 ","End":"14:38.920","Text":"That\u0027s what important to remember when we have a fixed end."},{"Start":"14:39.630 ","End":"14:46.340","Text":"Now let\u0027s talk about our fifth and final wave characteristic."},{"Start":"14:46.500 ","End":"14:50.005","Text":"Our fifth characteristic is that a pulse"},{"Start":"14:50.005 ","End":"14:54.490","Text":"propagating through different transmission mediums."},{"Start":"14:54.490 ","End":"14:59.200","Text":"Part of the wave is fixed end reflected."},{"Start":"14:59.200 ","End":"15:04.899","Text":"We have a reflection of Type 3 from the fourth characteristic."},{"Start":"15:04.899 ","End":"15:08.679","Text":"Another part of the wave is transferred to"},{"Start":"15:08.679 ","End":"15:13.990","Text":"the next transmission medium. What does this mean?"},{"Start":"15:13.990 ","End":"15:16.330","Text":"Let\u0027s say we have,"},{"Start":"15:16.330 ","End":"15:21.580","Text":"here is our hand that we\u0027re going to create the pulse on and here is"},{"Start":"15:21.580 ","End":"15:27.340","Text":"our first transmission medium where policy is going to propagate through it."},{"Start":"15:27.340 ","End":"15:31.735","Text":"Then our policy is going to get to another transmission medium."},{"Start":"15:31.735 ","End":"15:34.855","Text":"Let\u0027s say a thicker rope."},{"Start":"15:34.855 ","End":"15:39.145","Text":"Then this can carry on. It doesn\u0027t matter."},{"Start":"15:39.145 ","End":"15:41.830","Text":"This isn\u0027t the point of this characteristic."},{"Start":"15:41.830 ","End":"15:45.655","Text":"What exactly happens? Let\u0027s draw our wave."},{"Start":"15:45.655 ","End":"15:51.459","Text":"Here is our pulse that is propagating in this direction."},{"Start":"15:51.459 ","End":"15:54.250","Text":"At some stage, our pulse,"},{"Start":"15:54.250 ","End":"15:58.014","Text":"which has meant to look exactly the same over here,"},{"Start":"15:58.014 ","End":"16:01.569","Text":"is going to reach this point over here,"},{"Start":"16:01.569 ","End":"16:05.755","Text":"which is the end of a transmission medium Number 1,"},{"Start":"16:05.755 ","End":"16:07.510","Text":"at the end of the thin rope."},{"Start":"16:07.510 ","End":"16:13.030","Text":"That\u0027s right on the border of the second transmission medium or the thicker rope."},{"Start":"16:13.030 ","End":"16:17.320","Text":"First of all, what we\u0027re going to have is that part of"},{"Start":"16:17.320 ","End":"16:21.535","Text":"the wave is fixed end reflected. What does this mean?"},{"Start":"16:21.535 ","End":"16:26.379","Text":"This means that once our pulse or a wave reaches this section,"},{"Start":"16:26.379 ","End":"16:30.475","Text":"it\u0027s harder for it to carry on propagating this way."},{"Start":"16:30.475 ","End":"16:33.159","Text":"As far as the waves concerned,"},{"Start":"16:33.159 ","End":"16:37.600","Text":"it feels as if it\u0027s reached a fixed end. What does that mean?"},{"Start":"16:37.600 ","End":"16:43.465","Text":"That means that we\u0027re going to have a reflection in the x and y axes."},{"Start":"16:43.465 ","End":"16:47.360","Text":"Let\u0027s draw in blue and a dotted line."},{"Start":"16:49.260 ","End":"16:53.005","Text":"First, we\u0027re going to have a reflection in the y-axis,"},{"Start":"16:53.005 ","End":"16:54.565","Text":"which is this direction."},{"Start":"16:54.565 ","End":"16:57.969","Text":"Then we\u0027re going to have a reflection in the x-axis,"},{"Start":"16:57.969 ","End":"16:59.455","Text":"which is this direction."},{"Start":"16:59.455 ","End":"17:05.695","Text":"Our new wave is going to look something like so."},{"Start":"17:05.695 ","End":"17:08.409","Text":"If you don\u0027t understand how I got to this,"},{"Start":"17:08.409 ","End":"17:14.120","Text":"please go back to characteristic Number 4 where I explain it a bit more."},{"Start":"17:15.240 ","End":"17:19.270","Text":"That\u0027s part of the wave which is fixed and reflected and"},{"Start":"17:19.270 ","End":"17:22.990","Text":"another part of the wave is transferred."},{"Start":"17:22.990 ","End":"17:26.680","Text":"The second part of the wave is going to be over here,"},{"Start":"17:26.680 ","End":"17:34.135","Text":"where the wave is just going to carry on traveling in its original direction like so."},{"Start":"17:34.135 ","End":"17:38.410","Text":"This wave that was fixed and reflected is going to travel back to the origin."},{"Start":"17:38.410 ","End":"17:41.439","Text":"The other part of the wave which is just transferred,"},{"Start":"17:41.439 ","End":"17:47.020","Text":"is just transferred to the second transmission medium and it\u0027s just going to carry on"},{"Start":"17:47.020 ","End":"17:55.975","Text":"traveling pretty much in the same way but what do we have to notice over here is this."},{"Start":"17:55.975 ","End":"18:01.389","Text":"One, we learned about the energy of the wave and we saw that the energy of"},{"Start":"18:01.389 ","End":"18:08.210","Text":"the wave is proportional to its amplitude squared."},{"Start":"18:08.280 ","End":"18:15.849","Text":"Because our original wave has some amount of energy when our original wave splits into"},{"Start":"18:15.849 ","End":"18:23.530","Text":"the reflected wave and the transferred wave."},{"Start":"18:23.530 ","End":"18:27.310","Text":"Our energy, the original energy,"},{"Start":"18:27.310 ","End":"18:31.479","Text":"which is proportional to the original amplitude squared,"},{"Start":"18:31.479 ","End":"18:37.600","Text":"is going to be split into the energy of the reflected wave,"},{"Start":"18:37.600 ","End":"18:42.685","Text":"which is going to be proportional to some different amplitude squared."},{"Start":"18:42.685 ","End":"18:48.504","Text":"Let\u0027s write A_1^2 and"},{"Start":"18:48.504 ","End":"18:54.204","Text":"it\u0027s going to be split into the energy of the transmitted ray or wave,"},{"Start":"18:54.204 ","End":"19:01.850","Text":"which is proportional to the amplitude of that A_2 ^2."},{"Start":"19:02.250 ","End":"19:05.409","Text":"That means that what I drew over here,"},{"Start":"19:05.409 ","End":"19:09.684","Text":"my reflected ray is going to have a smaller amplitude,"},{"Start":"19:09.684 ","End":"19:11.980","Text":"so it\u0027s going to be a bit lower."},{"Start":"19:11.980 ","End":"19:14.080","Text":"Let\u0027s redraw this."},{"Start":"19:14.080 ","End":"19:17.215","Text":"It\u0027s going to have a smaller amplitude."},{"Start":"19:17.215 ","End":"19:19.999","Text":"Let\u0027s draw it like so."},{"Start":"19:20.460 ","End":"19:25.040","Text":"We can see it\u0027s smaller than this."},{"Start":"19:25.260 ","End":"19:31.525","Text":"Its amplitude is smaller than the amplitude of the original red wave."},{"Start":"19:31.525 ","End":"19:37.135","Text":"Of course, this amplitude over here will also change."},{"Start":"19:37.135 ","End":"19:42.550","Text":"It depends on the proportion of what\u0027s going on here but this might"},{"Start":"19:42.550 ","End":"19:47.890","Text":"have a different amplitude and might have slightly bigger than this or slightly smaller,"},{"Start":"19:47.890 ","End":"19:50.605","Text":"and it just carries on traveling."},{"Start":"19:50.605 ","End":"19:54.550","Text":"Now another important thing to note is"},{"Start":"19:54.550 ","End":"19:59.425","Text":"that because we\u0027re traveling between 2 different transmission mediums,"},{"Start":"19:59.425 ","End":"20:03.910","Text":"the density of each medium is going to be different,"},{"Start":"20:03.910 ","End":"20:08.170","Text":"which means that the velocity of each new wave."},{"Start":"20:08.170 ","End":"20:11.890","Text":"So the velocity of the reflected wave and"},{"Start":"20:11.890 ","End":"20:16.810","Text":"the velocity of the transmitted wave are going to be different."},{"Start":"20:16.810 ","End":"20:19.735","Text":"If we remember this equation over here,"},{"Start":"20:19.735 ","End":"20:24.475","Text":"and the velocity of the wave is equal to the square root of"},{"Start":"20:24.475 ","End":"20:29.875","Text":"the tension divided by the mass per unit length."},{"Start":"20:29.875 ","End":"20:32.019","Text":"The tension in both of"},{"Start":"20:32.019 ","End":"20:36.625","Text":"the transmission mediums is going to be the same because they\u0027re attached to one another."},{"Start":"20:36.625 ","End":"20:38.154","Text":"That doesn\u0027t change."},{"Start":"20:38.154 ","End":"20:44.965","Text":"However, the Mu of each transmission medium is going to be different."},{"Start":"20:44.965 ","End":"20:47.320","Text":"Because the Mu of transmission medium,"},{"Start":"20:47.320 ","End":"20:51.715","Text":"let\u0027s say this is transmission Medium 1 and this is transmission Medium 2."},{"Start":"20:51.715 ","End":"20:57.850","Text":"We can see that transmission Medium number 1 is thinner,"},{"Start":"20:57.850 ","End":"21:01.720","Text":"so we can assume that it\u0027s mass per unit length is"},{"Start":"21:01.720 ","End":"21:06.445","Text":"smaller than the mass per unit length of transmission Medium 2,"},{"Start":"21:06.445 ","End":"21:08.725","Text":"because it\u0027s much thicker."},{"Start":"21:08.725 ","End":"21:13.720","Text":"That means therefore, we can see that the velocity is"},{"Start":"21:13.720 ","End":"21:18.715","Text":"inversely proportional to the square root of this mass per unit length."},{"Start":"21:18.715 ","End":"21:23.270","Text":"Therefore, we can see that v_1,"},{"Start":"21:23.270 ","End":"21:27.135","Text":"the velocity of the reflected wave."},{"Start":"21:27.135 ","End":"21:30.240","Text":"Let\u0027s say v_R, the velocity of the reflected wave,"},{"Start":"21:30.240 ","End":"21:39.079","Text":"is therefore also going to be smaller than the velocity of the transmitted ray."},{"Start":"21:39.079 ","End":"21:43.720","Text":"That means that we can draw this in green."},{"Start":"21:43.720 ","End":"21:50.440","Text":"The reflected ray in 1 second will reach this area"},{"Start":"21:50.440 ","End":"21:58.120","Text":"over here but in the same time because the velocity of the transmitted ray is much less."},{"Start":"21:58.120 ","End":"22:05.335","Text":"The transmitted ray might just be here in 1 second."},{"Start":"22:05.335 ","End":"22:08.590","Text":"We can see that the reflected ray in this example"},{"Start":"22:08.590 ","End":"22:12.289","Text":"specifically is going to be traveling faster."},{"Start":"22:12.570 ","End":"22:17.605","Text":"I\u0027ve added these 2 points to our fifth characteristic."},{"Start":"22:17.605 ","End":"22:20.920","Text":"The amplitude is going to be decreased,"},{"Start":"22:20.920 ","End":"22:26.725","Text":"because the energy of the original wave is now split between 2 waves."},{"Start":"22:26.725 ","End":"22:34.180","Text":"Both of these waves are going to have a smaller amplitude than the original wave."},{"Start":"22:34.180 ","End":"22:38.785","Text":"The wave speed of the transmitted wave changes."},{"Start":"22:38.785 ","End":"22:43.569","Text":"Why does the wave speed only of the transmitted wave change?"},{"Start":"22:43.569 ","End":"22:45.955","Text":"Because we said that the wave speed"},{"Start":"22:45.955 ","End":"22:49.975","Text":"depends only on the properties of the transmission medium."},{"Start":"22:49.975 ","End":"22:55.149","Text":"If we start on the first transmission medium here it\u0027s the thinner one,"},{"Start":"22:55.149 ","End":"22:56.519","Text":"but it doesn\u0027t really matter,"},{"Start":"22:56.519 ","End":"22:59.199","Text":"if we start on some transmission medium,"},{"Start":"22:59.199 ","End":"23:01.555","Text":"that\u0027s our original transmission medium."},{"Start":"23:01.555 ","End":"23:04.989","Text":"Our wave is going to be traveling at a speed dependent on"},{"Start":"23:04.989 ","End":"23:10.045","Text":"that transmission medium but when it reaches the edge of the transmission medium,"},{"Start":"23:10.045 ","End":"23:13.180","Text":"the reflected ray is going to bounce back."},{"Start":"23:13.180 ","End":"23:14.455","Text":"It\u0027s reflected back."},{"Start":"23:14.455 ","End":"23:18.145","Text":"Notice that it\u0027s traveling on the same transmission medium."},{"Start":"23:18.145 ","End":"23:23.350","Text":"That means that its velocity is going to be exactly the same."},{"Start":"23:23.350 ","End":"23:27.850","Text":"However, the transmitted wave now moves from"},{"Start":"23:27.850 ","End":"23:34.015","Text":"the original transmission medium to the next transmission medium."},{"Start":"23:34.015 ","End":"23:38.155","Text":"Which means that it\u0027s traveling on a new transmission medium,"},{"Start":"23:38.155 ","End":"23:43.465","Text":"which means that its velocity is going to change."},{"Start":"23:43.465 ","End":"23:46.914","Text":"That\u0027s why the wave speed or the wave velocity of"},{"Start":"23:46.914 ","End":"23:50.574","Text":"only the transmitted wave is going to change,"},{"Start":"23:50.574 ","End":"23:57.070","Text":"whereas the reflected wave will have the same wave speed as the original wave."},{"Start":"23:57.070 ","End":"24:03.114","Text":"Up until now we\u0027ve been speaking about if we\u0027re moving from a certain medium to"},{"Start":"24:03.114 ","End":"24:06.295","Text":"a more dense medium but what happens"},{"Start":"24:06.295 ","End":"24:10.615","Text":"if we move from a certain medium to a less dense medium?"},{"Start":"24:10.615 ","End":"24:15.564","Text":"In that case, we\u0027re going to be using loose and reflection."},{"Start":"24:15.564 ","End":"24:20.979","Text":"If you already copied this out please add loose and reflection."},{"Start":"24:20.979 ","End":"24:26.740","Text":"Loose and reflection happens when we move from a dense medium to"},{"Start":"24:26.740 ","End":"24:30.610","Text":"a less dense medium and fixed end reflection is"},{"Start":"24:30.610 ","End":"24:35.170","Text":"when we move from a less dense medium to a more dense medium."},{"Start":"24:35.170 ","End":"24:38.950","Text":"Let\u0027s scroll down and draw this."},{"Start":"24:38.950 ","End":"24:41.605","Text":"Here\u0027s our hand creating the pulse."},{"Start":"24:41.605 ","End":"24:50.335","Text":"We\u0027re moving now from this dense medium to the less dense medium."},{"Start":"24:50.335 ","End":"24:57.130","Text":"Quick note. If the Mu or the mass per unit length of"},{"Start":"24:57.130 ","End":"25:00.609","Text":"the first transmission medium is less"},{"Start":"25:00.609 ","End":"25:05.350","Text":"than the mass per unit length of the second transmission medium i.e."},{"Start":"25:05.350 ","End":"25:10.045","Text":"the first transmission medium is less dense than the second transmission medium,"},{"Start":"25:10.045 ","End":"25:12.415","Text":"then we have fixed end reflection."},{"Start":"25:12.415 ","End":"25:15.264","Text":"Here we have if"},{"Start":"25:15.264 ","End":"25:24.950","Text":"the second transmission medium is less dense than the first transmission medium."},{"Start":"25:25.500 ","End":"25:29.499","Text":"Then we have loose and reflection,"},{"Start":"25:29.499 ","End":"25:33.430","Text":"which is what we\u0027re talking about now. Let\u0027s draw this."},{"Start":"25:33.430 ","End":"25:38.630","Text":"We have our pulse going out traveling like so."},{"Start":"25:39.180 ","End":"25:48.560","Text":"Our pulse reaches the end of our first transmission medium."},{"Start":"25:48.930 ","End":"25:56.110","Text":"Now we\u0027re going from a dense transmission medium to a less dense transmission medium,"},{"Start":"25:56.110 ","End":"25:59.020","Text":"which means that we\u0027re doing loose and reflection."},{"Start":"25:59.020 ","End":"26:00.670","Text":"What is loose and reflection?"},{"Start":"26:00.670 ","End":"26:03.565","Text":"It\u0027s just a reflection in the y-axis."},{"Start":"26:03.565 ","End":"26:07.359","Text":"In that case, if I use the idea of the gray arrows again,"},{"Start":"26:07.359 ","End":"26:14.775","Text":"I\u0027m going to have this type of reflection in this direction."},{"Start":"26:14.775 ","End":"26:19.710","Text":"That is going to be for the reflected ray,"},{"Start":"26:19.710 ","End":"26:24.135","Text":"of course, we\u0027re going to have a transmitted ray."},{"Start":"26:24.135 ","End":"26:25.815","Text":"Let\u0027s take a look."},{"Start":"26:25.815 ","End":"26:28.735","Text":"Our reflected ray, let\u0027s look."},{"Start":"26:28.735 ","End":"26:33.310","Text":"It\u0027s going to be reflected back on our first transmission medium."},{"Start":"26:33.310 ","End":"26:40.810","Text":"If this was the first section that hit our loose end with quotation marks."},{"Start":"26:40.810 ","End":"26:43.180","Text":"This side with the right angle."},{"Start":"26:43.180 ","End":"26:50.515","Text":"The side with the right angle is going to be the first to travel back to our origin."},{"Start":"26:50.515 ","End":"26:54.864","Text":"Before I draw this out we\u0027re going to remember that"},{"Start":"26:54.864 ","End":"26:59.605","Text":"the energy is proportional to the amplitude squared."},{"Start":"26:59.605 ","End":"27:07.404","Text":"Because my original wave is being split into 2 waves,"},{"Start":"27:07.404 ","End":"27:13.735","Text":"that means that each wave is going to have a lower amplitude than my original wave."},{"Start":"27:13.735 ","End":"27:18.729","Text":"That means that this end with a right angle is going to be the"},{"Start":"27:18.729 ","End":"27:25.000","Text":"first to travel back to the origin and the tail is going to be the last."},{"Start":"27:25.000 ","End":"27:29.410","Text":"However, the amplitude is going to be less."},{"Start":"27:29.410 ","End":"27:34.929","Text":"Of course, my reflected ray is traveling now in the opposite direction because it was"},{"Start":"27:34.929 ","End":"27:41.260","Text":"reflected in the y-axis because it\u0027s loose and reflection and it travels back."},{"Start":"27:41.260 ","End":"27:45.790","Text":"Then we also have my transmitted ray,"},{"Start":"27:45.790 ","End":"27:51.970","Text":"which is going to carry on in the new transmission medium."},{"Start":"27:51.970 ","End":"27:57.264","Text":"In this way, in this direction its amplitude is also going to change,"},{"Start":"27:57.264 ","End":"28:02.020","Text":"but it\u0027s carrying on in the same direction and it isn\u0027t reflected,"},{"Start":"28:02.020 ","End":"28:06.909","Text":"because it\u0027s transmitted and it just carries on like so."},{"Start":"28:06.909 ","End":"28:11.274","Text":"A quick little reminder as well that the"},{"Start":"28:11.274 ","End":"28:15.925","Text":"velocities or the wave speeds of each one of these subsequent waves."},{"Start":"28:15.925 ","End":"28:21.925","Text":"The reflected and transmitted ray is also going to be different, like what we saw."},{"Start":"28:21.925 ","End":"28:24.520","Text":"We saw that the amplitude has decreased and"},{"Start":"28:24.520 ","End":"28:28.165","Text":"the wave speed of the transmitted wave changes."},{"Start":"28:28.165 ","End":"28:32.980","Text":"Because we know that the density or the mass per"},{"Start":"28:32.980 ","End":"28:40.165","Text":"unit length of this new transmission medium is lower."},{"Start":"28:40.165 ","End":"28:45.189","Text":"We know that this wave is therefore going to be traveling"},{"Start":"28:45.189 ","End":"28:54.220","Text":"faster than the reflected wave."},{"Start":"28:54.220 ","End":"28:55.719","Text":"Here\u0027s our equation again."},{"Start":"28:55.719 ","End":"28:58.135","Text":"I\u0027m just going to scroll down to give us some more space."},{"Start":"28:58.135 ","End":"29:01.420","Text":"Here\u0027s our equation for our wave speed."},{"Start":"29:01.420 ","End":"29:08.350","Text":"The wave speed of our reflected wave is going to be exactly the same because it\u0027s"},{"Start":"29:08.350 ","End":"29:15.010","Text":"traveling on the exact same transmission medium and it\u0027s just going backwards."},{"Start":"29:15.010 ","End":"29:17.065","Text":"It\u0027s going to have,"},{"Start":"29:17.065 ","End":"29:19.840","Text":"the tension hasn\u0027t changed."},{"Start":"29:19.840 ","End":"29:22.915","Text":"This is going to be its velocity."},{"Start":"29:22.915 ","End":"29:30.324","Text":"Then the velocity of the transferred wave is going to increase."},{"Start":"29:30.324 ","End":"29:37.825","Text":"Why? Because the Mu of our second transmission medium is less."},{"Start":"29:37.825 ","End":"29:40.389","Text":"If the denominator is smaller,"},{"Start":"29:40.389 ","End":"29:44.750","Text":"that means that the velocity is going to be bigger."},{"Start":"29:45.570 ","End":"29:52.510","Text":"That means that we have this type of situation happening."},{"Start":"29:52.510 ","End":"29:56.259","Text":"If I draw the same diagram that I did over here with the green,"},{"Start":"29:56.259 ","End":"30:00.865","Text":"that means that if we wait a second,"},{"Start":"30:00.865 ","End":"30:05.484","Text":"in this case, when we\u0027re traveling from a denser medium to a less dense medium,"},{"Start":"30:05.484 ","End":"30:10.465","Text":"our transmitted wave would have moved this much."},{"Start":"30:10.465 ","End":"30:18.880","Text":"However, our reflected ray or wave would have moved this much."},{"Start":"30:21.080 ","End":"30:25.480","Text":"That is the end of our lesson."},{"Start":"30:25.480 ","End":"30:31.089","Text":"It\u0027s very important to remember when we\u0027re using loose end and when"},{"Start":"30:31.089 ","End":"30:33.639","Text":"using fixed end reflection when"},{"Start":"30:33.639 ","End":"30:37.885","Text":"our pulse is propagating through different transmission mediums."},{"Start":"30:37.885 ","End":"30:42.865","Text":"These are all of our wave characteristics."},{"Start":"30:42.865 ","End":"30:45.649","Text":"That\u0027s the end of this lesson."}],"ID":12461},{"Watched":false,"Name":"Exercise - Wave Calculations 1","Duration":"10m 30s","ChapterTopicVideoID":11994,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.055","Text":"Hello. In this lesson,"},{"Start":"00:02.055 ","End":"00:06.015","Text":"we\u0027re going to practice some of the stuff that we\u0027ve learned up until now."},{"Start":"00:06.015 ","End":"00:09.930","Text":"Here we\u0027re being told that a pulse moves rightwards,"},{"Start":"00:09.930 ","End":"00:12.263","Text":"so in this direction,"},{"Start":"00:12.263 ","End":"00:19.125","Text":"and its position is given at t=0 and at t=2 seconds."},{"Start":"00:19.125 ","End":"00:21.405","Text":"Question number 1 is,"},{"Start":"00:21.405 ","End":"00:24.940","Text":"what is the pulse\u0027s amplitude?"},{"Start":"00:25.130 ","End":"00:27.420","Text":"Before we begin answering,"},{"Start":"00:27.420 ","End":"00:29.250","Text":"we know that each square,"},{"Start":"00:29.250 ","End":"00:33.000","Text":"the height of the square is equal to 10 centimeters and the width"},{"Start":"00:33.000 ","End":"00:37.290","Text":"of the square is also equal to 10 centimeters."},{"Start":"00:37.290 ","End":"00:41.550","Text":"Let\u0027s begin by answering question number 1."},{"Start":"00:42.230 ","End":"00:47.750","Text":"First of all, let\u0027s remind ourselves what is an amplitude."},{"Start":"00:47.750 ","End":"00:52.580","Text":"The amplitude is the maximum particle deviation"},{"Start":"00:52.580 ","End":"00:56.060","Text":"from the point of equilibrium. What does that mean?"},{"Start":"00:56.060 ","End":"00:58.689","Text":"The 0 or the baseline,"},{"Start":"00:58.689 ","End":"01:00.937","Text":"we know is our point of equilibrium."},{"Start":"01:00.937 ","End":"01:06.057","Text":"So we\u0027re trying to find the maximum displacement from that."},{"Start":"01:06.057 ","End":"01:10.040","Text":"We can see that that\u0027s just going to be at the peak of the pulse."},{"Start":"01:10.040 ","End":"01:16.905","Text":"All of this over here is the amplitude."},{"Start":"01:16.905 ","End":"01:20.806","Text":"We can see that that\u0027s equal to 1, 2,"},{"Start":"01:20.806 ","End":"01:25.520","Text":"3 squares, where we said that the height of each square is 10 centimeters."},{"Start":"01:25.520 ","End":"01:36.185","Text":"We can see that the amplitude is simply equal to 3 times 10 centimeters,"},{"Start":"01:36.185 ","End":"01:42.140","Text":"which is equal to either 30 centimeters or usually we work in MKS,"},{"Start":"01:42.140 ","End":"01:49.890","Text":"which means that we\u0027re working in units of meters, so 0.3 meters."},{"Start":"01:50.480 ","End":"01:54.845","Text":"Now we\u0027re answering question number 2."},{"Start":"01:54.845 ","End":"01:56.420","Text":"Question number 2 is,"},{"Start":"01:56.420 ","End":"01:57.890","Text":"what is its velocity?"},{"Start":"01:57.890 ","End":"02:01.150","Text":"We\u0027re trying to find this pulse\u0027s velocity."},{"Start":"02:01.150 ","End":"02:03.505","Text":"We know that the width of each square"},{"Start":"02:03.505 ","End":"02:06.595","Text":"is 10 centimeters because that\u0027s what it says over here."},{"Start":"02:06.595 ","End":"02:09.790","Text":"When we\u0027re trying to find velocity,"},{"Start":"02:09.790 ","End":"02:13.870","Text":"we always have to measure the pulse from the same point."},{"Start":"02:13.870 ","End":"02:19.915","Text":"We\u0027re seeing where the same point originally was and where it is at our final moment."},{"Start":"02:19.915 ","End":"02:24.040","Text":"Usually, we measure this by taking"},{"Start":"02:24.040 ","End":"02:28.615","Text":"the position of the maximum or of the peak of the pulse."},{"Start":"02:28.615 ","End":"02:34.315","Text":"However, you can also measure its end over here or this end over here,"},{"Start":"02:34.315 ","End":"02:38.470","Text":"and then see where it is at time t=2 seconds,"},{"Start":"02:38.470 ","End":"02:42.995","Text":"but we\u0027re just going to measure where it is from the peak."},{"Start":"02:42.995 ","End":"02:47.598","Text":"The peak is over here, t=0 seconds,"},{"Start":"02:47.598 ","End":"02:54.430","Text":"and then we can see that at t=2 seconds the peak is over here."},{"Start":"02:54.430 ","End":"03:02.596","Text":"We can see if we go up like so,"},{"Start":"03:02.596 ","End":"03:05.865","Text":"that our peak has moved 1, 2,"},{"Start":"03:05.865 ","End":"03:09.890","Text":"3, 4 squares."},{"Start":"03:09.890 ","End":"03:15.945","Text":"In other words, our peak has moved 40 centimeters."},{"Start":"03:15.945 ","End":"03:19.180","Text":"Let\u0027s write out our equation for velocity and"},{"Start":"03:19.180 ","End":"03:22.059","Text":"it\u0027s the same equation that we\u0027ve seen in basic kinematics."},{"Start":"03:22.059 ","End":"03:26.350","Text":"Our velocity is our change in x,"},{"Start":"03:26.350 ","End":"03:28.374","Text":"so the distance traveled,"},{"Start":"03:28.374 ","End":"03:30.900","Text":"divided by our Delta t,"},{"Start":"03:30.900 ","End":"03:32.670","Text":"our change in time."},{"Start":"03:32.670 ","End":"03:34.910","Text":"Our change in x, as we saw,"},{"Start":"03:34.910 ","End":"03:36.845","Text":"is equal to 4 squares,"},{"Start":"03:36.845 ","End":"03:44.000","Text":"which is equal to 40 centimeters or 0.4 meters."},{"Start":"03:44.000 ","End":"03:46.310","Text":"Divided by our change in time,"},{"Start":"03:46.310 ","End":"03:50.320","Text":"we\u0027ve gotten from 0 seconds to 2 seconds."},{"Start":"03:50.320 ","End":"03:55.500","Text":"0.4 meters divided by 2 seconds."},{"Start":"03:55.500 ","End":"04:04.230","Text":"That means that our velocity is simply equal to 0.2 meters per second."},{"Start":"04:04.580 ","End":"04:09.375","Text":"Now let\u0027s answer question number 3."},{"Start":"04:09.375 ","End":"04:11.015","Text":"Question number 3 is,"},{"Start":"04:11.015 ","End":"04:19.955","Text":"what is the direction of travel of the rope particle located at point A at t=0 seconds."},{"Start":"04:19.955 ","End":"04:28.380","Text":"Here, we\u0027re on the graph of t=0 seconds and here is the particle at point A,"},{"Start":"04:28.380 ","End":"04:32.465","Text":"and we\u0027re trying to see what its direction of travel is."},{"Start":"04:32.465 ","End":"04:35.050","Text":"Let\u0027s see how we do this."},{"Start":"04:35.050 ","End":"04:39.215","Text":"First of all, we know that our wave is moving in the rightwards direction,"},{"Start":"04:39.215 ","End":"04:44.345","Text":"which means that our peak is moving also in this rightwards direction."},{"Start":"04:44.345 ","End":"04:48.470","Text":"So we know what it looks like a 2 seconds,"},{"Start":"04:48.470 ","End":"04:52.115","Text":"but let\u0027s look slightly after t=0."},{"Start":"04:52.115 ","End":"05:00.780","Text":"What we can see is that our peak at some stage is going to move right over here."},{"Start":"05:01.120 ","End":"05:06.350","Text":"It has just moved a little bit to this side and we can see that"},{"Start":"05:06.350 ","End":"05:11.950","Text":"this is placed exactly above our particle A."},{"Start":"05:12.770 ","End":"05:18.860","Text":"That means that a few moments after t=0 seconds,"},{"Start":"05:18.860 ","End":"05:25.800","Text":"our wave is going to look something like so."},{"Start":"05:25.800 ","End":"05:34.640","Text":"We can see in that case that our A for moving from this particle deviation of"},{"Start":"05:34.640 ","End":"05:38.645","Text":"10 centimeters is eventually going to move up"},{"Start":"05:38.645 ","End":"05:43.895","Text":"to a maximum particle deviation of 30 centimeters,"},{"Start":"05:43.895 ","End":"05:46.220","Text":"which is our amplitude."},{"Start":"05:46.220 ","End":"05:51.980","Text":"We can see that A is moving up to a maximum amplitude,"},{"Start":"05:51.980 ","End":"05:55.925","Text":"maximum particle deviation, which means that"},{"Start":"05:55.925 ","End":"06:01.620","Text":"the direction of travel of this point A is upwards."},{"Start":"06:02.270 ","End":"06:05.760","Text":"Point A is traveling upwards."},{"Start":"06:05.760 ","End":"06:12.440","Text":"It\u0027s very important to remember that we\u0027re talking about point A traveling upwards,"},{"Start":"06:12.440 ","End":"06:15.410","Text":"as in the wave isn\u0027t traveling upwards,"},{"Start":"06:15.410 ","End":"06:21.263","Text":"the wave as a whole or the pulse itself is traveling rightwards."},{"Start":"06:21.263 ","End":"06:26.240","Text":"But our point A in the pulse, our particle over,"},{"Start":"06:26.240 ","End":"06:36.795","Text":"here is moving upwards because we\u0027re dealing with a transverse wave."},{"Start":"06:36.795 ","End":"06:39.800","Text":"Remember that in a transverse wave,"},{"Start":"06:39.800 ","End":"06:43.580","Text":"our particle deviation is always at 90 degrees or"},{"Start":"06:43.580 ","End":"06:48.190","Text":"perpendicular to the direction of travel of the wave itself."},{"Start":"06:48.190 ","End":"06:53.360","Text":"Question 4 is, what is the direction of travel of the rope particles?"},{"Start":"06:53.360 ","End":"06:55.340","Text":"Again, we\u0027re dealing with the particle and not the wave"},{"Start":"06:55.340 ","End":"06:59.210","Text":"itself located at point B at t=0."},{"Start":"06:59.210 ","End":"07:03.380","Text":"Now we\u0027re looking at this point over here and literally by"},{"Start":"07:03.380 ","End":"07:07.995","Text":"using the exact same method that we did in question number 3,"},{"Start":"07:07.995 ","End":"07:09.915","Text":"we\u0027re going to answer this question."},{"Start":"07:09.915 ","End":"07:13.130","Text":"What we can see is that point B was"},{"Start":"07:13.130 ","End":"07:16.934","Text":"located at some kind of particle deviation of 10 centimeters."},{"Start":"07:16.934 ","End":"07:19.700","Text":"It was located 1 square up."},{"Start":"07:19.700 ","End":"07:21.580","Text":"A moment later,"},{"Start":"07:21.580 ","End":"07:29.150","Text":"we can see that our point B from being 1 square up is now at an amplitude of 0."},{"Start":"07:29.150 ","End":"07:33.890","Text":"It\u0027s now located at the base of the wave."},{"Start":"07:33.890 ","End":"07:38.390","Text":"We can see that our point B moved from particle displacement of"},{"Start":"07:38.390 ","End":"07:45.275","Text":"10 centimeters to a particle displacement of 0 or to its point of equilibrium,"},{"Start":"07:45.275 ","End":"07:50.490","Text":"which means that point B is traveling downwards."},{"Start":"07:50.990 ","End":"07:53.830","Text":"In order to answer question 3 and 4,"},{"Start":"07:53.830 ","End":"07:57.815","Text":"we just drew a snapshot of the wave a moment after."},{"Start":"07:57.815 ","End":"08:04.410","Text":"Then we could map where our points A and B would now be."},{"Start":"08:04.940 ","End":"08:08.680","Text":"Again, I\u0027m highlighting that it\u0027s point B,"},{"Start":"08:08.680 ","End":"08:10.210","Text":"which is traveling downwards,"},{"Start":"08:10.210 ","End":"08:11.530","Text":"not the pulse itself."},{"Start":"08:11.530 ","End":"08:15.200","Text":"The pulse itself is traveling rightwards."},{"Start":"08:16.020 ","End":"08:21.360","Text":"An easy way of just remembering how to answer question"},{"Start":"08:21.360 ","End":"08:26.875","Text":"3 and 4 or similar questions is if we have a snapshot of our waves."},{"Start":"08:26.875 ","End":"08:31.090","Text":"Let\u0027s look here at t=2 and let\u0027s say that we have these 2 points,"},{"Start":"08:31.090 ","End":"08:34.660","Text":"C and D. Of course, as we know,"},{"Start":"08:34.660 ","End":"08:39.890","Text":"our direction of travel in this case is in this rightwards direction."},{"Start":"08:39.890 ","End":"08:44.950","Text":"If our wave is traveling in the rightwards direction,"},{"Start":"08:44.950 ","End":"08:49.490","Text":"any point that comes after the wave or pulses"},{"Start":"08:49.490 ","End":"08:54.800","Text":"peak is going to be moving in the upwards direction."},{"Start":"08:54.800 ","End":"08:57.230","Text":"If I looked at point C,"},{"Start":"08:57.230 ","End":"08:59.210","Text":"it will be moving in the upward direction."},{"Start":"08:59.210 ","End":"09:00.410","Text":"If I look at this point,"},{"Start":"09:00.410 ","End":"09:02.225","Text":"at this point, at this point,"},{"Start":"09:02.225 ","End":"09:06.755","Text":"they\u0027re all going to be moving in the upwards direction because they come after"},{"Start":"09:06.755 ","End":"09:11.780","Text":"the peak point in relation to the direction of travel of the wave."},{"Start":"09:11.780 ","End":"09:16.385","Text":"However, any point that is located before"},{"Start":"09:16.385 ","End":"09:20.090","Text":"this peak point in relation to the direction of"},{"Start":"09:20.090 ","End":"09:24.210","Text":"the travel of the wave will be moving downwards."},{"Start":"09:24.210 ","End":"09:28.025","Text":"So our D, direction of travel, will be downwards."},{"Start":"09:28.025 ","End":"09:29.870","Text":"A point here will be downwards,"},{"Start":"09:29.870 ","End":"09:31.985","Text":"here will be downwards,"},{"Start":"09:31.985 ","End":"09:34.715","Text":"and so on and so forth."},{"Start":"09:34.715 ","End":"09:37.580","Text":"Now let\u0027s just for the sake of it,"},{"Start":"09:37.580 ","End":"09:42.695","Text":"say that here we have another wave and it looks like this,"},{"Start":"09:42.695 ","End":"09:46.235","Text":"and its direction of travel is in the opposite direction."},{"Start":"09:46.235 ","End":"09:49.185","Text":"Here we have our peak point."},{"Start":"09:49.185 ","End":"09:51.559","Text":"Let\u0027s take a look."},{"Start":"09:51.559 ","End":"09:56.623","Text":"This is point 1 and this is point 2."},{"Start":"09:56.623 ","End":"10:01.225","Text":"Try and see if you see in which direction will point 1 be going."},{"Start":"10:01.225 ","End":"10:07.550","Text":"Point 1 will be going in the upwards direction because it\u0027s located"},{"Start":"10:07.550 ","End":"10:14.305","Text":"after the peak in relation to the direction of travel of the wave."},{"Start":"10:14.305 ","End":"10:20.450","Text":"Point 2 will be moving downwards because it\u0027s"},{"Start":"10:20.450 ","End":"10:27.275","Text":"located before the peak in relation to the direction of travel of the wave."},{"Start":"10:27.275 ","End":"10:30.449","Text":"That\u0027s the end of the lesson."}],"ID":12462},{"Watched":false,"Name":"Exercise - Wave Calculation 2","Duration":"11m 57s","ChapterTopicVideoID":11995,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"Hello. In this lesson we\u0027re going to be answering this question."},{"Start":"00:04.725 ","End":"00:11.669","Text":"We have 2 disturbances which are propagating along a rope as shown in this diagram."},{"Start":"00:11.669 ","End":"00:17.685","Text":"We\u0027re being asked to sketch the disturbances at each of the following times."},{"Start":"00:17.685 ","End":"00:22.560","Text":"Here we have a picture of a disturbance at t=0."},{"Start":"00:22.560 ","End":"00:25.335","Text":"This wave is moving in this direction,"},{"Start":"00:25.335 ","End":"00:28.725","Text":"and this wave is moving in the opposite direction."},{"Start":"00:28.725 ","End":"00:37.040","Text":"We know that the width of each square is 20 centimeters and we\u0027re being told that"},{"Start":"00:37.040 ","End":"00:46.039","Text":"the velocity of the wave is equal to 10 centimeters per second."},{"Start":"00:46.039 ","End":"00:49.910","Text":"Now I\u0027ll remind you that the equation for the velocity"},{"Start":"00:49.910 ","End":"00:53.390","Text":"of a wave is equal to the square root of"},{"Start":"00:53.390 ","End":"01:02.375","Text":"the tension in the rope divided by the mass per unit length or the density of the rope."},{"Start":"01:02.375 ","End":"01:05.195","Text":"Because this is the same rope,"},{"Start":"01:05.195 ","End":"01:11.195","Text":"that means that the tension and the density of the rope is the same throughout."},{"Start":"01:11.195 ","End":"01:16.430","Text":"Which means, as we remember that the velocity of a wave or of"},{"Start":"01:16.430 ","End":"01:22.250","Text":"a disturbance depends only on the transmission medium which is propagating through."},{"Start":"01:22.250 ","End":"01:31.530","Text":"That means that this wave velocity is the same for this wave as it is for this wave."},{"Start":"01:32.390 ","End":"01:36.500","Text":"What we want to do is we want to sketch out what"},{"Start":"01:36.500 ","End":"01:40.460","Text":"these 2 waves are going to look like at the following times."},{"Start":"01:40.460 ","End":"01:44.134","Text":"Let\u0027s deal with the first question."},{"Start":"01:44.134 ","End":"01:49.005","Text":"This is the first question over here."},{"Start":"01:49.005 ","End":"01:54.730","Text":"A t=8 seconds."},{"Start":"01:56.570 ","End":"02:01.270","Text":"What we want to know is the distance that each wave has"},{"Start":"02:01.270 ","End":"02:08.335","Text":"traveled within this difference in time so within these 8 seconds."},{"Start":"02:08.335 ","End":"02:15.580","Text":"As we know, our equation is the kinematic equation for velocity,"},{"Start":"02:15.580 ","End":"02:19.600","Text":"which is the change in x divided by the change in time."},{"Start":"02:19.600 ","End":"02:24.920","Text":"What we want to know is we want to isolate out this change in x for each wave,"},{"Start":"02:24.920 ","End":"02:29.380","Text":"because each wave has the same velocity and we\u0027re using the same time frame."},{"Start":"02:29.380 ","End":"02:37.455","Text":"Delta x is equal to the velocity multiplied by Delta t. In that case,"},{"Start":"02:37.455 ","End":"02:41.330","Text":"all we have to do is we have to substitute in our velocity,"},{"Start":"02:41.330 ","End":"02:47.495","Text":"which we\u0027re being told is 10 centimeters per second multiplied by our Delta t,"},{"Start":"02:47.495 ","End":"02:50.140","Text":"which is 8 seconds."},{"Start":"02:50.140 ","End":"02:53.790","Text":"We therefore know that our Delta x,"},{"Start":"02:53.790 ","End":"02:57.710","Text":"the amount that each wave has moved,"},{"Start":"02:57.710 ","End":"03:02.820","Text":"is equal to 80 centimeters."},{"Start":"03:03.560 ","End":"03:08.675","Text":"Now we know that each square represents 20 centimeters,"},{"Start":"03:08.675 ","End":"03:14.675","Text":"so that means that each wave has to move along 4 squares,"},{"Start":"03:14.675 ","End":"03:21.125","Text":"because that would represent each wave moving in its direction by 80 centimeters."},{"Start":"03:21.125 ","End":"03:23.060","Text":"Let\u0027s go down to over here,"},{"Start":"03:23.060 ","End":"03:27.355","Text":"so the wave originally starts here and we can count 1,"},{"Start":"03:27.355 ","End":"03:30.165","Text":"2, 3, 4."},{"Start":"03:30.165 ","End":"03:33.275","Text":"Now the base of the wave,"},{"Start":"03:33.275 ","End":"03:34.955","Text":"this wave over here,"},{"Start":"03:34.955 ","End":"03:40.800","Text":"will be over here and then we can draw the diagram."},{"Start":"03:42.650 ","End":"03:45.800","Text":"This was here and now we\u0027re drawing it here,"},{"Start":"03:45.800 ","End":"03:49.920","Text":"so we can see that its amplitude is 3."},{"Start":"03:50.560 ","End":"03:53.555","Text":"Its width is 2,"},{"Start":"03:53.555 ","End":"03:58.410","Text":"and then we can just finish the line."},{"Start":"03:58.410 ","End":"04:02.376","Text":"Then this wave is traveling in the leftward direction,"},{"Start":"04:02.376 ","End":"04:05.908","Text":"so we can see that it\u0027s front corners over here,"},{"Start":"04:05.908 ","End":"04:09.620","Text":"and then it also moves 4 squares to the left,"},{"Start":"04:09.620 ","End":"04:13.425","Text":"so 1, 2, 3, 4."},{"Start":"04:13.425 ","End":"04:19.420","Text":"Then its amplitude is 1 and its width is 3."},{"Start":"04:23.870 ","End":"04:29.750","Text":"This is what our diagram looks like at 8 seconds."},{"Start":"04:29.750 ","End":"04:34.545","Text":"Now let\u0027s take a look at question number 2."},{"Start":"04:34.545 ","End":"04:39.030","Text":"Here at t=16 seconds."},{"Start":"04:39.030 ","End":"04:43.410","Text":"Again, we\u0027ve moved forward 8 seconds,"},{"Start":"04:43.410 ","End":"04:46.125","Text":"and now let\u0027s see where this is."},{"Start":"04:46.125 ","End":"04:53.045","Text":"Our Delta x is still going to be this exact equation,"},{"Start":"04:53.045 ","End":"04:54.950","Text":"which is equal to our velocity,"},{"Start":"04:54.950 ","End":"04:59.360","Text":"which is again still 10 centimeters per second,"},{"Start":"04:59.360 ","End":"05:02.105","Text":"multiplied by our change in time,"},{"Start":"05:02.105 ","End":"05:04.760","Text":"which is 8 seconds again."},{"Start":"05:04.760 ","End":"05:08.870","Text":"Again, we\u0027re going to get that we\u0027ve moved 80 centimeters,"},{"Start":"05:08.870 ","End":"05:13.100","Text":"which because each square has a width of 20 centimeters."},{"Start":"05:13.100 ","End":"05:18.230","Text":"Again, we\u0027re going to just move forward 4 squares."},{"Start":"05:18.230 ","End":"05:23.610","Text":"This corner, let\u0027s draw this over here,"},{"Start":"05:23.610 ","End":"05:25.115","Text":"starts off over here,"},{"Start":"05:25.115 ","End":"05:27.257","Text":"and then we move forward 4 squares,"},{"Start":"05:27.257 ","End":"05:29.810","Text":"so 1, 2, 3, 4,"},{"Start":"05:29.810 ","End":"05:32.315","Text":"then it has an amplitude of 3,"},{"Start":"05:32.315 ","End":"05:36.460","Text":"a width of 2 squares."},{"Start":"05:37.880 ","End":"05:41.985","Text":"Then we can draw this like so."},{"Start":"05:41.985 ","End":"05:43.725","Text":"Now what about this wave?"},{"Start":"05:43.725 ","End":"05:49.740","Text":"Its base is over here and then it moves forward 4 squares as well,"},{"Start":"05:49.740 ","End":"05:54.270","Text":"so 1, 2, 3, 4."},{"Start":"05:54.270 ","End":"06:01.395","Text":"Then it has an amplitude of 1 and a width of 3."},{"Start":"06:01.395 ","End":"06:07.460","Text":"Now we can see that our 2 waves are overlapping, so they\u0027ve met."},{"Start":"06:07.460 ","End":"06:11.360","Text":"As we remember from one of the first lessons in this chapter,"},{"Start":"06:11.360 ","End":"06:13.880","Text":"we have to use the idea of superposition,"},{"Start":"06:13.880 ","End":"06:21.585","Text":"the superposition principle, which means that we add up the 2 amplitudes."},{"Start":"06:21.585 ","End":"06:25.220","Text":"We know that this taller wave has an amplitude of"},{"Start":"06:25.220 ","End":"06:28.870","Text":"3 and the shorter wave has an amplitude of 1,"},{"Start":"06:28.870 ","End":"06:31.330","Text":"so the total amplitude,"},{"Start":"06:31.330 ","End":"06:37.960","Text":"A total is going to be equal to 3 plus 1,"},{"Start":"06:37.960 ","End":"06:41.412","Text":"which is equal to 4 squares."},{"Start":"06:41.412 ","End":"06:44.285","Text":"That\u0027s going to be in this area over here,"},{"Start":"06:44.285 ","End":"06:48.125","Text":"so let\u0027s first draw this with gray dots."},{"Start":"06:48.125 ","End":"06:53.630","Text":"We\u0027re going to have here and then we add on the amplitude of"},{"Start":"06:53.630 ","End":"07:01.190","Text":"the shorter wave up until here and then we follow through like this."},{"Start":"07:01.190 ","End":"07:05.710","Text":"Then we can carry on the rope over here."},{"Start":"07:05.710 ","End":"07:08.270","Text":"Now I\u0027m going to rub this out to make it a bit clearer,"},{"Start":"07:08.270 ","End":"07:10.685","Text":"but I hope you can see in the gray,"},{"Start":"07:10.685 ","End":"07:16.220","Text":"this is the outline of the resultant wave to the superposition."},{"Start":"07:16.220 ","End":"07:19.520","Text":"Here in the overlapping section,"},{"Start":"07:19.520 ","End":"07:21.755","Text":"we added on the 2 amplitudes."},{"Start":"07:21.755 ","End":"07:25.655","Text":"We took 1 plus 3,"},{"Start":"07:25.655 ","End":"07:29.550","Text":"and then we got this total amplitude of 4."},{"Start":"07:30.680 ","End":"07:38.465","Text":"I added in the the red dotted line representing this taller wave,"},{"Start":"07:38.465 ","End":"07:41.599","Text":"and the blue dotted lines representing the shorter wave."},{"Start":"07:41.599 ","End":"07:45.365","Text":"That it\u0027s clear how we did the superposition."},{"Start":"07:45.365 ","End":"07:47.990","Text":"Now let\u0027s answer question number 3,"},{"Start":"07:47.990 ","End":"07:53.310","Text":"which is at t=18 seconds."},{"Start":"07:53.330 ","End":"07:57.980","Text":"Now we can see that 2 seconds have gone by so I\u0027m just going to rub"},{"Start":"07:57.980 ","End":"08:02.105","Text":"out all of this and we\u0027re going to go over the same process again."},{"Start":"08:02.105 ","End":"08:03.725","Text":"Our Delta x,"},{"Start":"08:03.725 ","End":"08:07.220","Text":"how much we\u0027ve traveled in the 2 seconds is going to be our velocity,"},{"Start":"08:07.220 ","End":"08:10.220","Text":"which is 10 centimeters per second,"},{"Start":"08:10.220 ","End":"08:13.175","Text":"multiplied by our change in time,"},{"Start":"08:13.175 ","End":"08:14.825","Text":"which is 2 seconds,"},{"Start":"08:14.825 ","End":"08:18.995","Text":"so that is equal to 20 centimeters,"},{"Start":"08:18.995 ","End":"08:21.460","Text":"which as we see here,"},{"Start":"08:21.460 ","End":"08:24.750","Text":"each square has a width of 20 centimeters."},{"Start":"08:24.750 ","End":"08:31.450","Text":"Each wave is moving down 1 square."},{"Start":"08:31.820 ","End":"08:35.225","Text":"Here we can see that the red wave,"},{"Start":"08:35.225 ","End":"08:37.510","Text":"its edge is over here."},{"Start":"08:37.510 ","End":"08:41.990","Text":"We can move it down to this point over here and as we know,"},{"Start":"08:41.990 ","End":"08:43.790","Text":"the red wave is traveling in"},{"Start":"08:43.790 ","End":"08:49.984","Text":"this direction and the blue wave is traveling in this direction."},{"Start":"08:49.984 ","End":"08:51.560","Text":"Let\u0027s draw the red wave."},{"Start":"08:51.560 ","End":"08:53.360","Text":"I\u0027ll just carry on drawing it in red."},{"Start":"08:53.360 ","End":"08:58.935","Text":"If it\u0027s front end is on this square at t=16,"},{"Start":"08:58.935 ","End":"09:01.220","Text":"according to what we just calculated,"},{"Start":"09:01.220 ","End":"09:05.090","Text":"its front end is going to be now over here."},{"Start":"09:05.090 ","End":"09:12.590","Text":"Now let\u0027s draw it in a dotted line because we know that we\u0027re still using superposition."},{"Start":"09:12.590 ","End":"09:14.765","Text":"Then the blue wave,"},{"Start":"09:14.765 ","End":"09:18.710","Text":"we know that at t=16,"},{"Start":"09:18.710 ","End":"09:21.680","Text":"it\u0027s front edge was over here."},{"Start":"09:21.680 ","End":"09:25.789","Text":"But 2 seconds later it\u0027s moved 20 centimeters,"},{"Start":"09:25.789 ","End":"09:30.780","Text":"so it\u0027s moved along to 1 square to over here, it\u0027s front edge."},{"Start":"09:30.780 ","End":"09:35.270","Text":"I\u0027m just going to draw it also using a dotted line."},{"Start":"09:35.270 ","End":"09:38.630","Text":"Now we\u0027re going to superimpose both of the waves."},{"Start":"09:38.630 ","End":"09:41.330","Text":"We\u0027re going to add up the 2 amplitudes."},{"Start":"09:41.330 ","End":"09:46.490","Text":"We have that our total amplitude"},{"Start":"09:46.490 ","End":"09:49.850","Text":"at the highest point is equal to the amplitude of the red wave,"},{"Start":"09:49.850 ","End":"09:53.225","Text":"which is 3 plus amplitude of the blue wave, which is 1,"},{"Start":"09:53.225 ","End":"09:57.000","Text":"so we have an amplitude of 4 over here,"},{"Start":"09:57.320 ","End":"10:01.070","Text":"the whole place where they\u0027re overlapping."},{"Start":"10:01.070 ","End":"10:03.350","Text":"Then they stop overlapping,"},{"Start":"10:03.350 ","End":"10:09.260","Text":"so we can line our waves"},{"Start":"10:09.260 ","End":"10:17.640","Text":"like so and the rest of the rope looks like this."},{"Start":"10:17.640 ","End":"10:21.540","Text":"Finally, let\u0027s do Question 4."},{"Start":"10:21.540 ","End":"10:25.839","Text":"Question 4 is at t=22 seconds."},{"Start":"10:26.120 ","End":"10:29.470","Text":"Let\u0027s do this again."},{"Start":"10:30.500 ","End":"10:33.870","Text":"Our velocity is 10,"},{"Start":"10:33.870 ","End":"10:37.140","Text":"our Delta t is 22 minus 18,"},{"Start":"10:37.140 ","End":"10:42.435","Text":"which is 4, so that is equal to 40 centimeters."},{"Start":"10:42.435 ","End":"10:46.710","Text":"As we know, each square width is 20 centimeters,"},{"Start":"10:46.710 ","End":"10:56.925","Text":"so that means that we\u0027re moving in each direction 2 squares for each wave."},{"Start":"10:56.925 ","End":"11:02.805","Text":"A red wave, which its frontend was over here,"},{"Start":"11:02.805 ","End":"11:04.815","Text":"so 4 seconds later,"},{"Start":"11:04.815 ","End":"11:06.795","Text":"we set it move 2 squares,"},{"Start":"11:06.795 ","End":"11:08.330","Text":"so now it\u0027s over here."},{"Start":"11:08.330 ","End":"11:17.805","Text":"Let\u0027s draw it in red and our blue square,"},{"Start":"11:17.805 ","End":"11:22.670","Text":"so its front end used to be over here,"},{"Start":"11:22.670 ","End":"11:29.670","Text":"but now it\u0027s moved forward 2 squares so now it\u0027s going to be over here."},{"Start":"11:29.710 ","End":"11:33.720","Text":"We\u0027ll draw it in blue."},{"Start":"11:35.860 ","End":"11:43.530","Text":"Now we can draw the rest of the rope and of course"},{"Start":"11:43.530 ","End":"11:48.095","Text":"the waves come out from their interference"},{"Start":"11:48.095 ","End":"11:53.885","Text":"undisturbed and carry on traveling in the respective directions."},{"Start":"11:53.885 ","End":"11:56.970","Text":"That\u0027s the end of this lesson."}],"ID":12463},{"Watched":false,"Name":"Exercise - Wave Calculation 3","Duration":"7m 39s","ChapterTopicVideoID":11996,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"Hello, in this lesson we\u0027re going to be answering the following question."},{"Start":"00:03.930 ","End":"00:08.505","Text":"2 waves travel towards 1 another as described in the diagram."},{"Start":"00:08.505 ","End":"00:11.565","Text":"Sketch the disturbances at the following times,"},{"Start":"00:11.565 ","End":"00:13.590","Text":"t is equal to 8 seconds,"},{"Start":"00:13.590 ","End":"00:16.965","Text":"12 seconds, 13 seconds, and 16 seconds."},{"Start":"00:16.965 ","End":"00:20.235","Text":"This is what we have if t is equal to 0 seconds."},{"Start":"00:20.235 ","End":"00:24.510","Text":"We\u0027re being told that the width of each square here represents 1 meter,"},{"Start":"00:24.510 ","End":"00:30.165","Text":"and that the velocity of the waves is equal to 0.5 meter per second."},{"Start":"00:30.165 ","End":"00:33.585","Text":"Now, I\u0027m reminding you that the velocity of the wave is"},{"Start":"00:33.585 ","End":"00:37.615","Text":"only dependent on the transmission material,"},{"Start":"00:37.615 ","End":"00:39.139","Text":"or the transmission medium."},{"Start":"00:39.139 ","End":"00:44.705","Text":"Therefore, both of these waves are traveling the exact same velocity."},{"Start":"00:44.705 ","End":"00:46.535","Text":"In the previous lesson,"},{"Start":"00:46.535 ","End":"00:49.918","Text":"we saw that in order to see the position of the wave,"},{"Start":"00:49.918 ","End":"00:52.940","Text":"we\u0027re going to use the simple kinematic equation, that says,"},{"Start":"00:52.940 ","End":"01:00.285","Text":"that the distance traveled is equal to the velocity multiplied by the time difference."},{"Start":"01:00.285 ","End":"01:03.945","Text":"Let\u0027s answer question number 1,"},{"Start":"01:03.945 ","End":"01:07.170","Text":"which is at t is equal to 8 seconds."},{"Start":"01:07.170 ","End":"01:11.130","Text":"That means that there\u0027s a difference of 8 seconds."},{"Start":"01:11.130 ","End":"01:15.080","Text":"Our Delta x is going to be equal to our velocity,"},{"Start":"01:15.080 ","End":"01:16.655","Text":"which is 0.5,"},{"Start":"01:16.655 ","End":"01:19.280","Text":"multiplied by our Delta t,"},{"Start":"01:19.280 ","End":"01:21.560","Text":"which is equal to 8."},{"Start":"01:21.560 ","End":"01:26.735","Text":"So 0.5 times 8 is equal to 4 meters."},{"Start":"01:26.735 ","End":"01:30.890","Text":"So each wave is going to travel 4 meters in its direction of travel,"},{"Start":"01:30.890 ","End":"01:34.495","Text":"where we know that each square represents 1 meter."},{"Start":"01:34.495 ","End":"01:40.130","Text":"That means that each wave is going to move 4 squares forward."},{"Start":"01:40.130 ","End":"01:43.190","Text":"Let\u0027s take a look at this wave first."},{"Start":"01:43.190 ","End":"01:46.830","Text":"We can see that it\u0027s front begins over here,"},{"Start":"01:46.830 ","End":"01:48.825","Text":"t is equal to 0."},{"Start":"01:48.825 ","End":"01:52.170","Text":"Now it\u0027s going to move forward 4 squares,"},{"Start":"01:52.170 ","End":"01:55.005","Text":"so 1, 2, 3, 4."},{"Start":"01:55.005 ","End":"01:56.625","Text":"This is its front,"},{"Start":"01:56.625 ","End":"02:02.450","Text":"and we can see it has an amplitude of 3 and a width of 4."},{"Start":"02:02.450 ","End":"02:06.175","Text":"1, 2, 3, 4."},{"Start":"02:06.175 ","End":"02:09.035","Text":"Now let\u0027s look at this red wave."},{"Start":"02:09.035 ","End":"02:10.640","Text":"Obviously this isn\u0027t meant to be bumpy,"},{"Start":"02:10.640 ","End":"02:12.210","Text":"this has meant to be flat."},{"Start":"02:12.210 ","End":"02:16.970","Text":"This wave in red starts off over here."},{"Start":"02:16.970 ","End":"02:18.275","Text":"This is its front,"},{"Start":"02:18.275 ","End":"02:20.435","Text":"and it moves forward 4 squares,"},{"Start":"02:20.435 ","End":"02:24.335","Text":"so 1, 2, 3, 4."},{"Start":"02:24.335 ","End":"02:31.140","Text":"Now its front is here and it\u0027s facing downwards in this negative direction."},{"Start":"02:31.140 ","End":"02:34.590","Text":"Now we can draw this,"},{"Start":"02:34.590 ","End":"02:37.520","Text":"where the black wave is still traveling in this direction,"},{"Start":"02:37.520 ","End":"02:40.280","Text":"and the red wave is still traveling in this direction."},{"Start":"02:40.280 ","End":"02:43.350","Text":"Now let\u0027s answer question number 2."},{"Start":"02:43.350 ","End":"02:47.400","Text":"If t is equal to 12 seconds."},{"Start":"02:47.400 ","End":"02:50.010","Text":"Let\u0027s rub all of this out."},{"Start":"02:50.010 ","End":"02:55.625","Text":"Now, our velocity is still 0.5 meters per second."},{"Start":"02:55.625 ","End":"03:00.170","Text":"But now our Delta t is equal to 12 minus 8, which is 4,"},{"Start":"03:00.170 ","End":"03:05.475","Text":"so 0.5 times 4 is equal to 2 meters."},{"Start":"03:05.475 ","End":"03:08.630","Text":"Because each square has a width of 1 meter,"},{"Start":"03:08.630 ","End":"03:13.955","Text":"that means that each wave is going to move 2 squares in its direction of travel."},{"Start":"03:13.955 ","End":"03:17.495","Text":"Let\u0027s draw this over here."},{"Start":"03:17.495 ","End":"03:20.495","Text":"This wave at t is equal to 8,"},{"Start":"03:20.495 ","End":"03:22.175","Text":"it\u0027s front was over here."},{"Start":"03:22.175 ","End":"03:24.425","Text":"But now it\u0027s moving 2 squares,"},{"Start":"03:24.425 ","End":"03:26.630","Text":"1, 2, so its front will be here."},{"Start":"03:26.630 ","End":"03:29.465","Text":"Let\u0027s draw it. It has an amplitude of 3,"},{"Start":"03:29.465 ","End":"03:32.315","Text":"a width of 4, and there we go."},{"Start":"03:32.315 ","End":"03:34.760","Text":"Now let\u0027s deal with the red wave."},{"Start":"03:34.760 ","End":"03:36.725","Text":"If t is equal to 8,"},{"Start":"03:36.725 ","End":"03:38.465","Text":"its front was over here."},{"Start":"03:38.465 ","End":"03:40.505","Text":"Now its also moving 1,"},{"Start":"03:40.505 ","End":"03:43.580","Text":"2 squares in its direction of travel."},{"Start":"03:43.580 ","End":"03:46.830","Text":"Now it\u0027s located like so."},{"Start":"03:46.830 ","End":"03:49.740","Text":"Now directions of travel are still like this."},{"Start":"03:49.740 ","End":"03:53.505","Text":"Now let\u0027s answer question 3."},{"Start":"03:53.505 ","End":"03:58.935","Text":"Now we\u0027re dealing with t is equal to 13 seconds."},{"Start":"03:58.935 ","End":"04:01.635","Text":"Let\u0027s rub out all of this."},{"Start":"04:01.635 ","End":"04:09.595","Text":"Again, our velocity is 0.5 and our time difference is 1 second."},{"Start":"04:09.595 ","End":"04:14.160","Text":"0.5 times 1 is simply 0.5 or just,"},{"Start":"04:14.160 ","End":"04:15.660","Text":"we can write 1/2."},{"Start":"04:15.660 ","End":"04:17.175","Text":"We\u0027re moving 1/2 a meter."},{"Start":"04:17.175 ","End":"04:21.880","Text":"That means each shape is moving 1/2 a square."},{"Start":"04:21.880 ","End":"04:23.855","Text":"This is what it would look like."},{"Start":"04:23.855 ","End":"04:26.990","Text":"I\u0027m just trying to speed this up. You got the idea."},{"Start":"04:26.990 ","End":"04:32.155","Text":"It\u0027s moved 1/2 a square in each direction, each wave."},{"Start":"04:32.155 ","End":"04:35.115","Text":"Now let\u0027s answer question number 4."},{"Start":"04:35.115 ","End":"04:42.070","Text":"I changed the number to 25 seconds so that we can see the difference."},{"Start":"04:42.160 ","End":"04:45.469","Text":"Again, we\u0027re using the same equation."},{"Start":"04:45.469 ","End":"04:51.020","Text":"How much is each wave going to move in this time frame?"},{"Start":"04:51.020 ","End":"05:00.285","Text":"Our v is velocity is 0.5 and our delta t is now 25 minus 13, which is 12."},{"Start":"05:00.285 ","End":"05:03.979","Text":"1/2 times 12 is equal to 6 meters."},{"Start":"05:03.979 ","End":"05:06.305","Text":"Each square has a width of 1 meter,"},{"Start":"05:06.305 ","End":"05:13.890","Text":"so each wave is moving 6 squares in its direction of travel. Let\u0027s see."},{"Start":"05:13.890 ","End":"05:17.015","Text":"This wave begins in the middle over here."},{"Start":"05:17.015 ","End":"05:19.835","Text":"We move forward, 1, 2,"},{"Start":"05:19.835 ","End":"05:25.095","Text":"3, 4, 5, 6."},{"Start":"05:25.095 ","End":"05:28.305","Text":"Now we go 3 up,"},{"Start":"05:28.305 ","End":"05:30.690","Text":"I\u0027m going to do this in a dotted line actually."},{"Start":"05:30.690 ","End":"05:34.270","Text":"Because soon we\u0027re going to have some superposition."},{"Start":"05:34.270 ","End":"05:36.520","Text":"This is our black wave,"},{"Start":"05:36.520 ","End":"05:37.705","Text":"and it goes 1,"},{"Start":"05:37.705 ","End":"05:42.045","Text":"2, 3, 4, across."},{"Start":"05:42.045 ","End":"05:46.815","Text":"This is our larger wave, I can just draw this."},{"Start":"05:46.815 ","End":"05:52.865","Text":"Then our red wave is with its peak over here right now."},{"Start":"05:52.865 ","End":"05:55.065","Text":"It also moves 6,"},{"Start":"05:55.065 ","End":"05:57.210","Text":"but now in the leftward direction, so,"},{"Start":"05:57.210 ","End":"05:58.635","Text":"1, 2,"},{"Start":"05:58.635 ","End":"06:02.235","Text":"3, 4, 5, 6."},{"Start":"06:02.235 ","End":"06:05.570","Text":"Now its peak is over here."},{"Start":"06:05.570 ","End":"06:08.140","Text":"This is what it looks like."},{"Start":"06:08.140 ","End":"06:11.750","Text":"Now what we\u0027re going to do is in green,"},{"Start":"06:11.750 ","End":"06:14.630","Text":"we\u0027re going to draw the subsequent wave."},{"Start":"06:14.630 ","End":"06:17.255","Text":"We can see we start over here,"},{"Start":"06:17.255 ","End":"06:20.300","Text":"and we can see that it looks like so."},{"Start":"06:20.300 ","End":"06:22.220","Text":"Until we get over here,"},{"Start":"06:22.220 ","End":"06:28.790","Text":"where we can see that our red wave is overlapping with our black wave."},{"Start":"06:28.790 ","End":"06:37.075","Text":"Over here we have some triangle where its peak magnitude is negative 1."},{"Start":"06:37.075 ","End":"06:41.075","Text":"The magnitude of our black wave is 3."},{"Start":"06:41.075 ","End":"06:46.180","Text":"Our total amplitude is going to be equal"},{"Start":"06:46.180 ","End":"06:52.765","Text":"to the amplitude of the black wave 3 plus the amplitude of our red wave,"},{"Start":"06:52.765 ","End":"06:55.540","Text":"which is negative 1, but in a triangle."},{"Start":"06:55.540 ","End":"06:57.985","Text":"We\u0027ll just draw it in a triangle shape."},{"Start":"06:57.985 ","End":"07:03.435","Text":"This is equal to obviously, 2."},{"Start":"07:03.435 ","End":"07:08.550","Text":"Now we\u0027re going to draw it simply like so,"},{"Start":"07:08.550 ","End":"07:10.970","Text":"in this triangle way."},{"Start":"07:10.970 ","End":"07:17.405","Text":"Then we\u0027re just going to finish the outline of our resultant wave."},{"Start":"07:17.405 ","End":"07:22.060","Text":"This is what our wave looks like at the superposition."},{"Start":"07:22.060 ","End":"07:23.544","Text":"Then, of course,"},{"Start":"07:23.544 ","End":"07:25.150","Text":"in a few more seconds,"},{"Start":"07:25.150 ","End":"07:27.555","Text":"each wave is going to part,"},{"Start":"07:27.555 ","End":"07:31.279","Text":"where our red wave is going to carry on undisturbed in this direction."},{"Start":"07:31.279 ","End":"07:36.965","Text":"Our black wave is going to carry on undisturbed in this direction."},{"Start":"07:36.965 ","End":"07:40.140","Text":"That\u0027s the end of this lesson."}],"ID":12464},{"Watched":false,"Name":"Exercise - Wave Calculations 4","Duration":"19m 11s","ChapterTopicVideoID":11997,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"Hello. In this lesson,"},{"Start":"00:02.700 ","End":"00:04.830","Text":"we\u0027re going to be solving this question."},{"Start":"00:04.830 ","End":"00:09.200","Text":"We\u0027re being told that a triangular pulse reaches the end of the rope."},{"Start":"00:09.200 ","End":"00:14.910","Text":"Then we\u0027re asked to draw the pulse given these different situations."},{"Start":"00:14.910 ","End":"00:18.450","Text":"Before we look at the different questions,"},{"Start":"00:18.450 ","End":"00:20.670","Text":"let\u0027s draw our rope,"},{"Start":"00:20.670 ","End":"00:27.555","Text":"and then we have our pulse moving forwards like so."},{"Start":"00:27.555 ","End":"00:31.895","Text":"This is what we\u0027re dealing with and we\u0027re going to remember that"},{"Start":"00:31.895 ","End":"00:39.505","Text":"our energy is proportional to our amplitude^2."},{"Start":"00:39.505 ","End":"00:42.435","Text":"Now let\u0027s answer question number 1."},{"Start":"00:42.435 ","End":"00:46.970","Text":"We\u0027re trying to draw the pulse when the end of the rope is tied to the wall."},{"Start":"00:46.970 ","End":"00:50.420","Text":"As we know, if the end of the rope is tied to the wall,"},{"Start":"00:50.420 ","End":"00:59.870","Text":"then that means that we\u0027re dealing with a reflection in both the x and the y axis."},{"Start":"00:59.870 ","End":"01:06.320","Text":"In that case, let\u0027s draw our pulse or another pulse."},{"Start":"01:06.320 ","End":"01:07.865","Text":"This is our rope."},{"Start":"01:07.865 ","End":"01:10.180","Text":"Here, it\u0027s tied to the end of the wall."},{"Start":"01:10.180 ","End":"01:13.990","Text":"If our pulse is coming in this direction,"},{"Start":"01:13.990 ","End":"01:18.080","Text":"with the right angled side coming here first,"},{"Start":"01:18.080 ","End":"01:23.150","Text":"first of all, we know that it\u0027s going to bounce back with the right angle side,"},{"Start":"01:23.150 ","End":"01:26.420","Text":"also coming first back to the origin."},{"Start":"01:26.420 ","End":"01:29.240","Text":"That\u0027s our reflection in the y-axis."},{"Start":"01:29.240 ","End":"01:33.995","Text":"Then we know that because the end of the rope is tied to the wall,"},{"Start":"01:33.995 ","End":"01:37.055","Text":"that it\u0027s also going to be reflected in the x-axis."},{"Start":"01:37.055 ","End":"01:39.080","Text":"It\u0027s going to come back upside down."},{"Start":"01:39.080 ","End":"01:48.670","Text":"It\u0027s going to return looking like this and moving in this direction."},{"Start":"01:50.210 ","End":"01:53.600","Text":"This is what our pulse will look like."},{"Start":"01:53.600 ","End":"01:57.020","Text":"Here\u0027s our wall and the rope is tied to it, and of course,"},{"Start":"01:57.020 ","End":"02:01.430","Text":"the pulse will carry on moving back towards the origin."},{"Start":"02:01.430 ","End":"02:06.410","Text":"Now, of course, we know that the energy is proportional to the amplitude^2."},{"Start":"02:06.410 ","End":"02:10.040","Text":"If we know that the energy at the end,"},{"Start":"02:10.040 ","End":"02:13.670","Text":"final, is equal to the pulse\u0027s initial energy,"},{"Start":"02:13.670 ","End":"02:21.995","Text":"then we know that the final amplitude is equal to the initial amplitude."},{"Start":"02:21.995 ","End":"02:24.935","Text":"Now let\u0027s look at question 2."},{"Start":"02:24.935 ","End":"02:27.350","Text":"Here we\u0027re drawing the pulse."},{"Start":"02:27.350 ","End":"02:33.230","Text":"However, this time the end of the rope is draped over a pole."},{"Start":"02:33.680 ","End":"02:37.635","Text":"Here we have our rope."},{"Start":"02:37.635 ","End":"02:43.175","Text":"Then here we have our pole and it\u0027s just draped around."},{"Start":"02:43.175 ","End":"02:48.115","Text":"That means that the rope can move up and down in this y direction."},{"Start":"02:48.115 ","End":"02:52.895","Text":"Here our pulse is coming in this direction with the right-angled side,"},{"Start":"02:52.895 ","End":"02:57.945","Text":"first, arriving to the wall or the pole."},{"Start":"02:57.945 ","End":"03:01.860","Text":"If the rope is draped,"},{"Start":"03:01.860 ","End":"03:04.540","Text":"that means that it\u0027s a loose end,"},{"Start":"03:05.150 ","End":"03:08.580","Text":"when we\u0027re dealing with loose-end reflection,"},{"Start":"03:08.580 ","End":"03:16.735","Text":"we remember that in this case we only have a reflection of the wave in the y-axis."},{"Start":"03:16.735 ","End":"03:19.870","Text":"When we have a fixed end,"},{"Start":"03:19.870 ","End":"03:21.655","Text":"this was the fixed end,"},{"Start":"03:21.655 ","End":"03:27.150","Text":"so then we have x and y reflection."},{"Start":"03:27.150 ","End":"03:29.745","Text":"Here we just have y reflection."},{"Start":"03:29.745 ","End":"03:34.765","Text":"Our pulse is coming and the right-angled side is first."},{"Start":"03:34.765 ","End":"03:38.690","Text":"That means that in the reflected pulse,"},{"Start":"03:38.690 ","End":"03:41.740","Text":"the pulse that\u0027s traveling back to the origin,"},{"Start":"03:41.740 ","End":"03:47.150","Text":"the right-angled side is again going to be first,"},{"Start":"03:47.150 ","End":"03:53.850","Text":"and the tail is again going to be last and it will be traveling in this direction."},{"Start":"03:55.300 ","End":"03:58.985","Text":"These were 2 nice and easy ones."},{"Start":"03:58.985 ","End":"04:03.650","Text":"If you had a problem seeing what happened in examples 1 and 2,"},{"Start":"04:03.650 ","End":"04:06.560","Text":"then please go back to the previous video of"},{"Start":"04:06.560 ","End":"04:12.850","Text":"wave characteristics because there I explained the idea of reflection."},{"Start":"04:12.850 ","End":"04:15.900","Text":"Now let\u0027s go on to question number 3."},{"Start":"04:15.900 ","End":"04:17.500","Text":"Now, in question number 3,"},{"Start":"04:17.500 ","End":"04:21.175","Text":"we have a few things to take into consideration."},{"Start":"04:21.175 ","End":"04:23.835","Text":"We\u0027re going to do this a lot slower."},{"Start":"04:23.835 ","End":"04:31.105","Text":"Question number 3 is that the end of the rope is tied to another heavier rope."},{"Start":"04:31.105 ","End":"04:37.515","Text":"That means that we have our rope over here that the pulse is moving through,"},{"Start":"04:37.515 ","End":"04:44.700","Text":"and then it\u0027s attached to another heavy rope, like so."},{"Start":"04:45.950 ","End":"04:49.115","Text":"In that video, wave characteristics,"},{"Start":"04:49.115 ","End":"04:52.175","Text":"we spoke about 2 ropes tied together,"},{"Start":"04:52.175 ","End":"04:56.375","Text":"and we said that if we have a lighter rope attached to a heavier rope,"},{"Start":"04:56.375 ","End":"05:00.720","Text":"then that resembles a fixed end."},{"Start":"05:00.720 ","End":"05:04.385","Text":"If we have a heavier rope attached to a lighter rope,"},{"Start":"05:04.385 ","End":"05:07.160","Text":"then that resembles a loose end."},{"Start":"05:07.160 ","End":"05:09.433","Text":"In question 3, the rope,"},{"Start":"05:09.433 ","End":"05:10.490","Text":"it\u0027s a lighter rope,"},{"Start":"05:10.490 ","End":"05:12.568","Text":"attached to a heavier rope,"},{"Start":"05:12.568 ","End":"05:18.810","Text":"so that means that we can consider this a fixed end."},{"Start":"05:18.810 ","End":"05:22.620","Text":"This is going to help us to solve this."},{"Start":"05:22.620 ","End":"05:28.430","Text":"Let\u0027s draw, first of all, our pulse approaching like so."},{"Start":"05:28.430 ","End":"05:30.465","Text":"This is our original pulse."},{"Start":"05:30.465 ","End":"05:34.470","Text":"Now let\u0027s go on to draw our reflections."},{"Start":"05:34.470 ","End":"05:41.675","Text":"At some stage, our original pulse is going to reach this point over here."},{"Start":"05:41.675 ","End":"05:44.120","Text":"Let\u0027s draw it like so."},{"Start":"05:44.120 ","End":"05:48.760","Text":"We\u0027re imagining that this pulse looks exactly like this pulse."},{"Start":"05:48.760 ","End":"05:52.055","Text":"It\u0027s the same pulse just a few moments later."},{"Start":"05:52.055 ","End":"05:55.325","Text":"Now in blue, let\u0027s draw what happens."},{"Start":"05:55.325 ","End":"06:00.470","Text":"First of all, our original pulse has reached the end,"},{"Start":"06:00.470 ","End":"06:03.305","Text":"which we\u0027re saying is similar to a fixed end."},{"Start":"06:03.305 ","End":"06:05.630","Text":"In our lesson about wave characteristics,"},{"Start":"06:05.630 ","End":"06:09.050","Text":"we saw that in a situation like this,"},{"Start":"06:09.050 ","End":"06:13.550","Text":"original pulse is going to split into 2 pulses."},{"Start":"06:13.550 ","End":"06:18.590","Text":"One is a reflected pulse and another one is a transmitted pulse."},{"Start":"06:18.590 ","End":"06:26.535","Text":"We said that the reflected pulse is going to travel back to the origin."},{"Start":"06:26.535 ","End":"06:29.490","Text":"Now, what\u0027s it going to look like?"},{"Start":"06:29.490 ","End":"06:32.905","Text":"First of all, in questions 1 and 2,"},{"Start":"06:32.905 ","End":"06:39.095","Text":"we saw that this equation went for both of these cases."},{"Start":"06:39.095 ","End":"06:47.180","Text":"But here, we know that our energy at the end does not equal the energy at the beginning."},{"Start":"06:47.180 ","End":"06:50.225","Text":"Let\u0027s write this out over here."},{"Start":"06:50.225 ","End":"06:59.600","Text":"The energy of the original pulse is going to be much bigger than the pulse that"},{"Start":"06:59.600 ","End":"07:10.480","Text":"is reflected and then the pulse that is transmitted."},{"Start":"07:10.520 ","End":"07:16.355","Text":"E_1,f is the reflected pulse and E_2,f is the transmitted pulse."},{"Start":"07:16.355 ","End":"07:17.825","Text":"We know that the energy,"},{"Start":"07:17.825 ","End":"07:20.249","Text":"the original energy is much bigger,"},{"Start":"07:20.249 ","End":"07:25.265","Text":"and therefore, we can also write that for sure,"},{"Start":"07:25.265 ","End":"07:29.990","Text":"the amplitude of the transmitted pulse is going to"},{"Start":"07:29.990 ","End":"07:35.850","Text":"be less than the original pulse\u0027s amplitude."},{"Start":"07:36.380 ","End":"07:38.510","Text":"Now let\u0027s draw what happens."},{"Start":"07:38.510 ","End":"07:40.115","Text":"So we have a fixed end."},{"Start":"07:40.115 ","End":"07:43.250","Text":"First of all, we know that that means that we\u0027re going to have"},{"Start":"07:43.250 ","End":"07:48.325","Text":"a reflection in the y-axis and a reflection in the x-axis."},{"Start":"07:48.325 ","End":"07:51.860","Text":"What does that mean? That means that if this side with"},{"Start":"07:51.860 ","End":"07:54.740","Text":"a right angle reach the fixed end first,"},{"Start":"07:54.740 ","End":"07:59.000","Text":"it\u0027s also going to be the side that\u0027s going to first return back to"},{"Start":"07:59.000 ","End":"08:01.865","Text":"the origin and we know that it\u0027s returning"},{"Start":"08:01.865 ","End":"08:05.720","Text":"upside down because it\u0027s also reflected in the x-axis."},{"Start":"08:05.720 ","End":"08:08.260","Text":"We can draw it like this."},{"Start":"08:08.260 ","End":"08:11.420","Text":"We can remember that the amplitude of"},{"Start":"08:11.420 ","End":"08:15.395","Text":"this pulse is going to be less than the amplitude of the original pulse."},{"Start":"08:15.395 ","End":"08:20.720","Text":"That means that it\u0027s going to be just a little bit lower."},{"Start":"08:20.720 ","End":"08:23.990","Text":"We can see it has a lower amplitude than"},{"Start":"08:23.990 ","End":"08:30.335","Text":"the original pulse and it\u0027s going to be traveling back to the origin, like so."},{"Start":"08:30.335 ","End":"08:38.100","Text":"But we know that we have another wave or another pulse which is transmitted."},{"Start":"08:38.100 ","End":"08:41.345","Text":"Another thing that we have to take into account,"},{"Start":"08:41.345 ","End":"08:45.365","Text":"let\u0027s just scroll down to give us a little bit more space,"},{"Start":"08:45.365 ","End":"08:50.875","Text":"is that our velocity is only dependent on the transmission medium."},{"Start":"08:50.875 ","End":"08:53.195","Text":"We know that our velocity,"},{"Start":"08:53.195 ","End":"08:55.385","Text":"what\u0027s our equation for velocity of a wave?"},{"Start":"08:55.385 ","End":"09:01.895","Text":"It\u0027s equal to the square root of the tension in the rope,"},{"Start":"09:01.895 ","End":"09:03.785","Text":"which because they\u0027re tied together,"},{"Start":"09:03.785 ","End":"09:08.480","Text":"the tension in the lighter rope and in the heavier rope is the same,"},{"Start":"09:08.480 ","End":"09:10.550","Text":"divided by Mu,"},{"Start":"09:10.550 ","End":"09:13.935","Text":"which is the density of the rope or the mass per unit length."},{"Start":"09:13.935 ","End":"09:17.765","Text":"In the heavier rope, the mass per unit length is larger,"},{"Start":"09:17.765 ","End":"09:21.080","Text":"which means that the velocity of the wave traveling through"},{"Start":"09:21.080 ","End":"09:25.005","Text":"the heavier rope is going to be slower."},{"Start":"09:25.005 ","End":"09:28.745","Text":"What does that mean? Let\u0027s say that"},{"Start":"09:28.745 ","End":"09:35.360","Text":"the original wave has a length from its head until its tail of,"},{"Start":"09:35.360 ","End":"09:39.200","Text":"let\u0027s say that this is 10 centimeters."},{"Start":"09:39.200 ","End":"09:41.600","Text":"Then depending on what our Mu is,"},{"Start":"09:41.600 ","End":"09:45.515","Text":"we can say that in the heavier rope,"},{"Start":"09:45.515 ","End":"09:48.455","Text":"the velocity will be less."},{"Start":"09:48.455 ","End":"09:57.825","Text":"Let\u0027s say the original rope was moving at 10 centimeters per second."},{"Start":"09:57.825 ","End":"09:59.600","Text":"Then in the heavier rope,"},{"Start":"09:59.600 ","End":"10:03.050","Text":"we can say that the velocity of the pulse is equal to"},{"Start":"10:03.050 ","End":"10:07.745","Text":"5 centimeters per second because it\u0027s twice as dense."},{"Start":"10:07.745 ","End":"10:11.270","Text":"What does that mean? That means that the head of"},{"Start":"10:11.270 ","End":"10:16.140","Text":"the original pulse is going to reach our tied or our fixed end,"},{"Start":"10:16.250 ","End":"10:19.400","Text":"and then in 1 second,"},{"Start":"10:19.400 ","End":"10:21.608","Text":"it would have moved 10 centimeters."},{"Start":"10:21.608 ","End":"10:27.247","Text":"In 1 second, the tail is meant to be here,"},{"Start":"10:27.247 ","End":"10:29.480","Text":"but in the heavier rope,"},{"Start":"10:29.480 ","End":"10:33.223","Text":"our pulse can just move 5 centimeters per second,"},{"Start":"10:33.223 ","End":"10:37.850","Text":"which means that in the same time that it takes on"},{"Start":"10:37.850 ","End":"10:43.995","Text":"this section of the rope for the tail to go from here till here,"},{"Start":"10:43.995 ","End":"10:47.540","Text":"the head of this pulse would have only"},{"Start":"10:47.540 ","End":"10:52.025","Text":"moved half of that distance till around about here."},{"Start":"10:52.025 ","End":"10:55.400","Text":"Let\u0027s draw this transmitted pulse in red."},{"Start":"10:55.400 ","End":"11:03.700","Text":"What\u0027s happening is that our transmitted pulse is going to be squashed."},{"Start":"11:03.700 ","End":"11:07.060","Text":"The amplitude is also going to be a little bit less and"},{"Start":"11:07.060 ","End":"11:10.315","Text":"the pulse is also going to be squashed a little bit."},{"Start":"11:10.315 ","End":"11:14.410","Text":"Then of course it\u0027s going to be traveling in this direction and of course,"},{"Start":"11:14.410 ","End":"11:16.090","Text":"as time goes by,"},{"Start":"11:16.090 ","End":"11:19.420","Text":"so let\u0027s say we wait 2 seconds."},{"Start":"11:19.420 ","End":"11:24.400","Text":"Our reflected pulse will be approximately here,"},{"Start":"11:24.400 ","End":"11:26.379","Text":"traveling in this direction."},{"Start":"11:26.379 ","End":"11:28.150","Text":"However, in 2 seconds,"},{"Start":"11:28.150 ","End":"11:31.765","Text":"the velocity and the heavier rope is much less."},{"Start":"11:31.765 ","End":"11:38.755","Text":"In 2 seconds, it would have only moved 10 centimeters rather than,"},{"Start":"11:38.755 ","End":"11:41.960","Text":"over here, 20 centimeters."},{"Start":"11:43.650 ","End":"11:46.780","Text":"Let\u0027s just rub out this arrow,"},{"Start":"11:46.780 ","End":"11:51.860","Text":"this pulse would have only moved until maybe here."},{"Start":"11:51.960 ","End":"11:55.720","Text":"We can see that this change in density of the rope"},{"Start":"11:55.720 ","End":"11:59.995","Text":"affects the velocity that the pulse is traveling at,"},{"Start":"11:59.995 ","End":"12:06.550","Text":"but it also affects the shape of the pulse and it just squashes it and makes it shorter."},{"Start":"12:06.550 ","End":"12:14.420","Text":"Here, our new pulse is going to be only 5 centimeters."},{"Start":"12:14.970 ","End":"12:17.890","Text":"What we can see that in a case,"},{"Start":"12:17.890 ","End":"12:20.155","Text":"when a rope is tied to another rope,"},{"Start":"12:20.155 ","End":"12:25.810","Text":"we have to take into account if it\u0027s a heavier rope tied to a lighter rope or"},{"Start":"12:25.810 ","End":"12:28.840","Text":"lighter rope tied to a heavier rope and then we can"},{"Start":"12:28.840 ","End":"12:32.710","Text":"see if it\u0027s a loose end or like in this case it\u0027s a fixed end."},{"Start":"12:32.710 ","End":"12:37.390","Text":"Then we\u0027re going to always have a transmitted pulse and"},{"Start":"12:37.390 ","End":"12:42.550","Text":"a reflected pulse and the reflected pulse is obviously going to be dependent."},{"Start":"12:42.550 ","End":"12:48.680","Text":"What it looks like depending on if it\u0027s a fixed end or a loose end connection over here."},{"Start":"12:48.870 ","End":"12:56.695","Text":"The transmitted pulse potentially will be squashed"},{"Start":"12:56.695 ","End":"13:05.905","Text":"shorter if the rope that it\u0027s tied to is denser than the original rope."},{"Start":"13:05.905 ","End":"13:09.100","Text":"Of course, the change in density will always"},{"Start":"13:09.100 ","End":"13:12.670","Text":"affect the velocity of the transmitted pulse."},{"Start":"13:12.670 ","End":"13:16.330","Text":"Now let\u0027s take a look at question number 4."},{"Start":"13:16.330 ","End":"13:19.615","Text":"Question number 4 is similar to question number 3,"},{"Start":"13:19.615 ","End":"13:23.875","Text":"but this time the end of the rope is tied to a lighter rope."},{"Start":"13:23.875 ","End":"13:27.265","Text":"Let\u0027s scroll down to give us a little bit more space."},{"Start":"13:27.265 ","End":"13:32.545","Text":"That means that we have some heavier rope,"},{"Start":"13:32.545 ","End":"13:36.775","Text":"so this is our heavier rope and it\u0027s tied to a lighter rope."},{"Start":"13:36.775 ","End":"13:40.345","Text":"First of all, when we have a heavier rope tied to a lighter rope,"},{"Start":"13:40.345 ","End":"13:45.730","Text":"this can be thought of as similar to a loose end."},{"Start":"13:45.730 ","End":"13:47.890","Text":"In a loose end, like we saw,"},{"Start":"13:47.890 ","End":"13:54.040","Text":"our reflected pulse is only going to be reflected in the y-axis,"},{"Start":"13:54.040 ","End":"14:00.085","Text":"whereas the fixed end is reflected both in the y-axis and in the x-axis."},{"Start":"14:00.085 ","End":"14:02.380","Text":"Let\u0027s draw our original pulse."},{"Start":"14:02.380 ","End":"14:05.935","Text":"Let\u0027s say it looks like this and it\u0027s traveling in this direction."},{"Start":"14:05.935 ","End":"14:10.480","Text":"Once it gets to this end,"},{"Start":"14:10.480 ","End":"14:16.420","Text":"so this side with a right angle is the first that reaches the loose end."},{"Start":"14:16.420 ","End":"14:21.340","Text":"That means it\u0027s also going to be the first to return back to the origin."},{"Start":"14:21.340 ","End":"14:28.300","Text":"Remember we\u0027re having a reflection in the y-axis and what we had over"},{"Start":"14:28.300 ","End":"14:37.300","Text":"here with our energy and therefore amplitude also is correct for question number 4."},{"Start":"14:37.300 ","End":"14:41.230","Text":"The energy of the original pulse is going to be higher"},{"Start":"14:41.230 ","End":"14:45.159","Text":"than the reflected and transmitted subsequent pulses,"},{"Start":"14:45.159 ","End":"14:47.710","Text":"and therefore the amplitude of"},{"Start":"14:47.710 ","End":"14:53.005","Text":"the reflected pulse is going to be less than the amplitude of the original pulse."},{"Start":"14:53.005 ","End":"14:56.230","Text":"Let\u0027s draw the reflected pulse."},{"Start":"14:56.230 ","End":"15:01.735","Text":"We know that its amplitude is going to be less but aside from that,"},{"Start":"15:01.735 ","End":"15:03.790","Text":"it\u0027s going to be pretty similar,"},{"Start":"15:03.790 ","End":"15:09.190","Text":"just reflected back in the y-reflection."},{"Start":"15:09.190 ","End":"15:12.790","Text":"This is our reflective pulse because it\u0027s a loose end."},{"Start":"15:12.790 ","End":"15:15.865","Text":"Now what about our transmitted pulse?"},{"Start":"15:15.865 ","End":"15:17.485","Text":"Let\u0027s see over here."},{"Start":"15:17.485 ","End":"15:21.505","Text":"We saw that our velocity for the wave is dependent on the tension,"},{"Start":"15:21.505 ","End":"15:24.730","Text":"which is the same on both ropes because they\u0027re tied"},{"Start":"15:24.730 ","End":"15:28.735","Text":"to one another and the density of the ropes."},{"Start":"15:28.735 ","End":"15:32.470","Text":"The lighter rope is obviously going to have"},{"Start":"15:32.470 ","End":"15:38.140","Text":"a lower density and a lower density means a higher velocity."},{"Start":"15:38.140 ","End":"15:43.390","Text":"The velocity of the transmitted wave is going to be higher than"},{"Start":"15:43.390 ","End":"15:50.110","Text":"the velocity of the reflected wave and of the original wave in this case,"},{"Start":"15:50.110 ","End":"15:53.185","Text":"where we have a heavier rope attached to a lighter rope."},{"Start":"15:53.185 ","End":"15:57.100","Text":"What does that mean? If we look at our original wave,"},{"Start":"15:57.100 ","End":"16:00.850","Text":"let\u0027s say that our original wave,"},{"Start":"16:00.850 ","End":"16:02.755","Text":"again, let\u0027s draw this,"},{"Start":"16:02.755 ","End":"16:09.460","Text":"has a length of 10 centimeters and its velocity,"},{"Start":"16:09.460 ","End":"16:15.410","Text":"let\u0027s say, is 5 centimeters per second."},{"Start":"16:15.510 ","End":"16:20.980","Text":"Then let\u0027s say over here that the velocity is equal to 10 meters per"},{"Start":"16:20.980 ","End":"16:26.770","Text":"second because it\u0027s less dense so the pulse will travel here faster."},{"Start":"16:26.770 ","End":"16:32.020","Text":"The original wave, after 2 seconds,"},{"Start":"16:32.020 ","End":"16:37.510","Text":"the tail of the original wave would have reached this end over here."},{"Start":"16:37.510 ","End":"16:42.040","Text":"But the head of the wave or of the pulse would have"},{"Start":"16:42.040 ","End":"16:46.660","Text":"already moved into this new transmission medium,"},{"Start":"16:46.660 ","End":"16:51.805","Text":"the lighter rope and it moves forwards at a velocity of 10 meters per second."},{"Start":"16:51.805 ","End":"16:56.350","Text":"If in 2 seconds the tail has moved from here to here,"},{"Start":"16:56.350 ","End":"17:01.930","Text":"the head would have moved if this was 10 centimeters so around about here,"},{"Start":"17:01.930 ","End":"17:06.040","Text":"it would have moved double so it will be around about here."},{"Start":"17:06.040 ","End":"17:12.595","Text":"This would be where the new head will be."},{"Start":"17:12.595 ","End":"17:15.760","Text":"We can see if we draw this in red,"},{"Start":"17:15.760 ","End":"17:22.045","Text":"that the transmitted wave in this case is stretched."},{"Start":"17:22.045 ","End":"17:27.325","Text":"Now its amplitude is also going to be less than the original wave but aside from that,"},{"Start":"17:27.325 ","End":"17:28.480","Text":"it\u0027s stretched out,"},{"Start":"17:28.480 ","End":"17:32.335","Text":"like when we went into the denser medium,"},{"Start":"17:32.335 ","End":"17:34.975","Text":"the wave was squashed and became shorter."},{"Start":"17:34.975 ","End":"17:36.250","Text":"In the lighter medium,"},{"Start":"17:36.250 ","End":"17:38.650","Text":"because the front of the wave is going to move"},{"Start":"17:38.650 ","End":"17:42.625","Text":"faster than what the back of the wave is traveling,"},{"Start":"17:42.625 ","End":"17:47.125","Text":"it is going to stretch out the wave."},{"Start":"17:47.125 ","End":"17:50.380","Text":"Then of course, the velocity is going to be different."},{"Start":"17:50.380 ","End":"17:55.600","Text":"If we draw our transmitted wave after a few seconds,"},{"Start":"17:55.600 ","End":"17:59.245","Text":"it will be something like over here,"},{"Start":"17:59.245 ","End":"18:01.300","Text":"imagine with a smaller amplitude."},{"Start":"18:01.300 ","End":"18:06.310","Text":"Whereas if we draw our reflected wave after the same time frame,"},{"Start":"18:06.310 ","End":"18:09.070","Text":"because it\u0027s traveling at a slower velocity,"},{"Start":"18:09.070 ","End":"18:11.785","Text":"it will be something like over here."},{"Start":"18:11.785 ","End":"18:14.270","Text":"It wouldn\u0027t have moved much."},{"Start":"18:15.510 ","End":"18:18.550","Text":"Again, when we\u0027re dealing with this,"},{"Start":"18:18.550 ","End":"18:25.300","Text":"you always have to see if we\u0027re dealing with a fixed end similar situation or"},{"Start":"18:25.300 ","End":"18:28.630","Text":"a loose end situation and then you have to remember that there\u0027s"},{"Start":"18:28.630 ","End":"18:32.875","Text":"going to be both a reflected and the transmitted wave."},{"Start":"18:32.875 ","End":"18:36.310","Text":"Then you have to remember that the amplitudes of"},{"Start":"18:36.310 ","End":"18:39.430","Text":"the 2 subsequent waves are going to be"},{"Start":"18:39.430 ","End":"18:44.545","Text":"different and less than the amplitude of the original wave in both cases."},{"Start":"18:44.545 ","End":"18:49.330","Text":"To remember that the velocity of the waves changes"},{"Start":"18:49.330 ","End":"18:53.965","Text":"when they move through different transmission mediums and therefore,"},{"Start":"18:53.965 ","End":"19:00.400","Text":"if the wave is now traveling slower then the transmitted wave will be squashed."},{"Start":"19:00.400 ","End":"19:03.760","Text":"If the transmitted wave is traveling faster,"},{"Start":"19:03.760 ","End":"19:07.435","Text":"then it will be stretched out in length."},{"Start":"19:07.435 ","End":"19:10.730","Text":"That\u0027s the end of this lesson."}],"ID":12465},{"Watched":false,"Name":"Periodic Wave","Duration":"6m 27s","ChapterTopicVideoID":11998,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.995","Text":"Hello, in this lesson we\u0027re going to be learning about periodic waves."},{"Start":"00:04.995 ","End":"00:11.190","Text":"If I am once again holding my rope or my string over here,"},{"Start":"00:11.190 ","End":"00:13.905","Text":"and this is what it looks like."},{"Start":"00:13.905 ","End":"00:22.215","Text":"Now, if what I do is I move my hand up all the way,"},{"Start":"00:22.215 ","End":"00:28.155","Text":"and then I move it down all the way to here,"},{"Start":"00:28.155 ","End":"00:31.110","Text":"then what is going to happen?"},{"Start":"00:31.110 ","End":"00:33.660","Text":"Here\u0027s my hand."},{"Start":"00:33.660 ","End":"00:34.817","Text":"Up all the way,"},{"Start":"00:34.817 ","End":"00:36.165","Text":"I\u0027ll get something like this,"},{"Start":"00:36.165 ","End":"00:39.360","Text":"and then down all the way I\u0027ll get something like this,"},{"Start":"00:39.360 ","End":"00:44.685","Text":"and then back up to the end of the rope."},{"Start":"00:44.685 ","End":"00:49.985","Text":"What we can see is we have this sinusoidal shape,"},{"Start":"00:49.985 ","End":"00:53.830","Text":"and this is what is called a wave."},{"Start":"00:53.830 ","End":"00:55.766","Text":"This is a wave,"},{"Start":"00:55.766 ","End":"00:59.540","Text":"or we\u0027ve also referred to it as a pulse."},{"Start":"00:59.540 ","End":"01:02.810","Text":"Because we just have 1 oscillation."},{"Start":"01:02.810 ","End":"01:04.795","Text":"We start going up,"},{"Start":"01:04.795 ","End":"01:06.800","Text":"we reach a peak, we go down,"},{"Start":"01:06.800 ","End":"01:09.736","Text":"reach the bottom most value,"},{"Start":"01:09.736 ","End":"01:16.405","Text":"and then we carry on back up to our equilibrium point where our hand is, the same level."},{"Start":"01:16.405 ","End":"01:20.570","Text":"Now, if I do this a couple of times,"},{"Start":"01:20.570 ","End":"01:24.170","Text":"I move my hand up all the way,"},{"Start":"01:24.170 ","End":"01:27.875","Text":"down, then back up,"},{"Start":"01:27.875 ","End":"01:30.910","Text":"and then back down to here."},{"Start":"01:30.910 ","End":"01:32.835","Text":"What will my rope look like?"},{"Start":"01:32.835 ","End":"01:36.015","Text":"I have moved it up and down, up and down."},{"Start":"01:36.015 ","End":"01:38.085","Text":"I\u0027ll have 2 oscillations."},{"Start":"01:38.085 ","End":"01:41.015","Text":"I\u0027ll have up and down,"},{"Start":"01:41.015 ","End":"01:43.925","Text":"and up and down,"},{"Start":"01:43.925 ","End":"01:48.750","Text":"and then back to our equilibrium."},{"Start":"01:49.850 ","End":"01:55.210","Text":"Then these waves or these pulses or"},{"Start":"01:55.210 ","End":"02:01.830","Text":"these disturbances will carry on traveling or propagating down the rope."},{"Start":"02:01.970 ","End":"02:10.430","Text":"What we can see is if we move our hand up all the way and down all the way many,"},{"Start":"02:10.430 ","End":"02:13.245","Text":"many times, so we keep doing this."},{"Start":"02:13.245 ","End":"02:18.610","Text":"Let\u0027s do it in red to represent that we\u0027re doing this over and over again."},{"Start":"02:18.610 ","End":"02:24.500","Text":"Then what our rope will look like will just be like this."},{"Start":"02:24.500 ","End":"02:29.530","Text":"We\u0027re just going to have waves which are"},{"Start":"02:29.530 ","End":"02:36.110","Text":"periodic waves traveling down until infinity,"},{"Start":"02:36.110 ","End":"02:39.415","Text":"and they\u0027re all propagating along in the same direction."},{"Start":"02:39.415 ","End":"02:42.490","Text":"Obviously, all of these waves are meant to be"},{"Start":"02:42.490 ","End":"02:45.835","Text":"the same width and with the same amplitude,"},{"Start":"02:45.835 ","End":"02:49.420","Text":"both in the negative and positive directions."},{"Start":"02:49.420 ","End":"02:51.800","Text":"Obviously, because I was sketching it,"},{"Start":"02:51.800 ","End":"02:54.650","Text":"it doesn\u0027t exactly look like how it\u0027s meant to,"},{"Start":"02:54.650 ","End":"03:00.425","Text":"but just imagine that all of these waves look exactly the same."},{"Start":"03:00.425 ","End":"03:06.150","Text":"What we\u0027re looking at over here is a periodic wave."},{"Start":"03:06.350 ","End":"03:09.140","Text":"Here we have a diagram,"},{"Start":"03:09.140 ","End":"03:11.315","Text":"which is something that we\u0027ve seen before."},{"Start":"03:11.315 ","End":"03:17.765","Text":"If we just lifted our hand up and back down to our point of equilibrium,"},{"Start":"03:17.765 ","End":"03:22.355","Text":"then we would get a single pulse that would propagate along the rope."},{"Start":"03:22.355 ","End":"03:28.960","Text":"Then we can see that the pulse is propagating down and then it will reach the end."},{"Start":"03:28.960 ","End":"03:33.560","Text":"Over here, we can see that we\u0027ve lifted our hand up,"},{"Start":"03:33.560 ","End":"03:37.100","Text":"pulled it back down to this negative amplitude,"},{"Start":"03:37.100 ","End":"03:39.410","Text":"and then moved our hand back up,"},{"Start":"03:39.410 ","End":"03:43.140","Text":"and we\u0027ve done this continually over and over again."},{"Start":"03:43.140 ","End":"03:46.715","Text":"Then what we can see is that we have these periodic waves."},{"Start":"03:46.715 ","End":"03:52.450","Text":"Of course, they are also all traveling and propagating in this direction."},{"Start":"03:52.450 ","End":"03:58.683","Text":"1 way that scientists achieve these perfect sinusoidal waves,"},{"Start":"03:58.683 ","End":"04:02.095","Text":"is by taking a spring."},{"Start":"04:02.095 ","End":"04:06.515","Text":"Here\u0027s a spring and attaching it on 1 end to the ceiling,"},{"Start":"04:06.515 ","End":"04:10.540","Text":"and then the other end to some kind of mass."},{"Start":"04:10.540 ","End":"04:14.450","Text":"Then, attached to this mass over here,"},{"Start":"04:14.450 ","End":"04:16.610","Text":"we have our rope,"},{"Start":"04:16.610 ","End":"04:19.560","Text":"and imagine that it\u0027s straight."},{"Start":"04:19.700 ","End":"04:26.020","Text":"Then, if we pull our mass down,"},{"Start":"04:26.510 ","End":"04:32.105","Text":"then what will happen is our mass is going to move up and down, up and down,"},{"Start":"04:32.105 ","End":"04:33.830","Text":"up and down, up and down,"},{"Start":"04:33.830 ","End":"04:38.660","Text":"like so because the spring is just going to be bouncing up and down."},{"Start":"04:38.660 ","End":"04:44.720","Text":"Then, imagine that it bounces up and down to the exact same amplitude for infinity,"},{"Start":"04:44.720 ","End":"04:51.870","Text":"so if it reaches a maximum of this down and then it travels this way back up."},{"Start":"04:51.870 ","End":"04:54.380","Text":"The next time that it travels down,"},{"Start":"04:54.380 ","End":"04:59.015","Text":"it will still travel this way down and this way up."},{"Start":"04:59.015 ","End":"05:01.910","Text":"Then what will happen is,"},{"Start":"05:01.910 ","End":"05:03.364","Text":"and again, I\u0027m sketching."},{"Start":"05:03.364 ","End":"05:07.865","Text":"Imagine that this is what looking like a perfect sinusoidal wave."},{"Start":"05:07.865 ","End":"05:15.730","Text":"Then, our rope is going to do this way of formation."},{"Start":"05:15.730 ","End":"05:20.840","Text":"Where all of the spacings between the waves is equal."},{"Start":"05:20.840 ","End":"05:25.914","Text":"What I\u0027m talking about is this distance from peak to peak is the same,"},{"Start":"05:25.914 ","End":"05:28.764","Text":"the width of each wave is the same,"},{"Start":"05:28.764 ","End":"05:33.425","Text":"the amplitude of each wave in the positive direction is the same,"},{"Start":"05:33.425 ","End":"05:39.895","Text":"and the amplitude of each wave in the negative direction is also the same."},{"Start":"05:39.895 ","End":"05:42.620","Text":"A definition for a periodic wave,"},{"Start":"05:42.620 ","End":"05:44.300","Text":"and this isn\u0027t a formal definition,"},{"Start":"05:44.300 ","End":"05:46.685","Text":"but this is something that you can maybe keep in your minds."},{"Start":"05:46.685 ","End":"05:51.095","Text":"It\u0027s creating identical pulses at equal time intervals."},{"Start":"05:51.095 ","End":"05:54.656","Text":"That means that when we raise our arm,"},{"Start":"05:54.656 ","End":"05:58.040","Text":"so we raise it to some amplitude,"},{"Start":"05:58.040 ","End":"06:01.460","Text":"and when we drop our arm back down,"},{"Start":"06:01.460 ","End":"06:03.530","Text":"we drop it to a certain amplitude,"},{"Start":"06:03.530 ","End":"06:06.475","Text":"and we keep those amplitudes constant."},{"Start":"06:06.475 ","End":"06:11.195","Text":"We do that raising and dropping our arm at equal time intervals."},{"Start":"06:11.195 ","End":"06:13.505","Text":"Every second, or every half a second,"},{"Start":"06:13.505 ","End":"06:16.220","Text":"or every 1/10 of a second."},{"Start":"06:16.220 ","End":"06:18.470","Text":"That way, we\u0027re going to get"},{"Start":"06:18.470 ","End":"06:24.169","Text":"these identical pulses and they\u0027ll be at equal time intervals."},{"Start":"06:24.169 ","End":"06:27.210","Text":"That\u0027s the end of this lesson."}],"ID":12466},{"Watched":false,"Name":"Representing Periodic Waves","Duration":"20m 18s","ChapterTopicVideoID":11999,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.725","Text":"Hello. In this lesson,"},{"Start":"00:01.725 ","End":"00:05.145","Text":"we\u0027re going to learn how to represent periodic waves."},{"Start":"00:05.145 ","End":"00:07.275","Text":"A few lessons ago in this chapter,"},{"Start":"00:07.275 ","End":"00:10.140","Text":"we learned how to represent a wave,"},{"Start":"00:10.140 ","End":"00:11.850","Text":"but a single wave."},{"Start":"00:11.850 ","End":"00:19.065","Text":"Just 1 oscillation and we saw that in order to define where"},{"Start":"00:19.065 ","End":"00:26.730","Text":"each particle and a rope is in relation to the y-axis we had a function with 2 variables,"},{"Start":"00:26.730 ","End":"00:32.895","Text":"x and t. X being the position along the x-axis and t is time."},{"Start":"00:32.895 ","End":"00:36.455","Text":"Then we said that a function with 2 variables was difficult"},{"Start":"00:36.455 ","End":"00:40.774","Text":"to solve so we split this up into 2 different functions,"},{"Start":"00:40.774 ","End":"00:44.125","Text":"each dealing with just 1 variable."},{"Start":"00:44.125 ","End":"00:47.975","Text":"The first function was y as a function of x,"},{"Start":"00:47.975 ","End":"00:51.560","Text":"which was our position displacement function and"},{"Start":"00:51.560 ","End":"00:56.120","Text":"our second function was y as a function of time,"},{"Start":"00:56.120 ","End":"00:59.310","Text":"which was our time displacement function."},{"Start":"00:59.450 ","End":"01:02.450","Text":"Just a little reminder in"},{"Start":"01:02.450 ","End":"01:07.550","Text":"the position displacement function so our time was at a constant."},{"Start":"01:07.550 ","End":"01:10.190","Text":"If we see this as our periodic wave,"},{"Start":"01:10.190 ","End":"01:11.720","Text":"we\u0027ve taken a picture,"},{"Start":"01:11.720 ","End":"01:13.190","Text":"a snapshot in time,"},{"Start":"01:13.190 ","End":"01:17.540","Text":"at 10 seconds and then we look at"},{"Start":"01:17.540 ","End":"01:22.205","Text":"each position along the x-axis and its corresponding y values."},{"Start":"01:22.205 ","End":"01:25.865","Text":"Whereas when we take our time displacement function,"},{"Start":"01:25.865 ","End":"01:29.490","Text":"that means that our x is constant."},{"Start":"01:29.490 ","End":"01:32.420","Text":"That means that we\u0027re looking at one specific point,"},{"Start":"01:32.420 ","End":"01:35.540","Text":"let\u0027s say this point over here and we\u0027re looking"},{"Start":"01:35.540 ","End":"01:41.010","Text":"how this point moves up and down as time goes by."},{"Start":"01:41.360 ","End":"01:49.145","Text":"Now let\u0027s see how we can use this when representing the periodic waves."},{"Start":"01:49.145 ","End":"01:51.455","Text":"The first thing is again,"},{"Start":"01:51.455 ","End":"01:55.930","Text":"we\u0027re going to have a position displacement function."},{"Start":"01:55.930 ","End":"01:59.055","Text":"Let\u0027s draw our axes."},{"Start":"01:59.055 ","End":"02:03.410","Text":"Here is our y-axis,"},{"Start":"02:03.410 ","End":"02:06.620","Text":"and here is our x-axis."},{"Start":"02:06.620 ","End":"02:09.545","Text":"First, we\u0027re looking at our position displacement function,"},{"Start":"02:09.545 ","End":"02:13.400","Text":"which means that here we\u0027ve just taken a snapshot of our rope at"},{"Start":"02:13.400 ","End":"02:20.695","Text":"a certain time and now we\u0027re looking at what\u0027s going on at this specific time."},{"Start":"02:20.695 ","End":"02:27.695","Text":"What we can see is that depending on where we choose to start our looking."},{"Start":"02:27.695 ","End":"02:33.540","Text":"Let\u0027s say we begin by looking at this peak over here."},{"Start":"02:33.540 ","End":"02:39.150","Text":"We can see that this next peak the piece that is"},{"Start":"02:39.150 ","End":"02:45.800","Text":"between the 2 peaks is going to be repeated because this function,"},{"Start":"02:45.800 ","End":"02:47.990","Text":"this wave function is periodic,"},{"Start":"02:47.990 ","End":"02:49.580","Text":"it\u0027s given in the name."},{"Start":"02:49.580 ","End":"02:59.075","Text":"We can see that this exact pattern over here is going to be repeated over here."},{"Start":"02:59.075 ","End":"03:04.055","Text":"We go down to the same depth and back"},{"Start":"03:04.055 ","End":"03:09.935","Text":"up to our peak and we can see that this is just repeated and repeated."},{"Start":"03:09.935 ","End":"03:18.335","Text":"What we can see is that if I would take this whole section up until this peak,"},{"Start":"03:18.335 ","End":"03:21.335","Text":"and then on a computer, copy it,"},{"Start":"03:21.335 ","End":"03:24.695","Text":"and then paste it over here and over here,"},{"Start":"03:24.695 ","End":"03:29.825","Text":"we\u0027d see that we\u0027d get this exact same wave pattern."},{"Start":"03:29.825 ","End":"03:34.640","Text":"Now, of course, I didn\u0027t have to just measure that from peak to peak."},{"Start":"03:34.640 ","End":"03:37.340","Text":"I can measure from any other point on the wave."},{"Start":"03:37.340 ","End":"03:40.885","Text":"Let\u0027s say I want to measure from this over here."},{"Start":"03:40.885 ","End":"03:48.515","Text":"Here it\u0027s at 0 and then I can follow until I finish 1 oscillation."},{"Start":"03:48.515 ","End":"03:56.160","Text":"In our case, it would be until this point over here."},{"Start":"03:56.160 ","End":"04:00.310","Text":"I would just copy this whole section."},{"Start":"04:00.310 ","End":"04:03.370","Text":"Then I could just paste it, everything in green."},{"Start":"04:03.370 ","End":"04:08.860","Text":"I could copy it and paste it over and over again and we\u0027d get the same pattern."},{"Start":"04:08.860 ","End":"04:12.340","Text":"Now, something that\u0027s important to note is"},{"Start":"04:12.340 ","End":"04:15.910","Text":"that when we\u0027re measuring from this 0 point over here,"},{"Start":"04:15.910 ","End":"04:21.700","Text":"the end of the oscillation is over here, not over here."},{"Start":"04:21.700 ","End":"04:26.260","Text":"If I were to say that it was over here and then I would do"},{"Start":"04:26.260 ","End":"04:31.880","Text":"this copy-paste thing so what I would be doing is I\u0027d be copying this section."},{"Start":"04:31.880 ","End":"04:33.865","Text":"Then if I pasted it,"},{"Start":"04:33.865 ","End":"04:39.980","Text":"I would get a pattern like this which as we can see,"},{"Start":"04:39.980 ","End":"04:44.945","Text":"is just a bunch of bumps and it isn\u0027t this sinusoidal pattern."},{"Start":"04:44.945 ","End":"04:48.650","Text":"That\u0027s a way that can possibly help you to"},{"Start":"04:48.650 ","End":"04:53.330","Text":"know where the first period of the oscillation ends."},{"Start":"04:53.330 ","End":"04:57.770","Text":"What you want to do is you want to draw the first period such that if"},{"Start":"04:57.770 ","End":"05:03.980","Text":"now on a computer you were to take this and copy it and paste it here and here,"},{"Start":"05:03.980 ","End":"05:07.475","Text":"you would get this exact same pattern."},{"Start":"05:07.475 ","End":"05:10.475","Text":"It\u0027s very important if you\u0027re measuring from the 0,"},{"Start":"05:10.475 ","End":"05:14.570","Text":"you\u0027re not necessarily tracing back to the next 0 point,"},{"Start":"05:14.570 ","End":"05:18.599","Text":"but you\u0027re finishing off the oscillation up until here."},{"Start":"05:19.410 ","End":"05:25.660","Text":"We\u0027ve seen that we can measure the wave from the peak,"},{"Start":"05:25.660 ","End":"05:27.085","Text":"from the 0 point,"},{"Start":"05:27.085 ","End":"05:29.122","Text":"from this point over here."},{"Start":"05:29.122 ","End":"05:32.095","Text":"We can measure it from anywhere we want."},{"Start":"05:32.095 ","End":"05:35.230","Text":"Also from this peak in"},{"Start":"05:35.230 ","End":"05:38.290","Text":"the negative direction until the next peak in the negative direction"},{"Start":"05:38.290 ","End":"05:44.935","Text":"but what we have to do is each time we finish if we start at the peak,"},{"Start":"05:44.935 ","End":"05:48.340","Text":"then we have to finish at our next peak."},{"Start":"05:48.340 ","End":"05:50.380","Text":"If we start at the negative peak,"},{"Start":"05:50.380 ","End":"05:53.440","Text":"we have to finish at the next negative peak and if we"},{"Start":"05:53.440 ","End":"05:57.190","Text":"start at this 0 where our wave is going up,"},{"Start":"05:57.190 ","End":"06:01.990","Text":"then we have to finish at this arrow where our wave is again going to start going up."},{"Start":"06:01.990 ","End":"06:07.475","Text":"What we\u0027ll see is that the distances between all of these points is the same."},{"Start":"06:07.475 ","End":"06:12.725","Text":"Let\u0027s see in green the distance from peak to peak."},{"Start":"06:12.725 ","End":"06:18.005","Text":"This is the distance and this is also called the wavelength."},{"Start":"06:18.005 ","End":"06:20.525","Text":"Soon we\u0027re going to speak about it and it\u0027s"},{"Start":"06:20.525 ","End":"06:23.450","Text":"a certain length that measures from peak to peak."},{"Start":"06:23.450 ","End":"06:27.440","Text":"But what if I wanted to measure from this 0 point where the wave"},{"Start":"06:27.440 ","End":"06:32.415","Text":"goes up until the next 0 point where the wave begins going up?"},{"Start":"06:32.415 ","End":"06:36.560","Text":"If I look at the distance between these 2 points,"},{"Start":"06:36.560 ","End":"06:41.750","Text":"it\u0027s the same distance as the distance that we measured from peak to peak."},{"Start":"06:41.750 ","End":"06:46.640","Text":"The same if I chose any other 2 points to represent the oscillation or to"},{"Start":"06:46.640 ","End":"06:51.715","Text":"represent the period this distance will be the exact same thing."},{"Start":"06:51.715 ","End":"06:54.710","Text":"This is the definition of a wavelength,"},{"Start":"06:54.710 ","End":"06:56.450","Text":"which is this distance over here."},{"Start":"06:56.450 ","End":"07:01.745","Text":"The wavelength is denoted by the Greek letter Lambda,"},{"Start":"07:01.745 ","End":"07:04.615","Text":"and its units are in meters"},{"Start":"07:04.615 ","End":"07:10.460","Text":"and wavelength is just the distance over which a wave shape repeats."},{"Start":"07:10.460 ","End":"07:13.310","Text":"We can see that if it\u0027s from here to here,"},{"Start":"07:13.310 ","End":"07:17.240","Text":"this length over here is the wavelength represented by Lambda,"},{"Start":"07:17.240 ","End":"07:20.390","Text":"and the shape of our wave repeats."},{"Start":"07:20.390 ","End":"07:23.480","Text":"I can take this copy and paste over here,"},{"Start":"07:23.480 ","End":"07:24.980","Text":"and paste over here."},{"Start":"07:24.980 ","End":"07:28.250","Text":"Similarly, if I take from these two 0 points,"},{"Start":"07:28.250 ","End":"07:33.365","Text":"I can copy all of this S-shape and paste it over here,"},{"Start":"07:33.365 ","End":"07:38.254","Text":"and paste it over here and get the exact same shape as well."},{"Start":"07:38.254 ","End":"07:41.720","Text":"It\u0027s the wavelength that we\u0027re looking"},{"Start":"07:41.720 ","End":"07:46.435","Text":"for in order to use the position displacement function."},{"Start":"07:46.435 ","End":"07:49.505","Text":"Now let\u0027s look at the time displacement function."},{"Start":"07:49.505 ","End":"07:52.445","Text":"As we remember, if we can imagine we have"},{"Start":"07:52.445 ","End":"08:00.150","Text":"our oscillating wave moving and oscillating where the direction of travel is like so."},{"Start":"08:01.340 ","End":"08:04.730","Text":"Now what we want to see is the time displacement function,"},{"Start":"08:04.730 ","End":"08:08.930","Text":"which means that we\u0027re going to look at a particular point in"},{"Start":"08:08.930 ","End":"08:13.580","Text":"the oscillating wave and look how it moves as a function of time."},{"Start":"08:13.580 ","End":"08:17.720","Text":"Its position moving up and down as a function of time."},{"Start":"08:17.720 ","End":"08:23.370","Text":"This time, we can draw out the axes again."},{"Start":"08:23.370 ","End":"08:26.420","Text":"This is still going to be the y-axis,"},{"Start":"08:26.420 ","End":"08:28.220","Text":"where the amplitude is,"},{"Start":"08:28.220 ","End":"08:33.215","Text":"but this time, this axis is going to be the axis of time."},{"Start":"08:33.215 ","End":"08:37.580","Text":"Let\u0027s look at a video and see our point that is"},{"Start":"08:37.580 ","End":"08:42.200","Text":"moving up and down and then we\u0027ll understand this graph a little bit more."},{"Start":"08:42.200 ","End":"08:45.515","Text":"If we look at this green point over here,"},{"Start":"08:45.515 ","End":"08:48.860","Text":"we can see in a second that when the waves are going"},{"Start":"08:48.860 ","End":"08:52.130","Text":"to start oscillating this green point over here,"},{"Start":"08:52.130 ","End":"08:56.630","Text":"or it might be easier to look at this point over here it\u0027s going to move up and down."},{"Start":"08:56.630 ","End":"09:00.140","Text":"We can see the origin is moving up and down with respect to"},{"Start":"09:00.140 ","End":"09:04.210","Text":"time and so is that other green point just after it over here."},{"Start":"09:04.210 ","End":"09:08.000","Text":"That is exactly this graph over here."},{"Start":"09:08.000 ","End":"09:11.390","Text":"Our green point is moving up and down,"},{"Start":"09:11.390 ","End":"09:13.610","Text":"up and down with respect to time."},{"Start":"09:13.610 ","End":"09:15.305","Text":"Then when we plot that,"},{"Start":"09:15.305 ","End":"09:19.415","Text":"we\u0027re going to get this graph over here."},{"Start":"09:19.415 ","End":"09:24.590","Text":"What we can see is that at some point, at t=0,"},{"Start":"09:24.590 ","End":"09:29.300","Text":"let\u0027s say our point that we\u0027re looking at begins at the point of equilibrium,"},{"Start":"09:29.300 ","End":"09:32.906","Text":"and then slowly it rises up."},{"Start":"09:32.906 ","End":"09:36.469","Text":"If we\u0027re imagining the wave or the particles,"},{"Start":"09:36.469 ","End":"09:39.980","Text":"so it starts off over here and then slowly it"},{"Start":"09:39.980 ","End":"09:43.835","Text":"goes up to its peak amplitude so that\u0027s over here,"},{"Start":"09:43.835 ","End":"09:51.200","Text":"and then slowly it will start moving down to its peak amplitude,"},{"Start":"09:51.200 ","End":"09:55.760","Text":"but this time in the negative position and then it\u0027s going to repeat."},{"Start":"09:55.760 ","End":"10:00.980","Text":"It slowly goes up again to its peak amplitude and slowly goes down,"},{"Start":"10:00.980 ","End":"10:03.290","Text":"up and down, up and down,"},{"Start":"10:03.290 ","End":"10:11.989","Text":"relating to how fast our machine is creating these oscillations."},{"Start":"10:11.989 ","End":"10:15.910","Text":"What we can see is that when we\u0027re jumping from"},{"Start":"10:15.910 ","End":"10:21.475","Text":"this peak to this peak in the graph as a function of time."},{"Start":"10:21.475 ","End":"10:30.730","Text":"We can see that we\u0027re seeing when our particle on the rope is at its peak height,"},{"Start":"10:30.730 ","End":"10:33.369","Text":"its peak amplitude or just its amplitude."},{"Start":"10:33.369 ","End":"10:35.860","Text":"Then we can see the next time,"},{"Start":"10:35.860 ","End":"10:38.485","Text":"because we\u0027re going along the time axis."},{"Start":"10:38.485 ","End":"10:45.530","Text":"The next time, that our same point has reached its peak amplitude again."},{"Start":"10:46.560 ","End":"10:55.420","Text":"What we can say is that this distance over here on the time axis is the amount of time it"},{"Start":"10:55.420 ","End":"11:04.465","Text":"takes for a particle on the rope to complete 1 oscillation or to complete 1 period."},{"Start":"11:04.465 ","End":"11:13.465","Text":"We saw that our particle started off at the origin and then it moved up"},{"Start":"11:13.465 ","End":"11:17.650","Text":"to its peak amplitude or to its amplitude and then it moved"},{"Start":"11:17.650 ","End":"11:22.315","Text":"back down over here to it\u0027s negative amplitude,"},{"Start":"11:22.315 ","End":"11:27.010","Text":"then it went again back over here through its point of equilibrium,"},{"Start":"11:27.010 ","End":"11:29.620","Text":"and then back up to this peak amplitude."},{"Start":"11:29.620 ","End":"11:34.390","Text":"That\u0027s 1 period. This,"},{"Start":"11:34.390 ","End":"11:37.420","Text":"from peak to peak, that\u0027s 1 period."},{"Start":"11:37.420 ","End":"11:40.405","Text":"Here because we\u0027re measuring as a function of time,"},{"Start":"11:40.405 ","End":"11:47.480","Text":"we can see how long it takes for 1 oscillation or for 1 period."},{"Start":"11:48.270 ","End":"11:50.950","Text":"In the time displacement function,"},{"Start":"11:50.950 ","End":"11:55.420","Text":"this distance between peaks is called the period and it\u0027s denoted"},{"Start":"11:55.420 ","End":"12:00.175","Text":"by this capital letter T. T is of course,"},{"Start":"12:00.175 ","End":"12:01.420","Text":"a measurement of time,"},{"Start":"12:01.420 ","End":"12:03.430","Text":"so it\u0027s measured in seconds."},{"Start":"12:03.430 ","End":"12:06.340","Text":"The period, or the time period is"},{"Start":"12:06.340 ","End":"12:09.490","Text":"the time taken for the specific particle on the rope that"},{"Start":"12:09.490 ","End":"12:15.475","Text":"we\u0027re looking at to finish 1 complete oscillation or 1 complete round."},{"Start":"12:15.475 ","End":"12:20.200","Text":"From going from an equilibrium point back to the same equilibrium point,"},{"Start":"12:20.200 ","End":"12:25.405","Text":"or from being at its peak amplitude back to its peak amplitude,"},{"Start":"12:25.405 ","End":"12:27.745","Text":"and so on and so forth."},{"Start":"12:27.745 ","End":"12:30.895","Text":"When we\u0027re dealing with the time displacement function,"},{"Start":"12:30.895 ","End":"12:36.785","Text":"the distance from peak to peak is the period or the time period."},{"Start":"12:36.785 ","End":"12:44.000","Text":"Now another thing that we can find in our time displacements function is the frequency."},{"Start":"12:44.000 ","End":"12:52.510","Text":"The frequency denoted by the lowercase letter f is the number of cycles per second."},{"Start":"12:52.510 ","End":"12:57.445","Text":"Let\u0027s say we\u0027re measuring our cycles from this point of equilibrium over here."},{"Start":"12:57.445 ","End":"13:01.540","Text":"We can see that this is 1 cycle,"},{"Start":"13:01.540 ","End":"13:04.555","Text":"and then this is 2 cycles."},{"Start":"13:04.555 ","End":"13:10.615","Text":"Then let\u0027s say that this represents t of 1 second."},{"Start":"13:10.615 ","End":"13:14.920","Text":"What we can see is that we have 2 cycles per second."},{"Start":"13:14.920 ","End":"13:19.150","Text":"Our frequency f would be 2 per seconds."},{"Start":"13:19.150 ","End":"13:24.340","Text":"The units are per second or in hertz. It\u0027s the same thing."},{"Start":"13:24.340 ","End":"13:25.780","Text":"This is the SI unit,"},{"Start":"13:25.780 ","End":"13:27.295","Text":"but you can also use hertz."},{"Start":"13:27.295 ","End":"13:32.245","Text":"Let\u0027s say if this represented 1 second,"},{"Start":"13:32.245 ","End":"13:35.785","Text":"so if this was 1 and this was 1,"},{"Start":"13:35.785 ","End":"13:42.729","Text":"so we would have a frequency of 1 cycle or 1 oscillation per second."},{"Start":"13:42.729 ","End":"13:46.300","Text":"Let\u0027s cross this out,"},{"Start":"13:46.300 ","End":"13:49.930","Text":"this represented 1 second,"},{"Start":"13:49.930 ","End":"13:52.855","Text":"so we would have a frequency of 0.5."},{"Start":"13:52.855 ","End":"13:57.250","Text":"Because we can see that we only have 0.5 A cycle per second."},{"Start":"13:57.250 ","End":"14:03.040","Text":"What we can see from the units and also from understanding what frequency means,"},{"Start":"14:03.040 ","End":"14:07.375","Text":"the number of cycles per second or the number of oscillations per second,"},{"Start":"14:07.375 ","End":"14:13.795","Text":"it means that frequency is the inverse of the period or the time period."},{"Start":"14:13.795 ","End":"14:17.755","Text":"A very useful and important equation is that"},{"Start":"14:17.755 ","End":"14:23.620","Text":"f frequency is equal to the reciprocal of our time period,"},{"Start":"14:23.620 ","End":"14:30.820","Text":"so t. We can also write it as a time period t to the power of negative 1."},{"Start":"14:30.820 ","End":"14:33.055","Text":"These 2 mean the same thing."},{"Start":"14:33.055 ","End":"14:35.050","Text":"Copy out this equation."},{"Start":"14:35.050 ","End":"14:38.185","Text":"Now let\u0027s give few numbered examples."},{"Start":"14:38.185 ","End":"14:45.910","Text":"If we have that our time period t is equal to 2 seconds,"},{"Start":"14:45.910 ","End":"14:54.265","Text":"so that means that our frequency will be given as 0.5 hertz."},{"Start":"14:54.265 ","End":"14:58.930","Text":"Why? Our frequency is 1 divided by a time period."},{"Start":"14:58.930 ","End":"15:02.200","Text":"So 1 divided by 2 is 0.5."},{"Start":"15:02.200 ","End":"15:07.405","Text":"If our time period was equal to 10 seconds,"},{"Start":"15:07.405 ","End":"15:16.190","Text":"then our frequency would be equal to 1 divided by 10 hertz."},{"Start":"15:16.320 ","End":"15:26.200","Text":"What does that mean? Here we see that we have a whole oscillation every 2 seconds."},{"Start":"15:26.200 ","End":"15:31.180","Text":"Which means that every second we only have 1/2 an oscillation,"},{"Start":"15:31.180 ","End":"15:34.220","Text":"which was this situation over here."},{"Start":"15:35.580 ","End":"15:39.864","Text":"Here we can see that we only have 1/2 an oscillation."},{"Start":"15:39.864 ","End":"15:41.950","Text":"Here the period is 10 seconds,"},{"Start":"15:41.950 ","End":"15:45.325","Text":"which means that in 10 seconds we have a whole oscillation."},{"Start":"15:45.325 ","End":"15:48.054","Text":"Which means that if we look at 1 second,"},{"Start":"15:48.054 ","End":"15:53.065","Text":"only 1/10 of that oscillation would have been completed."},{"Start":"15:53.065 ","End":"15:59.785","Text":"That would mean only this section of the oscillation would be completed in 1 second."},{"Start":"15:59.785 ","End":"16:02.380","Text":"Because in order to complete all of this,"},{"Start":"16:02.380 ","End":"16:05.090","Text":"we need 10 seconds."},{"Start":"16:06.060 ","End":"16:11.980","Text":"Now what we\u0027re going to do is we\u0027re going to use this idea of period and frequency in"},{"Start":"16:11.980 ","End":"16:17.560","Text":"order to talk about the velocity or the speed of the wave,"},{"Start":"16:17.560 ","End":"16:22.100","Text":"which is something that we began talking about a few lessons ago."},{"Start":"16:23.280 ","End":"16:30.205","Text":"We know that the velocity of a wave that we were looking at before,"},{"Start":"16:30.205 ","End":"16:35.290","Text":"one of the equations that we used was just the regular kinematic equation,"},{"Start":"16:35.290 ","End":"16:37.915","Text":"which is a Delta x,"},{"Start":"16:37.915 ","End":"16:44.140","Text":"the distance traveled divided by Delta t, the time taken."},{"Start":"16:44.140 ","End":"16:46.045","Text":"Let\u0027s take a look."},{"Start":"16:46.045 ","End":"16:47.890","Text":"We know that we have Lambda,"},{"Start":"16:47.890 ","End":"16:49.330","Text":"which is the wavelength."},{"Start":"16:49.330 ","End":"16:51.610","Text":"We know that we have our time period,"},{"Start":"16:51.610 ","End":"16:54.700","Text":"and we know that we have our frequency."},{"Start":"16:54.700 ","End":"16:59.785","Text":"We want to see how this connects to the velocity of a wave."},{"Start":"16:59.785 ","End":"17:03.055","Text":"A Delta x is the distance traveled."},{"Start":"17:03.055 ","End":"17:07.015","Text":"Let\u0027s just draw this wave."},{"Start":"17:07.015 ","End":"17:11.365","Text":"It begins over here and it starts oscillating."},{"Start":"17:11.365 ","End":"17:14.560","Text":"Again, imagine that I\u0027m drawing this properly."},{"Start":"17:14.560 ","End":"17:18.745","Text":"We said that from this point to this point,"},{"Start":"17:18.745 ","End":"17:21.265","Text":"if this is our x,"},{"Start":"17:21.265 ","End":"17:27.385","Text":"so this is a wavelength Lambda."},{"Start":"17:27.385 ","End":"17:29.485","Text":"This is the length of the wave."},{"Start":"17:29.485 ","End":"17:34.720","Text":"What we can see is that if we look from this peak and we"},{"Start":"17:34.720 ","End":"17:40.630","Text":"want to see how far it travels until it\u0027s again at a peak."},{"Start":"17:40.630 ","End":"17:47.080","Text":"We go down all of this way and up here it\u0027s a peak again."},{"Start":"17:47.080 ","End":"17:54.070","Text":"We can see that this distance traveled is actually the wavelength. A change in x."},{"Start":"17:54.070 ","End":"17:55.915","Text":"We started at this x,"},{"Start":"17:55.915 ","End":"17:57.295","Text":"this was x1,"},{"Start":"17:57.295 ","End":"17:59.185","Text":"and we ended at this x,"},{"Start":"17:59.185 ","End":"18:05.875","Text":"where this is x2, until we got to the point that it\u0027s at a peak again."},{"Start":"18:05.875 ","End":"18:09.295","Text":"This delta x. Remember,"},{"Start":"18:09.295 ","End":"18:14.320","Text":"delta x is equal to x2 minus x1."},{"Start":"18:14.320 ","End":"18:18.985","Text":"That is equal to Lambda or wavelength."},{"Start":"18:18.985 ","End":"18:23.335","Text":"We can plug that in over here."},{"Start":"18:23.335 ","End":"18:25.270","Text":"What\u0027s it divided by?"},{"Start":"18:25.270 ","End":"18:28.540","Text":"Now let\u0027s just look at this as an axis."},{"Start":"18:28.540 ","End":"18:32.470","Text":"We know that our wave is oscillating up and down and we"},{"Start":"18:32.470 ","End":"18:37.180","Text":"know that we\u0027re specifically here measuring from the 2 peaks."},{"Start":"18:37.180 ","End":"18:44.825","Text":"We know from our time displacement function that the time taken from peak to peak."},{"Start":"18:44.825 ","End":"18:49.695","Text":"The time taken for a particle to be at the peak,"},{"Start":"18:49.695 ","End":"18:54.285","Text":"then to go down into this trough and then come back up to a peak."},{"Start":"18:54.285 ","End":"18:58.345","Text":"The time taken is period."},{"Start":"18:58.345 ","End":"19:00.610","Text":"It\u0027s also written here the time taken for"},{"Start":"19:00.610 ","End":"19:03.400","Text":"the particle on the rope to finish 1 complete oscillation,"},{"Start":"19:03.400 ","End":"19:05.515","Text":"which means from peak to peak."},{"Start":"19:05.515 ","End":"19:07.795","Text":"We can say that this is t1,"},{"Start":"19:07.795 ","End":"19:10.270","Text":"this point over here that the particles at a peak."},{"Start":"19:10.270 ","End":"19:12.760","Text":"Then when we draw it again a bit later,"},{"Start":"19:12.760 ","End":"19:16.435","Text":"it\u0027s again at a peak at time t2."},{"Start":"19:16.435 ","End":"19:24.410","Text":"A little reminder, our delta t is equal to t2 minus t1,"},{"Start":"19:24.410 ","End":"19:28.770","Text":"which as we know, is equal to our time period."},{"Start":"19:28.770 ","End":"19:32.355","Text":"We can plug in our time period over here."},{"Start":"19:32.355 ","End":"19:37.135","Text":"Then if we take all of this information,"},{"Start":"19:37.135 ","End":"19:42.670","Text":"we therefore know that the velocity of a wave can be written also as"},{"Start":"19:42.670 ","End":"19:50.145","Text":"the wavelength divided by the time period or whether using our equation for frequency,"},{"Start":"19:50.145 ","End":"19:54.270","Text":"because we know that frequency is the reciprocal of the time period."},{"Start":"19:54.270 ","End":"19:57.750","Text":"We can also write that the velocity of a wave is equal"},{"Start":"19:57.750 ","End":"20:03.010","Text":"to the wavelength multiplied by the frequency."},{"Start":"20:04.560 ","End":"20:07.615","Text":"Copy out these 2 equations."},{"Start":"20:07.615 ","End":"20:10.630","Text":"Remember what the definition of a wavelength,"},{"Start":"20:10.630 ","End":"20:13.730","Text":"period and frequency is."},{"Start":"20:13.860 ","End":"20:18.440","Text":"That is it. That is the end of this lesson."}],"ID":12467},{"Watched":false,"Name":"Standing Waves","Duration":"29m 5s","ChapterTopicVideoID":12000,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.905","Text":"Hello. In this lesson,"},{"Start":"00:01.905 ","End":"00:04.485","Text":"we\u0027re going to be speaking about standing waves."},{"Start":"00:04.485 ","End":"00:08.040","Text":"Another name for standing waves is stationary waves,"},{"Start":"00:08.040 ","End":"00:10.875","Text":"and both these names are interchangeable."},{"Start":"00:10.875 ","End":"00:16.755","Text":"Let\u0027s look at the second graph because the first one is missing the green wave."},{"Start":"00:16.755 ","End":"00:19.845","Text":"We\u0027re starting from over here."},{"Start":"00:19.845 ","End":"00:24.540","Text":"Someone sends out a red wave from over here,"},{"Start":"00:24.540 ","End":"00:29.795","Text":"and the red wave is propagating in the rightwards direction,"},{"Start":"00:29.795 ","End":"00:32.780","Text":"and we\u0027re imagining that this rope is endless."},{"Start":"00:32.780 ","End":"00:36.950","Text":"We\u0027re not dealing with the rope being attached to anything,"},{"Start":"00:36.950 ","End":"00:40.105","Text":"so we don\u0027t have loose ends or fixed end attachments."},{"Start":"00:40.105 ","End":"00:45.320","Text":"A rope is just some endless rope that\u0027s just flowing to infinity."},{"Start":"00:45.320 ","End":"00:49.760","Text":"We have the red wave propagating in this rightwards direction,"},{"Start":"00:49.760 ","End":"00:55.355","Text":"and then we have the green wave propagating in the opposite direction,"},{"Start":"00:55.355 ","End":"00:56.945","Text":"in the leftwards direction."},{"Start":"00:56.945 ","End":"00:59.890","Text":"The green wave is flowing like so."},{"Start":"00:59.890 ","End":"01:01.910","Text":"How do we get standing waves?"},{"Start":"01:01.910 ","End":"01:06.350","Text":"We get standing waves by having 2 identical waves which are"},{"Start":"01:06.350 ","End":"01:11.510","Text":"traveling towards one another and then they interfere with one another."},{"Start":"01:11.510 ","End":"01:16.730","Text":"Now, before we had a lesson on interference,"},{"Start":"01:16.730 ","End":"01:18.770","Text":"I suggest you watch it if you can\u0027t remember,"},{"Start":"01:18.770 ","End":"01:23.920","Text":"where we spoke about what happens if we have 1 pulse,"},{"Start":"01:23.920 ","End":"01:25.230","Text":"say like this,"},{"Start":"01:25.230 ","End":"01:27.080","Text":"traveling in this direction,"},{"Start":"01:27.080 ","End":"01:29.495","Text":"and another pulse, say like this,"},{"Start":"01:29.495 ","End":"01:31.633","Text":"traveling in the opposite direction,"},{"Start":"01:31.633 ","End":"01:33.094","Text":"and when these pulses meet,"},{"Start":"01:33.094 ","End":"01:35.110","Text":"they interfere with one another."},{"Start":"01:35.110 ","End":"01:37.320","Text":"What we have here is just many,"},{"Start":"01:37.320 ","End":"01:39.525","Text":"many pulses interfering with one another."},{"Start":"01:39.525 ","End":"01:42.600","Text":"We have some kind of constant interference."},{"Start":"01:42.600 ","End":"01:46.125","Text":"Here, we\u0027re just dealing with 1 moment of interference,"},{"Start":"01:46.125 ","End":"01:48.905","Text":"and here we\u0027re dealing with constant interference"},{"Start":"01:48.905 ","End":"01:53.100","Text":"because we just have a rope oscillating."},{"Start":"01:53.890 ","End":"01:58.880","Text":"Our red and green waves are going to interfere with one another,"},{"Start":"01:58.880 ","End":"02:03.215","Text":"and the interference that is generated is this purple wave."},{"Start":"02:03.215 ","End":"02:05.645","Text":"We\u0027re looking at a wave this color purple,"},{"Start":"02:05.645 ","End":"02:08.665","Text":"but we\u0027re looking at this second graph over here."},{"Start":"02:08.665 ","End":"02:13.190","Text":"What do we can see is this purple wave is"},{"Start":"02:13.190 ","End":"02:19.400","Text":"the product of the superposition of our red wave and of our green wave."},{"Start":"02:19.400 ","End":"02:22.100","Text":"If we remember what superposition means,"},{"Start":"02:22.100 ","End":"02:29.825","Text":"superposition means that we\u0027re just adding on the amplitudes of the 2 interfering waves."},{"Start":"02:29.825 ","End":"02:33.825","Text":"Let\u0027s start from, let\u0027s say,"},{"Start":"02:33.825 ","End":"02:36.150","Text":"over here at this point."},{"Start":"02:36.150 ","End":"02:38.325","Text":"We can see that at this point,"},{"Start":"02:38.325 ","End":"02:40.853","Text":"let\u0027s do this in black,"},{"Start":"02:40.853 ","End":"02:46.600","Text":"our red wave has some kind of negative amplitude over here."},{"Start":"02:46.670 ","End":"02:54.130","Text":"Let\u0027s go over here. Our green wave has a 0 amplitude at this exact point."},{"Start":"02:54.130 ","End":"02:59.030","Text":"We can see that our purple wave,"},{"Start":"02:59.030 ","End":"03:02.270","Text":"which is the product of the superposition between these 2 waves,"},{"Start":"03:02.270 ","End":"03:05.690","Text":"is simply going to have the amplitude of the red wave,"},{"Start":"03:05.690 ","End":"03:09.290","Text":"whatever it is, plus the amplitude of the green wave,"},{"Start":"03:09.290 ","End":"03:11.360","Text":"which at this point is 0."},{"Start":"03:11.360 ","End":"03:16.915","Text":"Our purple wave is going to have an amplitude over here."},{"Start":"03:16.915 ","End":"03:19.950","Text":"Then let\u0027s go over to this point."},{"Start":"03:19.950 ","End":"03:22.910","Text":"We can see at this point over here,"},{"Start":"03:22.910 ","End":"03:27.680","Text":"our red wave and our green wave have the same amplitude,"},{"Start":"03:27.680 ","End":"03:30.410","Text":"which is this section over here."},{"Start":"03:30.410 ","End":"03:33.400","Text":"It\u0027s approximately the size of 1/2 a square."},{"Start":"03:33.400 ","End":"03:39.170","Text":"That means that we\u0027re adding 1/2 a square of amplitude due to the amplitude of"},{"Start":"03:39.170 ","End":"03:45.140","Text":"the red wave plus 1/2 a square of amplitude due to the amplitude of the green wave,"},{"Start":"03:45.140 ","End":"03:47.180","Text":"so then we have a full square."},{"Start":"03:47.180 ","End":"03:50.805","Text":"That\u0027s why our purple wave over here,"},{"Start":"03:50.805 ","End":"03:54.170","Text":"which is the wave generated by"},{"Start":"03:54.170 ","End":"03:58.655","Text":"the superposition of the amplitudes of the red and green waves,"},{"Start":"03:58.655 ","End":"04:01.900","Text":"is this point over here."},{"Start":"04:01.900 ","End":"04:05.520","Text":"Then we can go over to,"},{"Start":"04:05.520 ","End":"04:07.230","Text":"let\u0027s say, this point over here."},{"Start":"04:07.230 ","End":"04:09.980","Text":"We can see that our green wave has some kind of amplitude,"},{"Start":"04:09.980 ","End":"04:12.350","Text":"but our red wave has a 0 amplitude."},{"Start":"04:12.350 ","End":"04:16.520","Text":"The green wave\u0027s amplitude plus 0 will just equal the green wave\u0027s amplitude."},{"Start":"04:16.520 ","End":"04:19.330","Text":"Then we\u0027ll have this point over here."},{"Start":"04:19.330 ","End":"04:22.065","Text":"Then let\u0027s look at this point over here."},{"Start":"04:22.065 ","End":"04:26.225","Text":"Our red wave has this amplitude in the positive direction,"},{"Start":"04:26.225 ","End":"04:31.160","Text":"and we can see that our green wave has the same amplitude just in the negative direction."},{"Start":"04:31.160 ","End":"04:32.855","Text":"If we have, let\u0027s say,"},{"Start":"04:32.855 ","End":"04:35.310","Text":"positive 1, negative 1,"},{"Start":"04:35.310 ","End":"04:40.655","Text":"so that\u0027s why the amplitude of our purple wave is here equal to 0."},{"Start":"04:40.655 ","End":"04:44.990","Text":"What do we do is we carry on going at different points of the waves,"},{"Start":"04:44.990 ","End":"04:48.305","Text":"adding up the amplitudes of"},{"Start":"04:48.305 ","End":"04:52.460","Text":"our green and red waves in order to get the superposition of the 2,"},{"Start":"04:52.460 ","End":"04:54.995","Text":"and then we get this purple wave,"},{"Start":"04:54.995 ","End":"05:00.600","Text":"which is the product of the interference between the red and green waves."},{"Start":"05:01.550 ","End":"05:09.395","Text":"What happens is that we have these points,"},{"Start":"05:09.395 ","End":"05:11.450","Text":"A, B,"},{"Start":"05:11.450 ","End":"05:14.075","Text":"C, and D over here."},{"Start":"05:14.075 ","End":"05:20.705","Text":"I\u0027ll actually do that in green, in blue."},{"Start":"05:20.705 ","End":"05:23.270","Text":"We have here our point A,"},{"Start":"05:23.270 ","End":"05:25.865","Text":"our point B, our point C,"},{"Start":"05:25.865 ","End":"05:30.245","Text":"and our point D. We can see that the amplitude of the purple wave,"},{"Start":"05:30.245 ","End":"05:34.580","Text":"because we\u0027re looking at the purple wave because it\u0027s the interference,"},{"Start":"05:34.580 ","End":"05:39.010","Text":"is at these points equal to 0."},{"Start":"05:41.060 ","End":"05:46.220","Text":"What we can see is that when we move on to this graph,"},{"Start":"05:46.220 ","End":"05:47.990","Text":"graph number 3,"},{"Start":"05:47.990 ","End":"05:50.300","Text":"up until now we were looking at graph number 2,"},{"Start":"05:50.300 ","End":"05:53.060","Text":"so now we\u0027re looking at graph number 3,"},{"Start":"05:53.060 ","End":"05:57.200","Text":"now what we can see is that our green wave obviously"},{"Start":"05:57.200 ","End":"06:01.460","Text":"is propagating in the leftwards direction,"},{"Start":"06:01.460 ","End":"06:06.390","Text":"and the red wave is propagating in the rightwards direction."},{"Start":"06:06.590 ","End":"06:11.085","Text":"Graph number 3 is a picture of the 2 waves,"},{"Start":"06:11.085 ","End":"06:15.165","Text":"a snapshot a few moments later."},{"Start":"06:15.165 ","End":"06:20.985","Text":"Our green wave has carried on propagating in this leftwards direction,"},{"Start":"06:20.985 ","End":"06:25.250","Text":"and our red wave has also propagated in the rightwards direction,"},{"Start":"06:25.250 ","End":"06:27.425","Text":"so they\u0027ve shifted a little bit."},{"Start":"06:27.425 ","End":"06:29.785","Text":"That\u0027s why we can see this different pattern."},{"Start":"06:29.785 ","End":"06:33.300","Text":"Now if we add on the amplitudes,"},{"Start":"06:33.300 ","End":"06:35.445","Text":"we superimpose these 2 waves."},{"Start":"06:35.445 ","End":"06:39.665","Text":"Here we can see that the green wave has a positive amplitude,"},{"Start":"06:39.665 ","End":"06:45.845","Text":"but the red wave at this exact point D has a negative amplitude,"},{"Start":"06:45.845 ","End":"06:49.005","Text":"but with the same magnitude."},{"Start":"06:49.005 ","End":"06:52.525","Text":"If we have a plus 1, minus 1,"},{"Start":"06:52.525 ","End":"07:00.805","Text":"we\u0027ll get that point D is equal to 0 on our superimposed purple wave that will soon join."},{"Start":"07:00.805 ","End":"07:04.925","Text":"Then let\u0027s look at this point over here."},{"Start":"07:04.925 ","End":"07:07.445","Text":"Here we have an amplitude,"},{"Start":"07:07.445 ","End":"07:10.940","Text":"the red wave has an amplitude of 0 and the green wave has an amplitude of 0,"},{"Start":"07:10.940 ","End":"07:17.165","Text":"so here we also have a stationary point."},{"Start":"07:17.165 ","End":"07:18.520","Text":"Here, again,"},{"Start":"07:18.520 ","End":"07:22.600","Text":"we have the same story as we had at point D. We have this 1/2 a"},{"Start":"07:22.600 ","End":"07:27.115","Text":"square in the positive direction and 1/2 a square amplitude in the negative direction."},{"Start":"07:27.115 ","End":"07:30.145","Text":"Again, we\u0027ll have a 0 amplitude here,"},{"Start":"07:30.145 ","End":"07:31.480","Text":"and so on and so forth."},{"Start":"07:31.480 ","End":"07:33.805","Text":"We can see that our interference wave,"},{"Start":"07:33.805 ","End":"07:35.650","Text":"in this type of situation,"},{"Start":"07:35.650 ","End":"07:39.565","Text":"we have complete destructive interference."},{"Start":"07:39.565 ","End":"07:44.210","Text":"Our purple wave is just going to be a flat line."},{"Start":"07:44.340 ","End":"07:47.155","Text":"Here we can see I\u0027ve drawn it in."},{"Start":"07:47.155 ","End":"07:51.640","Text":"Now what we can see that\u0027s important to note is also in this case,"},{"Start":"07:51.640 ","End":"07:53.740","Text":"point D is at 0,"},{"Start":"07:53.740 ","End":"07:56.350","Text":"C is at 0, B is at 0,"},{"Start":"07:56.350 ","End":"07:57.820","Text":"and A is at 0."},{"Start":"07:57.820 ","End":"07:59.995","Text":"The magnitude of the purple wave,"},{"Start":"07:59.995 ","End":"08:02.020","Text":"which is the product of the interference,"},{"Start":"08:02.020 ","End":"08:05.240","Text":"is always at 0 in this case as well."},{"Start":"08:05.790 ","End":"08:09.970","Text":"Now let\u0027s move on to graph number 4,"},{"Start":"08:09.970 ","End":"08:12.205","Text":"which was taken a moment later."},{"Start":"08:12.205 ","End":"08:16.270","Text":"Sometime later, it\u0027s another snapshot of our wave."},{"Start":"08:16.270 ","End":"08:18.880","Text":"Again, we have our red wave propagating in"},{"Start":"08:18.880 ","End":"08:23.920","Text":"the rightwards direction and our green wave propagating in our leftwards direction."},{"Start":"08:23.920 ","End":"08:29.960","Text":"Now let\u0027s see what our interference pattern looks like in this case."},{"Start":"08:30.000 ","End":"08:32.275","Text":"Let\u0027s look, so again,"},{"Start":"08:32.275 ","End":"08:38.576","Text":"our green wave has some positive amplitude and our red wave has a negative amplitude,"},{"Start":"08:38.576 ","End":"08:41.395","Text":"and both of these amplitudes have the same magnitude."},{"Start":"08:41.395 ","End":"08:43.615","Text":"Just 1 is positive and 1 is negative."},{"Start":"08:43.615 ","End":"08:46.135","Text":"We can see over here at point D,"},{"Start":"08:46.135 ","End":"08:49.885","Text":"the positive and the negative makes 0."},{"Start":"08:49.885 ","End":"08:52.870","Text":"Again, point D is stationary."},{"Start":"08:52.870 ","End":"08:59.440","Text":"Then over here we have the green and the red have the same amplitude each."},{"Start":"08:59.440 ","End":"09:02.020","Text":"Our interference wave, which is in purple,"},{"Start":"09:02.020 ","End":"09:05.185","Text":"will have a double amplitude, so it\u0027s up here."},{"Start":"09:05.185 ","End":"09:07.810","Text":"Let\u0027s look over here."},{"Start":"09:07.810 ","End":"09:12.790","Text":"At point C, we see that the red wave has some positive amplitude,"},{"Start":"09:12.790 ","End":"09:15.565","Text":"and the green wave has a negative amplitude,"},{"Start":"09:15.565 ","End":"09:18.820","Text":"but both of the amplitudes are of equal magnitude."},{"Start":"09:18.820 ","End":"09:21.820","Text":"That means that again,"},{"Start":"09:21.820 ","End":"09:25.330","Text":"point C is again at a 0."},{"Start":"09:25.330 ","End":"09:29.620","Text":"Then here we have the red and green have"},{"Start":"09:29.620 ","End":"09:34.585","Text":"the same amplitude in the negative direction with the same magnitude."},{"Start":"09:34.585 ","End":"09:38.790","Text":"Our interference wave is going to have a peak at"},{"Start":"09:38.790 ","End":"09:44.010","Text":"double the green wave\u0027s amplitude or double the red wave\u0027s amplitude,"},{"Start":"09:44.010 ","End":"09:45.450","Text":"so it has a peak."},{"Start":"09:45.450 ","End":"09:46.715","Text":"Then let\u0027s look at point B,"},{"Start":"09:46.715 ","End":"09:52.900","Text":"and we can see again the green and red have the same magnitude of amplitude,"},{"Start":"09:52.900 ","End":"09:54.850","Text":"however, with different signs."},{"Start":"09:54.850 ","End":"09:59.620","Text":"The green is a positive and the red is a negative so they cancel each other out,"},{"Start":"09:59.620 ","End":"10:02.950","Text":"and then point B again is a 0."},{"Start":"10:02.950 ","End":"10:06.310","Text":"Then we can do the same thing for point A."},{"Start":"10:06.310 ","End":"10:10.420","Text":"We can see that at these different times."},{"Start":"10:10.420 ","End":"10:14.155","Text":"It doesn\u0027t matter at what time we\u0027re looking."},{"Start":"10:14.155 ","End":"10:16.360","Text":"Points A, B, C,"},{"Start":"10:16.360 ","End":"10:22.210","Text":"and D are always going to be at the 0 mark."},{"Start":"10:22.210 ","End":"10:24.055","Text":"Now a few lessons ago,"},{"Start":"10:24.055 ","End":"10:28.180","Text":"we learned about the concept of nodes."},{"Start":"10:28.180 ","End":"10:30.970","Text":"These points A, B,"},{"Start":"10:30.970 ","End":"10:34.400","Text":"C, and D are all nodes."},{"Start":"10:34.950 ","End":"10:37.180","Text":"Nodes are very,"},{"Start":"10:37.180 ","End":"10:40.855","Text":"very important when dealing with standing waves."},{"Start":"10:40.855 ","End":"10:43.735","Text":"Whenever we have standing waves, we will have,"},{"Start":"10:43.735 ","End":"10:49.570","Text":"depending on the length of the rope and the frequency and wavelength of our waves,"},{"Start":"10:49.570 ","End":"10:55.420","Text":"but we will always have nodes when dealing with standing waves."},{"Start":"10:55.420 ","End":"10:58.120","Text":"When we have a node,"},{"Start":"10:58.120 ","End":"11:00.070","Text":"so let\u0027s say over here,"},{"Start":"11:00.070 ","End":"11:04.105","Text":"we can see that our purple wave has a peak,"},{"Start":"11:04.105 ","End":"11:05.485","Text":"it\u0027s at 0 with a node,"},{"Start":"11:05.485 ","End":"11:09.940","Text":"it peaks, then again 0 at another node and then another peak,"},{"Start":"11:09.940 ","End":"11:12.640","Text":"but in the negative direction, so a trough."},{"Start":"11:12.640 ","End":"11:18.340","Text":"This 0 point over here is called"},{"Start":"11:18.340 ","End":"11:26.240","Text":"a node and the point that has a peak is called an anti-node."},{"Start":"11:26.790 ","End":"11:29.905","Text":"When we\u0027re dealing with standing waves,"},{"Start":"11:29.905 ","End":"11:37.315","Text":"what we\u0027ll see is that if we draw out our axes, like so."},{"Start":"11:37.315 ","End":"11:42.730","Text":"Our standing wave will always look like so,"},{"Start":"11:42.730 ","End":"11:47.140","Text":"where our nodes are over here,"},{"Start":"11:47.140 ","End":"11:53.605","Text":"and the peak will just alternate"},{"Start":"11:53.605 ","End":"12:01.615","Text":"with a trough and the trough will alternate with the peak."},{"Start":"12:01.615 ","End":"12:05.560","Text":"What we\u0027ll see in a standing wave is that instead of saying"},{"Start":"12:05.560 ","End":"12:09.370","Text":"like before where we had some wave like so,"},{"Start":"12:09.370 ","End":"12:14.050","Text":"which was just propagating along the rope."},{"Start":"12:14.050 ","End":"12:19.270","Text":"What we\u0027ll see is our standing wave appears to be stationary or to be"},{"Start":"12:19.270 ","End":"12:25.660","Text":"standing and the nodes are always at the same place,"},{"Start":"12:25.660 ","End":"12:31.405","Text":"and the only thing that we\u0027ll see is that sometimes in between this node and this node,"},{"Start":"12:31.405 ","End":"12:36.520","Text":"we\u0027ll have a peak and sometimes will have a trough."},{"Start":"12:36.520 ","End":"12:41.620","Text":"But these points in red will always be stationary and it will"},{"Start":"12:41.620 ","End":"12:46.345","Text":"appear like our wave isn\u0027t moving and isn\u0027t propagating."},{"Start":"12:46.345 ","End":"12:49.000","Text":"This is what is called a standing wave,"},{"Start":"12:49.000 ","End":"12:53.470","Text":"and it\u0027s generated by sending one wave in"},{"Start":"12:53.470 ","End":"13:00.230","Text":"this direction and another wave in the opposite direction."},{"Start":"13:00.540 ","End":"13:06.505","Text":"If we send these 2 waves at the exact correct frequency,"},{"Start":"13:06.505 ","End":"13:10.585","Text":"then these 2 waves will interfere with one another,"},{"Start":"13:10.585 ","End":"13:14.530","Text":"causing us to have a standing wave."},{"Start":"13:14.530 ","End":"13:18.745","Text":"Now, you might have seen this if you play musical instruments,"},{"Start":"13:18.745 ","End":"13:21.070","Text":"if you pluck a guitar string at"},{"Start":"13:21.070 ","End":"13:26.195","Text":"the exact right position and in the exact right frequency,"},{"Start":"13:26.195 ","End":"13:31.120","Text":"then you might see on your guitar string a standing node and also if you get"},{"Start":"13:31.120 ","End":"13:36.355","Text":"a skipping rope and you stand on one end and a friend on the other end,"},{"Start":"13:36.355 ","End":"13:39.865","Text":"and you each try and make a wave on the skipping rope,"},{"Start":"13:39.865 ","End":"13:44.890","Text":"traveling in the opposite directions towards one another."},{"Start":"13:44.890 ","End":"13:49.450","Text":"If you and your friend are working at the correct frequency,"},{"Start":"13:49.450 ","End":"13:57.350","Text":"you might see on the skipping rope this standing wave phenomenon."},{"Start":"13:58.920 ","End":"14:01.240","Text":"In a standing wave,"},{"Start":"14:01.240 ","End":"14:08.605","Text":"we have our nodes which have amplitude or a particle displacement of 0 always."},{"Start":"14:08.605 ","End":"14:10.915","Text":"Then we have our anti-nodes,"},{"Start":"14:10.915 ","End":"14:14.920","Text":"which are points that move between"},{"Start":"14:14.920 ","End":"14:17.755","Text":"the maximum amplitude in the positive direction"},{"Start":"14:17.755 ","End":"14:21.985","Text":"0 and the maximum amplitude in the negative direction."},{"Start":"14:21.985 ","End":"14:24.910","Text":"If you\u0027re looking at a standing wave,"},{"Start":"14:24.910 ","End":"14:27.190","Text":"you won\u0027t see the wave propagating."},{"Start":"14:27.190 ","End":"14:32.020","Text":"It will look like it\u0027s a stationary wave where this section in"},{"Start":"14:32.020 ","End":"14:37.930","Text":"the anti-node is what is moving up and down and all the other points are stationary."},{"Start":"14:37.930 ","End":"14:44.309","Text":"Here we\u0027ll also be moving up and down and here as well but it\u0027s stationary."},{"Start":"14:44.309 ","End":"14:47.390","Text":"We spoke about imagining that"},{"Start":"14:47.390 ","End":"14:52.760","Text":"the rope that we\u0027re sending the green wave and red wave down,"},{"Start":"14:52.760 ","End":"14:54.600","Text":"isn\u0027t tied to an end."},{"Start":"14:54.600 ","End":"14:58.380","Text":"But if we imagine that our rope is tied to an end."},{"Start":"14:58.380 ","End":"15:03.280","Text":"If we imagine that our rope is always tied to point D,"},{"Start":"15:03.280 ","End":"15:05.515","Text":"so A, B, C,"},{"Start":"15:05.515 ","End":"15:12.205","Text":"D. Let\u0027s call this point D. If our rope was always tied to point D,"},{"Start":"15:12.205 ","End":"15:14.754","Text":"we\u0027ll have the exact same pattern,"},{"Start":"15:14.754 ","End":"15:21.730","Text":"and similarly, if our rope was always tied to point C over here,"},{"Start":"15:21.730 ","End":"15:25.510","Text":"we\u0027d also have this same pattern with"},{"Start":"15:25.510 ","End":"15:31.405","Text":"our anti-nodes over here in these 2 positions and our nodes over here."},{"Start":"15:31.405 ","End":"15:41.260","Text":"However, if what I would do is I would tie my rope to this point over here to"},{"Start":"15:41.260 ","End":"15:45.440","Text":"some kind of wall which is located at some distance"},{"Start":"15:45.440 ","End":"15:50.810","Text":"between the 2 nodes B and"},{"Start":"15:50.810 ","End":"15:56.840","Text":"C. If I had a wall or a pole and I tied my rope to it,"},{"Start":"15:56.840 ","End":"16:01.130","Text":"and its placement was between the nodes B and C,"},{"Start":"16:01.130 ","End":"16:03.590","Text":"then we wouldn\u0027t get the standing wave."},{"Start":"16:03.590 ","End":"16:06.290","Text":"It wouldn\u0027t work. Standing waves only"},{"Start":"16:06.290 ","End":"16:09.500","Text":"work if we either have a rope which isn\u0027t attached to"},{"Start":"16:09.500 ","End":"16:16.910","Text":"anything or if our rope is attached to the nodes."},{"Start":"16:20.130 ","End":"16:24.350","Text":"Let\u0027s put what we just said about tying our rope at"},{"Start":"16:24.350 ","End":"16:28.505","Text":"different places in mathematical terms."},{"Start":"16:28.505 ","End":"16:30.890","Text":"In order to get a standing wave,"},{"Start":"16:30.890 ","End":"16:36.050","Text":"we need our rope to be a certain length."},{"Start":"16:36.050 ","End":"16:39.530","Text":"Either the rope can be infinitely long,"},{"Start":"16:39.530 ","End":"16:41.645","Text":"in which case we don\u0027t have a problem."},{"Start":"16:41.645 ","End":"16:43.010","Text":"But as we know,"},{"Start":"16:43.010 ","End":"16:44.780","Text":"our rope is never infinitely long,"},{"Start":"16:44.780 ","End":"16:48.920","Text":"so we want to see how long it has to be and where"},{"Start":"16:48.920 ","End":"16:54.150","Text":"we have to tie it in order for us to get this standing or stationary wave."},{"Start":"16:54.330 ","End":"16:56.570","Text":"The equation goes like this."},{"Start":"16:56.570 ","End":"16:58.805","Text":"The length of the rope has to be n,"},{"Start":"16:58.805 ","End":"17:05.090","Text":"which n in mathematics represents some kind of whole number, so 1, 2, 3, 4."},{"Start":"17:05.090 ","End":"17:13.130","Text":"Those are whole numbers, multiplied by 1/2 of the wavelength,"},{"Start":"17:13.130 ","End":"17:16.140","Text":"where Lambda is the wavelength."},{"Start":"17:17.550 ","End":"17:19.570","Text":"Now we can see that,"},{"Start":"17:19.570 ","End":"17:22.075","Text":"let\u0027s see what this means."},{"Start":"17:22.075 ","End":"17:27.025","Text":"We said that as long as we tie the rope at 1 of the nodes,"},{"Start":"17:27.025 ","End":"17:31.090","Text":"we\u0027re still going to get this standing wave. Let\u0027s take a look."},{"Start":"17:31.090 ","End":"17:33.865","Text":"From this node over here,"},{"Start":"17:33.865 ","End":"17:37.945","Text":"A, let\u0027s give it a name. It\u0027s called A."},{"Start":"17:37.945 ","End":"17:41.590","Text":"If we go from A to B,"},{"Start":"17:41.590 ","End":"17:45.777","Text":"that is 1/2 a wavelength."},{"Start":"17:45.777 ","End":"17:48.235","Text":"Let\u0027s just scroll down a little bit."},{"Start":"17:48.235 ","End":"17:50.365","Text":"Whole numbers, let\u0027s write are 1,"},{"Start":"17:50.365 ","End":"17:54.350","Text":"2, 3, etc."},{"Start":"17:54.840 ","End":"18:01.585","Text":"If we go from A and tie our rope to B,"},{"Start":"18:01.585 ","End":"18:04.944","Text":"so we can see that we\u0027ve gone 1/2 a wavelength."},{"Start":"18:04.944 ","End":"18:08.230","Text":"If we\u0027re measuring our wavelength from A,"},{"Start":"18:08.230 ","End":"18:11.620","Text":"a whole wavelength is from A to C. Can you see that?"},{"Start":"18:11.620 ","End":"18:17.995","Text":"Because we\u0027re going from the 0 point until the section where the wave goes up."},{"Start":"18:17.995 ","End":"18:19.735","Text":"Here where the 0 point,"},{"Start":"18:19.735 ","End":"18:20.965","Text":"and the wave is going up,"},{"Start":"18:20.965 ","End":"18:23.080","Text":"the gradient is increasing."},{"Start":"18:23.080 ","End":"18:25.855","Text":"Now we can see we have 1/2 a wavelength."},{"Start":"18:25.855 ","End":"18:32.485","Text":"That means that we have 1 times a 1/2 of the wavelength. That will be fine."},{"Start":"18:32.485 ","End":"18:35.680","Text":"If we want to tie a rope to D, let\u0027s say,"},{"Start":"18:35.680 ","End":"18:39.160","Text":"so that means we have 1/2 a wavelength,"},{"Start":"18:39.160 ","End":"18:43.224","Text":"1 full wavelength, plus another 1/2 a wavelength."},{"Start":"18:43.224 ","End":"18:45.325","Text":"We have 1/2,"},{"Start":"18:45.325 ","End":"18:48.190","Text":"2 halves, 3 halves."},{"Start":"18:48.190 ","End":"18:50.035","Text":"3 is a whole number,"},{"Start":"18:50.035 ","End":"18:52.180","Text":"3 times 1/2 a wavelength."},{"Start":"18:52.180 ","End":"18:55.120","Text":"However, if we wanted a tie,"},{"Start":"18:55.120 ","End":"18:57.490","Text":"like we said before,"},{"Start":"18:57.490 ","End":"19:03.055","Text":"our rope to this bar over here,"},{"Start":"19:03.055 ","End":"19:05.575","Text":"so let\u0027s see what would happen."},{"Start":"19:05.575 ","End":"19:12.715","Text":"Then we can see that we 1/2 plus a 1/4 wavelength around about."},{"Start":"19:12.715 ","End":"19:16.315","Text":"We can see that 1/2 plus a 1/4 is 3/4."},{"Start":"19:16.315 ","End":"19:20.155","Text":"That means 3/4 times 1/2 Lambda,"},{"Start":"19:20.155 ","End":"19:23.300","Text":"but 3/4 isn\u0027t a whole number."},{"Start":"19:23.790 ","End":"19:29.630","Text":"That means that we won\u0027t get a standing wave."},{"Start":"19:30.810 ","End":"19:37.975","Text":"A way of looking at n is by just counting how many anti-nodes there are."},{"Start":"19:37.975 ","End":"19:39.910","Text":"Here\u0027s 1 anti-node,"},{"Start":"19:39.910 ","End":"19:41.590","Text":"here\u0027s another anti-node,"},{"Start":"19:41.590 ","End":"19:43.300","Text":"and the third anti-node,"},{"Start":"19:43.300 ","End":"19:45.355","Text":"and the fourth anti-node."},{"Start":"19:45.355 ","End":"19:47.875","Text":"In that case, the length,"},{"Start":"19:47.875 ","End":"19:50.260","Text":"how long will this rope be?"},{"Start":"19:50.260 ","End":"19:54.325","Text":"L is equal to n,"},{"Start":"19:54.325 ","End":"19:56.365","Text":"so the whole number, which has also,"},{"Start":"19:56.365 ","End":"20:00.670","Text":"we can say the number of anti-nodes here,"},{"Start":"20:00.670 ","End":"20:03.115","Text":"so the number of anti-nodes, a little note."},{"Start":"20:03.115 ","End":"20:05.110","Text":"We have 1 anti-node,"},{"Start":"20:05.110 ","End":"20:08.020","Text":"2 anti-node, 3 anti-node, and 4 anti-node,"},{"Start":"20:08.020 ","End":"20:12.955","Text":"so we have 4 multiplied by 1/2 of Lambda,"},{"Start":"20:12.955 ","End":"20:15.700","Text":"so that is equal to 2 Lambda."},{"Start":"20:15.700 ","End":"20:19.030","Text":"Then let\u0027s see exactly what that is."},{"Start":"20:19.030 ","End":"20:20.830","Text":"If we start from A,"},{"Start":"20:20.830 ","End":"20:24.220","Text":"so we have 1 wavelength,"},{"Start":"20:24.220 ","End":"20:25.900","Text":"so that\u0027s 1 Lambda."},{"Start":"20:25.900 ","End":"20:31.645","Text":"Then a second wavelength, that\u0027s 2 Lambda."},{"Start":"20:31.645 ","End":"20:37.150","Text":"We see that really this rope is of length 2 Lambda."},{"Start":"20:37.150 ","End":"20:41.360","Text":"That\u0027s how we can see how this equation makes sense and works."},{"Start":"20:41.430 ","End":"20:48.385","Text":"Now let\u0027s talk about how standing waves are generated physically in a lab."},{"Start":"20:48.385 ","End":"20:56.080","Text":"In a lab, we\u0027ll usually have our rope over here tied,"},{"Start":"20:56.080 ","End":"21:01.571","Text":"and it\u0027s usually attached to some kind of"},{"Start":"21:01.571 ","End":"21:09.160","Text":"loop or a wheel that has some mass attached to it."},{"Start":"21:09.160 ","End":"21:16.195","Text":"Then here, we have something that minimizes the friction."},{"Start":"21:16.195 ","End":"21:18.520","Text":"This is what this wheel really does,"},{"Start":"21:18.520 ","End":"21:21.610","Text":"it minimizes the friction with the help of this mass."},{"Start":"21:21.610 ","End":"21:24.220","Text":"This, you don\u0027t really need to know in detail about,"},{"Start":"21:24.220 ","End":"21:26.290","Text":"it looks something like this."},{"Start":"21:26.290 ","End":"21:31.765","Text":"Then we have some kind of machine that will lift the rope up like so,"},{"Start":"21:31.765 ","End":"21:35.350","Text":"and will push it back down like so."},{"Start":"21:35.350 ","End":"21:37.945","Text":"This will happen constantly."},{"Start":"21:37.945 ","End":"21:40.480","Text":"Our machine will move up and down, up and down,"},{"Start":"21:40.480 ","End":"21:44.770","Text":"up and down, and this will generate a wave."},{"Start":"21:44.770 ","End":"21:48.485","Text":"Then our wave will look something like so."},{"Start":"21:48.485 ","End":"21:55.275","Text":"That\u0027s 1 wavelength and then another wavelength."},{"Start":"21:55.275 ","End":"21:59.685","Text":"Imagine that again, this is a perfect sinusoidal wave,"},{"Start":"21:59.685 ","End":"22:02.445","Text":"not some weird sketch that I did."},{"Start":"22:02.445 ","End":"22:07.550","Text":"Here our end is tied."},{"Start":"22:07.550 ","End":"22:09.625","Text":"Or as we like to call it,"},{"Start":"22:09.625 ","End":"22:11.950","Text":"we have a fixed end over here."},{"Start":"22:11.950 ","End":"22:14.604","Text":"As we learned in previous lessons,"},{"Start":"22:14.604 ","End":"22:16.465","Text":"when we have a fixed end,"},{"Start":"22:16.465 ","End":"22:24.040","Text":"that means that we\u0027re going to have a reflection in the y-axis."},{"Start":"22:24.040 ","End":"22:26.260","Text":"We have a reflection in the y-axis,"},{"Start":"22:26.260 ","End":"22:30.340","Text":"which means that we\u0027re going to have the wave"},{"Start":"22:30.340 ","End":"22:37.540","Text":"returning like so, something like this."},{"Start":"22:37.540 ","End":"22:41.470","Text":"Again, imagine that I\u0027ve drawn this well."},{"Start":"22:41.470 ","End":"22:44.050","Text":"As we can see, our green wave is"},{"Start":"22:44.050 ","End":"22:49.105","Text":"propagating back to the source because it\u0027s a reflected wave,"},{"Start":"22:49.105 ","End":"22:54.865","Text":"whereas our red wave was the wave that our machine generated."},{"Start":"22:54.865 ","End":"22:58.360","Text":"This is how we generate a standing wave."},{"Start":"22:58.360 ","End":"23:00.760","Text":"Due to this reflection on the other end,"},{"Start":"23:00.760 ","End":"23:04.270","Text":"we\u0027re going to get 2 identical waves that"},{"Start":"23:04.270 ","End":"23:07.900","Text":"are propagating in opposite directions towards one another,"},{"Start":"23:07.900 ","End":"23:11.500","Text":"which then interfere with one another,"},{"Start":"23:11.500 ","End":"23:14.230","Text":"causing the standing wave."},{"Start":"23:14.230 ","End":"23:18.175","Text":"Now let\u0027s take a look at what happens with"},{"Start":"23:18.175 ","End":"23:23.005","Text":"standing waves when we\u0027re speaking about instruments,"},{"Start":"23:23.005 ","End":"23:28.820","Text":"specifically stringed instruments, such as guitars, and violins."},{"Start":"23:29.340 ","End":"23:32.410","Text":"Up until now, we\u0027ve been dealing with"},{"Start":"23:32.410 ","End":"23:36.580","Text":"this equation for the length of the string or of the rope,"},{"Start":"23:36.580 ","End":"23:42.265","Text":"which is equal to n multiplied by the 1/2 of Lambda."},{"Start":"23:42.265 ","End":"23:46.180","Text":"What we can see is that if we know the length of"},{"Start":"23:46.180 ","End":"23:51.920","Text":"our rope and we have a specific number of n\u0027s."},{"Start":"23:52.890 ","End":"23:58.000","Text":"We can see that our Lambda or wavelength is dependent on"},{"Start":"23:58.000 ","End":"24:05.275","Text":"n. Let\u0027s say that we have that n is equal to 1."},{"Start":"24:05.275 ","End":"24:08.770","Text":"If we multiply both sides by 2,"},{"Start":"24:08.770 ","End":"24:14.860","Text":"that means that we\u0027ll get that Lambda is equal to 2L."},{"Start":"24:14.860 ","End":"24:20.720","Text":"That means that the wavelength is equal to 2 times the length of our string."},{"Start":"24:21.990 ","End":"24:25.783","Text":"If n, I\u0027m reminding is the number of anti-nodes,"},{"Start":"24:25.783 ","End":"24:27.610","Text":"if n is equal to 2,"},{"Start":"24:27.610 ","End":"24:33.400","Text":"then we\u0027ll get that our wavelength is equal to the length of the string."},{"Start":"24:33.400 ","End":"24:36.670","Text":"If our n is equal to 3,"},{"Start":"24:36.670 ","End":"24:45.265","Text":"then we\u0027ll get that our wavelength is equal to 2/3 the length of the string."},{"Start":"24:45.265 ","End":"24:48.265","Text":"Of course, we can carry on."},{"Start":"24:48.265 ","End":"24:50.710","Text":"What we can see is that our wavelength is"},{"Start":"24:50.710 ","End":"24:55.910","Text":"dependent on the number of anti-nodes that we have."},{"Start":"24:57.810 ","End":"25:03.310","Text":"We\u0027ve seen that the velocity of a wave can be"},{"Start":"25:03.310 ","End":"25:09.850","Text":"expressed by writing it as the wavelength multiplied by the frequency."},{"Start":"25:09.850 ","End":"25:12.445","Text":"If we isolate out our wavelength,"},{"Start":"25:12.445 ","End":"25:15.565","Text":"so we\u0027ll get that our wavelength is equal to"},{"Start":"25:15.565 ","End":"25:20.410","Text":"the velocity of the wave divided by the frequency."},{"Start":"25:20.410 ","End":"25:27.625","Text":"Now, if we plug this into our equation L for the length of the string,"},{"Start":"25:27.625 ","End":"25:33.100","Text":"we\u0027ll see that L is equal to n multiplied by 1/2 of the wavelength,"},{"Start":"25:33.100 ","End":"25:37.360","Text":"where the wavelength is equal to v divided by"},{"Start":"25:37.360 ","End":"25:43.330","Text":"f. Now we can see that"},{"Start":"25:43.330 ","End":"25:49.450","Text":"the frequency is also dependent on n. The frequency,"},{"Start":"25:49.450 ","End":"25:51.655","Text":"which is what defines our wavelength,"},{"Start":"25:51.655 ","End":"25:55.610","Text":"is dependent on the number of anti-nodes."},{"Start":"25:56.970 ","End":"26:01.855","Text":"For every n, we have a specific frequency,"},{"Start":"26:01.855 ","End":"26:03.865","Text":"and this frequency,"},{"Start":"26:03.865 ","End":"26:07.760","Text":"we call it the normal mode."},{"Start":"26:08.360 ","End":"26:11.295","Text":"That\u0027s something important to know."},{"Start":"26:11.295 ","End":"26:14.940","Text":"Also, another thing to remember is that another way that we"},{"Start":"26:14.940 ","End":"26:19.420","Text":"describe the velocity of the wave was by the equation that"},{"Start":"26:19.420 ","End":"26:22.510","Text":"the velocity of the wave is equal to the square root of"},{"Start":"26:22.510 ","End":"26:27.475","Text":"the tension in the rope divided by Mu,"},{"Start":"26:27.475 ","End":"26:30.970","Text":"which is the density of the rope or string."},{"Start":"26:30.970 ","End":"26:37.285","Text":"That\u0027s for our v. Now we can substitute in our v over here"},{"Start":"26:37.285 ","End":"26:44.425","Text":"into this equation for L. We will get that L is equal to n,"},{"Start":"26:44.425 ","End":"26:46.600","Text":"multiplied by 1/2,"},{"Start":"26:46.600 ","End":"26:49.045","Text":"multiplied by v,"},{"Start":"26:49.045 ","End":"26:52.660","Text":"where v is equal to the square root of the tension in"},{"Start":"26:52.660 ","End":"26:57.085","Text":"the string divided by the density of the string,"},{"Start":"26:57.085 ","End":"27:01.090","Text":"divided by our normal mode, or a frequency,"},{"Start":"27:01.090 ","End":"27:05.665","Text":"which is dependent on n. Then we get this equation as well."},{"Start":"27:05.665 ","End":"27:08.305","Text":"What does this equation allow us to do?"},{"Start":"27:08.305 ","End":"27:12.175","Text":"Let\u0027s imagine I have my violin."},{"Start":"27:12.175 ","End":"27:16.150","Text":"I can see, let\u0027s say I want to get some kind of tone,"},{"Start":"27:16.150 ","End":"27:18.655","Text":"I want to play a note on my violin."},{"Start":"27:18.655 ","End":"27:20.785","Text":"If I\u0027m the violin maker,"},{"Start":"27:20.785 ","End":"27:23.905","Text":"so what I can do is I can change"},{"Start":"27:23.905 ","End":"27:30.685","Text":"3 different things and therefore change the tone or the note that I will play."},{"Start":"27:30.685 ","End":"27:34.360","Text":"The first thing I can do is dealing with Mu,"},{"Start":"27:34.360 ","End":"27:36.160","Text":"the density of the string."},{"Start":"27:36.160 ","End":"27:39.175","Text":"I can use different types of materials,"},{"Start":"27:39.175 ","End":"27:43.385","Text":"so different metals or different plastics to make the string,"},{"Start":"27:43.385 ","End":"27:48.465","Text":"and therefore, I will change the note that is played."},{"Start":"27:48.465 ","End":"27:51.870","Text":"Another way I can change the note that is played"},{"Start":"27:51.870 ","End":"27:55.365","Text":"is by changing the tension in the string."},{"Start":"27:55.365 ","End":"27:58.440","Text":"That\u0027s what you do when you turn the knobs"},{"Start":"27:58.440 ","End":"28:02.120","Text":"at the end of your violin or at the end of the guitar."},{"Start":"28:02.120 ","End":"28:05.440","Text":"You\u0027re changing the tension in the string,"},{"Start":"28:05.440 ","End":"28:09.220","Text":"and therefore changing the notes played."},{"Start":"28:09.220 ","End":"28:12.625","Text":"That\u0027s how you tune the guitar or the violin."},{"Start":"28:12.625 ","End":"28:17.980","Text":"Another way that we can do it is by changing the length of the string."},{"Start":"28:17.980 ","End":"28:22.480","Text":"If I have a shorter string or a longer string,"},{"Start":"28:22.480 ","End":"28:24.805","Text":"I\u0027ll have a different frequency played,"},{"Start":"28:24.805 ","End":"28:28.670","Text":"which corresponds to a different node being played."},{"Start":"28:29.610 ","End":"28:35.725","Text":"This has just a little side note of what we can do with standing waves."},{"Start":"28:35.725 ","End":"28:39.130","Text":"But what\u0027s important to remember, let\u0027s scroll up,"},{"Start":"28:39.130 ","End":"28:43.315","Text":"is 1, how standing waves are generally created."},{"Start":"28:43.315 ","End":"28:46.300","Text":"Instead of sending 2 waves together,"},{"Start":"28:46.300 ","End":"28:51.100","Text":"so we just use the idea of reflection of 1 wave and then we get standing waves."},{"Start":"28:51.100 ","End":"28:56.620","Text":"Really the most important is to remember"},{"Start":"28:56.620 ","End":"29:02.590","Text":"this equation and to remember what an anti-node is and what a node is."},{"Start":"29:02.590 ","End":"29:05.720","Text":"That\u0027s the end of this lesson."}],"ID":12468},{"Watched":false,"Name":"Exercise - Standing Waves","Duration":"18m 51s","ChapterTopicVideoID":12001,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.905","Text":"Hello. In this lesson,"},{"Start":"00:01.905 ","End":"00:06.330","Text":"we\u0027re going to be practicing what we learned in the previous lesson about standing waves."},{"Start":"00:06.330 ","End":"00:09.255","Text":"In this question, we have a rope AB,"},{"Start":"00:09.255 ","End":"00:11.570","Text":"which is 1 meter long,"},{"Start":"00:11.570 ","End":"00:16.710","Text":"and we have end A which is fixed to a pole,"},{"Start":"00:16.710 ","End":"00:25.511","Text":"and end B is attached to some wave generator."},{"Start":"00:25.511 ","End":"00:26.850","Text":"So this is end B,"},{"Start":"00:26.850 ","End":"00:30.390","Text":"and this entire length is 1 meter."},{"Start":"00:30.390 ","End":"00:32.280","Text":"The generator is switched on,"},{"Start":"00:32.280 ","End":"00:35.265","Text":"and the frequency is gradually increased."},{"Start":"00:35.265 ","End":"00:40.635","Text":"The frequency is recorded each time a standing wave is produced."},{"Start":"00:40.635 ","End":"00:43.385","Text":"If we remember how standing waves are produced,"},{"Start":"00:43.385 ","End":"00:46.364","Text":"so end B is producing some pulse,"},{"Start":"00:46.364 ","End":"00:49.790","Text":"let\u0027s draw it in blue, which, let\u0027s say,"},{"Start":"00:49.790 ","End":"00:51.620","Text":"it looks something like that,"},{"Start":"00:51.620 ","End":"00:56.075","Text":"which is going to travel all the way to end A."},{"Start":"00:56.075 ","End":"00:59.090","Text":"End A is a fixed end."},{"Start":"00:59.090 ","End":"01:04.698","Text":"We know that we\u0027re going to have some reflection,"},{"Start":"01:04.698 ","End":"01:06.725","Text":"and we know that that means that we\u0027re going to have"},{"Start":"01:06.725 ","End":"01:10.820","Text":"a reflection in both the x and y axes."},{"Start":"01:10.820 ","End":"01:15.575","Text":"So then our wave is going to be reflected back."},{"Start":"01:15.575 ","End":"01:20.775","Text":"We\u0027ll look in the x and y reflected direction."},{"Start":"01:20.775 ","End":"01:23.325","Text":"So it will look something like this."},{"Start":"01:23.325 ","End":"01:26.840","Text":"Then as we know, when we get to the right frequency,"},{"Start":"01:26.840 ","End":"01:28.865","Text":"we\u0027re going to have a standing wave,"},{"Start":"01:28.865 ","End":"01:33.000","Text":"which means that we\u0027re going to have nodes."},{"Start":"01:33.000 ","End":"01:36.025","Text":"If we draw this out,"},{"Start":"01:36.025 ","End":"01:39.320","Text":"where the nodes are,"},{"Start":"01:39.320 ","End":"01:45.470","Text":"these red points over here at the 0 mark."},{"Start":"01:45.470 ","End":"01:48.594","Text":"As we know, these points don\u0027t move."},{"Start":"01:48.594 ","End":"01:51.275","Text":"They remain stationary in the standing wave."},{"Start":"01:51.275 ","End":"01:55.265","Text":"As we know that these sections of maximum or minimum,"},{"Start":"01:55.265 ","End":"01:56.660","Text":"which are alternating,"},{"Start":"01:56.660 ","End":"02:00.685","Text":"and this is the section that moves this whole area."},{"Start":"02:00.685 ","End":"02:02.480","Text":"Let\u0027s mark it in blue."},{"Start":"02:02.480 ","End":"02:04.380","Text":"It\u0027s called the anti-node."},{"Start":"02:04.380 ","End":"02:06.920","Text":"So this is an anti-node. This is an anti-node,"},{"Start":"02:06.920 ","End":"02:08.945","Text":"and so on and so forth."},{"Start":"02:08.945 ","End":"02:14.220","Text":"As you can see, point B is also a node."},{"Start":"02:14.570 ","End":"02:19.300","Text":"The frequency is recorded each time we get a standing wave."},{"Start":"02:19.300 ","End":"02:22.005","Text":"Here we have a table."},{"Start":"02:22.005 ","End":"02:23.795","Text":"Question Number 1 is,"},{"Start":"02:23.795 ","End":"02:29.870","Text":"what is the wavelength denoted by the Greek letter Lambda of each standing wave?"},{"Start":"02:29.870 ","End":"02:32.450","Text":"Here we have, in this column,"},{"Start":"02:32.450 ","End":"02:34.145","Text":"the frequency recorded,"},{"Start":"02:34.145 ","End":"02:39.035","Text":"24 hertz, and here we have the shape of the wave."},{"Start":"02:39.035 ","End":"02:43.220","Text":"We know that the length of the rope is 1 meter."},{"Start":"02:43.220 ","End":"02:47.430","Text":"From this, we can calculate the wavelength."},{"Start":"02:48.230 ","End":"02:51.900","Text":"So this is our equation."},{"Start":"02:51.900 ","End":"02:54.595","Text":"Right now we\u0027re answering Question Number 1."},{"Start":"02:54.595 ","End":"02:56.240","Text":"Now we\u0027re going to fill it out in this table,"},{"Start":"02:56.240 ","End":"02:59.635","Text":"but just so you know which question we\u0027re answering."},{"Start":"02:59.635 ","End":"03:02.475","Text":"L is the length of the rope."},{"Start":"03:02.475 ","End":"03:06.650","Text":"Here we know that the length of the rope is 1 meter."},{"Start":"03:06.650 ","End":"03:12.745","Text":"N is the number of anti-nodes."},{"Start":"03:12.745 ","End":"03:15.180","Text":"In our first example,"},{"Start":"03:15.180 ","End":"03:16.815","Text":"that\u0027s going to be 1."},{"Start":"03:16.815 ","End":"03:18.665","Text":"Then we multiply it by 1/2,"},{"Start":"03:18.665 ","End":"03:20.180","Text":"multiplied by the wavelength,"},{"Start":"03:20.180 ","End":"03:23.105","Text":"which is what we\u0027re trying to calculate."},{"Start":"03:23.105 ","End":"03:28.135","Text":"What we can do is we can isolate out our wavelength,"},{"Start":"03:28.135 ","End":"03:37.179","Text":"and then we\u0027ll get that L divided by n multiplied by 2 is equal to the wavelength,"},{"Start":"03:37.179 ","End":"03:39.310","Text":"which is what we\u0027re trying to find."},{"Start":"03:39.310 ","End":"03:43.420","Text":"For the first question,"},{"Start":"03:43.420 ","End":"03:46.475","Text":"so L is equal to 1."},{"Start":"03:46.475 ","End":"03:48.360","Text":"Let\u0027s just sub that in."},{"Start":"03:48.360 ","End":"03:51.885","Text":"1 is going to be for each one of these examples."},{"Start":"03:51.885 ","End":"03:53.700","Text":"2 times 1 is 2,"},{"Start":"03:53.700 ","End":"03:58.605","Text":"so 2 divided by n is equal to Lambda."},{"Start":"03:58.605 ","End":"04:01.140","Text":"Let\u0027s find the wavelength of the first wave."},{"Start":"04:01.140 ","End":"04:05.175","Text":"We can see that we have 1 anti-node."},{"Start":"04:05.175 ","End":"04:08.015","Text":"Therefore, we\u0027re going to have,"},{"Start":"04:08.015 ","End":"04:10.190","Text":"for the first wave,"},{"Start":"04:10.190 ","End":"04:15.005","Text":"2 divided by 1 is going to be equal to Lambda."},{"Start":"04:15.005 ","End":"04:16.595","Text":"That\u0027s simply 2."},{"Start":"04:16.595 ","End":"04:19.470","Text":"So Lambda is equal to 2."},{"Start":"04:19.520 ","End":"04:22.650","Text":"Then for the second one,"},{"Start":"04:22.650 ","End":"04:30.560","Text":"we\u0027re going to have that Lambda is equal to 2 divided by the number of anti-nodes,"},{"Start":"04:30.560 ","End":"04:34.445","Text":"which is 2 divided 2,"},{"Start":"04:34.445 ","End":"04:37.285","Text":"which is equal to 1."},{"Start":"04:37.285 ","End":"04:41.855","Text":"We can write that in. Then for the third,"},{"Start":"04:41.855 ","End":"04:46.910","Text":"we\u0027re going to have that Lambda is equal"},{"Start":"04:46.910 ","End":"04:53.090","Text":"to 2 divided by the number of anti-nodes,"},{"Start":"04:53.090 ","End":"04:58.505","Text":"which is 1, 2, 3 so 2/3."},{"Start":"04:58.505 ","End":"05:02.525","Text":"That\u0027s our answer. Our fourth one,"},{"Start":"05:02.525 ","End":"05:06.620","Text":"Lambda is equal to 2 divided by the number of anti-nodes,"},{"Start":"05:06.620 ","End":"05:10.036","Text":"which is 1, 2, 3,"},{"Start":"05:10.036 ","End":"05:15.185","Text":"4, which is equal to 1/2."},{"Start":"05:15.185 ","End":"05:17.715","Text":"Let\u0027s just fill this in."},{"Start":"05:17.715 ","End":"05:20.075","Text":"So here we said we had 2/3,"},{"Start":"05:20.075 ","End":"05:23.250","Text":"and here we said we had 1/2."},{"Start":"05:24.470 ","End":"05:27.910","Text":"So this is the answer to question Number 1."},{"Start":"05:27.910 ","End":"05:30.545","Text":"It\u0027s right over here that this is question Number 1."},{"Start":"05:30.545 ","End":"05:34.025","Text":"Now let\u0027s answer Question Number 2, which is here."},{"Start":"05:34.025 ","End":"05:36.823","Text":"What is the value for 1 divided by Lambda,"},{"Start":"05:36.823 ","End":"05:40.400","Text":"so the reciprocal of the wavelength for each standing wave."},{"Start":"05:40.400 ","End":"05:48.520","Text":"Then we\u0027re going to draw a graph of frequency over here as a function of this column."},{"Start":"05:48.520 ","End":"05:51.290","Text":"Let\u0027s start about what we have over here."},{"Start":"05:52.280 ","End":"05:55.325","Text":"In order to answer question Number 2,"},{"Start":"05:55.325 ","End":"05:59.785","Text":"all we\u0027re doing is finding the reciprocal of what we found in this column."},{"Start":"05:59.785 ","End":"06:04.600","Text":"The reciprocal of 2 means 1 over 2, which is 1/2."},{"Start":"06:04.600 ","End":"06:09.695","Text":"The reciprocal of 1 is 1 divided by 1, which is 1."},{"Start":"06:09.695 ","End":"06:15.360","Text":"The reciprocal of 2/3 is 1 divided by 2/3,"},{"Start":"06:15.360 ","End":"06:19.785","Text":"which is simply 3 over 2 or 1-1/2."},{"Start":"06:19.785 ","End":"06:23.175","Text":"The reciprocal of 1/2 is 1 divided by 1/2,"},{"Start":"06:23.175 ","End":"06:25.960","Text":"which is simply 2."},{"Start":"06:26.100 ","End":"06:29.395","Text":"Now, before we begin drawing the graph,"},{"Start":"06:29.395 ","End":"06:32.305","Text":"let\u0027s see why we\u0027d want to draw the graph."},{"Start":"06:32.305 ","End":"06:34.495","Text":"We know that we have an equation,"},{"Start":"06:34.495 ","End":"06:40.180","Text":"which describes the relationship between wavelength and frequency and that is that"},{"Start":"06:40.180 ","End":"06:47.260","Text":"the wave speed is equal to the wavelength multiplied by the frequency."},{"Start":"06:47.260 ","End":"06:50.800","Text":"So if we isolate out the frequency because we are drawing"},{"Start":"06:50.800 ","End":"06:54.925","Text":"a graph of frequency as a function of 1 divided by Lambda,"},{"Start":"06:54.925 ","End":"07:05.040","Text":"so what we\u0027ll get is that frequency is equal to v multiplied by 1 divided by Lambda."},{"Start":"07:05.040 ","End":"07:09.950","Text":"What we can see is that f is"},{"Start":"07:09.950 ","End":"07:15.840","Text":"the y value of k because we are drawing graph of frequency as a function of this,"},{"Start":"07:16.250 ","End":"07:22.130","Text":"and 1 divided by Lambda is like our x value because it\u0027s as"},{"Start":"07:22.130 ","End":"07:27.176","Text":"a function of and then our V acts like M. So"},{"Start":"07:27.176 ","End":"07:31.985","Text":"what we can see is that we\u0027re going to have a linear relationship between"},{"Start":"07:31.985 ","End":"07:37.785","Text":"the frequency and our reciprocal of the wavelength."},{"Start":"07:37.785 ","End":"07:43.735","Text":"Then our v is the velocity of the wave or the speed of the wave,"},{"Start":"07:43.735 ","End":"07:49.040","Text":"and notice that Question 3 is asking us using this graph that we\u0027re about to"},{"Start":"07:49.040 ","End":"07:55.915","Text":"draw to calculate the propagation speed or the wave speed of each wave."},{"Start":"07:55.915 ","End":"08:00.950","Text":"So this is how we\u0027re going to find it through the gradient of the graph."},{"Start":"08:00.950 ","End":"08:10.400","Text":"This m represents the gradient of the graph or how steep this line is going to be,"},{"Start":"08:10.400 ","End":"08:12.830","Text":"and that is going to be our velocity."},{"Start":"08:12.830 ","End":"08:18.930","Text":"Now let\u0027s draw this graph."},{"Start":"08:19.190 ","End":"08:22.235","Text":"A few tips for drawing graphs."},{"Start":"08:22.235 ","End":"08:26.210","Text":"It\u0027s easiest if you\u0027re using square paper because"},{"Start":"08:26.210 ","End":"08:30.890","Text":"then you can really measure out the squares more accurately."},{"Start":"08:30.890 ","End":"08:39.260","Text":"Also, you have to use a ruler and a pen or pencil in order to draw your axes."},{"Start":"08:40.150 ","End":"08:44.375","Text":"Let\u0027s scroll down. Here I have my squared paper,"},{"Start":"08:44.375 ","End":"08:46.115","Text":"and I\u0027ve drawn,"},{"Start":"08:46.115 ","End":"08:50.060","Text":"with a ruler and pen, my axes."},{"Start":"08:50.060 ","End":"08:54.005","Text":"Over here I have my frequency,"},{"Start":"08:54.005 ","End":"08:57.715","Text":"which is given in Hertz."},{"Start":"08:57.715 ","End":"09:00.815","Text":"Over here in the x-axis,"},{"Start":"09:00.815 ","End":"09:05.985","Text":"I have my 1 divided by Lambda,"},{"Start":"09:05.985 ","End":"09:12.475","Text":"which is in units of 1 divided by meters or meters to the power of minus 1."},{"Start":"09:12.475 ","End":"09:14.865","Text":"Let\u0027s start with my frequency."},{"Start":"09:14.865 ","End":"09:17.295","Text":"I can see that I\u0027m going in jumps."},{"Start":"09:17.295 ","End":"09:21.060","Text":"I have 24, 45, 67, 88."},{"Start":"09:21.060 ","End":"09:23.880","Text":"Let\u0027s mark out every 25."},{"Start":"09:23.880 ","End":"09:26.332","Text":"If here I see as my origin 0,"},{"Start":"09:26.332 ","End":"09:28.580","Text":"so here I have 5, 10, 15, 20, 25, 30, 35, 40, 45, 50"},{"Start":"09:37.520 ","End":"09:43.343","Text":"over here. So each small square represents 5, 55, 60, 65, 70, 75, 80, 85,"},{"Start":"09:43.343 ","End":"09:52.189","Text":"90, 95, 100."},{"Start":"09:52.189 ","End":"10:01.600","Text":"Then I can see"},{"Start":"10:01.600 ","End":"10:06.520","Text":"that my wavelengths go from 1/2 until 2."},{"Start":"10:06.520 ","End":"10:10.670","Text":"Let\u0027s see how I can fit that in over here."},{"Start":"10:10.890 ","End":"10:15.489","Text":"I can see that I can just about fit it in every 4 squares."},{"Start":"10:15.489 ","End":"10:17.740","Text":"It doesn\u0027t really matter, but just to space it out,"},{"Start":"10:17.740 ","End":"10:19.585","Text":"it will be right on the edge of the arrow."},{"Start":"10:19.585 ","End":"10:22.465","Text":"If I have 1, 2, 3, 4,"},{"Start":"10:22.465 ","End":"10:26.080","Text":"so here I\u0027ll have 1/2,"},{"Start":"10:26.080 ","End":"10:28.795","Text":"1, 2, 3, 4."},{"Start":"10:28.795 ","End":"10:30.385","Text":"Here, I\u0027ll have 1,"},{"Start":"10:30.385 ","End":"10:33.415","Text":"1, 2, 3, 4."},{"Start":"10:33.415 ","End":"10:35.950","Text":"Here, I\u0027ll have 1.5,"},{"Start":"10:35.950 ","End":"10:37.555","Text":"which is 3 over 2,"},{"Start":"10:37.555 ","End":"10:40.540","Text":"and 1, 2, 3, 4."},{"Start":"10:40.540 ","End":"10:43.190","Text":"Here I\u0027ll have my 2."},{"Start":"10:45.060 ","End":"10:48.955","Text":"So these numbers are the x-axis,"},{"Start":"10:48.955 ","End":"10:51.205","Text":"and I\u0027ve just marked them out with a line."},{"Start":"10:51.205 ","End":"10:57.220","Text":"Now, what\u0027s important is to keep the spacing here constant."},{"Start":"10:57.220 ","End":"11:02.256","Text":"So now we can see that the spacing is constant in both of the axes,"},{"Start":"11:02.256 ","End":"11:04.480","Text":"and now we can begin plotting."},{"Start":"11:04.480 ","End":"11:09.940","Text":"So we see that the first frequency for a value of 1 divided by Lambda of a 1/2,"},{"Start":"11:09.940 ","End":"11:12.445","Text":"so that over here is 24."},{"Start":"11:12.445 ","End":"11:16.973","Text":"That\u0027s going to be just under this value for 25,"},{"Start":"11:16.973 ","End":"11:20.495","Text":"so let\u0027s plot it over here."},{"Start":"11:20.495 ","End":"11:24.660","Text":"Then for our value of 1 divided by Lambda 1,"},{"Start":"11:24.660 ","End":"11:27.570","Text":"we have a frequency of 45."},{"Start":"11:27.570 ","End":"11:30.285","Text":"So each small square we said is equal to 5."},{"Start":"11:30.285 ","End":"11:32.660","Text":"That means that we\u0027re plotting it over here."},{"Start":"11:32.660 ","End":"11:36.175","Text":"Then for 1.5 or 3 divided by 2,"},{"Start":"11:36.175 ","End":"11:39.640","Text":"we have a frequency of 67."},{"Start":"11:39.640 ","End":"11:41.335","Text":"That means 50,"},{"Start":"11:41.335 ","End":"11:44.230","Text":"55, 60, 65,"},{"Start":"11:44.230 ","End":"11:50.786","Text":"and then it\u0027s 67, so it\u0027s going to be just in the middle over here of these 2 squares."},{"Start":"11:50.786 ","End":"11:55.075","Text":"Then for reciprocal of the wavelength value of 2,"},{"Start":"11:55.075 ","End":"11:58.495","Text":"we have a frequency of 88."},{"Start":"11:58.495 ","End":"12:03.100","Text":"So here we have 75, 80,"},{"Start":"12:03.100 ","End":"12:09.790","Text":"85 and then 88 will be approximately over here."},{"Start":"12:09.790 ","End":"12:14.695","Text":"Now what we\u0027re going to do is we\u0027re going to draw with a ruler a straight line"},{"Start":"12:14.695 ","End":"12:20.455","Text":"that goes from the origin and through all of these points."},{"Start":"12:20.455 ","End":"12:23.260","Text":"Now we can see that we\u0027ve drawn this line,"},{"Start":"12:23.260 ","End":"12:25.945","Text":"and it\u0027s going through all of the points."},{"Start":"12:25.945 ","End":"12:27.880","Text":"Now what we want to do is we want to find"},{"Start":"12:27.880 ","End":"12:32.275","Text":"the gradient of this line or the gradient of this graph."},{"Start":"12:32.275 ","End":"12:34.240","Text":"How do we do this?"},{"Start":"12:34.240 ","End":"12:37.945","Text":"We\u0027re going to calculate rise over run."},{"Start":"12:37.945 ","End":"12:45.595","Text":"What do we do is we\u0027re going to choose 2 nice points which fit nicely on this line."},{"Start":"12:45.595 ","End":"12:48.115","Text":"Let\u0027s take, let\u0027s say,"},{"Start":"12:48.115 ","End":"12:51.640","Text":"this point and this point."},{"Start":"12:51.640 ","End":"12:57.295","Text":"This point we can see is that its x value is 3 divided by 2,"},{"Start":"12:57.295 ","End":"13:00.685","Text":"and its y value was 67."},{"Start":"13:00.685 ","End":"13:07.795","Text":"This point is at x value 1 and y value 45."},{"Start":"13:07.795 ","End":"13:09.715","Text":"Now what we\u0027re going to do,"},{"Start":"13:09.715 ","End":"13:17.930","Text":"we know that our equation is f is equal to v multiplied by 1 divided by Lambda."},{"Start":"13:18.660 ","End":"13:24.445","Text":"So we saw that that is equal to y is equal to mx."},{"Start":"13:24.445 ","End":"13:27.595","Text":"We want to find what m is because that\u0027s our gradient."},{"Start":"13:27.595 ","End":"13:29.110","Text":"So we\u0027re going to isolate it out."},{"Start":"13:29.110 ","End":"13:34.015","Text":"We see that we get that the gradient is equal to y divided by x,"},{"Start":"13:34.015 ","End":"13:37.030","Text":"which is rise over run."},{"Start":"13:37.030 ","End":"13:39.610","Text":"Now in order to make this more accurate,"},{"Start":"13:39.610 ","End":"13:45.475","Text":"we\u0027re going to use instead our Delta y divided by Delta x."},{"Start":"13:45.475 ","End":"13:48.610","Text":"Therefore, our gradient m,"},{"Start":"13:48.610 ","End":"13:50.830","Text":"which is equal to our wave speed,"},{"Start":"13:50.830 ","End":"13:53.725","Text":"is equal to our change in y,"},{"Start":"13:53.725 ","End":"14:02.065","Text":"which is 67 minus 45 divided by our change in x,"},{"Start":"14:02.065 ","End":"14:05.770","Text":"which is 3 over 2 minus 1."},{"Start":"14:05.770 ","End":"14:09.220","Text":"Now if we plug this into our calculator,"},{"Start":"14:09.220 ","End":"14:17.990","Text":"we will get 22 divided by 1/2 which is simply equal to 44."},{"Start":"14:18.780 ","End":"14:22.030","Text":"First of all, we\u0027ve drawn our graph,"},{"Start":"14:22.030 ","End":"14:25.450","Text":"and now we\u0027ve started actually answering Question Number 3,"},{"Start":"14:25.450 ","End":"14:27.145","Text":"so let\u0027s scroll back up."},{"Start":"14:27.145 ","End":"14:29.770","Text":"We can see that Question Number 3 is using the graph,"},{"Start":"14:29.770 ","End":"14:34.015","Text":"calculate the propagation speed or the wave speed of each wave."},{"Start":"14:34.015 ","End":"14:36.610","Text":"What we\u0027ve done is from the graph,"},{"Start":"14:36.610 ","End":"14:40.990","Text":"we\u0027ve worked out our gradient m,"},{"Start":"14:40.990 ","End":"14:45.610","Text":"and we said that we know from what we did over here, rearranging,"},{"Start":"14:45.610 ","End":"14:49.960","Text":"we found out that our gradient is equal to our wave speed,"},{"Start":"14:49.960 ","End":"14:52.030","Text":"which was equal to 44 and,"},{"Start":"14:52.030 ","End":"14:55.300","Text":"of course, speed or velocity is in meters per second."},{"Start":"14:55.300 ","End":"14:58.480","Text":"So our answer therefore to Question number 3,"},{"Start":"14:58.480 ","End":"15:05.810","Text":"v is equal to 44 meters per second."},{"Start":"15:06.240 ","End":"15:10.120","Text":"Now let\u0027s go on to Question Number 4."},{"Start":"15:10.120 ","End":"15:14.140","Text":"The frequency of the wave generator is further increased."},{"Start":"15:14.140 ","End":"15:17.710","Text":"We got to 88, and now we\u0027re increasing the frequency more."},{"Start":"15:17.710 ","End":"15:23.140","Text":"What will be the next frequency to produce a standing wave?"},{"Start":"15:23.140 ","End":"15:28.555","Text":"What I\u0027m going to do is I\u0027m going to rub out the graph that we did for Question 2."},{"Start":"15:28.555 ","End":"15:30.700","Text":"If you need to look at it again,"},{"Start":"15:30.700 ","End":"15:32.418","Text":"please rewind the video."},{"Start":"15:32.418 ","End":"15:35.330","Text":"Now let\u0027s answer Question Number 4."},{"Start":"15:35.370 ","End":"15:39.430","Text":"Again, we know that we have our equation of L,"},{"Start":"15:39.430 ","End":"15:41.695","Text":"the length of the rope or the string,"},{"Start":"15:41.695 ","End":"15:43.825","Text":"which is equal to n,"},{"Start":"15:43.825 ","End":"15:45.385","Text":"the number of anti-nodes,"},{"Start":"15:45.385 ","End":"15:49.855","Text":"multiplied by 1/2 times the wavelength."},{"Start":"15:49.855 ","End":"15:56.950","Text":"Now we want to see at which frequency we\u0027re going to get another standing wave."},{"Start":"15:56.950 ","End":"16:00.970","Text":"First of all, we know that L is equal to 1 meter."},{"Start":"16:00.970 ","End":"16:03.100","Text":"Let\u0027s just plug that in."},{"Start":"16:03.100 ","End":"16:06.050","Text":"1, which is equal to n,"},{"Start":"16:06.050 ","End":"16:07.825","Text":"let\u0027s write that in a second,"},{"Start":"16:07.825 ","End":"16:10.390","Text":"multiplied by 1/2,"},{"Start":"16:10.390 ","End":"16:14.275","Text":"multiplied by the wavelength."},{"Start":"16:14.275 ","End":"16:17.335","Text":"So n, we already know that at 88,"},{"Start":"16:17.335 ","End":"16:18.550","Text":"we had 1,"},{"Start":"16:18.550 ","End":"16:19.765","Text":"2, 3,"},{"Start":"16:19.765 ","End":"16:22.630","Text":"4 anti-nodes and at 67,"},{"Start":"16:22.630 ","End":"16:25.825","Text":"which was the frequency down from 88,"},{"Start":"16:25.825 ","End":"16:27.160","Text":"that made a standing wave."},{"Start":"16:27.160 ","End":"16:29.425","Text":"We had 1, 2, 3."},{"Start":"16:29.425 ","End":"16:32.320","Text":"Therefore we can know from this pattern that"},{"Start":"16:32.320 ","End":"16:36.640","Text":"the next standing wave will just have 1 extra anti-node,"},{"Start":"16:36.640 ","End":"16:38.290","Text":"which means that if at 88 we had"},{"Start":"16:38.290 ","End":"16:43.480","Text":"4 anti-nodes at the next frequency which we\u0027re trying to calculate,"},{"Start":"16:43.480 ","End":"16:46.960","Text":"we\u0027ll have 5 anti-nodes."},{"Start":"16:46.960 ","End":"16:53.095","Text":"N is the amount of anti-nodes we know we need to look for something with 5 anti-nodes."},{"Start":"16:53.095 ","End":"16:56.860","Text":"First, we\u0027re going to isolate out this Lambda,"},{"Start":"16:56.860 ","End":"16:58.690","Text":"which is the wavelength."},{"Start":"16:58.690 ","End":"17:08.620","Text":"We\u0027ll get that our wavelength Lambda is equal to 1 multiplied by 2 is 2 divided by 5."},{"Start":"17:08.620 ","End":"17:14.815","Text":"2 divided by 5 is equal to 0.4 meters."},{"Start":"17:14.815 ","End":"17:19.765","Text":"This is the wavelength of this wave and now we want to find the frequency."},{"Start":"17:19.765 ","End":"17:24.910","Text":"We know from our equation before that the velocity is equal"},{"Start":"17:24.910 ","End":"17:30.160","Text":"to the wavelength multiplied by the frequency."},{"Start":"17:30.160 ","End":"17:32.560","Text":"We already know that the velocity of the wave,"},{"Start":"17:32.560 ","End":"17:34.420","Text":"which we learned from previous lessons,"},{"Start":"17:34.420 ","End":"17:40.880","Text":"is only dependent on the medium with which the wave is propagating through."},{"Start":"17:41.010 ","End":"17:45.535","Text":"Because the wave is still propagating through the same rope,"},{"Start":"17:45.535 ","End":"17:49.300","Text":"that means that the velocity is going to be constant."},{"Start":"17:49.300 ","End":"17:56.320","Text":"Therefore, we can plug in that the velocity is equal to 44 meters per second."},{"Start":"17:56.320 ","End":"17:58.555","Text":"We calculated this in the previous question,"},{"Start":"17:58.555 ","End":"18:00.520","Text":"which is equal to the wavelength,"},{"Start":"18:00.520 ","End":"18:05.770","Text":"which is 0.4 multiplied by the frequency."},{"Start":"18:05.770 ","End":"18:10.780","Text":"Our question is to find the frequency. Let\u0027s isolate that."},{"Start":"18:10.780 ","End":"18:19.270","Text":"The frequency is going to be simply equal to 44 divided by 0.4,"},{"Start":"18:19.270 ","End":"18:22.885","Text":"which if we plug into our calculator,"},{"Start":"18:22.885 ","End":"18:29.140","Text":"is simply going to be equal to 110 hertz."},{"Start":"18:29.140 ","End":"18:32.110","Text":"This is the answer to Question Number 4."},{"Start":"18:32.110 ","End":"18:38.680","Text":"The next frequency where we\u0027ll have a standing wave is at 110 hertz."},{"Start":"18:38.680 ","End":"18:42.760","Text":"Here is the answer to Question 3, 2, and 1,"},{"Start":"18:42.760 ","End":"18:44.395","Text":"and of course for Question 2,"},{"Start":"18:44.395 ","End":"18:48.090","Text":"we also had a graph which I rubbed out already."},{"Start":"18:48.090 ","End":"18:50.980","Text":"That\u0027s the end of this lesson."}],"ID":12469},{"Watched":false,"Name":"Differences Between Standing and Travelling Waves","Duration":"13m 10s","ChapterTopicVideoID":12002,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.770","Text":"Hello. In this video,"},{"Start":"00:01.770 ","End":"00:03.944","Text":"we\u0027re going to be speaking about the difference"},{"Start":"00:03.944 ","End":"00:06.900","Text":"between standing waves and traveling waves."},{"Start":"00:06.900 ","End":"00:10.050","Text":"Here we\u0027ve taken a picture of a standing wave and here we\u0027ve"},{"Start":"00:10.050 ","End":"00:13.245","Text":"taken a picture of a traveling wave at certain times,"},{"Start":"00:13.245 ","End":"00:15.270","Text":"such that at this exact time,"},{"Start":"00:15.270 ","End":"00:19.330","Text":"the standing wave and the traveling wave look exactly the same."},{"Start":"00:19.330 ","End":"00:23.350","Text":"What exactly is the difference between the 2?"},{"Start":"00:23.450 ","End":"00:27.015","Text":"Let\u0027s start off with the traveling wave."},{"Start":"00:27.015 ","End":"00:29.620","Text":"Here we can see at this time,"},{"Start":"00:29.620 ","End":"00:35.195","Text":"we have our amplitude over here at positive 15"},{"Start":"00:35.195 ","End":"00:42.035","Text":"and a negative 15 at this exact time but we know that a few moments later,"},{"Start":"00:42.035 ","End":"00:45.325","Text":"our wave would have propagated,"},{"Start":"00:45.325 ","End":"00:53.130","Text":"such that amplitudes or peak points would have shifted."},{"Start":"00:53.130 ","End":"00:57.820","Text":"Now the peak is over here and over here."},{"Start":"00:57.820 ","End":"01:02.340","Text":"We can see, and then this will carry on down here."},{"Start":"01:02.930 ","End":"01:10.205","Text":"What we can see, is that each point has a different particle displacement."},{"Start":"01:10.205 ","End":"01:16.880","Text":"At 1 point the particle located at point 0 over here is that an amplitude of 0."},{"Start":"01:16.880 ","End":"01:21.915","Text":"Then the next moment it has some negative amplitude."},{"Start":"01:21.915 ","End":"01:25.025","Text":"For this value, over here,"},{"Start":"01:25.025 ","End":"01:27.035","Text":"let\u0027s draw it in blue."},{"Start":"01:27.035 ","End":"01:32.870","Text":"Where suddenly the particle over here on the rope is at its peak amplitude of 15,"},{"Start":"01:32.870 ","End":"01:35.540","Text":"and then we can see that a few moments later,"},{"Start":"01:35.540 ","End":"01:39.615","Text":"it\u0027s moved downwards to this point over here,"},{"Start":"01:39.615 ","End":"01:43.235","Text":"and a different particle located over here at half,"},{"Start":"01:43.235 ","End":"01:46.340","Text":"is now at the peak amplitude."},{"Start":"01:46.340 ","End":"01:50.160","Text":"This is what we can see for traveling waves."},{"Start":"01:50.470 ","End":"01:54.035","Text":"Now when we\u0027re dealing with standing waves,"},{"Start":"01:54.035 ","End":"01:59.390","Text":"every single particle has its own amplitude."},{"Start":"01:59.390 ","End":"02:03.455","Text":"As we know, we have nodes in a standing wave."},{"Start":"02:03.455 ","End":"02:05.900","Text":"This is a node, this is a node, this,"},{"Start":"02:05.900 ","End":"02:08.610","Text":"this, this and this, all nodes."},{"Start":"02:08.610 ","End":"02:11.190","Text":"As we know, these points don\u0027t move,"},{"Start":"02:11.190 ","End":"02:16.295","Text":"they always stay at this 0 mark over here but what we can see,"},{"Start":"02:16.295 ","End":"02:19.910","Text":"is that right now at this time over here,"},{"Start":"02:19.910 ","End":"02:25.325","Text":"we have some peak where over here at a quarter,"},{"Start":"02:25.325 ","End":"02:30.520","Text":"we have a peak amplitude of 15 centimeters."},{"Start":"02:30.520 ","End":"02:33.750","Text":"This is the peak of the entire wave."},{"Start":"02:33.750 ","End":"02:42.170","Text":"However, each one of these points is currently also at its individual peak."},{"Start":"02:42.170 ","End":"02:46.850","Text":"This point over here is at a peak of 5 centimeters,"},{"Start":"02:46.850 ","End":"02:53.910","Text":"and we will never get this point to be with the larger amplitude,"},{"Start":"02:53.910 ","End":"02:56.810","Text":"because the wave is standing at stationary."},{"Start":"02:56.810 ","End":"03:03.305","Text":"Now as we know, a few moments later our standing wave will look like this."},{"Start":"03:03.305 ","End":"03:07.835","Text":"We can see that our peak amplitude has moved down over here,"},{"Start":"03:07.835 ","End":"03:10.940","Text":"and this point over here that has a peak amplitude of"},{"Start":"03:10.940 ","End":"03:14.480","Text":"5 now has a slightly lower amplitude,"},{"Start":"03:14.480 ","End":"03:17.540","Text":"which is located over here and maybe 3."},{"Start":"03:17.540 ","End":"03:19.595","Text":"A few moments later,"},{"Start":"03:19.595 ","End":"03:25.260","Text":"we can see that we have an even smaller peak amplitude."},{"Start":"03:25.580 ","End":"03:32.975","Text":"This point over here has an even smaller peak amplitude of its own of approximately 1."},{"Start":"03:32.975 ","End":"03:34.945","Text":"Then a few moments later,"},{"Start":"03:34.945 ","End":"03:39.845","Text":"our standing wave will move into the negative realm and it will be here,"},{"Start":"03:39.845 ","End":"03:43.265","Text":"and then it will move down here,"},{"Start":"03:43.265 ","End":"03:45.320","Text":"and then eventually,"},{"Start":"03:45.320 ","End":"03:50.675","Text":"it too will reach its peak amplitude just in the negative direction,"},{"Start":"03:50.675 ","End":"03:52.819","Text":"so it will be at negative 15."},{"Start":"03:52.819 ","End":"03:54.680","Text":"This point will again,"},{"Start":"03:54.680 ","End":"03:59.600","Text":"be located over here at its peak amplitude in the negative realm,"},{"Start":"03:59.600 ","End":"04:01.215","Text":"which is at negative 5."},{"Start":"04:01.215 ","End":"04:05.630","Text":"We can see that we have a mirror image in the x-axis,"},{"Start":"04:05.630 ","End":"04:09.650","Text":"and we can see that every single point has its own peak,"},{"Start":"04:09.650 ","End":"04:12.950","Text":"so this point over here will never ever"},{"Start":"04:12.950 ","End":"04:17.300","Text":"reach the peak amplitude of the entire wave, which is at 15,"},{"Start":"04:17.300 ","End":"04:21.110","Text":"it will always reach just maybe a 10 or 11,"},{"Start":"04:21.110 ","End":"04:26.810","Text":"whatever this value is and then the negative section it will also be around 11."},{"Start":"04:26.810 ","End":"04:29.150","Text":"Whereas what we saw with the traveling wave,"},{"Start":"04:29.150 ","End":"04:33.650","Text":"is at every single point will eventually be the peak."},{"Start":"04:33.650 ","End":"04:36.335","Text":"Here, this point was the peak,"},{"Start":"04:36.335 ","End":"04:37.805","Text":"a few moments later,"},{"Start":"04:37.805 ","End":"04:45.050","Text":"a point at half located at the value half down the rope will be at a peak."},{"Start":"04:45.050 ","End":"04:47.300","Text":"Then a little bit later,"},{"Start":"04:47.300 ","End":"04:50.500","Text":"a point valued at three quarters will be at the peak in"},{"Start":"04:50.500 ","End":"04:53.820","Text":"the negative direction, and so on."},{"Start":"04:53.820 ","End":"04:55.670","Text":"We can say that as time goes by,"},{"Start":"04:55.670 ","End":"05:01.865","Text":"every single particle on the node will reach the peak amplitude of 15."},{"Start":"05:01.865 ","End":"05:03.680","Text":"Whereas in a standing wave,"},{"Start":"05:03.680 ","End":"05:11.545","Text":"only 1 single point in every single anti-node will reach this peak amplitude."},{"Start":"05:11.545 ","End":"05:16.625","Text":"All the other points also have peak amplitudes for them"},{"Start":"05:16.625 ","End":"05:22.950","Text":"that they\u0027ll never get higher and that is their individual peak amplitude."},{"Start":"05:23.810 ","End":"05:27.050","Text":"Here we can see that in a standing wave,"},{"Start":"05:27.050 ","End":"05:32.510","Text":"each individual point has its own peak amplitude that it will reach,"},{"Start":"05:32.510 ","End":"05:34.235","Text":"whereas in a traveling wave,"},{"Start":"05:34.235 ","End":"05:39.770","Text":"each point will reach the peak amplitude of the entire wave."},{"Start":"05:39.770 ","End":"05:44.600","Text":"Each point will reach this maximum of 15 in a traveling wave,"},{"Start":"05:44.600 ","End":"05:50.450","Text":"in the standing wave only 1 point per anti-node will reach this peak amplitude,"},{"Start":"05:50.450 ","End":"05:55.330","Text":"and all the other points will reach their own amplitude."},{"Start":"05:55.330 ","End":"05:59.165","Text":"Now for the next difference between them,"},{"Start":"05:59.165 ","End":"06:01.220","Text":"in a standing wave,"},{"Start":"06:01.220 ","End":"06:08.810","Text":"we saw that this point over here reaches its peak amplitude and this point over"},{"Start":"06:08.810 ","End":"06:12.980","Text":"here reaches its own individual peak amplitude"},{"Start":"06:12.980 ","End":"06:16.865","Text":"that we know this point over here will never have a higher value,"},{"Start":"06:16.865 ","End":"06:19.910","Text":"and so on and so forth for all the points in"},{"Start":"06:19.910 ","End":"06:23.315","Text":"this section of the wave or in this anti-node."},{"Start":"06:23.315 ","End":"06:26.840","Text":"Now what we can see is that all of these points will reach"},{"Start":"06:26.840 ","End":"06:32.440","Text":"their own individual peak amplitudes at the exact same time."},{"Start":"06:32.440 ","End":"06:36.665","Text":"Because this whole section is moving in unison,"},{"Start":"06:36.665 ","End":"06:42.335","Text":"so we can just imagine that as soon as this point over here reaches its peak,"},{"Start":"06:42.335 ","End":"06:46.940","Text":"this point over here at the exact same time will reach its own peak,"},{"Start":"06:46.940 ","End":"06:49.670","Text":"and so will this point at the exact same time,"},{"Start":"06:49.670 ","End":"06:51.995","Text":"and this point at the exact same time."},{"Start":"06:51.995 ","End":"06:54.560","Text":"However, with the traveling wave,"},{"Start":"06:54.560 ","End":"06:59.715","Text":"each point will reach the peak amplitude of the wave,"},{"Start":"06:59.715 ","End":"07:02.385","Text":"but this will happen at different times."},{"Start":"07:02.385 ","End":"07:04.190","Text":"Let\u0027s say at 0 seconds,"},{"Start":"07:04.190 ","End":"07:11.730","Text":"we can see that this point over here at this value is at the peak amplitude."},{"Start":"07:11.730 ","End":"07:16.594","Text":"Then we can say that 1 second later this point"},{"Start":"07:16.594 ","End":"07:21.625","Text":"over here that was at a half is now at the peak amplitude,"},{"Start":"07:21.625 ","End":"07:27.140","Text":"and it\u0027s the same peak amplitude of 15 centimeters just at a different time."},{"Start":"07:27.140 ","End":"07:32.680","Text":"Then we can say that a second or 2 seconds later,"},{"Start":"07:32.680 ","End":"07:38.885","Text":"this point over here at 1 is at the peak amplitude just in the negative section."},{"Start":"07:38.885 ","End":"07:45.410","Text":"What we can see or we can just carry on drawing this wave as it keeps on going,"},{"Start":"07:45.410 ","End":"07:47.860","Text":"so a moment later,"},{"Start":"07:47.860 ","End":"07:52.400","Text":"this point over here will be at the peak amplitude,"},{"Start":"07:52.400 ","End":"07:59.570","Text":"and then a moment later we have this at the peak amplitude and a moment later this is at,"},{"Start":"07:59.570 ","End":"08:02.410","Text":"let\u0027s say 4 seconds,"},{"Start":"08:02.410 ","End":"08:07.340","Text":"so now this point over here is at the peak amplitude,"},{"Start":"08:07.340 ","End":"08:10.715","Text":"and they\u0027re always at the same peak amplitude of 15,"},{"Start":"08:10.715 ","End":"08:13.345","Text":"just at different times."},{"Start":"08:13.345 ","End":"08:17.510","Text":"Each point will reach this peak amplitude of"},{"Start":"08:17.510 ","End":"08:21.650","Text":"15 just at a different time in a traveling wave,"},{"Start":"08:21.650 ","End":"08:23.105","Text":"and in a standing wave,"},{"Start":"08:23.105 ","End":"08:24.575","Text":"each point will reach"},{"Start":"08:24.575 ","End":"08:31.790","Text":"its own individual peak amplitude that it will never rise higher than that 1,"},{"Start":"08:31.790 ","End":"08:34.370","Text":"but all of these points will reach"},{"Start":"08:34.370 ","End":"08:39.570","Text":"their individual peak amplitudes at the exact same time."},{"Start":"08:39.670 ","End":"08:44.825","Text":"The next difference between standing waves and traveling waves,"},{"Start":"08:44.825 ","End":"08:47.960","Text":"the direction of travel of the points."},{"Start":"08:47.960 ","End":"08:49.775","Text":"In a standing wave,"},{"Start":"08:49.775 ","End":"08:52.490","Text":"we know that between 2 nodes,"},{"Start":"08:52.490 ","End":"08:58.515","Text":"all the points in the rope travel in the same direction. Let\u0027s draw that out."},{"Start":"08:58.515 ","End":"08:59.880","Text":"Here we have a node,"},{"Start":"08:59.880 ","End":"09:01.055","Text":"here we have a node,"},{"Start":"09:01.055 ","End":"09:03.835","Text":"here we have a node, and here we have a node."},{"Start":"09:03.835 ","End":"09:09.050","Text":"If we draw a moment later what this anti-node will look like,"},{"Start":"09:09.050 ","End":"09:11.615","Text":"it will be at this type of amplitude,"},{"Start":"09:11.615 ","End":"09:17.210","Text":"this anti-node will be at this amplitude,"},{"Start":"09:17.210 ","End":"09:22.720","Text":"and this anti-node will look like so."},{"Start":"09:22.720 ","End":"09:27.890","Text":"Between 2 nodes, all the points in the rope travel in the same direction."},{"Start":"09:27.890 ","End":"09:30.770","Text":"Here\u0027s 1 node and here\u0027s another node."},{"Start":"09:30.770 ","End":"09:36.300","Text":"We can see that all the points traveled downwards."},{"Start":"09:36.500 ","End":"09:41.195","Text":"Every single point has a smaller amplitude now,"},{"Start":"09:41.195 ","End":"09:43.385","Text":"all the points travel downwards."},{"Start":"09:43.385 ","End":"09:46.715","Text":"Now here let\u0027s look at this node and this node."},{"Start":"09:46.715 ","End":"09:54.540","Text":"We\u0027re looking between these nodes and all of these points traveled upwards."},{"Start":"09:54.540 ","End":"09:57.960","Text":"Then here is a node and here is a node,"},{"Start":"09:57.960 ","End":"10:01.250","Text":"and now we can see that between these 2 nodes,"},{"Start":"10:01.250 ","End":"10:05.030","Text":"all the points have traveled downwards."},{"Start":"10:05.030 ","End":"10:10.295","Text":"In a traveling wave between the positive and negative peak amplitudes,"},{"Start":"10:10.295 ","End":"10:14.815","Text":"the points in the rope will travel in the same direction."},{"Start":"10:14.815 ","End":"10:17.135","Text":"Here we have a traveling wave,"},{"Start":"10:17.135 ","End":"10:21.425","Text":"and here we have our positive peak amplitude,"},{"Start":"10:21.425 ","End":"10:26.510","Text":"and here we have a negative peak amplitude at this specific time."},{"Start":"10:26.510 ","End":"10:29.960","Text":"Then we know that a moment later,"},{"Start":"10:29.960 ","End":"10:34.175","Text":"our wave will look something like this,"},{"Start":"10:34.175 ","End":"10:39.050","Text":"it will be shifted slightly like so."},{"Start":"10:39.050 ","End":"10:43.625","Text":"Now we\u0027ll look around about something like this."},{"Start":"10:43.625 ","End":"10:50.585","Text":"What we can see between this peak amplitude and this negative amplitude,"},{"Start":"10:50.585 ","End":"10:56.080","Text":"that all of our points have moved in the same direction."},{"Start":"10:56.080 ","End":"11:00.125","Text":"We can see that this point over here used to be at the 0,"},{"Start":"11:00.125 ","End":"11:03.275","Text":"but now it itself is at a peak,"},{"Start":"11:03.275 ","End":"11:07.669","Text":"and at this point over here used to be at the peak in the negative direction,"},{"Start":"11:07.669 ","End":"11:12.540","Text":"negative 15, and now it\u0027s located at 0."},{"Start":"11:12.540 ","End":"11:19.720","Text":"We can see that all of the points in between have also increased."},{"Start":"11:19.720 ","End":"11:24.935","Text":"Whereas we can see that all of the points over here before"},{"Start":"11:24.935 ","End":"11:33.990","Text":"this peak amplitude used to be here and now they\u0027ve moved down here, and so on."},{"Start":"11:34.510 ","End":"11:37.655","Text":"Then if we look in this side,"},{"Start":"11:37.655 ","End":"11:39.335","Text":"if we carry this on,"},{"Start":"11:39.335 ","End":"11:44.351","Text":"so here we had a negative and here we had a positive peak amplitude."},{"Start":"11:44.351 ","End":"11:46.625","Text":"Then we can see also over here,"},{"Start":"11:46.625 ","End":"11:50.750","Text":"this was at a positive direction and now it\u0027s at the negative peak amplitude,"},{"Start":"11:50.750 ","End":"11:56.660","Text":"this was at the peak amplitude and now it\u0027s gone into the negative direction,"},{"Start":"11:56.660 ","End":"12:03.219","Text":"and all of these points a moment later have just moved down and shifted."},{"Start":"12:03.219 ","End":"12:06.050","Text":"Then again, between these 2 points,"},{"Start":"12:06.050 ","End":"12:11.545","Text":"we can see that this particle used to be over here and now it\u0027s moved up here."},{"Start":"12:11.545 ","End":"12:16.145","Text":"This seems to be at the negative peak amplitude and now it\u0027s equal to 0,"},{"Start":"12:16.145 ","End":"12:21.145","Text":"and then all the points in between have also increased."},{"Start":"12:21.145 ","End":"12:24.380","Text":"In this lesson,"},{"Start":"12:24.380 ","End":"12:28.355","Text":"we\u0027ve seen the differences between the standing wave and the traveling wave."},{"Start":"12:28.355 ","End":"12:33.394","Text":"The first difference is about"},{"Start":"12:33.394 ","End":"12:40.250","Text":"the amplitudes in traveling waves and standing waves."},{"Start":"12:40.250 ","End":"12:43.895","Text":"The second was about when we reach"},{"Start":"12:43.895 ","End":"12:49.115","Text":"these different amplitudes in the standing wave and then the traveling wave."},{"Start":"12:49.115 ","End":"12:55.310","Text":"The third point was the direction of travel of all of the particles."},{"Start":"12:55.310 ","End":"12:58.775","Text":"We can see that the standing wave in almost all of these,"},{"Start":"12:58.775 ","End":"13:02.149","Text":"is exactly the opposite or very different"},{"Start":"13:02.149 ","End":"13:07.895","Text":"to what is happening to each particle when dealing with a traveling wave."},{"Start":"13:07.895 ","End":"13:10.920","Text":"That\u0027s the end of this lesson."}],"ID":12470},{"Watched":false,"Name":"Exercise - Wave Calculations 5","Duration":"13m 5s","ChapterTopicVideoID":12003,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.710","Text":"Hello. In this question we have a wave propagating through a rope,"},{"Start":"00:04.710 ","End":"00:08.445","Text":"and we have 2 graphs representing this wave."},{"Start":"00:08.445 ","End":"00:11.370","Text":"The first is the time displacement graph."},{"Start":"00:11.370 ","End":"00:14.580","Text":"Here we\u0027re following a certain point on the rope and"},{"Start":"00:14.580 ","End":"00:18.165","Text":"seeing its position as a function of time,"},{"Start":"00:18.165 ","End":"00:23.955","Text":"and we have the position displacement graph of the rope for time,"},{"Start":"00:23.955 ","End":"00:31.815","Text":"t= 0, where here at t=0 we\u0027ve taken a snapshot of what the rope looks like."},{"Start":"00:31.815 ","End":"00:35.025","Text":"Now we\u0027re going to answer these 5 questions."},{"Start":"00:35.025 ","End":"00:36.810","Text":"Question number 1 is,"},{"Start":"00:36.810 ","End":"00:39.045","Text":"what is the wave\u0027s amplitude?"},{"Start":"00:39.045 ","End":"00:42.788","Text":"This we can answer by looking at either 1 of the graphs,"},{"Start":"00:42.788 ","End":"00:46.550","Text":"either we can look at this position displacement graph,"},{"Start":"00:46.550 ","End":"00:50.120","Text":"where here we can see that we have a snapshot of"},{"Start":"00:50.120 ","End":"00:54.500","Text":"the wave and we can see that it has a peak at 15 centimeters."},{"Start":"00:54.500 ","End":"00:58.730","Text":"Then we can say that the amplitude is equal to 15 centimeters"},{"Start":"00:58.730 ","End":"01:05.015","Text":"or 0.15 meters if we\u0027re going with mks."},{"Start":"01:05.015 ","End":"01:09.980","Text":"But we can also get this amplitude from my time displacement graph."},{"Start":"01:09.980 ","End":"01:11.855","Text":"A time displacement graph is,"},{"Start":"01:11.855 ","End":"01:14.420","Text":"as we said, following a particular point,"},{"Start":"01:14.420 ","End":"01:20.870","Text":"as it oscillates up and down due to the wave passing through."},{"Start":"01:20.870 ","End":"01:26.335","Text":"We can see that point at t=0,"},{"Start":"01:26.335 ","End":"01:31.310","Text":"a particle is that an amplitude or a particle displacement of 0,"},{"Start":"01:31.310 ","End":"01:33.515","Text":"and as the time goes by,"},{"Start":"01:33.515 ","End":"01:35.765","Text":"the wave propagates through the rope,"},{"Start":"01:35.765 ","End":"01:41.690","Text":"and a particular point on the rope goes up higher and higher and"},{"Start":"01:41.690 ","End":"01:44.000","Text":"higher to higher displacement until it reaches"},{"Start":"01:44.000 ","End":"01:47.810","Text":"this maximum displacement of 15 centimeters,"},{"Start":"01:47.810 ","End":"01:49.944","Text":"which is what we got here."},{"Start":"01:49.944 ","End":"01:53.710","Text":"Then it drops down again as the wave passes through,"},{"Start":"01:53.710 ","End":"01:57.230","Text":"and then it goes further to its peak in the negative direction and so"},{"Start":"01:57.230 ","End":"02:01.450","Text":"can see how the particle moves as a function of time."},{"Start":"02:01.450 ","End":"02:06.095","Text":"Here we can see again it reaches a peak position of 15 centimeters,"},{"Start":"02:06.095 ","End":"02:08.520","Text":"which is the amplitude."},{"Start":"02:08.950 ","End":"02:11.990","Text":"Now let\u0027s look at Question number 2."},{"Start":"02:11.990 ","End":"02:15.330","Text":"What is the wavelength of the wave?"},{"Start":"02:16.280 ","End":"02:19.415","Text":"The wavelength, as we know,"},{"Start":"02:19.415 ","End":"02:27.740","Text":"is the distance that we travel when the wave will make 1 copy of itself,"},{"Start":"02:27.740 ","End":"02:30.620","Text":"such that if we would cut out that piece of the wave,"},{"Start":"02:30.620 ","End":"02:33.455","Text":"we could just copy and paste it along,"},{"Start":"02:33.455 ","End":"02:37.210","Text":"and then we can get the exact same wave pattern."},{"Start":"02:37.210 ","End":"02:40.180","Text":"Usually we measure it from peak to peak."},{"Start":"02:40.180 ","End":"02:45.225","Text":"We can go from here until here."},{"Start":"02:45.225 ","End":"02:54.215","Text":"But now I can see that this is probably 0.25 and this is probably 1.25 in which case,"},{"Start":"02:54.215 ","End":"02:58.310","Text":"this distance over here is the wavelength,"},{"Start":"02:58.310 ","End":"03:02.330","Text":"which is approximately equal to 1 centimeter."},{"Start":"03:02.330 ","End":"03:05.980","Text":"However, we can also choose a different point."},{"Start":"03:05.980 ","End":"03:08.060","Text":"Let\u0027s do that in green."},{"Start":"03:08.060 ","End":"03:09.275","Text":"We can choose from here,"},{"Start":"03:09.275 ","End":"03:10.700","Text":"and then we follow the wave."},{"Start":"03:10.700 ","End":"03:16.294","Text":"Now notice we don\u0027t stop over here because if we would just copy and paste this section,"},{"Start":"03:16.294 ","End":"03:18.455","Text":"we would just get a bunch of hills,"},{"Start":"03:18.455 ","End":"03:20.615","Text":"but we want the whole wave shape."},{"Start":"03:20.615 ","End":"03:25.540","Text":"We go from the 0 and then we see the gradient is going up,"},{"Start":"03:25.540 ","End":"03:30.050","Text":"then we get to the 0 and we see that the gradient is going down so we don\u0027t stop there."},{"Start":"03:30.050 ","End":"03:32.110","Text":"Then we again get to the 0,"},{"Start":"03:32.110 ","End":"03:33.650","Text":"and we see that here again,"},{"Start":"03:33.650 ","End":"03:36.805","Text":"the gradient is going up again."},{"Start":"03:36.805 ","End":"03:39.620","Text":"We can see that if we take this whole section and"},{"Start":"03:39.620 ","End":"03:42.320","Text":"do copy and paste and paste it over here,"},{"Start":"03:42.320 ","End":"03:44.450","Text":"we\u0027ll get our wave pattern again."},{"Start":"03:44.450 ","End":"03:47.570","Text":"Now we can see that this distance over"},{"Start":"03:47.570 ","End":"03:50.990","Text":"here is going to be the same distance as this over here,"},{"Start":"03:50.990 ","End":"03:52.864","Text":"and that\u0027s 1 wavelength."},{"Start":"03:52.864 ","End":"03:57.930","Text":"We can see it goes from 0-1 centimeter."},{"Start":"03:58.550 ","End":"04:02.505","Text":"Our wavelength, denoted by the Greek letter Lambda,"},{"Start":"04:02.505 ","End":"04:10.560","Text":"is equal to 1 centimeter or 0.01 meters."},{"Start":"04:10.910 ","End":"04:18.035","Text":"Question number 3 is what is the period of the wave?"},{"Start":"04:18.035 ","End":"04:24.660","Text":"The period is how long it takes for a wave to complete 1 oscillation."},{"Start":"04:25.010 ","End":"04:29.855","Text":"We can see if we go from the same point so from here,"},{"Start":"04:29.855 ","End":"04:34.940","Text":"this represents 1 oscillation when our particle starts at"},{"Start":"04:34.940 ","End":"04:39.170","Text":"some amplitude or some particle displacement"},{"Start":"04:39.170 ","End":"04:42.081","Text":"and returns to that same particle displacement."},{"Start":"04:42.081 ","End":"04:48.080","Text":"That\u0027s 1 oscillation. First of all,"},{"Start":"04:48.080 ","End":"04:50.105","Text":"we have to use the time displacement graph,"},{"Start":"04:50.105 ","End":"04:51.975","Text":"the clue is in the name."},{"Start":"04:51.975 ","End":"04:57.219","Text":"We can see that the time period is 4 seconds,"},{"Start":"04:57.219 ","End":"04:58.575","Text":"so we can already write that."},{"Start":"04:58.575 ","End":"05:03.485","Text":"The time period is denoted by T and it\u0027s equal to 4 seconds."},{"Start":"05:03.485 ","End":"05:06.272","Text":"But we could also measure this from another point,"},{"Start":"05:06.272 ","End":"05:08.650","Text":"let\u0027s say from peak to peak."},{"Start":"05:08.650 ","End":"05:13.490","Text":"Again, our particle was at a maximum displacement,"},{"Start":"05:13.490 ","End":"05:17.690","Text":"and then it returned to the maximum displacement which"},{"Start":"05:17.690 ","End":"05:23.360","Text":"means that this is again a time period because 1 full oscillation was completed."},{"Start":"05:23.360 ","End":"05:29.969","Text":"We can see that again it\u0027s 4 seconds because we have the change in time is 5 minus 1,"},{"Start":"05:29.969 ","End":"05:32.635","Text":"which is 4 seconds."},{"Start":"05:32.635 ","End":"05:36.150","Text":"Now let\u0027s answer Question 4."},{"Start":"05:36.150 ","End":"05:40.950","Text":"What is the velocity of the wave?"},{"Start":"05:40.950 ","End":"05:45.980","Text":"As we know, an equation for the velocity of the wave V is equal"},{"Start":"05:45.980 ","End":"05:52.250","Text":"to wavelength Lambda multiplied by the frequency."},{"Start":"05:52.250 ","End":"05:58.977","Text":"As we know, our frequency is equal to 1 divided by our time period,"},{"Start":"05:58.977 ","End":"06:03.670","Text":"our frequency is the reciprocal of the time period."},{"Start":"06:03.670 ","End":"06:09.410","Text":"Therefore, our velocity is going to be equal to our wavelength,"},{"Start":"06:09.410 ","End":"06:13.535","Text":"which is equal to 0.01 meters,"},{"Start":"06:13.535 ","End":"06:16.505","Text":"multiplied by our frequency,"},{"Start":"06:16.505 ","End":"06:19.639","Text":"which is the reciprocal of the time period,"},{"Start":"06:19.639 ","End":"06:25.630","Text":"so 1 divided by the time period is 1 divided by 4."},{"Start":"06:25.630 ","End":"06:28.985","Text":"Once we plug that into our calculator,"},{"Start":"06:28.985 ","End":"06:31.370","Text":"we\u0027ll get that the velocity is equal to"},{"Start":"06:31.370 ","End":"06:39.917","Text":"0.0025 meters per second."},{"Start":"06:39.917 ","End":"06:43.850","Text":"Or so that it\u0027s something that we can understand a bit more,"},{"Start":"06:43.850 ","End":"06:53.370","Text":"we\u0027ll get that the velocity of the wave is equal to 0.25 centimeters per second."},{"Start":"06:54.220 ","End":"06:57.750","Text":"Now the last question that we have is 5."},{"Start":"06:57.750 ","End":"07:05.350","Text":"Which point or points on the rope could the time displacement graph represent?"},{"Start":"07:05.350 ","End":"07:08.270","Text":"First of all,"},{"Start":"07:08.270 ","End":"07:13.590","Text":"our position displacement graph is for the time t is equal to 0."},{"Start":"07:13.590 ","End":"07:16.550","Text":"That we\u0027re given in the question."},{"Start":"07:16.550 ","End":"07:20.015","Text":"Now we can see t is equal to 0,"},{"Start":"07:20.015 ","End":"07:29.783","Text":"our particle on the rope that we\u0027re watching is at 0 amplitude."},{"Start":"07:29.783 ","End":"07:33.365","Text":"Also over here in our time displacement graph,"},{"Start":"07:33.365 ","End":"07:35.810","Text":"if we go to t is equal to 0,"},{"Start":"07:35.810 ","End":"07:40.427","Text":"we can see that the particle that we\u0027re watching is also at the origin,"},{"Start":"07:40.427 ","End":"07:43.355","Text":"it also has the specific particle,"},{"Start":"07:43.355 ","End":"07:46.010","Text":"also has the position of 0."},{"Start":"07:46.010 ","End":"07:51.274","Text":"First of all, let\u0027s mark out our possible particles."},{"Start":"07:51.274 ","End":"07:54.935","Text":"We know that now we\u0027re looking at points on the rope,"},{"Start":"07:54.935 ","End":"08:00.600","Text":"that at t is equal to 0 is at the 0 position."},{"Start":"08:00.600 ","End":"08:04.025","Text":"We\u0027re looking at our point over here,"},{"Start":"08:04.025 ","End":"08:05.950","Text":"right at the beginning of the rope,"},{"Start":"08:05.950 ","End":"08:09.770","Text":"we\u0027re looking at a point half a centimeter down,"},{"Start":"08:09.770 ","End":"08:13.625","Text":"1 centimeter down, 1.5 centimeters down,"},{"Start":"08:13.625 ","End":"08:20.050","Text":"2 centimeters down, and approximately 2.5 centimeters down."},{"Start":"08:20.050 ","End":"08:24.685","Text":"Because all of these points are at an amplitude of 0,"},{"Start":"08:24.685 ","End":"08:27.170","Text":"so all of these points at this stage in"},{"Start":"08:27.170 ","End":"08:32.390","Text":"our calculation could be what we\u0027re looking at and watching"},{"Start":"08:32.390 ","End":"08:35.562","Text":"the movements of in our time displacement graph"},{"Start":"08:35.562 ","End":"08:40.925","Text":"because all of these points at t is equal to 0, are at 0."},{"Start":"08:40.925 ","End":"08:45.995","Text":"However, some of these points we can ignore."},{"Start":"08:45.995 ","End":"08:49.880","Text":"Let\u0027s look at which ones we can ignore."},{"Start":"08:49.880 ","End":"08:53.195","Text":"When we\u0027re looking at the time displacement graph"},{"Start":"08:53.195 ","End":"08:56.585","Text":"so we know that in this time displacement graph,"},{"Start":"08:56.585 ","End":"09:01.130","Text":"we\u0027re looking at a specific particle or a specific point on"},{"Start":"09:01.130 ","End":"09:06.485","Text":"the rope and seeing how it progresses through time."},{"Start":"09:06.485 ","End":"09:11.555","Text":"What we can see is that our particle begins at an amplitude of 0,"},{"Start":"09:11.555 ","End":"09:13.700","Text":"but as time goes by,"},{"Start":"09:13.700 ","End":"09:21.270","Text":"our particle moves upwards in the y-direction or moves in the positive y-direction."},{"Start":"09:21.920 ","End":"09:26.705","Text":"Now let\u0027s look at our position displacement graph."},{"Start":"09:26.705 ","End":"09:29.120","Text":"We can see that this graph is,"},{"Start":"09:29.120 ","End":"09:30.230","Text":"as we\u0027re being told,"},{"Start":"09:30.230 ","End":"09:32.800","Text":"given for t is equal to 0."},{"Start":"09:32.800 ","End":"09:39.410","Text":"Let\u0027s take a look at what the graph looks like a moment after t is equal to 0."},{"Start":"09:39.410 ","End":"09:47.095","Text":"As we know, our graph will have just shifted slightly so now the peak will be here,"},{"Start":"09:47.095 ","End":"09:50.425","Text":"and then it will pass through the 0 over here,"},{"Start":"09:50.425 ","End":"09:53.789","Text":"and then the negative peak will be over here,"},{"Start":"09:53.789 ","End":"09:57.925","Text":"that will return up all the way to here,"},{"Start":"09:57.925 ","End":"10:03.693","Text":"and then it will move back down to here so"},{"Start":"10:03.693 ","End":"10:11.555","Text":"everything is just being shifted like so."},{"Start":"10:11.555 ","End":"10:15.403","Text":"Then we can continue this back down over here."},{"Start":"10:15.403 ","End":"10:16.810","Text":"Perfect."},{"Start":"10:16.810 ","End":"10:21.475","Text":"Then we can continue this to"},{"Start":"10:21.475 ","End":"10:27.945","Text":"this point over here, something like so."},{"Start":"10:27.945 ","End":"10:31.100","Text":"Now what we can see is that as time went by,"},{"Start":"10:31.100 ","End":"10:33.410","Text":"our point at 0,"},{"Start":"10:33.410 ","End":"10:40.070","Text":"so the specific particle of the rope that is located at 0"},{"Start":"10:40.070 ","End":"10:47.305","Text":"so right at the end of the rope went from an amplitude of 0 and moved downwards."},{"Start":"10:47.305 ","End":"10:53.535","Text":"It got a negative amplitude which can be, as we know,"},{"Start":"10:53.535 ","End":"10:57.740","Text":"that our particle that we\u0027re specifically looking at has"},{"Start":"10:57.740 ","End":"11:04.080","Text":"an increasing amplitude so we can cancel out this point over here."},{"Start":"11:04.360 ","End":"11:11.155","Text":"Then we can look at this point and what we can see is that this point started at 0,"},{"Start":"11:11.155 ","End":"11:14.345","Text":"but then we can see that a few moments later,"},{"Start":"11:14.345 ","End":"11:20.840","Text":"it\u0027s now at the peak amplitude so we can see that this point has moved up."},{"Start":"11:20.840 ","End":"11:25.550","Text":"We can see that this point is"},{"Start":"11:25.550 ","End":"11:32.485","Text":"a possible point on the rope that the time displacement graph could be representing."},{"Start":"11:32.485 ","End":"11:35.910","Text":"Then let\u0027s look again at this point."},{"Start":"11:35.910 ","End":"11:38.190","Text":"Again, it starts at 0,"},{"Start":"11:38.190 ","End":"11:41.480","Text":"and then it moves down to the negative section,"},{"Start":"11:41.480 ","End":"11:43.745","Text":"so it\u0027s amplitude decreases."},{"Start":"11:43.745 ","End":"11:47.250","Text":"This point we\u0027re also not looking"},{"Start":"11:47.250 ","End":"11:51.245","Text":"at because the time displacement graph doesn\u0027t represent this."},{"Start":"11:51.245 ","End":"11:54.250","Text":"Now let\u0027s look at this point over here."},{"Start":"11:54.250 ","End":"11:56.915","Text":"Again, it goes from a zero amplitude,"},{"Start":"11:56.915 ","End":"12:00.920","Text":"and then its amplitude increases to the peak amplitude,"},{"Start":"12:00.920 ","End":"12:05.465","Text":"or the peak particle displacement of 15 centimeters."},{"Start":"12:05.465 ","End":"12:10.010","Text":"We can see that our amplitude here is increasing so this is"},{"Start":"12:10.010 ","End":"12:14.920","Text":"also a possible point that our time displacement graph is representing."},{"Start":"12:14.920 ","End":"12:16.935","Text":"Then we can just carry on,"},{"Start":"12:16.935 ","End":"12:18.680","Text":"and we can see that again,"},{"Start":"12:18.680 ","End":"12:24.155","Text":"this points amplitude is decreasing so we\u0027re not talking about this point,"},{"Start":"12:24.155 ","End":"12:29.030","Text":"and this point its amplitude is increasing to the peak again"},{"Start":"12:29.030 ","End":"12:35.150","Text":"so our time displacement graph could also be representing this point on the rope."},{"Start":"12:35.150 ","End":"12:41.570","Text":"The points that our time displacement graph could be"},{"Start":"12:41.570 ","End":"12:47.600","Text":"representing are the points located at half a centimeter,"},{"Start":"12:47.600 ","End":"12:51.240","Text":"at 1.5 centimeters,"},{"Start":"12:51.240 ","End":"12:53.535","Text":"1.5 is also 3 over 2,"},{"Start":"12:53.535 ","End":"12:58.620","Text":"and at 2.5 or in fraction form,"},{"Start":"12:58.620 ","End":"13:02.110","Text":"5 over 2 centimeters."},{"Start":"13:03.020 ","End":"13:06.610","Text":"That\u0027s the end of this lesson."}],"ID":12471},{"Watched":false,"Name":"Exercise - Wave Calculations 6","Duration":"9m 8s","ChapterTopicVideoID":12004,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.644","Text":"Hello, before you are the time, displacement,"},{"Start":"00:03.644 ","End":"00:07.755","Text":"and position displacement graphs from the previous exercise,"},{"Start":"00:07.755 ","End":"00:10.770","Text":"the period and wavelength of the wave in"},{"Start":"00:10.770 ","End":"00:13.785","Text":"the previous exercise given in these two graphs,"},{"Start":"00:13.785 ","End":"00:17.760","Text":"it was calculated to be a period of 4 seconds and"},{"Start":"00:17.760 ","End":"00:22.605","Text":"a wavelength of 0.01 meters, respectively."},{"Start":"00:22.605 ","End":"00:26.565","Text":"The frequency of the wave generator is now doubled."},{"Start":"00:26.565 ","End":"00:31.929","Text":"Draw the new time displacement and position displacement graphs."},{"Start":"00:32.630 ","End":"00:35.390","Text":"So let\u0027s answer this."},{"Start":"00:35.390 ","End":"00:40.515","Text":"We\u0027re being told that the mu frequency fn"},{"Start":"00:40.515 ","End":"00:46.730","Text":"is now equal to double the original or initial frequency."},{"Start":"00:46.730 ","End":"00:51.124","Text":"So 2fi initial frequency."},{"Start":"00:51.124 ","End":"01:00.020","Text":"So we remember or we\u0027re given in the question that our initial time periods T_i."},{"Start":"01:00.020 ","End":"01:05.535","Text":"Let\u0027s call it that, is equal to 4 seconds."},{"Start":"01:05.535 ","End":"01:09.715","Text":"Which means that the initial frequency,"},{"Start":"01:09.715 ","End":"01:11.645","Text":"f_i, as we know,"},{"Start":"01:11.645 ","End":"01:15.170","Text":"is equal to the reciprocal of the time periods."},{"Start":"01:15.170 ","End":"01:16.700","Text":"So 1 over T_i,"},{"Start":"01:16.700 ","End":"01:20.105","Text":"which is equal to 1-quarter,"},{"Start":"01:20.105 ","End":"01:23.070","Text":"1 divided by 4."},{"Start":"01:23.390 ","End":"01:26.280","Text":"So now, therefore,"},{"Start":"01:26.280 ","End":"01:31.790","Text":"we can go on and therefore say that our new frequency,"},{"Start":"01:31.790 ","End":"01:34.640","Text":"which is equal to twice,"},{"Start":"01:34.640 ","End":"01:42.605","Text":"our initial frequency, is therefore equal to 2 multiplied by 1/4,"},{"Start":"01:42.605 ","End":"01:47.820","Text":"which is equal to 1.5 hertz."},{"Start":"01:50.930 ","End":"01:57.745","Text":"Now we know that our final or our new frequency is 1/2 hertz."},{"Start":"01:57.745 ","End":"02:02.585","Text":"So now we can find out what our new time period is."},{"Start":"02:02.585 ","End":"02:09.770","Text":"So T_n, and this is how we\u0027re going to draw this time displacement graph is equal to,"},{"Start":"02:09.770 ","End":"02:13.220","Text":"as we know, the reciprocal of our frequency."},{"Start":"02:13.220 ","End":"02:15.905","Text":"Here we\u0027re dealing with our new frequency."},{"Start":"02:15.905 ","End":"02:21.740","Text":"So the reciprocal of 1/2 is 2 and our time period is given in seconds."},{"Start":"02:21.740 ","End":"02:24.830","Text":"So now we know if we draw this in red,"},{"Start":"02:24.830 ","End":"02:28.590","Text":"that if our original time period,"},{"Start":"02:28.590 ","End":"02:34.445","Text":"and we remember it\u0027s the time taken to complete one complete oscillation."},{"Start":"02:34.445 ","End":"02:37.760","Text":"So if this was our initial time period,"},{"Start":"02:37.760 ","End":"02:40.595","Text":"4 seconds, which we can see over here."},{"Start":"02:40.595 ","End":"02:46.955","Text":"Then, now we know that our new time period is 2 seconds."},{"Start":"02:46.955 ","End":"02:52.730","Text":"What we\u0027re going to be doing is instead of finishing a full oscillation in 4 seconds,"},{"Start":"02:52.730 ","End":"02:56.060","Text":"we\u0027re going to be finishing a full oscillation in 2 seconds."},{"Start":"02:56.060 ","End":"03:02.840","Text":"So that means that our wave is going to"},{"Start":"03:02.840 ","End":"03:05.180","Text":"look something"},{"Start":"03:05.180 ","End":"03:13.290","Text":"like this."},{"Start":"03:13.290 ","End":"03:18.815","Text":"Now I\u0027m going to draw a new time period in red for this entire graph."},{"Start":"03:18.815 ","End":"03:27.315","Text":"So again, we\u0027re completing 1 full oscillation every 2 seconds."},{"Start":"03:27.315 ","End":"03:33.729","Text":"So we\u0027re going like this and like this."},{"Start":"03:37.220 ","End":"03:42.735","Text":"Here we have 3 oscillations by 6 seconds."},{"Start":"03:42.735 ","End":"03:47.255","Text":"Then we\u0027re going into a fourth oscillation by 8 seconds."},{"Start":"03:47.255 ","End":"03:55.660","Text":"We finished our fifth oscillation by 10 seconds, and so on."},{"Start":"03:59.510 ","End":"04:03.920","Text":"This is our new time displacement graph."},{"Start":"04:03.920 ","End":"04:08.870","Text":"So we now have a time period of 2 seconds."},{"Start":"04:08.870 ","End":"04:13.975","Text":"So what we can see is that the entire graph has just been squashed a little bit."},{"Start":"04:13.975 ","End":"04:15.780","Text":"It\u0027s been squashed by 2."},{"Start":"04:15.780 ","End":"04:21.540","Text":"It\u0027s now double as squashed or as condensed."},{"Start":"04:21.540 ","End":"04:23.940","Text":"Now what we\u0027re going to look,"},{"Start":"04:23.940 ","End":"04:26.775","Text":"so that on said that the new time displacement."},{"Start":"04:26.775 ","End":"04:30.985","Text":"Now we\u0027re going to deal with a position displacement graph, this one over here."},{"Start":"04:30.985 ","End":"04:36.875","Text":"What do we want to see now is if this graph is going to change."},{"Start":"04:36.875 ","End":"04:41.330","Text":"Let\u0027s remember our equation for velocity of a wave."},{"Start":"04:41.330 ","End":"04:45.125","Text":"So one of the equations was given as the square root"},{"Start":"04:45.125 ","End":"04:49.415","Text":"of the tension in the rope or the string,"},{"Start":"04:49.415 ","End":"04:54.250","Text":"divided by the density of the rope or string."},{"Start":"04:54.250 ","End":"04:58.490","Text":"So we know that our velocity of a wave is"},{"Start":"04:58.490 ","End":"05:03.170","Text":"dependent simply on how taught the string is and the material from"},{"Start":"05:03.170 ","End":"05:08.570","Text":"which the string or the rope is made and that was equal"},{"Start":"05:08.570 ","End":"05:14.585","Text":"to our wavelength multiplied by our frequency,"},{"Start":"05:14.585 ","End":"05:19.320","Text":"which is now going to be our new frequency over here."},{"Start":"05:20.150 ","End":"05:24.020","Text":"We know that velocity is only dependent on the rope"},{"Start":"05:24.020 ","End":"05:27.635","Text":"itself and our rope has remained the same,"},{"Start":"05:27.635 ","End":"05:29.990","Text":"just we know that our frequency has doubled."},{"Start":"05:29.990 ","End":"05:32.375","Text":"So our f frequency has doubled."},{"Start":"05:32.375 ","End":"05:35.810","Text":"But our velocity, it remains the same because it\u0027s the same rope."},{"Start":"05:35.810 ","End":"05:40.115","Text":"So in order to keep Lambda f constant,"},{"Start":"05:40.115 ","End":"05:41.915","Text":"if f has doubled,"},{"Start":"05:41.915 ","End":"05:47.490","Text":"that means that Lambda or wavelength has to have."},{"Start":"05:47.750 ","End":"05:54.665","Text":"Because then if I wavelength is divided by 2 and our frequency is multiplied by 2,"},{"Start":"05:54.665 ","End":"05:58.040","Text":"then the 2s, the having and the doubling will"},{"Start":"05:58.040 ","End":"06:02.035","Text":"cancel out and our velocity will remain the same."},{"Start":"06:02.035 ","End":"06:06.590","Text":"What we can see is that our wavelength has 2 1/2."},{"Start":"06:06.590 ","End":"06:08.360","Text":"So what did we see originally?"},{"Start":"06:08.360 ","End":"06:14.030","Text":"We saw that our wavelength is from 1 point on a displacement."},{"Start":"06:14.030 ","End":"06:16.910","Text":"I positioned displacement graph over here."},{"Start":"06:16.910 ","End":"06:20.480","Text":"Let\u0027s say from this position up until this position,"},{"Start":"06:20.480 ","End":"06:21.830","Text":"this was our wavelength."},{"Start":"06:21.830 ","End":"06:26.160","Text":"We saw that that\u0027s equal to 1 centimeter."},{"Start":"06:26.540 ","End":"06:33.485","Text":"Now we have to have our wavelength in order to keep this constant."},{"Start":"06:33.485 ","End":"06:36.245","Text":"So let\u0027s draw this new wavelength in red."},{"Start":"06:36.245 ","End":"06:39.590","Text":"Instead of our wavelength being 1 centimeter,"},{"Start":"06:39.590 ","End":"06:43.205","Text":"it\u0027s now going to be 1/2 a centimeter."},{"Start":"06:43.205 ","End":"06:47.420","Text":"So that means that we\u0027re going to draw this just like in the frequency,"},{"Start":"06:47.420 ","End":"06:53.690","Text":"which means that we have to complete 1 oscillation in order to represent our wavelength."},{"Start":"06:53.690 ","End":"07:03.290","Text":"So that means that we\u0027re going to have to draw something like this."},{"Start":"07:03.290 ","End":"07:08.995","Text":"Then we can copy and paste this over and over again every 1/2 a wavelength."},{"Start":"07:08.995 ","End":"07:15.290","Text":"So let\u0027s do this again and then imagine that I\u0027m drawing this nicely."},{"Start":"07:15.290 ","End":"07:22.940","Text":"Imagine that all of these oscillations and drawing evenly and equally."},{"Start":"07:23.990 ","End":"07:28.735","Text":"Every half a centimeter I have a full wavelength."},{"Start":"07:28.735 ","End":"07:31.885","Text":"So if I go from peak to peak,"},{"Start":"07:31.885 ","End":"07:33.220","Text":"we can see, well,"},{"Start":"07:33.220 ","End":"07:35.770","Text":"we can\u0027t really see it from this diagram,"},{"Start":"07:35.770 ","End":"07:39.080","Text":"but it\u0027s meant to be half a centimeter."},{"Start":"07:39.600 ","End":"07:46.580","Text":"We\u0027re going like this."},{"Start":"07:51.590 ","End":"07:56.045","Text":"What we can see in our position displacement graph,"},{"Start":"07:56.045 ","End":"07:58.085","Text":"the same thing has happened."},{"Start":"07:58.085 ","End":"08:01.085","Text":"So now we\u0027re looking at the red oscillation."},{"Start":"08:01.085 ","End":"08:09.245","Text":"This is our new graph and we can see that our wavelength is just 1/2 a centimeter long,"},{"Start":"08:09.245 ","End":"08:11.900","Text":"because we\u0027ve had to have a wavelength."},{"Start":"08:11.900 ","End":"08:14.135","Text":"So we can say that our Lambda n,"},{"Start":"08:14.135 ","End":"08:20.985","Text":"Lambda nu is equal to 1/2 of i Lambda i,"},{"Start":"08:20.985 ","End":"08:23.955","Text":"Lambda i is 1 centimeter,"},{"Start":"08:23.955 ","End":"08:29.000","Text":"which is equal to 1/2 a centimeter."},{"Start":"08:29.000 ","End":"08:32.540","Text":"So we can see that our graph has just been squished,"},{"Start":"08:32.540 ","End":"08:36.980","Text":"were made more condensed by a factor of 2,"},{"Start":"08:36.980 ","End":"08:41.160","Text":"just like in our time displacement graph."},{"Start":"08:42.470 ","End":"08:49.850","Text":"What do we want to take from this lesson is that if we double the frequency of any wave,"},{"Start":"08:49.850 ","End":"08:54.320","Text":"then that means that we\u0027re going to be having the wavelength of that wave."},{"Start":"08:54.320 ","End":"08:57.275","Text":"Or if we triple the frequency of any wave,"},{"Start":"08:57.275 ","End":"09:01.790","Text":"then we\u0027re going to have a smaller wavelength by a factor of 3."},{"Start":"09:01.790 ","End":"09:05.495","Text":"So our new wavelength would be a third of the original wavelength."},{"Start":"09:05.495 ","End":"09:08.340","Text":"That\u0027s the end of this lesson."}],"ID":12472},{"Watched":false,"Name":"Exercise - Wave Calculations 7","Duration":"26m 51s","ChapterTopicVideoID":12005,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.905","Text":"Hello. In this lesson,"},{"Start":"00:01.905 ","End":"00:04.620","Text":"we\u0027re going to be answering the following question."},{"Start":"00:04.620 ","End":"00:08.010","Text":"Here we have two images of waves,"},{"Start":"00:08.010 ","End":"00:11.505","Text":"one is a traveling wave and one is a standing wave."},{"Start":"00:11.505 ","End":"00:15.990","Text":"Let\u0027s say that this is the traveling wave and this is the standing wave."},{"Start":"00:15.990 ","End":"00:22.180","Text":"Question Number 1 is to calculate the wavelength of each wave."},{"Start":"00:22.550 ","End":"00:26.295","Text":"We\u0027ve already seen that in a traveling wave,"},{"Start":"00:26.295 ","End":"00:30.660","Text":"we can measure the wavelength from peak to peak,"},{"Start":"00:30.660 ","End":"00:39.630","Text":"or from any similar point where we can copy and paste it and repeat our wave pattern."},{"Start":"00:39.630 ","End":"00:42.830","Text":"Let\u0027s this time measure it from peak to peak."},{"Start":"00:42.830 ","End":"00:45.110","Text":"Here in our traveling wave,"},{"Start":"00:45.110 ","End":"00:49.735","Text":"we can see that this is a wavelength,"},{"Start":"00:49.735 ","End":"00:53.235","Text":"Lambda t, wavelength of the traveling wave."},{"Start":"00:53.235 ","End":"00:56.270","Text":"Now we can give it some value,"},{"Start":"00:56.270 ","End":"00:59.405","Text":"so we don\u0027t have some measuring systems."},{"Start":"00:59.405 ","End":"01:02.920","Text":"Let\u0027s just say that it\u0027s equal to 80 centimeters."},{"Start":"01:02.920 ","End":"01:08.810","Text":"We can just say that this distance is 80 centimeters so the wavelength of"},{"Start":"01:08.810 ","End":"01:12.680","Text":"the traveling wave is from peak to peak as we know"},{"Start":"01:12.680 ","End":"01:17.090","Text":"and we\u0027ve just said arbitrarily that this is 80 centimeters."},{"Start":"01:17.090 ","End":"01:19.245","Text":"It doesn\u0027t really matter."},{"Start":"01:19.245 ","End":"01:23.360","Text":"Now let\u0027s take a look at our standing wave."},{"Start":"01:23.360 ","End":"01:25.540","Text":"With our standing wave,"},{"Start":"01:25.540 ","End":"01:31.540","Text":"we know that it\u0027s formed by one wave of a certain wavelength"},{"Start":"01:31.540 ","End":"01:39.405","Text":"traveling in one direction and another wave of a similar wavelength,"},{"Start":"01:39.405 ","End":"01:43.420","Text":"or of the same wavelength traveling in the opposite direction."},{"Start":"01:43.420 ","End":"01:46.645","Text":"Then they meet and interfere with one another,"},{"Start":"01:46.645 ","End":"01:49.730","Text":"causing a standing wave."},{"Start":"01:50.750 ","End":"01:53.655","Text":"What we do know however,"},{"Start":"01:53.655 ","End":"01:57.360","Text":"is that in a standing wave,"},{"Start":"01:57.360 ","End":"02:00.050","Text":"between each two points known as nodes,"},{"Start":"02:00.050 ","End":"02:01.860","Text":"this point and this point,"},{"Start":"02:01.860 ","End":"02:03.900","Text":"these are both nodes."},{"Start":"02:03.900 ","End":"02:05.840","Text":"All of the 0 points,"},{"Start":"02:05.840 ","End":"02:11.630","Text":"we learned that these are nodes and they remain with a 0 value always."},{"Start":"02:11.630 ","End":"02:16.010","Text":"We said that between these two gray points,"},{"Start":"02:16.010 ","End":"02:19.150","Text":"we always have half a wavelength."},{"Start":"02:19.150 ","End":"02:22.925","Text":"This is what we learned in the previous lesson."},{"Start":"02:22.925 ","End":"02:27.815","Text":"That means that if we go from, let\u0027s say,"},{"Start":"02:27.815 ","End":"02:34.399","Text":"this point, and then we do this is half a wavelength."},{"Start":"02:34.399 ","End":"02:36.650","Text":"Then to complete a full oscillation,"},{"Start":"02:36.650 ","End":"02:38.015","Text":"it\u0027s up until here,"},{"Start":"02:38.015 ","End":"02:40.360","Text":"which is a full wavelength."},{"Start":"02:40.360 ","End":"02:44.870","Text":"It\u0027s twice half a wavelength and as we know,"},{"Start":"02:44.870 ","End":"02:49.565","Text":"that\u0027s also the same as measuring from peak to peak."},{"Start":"02:49.565 ","End":"02:55.340","Text":"We can say that this is one full wavelength of the standing wave,"},{"Start":"02:55.340 ","End":"03:00.240","Text":"and it\u0027s the same as this distance over here."},{"Start":"03:00.240 ","End":"03:06.260","Text":"What we can see is that the wavelength of the traveling wave and of"},{"Start":"03:06.260 ","End":"03:12.440","Text":"the standing wave are both equal to the same thing, which is here."},{"Start":"03:12.440 ","End":"03:15.710","Text":"We\u0027ve just given it a value of 80 centimeters."},{"Start":"03:15.710 ","End":"03:18.590","Text":"Now let\u0027s look at Question Number 2."},{"Start":"03:18.590 ","End":"03:23.945","Text":"Question Number 2 is to sketch the rope a quarter of a period later."},{"Start":"03:23.945 ","End":"03:29.480","Text":"As we know, a period is the time taken to complete one oscillation."},{"Start":"03:29.480 ","End":"03:31.955","Text":"If we look at a particle in the rope,"},{"Start":"03:31.955 ","End":"03:34.640","Text":"the time taken from it to be at a peak,"},{"Start":"03:34.640 ","End":"03:36.230","Text":"if it starts at a peak,"},{"Start":"03:36.230 ","End":"03:42.390","Text":"to then go all the way down and return to its original position."},{"Start":"03:42.390 ","End":"03:45.890","Text":"Let\u0027s deal with our traveling wave first and let\u0027s"},{"Start":"03:45.890 ","End":"03:49.130","Text":"just say that it\u0027s traveling in the rightwards direction."},{"Start":"03:49.130 ","End":"03:52.280","Text":"It could also be traveling in the leftward direction and"},{"Start":"03:52.280 ","End":"03:55.935","Text":"then everything would be shifting to the left."},{"Start":"03:55.935 ","End":"03:58.905","Text":"If we\u0027re traveling in the rightwards direction,"},{"Start":"03:58.905 ","End":"04:03.740","Text":"that means that our peaks are"},{"Start":"04:03.740 ","End":"04:09.920","Text":"going to also be moving if we watch the wave oscillates and travel,"},{"Start":"04:09.920 ","End":"04:14.030","Text":"there\u0027ll be shifting in the rightwards direction."},{"Start":"04:14.030 ","End":"04:17.480","Text":"Let\u0027s take a look at what happens."},{"Start":"04:17.480 ","End":"04:22.800","Text":"Here we have our particle at the peak."},{"Start":"04:23.450 ","End":"04:28.590","Text":"After one quarter of a period,"},{"Start":"04:28.590 ","End":"04:34.505","Text":"a particle will have moved down until the point of equilibrium, the 0 point."},{"Start":"04:34.505 ","End":"04:38.120","Text":"Then after another quarter of a period,"},{"Start":"04:38.120 ","End":"04:43.445","Text":"it will move further down to its minimum point, it\u0027s trough."},{"Start":"04:43.445 ","End":"04:46.385","Text":"Then after another quarter of a wavelength,"},{"Start":"04:46.385 ","End":"04:49.460","Text":"it will move back up to its point of"},{"Start":"04:49.460 ","End":"04:54.680","Text":"equilibrium and after another quarter of a wavelength so in the fourth-quarter,"},{"Start":"04:54.680 ","End":"04:57.725","Text":"which is after one quarter of a period,"},{"Start":"04:57.725 ","End":"05:02.030","Text":"so after one full period,"},{"Start":"05:02.030 ","End":"05:05.365","Text":"it will move back up to its original position."},{"Start":"05:05.365 ","End":"05:10.000","Text":"We can see that this is how the particle will move."},{"Start":"05:10.000 ","End":"05:14.300","Text":"What we\u0027ve seen is that in one quarter of a period,"},{"Start":"05:14.300 ","End":"05:19.430","Text":"our particle will move from the peak to the point of equilibrium."},{"Start":"05:19.430 ","End":"05:28.200","Text":"Let\u0027s draw it. This particle and the peak will travel down to be over here."},{"Start":"05:29.600 ","End":"05:33.515","Text":"Now because we know that our wave is moving rightwards,"},{"Start":"05:33.515 ","End":"05:37.070","Text":"that means that our peak has to be moving rightwards."},{"Start":"05:37.070 ","End":"05:41.540","Text":"Then we can now look at this particle over here of the rope,"},{"Start":"05:41.540 ","End":"05:44.345","Text":"which is located at the point of equilibrium."},{"Start":"05:44.345 ","End":"05:47.015","Text":"After one quarter of a period,"},{"Start":"05:47.015 ","End":"05:52.155","Text":"it would have moved up to this position over here."},{"Start":"05:52.155 ","End":"05:54.200","Text":"Then after full period,"},{"Start":"05:54.200 ","End":"05:56.735","Text":"it would have moved back down to the equilibrium,"},{"Start":"05:56.735 ","End":"06:02.210","Text":"down to a minus and back up to the equilibrium to complete its oscillation."},{"Start":"06:02.210 ","End":"06:05.270","Text":"Now, let\u0027s look at this trough,"},{"Start":"06:05.270 ","End":"06:08.315","Text":"this minimum point over here."},{"Start":"06:08.315 ","End":"06:10.535","Text":"Because our wave is moving rightwards,"},{"Start":"06:10.535 ","End":"06:12.365","Text":"we know that this peak,"},{"Start":"06:12.365 ","End":"06:15.080","Text":"just like this peak over here move rightwards."},{"Start":"06:15.080 ","End":"06:19.670","Text":"So this peak in the negative direction also has to move rightwards."},{"Start":"06:19.670 ","End":"06:23.690","Text":"That means that this point over here will be"},{"Start":"06:23.690 ","End":"06:29.750","Text":"moving one quarter of a period and it will be positioned over here and"},{"Start":"06:29.750 ","End":"06:34.295","Text":"that this point over here"},{"Start":"06:34.295 ","End":"06:40.215","Text":"at the equilibrium will move down to a negative."},{"Start":"06:40.215 ","End":"06:43.940","Text":"What we can see in actual fact,"},{"Start":"06:43.940 ","End":"06:45.365","Text":"if I draw this in gray,"},{"Start":"06:45.365 ","End":"06:52.505","Text":"is that our peak has moved right and so has our peak in the negative direction."},{"Start":"06:52.505 ","End":"06:56.430","Text":"Now if we join all of these points up,"},{"Start":"06:56.980 ","End":"07:01.500","Text":"we\u0027ll get a wave like so."},{"Start":"07:01.500 ","End":"07:03.870","Text":"Then just to complete this,"},{"Start":"07:03.870 ","End":"07:08.225","Text":"we have a peak so just like in this peak the point move down,"},{"Start":"07:08.225 ","End":"07:09.860","Text":"so in this peak,"},{"Start":"07:09.860 ","End":"07:12.900","Text":"the point will also move down."},{"Start":"07:12.900 ","End":"07:17.270","Text":"There we\u0027ve completed one full oscillation where we"},{"Start":"07:17.270 ","End":"07:21.470","Text":"can see our amplitudes is moving rightwards,"},{"Start":"07:21.470 ","End":"07:26.160","Text":"which is what we would expect with the direction of the wave traveling."},{"Start":"07:26.660 ","End":"07:28.760","Text":"Now with the traveling wave,"},{"Start":"07:28.760 ","End":"07:30.320","Text":"let\u0027s just complete it."},{"Start":"07:30.320 ","End":"07:37.580","Text":"What we can see is that everywhere each point of the wave is exactly the same,"},{"Start":"07:37.580 ","End":"07:40.880","Text":"it\u0027s only been shifted in the direction of travel of"},{"Start":"07:40.880 ","End":"07:46.829","Text":"the wave by one quarter of a wavelength."},{"Start":"07:47.060 ","End":"07:54.796","Text":"Imagine again that I\u0027ve drawn this like I\u0027m supposed to."},{"Start":"07:54.796 ","End":"08:02.050","Text":"This is a traveling wave 1/4 of a period later."},{"Start":"08:02.050 ","End":"08:05.750","Text":"Now let\u0027s take a look at our standing wave."},{"Start":"08:06.330 ","End":"08:09.385","Text":"In a standing wave, as we know,"},{"Start":"08:09.385 ","End":"08:13.600","Text":"we have nodes and we know that these nodes which are"},{"Start":"08:13.600 ","End":"08:18.175","Text":"located here at the point of equilibrium don\u0027t ever move,"},{"Start":"08:18.175 ","End":"08:22.135","Text":"and they\u0027re always located here at 0."},{"Start":"08:22.135 ","End":"08:24.190","Text":"What happens on the standing wave?"},{"Start":"08:24.190 ","End":"08:27.670","Text":"Well, let\u0027s look at a particle at the peak."},{"Start":"08:27.670 ","End":"08:32.860","Text":"We know that after one whole period,"},{"Start":"08:32.860 ","End":"08:36.880","Text":"the particle that is located initially at the peak will"},{"Start":"08:36.880 ","End":"08:41.260","Text":"return after full period to be once again at the peak."},{"Start":"08:41.260 ","End":"08:46.510","Text":"Therefore, we know that after 1/4 of a period,"},{"Start":"08:46.510 ","End":"08:49.315","Text":"the particle will be at the point of equilibrium."},{"Start":"08:49.315 ","End":"08:52.240","Text":"After 1/2 a period,"},{"Start":"08:52.240 ","End":"08:55.929","Text":"the particle will be at the trough."},{"Start":"08:55.929 ","End":"08:58.975","Text":"After 3/4 of a period,"},{"Start":"08:58.975 ","End":"09:01.825","Text":"the particle will return to the point of equilibrium,"},{"Start":"09:01.825 ","End":"09:05.095","Text":"and after 4th quarters of a period,"},{"Start":"09:05.095 ","End":"09:08.890","Text":"aka a full period will complete"},{"Start":"09:08.890 ","End":"09:14.500","Text":"one full oscillation and our particle will return to be at the peak."},{"Start":"09:14.500 ","End":"09:22.285","Text":"In that case, we saw that a particle that starts at the peak after 1/4 of a wavelength,"},{"Start":"09:22.285 ","End":"09:25.750","Text":"it will be at this point of equilibrium,"},{"Start":"09:25.750 ","End":"09:29.450","Text":"sorry, after 1/4 of a period."},{"Start":"09:30.570 ","End":"09:35.260","Text":"Now we know that each individual particle of a rope in"},{"Start":"09:35.260 ","End":"09:40.960","Text":"a standing wave has its own particular, an individual amplitude."},{"Start":"09:40.960 ","End":"09:44.350","Text":"If we look at a particle located over here,"},{"Start":"09:44.350 ","End":"09:47.710","Text":"it is right now at its peak amplitude."},{"Start":"09:47.710 ","End":"09:49.915","Text":"Now let\u0027s do the same thing."},{"Start":"09:49.915 ","End":"09:52.240","Text":"After 1/4 of a period,"},{"Start":"09:52.240 ","End":"09:53.530","Text":"which is what we\u0027re doing,"},{"Start":"09:53.530 ","End":"09:57.850","Text":"this particle will be located here,"},{"Start":"09:57.850 ","End":"10:01.300","Text":"and after 1.5 of a period,"},{"Start":"10:01.300 ","End":"10:08.680","Text":"our particle will be located a mirror image of itself just in the negative direction."},{"Start":"10:08.680 ","End":"10:13.180","Text":"It will have the same magnitude just in the negative direction."},{"Start":"10:13.180 ","End":"10:15.400","Text":"Then after 3/4 of a period,"},{"Start":"10:15.400 ","End":"10:19.675","Text":"it will go back to the point of equilibrium and after one full period,"},{"Start":"10:19.675 ","End":"10:23.510","Text":"it will be back at its peak amplitude over here."},{"Start":"10:23.880 ","End":"10:27.430","Text":"What we can see is that this particle too,"},{"Start":"10:27.430 ","End":"10:31.780","Text":"that starts over here after 1/4 of a period,"},{"Start":"10:31.780 ","End":"10:35.500","Text":"it\u0027s back at this point over here."},{"Start":"10:35.500 ","End":"10:40.825","Text":"Then we can look over here at this point here,"},{"Start":"10:40.825 ","End":"10:42.984","Text":"and after 1/4 of a wavelength,"},{"Start":"10:42.984 ","End":"10:46.180","Text":"it will move up to this point over here,"},{"Start":"10:46.180 ","End":"10:48.625","Text":"1/4 of a period, sorry."},{"Start":"10:48.625 ","End":"10:52.435","Text":"Will this particle after 1/4 of a period move here."},{"Start":"10:52.435 ","End":"11:02.185","Text":"What we can see is that all of the particles move to this point of equilibrium."},{"Start":"11:02.185 ","End":"11:06.250","Text":"Then if I draw in red on a standing wave,"},{"Start":"11:06.250 ","End":"11:11.170","Text":"the rope 1/4 of a period after the initial picture was taken,"},{"Start":"11:11.170 ","End":"11:14.710","Text":"this is what the wave will look like."},{"Start":"11:14.710 ","End":"11:18.565","Text":"It will be a flat line at the point of equilibrium."},{"Start":"11:18.565 ","End":"11:23.410","Text":"A quarter of a period later on the standing wave we\u0027ll just see"},{"Start":"11:23.410 ","End":"11:28.885","Text":"a flat rope as if no wave is present."},{"Start":"11:28.885 ","End":"11:33.115","Text":"Now, let\u0027s answer Question Number 3 and we\u0027re being"},{"Start":"11:33.115 ","End":"11:36.970","Text":"asked to sketch the rope 1/2 a period later."},{"Start":"11:36.970 ","End":"11:39.865","Text":"I\u0027m going to rub all of this out."},{"Start":"11:39.865 ","End":"11:42.880","Text":"Here we\u0027ve seen what happens after 1/4 period,"},{"Start":"11:42.880 ","End":"11:48.530","Text":"so we\u0027re going to do the exact same calculation just for 1/2 a period."},{"Start":"11:49.890 ","End":"11:53.920","Text":"Let\u0027s deal with the traveling wave first."},{"Start":"11:53.920 ","End":"11:55.720","Text":"Again, our traveling wave is still in"},{"Start":"11:55.720 ","End":"11:59.605","Text":"the rightwards direction and we saw previously that"},{"Start":"11:59.605 ","End":"12:06.910","Text":"this point over here at the peak after 1/4 of a period will be at the 0 point,"},{"Start":"12:06.910 ","End":"12:11.740","Text":"and after 1/2 a period will be at this minus point."},{"Start":"12:11.740 ","End":"12:13.570","Text":"In the traveling wave,"},{"Start":"12:13.570 ","End":"12:17.440","Text":"we\u0027re going to have a point over here,"},{"Start":"12:17.440 ","End":"12:22.690","Text":"and then this minus point is going to be after 1/4 of"},{"Start":"12:22.690 ","End":"12:29.500","Text":"a period over here and after 1/2 a period over here at a peak."},{"Start":"12:29.500 ","End":"12:34.780","Text":"Then again, this peak after 1/4 of a period will be at the"},{"Start":"12:34.780 ","End":"12:41.365","Text":"0 and after 1/2 a period will be at the minus amplitude."},{"Start":"12:41.365 ","End":"12:50.890","Text":"Now what we can see is that we\u0027re going to have this type of shape over here."},{"Start":"12:50.890 ","End":"12:53.390","Text":"I\u0027m actually going to draw it in red."},{"Start":"12:54.390 ","End":"12:59.994","Text":"We have our trough over here going into a 0,"},{"Start":"12:59.994 ","End":"13:01.525","Text":"up to a peak,"},{"Start":"13:01.525 ","End":"13:05.035","Text":"back down to a 0, and back down to a trough."},{"Start":"13:05.035 ","End":"13:08.875","Text":"That is one complete oscillation,"},{"Start":"13:08.875 ","End":"13:12.140","Text":"and then we\u0027re going to copy this."},{"Start":"13:13.170 ","End":"13:20.350","Text":"It\u0027s going like so"},{"Start":"13:20.350 ","End":"13:25.150","Text":"and our wave is moving in the rightwards direction."},{"Start":"13:25.150 ","End":"13:29.860","Text":"What we can see is that it\u0027s as if a wave has been"},{"Start":"13:29.860 ","End":"13:36.230","Text":"reflected in the x-axis but what has actually happened?"},{"Start":"13:36.240 ","End":"13:44.725","Text":"All that\u0027s happened is that our point over here in this trough has moved over here."},{"Start":"13:44.725 ","End":"13:48.145","Text":"It\u0027s just moved rightwards 1/2 a period,"},{"Start":"13:48.145 ","End":"13:50.965","Text":"or as we can see here, 1/2 a wavelength."},{"Start":"13:50.965 ","End":"13:53.590","Text":"This peak over here has also,"},{"Start":"13:53.590 ","End":"13:55.360","Text":"because our wave is moving rightwards,"},{"Start":"13:55.360 ","End":"13:58.150","Text":"moved to this point over here,"},{"Start":"13:58.150 ","End":"14:01.270","Text":"has just moved 1/2 a period later."},{"Start":"14:01.270 ","End":"14:03.370","Text":"We\u0027re seeing a snapshot of 1/2 a period later,"},{"Start":"14:03.370 ","End":"14:08.215","Text":"so it\u0027s moved 1/2 a wavelength and similarly here."},{"Start":"14:08.215 ","End":"14:18.385","Text":"All of our points have just shifted 1/2 a wavelength matching to this 1/2 a period shot."},{"Start":"14:18.385 ","End":"14:22.735","Text":"What\u0027s going to happen here with our standing wave?"},{"Start":"14:22.735 ","End":"14:29.680","Text":"Similarly, what we saw is that if we take a point at a peak after 1/4 of a wavelength,"},{"Start":"14:29.680 ","End":"14:32.470","Text":"our point was located at the 0,"},{"Start":"14:32.470 ","End":"14:34.630","Text":"sorry, 1/4 of a period,"},{"Start":"14:34.630 ","End":"14:41.245","Text":"and after 1/2 a period our particle was located over here."},{"Start":"14:41.245 ","End":"14:45.340","Text":"Let\u0027s draw that point in red and then"},{"Start":"14:45.340 ","End":"14:49.510","Text":"we saw that with this trough after 1/4 of a period,"},{"Start":"14:49.510 ","End":"14:53.635","Text":"the particle is located over here and after 1/2 a period,"},{"Start":"14:53.635 ","End":"14:56.125","Text":"the particle is located over here."},{"Start":"14:56.125 ","End":"14:57.970","Text":"Let\u0027s draw that in red."},{"Start":"14:57.970 ","End":"15:00.085","Text":"Then we can see that that happens here."},{"Start":"15:00.085 ","End":"15:02.200","Text":"Also this peak 1/4 of a period,"},{"Start":"15:02.200 ","End":"15:05.395","Text":"it\u0027s here, another 1/4 period, it\u0027s here."},{"Start":"15:05.395 ","End":"15:09.010","Text":"Of course, I know it\u0027s the points in gray aren\u0027t moving."},{"Start":"15:09.010 ","End":"15:11.275","Text":"Now we can just fill this in,"},{"Start":"15:11.275 ","End":"15:13.450","Text":"1/4 of a period moves up,"},{"Start":"15:13.450 ","End":"15:15.340","Text":"1/4 of a period again,"},{"Start":"15:15.340 ","End":"15:16.675","Text":"so 1/2 a period,"},{"Start":"15:16.675 ","End":"15:18.490","Text":"it\u0027s back up here."},{"Start":"15:18.490 ","End":"15:22.600","Text":"This 1/4 of a period moves down here and another 1/4,"},{"Start":"15:22.600 ","End":"15:25.075","Text":"so that\u0027s 1/2 a period, it\u0027s down here."},{"Start":"15:25.075 ","End":"15:29.860","Text":"What we can see is that we\u0027ll"},{"Start":"15:29.860 ","End":"15:39.895","Text":"have a standing wave that looks like so."},{"Start":"15:39.895 ","End":"15:45.454","Text":"Just complete this sketch."},{"Start":"15:45.454 ","End":"15:51.300","Text":"What we can see here is that our nodes are in the exact same position,"},{"Start":"15:51.300 ","End":"15:53.340","Text":"and in a standing wave,"},{"Start":"15:53.340 ","End":"15:58.665","Text":"a wave has simply been reflected in the x axes."},{"Start":"15:58.665 ","End":"16:05.025","Text":"What we can see is that our traveling wave in a standing wave look very similar."},{"Start":"16:05.025 ","End":"16:10.710","Text":"However, the difference is how the particles have moved."},{"Start":"16:10.710 ","End":"16:12.360","Text":"Here we can see,"},{"Start":"16:12.360 ","End":"16:14.190","Text":"well not the particles,"},{"Start":"16:14.190 ","End":"16:19.440","Text":"but rather how the wave has traveled through the particles."},{"Start":"16:19.440 ","End":"16:21.225","Text":"In the traveling waves,"},{"Start":"16:21.225 ","End":"16:27.105","Text":"we saw that the wave was just shifting constantly to the right."},{"Start":"16:27.105 ","End":"16:32.130","Text":"Here it\u0027s shifted to the right by 0.5 a wavelength but what do we can"},{"Start":"16:32.130 ","End":"16:38.250","Text":"see in our standing wave is that the wave isn\u0027t shifting to the right,"},{"Start":"16:38.250 ","End":"16:44.985","Text":"but rather it\u0027s just moving in a different direction."},{"Start":"16:44.985 ","End":"16:49.005","Text":"If our particle started over here,"},{"Start":"16:49.005 ","End":"16:52.260","Text":"it\u0027s now moved down here,"},{"Start":"16:52.260 ","End":"16:54.480","Text":"and if it started down here,"},{"Start":"16:54.480 ","End":"16:58.230","Text":"it\u0027s now moved to be up here."},{"Start":"16:58.230 ","End":"17:04.590","Text":"Here we can see that the peaks were just shifting rightwards but here we can see that"},{"Start":"17:04.590 ","End":"17:11.920","Text":"our peaks are just moving up and down and alternating in the y-direction."},{"Start":"17:13.220 ","End":"17:18.900","Text":"In other words, in the traveling wave will just see a continuation"},{"Start":"17:18.900 ","End":"17:24.135","Text":"of a traveling wave just moving and shifting to the right."},{"Start":"17:24.135 ","End":"17:28.515","Text":"Just traveling and the rightwards direction and the peaks will just mean moving."},{"Start":"17:28.515 ","End":"17:31.425","Text":"Conversely in our standing waves,"},{"Start":"17:31.425 ","End":"17:38.040","Text":"we\u0027re going to see the particle of the rope just alternating from being located at"},{"Start":"17:38.040 ","End":"17:44.940","Text":"a peak to then be located at a trough and then back at a peak and back in a trough."},{"Start":"17:44.940 ","End":"17:49.755","Text":"What we\u0027re going to see is our waves just bouncing up and down,"},{"Start":"17:49.755 ","End":"17:55.570","Text":"alternating between when they\u0027re at peak and when they\u0027re at a trough."},{"Start":"17:56.840 ","End":"18:00.960","Text":"Let\u0027s clarify this a bit by showing what"},{"Start":"18:00.960 ","End":"18:06.720","Text":"these 2 waves would look like after 1/8 of a period."},{"Start":"18:06.720 ","End":"18:09.135","Text":"Let\u0027s look at the standing wave."},{"Start":"18:09.135 ","End":"18:12.480","Text":"After 1/8 of a period, let\u0027s see,"},{"Start":"18:12.480 ","End":"18:19.194","Text":"we saw that a course over period would bring the particle from a peak down to the 0."},{"Start":"18:19.194 ","End":"18:23.010","Text":"1/8 is 0.5 of a quarter so that means"},{"Start":"18:23.010 ","End":"18:27.520","Text":"that the peak will move down to this point over here."},{"Start":"18:27.520 ","End":"18:30.004","Text":"Then similarly the trough,"},{"Start":"18:30.004 ","End":"18:34.235","Text":"this point over here after crossover period was over here,"},{"Start":"18:34.235 ","End":"18:38.290","Text":"so after 1/8 of a period, it\u0027ll be over here."},{"Start":"18:38.290 ","End":"18:43.410","Text":"What we\u0027ll see is that we have points like so and of course,"},{"Start":"18:43.410 ","End":"18:46.935","Text":"our nodes are never moving."},{"Start":"18:46.935 ","End":"18:52.740","Text":"Then what we\u0027ll have after 1/8 of a period is"},{"Start":"18:52.740 ","End":"18:59.955","Text":"simply a standing wave where it\u0027s amplitude is just much less."},{"Start":"18:59.955 ","End":"19:04.800","Text":"We\u0027ll have a standing wave that looks like this,"},{"Start":"19:04.800 ","End":"19:09.240","Text":"which is just a shallower wave."},{"Start":"19:09.240 ","End":"19:11.235","Text":"Something like this."},{"Start":"19:11.235 ","End":"19:13.050","Text":"This points will move down,"},{"Start":"19:13.050 ","End":"19:15.250","Text":"this will move up."},{"Start":"19:15.560 ","End":"19:22.410","Text":"Our wave will just be shallower and it\u0027s just going to oscillate here to the 0,"},{"Start":"19:22.410 ","End":"19:25.875","Text":"to here to the peak and back."},{"Start":"19:25.875 ","End":"19:28.290","Text":"Conversely, on the traveling wave,"},{"Start":"19:28.290 ","End":"19:31.125","Text":"1/8 of a period later,"},{"Start":"19:31.125 ","End":"19:36.660","Text":"our peaks will just move 1/8 of a wavelength in the right direction,"},{"Start":"19:36.660 ","End":"19:38.040","Text":"in the direction of travel."},{"Start":"19:38.040 ","End":"19:39.915","Text":"This peak will be here."},{"Start":"19:39.915 ","End":"19:43.720","Text":"This arrow point will have gone up a little bit."},{"Start":"19:44.000 ","End":"19:48.670","Text":"This peak will be over here."},{"Start":"19:49.760 ","End":"19:53.940","Text":"This trough will be over here."},{"Start":"19:53.940 ","End":"19:59.320","Text":"This 0 point will be here."},{"Start":"19:59.320 ","End":"20:01.920","Text":"What we can do,"},{"Start":"20:01.920 ","End":"20:06.420","Text":"is we can just simply connect"},{"Start":"20:06.420 ","End":"20:09.010","Text":"these points,"},{"Start":"20:09.430 ","End":"20:17.580","Text":"like so."},{"Start":"20:17.580 ","End":"20:25.965","Text":"The difference that we can see in the traveling wave and in the standing wave,"},{"Start":"20:25.965 ","End":"20:27.915","Text":"can imagine I drew this much better,"},{"Start":"20:27.915 ","End":"20:35.910","Text":"is simply that here our peaks are just moving continually in the right direction."},{"Start":"20:35.910 ","End":"20:40.920","Text":"The peaks are just shifting slightly more to the right."},{"Start":"20:40.920 ","End":"20:46.515","Text":"Whereas here amplitudes are bobbing up and down,"},{"Start":"20:46.515 ","End":"20:54.555","Text":"and we don\u0027t see any shift in the wavelength or in the possession of the peaks."},{"Start":"20:54.555 ","End":"20:57.720","Text":"If the peak is here a little bit later,"},{"Start":"20:57.720 ","End":"20:58.950","Text":"the peak might be lower,"},{"Start":"20:58.950 ","End":"21:01.710","Text":"but it\u0027s at the same x position."},{"Start":"21:01.710 ","End":"21:07.120","Text":"Whereas here our peak is constantly changing x position."},{"Start":"21:08.270 ","End":"21:12.855","Text":"I\u0027ve written a small note for our Questions 2 and 3,"},{"Start":"21:12.855 ","End":"21:15.300","Text":"that when we\u0027re dealing with a traveling wave."},{"Start":"21:15.300 ","End":"21:17.865","Text":"If we just look at the peak,"},{"Start":"21:17.865 ","End":"21:21.690","Text":"we can see that they\u0027re just changing the x position,"},{"Start":"21:21.690 ","End":"21:28.950","Text":"so their amplitude or their y position remains constant but their x position is changing."},{"Start":"21:28.950 ","End":"21:37.893","Text":"They move from this x position over here in red to this x position over here."},{"Start":"21:37.893 ","End":"21:43.395","Text":"They\u0027ll move later depending on how long we wait to here, to here, to here."},{"Start":"21:43.395 ","End":"21:47.010","Text":"All the peaks are constantly moving along the x-axis."},{"Start":"21:47.010 ","End":"21:49.710","Text":"Whereas when we\u0027re dealing with a standing wave,"},{"Start":"21:49.710 ","End":"21:52.935","Text":"the peaks change y position."},{"Start":"21:52.935 ","End":"21:58.410","Text":"The peak is always going to be at this specific x position."},{"Start":"21:58.410 ","End":"22:03.495","Text":"However, we can see that its original y position is over here,"},{"Start":"22:03.495 ","End":"22:07.437","Text":"and then it moves down to this y position over here."},{"Start":"22:07.437 ","End":"22:12.945","Text":"Then it moves down to the 0 and then it will move down to a y position."},{"Start":"22:12.945 ","End":"22:15.690","Text":"Then the negative until it reaches"},{"Start":"22:15.690 ","End":"22:24.261","Text":"its negative y position over here."},{"Start":"22:24.261 ","End":"22:28.935","Text":"It goes like so."},{"Start":"22:28.935 ","End":"22:31.350","Text":"This is a little note that might help."},{"Start":"22:31.350 ","End":"22:36.525","Text":"That answer Question Number 4."},{"Start":"22:36.525 ","End":"22:39.810","Text":"Choose one point to the left and one"},{"Start":"22:39.810 ","End":"22:43.500","Text":"to the right of the amplitude so that means of the peak."},{"Start":"22:43.500 ","End":"22:47.320","Text":"Draw these points direction of travel."},{"Start":"22:47.690 ","End":"22:51.840","Text":"First, let\u0027s look at our traveling wave."},{"Start":"22:51.840 ","End":"22:57.390","Text":"Let\u0027s choose this point to the left of our peak,"},{"Start":"22:57.390 ","End":"22:59.220","Text":"and this is point a,"},{"Start":"22:59.220 ","End":"23:05.370","Text":"and let\u0027s choose this point to the right of our peak, b."},{"Start":"23:05.370 ","End":"23:08.670","Text":"Now what we can do is to make this easier,"},{"Start":"23:08.670 ","End":"23:13.320","Text":"we can draw this wave a moment later."},{"Start":"23:13.320 ","End":"23:17.265","Text":"Because we know that our wave is traveling to the right a moment later,"},{"Start":"23:17.265 ","End":"23:18.795","Text":"peak will be over here,"},{"Start":"23:18.795 ","End":"23:21.510","Text":"and this trough will be over here,"},{"Start":"23:21.510 ","End":"23:25.800","Text":"and this trough will be over here."},{"Start":"23:25.800 ","End":"23:29.620","Text":"If we draw this,"},{"Start":"23:31.070 ","End":"23:37.360","Text":"we\u0027ll get something along these lines."},{"Start":"23:38.420 ","End":"23:42.990","Text":"What we can see is that point A,"},{"Start":"23:42.990 ","End":"23:45.465","Text":"which was once over here,"},{"Start":"23:45.465 ","End":"23:48.195","Text":"will now be over here."},{"Start":"23:48.195 ","End":"23:51.705","Text":"It\u0027s moved downwards, and that point B,"},{"Start":"23:51.705 ","End":"23:53.445","Text":"which was once over here,"},{"Start":"23:53.445 ","End":"23:57.225","Text":"has now moved to be at this point over here."},{"Start":"23:57.225 ","End":"23:59.520","Text":"It\u0027s now moved to the peak."},{"Start":"23:59.520 ","End":"24:02.385","Text":"It\u0027s moved upwards."},{"Start":"24:02.385 ","End":"24:06.000","Text":"What we can see and we\u0027ve already spoken about this,"},{"Start":"24:06.000 ","End":"24:08.190","Text":"is that in traveling waves,"},{"Start":"24:08.190 ","End":"24:15.090","Text":"every single point that comes before the peak,"},{"Start":"24:15.090 ","End":"24:17.100","Text":"depending on the direction of travel."},{"Start":"24:17.100 ","End":"24:21.480","Text":"Before the peak, if the direction of travel is in the rightwards direction,"},{"Start":"24:21.480 ","End":"24:24.625","Text":"that will be point B because it will reach,"},{"Start":"24:24.625 ","End":"24:26.390","Text":"if we have some barrier,"},{"Start":"24:26.390 ","End":"24:28.610","Text":"point B will reach first."},{"Start":"24:28.610 ","End":"24:35.325","Text":"Every point before the peak will move upwards,"},{"Start":"24:35.325 ","End":"24:37.740","Text":"and every point after the peak,"},{"Start":"24:37.740 ","End":"24:39.090","Text":"so that\u0027s A,"},{"Start":"24:39.090 ","End":"24:46.000","Text":"it\u0027s located after the peak relative to the direction of travel will move downwards."},{"Start":"24:46.460 ","End":"24:52.920","Text":"Here I\u0027ve written a little note in traveling waves relative to the direction of travel."},{"Start":"24:52.920 ","End":"24:56.414","Text":"The points before the amplitude move upwards"},{"Start":"24:56.414 ","End":"25:00.705","Text":"and the points after the amplitude move downwards."},{"Start":"25:00.705 ","End":"25:04.650","Text":"Now, let\u0027s take a look at the standing wave."},{"Start":"25:04.650 ","End":"25:09.285","Text":"We\u0027ve already seen in a previous video that we have our nodes,"},{"Start":"25:09.285 ","End":"25:13.992","Text":"and we already saw that between every two nodes,"},{"Start":"25:13.992 ","End":"25:16.275","Text":"let\u0027s say between these two nodes,"},{"Start":"25:16.275 ","End":"25:21.300","Text":"all of the points move in the same direction together."},{"Start":"25:21.300 ","End":"25:25.515","Text":"Here specifically, if we choose,"},{"Start":"25:25.515 ","End":"25:27.885","Text":"let\u0027s say like we did before."},{"Start":"25:27.885 ","End":"25:29.985","Text":"Here we have a point A,"},{"Start":"25:29.985 ","End":"25:32.649","Text":"which is after the amplitude."},{"Start":"25:32.649 ","End":"25:34.935","Text":"Here we have our point B,"},{"Start":"25:34.935 ","End":"25:37.034","Text":"which is before the amplitude,"},{"Start":"25:37.034 ","End":"25:43.310","Text":"so what we can see is that B is going to be moving downwards towards the 0 line."},{"Start":"25:43.310 ","End":"25:46.385","Text":"Then to the other side, and similarly,"},{"Start":"25:46.385 ","End":"25:51.845","Text":"A is going to be moving downwards towards the 0 line and then towards the negative side."},{"Start":"25:51.845 ","End":"25:57.300","Text":"In actual fact, what we can see is that all the points between two nodes,"},{"Start":"25:57.300 ","End":"26:01.815","Text":"and we\u0027ve already seen this move in the same direction."},{"Start":"26:01.815 ","End":"26:04.740","Text":"All of these points between these two nodes will"},{"Start":"26:04.740 ","End":"26:07.890","Text":"be moving upwards between these two nodes."},{"Start":"26:07.890 ","End":"26:13.868","Text":"All the points will be moving downwards between these two nodes,"},{"Start":"26:13.868 ","End":"26:18.220","Text":"all the points will be moving upwards."},{"Start":"26:18.620 ","End":"26:20.835","Text":"In the standing wave,"},{"Start":"26:20.835 ","End":"26:25.800","Text":"all the points between two nodes move together in the same direction,"},{"Start":"26:25.800 ","End":"26:31.170","Text":"and the direction is either up or down depending if they\u0027re located at the peak,"},{"Start":"26:31.170 ","End":"26:33.975","Text":"they\u0027ll move down if the 0 line,"},{"Start":"26:33.975 ","End":"26:36.375","Text":"they move up or down depending on where they were before."},{"Start":"26:36.375 ","End":"26:39.910","Text":"If they\u0027re at a trough, they\u0027ll move up."},{"Start":"26:40.430 ","End":"26:44.265","Text":"I hope this exercise helped to highlight"},{"Start":"26:44.265 ","End":"26:48.855","Text":"the differences between a traveling and a standing wave."},{"Start":"26:48.855 ","End":"26:51.880","Text":"That\u0027s the end of this lesson."}],"ID":12473},{"Watched":false,"Name":"Exercise - Finding Frequency","Duration":"5m 38s","ChapterTopicVideoID":12006,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"Hello. In this question,"},{"Start":"00:02.040 ","End":"00:04.650","Text":"we\u0027re being told that a wave propagates through a rope with"},{"Start":"00:04.650 ","End":"00:07.620","Text":"a velocity of 25 meters per second."},{"Start":"00:07.620 ","End":"00:10.665","Text":"The rope is tied to 2 posts,"},{"Start":"00:10.665 ","End":"00:12.255","Text":"which are 3 meters apart."},{"Start":"00:12.255 ","End":"00:16.850","Text":"At what frequency must the rope be"},{"Start":"00:16.850 ","End":"00:21.770","Text":"swung in order to produce a standing wave with 12 nodes?"},{"Start":"00:21.770 ","End":"00:23.390","Text":"Here are options."},{"Start":"00:23.390 ","End":"00:29.035","Text":"We want to find out which 1 from a-e is the right answer."},{"Start":"00:29.035 ","End":"00:33.260","Text":"Some equations that we have for standing waves."},{"Start":"00:33.260 ","End":"00:40.840","Text":"We know that the wavelength of a standing wave is dependent on the amount of antinodes."},{"Start":"00:40.840 ","End":"00:47.145","Text":"Remember, n is for anti nodes,"},{"Start":"00:47.145 ","End":"00:57.460","Text":"and this is equal to twice the length of the rope divided by the number of antinodes."},{"Start":"00:57.460 ","End":"01:02.090","Text":"We also know that the frequency of a standing wave is"},{"Start":"01:02.090 ","End":"01:06.530","Text":"also dependent on the amount of antinodes and that,"},{"Start":"01:06.530 ","End":"01:11.180","Text":"that is equal to the number of antinodes multiplied"},{"Start":"01:11.180 ","End":"01:17.970","Text":"by the velocity divided by twice the length of the rope."},{"Start":"01:17.990 ","End":"01:21.200","Text":"What information are we being given?"},{"Start":"01:21.200 ","End":"01:28.205","Text":"We know that the velocity is equal to 25 meters per second."},{"Start":"01:28.205 ","End":"01:31.865","Text":"We\u0027re given that in our question and our L,"},{"Start":"01:31.865 ","End":"01:34.044","Text":"the length of the rope."},{"Start":"01:34.044 ","End":"01:37.890","Text":"It\u0027s tied between 2 posts that are 3 meters apart,"},{"Start":"01:37.890 ","End":"01:42.690","Text":"so that means that the length of our rope is 3 meters."},{"Start":"01:42.690 ","End":"01:47.630","Text":"Now we\u0027re being told that we want to find the frequency that we have to swing"},{"Start":"01:47.630 ","End":"01:52.250","Text":"the rope in such that there\u0027ll be a standing wave with 12 nodes."},{"Start":"01:52.250 ","End":"01:54.385","Text":"Let\u0027s see what that looks like."},{"Start":"01:54.385 ","End":"01:58.190","Text":"We have a post over here,"},{"Start":"01:58.190 ","End":"02:01.010","Text":"and we\u0027ll draw the second post in a second."},{"Start":"02:01.010 ","End":"02:02.810","Text":"Unless we\u0027re told otherwise,"},{"Start":"02:02.810 ","End":"02:07.760","Text":"the 12 nodes includes the nodes right at"},{"Start":"02:07.760 ","End":"02:14.640","Text":"the beginning where our rope is tied to the first post and to the second post."},{"Start":"02:14.860 ","End":"02:18.880","Text":"Unless we\u0027re told otherwise. This is node number 1,"},{"Start":"02:18.880 ","End":"02:22.950","Text":"2, 3, 4,"},{"Start":"02:22.950 ","End":"02:27.150","Text":"5, 6, 7,"},{"Start":"02:27.150 ","End":"02:29.620","Text":"8, 9,"},{"Start":"02:29.620 ","End":"02:33.755","Text":"10, 11, and 12."},{"Start":"02:33.755 ","End":"02:39.110","Text":"Because our nodes are included in what\u0027s tied into the rope."},{"Start":"02:39.110 ","End":"02:44.345","Text":"Here we put our next post."},{"Start":"02:44.345 ","End":"02:47.705","Text":"Now we have 12 nodes."},{"Start":"02:47.705 ","End":"02:52.710","Text":"Now we want to draw a standing wave that looks something like this."},{"Start":"02:55.850 ","End":"03:02.345","Text":"Where of course, all of these waves are the same,"},{"Start":"03:02.345 ","End":"03:08.080","Text":"they have the same width and the same amplitude."},{"Start":"03:08.080 ","End":"03:11.570","Text":"In order to use these equations,"},{"Start":"03:11.570 ","End":"03:14.405","Text":"we need to know what the number of antinodes,"},{"Start":"03:14.405 ","End":"03:16.100","Text":"not the number of nodes."},{"Start":"03:16.100 ","End":"03:18.635","Text":"How many antinodes do we have?"},{"Start":"03:18.635 ","End":"03:20.420","Text":"Let\u0027s mark them in red."},{"Start":"03:20.420 ","End":"03:22.475","Text":"This is 1 antinode,"},{"Start":"03:22.475 ","End":"03:25.320","Text":"this is another antinode,"},{"Start":"03:25.320 ","End":"03:27.930","Text":"3 antinode, 4,"},{"Start":"03:27.930 ","End":"03:31.665","Text":"5, 6, 7,"},{"Start":"03:31.665 ","End":"03:33.930","Text":"8, 9,"},{"Start":"03:33.930 ","End":"03:36.930","Text":"10, and 11."},{"Start":"03:36.930 ","End":"03:44.520","Text":"What we can see is that if we have, let\u0027s say,"},{"Start":"03:44.520 ","End":"03:54.550","Text":"n antinodes, that means that we have n plus 1 nodes."},{"Start":"04:00.290 ","End":"04:02.930","Text":"This is something that\u0027s useful to"},{"Start":"04:02.930 ","End":"04:05.750","Text":"remember so that you don\u0027t always have to draw this out."},{"Start":"04:05.750 ","End":"04:10.820","Text":"Let\u0027s say I want to see how many antinodes I would have if I had,"},{"Start":"04:10.820 ","End":"04:13.620","Text":"let\u0027s say 6 nodes."},{"Start":"04:13.620 ","End":"04:15.345","Text":"Here\u0027s 1, 2,"},{"Start":"04:15.345 ","End":"04:16.635","Text":"3, 4,"},{"Start":"04:16.635 ","End":"04:20.505","Text":"5, 6 nodes would end over here."},{"Start":"04:20.505 ","End":"04:22.740","Text":"I can see that I have 1, 2, 3,"},{"Start":"04:22.740 ","End":"04:26.430","Text":"4, 5 antinodes."},{"Start":"04:26.430 ","End":"04:30.200","Text":"Now we know that n is the number of antinodes."},{"Start":"04:30.200 ","End":"04:35.090","Text":"I know that I have 11 antinodes."},{"Start":"04:35.090 ","End":"04:38.725","Text":"Now because I\u0027m trying to find the frequency,"},{"Start":"04:38.725 ","End":"04:44.645","Text":"I can plug in that the frequency is equal to n, which is 11,"},{"Start":"04:44.645 ","End":"04:46.940","Text":"multiplied by the velocity,"},{"Start":"04:46.940 ","End":"04:49.745","Text":"which is 25 meters per second,"},{"Start":"04:49.745 ","End":"04:53.040","Text":"divided by twice the length of the rope,"},{"Start":"04:53.040 ","End":"04:57.075","Text":"so 2 times 3 meters."},{"Start":"04:57.075 ","End":"05:01.295","Text":"If I plug that in to the calculator,"},{"Start":"05:01.295 ","End":"05:10.220","Text":"I\u0027ll get that that is equal to 45.8 Hertz."},{"Start":"05:10.220 ","End":"05:12.145","Text":"The answer is a."},{"Start":"05:12.145 ","End":"05:14.175","Text":"This is the answer."},{"Start":"05:14.175 ","End":"05:17.070","Text":"Remember this equation,"},{"Start":"05:17.070 ","End":"05:20.135","Text":"and also this over here."},{"Start":"05:20.135 ","End":"05:24.860","Text":"First of all, remember that n represents the number of antinodes,"},{"Start":"05:24.860 ","End":"05:26.810","Text":"so n is antinodes."},{"Start":"05:26.810 ","End":"05:29.750","Text":"Remember that if you have n antinodes,"},{"Start":"05:29.750 ","End":"05:34.150","Text":"that means you have n plus 1 nodes."},{"Start":"05:35.450 ","End":"05:38.920","Text":"That\u0027s the end of this lesson."}],"ID":12474},{"Watched":false,"Name":"Exercise - Finding Velocity","Duration":"3m 54s","ChapterTopicVideoID":12007,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"Hello. In this question, a string,"},{"Start":"00:02.910 ","End":"00:04.980","Text":"of length 90 centimeters,"},{"Start":"00:04.980 ","End":"00:07.965","Text":"is tied to posts at both ends."},{"Start":"00:07.965 ","End":"00:12.375","Text":"When the string is swung at a frequency of 150 hertz,"},{"Start":"00:12.375 ","End":"00:16.350","Text":"a standing wave with 8 nodes is formed."},{"Start":"00:16.350 ","End":"00:20.380","Text":"What is the velocity of the wave?"},{"Start":"00:20.750 ","End":"00:27.630","Text":"In this question, we\u0027re being asked to find the velocity and let\u0027s see what we have."},{"Start":"00:27.630 ","End":"00:35.595","Text":"The length of the string we\u0027re being told is 90 centimeters and we know"},{"Start":"00:35.595 ","End":"00:43.910","Text":"that the frequency that it\u0027s being swung is equal to 150 hertz."},{"Start":"00:43.910 ","End":"00:49.460","Text":"Another piece of information that we have is that the standing wave has 8 nodes."},{"Start":"00:49.460 ","End":"00:51.380","Text":"Now unless we\u0027re being told otherwise,"},{"Start":"00:51.380 ","End":"00:59.027","Text":"these 8 nodes include the 2 nodes that connect the string to the posts."},{"Start":"00:59.027 ","End":"01:03.110","Text":"As we learned, if we have a number of nodes,"},{"Start":"01:03.110 ","End":"01:07.425","Text":"then we have that number minus 1 anti-nodes,"},{"Start":"01:07.425 ","End":"01:09.530","Text":"which is what we need we always deal with"},{"Start":"01:09.530 ","End":"01:13.430","Text":"anti-nodes in our calculations with standing waves."},{"Start":"01:13.430 ","End":"01:19.035","Text":"If we have 8 nodes then 8 minus 1 is 7 anti-nodes,"},{"Start":"01:19.035 ","End":"01:21.300","Text":"so n, which is,"},{"Start":"01:21.300 ","End":"01:22.575","Text":"I\u0027m reminding you,"},{"Start":"01:22.575 ","End":"01:27.970","Text":"anti-nodes is equal to 7."},{"Start":"01:28.610 ","End":"01:31.025","Text":"Just as a reminder,"},{"Start":"01:31.025 ","End":"01:35.240","Text":"with the issue with the anti-nodes and the nodes so were being told that"},{"Start":"01:35.240 ","End":"01:43.942","Text":"we have 8 nodes so that\u0027s 1,2,"},{"Start":"01:43.942 ","End":"01:45.822","Text":"3, 4, 5, 6, 7,"},{"Start":"01:45.822 ","End":"01:48.770","Text":"8 and because we weren\u0027t told, otherwise,"},{"Start":"01:48.770 ","End":"01:52.410","Text":"these 8 nodes include"},{"Start":"01:52.410 ","End":"01:59.735","Text":"the 2 nodes at the end of the string and then if we draw in a wave,"},{"Start":"01:59.735 ","End":"02:02.105","Text":"imagine that I\u0027m drawing this."},{"Start":"02:02.105 ","End":"02:12.005","Text":"Well, so each peak is considered an anti-node, I\u0027m reminding you."},{"Start":"02:12.005 ","End":"02:17.285","Text":"Now we can count the number of anti-nodes so this is 1, this is 2,"},{"Start":"02:17.285 ","End":"02:20.520","Text":"this is 3, this is 4, this is 5,"},{"Start":"02:20.520 ","End":"02:24.165","Text":"6, and 7, 7 anti-nodes."},{"Start":"02:24.165 ","End":"02:28.460","Text":"The equation that we need to use is our usual equation for"},{"Start":"02:28.460 ","End":"02:33.755","Text":"our frequency that is dependent on the number of anti-nodes so we can add an n over here,"},{"Start":"02:33.755 ","End":"02:35.795","Text":"because our frequency with a standing wave"},{"Start":"02:35.795 ","End":"02:38.510","Text":"depends on the number of anti-nodes and that is"},{"Start":"02:38.510 ","End":"02:43.730","Text":"equal to the number of anti-nodes multiplied by the velocity,"},{"Start":"02:43.730 ","End":"02:45.320","Text":"which is what we\u0027re trying to find,"},{"Start":"02:45.320 ","End":"02:49.210","Text":"divided by twice the length of the string."},{"Start":"02:49.210 ","End":"02:53.595","Text":"Because we want to find the velocity let\u0027s isolate that"},{"Start":"02:53.595 ","End":"02:58.055","Text":"out and then we\u0027ll get that the velocity is equal"},{"Start":"02:58.055 ","End":"03:02.465","Text":"to 2L multiplied by"},{"Start":"03:02.465 ","End":"03:08.700","Text":"our frequency divided by the number of anti-nodes."},{"Start":"03:08.700 ","End":"03:11.255","Text":"Now all we have to do is we have to plug this in"},{"Start":"03:11.255 ","End":"03:14.090","Text":"so we\u0027ll have 2 multiply it by the length of the string,"},{"Start":"03:14.090 ","End":"03:16.115","Text":"which is 90 centimeters."},{"Start":"03:16.115 ","End":"03:18.110","Text":"I want to work in MKS,"},{"Start":"03:18.110 ","End":"03:26.930","Text":"so convert this to meters so 90 centimeters is 0.9 meters multiplied by our frequency,"},{"Start":"03:26.930 ","End":"03:29.510","Text":"which is a 150 Hertz,"},{"Start":"03:29.510 ","End":"03:34.510","Text":"divided by the number of anti-nodes which we worked out is equal to 7."},{"Start":"03:34.510 ","End":"03:36.995","Text":"Once we plug this into our calculator,"},{"Start":"03:36.995 ","End":"03:43.010","Text":"we\u0027ll get that the velocity of the standing wave in the string is equal to"},{"Start":"03:43.010 ","End":"03:50.700","Text":"38.57 meters per second."},{"Start":"03:50.770 ","End":"03:54.270","Text":"That\u0027s the end of our lesson."}],"ID":12475},{"Watched":false,"Name":"Exercise - Finding Velocity 2","Duration":"2m 27s","ChapterTopicVideoID":12008,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.415","Text":"Hello. In this lesson we\u0027re going to be answering this question."},{"Start":"00:04.415 ","End":"00:08.430","Text":"A rope is swung at a frequency of 100 Hertz."},{"Start":"00:08.430 ","End":"00:13.485","Text":"The rope is 3 meters long and a standing wave of 5 nodes is formed."},{"Start":"00:13.485 ","End":"00:16.020","Text":"What is the velocity of the wave?"},{"Start":"00:16.020 ","End":"00:18.900","Text":"We\u0027re trying to find the velocity of"},{"Start":"00:18.900 ","End":"00:22.305","Text":"the wave and we know that our equation for the velocity,"},{"Start":"00:22.305 ","End":"00:25.695","Text":"when you\u0027re dealing with a standing wave is equal to"},{"Start":"00:25.695 ","End":"00:30.735","Text":"2 times the length of the rope multiplied by the frequency,"},{"Start":"00:30.735 ","End":"00:32.850","Text":"which as we know, is dependent on the number of"},{"Start":"00:32.850 ","End":"00:38.475","Text":"antinodes divided by the number of antinodes."},{"Start":"00:38.475 ","End":"00:42.870","Text":"In this, we have to remember our antinodes and our nodes."},{"Start":"00:42.870 ","End":"00:52.235","Text":"Now we know from the question that our frequency is equal to 100 Hertz."},{"Start":"00:52.235 ","End":"00:56.585","Text":"We also know that the length of the rope is equal to"},{"Start":"00:56.585 ","End":"01:02.610","Text":"3 meters and we have a standing wave of 5 nodes."},{"Start":"01:02.610 ","End":"01:04.250","Text":"That means that n,"},{"Start":"01:04.250 ","End":"01:05.765","Text":"the number of antinodes,"},{"Start":"01:05.765 ","End":"01:09.890","Text":"is equal to 5 minus 1, which is 4."},{"Start":"01:09.890 ","End":"01:11.720","Text":"Let\u0027s just draw that out."},{"Start":"01:11.720 ","End":"01:18.380","Text":"Here are our 5 nodes and unless we\u0027re told otherwise,"},{"Start":"01:18.380 ","End":"01:25.690","Text":"these 5 nodes also include the 2 nodes that connect the rope to the poles."},{"Start":"01:25.690 ","End":"01:29.029","Text":"Now if we draw a nice standing wave,"},{"Start":"01:29.029 ","End":"01:32.670","Text":"we have a wave going like so."},{"Start":"01:32.800 ","End":"01:36.965","Text":"Now let\u0027s count how many antinodes there are."},{"Start":"01:36.965 ","End":"01:41.420","Text":"I\u0027m reminding you that the antinodes are where we have peaks."},{"Start":"01:41.420 ","End":"01:43.285","Text":"Here\u0027s 1 peak,"},{"Start":"01:43.285 ","End":"01:47.265","Text":"2 peaks, 3 peaks, and 4 peaks."},{"Start":"01:47.265 ","End":"01:49.615","Text":"We have 4 antinodes."},{"Start":"01:49.615 ","End":"01:54.005","Text":"Now, all that\u0027s left to do is to plug all of this into our equation."},{"Start":"01:54.005 ","End":"01:58.415","Text":"Velocity is equal to 2 multiplied by the length of the rope,"},{"Start":"01:58.415 ","End":"02:00.020","Text":"which is 3 meters,"},{"Start":"02:00.020 ","End":"02:01.955","Text":"multiplied by the frequency,"},{"Start":"02:01.955 ","End":"02:03.740","Text":"which is 100 Hertz,"},{"Start":"02:03.740 ","End":"02:07.695","Text":"divided by the number of antinodes, which is 4."},{"Start":"02:07.695 ","End":"02:10.595","Text":"Once we plug all of this into the calculator,"},{"Start":"02:10.595 ","End":"02:14.300","Text":"we\u0027ll get that the velocity of the wave propagating through"},{"Start":"02:14.300 ","End":"02:20.240","Text":"the rope is equal to 150 meters per second."},{"Start":"02:20.990 ","End":"02:24.125","Text":"This is the answer to our question,"},{"Start":"02:24.125 ","End":"02:27.360","Text":"and this is the end of this lesson."}],"ID":12476},{"Watched":false,"Name":"Exercise - Guitar String","Duration":"3m 48s","ChapterTopicVideoID":12009,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.495","Text":"Hello. A guitar string plays a note which has a frequency of 300 hertz."},{"Start":"00:06.495 ","End":"00:13.020","Text":"We want to change the pitch or the frequency to now be at 900 hertz."},{"Start":"00:13.020 ","End":"00:15.090","Text":"How can this be done?"},{"Start":"00:15.090 ","End":"00:20.850","Text":"First of all we know that our equation for frequency, fn,"},{"Start":"00:20.850 ","End":"00:24.435","Text":"is equal to the number of anti-nodes"},{"Start":"00:24.435 ","End":"00:29.445","Text":"multiplied by the velocity of the wave propagating through,"},{"Start":"00:29.445 ","End":"00:33.990","Text":"divided by twice the length of the rope."},{"Start":"00:33.990 ","End":"00:44.175","Text":"We want our new frequency to be 3 times our original frequency."},{"Start":"00:44.175 ","End":"00:47.360","Text":"So fn final."},{"Start":"00:47.360 ","End":"00:54.135","Text":"Because 300 hertz multiplied by 3 is 900 hertz."},{"Start":"00:54.135 ","End":"00:56.910","Text":"We want to triple our fn,"},{"Start":"00:56.910 ","End":"00:59.205","Text":"let\u0027s see what we can do."},{"Start":"00:59.205 ","End":"01:06.860","Text":"Our n, our number of anti-nodes can change because we\u0027re playing a specific note,"},{"Start":"01:06.860 ","End":"01:11.200","Text":"so n will remain constant."},{"Start":"01:11.200 ","End":"01:13.490","Text":"What can we change?"},{"Start":"01:13.490 ","End":"01:23.290","Text":"We can change our velocity in the string and we can change the length of the string."},{"Start":"01:23.290 ","End":"01:29.950","Text":"Let\u0027s first look at the idea of changing the length."},{"Start":"01:29.950 ","End":"01:34.610","Text":"We can see that the length is located in the denominator."},{"Start":"01:34.610 ","End":"01:44.180","Text":"That means that if we divide the denominator by 3,"},{"Start":"01:44.180 ","End":"01:48.020","Text":"we\u0027ll be multiplying this whole value by 3."},{"Start":"01:48.020 ","End":"01:54.155","Text":"Meaning that we will increase our frequency by a factor of 3,"},{"Start":"01:54.155 ","End":"01:56.270","Text":"which is exactly what we want."},{"Start":"01:56.270 ","End":"02:06.710","Text":"If our length final is equal to 1/3 our initial length,"},{"Start":"02:06.710 ","End":"02:12.155","Text":"then we will manage to get this 900 hertz frequency."},{"Start":"02:12.155 ","End":"02:15.985","Text":"Then, what about velocity?"},{"Start":"02:15.985 ","End":"02:20.030","Text":"As we know an equation for the velocity is equal"},{"Start":"02:20.030 ","End":"02:24.275","Text":"to the square root of the tension in the string,"},{"Start":"02:24.275 ","End":"02:28.055","Text":"divided by the string\u0027s density."},{"Start":"02:28.055 ","End":"02:29.780","Text":"The density of course,"},{"Start":"02:29.780 ","End":"02:32.150","Text":"we can\u0027t change unless we\u0027re switching strings,"},{"Start":"02:32.150 ","End":"02:34.745","Text":"but we\u0027re assuming that we\u0027re using the same string."},{"Start":"02:34.745 ","End":"02:38.750","Text":"What we can do is, we can increase the tension."},{"Start":"02:38.750 ","End":"02:42.950","Text":"We can see that the velocity is located in the numerator,"},{"Start":"02:42.950 ","End":"02:45.755","Text":"so all we have to do is increase the tension."},{"Start":"02:45.755 ","End":"02:50.070","Text":"But because the tension is located inside the square root sign,"},{"Start":"02:50.070 ","End":"02:55.340","Text":"in order to increase our value of velocity by a factor of 3,"},{"Start":"02:55.340 ","End":"03:00.400","Text":"that means we have to increase our tension by a factor of 9."},{"Start":"03:00.400 ","End":"03:02.700","Text":"If we have 9 Tau,"},{"Start":"03:02.700 ","End":"03:07.425","Text":"then the square root of 9 Tau will be 3 root Tau,"},{"Start":"03:07.425 ","End":"03:10.270","Text":"where Tau is the tension."},{"Start":"03:11.240 ","End":"03:21.390","Text":"If our new tension or our final tension is equal to 9 times our initial tension,"},{"Start":"03:21.390 ","End":"03:30.425","Text":"then we will get an increase by a factor of 3 for this value over here."},{"Start":"03:30.425 ","End":"03:33.485","Text":"This is the answer to this question."},{"Start":"03:33.485 ","End":"03:38.945","Text":"Either we decrease or we shorten the length of the string by a factor of"},{"Start":"03:38.945 ","End":"03:45.845","Text":"3 or we increase the tension in the string by a factor of 9."},{"Start":"03:45.845 ","End":"03:48.720","Text":"That\u0027s the end of this lesson."}],"ID":12477},{"Watched":false,"Name":"Two Dimensional Waves 1","Duration":"9m 38s","ChapterTopicVideoID":12010,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.010","Text":"Hello. In this lesson,"},{"Start":"00:02.010 ","End":"00:04.830","Text":"we\u0027re going to be speaking about 2-dimensional waves."},{"Start":"00:04.830 ","End":"00:06.300","Text":"Now in previous lessons,"},{"Start":"00:06.300 ","End":"00:08.280","Text":"we were speaking about 1-dimensional waves."},{"Start":"00:08.280 ","End":"00:11.760","Text":"The reason that we were talking about that is because we were"},{"Start":"00:11.760 ","End":"00:15.810","Text":"doing a preparation to when we\u0027re going to speak about 3-dimensional waves,"},{"Start":"00:15.810 ","End":"00:18.810","Text":"which is specifically dealing with light."},{"Start":"00:18.810 ","End":"00:21.985","Text":"We will see that light has wave-like properties."},{"Start":"00:21.985 ","End":"00:25.020","Text":"Now we\u0027re going to speak about 2-dimensional waves,"},{"Start":"00:25.020 ","End":"00:28.110","Text":"which is again going to give us further understanding and prepare"},{"Start":"00:28.110 ","End":"00:31.860","Text":"us for when we speak about waves in 3D."},{"Start":"00:31.860 ","End":"00:34.515","Text":"In order to speak about 2-dimensional waves,"},{"Start":"00:34.515 ","End":"00:38.020","Text":"we\u0027re going to speak about waves in water."},{"Start":"00:38.150 ","End":"00:40.410","Text":"How are we going to do that?"},{"Start":"00:40.410 ","End":"00:43.580","Text":"If we take some shallow basin."},{"Start":"00:43.580 ","End":"00:45.920","Text":"Let\u0027s draw it over here."},{"Start":"00:45.920 ","End":"00:55.305","Text":"Here we have our shallow basin and we fill it inside with water."},{"Start":"00:55.305 ","End":"01:00.830","Text":"All of this is water. What we\u0027ll see is that we can make waves which will only"},{"Start":"01:00.830 ","End":"01:06.500","Text":"travel on the surface of the water and they\u0027ll travel either in this direction,"},{"Start":"01:06.500 ","End":"01:12.395","Text":"in the x direction or in this direction in the y direction."},{"Start":"01:12.395 ","End":"01:15.770","Text":"We can see that there are 2 dimensions."},{"Start":"01:15.770 ","End":"01:18.950","Text":"We\u0027re not dealing with 3-dimensions because the water isn\u0027t"},{"Start":"01:18.950 ","End":"01:24.065","Text":"moving up or down in the z direction."},{"Start":"01:24.065 ","End":"01:28.340","Text":"This has water waves moving in 2 dimensions."},{"Start":"01:28.340 ","End":"01:30.155","Text":"Now, light waves,"},{"Start":"01:30.155 ","End":"01:34.370","Text":"if we have a bulb that is emitting light,"},{"Start":"01:34.370 ","End":"01:38.045","Text":"we know that the light goes out radially,"},{"Start":"01:38.045 ","End":"01:40.730","Text":"which means that light is in 3-dimensions."},{"Start":"01:40.730 ","End":"01:44.760","Text":"That we will speak about a bit later in this course."},{"Start":"01:45.500 ","End":"01:49.760","Text":"What we\u0027re going to speak about now is the different types of"},{"Start":"01:49.760 ","End":"01:54.170","Text":"2-dimensional waves that we can form in this water basin."},{"Start":"01:54.170 ","End":"01:55.505","Text":"Now before we do that,"},{"Start":"01:55.505 ","End":"02:00.620","Text":"let\u0027s just learn some new terms that will help us to explain what is going on."},{"Start":"02:00.620 ","End":"02:05.140","Text":"The first term that we\u0027re going to learn about is called the wavefront."},{"Start":"02:05.140 ","End":"02:13.295","Text":"The wavefront is the set of points of the wave that reach their amplitude simultaneously."},{"Start":"02:13.295 ","End":"02:17.270","Text":"Remember that amplitude can be in the positive direction."},{"Start":"02:17.270 ","End":"02:20.615","Text":"If we have a wave like so,"},{"Start":"02:20.615 ","End":"02:22.090","Text":"this is the amplitude,"},{"Start":"02:22.090 ","End":"02:25.640","Text":"but this is also the amplitude is in the negative direction"},{"Start":"02:25.640 ","End":"02:30.480","Text":"because they have the same magnitude."},{"Start":"02:31.030 ","End":"02:36.230","Text":"If we draw a basin from a bird\u0027s eye view,"},{"Start":"02:36.230 ","End":"02:40.865","Text":"we\u0027re just looking at the basin from up top."},{"Start":"02:40.865 ","End":"02:46.815","Text":"If we start a wave from this point over here,"},{"Start":"02:46.815 ","End":"02:49.265","Text":"we\u0027ll see that a few moments later,"},{"Start":"02:49.265 ","End":"02:53.745","Text":"the amplitude of the wave will be over here."},{"Start":"02:53.745 ","End":"02:57.470","Text":"All of these points represent the wavefront."},{"Start":"02:57.470 ","End":"02:58.880","Text":"Because they\u0027re all of these points,"},{"Start":"02:58.880 ","End":"03:03.170","Text":"a set of points of the wave that reach their amplitude simultaneously."},{"Start":"03:03.170 ","End":"03:07.310","Text":"All of these points are right now at their amplitude at the same time."},{"Start":"03:07.310 ","End":"03:09.545","Text":"A few moments later,"},{"Start":"03:09.545 ","End":"03:12.245","Text":"the amplitude will be over here."},{"Start":"03:12.245 ","End":"03:15.590","Text":"All of these points represent the wavefront because they\u0027re"},{"Start":"03:15.590 ","End":"03:20.060","Text":"the new set of points of the wave that reach the amplitudes simultaneously."},{"Start":"03:20.060 ","End":"03:22.280","Text":"Then a few moments later,"},{"Start":"03:22.280 ","End":"03:24.035","Text":"this is the wavefront,"},{"Start":"03:24.035 ","End":"03:26.645","Text":"and then this is the wavefront,"},{"Start":"03:26.645 ","End":"03:29.610","Text":"and then this is the wavefront."},{"Start":"03:29.870 ","End":"03:35.155","Text":"Now, this example over here is the example of a straight wavefront,"},{"Start":"03:35.155 ","End":"03:40.565","Text":"which we could create by dropping some rod straight into the water."},{"Start":"03:40.565 ","End":"03:46.190","Text":"Another type of wavefront that we can create is a circular wavefront."},{"Start":"03:46.440 ","End":"03:50.245","Text":"When creating a circular type of wavefront,"},{"Start":"03:50.245 ","End":"03:56.190","Text":"we can take some points like the end of the rod and just dip in the end of the rod."},{"Start":"03:56.190 ","End":"03:59.395","Text":"Here we had the rod lengthwise which was dropped in."},{"Start":"03:59.395 ","End":"04:03.705","Text":"Here, we\u0027re putting just the tip of the rod over here."},{"Start":"04:03.705 ","End":"04:07.290","Text":"We\u0027re going to get a circular wavefront."},{"Start":"04:07.290 ","End":"04:12.930","Text":"We\u0027ll see that a few moments later a wavefront will be over here."},{"Start":"04:12.930 ","End":"04:14.645","Text":"These are all the points,"},{"Start":"04:14.645 ","End":"04:18.860","Text":"the set of points of the wave that reach their amplitude simultaneously."},{"Start":"04:18.860 ","End":"04:24.320","Text":"Then the wavefront a few moments later will be like this."},{"Start":"04:24.320 ","End":"04:27.420","Text":"Imagine that all of these circles,"},{"Start":"04:28.750 ","End":"04:35.195","Text":"each point is equidistant from the center over here from the origin."},{"Start":"04:35.195 ","End":"04:37.550","Text":"Then a few moments later,"},{"Start":"04:37.550 ","End":"04:41.465","Text":"the circular wavefront,"},{"Start":"04:41.465 ","End":"04:45.070","Text":"will be over here like so."},{"Start":"04:46.430 ","End":"04:50.055","Text":"Imagine that this is a perfect circle."},{"Start":"04:50.055 ","End":"04:56.740","Text":"Each one of these dotted cycles is a wavefront."},{"Start":"04:56.960 ","End":"05:01.095","Text":"The next term is the wavevector."},{"Start":"05:01.095 ","End":"05:04.355","Text":"We\u0027re not going to go into too much detail about the wavevector."},{"Start":"05:04.355 ","End":"05:10.395","Text":"But the wavevector is simply the vector which describes the specific wave."},{"Start":"05:10.395 ","End":"05:13.790","Text":"We\u0027re just going to focus on it being just a vector,"},{"Start":"05:13.790 ","End":"05:19.765","Text":"which means it has size and direction like all other normal vectors."},{"Start":"05:19.765 ","End":"05:23.615","Text":"The direction of the wavevector"},{"Start":"05:23.615 ","End":"05:29.395","Text":"simply describes the direction that the wavefront will travel in."},{"Start":"05:29.395 ","End":"05:32.600","Text":"How do we know that our wavefront will be here after"},{"Start":"05:32.600 ","End":"05:35.690","Text":"a few moments and then here and then here and then here and here."},{"Start":"05:35.690 ","End":"05:37.400","Text":"In the circular wavefront,"},{"Start":"05:37.400 ","End":"05:40.235","Text":"how do we know that our first cycle will be here,"},{"Start":"05:40.235 ","End":"05:41.660","Text":"our next one here,"},{"Start":"05:41.660 ","End":"05:44.050","Text":"and so on and so forth."},{"Start":"05:44.050 ","End":"05:49.220","Text":"The direction that a wavefront will travel is given by the direction of the wavevector,"},{"Start":"05:49.220 ","End":"05:54.580","Text":"and it\u0027s always in the direction normal to the wavefront."},{"Start":"05:54.580 ","End":"05:58.100","Text":"Let\u0027s take a look over here on the street wavefront,"},{"Start":"05:58.100 ","End":"06:02.210","Text":"and I\u0027ll draw the direction of the wavevector in red."},{"Start":"06:02.210 ","End":"06:07.130","Text":"Here, we plunged our entire rod into our water basin."},{"Start":"06:07.130 ","End":"06:11.849","Text":"The direction normal to this interference that we\u0027ve created,"},{"Start":"06:11.849 ","End":"06:14.910","Text":"so normal, I\u0027m reminding us that 90 degrees."},{"Start":"06:14.910 ","End":"06:18.280","Text":"That\u0027s going to be in this direction."},{"Start":"06:20.300 ","End":"06:26.650","Text":"Of course also in the opposite direction, like so."},{"Start":"06:26.650 ","End":"06:30.350","Text":"That means that our wavefront is going to be traveling in"},{"Start":"06:30.350 ","End":"06:35.340","Text":"a straight line along this positive x-direction."},{"Start":"06:35.530 ","End":"06:39.120","Text":"Our wavefront will reach here."},{"Start":"06:39.120 ","End":"06:43.670","Text":"The direction that our wave will carry on traveling in is like"},{"Start":"06:43.670 ","End":"06:50.020","Text":"so because it\u0027s in the direction normal to the wavefront at 90 degrees to the wavefront."},{"Start":"06:50.020 ","End":"06:53.445","Text":"Similarly here, in our circular wavefront."},{"Start":"06:53.445 ","End":"06:59.550","Text":"We had some points that we plunged into the water and it caused a disturbance."},{"Start":"07:01.670 ","End":"07:05.240","Text":"The direction that our wavefront will travel is in"},{"Start":"07:05.240 ","End":"07:08.405","Text":"the direction normal to the wavefront,"},{"Start":"07:08.405 ","End":"07:11.255","Text":"which at the beginning started off as this interference."},{"Start":"07:11.255 ","End":"07:17.580","Text":"Normal to a point is just going to be out in the radial direction."},{"Start":"07:17.900 ","End":"07:25.400","Text":"We can imagine lots of arrows going out like this and the radial direction like so."},{"Start":"07:25.400 ","End":"07:28.070","Text":"Now we can see that the wavefront will move out over"},{"Start":"07:28.070 ","End":"07:31.820","Text":"here because these are all normal to the point."},{"Start":"07:31.820 ","End":"07:34.380","Text":"The wavefront will be over here."},{"Start":"07:34.380 ","End":"07:40.370","Text":"Then again, our wavefront will carry on moving in the direction normal."},{"Start":"07:40.370 ","End":"07:46.610","Text":"Each point, these arrows are normal to each one of these points."},{"Start":"07:46.610 ","End":"07:55.190","Text":"Then we can see that the wavefront will move out to this second dotted line over here."},{"Start":"07:55.190 ","End":"08:05.790","Text":"Again normal to that is out again in the radial direction, like so."},{"Start":"08:06.050 ","End":"08:08.285","Text":"That\u0027s how we know"},{"Start":"08:08.285 ","End":"08:12.560","Text":"what the direction of the wavevector will be or the direction"},{"Start":"08:12.560 ","End":"08:18.010","Text":"of the propagation of the amplitude of the wave."},{"Start":"08:18.200 ","End":"08:22.190","Text":"Another way with a secular wavefront is to draw"},{"Start":"08:22.190 ","End":"08:25.660","Text":"the tangent to the circle at a specific point."},{"Start":"08:25.660 ","End":"08:30.845","Text":"That\u0027s a straight line which just touches one of the points on the circle."},{"Start":"08:30.845 ","End":"08:35.560","Text":"Then we draw an arrow which is perpendicular to the tangent."},{"Start":"08:35.560 ","End":"08:42.410","Text":"That would be here. Then if we keep doing this along different points of the circle,"},{"Start":"08:42.410 ","End":"08:45.095","Text":"we draw the tangent at that point."},{"Start":"08:45.095 ","End":"08:50.950","Text":"That can also help to draw the direction of the wavevector."},{"Start":"08:50.950 ","End":"08:57.905","Text":"It\u0027s always going to be perpendicular to this tangent to the circle at a specific point."},{"Start":"08:57.905 ","End":"09:00.470","Text":"We could have done this anywhere over here."},{"Start":"09:00.470 ","End":"09:07.730","Text":"That means that this red vector is also perpendicular to this point over here,"},{"Start":"09:07.730 ","End":"09:10.445","Text":"which we drew the tangent for."},{"Start":"09:10.445 ","End":"09:13.190","Text":"We learned the term wavefront."},{"Start":"09:13.190 ","End":"09:15.304","Text":"We learned the term wavevector,"},{"Start":"09:15.304 ","End":"09:17.960","Text":"which is just the vector which describes the wave."},{"Start":"09:17.960 ","End":"09:23.600","Text":"What\u0027s important to remember is that like any vector has size and direction."},{"Start":"09:23.600 ","End":"09:27.950","Text":"Then we learned about the direction that the wavevector will point,"},{"Start":"09:27.950 ","End":"09:35.165","Text":"which will allow us to know in which direction our wavefront is going to be traveling."},{"Start":"09:35.165 ","End":"09:38.130","Text":"That\u0027s the end of this lesson."}],"ID":12478},{"Watched":false,"Name":"Wave Tanks","Duration":"21m 49s","ChapterTopicVideoID":12011,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"Hello. In the previous lesson,"},{"Start":"00:02.040 ","End":"00:04.530","Text":"we started speaking about two-dimensional waves,"},{"Start":"00:04.530 ","End":"00:06.975","Text":"and specifically about water waves."},{"Start":"00:06.975 ","End":"00:12.548","Text":"Water waves are generated in wave tanks or wave basins,"},{"Start":"00:12.548 ","End":"00:14.295","Text":"and we can observe them."},{"Start":"00:14.295 ","End":"00:18.225","Text":"We have to observe them in a very specific way in"},{"Start":"00:18.225 ","End":"00:22.140","Text":"order to get accurate results and accurate conclusions."},{"Start":"00:22.140 ","End":"00:25.725","Text":"In this lesson we\u0027re going to see how wave tanks"},{"Start":"00:25.725 ","End":"00:30.504","Text":"work in order for us to understand the two-dimensional wave better,"},{"Start":"00:30.504 ","End":"00:33.195","Text":"and then in lessons that will come,"},{"Start":"00:33.195 ","End":"00:39.655","Text":"it will be easier for us to understand how these calculations and observations are made."},{"Start":"00:39.655 ","End":"00:47.225","Text":"Here is a picture of a wave tank or a ripple tank or a wave basin."},{"Start":"00:47.225 ","End":"00:50.345","Text":"Now we\u0027re going to see how this works."},{"Start":"00:50.345 ","End":"00:53.735","Text":"The first thing that we have,"},{"Start":"00:53.735 ","End":"00:57.810","Text":"let\u0027s mark it in green,"},{"Start":"00:57.810 ","End":"01:02.945","Text":"this type of frame over here which is labeled with a sponge beach."},{"Start":"01:02.945 ","End":"01:08.630","Text":"We have this square frame going like so."},{"Start":"01:08.630 ","End":"01:11.900","Text":"Inside this square frame,"},{"Start":"01:11.900 ","End":"01:13.850","Text":"and obviously it goes all around,"},{"Start":"01:13.850 ","End":"01:23.730","Text":"we have approximately 2-3 millimeters of water."},{"Start":"01:27.700 ","End":"01:31.685","Text":"Now we have this straight wave dipper,"},{"Start":"01:31.685 ","End":"01:33.920","Text":"which is this over here."},{"Start":"01:33.920 ","End":"01:37.415","Text":"This over here in blue,"},{"Start":"01:37.415 ","End":"01:40.820","Text":"I\u0027ve just traced out, is the straight wave dipper."},{"Start":"01:40.820 ","End":"01:45.445","Text":"This is what causes or generates the wave."},{"Start":"01:45.445 ","End":"01:50.075","Text":"As we can see, it\u0027s connected to these rubber bands,"},{"Start":"01:50.075 ","End":"01:54.095","Text":"which I connected to some kind of electricity."},{"Start":"01:54.095 ","End":"01:58.385","Text":"What happens is that this contraption"},{"Start":"01:58.385 ","End":"02:03.535","Text":"lifts and drops back down this straight wave dipper."},{"Start":"02:03.535 ","End":"02:07.600","Text":"This will generate straight waves."},{"Start":"02:07.600 ","End":"02:09.520","Text":"It goes up and down, up and down,"},{"Start":"02:09.520 ","End":"02:11.990","Text":"up and down, this whole contraption over here."},{"Start":"02:11.990 ","End":"02:14.055","Text":"Then waves are generated."},{"Start":"02:14.055 ","End":"02:18.175","Text":"Then, because we\u0027re dealing with a straight wave dipper specifically over here,"},{"Start":"02:18.175 ","End":"02:20.560","Text":"so we can see that few moments later,"},{"Start":"02:20.560 ","End":"02:23.350","Text":"wavefront will move over here."},{"Start":"02:23.350 ","End":"02:25.345","Text":"Then a few moments later,"},{"Start":"02:25.345 ","End":"02:28.795","Text":"a wavefront will travel over here,"},{"Start":"02:28.795 ","End":"02:30.835","Text":"and so on and so forth."},{"Start":"02:30.835 ","End":"02:34.865","Text":"As we can see, we have these little bubbles over here."},{"Start":"02:34.865 ","End":"02:39.235","Text":"If we push these down instead of the straight wave dipper,"},{"Start":"02:39.235 ","End":"02:41.440","Text":"then we\u0027ll get circular waves,"},{"Start":"02:41.440 ","End":"02:43.540","Text":"which is what we saw in the previous lesson,"},{"Start":"02:43.540 ","End":"02:49.500","Text":"where we have circular waves and then we have circular wave fronts."},{"Start":"02:49.670 ","End":"02:56.735","Text":"Next, what we have is this blank white sheet of paper."},{"Start":"02:56.735 ","End":"02:58.535","Text":"I know it has stripes on it right now,"},{"Start":"02:58.535 ","End":"03:00.695","Text":"but ignore that for the meantime."},{"Start":"03:00.695 ","End":"03:02.515","Text":"We\u0027ll speak about it later."},{"Start":"03:02.515 ","End":"03:05.210","Text":"That\u0027s just the way friend pattern that we\u0027ll see,"},{"Start":"03:05.210 ","End":"03:07.160","Text":"but we\u0027ll explain this soon."},{"Start":"03:07.160 ","End":"03:14.500","Text":"This, what we have in this red square over here is a white screen."},{"Start":"03:14.500 ","End":"03:16.650","Text":"Let\u0027s just label that."},{"Start":"03:16.650 ","End":"03:19.449","Text":"This is the white screen."},{"Start":"03:20.150 ","End":"03:24.230","Text":"Here we have just a white screen over here."},{"Start":"03:24.230 ","End":"03:27.395","Text":"Then we can see that here we have a lamp house."},{"Start":"03:27.395 ","End":"03:31.105","Text":"Let\u0027s just highlight this in yellow."},{"Start":"03:31.105 ","End":"03:33.035","Text":"Here we have our lamp house,"},{"Start":"03:33.035 ","End":"03:35.270","Text":"and that, of course, what does it do?"},{"Start":"03:35.270 ","End":"03:39.610","Text":"It shines light onto our water tank."},{"Start":"03:39.610 ","End":"03:44.860","Text":"This basin with our 2-3 millimeters of water inside,"},{"Start":"03:44.860 ","End":"03:48.035","Text":"and then we can see all of the wave fronts."},{"Start":"03:48.035 ","End":"03:49.700","Text":"Here there are straight wavefront,"},{"Start":"03:49.700 ","End":"03:53.150","Text":"so as we said, it looks something like this."},{"Start":"03:53.150 ","End":"03:58.958","Text":"Then because of the light shining directly onto this water tank,"},{"Start":"03:58.958 ","End":"04:03.800","Text":"so then we\u0027ll see the wave pattern on the white screen."},{"Start":"04:03.800 ","End":"04:05.600","Text":"This is these lines."},{"Start":"04:05.600 ","End":"04:09.770","Text":"We see the shadow of the peaks of"},{"Start":"04:09.770 ","End":"04:14.780","Text":"the wavefront or the shadow of the wavefront over here on our white screen."},{"Start":"04:14.780 ","End":"04:20.790","Text":"These lines are the shadows of these dotted lines over here."},{"Start":"04:22.790 ","End":"04:30.020","Text":"Now what we want to do is we want to know how the shadows are formed exactly."},{"Start":"04:30.020 ","End":"04:35.230","Text":"We know that light is hitting the wavefronts or the water in general."},{"Start":"04:35.230 ","End":"04:37.470","Text":"Then we see these shadows."},{"Start":"04:37.470 ","End":"04:39.410","Text":"Why exactly is that?"},{"Start":"04:39.410 ","End":"04:43.055","Text":"What are we going to do now is we\u0027re going to look at"},{"Start":"04:43.055 ","End":"04:48.095","Text":"the first interference created by the straight wave differ."},{"Start":"04:48.095 ","End":"04:50.220","Text":"So this in blue."},{"Start":"04:50.320 ","End":"04:54.445","Text":"Let\u0027s imagine that here is"},{"Start":"04:54.445 ","End":"05:00.610","Text":"our eye and we\u0027re looking at what is going on over here. This is an eye."},{"Start":"05:00.610 ","End":"05:05.980","Text":"Let\u0027s see what happens. A straight wave dipper goes down and up."},{"Start":"05:05.980 ","End":"05:10.270","Text":"Then we will see that we\u0027ll get some interference,"},{"Start":"05:10.270 ","End":"05:12.175","Text":"a wave like so."},{"Start":"05:12.175 ","End":"05:17.140","Text":"Now we saw that we have light waves."},{"Start":"05:17.140 ","End":"05:20.110","Text":"Let\u0027s draw the light waves in blue so that it\u0027s clearer."},{"Start":"05:20.110 ","End":"05:27.790","Text":"We have our light waves from our lamphouse coming down like so,"},{"Start":"05:27.790 ","End":"05:32.570","Text":"straight down towards the wave."},{"Start":"05:32.700 ","End":"05:40.030","Text":"Now let\u0027s draw the direction of the light waves as they hit the water."},{"Start":"05:40.030 ","End":"05:45.220","Text":"This light wave is going to carry on and then it\u0027s going to hit the wave over here."},{"Start":"05:45.220 ","End":"05:52.450","Text":"We can see that it\u0027s angle when it comes into contact with this water wave, is 0."},{"Start":"05:52.450 ","End":"05:57.475","Text":"It\u0027s just going to carry on straight over to here,"},{"Start":"05:57.475 ","End":"06:01.645","Text":"and let\u0027s just draw over here."},{"Start":"06:01.645 ","End":"06:05.150","Text":"This is our white screen in red."},{"Start":"06:05.880 ","End":"06:09.850","Text":"This light wave, which the light wave I\u0027ve drawn"},{"Start":"06:09.850 ","End":"06:12.880","Text":"in blue and the black by these water waves."},{"Start":"06:12.880 ","End":"06:15.760","Text":"I\u0027ve mixed up the colors a bit, but never mind."},{"Start":"06:15.760 ","End":"06:20.405","Text":"Then the light wave will reach this point over here."},{"Start":"06:20.405 ","End":"06:25.700","Text":"But now let\u0027s look at this wave that\u0027s slightly more over here."},{"Start":"06:25.700 ","End":"06:32.120","Text":"It will carry on and hit our water wave over here at this point."},{"Start":"06:32.120 ","End":"06:35.990","Text":"What we can see is that the angle that our light wave"},{"Start":"06:35.990 ","End":"06:41.820","Text":"comes into contact with our water wave is not an angle of 0."},{"Start":"06:41.990 ","End":"06:45.650","Text":"We learned in optical geometry,"},{"Start":"06:45.650 ","End":"06:47.720","Text":"which was a few chapters ago,"},{"Start":"06:47.720 ","End":"06:54.110","Text":"what happens when a light wave travels between a less dense medium of air"},{"Start":"06:54.110 ","End":"07:00.440","Text":"over here into a more dense medium of water over here."},{"Start":"07:00.440 ","End":"07:07.085","Text":"We learned that we have our normal to the wave at this point,"},{"Start":"07:07.085 ","End":"07:09.871","Text":"the normal to the water wave,"},{"Start":"07:09.871 ","End":"07:12.835","Text":"so it\u0027s at 90 degrees to the water wave."},{"Start":"07:12.835 ","End":"07:18.830","Text":"We learned that when a wave goes from a less dense medium to a more dense medium,"},{"Start":"07:18.830 ","End":"07:23.240","Text":"the direction of travel of our light wave is going"},{"Start":"07:23.240 ","End":"07:27.755","Text":"to change and it\u0027s going to move closer."},{"Start":"07:27.755 ","End":"07:32.330","Text":"It\u0027s going to get closer to this normal line over here."},{"Start":"07:32.330 ","End":"07:36.530","Text":"That means that instead of traveling in a straight line,"},{"Start":"07:36.530 ","End":"07:41.195","Text":"it\u0027s going to get closer to the normal wave,"},{"Start":"07:41.195 ","End":"07:49.080","Text":"which means it will carry on traveling until it as well reaches this point over here."},{"Start":"07:50.350 ","End":"07:55.550","Text":"Now, what about the light wave that\u0027s coming from here?"},{"Start":"07:55.550 ","End":"07:57.755","Text":"It\u0027s going to carry on, carry on."},{"Start":"07:57.755 ","End":"08:01.505","Text":"Again, it\u0027s going to hit the water wave."},{"Start":"08:01.505 ","End":"08:05.330","Text":"It\u0027s again going to have some angle."},{"Start":"08:05.330 ","End":"08:09.295","Text":"It\u0027s not hitting at exactly 90 degrees."},{"Start":"08:09.295 ","End":"08:14.270","Text":"Again, we\u0027ll draw the normal line to this point over here,"},{"Start":"08:14.270 ","End":"08:17.215","Text":"which goes approximately like this."},{"Start":"08:17.215 ","End":"08:19.010","Text":"This is the normal line."},{"Start":"08:19.010 ","End":"08:21.770","Text":"Again, we\u0027re going from a less dense medium of"},{"Start":"08:21.770 ","End":"08:25.250","Text":"air through to a more dense medium of water."},{"Start":"08:25.250 ","End":"08:33.070","Text":"That means that our light ray is going to bend slightly towards the normal."},{"Start":"08:33.070 ","End":"08:35.315","Text":"Instead of traveling straight down like this,"},{"Start":"08:35.315 ","End":"08:41.105","Text":"it\u0027s going to bend so that its angle with the normal is going to be smaller."},{"Start":"08:41.105 ","End":"08:48.065","Text":"Then it bends inwards and it travels again to this point over here."},{"Start":"08:48.065 ","End":"08:53.915","Text":"What we\u0027re going to see is that we\u0027re going to have a light patch under,"},{"Start":"08:53.915 ","End":"08:58.355","Text":"where our amplitude of the water wave is."},{"Start":"08:58.355 ","End":"09:01.250","Text":"Here we see the amplitude of the water wave."},{"Start":"09:01.250 ","End":"09:08.730","Text":"Over here we\u0027ll have a light patch right directly underneath."},{"Start":"09:09.670 ","End":"09:19.505","Text":"What we can say is that this amplitude over here in the water wave acts like a lens."},{"Start":"09:19.505 ","End":"09:23.015","Text":"It concentrates the light below it,"},{"Start":"09:23.015 ","End":"09:26.880","Text":"meaning that there\u0027ll be a light patch over here."},{"Start":"09:28.090 ","End":"09:38.720","Text":"If we look at the light patch made from this first wave front,"},{"Start":"09:38.720 ","End":"09:42.365","Text":"that will correspond to a light patch over here."},{"Start":"09:42.365 ","End":"09:46.310","Text":"So imagine that this line is not a shadow,"},{"Start":"09:46.310 ","End":"09:47.930","Text":"but rather a patch of light,"},{"Start":"09:47.930 ","End":"09:52.110","Text":"and the white is the shadow section."},{"Start":"09:52.590 ","End":"09:56.695","Text":"Then this wavefront that we have over here,"},{"Start":"09:56.695 ","End":"09:58.525","Text":"the same thing will happen."},{"Start":"09:58.525 ","End":"10:00.159","Text":"Where we have the wavefront,"},{"Start":"10:00.159 ","End":"10:03.970","Text":"which is at the peak or the amplitude of the water wave,"},{"Start":"10:03.970 ","End":"10:06.085","Text":"will have a light patch underneath."},{"Start":"10:06.085 ","End":"10:10.660","Text":"Then we can consider this line over here as a line of"},{"Start":"10:10.660 ","End":"10:15.925","Text":"light that corresponds to this wavefront"},{"Start":"10:15.925 ","End":"10:19.720","Text":"over here and then this wavefront over here will"},{"Start":"10:19.720 ","End":"10:25.840","Text":"correspond to this patch of light or line of light over here,"},{"Start":"10:25.840 ","End":"10:28.225","Text":"and so on and so forth."},{"Start":"10:28.225 ","End":"10:32.230","Text":"Here we also have a wavefront which acts as a lens,"},{"Start":"10:32.230 ","End":"10:35.800","Text":"so it concentrates all the lights over here,"},{"Start":"10:35.800 ","End":"10:42.380","Text":"and then we\u0027ll see a patch of light over here on the white screen."},{"Start":"10:43.110 ","End":"10:49.585","Text":"Now let\u0027s see what happens to this dip or this trough in the water wave."},{"Start":"10:49.585 ","End":"10:51.970","Text":"Let\u0027s carry on with our wave."},{"Start":"10:51.970 ","End":"10:57.400","Text":"Here, let\u0027s say we have this light wave or light ray over here that"},{"Start":"10:57.400 ","End":"11:03.310","Text":"travels down and hits right at the bottom of a water wave."},{"Start":"11:03.310 ","End":"11:08.755","Text":"It has an angle of 0 relative to the normal."},{"Start":"11:08.755 ","End":"11:14.440","Text":"It carries on down to this point over here and hits this point."},{"Start":"11:14.440 ","End":"11:16.810","Text":"But what about this wave over here?"},{"Start":"11:16.810 ","End":"11:18.783","Text":"It carries on down,"},{"Start":"11:18.783 ","End":"11:24.835","Text":"and it hits over here where it\u0027s angle with the wave isn\u0027t equal to 0."},{"Start":"11:24.835 ","End":"11:27.610","Text":"If we draw the normal at this point,"},{"Start":"11:27.610 ","End":"11:30.440","Text":"it\u0027s going to be something like this."},{"Start":"11:30.960 ","End":"11:34.570","Text":"Again, our light ray is going"},{"Start":"11:34.570 ","End":"11:39.145","Text":"from a less dense medium of air to a more dense medium of water."},{"Start":"11:39.145 ","End":"11:43.300","Text":"Again, it\u0027s going to change its direction slightly to have"},{"Start":"11:43.300 ","End":"11:47.730","Text":"a smaller angle with respect to the normal,"},{"Start":"11:47.730 ","End":"11:50.830","Text":"so then it will carry on traveling,"},{"Start":"11:50.830 ","End":"11:55.630","Text":"and it will hit our white screen over here."},{"Start":"11:55.630 ","End":"12:02.455","Text":"Then similarly, let\u0027s draw another light ray over here and this 1 also travels down,"},{"Start":"12:02.455 ","End":"12:05.020","Text":"and it hits our water wave over here,"},{"Start":"12:05.020 ","End":"12:08.020","Text":"and we can again see that it\u0027s not at 90 degrees,"},{"Start":"12:08.020 ","End":"12:11.695","Text":"so we have a normal over here which goes like this."},{"Start":"12:11.695 ","End":"12:14.650","Text":"Again, our light ray is traveling from"},{"Start":"12:14.650 ","End":"12:18.100","Text":"a less dense medium of air to a more dense medium of water."},{"Start":"12:18.100 ","End":"12:21.370","Text":"It\u0027s going to also change its direction"},{"Start":"12:21.370 ","End":"12:26.230","Text":"slightly so that it\u0027s at a smaller angle relative to the normal;"},{"Start":"12:26.230 ","End":"12:30.550","Text":"so relative to this gray muck over here and so it is"},{"Start":"12:30.550 ","End":"12:36.055","Text":"also going to slightly diffract and travel to this point over here."},{"Start":"12:36.055 ","End":"12:42.100","Text":"What we can see is that when we have a peak in the positive direction,"},{"Start":"12:42.100 ","End":"12:45.330","Text":"so a slightly higher point in the water,"},{"Start":"12:45.330 ","End":"12:51.970","Text":"we get a light patch over here because our peak acts as a concentrating lens."},{"Start":"12:51.970 ","End":"12:56.230","Text":"All the light rays are concentrated to right underneath it."},{"Start":"12:56.230 ","End":"12:58.284","Text":"However, when we have a trough,"},{"Start":"12:58.284 ","End":"13:01.855","Text":"this acts as a diffusing lens."},{"Start":"13:01.855 ","End":"13:08.605","Text":"All the light rays traveling through the trough are going to diffuse outwards,"},{"Start":"13:08.605 ","End":"13:12.070","Text":"meaning that there\u0027s going to be less density of"},{"Start":"13:12.070 ","End":"13:16.900","Text":"light rays reaching the white screen under the trough."},{"Start":"13:16.900 ","End":"13:20.680","Text":"Then we can see that in this area here,"},{"Start":"13:20.680 ","End":"13:23.005","Text":"let\u0027s mark it with gray."},{"Start":"13:23.005 ","End":"13:25.360","Text":"In the area of the trough,"},{"Start":"13:25.360 ","End":"13:30.050","Text":"we\u0027re going to have a dark patch."},{"Start":"13:31.020 ","End":"13:37.790","Text":"This dark patch corresponds to where we see all these white lines over here."},{"Start":"13:38.370 ","End":"13:42.160","Text":"Let\u0027s mark it in green."},{"Start":"13:42.160 ","End":"13:51.280","Text":"If we look here, we have all of this space between the 2 peaks or the 2 wavefronts."},{"Start":"13:51.280 ","End":"13:55.720","Text":"Let\u0027s just say, see again that this is 1 wavefront and this is"},{"Start":"13:55.720 ","End":"14:01.120","Text":"another wavefront and another wavefront and another 1, and another 1."},{"Start":"14:01.120 ","End":"14:03.700","Text":"In the spaces between,"},{"Start":"14:03.700 ","End":"14:05.830","Text":"all this whites-pace over here,"},{"Start":"14:05.830 ","End":"14:10.625","Text":"this is where the trough of the wave is so that corresponds"},{"Start":"14:10.625 ","End":"14:17.220","Text":"to this white space over here between our patches of light."},{"Start":"14:17.220 ","End":"14:23.470","Text":"In green, this is our dark patches which are made from the trough of the wave."},{"Start":"14:23.470 ","End":"14:25.405","Text":"When we have a peak,"},{"Start":"14:25.405 ","End":"14:27.955","Text":"so that means when we\u0027re at our wavefront,"},{"Start":"14:27.955 ","End":"14:30.970","Text":"we have a patch of light on our widescreen."},{"Start":"14:30.970 ","End":"14:35.155","Text":"Whereas when we\u0027re at our trough over here,"},{"Start":"14:35.155 ","End":"14:39.985","Text":"so we have a patch of dark on our white screen."},{"Start":"14:39.985 ","End":"14:42.610","Text":"Now let\u0027s speak about this lamp."},{"Start":"14:42.610 ","End":"14:46.690","Text":"This lamp is in fact a stroboscope."},{"Start":"14:46.690 ","End":"14:53.620","Text":"The stroboscope shines momentary bright light at different intervals."},{"Start":"14:53.620 ","End":"14:55.158","Text":"It will shine light,"},{"Start":"14:55.158 ","End":"14:58.165","Text":"and then it will go dark and shine light again and back to dark."},{"Start":"14:58.165 ","End":"15:03.355","Text":"A stroboscope is simply just like a flashing light."},{"Start":"15:03.355 ","End":"15:10.630","Text":"Let\u0027s imagine that our stroboscope is flashing light or emitting light at a frequency,"},{"Start":"15:10.630 ","End":"15:16.075","Text":"so let\u0027s say that is at a frequency of 1 hertz."},{"Start":"15:16.075 ","End":"15:20.635","Text":"That means that there\u0027s 1 flash every second."},{"Start":"15:20.635 ","End":"15:28.630","Text":"Then let\u0027s say that over here we have our straight wave dipper,"},{"Start":"15:28.630 ","End":"15:32.020","Text":"which is making waves so we\u0027re going up,"},{"Start":"15:32.020 ","End":"15:35.305","Text":"down, and up and down with this."},{"Start":"15:35.305 ","End":"15:38.170","Text":"As we know, we\u0027re going to have our wavefront,"},{"Start":"15:38.170 ","End":"15:40.833","Text":"which will start over here,"},{"Start":"15:40.833 ","End":"15:45.850","Text":"and then it will move in this direction to be over here."},{"Start":"15:45.850 ","End":"15:52.450","Text":"It will move a moment later to be over here and then a new wavefront will"},{"Start":"15:52.450 ","End":"16:00.790","Text":"form from the wave dipper and then this first wave will move over here."},{"Start":"16:00.790 ","End":"16:04.270","Text":"The previous wave will replace the wavefront over here,"},{"Start":"16:04.270 ","End":"16:07.990","Text":"and the new wave will be made over here and so on and so forth,"},{"Start":"16:07.990 ","End":"16:11.905","Text":"where each new wave is produced every second,"},{"Start":"16:11.905 ","End":"16:14.620","Text":"and its wavefront moves forward and forward and"},{"Start":"16:14.620 ","End":"16:18.790","Text":"slowly takes the place of its previous wave."},{"Start":"16:18.790 ","End":"16:22.660","Text":"Eventually our first wave that was created will be over here."},{"Start":"16:22.660 ","End":"16:25.720","Text":"The wave that was created after it will be over here,"},{"Start":"16:25.720 ","End":"16:27.955","Text":"the wave after it will be over here,"},{"Start":"16:27.955 ","End":"16:30.295","Text":"the wave after that 1 will be over here,"},{"Start":"16:30.295 ","End":"16:34.150","Text":"and then the newest wave will be formed over here."},{"Start":"16:34.150 ","End":"16:41.440","Text":"All of this is of course happening whilst the stroboscope is flashing light."},{"Start":"16:41.440 ","End":"16:47.320","Text":"What we\u0027ll see and let\u0027s just draw it in red is that if we have waves,"},{"Start":"16:47.320 ","End":"16:49.750","Text":"this is our first wave, second wave,"},{"Start":"16:49.750 ","End":"16:53.530","Text":"and third wave so when the light is on,"},{"Start":"16:53.530 ","End":"16:55.015","Text":"we\u0027ll see this picture."},{"Start":"16:55.015 ","End":"16:59.388","Text":"Then the stroboscope will switch off so the light will be off,"},{"Start":"16:59.388 ","End":"17:00.805","Text":"and so we won\u0027t see,"},{"Start":"17:00.805 ","End":"17:07.720","Text":"but wave number 1 will move to this position over here so 1 will be over here."},{"Start":"17:07.720 ","End":"17:09.370","Text":"Wave number 2,"},{"Start":"17:09.370 ","End":"17:14.905","Text":"will move over here and wave number 3 will move over here,"},{"Start":"17:14.905 ","End":"17:20.140","Text":"and then we\u0027ll have a new wave forming over here called wave number 4."},{"Start":"17:20.140 ","End":"17:23.785","Text":"Then our stroboscope will switch back on again,"},{"Start":"17:23.785 ","End":"17:26.500","Text":"and then we\u0027ll see this picture."},{"Start":"17:26.500 ","End":"17:32.080","Text":"Then our stroboscope will switch back off and our waves will carry on moving."},{"Start":"17:32.080 ","End":"17:36.157","Text":"Then we\u0027ll have wave number 1 which has moved here,"},{"Start":"17:36.157 ","End":"17:38.635","Text":"then wave number 2 will be here,"},{"Start":"17:38.635 ","End":"17:40.893","Text":"and this will be 3, and this will be 4,"},{"Start":"17:40.893 ","End":"17:43.210","Text":"and this will be wave number 5."},{"Start":"17:43.210 ","End":"17:45.745","Text":"Then the stroboscope will switch back on,"},{"Start":"17:45.745 ","End":"17:47.860","Text":"and we\u0027ll see this picture again."},{"Start":"17:47.860 ","End":"17:51.715","Text":"It looks like we have the same wavefronts,"},{"Start":"17:51.715 ","End":"17:53.425","Text":"and it looks like nothing has moved."},{"Start":"17:53.425 ","End":"17:57.235","Text":"But we know because of how our wave tank works,"},{"Start":"17:57.235 ","End":"18:03.970","Text":"we know that our waves are moving and traveling through the water and the wave tank."},{"Start":"18:03.970 ","End":"18:08.455","Text":"The stroboscope is helping us to see,"},{"Start":"18:08.455 ","End":"18:12.160","Text":"it\u0027s like an optical illusion type thing, not exactly,"},{"Start":"18:12.160 ","End":"18:17.065","Text":"but it\u0027s helping us to see as if everything here is static"},{"Start":"18:17.065 ","End":"18:23.185","Text":"and then that\u0027s how we can see these lines and take a picture of them very easily."},{"Start":"18:23.185 ","End":"18:27.700","Text":"In a water tank or a wave tank,"},{"Start":"18:27.700 ","End":"18:31.180","Text":"what is commonly done is that the white screen instead of"},{"Start":"18:31.180 ","End":"18:34.705","Text":"being placed under the wave tank,"},{"Start":"18:34.705 ","End":"18:37.750","Text":"often it\u0027s placed over here."},{"Start":"18:37.750 ","End":"18:41.250","Text":"The white screen would we like so,"},{"Start":"18:41.250 ","End":"18:46.500","Text":"and then in order to see this wave pattern that we see over here,"},{"Start":"18:46.500 ","End":"18:56.600","Text":"a mirror is placed at a 45-degree angle to the water basin itself."},{"Start":"18:56.610 ","End":"18:58.944","Text":"If you can imagine,"},{"Start":"18:58.944 ","End":"19:00.280","Text":"a mirror over here,"},{"Start":"19:00.280 ","End":"19:03.550","Text":"this green at 45 degrees,"},{"Start":"19:03.550 ","End":"19:11.080","Text":"and then a white screen becomes this red section over here."},{"Start":"19:11.080 ","End":"19:15.955","Text":"This is what a common wave tank looks like today."},{"Start":"19:15.955 ","End":"19:19.480","Text":"What we have here is a picture of"},{"Start":"19:19.480 ","End":"19:26.650","Text":"a wave tank and here we can see that we have something over here,"},{"Start":"19:26.650 ","End":"19:30.940","Text":"let\u0027s draw it in blue,"},{"Start":"19:30.940 ","End":"19:33.460","Text":"that moves up and down like silver here,"},{"Start":"19:33.460 ","End":"19:35.215","Text":"and it\u0027s attached to this arm,"},{"Start":"19:35.215 ","End":"19:37.900","Text":"which is attached to overhear the thing that"},{"Start":"19:37.900 ","End":"19:42.100","Text":"creates the wave or creates the disturbance in the water."},{"Start":"19:42.100 ","End":"19:46.520","Text":"As this little screw over here moves up and down,"},{"Start":"19:46.520 ","End":"19:52.540","Text":"a wave maker over here moves up and down as well."},{"Start":"19:52.540 ","End":"19:59.090","Text":"This is the wave maker and over here we can see at 45 degrees,"},{"Start":"19:59.090 ","End":"20:04.450","Text":"we have our mirror that we were talking about before,"},{"Start":"20:04.450 ","End":"20:06.070","Text":"and then over here,"},{"Start":"20:06.070 ","End":"20:07.525","Text":"let\u0027s draw it in red,"},{"Start":"20:07.525 ","End":"20:16.655","Text":"the outline, we can see that this is our screen where we see the wave pattern."},{"Start":"20:16.655 ","End":"20:20.074","Text":"Here we have our lamp,"},{"Start":"20:20.074 ","End":"20:21.140","Text":"which as we know,"},{"Start":"20:21.140 ","End":"20:24.455","Text":"is a stroboscope and so as we already said,"},{"Start":"20:24.455 ","End":"20:27.470","Text":"if we coordinate between the frequency of"},{"Start":"20:27.470 ","End":"20:31.250","Text":"the stroboscope and the frequency of the wave maker,"},{"Start":"20:31.250 ","End":"20:36.005","Text":"then we\u0027ll see the wave that\u0027s if it\u0027s stationary,"},{"Start":"20:36.005 ","End":"20:38.810","Text":"because we won\u0027t see the wave moving,"},{"Start":"20:38.810 ","End":"20:40.105","Text":"and we\u0027ll just see it\u0027s stationary."},{"Start":"20:40.105 ","End":"20:43.240","Text":"If we make the wave maker 1 hertz,"},{"Start":"20:43.240 ","End":"20:47.540","Text":"so that\u0027s 1 wave per second or 2 hertz 2 waves per second,"},{"Start":"20:47.540 ","End":"20:49.160","Text":"or however many you want,"},{"Start":"20:49.160 ","End":"20:54.395","Text":"make the stroboscope also flashlight at the same frequency,"},{"Start":"20:54.395 ","End":"20:57.105","Text":"and we\u0027ll see the wave as stationary."},{"Start":"20:57.105 ","End":"20:59.810","Text":"As we can see let\u0027s draw it in green."},{"Start":"20:59.810 ","End":"21:05.735","Text":"We have our lights coming out of our lamp and it will travel"},{"Start":"21:05.735 ","End":"21:12.605","Text":"downwards to here until it hits our mirror in the middle here."},{"Start":"21:12.605 ","End":"21:15.769","Text":"Then because the mirror is at 45 degrees,"},{"Start":"21:15.769 ","End":"21:23.960","Text":"then our light ray is going to travel like so and hit the screen and that\u0027s how we"},{"Start":"21:23.960 ","End":"21:27.950","Text":"see the wave pattern made in the wave tank up here on"},{"Start":"21:27.950 ","End":"21:33.120","Text":"the white screen placed at 90 degrees to the wave tank."},{"Start":"21:33.120 ","End":"21:36.545","Text":"In this lesson, we learned how a wave tank"},{"Start":"21:36.545 ","End":"21:40.310","Text":"works with all of the different bits and pieces that make"},{"Start":"21:40.310 ","End":"21:47.585","Text":"it up and how exactly we get the image of the wave on the screen."},{"Start":"21:47.585 ","End":"21:50.550","Text":"That\u0027s the end of this lesson."}],"ID":12479},{"Watched":false,"Name":"Reflection in Wave Tanks","Duration":"30m 23s","ChapterTopicVideoID":12012,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.785","Text":"Hello. In this lesson,"},{"Start":"00:01.785 ","End":"00:05.969","Text":"we\u0027re going to be dealing with reflection and specifically dealing with"},{"Start":"00:05.969 ","End":"00:10.875","Text":"a wave tank where we have a wave hitting some obstacle,"},{"Start":"00:10.875 ","End":"00:13.545","Text":"and then we want to see what will happen."},{"Start":"00:13.545 ","End":"00:20.938","Text":"Here, this blue rectangle is our wave tank,"},{"Start":"00:20.938 ","End":"00:28.020","Text":"and this black line over here is our obstacle."},{"Start":"00:28.820 ","End":"00:33.230","Text":"Now, we\u0027re going to take a straight wave,"},{"Start":"00:33.230 ","End":"00:37.992","Text":"a wavefront, let\u0027s say this over here,"},{"Start":"00:37.992 ","End":"00:43.715","Text":"and what we want to see is once it reaches the obstacle,"},{"Start":"00:43.715 ","End":"00:47.315","Text":"what it will look like when it is reflected back."},{"Start":"00:47.315 ","End":"00:52.230","Text":"What we\u0027re going to do is we\u0027re going to label this end side A,"},{"Start":"00:52.230 ","End":"00:55.550","Text":"we\u0027re going to label this end C,"},{"Start":"00:55.550 ","End":"00:59.359","Text":"and then right in the middle over here,"},{"Start":"00:59.359 ","End":"01:02.095","Text":"this is going to be point B."},{"Start":"01:02.095 ","End":"01:07.160","Text":"We know that our wavefront is traveling like so."},{"Start":"01:07.160 ","End":"01:09.860","Text":"We know that a few moments later,"},{"Start":"01:09.860 ","End":"01:12.205","Text":"we\u0027re going to have point C,"},{"Start":"01:12.205 ","End":"01:14.610","Text":"straight and black,"},{"Start":"01:14.610 ","End":"01:16.739","Text":"point C over here,"},{"Start":"01:16.739 ","End":"01:19.889","Text":"point B over here,"},{"Start":"01:19.889 ","End":"01:24.055","Text":"and point A over here."},{"Start":"01:24.055 ","End":"01:27.529","Text":"This is exactly what we would expect."},{"Start":"01:27.529 ","End":"01:32.742","Text":"Now, let\u0027s imagine that our wave carries on traveling,"},{"Start":"01:32.742 ","End":"01:35.870","Text":"and then let\u0027s see what\u0027s going to happen once it"},{"Start":"01:35.870 ","End":"01:40.560","Text":"hits the obstacle and how it will be reflected back."},{"Start":"01:40.670 ","End":"01:43.515","Text":"Let\u0027s start with A."},{"Start":"01:43.515 ","End":"01:47.513","Text":"Point A is going to travel from here, well,"},{"Start":"01:47.513 ","End":"01:51.910","Text":"really from here, passing through this point over here,"},{"Start":"01:51.910 ","End":"02:01.375","Text":"and carrying on in this direction until it reaches this point over here."},{"Start":"02:01.375 ","End":"02:02.875","Text":"At this point over here,"},{"Start":"02:02.875 ","End":"02:05.425","Text":"it\u0027s going to reflect back."},{"Start":"02:05.425 ","End":"02:07.060","Text":"Remember how it reflects back,"},{"Start":"02:07.060 ","End":"02:11.300","Text":"it reflects back at the same angle that it hit."},{"Start":"02:11.300 ","End":"02:15.110","Text":"Here we can see it hit at 45 degrees."},{"Start":"02:15.110 ","End":"02:18.845","Text":"This angle is 45 degrees."},{"Start":"02:18.845 ","End":"02:23.580","Text":"That means it\u0027s going to be reflected also at 45 degrees,"},{"Start":"02:23.580 ","End":"02:28.255","Text":"so this angle over here is also 45 degrees."},{"Start":"02:28.255 ","End":"02:32.388","Text":"It\u0027s going to continue like so,"},{"Start":"02:32.388 ","End":"02:35.153","Text":"traveling like this,"},{"Start":"02:35.153 ","End":"02:38.085","Text":"and then let\u0027s stop over here."},{"Start":"02:38.085 ","End":"02:40.440","Text":"This is our point A."},{"Start":"02:40.440 ","End":"02:44.200","Text":"Now, let\u0027s see what\u0027s going to happen with our point B."},{"Start":"02:44.200 ","End":"02:48.567","Text":"It starts traveling from here to here,"},{"Start":"02:48.567 ","End":"02:51.400","Text":"and then it carries on."},{"Start":"02:51.400 ","End":"02:55.459","Text":"It\u0027s traveling in a straight line"},{"Start":"02:55.459 ","End":"02:58.700","Text":"because this is a straight wavefront and then we can see"},{"Start":"02:58.700 ","End":"03:08.085","Text":"that point B is going to hit the obstacle at this position and the obstacle over here."},{"Start":"03:08.085 ","End":"03:12.785","Text":"What we can see is when we\u0027re dealing with a straight wavefront,"},{"Start":"03:12.785 ","End":"03:19.060","Text":"each point along the wavefront is going to hit the obstacle at a different position."},{"Start":"03:19.060 ","End":"03:24.544","Text":"Again, where the angle\u0027s point B is going to hit exactly like point A at the same angle,"},{"Start":"03:24.544 ","End":"03:27.005","Text":"which as we said was 45 degrees."},{"Start":"03:27.005 ","End":"03:30.470","Text":"It\u0027s going to be reflected also at 45 degrees."},{"Start":"03:30.470 ","End":"03:36.439","Text":"It\u0027s going to be traveling in this direction like so and carrying"},{"Start":"03:36.439 ","End":"03:43.559","Text":"on up until it reaches this point over here."},{"Start":"03:44.120 ","End":"03:48.924","Text":"Then we have point C. Point C starts from over here"},{"Start":"03:48.924 ","End":"03:53.403","Text":"and it travels and a moment later it reaches this point like we expected,"},{"Start":"03:53.403 ","End":"03:57.900","Text":"and then it continues traveling straight."},{"Start":"03:57.900 ","End":"04:02.370","Text":"We can see it\u0027s carrying on and continuing."},{"Start":"04:02.370 ","End":"04:05.290","Text":"Then we can see that point C of"},{"Start":"04:05.290 ","End":"04:10.053","Text":"our original wave is going to hit the obstacle over here,"},{"Start":"04:10.053 ","End":"04:14.155","Text":"and only over here is it going to be reflected off."},{"Start":"04:14.155 ","End":"04:17.583","Text":"Again, it hit at 45 degrees like point A and B,"},{"Start":"04:17.583 ","End":"04:23.746","Text":"so it\u0027s going to be reflected at 45 degrees as well and"},{"Start":"04:23.746 ","End":"04:32.190","Text":"it\u0027s just going to travel straight until it reaches this point over here."},{"Start":"04:32.720 ","End":"04:40.195","Text":"Now, we can draw in our reflected wave and it\u0027s going to go like so."},{"Start":"04:40.195 ","End":"04:46.345","Text":"Now, we can see that it\u0027s at the same angle because we can see that from A to B,"},{"Start":"04:46.345 ","End":"04:50.542","Text":"we go down to and across to,"},{"Start":"04:50.542 ","End":"04:51.959","Text":"and from B to C,"},{"Start":"04:51.959 ","End":"04:55.670","Text":"we also go down to and across to."},{"Start":"04:55.670 ","End":"04:57.889","Text":"From our original wave, from A to B,"},{"Start":"04:57.889 ","End":"05:00.688","Text":"we went down to and across to,"},{"Start":"05:00.688 ","End":"05:04.430","Text":"and from B to C, we also went down to and across to."},{"Start":"05:04.430 ","End":"05:06.830","Text":"I\u0027ll just draw that in gray,"},{"Start":"05:06.830 ","End":"05:13.783","Text":"so down to and across to from A to B and the same from B to C. Here again,"},{"Start":"05:13.783 ","End":"05:20.750","Text":"down to and across to from A to B and the same from B to"},{"Start":"05:20.750 ","End":"05:28.175","Text":"C. What we can see now is that we have our reflected wave,"},{"Start":"05:28.175 ","End":"05:32.762","Text":"which is maintaining its angle,"},{"Start":"05:32.762 ","End":"05:35.280","Text":"so the angle that it\u0027s pointing at."},{"Start":"05:35.280 ","End":"05:42.619","Text":"The only difference is our A has changed sides supposedly and so has our"},{"Start":"05:42.619 ","End":"05:50.734","Text":"C. What we can see is that in the wave right at the start before it was reflected,"},{"Start":"05:50.734 ","End":"05:57.299","Text":"point A was on the right and after the reflection,"},{"Start":"05:57.299 ","End":"06:02.644","Text":"we can see that point A is at the left corner now."},{"Start":"06:02.644 ","End":"06:06.874","Text":"At first it was in the right corner and now it\u0027s on the left corner of the wave."},{"Start":"06:06.874 ","End":"06:09.469","Text":"The shape of the wave itself is the same."},{"Start":"06:09.469 ","End":"06:14.609","Text":"However, it looks as if our points have switched."},{"Start":"06:14.860 ","End":"06:22.385","Text":"Now, notice also that each point on the wavefront acts like a ray of light."},{"Start":"06:22.385 ","End":"06:27.499","Text":"Just how we learned how rays of light behave upon reflection,"},{"Start":"06:27.499 ","End":"06:33.385","Text":"so they travel all the way until the mirror or the object,"},{"Start":"06:33.385 ","End":"06:39.355","Text":"and then they\u0027re reflected back at the same angle that they reached the obstacle at."},{"Start":"06:39.355 ","End":"06:43.010","Text":"Every point on the wavefront behaves like that."},{"Start":"06:43.010 ","End":"06:44.495","Text":"A behaved like that,"},{"Start":"06:44.495 ","End":"06:45.844","Text":"B behaved like that,"},{"Start":"06:45.844 ","End":"06:48.324","Text":"and C behaved like that."},{"Start":"06:48.324 ","End":"06:50.369","Text":"Just as a note,"},{"Start":"06:50.369 ","End":"06:51.630","Text":"as I made a mistake,"},{"Start":"06:51.630 ","End":"06:56.210","Text":"the angle of the ray is relative to the normal."},{"Start":"06:56.210 ","End":"06:58.565","Text":"If this is our obstacle,"},{"Start":"06:58.565 ","End":"07:01.241","Text":"then we take the normal to the obstacle,"},{"Start":"07:01.241 ","End":"07:04.970","Text":"so that\u0027s some line that\u0027s at 90 degrees to"},{"Start":"07:04.970 ","End":"07:10.130","Text":"our obstacle and then this is the angle that we\u0027re measuring."},{"Start":"07:10.130 ","End":"07:12.620","Text":"Here in this example specifically,"},{"Start":"07:12.620 ","End":"07:16.444","Text":"we\u0027re still dealing with a 45-degree angle."},{"Start":"07:16.444 ","End":"07:20.869","Text":"However, we measure the angle from the normal and not from this side over here,"},{"Start":"07:20.869 ","End":"07:23.520","Text":"which is what I did before."},{"Start":"07:26.780 ","End":"07:32.118","Text":"If this was the ray coming in at 45 degrees,"},{"Start":"07:32.118 ","End":"07:33.990","Text":"so the reflected ray,"},{"Start":"07:33.990 ","End":"07:37.519","Text":"so that\u0027s this over here,"},{"Start":"07:37.519 ","End":"07:41.779","Text":"will be leaving this angle here,"},{"Start":"07:41.779 ","End":"07:45.460","Text":"which is also 45 degrees."},{"Start":"07:45.460 ","End":"07:47.625","Text":"I\u0027m just going to do that again."},{"Start":"07:47.625 ","End":"07:51.182","Text":"Where point B hits the obstacle,"},{"Start":"07:51.182 ","End":"07:55.069","Text":"so we take a line that\u0027s at 90 degrees to that and then we"},{"Start":"07:55.069 ","End":"07:59.605","Text":"can see we have the ray coming in and the reflected ray."},{"Start":"07:59.605 ","End":"08:05.090","Text":"This over here is 45 degrees and this"},{"Start":"08:05.090 ","End":"08:10.534","Text":"over here is also 45 degrees and similarly for point C,"},{"Start":"08:10.534 ","End":"08:12.560","Text":"here is our normal,"},{"Start":"08:12.560 ","End":"08:17.768","Text":"here is 45-degree angle that the ray is coming in at,"},{"Start":"08:17.768 ","End":"08:21.290","Text":"and the reflected ray is leaving at"},{"Start":"08:21.290 ","End":"08:26.580","Text":"an angle of 45 degrees relative to the normal as well."},{"Start":"08:27.650 ","End":"08:31.625","Text":"What we can see is we have the incident angle,"},{"Start":"08:31.625 ","End":"08:36.810","Text":"which is the angle between the incident ray and the normal."},{"Start":"08:36.830 ","End":"08:39.724","Text":"If we redraw this,"},{"Start":"08:39.724 ","End":"08:45.085","Text":"so if here we have our obstacle,"},{"Start":"08:45.085 ","End":"08:50.529","Text":"if we have a ray coming in like so."},{"Start":"08:50.529 ","End":"08:57.859","Text":"This ray is our incident ray and"},{"Start":"08:57.859 ","End":"09:06.845","Text":"then we would have our normal over here,"},{"Start":"09:06.845 ","End":"09:16.595","Text":"and then we would have our reflected ray over here traveling in this direction."},{"Start":"09:16.595 ","End":"09:21.948","Text":"Then we could say that the angle between the incident ray and the normal,"},{"Start":"09:21.948 ","End":"09:24.080","Text":"so that\u0027s this over here,"},{"Start":"09:24.080 ","End":"09:25.600","Text":"is our incident angle,"},{"Start":"09:25.600 ","End":"09:29.660","Text":"so let\u0027s call it Theta i for incident angle."},{"Start":"09:29.660 ","End":"09:34.735","Text":"Then the reflected angle is the angle between the reflected ray and the normal."},{"Start":"09:34.735 ","End":"09:36.190","Text":"That would be this,"},{"Start":"09:36.190 ","End":"09:39.574","Text":"so let\u0027s call that Theta t,"},{"Start":"09:39.574 ","End":"09:44.200","Text":"Theta r for reflected."},{"Start":"09:44.870 ","End":"09:52.750","Text":"These are the definitions if we\u0027re looking at light waves."},{"Start":"09:52.900 ","End":"09:57.409","Text":"However, right now we\u0027re speaking about water waves."},{"Start":"09:57.409 ","End":"09:59.749","Text":"We\u0027re dealing with a wave tank."},{"Start":"09:59.749 ","End":"10:03.470","Text":"We have slightly different definitions."},{"Start":"10:03.470 ","End":"10:09.100","Text":"Copy out this diagram for light waves and now I\u0027m going to rub it out."},{"Start":"10:09.320 ","End":"10:11.719","Text":"Here we have the definition for"},{"Start":"10:11.719 ","End":"10:14.659","Text":"water waves and this is what you have to remember right now."},{"Start":"10:14.659 ","End":"10:21.640","Text":"The incident angle is the angle between the incident wavefront and the obstacle."},{"Start":"10:21.640 ","End":"10:23.835","Text":"Let\u0027s draw this in red."},{"Start":"10:23.835 ","End":"10:28.594","Text":"Here we can see our point A of the wavefront carries on and"},{"Start":"10:28.594 ","End":"10:33.714","Text":"then it hits or is incident on the obstacle at this point over here."},{"Start":"10:33.714 ","End":"10:35.835","Text":"This is our point A,"},{"Start":"10:35.835 ","End":"10:38.984","Text":"and then we know that this is our point C,"},{"Start":"10:38.984 ","End":"10:42.502","Text":"and this is what our wave will look like,"},{"Start":"10:42.502 ","End":"10:46.535","Text":"where over here we have our point B."},{"Start":"10:46.535 ","End":"10:52.099","Text":"What we\u0027re going to have is that"},{"Start":"10:52.099 ","End":"10:57.170","Text":"our incident angle is going to be the angle between our wavefronts."},{"Start":"10:57.170 ","End":"10:58.895","Text":"Let\u0027s draw this in black."},{"Start":"10:58.895 ","End":"11:01.399","Text":"That\u0027s the angle between the wavefront,"},{"Start":"11:01.399 ","End":"11:06.345","Text":"this is our wavefront and our obstacle."},{"Start":"11:06.345 ","End":"11:11.070","Text":"That\u0027s this over here, our obstacle."},{"Start":"11:11.070 ","End":"11:17.609","Text":"Then this will be our Theta i."},{"Start":"11:17.609 ","End":"11:24.707","Text":"This is our angle, our incident angle."},{"Start":"11:24.707 ","End":"11:30.325","Text":"Now, let\u0027s go back to our definition for light."},{"Start":"11:30.325 ","End":"11:33.489","Text":"We remember that we had our normal over here,"},{"Start":"11:33.489 ","End":"11:38.185","Text":"and then we had our incident angle."},{"Start":"11:38.185 ","End":"11:41.815","Text":"Let\u0027s call it Theta 1 this time."},{"Start":"11:41.815 ","End":"11:43.780","Text":"This is Theta 1,"},{"Start":"11:43.780 ","End":"11:47.350","Text":"and then we had our reflected angle over here,"},{"Start":"11:47.350 ","End":"11:49.790","Text":"let\u0027s call it Theta 2."},{"Start":"11:50.910 ","End":"11:56.470","Text":"Then Theta i is our incident angle when we\u0027re dealing with water waves,"},{"Start":"11:56.470 ","End":"12:02.000","Text":"and Theta 1 is our incident angle when we\u0027re dealing with light waves."},{"Start":"12:02.910 ","End":"12:13.000","Text":"Now what we can see is if we draw a straight line from here,"},{"Start":"12:15.600 ","End":"12:25.729","Text":"then this angle over here made with our incident ray is going to be 90 degrees."},{"Start":"12:26.550 ","End":"12:29.380","Text":"Why is it 90 degrees?"},{"Start":"12:29.380 ","End":"12:32.890","Text":"Because we know that our wavefront is always"},{"Start":"12:32.890 ","End":"12:38.170","Text":"perpendicular or at 90 degrees to its direction of travel."},{"Start":"12:38.170 ","End":"12:41.650","Text":"This is its direction of travel, like so,"},{"Start":"12:41.650 ","End":"12:43.555","Text":"and this is the wavefront,"},{"Start":"12:43.555 ","End":"12:49.369","Text":"and they\u0027re always at 90 degrees."},{"Start":"12:49.770 ","End":"12:56.589","Text":"Then we can draw another angle of 90 degrees if we"},{"Start":"12:56.589 ","End":"13:03.279","Text":"go along our obstacle and the normal to the obstacle,"},{"Start":"13:03.279 ","End":"13:07.690","Text":"so we know that the angle between the normal of the obstacle"},{"Start":"13:07.690 ","End":"13:12.980","Text":"and the obstacle is also going to be 90 degrees."},{"Start":"13:14.040 ","End":"13:19.029","Text":"Now, we\u0027re dealing with the angle between the blue so"},{"Start":"13:19.029 ","End":"13:23.590","Text":"that\u0027s also 90 degrees and that\u0027s just the definition of the normal."},{"Start":"13:23.590 ","End":"13:30.200","Text":"The normal is always going to be 90 degrees to whatever it is normal of."},{"Start":"13:30.270 ","End":"13:35.890","Text":"Now what we can see is we have this angle Theta 2 in this area over here."},{"Start":"13:35.890 ","End":"13:38.500","Text":"Let\u0027s cover it in black lines,"},{"Start":"13:38.500 ","End":"13:40.749","Text":"so we can see it a little bit more clearly."},{"Start":"13:40.749 ","End":"13:43.779","Text":"You can see that over here in this space,"},{"Start":"13:43.779 ","End":"13:46.910","Text":"we have Theta 2."},{"Start":"13:48.960 ","End":"13:51.160","Text":"We\u0027ve called it Theta 2."},{"Start":"13:51.160 ","End":"13:52.419","Text":"It can be anything else,"},{"Start":"13:52.419 ","End":"13:54.760","Text":"you can call it x or whatever,"},{"Start":"13:54.760 ","End":"13:57.260","Text":"and this is Theta 2."},{"Start":"13:57.690 ","End":"14:07.104","Text":"Now what we can see that in order to make the 90-degree angle between the green lines,"},{"Start":"14:07.104 ","End":"14:11.079","Text":"let\u0027s write it out so for the green lines,"},{"Start":"14:11.079 ","End":"14:17.830","Text":"so we can see that 90 is equal to our Theta 1,"},{"Start":"14:17.830 ","End":"14:22.090","Text":"which is our incident angle when dealing with light waves,"},{"Start":"14:22.090 ","End":"14:26.479","Text":"so Theta 1 plus our Theta 2."},{"Start":"14:27.180 ","End":"14:29.980","Text":"So that\u0027s a 90 degrees."},{"Start":"14:29.980 ","End":"14:34.975","Text":"Now let\u0027s take a look at for the blue line."},{"Start":"14:34.975 ","End":"14:40.390","Text":"This is also 90 degrees so now we\u0027re dealing with the 90-degree angle for the blue,"},{"Start":"14:40.390 ","End":"14:46.450","Text":"and we can see that 90 degrees is equal to"},{"Start":"14:46.450 ","End":"14:54.520","Text":"our incident angle Theta i from our water wave."},{"Start":"14:54.520 ","End":"14:57.355","Text":"Theta i is this over here."},{"Start":"14:57.355 ","End":"15:05.060","Text":"Our incident angle for our water wave plus this Theta 2 over here."},{"Start":"15:07.440 ","End":"15:12.265","Text":"This we can see it very easily from the diagram."},{"Start":"15:12.265 ","End":"15:18.320","Text":"What we can see is that if we isolate out our Theta 2,"},{"Start":"15:19.170 ","End":"15:28.675","Text":"so our Theta 2 over here is equal to 90 degrees minus our Theta 1,"},{"Start":"15:28.675 ","End":"15:38.260","Text":"and over here from the blue our Theta 2 is equal to 90 degrees minus Theta i."},{"Start":"15:38.260 ","End":"15:41.499","Text":"Now, if we rearrange everything,"},{"Start":"15:41.499 ","End":"15:44.095","Text":"we\u0027ll get that our Theta 1,"},{"Start":"15:44.095 ","End":"15:47.979","Text":"which is our incident angle when we\u0027re dealing with light waves,"},{"Start":"15:47.979 ","End":"15:54.265","Text":"is equal to our Theta i,"},{"Start":"15:54.265 ","End":"15:59.004","Text":"which is our incident angle when dealing with water waves."},{"Start":"15:59.004 ","End":"16:03.430","Text":"Because from this equation here we get that 90 minus Theta 1,"},{"Start":"16:03.430 ","End":"16:07.555","Text":"we can cross this out, is equal to 90 minus Theta i."},{"Start":"16:07.555 ","End":"16:16.420","Text":"We can simplify both sides by subtracting 90 and multiplying both sides by negative 1,"},{"Start":"16:16.420 ","End":"16:18.295","Text":"and then we\u0027ll get this."},{"Start":"16:18.295 ","End":"16:21.039","Text":"What do, we can see is that the definition,"},{"Start":"16:21.039 ","End":"16:24.114","Text":"or in mathematical terms, the incident angle,"},{"Start":"16:24.114 ","End":"16:31.960","Text":"whether we\u0027re speaking in terms of light waves or in terms of water waves is the same."},{"Start":"16:32.610 ","End":"16:39.355","Text":"What\u0027s important to note is that the definition in words for"},{"Start":"16:39.355 ","End":"16:42.579","Text":"the incident angle and the reflected angle when dealing with"},{"Start":"16:42.579 ","End":"16:46.510","Text":"light waves or with water waves is slightly different."},{"Start":"16:46.510 ","End":"16:49.885","Text":"However, in mathematical terms,"},{"Start":"16:49.885 ","End":"16:53.150","Text":"they are the same."},{"Start":"16:54.060 ","End":"16:57.579","Text":"A question that we can be asked when dealing with"},{"Start":"16:57.579 ","End":"17:00.835","Text":"reflections so we have the exact same diagram,"},{"Start":"17:00.835 ","End":"17:06.534","Text":"is what does the wavefront look like when half of it has hit the obstacle,"},{"Start":"17:06.534 ","End":"17:09.205","Text":"but the other half has not?"},{"Start":"17:09.205 ","End":"17:11.214","Text":"As we can see here,"},{"Start":"17:11.214 ","End":"17:19.240","Text":"half of the wavefront is from A to B and the other half is from B to C. Just like before,"},{"Start":"17:19.240 ","End":"17:21.699","Text":"let\u0027s draw it in black,"},{"Start":"17:21.699 ","End":"17:24.145","Text":"how the wave will move forward."},{"Start":"17:24.145 ","End":"17:31.900","Text":"We know that point A is going to travel forward like so in this straight line,"},{"Start":"17:31.900 ","End":"17:35.815","Text":"and eventually, it will hit the obstacle over here,"},{"Start":"17:35.815 ","End":"17:39.895","Text":"and then it will be reflected back."},{"Start":"17:39.895 ","End":"17:46.010","Text":"Let\u0027s draw the line that it will be reflected back in, in red."},{"Start":"17:46.740 ","End":"17:49.944","Text":"It will be reflected like so."},{"Start":"17:49.944 ","End":"17:52.990","Text":"Now let\u0027s deal with point B."},{"Start":"17:52.990 ","End":"17:59.454","Text":"We\u0027re dealing with the case where point A has reached the barrier or the obstacle,"},{"Start":"17:59.454 ","End":"18:02.800","Text":"and point B has also reached it because we"},{"Start":"18:02.800 ","End":"18:06.670","Text":"need that half of the wavefront has hit the obstacle."},{"Start":"18:06.670 ","End":"18:09.610","Text":"Point B is traveling like so,"},{"Start":"18:09.610 ","End":"18:11.470","Text":"and we know that all of the points on"},{"Start":"18:11.470 ","End":"18:15.805","Text":"the wavefront will hit the obstacle at different times."},{"Start":"18:15.805 ","End":"18:22.040","Text":"Point B will hit over here,"},{"Start":"18:22.530 ","End":"18:26.139","Text":"and we know that point B has hit the obstacle,"},{"Start":"18:26.139 ","End":"18:27.790","Text":"but it hasn\u0027t yet reflected,"},{"Start":"18:27.790 ","End":"18:33.580","Text":"and point C will travel from where it is over here,"},{"Start":"18:33.580 ","End":"18:36.760","Text":"straight like so,"},{"Start":"18:36.760 ","End":"18:44.590","Text":"and we know that C has not yet hit the obstacle,"},{"Start":"18:44.590 ","End":"18:48.310","Text":"so it will be located over here."},{"Start":"18:48.310 ","End":"18:52.760","Text":"This is where our point C is."},{"Start":"18:53.640 ","End":"18:55.944","Text":"Point B is here."},{"Start":"18:55.944 ","End":"18:58.629","Text":"We know that point and it\u0027s hit the obstacle."},{"Start":"18:58.629 ","End":"19:01.360","Text":"Point C has not yet reached obstacles,"},{"Start":"19:01.360 ","End":"19:03.490","Text":"so it\u0027s located over here."},{"Start":"19:03.490 ","End":"19:08.320","Text":"Now let\u0027s see where point A is exactly."},{"Start":"19:08.320 ","End":"19:09.730","Text":"We know it\u0027s been reflected,"},{"Start":"19:09.730 ","End":"19:12.310","Text":"but where along the line hasn\u0027t been reflected?"},{"Start":"19:12.310 ","End":"19:19.570","Text":"Let\u0027s say that point A moves one square down and one square cross,"},{"Start":"19:19.570 ","End":"19:24.265","Text":"or two squares down and two squares across every second."},{"Start":"19:24.265 ","End":"19:27.430","Text":"Then we\u0027ll have 1 second,"},{"Start":"19:27.430 ","End":"19:32.815","Text":"2 seconds, 3 seconds, and 4 seconds."},{"Start":"19:32.815 ","End":"19:36.550","Text":"Point A within 4 seconds has hit the obstacle."},{"Start":"19:36.550 ","End":"19:38.170","Text":"Point B, as we know,"},{"Start":"19:38.170 ","End":"19:42.910","Text":"is traveling with the same velocity as point A,"},{"Start":"19:42.910 ","End":"19:44.590","Text":"but because it\u0027s at an angle,"},{"Start":"19:44.590 ","End":"19:47.905","Text":"it\u0027s going to reach the obstacle at a different time."},{"Start":"19:47.905 ","End":"19:52.629","Text":"Again, down two and across two so 1 second,"},{"Start":"19:52.629 ","End":"19:55.855","Text":"2 seconds, 3 seconds,"},{"Start":"19:55.855 ","End":"19:59.410","Text":"4 seconds, 5 seconds."},{"Start":"19:59.410 ","End":"20:02.485","Text":"Point A takes 4 seconds to hit the obstacle."},{"Start":"20:02.485 ","End":"20:05.814","Text":"Point B takes 5 seconds to hit the obstacle,"},{"Start":"20:05.814 ","End":"20:09.655","Text":"and then point C will take 6 seconds to hit the obstacle."},{"Start":"20:09.655 ","End":"20:16.455","Text":"Between point A hitting the obstacle and point B hitting the obstacle,"},{"Start":"20:16.455 ","End":"20:19.650","Text":"1 second has gone by."},{"Start":"20:19.650 ","End":"20:23.969","Text":"Because this took 5 seconds and point A took 4 seconds,"},{"Start":"20:23.969 ","End":"20:26.265","Text":"so 5 minus 4 is 1."},{"Start":"20:26.265 ","End":"20:28.290","Text":"1 second has gone by,"},{"Start":"20:28.290 ","End":"20:31.755","Text":"and we know that once point A has hit the obstacle,"},{"Start":"20:31.755 ","End":"20:36.960","Text":"it\u0027s reflected back in a slightly different direction, it\u0027s reflected."},{"Start":"20:36.960 ","End":"20:40.370","Text":"However, it\u0027s still traveling with the same velocity."},{"Start":"20:40.370 ","End":"20:47.095","Text":"We said that it hits the obstacle and then after 1 second,"},{"Start":"20:47.095 ","End":"20:48.729","Text":"B hits the obstacle,"},{"Start":"20:48.729 ","End":"20:51.055","Text":"and that\u0027s where our diagram is meant to end."},{"Start":"20:51.055 ","End":"20:56.260","Text":"Point A is moving two squares down and two squares across."},{"Start":"20:56.260 ","End":"20:58.180","Text":"That means that point A,"},{"Start":"20:58.180 ","End":"21:01.134","Text":"which is reflected, is going to be over here."},{"Start":"21:01.134 ","End":"21:07.270","Text":"It hit the obstacle and then we move two down and two across."},{"Start":"21:07.270 ","End":"21:13.645","Text":"Point A, the reflected ray is going to be here like so."},{"Start":"21:13.645 ","End":"21:20.800","Text":"Now, before we\u0027re going to draw what our wave looks like,"},{"Start":"21:20.800 ","End":"21:22.644","Text":"let\u0027s draw it in green."},{"Start":"21:22.644 ","End":"21:26.485","Text":"We know that B and C are joined like so,"},{"Start":"21:26.485 ","End":"21:32.070","Text":"and A is going to be joined like this to B."},{"Start":"21:32.070 ","End":"21:36.565","Text":"This is the answer to the question."},{"Start":"21:36.565 ","End":"21:41.665","Text":"If we have a wavefront traveling towards some obstacle."},{"Start":"21:41.665 ","End":"21:45.969","Text":"A water wave, if half of the wavefront has hit the obstacle,"},{"Start":"21:45.969 ","End":"21:47.920","Text":"but the other half still has not,"},{"Start":"21:47.920 ","End":"21:54.350","Text":"this is what a wave at that moment will look like."},{"Start":"21:54.360 ","End":"21:58.315","Text":"We can see here specifically it\u0027s at a right angle,"},{"Start":"21:58.315 ","End":"21:59.830","Text":"but it doesn\u0027t have to be,"},{"Start":"21:59.830 ","End":"22:07.490","Text":"but the wave will be some triangular shape like so."},{"Start":"22:08.270 ","End":"22:15.210","Text":"Here we can see a picture from real life wave tank."},{"Start":"22:15.210 ","End":"22:17.610","Text":"What we can see over here,"},{"Start":"22:17.610 ","End":"22:20.010","Text":"let\u0027s do it in green,"},{"Start":"22:20.010 ","End":"22:23.300","Text":"is this is our obstacle."},{"Start":"22:23.300 ","End":"22:25.615","Text":"In green, we have our obstacle,"},{"Start":"22:25.615 ","End":"22:27.190","Text":"and then we have our wave."},{"Start":"22:27.190 ","End":"22:31.045","Text":"Here\u0027s our first wavefront and another wavefront,"},{"Start":"22:31.045 ","End":"22:32.965","Text":"and another wavefront,"},{"Start":"22:32.965 ","End":"22:36.955","Text":"and another wavefront like so and so on and so forth."},{"Start":"22:36.955 ","End":"22:40.660","Text":"Then what we can see is that over here,"},{"Start":"22:40.660 ","End":"22:45.580","Text":"we have a wavefront which is still traveling in this direction, however,"},{"Start":"22:45.580 ","End":"22:51.145","Text":"some of it has hit the obstacle."},{"Start":"22:51.145 ","End":"22:55.240","Text":"What we can see is that the point that would be here has hit"},{"Start":"22:55.240 ","End":"22:59.395","Text":"the obstacle and it\u0027s gone and moved a bit to the side."},{"Start":"22:59.395 ","End":"23:01.555","Text":"We can see that soon,"},{"Start":"23:01.555 ","End":"23:06.970","Text":"this point over here is going to move to this section over here,"},{"Start":"23:06.970 ","End":"23:09.985","Text":"and then it\u0027s going to be reflected at the same angle."},{"Start":"23:09.985 ","End":"23:15.680","Text":"It\u0027s going to be reflected to this point over here."},{"Start":"23:16.770 ","End":"23:19.270","Text":"We can draw that."},{"Start":"23:19.270 ","End":"23:26.080","Text":"Then eventually, this wavefront is going to be here and"},{"Start":"23:26.080 ","End":"23:29.169","Text":"this point is soon going to reach"},{"Start":"23:29.169 ","End":"23:35.239","Text":"the obstacle and then it\u0027s going to be reflected like so."},{"Start":"23:36.000 ","End":"23:40.390","Text":"Then what we can see is that we\u0027ll have lots of"},{"Start":"23:40.390 ","End":"23:44.815","Text":"wavefronts which are still moving forwards,"},{"Start":"23:44.815 ","End":"23:48.750","Text":"and the reflected parts of"},{"Start":"23:48.750 ","End":"23:57.174","Text":"the wavefront that have already hit the obstacle and been reflected."},{"Start":"23:57.174 ","End":"23:58.570","Text":"Now what we can see is that"},{"Start":"23:58.570 ","End":"24:03.639","Text":"our reflected waves are slightly more blurred and that\u0027s because some of"},{"Start":"24:03.639 ","End":"24:11.740","Text":"the energy from the water wave hitting the obstacle will cause some energy loss,"},{"Start":"24:11.740 ","End":"24:18.490","Text":"so the reflected waves will have slightly less energy."},{"Start":"24:18.620 ","End":"24:24.485","Text":"What we can see is that our obstacle is at 45 degrees."},{"Start":"24:24.485 ","End":"24:31.839","Text":"We can see that our reflected rays are going to be reflected like so and they\u0027re"},{"Start":"24:31.839 ","End":"24:35.170","Text":"traveling in this direction whereas our incident rays are"},{"Start":"24:35.170 ","End":"24:39.860","Text":"traveling in this direction like so."},{"Start":"24:40.020 ","End":"24:44.274","Text":"I know it\u0027s a bit hard to get your head around it,"},{"Start":"24:44.274 ","End":"24:48.730","Text":"but it is something that\u0027s difficult to understand."},{"Start":"24:48.730 ","End":"24:54.609","Text":"Just remember that as different parts of the wavefront hit the obstacle,"},{"Start":"24:54.609 ","End":"24:57.174","Text":"each point is going to be"},{"Start":"24:57.174 ","End":"25:03.715","Text":"reflected and then we\u0027ll get a shape just like what we did over here."},{"Start":"25:03.715 ","End":"25:07.555","Text":"Now we can see that we get this lattice shape."},{"Start":"25:07.555 ","End":"25:12.999","Text":"If I rub everything out we can see that on this diagram,"},{"Start":"25:12.999 ","End":"25:15.175","Text":"the brightest points,"},{"Start":"25:15.175 ","End":"25:19.075","Text":"so let\u0027s say this point over here or here,"},{"Start":"25:19.075 ","End":"25:24.804","Text":"and so on are the points where the wavefront of"},{"Start":"25:24.804 ","End":"25:32.500","Text":"our reflected wave hits the wavefront of our incident wave."},{"Start":"25:32.500 ","End":"25:36.550","Text":"If I draw it again, here we have an incident,"},{"Start":"25:36.550 ","End":"25:40.825","Text":"and then here I have my reflected, and also here."},{"Start":"25:40.825 ","End":"25:44.950","Text":"They go through, they meet at these points over here,"},{"Start":"25:44.950 ","End":"25:49.160","Text":"which are our brightest points in the diagram."},{"Start":"25:50.340 ","End":"25:56.230","Text":"This is what reflection looks like when we have a straight wavefront."},{"Start":"25:56.230 ","End":"26:01.310","Text":"Now let\u0027s talk about what happens when we have a circular wavefront."},{"Start":"26:01.730 ","End":"26:05.444","Text":"Here we have our circular wave,"},{"Start":"26:05.444 ","End":"26:09.780","Text":"which is generated by this pin hitting the water"},{"Start":"26:09.780 ","End":"26:14.925","Text":"at some constant frequency causing a circular wave."},{"Start":"26:14.925 ","End":"26:18.125","Text":"We know that our circular wave is periodical,"},{"Start":"26:18.125 ","End":"26:22.495","Text":"so here is our circular wavefront,"},{"Start":"26:22.495 ","End":"26:23.875","Text":"so our latest one."},{"Start":"26:23.875 ","End":"26:25.600","Text":"Ignore this."},{"Start":"26:25.600 ","End":"26:27.385","Text":"I\u0027ll explain it afterwards."},{"Start":"26:27.385 ","End":"26:35.755","Text":"Then the circular wavefront generated right before that 1 is located over here."},{"Start":"26:35.755 ","End":"26:37.479","Text":"It\u0027s traveled out a bit more."},{"Start":"26:37.479 ","End":"26:41.079","Text":"The circular wavefront generated right before that one"},{"Start":"26:41.079 ","End":"26:45.084","Text":"has also traveled a bit further and it looks like so,"},{"Start":"26:45.084 ","End":"26:48.070","Text":"and so on and so forth."},{"Start":"26:48.070 ","End":"26:58.190","Text":"Now let\u0027s go back to the fact that we have over here some barrier or some obstacle."},{"Start":"26:59.100 ","End":"27:04.195","Text":"Let\u0027s imagine again that we have our circular wavefront,"},{"Start":"27:04.195 ","End":"27:10.060","Text":"which is arriving at the barrier like so."},{"Start":"27:10.060 ","End":"27:11.920","Text":"This is our circular wavefront,"},{"Start":"27:11.920 ","End":"27:14.484","Text":"not too elegantly drawn."},{"Start":"27:14.484 ","End":"27:19.375","Text":"What we can see is that this point over here hits the barrier"},{"Start":"27:19.375 ","End":"27:24.265","Text":"first and so it will travel upwards like this."},{"Start":"27:24.265 ","End":"27:26.754","Text":"Then slightly later on,"},{"Start":"27:26.754 ","End":"27:30.040","Text":"this point over here will hit"},{"Start":"27:30.040 ","End":"27:35.065","Text":"the barrier and then it too will be reflected and travel upwards,"},{"Start":"27:35.065 ","End":"27:39.385","Text":"but by that time, this point over here would have traveled further."},{"Start":"27:39.385 ","End":"27:41.469","Text":"Similarly with this point,"},{"Start":"27:41.469 ","End":"27:44.050","Text":"it hits the barrier after this point over here,"},{"Start":"27:44.050 ","End":"27:46.570","Text":"so it still has to travel like this."},{"Start":"27:46.570 ","End":"27:48.520","Text":"Then a little while later,"},{"Start":"27:48.520 ","End":"27:52.000","Text":"it too is reflected back up just like this."},{"Start":"27:52.000 ","End":"27:54.140","Text":"It\u0027s a mirror image."},{"Start":"27:55.080 ","End":"27:57.759","Text":"It\u0027s the same for all of these points."},{"Start":"27:57.759 ","End":"28:04.660","Text":"They travel a bit further and then they travel back up like so when they\u0027re reflected."},{"Start":"28:04.660 ","End":"28:08.034","Text":"What we can see is that we\u0027ll get"},{"Start":"28:08.034 ","End":"28:16.525","Text":"some reflection that looks like this if I would have drawn it properly,"},{"Start":"28:16.525 ","End":"28:21.100","Text":"which is exactly what we can see in our reflected wavefronts."},{"Start":"28:21.100 ","End":"28:27.550","Text":"This is another reflected wavefront and another 1 over here,"},{"Start":"28:27.550 ","End":"28:30.010","Text":"and another 1 over here."},{"Start":"28:30.010 ","End":"28:35.785","Text":"What we get is if we had some circle traveling towards the barrier,"},{"Start":"28:35.785 ","End":"28:40.630","Text":"so the tip of the circle which reaches the barrier"},{"Start":"28:40.630 ","End":"28:45.490","Text":"first is reflected obviously first and it starts traveling this way,"},{"Start":"28:45.490 ","End":"28:49.689","Text":"and then all the subsequent points on the circumference of the circle."},{"Start":"28:49.689 ","End":"28:53.170","Text":"Now, maybe if it will make it easier for you to understand,"},{"Start":"28:53.170 ","End":"28:57.100","Text":"if you take a piece of paper and you cut it into a circle and"},{"Start":"28:57.100 ","End":"29:03.070","Text":"then you bring it up to a wall or some pencil barrier that you make,"},{"Start":"29:03.070 ","End":"29:07.989","Text":"so you\u0027ll have a circle traveling towards"},{"Start":"29:07.989 ","End":"29:13.960","Text":"your barrier and as the circle is closer to the barrier,"},{"Start":"29:13.960 ","End":"29:16.885","Text":"you\u0027ll start folding the circle in."},{"Start":"29:16.885 ","End":"29:18.835","Text":"A few moments later,"},{"Start":"29:18.835 ","End":"29:22.839","Text":"you\u0027ll have something that looks like this,"},{"Start":"29:22.839 ","End":"29:28.945","Text":"and then you\u0027ll have a section of the cycle that is reflected which will be folded in."},{"Start":"29:28.945 ","End":"29:33.835","Text":"If you fold the top third of the circle in,"},{"Start":"29:33.835 ","End":"29:37.255","Text":"this is your reflected here in blue."},{"Start":"29:37.255 ","End":"29:38.680","Text":"I switched the colors."},{"Start":"29:38.680 ","End":"29:41.185","Text":"Here in blue, you have your reflected,"},{"Start":"29:41.185 ","End":"29:42.534","Text":"and here in green,"},{"Start":"29:42.534 ","End":"29:43.990","Text":"you have your incident."},{"Start":"29:43.990 ","End":"29:47.635","Text":"As you move the circle closer and closer,"},{"Start":"29:47.635 ","End":"29:52.735","Text":"so then half the circle still hasn\u0027t hit,"},{"Start":"29:52.735 ","End":"29:54.850","Text":"but half the circle has it,"},{"Start":"29:54.850 ","End":"29:57.955","Text":"so here you\u0027ll get your reflected on the same line."},{"Start":"29:57.955 ","End":"30:02.920","Text":"Now it\u0027s as if you folded the paper in half and so on and so forth."},{"Start":"30:02.920 ","End":"30:12.864","Text":"Then you\u0027ll get your reflected waves traveling like so outwards like this."},{"Start":"30:12.864 ","End":"30:18.564","Text":"I don\u0027t know if that confused you or made it maybe a little bit easier to understand."},{"Start":"30:18.564 ","End":"30:21.265","Text":"If it confuse you, just ignore that."},{"Start":"30:21.265 ","End":"30:24.290","Text":"That\u0027s the end of our lesson."}],"ID":12480},{"Watched":false,"Name":"Water Waves Travelling Through Different Mediums","Duration":"15m 17s","ChapterTopicVideoID":12013,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.815","Text":"Hello, in this lesson,"},{"Start":"00:01.815 ","End":"00:04.080","Text":"we\u0027re going to be learning how water waves can"},{"Start":"00:04.080 ","End":"00:07.545","Text":"travel through different transmission mediums."},{"Start":"00:07.545 ","End":"00:11.220","Text":"We learned in this section about light waves,"},{"Start":"00:11.220 ","End":"00:13.485","Text":"that we have refraction."},{"Start":"00:13.485 ","End":"00:16.365","Text":"That\u0027s when light is traveling through"},{"Start":"00:16.365 ","End":"00:21.285","Text":"different transmission mediums so we get this physical effect."},{"Start":"00:21.285 ","End":"00:25.950","Text":"Right now we\u0027re going to be speaking about this physical factor fraction,"},{"Start":"00:25.950 ","End":"00:30.540","Text":"but in terms of water waves or in terms of 2D waves."},{"Start":"00:30.540 ","End":"00:36.570","Text":"If we remember, light is an example of a 3-dimensional wave,"},{"Start":"00:36.570 ","End":"00:39.960","Text":"and water is an example of a 2-dimensional wave."},{"Start":"00:39.960 ","End":"00:43.875","Text":"A little note on transmission mediums."},{"Start":"00:43.875 ","End":"00:47.835","Text":"That\u0027s the substance that can propagate energy waves."},{"Start":"00:47.835 ","End":"00:52.410","Text":"It just means the material that waves pass through."},{"Start":"00:52.410 ","End":"00:55.815","Text":"That can be a spring, water,"},{"Start":"00:55.815 ","End":"00:59.660","Text":"a rope, and with light for example,"},{"Start":"00:59.660 ","End":"01:03.085","Text":"that can also be even just a vacuum."},{"Start":"01:03.085 ","End":"01:11.610","Text":"We remember when we were dealing with lights and with refraction that we had n,"},{"Start":"01:12.100 ","End":"01:15.790","Text":"which is of course their refractive index."},{"Start":"01:15.790 ","End":"01:19.445","Text":"We said that it\u0027s equal to the speed of light,"},{"Start":"01:19.445 ","End":"01:22.780","Text":"c divided by v,"},{"Start":"01:22.780 ","End":"01:30.090","Text":"where v is the speed of light traveling through that specific transmission medium."},{"Start":"01:30.740 ","End":"01:34.250","Text":"C is the speed of light in a vacuum"},{"Start":"01:34.250 ","End":"01:37.535","Text":"and v is the speed of light in the transmission medium."},{"Start":"01:37.535 ","End":"01:42.595","Text":"For instance, the speed of light traveling through water."},{"Start":"01:42.595 ","End":"01:47.060","Text":"In previous lessons, we already learned about"},{"Start":"01:47.060 ","End":"01:53.515","Text":"our 1-dimensional wave where we use the example of a spring."},{"Start":"01:53.515 ","End":"01:57.110","Text":"If you remember, we gave an example of"},{"Start":"01:57.110 ","End":"02:00.850","Text":"a thin spring and then we had a wave passing through it."},{"Start":"02:00.850 ","End":"02:05.060","Text":"Then our spring turned"},{"Start":"02:05.060 ","End":"02:12.800","Text":"into a slightly different transmission medium which was a thicker spring."},{"Start":"02:12.800 ","End":"02:18.965","Text":"We had our wave passing from a thin spring and into a thick spring."},{"Start":"02:18.965 ","End":"02:24.870","Text":"We saw that if this is transmission medium a,"},{"Start":"02:24.870 ","End":"02:28.755","Text":"then this would be transmission medium b."},{"Start":"02:28.755 ","End":"02:33.405","Text":"We saw that the velocity of the wave traveling through"},{"Start":"02:33.405 ","End":"02:36.260","Text":"transmission medium a would be"},{"Start":"02:36.260 ","End":"02:41.210","Text":"different to the velocity of the wave traveling through transmission medium b."},{"Start":"02:41.210 ","End":"02:45.920","Text":"Because if we remember our equation for the velocity was equal to"},{"Start":"02:45.920 ","End":"02:51.845","Text":"the square root of the tension in the spring or in the rope,"},{"Start":"02:51.845 ","End":"02:58.590","Text":"divided by Mu, which was the density per unit length."},{"Start":"02:58.640 ","End":"03:02.150","Text":"We remembered that the velocity of wave is only"},{"Start":"03:02.150 ","End":"03:06.815","Text":"dependent on how taut the rope or the spring is,"},{"Start":"03:06.815 ","End":"03:08.770","Text":"and the density of the spring."},{"Start":"03:08.770 ","End":"03:13.010","Text":"We could see that the velocity of the wave would change as it"},{"Start":"03:13.010 ","End":"03:18.550","Text":"switches from transmission medium a to transmission medium b."},{"Start":"03:18.550 ","End":"03:23.535","Text":"What exactly happens in a 2D wave"},{"Start":"03:23.535 ","End":"03:29.960","Text":"where we have specifically been speaking about water waves?"},{"Start":"03:31.470 ","End":"03:38.930","Text":"In water, the velocity of the water wave is dependent on the depth of the water."},{"Start":"03:38.930 ","End":"03:41.080","Text":"The deeper the water,"},{"Start":"03:41.080 ","End":"03:43.405","Text":"the faster the wave will travel."},{"Start":"03:43.405 ","End":"03:48.520","Text":"Now, this is only correct for small depths of up to a few centimeters."},{"Start":"03:48.520 ","End":"03:54.945","Text":"If you take the depth of an ocean then this will not be correct."},{"Start":"03:54.945 ","End":"03:58.465","Text":"We\u0027re specifically speaking about this when we\u0027re dealing with"},{"Start":"03:58.465 ","End":"04:03.680","Text":"depths that are relevant for testing inside the lab."},{"Start":"04:04.250 ","End":"04:09.140","Text":"How can I use this fact to test what happens to"},{"Start":"04:09.140 ","End":"04:14.435","Text":"water waves when they travel through different transmission mediums."},{"Start":"04:14.435 ","End":"04:21.285","Text":"Let\u0027s scroll down and let\u0027s draw our tank."},{"Start":"04:21.285 ","End":"04:25.420","Text":"Here I have my water tank,"},{"Start":"04:25.420 ","End":"04:32.070","Text":"and here I can have my water level."},{"Start":"04:32.070 ","End":"04:35.020","Text":"It\u0027s like so."},{"Start":"04:35.720 ","End":"04:42.650","Text":"Then over here I\u0027ll play some wave maker that will move up and down and cause"},{"Start":"04:42.650 ","End":"04:50.500","Text":"a disturbance that will cause some wave to travel through."},{"Start":"04:50.500 ","End":"05:00.470","Text":"Now I have a wave traveling like so."},{"Start":"05:00.670 ","End":"05:03.755","Text":"If I\u0027ve drawn this correctly,"},{"Start":"05:03.755 ","End":"05:08.885","Text":"we know that the wavelength between each peak,"},{"Start":"05:08.885 ","End":"05:12.920","Text":"so the distance between each peak is going to be the wavelength,"},{"Start":"05:12.920 ","End":"05:15.425","Text":"and this is a constant."},{"Start":"05:15.425 ","End":"05:19.080","Text":"If we call this lambda 1 from this peak to this peak,"},{"Start":"05:19.080 ","End":"05:24.360","Text":"so the distance from this peak to this peak will also be lambda 1."},{"Start":"05:24.980 ","End":"05:34.175","Text":"Of course, that the amplitude is the same over here as it is over here,"},{"Start":"05:34.175 ","End":"05:37.710","Text":"over here as well, and over here."},{"Start":"05:37.710 ","End":"05:41.585","Text":"My drawing, my waves are meant to be of equal size."},{"Start":"05:41.585 ","End":"05:48.670","Text":"Then we can see that here we have some depth."},{"Start":"05:48.670 ","End":"05:53.049","Text":"If over here, I play some block"},{"Start":"05:53.049 ","End":"05:57.830","Text":"made out of any material but let\u0027s say metal so that it won\u0027t float upwards."},{"Start":"05:57.830 ","End":"05:59.855","Text":"Something that will stay down here,"},{"Start":"05:59.855 ","End":"06:02.950","Text":"that we can see that I\u0027ve changed the depth."},{"Start":"06:02.950 ","End":"06:07.175","Text":"Now my depth is approximately half."},{"Start":"06:07.175 ","End":"06:14.950","Text":"That means that here our water wave was traveling with a faster velocity,"},{"Start":"06:14.950 ","End":"06:17.644","Text":"and here because the depth has changed,"},{"Start":"06:17.644 ","End":"06:24.960","Text":"I know that my water wave will be traveling at a slow or a slower velocity."},{"Start":"06:25.490 ","End":"06:29.570","Text":"A question can be asked given this diagram,"},{"Start":"06:29.570 ","End":"06:32.935","Text":"that is what will the wave look like?"},{"Start":"06:32.935 ","End":"06:37.130","Text":"First of all, we know that we have an equation"},{"Start":"06:37.130 ","End":"06:41.690","Text":"connecting our wavelength and our velocity of"},{"Start":"06:41.690 ","End":"06:45.815","Text":"a wave and that is the velocity is equal to"},{"Start":"06:45.815 ","End":"06:50.165","Text":"the wavelength of the wave multiplied by its frequency."},{"Start":"06:50.165 ","End":"06:53.660","Text":"This is correct when we\u0027re dealing with 1D waves,"},{"Start":"06:53.660 ","End":"06:55.355","Text":"with 2D waves,"},{"Start":"06:55.355 ","End":"06:58.500","Text":"and with 3D waves."},{"Start":"06:58.970 ","End":"07:05.360","Text":"Let\u0027s call the area where the wave is moving faster."},{"Start":"07:05.360 ","End":"07:09.435","Text":"Let\u0027s call this transmission medium a."},{"Start":"07:09.435 ","End":"07:13.925","Text":"Then where the wave is moving slower because the depth has changed,"},{"Start":"07:13.925 ","End":"07:17.770","Text":"let\u0027s call this transmission medium b."},{"Start":"07:17.770 ","End":"07:20.460","Text":"In transmission medium a,"},{"Start":"07:20.460 ","End":"07:22.080","Text":"we know the velocity,"},{"Start":"07:22.080 ","End":"07:24.835","Text":"so let\u0027s call this a."},{"Start":"07:24.835 ","End":"07:26.165","Text":"We know the velocity,"},{"Start":"07:26.165 ","End":"07:28.730","Text":"and it\u0027s the fast velocity,"},{"Start":"07:28.730 ","End":"07:33.450","Text":"and we also know the frequency."},{"Start":"07:34.800 ","End":"07:41.650","Text":"Because we are deciding it based on this machine over here."},{"Start":"07:41.650 ","End":"07:49.255","Text":"Therefore, we can simplify to find out what our Lambda is or what our wavelength is,"},{"Start":"07:49.255 ","End":"07:53.395","Text":"which is what we defined over here as Lambda 1."},{"Start":"07:53.395 ","End":"07:57.410","Text":"Here we have our wavelength."},{"Start":"07:58.620 ","End":"08:06.070","Text":"Now, what is going to happen when we move through transmission medium B?"},{"Start":"08:06.070 ","End":"08:09.640","Text":"First of all, from what we said over here,"},{"Start":"08:09.640 ","End":"08:16.225","Text":"we know that the water is now more shallow in transmission medium B,"},{"Start":"08:16.225 ","End":"08:23.005","Text":"and therefore, the velocity is going to be slower, we already said that."},{"Start":"08:23.005 ","End":"08:26.690","Text":"Our V is slower,"},{"Start":"08:27.210 ","End":"08:31.015","Text":"but we\u0027re still using the same equation."},{"Start":"08:31.015 ","End":"08:33.040","Text":"This is equal to Lambda,"},{"Start":"08:33.040 ","End":"08:36.505","Text":"the wavelength, multiplied by the frequency."},{"Start":"08:36.505 ","End":"08:40.705","Text":"We know that our value for V has to be reduced,"},{"Start":"08:40.705 ","End":"08:43.210","Text":"and now we need to balance out this side."},{"Start":"08:43.210 ","End":"08:49.550","Text":"We want to know if the wavelength changes or if the frequency changes."},{"Start":"08:49.620 ","End":"08:56.545","Text":"We know that our wave maker is moving up and down with some frequency."},{"Start":"08:56.545 ","End":"09:01.285","Text":"Let\u0027s say it has a frequency of 1 hertz."},{"Start":"09:01.285 ","End":"09:05.290","Text":"That means 1 full wave per second."},{"Start":"09:05.290 ","End":"09:09.430","Text":"We can see that every second we have another wave and"},{"Start":"09:09.430 ","End":"09:13.990","Text":"it keeps on traveling until it gets to this area,"},{"Start":"09:13.990 ","End":"09:18.970","Text":"which is where the transmission medium changes from A to B."},{"Start":"09:18.970 ","End":"09:25.735","Text":"We can see that every second another wave is going to go past this line over here."},{"Start":"09:25.735 ","End":"09:29.860","Text":"Which means that after this line over here,"},{"Start":"09:29.860 ","End":"09:34.240","Text":"we\u0027re also going to have a similar frequency."},{"Start":"09:34.240 ","End":"09:38.830","Text":"There\u0027s no reason why the frequency would change at this imaginary line."},{"Start":"09:38.830 ","End":"09:42.160","Text":"We\u0027re still going to have waves going at a frequency of"},{"Start":"09:42.160 ","End":"09:47.305","Text":"1 hertz over here as well. Let\u0027s write that in."},{"Start":"09:47.305 ","End":"09:54.025","Text":"Nothing changes, so the frequency stays the same."},{"Start":"09:54.025 ","End":"09:57.985","Text":"There\u0027s nothing that would cause it to change."},{"Start":"09:57.985 ","End":"10:05.720","Text":"Therefore, that means that our wavelength becomes smaller."},{"Start":"10:05.840 ","End":"10:10.740","Text":"Our wavelength is reduced. What does that mean?"},{"Start":"10:10.740 ","End":"10:16.970","Text":"That means that the waves passing through are going to be more squashed."},{"Start":"10:16.970 ","End":"10:21.745","Text":"Instead of having this long wavelength,"},{"Start":"10:21.745 ","End":"10:26.005","Text":"the amplitude is going to be exactly the same."},{"Start":"10:26.005 ","End":"10:34.915","Text":"However, the distance from peak to peak will be less."},{"Start":"10:34.915 ","End":"10:38.905","Text":"I\u0027m drawing in blue that the amplitude"},{"Start":"10:38.905 ","End":"10:44.360","Text":"is the exact same amplitude as we had in transmission medium A."},{"Start":"10:44.400 ","End":"10:46.900","Text":"This is the same size."},{"Start":"10:46.900 ","End":"10:52.450","Text":"The only thing that has changed is this distance from peak to peak."},{"Start":"10:52.450 ","End":"10:54.190","Text":"We have a different wavelength,"},{"Start":"10:54.190 ","End":"10:56.780","Text":"let\u0027s call it Lambda 2."},{"Start":"10:57.120 ","End":"10:59.680","Text":"Let\u0027s draw this tank,"},{"Start":"10:59.680 ","End":"11:03.100","Text":"this exact same tank from a bird\u0027s eye view,"},{"Start":"11:03.100 ","End":"11:09.850","Text":"and let\u0027s see what the wave fronts will look like."},{"Start":"11:09.850 ","End":"11:15.040","Text":"Let\u0027s imagine that up until here we had"},{"Start":"11:15.040 ","End":"11:20.695","Text":"transmission medium A and here we had transmission medium B."},{"Start":"11:20.695 ","End":"11:25.975","Text":"As we know, transmission medium A is the deep end,"},{"Start":"11:25.975 ","End":"11:30.550","Text":"and transmission medium B is the shallow end."},{"Start":"11:30.550 ","End":"11:33.730","Text":"And we know that in the deep end we\u0027re moving"},{"Start":"11:33.730 ","End":"11:37.555","Text":"fast and in the shallow end we\u0027re moving slow."},{"Start":"11:37.555 ","End":"11:42.070","Text":"Our wavefronts in the deep end would look like so."},{"Start":"11:42.070 ","End":"11:47.530","Text":"Here\u0027s a wavefront, here\u0027s another wavefront, and another wavefront."},{"Start":"11:47.530 ","End":"11:53.830","Text":"We can see that we have this wavelength over here of Lambda 1."},{"Start":"11:53.830 ","End":"11:59.935","Text":"Also, the distance from here to here is of course the same wavelength for Lambda 1."},{"Start":"11:59.935 ","End":"12:04.480","Text":"However, once we pass into the shallow end or transmission medium B,"},{"Start":"12:04.480 ","End":"12:06.700","Text":"our wave starts moving slower,"},{"Start":"12:06.700 ","End":"12:10.720","Text":"which as we saw, means that we get a reduced wavelength."},{"Start":"12:10.720 ","End":"12:15.240","Text":"That means that here we\u0027ll have a wavefront."},{"Start":"12:15.240 ","End":"12:20.370","Text":"Then here we\u0027ll have another wavefront and another wavefront."},{"Start":"12:20.370 ","End":"12:25.390","Text":"We can see that the wavefronts or just squashing a little bit more."},{"Start":"12:25.390 ","End":"12:29.140","Text":"Imagine obviously that these waves are moving at"},{"Start":"12:29.140 ","End":"12:34.780","Text":"right angles to the sides of the tank and that they\u0027re completely straight lines."},{"Start":"12:34.780 ","End":"12:38.740","Text":"Here we can see that we have this wavelength Lambda 2"},{"Start":"12:38.740 ","End":"12:43.435","Text":"and here we also have this wavelength Lambda 2."},{"Start":"12:43.435 ","End":"12:48.085","Text":"Now if we\u0027re just going to liken these water waves too,"},{"Start":"12:48.085 ","End":"12:50.946","Text":"if we\u0027re dealing with light,"},{"Start":"12:50.946 ","End":"12:56.040","Text":"we know that our water waves are traveling in this direction like so in the red."},{"Start":"12:56.040 ","End":"13:01.275","Text":"But let\u0027s imagine that this was a light ray or a wave of light."},{"Start":"13:01.275 ","End":"13:07.450","Text":"If this was our border between one transmission medium to another,"},{"Start":"13:07.450 ","End":"13:12.940","Text":"where we\u0027re going from a more dense medium to a less dense medium,"},{"Start":"13:12.940 ","End":"13:14.607","Text":"or vice versa,"},{"Start":"13:14.607 ","End":"13:22.750","Text":"we know that we have to find the normal to our obstacle or our border which is here."},{"Start":"13:22.750 ","End":"13:29.275","Text":"Then the light wave will travel straight on parallel to the normal line,"},{"Start":"13:29.275 ","End":"13:39.310","Text":"which means that the angle or the incident angle is going to be equal to 0."},{"Start":"13:39.310 ","End":"13:44.350","Text":"That means that the wave of light after it passes,"},{"Start":"13:44.350 ","End":"13:47.635","Text":"or let\u0027s call this 1 after it passes through"},{"Start":"13:47.635 ","End":"13:52.135","Text":"this change in density between one medium to another,"},{"Start":"13:52.135 ","End":"13:57.400","Text":"the light is going to be carrying on traveling in"},{"Start":"13:57.400 ","End":"14:04.240","Text":"the same direction with a refracted angle of 0."},{"Start":"14:04.240 ","End":"14:08.020","Text":"It\u0027s just going to carry on straight, nothing changes."},{"Start":"14:08.020 ","End":"14:13.660","Text":"In that way, we can see that this is similar to the water waves that are traveling in"},{"Start":"14:13.660 ","End":"14:21.500","Text":"this straight wavefront fashion and they just keep moving in the same direction."},{"Start":"14:21.720 ","End":"14:24.114","Text":"This, of course,"},{"Start":"14:24.114 ","End":"14:29.275","Text":"comes from Snell\u0027s law."},{"Start":"14:29.275 ","End":"14:32.470","Text":"You can look it up on the internet,"},{"Start":"14:32.470 ","End":"14:36.145","Text":"but we will speak about it later on in this chapter."},{"Start":"14:36.145 ","End":"14:37.990","Text":"With these water waves,"},{"Start":"14:37.990 ","End":"14:42.310","Text":"we can see that if we remember the definition of the angle"},{"Start":"14:42.310 ","End":"14:47.260","Text":"between a wavefront and an obstacle is just that,"},{"Start":"14:47.260 ","End":"14:49.240","Text":"the angle between the wavefront and the obstacle."},{"Start":"14:49.240 ","End":"14:55.630","Text":"We can see that the angle is 0 because the wavefront is parallel to the obstacle."},{"Start":"14:55.630 ","End":"14:59.020","Text":"That means that the incident angle,"},{"Start":"14:59.020 ","End":"15:02.830","Text":"let\u0027s call that Theta 1 tag of the water wave is going to be equal to 0."},{"Start":"15:02.830 ","End":"15:11.515","Text":"Therefore, the refracted angle of the water wave Theta 2 tag will also be 0."},{"Start":"15:11.515 ","End":"15:14.515","Text":"This is again Snell\u0027s law."},{"Start":"15:14.515 ","End":"15:17.840","Text":"That\u0027s the end of this lesson."}],"ID":12481},{"Watched":false,"Name":"Derivation Two Dimentional Wave","Duration":"25m 24s","ChapterTopicVideoID":12014,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.755","Text":"Hello. In this lesson,"},{"Start":"00:01.755 ","End":"00:08.595","Text":"we\u0027re going to be speaking about water waves that move between one medium to another."},{"Start":"00:08.595 ","End":"00:15.450","Text":"Here, we can say that this is transmission medium number 1 and that here,"},{"Start":"00:15.450 ","End":"00:18.405","Text":"on this side of this line,"},{"Start":"00:18.405 ","End":"00:21.585","Text":"we have transmission medium number 2."},{"Start":"00:21.585 ","End":"00:26.430","Text":"This line represents a change in depth of our water tank."},{"Start":"00:26.430 ","End":"00:31.740","Text":"Let\u0027s say that transmission medium number 1 is the deep end of"},{"Start":"00:31.740 ","End":"00:37.965","Text":"the tank and transmission medium number 2 is the shallow end of the tank."},{"Start":"00:37.965 ","End":"00:41.320","Text":"What we know is that if we\u0027re in the deep end,"},{"Start":"00:41.320 ","End":"00:45.825","Text":"then the velocity of the wave is fast,"},{"Start":"00:45.825 ","End":"00:49.055","Text":"so we have a wave traveling fast,"},{"Start":"00:49.055 ","End":"00:52.010","Text":"and we know that in the shallow end,"},{"Start":"00:52.010 ","End":"00:56.430","Text":"our wave is going to travel slower,"},{"Start":"00:56.430 ","End":"00:59.100","Text":"so here, we have a slow wave."},{"Start":"00:59.100 ","End":"01:03.140","Text":"That\u0027s what\u0027s happening in transmission medium number 1 and transmission medium number 2."},{"Start":"01:03.140 ","End":"01:09.735","Text":"This line represents the change in depth."},{"Start":"01:09.735 ","End":"01:15.625","Text":"If we start off with a wave and let\u0027s draw it over here."},{"Start":"01:15.625 ","End":"01:17.990","Text":"Let\u0027s call this point,"},{"Start":"01:17.990 ","End":"01:22.859","Text":"point A_naught, then this point over here,"},{"Start":"01:22.859 ","End":"01:25.200","Text":"so 4 squares across, is B_naught,"},{"Start":"01:25.200 ","End":"01:29.025","Text":"then another 4 squares across, C_naught,"},{"Start":"01:29.025 ","End":"01:31.200","Text":"another 4 squares across, D_naught,"},{"Start":"01:31.200 ","End":"01:35.070","Text":"and right at the end, E_naught."},{"Start":"01:35.070 ","End":"01:39.545","Text":"As we know, our wave is traveling in this direction"},{"Start":"01:39.545 ","End":"01:45.071","Text":"perpendicular to its wavefront and then a few moments later, so 1,"},{"Start":"01:45.071 ","End":"01:47.375","Text":"2, 3, 4 squares later,"},{"Start":"01:47.375 ","End":"01:51.590","Text":"our point A_naught is going"},{"Start":"01:51.590 ","End":"01:56.510","Text":"to come into contact at the barrier between the change in depth,"},{"Start":"01:56.510 ","End":"01:58.775","Text":"so let\u0027s call this point A_1."},{"Start":"01:58.775 ","End":"02:05.220","Text":"Of course, this is what the rest of the wavefront is going to look like."},{"Start":"02:05.220 ","End":"02:09.405","Text":"It still hasn\u0027t come in contact with the barrier."},{"Start":"02:09.405 ","End":"02:11.985","Text":"Here we have B_1,"},{"Start":"02:11.985 ","End":"02:14.295","Text":"here we have C_1,"},{"Start":"02:14.295 ","End":"02:16.635","Text":"here we have D_1,"},{"Start":"02:16.635 ","End":"02:19.410","Text":"and here we have E_1."},{"Start":"02:19.410 ","End":"02:22.460","Text":"A few moments later,"},{"Start":"02:22.460 ","End":"02:26.325","Text":"let\u0027s say that C has touched the barrier,"},{"Start":"02:26.325 ","End":"02:28.500","Text":"so we\u0027ve moved up 1, 2, 3,"},{"Start":"02:28.500 ","End":"02:32.160","Text":"4, 5 squares."},{"Start":"02:32.160 ","End":"02:35.700","Text":"This we can call already C_2."},{"Start":"02:35.700 ","End":"02:41.910","Text":"As we know, if it hasn\u0027t passed the barrier this side of"},{"Start":"02:41.910 ","End":"02:44.906","Text":"the wavefront is going to be traveling at"},{"Start":"02:44.906 ","End":"02:49.855","Text":"the exact same velocity that it was this entire time."},{"Start":"02:49.855 ","End":"02:53.175","Text":"However, what\u0027s going to happen to points A"},{"Start":"02:53.175 ","End":"02:56.870","Text":"and B once they\u0027ve passed into the shallow end?"},{"Start":"02:56.870 ","End":"02:58.715","Text":"We know that in the shallow end,"},{"Start":"02:58.715 ","End":"03:02.314","Text":"the wave is traveling with a slower velocity."},{"Start":"03:02.314 ","End":"03:08.200","Text":"If our original wavefront which is traveling in the deep end,"},{"Start":"03:08.200 ","End":"03:10.025","Text":"so it has a faster velocity,"},{"Start":"03:10.025 ","End":"03:11.660","Text":"managed to travel 1,"},{"Start":"03:11.660 ","End":"03:13.280","Text":"2, 3, 4,"},{"Start":"03:13.280 ","End":"03:18.335","Text":"5 squares up so we know that A and B would have traveled less."},{"Start":"03:18.335 ","End":"03:23.060","Text":"A, because it was the first point to reach the shallow end,"},{"Start":"03:23.060 ","End":"03:26.300","Text":"it\u0027s been traveling slower for longer."},{"Start":"03:26.300 ","End":"03:28.955","Text":"It would have only moved up, let\u0027s say,"},{"Start":"03:28.955 ","End":"03:32.640","Text":"in this whole time, 2.5 squares."},{"Start":"03:32.640 ","End":"03:40.190","Text":"Then B had almost 3 squares where it was traveling at the same velocity as C,"},{"Start":"03:40.190 ","End":"03:42.675","Text":"D, and E. Then afterwards,"},{"Start":"03:42.675 ","End":"03:44.595","Text":"it too hit the slow end,"},{"Start":"03:44.595 ","End":"03:48.210","Text":"so it also had to slow down."},{"Start":"03:48.210 ","End":"03:51.260","Text":"It will be located over here."},{"Start":"03:51.260 ","End":"03:55.159","Text":"Then if I draw what our new wave will look like,"},{"Start":"03:55.159 ","End":"03:58.790","Text":"so let\u0027s call this A_2 and B_2,"},{"Start":"03:58.790 ","End":"04:02.294","Text":"our wavefront will look like this."},{"Start":"04:02.294 ","End":"04:06.754","Text":"It will change shape slightly as we can see over here."},{"Start":"04:06.754 ","End":"04:12.650","Text":"It\u0027s no longer a straight wavefront but we have some bend due to the refraction."},{"Start":"04:12.650 ","End":"04:15.183","Text":"What\u0027s important to note is,"},{"Start":"04:15.183 ","End":"04:19.985","Text":"let\u0027s draw it here as a note so a note on the side,"},{"Start":"04:19.985 ","End":"04:24.020","Text":"if our change in depth was this straight line over here and"},{"Start":"04:24.020 ","End":"04:30.610","Text":"our wavefront coming forward was this straight wavefront over here,"},{"Start":"04:30.610 ","End":"04:35.855","Text":"so as we know the direction of travel is perpendicular to the wavefront,"},{"Start":"04:35.855 ","End":"04:39.640","Text":"we can see that every single point on this wavefront will"},{"Start":"04:39.640 ","End":"04:43.790","Text":"hit this obstacle barrier or change in depth."},{"Start":"04:43.790 ","End":"04:44.990","Text":"In this case,"},{"Start":"04:44.990 ","End":"04:50.515","Text":"will hit this change in depth or will hit the shallow end at the same time."},{"Start":"04:50.515 ","End":"04:54.755","Text":"Every single point will slow down at the same time."},{"Start":"04:54.755 ","End":"04:58.805","Text":"That means that a moment before it hits,"},{"Start":"04:58.805 ","End":"05:02.570","Text":"the wavefront will look like so."},{"Start":"05:02.570 ","End":"05:04.865","Text":"Then a moment after it hits,"},{"Start":"05:04.865 ","End":"05:09.260","Text":"every single point reaches the shallow end at the same time."},{"Start":"05:09.260 ","End":"05:12.985","Text":"The wavefront will look exactly the same,"},{"Start":"05:12.985 ","End":"05:16.370","Text":"it will still be a straight line because every single point on"},{"Start":"05:16.370 ","End":"05:20.870","Text":"the wavefront slows down simultaneously."},{"Start":"05:20.870 ","End":"05:29.540","Text":"This is what happens if our change in transmission medium is parallel to our wavefront."},{"Start":"05:29.540 ","End":"05:32.090","Text":"However, here, we can see that our change in"},{"Start":"05:32.090 ","End":"05:37.890","Text":"transmission medium is at some angle to our wavefront."},{"Start":"05:37.890 ","End":"05:43.070","Text":"We can see that every single point in our wavefront does not reach"},{"Start":"05:43.070 ","End":"05:49.025","Text":"the shallow end or does not reach the second transmission medium at the same time,"},{"Start":"05:49.025 ","End":"05:55.955","Text":"which means that each point on the wavefront slows down at a different moment in time."},{"Start":"05:55.955 ","End":"05:59.450","Text":"First, our point A will slow down,"},{"Start":"05:59.450 ","End":"06:01.055","Text":"then our point B,"},{"Start":"06:01.055 ","End":"06:03.020","Text":"then our point C, then D,"},{"Start":"06:03.020 ","End":"06:04.805","Text":"and then only finally,"},{"Start":"06:04.805 ","End":"06:07.115","Text":"will our point E slowed down."},{"Start":"06:07.115 ","End":"06:12.320","Text":"We can see that each point on the wavefront is going to reach"},{"Start":"06:12.320 ","End":"06:15.560","Text":"this transmission medium at a different time and"},{"Start":"06:15.560 ","End":"06:18.755","Text":"that is what causes this change in shape,"},{"Start":"06:18.755 ","End":"06:25.489","Text":"because in the same time it took point C to travel these 5 squares,"},{"Start":"06:25.489 ","End":"06:29.870","Text":"point A and B traveled less because now their velocity was"},{"Start":"06:29.870 ","End":"06:34.720","Text":"reduced because they\u0027ve entered the shallow end or transmission medium 2."},{"Start":"06:34.720 ","End":"06:37.925","Text":"Now, we can see that at this section of the wavefront,"},{"Start":"06:37.925 ","End":"06:40.925","Text":"it\u0027s still traveling in the same direction."},{"Start":"06:40.925 ","End":"06:45.570","Text":"However, here, it\u0027s traveling normal to the wavefront,"},{"Start":"06:45.570 ","End":"06:50.575","Text":"so this side is traveling in this slanted direction."},{"Start":"06:50.575 ","End":"06:53.840","Text":"Now, you can see this happening also in real life."},{"Start":"06:53.840 ","End":"06:58.580","Text":"Let\u0027s say if you have a car in your day-to-day life,"},{"Start":"06:58.580 ","End":"07:02.000","Text":"if you have a car in here, four wheels,"},{"Start":"07:02.000 ","End":"07:07.705","Text":"and then you reach some barrier so here you have road,"},{"Start":"07:07.705 ","End":"07:14.680","Text":"you\u0027re driving on concrete or whatever it might be and here you have sand."},{"Start":"07:14.680 ","End":"07:20.525","Text":"You can see that this wheel we\u0027ll reach the sand faster."},{"Start":"07:20.525 ","End":"07:22.970","Text":"Then, because it\u0027s sand,"},{"Start":"07:22.970 ","End":"07:25.310","Text":"the wheel will slip in the sand."},{"Start":"07:25.310 ","End":"07:29.260","Text":"Therefore what you\u0027ll get is that your car,"},{"Start":"07:29.260 ","End":"07:32.170","Text":"from traveling in this direction,"},{"Start":"07:32.170 ","End":"07:36.100","Text":"will then be located like so where"},{"Start":"07:36.100 ","End":"07:40.450","Text":"this wheel would have traveled faster because it was on the road and it wasn\u0027t slipping."},{"Start":"07:40.450 ","End":"07:45.280","Text":"The tire had a grip on the road when it doesn\u0027t have a grip on the sand and"},{"Start":"07:45.280 ","End":"07:50.690","Text":"then you can see that now your direction of travel will be in this direction."},{"Start":"07:50.690 ","End":"07:54.774","Text":"Now that we\u0027ve understood the idea of what is happening"},{"Start":"07:54.774 ","End":"07:59.350","Text":"when our wave reaches this change in transmission medium,"},{"Start":"07:59.350 ","End":"08:03.155","Text":"let\u0027s do a mathematical derivation."},{"Start":"08:03.155 ","End":"08:08.420","Text":"We\u0027re back to this original drawing and now what I want to do is I want to"},{"Start":"08:08.420 ","End":"08:14.810","Text":"draw some triangles so that we can see and do some calculations."},{"Start":"08:14.810 ","End":"08:20.565","Text":"This triangle between connecting C_2 to C_1,"},{"Start":"08:20.565 ","End":"08:23.975","Text":"so this triangle over here,"},{"Start":"08:23.975 ","End":"08:28.325","Text":"where A_1 has hit the barrier, C_2 and C_1."},{"Start":"08:28.325 ","End":"08:34.890","Text":"Here we have a triangle and I\u0027m going to call this angle over here Theta_1."},{"Start":"08:34.890 ","End":"08:38.105","Text":"Then the next triangle I want us to look at."},{"Start":"08:38.105 ","End":"08:39.920","Text":"I\u0027m going to draw this in green."},{"Start":"08:39.920 ","End":"08:46.610","Text":"This is connecting A_2 to A_1 and all of this together,"},{"Start":"08:46.610 ","End":"08:49.040","Text":"so B_2 to C_2,"},{"Start":"08:49.040 ","End":"08:52.040","Text":"so it\u0027s this triangle encased in green,"},{"Start":"08:52.040 ","End":"08:53.420","Text":"I hope you can see it."},{"Start":"08:53.420 ","End":"08:56.540","Text":"Then, this angle over here,"},{"Start":"08:56.540 ","End":"09:00.350","Text":"so this angle, I\u0027m going to call Theta_2."},{"Start":"09:00.350 ","End":"09:03.635","Text":"The first thing I want to note is that our triangle,"},{"Start":"09:03.635 ","End":"09:05.705","Text":"A_1, C_1, C_2,"},{"Start":"09:05.705 ","End":"09:07.580","Text":"so that\u0027s this triangle in red,"},{"Start":"09:07.580 ","End":"09:11.390","Text":"is a right-angled triangle."},{"Start":"09:11.390 ","End":"09:15.980","Text":"Let\u0027s write that. We have triangle A_1, C_1,"},{"Start":"09:15.980 ","End":"09:21.680","Text":"C_2 is a right-angled triangle."},{"Start":"09:21.680 ","End":"09:23.300","Text":"Why is this?"},{"Start":"09:23.300 ","End":"09:25.655","Text":"We know that this line A_1,"},{"Start":"09:25.655 ","End":"09:27.050","Text":"B_1, C_1,"},{"Start":"09:27.050 ","End":"09:29.660","Text":"so let\u0027s write this over here,"},{"Start":"09:29.660 ","End":"09:32.675","Text":"so the line A_1, B_1,"},{"Start":"09:32.675 ","End":"09:36.170","Text":"C_1 is equal to, what is it?"},{"Start":"09:36.170 ","End":"09:37.760","Text":"This is the wavefrond."},{"Start":"09:37.760 ","End":"09:40.640","Text":"Then we know that our red line,"},{"Start":"09:40.640 ","End":"09:42.035","Text":"C_1, C_2,"},{"Start":"09:42.035 ","End":"09:48.750","Text":"so that\u0027s this line over here is the direction of travel of the wavefront."},{"Start":"09:48.750 ","End":"09:52.270","Text":"We know by definition that the direction of travel of"},{"Start":"09:52.270 ","End":"09:56.689","Text":"the is going to be perpendicular to the wavefrond."},{"Start":"09:56.689 ","End":"10:04.220","Text":"Perpendicular means that right angles or at a 90-degree angle to it."},{"Start":"10:04.220 ","End":"10:07.220","Text":"Now we can see that A_1, C_1,"},{"Start":"10:07.220 ","End":"10:11.750","Text":"C_2, this red triangle is a right-angled triangle."},{"Start":"10:11.750 ","End":"10:17.570","Text":"We know that these two lines are perpendicular,"},{"Start":"10:17.570 ","End":"10:20.180","Text":"so a lot of the time of perpendicular angle is"},{"Start":"10:20.180 ","End":"10:26.130","Text":"represented like so because we can see that this line is perpendicular to this one."},{"Start":"10:26.410 ","End":"10:30.890","Text":"Now if we remember our trig identities,"},{"Start":"10:30.890 ","End":"10:35.390","Text":"so we have SOH CAH TOA, so what is this?"},{"Start":"10:35.390 ","End":"10:39.080","Text":"If you don\u0027t know this, this is an easy way to remember trig identities,"},{"Start":"10:39.080 ","End":"10:46.790","Text":"so SOH is sine of an angle is equal to the opposite divided by the hypotenuse,"},{"Start":"10:46.790 ","End":"10:51.455","Text":"cosine of an angle is equal to the adjacent over the hypotenuse,"},{"Start":"10:51.455 ","End":"10:56.795","Text":"tan of an angle is equal to the opposite side over the adjacent side."},{"Start":"10:56.795 ","End":"11:04.565","Text":"So, here we can see in this triangle that we\u0027re going to use sine of the angle."},{"Start":"11:04.565 ","End":"11:07.580","Text":"Sine of Theta_1,"},{"Start":"11:07.580 ","End":"11:12.335","Text":"this is the angle is equal to the opposite side,"},{"Start":"11:12.335 ","End":"11:14.510","Text":"so that\u0027s our lines, C_1,"},{"Start":"11:14.510 ","End":"11:17.390","Text":"C_2, so C_1, C_2,"},{"Start":"11:17.390 ","End":"11:18.890","Text":"this is the opposite side,"},{"Start":"11:18.890 ","End":"11:20.345","Text":"this distance over here,"},{"Start":"11:20.345 ","End":"11:23.195","Text":"divided by the hypotenuse."},{"Start":"11:23.195 ","End":"11:29.640","Text":"The hypotenuse is the line joining A_1 to C_2."},{"Start":"11:31.390 ","End":"11:38.045","Text":"Now, what does C_1, C_2 represent?"},{"Start":"11:38.045 ","End":"11:40.400","Text":"We can see that the line C_1,"},{"Start":"11:40.400 ","End":"11:47.420","Text":"C_2 connects the 2 positions that the wave is at, at different times."},{"Start":"11:47.420 ","End":"11:50.300","Text":"So, we can say that C_1,"},{"Start":"11:50.300 ","End":"11:55.909","Text":"C_2 is the distance traveled by the wavefrond and some time,"},{"Start":"11:55.909 ","End":"12:05.240","Text":"Delta t. and all of this is divided by A_1, C_2."},{"Start":"12:05.240 ","End":"12:09.110","Text":"What is the distance traveled?"},{"Start":"12:09.110 ","End":"12:12.320","Text":"This is equal to, as we know,"},{"Start":"12:12.320 ","End":"12:22.310","Text":"the velocity multiplied by the timeframe Delta t,"},{"Start":"12:22.310 ","End":"12:26.150","Text":"divided by A_1, C_2,"},{"Start":"12:26.150 ","End":"12:31.710","Text":"because we know that distance is velocity multiplied by time."},{"Start":"12:32.620 ","End":"12:36.080","Text":"Now, let\u0027s look at this triangle, A_1,"},{"Start":"12:36.080 ","End":"12:37.850","Text":"A_2, C_2,"},{"Start":"12:37.850 ","End":"12:39.965","Text":"so that\u0027s the triangle in green."},{"Start":"12:39.965 ","End":"12:46.670","Text":"I should have drawn my A_2 slightly more over here."},{"Start":"12:46.670 ","End":"12:49.070","Text":"This is meant to be in my triangle,"},{"Start":"12:49.070 ","End":"12:54.215","Text":"so let\u0027s draw it now in black."},{"Start":"12:54.215 ","End":"12:56.435","Text":"This is my triangle,"},{"Start":"12:56.435 ","End":"12:59.730","Text":"and this angle over here is Theta_2."},{"Start":"12:59.830 ","End":"13:07.955","Text":"Now, we\u0027re looking at triangle A_1, A_2, C_2."},{"Start":"13:07.955 ","End":"13:11.915","Text":"First of all, we can see that this is also going to be"},{"Start":"13:11.915 ","End":"13:15.980","Text":"a right angled triangle for the exact same reason that A_1,"},{"Start":"13:15.980 ","End":"13:19.800","Text":"C1, C2 was a right-angled triangle."},{"Start":"13:20.160 ","End":"13:25.390","Text":"We can see that this is the direction of travel of"},{"Start":"13:25.390 ","End":"13:30.070","Text":"the wave in this direction over here and that it has"},{"Start":"13:30.070 ","End":"13:39.565","Text":"to be perpendicular to the wavefrond, from the definition."},{"Start":"13:39.565 ","End":"13:44.390","Text":"So, that means that this angle over here is a right angle."},{"Start":"13:45.310 ","End":"13:51.050","Text":"A_1, A_2 is perpendicular to A_2, C_2,"},{"Start":"13:51.050 ","End":"13:52.535","Text":"the wavefrond over here,"},{"Start":"13:52.535 ","End":"13:57.420","Text":"and that\u0027s just from definition of how this works."},{"Start":"13:57.640 ","End":"14:02.630","Text":"First of all, we know that the time taken for the point"},{"Start":"14:02.630 ","End":"14:06.845","Text":"on the wavefrond C to reach from position 1 to position 2,"},{"Start":"14:06.845 ","End":"14:09.125","Text":"we know it took a time of Delta t,"},{"Start":"14:09.125 ","End":"14:10.910","Text":"and then that same time,"},{"Start":"14:10.910 ","End":"14:18.050","Text":"point A move from position one over here to position 2 over here."},{"Start":"14:18.050 ","End":"14:21.740","Text":"We know that the same amount of time went by you,"},{"Start":"14:21.740 ","End":"14:26.990","Text":"it\u0027s just traveling with a slower velocity because it\u0027s now in the shallow end."},{"Start":"14:26.990 ","End":"14:29.615","Text":"The wave is traveling slower."},{"Start":"14:29.615 ","End":"14:36.185","Text":"However, it traveled this distance in the same time that C traveled this distance."},{"Start":"14:36.185 ","End":"14:44.555","Text":"Again, we\u0027re going to use our sign trig identity."},{"Start":"14:44.555 ","End":"14:47.870","Text":"Sine of our angle Theta_2,"},{"Start":"14:47.870 ","End":"14:50.375","Text":"is equal to the opposite side,"},{"Start":"14:50.375 ","End":"14:52.790","Text":"so that\u0027s our side, A_1,"},{"Start":"14:52.790 ","End":"14:59.315","Text":"A_2 divided by our hypotenuse."},{"Start":"14:59.315 ","End":"15:03.725","Text":"Our hypotenuse, as we know, is A_1,"},{"Start":"15:03.725 ","End":"15:07.355","Text":"C_2, the hypotenuse is the long side,"},{"Start":"15:07.355 ","End":"15:09.800","Text":"the side opposite to the right angles."},{"Start":"15:09.800 ","End":"15:13.730","Text":"So, that\u0027s A_1, C_2."},{"Start":"15:13.730 ","End":"15:16.610","Text":"Again, A_1 to A_2,"},{"Start":"15:16.610 ","End":"15:20.644","Text":"we know that this is the direction of travel of the wavefront,"},{"Start":"15:20.644 ","End":"15:23.690","Text":"and we know that this line A_1,"},{"Start":"15:23.690 ","End":"15:27.540","Text":"A_2 represents the distance"},{"Start":"15:28.600 ","End":"15:37.025","Text":"traveled by the wave-front in time,"},{"Start":"15:37.025 ","End":"15:40.205","Text":"Delta t, and it\u0027s the same Delta t,"},{"Start":"15:40.205 ","End":"15:43.025","Text":"and divided by A_1,"},{"Start":"15:43.025 ","End":"15:45.875","Text":"C_2, which is the hypotenuse."},{"Start":"15:45.875 ","End":"15:52.340","Text":"The distance traveled in time Delta t is simply going to be equal to the velocity,"},{"Start":"15:52.340 ","End":"15:57.215","Text":"so now it\u0027s the velocity in the transmission medium 2,"},{"Start":"15:57.215 ","End":"16:05.100","Text":"multiplied by the same Delta t and divided by A_1, C_2."},{"Start":"16:06.340 ","End":"16:13.470","Text":"Let\u0027s call these equations 1 and this equation 2."},{"Start":"16:13.510 ","End":"16:18.575","Text":"Let\u0027s scroll down to make a little bit of extra space."},{"Start":"16:18.575 ","End":"16:25.700","Text":"Now what I\u0027m going to do is I\u0027m going to divide equation 1 by equation 2,"},{"Start":"16:25.700 ","End":"16:29.375","Text":"so we\u0027re dividing equation 1 by equation 2."},{"Start":"16:29.375 ","End":"16:34.700","Text":"So, we\u0027re going to have equation 1 on the left side of the equals sign,"},{"Start":"16:34.700 ","End":"16:38.360","Text":"so equation 1 is sine of Theta_1,"},{"Start":"16:38.360 ","End":"16:40.700","Text":"divided by equation 2,"},{"Start":"16:40.700 ","End":"16:44.280","Text":"which is sine of theta."},{"Start":"16:45.370 ","End":"16:50.390","Text":"We have that this is equal to V_1 Delta t divided by A_1,"},{"Start":"16:50.390 ","End":"16:53.420","Text":"C_2, and all of this is going to be"},{"Start":"16:53.420 ","End":"16:57.650","Text":"divided by the right side of the equals sign of equation number 2."},{"Start":"16:57.650 ","End":"17:04.895","Text":"We\u0027ll have V_1 Delta t divided by A_1, C_2,"},{"Start":"17:04.895 ","End":"17:08.345","Text":"so this is the right side of the equals side of equation 1,"},{"Start":"17:08.345 ","End":"17:16.730","Text":"divided by this V_2 Delta t divided by A_1, C_2."},{"Start":"17:16.730 ","End":"17:19.010","Text":"Now, as we can see,"},{"Start":"17:19.010 ","End":"17:20.960","Text":"let\u0027s draw it in green."},{"Start":"17:20.960 ","End":"17:24.080","Text":"Delta t divided by A_1,"},{"Start":"17:24.080 ","End":"17:30.380","Text":"C_2 appears in the numerator and also in the denominator,"},{"Start":"17:30.380 ","End":"17:32.470","Text":"Delta t, A_1, C_2, Delta t, A_1,"},{"Start":"17:32.470 ","End":"17:36.425","Text":"C_2, which means that we can cancel them out."},{"Start":"17:36.425 ","End":"17:38.510","Text":"We can call this, let\u0027s say x,"},{"Start":"17:38.510 ","End":"17:41.705","Text":"and then we have V_1x divided by V_2x,"},{"Start":"17:41.705 ","End":"17:43.715","Text":"which means that the x is cancel out,"},{"Start":"17:43.715 ","End":"17:49.235","Text":"which means that this is equal to V_1 divided by V_2."},{"Start":"17:49.235 ","End":"18:00.190","Text":"Now we have that sine Theta_1 divided by sine Theta_2 is equal to V_1 divided by V_2."},{"Start":"18:00.190 ","End":"18:06.040","Text":"Now we can see that we have the relationship between the velocities"},{"Start":"18:06.040 ","End":"18:12.700","Text":"in each transmission medium and the direction of travel in each transmission medium."},{"Start":"18:12.700 ","End":"18:17.890","Text":"If you get a question and you\u0027re given the velocity and transmission medium number 1,"},{"Start":"18:17.890 ","End":"18:19.990","Text":"the velocity and transmission medium number 2,"},{"Start":"18:19.990 ","End":"18:23.545","Text":"and the direction of travel into transmission medium number 1,"},{"Start":"18:23.545 ","End":"18:29.140","Text":"you can find the direction of travel in transmission medium number 2 and vice versa."},{"Start":"18:29.140 ","End":"18:32.994","Text":"If you have any 3 of these 4 values,"},{"Start":"18:32.994 ","End":"18:35.755","Text":"you can calculate the fourth value."},{"Start":"18:35.755 ","End":"18:38.920","Text":"This is an important equation to know."},{"Start":"18:38.920 ","End":"18:41.050","Text":"Write it in your equation sheets."},{"Start":"18:41.050 ","End":"18:49.150","Text":"Also, I\u0027m going to now show you how this equation is related to Snell\u0027s law,"},{"Start":"18:49.150 ","End":"18:52.640","Text":"which speaks about light."},{"Start":"18:53.250 ","End":"18:56.305","Text":"If we remember from Snell\u0027s law,"},{"Start":"18:56.305 ","End":"19:02.560","Text":"the refractive index n was equal to the speed of light in vacuum"},{"Start":"19:02.560 ","End":"19:10.640","Text":"divided by the speed of light traveling through a specific transmission medium."},{"Start":"19:10.640 ","End":"19:14.200","Text":"If I wanted to isolate out this v,"},{"Start":"19:14.200 ","End":"19:17.950","Text":"the velocity or the speed of light through the specific medium,"},{"Start":"19:17.950 ","End":"19:24.490","Text":"this would be equal to the speed of light in a vacuum divided by the refractive index."},{"Start":"19:24.490 ","End":"19:30.385","Text":"Now let\u0027s substitute this into our equation that we just derived."},{"Start":"19:30.385 ","End":"19:37.975","Text":"That means that we\u0027ll have sine of Theta_1 divided by sine of Theta_2."},{"Start":"19:37.975 ","End":"19:39.910","Text":"This is equal to V_1."},{"Start":"19:39.910 ","End":"19:43.105","Text":"That\u0027s going to be equal to C, which never changes,"},{"Start":"19:43.105 ","End":"19:48.025","Text":"divided by n_1, then V_2."},{"Start":"19:48.025 ","End":"19:50.065","Text":"This is all divided by V_2,"},{"Start":"19:50.065 ","End":"19:54.730","Text":"which will equal to c divided by n_2,"},{"Start":"19:54.730 ","End":"19:58.570","Text":"the refractive index of the second material."},{"Start":"19:58.570 ","End":"20:02.530","Text":"Then, if we rearrange this,"},{"Start":"20:02.530 ","End":"20:06.655","Text":"we can see that the C\u0027s cancel out and our denominators"},{"Start":"20:06.655 ","End":"20:09.570","Text":"switch with each other because 1 denominators"},{"Start":"20:09.570 ","End":"20:12.750","Text":"in the numerator and the denominator and the denominator."},{"Start":"20:12.750 ","End":"20:18.610","Text":"We\u0027ll get that sine Theta_1 divided by sine Theta_2 is equal to n_2,"},{"Start":"20:18.610 ","End":"20:22.600","Text":"the refractive index in the second transmission medium,"},{"Start":"20:22.600 ","End":"20:24.145","Text":"divided by n_1,"},{"Start":"20:24.145 ","End":"20:27.805","Text":"the refractive index in the first transmission medium."},{"Start":"20:27.805 ","End":"20:32.650","Text":"Now let\u0027s get our n_1 on the same side of the equation as"},{"Start":"20:32.650 ","End":"20:38.620","Text":"our Theta_1 and our n_2 on the same side of the equation as our Theta_2."},{"Start":"20:38.620 ","End":"20:44.410","Text":"We\u0027ll multiply both sides by n_1 and both sides by sine Theta_2."},{"Start":"20:44.410 ","End":"20:49.930","Text":"Then we\u0027ll get that n_1 sine of Theta_1 is"},{"Start":"20:49.930 ","End":"20:55.420","Text":"equal to n_2 sine of Theta_2."},{"Start":"20:55.420 ","End":"21:02.120","Text":"Of course, this is exactly Snell\u0027s law."},{"Start":"21:03.750 ","End":"21:06.205","Text":"Now a little note,"},{"Start":"21:06.205 ","End":"21:12.730","Text":"water doesn\u0027t have a refractive index for water waves specifically."},{"Start":"21:12.730 ","End":"21:16.030","Text":"There\u0027s no refractive index for how a water wave"},{"Start":"21:16.030 ","End":"21:20.005","Text":"will bend when it passes from one medium to another."},{"Start":"21:20.005 ","End":"21:27.820","Text":"But what this does show is that light behaves like water waves."},{"Start":"21:27.820 ","End":"21:32.620","Text":"This is showing the wave properties of light and how we can look at"},{"Start":"21:32.620 ","End":"21:38.425","Text":"water waves and from that learn how light works."},{"Start":"21:38.425 ","End":"21:42.530","Text":"We can see the similarity between these two."},{"Start":"21:43.350 ","End":"21:46.870","Text":"The last thing that we\u0027re going to speak about,"},{"Start":"21:46.870 ","End":"21:50.275","Text":"although I doubt that you\u0027ll be asked this in any exam,"},{"Start":"21:50.275 ","End":"21:54.610","Text":"is when we\u0027re speaking about the dispersion of light and"},{"Start":"21:54.610 ","End":"21:58.990","Text":"how we can speak about the dispersion of water waves."},{"Start":"21:58.990 ","End":"22:04.120","Text":"Just to remind you what dispersion is, here\u0027s a definition."},{"Start":"22:04.120 ","End":"22:08.830","Text":"Dispersion is when white light is split as"},{"Start":"22:08.830 ","End":"22:13.795","Text":"a consequence of refraction into its different colored components."},{"Start":"22:13.795 ","End":"22:18.070","Text":"Each color, which is a different wavelength as we know,"},{"Start":"22:18.070 ","End":"22:21.205","Text":"different colors have different wavelengths,"},{"Start":"22:21.205 ","End":"22:24.025","Text":"has a different refractive index."},{"Start":"22:24.025 ","End":"22:27.294","Text":"You might have seen this diagram somewhere."},{"Start":"22:27.294 ","End":"22:31.585","Text":"An example is on the cover of one of Pink Floyd\u0027s albums."},{"Start":"22:31.585 ","End":"22:35.425","Text":"We can see that white light enters a prism."},{"Start":"22:35.425 ","End":"22:42.750","Text":"Because white light is made up of different colors that together make white light,"},{"Start":"22:42.750 ","End":"22:46.185","Text":"inside the Glass Prism because each color,"},{"Start":"22:46.185 ","End":"22:50.160","Text":"each wavelength has a different refractive index,"},{"Start":"22:50.160 ","End":"22:53.340","Text":"the prism splits the white light due to"},{"Start":"22:53.340 ","End":"22:57.165","Text":"this refractive index difference between each wavelength."},{"Start":"22:57.165 ","End":"23:03.245","Text":"Then we can see this rainbow and we can see all the colors which make up white light."},{"Start":"23:03.245 ","End":"23:08.420","Text":"It turns out that water waves exhibit the same property."},{"Start":"23:09.180 ","End":"23:15.320","Text":"Let\u0027s take a wave tank over here."},{"Start":"23:15.480 ","End":"23:21.805","Text":"Let\u0027s put in this over here as a barrier."},{"Start":"23:21.805 ","End":"23:29.690","Text":"Now let\u0027s send in waves with a straight wavefront with a certain wavelength."},{"Start":"23:30.840 ","End":"23:33.640","Text":"As the wavefrond passes,"},{"Start":"23:33.640 ","End":"23:41.035","Text":"this barrier we\u0027ll get something that looks like this."},{"Start":"23:41.035 ","End":"23:46.045","Text":"We\u0027ll get some change in the angle."},{"Start":"23:46.045 ","End":"23:50.695","Text":"Now if I take the exact same wave tank,"},{"Start":"23:50.695 ","End":"23:54.850","Text":"so if I\u0027m drawing it now and it doesn\u0027t look exactly identical."},{"Start":"23:54.850 ","End":"23:57.370","Text":"Imagine that it is identical."},{"Start":"23:57.370 ","End":"24:00.789","Text":"The exact same wave tank with the exact same barrier,"},{"Start":"24:00.789 ","End":"24:06.325","Text":"where the angle of the barrier Theta is exactly the same over here."},{"Start":"24:06.325 ","End":"24:07.810","Text":"This is also Theta."},{"Start":"24:07.810 ","End":"24:15.380","Text":"But this time I send in water waves with a slightly different wavelength."},{"Start":"24:15.570 ","End":"24:22.255","Text":"We can see that this is a different wavelength and it\u0027s still a straight wave."},{"Start":"24:22.255 ","End":"24:28.375","Text":"We\u0027ll see that once the waves pass the barrier,"},{"Start":"24:28.375 ","End":"24:36.110","Text":"they\u0027ll have a different angle of refraction."},{"Start":"24:36.210 ","End":"24:38.650","Text":"If I draw this,"},{"Start":"24:38.650 ","End":"24:42.955","Text":"let\u0027s say that this angle is Alpha,"},{"Start":"24:42.955 ","End":"24:50.200","Text":"the angle of refraction of the different wavelength will be, let\u0027s say Gamma."},{"Start":"24:50.200 ","End":"24:58.310","Text":"We get different angles of refraction if we change the wavelength of the water waves."},{"Start":"24:58.680 ","End":"25:03.070","Text":"We can see that water waves exhibit this discretion that"},{"Start":"25:03.070 ","End":"25:06.700","Text":"we saw in light with different wavelengths."},{"Start":"25:06.700 ","End":"25:09.370","Text":"Here we have a larger wavelength and here"},{"Start":"25:09.370 ","End":"25:15.560","Text":"a smaller wavelength will have a different refractive angle."},{"Start":"25:16.170 ","End":"25:18.970","Text":"I don\u0027t think you\u0027ll be asked about this,"},{"Start":"25:18.970 ","End":"25:22.120","Text":"but this is something interesting to note."},{"Start":"25:22.120 ","End":"25:24.980","Text":"That\u0027s the end of this lesson."}],"ID":12482},{"Watched":false,"Name":"Exercise- Reflection","Duration":"12m 37s","ChapterTopicVideoID":12015,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"Hello. In this lesson,"},{"Start":"00:01.845 ","End":"00:04.275","Text":"we\u0027re going to be answering the following question."},{"Start":"00:04.275 ","End":"00:06.150","Text":"A straight wavefront A_0,"},{"Start":"00:06.150 ","End":"00:09.235","Text":"B_0 travels towards a barrier."},{"Start":"00:09.235 ","End":"00:14.430","Text":"Question number 1 is to draw the direction of travel of the wave."},{"Start":"00:14.430 ","End":"00:20.445","Text":"We know that the direction of travel of the wave is perpendicular to the wavefront."},{"Start":"00:20.445 ","End":"00:22.335","Text":"Let\u0027s measure from here."},{"Start":"00:22.335 ","End":"00:24.495","Text":"We can see that from here,"},{"Start":"00:24.495 ","End":"00:26.778","Text":"if we start at this point, we go 1, 2,"},{"Start":"00:26.778 ","End":"00:31.995","Text":"3 across and 1, 2 up."},{"Start":"00:31.995 ","End":"00:35.310","Text":"If we have 3 across and 2 up,"},{"Start":"00:35.310 ","End":"00:38.760","Text":"in order to get the arrow perpendicular to this,"},{"Start":"00:38.760 ","End":"00:43.155","Text":"we have to go 3 up and 2 across."},{"Start":"00:43.155 ","End":"00:45.180","Text":"The direction is the opposite."},{"Start":"00:45.180 ","End":"00:46.650","Text":"If we start from here,"},{"Start":"00:46.650 ","End":"00:48.474","Text":"so 3 up is 1,"},{"Start":"00:48.474 ","End":"00:52.020","Text":"2, 3, and 2 across is 1, 2."},{"Start":"00:52.020 ","End":"00:56.210","Text":"We\u0027re going to draw an arrow between these 2 points,"},{"Start":"00:56.210 ","End":"01:00.260","Text":"and this is the direction of travel of the wave."},{"Start":"01:00.260 ","End":"01:04.340","Text":"Now we\u0027re going to answer Question number 2."},{"Start":"01:04.340 ","End":"01:07.610","Text":"What will the wave look like after hitting the barrier?"},{"Start":"01:07.610 ","End":"01:14.075","Text":"Each point A_0 and B_0 is going to travel in the same direction of travel as our arrow,"},{"Start":"01:14.075 ","End":"01:18.245","Text":"which is 2 across and 3 up until it hits the barrier,"},{"Start":"01:18.245 ","End":"01:22.195","Text":"and then each point is going to be reflected back."},{"Start":"01:22.195 ","End":"01:26.514","Text":"Let\u0027s do A_0, so it goes 2 across and 1,"},{"Start":"01:26.514 ","End":"01:28.170","Text":"2, 3 up."},{"Start":"01:28.170 ","End":"01:30.570","Text":"This is after 1 second, let\u0027s say,"},{"Start":"01:30.570 ","End":"01:32.580","Text":"and 2 across and 1,"},{"Start":"01:32.580 ","End":"01:34.230","Text":"2, 3 up."},{"Start":"01:34.230 ","End":"01:36.195","Text":"Again, 2 across and 1,"},{"Start":"01:36.195 ","End":"01:37.200","Text":"2, 3 up."},{"Start":"01:37.200 ","End":"01:38.400","Text":"This is 3 seconds,"},{"Start":"01:38.400 ","End":"01:39.510","Text":"2 across and 1, 2,"},{"Start":"01:39.510 ","End":"01:43.140","Text":"3 up 2 a cross and 1,"},{"Start":"01:43.140 ","End":"01:45.225","Text":"2, 3 up."},{"Start":"01:45.225 ","End":"01:48.560","Text":"Here point A_0 hits the barrier,"},{"Start":"01:48.560 ","End":"01:51.245","Text":"and now it\u0027s going to be reflected."},{"Start":"01:51.245 ","End":"01:53.255","Text":"Again, we go 2 across,"},{"Start":"01:53.255 ","End":"01:55.130","Text":"but this time because it\u0027s being reflected back,"},{"Start":"01:55.130 ","End":"01:57.404","Text":"we go 3 down, so 1,"},{"Start":"01:57.404 ","End":"01:59.385","Text":"2, 3 down,"},{"Start":"01:59.385 ","End":"02:01.488","Text":"and then 2 across and 1,"},{"Start":"02:01.488 ","End":"02:04.215","Text":"2, 3 down."},{"Start":"02:04.215 ","End":"02:10.095","Text":"This has been 1, 2, 3, 4, 5, 6, 7 seconds."},{"Start":"02:10.095 ","End":"02:11.715","Text":"So let\u0027s do 8 seconds,"},{"Start":"02:11.715 ","End":"02:13.572","Text":"so 2 across and 1,"},{"Start":"02:13.572 ","End":"02:17.150","Text":"2, 3 down."},{"Start":"02:17.150 ","End":"02:20.810","Text":"This is where our point will be after 8 seconds."},{"Start":"02:20.810 ","End":"02:24.440","Text":"Now let\u0027s see what will happen to B_0 after 8 seconds."},{"Start":"02:24.440 ","End":"02:27.950","Text":"It\u0027s traveling in the exact same direction as A_0,"},{"Start":"02:27.950 ","End":"02:32.590","Text":"which has the exact same direction as this arrow."},{"Start":"02:32.590 ","End":"02:38.106","Text":"B_0 is also going 2 across and 1, 2,"},{"Start":"02:38.106 ","End":"02:41.015","Text":"3 up, so this is after 1 second,"},{"Start":"02:41.015 ","End":"02:43.256","Text":"2 across and 1, 2,"},{"Start":"02:43.256 ","End":"02:49.225","Text":"3 up, 2 seconds,"},{"Start":"02:49.225 ","End":"02:51.748","Text":"2 across and 1,"},{"Start":"02:51.748 ","End":"02:53.210","Text":"2, 3 up, 3, 1, 2, 3 this is after 1, 2,"},{"Start":"02:53.210 ","End":"02:56.165","Text":"3, 4 seconds,"},{"Start":"02:56.165 ","End":"02:58.675","Text":"2 across and 1, 2,"},{"Start":"02:58.675 ","End":"03:02.216","Text":"3 up 2 across and 1, 2,"},{"Start":"03:02.216 ","End":"03:06.752","Text":"3 up, 2 across and 1,"},{"Start":"03:06.752 ","End":"03:08.690","Text":"2, 3 up."},{"Start":"03:08.690 ","End":"03:13.455","Text":"Here, A hit the barrier,"},{"Start":"03:13.455 ","End":"03:15.960","Text":"and here B hit the barrier."},{"Start":"03:15.960 ","End":"03:20.465","Text":"Now point B is also going to be reflected back so we want to get"},{"Start":"03:20.465 ","End":"03:27.215","Text":"B\u0027s position after 8 seconds because we looked at A\u0027s position after 8 seconds over here."},{"Start":"03:27.215 ","End":"03:30.816","Text":"So 1, 2,"},{"Start":"03:30.816 ","End":"03:32.575","Text":"3, 4, 5, 6, 7,"},{"Start":"03:32.575 ","End":"03:35.690","Text":"so 1 more second so 2 a cross,"},{"Start":"03:35.690 ","End":"03:42.630","Text":"but now it\u0027s being reflected back so that means 3 down this time so 1, 2, 3."},{"Start":"03:43.340 ","End":"03:48.570","Text":"So B after 8 seconds will be over here."},{"Start":"03:48.570 ","End":"03:51.625","Text":"Now if we just draw this wavefront,"},{"Start":"03:51.625 ","End":"03:59.930","Text":"this is a wavefront once it\u0027s being reflected back after 8 seconds."},{"Start":"03:59.930 ","End":"04:04.055","Text":"What we can see is that point A was originally higher,"},{"Start":"04:04.055 ","End":"04:06.085","Text":"but after our reflection,"},{"Start":"04:06.085 ","End":"04:08.160","Text":"point A is now the lowest,"},{"Start":"04:08.160 ","End":"04:10.940","Text":"and B which was the lowest after reflection,"},{"Start":"04:10.940 ","End":"04:13.980","Text":"is now the highest."},{"Start":"04:14.840 ","End":"04:17.610","Text":"This is the answer to Question 2."},{"Start":"04:17.610 ","End":"04:19.330","Text":"Now let\u0027s see Question 3."},{"Start":"04:19.330 ","End":"04:24.760","Text":"Draw the incident angle and reflected angle as defined for light."},{"Start":"04:24.760 ","End":"04:27.800","Text":"Let\u0027s draw this in blue."},{"Start":"04:28.310 ","End":"04:34.640","Text":"With light, we just consider each point as a light ray, a ray of light."},{"Start":"04:34.640 ","End":"04:39.290","Text":"A_0 would continue in this direction,"},{"Start":"04:39.290 ","End":"04:45.785","Text":"like so until it reaches this point where it meets the barrier and is reflected."},{"Start":"04:45.785 ","End":"04:56.170","Text":"This is the incident ray and then this over here is the reflected ray."},{"Start":"04:56.170 ","End":"05:01.635","Text":"Let\u0027s do i for incident ray and r for reflected ray."},{"Start":"05:01.635 ","End":"05:05.690","Text":"As we know, we define the incident angle and reflected"},{"Start":"05:05.690 ","End":"05:12.090","Text":"angle with respect to the normal to the barrier."},{"Start":"05:12.090 ","End":"05:16.220","Text":"The normal to the barrier means some line which is perpendicular"},{"Start":"05:16.220 ","End":"05:20.420","Text":"to the barrier so a line which is at 90 degrees to the barrier."},{"Start":"05:20.420 ","End":"05:26.225","Text":"The incident angle is the angle between the incident ray,"},{"Start":"05:26.225 ","End":"05:30.160","Text":"which is this over here, and the normal."},{"Start":"05:30.160 ","End":"05:36.365","Text":"This would be Theta_i and then the reflected angle is"},{"Start":"05:36.365 ","End":"05:43.235","Text":"the angle between the normal so this line over here and the reflected ray,"},{"Start":"05:43.235 ","End":"05:45.390","Text":"which is this ray over here."},{"Start":"05:45.390 ","End":"05:49.275","Text":"This is Theta_r, the reflected angle."},{"Start":"05:49.275 ","End":"05:54.005","Text":"Of course, for point B it will be the exact same thing."},{"Start":"05:54.005 ","End":"05:58.910","Text":"I\u0027m not going to draw it because it will make this drawing a little bit messy."},{"Start":"05:58.910 ","End":"06:02.870","Text":"But again, this over here connecting these dots is"},{"Start":"06:02.870 ","End":"06:07.010","Text":"the incident ray and this is the reflected ray,"},{"Start":"06:07.010 ","End":"06:11.240","Text":"and then we\u0027ll draw a line at 90 degrees to the barrier."},{"Start":"06:11.240 ","End":"06:16.295","Text":"Then Theta_i will be the angle between the incident ray and the"},{"Start":"06:16.295 ","End":"06:23.920","Text":"normal and our Theta_r will be the angle between a reflected ray and the normal."},{"Start":"06:27.140 ","End":"06:30.425","Text":"This was the answer to Question number 3."},{"Start":"06:30.425 ","End":"06:35.430","Text":"This is for light and Question number 4 is to draw."},{"Start":"06:36.320 ","End":"06:40.420","Text":"Let\u0027s do this in red."},{"Start":"06:40.880 ","End":"06:47.225","Text":"First of all, when point A hits the barrier,"},{"Start":"06:47.225 ","End":"06:55.930","Text":"we know that point B is located over here."},{"Start":"06:56.440 ","End":"07:01.670","Text":"This is what the wavefront looks like when A has hit the barrier,"},{"Start":"07:01.670 ","End":"07:03.860","Text":"but before it\u0027s reflected."},{"Start":"07:03.860 ","End":"07:06.110","Text":"We know that with water,"},{"Start":"07:06.110 ","End":"07:15.420","Text":"the incident angle is defined as the angle between the incident wave and the barrier."},{"Start":"07:15.420 ","End":"07:21.800","Text":"This is the incident wave as it\u0027s about to be reflected from the barrier and so"},{"Start":"07:21.800 ","End":"07:30.500","Text":"this angle is Theta_i and then tag because we\u0027re defining it for water here,"},{"Start":"07:30.500 ","End":"07:33.425","Text":"Theta_i was defined for light,"},{"Start":"07:33.425 ","End":"07:36.970","Text":"and here our Theta_i is for water so I put a tag."},{"Start":"07:36.970 ","End":"07:41.329","Text":"This is our incident angle and our reflected angle."},{"Start":"07:41.329 ","End":"07:45.800","Text":"Let\u0027s draw this in green,"},{"Start":"07:45.800 ","End":"07:53.040","Text":"is going to be the moment the entire wavefront has been reflected."},{"Start":"07:53.040 ","End":"07:56.120","Text":"That\u0027s when the last point on the wavefront is being reflected,"},{"Start":"07:56.120 ","End":"07:58.530","Text":"which as we know is B."},{"Start":"07:58.530 ","End":"08:01.215","Text":"That happened after 7 seconds,"},{"Start":"08:01.215 ","End":"08:04.370","Text":"so our wavefront is going to begin here and then we have"},{"Start":"08:04.370 ","End":"08:07.490","Text":"to find where point A is after 7 seconds."},{"Start":"08:07.490 ","End":"08:09.426","Text":"So 1, 2, 3,"},{"Start":"08:09.426 ","End":"08:11.885","Text":"4, 5, 6, 7."},{"Start":"08:11.885 ","End":"08:14.880","Text":"Point A is over here."},{"Start":"08:15.410 ","End":"08:21.130","Text":"This is our reflected wavefront."},{"Start":"08:21.440 ","End":"08:28.370","Text":"Our incident wavefront is defined as the wavefront that we"},{"Start":"08:28.370 ","End":"08:35.750","Text":"have when the first point of the wavefront hits the barrier to be reflected."},{"Start":"08:35.750 ","End":"08:38.900","Text":"A reflected wavefront is defined as when"},{"Start":"08:38.900 ","End":"08:44.905","Text":"the last point on the wavefront has hit the barrier to be reflected."},{"Start":"08:44.905 ","End":"08:47.390","Text":"As we know with water waves,"},{"Start":"08:47.390 ","End":"08:50.750","Text":"the reflected angle is defined as the angle"},{"Start":"08:50.750 ","End":"08:55.410","Text":"between the reflected wavefront and the barrier."},{"Start":"08:55.700 ","End":"09:00.210","Text":"This is Theta_i tag."},{"Start":"09:00.210 ","End":"09:07.160","Text":"Theta_i over here was for the reflected angle when dealing with light waves."},{"Start":"09:07.160 ","End":"09:12.545","Text":"Here, Theta_i tag is the reflected angle when dealing with water waves."},{"Start":"09:12.545 ","End":"09:14.225","Text":"Now as we know,"},{"Start":"09:14.225 ","End":"09:20.180","Text":"Theta_i is equal to Theta_i tag."},{"Start":"09:20.180 ","End":"09:21.515","Text":"We saw that before."},{"Start":"09:21.515 ","End":"09:26.690","Text":"The incident angle for light waves mathematically is equal to"},{"Start":"09:26.690 ","End":"09:34.680","Text":"the incident angle for water waves and similarly for the reflected waves."},{"Start":"09:35.840 ","End":"09:38.420","Text":"The definition is different,"},{"Start":"09:38.420 ","End":"09:41.570","Text":"but if we put the numbers of the angles,"},{"Start":"09:41.570 ","End":"09:47.430","Text":"so the size of the angles will see that these sizes are the same."},{"Start":"09:48.520 ","End":"09:51.440","Text":"Now let\u0027s answer Question number 5."},{"Start":"09:51.440 ","End":"09:56.285","Text":"Draw the wavefront the moment the middle of the wavefront reaches the barrier."},{"Start":"09:56.285 ","End":"09:59.180","Text":"I\u0027m going to rub out some of what is on"},{"Start":"09:59.180 ","End":"10:03.900","Text":"this board so that we can answer this question in a neat way."},{"Start":"10:04.610 ","End":"10:08.330","Text":"Now what we\u0027re trying to do is we\u0027re trying to see"},{"Start":"10:08.330 ","End":"10:11.060","Text":"what our wave will look like when this point here,"},{"Start":"10:11.060 ","End":"10:13.595","Text":"which is approximately in the middle,"},{"Start":"10:13.595 ","End":"10:19.230","Text":"will reach the barrier and just before it gets reflected."},{"Start":"10:19.230 ","End":"10:23.095","Text":"What does that mean? If this point has reached the barrier,"},{"Start":"10:23.095 ","End":"10:27.365","Text":"let\u0027s call this point M because it\u0027s in the middle,"},{"Start":"10:27.365 ","End":"10:29.040","Text":"this point over here."},{"Start":"10:29.040 ","End":"10:32.945","Text":"That means that point A has already been reflected."},{"Start":"10:32.945 ","End":"10:38.080","Text":"M is of course at the barrier and B has not yet been reflected."},{"Start":"10:38.080 ","End":"10:41.620","Text":"I\u0027m going to draw this again in red so that it stands out."},{"Start":"10:41.620 ","End":"10:48.470","Text":"Let\u0027s follow point M and see how many seconds it takes to reach the barrier."},{"Start":"10:48.470 ","End":"10:51.920","Text":"We go 2 across and 1, 2, 3 up."},{"Start":"10:51.920 ","End":"10:54.165","Text":"This is 1 second,"},{"Start":"10:54.165 ","End":"10:55.614","Text":"2 across and 1,"},{"Start":"10:55.614 ","End":"10:57.870","Text":"2, 3 up, 2 seconds,"},{"Start":"10:57.870 ","End":"11:00.345","Text":"2 across and 1, 2, 3 up,"},{"Start":"11:00.345 ","End":"11:04.200","Text":"3 2 across, and 1, 2,"},{"Start":"11:04.200 ","End":"11:07.908","Text":"3 up 2 across, 1,"},{"Start":"11:07.908 ","End":"11:12.570","Text":"2, 3, 2 across 1, 2, 3."},{"Start":"11:12.570 ","End":"11:16.450","Text":"Here we can see 1 second,"},{"Start":"11:17.300 ","End":"11:20.265","Text":"2, 3, 4, 5, 6."},{"Start":"11:20.265 ","End":"11:26.205","Text":"It takes point M 6 seconds to reach the barrier."},{"Start":"11:26.205 ","End":"11:28.860","Text":"Where is point A after 6 seconds?"},{"Start":"11:28.860 ","End":"11:31.420","Text":"So this is 1 second,"},{"Start":"11:31.640 ","End":"11:34.680","Text":"2, 3, 4, 5, 6."},{"Start":"11:34.680 ","End":"11:37.335","Text":"Point A is over here."},{"Start":"11:37.335 ","End":"11:39.495","Text":"Let\u0027s call this A tag."},{"Start":"11:39.495 ","End":"11:43.425","Text":"This is the M tag and where is point B after 6 seconds?"},{"Start":"11:43.425 ","End":"11:44.962","Text":"B is at 1, 2,"},{"Start":"11:44.962 ","End":"11:47.260","Text":"3, 4, 5, 6."},{"Start":"11:50.090 ","End":"11:53.985","Text":"This is B tag."},{"Start":"11:53.985 ","End":"11:56.990","Text":"Now we can draw our wave."},{"Start":"11:56.990 ","End":"11:59.120","Text":"Let\u0027s continue it in red."},{"Start":"11:59.120 ","End":"12:06.400","Text":"A tag will join to M tag and M tag will join to B tag."},{"Start":"12:06.400 ","End":"12:10.205","Text":"What we can see is that A has been reflected,"},{"Start":"12:10.205 ","End":"12:12.275","Text":"M has just hit the barrier,"},{"Start":"12:12.275 ","End":"12:17.090","Text":"which is the question when the middle of the wavefront hits the barrier,"},{"Start":"12:17.090 ","End":"12:20.645","Text":"and B has not yet been reflected."},{"Start":"12:20.645 ","End":"12:26.900","Text":"We can see we get this upside-down V-shape for a wavefront,"},{"Start":"12:26.900 ","End":"12:33.270","Text":"where we can see that half of it has been reflected and half of it has not yet been."},{"Start":"12:34.450 ","End":"12:37.950","Text":"That\u0027s the end of this lesson."}],"ID":12483},{"Watched":false,"Name":"Exercise - Waves Travelling Through Different Mediums","Duration":"6m 23s","ChapterTopicVideoID":12016,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.755","Text":"Hello. In this lesson,"},{"Start":"00:01.755 ","End":"00:04.305","Text":"we\u0027re going to be answering the following question."},{"Start":"00:04.305 ","End":"00:08.925","Text":"A wave generator, on the left-hand side of the wave tank,"},{"Start":"00:08.925 ","End":"00:15.180","Text":"generates waves with a straight wavefront at a frequency of 4 hertz."},{"Start":"00:15.180 ","End":"00:17.985","Text":"That\u0027s 4 peaks per second."},{"Start":"00:17.985 ","End":"00:24.330","Text":"One side of the wave tank is a deep and the other side is shallow."},{"Start":"00:24.330 ","End":"00:28.530","Text":"The velocity of the wave at the deep end is"},{"Start":"00:28.530 ","End":"00:36.480","Text":"V_1 and it is equal to 20 centimeters per second."},{"Start":"00:36.480 ","End":"00:40.230","Text":"Question number 1 that we\u0027re going to answer is what"},{"Start":"00:40.230 ","End":"00:43.920","Text":"is the velocity of the wave at the shallow end?"},{"Start":"00:43.920 ","End":"00:48.350","Text":"Two lessons ago, we derived this equation."},{"Start":"00:48.350 ","End":"00:51.065","Text":"If you don\u0027t remember it, you can go back to your lessons."},{"Start":"00:51.065 ","End":"00:53.760","Text":"That was that V_1,"},{"Start":"00:53.760 ","End":"00:55.505","Text":"the velocity in 1 end,"},{"Start":"00:55.505 ","End":"00:56.780","Text":"divided by V_2,"},{"Start":"00:56.780 ","End":"00:58.610","Text":"the velocity and the other end,"},{"Start":"00:58.610 ","End":"01:03.800","Text":"is equal to sine of the angle Theta_1,"},{"Start":"01:03.800 ","End":"01:11.135","Text":"that\u0027s the angle that the wavefront hits our barrier or our change in depth,"},{"Start":"01:11.135 ","End":"01:20.180","Text":"divided by sine of the angle that our wavefront leaves our barrier or a change in depth."},{"Start":"01:20.360 ","End":"01:24.935","Text":"Here we\u0027ll keep our units in centimeters per second."},{"Start":"01:24.935 ","End":"01:30.410","Text":"V_1 is 20 centimeters per second divided by V_2,"},{"Start":"01:30.410 ","End":"01:32.345","Text":"which is what we\u0027re trying to find,"},{"Start":"01:32.345 ","End":"01:37.700","Text":"which is equal to sine of the angle that our wavefront hits,"},{"Start":"01:37.700 ","End":"01:41.180","Text":"our change in depth, that\u0027s 30 degrees."},{"Start":"01:41.180 ","End":"01:44.360","Text":"Sine of 30 degrees divided by sine of"},{"Start":"01:44.360 ","End":"01:48.650","Text":"the angle at which our wavefront leaves our change in depth,"},{"Start":"01:48.650 ","End":"01:52.885","Text":"which is over here we can see 20 degrees."},{"Start":"01:52.885 ","End":"01:59.915","Text":"Now if we isolate out our V_2 and plug everything into a calculator,"},{"Start":"01:59.915 ","End":"02:09.880","Text":"we\u0027ll get that our V_2=13.7 centimeters per second."},{"Start":"02:09.880 ","End":"02:12.870","Text":"That\u0027s the answer to Question number 1."},{"Start":"02:12.870 ","End":"02:16.195","Text":"Now, let\u0027s answer Question number 2."},{"Start":"02:16.195 ","End":"02:20.225","Text":"What is the wavelength of the wave at the deep end?"},{"Start":"02:20.225 ","End":"02:25.765","Text":"We\u0027re trying to find this wavelength over here, Lambda_1."},{"Start":"02:25.765 ","End":"02:32.060","Text":"We remember from previous lessons that our equation linking the velocity, the frequency,"},{"Start":"02:32.060 ","End":"02:36.770","Text":"and the wavelength is V is equal to Lambda"},{"Start":"02:36.770 ","End":"02:43.595","Text":"multiplied by f. Here we\u0027re using our velocity in the deep end."},{"Start":"02:43.595 ","End":"02:52.300","Text":"We\u0027re using, therefore that V_1 is equal to Lambda_1 f_1."},{"Start":"02:52.300 ","End":"02:57.514","Text":"Our V is 20 centimeters per second,"},{"Start":"02:57.514 ","End":"03:00.050","Text":"which is equal to our wavelength, Lambda_1,"},{"Start":"03:00.050 ","End":"03:01.670","Text":"which is what we\u0027re trying to find,"},{"Start":"03:01.670 ","End":"03:04.085","Text":"multiplied by our frequency,"},{"Start":"03:04.085 ","End":"03:08.575","Text":"which we\u0027re being told is equal to 4 hertz."},{"Start":"03:08.575 ","End":"03:12.750","Text":"Therefore, we can isolate out our Lambda_1,"},{"Start":"03:12.750 ","End":"03:16.860","Text":"so we divide both sides by 4 and therefore,"},{"Start":"03:16.860 ","End":"03:20.396","Text":"I will get that Lambda_1=5,"},{"Start":"03:20.396 ","End":"03:26.225","Text":"and then units because we\u0027re using the velocity is centimeters per second."},{"Start":"03:26.225 ","End":"03:30.355","Text":"Our wavelength is 5 centimeters."},{"Start":"03:30.355 ","End":"03:33.165","Text":"That\u0027s the answer to Question 2."},{"Start":"03:33.165 ","End":"03:35.880","Text":"Now let\u0027s answer Question 3."},{"Start":"03:35.880 ","End":"03:40.985","Text":"Question 3 is what is the wavelength of the wave at the shallow end?"},{"Start":"03:40.985 ","End":"03:43.720","Text":"Now we\u0027re trying to find Lambda_2."},{"Start":"03:43.720 ","End":"03:46.010","Text":"We\u0027re going to use the same equation here,"},{"Start":"03:46.010 ","End":"03:48.110","Text":"but now with using our numbers too."},{"Start":"03:48.110 ","End":"03:53.785","Text":"Here we\u0027re going to use that V_2 is equal to Lambda_2 f_2."},{"Start":"03:53.785 ","End":"03:59.120","Text":"Now of course we know that our frequency remains the same."},{"Start":"03:59.120 ","End":"04:03.850","Text":"Let\u0027s write over here that f_1 is equal to f_2."},{"Start":"04:03.850 ","End":"04:07.755","Text":"Our V_2 we calculated in Question number 1."},{"Start":"04:07.755 ","End":"04:12.185","Text":"V_2=13.7 centimeters per second,"},{"Start":"04:12.185 ","End":"04:13.850","Text":"which is equal to Lambda_2,"},{"Start":"04:13.850 ","End":"04:15.335","Text":"which is what we\u0027re trying to find,"},{"Start":"04:15.335 ","End":"04:17.540","Text":"multiplied by our frequency,"},{"Start":"04:17.540 ","End":"04:19.750","Text":"which is 4 hertz."},{"Start":"04:19.750 ","End":"04:23.090","Text":"Therefore, we can isolate out our Lambda_2."},{"Start":"04:23.090 ","End":"04:26.450","Text":"We\u0027re going to divide both sides by 4 and then we\u0027ll get"},{"Start":"04:26.450 ","End":"04:33.570","Text":"that our Lambda_2=3.42 centimeters."},{"Start":"04:34.570 ","End":"04:38.180","Text":"That answers Question number 3."},{"Start":"04:38.180 ","End":"04:42.680","Text":"Now what we want to do is we want to answer Question number 4,"},{"Start":"04:42.680 ","End":"04:48.310","Text":"which is to add 2 more wavefronts to the shallow end of the diagram."},{"Start":"04:48.310 ","End":"04:50.660","Text":"Let\u0027s draw this in black."},{"Start":"04:50.660 ","End":"04:53.465","Text":"Now, what they want to show here is 2 things."},{"Start":"04:53.465 ","End":"04:57.350","Text":"Number 1, we want to show that we know that"},{"Start":"04:57.350 ","End":"05:03.630","Text":"the wavefront is going to be traveling perpendicular to this first wavefront."},{"Start":"05:04.040 ","End":"05:07.120","Text":"If we take some line,"},{"Start":"05:07.120 ","End":"05:09.590","Text":"it\u0027s going to be perpendicular over here,"},{"Start":"05:09.590 ","End":"05:14.300","Text":"and then we can draw our new wavefront here."},{"Start":"05:14.300 ","End":"05:16.970","Text":"We can see that it\u0027s going to be perpendicular to this wavefront,"},{"Start":"05:16.970 ","End":"05:20.540","Text":"which means that it\u0027s going to be not exactly straight,"},{"Start":"05:20.540 ","End":"05:27.380","Text":"but traveling at a bit of an angle because here we have a bit of a slant."},{"Start":"05:27.380 ","End":"05:33.260","Text":"Here\u0027s our first wavefront that we\u0027re adding in, and of course,"},{"Start":"05:33.260 ","End":"05:36.630","Text":"this length over here is Lambda_2,"},{"Start":"05:36.630 ","End":"05:38.010","Text":"which as we can see,"},{"Start":"05:38.010 ","End":"05:40.395","Text":"Lambda_2 is smaller than Lambda_1."},{"Start":"05:40.395 ","End":"05:44.585","Text":"We\u0027re going to be drawing these new wavefronts slightly closer to one another."},{"Start":"05:44.585 ","End":"05:50.415","Text":"Then again, we\u0027re drawing another line 90 degrees."},{"Start":"05:50.415 ","End":"05:53.540","Text":"Imagine that this is straight and at 90 degrees to"},{"Start":"05:53.540 ","End":"05:58.215","Text":"the wavefront and then we can draw our second wavefront."},{"Start":"05:58.215 ","End":"06:01.545","Text":"Of course, this distance over here is meant to be Lambda_2 as well,"},{"Start":"06:01.545 ","End":"06:03.500","Text":"it\u0027s meant to be the same distance."},{"Start":"06:03.500 ","End":"06:07.115","Text":"That\u0027s what we\u0027re checking that you remember to draw"},{"Start":"06:07.115 ","End":"06:11.840","Text":"your new wavefront at a 90-degree angle to the previous wavefront and"},{"Start":"06:11.840 ","End":"06:15.530","Text":"that you remember that the distance"},{"Start":"06:15.530 ","End":"06:20.420","Text":"between each wavefronts is going to be less based on our calculations."},{"Start":"06:20.420 ","End":"06:23.340","Text":"That\u0027s the end of our lesson."}],"ID":12484},{"Watched":false,"Name":"Energy and Amplitude","Duration":"20m 53s","ChapterTopicVideoID":12017,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.965","Text":"Hello. In this lesson,"},{"Start":"00:01.965 ","End":"00:06.660","Text":"we\u0027re going to be learning about the energy and amplitude of waves."},{"Start":"00:06.660 ","End":"00:11.835","Text":"We\u0027ve already learned that the energy of a wave"},{"Start":"00:11.835 ","End":"00:16.890","Text":"is proportional to the amplitude of the wave squared."},{"Start":"00:16.890 ","End":"00:18.375","Text":"This we already know,"},{"Start":"00:18.375 ","End":"00:23.490","Text":"and in this lesson we\u0027re going to be learning about this in slightly more detail."},{"Start":"00:23.490 ","End":"00:26.610","Text":"Let\u0027s recap what happens to our different waves."},{"Start":"00:26.610 ","End":"00:29.460","Text":"We saw that in a 1-dimensional wave,"},{"Start":"00:29.460 ","End":"00:32.145","Text":"we have our transmission medium,"},{"Start":"00:32.145 ","End":"00:36.570","Text":"and that if we have some wave like this passing through,"},{"Start":"00:36.570 ","End":"00:39.995","Text":"so a pulse, so a few moments later,"},{"Start":"00:39.995 ","End":"00:42.770","Text":"the pulse will be exactly identical."},{"Start":"00:42.770 ","End":"00:43.925","Text":"What does that mean?"},{"Start":"00:43.925 ","End":"00:51.095","Text":"It means that its amplitude will be the same and its width will be the same."},{"Start":"00:51.095 ","End":"00:55.955","Text":"We can see that the shape doesn\u0027t change."},{"Start":"00:55.955 ","End":"01:04.745","Text":"If this pulse over here is at t_1 and a few moments later pulse is over here at t_2,"},{"Start":"01:04.745 ","End":"01:06.950","Text":"so if the shape doesn\u0027t change,"},{"Start":"01:06.950 ","End":"01:13.960","Text":"that means that the energy at 0.1 is equal to the energy at 0.2."},{"Start":"01:13.960 ","End":"01:19.940","Text":"Therefore, if the energy at 0.1 is equal to the energy of 0.2 from this equation here,"},{"Start":"01:19.940 ","End":"01:24.590","Text":"that therefore means that the amplitude squared at"},{"Start":"01:24.590 ","End":"01:29.925","Text":"0.1 is equal to the amplitude squared at 0.2,"},{"Start":"01:29.925 ","End":"01:36.800","Text":"and therefore, we can say that the amplitude at 0.1 is equal to the amplitude at 0.2,"},{"Start":"01:36.800 ","End":"01:40.110","Text":"which is what we already said."},{"Start":"01:40.150 ","End":"01:46.475","Text":"Now, this is of course true if we have no energy loss."},{"Start":"01:46.475 ","End":"01:50.040","Text":"Now let\u0027s take a look at 2D waves."},{"Start":"01:50.040 ","End":"01:55.730","Text":"Here we\u0027ve been speaking for the past few lessons about water waves."},{"Start":"01:55.730 ","End":"02:00.245","Text":"Let\u0027s take our first case over here,"},{"Start":"02:00.245 ","End":"02:02.960","Text":"where we have a straight wavefront."},{"Start":"02:02.960 ","End":"02:07.070","Text":"Let\u0027s call this a, so here we\u0027re dealing with a straight wave front,"},{"Start":"02:07.070 ","End":"02:11.090","Text":"so we know that if this is what our wavefront looks like at t_1,"},{"Start":"02:11.090 ","End":"02:14.180","Text":"so a few moments later at t_2,"},{"Start":"02:14.180 ","End":"02:19.355","Text":"a wave front will look exactly the same and it will carry on looking exactly the same."},{"Start":"02:19.355 ","End":"02:21.920","Text":"Of course, this is if we have no energy loss."},{"Start":"02:21.920 ","End":"02:23.630","Text":"In a 2D wave,"},{"Start":"02:23.630 ","End":"02:29.010","Text":"if we have no energy loss with a straight wavefront, so E_1,"},{"Start":"02:29.010 ","End":"02:33.400","Text":"the energy at t_1 will be equal to our energy at t_2,"},{"Start":"02:33.400 ","End":"02:39.130","Text":"and therefore, again, the amplitude squared of each 1 will be the same,"},{"Start":"02:39.130 ","End":"02:45.200","Text":"and therefore we can say that the amplitude of the 2D wave"},{"Start":"02:45.200 ","End":"02:52.730","Text":"at time t_1 will equal to the amplitude of the 2D wave at time number 2."},{"Start":"02:53.120 ","End":"02:58.180","Text":"This is very important because this is with a straight wavefront."},{"Start":"02:58.180 ","End":"03:02.020","Text":"But now let\u0027s deal with the second case."},{"Start":"03:02.020 ","End":"03:03.720","Text":"This is case number 2,"},{"Start":"03:03.720 ","End":"03:09.820","Text":"and here we\u0027re dealing with a circular wavefront."},{"Start":"03:10.610 ","End":"03:18.095","Text":"Here again, let\u0027s speak about the case where our energy isn\u0027t being lost."},{"Start":"03:18.095 ","End":"03:23.910","Text":"Our energy E_1 is equal to our energy E_2."},{"Start":"03:23.910 ","End":"03:25.620","Text":"We have no energy loss."},{"Start":"03:25.620 ","End":"03:27.645","Text":"Let\u0027s say at t_1,"},{"Start":"03:27.645 ","End":"03:29.670","Text":"this is what our wave front looks like,"},{"Start":"03:29.670 ","End":"03:31.935","Text":"so let\u0027s call this at t_1,"},{"Start":"03:31.935 ","End":"03:34.305","Text":"and then at t_2,"},{"Start":"03:34.305 ","End":"03:37.245","Text":"our wavefront looks like this,"},{"Start":"03:37.245 ","End":"03:39.375","Text":"it\u0027s spread out,"},{"Start":"03:39.375 ","End":"03:42.225","Text":"so this is at t_2,"},{"Start":"03:42.225 ","End":"03:44.440","Text":"and then as we know at t_3,"},{"Start":"03:44.440 ","End":"03:47.580","Text":"it will look something like so."},{"Start":"03:48.980 ","End":"03:52.335","Text":"Just circular wavefronts moving out."},{"Start":"03:52.335 ","End":"03:55.320","Text":"This is our point t_2,"},{"Start":"03:55.320 ","End":"03:56.540","Text":"we won\u0027t look at t_3."},{"Start":"03:56.540 ","End":"04:02.440","Text":"In the meantime, we\u0027re just focusing on the differences between t_1 and t_2."},{"Start":"04:02.440 ","End":"04:04.630","Text":"What\u0027s important to note here,"},{"Start":"04:04.630 ","End":"04:07.970","Text":"is that the energy is the same."},{"Start":"04:07.970 ","End":"04:10.804","Text":"We\u0027re not losing any energy."},{"Start":"04:10.804 ","End":"04:14.705","Text":"However, what\u0027s different here in the circular wavefront,"},{"Start":"04:14.705 ","End":"04:17.990","Text":"is that the energy of the circular wave front at"},{"Start":"04:17.990 ","End":"04:22.475","Text":"t_1 is spread across a certain circumference."},{"Start":"04:22.475 ","End":"04:24.740","Text":"Let\u0027s call it C_1."},{"Start":"04:24.740 ","End":"04:29.305","Text":"Whereas the energy for wavefront at t_2,"},{"Start":"04:29.305 ","End":"04:31.590","Text":"so over here, so E_2,"},{"Start":"04:31.590 ","End":"04:35.420","Text":"it\u0027s the same energy because we said that we don\u0027t have any energy loss."},{"Start":"04:35.420 ","End":"04:41.075","Text":"However, this energy is spread out across a much larger circumference,"},{"Start":"04:41.075 ","End":"04:43.535","Text":"as we can see over here."},{"Start":"04:43.535 ","End":"04:45.545","Text":"What does this mean,"},{"Start":"04:45.545 ","End":"04:48.750","Text":"when we\u0027re speaking about the amplitude?"},{"Start":"04:49.600 ","End":"04:54.530","Text":"We\u0027re remembering that the total energy is being saved."},{"Start":"04:54.530 ","End":"04:56.210","Text":"We\u0027re not losing any energy."},{"Start":"04:56.210 ","End":"05:00.560","Text":"However, let\u0027s take a look at the energy at every single point."},{"Start":"05:00.560 ","End":"05:02.555","Text":"Let\u0027s look at the moment t_1,"},{"Start":"05:02.555 ","End":"05:06.095","Text":"the energy of this point over here in red,"},{"Start":"05:06.095 ","End":"05:07.415","Text":"that little dot,"},{"Start":"05:07.415 ","End":"05:09.325","Text":"and then let\u0027s look at t_2,"},{"Start":"05:09.325 ","End":"05:14.500","Text":"the energy at this specific point on the wavefront in blue."},{"Start":"05:15.050 ","End":"05:17.445","Text":"What we can say,"},{"Start":"05:17.445 ","End":"05:18.960","Text":"let\u0027s call it tags."},{"Start":"05:18.960 ","End":"05:23.580","Text":"The energy of a point on the wave front at t_1,"},{"Start":"05:23.580 ","End":"05:27.405","Text":"so let\u0027s call that E_1 tag."},{"Start":"05:27.405 ","End":"05:31.535","Text":"This is the energy of that specific red point."},{"Start":"05:31.535 ","End":"05:36.770","Text":"Then if we multiply that energy of that specific point,"},{"Start":"05:36.770 ","End":"05:42.975","Text":"multiplied by the circumference of the circle,"},{"Start":"05:42.975 ","End":"05:45.630","Text":"so the circumference of the wavefront,"},{"Start":"05:45.630 ","End":"05:47.510","Text":"but we\u0027re dealing with a circular wavefront,"},{"Start":"05:47.510 ","End":"05:51.830","Text":"so we multiply it by 2Pi multiplied by that radius,"},{"Start":"05:51.830 ","End":"05:54.570","Text":"so let\u0027s call that radius r_1."},{"Start":"05:55.130 ","End":"05:59.995","Text":"We know this, what is this all equal to?"},{"Start":"05:59.995 ","End":"06:02.635","Text":"This is simply equal to E_1,"},{"Start":"06:02.635 ","End":"06:07.205","Text":"the total energy of this circular wavefront."},{"Start":"06:07.205 ","End":"06:09.360","Text":"We know that this is equal to E_2,"},{"Start":"06:09.360 ","End":"06:10.835","Text":"so what is E_2?"},{"Start":"06:10.835 ","End":"06:17.110","Text":"E_2 is made up of the energy at this specific point that we drew in blue,"},{"Start":"06:17.110 ","End":"06:20.410","Text":"so let\u0027s call that the energy E_2 tag."},{"Start":"06:20.410 ","End":"06:26.950","Text":"That\u0027s the specific energy of this blue point multiplied by the circumference."},{"Start":"06:26.950 ","End":"06:31.930","Text":"We have this point times the circumference of this circle."},{"Start":"06:31.930 ","End":"06:37.450","Text":"That is equal to 2Pi multiplied by the radius of that circle,"},{"Start":"06:37.450 ","End":"06:39.680","Text":"which is r_2,"},{"Start":"06:39.680 ","End":"06:46.535","Text":"because we have this point multiplied by all of the points that make up this circle."},{"Start":"06:46.535 ","End":"06:54.020","Text":"That\u0027s 2Pir_2, and obviously all of this together is equal to the total energy E_2."},{"Start":"06:54.020 ","End":"06:59.420","Text":"We know that E_1 is equal to E_2 because we said that we don\u0027t have any energy loss."},{"Start":"06:59.420 ","End":"07:04.420","Text":"Now what we can do, is we can divide both sides by 2Pi,"},{"Start":"07:04.420 ","End":"07:11.210","Text":"and then if we want to get our E_1 tag on the same side as our E_2 tag, so we\u0027ll get,"},{"Start":"07:11.210 ","End":"07:18.240","Text":"if we divide both sides by 2Pi and divide both sides by E_2 tag."},{"Start":"07:18.240 ","End":"07:26.370","Text":"We\u0027ll get that E_1 tag divided by E_2 tag is equal to r_2,"},{"Start":"07:26.370 ","End":"07:31.455","Text":"and then we\u0027ll divide both sides by r_1 divided by r_1."},{"Start":"07:31.455 ","End":"07:34.940","Text":"We can see that the relationship between"},{"Start":"07:34.940 ","End":"07:38.690","Text":"the energy at a specific point on the wavefront,"},{"Start":"07:38.690 ","End":"07:46.824","Text":"on a circular wavefront in 2D waves is inversely proportional to its radius."},{"Start":"07:46.824 ","End":"07:51.375","Text":"Let\u0027s scroll down a little bit over here."},{"Start":"07:51.375 ","End":"07:57.540","Text":"What we can see if we flip these around,"},{"Start":"07:57.540 ","End":"08:00.530","Text":"so let\u0027s write this in this formation,"},{"Start":"08:00.530 ","End":"08:02.450","Text":"that E_2 tag,"},{"Start":"08:02.450 ","End":"08:13.300","Text":"so the energy at this point over here divided by E_1 tag is equal to r_1 divided by r_2."},{"Start":"08:14.030 ","End":"08:17.315","Text":"This is the same equation."},{"Start":"08:17.315 ","End":"08:18.860","Text":"I just took the reciprocal,"},{"Start":"08:18.860 ","End":"08:21.365","Text":"but it\u0027s equal to the same thing."},{"Start":"08:21.365 ","End":"08:27.645","Text":"Now we can see that the larger my radius r_2,"},{"Start":"08:27.645 ","End":"08:30.715","Text":"so we know that this is the radius r_1,"},{"Start":"08:30.715 ","End":"08:33.590","Text":"and this is the radius r_2."},{"Start":"08:33.590 ","End":"08:36.065","Text":"The larger my radius r_2,"},{"Start":"08:36.065 ","End":"08:38.860","Text":"the smaller this energy,"},{"Start":"08:38.860 ","End":"08:43.925","Text":"the energy that each point in my circular wavefront will have."},{"Start":"08:43.925 ","End":"08:49.340","Text":"Let\u0027s say that my radius increases in"},{"Start":"08:49.340 ","End":"08:56.835","Text":"size by a factor of 4,"},{"Start":"08:56.835 ","End":"09:00.895","Text":"so then that will mean that by energy,"},{"Start":"09:00.895 ","End":"09:04.890","Text":"at any point on that circular wavefront,"},{"Start":"09:04.890 ","End":"09:08.630","Text":"so my energy at that point is going to"},{"Start":"09:08.630 ","End":"09:16.440","Text":"decrease by a factor of 4."},{"Start":"09:16.440 ","End":"09:21.250","Text":"Or we can write it either it\u0027s going to decrease by a factor of 4,"},{"Start":"09:21.250 ","End":"09:24.520","Text":"or if my original,"},{"Start":"09:24.520 ","End":"09:31.710","Text":"if I had E_1 tag was equal to some energy."},{"Start":"09:31.710 ","End":"09:35.605","Text":"So then if I increase my radius by 4,"},{"Start":"09:35.605 ","End":"09:40.300","Text":"so then my E_2 tag at my second point,"},{"Start":"09:40.300 ","End":"09:42.510","Text":"is going to be E,"},{"Start":"09:42.510 ","End":"09:48.140","Text":"so I\u0027m decreasing it by a factor of 4 divided by 4."},{"Start":"09:48.140 ","End":"09:54.885","Text":"What we can see is that the total energy is being conserved,"},{"Start":"09:54.885 ","End":"09:58.950","Text":"but the energy of every single point on a circular wavefront is"},{"Start":"09:58.950 ","End":"10:03.247","Text":"not being conserved because it\u0027s dependent on the radius,"},{"Start":"10:03.247 ","End":"10:05.100","Text":"because the larger the radius,"},{"Start":"10:05.100 ","End":"10:09.300","Text":"the larger the circumference which means that we have"},{"Start":"10:09.300 ","End":"10:16.390","Text":"a larger length that this exact same energy has to be spread across."},{"Start":"10:16.790 ","End":"10:22.905","Text":"Let\u0027s rub this out and let\u0027s see what this means when we\u0027re dealing with the amplitude."},{"Start":"10:22.905 ","End":"10:30.915","Text":"What we know is that our energy is directly proportional to the amplitude squared."},{"Start":"10:30.915 ","End":"10:35.160","Text":"But what we\u0027ve seen over here is that"},{"Start":"10:35.160 ","End":"10:39.570","Text":"our energy in a circular 2-Dimensional wavefront is that"},{"Start":"10:39.570 ","End":"10:48.705","Text":"our energy is directly proportional to 1 divided by r,"},{"Start":"10:48.705 ","End":"10:52.725","Text":"in a circular 2-Dimensional wavefront."},{"Start":"10:52.725 ","End":"10:56.865","Text":"Therefore, we can say that our amplitude squared"},{"Start":"10:56.865 ","End":"11:01.510","Text":"is directly proportional to 1 divided by r."},{"Start":"11:01.510 ","End":"11:10.605","Text":"So our amplitude squared is inversely proportional to r. Therefore,"},{"Start":"11:10.605 ","End":"11:12.450","Text":"if we take the square root of both sides,"},{"Start":"11:12.450 ","End":"11:19.140","Text":"we get that our amplitude of our wave is proportional to 1 divided by"},{"Start":"11:19.140 ","End":"11:26.580","Text":"the square root of r. Let\u0027s use some number."},{"Start":"11:26.580 ","End":"11:32.205","Text":"Let\u0027s use this example again where we saw that our radius increases by a factor of 4."},{"Start":"11:32.205 ","End":"11:35.058","Text":"We saw that 1,"},{"Start":"11:35.058 ","End":"11:38.220","Text":"our energy at each point will decrease by a factor of 4,"},{"Start":"11:38.220 ","End":"11:41.400","Text":"but what does that mean for the amplitude?"},{"Start":"11:41.400 ","End":"11:44.205","Text":"We saw if we\u0027re using this,"},{"Start":"11:44.205 ","End":"11:48.210","Text":"that our amplitude is directly proportional to 1 divided"},{"Start":"11:48.210 ","End":"11:52.170","Text":"by the square root of r. If we\u0027re using this example,"},{"Start":"11:52.170 ","End":"11:54.630","Text":"where r increases by a factor of 4,"},{"Start":"11:54.630 ","End":"12:00.396","Text":"so our amplitude will be directly proportional to 1 divided by the square root of 4"},{"Start":"12:00.396 ","End":"12:04.590","Text":"which therefore means that our amplitude will be"},{"Start":"12:04.590 ","End":"12:09.200","Text":"directly proportional to 1/2 so our amplitude,"},{"Start":"12:09.200 ","End":"12:14.050","Text":"will halve in size or in magnitude."},{"Start":"12:14.050 ","End":"12:18.720","Text":"Now we can see that in 2-Dimensional circular wavefronts,"},{"Start":"12:18.720 ","End":"12:24.495","Text":"we can see that the energy is directly proportional to 1 divided by r,"},{"Start":"12:24.495 ","End":"12:26.820","Text":"and that the amplitude of the wave is directly"},{"Start":"12:26.820 ","End":"12:29.520","Text":"proportional to 1 divided by the square root of"},{"Start":"12:29.520 ","End":"12:37.265","Text":"r. Now what we\u0027re going to do is we\u0027re going to speak about 3-Dimensional waves,"},{"Start":"12:37.265 ","End":"12:41.820","Text":"where an example of a 3-Dimensional wave is light."},{"Start":"12:42.890 ","End":"12:48.675","Text":"As we said, examples of 3-Dimensional waves, are light,"},{"Start":"12:48.675 ","End":"12:55.420","Text":"sound, and shock from a blast or whatever that might be."},{"Start":"12:55.430 ","End":"13:00.960","Text":"We know that our energy is still going to be proportional"},{"Start":"13:00.960 ","End":"13:07.180","Text":"to the amplitude squared and remember we\u0027re speaking about the energy total."},{"Start":"13:07.190 ","End":"13:13.350","Text":"Here what we\u0027re going to be doing is we\u0027re going to be using"},{"Start":"13:13.350 ","End":"13:22.095","Text":"the same derivation idea that we did with our circular 2-Dimensional wave however,"},{"Start":"13:22.095 ","End":"13:25.965","Text":"now we\u0027re going to be turning this into 3-Dimensions."},{"Start":"13:25.965 ","End":"13:27.690","Text":"How does light, sound,"},{"Start":"13:27.690 ","End":"13:31.575","Text":"and shock and whatever it might be propagates"},{"Start":"13:31.575 ","End":"13:36.175","Text":"through air or whichever transmission medium it might be,"},{"Start":"13:36.175 ","End":"13:39.585","Text":"so it propagates in a sphere."},{"Start":"13:39.585 ","End":"13:42.820","Text":"If we draw this now,"},{"Start":"13:43.340 ","End":"13:52.425","Text":"just like in a 2-Dimensional circular wavefront will have this over here at time t_1."},{"Start":"13:52.425 ","End":"13:55.500","Text":"But remember instead of it being a circle in"},{"Start":"13:55.500 ","End":"13:58.365","Text":"2-Dimensions that we\u0027re doing with 3-Dimensions,"},{"Start":"13:58.365 ","End":"14:01.080","Text":"so it\u0027s going to be a sphere."},{"Start":"14:01.080 ","End":"14:06.105","Text":"Then a t_2 we\u0027ll have a wavefront,"},{"Start":"14:06.105 ","End":"14:14.175","Text":"like so, so 3-Dimensional wavefronts or spherical wavefront that has moved further."},{"Start":"14:14.175 ","End":"14:18.285","Text":"Here we\u0027re going to have our inner radius,"},{"Start":"14:18.285 ","End":"14:22.960","Text":"and here we\u0027re going to have our outer radius, r_2."},{"Start":"14:23.270 ","End":"14:27.060","Text":"Then, of course, there\u0027ll be a few moments later,"},{"Start":"14:27.060 ","End":"14:31.560","Text":"a subsequent spherical wavefront out here."},{"Start":"14:31.560 ","End":"14:33.135","Text":"But we\u0027re not going to speak about that now,"},{"Start":"14:33.135 ","End":"14:38.760","Text":"we\u0027re speaking just about the wavefront at t_1 and the wavefront at t_2."},{"Start":"14:38.760 ","End":"14:42.405","Text":"Just like with our 2-Dimensional circular wavefront,"},{"Start":"14:42.405 ","End":"14:45.645","Text":"let\u0027s write out like so."},{"Start":"14:45.645 ","End":"14:54.485","Text":"Here we\u0027re going to write down the energy of this point over here as E\u0027_1,"},{"Start":"14:54.485 ","End":"14:58.835","Text":"of that little point over here on our inner spherical wavefront,"},{"Start":"14:58.835 ","End":"15:03.630","Text":"and then this point over here in blue is another little point and the energy of"},{"Start":"15:03.630 ","End":"15:09.630","Text":"this point will be E\u0027_2 so it\u0027s just the energy of this point over here."},{"Start":"15:09.630 ","End":"15:13.830","Text":"Now if we\u0027re dealing with the case of 0 energy loss,"},{"Start":"15:13.830 ","End":"15:18.765","Text":"that means that the total energy at t_1 is going to"},{"Start":"15:18.765 ","End":"15:24.210","Text":"equal to the total energy at t_2 over here."},{"Start":"15:24.210 ","End":"15:27.525","Text":"As we saw with the 2-Dimensional wave,"},{"Start":"15:27.525 ","End":"15:34.470","Text":"even if the total energy at time t_1 is equal to the total energy at time t_2,"},{"Start":"15:34.470 ","End":"15:37.680","Text":"because that same energy is split when we\u0027re dealing with"},{"Start":"15:37.680 ","End":"15:42.255","Text":"2-Dimensions over a different length,"},{"Start":"15:42.255 ","End":"15:44.535","Text":"so a different circumference."},{"Start":"15:44.535 ","End":"15:48.900","Text":"Here the total energy is going to have to be"},{"Start":"15:48.900 ","End":"15:53.475","Text":"spread out over a different surface area because we\u0027re dealing with 3D,"},{"Start":"15:53.475 ","End":"15:56.290","Text":"so we\u0027re dealing with a surface area."},{"Start":"15:57.350 ","End":"16:01.485","Text":"E_1, the energy at time t_1,"},{"Start":"16:01.485 ","End":"16:03.413","Text":"so let\u0027s see what that\u0027s equal to,"},{"Start":"16:03.413 ","End":"16:09.375","Text":"so that\u0027s equal to this energy at this specific point so that\u0027s E\u0027_1,"},{"Start":"16:09.375 ","End":"16:14.295","Text":"and then multiplied by this point all over everywhere"},{"Start":"16:14.295 ","End":"16:19.920","Text":"along the surface area of the sphere at time t_1."},{"Start":"16:19.920 ","End":"16:22.380","Text":"What\u0027s the surface area?"},{"Start":"16:22.380 ","End":"16:30.180","Text":"That is equal to multiplied by Pir^2 so here it"},{"Start":"16:30.180 ","End":"16:33.870","Text":"will be r_1^2 and then we also"},{"Start":"16:33.870 ","End":"16:38.620","Text":"have to multiply this by 4 to make this surface area of a sphere."},{"Start":"16:41.000 ","End":"16:46.065","Text":"Then we have this point making up the entire surface area of this sphere."},{"Start":"16:46.065 ","End":"16:48.900","Text":"This is of course, because we have no energy loss equal"},{"Start":"16:48.900 ","End":"16:52.665","Text":"to the energy at time t_2, so E_2."},{"Start":"16:52.665 ","End":"16:54.450","Text":"What is E_2 equal to?"},{"Start":"16:54.450 ","End":"16:58.980","Text":"E_2 is equal to the energy at this specific point so that we called"},{"Start":"16:58.980 ","End":"17:05.535","Text":"E\u0027_2 multiplied by the surface area of the sphere at time t_2."},{"Start":"17:05.535 ","End":"17:07.650","Text":"The surface area of this larger sphere."},{"Start":"17:07.650 ","End":"17:10.450","Text":"We\u0027re going to multiply this by 4Pir^2,"},{"Start":"17:10.580 ","End":"17:17.340","Text":"where now our radius is bigger so now it\u0027s r_2^2."},{"Start":"17:17.340 ","End":"17:21.420","Text":"Now, just like what we did before with the 2-Dimensional circular wave,"},{"Start":"17:21.420 ","End":"17:25.380","Text":"so we\u0027re going to do this with a 3-Dimensional spherical wave."},{"Start":"17:25.380 ","End":"17:33.420","Text":"We can cross out Pi and 4 from both sides and now what we can get is if"},{"Start":"17:33.420 ","End":"17:41.811","Text":"we divide both sides by E\u0027_1 and by r_2^2 we\u0027ll get that E\u0027_2,"},{"Start":"17:41.811 ","End":"17:47.130","Text":"so the energy at this point over here in blue divided by E\u0027_1,"},{"Start":"17:47.130 ","End":"17:50.295","Text":"the energy at this point here in red,"},{"Start":"17:50.295 ","End":"18:00.810","Text":"is simply equal to r_1^2 divided by r_2^2."},{"Start":"18:00.810 ","End":"18:07.440","Text":"We get this inverse relationship again but now because we\u0027ve gone up in a dimension,"},{"Start":"18:07.440 ","End":"18:12.210","Text":"so our radiuses also go up in a dimension whereas before we just had"},{"Start":"18:12.210 ","End":"18:17.385","Text":"r by a factor of r so here we have by a factor of r^2."},{"Start":"18:17.385 ","End":"18:21.990","Text":"Now we can see that the energy at a specific point on the sphere,"},{"Start":"18:21.990 ","End":"18:28.950","Text":"so a specific point is directly proportional to 1 divided"},{"Start":"18:28.950 ","End":"18:36.855","Text":"by r^2 so the radius from the center of the sphere to that point."},{"Start":"18:36.855 ","End":"18:42.090","Text":"Let\u0027s imagine that our radius,"},{"Start":"18:42.090 ","End":"18:48.960","Text":"so r is increased by a factor of 4."},{"Start":"18:48.960 ","End":"18:51.480","Text":"What will that mean?"},{"Start":"18:51.480 ","End":"18:55.800","Text":"Therefore, our energy at"},{"Start":"18:55.800 ","End":"19:03.630","Text":"that specific point will decrease by a factor of 4^2,"},{"Start":"19:03.630 ","End":"19:05.790","Text":"or by a factor of 16."},{"Start":"19:05.790 ","End":"19:13.095","Text":"Now let\u0027s see what this means when we\u0027re speaking about the amplitude of the wavefront."},{"Start":"19:13.095 ","End":"19:20.639","Text":"First of all, we know that our energy is proportional to our amplitude squared."},{"Start":"19:20.639 ","End":"19:27.625","Text":"We saw from here that that is proportional to 1 divided by r^2."},{"Start":"19:27.625 ","End":"19:30.630","Text":"Therefore, we see that our amplitude squared is"},{"Start":"19:30.630 ","End":"19:34.110","Text":"proportional to 1 divided by r^2, or in other words,"},{"Start":"19:34.110 ","End":"19:38.100","Text":"our amplitude is proportional to 1 divided by"},{"Start":"19:38.100 ","End":"19:45.215","Text":"r. If again we look at this example where our radius has increased by a factor of 4,"},{"Start":"19:45.215 ","End":"19:49.625","Text":"we already saw that the energy at that specific point,"},{"Start":"19:49.625 ","End":"19:55.855","Text":"it\u0027s right rather E\u0027 will have decreased by a factor of 4^2,"},{"Start":"19:55.855 ","End":"19:58.155","Text":"so by a factor of 16."},{"Start":"19:58.155 ","End":"20:00.410","Text":"Then if we look at our amplitude,"},{"Start":"20:00.410 ","End":"20:05.410","Text":"if our energy has decreased by a factor 4^2,"},{"Start":"20:05.410 ","End":"20:12.510","Text":"so our amplitude will be therefore proportional to 1 divided by the radius,"},{"Start":"20:12.510 ","End":"20:21.465","Text":"so 1 divided by a factor of 4 so our amplitude is just 1/4 of the original amplitude."},{"Start":"20:21.465 ","End":"20:26.075","Text":"What this explains is why the further we are from a light source,"},{"Start":"20:26.075 ","End":"20:30.470","Text":"the dimmer it looks because its energy or its amplitude"},{"Start":"20:30.470 ","End":"20:35.600","Text":"is decreasing as we go further away from the light source itself."},{"Start":"20:35.600 ","End":"20:37.817","Text":"That is the end of the lesson,"},{"Start":"20:37.817 ","End":"20:43.790","Text":"these are the 2 equations to take away with you for this section in 3D waves,"},{"Start":"20:43.790 ","End":"20:47.030","Text":"and the next question or the next lesson we\u0027ll be"},{"Start":"20:47.030 ","End":"20:51.290","Text":"dealing with answering a question of 3D waves."},{"Start":"20:51.290 ","End":"20:53.880","Text":"That\u0027s the end of this lesson."}],"ID":12485},{"Watched":false,"Name":"Exercise - Energy and Amplitude","Duration":"20m 40s","ChapterTopicVideoID":12018,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.995","Text":"Hello. In the last lesson,"},{"Start":"00:01.995 ","End":"00:07.965","Text":"we spoke about 2-dimensional circular wavefronts and 3-dimensional spherical wavefronts."},{"Start":"00:07.965 ","End":"00:13.080","Text":"In this lesson, we\u0027re going to be answering practice questions which deal"},{"Start":"00:13.080 ","End":"00:18.156","Text":"with the relationship between energy and amplitude of these wavefronts."},{"Start":"00:18.156 ","End":"00:23.370","Text":"Our first question is dealing with 2D and in a few more minutes,"},{"Start":"00:23.370 ","End":"00:24.735","Text":"once we\u0027ve answered this question,"},{"Start":"00:24.735 ","End":"00:28.245","Text":"we\u0027re going to answer a question dealing with 3D."},{"Start":"00:28.245 ","End":"00:32.926","Text":"A 2-dimensional circular wavefront propagates through a wave tank,"},{"Start":"00:32.926 ","End":"00:38.150","Text":"its amplitude was 1 centimeter when its radius was 3 centimeters."},{"Start":"00:38.150 ","End":"00:41.300","Text":"What will be the energy of a point on the wavefront"},{"Start":"00:41.300 ","End":"00:44.960","Text":"when its radius will be 15 centimeters?"},{"Start":"00:44.960 ","End":"00:49.720","Text":"That\u0027s question number 1 that we\u0027re going to answer now."},{"Start":"00:51.920 ","End":"01:00.590","Text":"Let\u0027s call this the original energy of the wave when the radius is 3 centimeters."},{"Start":"01:00.590 ","End":"01:05.420","Text":"In the previous lesson, we spoke about that if the original energy of"},{"Start":"01:05.420 ","End":"01:10.740","Text":"the wavefront is equal to some value and we don\u0027t have energy loss,"},{"Start":"01:10.740 ","End":"01:14.045","Text":"so the energy of the final wave front,"},{"Start":"01:14.045 ","End":"01:18.140","Text":"so that\u0027s the wavefront of 15 centimeter radius will be the"},{"Start":"01:18.140 ","End":"01:24.230","Text":"same and here we said that the total energy will be the same."},{"Start":"01:24.230 ","End":"01:28.175","Text":"But we know that if our radius is increasing,"},{"Start":"01:28.175 ","End":"01:33.900","Text":"that means that this same amount of energy has to be spread along in 2 dimensions,"},{"Start":"01:33.900 ","End":"01:37.655","Text":"it has to be spread along a longer total length,"},{"Start":"01:37.655 ","End":"01:40.550","Text":"which means that each point in a circular"},{"Start":"01:40.550 ","End":"01:44.500","Text":"wavefront will have a smaller portion of energy."},{"Start":"01:44.500 ","End":"01:49.430","Text":"Each point will have a smaller portion of energy because"},{"Start":"01:49.430 ","End":"01:53.600","Text":"the same energy has to be shared among a longer length."},{"Start":"01:53.600 ","End":"01:58.040","Text":"However, the total energy of the original wavefront or"},{"Start":"01:58.040 ","End":"02:03.475","Text":"the initial wavefront and the final wavefront will be the same."},{"Start":"02:03.475 ","End":"02:09.410","Text":"Then we remember that we said that the energy of every single specific points,"},{"Start":"02:09.410 ","End":"02:10.930","Text":"so we call that E tag,"},{"Start":"02:10.930 ","End":"02:19.670","Text":"is proportional in 2-dimensions to 1 divided by r. We said that we have to find"},{"Start":"02:19.670 ","End":"02:24.890","Text":"the factor by which the radius has increased and then we\u0027ll know that"},{"Start":"02:24.890 ","End":"02:31.150","Text":"the energy of every single point has decreased by that exact same factor."},{"Start":"02:31.150 ","End":"02:33.870","Text":"First of all, we can say that our original or"},{"Start":"02:33.870 ","End":"02:40.664","Text":"our initial radius was equal to 3 centimeters and our final radius,"},{"Start":"02:40.664 ","End":"02:45.510","Text":"so r_f is given to us as 15 centimeters."},{"Start":"02:45.510 ","End":"02:50.755","Text":"How are we going to find the factor by which our radius has changed?"},{"Start":"02:50.755 ","End":"02:54.250","Text":"Let\u0027s write r factor."},{"Start":"02:54.280 ","End":"02:59.720","Text":"That is going to be equal to our final radius 15,"},{"Start":"02:59.720 ","End":"03:01.700","Text":"divided by our initial radius,"},{"Start":"03:01.700 ","End":"03:04.945","Text":"3, which is equal to 5."},{"Start":"03:04.945 ","End":"03:10.370","Text":"What we can see is that our radius has increased by a factor of 5,"},{"Start":"03:10.370 ","End":"03:18.980","Text":"which means that our final energy"},{"Start":"03:18.980 ","End":"03:25.196","Text":"at every single point is going to be proportional by 1 divided by this r factor,"},{"Start":"03:25.196 ","End":"03:35.610","Text":"so 1 divided by 5 of our initial energy at every single point."},{"Start":"03:36.100 ","End":"03:39.680","Text":"That means that if we looked at"},{"Start":"03:39.680 ","End":"03:46.870","Text":"this circular wavefront of radius 3 centimeters, so r_i,"},{"Start":"03:46.870 ","End":"03:54.710","Text":"and then we look at every single point on this circular wavefront of radius r_f,"},{"Start":"03:54.710 ","End":"03:56.707","Text":"which is 15 centimeters,"},{"Start":"03:56.707 ","End":"04:00.215","Text":"the energy of every single point"},{"Start":"04:00.215 ","End":"04:06.979","Text":"on our initial wave is going to be some value."},{"Start":"04:06.979 ","End":"04:11.750","Text":"But then when we look at the energy of every single point on our final wave that"},{"Start":"04:11.750 ","End":"04:17.055","Text":"has a radius 5 times as big as our original wave,"},{"Start":"04:17.055 ","End":"04:23.210","Text":"so the energy of every single blue point is going to be 1/5 the energy of"},{"Start":"04:23.210 ","End":"04:27.575","Text":"every single red point so 1/5 the energy of"},{"Start":"04:27.575 ","End":"04:34.400","Text":"every point on our original wavefront."},{"Start":"04:35.420 ","End":"04:38.255","Text":"This is the answer to question number 1,"},{"Start":"04:38.255 ","End":"04:40.700","Text":"now let\u0027s answer question number 2."},{"Start":"04:40.700 ","End":"04:44.345","Text":"What will be the amplitude of this wavefront?"},{"Start":"04:44.345 ","End":"04:49.145","Text":"Speaking about the new wavefront with a radius of 15 centimeters."},{"Start":"04:49.145 ","End":"04:54.425","Text":"We remember the relationship between energy and amplitude,"},{"Start":"04:54.425 ","End":"04:57.850","Text":"and we know that our energy,"},{"Start":"04:57.850 ","End":"05:01.130","Text":"so here we were using the energy of every single point,"},{"Start":"05:01.130 ","End":"05:04.160","Text":"but obviously the wavefront itself"},{"Start":"05:04.160 ","End":"05:07.475","Text":"or every point on the wavefront will have the same amplitude."},{"Start":"05:07.475 ","End":"05:09.815","Text":"When we\u0027re speaking about amplitude,"},{"Start":"05:09.815 ","End":"05:12.860","Text":"the amplitude of this specific blue point,"},{"Start":"05:12.860 ","End":"05:17.240","Text":"or if we\u0027re speaking of the amplitude of the entire circular wavefront, it\u0027s the same,"},{"Start":"05:17.240 ","End":"05:18.785","Text":"it will have the same value,"},{"Start":"05:18.785 ","End":"05:22.410","Text":"specifically in a 2D circular wavefront."},{"Start":"05:22.410 ","End":"05:30.520","Text":"We said that our energy is proportional to our amplitude squared always."},{"Start":"05:30.520 ","End":"05:33.540","Text":"Here we can just rub off this tag,"},{"Start":"05:33.540 ","End":"05:34.760","Text":"it doesn\u0027t make a difference,"},{"Start":"05:34.760 ","End":"05:36.290","Text":"as we just said before,"},{"Start":"05:36.290 ","End":"05:41.750","Text":"and we know from the previous question that our energy is"},{"Start":"05:41.750 ","End":"05:48.450","Text":"proportional to 1 divided by our radius,"},{"Start":"05:48.450 ","End":"05:51.635","Text":"where here, we\u0027re speaking specifically about"},{"Start":"05:51.635 ","End":"05:56.045","Text":"the factor by which the radius was increased or decreased."},{"Start":"05:56.045 ","End":"05:58.940","Text":"Therefore, we can say,"},{"Start":"05:58.940 ","End":"06:04.820","Text":"if the amplitude squared is proportional to 1 divided by radius in a 2D wave,"},{"Start":"06:04.820 ","End":"06:09.830","Text":"we can say therefore that the amplitude is proportional to 1 divided"},{"Start":"06:09.830 ","End":"06:14.960","Text":"by the square root of the radius and of course,"},{"Start":"06:14.960 ","End":"06:19.425","Text":"we\u0027re speaking about the factor by which the radius has been increased."},{"Start":"06:19.425 ","End":"06:25.910","Text":"Here, we\u0027re dealing with a factor increase of 5 so that means that our amplitude will be"},{"Start":"06:25.910 ","End":"06:33.000","Text":"proportional to 1 divided by the square root of 5."},{"Start":"06:36.170 ","End":"06:41.335","Text":"Our amplitude is proportional to this change in factor."},{"Start":"06:41.335 ","End":"06:46.420","Text":"What are we going to do is we\u0027re going to look at our initial amplitude A_i,"},{"Start":"06:46.420 ","End":"06:50.850","Text":"which is given to us as 1 centimeter."},{"Start":"06:50.850 ","End":"06:53.395","Text":"Here from this equation,"},{"Start":"06:53.395 ","End":"06:55.615","Text":"we know that our final amplitude,"},{"Start":"06:55.615 ","End":"07:01.290","Text":"A_f is going to be equal to 1 divided by"},{"Start":"07:01.290 ","End":"07:08.355","Text":"the square root of 5 times our initial amplitude."},{"Start":"07:08.355 ","End":"07:10.630","Text":"Just like we were speaking about"},{"Start":"07:10.630 ","End":"07:16.030","Text":"our energy change for every single point in the wavefront,"},{"Start":"07:16.030 ","End":"07:21.785","Text":"here we\u0027re speaking about how the amplitude is changing"},{"Start":"07:21.785 ","End":"07:29.545","Text":"its magnitude and it\u0027s changing its magnitude by a factor of 1 divided by root 5."},{"Start":"07:29.545 ","End":"07:36.975","Text":"That means that our final amplitude of this wavefront with a radius of 15 centimeters,"},{"Start":"07:36.975 ","End":"07:40.475","Text":"its amplitude is going to be the original amplitude,"},{"Start":"07:40.475 ","End":"07:46.150","Text":"which was 1 centimeter multiplied by this factor change."},{"Start":"07:46.150 ","End":"07:49.610","Text":"Therefore, once we plug this into our calculator,"},{"Start":"07:49.610 ","End":"07:51.335","Text":"this is going to be equal to,"},{"Start":"07:51.335 ","End":"07:58.335","Text":"so 1 divided by root 5 multiplied by 1,"},{"Start":"07:58.335 ","End":"08:03.425","Text":"so this is simply going to be equal to"},{"Start":"08:03.425 ","End":"08:10.350","Text":"in the calculator 0.45 centimeters."},{"Start":"08:11.180 ","End":"08:14.065","Text":"This is the answer to question 2,"},{"Start":"08:14.065 ","End":"08:18.245","Text":"when we\u0027re dealing with 2D circular wavefronts."},{"Start":"08:18.245 ","End":"08:25.100","Text":"What\u0027s important to remember is that every time we find this change, this relationship,"},{"Start":"08:25.100 ","End":"08:28.820","Text":"so we\u0027re finding the relationship in the change in factor,"},{"Start":"08:28.820 ","End":"08:30.260","Text":"which means that we always,"},{"Start":"08:30.260 ","End":"08:35.645","Text":"in order to find the final energy or the final amplitude,"},{"Start":"08:35.645 ","End":"08:41.435","Text":"we have to multiply this factor that we found by the original."},{"Start":"08:41.435 ","End":"08:44.585","Text":"If it\u0027s the final energy we\u0027re multiplying by factor of"},{"Start":"08:44.585 ","End":"08:48.740","Text":"1/5 the original energy of each point and for the amplitude,"},{"Start":"08:48.740 ","End":"08:53.845","Text":"the final amplitude will be multiplying by this factor of here,"},{"Start":"08:53.845 ","End":"08:58.245","Text":"1 divided by root 5 multiplied by the original amplitude,"},{"Start":"08:58.245 ","End":"09:01.665","Text":"so this is really what\u0027s important to remember."},{"Start":"09:01.665 ","End":"09:09.644","Text":"Now we\u0027re going to answer a question dealing with 3-dimensional spherical waves."},{"Start":"09:09.644 ","End":"09:13.135","Text":"This is the next question where here,"},{"Start":"09:13.135 ","End":"09:17.140","Text":"we\u0027re dealing with a 3 dimensional wave."},{"Start":"09:17.140 ","End":"09:18.695","Text":"Because we\u0027re dealing with,"},{"Start":"09:18.695 ","End":"09:20.760","Text":"we\u0027re being told that a Lambda is lit,"},{"Start":"09:20.760 ","End":"09:22.440","Text":"which means that we\u0027re dealing with light,"},{"Start":"09:22.440 ","End":"09:25.565","Text":"and light is a 3-dimensional wave."},{"Start":"09:25.565 ","End":"09:32.635","Text":"A lamp is lit and at a distance d is equal to 2 meters away from the lamp."},{"Start":"09:32.635 ","End":"09:39.490","Text":"The light intensity is B and the amplitude of the wave front is A."},{"Start":"09:39.490 ","End":"09:43.780","Text":"First of all, something to know when we\u0027re dealing with the word,"},{"Start":"09:43.780 ","End":"09:46.165","Text":"when we see the word light intensity,"},{"Start":"09:46.165 ","End":"09:50.890","Text":"that just means the energy E. We\u0027re being told that"},{"Start":"09:50.890 ","End":"09:56.440","Text":"our energy is B and the amplitude of the wave front is A."},{"Start":"09:56.440 ","End":"09:58.450","Text":"Question number 1 is,"},{"Start":"09:58.450 ","End":"10:05.905","Text":"what was the intensity of the light at a distance of 1/2 a meter from the lamp?"},{"Start":"10:05.905 ","End":"10:09.850","Text":"Again, here we\u0027re being asked about intensity."},{"Start":"10:09.850 ","End":"10:14.875","Text":"That means that we\u0027re trying to find E. Specifically,"},{"Start":"10:14.875 ","End":"10:18.280","Text":"we know that the total energy."},{"Start":"10:18.280 ","End":"10:22.660","Text":"Let\u0027s say that this is E_i, and here,"},{"Start":"10:22.660 ","End":"10:25.060","Text":"when we\u0027re at half a meter away,"},{"Start":"10:25.060 ","End":"10:26.350","Text":"let\u0027s say that\u0027s E_f."},{"Start":"10:26.350 ","End":"10:27.924","Text":"We\u0027re going backwards."},{"Start":"10:27.924 ","End":"10:34.330","Text":"We\u0027re located at 2 meters and now we want to know what happened 1/2 a meter away."},{"Start":"10:34.330 ","End":"10:36.475","Text":"We\u0027re going backwards, but it doesn\u0027t matter."},{"Start":"10:36.475 ","End":"10:39.850","Text":"We\u0027re still going to say that this is the I because this was given in our question,"},{"Start":"10:39.850 ","End":"10:42.040","Text":"and that E_f is what we\u0027re trying to find,"},{"Start":"10:42.040 ","End":"10:44.560","Text":"but it doesn\u0027t make a difference."},{"Start":"10:44.560 ","End":"10:50.455","Text":"We know that the energy of our 3-dimensional spherical wave front,"},{"Start":"10:50.455 ","End":"10:52.150","Text":"when it\u0027s 2 meters away,"},{"Start":"10:52.150 ","End":"10:55.135","Text":"the total energy is going to be equal to"},{"Start":"10:55.135 ","End":"11:01.970","Text":"the total energy of 3-dimensional spherical wave front when it\u0027s half a meter away."},{"Start":"11:02.100 ","End":"11:05.965","Text":"First of all, we know that what is the difference here?"},{"Start":"11:05.965 ","End":"11:11.440","Text":"Here we\u0027re being asked that the intensity of each point."},{"Start":"11:11.440 ","End":"11:15.380","Text":"Just like with a 2-dimensional wave."},{"Start":"11:15.450 ","End":"11:22.915","Text":"If we had this as a radius of 2 meters,"},{"Start":"11:22.915 ","End":"11:26.020","Text":"we said that each point is going to have"},{"Start":"11:26.020 ","End":"11:33.125","Text":"some portion of this light intensity B, of this energy."},{"Start":"11:33.125 ","End":"11:36.540","Text":"Each point is going to have some portion of this."},{"Start":"11:36.540 ","End":"11:41.925","Text":"But then, when we\u0027re located somewhere over here,"},{"Start":"11:41.925 ","End":"11:46.130","Text":"where we know that our radius is 1/2 a meter."},{"Start":"11:46.130 ","End":"11:51.610","Text":"Each point over here is going to have a portion of this energy,"},{"Start":"11:51.610 ","End":"11:54.280","Text":"because we\u0027re told that no energy is lost."},{"Start":"11:54.280 ","End":"12:01.645","Text":"But because now the same exact energy is spread across a smaller area,"},{"Start":"12:01.645 ","End":"12:05.170","Text":"we know that each blue point is going to have"},{"Start":"12:05.170 ","End":"12:08.815","Text":"a larger portion of energy than each red point,"},{"Start":"12:08.815 ","End":"12:14.230","Text":"because here the same exact energy is spread over a much larger surface area."},{"Start":"12:14.230 ","End":"12:16.105","Text":"When we\u0027re 2 meters away,"},{"Start":"12:16.105 ","End":"12:19.045","Text":"than when we\u0027re half a meter away."},{"Start":"12:19.045 ","End":"12:25.150","Text":"Now we\u0027re going to remember that when dealing with 3-dimensional spherical waves,"},{"Start":"12:25.150 ","End":"12:28.720","Text":"we saw that our energy per point has"},{"Start":"12:28.720 ","End":"12:35.560","Text":"a relationship which is proportional to 1 divided by r squared."},{"Start":"12:35.560 ","End":"12:39.400","Text":"Let\u0027s just write that this is when we\u0027re dealing with 3D waves,"},{"Start":"12:39.400 ","End":"12:41.410","Text":"but now we\u0027re working backwards,"},{"Start":"12:41.410 ","End":"12:46.450","Text":"so we know what our energy is when we\u0027re 2 meters away,"},{"Start":"12:46.450 ","End":"12:50.185","Text":"and now we want to know what our energy is when we\u0027re 1/2 a meter away."},{"Start":"12:50.185 ","End":"12:53.814","Text":"Now the most important thing to remember here is that,"},{"Start":"12:53.814 ","End":"12:56.980","Text":"when we\u0027re using this r over here,"},{"Start":"12:56.980 ","End":"13:04.165","Text":"we\u0027re speaking about the factor by which r has increased or decreased."},{"Start":"13:04.165 ","End":"13:06.235","Text":"In the previous question,"},{"Start":"13:06.235 ","End":"13:08.020","Text":"dealing with 2D waves,"},{"Start":"13:08.020 ","End":"13:13.075","Text":"we saw that we had our original radius and then our radius increased."},{"Start":"13:13.075 ","End":"13:15.520","Text":"In this question, we\u0027re just going backwards."},{"Start":"13:15.520 ","End":"13:18.295","Text":"That\u0027s something that we should look at where we have"},{"Start":"13:18.295 ","End":"13:22.810","Text":"a radius and then we\u0027re going backwards to a smaller radius."},{"Start":"13:22.810 ","End":"13:27.175","Text":"We have to find this change in factor."},{"Start":"13:27.175 ","End":"13:30.918","Text":"This is what is very important to remember this change in factor."},{"Start":"13:30.918 ","End":"13:35.859","Text":"We\u0027re not multiplying by 1 divided by 1/2 a meter squared."},{"Start":"13:35.859 ","End":"13:43.310","Text":"We\u0027re multiplying by 1 divided by the factor change of the radius squared."},{"Start":"13:44.040 ","End":"13:51.430","Text":"Our original radius, which we just designated as r_i,"},{"Start":"13:51.430 ","End":"13:54.400","Text":"was 2 meters,"},{"Start":"13:54.400 ","End":"13:57.295","Text":"and our final radius,"},{"Start":"13:57.295 ","End":"14:00.145","Text":"which we just designated r_f,"},{"Start":"14:00.145 ","End":"14:03.370","Text":"is 1/2 a meter."},{"Start":"14:03.370 ","End":"14:09.265","Text":"Before, when we were going from smaller radius to larger radius,"},{"Start":"14:09.265 ","End":"14:12.065","Text":"we said that our r_fac."},{"Start":"14:12.065 ","End":"14:17.700","Text":"The factor that our i has been increased was r_f divided by r_i."},{"Start":"14:17.700 ","End":"14:24.005","Text":"However, now, because we\u0027re going from a larger radius to a smaller radius,"},{"Start":"14:24.005 ","End":"14:28.150","Text":"will do r_i divided by r_f."},{"Start":"14:28.150 ","End":"14:38.395","Text":"If we\u0027re going from large r to small r will do r_i divided by r_f,"},{"Start":"14:38.395 ","End":"14:46.255","Text":"where our r_i and r_f are just arbitrary names that we gave to these values."},{"Start":"14:46.255 ","End":"14:51.370","Text":"r_i is 2 meters divided by r_f,"},{"Start":"14:51.370 ","End":"14:56.210","Text":"which is 0.5 or 1/2."},{"Start":"14:56.820 ","End":"15:03.340","Text":"2 divided by 1.5 is equal to 2 multiplied by 2,"},{"Start":"15:03.340 ","End":"15:05.755","Text":"which is equal to 4."},{"Start":"15:05.755 ","End":"15:09.355","Text":"We can see that our change in radius"},{"Start":"15:09.355 ","End":"15:13.555","Text":"from 1/2 a meter to 2 meters has changed by a factor of 4."},{"Start":"15:13.555 ","End":"15:19.600","Text":"However, of course, and in questioned we\u0027re going from 2 meters to 1/2 a meter."},{"Start":"15:19.600 ","End":"15:24.400","Text":"Now we\u0027re going to have to change around how we equate this."},{"Start":"15:24.400 ","End":"15:29.215","Text":"That means that our initial energy per point,"},{"Start":"15:29.215 ","End":"15:32.260","Text":"because it has a larger radius,"},{"Start":"15:32.260 ","End":"15:35.260","Text":"so a larger surface area, as we know,"},{"Start":"15:35.260 ","End":"15:44.965","Text":"it\u0027s proportional to 1 divided by r_fac squared of our smaller wavefront."},{"Start":"15:44.965 ","End":"15:47.950","Text":"Our wavefront with a smaller radius,"},{"Start":"15:47.950 ","End":"15:57.640","Text":"which here specifically we called E_f tag of this point over here."},{"Start":"15:57.640 ","End":"16:02.365","Text":"What\u0027s important to remember is that every point on"},{"Start":"16:02.365 ","End":"16:07.555","Text":"the larger wavefront is going to have an energy serving"},{"Start":"16:07.555 ","End":"16:13.150","Text":"proportional to 1 divided by the factor change in the radius"},{"Start":"16:13.150 ","End":"16:19.224","Text":"squared of each energy point on the wavefront with a smaller radius."},{"Start":"16:19.224 ","End":"16:22.540","Text":"Here, it\u0027s labeled with an I and an F,"},{"Start":"16:22.540 ","End":"16:26.935","Text":"but we could have just labeled it larger radius and smaller radius."},{"Start":"16:26.935 ","End":"16:31.630","Text":"Now what we want to isolate is this E_f tag because we want to find"},{"Start":"16:31.630 ","End":"16:34.300","Text":"how much energy is on each of"},{"Start":"16:34.300 ","End":"16:39.050","Text":"these blue points because we\u0027re being asked about the radius of 1/2 a meter."},{"Start":"16:39.060 ","End":"16:42.100","Text":"Therefore, just like normal algebra,"},{"Start":"16:42.100 ","End":"16:46.105","Text":"we can say that our E_f tag will be proportional,"},{"Start":"16:46.105 ","End":"16:51.640","Text":"so we\u0027ll multiply both sides by this r_fac squared."},{"Start":"16:51.640 ","End":"16:54.640","Text":"Then we\u0027ll get our E_f tag."},{"Start":"16:54.640 ","End":"17:01.270","Text":"r_fac squared multiplied by our E_i tag."},{"Start":"17:01.270 ","End":"17:03.460","Text":"This is equal to,"},{"Start":"17:03.460 ","End":"17:05.064","Text":"so what\u0027s our r factor,"},{"Start":"17:05.064 ","End":"17:07.030","Text":"our r factor we worked out over here,"},{"Start":"17:07.030 ","End":"17:10.255","Text":"which is equal to 4 squared,"},{"Start":"17:10.255 ","End":"17:14.320","Text":"4 squared multiplied by E_i tag."},{"Start":"17:14.320 ","End":"17:22.435","Text":"E_i tag is the energy portion per red dots on the surface area of the larger wavefront."},{"Start":"17:22.435 ","End":"17:29.680","Text":"Here, that 4 squared is 16 and E_i tag, we were told."},{"Start":"17:29.680 ","End":"17:36.550","Text":"The light intensity of the larger wavefront was just given as B."},{"Start":"17:36.550 ","End":"17:44.830","Text":"Now this is going to be our energy portion per blue dot."},{"Start":"17:44.830 ","End":"17:51.485","Text":"The energy portion per point on the smaller wavefront."},{"Start":"17:51.485 ","End":"17:54.195","Text":"That\u0027s the answer to question 1,"},{"Start":"17:54.195 ","End":"17:59.370","Text":"and now let\u0027s just answer question 2 and question 2 will now be much easier."},{"Start":"17:59.370 ","End":"18:02.715","Text":"What was the amplitude of the wave at that same distance?"},{"Start":"18:02.715 ","End":"18:09.490","Text":"We\u0027re being asked the amplitude of the wavefront at D is equal to 1/2 a meter."},{"Start":"18:09.490 ","End":"18:12.850","Text":"Again, I\u0027m going to remind you that when we\u0027re speaking about amplitude,"},{"Start":"18:12.850 ","End":"18:14.380","Text":"if we\u0027re speaking about the amplitude of"},{"Start":"18:14.380 ","End":"18:17.275","Text":"each point or the amplitude of the wavefront itself,"},{"Start":"18:17.275 ","End":"18:19.630","Text":"it makes no difference."},{"Start":"18:19.630 ","End":"18:21.710","Text":"Whereas with energy it does make a difference,"},{"Start":"18:21.710 ","End":"18:25.010","Text":"but with the amplitude will get the same value."},{"Start":"18:25.010 ","End":"18:30.290","Text":"As we know, if we\u0027re dealing with a larger wavefront,"},{"Start":"18:30.290 ","End":"18:33.965","Text":"the amplitude of the larger wavefront,"},{"Start":"18:33.965 ","End":"18:35.810","Text":"which here we called I,"},{"Start":"18:35.810 ","End":"18:37.955","Text":"is proportional when we\u0027re dealing with"},{"Start":"18:37.955 ","End":"18:44.330","Text":"a 3-dimensional circular wavefront to 1 divided by I, where again,"},{"Start":"18:44.330 ","End":"18:54.190","Text":"I\u0027m reminding you of this is the r_fac multiplied by the original amplitude."},{"Start":"18:54.190 ","End":"18:58.955","Text":"Or here, it will be multiplied by the amplitude of the smaller wavefront."},{"Start":"18:58.955 ","End":"19:01.804","Text":"Where here, the amplitude of the smaller wavefront,"},{"Start":"19:01.804 ","End":"19:05.700","Text":"we\u0027re just calling A_f."},{"Start":"19:08.160 ","End":"19:10.525","Text":"What we\u0027re trying to find though,"},{"Start":"19:10.525 ","End":"19:12.330","Text":"is our A_f,"},{"Start":"19:12.330 ","End":"19:16.085","Text":"so the amplitude of our smaller wavefront."},{"Start":"19:16.085 ","End":"19:21.395","Text":"Again, all we\u0027re going to do is we\u0027re going to use algebra like we did over here."},{"Start":"19:21.395 ","End":"19:25.595","Text":"We\u0027re going to isolate out our A_f because that\u0027s what we\u0027re trying to find,"},{"Start":"19:25.595 ","End":"19:30.175","Text":"and we\u0027re just going to multiply both sides by this r_fac."},{"Start":"19:30.175 ","End":"19:37.790","Text":"It\u0027s proportional to r_fac multiplied by the amplitude of the A_i,"},{"Start":"19:37.790 ","End":"19:41.620","Text":"which over here is the larger wavefront."},{"Start":"19:41.620 ","End":"19:42.935","Text":"Now let\u0027s do that."},{"Start":"19:42.935 ","End":"19:46.580","Text":"A_f is going to be proportional to r_fac,"},{"Start":"19:46.580 ","End":"19:48.785","Text":"which we calculated over here."},{"Start":"19:48.785 ","End":"19:55.715","Text":"Radius change, it\u0027s changed by a factor of 4 multiplied by our original amplitude,"},{"Start":"19:55.715 ","End":"19:58.290","Text":"which was given as A."},{"Start":"20:00.690 ","End":"20:03.700","Text":"This is our answer to question number 2,"},{"Start":"20:03.700 ","End":"20:08.090","Text":"and we can say that our final amplitude,"},{"Start":"20:08.090 ","End":"20:11.780","Text":"or the amplitude of the smaller wavefront,"},{"Start":"20:11.780 ","End":"20:13.490","Text":"because that\u0027s what we\u0027re being asked,"},{"Start":"20:13.490 ","End":"20:20.660","Text":"is proportional to 4 times the amplitude of our original or initial wavefront,"},{"Start":"20:20.660 ","End":"20:25.255","Text":"which was add a further away radius."},{"Start":"20:25.255 ","End":"20:29.510","Text":"We can see that as we get closer to the light source,"},{"Start":"20:29.510 ","End":"20:37.265","Text":"our amplitude is bigger and the intensity of light is also bigger."},{"Start":"20:37.265 ","End":"20:40.350","Text":"That\u0027s the end of this question."}],"ID":12486},{"Watched":false,"Name":"Intro Water Wave Interference","Duration":"8m 37s","ChapterTopicVideoID":12019,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.875","Text":"Hello. In this lesson,"},{"Start":"00:01.875 ","End":"00:06.885","Text":"we\u0027re going to be speaking about interference of water waves."},{"Start":"00:06.885 ","End":"00:13.575","Text":"We already saw that when we want to make circular wave fronts with the water waves,"},{"Start":"00:13.575 ","End":"00:19.365","Text":"so we have some kind of needle or some kind of pin that looks like this, let\u0027s say,"},{"Start":"00:19.365 ","End":"00:26.265","Text":"and it hits the water surface like so moving up and down hitting the water surface."},{"Start":"00:26.265 ","End":"00:30.140","Text":"Then some wave pattern will form,"},{"Start":"00:30.140 ","End":"00:32.795","Text":"which looks like this image."},{"Start":"00:32.795 ","End":"00:37.460","Text":"Here we can see the circular wave pattern and here we"},{"Start":"00:37.460 ","End":"00:41.815","Text":"can see the needle or the pin hitting the water."},{"Start":"00:41.815 ","End":"00:45.555","Text":"We can see over here in red,"},{"Start":"00:45.555 ","End":"00:47.955","Text":"we have the first wave front."},{"Start":"00:47.955 ","End":"00:49.460","Text":"If we wait a few moments,"},{"Start":"00:49.460 ","End":"00:54.530","Text":"this first wave front will travel over here."},{"Start":"00:54.530 ","End":"00:57.920","Text":"But then, because the needle keeps hitting the surface of"},{"Start":"00:57.920 ","End":"01:01.345","Text":"the water a new wave front will form over here."},{"Start":"01:01.345 ","End":"01:06.340","Text":"Then this wave front will move over here."},{"Start":"01:06.900 ","End":"01:12.430","Text":"The previous or the second wave front will move to this position over here."},{"Start":"01:12.430 ","End":"01:16.330","Text":"A new third wave front will form over here."},{"Start":"01:16.330 ","End":"01:20.270","Text":"This happens over and over again."},{"Start":"01:20.420 ","End":"01:24.610","Text":"Now let\u0027s look at a diagram where we have"},{"Start":"01:24.610 ","End":"01:30.590","Text":"2 circular wave fronts that are interfering with one another."},{"Start":"01:32.090 ","End":"01:39.780","Text":"Over here, we can see we have 2 sources for circular wave front."},{"Start":"01:39.780 ","End":"01:42.240","Text":"Let\u0027s call this source S_1."},{"Start":"01:42.240 ","End":"01:46.360","Text":"We can see here\u0027s the first secular wave front."},{"Start":"01:46.360 ","End":"01:52.695","Text":"Then our first circular wave front moves over here."},{"Start":"01:52.695 ","End":"01:55.335","Text":"Then a new circular wave front is formed."},{"Start":"01:55.335 ","End":"02:00.875","Text":"Then these 2 circular wave fronts move up 1 as well and another wave front is formed,"},{"Start":"02:00.875 ","End":"02:02.870","Text":"and so on and so forth."},{"Start":"02:02.870 ","End":"02:07.820","Text":"Here we can see exactly what we"},{"Start":"02:07.820 ","End":"02:13.940","Text":"would expect to see with these circular wave fronts."},{"Start":"02:13.940 ","End":"02:17.665","Text":"Then here we have source number 2."},{"Start":"02:17.665 ","End":"02:20.810","Text":"Here again, our first wave front"},{"Start":"02:20.810 ","End":"02:24.130","Text":"and then it moves over here and a new wave front has formed."},{"Start":"02:24.130 ","End":"02:28.910","Text":"Then these wave fronts move up 1 and another wave front has formed."},{"Start":"02:28.910 ","End":"02:32.780","Text":"Then we can see that we have all of"},{"Start":"02:32.780 ","End":"02:38.850","Text":"these circular wave fronts just like we would expect to see."},{"Start":"02:38.950 ","End":"02:42.770","Text":"All of this is as expected and we can see how"},{"Start":"02:42.770 ","End":"02:47.585","Text":"the wave fronts from the 2 different sources are interacting."},{"Start":"02:47.585 ","End":"02:51.740","Text":"Now I\u0027m going to rub out all of this and I\u0027m going to show you"},{"Start":"02:51.740 ","End":"02:55.685","Text":"something that we didn\u0027t expect,"},{"Start":"02:55.685 ","End":"02:59.430","Text":"and we\u0027re going to speak about it a bit later."},{"Start":"02:59.830 ","End":"03:02.015","Text":"If we look carefully,"},{"Start":"03:02.015 ","End":"03:07.240","Text":"we can see in the picture lines going through the water."},{"Start":"03:07.240 ","End":"03:09.690","Text":"I\u0027m going to draw this in purple."},{"Start":"03:09.690 ","End":"03:11.780","Text":"If you look very carefully,"},{"Start":"03:11.780 ","End":"03:17.715","Text":"you\u0027ll see that there\u0027s some line over here going like so."},{"Start":"03:17.715 ","End":"03:25.835","Text":"Then to its left we can also see this line over here and another line over here."},{"Start":"03:25.835 ","End":"03:31.245","Text":"There\u0027s slightly blended gray lines like so."},{"Start":"03:31.245 ","End":"03:35.540","Text":"We can see this all throughout the picture if you look very carefully."},{"Start":"03:35.540 ","End":"03:38.285","Text":"Also to the right over here."},{"Start":"03:38.285 ","End":"03:42.170","Text":"Here we have another one, another one,"},{"Start":"03:42.170 ","End":"03:45.580","Text":"here, another,"},{"Start":"03:46.100 ","End":"03:49.440","Text":"and so on like so."},{"Start":"03:49.440 ","End":"03:52.295","Text":"This effect we\u0027ll look at later."},{"Start":"03:52.295 ","End":"03:58.310","Text":"But now let\u0027s remind ourselves a little bit about what we\u0027ve learned in previous lessons."},{"Start":"03:58.310 ","End":"04:04.835","Text":"We saw in previous lessons that if we have a crest of the wave,"},{"Start":"04:04.835 ","End":"04:09.815","Text":"then it acts as some kind of concentrating lens."},{"Start":"04:09.815 ","End":"04:14.300","Text":"We saw that all the light that passes through the crest of"},{"Start":"04:14.300 ","End":"04:20.330","Text":"the wave will cause a bright spots over here under the crest."},{"Start":"04:20.330 ","End":"04:24.270","Text":"Conversely, we saw that when we have a trough,"},{"Start":"04:24.590 ","End":"04:29.715","Text":"so the trough acts as a dispersion lens."},{"Start":"04:29.715 ","End":"04:32.015","Text":"Under where the trough is,"},{"Start":"04:32.015 ","End":"04:34.800","Text":"we\u0027ll have a patch of dark."},{"Start":"04:35.660 ","End":"04:39.270","Text":"This is what we can see over here."},{"Start":"04:39.270 ","End":"04:45.545","Text":"Let\u0027s draw in green or rather let\u0027s draw in red, the light."},{"Start":"04:45.545 ","End":"04:47.795","Text":"We can see all of this as light."},{"Start":"04:47.795 ","End":"04:52.760","Text":"All of this is the crest of the wave."},{"Start":"04:52.760 ","End":"04:55.160","Text":"Then as we would expect,"},{"Start":"04:55.160 ","End":"04:58.385","Text":"after each peak, we have a trough."},{"Start":"04:58.385 ","End":"05:04.670","Text":"Here we have 1 trough that we can see in dark patches."},{"Start":"05:04.670 ","End":"05:05.960","Text":"Here is another one."},{"Start":"05:05.960 ","End":"05:07.850","Text":"After this other peak in red,"},{"Start":"05:07.850 ","End":"05:11.210","Text":"we have another blue trough over here because it\u0027s dark."},{"Start":"05:11.210 ","End":"05:16.150","Text":"Similarly for the wave fronts coming out of source 2."},{"Start":"05:16.150 ","End":"05:19.295","Text":"Let\u0027s draw the crests of the wave or the peaks."},{"Start":"05:19.295 ","End":"05:22.100","Text":"Here this is a peak because it\u0027s light."},{"Start":"05:22.100 ","End":"05:25.835","Text":"All of this is light patches, are peaks."},{"Start":"05:25.835 ","End":"05:29.095","Text":"Then similarly, let\u0027s draw the trough in blue."},{"Start":"05:29.095 ","End":"05:34.710","Text":"All of these dark patches are the trough."},{"Start":"05:35.930 ","End":"05:41.900","Text":"Now let\u0027s look at what happens at this point over here."},{"Start":"05:41.900 ","End":"05:44.750","Text":"This gray point over here,"},{"Start":"05:44.750 ","End":"05:51.350","Text":"which is right where our peak suddenly turns into a trough,"},{"Start":"05:51.350 ","End":"05:56.540","Text":"where our wave goes from being at a high to a low."},{"Start":"05:56.540 ","End":"06:01.740","Text":"Here we can see it\u0027s at the 0, at the baseline."},{"Start":"06:01.790 ","End":"06:03.885","Text":"Let\u0027s draw that over here."},{"Start":"06:03.885 ","End":"06:08.345","Text":"What if all of the water in the wave tank was like that?"},{"Start":"06:08.345 ","End":"06:10.160","Text":"All of the water was level."},{"Start":"06:10.160 ","End":"06:13.430","Text":"They weren\u0027t higher and lower points."},{"Start":"06:13.430 ","End":"06:16.835","Text":"Then the light traveling through the water"},{"Start":"06:16.835 ","End":"06:20.150","Text":"to the bottom of the tank would just travel like"},{"Start":"06:20.150 ","End":"06:27.635","Text":"so straight down because the angle that it\u0027s hitting is to the normal."},{"Start":"06:27.635 ","End":"06:29.965","Text":"The light won\u0027t bend."},{"Start":"06:29.965 ","End":"06:36.335","Text":"Then another light ray will travel through as well."},{"Start":"06:36.335 ","End":"06:41.990","Text":"Again, it will hit at 90 degrees and travels straight through"},{"Start":"06:41.990 ","End":"06:44.840","Text":"and so on and so forth with all of"},{"Start":"06:44.840 ","End":"06:48.125","Text":"the waves that travel through this gray section over here."},{"Start":"06:48.125 ","End":"06:54.110","Text":"What we can see is that we aren\u0027t getting some concentration of light,"},{"Start":"06:54.110 ","End":"06:57.850","Text":"but we\u0027re also not getting a dispersion of light."},{"Start":"06:57.850 ","End":"07:04.700","Text":"That means that we\u0027re going to see some patch that isn\u0027t bright white,"},{"Start":"07:04.700 ","End":"07:09.305","Text":"but it also isn\u0027t dark gray or dark black."},{"Start":"07:09.305 ","End":"07:12.350","Text":"The color that we\u0027re going to see is somewhere between"},{"Start":"07:12.350 ","End":"07:17.165","Text":"this bright white color and this dark gray color."},{"Start":"07:17.165 ","End":"07:23.170","Text":"What we\u0027ll see is some patch of light gray."},{"Start":"07:24.740 ","End":"07:32.885","Text":"This is what happens when we\u0027re at this base 0 over here,"},{"Start":"07:32.885 ","End":"07:36.710","Text":"where our water level is level."},{"Start":"07:36.710 ","End":"07:43.595","Text":"Now we can see the same peak and trough action happening over here,"},{"Start":"07:43.595 ","End":"07:48.545","Text":"where we have our diagram with just one wave source."},{"Start":"07:48.545 ","End":"07:51.710","Text":"Here in red we can see the latest patches which"},{"Start":"07:51.710 ","End":"07:55.230","Text":"we know are the peaks which we\u0027ve already shown."},{"Start":"07:55.230 ","End":"07:58.010","Text":"Then in blue we can see how troughs,"},{"Start":"07:58.010 ","End":"08:04.035","Text":"which are the dark patches due to the troughs diffusing the light."},{"Start":"08:04.035 ","End":"08:05.795","Text":"Here we have a trough."},{"Start":"08:05.795 ","End":"08:08.600","Text":"Here we have a second one."},{"Start":"08:08.600 ","End":"08:12.960","Text":"Here we have a third one."},{"Start":"08:13.820 ","End":"08:16.460","Text":"That\u0027s the end of this lesson."},{"Start":"08:16.460 ","End":"08:20.930","Text":"What we\u0027re going to do in the next lesson is we\u0027re going to carry on looking at"},{"Start":"08:20.930 ","End":"08:26.670","Text":"this diagram or this picture rather in much more detail."},{"Start":"08:26.670 ","End":"08:29.735","Text":"We\u0027re going to look at what\u0027s going on in the light patches,"},{"Start":"08:29.735 ","End":"08:31.445","Text":"in the dark patches,"},{"Start":"08:31.445 ","End":"08:34.775","Text":"and in this gray area."},{"Start":"08:34.775 ","End":"08:37.950","Text":"That\u0027s the end of this lesson."}],"ID":12487},{"Watched":false,"Name":"Constructive Interference Water Wave","Duration":"32m 3s","ChapterTopicVideoID":12020,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.280","Text":"Hello. Here\u0027s the picture that we saw previously,"},{"Start":"00:05.280 ","End":"00:10.095","Text":"where we have 2 sources creating circular wave fronts."},{"Start":"00:10.095 ","End":"00:12.367","Text":"Here we have our source,"},{"Start":"00:12.367 ","End":"00:15.870","Text":"S_1, and here we have our source, S_2."},{"Start":"00:15.870 ","End":"00:19.229","Text":"Now, S_1 and S_2 are creating waves"},{"Start":"00:19.229 ","End":"00:22.935","Text":"with the same frequency and with the same wavelength,"},{"Start":"00:22.935 ","End":"00:25.860","Text":"and they have the same phase."},{"Start":"00:25.860 ","End":"00:27.690","Text":"Let\u0027s write that down here."},{"Start":"00:27.690 ","End":"00:33.090","Text":"We can say the wavelength from Source 1 is equal to the wavelength of Source."},{"Start":"00:33.090 ","End":"00:37.215","Text":"The frequency of Source 1 is equal to the frequency of Source 2,"},{"Start":"00:37.215 ","End":"00:43.010","Text":"and the phase therefore is also going to be the same."},{"Start":"00:43.010 ","End":"00:47.750","Text":"The first question that we want to ask before we look at everything else"},{"Start":"00:47.750 ","End":"00:53.405","Text":"is why do we see certain areas that have extremely bright patches?"},{"Start":"00:53.405 ","End":"00:57.540","Text":"Let\u0027s say this point over here is extremely bright,"},{"Start":"00:57.540 ","End":"01:01.730","Text":"and of course there are other points such as this one or this one and so on,"},{"Start":"01:01.730 ","End":"01:04.495","Text":"but let\u0027s speak about this point specifically."},{"Start":"01:04.495 ","End":"01:07.770","Text":"Let\u0027s take a look at this."},{"Start":"01:07.770 ","End":"01:12.915","Text":"We\u0027re going to draw this in green, our first wave."},{"Start":"01:12.915 ","End":"01:17.640","Text":"From S_1, we have at time t=0."},{"Start":"01:17.640 ","End":"01:20.175","Text":"At time t=0,"},{"Start":"01:20.175 ","End":"01:26.299","Text":"a wave is formed from S_1 and it travels in all directions,"},{"Start":"01:26.299 ","End":"01:28.160","Text":"because this is a circular wave front,"},{"Start":"01:28.160 ","End":"01:30.019","Text":"so it travels in the radial direction."},{"Start":"01:30.019 ","End":"01:33.175","Text":"At some stage the wave front,"},{"Start":"01:33.175 ","End":"01:34.505","Text":"or part of the wave front,"},{"Start":"01:34.505 ","End":"01:38.434","Text":"is going to reach this exact point over here."},{"Start":"01:38.434 ","End":"01:42.065","Text":"Now similarly, let\u0027s draw this in blue for S_2."},{"Start":"01:42.065 ","End":"01:45.935","Text":"S_2 is also producing or generating"},{"Start":"01:45.935 ","End":"01:50.930","Text":"circular wave fronts that are also going to travel out in the radial direction."},{"Start":"01:50.930 ","End":"01:55.850","Text":"Because we know that their wavelength and their frequency and"},{"Start":"01:55.850 ","End":"02:00.500","Text":"also their phase is exactly the same as the waves produced by S_1,"},{"Start":"02:00.500 ","End":"02:03.485","Text":"we know that at the same time, t=0,"},{"Start":"02:03.485 ","End":"02:05.780","Text":"the wave front produced by S_2,"},{"Start":"02:05.780 ","End":"02:07.910","Text":"so part of it is going to travel,"},{"Start":"02:07.910 ","End":"02:13.685","Text":"and at the exact same moment that this part of the wave front of S_1 reaches this point,"},{"Start":"02:13.685 ","End":"02:18.810","Text":"this part of the wave front of S_2 will reach this point as well."},{"Start":"02:18.810 ","End":"02:21.609","Text":"We can see that there\u0027s a light patch over here."},{"Start":"02:21.609 ","End":"02:27.665","Text":"As we know, the light patch is generated from a peak in the wave,"},{"Start":"02:27.665 ","End":"02:29.419","Text":"because the peak in the wave,"},{"Start":"02:29.419 ","End":"02:36.414","Text":"so something like this acts like a concentrating lens."},{"Start":"02:36.414 ","End":"02:41.915","Text":"Let\u0027s say that the wave front from S_1,"},{"Start":"02:41.915 ","End":"02:43.849","Text":"reaches this point over here,"},{"Start":"02:43.849 ","End":"02:46.925","Text":"where it\u0027s at a peak, and similarly,"},{"Start":"02:46.925 ","End":"02:51.425","Text":"the wave front from S_2 reaches this point over here, also at a peak."},{"Start":"02:51.425 ","End":"02:53.240","Text":"If S_1 is at a peak at this point,"},{"Start":"02:53.240 ","End":"02:57.904","Text":"then so is S_2, because of the same wavelength and the same phase."},{"Start":"02:57.904 ","End":"03:01.715","Text":"Then we have our green and our blue waves,"},{"Start":"03:01.715 ","End":"03:06.909","Text":"1 on top of the other and imagine that they\u0027re the exact same shape and size."},{"Start":"03:06.909 ","End":"03:10.725","Text":"We know that we\u0027re going to have superposition."},{"Start":"03:10.725 ","End":"03:12.795","Text":"What type of superposition?"},{"Start":"03:12.795 ","End":"03:15.770","Text":"We\u0027re going to have constructive interference"},{"Start":"03:15.770 ","End":"03:19.610","Text":"because they\u0027ve both arrived in phase with one another,"},{"Start":"03:19.610 ","End":"03:24.600","Text":"and so they\u0027re going to build one on top of the other for constructive interference,"},{"Start":"03:24.600 ","End":"03:31.009","Text":"and we\u0027re going to add their amplitudes to get a very, very tall wave."},{"Start":"03:31.009 ","End":"03:36.920","Text":"This very, very tall wave we know is going to concentrate the light even more."},{"Start":"03:36.920 ","End":"03:39.424","Text":"The taller the peak,"},{"Start":"03:39.424 ","End":"03:43.880","Text":"the more concentrated the light will be underneath the peak,"},{"Start":"03:43.880 ","End":"03:48.875","Text":"the more concentrated the light patch under the peak will be."},{"Start":"03:48.875 ","End":"03:50.479","Text":"Just like we said,"},{"Start":"03:50.479 ","End":"03:54.169","Text":"when a light ray hits this section over here,"},{"Start":"03:54.169 ","End":"03:57.770","Text":"it will bend inwards, and similarly,"},{"Start":"03:57.770 ","End":"04:00.079","Text":"over here it will bend inwards,"},{"Start":"04:00.079 ","End":"04:02.750","Text":"and then we\u0027re going to have lots and lots of"},{"Start":"04:02.750 ","End":"04:06.335","Text":"rays that are bending towards this point over here,"},{"Start":"04:06.335 ","End":"04:09.019","Text":"making this light patch extremely bright,"},{"Start":"04:09.019 ","End":"04:10.865","Text":"which is what we saw here."},{"Start":"04:10.865 ","End":"04:14.405","Text":"This bright patch over here was generated"},{"Start":"04:14.405 ","End":"04:19.235","Text":"by S_1 and S_2 being of the same wavelength and in phase,"},{"Start":"04:19.235 ","End":"04:23.374","Text":"and therefore reaching this point both when they\u0027re at a peak."},{"Start":"04:23.374 ","End":"04:28.450","Text":"Then there was complete constructive interference."},{"Start":"04:28.450 ","End":"04:31.700","Text":"As we know, the higher the amplitude,"},{"Start":"04:31.700 ","End":"04:35.135","Text":"and here we can see a very high amplitude,"},{"Start":"04:35.135 ","End":"04:39.005","Text":"the brighter the light we will see underneath."},{"Start":"04:39.005 ","End":"04:41.120","Text":"Now let\u0027s show this again."},{"Start":"04:41.120 ","End":"04:43.864","Text":"Let\u0027s rub out these green and blue marks."},{"Start":"04:43.864 ","End":"04:48.714","Text":"Now, let\u0027s take a look at this bright patch over here."},{"Start":"04:48.714 ","End":"04:54.675","Text":"Again, we have our wave front coming from a wave generator S_1,"},{"Start":"04:54.675 ","End":"04:57.559","Text":"and it will take"},{"Start":"04:57.559 ","End":"04:59.509","Text":"a longer time than the wave front to"},{"Start":"04:59.509 ","End":"05:01.820","Text":"reach this point because this point is further away,"},{"Start":"05:01.820 ","End":"05:06.620","Text":"but either way it will eventually reach this point at a certain time."},{"Start":"05:06.620 ","End":"05:08.810","Text":"Then from S_2,"},{"Start":"05:08.810 ","End":"05:18.415","Text":"the wave front generated will also travel towards this exact same point."},{"Start":"05:18.415 ","End":"05:25.370","Text":"Now, we know that these 2 distances are exactly the same."},{"Start":"05:25.370 ","End":"05:30.960","Text":"This distance is exactly the same as this distance."},{"Start":"05:30.960 ","End":"05:33.860","Text":"Because of the same wavelength and the same phase,"},{"Start":"05:33.860 ","End":"05:39.150","Text":"they\u0027re going to arrive together at the same phase over here."},{"Start":"05:39.280 ","End":"05:42.814","Text":"We\u0027re going to get some peak,"},{"Start":"05:42.814 ","End":"05:45.049","Text":"like so because it\u0027s a light patch,"},{"Start":"05:45.049 ","End":"05:49.910","Text":"so we know it\u0027s going to be a peak as we described above from Source 1,"},{"Start":"05:49.910 ","End":"05:51.739","Text":"and at the exact same time,"},{"Start":"05:51.739 ","End":"05:55.715","Text":"the exact same wave is going to arrive,"},{"Start":"05:55.715 ","End":"05:57.305","Text":"but from Source 2."},{"Start":"05:57.305 ","End":"06:02.750","Text":"Then again, we\u0027re going to get complete constructive interference."},{"Start":"06:02.750 ","End":"06:06.140","Text":"We\u0027re just going to superimpose these 2,"},{"Start":"06:06.140 ","End":"06:14.570","Text":"and we\u0027re just going to add up their amplitudes and get this final amplitude over here."},{"Start":"06:14.570 ","End":"06:21.880","Text":"Now, I\u0027ve on purpose drawn these peaks for this second point over here,"},{"Start":"06:21.880 ","End":"06:26.895","Text":"lower than the peaks for Point 1 over here."},{"Start":"06:26.895 ","End":"06:31.744","Text":"Why is the amplitude of each individual wave front from S_1 and"},{"Start":"06:31.744 ","End":"06:37.755","Text":"S_2 smaller at Point 2 than at Point 1?"},{"Start":"06:37.755 ","End":"06:41.659","Text":"That is because if we remember from previous lessons,"},{"Start":"06:41.659 ","End":"06:47.794","Text":"we saw that the amplitude is proportional to the radius."},{"Start":"06:47.794 ","End":"06:51.605","Text":"Even if the energy of each wave front,"},{"Start":"06:51.605 ","End":"06:53.719","Text":"each one of these circles,"},{"Start":"06:53.719 ","End":"06:57.304","Text":"the total energy of the circle is going to be the same."},{"Start":"06:57.304 ","End":"07:01.489","Text":"The energy has to be divided because we\u0027re dealing with 2 dimensional waves"},{"Start":"07:01.489 ","End":"07:05.914","Text":"here along a larger circumference or larger length,"},{"Start":"07:05.914 ","End":"07:13.020","Text":"which means that the energy of each specific point along these wave fronts is"},{"Start":"07:13.020 ","End":"07:19.999","Text":"going to be smaller than the energy of each specific point along this wave front."},{"Start":"07:19.999 ","End":"07:25.460","Text":"We know that our energy is proportional to the amplitude squared."},{"Start":"07:25.460 ","End":"07:35.360","Text":"That means that if the energy here is going to be slightly less than the energy here,"},{"Start":"07:35.360 ","End":"07:38.390","Text":"that means that the amplitude over"},{"Start":"07:38.390 ","End":"07:44.279","Text":"here will be less than the amplitude of this wave front."},{"Start":"07:44.630 ","End":"07:47.045","Text":"What we\u0027ve spoken about here,"},{"Start":"07:47.045 ","End":"07:50.150","Text":"about this constructive interference is the same,"},{"Start":"07:50.150 ","End":"07:51.860","Text":"not just for these 2 points,"},{"Start":"07:51.860 ","End":"07:57.439","Text":"but also for this point over here and here and here and here and so on."},{"Start":"07:57.439 ","End":"08:06.175","Text":"Also, a similar thing happens for all the other light points that you see around."},{"Start":"08:06.175 ","End":"08:12.874","Text":"But this constructive interference happens not just where we have the light patches,"},{"Start":"08:12.874 ","End":"08:16.085","Text":"but also where we have the dark patches."},{"Start":"08:16.085 ","End":"08:21.365","Text":"For instance, over here between the square of light,"},{"Start":"08:21.365 ","End":"08:24.005","Text":"we have this dark patch over here, and here,"},{"Start":"08:24.005 ","End":"08:27.739","Text":"we also have constructive interference."},{"Start":"08:27.739 ","End":"08:31.499","Text":"Let\u0027s see what happens over here."},{"Start":"08:31.550 ","End":"08:40.594","Text":"We know that this over here is going to be some crest and the wave,"},{"Start":"08:40.594 ","End":"08:42.710","Text":"and this is another crest."},{"Start":"08:42.710 ","End":"08:46.984","Text":"That means that this line over here,"},{"Start":"08:46.984 ","End":"08:49.234","Text":"has to be a trough."},{"Start":"08:49.234 ","End":"08:54.045","Text":"Also over here we\u0027re going to have a trough as well in black."},{"Start":"08:54.045 ","End":"08:59.495","Text":"We can go over it over here as well. What does that mean?"},{"Start":"08:59.495 ","End":"09:03.139","Text":"That means that to this point over here,"},{"Start":"09:03.139 ","End":"09:08.510","Text":"we get a trough arriving from Source Number 1 and the wave front,"},{"Start":"09:08.510 ","End":"09:14.905","Text":"and similarly, we get a trough arriving from Source Number 2\u0027s wave front."},{"Start":"09:14.905 ","End":"09:19.580","Text":"These 2 both of the same wavelength and they\u0027re both also in phase,"},{"Start":"09:19.580 ","End":"09:23.239","Text":"so they arrive at this point at the exact same time,"},{"Start":"09:23.239 ","End":"09:25.430","Text":"just like when we were dealing with the crest,"},{"Start":"09:25.430 ","End":"09:26.990","Text":"so with the peaks."},{"Start":"09:26.990 ","End":"09:31.155","Text":"That means that we\u0027re going to superimpose their amplitudes."},{"Start":"09:31.155 ","End":"09:35.900","Text":"We\u0027re going to get complete constructive interference and we\u0027re going to get"},{"Start":"09:35.900 ","End":"09:41.990","Text":"a larger amplitude just in the negative direction."},{"Start":"09:41.990 ","End":"09:45.139","Text":"As we know from what we\u0027ve seen in previous lessons,"},{"Start":"09:45.139 ","End":"09:47.014","Text":"when we have a trough,"},{"Start":"09:47.014 ","End":"09:52.330","Text":"the light is deviated away from underneath."},{"Start":"09:52.330 ","End":"09:56.090","Text":"A trough acts like a dispersion lens,"},{"Start":"09:56.090 ","End":"09:58.609","Text":"which means that right under the trough,"},{"Start":"09:58.609 ","End":"10:01.550","Text":"we\u0027re going to have a dark patch."},{"Start":"10:01.550 ","End":"10:05.510","Text":"Similarly, just like a higher amplitude in"},{"Start":"10:05.510 ","End":"10:12.080","Text":"the positive direction means that more light is concentrated right under the peak."},{"Start":"10:12.080 ","End":"10:13.655","Text":"In the same way,"},{"Start":"10:13.655 ","End":"10:17.645","Text":"the larger the amplitude in the negative direction,"},{"Start":"10:17.645 ","End":"10:23.210","Text":"the more the light rays are deviated away from the area underneath,"},{"Start":"10:23.210 ","End":"10:27.079","Text":"which means the darker the patch underneath."},{"Start":"10:27.079 ","End":"10:31.989","Text":"Now, of course, all of these wave fronts we know are moving,"},{"Start":"10:31.989 ","End":"10:36.320","Text":"and so if here we have a bright spot and here we have a dark spot,"},{"Start":"10:36.320 ","End":"10:37.760","Text":"a moment later,"},{"Start":"10:37.760 ","End":"10:39.949","Text":"a bright spot will be here and"},{"Start":"10:39.949 ","End":"10:43.880","Text":"our dark spot will have switched this bright spot over here."},{"Start":"10:43.880 ","End":"10:50.405","Text":"But what\u0027s important to remember is that both at the bright spot and at the dark spot,"},{"Start":"10:50.405 ","End":"10:54.560","Text":"we have complete constructive interference."},{"Start":"10:54.560 ","End":"11:00.875","Text":"In the bright spot it\u0027s complete constructive interference in the positive direction,"},{"Start":"11:00.875 ","End":"11:06.960","Text":"and at the dark spot is complete constructive interference in the negative direction."},{"Start":"11:07.070 ","End":"11:12.500","Text":"In actual fact, we can see that here we have constructive interference,"},{"Start":"11:12.500 ","End":"11:15.120","Text":"here constructive, constructive,"},{"Start":"11:15.120 ","End":"11:21.315","Text":"also here we have constructive interference and so on and so forth,"},{"Start":"11:21.315 ","End":"11:24.305","Text":"and also this point over here,"},{"Start":"11:24.305 ","End":"11:30.474","Text":"over here, and all of these subsequent points over here."},{"Start":"11:30.474 ","End":"11:40.590","Text":"What we can actually see is that we have a line over here going straight down the middle,"},{"Start":"11:40.750 ","End":"11:46.400","Text":"where every single point along this line is"},{"Start":"11:46.400 ","End":"11:53.014","Text":"experiencing complete constructive interference."},{"Start":"11:53.014 ","End":"12:02.049","Text":"If I choose some random point over here, P. Mathematically,"},{"Start":"12:02.049 ","End":"12:08.035","Text":"in order to show that this point P lies on this maximum line,"},{"Start":"12:08.035 ","End":"12:12.009","Text":"so let\u0027s call this max line,"},{"Start":"12:12.009 ","End":"12:14.260","Text":"this red over here."},{"Start":"12:14.260 ","End":"12:19.705","Text":"The condition to prove that P lies on this maximum line,"},{"Start":"12:19.705 ","End":"12:26.680","Text":"so this line that goes right through the middle between the 2 sources."},{"Start":"12:26.680 ","End":"12:36.010","Text":"That means that the distance from source 1 to point P has to be equal"},{"Start":"12:36.010 ","End":"12:45.880","Text":"to the distance from source 2 to P. Because if these distances are the same,"},{"Start":"12:45.880 ","End":"12:53.215","Text":"then that means that P lies right in the middle on the maximum line,"},{"Start":"12:53.215 ","End":"12:58.405","Text":"that lies right between sources 1 and 2."},{"Start":"12:58.405 ","End":"13:01.780","Text":"Then we know that this is the maximum line, and on this line,"},{"Start":"13:01.780 ","End":"13:09.215","Text":"every single point on this line is experiencing complete constructive interference."},{"Start":"13:09.215 ","End":"13:13.270","Text":"This is the condition."},{"Start":"13:14.810 ","End":"13:26.170","Text":"Another way of writing this is saying that S_1P minus S_2P is equal to 0."},{"Start":"13:28.590 ","End":"13:32.469","Text":"This is the condition for our maximum line."},{"Start":"13:32.469 ","End":"13:34.900","Text":"Or if we\u0027re going to be a little bit more specific,"},{"Start":"13:34.900 ","End":"13:36.565","Text":"the maximum line,"},{"Start":"13:36.565 ","End":"13:39.685","Text":"so this line that goes right down the middle,"},{"Start":"13:39.685 ","End":"13:44.529","Text":"where we have along this entire line at every single point on this entire line,"},{"Start":"13:44.529 ","End":"13:48.010","Text":"we have complete constructive interference."},{"Start":"13:48.010 ","End":"13:53.590","Text":"We can also call this the zeroth order maximum."},{"Start":"13:53.590 ","End":"13:57.550","Text":"The zeroth order maximum refers to"},{"Start":"13:57.550 ","End":"14:01.465","Text":"this line where we have complete constructive interference."},{"Start":"14:01.465 ","End":"14:03.369","Text":"Why is it called zeroth order?"},{"Start":"14:03.369 ","End":"14:06.490","Text":"We\u0027ll speak about it a little bit more in detail soon,"},{"Start":"14:06.490 ","End":"14:08.349","Text":"but it comes from this over here,"},{"Start":"14:08.349 ","End":"14:12.400","Text":"that S_1P minus S_2P is equal to 0."},{"Start":"14:12.400 ","End":"14:14.253","Text":"It\u0027s to do with this,"},{"Start":"14:14.253 ","End":"14:17.000","Text":"and we\u0027ll speak about it soon."},{"Start":"14:18.740 ","End":"14:25.379","Text":"Here we\u0027ve spoken about our line where we have complete constructive interference."},{"Start":"14:25.379 ","End":"14:28.514","Text":"Now I\u0027m going to rub out everything on the screen,"},{"Start":"14:28.514 ","End":"14:34.870","Text":"and we\u0027re going to speak about other points of interest in this image."},{"Start":"14:36.500 ","End":"14:41.799","Text":"We spoke about the line that goes straight through the middle."},{"Start":"14:41.799 ","End":"14:47.210","Text":"But now, let\u0027s speak about a different bright spot."},{"Start":"14:47.210 ","End":"14:56.079","Text":"Let\u0027s take a look at this spot over here."},{"Start":"14:56.079 ","End":"14:59.604","Text":"We can see that here we still have a bright spot."},{"Start":"14:59.604 ","End":"15:01.360","Text":"Now, if we look,"},{"Start":"15:01.360 ","End":"15:07.809","Text":"we can see that the distance from S_1 to"},{"Start":"15:07.809 ","End":"15:15.955","Text":"this point is different than the distance from S_2 to this point."},{"Start":"15:15.955 ","End":"15:22.000","Text":"Now it might not be the"},{"Start":"15:22.000 ","End":"15:28.330","Text":"most clear that this point is slightly closer to S_1 than to S_2,"},{"Start":"15:28.330 ","End":"15:30.760","Text":"but we can show the same thing if we look at,"},{"Start":"15:30.760 ","End":"15:33.205","Text":"let\u0027s say, this point over here."},{"Start":"15:33.205 ","End":"15:36.805","Text":"We also have some form of constructive interference."},{"Start":"15:36.805 ","End":"15:38.784","Text":"We still have a bright spot over here."},{"Start":"15:38.784 ","End":"15:43.795","Text":"But here we can see that very clearly,"},{"Start":"15:43.795 ","End":"15:49.850","Text":"it\u0027s much closer to S_1 than it is to S_2."},{"Start":"15:50.340 ","End":"15:56.620","Text":"We can see that this goes for the same thing as this point over here,"},{"Start":"15:56.620 ","End":"16:00.250","Text":"but we\u0027re going to speak again about this point over here,"},{"Start":"16:00.250 ","End":"16:01.554","Text":"so I\u0027ll rub out this point."},{"Start":"16:01.554 ","End":"16:04.630","Text":"But what\u0027s important to note is that this bright spot over"},{"Start":"16:04.630 ","End":"16:08.365","Text":"here is closer to S_1 than to S_2."},{"Start":"16:08.365 ","End":"16:13.850","Text":"As in, it\u0027s not on the line going straight through the middle."},{"Start":"16:15.480 ","End":"16:20.334","Text":"What we can see from this bright patch is that, 1,"},{"Start":"16:20.334 ","End":"16:22.975","Text":"we know that there\u0027s going to be a crest over here,"},{"Start":"16:22.975 ","End":"16:27.460","Text":"which means that we know that there\u0027s constructive interference."},{"Start":"16:27.460 ","End":"16:30.744","Text":"How can there be constructive interference if"},{"Start":"16:30.744 ","End":"16:36.460","Text":"this point is closer to source 1 than it is to source 2?"},{"Start":"16:36.460 ","End":"16:38.980","Text":"Let\u0027s draw this in color."},{"Start":"16:38.980 ","End":"16:42.189","Text":"We can see that our wave from source 1,"},{"Start":"16:42.189 ","End":"16:44.020","Text":"in order to get to this point over here,"},{"Start":"16:44.020 ","End":"16:47.229","Text":"has traveled 1 wavelength,"},{"Start":"16:47.229 ","End":"16:50.605","Text":"2 wavelengths, 3 wavelengths,"},{"Start":"16:50.605 ","End":"16:56.680","Text":"4 and 5 wavelengths to get to this point."},{"Start":"16:56.680 ","End":"17:01.930","Text":"Then it reaches this point when it\u0027s at a crest."},{"Start":"17:01.930 ","End":"17:07.750","Text":"Now let\u0027s look at the wave front coming from source 2."},{"Start":"17:07.750 ","End":"17:12.479","Text":"Here it\u0027s traveled 1 wavelength, 2 wavelengths."},{"Start":"17:12.479 ","End":"17:14.370","Text":"I\u0027m counting from peak to peak,"},{"Start":"17:14.370 ","End":"17:15.615","Text":"so from crest to crest,"},{"Start":"17:15.615 ","End":"17:17.454","Text":"or in the diagram,"},{"Start":"17:17.454 ","End":"17:20.130","Text":"from white circle to white circle."},{"Start":"17:20.130 ","End":"17:21.510","Text":"Again, 1,"},{"Start":"17:21.510 ","End":"17:23.655","Text":"2, 3,"},{"Start":"17:23.655 ","End":"17:28.590","Text":"4, 5, 6 wavelengths."},{"Start":"17:28.590 ","End":"17:31.505","Text":"We saw S_2,"},{"Start":"17:31.505 ","End":"17:37.150","Text":"the wave front has traveled 6 wavelengths in order to arrive at this bright patch,"},{"Start":"17:37.150 ","End":"17:38.792","Text":"and it also arrives,"},{"Start":"17:38.792 ","End":"17:40.449","Text":"because we can see that it\u0027s a light patch,"},{"Start":"17:40.449 ","End":"17:43.555","Text":"it also arrives at a crest."},{"Start":"17:43.555 ","End":"17:48.564","Text":"What has happened if I try and draw this is that"},{"Start":"17:48.564 ","End":"17:55.090","Text":"our green wave coming from S_1 has traveled from peak to peak,"},{"Start":"17:55.090 ","End":"17:59.810","Text":"so 1, 2,"},{"Start":"18:00.990 ","End":"18:06.558","Text":"3, 4, 5,"},{"Start":"18:06.558 ","End":"18:08.559","Text":"and then arrived at the bright patch,"},{"Start":"18:08.559 ","End":"18:09.880","Text":"1, 2, 3,"},{"Start":"18:09.880 ","End":"18:11.905","Text":"4, 5 wavelengths."},{"Start":"18:11.905 ","End":"18:14.688","Text":"Then for my source 2,"},{"Start":"18:14.688 ","End":"18:18.505","Text":"the wave has come also from peak to peak,"},{"Start":"18:18.505 ","End":"18:21.730","Text":"so 1, 2,"},{"Start":"18:21.730 ","End":"18:24.960","Text":"3,"},{"Start":"18:24.960 ","End":"18:32.755","Text":"4, 5, 6."},{"Start":"18:32.755 ","End":"18:35.875","Text":"Now I just have to move them together."},{"Start":"18:35.875 ","End":"18:40.254","Text":"To arrive at a peak at this exact point."},{"Start":"18:40.254 ","End":"18:45.985","Text":"Ignore that over here the green and blue are touching."},{"Start":"18:45.985 ","End":"18:50.020","Text":"The important point is that here they are touching."},{"Start":"18:50.020 ","End":"18:59.679","Text":"Here, what we have are 2 peaks touching,"},{"Start":"18:59.679 ","End":"19:03.235","Text":"or arriving at the same point at the same time."},{"Start":"19:03.235 ","End":"19:11.360","Text":"Therefore, we have constructive interference."},{"Start":"19:13.740 ","End":"19:17.245","Text":"Then we get this bright patch."},{"Start":"19:17.245 ","End":"19:22.510","Text":"What has, in fact, happened is that our wave from source 1 has traveled"},{"Start":"19:22.510 ","End":"19:27.925","Text":"5 wavelengths and has arrived at this point in the crest, like so."},{"Start":"19:27.925 ","End":"19:34.735","Text":"Then our wave front from source 2 has traveled a slightly further distance,"},{"Start":"19:34.735 ","End":"19:39.564","Text":"but still an even multiple of wavelengths,"},{"Start":"19:39.564 ","End":"19:42.565","Text":"so a whole number multiplied of wavelengths, so 6,"},{"Start":"19:42.565 ","End":"19:51.280","Text":"and arrived also with a crest at this point over here."},{"Start":"19:51.280 ","End":"19:55.195","Text":"Then as we know, when we superimpose these 2 waves,"},{"Start":"19:55.195 ","End":"19:59.230","Text":"we\u0027re just going to add up their amplitudes,"},{"Start":"19:59.230 ","End":"20:03.579","Text":"and then we\u0027re going to get this bright patch,"},{"Start":"20:03.579 ","End":"20:06.624","Text":"where we know when we have a crest,"},{"Start":"20:06.624 ","End":"20:10.674","Text":"we\u0027re going to have a light patch underneath,"},{"Start":"20:10.674 ","End":"20:13.250","Text":"like we spoke about that before."},{"Start":"20:13.380 ","End":"20:18.160","Text":"What\u0027s in fact happened is that the wave front"},{"Start":"20:18.160 ","End":"20:22.885","Text":"from source 2 started traveling and it traveled 1 wavelength,"},{"Start":"20:22.885 ","End":"20:28.120","Text":"and then a wave front from source 1 that was released at the same time"},{"Start":"20:28.120 ","End":"20:33.819","Text":"that the wave front from source 2 had already traveled 1 wavelength started traveling."},{"Start":"20:33.819 ","End":"20:38.155","Text":"Then they arrived at the same time to this point."},{"Start":"20:38.155 ","End":"20:44.274","Text":"That is how we get complete constructive interference over here as well,"},{"Start":"20:44.274 ","End":"20:47.635","Text":"even though it\u0027s not at the midpoint."},{"Start":"20:47.635 ","End":"20:50.740","Text":"What\u0027s important to remember is that"},{"Start":"20:50.740 ","End":"20:54.925","Text":"the wavelength of source 1 is equal to the wavelength of source 2,"},{"Start":"20:54.925 ","End":"20:58.310","Text":"and that they\u0027re in the same phase."},{"Start":"21:00.360 ","End":"21:03.745","Text":"Their phases are the same, otherwise,"},{"Start":"21:03.745 ","End":"21:08.030","Text":"this wouldn\u0027t necessarily be the case."},{"Start":"21:09.120 ","End":"21:17.530","Text":"What we can see is that the wave front coming from S_1 traveled 5 wavelengths,"},{"Start":"21:17.530 ","End":"21:20.319","Text":"Lambda is a sign for wavelengths,"},{"Start":"21:20.319 ","End":"21:28.069","Text":"and that the wave front from S_2 traveled 6 wavelengths."},{"Start":"21:28.069 ","End":"21:33.370","Text":"We can see that the change in distance of"},{"Start":"21:33.370 ","End":"21:40.750","Text":"this point from S_1 to S_2 is 6 Lambda minus 5 Lambda,"},{"Start":"21:40.750 ","End":"21:43.165","Text":"which is equal to Lambda."},{"Start":"21:43.165 ","End":"21:50.050","Text":"This is the distance that S_2 has to travel further in order to get to this point."},{"Start":"21:50.050 ","End":"21:54.955","Text":"What if we look at this light patch over here?"},{"Start":"21:54.955 ","End":"21:56.470","Text":"Let\u0027s take a look."},{"Start":"21:56.470 ","End":"22:03.149","Text":"From S_1 we have to travel 1,"},{"Start":"22:03.149 ","End":"22:04.305","Text":"2, 3, 4, 5,"},{"Start":"22:04.305 ","End":"22:08.440","Text":"6 wavelengths and from S_2 to get to the same point,"},{"Start":"22:08.440 ","End":"22:10.029","Text":"and both of them, of course,"},{"Start":"22:10.029 ","End":"22:11.920","Text":"arrive with a crest at a peak,"},{"Start":"22:11.920 ","End":"22:20.559","Text":"we have to travel 1, 2, 3, 4, 5, 6, 7 wavelengths."},{"Start":"22:20.559 ","End":"22:25.510","Text":"Now from S_2 we have 7 wavelengths,"},{"Start":"22:25.510 ","End":"22:26.859","Text":"for this point over here,"},{"Start":"22:26.859 ","End":"22:30.880","Text":"and from S_1 we have 6 wavelengths."},{"Start":"22:30.880 ","End":"22:37.675","Text":"Again, the distance is equal to 7 Lambda minus 6 Lambda,"},{"Start":"22:37.675 ","End":"22:40.780","Text":"which is again equal to Lambda."},{"Start":"22:40.780 ","End":"22:45.624","Text":"Now, if we look at all of these points over here,"},{"Start":"22:45.624 ","End":"22:49.420","Text":"all of these bright patches, like so,"},{"Start":"22:49.420 ","End":"22:56.635","Text":"if we check the distance to each one of these points from S_1 and then from S_2,"},{"Start":"22:56.635 ","End":"23:02.125","Text":"we\u0027ll see that the difference is always going to be 1 Lambda."},{"Start":"23:02.125 ","End":"23:06.894","Text":"We can see that the condition for this line,"},{"Start":"23:06.894 ","End":"23:09.535","Text":"so I\u0027m going to join all of these points,"},{"Start":"23:09.535 ","End":"23:13.659","Text":"the condition for all of the points on this line is that"},{"Start":"23:13.659 ","End":"23:20.065","Text":"the distance from S_1 to some arbitrary point P on this line,"},{"Start":"23:20.065 ","End":"23:21.490","Text":"so that the distance"},{"Start":"23:21.490 ","End":"23:31.060","Text":"from S_2 to P"},{"Start":"23:31.060 ","End":"23:32.889","Text":"minus S_1 to P,"},{"Start":"23:32.889 ","End":"23:35.935","Text":"because S_2 is further away,"},{"Start":"23:35.935 ","End":"23:38.500","Text":"otherwise we\u0027ll get a negative number."},{"Start":"23:38.500 ","End":"23:44.169","Text":"The distance from S_2 this point minus the distance from S_1 to"},{"Start":"23:44.169 ","End":"23:50.920","Text":"this point is always going to be 1 multiplied by Lambda."},{"Start":"23:50.920 ","End":"23:54.955","Text":"Let\u0027s just try with this arbitrary point."},{"Start":"23:54.955 ","End":"23:57.522","Text":"We can see from S_1 we\u0027re going to travel 1,"},{"Start":"23:57.522 ","End":"23:59.539","Text":"2, 3, 4, 5, 6,"},{"Start":"23:59.539 ","End":"24:04.855","Text":"7, 8, 9 wavelengths."},{"Start":"24:04.855 ","End":"24:06.968","Text":"From S_2 are going to travel 1,"},{"Start":"24:06.968 ","End":"24:09.038","Text":"2, 3, 4, 5, 6, 7,"},{"Start":"24:09.038 ","End":"24:14.870","Text":"8, 9, 10 wavelengths."},{"Start":"24:14.870 ","End":"24:21.504","Text":"Here we can see that the distance from S_2 to P is 10 wavelengths,"},{"Start":"24:21.504 ","End":"24:28.070","Text":"and that the distance from S_1 to P was 9 wavelengths."},{"Start":"24:28.710 ","End":"24:36.910","Text":"Therefore 10 Lambda minus 9 Lambda is 1 Lambda or just Lambda."},{"Start":"24:36.910 ","End":"24:43.405","Text":"This is the condition that defines all of the points on this line over here."},{"Start":"24:43.405 ","End":"24:48.940","Text":"Of course, the same goes for the dark patches."},{"Start":"24:48.940 ","End":"24:53.905","Text":"Just like we saw with the 0th order maximum,"},{"Start":"24:53.905 ","End":"24:56.934","Text":"so also over here where the dark patches are,"},{"Start":"24:56.934 ","End":"25:01.615","Text":"we also get constructive interference."},{"Start":"25:01.615 ","End":"25:05.539","Text":"Let\u0027s see how this happens."},{"Start":"25:05.610 ","End":"25:08.724","Text":"I\u0027ve made some more space."},{"Start":"25:08.724 ","End":"25:12.160","Text":"We\u0027re going now to this point over here."},{"Start":"25:12.160 ","End":"25:14.185","Text":"Let\u0027s mark it in black,"},{"Start":"25:14.185 ","End":"25:16.615","Text":"where we have a dark patch."},{"Start":"25:16.615 ","End":"25:23.430","Text":"From Source 1 the distance is 1,"},{"Start":"25:23.430 ","End":"25:25.610","Text":"2, 3, 4, 5, 6, 7."},{"Start":"25:26.310 ","End":"25:29.200","Text":"We\u0027ve gone from peak to peak 7,"},{"Start":"25:29.200 ","End":"25:31.465","Text":"and then this is at a trough."},{"Start":"25:31.465 ","End":"25:36.230","Text":"That means that we\u0027re traveling half a wavelength."},{"Start":"25:36.300 ","End":"25:40.990","Text":"Let me just draw this wave out for you."},{"Start":"25:40.990 ","End":"25:48.205","Text":"As we remember, if this is what a wave looks like from peak to peak,,"},{"Start":"25:48.205 ","End":"25:50.920","Text":"this is 1 wavelength,"},{"Start":"25:50.920 ","End":"25:54.204","Text":"Then if we want to go from peak to trough,"},{"Start":"25:54.204 ","End":"25:56.740","Text":"that\u0027s exactly right in the middle."},{"Start":"25:56.740 ","End":"25:59.440","Text":"From peak to trough is half wavelength."},{"Start":"25:59.440 ","End":"26:03.940","Text":"Here we saw that from Source 1, let\u0027s write it here,"},{"Start":"26:03.940 ","End":"26:10.315","Text":"S_1, we\u0027ve traveled 7.5 wavelengths."},{"Start":"26:10.315 ","End":"26:13.669","Text":"Now let\u0027s see from Source 2 to this trough over here,"},{"Start":"26:13.669 ","End":"26:15.085","Text":"to this dark patch."},{"Start":"26:15.085 ","End":"26:18.537","Text":"We\u0027ve traveled 1, 2, 3, 4, 5, 6, 7,"},{"Start":"26:18.537 ","End":"26:27.640","Text":"8 to this peak and then we go to a trough, so that\u0027s 8.5."},{"Start":"26:27.640 ","End":"26:30.985","Text":"From S_2 to this dark patch,"},{"Start":"26:30.985 ","End":"26:34.750","Text":"we\u0027ve traveled 8.5 wavelengths."},{"Start":"26:34.750 ","End":"26:37.390","Text":"Again, if we plug this into this equation,"},{"Start":"26:37.390 ","End":"26:40.060","Text":"so S_2 to this point,"},{"Start":"26:40.060 ","End":"26:42.880","Text":"which now we\u0027re calling this point P,"},{"Start":"26:42.880 ","End":"26:49.795","Text":"so S_2 to P is 8.5 Lambda minus S_1 to P,"},{"Start":"26:49.795 ","End":"26:53.320","Text":"which is 7.5 Lambda,"},{"Start":"26:53.320 ","End":"26:59.950","Text":"and 8.5 minus 7.5 is again Lambda or 1 multiplied by Lambda."},{"Start":"26:59.950 ","End":"27:05.664","Text":"We can see that every single point along this line obeys this condition."},{"Start":"27:05.664 ","End":"27:11.080","Text":"This line is called the first order maximum,"},{"Start":"27:11.080 ","End":"27:17.185","Text":"because we can see first Number 1 multiplied by Lambda."},{"Start":"27:17.185 ","End":"27:20.289","Text":"This is where these names come from."},{"Start":"27:20.289 ","End":"27:25.690","Text":"I\u0027m just going to rub stuff out so that we can write this a little bit more neatly."},{"Start":"27:25.690 ","End":"27:30.610","Text":"The way we call these lines is dependent on the change in"},{"Start":"27:30.610 ","End":"27:35.830","Text":"wavelength between the distances from the two sources."},{"Start":"27:35.830 ","End":"27:40.224","Text":"If we take some arbitrary point P on the line,"},{"Start":"27:40.224 ","End":"27:42.580","Text":"and then we take the distance from S_2 to"},{"Start":"27:42.580 ","End":"27:46.135","Text":"that point minus the distance from S_1 to that point,"},{"Start":"27:46.135 ","End":"27:49.825","Text":"if it equals 0 or 0 multiplied by Lambda,"},{"Start":"27:49.825 ","End":"27:52.435","Text":"then we call this the 0th order."},{"Start":"27:52.435 ","End":"27:54.955","Text":"If the distance,"},{"Start":"27:54.955 ","End":"28:01.045","Text":"the change in distance between these two points is 1 multiplied by Lambda,"},{"Start":"28:01.045 ","End":"28:04.045","Text":"so this is called the first-order."},{"Start":"28:04.045 ","End":"28:08.065","Text":"If we\u0027re going to have that the difference is 2 Lambda"},{"Start":"28:08.065 ","End":"28:12.295","Text":"it will be second-order and so on and so forth."},{"Start":"28:12.295 ","End":"28:17.890","Text":"What we can see is that along this line over here,"},{"Start":"28:17.890 ","End":"28:21.984","Text":"where we said it was right through the middle,"},{"Start":"28:21.984 ","End":"28:25.720","Text":"here we have our 0th order."},{"Start":"28:25.720 ","End":"28:30.145","Text":"Then when we go through all the other maximums over here,"},{"Start":"28:30.145 ","End":"28:32.515","Text":"this is our first order."},{"Start":"28:32.515 ","End":"28:36.985","Text":"Then when we go through or the maximum points over here,"},{"Start":"28:36.985 ","End":"28:40.150","Text":"this is our second-order maximum."},{"Start":"28:40.150 ","End":"28:44.230","Text":"Then all the maximum constructive points over here,"},{"Start":"28:44.230 ","End":"28:46.720","Text":"this is our third order."},{"Start":"28:46.720 ","End":"28:51.339","Text":"Then through all the maximum points over here,"},{"Start":"28:51.339 ","End":"28:53.935","Text":"this is our fourth order."},{"Start":"28:53.935 ","End":"28:58.029","Text":"Then if we go through all our maximums over here,"},{"Start":"28:58.029 ","End":"29:02.754","Text":"this will be our fifth order maximum."},{"Start":"29:02.754 ","End":"29:07.450","Text":"Of course, there are more orders of maximum over here,"},{"Start":"29:07.450 ","End":"29:10.659","Text":"but it\u0027s a little bit more difficult and tricky to spot them,"},{"Start":"29:10.659 ","End":"29:12.430","Text":"but you get the general picture,"},{"Start":"29:12.430 ","End":"29:14.679","Text":"and the same goes for over this side."},{"Start":"29:14.679 ","End":"29:18.550","Text":"If we take all of the pluses or all of the areas of"},{"Start":"29:18.550 ","End":"29:23.650","Text":"bright and dark patches over here where we have constructive interference,"},{"Start":"29:23.650 ","End":"29:29.020","Text":"so this is also the first order maximum just on the other side."},{"Start":"29:29.020 ","End":"29:32.890","Text":"Then obviously, if we keep the equation like this,"},{"Start":"29:32.890 ","End":"29:36.640","Text":"we\u0027ll get a difference of negative Lambda over here,"},{"Start":"29:36.640 ","End":"29:38.335","Text":"or you could just switch these around,"},{"Start":"29:38.335 ","End":"29:40.915","Text":"multiply both sides by negative 1."},{"Start":"29:40.915 ","End":"29:44.290","Text":"These points are obviously along this line closer"},{"Start":"29:44.290 ","End":"29:47.725","Text":"to Source Number 2 than to Source Number 1."},{"Start":"29:47.725 ","End":"29:52.390","Text":"We\u0027ll put Source 1 to point P minus Source 2 to point P,"},{"Start":"29:52.390 ","End":"29:54.700","Text":"and then we\u0027ll get a difference of Lambda."},{"Start":"29:54.700 ","End":"29:58.885","Text":"Then here along all of these bright and dark patches,"},{"Start":"29:58.885 ","End":"30:02.605","Text":"this is the second-order along all of here."},{"Start":"30:02.605 ","End":"30:05.590","Text":"This is our third order."},{"Start":"30:05.590 ","End":"30:07.599","Text":"Of course, we can carry on fourth,"},{"Start":"30:07.599 ","End":"30:09.080","Text":"fifth, and so on."},{"Start":"30:09.080 ","End":"30:13.740","Text":"In general, the condition for the nth order maximum,"},{"Start":"30:13.740 ","End":"30:16.035","Text":"that could be the 10th order,"},{"Start":"30:16.035 ","End":"30:19.665","Text":"the 100th order, the first order maximum,"},{"Start":"30:19.665 ","End":"30:27.070","Text":"it\u0027s the absolute value of S_2 to P minus S_1 to P. That"},{"Start":"30:27.070 ","End":"30:34.555","Text":"will equal to some number multiplied by Lambda, the wavelength."},{"Start":"30:34.555 ","End":"30:40.075","Text":"We have the absolute value just to cut out the problem that we had over here,"},{"Start":"30:40.075 ","End":"30:47.290","Text":"whether we\u0027re looking at the first-order maximum that\u0027s closer to the S_1 source,"},{"Start":"30:47.290 ","End":"30:50.170","Text":"or we\u0027re looking at the first-order maximum,"},{"Start":"30:50.170 ","End":"30:53.800","Text":"which is closer to the S_2 source."},{"Start":"30:53.800 ","End":"30:56.289","Text":"In order to generalize,"},{"Start":"30:56.289 ","End":"31:00.685","Text":"we know that each one has a difference of 1 wavelength."},{"Start":"31:00.685 ","End":"31:03.429","Text":"If 1 Lambda, just one is a difference of"},{"Start":"31:03.429 ","End":"31:07.945","Text":"1 Lambda to the left and one is a difference of 1 Lambda to the right."},{"Start":"31:07.945 ","End":"31:12.220","Text":"In order to not have to deal with these negative numbers,"},{"Start":"31:12.220 ","End":"31:14.065","Text":"because we know what this means,"},{"Start":"31:14.065 ","End":"31:16.720","Text":"we put the absolute value over here and then"},{"Start":"31:16.720 ","End":"31:20.230","Text":"whatever comes out will always be a positive number."},{"Start":"31:20.230 ","End":"31:21.909","Text":"This will equal to n,"},{"Start":"31:21.909 ","End":"31:24.490","Text":"which is some whole number,"},{"Start":"31:24.490 ","End":"31:27.453","Text":"n is equal to 0,1,"},{"Start":"31:27.453 ","End":"31:35.050","Text":"2 and so on to infinity multiplied by Lambda, the wavelength."},{"Start":"31:35.050 ","End":"31:37.705","Text":"This is what\u0027s important to remember."},{"Start":"31:37.705 ","End":"31:46.389","Text":"What we\u0027ve seen in this lesson is an explanation for some of what we see in this image,"},{"Start":"31:46.389 ","End":"31:52.315","Text":"where we have two sources creating circular wave fronts in a wave tank."},{"Start":"31:52.315 ","End":"32:00.804","Text":"We\u0027ve seen the conditions for finding these maximum lines of nth order."},{"Start":"32:00.804 ","End":"32:03.949","Text":"That\u0027s the end of this lesson."}],"ID":12488},{"Watched":false,"Name":"Number of Maximum Lines","Duration":"14m 1s","ChapterTopicVideoID":12021,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.595","Text":"Hello. In the previous lesson,"},{"Start":"00:02.595 ","End":"00:09.668","Text":"we saw this wave tank where we had 2 sources creating circular wave front waves,"},{"Start":"00:09.668 ","End":"00:14.865","Text":"and we saw how we could define the order of maximum lines."},{"Start":"00:14.865 ","End":"00:16.980","Text":"Here we had the zeroth-order,"},{"Start":"00:16.980 ","End":"00:20.310","Text":"and on either side of that we have the first-order of maximum,"},{"Start":"00:20.310 ","End":"00:22.200","Text":"Then after that second-order of maximum,"},{"Start":"00:22.200 ","End":"00:25.560","Text":"third-order of maximum, fourth-order of maximum,"},{"Start":"00:25.560 ","End":"00:27.370","Text":"and so on and so forth."},{"Start":"00:27.370 ","End":"00:29.885","Text":"We saw that the equation,"},{"Start":"00:29.885 ","End":"00:36.320","Text":"in order to define these lines of nth-order maximum,"},{"Start":"00:36.320 ","End":"00:37.820","Text":"we use this equation,"},{"Start":"00:37.820 ","End":"00:42.950","Text":"which was the distance from source to some arbitrary point"},{"Start":"00:42.950 ","End":"00:49.460","Text":"along the line minus the distance from S_1 to that same arbitrary point."},{"Start":"00:49.460 ","End":"00:52.775","Text":"Then we took the absolute value of that,"},{"Start":"00:52.775 ","End":"01:00.210","Text":"and then we found that this was some multiple n multiplied by Lambda, the wavelength."},{"Start":"01:00.210 ","End":"01:02.870","Text":"Then by knowing what this n is,"},{"Start":"01:02.870 ","End":"01:05.810","Text":"we knew if it was the 0th, 1st,"},{"Start":"01:05.810 ","End":"01:09.820","Text":"2nd, 3rd, or 4th order of maximum."},{"Start":"01:09.820 ","End":"01:13.550","Text":"The question that we want to answer in this lesson is"},{"Start":"01:13.550 ","End":"01:18.185","Text":"how many maximums can we get in a wave tank?"},{"Start":"01:18.185 ","End":"01:22.280","Text":"I\u0027m just going to rub out everything that we have on this picture."},{"Start":"01:22.280 ","End":"01:29.795","Text":"Now, what I want to know is that if I choose this point over here, point P,"},{"Start":"01:29.795 ","End":"01:37.640","Text":"and then I look at the difference from S_1 to S_2,"},{"Start":"01:37.640 ","End":"01:41.465","Text":"I want to know the maximum number of"},{"Start":"01:41.465 ","End":"01:50.130","Text":"n\u0027s that I will manage to find in a wave tank."},{"Start":"01:51.260 ","End":"01:54.510","Text":"We have this equation over here,"},{"Start":"01:54.510 ","End":"01:59.315","Text":"and just because here we can see that S_2 is longer than S_1,"},{"Start":"01:59.315 ","End":"02:01.860","Text":"so that anyway, get a positive value over here."},{"Start":"02:01.860 ","End":"02:05.180","Text":"Let\u0027s just take out just for now for ease of calculation,"},{"Start":"02:05.180 ","End":"02:08.620","Text":"these absolute value signs."},{"Start":"02:08.620 ","End":"02:10.970","Text":"We have this equation,"},{"Start":"02:10.970 ","End":"02:12.410","Text":"this is our first equation,"},{"Start":"02:12.410 ","End":"02:15.215","Text":"and the second rule that we know,"},{"Start":"02:15.215 ","End":"02:23.475","Text":"is whenever we have any type of triangle,"},{"Start":"02:23.475 ","End":"02:25.950","Text":"let\u0027s say like so,"},{"Start":"02:25.950 ","End":"02:31.195","Text":"we know that the sum of 2 any 2 sides,"},{"Start":"02:31.195 ","End":"02:33.980","Text":"always has to be larger,"},{"Start":"02:33.980 ","End":"02:36.985","Text":"than the length of 1 side."},{"Start":"02:36.985 ","End":"02:40.900","Text":"This is true for every single triangle."},{"Start":"02:42.320 ","End":"02:49.115","Text":"The sum of any 2 sides of a triangle is always larger than the third."},{"Start":"02:49.115 ","End":"02:56.225","Text":"If we take this length and add it on to this length, any 2 sides,"},{"Start":"02:56.225 ","End":"03:00.020","Text":"the sum of these 2 lengths added together is always going to be"},{"Start":"03:00.020 ","End":"03:04.375","Text":"bigger than the length of the third side."},{"Start":"03:04.375 ","End":"03:07.875","Text":"We can call this point S_1,"},{"Start":"03:07.875 ","End":"03:12.000","Text":"and we call this point S_2."},{"Start":"03:12.000 ","End":"03:17.510","Text":"Here we have our point P. We know that this length over here,"},{"Start":"03:17.510 ","End":"03:20.225","Text":"we\u0027ve already called it S_1P,"},{"Start":"03:20.225 ","End":"03:22.340","Text":"and that this length over here,"},{"Start":"03:22.340 ","End":"03:26.310","Text":"we\u0027ve already called it S_2P."},{"Start":"03:27.010 ","End":"03:31.220","Text":"Then this side over here is just the distance"},{"Start":"03:31.220 ","End":"03:36.575","Text":"between the 2 wave generators or between the 2 wave sources."},{"Start":"03:36.575 ","End":"03:43.020","Text":"Then this side, so the distance between the 2 sources we can call d. Let\u0027s"},{"Start":"03:43.020 ","End":"03:49.175","Text":"call this distance over here d. Then what we can write,"},{"Start":"03:49.175 ","End":"03:55.245","Text":"we can write this equation that S_1P,"},{"Start":"03:55.245 ","End":"03:56.970","Text":"so that\u0027s this side,"},{"Start":"03:56.970 ","End":"04:00.700","Text":"plus d, this side."},{"Start":"04:00.700 ","End":"04:05.840","Text":"From this law, we know that this is always going"},{"Start":"04:05.840 ","End":"04:10.235","Text":"to be larger or equal to the third side,"},{"Start":"04:10.235 ","End":"04:14.550","Text":"and the third side over here is S_2P."},{"Start":"04:15.350 ","End":"04:17.640","Text":"Now we can rearrange this,"},{"Start":"04:17.640 ","End":"04:23.630","Text":"we can minus d from both sides and add S_2P to both sides,"},{"Start":"04:23.630 ","End":"04:31.980","Text":"and then we\u0027ll get that S_2P minus"},{"Start":"04:31.980 ","End":"04:36.350","Text":"S_1P is always smaller or equal"},{"Start":"04:36.350 ","End":"04:43.175","Text":"to d. We\u0027ve just rearranged the original equation that we wrote."},{"Start":"04:43.175 ","End":"04:52.120","Text":"Now we know that S_2P minus S_1P is this equation over here."},{"Start":"04:52.120 ","End":"04:55.485","Text":"Let\u0027s just draw a little arrow."},{"Start":"04:55.485 ","End":"04:58.275","Text":"This is exactly this."},{"Start":"04:58.275 ","End":"05:03.390","Text":"S_2P minus S_1P is equal to n Lambda."},{"Start":"05:03.390 ","End":"05:08.795","Text":"Therefore, we can write that n Lambda is always smaller or equal to"},{"Start":"05:08.795 ","End":"05:15.005","Text":"d. Then what we\u0027re trying to find is the maximum n,"},{"Start":"05:15.005 ","End":"05:16.220","Text":"this is what we\u0027re trying to find."},{"Start":"05:16.220 ","End":"05:18.340","Text":"How many maximums can we get an a wave tank?"},{"Start":"05:18.340 ","End":"05:22.940","Text":"The maximum n. What we\u0027re going to do is we\u0027re going to isolate out"},{"Start":"05:22.940 ","End":"05:28.430","Text":"therefore this n. We\u0027re just going to divide both sides by Lambda,"},{"Start":"05:28.430 ","End":"05:36.700","Text":"and therefore will get that n is always smaller or equal to d divided by Lambda,"},{"Start":"05:36.700 ","End":"05:39.950","Text":"where d is the distance between the 2 sources,"},{"Start":"05:39.950 ","End":"05:43.380","Text":"and Lambda is the wavelength."},{"Start":"05:43.460 ","End":"05:47.850","Text":"The maximum number of n\u0027s,"},{"Start":"05:47.850 ","End":"05:51.260","Text":"of orders of maximum that I can get,"},{"Start":"05:51.260 ","End":"05:56.970","Text":"is bound by the value d divided by Lambda."},{"Start":"06:00.380 ","End":"06:02.560","Text":"Now, something very,"},{"Start":"06:02.560 ","End":"06:10.730","Text":"very important to note is that this n always has to be a whole number."},{"Start":"06:10.730 ","End":"06:13.935","Text":"That means, not a fraction,"},{"Start":"06:13.935 ","End":"06:18.235","Text":"so not 4.5, not 7.6,"},{"Start":"06:18.235 ","End":"06:19.760","Text":"nothing like that either."},{"Start":"06:19.760 ","End":"06:21.075","Text":"0, 1,"},{"Start":"06:21.075 ","End":"06:22.260","Text":"2, 3, 4,"},{"Start":"06:22.260 ","End":"06:25.005","Text":"5, so a whole number."},{"Start":"06:25.005 ","End":"06:27.159","Text":"This is super, super important."},{"Start":"06:27.159 ","End":"06:30.400","Text":"I\u0027m also going to just circle this in red."},{"Start":"06:30.400 ","End":"06:34.935","Text":"Let\u0027s give a little example."},{"Start":"06:34.935 ","End":"06:37.620","Text":"Let\u0027s say that my d,"},{"Start":"06:37.620 ","End":"06:45.885","Text":"my distance between the 2 wave sources is equal to 7.5 centimeters."},{"Start":"06:45.885 ","End":"06:53.940","Text":"Then let\u0027s say that the wavelength of the waves is equal to 2 centimeters."},{"Start":"06:53.940 ","End":"07:00.960","Text":"Therefore, my n has to be bound by d,"},{"Start":"07:00.960 ","End":"07:04.995","Text":"so 7.5 divided by Lambda,"},{"Start":"07:04.995 ","End":"07:06.952","Text":"so divided by 2."},{"Start":"07:06.952 ","End":"07:13.625","Text":"7.5 divided by 2 is equal to 3.75."},{"Start":"07:13.625 ","End":"07:18.290","Text":"But n has to be a whole number."},{"Start":"07:18.700 ","End":"07:24.885","Text":"Here it\u0027s super important, exclamation marks."},{"Start":"07:24.885 ","End":"07:26.600","Text":"N has to be a whole number."},{"Start":"07:26.600 ","End":"07:30.640","Text":"What is the maximum number of n\u0027s?"},{"Start":"07:30.640 ","End":"07:36.475","Text":"N will be 3."},{"Start":"07:36.475 ","End":"07:44.155","Text":"Therefore the maximum amount of n\u0027s n will be bound by the number 3."},{"Start":"07:44.155 ","End":"07:46.120","Text":"Why is that?"},{"Start":"07:46.120 ","End":"07:49.000","Text":"Or n will be equal to 3,"},{"Start":"07:49.000 ","End":"07:54.850","Text":"the maximum amount of orders of maximum lines or"},{"Start":"07:54.850 ","End":"08:02.890","Text":"the maximum lines that we will be able to join this specific wave tank will be 3 lines."},{"Start":"08:03.120 ","End":"08:06.740","Text":"Because of course, over here we can\u0027t round up,"},{"Start":"08:06.740 ","End":"08:08.755","Text":"because if we round up that will be 4,"},{"Start":"08:08.755 ","End":"08:17.465","Text":"and we know that n can\u0027t ever reach 4 because it\u0027s always smaller or equal to 3.75."},{"Start":"08:17.465 ","End":"08:19.644","Text":"So we always have to round down."},{"Start":"08:19.644 ","End":"08:24.085","Text":"So if you get that n is smaller than equal to 2.1,"},{"Start":"08:24.085 ","End":"08:26.750","Text":"then n will have to be equal to 2."},{"Start":"08:26.750 ","End":"08:32.240","Text":"You always round down. If n is smaller or equal to 5.9,"},{"Start":"08:32.240 ","End":"08:35.770","Text":"so n will always be equal to 5."},{"Start":"08:35.770 ","End":"08:38.410","Text":"You always round down."},{"Start":"08:38.820 ","End":"08:41.990","Text":"All right. What does this mean?"},{"Start":"08:41.990 ","End":"08:48.310","Text":"This means that this is the maximum number of order."},{"Start":"08:48.310 ","End":"08:51.680","Text":"If we look over here,"},{"Start":"08:51.680 ","End":"08:54.460","Text":"or if we have some wave,"},{"Start":"08:54.460 ","End":"08:58.875","Text":"so we already said we have the 0th order, the 1st order,"},{"Start":"08:58.875 ","End":"09:01.715","Text":"2nd order, and up until the third-order,"},{"Start":"09:01.715 ","End":"09:03.250","Text":"this is what this means,"},{"Start":"09:03.250 ","End":"09:04.955","Text":"up until the third-order."},{"Start":"09:04.955 ","End":"09:14.920","Text":"But then what do we want to know is how many of these maximum lines will we see."},{"Start":"09:16.520 ","End":"09:20.595","Text":"Here we have up until the third-order."},{"Start":"09:20.595 ","End":"09:30.490","Text":"We have 3 orders to the left."},{"Start":"09:31.250 ","End":"09:35.020","Text":"Then we know we also have the mirror image."},{"Start":"09:35.020 ","End":"09:37.120","Text":"If we have 3 orders to the left,"},{"Start":"09:37.120 ","End":"09:39.695","Text":"we always have the same amount,"},{"Start":"09:39.695 ","End":"09:44.050","Text":"so 3 orders to the right,"},{"Start":"09:44.120 ","End":"09:48.685","Text":"and then this includes the first, second and third-order."},{"Start":"09:48.685 ","End":"09:51.850","Text":"But we also always have"},{"Start":"09:51.850 ","End":"09:59.590","Text":"the zeroth-order right in the middle."},{"Start":"09:59.590 ","End":"10:06.175","Text":"Therefore, how many maximum lines will we see in total in the tank if we get that n=3?"},{"Start":"10:06.175 ","End":"10:11.455","Text":"We have here 3 plus 3 plus 1,"},{"Start":"10:11.455 ","End":"10:14.550","Text":"so that\u0027s 7 lines,"},{"Start":"10:14.550 ","End":"10:17.969","Text":"so 7 maximum lines."},{"Start":"10:17.969 ","End":"10:19.690","Text":"Let\u0027s go back to the picture."},{"Start":"10:19.690 ","End":"10:21.485","Text":"We know that we have 7 lines,"},{"Start":"10:21.485 ","End":"10:25.209","Text":"so let\u0027s say that that is what we had over here,"},{"Start":"10:25.209 ","End":"10:28.450","Text":"so this distance d over here was, as we said,"},{"Start":"10:28.450 ","End":"10:31.975","Text":"7.5 centimeters, and the wavelength of"},{"Start":"10:31.975 ","End":"10:39.240","Text":"each source was creating waves of wavelength 2 centimeters,"},{"Start":"10:39.240 ","End":"10:43.840","Text":"so we\u0027ll always have over here the zeroth-order."},{"Start":"10:43.840 ","End":"10:45.820","Text":"Let\u0027s just draw it over here,"},{"Start":"10:45.820 ","End":"10:47.545","Text":"so this is the zeroth-order,"},{"Start":"10:47.545 ","End":"10:51.835","Text":"and then we\u0027ll have first-order to the left,"},{"Start":"10:51.835 ","End":"10:54.050","Text":"first-order to the right,"},{"Start":"10:54.050 ","End":"10:56.505","Text":"second-order to the left,"},{"Start":"10:56.505 ","End":"10:58.620","Text":"second-order to the right,"},{"Start":"10:58.620 ","End":"10:59.985","Text":"and then we said,"},{"Start":"10:59.985 ","End":"11:04.920","Text":"given these numbers, we can only have a maximum of 3 orders."},{"Start":"11:04.920 ","End":"11:08.210","Text":"Then we\u0027d have third-order to the left,"},{"Start":"11:08.210 ","End":"11:10.425","Text":"and third-order to the right."},{"Start":"11:10.425 ","End":"11:13.895","Text":"This is our first, second,"},{"Start":"11:13.895 ","End":"11:19.165","Text":"and third, and here we also have the exact same."},{"Start":"11:19.165 ","End":"11:21.270","Text":"Then we have 3, 4,"},{"Start":"11:21.270 ","End":"11:23.535","Text":"5, 6, 7."},{"Start":"11:23.535 ","End":"11:26.165","Text":"So 7 total lines,"},{"Start":"11:26.165 ","End":"11:29.310","Text":"but 3 orders,"},{"Start":"11:29.310 ","End":"11:32.710","Text":"and that\u0027s what we got over here."},{"Start":"11:32.710 ","End":"11:37.075","Text":"There\u0027s an easy way to compute this."},{"Start":"11:37.075 ","End":"11:40.995","Text":"How many maximum lines will we see in the tank?"},{"Start":"11:40.995 ","End":"11:46.150","Text":"The equation for that is 2n,"},{"Start":"11:46.150 ","End":"11:54.220","Text":"so 2 multiplied by the value of n that we got from this equation over here, plus 1."},{"Start":"11:55.730 ","End":"11:58.325","Text":"If our n was 3,"},{"Start":"11:58.325 ","End":"12:00.260","Text":"so 2 times 3 is 6,"},{"Start":"12:00.260 ","End":"12:01.735","Text":"plus 1 is 7,"},{"Start":"12:01.735 ","End":"12:04.720","Text":"and that\u0027s what we got, 7 lines."},{"Start":"12:04.830 ","End":"12:07.630","Text":"Then if we had,"},{"Start":"12:07.630 ","End":"12:09.610","Text":"let\u0027s say that n=7,"},{"Start":"12:09.610 ","End":"12:11.320","Text":"so 2 times 7 is 14,"},{"Start":"12:11.320 ","End":"12:13.070","Text":"plus 1 is 15."},{"Start":"12:13.070 ","End":"12:17.925","Text":"If we get a wave tank that we can see up to the seventh-order,"},{"Start":"12:17.925 ","End":"12:23.174","Text":"that means that in total the number of maximum lines that we\u0027ll see is 15."},{"Start":"12:23.174 ","End":"12:26.235","Text":"Now, something that\u0027s interesting to note,"},{"Start":"12:26.235 ","End":"12:28.545","Text":"let me just rub all of this out."},{"Start":"12:28.545 ","End":"12:35.825","Text":"Is that we saw that here we have all of our maximum points."},{"Start":"12:35.825 ","End":"12:38.145","Text":"Then in the next lesson,"},{"Start":"12:38.145 ","End":"12:41.290","Text":"we will see few lessons ago we spoke"},{"Start":"12:41.290 ","End":"12:45.820","Text":"about these blurry gray lines over here that we could see."},{"Start":"12:45.820 ","End":"12:52.754","Text":"We can see them also over here and here, and also here."},{"Start":"12:52.754 ","End":"12:55.570","Text":"We can see all of these blurry gray lines,"},{"Start":"12:55.570 ","End":"13:00.273","Text":"and we can see that in the middle of each of these blurry gray lines,"},{"Start":"13:00.273 ","End":"13:02.395","Text":"so what I marked in blue,"},{"Start":"13:02.395 ","End":"13:07.435","Text":"we have these maximum points of constructive interference,"},{"Start":"13:07.435 ","End":"13:09.795","Text":"that we were just speaking about."},{"Start":"13:09.795 ","End":"13:12.390","Text":"Here we have our first-order in black,"},{"Start":"13:12.390 ","End":"13:14.830","Text":"second-order in black, third in black,"},{"Start":"13:14.830 ","End":"13:16.375","Text":"and fourth-order in black."},{"Start":"13:16.375 ","End":"13:17.855","Text":"In the next lesson,"},{"Start":"13:17.855 ","End":"13:22.540","Text":"we\u0027re going to be speaking about what these gray blurred lines are,"},{"Start":"13:22.540 ","End":"13:25.997","Text":"and why in the middle of these gray blurred lines,"},{"Start":"13:25.997 ","End":"13:29.690","Text":"we have this constructive interference?"},{"Start":"13:29.690 ","End":"13:32.435","Text":"That\u0027s the end of this lesson."},{"Start":"13:32.435 ","End":"13:37.960","Text":"What\u0027s important to remember from this lesson is this equation over here,"},{"Start":"13:37.960 ","End":"13:40.058","Text":"knowing that n has to be a whole number,"},{"Start":"13:40.058 ","End":"13:41.985","Text":"so we always round down."},{"Start":"13:41.985 ","End":"13:48.395","Text":"That gives us the maximum order that we can get,"},{"Start":"13:48.395 ","End":"13:51.815","Text":"and in order to know how many lines we will actually be able to"},{"Start":"13:51.815 ","End":"13:56.009","Text":"draw or how many we will see in the wave tank,"},{"Start":"13:56.009 ","End":"14:01.780","Text":"this is the equation to remember; 2n plus 1."}],"ID":12489},{"Watched":false,"Name":"Minimum Lines","Duration":"26m 25s","ChapterTopicVideoID":12022,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"Hello. We\u0027ve spent the past couple of videos speaking about"},{"Start":"00:03.960 ","End":"00:10.200","Text":"the maximum lines where we saw that we have complete constructive interference."},{"Start":"00:10.200 ","End":"00:14.685","Text":"In this lesson, we\u0027re going to be speaking about these gray blurry lines,"},{"Start":"00:14.685 ","End":"00:17.600","Text":"which we spoke about a few lessons ago,"},{"Start":"00:17.600 ","End":"00:20.640","Text":"and that we said that we\u0027ll get back to you at a later lesson."},{"Start":"00:20.640 ","End":"00:24.510","Text":"This is the later lesson and these are the gray blurry lines I\u0027m talking about."},{"Start":"00:24.510 ","End":"00:27.340","Text":"You can see this line over here,"},{"Start":"00:27.500 ","End":"00:30.749","Text":"it\u0027s a bit of a gray blurry line."},{"Start":"00:30.749 ","End":"00:36.780","Text":"Here we can see that there\u0027s more and we\u0027ll notice that in-between"},{"Start":"00:36.780 ","End":"00:43.550","Text":"every 2 gray blurry lines or in-between every 2 of these pink lines,"},{"Start":"00:43.550 ","End":"00:47.315","Text":"we can see that we have our points of"},{"Start":"00:47.315 ","End":"00:53.330","Text":"maximum or a line of maximum or where we have complete constructive interference."},{"Start":"00:53.330 ","End":"00:54.965","Text":"Also on the other side,"},{"Start":"00:54.965 ","End":"00:58.690","Text":"we have these gray blurry lines as well."},{"Start":"00:58.690 ","End":"01:00.885","Text":"I\u0027m just going to draw all of them in,"},{"Start":"01:00.885 ","End":"01:06.130","Text":"I hope that you can see along which lines I\u0027m drawing them."},{"Start":"01:06.290 ","End":"01:10.630","Text":"Here we have all of these lines."},{"Start":"01:11.120 ","End":"01:20.400","Text":"Now let\u0027s rub out all of these lines and see what is causing this effect."},{"Start":"01:20.780 ","End":"01:27.530","Text":"Let\u0027s take a look at one of the central points over here on one of the gray lines."},{"Start":"01:27.530 ","End":"01:30.190","Text":"Let\u0027s take a look at this point over here,"},{"Start":"01:30.190 ","End":"01:31.805","Text":"and as we can see,"},{"Start":"01:31.805 ","End":"01:37.760","Text":"it has a gray tone and it\u0027s located on one of these blurry gray lines."},{"Start":"01:37.760 ","End":"01:40.580","Text":"Now let\u0027s just do a recap."},{"Start":"01:40.580 ","End":"01:45.605","Text":"We saw that when a wave is coming in at some peak,"},{"Start":"01:45.605 ","End":"01:51.545","Text":"this peak acts as some concentrating lens."},{"Start":"01:51.545 ","End":"01:55.850","Text":"Then that every light ray that hits"},{"Start":"01:55.850 ","End":"02:02.115","Text":"it gets concentrated inwards like so,"},{"Start":"02:02.115 ","End":"02:11.685","Text":"and that results in a bright patch of light under the peak."},{"Start":"02:11.685 ","End":"02:17.310","Text":"We also spoke about when we get a trough,"},{"Start":"02:17.380 ","End":"02:20.915","Text":"which means this low patch."},{"Start":"02:20.915 ","End":"02:23.015","Text":"We saw that the trough,"},{"Start":"02:23.015 ","End":"02:29.890","Text":"it actually diffuses the light outwards."},{"Start":"02:30.230 ","End":"02:37.845","Text":"We can see that the light will be diffused out,"},{"Start":"02:37.845 ","End":"02:43.705","Text":"and then what we saw is that we get a dark patch underneath."},{"Start":"02:43.705 ","End":"02:47.825","Text":"Previously, we spoke about this area over here"},{"Start":"02:47.825 ","End":"02:53.705","Text":"where it\u0027s approximately water level with an amplitude of 0."},{"Start":"02:53.705 ","End":"02:57.500","Text":"Then we saw that the light waves or"},{"Start":"02:57.500 ","End":"03:03.289","Text":"the light rays will travel towards this water level area."},{"Start":"03:03.289 ","End":"03:08.570","Text":"Then because they\u0027re hitting at an angle that is the same as the normal,"},{"Start":"03:08.570 ","End":"03:11.590","Text":"they have a 0 angle relative to the normal,"},{"Start":"03:11.590 ","End":"03:15.425","Text":"the light rays will just travel all the way through."},{"Start":"03:15.425 ","End":"03:17.660","Text":"Which means that we\u0027re not getting"},{"Start":"03:17.660 ","End":"03:21.080","Text":"concentrated light and we\u0027re not getting diffused light."},{"Start":"03:21.080 ","End":"03:23.975","Text":"We\u0027re not getting a bright or a dark spot."},{"Start":"03:23.975 ","End":"03:25.945","Text":"We\u0027re getting somewhere in the middle,"},{"Start":"03:25.945 ","End":"03:27.320","Text":"or in other words,"},{"Start":"03:27.320 ","End":"03:31.315","Text":"we\u0027re getting a gray spot underneath."},{"Start":"03:31.315 ","End":"03:35.390","Text":"If we were just looking at our water tank and we didn\u0027t"},{"Start":"03:35.390 ","End":"03:39.485","Text":"have Source 1 and Source 2 generating waves,"},{"Start":"03:39.485 ","End":"03:42.095","Text":"and we kept the tank completely still,"},{"Start":"03:42.095 ","End":"03:45.110","Text":"all of the water will be level."},{"Start":"03:45.110 ","End":"03:48.365","Text":"All of the water will have an amplitude of 0"},{"Start":"03:48.365 ","End":"03:52.160","Text":"because there won\u0027t be any waves causing peaks or troughs."},{"Start":"03:52.160 ","End":"03:56.365","Text":"In that case, if we would look at the entire tank,"},{"Start":"03:56.365 ","End":"04:00.070","Text":"it would just be gray."},{"Start":"04:03.170 ","End":"04:07.095","Text":"We\u0027ll have a gray tank."},{"Start":"04:07.095 ","End":"04:08.985","Text":"Now, what do we want to see,"},{"Start":"04:08.985 ","End":"04:10.420","Text":"our new question is,"},{"Start":"04:10.420 ","End":"04:14.315","Text":"why is this point over here gray?"},{"Start":"04:14.315 ","End":"04:19.335","Text":"Now we understand why there is gray or how it happens,"},{"Start":"04:19.335 ","End":"04:22.630","Text":"and it\u0027s because the water here has an amplitude of 0,"},{"Start":"04:22.630 ","End":"04:25.690","Text":"and now we want to see how that happens."},{"Start":"04:25.690 ","End":"04:29.015","Text":"Let\u0027s start over here Source 1."},{"Start":"04:29.015 ","End":"04:35.720","Text":"We can see that the wave coming from Source 1 in order to get to this point, travels 1,"},{"Start":"04:35.720 ","End":"04:38.475","Text":"2, 3, 4,"},{"Start":"04:38.475 ","End":"04:43.960","Text":"5, 6, 7 wavelengths."},{"Start":"04:45.670 ","End":"04:52.820","Text":"The wave from Source 1 travels 7 wavelengths in order to get to this point."},{"Start":"04:52.820 ","End":"04:57.530","Text":"Now what we can see is that this point is located between"},{"Start":"04:57.530 ","End":"05:03.920","Text":"this peak over here or this wavefront over here and this wavefront over here."},{"Start":"05:03.920 ","End":"05:08.265","Text":"We can see that. Now let\u0027s see from Source 2"},{"Start":"05:08.265 ","End":"05:13.685","Text":"a wave coming from Source 2 how far it has to travel to get to this point, 1,"},{"Start":"05:13.685 ","End":"05:15.455","Text":"2, 3,"},{"Start":"05:15.455 ","End":"05:18.320","Text":"4, 5, 6,"},{"Start":"05:18.320 ","End":"05:25.625","Text":"7, but our point is located somewhere in the middle and 0.5."},{"Start":"05:25.625 ","End":"05:33.750","Text":"A wave from source number 2 has to travel 7.5 wavelengths."},{"Start":"05:35.030 ","End":"05:39.240","Text":"Now we can see that our point is"},{"Start":"05:39.240 ","End":"05:47.460","Text":"located if we\u0027re looking at this wavefront coming from Source 1."},{"Start":"05:47.460 ","End":"05:50.650","Text":"Let\u0027s just draw that in green over here so"},{"Start":"05:50.650 ","End":"05:53.800","Text":"we can see this wavefront and it\u0027s on a light patch."},{"Start":"05:53.800 ","End":"05:56.980","Text":"We can see that this point is, on the one hand,"},{"Start":"05:56.980 ","End":"06:03.970","Text":"located on some peak wavefront coming from Source 1."},{"Start":"06:03.970 ","End":"06:05.305","Text":"But on the other hand,"},{"Start":"06:05.305 ","End":"06:07.780","Text":"if we\u0027re looking at Source 2,"},{"Start":"06:07.780 ","End":"06:13.524","Text":"we can see that Source 2 where this red point is located,"},{"Start":"06:13.524 ","End":"06:22.980","Text":"is located along this dark area over here between 2 wavefronts coming from Source 2."},{"Start":"06:22.980 ","End":"06:28.540","Text":"These 2 white patches are wavefronts with their peaks."},{"Start":"06:28.540 ","End":"06:35.140","Text":"Here this dark area we can see is just going to be the minimum or"},{"Start":"06:35.140 ","End":"06:42.620","Text":"the trough of 1 of the wavefronts or 1 of the waves coming from source s_2."},{"Start":"06:42.960 ","End":"06:46.525","Text":"Also, we can see that from the numbers,"},{"Start":"06:46.525 ","End":"06:51.110","Text":"7 whole wavefronts are traveled by the green."},{"Start":"06:51.440 ","End":"06:53.960","Text":"If we draw that over here,"},{"Start":"06:53.960 ","End":"06:56.795","Text":"we have starting from a peak,"},{"Start":"06:56.795 ","End":"07:00.160","Text":"1, 2,"},{"Start":"07:00.160 ","End":"07:03.440","Text":"3, 4,"},{"Start":"07:03.440 ","End":"07:10.015","Text":"5, 6, and 7."},{"Start":"07:10.015 ","End":"07:18.940","Text":"Finish it you get at a peak let\u0027s just move a bit to the side."},{"Start":"07:18.940 ","End":"07:24.070","Text":"We can see from source 2 we have wavefronts traveling also"},{"Start":"07:24.070 ","End":"07:29.665","Text":"to this point over here so our red point is located over here at the peak."},{"Start":"07:29.665 ","End":"07:32.485","Text":"However, from source 2 in blue,"},{"Start":"07:32.485 ","End":"07:36.140","Text":"we\u0027re traveling 7.5 wavefronts."},{"Start":"07:36.930 ","End":"07:39.805","Text":"Again let\u0027s start from a peak."},{"Start":"07:39.805 ","End":"07:46.217","Text":"We\u0027re going 1,"},{"Start":"07:46.217 ","End":"07:50.131","Text":"2, 3,"},{"Start":"07:50.131 ","End":"07:54.045","Text":"4, 5,"},{"Start":"07:54.045 ","End":"07:57.274","Text":"6, 7,"},{"Start":"07:57.274 ","End":"07:59.690","Text":"so here we have,"},{"Start":"08:05.040 ","End":"08:13.300","Text":"1, 2, 3, 4, 5, 6, 7 and then we have our 1/2 so from a peak it drops down to here."},{"Start":"08:13.300 ","End":"08:20.470","Text":"Then we can see that the red point relative Test 2 is that a trough."},{"Start":"08:20.470 ","End":"08:25.345","Text":"Now we\u0027re going to super impose all of this."},{"Start":"08:25.345 ","End":"08:33.880","Text":"Let\u0027s just draw the line for 0 over here and then we can see that of course the green and"},{"Start":"08:33.880 ","End":"08:38.560","Text":"the blue waves are identical so here we have an amplitude of A and"},{"Start":"08:38.560 ","End":"08:43.315","Text":"here we have an amplitude of A just in the negative direction and similarly,"},{"Start":"08:43.315 ","End":"08:46.330","Text":"here we have an amplitude of A and"},{"Start":"08:46.330 ","End":"08:49.914","Text":"here we have an amplitude of A just in the negative direction."},{"Start":"08:49.914 ","End":"08:55.195","Text":"What we can see that over here on the green we go up A,"},{"Start":"08:55.195 ","End":"08:58.615","Text":"but over here in the blue we have to go down A."},{"Start":"08:58.615 ","End":"09:02.095","Text":"Now when we super impose,"},{"Start":"09:02.095 ","End":"09:07.585","Text":"so here the point is going to rest on A plus,"},{"Start":"09:07.585 ","End":"09:12.940","Text":"minus A, we\u0027re superimposing these 2 amplitudes."},{"Start":"09:12.940 ","End":"09:18.700","Text":"So A plus minus A is equal to 0 so therefore we can"},{"Start":"09:18.700 ","End":"09:26.810","Text":"see that in actual fact our red point will lay at an amplitude of 0."},{"Start":"09:27.120 ","End":"09:30.910","Text":"That means that we\u0027re not located at"},{"Start":"09:30.910 ","End":"09:34.870","Text":"a peak so we won\u0027t have a light patch and we\u0027re not located at a trough so we"},{"Start":"09:34.870 ","End":"09:40.570","Text":"won\u0027t have a dark patch and that is why we get this gray color and this is"},{"Start":"09:40.570 ","End":"09:48.290","Text":"called complete destructive interference."},{"Start":"09:51.720 ","End":"09:57.250","Text":"Now we\u0027ve said that this is complete destructive interference,"},{"Start":"09:57.250 ","End":"09:59.875","Text":"so let\u0027s speak about another thing."},{"Start":"09:59.875 ","End":"10:05.605","Text":"What we\u0027ll see if we\u0027re not looking at a still or a photograph"},{"Start":"10:05.605 ","End":"10:08.980","Text":"of our wave tank but rather we\u0027re watching a video"},{"Start":"10:08.980 ","End":"10:12.610","Text":"or we\u0027re watching this in live how the waves continue."},{"Start":"10:12.610 ","End":"10:17.755","Text":"We\u0027ll see that every single point along these blurry gray lines,"},{"Start":"10:17.755 ","End":"10:24.430","Text":"so these points over here that we marked will always appear stationary."},{"Start":"10:24.430 ","End":"10:28.854","Text":"These lines will never move and they always stay in the same position."},{"Start":"10:28.854 ","End":"10:33.400","Text":"However if we look at the points in-between these lines or at"},{"Start":"10:33.400 ","End":"10:39.505","Text":"the maximum lines we\u0027ll see that these will move onwards like this."},{"Start":"10:39.505 ","End":"10:45.160","Text":"Each point over here let\u0027s say this will slowly move to here and then to"},{"Start":"10:45.160 ","End":"10:48.490","Text":"here and to here and we\u0027ll keep moving and you\u0027ll see"},{"Start":"10:48.490 ","End":"10:52.910","Text":"movement between these gray blurry lines."},{"Start":"10:53.100 ","End":"10:57.400","Text":"Why does this remain the same or stationary?"},{"Start":"10:57.400 ","End":"11:00.535","Text":"Why is every point on 1 of these"},{"Start":"11:00.535 ","End":"11:04.750","Text":"now colored in purple but in general these gray blurry lines?"},{"Start":"11:04.750 ","End":"11:07.555","Text":"Why is it always 0 over here?"},{"Start":"11:07.555 ","End":"11:10.105","Text":"Let\u0027s draw this again."},{"Start":"11:10.105 ","End":"11:18.610","Text":"We know that at the beginning we had a case like so where on"},{"Start":"11:18.610 ","End":"11:23.170","Text":"our green line our green line reached this point with a peak but"},{"Start":"11:23.170 ","End":"11:28.150","Text":"then that our blue line reach this point at a trough."},{"Start":"11:28.150 ","End":"11:32.620","Text":"Then therefore, our point will lie over here when we"},{"Start":"11:32.620 ","End":"11:38.450","Text":"superimpose because we have complete destructive interference so it\u0027ll be on the 0 line."},{"Start":"11:38.760 ","End":"11:42.805","Text":"But as we know a moment later,"},{"Start":"11:42.805 ","End":"11:46.040","Text":"so we can draw it like this."},{"Start":"11:46.380 ","End":"11:52.478","Text":"Our blue point over here will move to a peak over here,"},{"Start":"11:52.478 ","End":"12:00.795","Text":"whereas our green wave will move from a peak over here to a trough over here."},{"Start":"12:00.795 ","End":"12:05.430","Text":"Again, we\u0027re going to have complete destructive interference because"},{"Start":"12:05.430 ","End":"12:10.060","Text":"we have a peak and trough and our pointing again will be on the 0 line."},{"Start":"12:10.060 ","End":"12:13.540","Text":"In fact we can carry this on with our green,"},{"Start":"12:13.540 ","End":"12:18.614","Text":"a moment later we\u0027ll reach a peak and then a trough,"},{"Start":"12:18.614 ","End":"12:22.705","Text":"and similarly with our blue,"},{"Start":"12:22.705 ","End":"12:24.325","Text":"here it\u0027s at a trough,"},{"Start":"12:24.325 ","End":"12:28.990","Text":"here it\u0027s at a peak."},{"Start":"12:28.990 ","End":"12:34.660","Text":"Always, our point will be located at"},{"Start":"12:34.660 ","End":"12:42.160","Text":"the 0 mark so we can see that it just stays as a line with 0 amplitude."},{"Start":"12:42.160 ","End":"12:44.305","Text":"Just as a little note,"},{"Start":"12:44.305 ","End":"12:48.580","Text":"when 2 waves come completely opposite,"},{"Start":"12:48.580 ","End":"12:51.805","Text":"so when 1 wave reaches a certain point,"},{"Start":"12:51.805 ","End":"12:57.130","Text":"when the wave is at a peak and another wave reaches that same point."},{"Start":"12:57.130 ","End":"12:59.770","Text":"We can say, let\u0027s say this line over here,"},{"Start":"12:59.770 ","End":"13:03.775","Text":"when the wave is at a trough or at a minimum,"},{"Start":"13:03.775 ","End":"13:08.720","Text":"so we call these 2 waves out of phase."},{"Start":"13:09.090 ","End":"13:14.110","Text":"We\u0027ll speak a little bit more about waves in and out of phase"},{"Start":"13:14.110 ","End":"13:19.720","Text":"soon but just know if 2 waves reach the same point at the same time,"},{"Start":"13:19.720 ","End":"13:25.360","Text":"where 1 wave is at a peak or the maximum and the other wave is at a minimum"},{"Start":"13:25.360 ","End":"13:27.460","Text":"then we will have"},{"Start":"13:27.460 ","End":"13:32.500","Text":"complete destructive interference because these 2 waves are out of phase."},{"Start":"13:32.500 ","End":"13:40.150","Text":"Now another important point is every place where we have this 0 amplitude,"},{"Start":"13:40.150 ","End":"13:45.415","Text":"over here, here, or in fact every single point along"},{"Start":"13:45.415 ","End":"13:50.455","Text":"1 of these purplish pink lines is a node."},{"Start":"13:50.455 ","End":"13:53.200","Text":"Remember when we spoke about nodes when dealing with"},{"Start":"13:53.200 ","End":"13:58.120","Text":"standing waves they were areas of the wave that were stationary,"},{"Start":"13:58.120 ","End":"14:03.290","Text":"who had 0 amplitude and they always remained at the exact same point."},{"Start":"14:03.930 ","End":"14:09.910","Text":"These points over here and this point in red and every single point on 1 of"},{"Start":"14:09.910 ","End":"14:16.315","Text":"these lines is a node exactly because of that definition."},{"Start":"14:16.315 ","End":"14:20.590","Text":"Now I want us to look at another point over here,"},{"Start":"14:20.590 ","End":"14:29.125","Text":"so now let\u0027s look at a point that\u0027s located over here on our line of 0 amplitude."},{"Start":"14:29.125 ","End":"14:35.480","Text":"Then we can see that now our green wave,"},{"Start":"14:35.480 ","End":"14:39.875","Text":"our wave coming from source Number 1 has now traveled,"},{"Start":"14:39.875 ","End":"14:45.755","Text":"if here it was 7 wavelengths so here it will be 7.5 wavelengths."},{"Start":"14:45.755 ","End":"14:53.120","Text":"Because here we\u0027re located at a trough on the wave front coming from S1 at a minimum"},{"Start":"14:53.120 ","End":"14:55.355","Text":"because we can see that it\u0027s between"},{"Start":"14:55.355 ","End":"15:00.365","Text":"this wave front and this wave front which means that it\u0027s at a minimum."},{"Start":"15:00.365 ","End":"15:03.410","Text":"We\u0027re looking at this red point over here."},{"Start":"15:03.410 ","End":"15:07.940","Text":"Then coming from source Number 2 what we can see is"},{"Start":"15:07.940 ","End":"15:14.390","Text":"that before we were located 7.5 wavelengths but now we\u0027re at a peak again,"},{"Start":"15:14.390 ","End":"15:18.825","Text":"so 7.5 plus 1/2 a wavelength is 8."},{"Start":"15:18.825 ","End":"15:23.225","Text":"The wavelength coming from source 2 has traveled 8 wavelengths."},{"Start":"15:23.225 ","End":"15:26.840","Text":"Again we can see that there\u0027s a difference of 1/2"},{"Start":"15:26.840 ","End":"15:32.915","Text":"a wavelength between the 2 which means that they\u0027re out of phase."},{"Start":"15:32.915 ","End":"15:35.990","Text":"That means that if the blue wave is coming"},{"Start":"15:35.990 ","End":"15:38.930","Text":"over here specifically at a peak we can see that"},{"Start":"15:38.930 ","End":"15:46.475","Text":"the green wave is coming 7.5 which means that it\u0027s coming at a trough."},{"Start":"15:46.475 ","End":"15:51.185","Text":"Now we can see that when we join up these 2 amplitudes"},{"Start":"15:51.185 ","End":"15:56.390","Text":"our point is again going to be at a 0 point."},{"Start":"15:56.390 ","End":"16:01.590","Text":"This point, this other arbitrary point that we chose is also a node."},{"Start":"16:02.390 ","End":"16:11.130","Text":"This blurry gray line over here that we\u0027ve drawn in pink is called a minimum line."},{"Start":"16:11.130 ","End":"16:17.180","Text":"Before we dealt with maximum lines so here we have a minimum line and this is"},{"Start":"16:17.180 ","End":"16:24.800","Text":"called a first-order minimum line."},{"Start":"16:25.890 ","End":"16:31.370","Text":"Another first-order minimum line is on the other side over"},{"Start":"16:31.370 ","End":"16:36.280","Text":"here let\u0027s just draw it in the same color"},{"Start":"16:36.280 ","End":"16:46.158","Text":"so this line over here 2 is a first-order minimum line."},{"Start":"16:46.158 ","End":"16:52.270","Text":"Now sometimes these minimum lines will be called nodal lines."},{"Start":"16:52.270 ","End":"17:00.010","Text":"But generally they\u0027re called the minimum line and these 2 are called the first order."},{"Start":"17:00.010 ","End":"17:05.750","Text":"Now let\u0027s write up the condition for a first order minimum line."},{"Start":"17:06.270 ","End":"17:11.830","Text":"The condition for a first order minimum line is if"},{"Start":"17:11.830 ","End":"17:16.390","Text":"we take this distance S_2 to our point P over here,"},{"Start":"17:16.390 ","End":"17:23.950","Text":"let\u0027s call this point P. S_2 to P minus the distance from source 1 to P,"},{"Start":"17:23.950 ","End":"17:33.265","Text":"so minus S_1P is going to be equal to half a wavelength."},{"Start":"17:33.265 ","End":"17:36.685","Text":"We see that the difference is always half a wavelength."},{"Start":"17:36.685 ","End":"17:41.590","Text":"Then we know that they\u0027re out-of-phase and we\u0027ll get complete destructive interference."},{"Start":"17:41.590 ","End":"17:46.720","Text":"Of course, if we add on these absolute value signs,"},{"Start":"17:46.720 ","End":"17:53.090","Text":"they will give us the onset to the other first order minimum line."},{"Start":"17:54.960 ","End":"17:58.585","Text":"Here we can see that this is for first order,"},{"Start":"17:58.585 ","End":"18:01.645","Text":"but what happens with the minimum lines of second,"},{"Start":"18:01.645 ","End":"18:04.195","Text":"third, fourth, and so on order?"},{"Start":"18:04.195 ","End":"18:11.240","Text":"What I\u0027m going to do is I\u0027m going to rub out everything and let\u0027s find out."},{"Start":"18:11.820 ","End":"18:16.315","Text":"Now let\u0027s try and choose another point."},{"Start":"18:16.315 ","End":"18:22.825","Text":"Let\u0027s choose this point over here."},{"Start":"18:22.825 ","End":"18:29.695","Text":"We can see that it\u0027s located on a peak of a wavefront coming from source S_1."},{"Start":"18:29.695 ","End":"18:34.465","Text":"However, it\u0027s located in the middle of 2 wavefronts."},{"Start":"18:34.465 ","End":"18:38.590","Text":"The 2 peaks of 2 wavefronts coming from S_2."},{"Start":"18:38.590 ","End":"18:42.835","Text":"We can already see that we\u0027re going to have a difference in wavelength."},{"Start":"18:42.835 ","End":"18:45.310","Text":"Here we can see from green,"},{"Start":"18:45.310 ","End":"18:49.003","Text":"the distance traveled to this point is 1, 2, 3,"},{"Start":"18:49.003 ","End":"18:55.240","Text":"4, 5, 6 wavelengths."},{"Start":"18:55.240 ","End":"18:57.769","Text":"Now let\u0027s look from the blue source."},{"Start":"18:57.769 ","End":"19:01.933","Text":"You can see it\u0027s 1,"},{"Start":"19:01.933 ","End":"19:03.294","Text":"2, 3, 4, 5, 6,"},{"Start":"19:03.294 ","End":"19:11.390","Text":"7, and a little bit, so 7.5 wavelengths."},{"Start":"19:11.390 ","End":"19:18.930","Text":"Now we can see that 7.5 minus 6 is equal to 1.5."},{"Start":"19:18.930 ","End":"19:21.889","Text":"Or if we put this in fraction form,"},{"Start":"19:21.889 ","End":"19:23.815","Text":"it\u0027s 3 over 2."},{"Start":"19:23.815 ","End":"19:25.585","Text":"Let\u0027s just write this out,"},{"Start":"19:25.585 ","End":"19:34.045","Text":"7.5 wavelengths minus 6 wavelengths is equal to 1.5."},{"Start":"19:34.045 ","End":"19:37.585","Text":"This is wavelength, 1.5 wavelengths,"},{"Start":"19:37.585 ","End":"19:43.790","Text":"which in fraction form is 3 divided by 2 wavelengths."},{"Start":"19:45.090 ","End":"19:53.530","Text":"Now what we can see is that this point P lies on this line over here."},{"Start":"19:53.530 ","End":"20:04.060","Text":"What we can see is that this is the second order minimum line."},{"Start":"20:04.060 ","End":"20:07.450","Text":"Here\u0027s 1 of the first orders and here\u0027s another second order,"},{"Start":"20:07.450 ","End":"20:11.005","Text":"and here, 1 after it is the second order."},{"Start":"20:11.005 ","End":"20:13.300","Text":"Now over here we have a first order,"},{"Start":"20:13.300 ","End":"20:17.050","Text":"so here will be the second order."},{"Start":"20:17.050 ","End":"20:21.985","Text":"Then let\u0027s just draw that in pink."},{"Start":"20:21.985 ","End":"20:24.145","Text":"Not to get confused."},{"Start":"20:24.145 ","End":"20:29.020","Text":"Then over here, the next line down will be"},{"Start":"20:29.020 ","End":"20:36.040","Text":"our third order minimum and also over here we\u0027ll have our third order minimum,"},{"Start":"20:36.040 ","End":"20:38.810","Text":"and so on and so forth."},{"Start":"20:39.240 ","End":"20:43.160","Text":"Let\u0027s write this down."},{"Start":"20:43.950 ","End":"20:53.890","Text":"Here, this we saw is the second order minimum because it sits on this line."},{"Start":"20:53.890 ","End":"20:56.770","Text":"Then if we had another point,"},{"Start":"20:56.770 ","End":"21:06.580","Text":"we\u0027d get that our S_2P minus S_1P in order to land on a third order minimum."},{"Start":"21:06.580 ","End":"21:13.990","Text":"We\u0027ll get that that is equal to 2.5 Lambda."},{"Start":"21:13.990 ","End":"21:17.755","Text":"Or in other words, 5 divided by 2 Lambda."},{"Start":"21:17.755 ","End":"21:25.385","Text":"This would be for a point sitting on the third order minimum line,"},{"Start":"21:25.385 ","End":"21:27.555","Text":"1 of these 2."},{"Start":"21:27.555 ","End":"21:30.030","Text":"Because we\u0027re dealing with physics,"},{"Start":"21:30.030 ","End":"21:36.780","Text":"we want to find some rule that will work and define any minimum line,"},{"Start":"21:36.780 ","End":"21:38.880","Text":"be it from first order, second order,"},{"Start":"21:38.880 ","End":"21:43.985","Text":"third order, or if there is a case, 10th order."},{"Start":"21:43.985 ","End":"21:46.600","Text":"Let\u0027s write this down."},{"Start":"21:46.600 ","End":"21:49.430","Text":"Let\u0027s scroll a little bit more."},{"Start":"21:50.580 ","End":"21:55.795","Text":"This is the condition for nth order minimum line."},{"Start":"21:55.795 ","End":"22:03.355","Text":"We\u0027ll have that S_2P minus S_1P."},{"Start":"22:03.355 ","End":"22:11.020","Text":"The distance from S_2 to the point minus the distance from S_1 to the point has to"},{"Start":"22:11.020 ","End":"22:19.060","Text":"be equal to n minus 1/2 Lambda."},{"Start":"22:19.060 ","End":"22:22.690","Text":"Of course, if we put these absolute value signs in,"},{"Start":"22:22.690 ","End":"22:26.605","Text":"we\u0027ll find the lines on either side."},{"Start":"22:26.605 ","End":"22:34.820","Text":"We\u0027ll find the 2 second orders or the 2 third order lines."},{"Start":"22:35.070 ","End":"22:39.700","Text":"This will define anything. Let\u0027s just see."},{"Start":"22:39.700 ","End":"22:42.280","Text":"Let\u0027s say if n is 2,"},{"Start":"22:42.280 ","End":"22:45.232","Text":"so n is equal to 2,"},{"Start":"22:45.232 ","End":"22:54.565","Text":"therefore the difference in wavelengths 2 minus 1/2 will be 3 over 2 wavelengths."},{"Start":"22:54.565 ","End":"22:58.570","Text":"Here we see n is the nth order."},{"Start":"22:58.570 ","End":"23:00.490","Text":"Here we saw it\u0027s the second order."},{"Start":"23:00.490 ","End":"23:06.070","Text":"We can see that we got 3 over 2 Lambda as our difference in wavelengths."},{"Start":"23:06.070 ","End":"23:08.005","Text":"Let\u0027s say n is equal to 3."},{"Start":"23:08.005 ","End":"23:10.960","Text":"That means that we\u0027re on the third order line."},{"Start":"23:10.960 ","End":"23:12.940","Text":"The change in wavelength,"},{"Start":"23:12.940 ","End":"23:19.645","Text":"so 3 minus 1/2 is simply going to be 5 divided by 2 Lambda,"},{"Start":"23:19.645 ","End":"23:22.250","Text":"which is exactly this."},{"Start":"23:23.340 ","End":"23:27.340","Text":"You can also copy out this equation over here."},{"Start":"23:27.340 ","End":"23:32.080","Text":"However, this is just for the case of the first order minimum."},{"Start":"23:32.080 ","End":"23:36.775","Text":"But here we can find any order minimum line."},{"Start":"23:36.775 ","End":"23:39.175","Text":"Let\u0027s say if we wanted to find the first order,"},{"Start":"23:39.175 ","End":"23:42.700","Text":"so that would mean that n is equal to 1,"},{"Start":"23:42.700 ","End":"23:45.265","Text":"and then n minus 1/2."},{"Start":"23:45.265 ","End":"23:48.460","Text":"1 minus 1/2 is 1/2 Lambda,"},{"Start":"23:48.460 ","End":"23:51.410","Text":"which is exactly what we got over here."},{"Start":"23:51.810 ","End":"23:58.630","Text":"This is the important equation to remember and to write down in your equation sheets,"},{"Start":"23:58.630 ","End":"24:04.340","Text":"this is correct for any order of minimum line."},{"Start":"24:04.770 ","End":"24:10.135","Text":"Copy this out and remember that these represent the node lines."},{"Start":"24:10.135 ","End":"24:13.510","Text":"These lines are always with an amplitude of"},{"Start":"24:13.510 ","End":"24:17.740","Text":"0 and they\u0027re stationary just like a regular node."},{"Start":"24:17.740 ","End":"24:21.880","Text":"Now let\u0027s just scroll back up to the diagram."},{"Start":"24:21.880 ","End":"24:25.945","Text":"What I want us to just go through are all of the lines."},{"Start":"24:25.945 ","End":"24:29.785","Text":"Here we have our first order minimum lines."},{"Start":"24:29.785 ","End":"24:34.465","Text":"Here we said we had our second order and another second order."},{"Start":"24:34.465 ","End":"24:38.080","Text":"Here\u0027s 2 and here\u0027s another 2 then here we can see"},{"Start":"24:38.080 ","End":"24:41.695","Text":"we have our third order and our third order."},{"Start":"24:41.695 ","End":"24:48.040","Text":"Here we have our fourth order and the second fourth order, the fifth order,"},{"Start":"24:48.040 ","End":"24:50.690","Text":"and the second fifth order,"},{"Start":"24:51.780 ","End":"24:54.830","Text":"6, 7, 7,"},{"Start":"24:55.830 ","End":"24:58.105","Text":"and so on and so forth."},{"Start":"24:58.105 ","End":"25:03.985","Text":"Now, if you were so inclined to find the lines"},{"Start":"25:03.985 ","End":"25:10.735","Text":"of maximum so where we have complete constructive interference."},{"Start":"25:10.735 ","End":"25:13.990","Text":"Between every 2 minimum lines,"},{"Start":"25:13.990 ","End":"25:17.800","Text":"we have a line of maximum or a maximum line."},{"Start":"25:17.800 ","End":"25:21.010","Text":"Here we have our maximum of zeroth order."},{"Start":"25:21.010 ","End":"25:24.610","Text":"Here we have our maximum of first order and another maximum of"},{"Start":"25:24.610 ","End":"25:28.240","Text":"first order maximum, so 1, 1,"},{"Start":"25:28.240 ","End":"25:30.115","Text":"maximum of second order,"},{"Start":"25:30.115 ","End":"25:32.305","Text":"another maximum of second order,"},{"Start":"25:32.305 ","End":"25:34.915","Text":"maximum of third order,"},{"Start":"25:34.915 ","End":"25:38.290","Text":"and another maximum of third order."},{"Start":"25:38.290 ","End":"25:42.790","Text":"Now the drawing is beginning to be a little confusing."},{"Start":"25:42.790 ","End":"25:47.350","Text":"But the important part is to remember that between every 2 minimum lines,"},{"Start":"25:47.350 ","End":"25:50.360","Text":"we have a maximum line."},{"Start":"25:51.780 ","End":"25:55.450","Text":"I suggest as a practice to look at"},{"Start":"25:55.450 ","End":"25:59.470","Text":"this image and try and notice where you can see the maximum lines,"},{"Start":"25:59.470 ","End":"26:01.465","Text":"where you can see the minimum lines,"},{"Start":"26:01.465 ","End":"26:08.410","Text":"and to work out of which order the maximum or minimum line is and how to"},{"Start":"26:08.410 ","End":"26:16.155","Text":"count or how to calculate the distance from point S_1 to your point P,"},{"Start":"26:16.155 ","End":"26:23.005","Text":"and from S_2 to your P. Remember to use this equation over here."},{"Start":"26:23.005 ","End":"26:26.480","Text":"That is the end of our lesson."}],"ID":12490},{"Watched":false,"Name":"Number of Minimum Lines","Duration":"14m 33s","ChapterTopicVideoID":12023,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.815","Text":"Hello. In this lesson,"},{"Start":"00:01.815 ","End":"00:06.840","Text":"we\u0027re going to see how many minimum lines we can get in this type"},{"Start":"00:06.840 ","End":"00:13.005","Text":"of interference pattern from a 2-dimensional wave going through a wave tank."},{"Start":"00:13.005 ","End":"00:16.650","Text":"Now we previously saw this type of question when dealing with"},{"Start":"00:16.650 ","End":"00:20.255","Text":"how many maximum lines we can calculate."},{"Start":"00:20.255 ","End":"00:23.610","Text":"Now we\u0027re saying how many minimum lines there are,"},{"Start":"00:23.610 ","End":"00:30.090","Text":"and I\u0027m reminding you the minimum lines are these blurry gray lines."},{"Start":"00:30.090 ","End":"00:33.780","Text":"What are we going to be looking at is this point over here."},{"Start":"00:33.780 ","End":"00:38.165","Text":"Let\u0027s call it P. Here\u0027s our S_1."},{"Start":"00:38.165 ","End":"00:43.280","Text":"Here we generate circular 2D waves through the rave tank."},{"Start":"00:43.280 ","End":"00:44.870","Text":"Here is S_2,"},{"Start":"00:44.870 ","End":"00:48.070","Text":"the second wave generator."},{"Start":"00:48.070 ","End":"00:55.605","Text":"This distance over here from S_1 to point P,"},{"Start":"00:55.605 ","End":"01:00.300","Text":"we already called S_1P in the previous video."},{"Start":"01:00.300 ","End":"01:06.994","Text":"The distance from point P to S_2,"},{"Start":"01:06.994 ","End":"01:11.475","Text":"this line over here,"},{"Start":"01:11.475 ","End":"01:18.413","Text":"we know is called S_2P."},{"Start":"01:18.413 ","End":"01:21.611","Text":"We called this distance over here,"},{"Start":"01:21.611 ","End":"01:26.795","Text":"so the distance between the 2 wave sources or the 2 wave generators"},{"Start":"01:26.795 ","End":"01:33.120","Text":"is d. What we\u0027re going to do,"},{"Start":"01:33.120 ","End":"01:36.290","Text":"and what we\u0027re going to use in order to answer"},{"Start":"01:36.290 ","End":"01:40.505","Text":"this is our triangle rule which we looked in the video,"},{"Start":"01:40.505 ","End":"01:43.325","Text":"dealing with how many maximum lines there are."},{"Start":"01:43.325 ","End":"01:47.960","Text":"The triangle rule states the sum of"},{"Start":"01:47.960 ","End":"01:53.980","Text":"2 sides of a triangle is always greater than the third side length."},{"Start":"01:53.980 ","End":"01:56.475","Text":"Let\u0027s see what that means."},{"Start":"01:56.475 ","End":"01:58.594","Text":"Here we have our point S_1,"},{"Start":"01:58.594 ","End":"02:00.595","Text":"here we have our point S_2,"},{"Start":"02:00.595 ","End":"02:09.540","Text":"and somewhere over here we have our point P. This is what our triangle looks like,"},{"Start":"02:09.540 ","End":"02:11.115","Text":"so here is S_1,"},{"Start":"02:11.115 ","End":"02:14.330","Text":"here is S_2, and here\u0027s P,"},{"Start":"02:14.330 ","End":"02:18.980","Text":"where this length is d. This is S_1P,"},{"Start":"02:18.980 ","End":"02:23.510","Text":"and this is S_2P."},{"Start":"02:23.510 ","End":"02:32.700","Text":"If so, then let\u0027s add up"},{"Start":"02:32.700 ","End":"02:37.920","Text":"our side S_1P plus"},{"Start":"02:37.920 ","End":"02:44.615","Text":"our side d. Then we know this is always greater than the third side\u0027s length,"},{"Start":"02:44.615 ","End":"02:47.390","Text":"so this is always greater than our third side\u0027s length,"},{"Start":"02:47.390 ","End":"02:50.610","Text":"which here is S_2P."},{"Start":"02:51.770 ","End":"02:58.285","Text":"Now what do we want to do is we want to isolate out our"},{"Start":"02:58.285 ","End":"03:06.285","Text":"d. We\u0027re going to subtract d from both sides and subtract S_2P from both sides."},{"Start":"03:06.285 ","End":"03:13.080","Text":"We\u0027re going to have S_1P minus S_2P and this is greater"},{"Start":"03:13.080 ","End":"03:20.450","Text":"than negative d. Now we want this to be a positive number,"},{"Start":"03:20.450 ","End":"03:25.715","Text":"sorry, so we\u0027re going to multiply both sides by negative 1."},{"Start":"03:25.715 ","End":"03:30.675","Text":"Therefore, we\u0027re going to get that S_2P minus"},{"Start":"03:30.675 ","End":"03:38.965","Text":"S_1P is always smaller than d. If you multiply by negative 1,"},{"Start":"03:38.965 ","End":"03:42.935","Text":"you have to switch around the side of the inequality."},{"Start":"03:42.935 ","End":"03:46.535","Text":"If here, the left side is greater than the right side,"},{"Start":"03:46.535 ","End":"03:48.680","Text":"when you multiply by minus 1,"},{"Start":"03:48.680 ","End":"03:53.320","Text":"it becomes the left side is smaller than the right side."},{"Start":"03:53.320 ","End":"03:57.650","Text":"Now, we can put over here,"},{"Start":"03:57.650 ","End":"04:01.745","Text":"we know that d is always going to be a positive number and here we can put"},{"Start":"04:01.745 ","End":"04:10.995","Text":"our absolute value signs to show that if this P was married to this side of the image,"},{"Start":"04:10.995 ","End":"04:15.390","Text":"the distances were exactly the same, just switched."},{"Start":"04:16.060 ","End":"04:20.325","Text":"This inequality would still be correct."},{"Start":"04:20.325 ","End":"04:23.320","Text":"Now, we also know,"},{"Start":"04:23.320 ","End":"04:25.170","Text":"let\u0027s do a star here,"},{"Start":"04:25.170 ","End":"04:30.298","Text":"that S_2P minus S_1P,"},{"Start":"04:30.298 ","End":"04:32.785","Text":"we saw this in the previous lesson,"},{"Start":"04:32.785 ","End":"04:38.480","Text":"is equal to the distance between them."},{"Start":"04:38.480 ","End":"04:41.410","Text":"When we\u0027re dealing with a minimum line,"},{"Start":"04:41.410 ","End":"04:46.085","Text":"so a point that lies on 1 of these blurry gray lines so easy"},{"Start":"04:46.085 ","End":"04:51.365","Text":"equal to n minus 1/2 multiplied by Lambda."},{"Start":"04:51.365 ","End":"04:54.810","Text":"N minus 1/2 wavelength."},{"Start":"04:55.240 ","End":"05:00.765","Text":"In that case, we can plug in this n minus"},{"Start":"05:00.765 ","End":"05:06.010","Text":"1/2 Lambda into this section to this side of the inequality."},{"Start":"05:06.010 ","End":"05:14.350","Text":"Therefore, we\u0027ll get that n minus 1/2 multiplied by Lambda is"},{"Start":"05:14.350 ","End":"05:23.380","Text":"smaller than d. Now let\u0027s scroll down a little bit more in order to have some more space."},{"Start":"05:23.380 ","End":"05:30.550","Text":"Now what we can do is we can isolate this n because the n is the total number of"},{"Start":"05:30.550 ","End":"05:37.960","Text":"minimum lines that we can get in a given image given our wave generators."},{"Start":"05:37.960 ","End":"05:41.270","Text":"We want to isolate this out because that\u0027s the answer."},{"Start":"05:41.270 ","End":"05:47.990","Text":"We\u0027re going to divide both sides by Lambda and add a half to both sides,"},{"Start":"05:47.990 ","End":"05:53.780","Text":"and then what we\u0027ll get is that n has to be smaller than or equal"},{"Start":"05:53.780 ","End":"06:01.420","Text":"to d divided by Lambda plus 1/2."},{"Start":"06:01.420 ","End":"06:07.770","Text":"This is the answer to how many minimum lines there can be."},{"Start":"06:08.270 ","End":"06:15.230","Text":"This has the greatest number of minimum lines and just like in the previous question,"},{"Start":"06:15.230 ","End":"06:19.040","Text":"2 lessons ago, we saw that 1,"},{"Start":"06:19.040 ","End":"06:23.460","Text":"n has to be a whole number."},{"Start":"06:24.430 ","End":"06:29.795","Text":"Also that means that if n has to be whole number,"},{"Start":"06:29.795 ","End":"06:35.190","Text":"if we get the d divided by Lambda plus a half isn\u0027t a whole number,"},{"Start":"06:35.190 ","End":"06:45.183","Text":"let\u0027s say if it\u0027s equal to 5.1 or 5.9 or 10.8 and so on and so forth,"},{"Start":"06:45.183 ","End":"06:49.820","Text":"we always have to round down because n can never exceed that."},{"Start":"06:49.820 ","End":"06:55.535","Text":"If we let say have that n is smaller or equal to 5.9,"},{"Start":"06:55.535 ","End":"07:00.680","Text":"then our answer will be that n=5."},{"Start":"07:00.680 ","End":"07:04.235","Text":"We can never say that it is 6,"},{"Start":"07:04.235 ","End":"07:06.965","Text":"because 6 is greater than 5.9,"},{"Start":"07:06.965 ","End":"07:14.323","Text":"and this equation here is the upper limit."},{"Start":"07:14.323 ","End":"07:17.410","Text":"Now, something to note over here."},{"Start":"07:17.410 ","End":"07:22.955","Text":"This is the greatest number of minimum lines in half of the image."},{"Start":"07:22.955 ","End":"07:27.430","Text":"We can do 1/2 the greatest number of minimum lines."},{"Start":"07:27.430 ","End":"07:30.115","Text":"Why is this? Let\u0027s draw this in blue."},{"Start":"07:30.115 ","End":"07:34.150","Text":"We already saw that in the previous lesson"},{"Start":"07:34.150 ","End":"07:39.790","Text":"that this is a minimum line of first-order and similarly,"},{"Start":"07:39.790 ","End":"07:44.710","Text":"this is a minimum line of first-order so 1 and 1,"},{"Start":"07:44.710 ","End":"07:49.570","Text":"and this is the minimum line of second-order and that this over here,"},{"Start":"07:49.570 ","End":"07:51.130","Text":"the mirror image over here,"},{"Start":"07:51.130 ","End":"07:55.810","Text":"is also minimum line of second-order just on the other side."},{"Start":"07:55.810 ","End":"07:59.260","Text":"If let\u0027s say we get that n=3,"},{"Start":"07:59.260 ","End":"08:05.065","Text":"then we\u0027ll have 1 over here, 2 and 3."},{"Start":"08:05.065 ","End":"08:07.253","Text":"But then if we have 1, 2,"},{"Start":"08:07.253 ","End":"08:09.280","Text":"3 on the left-hand side,"},{"Start":"08:09.280 ","End":"08:11.307","Text":"then we must also have 1, 2,"},{"Start":"08:11.307 ","End":"08:15.580","Text":"3 on the right-hand side."},{"Start":"08:15.580 ","End":"08:21.355","Text":"That\u0027s why this equation will give us 1/2 the greatest number of minimum lines."},{"Start":"08:21.355 ","End":"08:24.865","Text":"In order if they\u0027re asking us in a whole image,"},{"Start":"08:24.865 ","End":"08:27.100","Text":"how many minimum lines are there,"},{"Start":"08:27.100 ","End":"08:29.050","Text":"we multiply this by 2."},{"Start":"08:29.050 ","End":"08:30.625","Text":"If our answer is 3,"},{"Start":"08:30.625 ","End":"08:33.700","Text":"we\u0027ll have 1,"},{"Start":"08:33.700 ","End":"08:35.140","Text":"2, 3, 4, 5, 6."},{"Start":"08:35.140 ","End":"08:40.405","Text":"If our answer was that n is 4 so"},{"Start":"08:40.405 ","End":"08:47.750","Text":"4 times 2 is 8 and then we can see that we have 8 minimum lines."},{"Start":"08:47.790 ","End":"08:54.430","Text":"Let\u0027s just illustrate this within little numbered example."},{"Start":"08:54.430 ","End":"09:03.355","Text":"Let\u0027s say that the distance d between the 2 wave generators is equal to 5 centimeters."},{"Start":"09:03.355 ","End":"09:05.590","Text":"Let\u0027s say that the wavelength,"},{"Start":"09:05.590 ","End":"09:08.230","Text":"so the distance from peak to peak, let\u0027s say,"},{"Start":"09:08.230 ","End":"09:12.835","Text":"of the waves coming out of S_1 and coming out of S_2"},{"Start":"09:12.835 ","End":"09:18.610","Text":"so Lambda is equal to 1.5 centimeters."},{"Start":"09:18.610 ","End":"09:28.555","Text":"Then we can say that n is smaller or equal to d divided by Lambda minus a 1/2,"},{"Start":"09:28.555 ","End":"09:32.560","Text":"where d is 5 divided by Lambda,"},{"Start":"09:32.560 ","End":"09:39.010","Text":"which is 1.5 minus 1/2 so minus 0.5."},{"Start":"09:39.010 ","End":"09:41.530","Text":"If we work this out,"},{"Start":"09:41.530 ","End":"09:46.250","Text":"we\u0027ll get that this is equal to 3.83."},{"Start":"09:47.460 ","End":"09:52.195","Text":"N is smaller than 3.83."},{"Start":"09:52.195 ","End":"09:54.220","Text":"As we said over here,"},{"Start":"09:54.220 ","End":"09:56.710","Text":"n is always a whole number."},{"Start":"09:56.710 ","End":"10:01.270","Text":"It doesn\u0027t matter that 3.83 is closer to 4."},{"Start":"10:01.270 ","End":"10:08.005","Text":"Well, usually you\u0027re maybe used to rounding up if we\u0027re located at 0.5 or above."},{"Start":"10:08.005 ","End":"10:12.670","Text":"But here, because this is the upper bound, the upper limit,"},{"Start":"10:12.670 ","End":"10:23.470","Text":"we know therefore that n has to be equal to 3 because we round down always."},{"Start":"10:23.470 ","End":"10:31.975","Text":"Therefore, total number of minimum lines in the image."},{"Start":"10:31.975 ","End":"10:35.770","Text":"As we said, we have 3 to the left,"},{"Start":"10:35.770 ","End":"10:37.930","Text":"1, 2, 3."},{"Start":"10:37.930 ","End":"10:40.780","Text":"Then we also are going to have the mirror image,"},{"Start":"10:40.780 ","End":"10:42.010","Text":"so 3 to the right,"},{"Start":"10:42.010 ","End":"10:43.930","Text":"1, 2, 3."},{"Start":"10:43.930 ","End":"10:49.525","Text":"Therefore we have 3 plus 3 or 3 times 2,"},{"Start":"10:49.525 ","End":"10:52.490","Text":"which is equal to 6."},{"Start":"10:52.770 ","End":"10:57.440","Text":"This will be the total number of minimum lines in the image."},{"Start":"10:57.440 ","End":"11:04.810","Text":"I\u0027m sorry, of course here this is d divided by Lambda plus 1/2."},{"Start":"11:04.810 ","End":"11:09.625","Text":"This is 5 divided by 1.5 plus 1/2."},{"Start":"11:09.625 ","End":"11:15.340","Text":"Now, let\u0027s just look at another example."},{"Start":"11:15.340 ","End":"11:21.655","Text":"Let\u0027s say now that our d=5.5 centimeters,"},{"Start":"11:21.655 ","End":"11:26.590","Text":"the distance between our 2 wave generators and that the wavelength of the waves that"},{"Start":"11:26.590 ","End":"11:32.365","Text":"each generator is generating is still 1.5 centimeters like before."},{"Start":"11:32.365 ","End":"11:37.780","Text":"Then we know that n is smaller or equal to d divided by Lambda,"},{"Start":"11:37.780 ","End":"11:45.760","Text":"so that\u0027s 5.5 divided by 1.5 plus a 1/2."},{"Start":"11:45.760 ","End":"11:49.615","Text":"Now, if we plug this into a calculator,"},{"Start":"11:49.615 ","End":"11:54.350","Text":"we\u0027ll get that this is equal to 4.17."},{"Start":"11:54.510 ","End":"11:58.135","Text":"N is smaller or equal to 4.17."},{"Start":"11:58.135 ","End":"12:01.270","Text":"Again, n has to be a whole number."},{"Start":"12:01.270 ","End":"12:04.600","Text":"Therefore n is equal to,"},{"Start":"12:04.600 ","End":"12:06.145","Text":"so what did we say?"},{"Start":"12:06.145 ","End":"12:08.725","Text":"We always round down."},{"Start":"12:08.725 ","End":"12:12.380","Text":"Therefore n=4."},{"Start":"12:16.770 ","End":"12:24.715","Text":"Let\u0027s say total number of men lines in the image that we\u0027ll see."},{"Start":"12:24.715 ","End":"12:30.385","Text":"It\u0027s 2n or you could also say that it\u0027s n plus n,"},{"Start":"12:30.385 ","End":"12:35.290","Text":"which is equal to 4 plus 4,"},{"Start":"12:35.290 ","End":"12:38.125","Text":"which is equal to 8."},{"Start":"12:38.125 ","End":"12:40.450","Text":"We remember if n=4,"},{"Start":"12:40.450 ","End":"12:43.420","Text":"that means we have first-order, second-order,"},{"Start":"12:43.420 ","End":"12:46.840","Text":"third-order, fourth-order on the left-hand side"},{"Start":"12:46.840 ","End":"12:50.440","Text":"and we have the exact same thing always on the right hand side."},{"Start":"12:50.440 ","End":"12:54.565","Text":"First-order, second-order, third-order, fourth-order,"},{"Start":"12:54.565 ","End":"12:56.861","Text":"1, 2, 3, 4, 5,"},{"Start":"12:56.861 ","End":"13:00.260","Text":"6, 7, 8."},{"Start":"13:00.270 ","End":"13:05.630","Text":"We have 8 minimum lines in the image."},{"Start":"13:06.450 ","End":"13:09.985","Text":"It\u0027s important to remember this example,"},{"Start":"13:09.985 ","End":"13:13.120","Text":"to remember that n over here represents"},{"Start":"13:13.120 ","End":"13:17.905","Text":"1/2 of the amount of minimum lines that will find."},{"Start":"13:17.905 ","End":"13:21.009","Text":"Similarly, when we\u0027re trying to find maximum lines,"},{"Start":"13:21.009 ","End":"13:30.295","Text":"remember that n for the maximum lines represented 1/2, but minus 1."},{"Start":"13:30.295 ","End":"13:34.225","Text":"In order to find the total and the image of maximum lines."},{"Start":"13:34.225 ","End":"13:40.120","Text":"I\u0027m just reminding you the equation for maximum line so write it like this,"},{"Start":"13:40.120 ","End":"13:44.545","Text":"is given as just d divided by Lambda."},{"Start":"13:44.545 ","End":"13:47.815","Text":"The total maximum lines,"},{"Start":"13:47.815 ","End":"13:57.115","Text":"so total max lines was equal to 2n plus 1 for maximum lines,"},{"Start":"13:57.115 ","End":"14:03.595","Text":"whereas the total minimum lines is equal to 2n."},{"Start":"14:03.595 ","End":"14:09.580","Text":"Remember these differences I\u0027ll put these 2 together because they represent"},{"Start":"14:09.580 ","End":"14:12.895","Text":"maximum lines and this represents the minimum lines"},{"Start":"14:12.895 ","End":"14:17.960","Text":"in both examples and always has to be a whole number."},{"Start":"14:18.690 ","End":"14:20.755","Text":"In the next lesson,"},{"Start":"14:20.755 ","End":"14:22.930","Text":"we\u0027ll go through some examples."},{"Start":"14:22.930 ","End":"14:31.300","Text":"Please do write these equations and also this one in your equation sheets."},{"Start":"14:31.300 ","End":"14:34.010","Text":"That\u0027s the end of the lesson."}],"ID":12491},{"Watched":false,"Name":"Exercise - Finding Wavelength","Duration":"13m 52s","ChapterTopicVideoID":12024,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.025","Text":"Hello. In this question,"},{"Start":"00:02.025 ","End":"00:05.190","Text":"we\u0027re being asked to find the wavelength of"},{"Start":"00:05.190 ","End":"00:10.176","Text":"the wave given the following interference pattern."},{"Start":"00:10.176 ","End":"00:14.430","Text":"We\u0027re being told that we have a point P that\u0027s located here."},{"Start":"00:14.430 ","End":"00:20.115","Text":"Now we can see that P is located on a bright patch."},{"Start":"00:20.115 ","End":"00:27.750","Text":"We know that if we have either a very bright patch or a very dark patch,"},{"Start":"00:27.750 ","End":"00:33.015","Text":"that means that we\u0027re dealing with constructive interference."},{"Start":"00:33.015 ","End":"00:35.430","Text":"If we have the blurry gray,"},{"Start":"00:35.430 ","End":"00:37.935","Text":"we\u0027re dealing with destructive interference."},{"Start":"00:37.935 ","End":"00:47.050","Text":"Now when we have constructive interference, what does this mean?"},{"Start":"00:47.050 ","End":"00:48.325","Text":"We\u0027ll write it over here."},{"Start":"00:48.325 ","End":"00:58.093","Text":"Constructive interference means that we\u0027re dealing with maximum lines."},{"Start":"00:58.093 ","End":"01:02.980","Text":"Remember, that means not the minimum lines."},{"Start":"01:02.980 ","End":"01:07.210","Text":"The minimum lines are the blurry gray lines that we can see running through the images,"},{"Start":"01:07.210 ","End":"01:08.950","Text":"but rather the maximum lines,"},{"Start":"01:08.950 ","End":"01:12.430","Text":"the area in-between, the blurry gray lines which"},{"Start":"01:12.430 ","End":"01:17.004","Text":"represent these extremely dark or extremely light points."},{"Start":"01:17.004 ","End":"01:18.745","Text":"This light, very dark,"},{"Start":"01:18.745 ","End":"01:20.960","Text":"very light, very dark points."},{"Start":"01:20.960 ","End":"01:25.109","Text":"This is where we have the constructive interference."},{"Start":"01:25.109 ","End":"01:30.455","Text":"Point P is sitting on one of these bright points."},{"Start":"01:30.455 ","End":"01:33.320","Text":"All over here we can see it."},{"Start":"01:33.320 ","End":"01:38.165","Text":"We\u0027re being told that the distance from S_1 to P,"},{"Start":"01:38.165 ","End":"01:46.970","Text":"so that\u0027s this distance over here is 7 centimeters and that the distance from S_2 to P,"},{"Start":"01:46.970 ","End":"01:54.520","Text":"so that\u0027s this distance over here is 13 centimeters."},{"Start":"01:54.520 ","End":"02:03.030","Text":"We know that the equation for a maximum lines is given as the absolute value of S_2P"},{"Start":"02:03.030 ","End":"02:12.951","Text":"minus S_1P is equal to n Lambda."},{"Start":"02:12.951 ","End":"02:19.160","Text":"We can work this out because S_2P and S_1P are both positive numbers,"},{"Start":"02:19.160 ","End":"02:22.840","Text":"and S_1P is smaller than S_2P."},{"Start":"02:22.840 ","End":"02:26.735","Text":"That means that we can just work it out without the absolute value signs."},{"Start":"02:26.735 ","End":"02:33.875","Text":"Because we can see that the answer inside the absolute value signs is a positive."},{"Start":"02:33.875 ","End":"02:40.880","Text":"That means that S_2P is 13 minus S_1P,"},{"Start":"02:40.880 ","End":"02:42.215","Text":"which is 7,"},{"Start":"02:42.215 ","End":"02:47.595","Text":"is equal to 6 centimeters."},{"Start":"02:47.595 ","End":"02:52.355","Text":"We also know that this is equal to,"},{"Start":"02:52.355 ","End":"02:54.245","Text":"from this equation over here,"},{"Start":"02:54.245 ","End":"02:59.140","Text":"n multiplied by Lambda, the wavelength."},{"Start":"02:59.140 ","End":"03:06.230","Text":"Now we can see that we have n and we have Lambda,"},{"Start":"03:06.230 ","End":"03:08.360","Text":"where Lambda is what we\u0027re trying to find out."},{"Start":"03:08.360 ","End":"03:10.130","Text":"Let\u0027s see what n is."},{"Start":"03:10.130 ","End":"03:11.945","Text":"Let\u0027s make it in blue."},{"Start":"03:11.945 ","End":"03:17.405","Text":"What we can see here is we have the middle line over here,"},{"Start":"03:17.405 ","End":"03:19.003","Text":"right in the center,"},{"Start":"03:19.003 ","End":"03:24.270","Text":"and this is our maximum line of 0th order."},{"Start":"03:24.270 ","End":"03:31.340","Text":"Then over here we have after that the maximum line of first order."},{"Start":"03:31.340 ","End":"03:33.245","Text":"Then after that,"},{"Start":"03:33.245 ","End":"03:37.946","Text":"we have our maximum line of second order."},{"Start":"03:37.946 ","End":"03:40.430","Text":"Then right over here,"},{"Start":"03:40.430 ","End":"03:43.835","Text":"which is exactly where our point P is,"},{"Start":"03:43.835 ","End":"03:47.150","Text":"we have maximum line of third order."},{"Start":"03:47.150 ","End":"03:50.540","Text":"Therefore, that means"},{"Start":"03:50.540 ","End":"03:58.745","Text":"that n=3 because our point P lies on the maximum line of third order."},{"Start":"03:58.745 ","End":"04:06.510","Text":"Therefore, we have that 6=3 times Lambda."},{"Start":"04:06.510 ","End":"04:09.155","Text":"Therefore, if we isolate out Lambda,"},{"Start":"04:09.155 ","End":"04:15.760","Text":"Lambda is equal to 6 divided by 3, or alternatively 2."},{"Start":"04:15.760 ","End":"04:20.550","Text":"We\u0027re specifically dealing with centimeters."},{"Start":"04:21.420 ","End":"04:31.040","Text":"That means that the wavelength is equal to 2 centimeters."},{"Start":"04:31.040 ","End":"04:36.110","Text":"Now, another type of question that they can ask us is when P is,"},{"Start":"04:36.110 ","End":"04:39.365","Text":"let\u0027s say, located on a minimum line."},{"Start":"04:39.365 ","End":"04:42.800","Text":"Along one of these blurry gray lines,"},{"Start":"04:42.800 ","End":"04:46.609","Text":"and that we\u0027re not given these values."},{"Start":"04:46.609 ","End":"04:48.650","Text":"All we\u0027re given is the image."},{"Start":"04:48.650 ","End":"04:52.055","Text":"Let\u0027s erase all of this."},{"Start":"04:52.055 ","End":"04:58.065","Text":"We\u0027re given this time, this image,"},{"Start":"04:58.065 ","End":"05:08.060","Text":"and then let\u0027s say that we\u0027re dealing with this point over here along the gray line."},{"Start":"05:08.060 ","End":"05:10.940","Text":"Let\u0027s say right over here."},{"Start":"05:10.940 ","End":"05:17.990","Text":"This is our point P. Then what we\u0027ll do,"},{"Start":"05:17.990 ","End":"05:22.940","Text":"so we\u0027ve just chosen this random point P on our image and we\u0027ve"},{"Start":"05:22.940 ","End":"05:28.235","Text":"labeled this wave generator S_1 and this wave generator S_2."},{"Start":"05:28.235 ","End":"05:30.650","Text":"On a plain image, you could have also chosen"},{"Start":"05:30.650 ","End":"05:33.865","Text":"this point over here as P. It doesn\u0027t really matter,"},{"Start":"05:33.865 ","End":"05:39.320","Text":"or any other point as long as it\u0027s along the minimum lines."},{"Start":"05:39.320 ","End":"05:42.545","Text":"Because now we\u0027re going to deal with working"},{"Start":"05:42.545 ","End":"05:47.080","Text":"out the wavelength when we\u0027re using our equation for the minimum line."},{"Start":"05:47.080 ","End":"05:50.840","Text":"Then what we would do is we would measure with a ruler."},{"Start":"05:50.840 ","End":"05:56.180","Text":"This line over here joining S_1 to P. Let\u0027s say we measure it"},{"Start":"05:56.180 ","End":"06:01.624","Text":"with a ruler and it comes out as 9.2 centimeters."},{"Start":"06:01.624 ","End":"06:08.120","Text":"Then we measure this line between point P and point S_2,"},{"Start":"06:08.120 ","End":"06:09.500","Text":"and it comes out as,"},{"Start":"06:09.500 ","End":"06:13.185","Text":"let\u0027s say, 12.8 centimeters."},{"Start":"06:13.185 ","End":"06:21.705","Text":"Because we see that our point P is on a blurry gray line."},{"Start":"06:21.705 ","End":"06:26.140","Text":"So P is on gray line."},{"Start":"06:26.230 ","End":"06:31.830","Text":"That means that it\u0027s on a minimum line,"},{"Start":"06:32.360 ","End":"06:44.250","Text":"and that is because there has been destructive interference."},{"Start":"06:46.300 ","End":"06:57.155","Text":"If so, we\u0027ve seen in previous videos that the absolute value of S_2P"},{"Start":"06:57.155 ","End":"07:01.250","Text":"minus S_1P is equal"},{"Start":"07:01.250 ","End":"07:08.745","Text":"to n minus a 1/2 multiplied by Lambda."},{"Start":"07:08.745 ","End":"07:16.500","Text":"This is what we\u0027ve seen when our point P is on one of these gray lines."},{"Start":"07:16.900 ","End":"07:26.990","Text":"We can see that S_2P minus S_1P is going to be a positive number,"},{"Start":"07:26.990 ","End":"07:31.205","Text":"which means that we can just work out this equation without the absolute value."},{"Start":"07:31.205 ","End":"07:41.750","Text":"The absolute value is only if we subtracted S_1P from S_2P and we got a negative number."},{"Start":"07:41.750 ","End":"07:45.045","Text":"Let\u0027s do this,"},{"Start":"07:45.045 ","End":"07:55.925","Text":"S_2P minus S_1P is equal to 12.8 minus 9.2,"},{"Start":"07:55.925 ","End":"08:01.235","Text":"which is equal to 3.6."},{"Start":"08:01.235 ","End":"08:03.170","Text":"Now as we know,"},{"Start":"08:03.170 ","End":"08:09.135","Text":"this is equal to the right side of this equation."},{"Start":"08:09.135 ","End":"08:16.980","Text":"So this is equal to n minus 1/2 multiplied by Lambda."},{"Start":"08:16.980 ","End":"08:19.860","Text":"What we\u0027re trying to find is this Lambda."},{"Start":"08:19.860 ","End":"08:23.090","Text":"Let\u0027s figure out from our image what our n is."},{"Start":"08:23.090 ","End":"08:25.160","Text":"Let\u0027s draw this in blue."},{"Start":"08:25.160 ","End":"08:29.105","Text":"We saw that when we\u0027re dealing with minimum lines,"},{"Start":"08:29.105 ","End":"08:33.780","Text":"this is first order minimum,"},{"Start":"08:33.780 ","End":"08:38.315","Text":"and this is also,"},{"Start":"08:38.315 ","End":"08:40.310","Text":"by the way, first order minimum,"},{"Start":"08:40.310 ","End":"08:41.705","Text":"but from the other side."},{"Start":"08:41.705 ","End":"08:45.470","Text":"This is second order minimum,"},{"Start":"08:45.470 ","End":"08:49.805","Text":"and here we have third order minimum."},{"Start":"08:49.805 ","End":"08:56.405","Text":"Point P lies exactly on this gray line of third order minimum."},{"Start":"08:56.405 ","End":"09:00.740","Text":"That means that n over here is equal to 3."},{"Start":"09:00.740 ","End":"09:06.259","Text":"Therefore, we have that 3.6 is equal to 3"},{"Start":"09:06.259 ","End":"09:12.641","Text":"minus 1/2 multiplied by Lambda."},{"Start":"09:12.641 ","End":"09:16.465","Text":"Then what we therefore have is"},{"Start":"09:16.465 ","End":"09:24.035","Text":"3.6 is equal to 2.5 multiplied by Lambda."},{"Start":"09:24.035 ","End":"09:29.645","Text":"Now what we\u0027re going to do is we\u0027re going to isolate out our Lambda,"},{"Start":"09:29.645 ","End":"09:31.460","Text":"because this is what we\u0027re trying to find."},{"Start":"09:31.460 ","End":"09:37.790","Text":"Lambda is equal to 3.6 divided by 2.5,"},{"Start":"09:37.790 ","End":"09:40.445","Text":"which if we plug this into a calculator,"},{"Start":"09:40.445 ","End":"09:47.770","Text":"is equal to 1.44 centimeters."},{"Start":"09:47.900 ","End":"09:54.020","Text":"This is the answer for what is the wavelength"},{"Start":"09:54.020 ","End":"10:03.455","Text":"just given the image and finding or choosing some arbitrary point P,"},{"Start":"10:03.455 ","End":"10:07.835","Text":"noting whether it lies on a minimum line or on a maximum line."},{"Start":"10:07.835 ","End":"10:15.410","Text":"Then just taking a ruler and measuring these distances between P and S_1 and S_2,"},{"Start":"10:15.410 ","End":"10:17.600","Text":"and knowing that because it\u0027s on a minimum line,"},{"Start":"10:17.600 ","End":"10:22.800","Text":"we use this version of our equation and that is it."},{"Start":"10:23.120 ","End":"10:29.285","Text":"In conclusion, in order to answer these types of questions,"},{"Start":"10:29.285 ","End":"10:33.220","Text":"the first thing you want to do is you want to see if"},{"Start":"10:33.220 ","End":"10:39.720","Text":"the point is located on a minimum line or a maximum line."},{"Start":"10:40.490 ","End":"10:46.025","Text":"If we know if the point is on the minimum or maximum line,"},{"Start":"10:46.025 ","End":"10:48.260","Text":"if it\u0027s a minimum line,"},{"Start":"10:48.260 ","End":"10:50.060","Text":"then it\u0027s destructive interference,"},{"Start":"10:50.060 ","End":"10:51.583","Text":"or if it\u0027s on the maximum,"},{"Start":"10:51.583 ","End":"10:56.050","Text":"constructive interference, then we\u0027ll know which equation to use."},{"Start":"10:56.050 ","End":"11:04.250","Text":"If this absolute value sign is equal to n multiplied by Lambda,"},{"Start":"11:04.250 ","End":"11:08.173","Text":"or if it\u0027s equal to n minus a 1/2 multiplied by Lambda,"},{"Start":"11:08.173 ","End":"11:12.440","Text":"and then point number 2 is to"},{"Start":"11:12.440 ","End":"11:18.725","Text":"find the value for n. This is very, very easy."},{"Start":"11:18.725 ","End":"11:25.048","Text":"You just see on which order of the minimum or maximum line the point is lying on,"},{"Start":"11:25.048 ","End":"11:28.640","Text":"and that is equal to n. Here we saw with the minimum,"},{"Start":"11:28.640 ","End":"11:31.565","Text":"you have to find the first order,"},{"Start":"11:31.565 ","End":"11:33.770","Text":"and then the next gray line is second,"},{"Start":"11:33.770 ","End":"11:37.520","Text":"next is third, and here specifically point P was lying on the third."},{"Start":"11:37.520 ","End":"11:39.590","Text":"But if it was here, this would be the fourth,"},{"Start":"11:39.590 ","End":"11:40.660","Text":"fifth, and so on,"},{"Start":"11:40.660 ","End":"11:46.025","Text":"and the same goes for if the point lies on the other side of the image."},{"Start":"11:46.025 ","End":"11:50.435","Text":"Of course, remember if you\u0027re dealing with a maximum line,"},{"Start":"11:50.435 ","End":"11:53.525","Text":"that there\u0027s the 0th order as well."},{"Start":"11:53.525 ","End":"11:55.870","Text":"On a maximum line,"},{"Start":"11:55.870 ","End":"11:59.260","Text":"this would be the 0th order."},{"Start":"11:59.260 ","End":"12:05.530","Text":"That is all you have to do in order to solve these types of questions."},{"Start":"12:05.530 ","End":"12:08.610","Text":"By the way, a little note,"},{"Start":"12:08.610 ","End":"12:14.270","Text":"working out the wavelength via these equations is much,"},{"Start":"12:14.270 ","End":"12:23.075","Text":"much more reliable and accurate than trying to measure out this distance over here,"},{"Start":"12:23.075 ","End":"12:26.270","Text":"the distance between each bright patch."},{"Start":"12:26.270 ","End":"12:29.885","Text":"Because as we can see, the bright patch is very wide,"},{"Start":"12:29.885 ","End":"12:32.675","Text":"so does that mean that we\u0027re measuring from here,"},{"Start":"12:32.675 ","End":"12:35.520","Text":"or from here, or from the middle?"},{"Start":"12:35.520 ","End":"12:37.215","Text":"It\u0027s very inaccurate."},{"Start":"12:37.215 ","End":"12:41.455","Text":"Using this equation is much more accurate and reliable."},{"Start":"12:41.455 ","End":"12:47.810","Text":"An important note that some students sometimes do is they think that they can work out"},{"Start":"12:47.810 ","End":"12:51.200","Text":"the wavelength via taking this maximum point and"},{"Start":"12:51.200 ","End":"12:55.250","Text":"this maximum point and finding the distance between the two."},{"Start":"12:55.250 ","End":"12:57.125","Text":"This is not correct,"},{"Start":"12:57.125 ","End":"13:04.610","Text":"because this point over here refers to a maximum on this wavefront,"},{"Start":"13:04.610 ","End":"13:08.465","Text":"and although this point over here refers to a maximum on the next wavefront,"},{"Start":"13:08.465 ","End":"13:11.240","Text":"we can see that it\u0027s shifted up one."},{"Start":"13:11.240 ","End":"13:19.805","Text":"We can see that we\u0027re measuring this diamond shape rather than this perpendicular."},{"Start":"13:19.805 ","End":"13:27.900","Text":"If anything, we have to make sure that we\u0027re measuring perpendicular to each wavefront,"},{"Start":"13:27.900 ","End":"13:31.440","Text":"and not coming in at a 45-degree angle."},{"Start":"13:31.440 ","End":"13:36.185","Text":"Definitely, don\u0027t measure from point to point like this."},{"Start":"13:36.185 ","End":"13:41.029","Text":"If anything, measure from wavefront to wavefront,"},{"Start":"13:41.029 ","End":"13:44.389","Text":"keeping these lines perpendicular."},{"Start":"13:44.389 ","End":"13:48.065","Text":"However, use this equation."},{"Start":"13:48.065 ","End":"13:50.108","Text":"It\u0027s much better."},{"Start":"13:50.108 ","End":"13:53.070","Text":"That\u0027s the end of this lesson."}],"ID":12492},{"Watched":false,"Name":"Generating an Interference Pattern Diffraction","Duration":"14m ","ChapterTopicVideoID":12025,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.695","Text":"Hello. In this lesson,"},{"Start":"00:01.695 ","End":"00:04.695","Text":"we\u0027re going to be speaking about how to physically or"},{"Start":"00:04.695 ","End":"00:08.220","Text":"actually generate an interference pattern."},{"Start":"00:08.220 ","End":"00:11.115","Text":"Later on, in this course,"},{"Start":"00:11.115 ","End":"00:13.755","Text":"this is going to come in,"},{"Start":"00:13.755 ","End":"00:15.975","Text":"also useful to us."},{"Start":"00:15.975 ","End":"00:18.570","Text":"Here\u0027s the interference pattern that we\u0027ve been"},{"Start":"00:18.570 ","End":"00:20.835","Text":"looking at for the past couple of lessons."},{"Start":"00:20.835 ","End":"00:24.525","Text":"Now let\u0027s speak about how it\u0027s generated."},{"Start":"00:24.525 ","End":"00:29.955","Text":"Let\u0027s look at this image and imagine that we\u0027re looking at it from the side."},{"Start":"00:29.955 ","End":"00:34.665","Text":"Here we have our water level,"},{"Start":"00:34.665 ","End":"00:38.140","Text":"and then here,"},{"Start":"00:39.140 ","End":"00:42.330","Text":"let\u0027s draw it in green,"},{"Start":"00:42.330 ","End":"00:45.135","Text":"we have this over here,"},{"Start":"00:45.135 ","End":"00:47.270","Text":"a pin over here,"},{"Start":"00:47.270 ","End":"00:50.825","Text":"which goes and dips into the water,"},{"Start":"00:50.825 ","End":"00:54.155","Text":"generating the circular wavefront."},{"Start":"00:54.155 ","End":"00:57.980","Text":"Over here in red,"},{"Start":"00:57.980 ","End":"01:01.045","Text":"we have our second source."},{"Start":"01:01.045 ","End":"01:04.940","Text":"This is also some pin that dips into the water over"},{"Start":"01:04.940 ","End":"01:09.755","Text":"here and causes these circular wavefronts."},{"Start":"01:09.755 ","End":"01:13.320","Text":"In actual fact, we know that"},{"Start":"01:13.320 ","End":"01:19.190","Text":"these 2 pins are dipping usually at the exact same frequency."},{"Start":"01:19.190 ","End":"01:22.610","Text":"That means that they dip at the exact same time."},{"Start":"01:22.610 ","End":"01:24.635","Text":"How do we do that?"},{"Start":"01:24.635 ","End":"01:29.200","Text":"These 2 pins are actually joined together."},{"Start":"01:29.200 ","End":"01:35.775","Text":"We have some machine here which is attached to the computer."},{"Start":"01:35.775 ","End":"01:41.555","Text":"Then the computer tells it what frequency to dip into the water."},{"Start":"01:41.555 ","End":"01:46.910","Text":"Then, every time that this block moves down,"},{"Start":"01:46.910 ","End":"01:52.500","Text":"both pins dip into the water at the exact same time."},{"Start":"01:52.940 ","End":"01:57.950","Text":"Then this green pin is starting over here and it steps over here and it"},{"Start":"01:57.950 ","End":"02:02.450","Text":"forms this wavefront and then this one and then the next one."},{"Start":"02:02.450 ","End":"02:08.270","Text":"The red pin dips in over here and then it forms this wavefront,"},{"Start":"02:08.270 ","End":"02:10.655","Text":"then this one, and then the next one,"},{"Start":"02:10.655 ","End":"02:12.860","Text":"and so on and so forth."},{"Start":"02:12.860 ","End":"02:15.155","Text":"Then after just a few seconds,"},{"Start":"02:15.155 ","End":"02:20.250","Text":"we\u0027ll get this interference pattern in the wave tank."},{"Start":"02:20.710 ","End":"02:24.755","Text":"Then if we draw this over here in bird\u0027s eye view,"},{"Start":"02:24.755 ","End":"02:33.735","Text":"here we have the 2 pins joined to 1 another via this sheet."},{"Start":"02:33.735 ","End":"02:41.945","Text":"This is the first option of how to generate waves, this interference pattern."},{"Start":"02:41.945 ","End":"02:45.630","Text":"Now let\u0027s look at the second way."},{"Start":"02:46.010 ","End":"02:52.430","Text":"Now we\u0027re speaking about the second way to generate this interference pattern."},{"Start":"02:52.430 ","End":"02:55.400","Text":"This second way is the,"},{"Start":"02:55.400 ","End":"02:59.185","Text":"shall we say smarter way."},{"Start":"02:59.185 ","End":"03:03.200","Text":"What we\u0027re going to speak about now is going to come in very,"},{"Start":"03:03.200 ","End":"03:06.110","Text":"very useful later in the chapter."},{"Start":"03:06.110 ","End":"03:14.310","Text":"The second way of generating this interference pattern is by way of diffraction."},{"Start":"03:19.580 ","End":"03:27.445","Text":"Diffraction speaks about the behavior of a wave when it reaches some obstacle,"},{"Start":"03:27.445 ","End":"03:30.085","Text":"such as slate hole."},{"Start":"03:30.085 ","End":"03:32.964","Text":"Another word for hole is aperture."},{"Start":"03:32.964 ","End":"03:40.670","Text":"Diffraction describes what the wave will do when it reaches this obstacle."},{"Start":"03:40.670 ","End":"03:46.840","Text":"In diffraction, the wave will bend around this obstacle or hole that it"},{"Start":"03:46.840 ","End":"03:53.155","Text":"has reached so that the wave reaches an area that it shouldn\u0027t reach."},{"Start":"03:53.155 ","End":"03:54.970","Text":"You can liken this to,"},{"Start":"03:54.970 ","End":"03:57.365","Text":"for instance, a shadow."},{"Start":"03:57.365 ","End":"03:59.280","Text":"What do I mean by that?"},{"Start":"03:59.280 ","End":"04:05.065","Text":"Let\u0027s speak about it in a little bit more detail right now."},{"Start":"04:05.065 ","End":"04:13.650","Text":"Let\u0027s imagine that we have a wave tank over here, like so."},{"Start":"04:14.390 ","End":"04:22.250","Text":"Then this wave tank, we are generating straight wavelengths."},{"Start":"04:22.250 ","End":"04:28.355","Text":"Here is our wave generator and"},{"Start":"04:28.355 ","End":"04:35.210","Text":"here are straight wavefronts traveling like so."},{"Start":"04:35.210 ","End":"04:38.555","Text":"Now in the wave tank,"},{"Start":"04:38.555 ","End":"04:41.165","Text":"we play some kind of barrier,"},{"Start":"04:41.165 ","End":"04:48.615","Text":"which is just some slate or some aperture in it."},{"Start":"04:48.615 ","End":"04:55.150","Text":"What we would expect is that only this section of"},{"Start":"04:55.150 ","End":"04:58.360","Text":"the wave that can fit through this aperture will"},{"Start":"04:58.360 ","End":"05:02.635","Text":"come through and will carry on traveling like so."},{"Start":"05:02.635 ","End":"05:05.975","Text":"Obviously, the wavelength doesn\u0027t change."},{"Start":"05:05.975 ","End":"05:10.280","Text":"It will look something like this."},{"Start":"05:10.280 ","End":"05:12.370","Text":"This is what we would expect."},{"Start":"05:12.370 ","End":"05:13.690","Text":"But in actual fact,"},{"Start":"05:13.690 ","End":"05:17.170","Text":"what we\u0027ll show now is diffraction occurs,"},{"Start":"05:17.170 ","End":"05:20.350","Text":"which means that the wave will bend around"},{"Start":"05:20.350 ","End":"05:24.175","Text":"the obstacle so that the wave reaches an area that it shouldn\u0027t reach."},{"Start":"05:24.175 ","End":"05:27.870","Text":"We\u0027ve already said that this area,"},{"Start":"05:27.870 ","End":"05:31.145","Text":"we wouldn\u0027t expect the wave to reach,"},{"Start":"05:31.145 ","End":"05:33.020","Text":"this area in gray."},{"Start":"05:33.020 ","End":"05:35.450","Text":"Because here we have a barrier."},{"Start":"05:35.450 ","End":"05:38.045","Text":"What we\u0027ll see is that due to diffraction,"},{"Start":"05:38.045 ","End":"05:43.290","Text":"the wave actually will reach this area, this gray area."},{"Start":"05:44.810 ","End":"05:50.850","Text":"What we\u0027ll see, and this is obviously a sketch,"},{"Start":"05:50.850 ","End":"05:55.610","Text":"is we\u0027ll see that a small part of wave will travel through here,"},{"Start":"05:55.610 ","End":"06:00.290","Text":"and then the wave will bend around such that"},{"Start":"06:00.290 ","End":"06:05.265","Text":"it will carry on traveling with this circular wavefront."},{"Start":"06:05.265 ","End":"06:11.440","Text":"What we\u0027ll create is circular wavefront, like so."},{"Start":"06:12.560 ","End":"06:18.770","Text":"Here, we have a picture from a wave tank of this exactly happening."},{"Start":"06:18.770 ","End":"06:21.395","Text":"Here, let\u0027s liken this to this picture."},{"Start":"06:21.395 ","End":"06:27.480","Text":"Here we can see our straight wavefronts traveling."},{"Start":"06:27.480 ","End":"06:34.530","Text":"Here we can see we have our barrier and"},{"Start":"06:34.530 ","End":"06:42.140","Text":"here we can see that now circular wavefronts are being formed."},{"Start":"06:42.140 ","End":"06:43.940","Text":"Here\u0027s the first wavefront,"},{"Start":"06:43.940 ","End":"06:45.660","Text":"the second wavefront,"},{"Start":"06:45.660 ","End":"06:49.890","Text":"the third, and so on."},{"Start":"06:49.890 ","End":"06:55.495","Text":"Now here\u0027s an image of this happening in real life,"},{"Start":"06:55.495 ","End":"06:57.930","Text":"not in a lab."},{"Start":"06:57.930 ","End":"07:02.810","Text":"This is a picture of a river or something like that,"},{"Start":"07:02.810 ","End":"07:05.930","Text":"and in here we can see the side of the bridge."},{"Start":"07:05.930 ","End":"07:09.590","Text":"We can see that the bridge is acting like a barrier."},{"Start":"07:09.590 ","End":"07:13.725","Text":"Here we have part of the barrier and going down,"},{"Start":"07:13.725 ","End":"07:15.480","Text":"and carrying on to this side,"},{"Start":"07:15.480 ","End":"07:17.970","Text":"we have the other side of the barrier."},{"Start":"07:17.970 ","End":"07:23.040","Text":"Then we know that we\u0027re going to have some water waves traveling towards"},{"Start":"07:23.040 ","End":"07:28.545","Text":"this and then the water waves travel through this barrier,"},{"Start":"07:28.545 ","End":"07:35.170","Text":"and then they form these circular waves that we can see over here that\u0027s spread out."},{"Start":"07:35.630 ","End":"07:40.740","Text":"We can see it over here like so."},{"Start":"07:40.740 ","End":"07:48.075","Text":"Can you see we have these curving waves traveling outwards like so?"},{"Start":"07:48.075 ","End":"07:50.685","Text":"Here you can see this as in real life."},{"Start":"07:50.685 ","End":"07:54.255","Text":"Here, this is an image from the lab."},{"Start":"07:54.255 ","End":"07:59.670","Text":"What do we can see is that our wavefronts aren\u0027t just traveling here,"},{"Start":"07:59.670 ","End":"08:06.755","Text":"just small sections of straight wave front traveling like so."},{"Start":"08:06.755 ","End":"08:10.670","Text":"But rather in this gray region that we"},{"Start":"08:10.670 ","End":"08:16.170","Text":"previously described as the shadow or the geometrical shadow."},{"Start":"08:16.750 ","End":"08:19.910","Text":"If let\u0027s say we shine light over here,"},{"Start":"08:19.910 ","End":"08:23.600","Text":"so we would expect a shadow in this region over here,"},{"Start":"08:23.600 ","End":"08:28.470","Text":"and only over here for our waves to pass,"},{"Start":"08:28.470 ","End":"08:31.890","Text":"but all of this region over here and here would be a shadow."},{"Start":"08:31.890 ","End":"08:34.605","Text":"Here we can see that we have diffraction."},{"Start":"08:34.605 ","End":"08:41.550","Text":"The wave bends around the obstacle or hole so that the wave reaches areas that it"},{"Start":"08:41.550 ","End":"08:51.940","Text":"shouldn\u0027t reach into this geometrical shadow area."},{"Start":"08:54.830 ","End":"08:58.140","Text":"Now we\u0027ve spoken about diffraction,"},{"Start":"08:58.140 ","End":"08:59.470","Text":"what it looks like,"},{"Start":"08:59.470 ","End":"09:02.024","Text":"so let\u0027s go back to our original question,"},{"Start":"09:02.024 ","End":"09:06.885","Text":"which was about generating an interference pattern."},{"Start":"09:06.885 ","End":"09:11.190","Text":"How can I use this idea of diffraction to generate an interference pattern?"},{"Start":"09:11.190 ","End":"09:16.990","Text":"I\u0027m just going to rub this out to give us a little bit more space."},{"Start":"09:17.240 ","End":"09:19.380","Text":"How can I do that?"},{"Start":"09:19.380 ","End":"09:25.545","Text":"Let\u0027s imagine that this wave tank continues down until over"},{"Start":"09:25.545 ","End":"09:32.115","Text":"here and then I have my wave generator over here,"},{"Start":"09:32.115 ","End":"09:33.690","Text":"just like I had over here,"},{"Start":"09:33.690 ","End":"09:39.255","Text":"which generates these straight wavefronts."},{"Start":"09:39.255 ","End":"09:44.170","Text":"I have my wavefronts traveling like so,"},{"Start":"09:44.210 ","End":"09:47.400","Text":"and they\u0027re traveling,"},{"Start":"09:47.400 ","End":"09:54.720","Text":"and traveling, and of course,"},{"Start":"09:54.720 ","End":"09:57.120","Text":"they\u0027re all the same length."},{"Start":"09:57.120 ","End":"10:01.410","Text":"Then, let\u0027s say that I have my barrier."},{"Start":"10:01.410 ","End":"10:07.440","Text":"That means that I can have a barrier over here and a barrier over here."},{"Start":"10:07.440 ","End":"10:10.155","Text":"I have a barrier with 2 holes,"},{"Start":"10:10.155 ","End":"10:16.425","Text":"so then I\u0027ll get what we saw in this image or this image twice."},{"Start":"10:16.425 ","End":"10:22.890","Text":"Then I can call this hole S1 and this hole S2."},{"Start":"10:22.890 ","End":"10:27.150","Text":"Just like we saw over here and then the diagrams before,"},{"Start":"10:27.150 ","End":"10:30.510","Text":"then we\u0027ll get in each hole, diffraction,"},{"Start":"10:30.510 ","End":"10:36.480","Text":"and we\u0027ll get these circular wavefronts traveling through like"},{"Start":"10:36.480 ","End":"10:44.430","Text":"so from S1 and the exact same ones traveling through S2."},{"Start":"10:44.430 ","End":"10:52.230","Text":"What\u0027s important to note is that the wavelength of my straight wavefront over here"},{"Start":"10:52.230 ","End":"10:56.400","Text":"Lambda dictates what my wavelength is going to be after"},{"Start":"10:56.400 ","End":"11:01.485","Text":"the barrier and it\u0027s the exact same wavelength."},{"Start":"11:01.485 ","End":"11:03.885","Text":"This over here is Lambda."},{"Start":"11:03.885 ","End":"11:06.420","Text":"Let\u0027s draw it in red so that it\u0027s clear."},{"Start":"11:06.420 ","End":"11:09.420","Text":"This over here is going to be Lambda."},{"Start":"11:09.420 ","End":"11:17.220","Text":"This also over here is going to be Lambda and it\u0027s not meant to change."},{"Start":"11:17.220 ","End":"11:21.480","Text":"It\u0027s the exact same wavelength."},{"Start":"11:21.480 ","End":"11:31.920","Text":"The next important point to note is that when the wave arrives at the barrier over here,"},{"Start":"11:31.920 ","End":"11:36.570","Text":"so we know that the circular wave fronts"},{"Start":"11:36.570 ","End":"11:41.080","Text":"that will be formed after the barrier will be in phase,"},{"Start":"11:41.080 ","End":"11:44.260","Text":"so we can write in phase."},{"Start":"11:44.390 ","End":"11:46.560","Text":"What is in phase mean?"},{"Start":"11:46.560 ","End":"11:50.910","Text":"It means that here we know that we\u0027re going to"},{"Start":"11:50.910 ","End":"11:57.225","Text":"get a peak over here when this wavefront arrives and because it\u0027s a straight wavefront,"},{"Start":"11:57.225 ","End":"12:01.485","Text":"then this whole blue line is a peak."},{"Start":"12:01.485 ","End":"12:05.270","Text":"The peaks are going to reach the barrier at the same time."},{"Start":"12:05.270 ","End":"12:09.915","Text":"That means if a peak reaches the barrier over here at the same time,"},{"Start":"12:09.915 ","End":"12:15.660","Text":"a peak will reach this hole in the barrier at S2 over here."},{"Start":"12:15.660 ","End":"12:19.980","Text":"Then afterwards where we have a trough,"},{"Start":"12:19.980 ","End":"12:23.535","Text":"so let\u0027s show a trough,"},{"Start":"12:23.535 ","End":"12:27.645","Text":"the negative point of the wave with a dotted line."},{"Start":"12:27.645 ","End":"12:32.325","Text":"When the trough reaches the barrier because it\u0027s a straight wavefront,"},{"Start":"12:32.325 ","End":"12:38.400","Text":"so the trough is going to reach the holes in the barrier at the exact same time as well."},{"Start":"12:38.400 ","End":"12:44.110","Text":"We\u0027ll get a trough over in S1 and the trough over in S2."},{"Start":"12:44.720 ","End":"12:53.580","Text":"Therefore we can see that the 2 circular wavefronts that will be formed from each hole or"},{"Start":"12:53.580 ","End":"12:57.720","Text":"aperture in the barrier will be in phase and we already"},{"Start":"12:57.720 ","End":"13:03.000","Text":"spoke about having the same wavelength."},{"Start":"13:03.000 ","End":"13:07.050","Text":"These are the 2 things that are"},{"Start":"13:07.050 ","End":"13:11.325","Text":"imperative to have when generating an interference pattern."},{"Start":"13:11.325 ","End":"13:15.780","Text":"The waves always have to be in phase with one another."},{"Start":"13:15.780 ","End":"13:19.260","Text":"At least that these starting points where the 2 sources"},{"Start":"13:19.260 ","End":"13:23.680","Text":"are and they have to have the same wavelength."},{"Start":"13:24.140 ","End":"13:30.030","Text":"Diffraction is the second way of generating an interference pattern"},{"Start":"13:30.030 ","End":"13:35.970","Text":"in a wave tank and diffraction isn\u0027t just used with water waves,"},{"Start":"13:35.970 ","End":"13:39.690","Text":"it\u0027s also used with light waves and diffraction is"},{"Start":"13:39.690 ","End":"13:44.940","Text":"a very important phenomenon in physics."},{"Start":"13:44.940 ","End":"13:50.745","Text":"Soon we\u0027ll see how the discovery or the understanding of diffraction"},{"Start":"13:50.745 ","End":"13:57.915","Text":"led to a lot of understanding about waves in physics."},{"Start":"13:57.915 ","End":"14:01.030","Text":"That\u0027s the end of this lesson."}],"ID":12493},{"Watched":false,"Name":"Exercise - Diffraction 1","Duration":"20m 58s","ChapterTopicVideoID":12026,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.800","Text":"Hello. In this lesson,"},{"Start":"00:01.800 ","End":"00:04.965","Text":"we\u0027ve been given an exercise where we have"},{"Start":"00:04.965 ","End":"00:13.155","Text":"2 wave generators that are forming circular waves with the same wavelength and figures."},{"Start":"00:13.155 ","End":"00:18.480","Text":"The dotted lines represent troughs in the wave,"},{"Start":"00:18.480 ","End":"00:22.650","Text":"and the solid lines represent peaks."},{"Start":"00:22.650 ","End":"00:26.740","Text":"Find the maximum and minimum lines?"},{"Start":"00:27.980 ","End":"00:34.080","Text":"Let\u0027s start with the maximum lines."},{"Start":"00:34.080 ","End":"00:44.520","Text":"We know that maximum lines are formed by constructive interference."},{"Start":"00:44.520 ","End":"00:46.875","Text":"What does that mean?"},{"Start":"00:46.875 ","End":"00:51.210","Text":"If we have a blue wave,"},{"Start":"00:51.210 ","End":"00:57.499","Text":"so if our blue wave is at a peak and at the exact same point,"},{"Start":"00:57.499 ","End":"00:59.300","Text":"and at the exact same time,"},{"Start":"00:59.300 ","End":"01:01.430","Text":"our green wave is also at a peak."},{"Start":"01:01.430 ","End":"01:05.405","Text":"We know that we\u0027re going to superimpose these 2 waves."},{"Start":"01:05.405 ","End":"01:07.745","Text":"That means add-on their amplitude."},{"Start":"01:07.745 ","End":"01:12.485","Text":"Then we\u0027re going to get this much larger peak."},{"Start":"01:12.485 ","End":"01:18.240","Text":"We have constructive or complete constructive specifically over here."},{"Start":"01:19.270 ","End":"01:22.280","Text":"Interference."},{"Start":"01:22.280 ","End":"01:26.165","Text":"We also know that we\u0027ve seen from previous lessons"},{"Start":"01:26.165 ","End":"01:30.005","Text":"that if the blue wave is at a particular point,"},{"Start":"01:30.005 ","End":"01:32.480","Text":"at a trough, and similarly,"},{"Start":"01:32.480 ","End":"01:36.815","Text":"the green wave arrives at the same point at the same time also at a trough."},{"Start":"01:36.815 ","End":"01:40.355","Text":"Again, we\u0027re going to superimpose these 2 waves"},{"Start":"01:40.355 ","End":"01:44.540","Text":"and add up their amplitudes and get a larger peak."},{"Start":"01:44.540 ","End":"01:51.330","Text":"This, too, is an example of complete constructive interference."},{"Start":"01:51.370 ","End":"01:57.395","Text":"We know that at every single point along the max lines or the maximum lines,"},{"Start":"01:57.395 ","End":"02:01.370","Text":"we\u0027re going to have this complete constructive interference"},{"Start":"02:01.370 ","End":"02:05.500","Text":"and that each of these points are"},{"Start":"02:05.500 ","End":"02:08.540","Text":"moving between being a complete constructive interference in"},{"Start":"02:08.540 ","End":"02:14.630","Text":"the positive direction and complete constructive interference in the negative direction."},{"Start":"02:14.630 ","End":"02:17.725","Text":"That\u0027s because at 1 moment,"},{"Start":"02:17.725 ","End":"02:20.990","Text":"waves will be both at a peak."},{"Start":"02:20.990 ","End":"02:23.390","Text":"But then if we wait half a wavelength,"},{"Start":"02:23.390 ","End":"02:27.515","Text":"the waves will be at a trough and then another half a wavelength,"},{"Start":"02:27.515 ","End":"02:30.425","Text":"and we\u0027ll be back at this peak."},{"Start":"02:30.425 ","End":"02:33.785","Text":"But what we know is that at every single point,"},{"Start":"02:33.785 ","End":"02:36.560","Text":"we have complete constructive interference."},{"Start":"02:36.560 ","End":"02:41.305","Text":"Just the question is if it\u0027s to the positive or negative direction?"},{"Start":"02:41.305 ","End":"02:42.735","Text":"But that\u0027s what\u0027s going on,"},{"Start":"02:42.735 ","End":"02:45.100","Text":"on the max lines."},{"Start":"02:45.650 ","End":"02:53.565","Text":"Let\u0027s find the maximum lines now that we\u0027ve remembered what a maximum line represents."},{"Start":"02:53.565 ","End":"02:56.325","Text":"Let\u0027s call the pink wavefronts."},{"Start":"02:56.325 ","End":"02:59.615","Text":"That\u0027s this source over here, S1."},{"Start":"02:59.615 ","End":"03:06.185","Text":"Then let\u0027s call the source that creates the green wavefronts."},{"Start":"03:06.185 ","End":"03:09.265","Text":"Let\u0027s call it S_2."},{"Start":"03:09.265 ","End":"03:13.170","Text":"Now in red, let\u0027s try and locate these points."},{"Start":"03:13.170 ","End":"03:17.270","Text":"First let\u0027s look at where both of our waves,"},{"Start":"03:17.270 ","End":"03:19.700","Text":"both pink and green,"},{"Start":"03:19.700 ","End":"03:22.520","Text":"are at their maximum points."},{"Start":"03:22.520 ","End":"03:25.640","Text":"First of all, we can see that right here in the middle,"},{"Start":"03:25.640 ","End":"03:27.810","Text":"hit the midpoint between the 2."},{"Start":"03:27.810 ","End":"03:31.385","Text":"We have the maximum of the pink, over here,"},{"Start":"03:31.385 ","End":"03:36.870","Text":"meets the maximum of the green over here."},{"Start":"03:36.870 ","End":"03:42.530","Text":"As we said, the solid lines are the peaks or the maximums of the wavefront,"},{"Start":"03:42.530 ","End":"03:46.505","Text":"and the dotted lines are the minimum points or the troughs."},{"Start":"03:46.505 ","End":"03:49.070","Text":"Then we can go up again, and here again,"},{"Start":"03:49.070 ","End":"03:52.250","Text":"we have a peak and a peak meeting."},{"Start":"03:52.250 ","End":"03:55.640","Text":"This is also along the maximum line and so is this peak,"},{"Start":"03:55.640 ","End":"03:58.265","Text":"and peak meeting, and so on."},{"Start":"03:58.265 ","End":"04:01.580","Text":"Similarly, on the other side, peak and peak,"},{"Start":"04:01.580 ","End":"04:03.905","Text":"peak and peak, peak and peak,"},{"Start":"04:03.905 ","End":"04:06.295","Text":"peak and peak, and peak and peak."},{"Start":"04:06.295 ","End":"04:08.015","Text":"Then we also said that"},{"Start":"04:08.015 ","End":"04:15.610","Text":"all this completely constructive interference is happening when 2 minimums meet."},{"Start":"04:15.680 ","End":"04:19.185","Text":"The dotted lines represents the troughs."},{"Start":"04:19.185 ","End":"04:23.945","Text":"Here we have a pink dotted line meeting a green dotted line at this point over here."},{"Start":"04:23.945 ","End":"04:26.795","Text":"This is also complete constructive interference."},{"Start":"04:26.795 ","End":"04:30.515","Text":"Similarly, here and here,"},{"Start":"04:30.515 ","End":"04:32.390","Text":"and here, and here,"},{"Start":"04:32.390 ","End":"04:35.030","Text":"and also in the mirror."},{"Start":"04:35.030 ","End":"04:38.285","Text":"All the way down here as well."},{"Start":"04:38.285 ","End":"04:39.635","Text":"Now, as we know,"},{"Start":"04:39.635 ","End":"04:43.790","Text":"all of these points on the maximum line do move because,"},{"Start":"04:43.790 ","End":"04:50.845","Text":"as we saw, will have both waves coming together with exact same phase and wavelength."},{"Start":"04:50.845 ","End":"04:53.615","Text":"At every single point on this line,"},{"Start":"04:53.615 ","End":"04:57.665","Text":"we\u0027re going to have some form of constructive interference."},{"Start":"04:57.665 ","End":"05:02.300","Text":"If here we have constructive interference and the positive direction here,"},{"Start":"05:02.300 ","End":"05:07.165","Text":"both waves will go down into the negative zone"},{"Start":"05:07.165 ","End":"05:13.380","Text":"of their amplitude and then they\u0027ll both reach the minimum points,"},{"Start":"05:13.380 ","End":"05:17.345","Text":"or their traps, at the same time and at the same point."},{"Start":"05:17.345 ","End":"05:23.570","Text":"That means that we can see that anywhere along this dotted line."},{"Start":"05:23.570 ","End":"05:31.670","Text":"These points is part of the maximum lines so that means here if we waste a few moments,"},{"Start":"05:31.670 ","End":"05:37.085","Text":"will have either a peak and the peak or trough to trough and here,"},{"Start":"05:37.085 ","End":"05:39.980","Text":"and here, and here and here,"},{"Start":"05:39.980 ","End":"05:41.810","Text":"and here and so on."},{"Start":"05:41.810 ","End":"05:49.295","Text":"What we can see is that this whole line is a maximum line and specifically,"},{"Start":"05:49.295 ","End":"05:55.530","Text":"this is a maximum line of zeroth-order."},{"Start":"05:56.450 ","End":"06:01.130","Text":"We\u0027ve already spoken about the maximum line and"},{"Start":"06:01.130 ","End":"06:07.915","Text":"the zeroth-order maximum line and what defines it mathematically, but visually."},{"Start":"06:07.915 ","End":"06:12.950","Text":"Just looking at this image, we can see that it\u0027s the zeroth-order because it goes right"},{"Start":"06:12.950 ","End":"06:18.450","Text":"between where it\u0027s equidistant to both wave sources."},{"Start":"06:19.160 ","End":"06:23.310","Text":"What is the next maximum line?"},{"Start":"06:23.310 ","End":"06:25.440","Text":"Let\u0027s look over here."},{"Start":"06:25.440 ","End":"06:29.070","Text":"Again, we\u0027re looking for peak and peak."},{"Start":"06:29.070 ","End":"06:30.750","Text":"Here it\u0027s pink and green,"},{"Start":"06:30.750 ","End":"06:34.335","Text":"peak and peak, peak and peak, peak."},{"Start":"06:34.335 ","End":"06:40.380","Text":"Solid lines together and then we also go on this side."},{"Start":"06:40.380 ","End":"06:43.515","Text":"Here we have solid pink and green together."},{"Start":"06:43.515 ","End":"06:48.917","Text":"Peaks and of course,"},{"Start":"06:48.917 ","End":"06:53.780","Text":"we know that also v\u0027s where we have"},{"Start":"06:53.780 ","End":"06:59.480","Text":"trough to trough also represents this complete constructive interference."},{"Start":"06:59.480 ","End":"07:03.455","Text":"That\u0027s pink dotted line to green dotted line here."},{"Start":"07:03.455 ","End":"07:08.865","Text":"Trough to trough to trough over here and on this side,"},{"Start":"07:08.865 ","End":"07:11.670","Text":"trough to trough, trough to trough."},{"Start":"07:11.670 ","End":"07:14.210","Text":"All of these points,"},{"Start":"07:14.210 ","End":"07:18.565","Text":"and then we can connect all of these dots."},{"Start":"07:18.565 ","End":"07:22.820","Text":"What we can see is that this 2 is a maximum line,"},{"Start":"07:22.820 ","End":"07:27.170","Text":"specifically a maximum line of first-order."},{"Start":"07:27.170 ","End":"07:32.045","Text":"What\u0027s just important to note that even though it\u0027s called a maximum line,"},{"Start":"07:32.045 ","End":"07:35.335","Text":"it doesn\u0027t necessarily have to be straight."},{"Start":"07:35.335 ","End":"07:38.975","Text":"We can see that there\u0027s a little bit of a bend away here"},{"Start":"07:38.975 ","End":"07:44.900","Text":"and if there\u0027s a maximum line of order number 1 to the right,"},{"Start":"07:44.900 ","End":"07:48.070","Text":"then there\u0027s also going to be 1 to the left."},{"Start":"07:48.070 ","End":"07:50.900","Text":"Let\u0027s just draw this out quickly."},{"Start":"07:50.900 ","End":"07:52.880","Text":"Here we have trough to trough,"},{"Start":"07:52.880 ","End":"07:54.320","Text":"peak to peak, trough to trough,"},{"Start":"07:54.320 ","End":"07:58.949","Text":"peak to peak, just going like so."},{"Start":"07:58.949 ","End":"08:01.535","Text":"Then it bends a little bit up to here,"},{"Start":"08:01.535 ","End":"08:02.945","Text":"trough to trough, peak to peak,"},{"Start":"08:02.945 ","End":"08:05.735","Text":"trough to trough, peak to peak."},{"Start":"08:05.735 ","End":"08:10.230","Text":"Now I\u0027m just following this line, straight like so."},{"Start":"08:10.230 ","End":"08:14.210","Text":"This is also a maximum of first-order."},{"Start":"08:14.210 ","End":"08:18.375","Text":"Now let\u0027s look at the next 1 along."},{"Start":"08:18.375 ","End":"08:21.005","Text":"Here we can see we have a trough to trough."},{"Start":"08:21.005 ","End":"08:22.550","Text":"This is a point and peak to peak,"},{"Start":"08:22.550 ","End":"08:25.100","Text":"and trough to trough, and peak to peak, trough,"},{"Start":"08:25.100 ","End":"08:26.585","Text":"peak, trough, peak,"},{"Start":"08:26.585 ","End":"08:30.435","Text":"trough, peak, trough, peak."},{"Start":"08:30.435 ","End":"08:33.120","Text":"Then we see also here,"},{"Start":"08:33.120 ","End":"08:37.535","Text":"we have this here,"},{"Start":"08:37.535 ","End":"08:41.135","Text":"peak to peak, peak to peak, trough,"},{"Start":"08:41.135 ","End":"08:45.830","Text":"peak, trough, peak, trough, peak."},{"Start":"08:46.840 ","End":"08:50.675","Text":"Just like so, putting in all the points."},{"Start":"08:50.675 ","End":"08:53.420","Text":"Now, we can again draw this."},{"Start":"08:53.420 ","End":"08:54.875","Text":"Now we can see again,"},{"Start":"08:54.875 ","End":"08:56.540","Text":"there\u0027s a bit of a curve in here,"},{"Start":"08:56.540 ","End":"09:00.335","Text":"but it\u0027s still a maximum line even though it\u0027s not straight."},{"Start":"09:00.335 ","End":"09:06.610","Text":"This we can see it\u0027s going to be a maximum line of second-order."},{"Start":"09:07.000 ","End":"09:11.225","Text":"Of course, if here we have a maximum line of second-order."},{"Start":"09:11.225 ","End":"09:15.245","Text":"Similarly here, we will have a maximum allowance for second order."},{"Start":"09:15.245 ","End":"09:17.675","Text":"Here we have peak to peak, trough to trough."},{"Start":"09:17.675 ","End":"09:21.980","Text":"We\u0027re just finding the next line down where we"},{"Start":"09:21.980 ","End":"09:27.745","Text":"have solid lines with solid lines and dotted lines with dotted lines."},{"Start":"09:27.745 ","End":"09:32.440","Text":"Then we just follow those out."},{"Start":"09:32.840 ","End":"09:36.375","Text":"I\u0027m just marking all of the points and of course,"},{"Start":"09:36.375 ","End":"09:40.440","Text":"we know that the points or so are in-between,"},{"Start":"09:40.440 ","End":"09:48.060","Text":"so any point on this red line is still on the maximum line."},{"Start":"09:49.350 ","End":"09:52.825","Text":"Now let\u0027s find the next 1."},{"Start":"09:52.825 ","End":"09:59.420","Text":"We have this point here and here, and so on."},{"Start":"09:59.420 ","End":"10:01.355","Text":"I\u0027m drawing it a bit faster."},{"Start":"10:01.355 ","End":"10:07.475","Text":"Obviously, all of these lines are meant to be drawing much nicer than what I\u0027m doing."},{"Start":"10:07.475 ","End":"10:13.535","Text":"Here I\u0027ve searched again on the other side for where my 2 solid lines meet,"},{"Start":"10:13.535 ","End":"10:15.650","Text":"2 dotted lines, solid lines."},{"Start":"10:15.650 ","End":"10:19.550","Text":"Where we have peak to peak or trough to trough."},{"Start":"10:19.550 ","End":"10:25.260","Text":"Now we can connect all of this over here."},{"Start":"10:25.260 ","End":"10:30.295","Text":"Remember to always go around where the source is."},{"Start":"10:30.295 ","End":"10:35.095","Text":"This is our maximum line of third order."},{"Start":"10:35.095 ","End":"10:40.440","Text":"Of course, it\u0027s symmetrical so we also have this on this side over here."},{"Start":"10:40.440 ","End":"10:50.440","Text":"It\u0027s like I\u0027m just searching for all the points that match each other. Not that point."},{"Start":"10:50.440 ","End":"10:58.080","Text":"Here, where all of the lines match up."},{"Start":"10:58.080 ","End":"11:03.150","Text":"Then I\u0027ll just join this up."},{"Start":"11:03.150 ","End":"11:09.075","Text":"Of course, imagine that all the points sit on the line."},{"Start":"11:09.075 ","End":"11:13.665","Text":"This is maximum of 3rd order just from the other side."},{"Start":"11:13.665 ","End":"11:16.980","Text":"Now, we\u0027ve seen what it looks like visually and how you can"},{"Start":"11:16.980 ","End":"11:20.955","Text":"calculate this visually to join the maximum lines."},{"Start":"11:20.955 ","End":"11:23.564","Text":"But let\u0027s say you want to check yourself."},{"Start":"11:23.564 ","End":"11:27.255","Text":"Let\u0027s choose this point over here,"},{"Start":"11:27.255 ","End":"11:32.460","Text":"and let\u0027s call it p. We can see that this point is located."},{"Start":"11:32.460 ","End":"11:36.165","Text":"We can see it\u0027s on a minimum,"},{"Start":"11:36.165 ","End":"11:38.865","Text":"of the pink and a minimum of the green."},{"Start":"11:38.865 ","End":"11:41.760","Text":"First of all, that\u0027s a trough and a trough."},{"Start":"11:41.760 ","End":"11:44.490","Text":"We can see it\u0027s complete constructive interference,"},{"Start":"11:44.490 ","End":"11:46.800","Text":"which means it has to sit on a maximum line,"},{"Start":"11:46.800 ","End":"11:50.460","Text":"which is exactly what we can see it as doing."},{"Start":"11:50.460 ","End":"11:52.620","Text":"First of all, that\u0027s green."},{"Start":"11:52.620 ","End":"11:57.010","Text":"But now, we\u0027re going to count wavelengths from the minimum line."},{"Start":"11:57.020 ","End":"12:04.110","Text":"Here we have 1 trough to 2."},{"Start":"12:04.110 ","End":"12:09.165","Text":"Here, this is 1 wavelength and this is 2 wavelengths."},{"Start":"12:09.165 ","End":"12:14.445","Text":"Then from the green we can see that this is 1 wavelength. Trough to trough."},{"Start":"12:14.445 ","End":"12:20.820","Text":"2 wavelengths, 3 wavelengths, 4 wavelengths."},{"Start":"12:20.820 ","End":"12:28.470","Text":"We can see that the distance S_2 to P minus S_1 to P,"},{"Start":"12:28.470 ","End":"12:30.674","Text":"remember this was the equation,"},{"Start":"12:30.674 ","End":"12:36.315","Text":"is equal to 4 minus 2,"},{"Start":"12:36.315 ","End":"12:38.445","Text":"which is equal to 2."},{"Start":"12:38.445 ","End":"12:42.510","Text":"Of course, this is 4 Lambda minus 2 Lambda."},{"Start":"12:42.510 ","End":"12:45.420","Text":"We have a difference of 2 Lambda,"},{"Start":"12:45.420 ","End":"12:49.080","Text":"which is exactly the mathematical definition or"},{"Start":"12:49.080 ","End":"12:54.015","Text":"condition for a maximum line of 2nd order."},{"Start":"12:54.015 ","End":"12:57.330","Text":"This is exactly where at point P lies."},{"Start":"12:57.330 ","End":"13:01.630","Text":"That\u0027s how we could test that out."},{"Start":"13:02.240 ","End":"13:07.710","Text":"Now we can see that we found all of the maximum lines."},{"Start":"13:07.710 ","End":"13:12.735","Text":"We can just check this visually by seeing that there\u0027s no more points"},{"Start":"13:12.735 ","End":"13:18.405","Text":"where we have a peak meeting a peak and the minimum meeting a minimum."},{"Start":"13:18.405 ","End":"13:19.860","Text":"In this area over here,"},{"Start":"13:19.860 ","End":"13:22.060","Text":"on this area over here."},{"Start":"13:22.190 ","End":"13:28.964","Text":"Now let\u0027s move on to finding our minimum lines."},{"Start":"13:28.964 ","End":"13:39.280","Text":"What were our minimum lines when we have complete destructive interference?"},{"Start":"13:39.830 ","End":"13:43.155","Text":"We\u0027ve already seen what this means."},{"Start":"13:43.155 ","End":"13:48.690","Text":"That means that if our blue wave comes in at a peak,"},{"Start":"13:48.690 ","End":"13:55.800","Text":"complete destructive interference means that our green wave comes in at a trough."},{"Start":"13:55.800 ","End":"14:00.450","Text":"Then that means that when we superimpose the 2,"},{"Start":"14:00.450 ","End":"14:02.655","Text":"so when we add on the 2 amplitudes,"},{"Start":"14:02.655 ","End":"14:06.285","Text":"so we know that the blue has an amplitude in the positive direction of"},{"Start":"14:06.285 ","End":"14:10.740","Text":"A and the green has an amplitude in the negative direction of A."},{"Start":"14:10.740 ","End":"14:15.810","Text":"Therefore we have A plus minus A which is equal to 0."},{"Start":"14:15.810 ","End":"14:18.735","Text":"Our point will be on a minimum point."},{"Start":"14:18.735 ","End":"14:26.640","Text":"This will happen also if our green comes at a peak,"},{"Start":"14:26.640 ","End":"14:28.350","Text":"but our blue,"},{"Start":"14:28.350 ","End":"14:30.645","Text":"comes at a trough."},{"Start":"14:30.645 ","End":"14:36.000","Text":"Again, these will cancel each other out completely."},{"Start":"14:36.000 ","End":"14:39.220","Text":"We\u0027ll have a point on the 0 line."},{"Start":"14:39.290 ","End":"14:45.840","Text":"We know that every single point that lies on the minimum line obeys this."},{"Start":"14:45.840 ","End":"14:50.985","Text":"We also said that a minimum line is also called a nodal line,"},{"Start":"14:50.985 ","End":"14:55.965","Text":"because as opposed to the maximum line where our points are moving,"},{"Start":"14:55.965 ","End":"14:58.515","Text":"we know that at the minimum line,"},{"Start":"14:58.515 ","End":"15:00.840","Text":"every point is stationery."},{"Start":"15:00.840 ","End":"15:07.545","Text":"Every point doesn\u0027t move and it remains on the 0 and it\u0027s not moving."},{"Start":"15:07.545 ","End":"15:11.730","Text":"Minimum lines happen when we have peak to trough and"},{"Start":"15:11.730 ","End":"15:16.510","Text":"maximum lines happen when we have peak to peak or trough to trough."},{"Start":"15:17.900 ","End":"15:21.540","Text":"Now let\u0027s rub out this blue,"},{"Start":"15:21.540 ","End":"15:23.145","Text":"I have this a bit neater."},{"Start":"15:23.145 ","End":"15:25.440","Text":"Now let\u0027s draw out our minimum line."},{"Start":"15:25.440 ","End":"15:27.450","Text":"Let\u0027s draw this in black."},{"Start":"15:27.450 ","End":"15:30.390","Text":"If we go to 1 side,"},{"Start":"15:30.390 ","End":"15:35.630","Text":"either to the left or to the right of the zeroth-order maximum line,"},{"Start":"15:35.630 ","End":"15:41.730","Text":"here we will find our 1st order minimum line."},{"Start":"15:42.200 ","End":"15:45.450","Text":"We already saw how we do it mathematically,"},{"Start":"15:45.450 ","End":"15:47.430","Text":"but now visually, this is all we do."},{"Start":"15:47.430 ","End":"15:50.850","Text":"We go to 1 side of the maximum line and"},{"Start":"15:50.850 ","End":"15:54.720","Text":"we find everywhere where we have a peak to trough."},{"Start":"15:54.720 ","End":"15:59.920","Text":"That means everywhere where we have a solid pink line to a dotted green line."},{"Start":"15:59.920 ","End":"16:01.880","Text":"This is a minimum point,"},{"Start":"16:01.880 ","End":"16:07.185","Text":"or a solid green line to a dotted pink line, another minimum point."},{"Start":"16:07.185 ","End":"16:12.320","Text":"We\u0027re looking for a pink and the green to meet where 1"},{"Start":"16:12.320 ","End":"16:18.075","Text":"of the wavefronts is a solid line and the other 1 is a dotted line."},{"Start":"16:18.075 ","End":"16:22.005","Text":"Because that means that they\u0027re meeting peak to trough."},{"Start":"16:22.005 ","End":"16:26.925","Text":"That means that we have complete destructive interference."},{"Start":"16:26.925 ","End":"16:32.800","Text":"Then of course, we have these points over here as well."},{"Start":"16:33.530 ","End":"16:36.090","Text":"Again, this is a line,"},{"Start":"16:36.090 ","End":"16:38.715","Text":"it doesn\u0027t have to be a straight line."},{"Start":"16:38.715 ","End":"16:41.640","Text":"Now we can connect all of these dots."},{"Start":"16:41.640 ","End":"16:44.325","Text":"We know that all of these dots are stationary points."},{"Start":"16:44.325 ","End":"16:46.710","Text":"All of these dots are nodes."},{"Start":"16:46.710 ","End":"16:51.900","Text":"This is a minimum line of 1st order."},{"Start":"16:51.900 ","End":"16:53.475","Text":"1st order minimum line."},{"Start":"16:53.475 ","End":"16:57.570","Text":"Of course, we have the mirror image of it over here."},{"Start":"16:57.570 ","End":"17:00.960","Text":"We\u0027re just finding all of the peak to trough,"},{"Start":"17:00.960 ","End":"17:05.770","Text":"like so and similarly on the other side."},{"Start":"17:07.730 ","End":"17:13.830","Text":"Then we\u0027re just going to connect all of them together."},{"Start":"17:13.830 ","End":"17:17.685","Text":"This is also a 1st order minimum line."},{"Start":"17:17.685 ","End":"17:19.050","Text":"Now we\u0027re looking for,"},{"Start":"17:19.050 ","End":"17:21.045","Text":"the 2nd order minimum line."},{"Start":"17:21.045 ","End":"17:25.170","Text":"We know already from previous lessons that the 2nd order minimum line is going to"},{"Start":"17:25.170 ","End":"17:29.265","Text":"lie between the 1st and 2nd order maximum line."},{"Start":"17:29.265 ","End":"17:33.570","Text":"Again, we\u0027re just looking between here everywhere where we"},{"Start":"17:33.570 ","End":"17:38.235","Text":"have a dotted line meeting a solid line of a different color."},{"Start":"17:38.235 ","End":"17:47.100","Text":"That\u0027s just all along here and over here as well."},{"Start":"17:47.100 ","End":"17:50.475","Text":"I\u0027m just marking all of this out."},{"Start":"17:50.475 ","End":"17:58.350","Text":"Now we\u0027re just going to connect the dots as accurately as I can freestyle."},{"Start":"17:58.350 ","End":"18:01.785","Text":"This is a 2nd order minimum line."},{"Start":"18:01.785 ","End":"18:05.535","Text":"Of course, we have the same thing on the other side, the mirror image."},{"Start":"18:05.535 ","End":"18:09.960","Text":"I\u0027m just going to straight away just freestyle this"},{"Start":"18:09.960 ","End":"18:15.210","Text":"over here because you\u0027ve probably gotten the gist of my dots."},{"Start":"18:15.210 ","End":"18:17.040","Text":"Please, if you don\u0027t understand,"},{"Start":"18:17.040 ","End":"18:20.700","Text":"pause the video and try and do it yourself."},{"Start":"18:20.700 ","End":"18:23.610","Text":"Here this is meant to be just a single line."},{"Start":"18:23.610 ","End":"18:26.955","Text":"This is also a 2nd order minimum line."},{"Start":"18:26.955 ","End":"18:29.775","Text":"Then here we\u0027ll have our 3rd order minimum lines."},{"Start":"18:29.775 ","End":"18:35.580","Text":"We can see that it\u0027s just going through all of these points where"},{"Start":"18:35.580 ","End":"18:42.480","Text":"a trough of 1 color meets the peak of another color,"},{"Start":"18:42.480 ","End":"18:48.850","Text":"causing complete deconstructive interference."},{"Start":"18:51.320 ","End":"18:56.700","Text":"This is a 3rd order minimum line."},{"Start":"18:56.700 ","End":"19:01.320","Text":"Similarly, we have this on this side as well."},{"Start":"19:01.320 ","End":"19:05.490","Text":"I\u0027m just again going to sketch"},{"Start":"19:05.490 ","End":"19:13.155","Text":"this, like so."},{"Start":"19:13.155 ","End":"19:16.125","Text":"This is also a 3rd on a minimum line."},{"Start":"19:16.125 ","End":"19:19.305","Text":"Then we can see that\u0027s it."},{"Start":"19:19.305 ","End":"19:26.520","Text":"Over here we can see that we have some constructive destructive interference going on."},{"Start":"19:26.520 ","End":"19:32.955","Text":"But we can see that our wave fronts on crossing over with 1 another."},{"Start":"19:32.955 ","End":"19:35.565","Text":"Here we could really see an x."},{"Start":"19:35.565 ","End":"19:39.165","Text":"Here we don\u0027t see an x or on this side."},{"Start":"19:39.165 ","End":"19:40.545","Text":"That means that\u0027s it."},{"Start":"19:40.545 ","End":"19:45.630","Text":"We have up to order 3 in the minimum lines."},{"Start":"19:45.630 ","End":"19:51.720","Text":"Of course, what you can do is you can choose some points on the minimum line."},{"Start":"19:51.720 ","End":"19:53.490","Text":"Let\u0027s say this point."},{"Start":"19:53.490 ","End":"19:56.740","Text":"Let\u0027s call this point A."},{"Start":"19:57.350 ","End":"20:03.570","Text":"Just like we did a sanity check with this point P over here on the maximum line."},{"Start":"20:03.570 ","End":"20:06.060","Text":"You can do a sanity check with this point A on"},{"Start":"20:06.060 ","End":"20:09.885","Text":"the minimum line with the equation that we had,"},{"Start":"20:09.885 ","End":"20:18.580","Text":"which was the absolute value of S_2 P minus S_1 P or here instead it would be A."},{"Start":"20:19.160 ","End":"20:26.265","Text":"Has to be equal to n minus a 1/2 multiplied by Lambda."},{"Start":"20:26.265 ","End":"20:31.920","Text":"If you manage to get that n is equal to here,"},{"Start":"20:31.920 ","End":"20:34.650","Text":"this lies on the 3rd order minimum line."},{"Start":"20:34.650 ","End":"20:39.135","Text":"If you manage to solve this equation and get that n is equal to 3,"},{"Start":"20:39.135 ","End":"20:42.780","Text":"then you know that you\u0027ve done a great job."},{"Start":"20:42.780 ","End":"20:48.660","Text":"Also, you can see that here we have the peak of the green with a trough of the pink."},{"Start":"20:48.660 ","End":"20:51.960","Text":"That also confirms that it is in fact a minimum line."},{"Start":"20:51.960 ","End":"20:55.480","Text":"Then we just have to check that it\u0027s of the right order."},{"Start":"20:55.700 ","End":"20:59.110","Text":"That\u0027s the end of this lesson."}],"ID":12494},{"Watched":false,"Name":"Exercise - Diffraction 2","Duration":"17m 47s","ChapterTopicVideoID":12027,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.995","Text":"Hello. In this lesson,"},{"Start":"00:01.995 ","End":"00:04.935","Text":"we\u0027re going to be answering the following question."},{"Start":"00:04.935 ","End":"00:08.625","Text":"We are given a wave tank with 2 wave generators,"},{"Start":"00:08.625 ","End":"00:12.420","Text":"a distance of 7 meters away from each other."},{"Start":"00:12.420 ","End":"00:18.525","Text":"The waves generated are of the same phase and have a frequency of 20 hertz."},{"Start":"00:18.525 ","End":"00:25.660","Text":"The waves have a velocity of 25 centimeters per second."},{"Start":"00:25.660 ","End":"00:31.245","Text":"Question number 1 is to find the wavelength of the waves."},{"Start":"00:31.245 ","End":"00:33.015","Text":"Let\u0027s answer this over here."},{"Start":"00:33.015 ","End":"00:40.945","Text":"We know that we have the equation that v=f multiplied by Lambda."},{"Start":"00:40.945 ","End":"00:43.970","Text":"We know what v is and we know what f is,"},{"Start":"00:43.970 ","End":"00:46.670","Text":"and Lambda is what we\u0027re trying to find."},{"Start":"00:46.670 ","End":"00:49.325","Text":"Therefore we can isolate out Lambda."},{"Start":"00:49.325 ","End":"00:53.495","Text":"Lambda is equal to v divided by f,"},{"Start":"00:53.495 ","End":"00:55.060","Text":"which in my case,"},{"Start":"00:55.060 ","End":"00:59.269","Text":"v is 25 centimeters per second,"},{"Start":"00:59.269 ","End":"01:03.350","Text":"divided by f, which is 20 hertz."},{"Start":"01:03.350 ","End":"01:06.875","Text":"If we plug this into our calculator,"},{"Start":"01:06.875 ","End":"01:14.605","Text":"we\u0027ll get that Lambda is equal to 1.2 centimeters."},{"Start":"01:14.605 ","End":"01:18.525","Text":"Now let\u0027s answer Question 2."},{"Start":"01:18.525 ","End":"01:24.680","Text":"Question 2 is asking if the following points are located on maximum lines,"},{"Start":"01:24.680 ","End":"01:27.560","Text":"minimum lines or on neither."},{"Start":"01:27.560 ","End":"01:31.520","Text":"What we\u0027re given is the distance from the sources."},{"Start":"01:31.520 ","End":"01:34.625","Text":"Let\u0027s say that a is called point a,"},{"Start":"01:34.625 ","End":"01:39.030","Text":"located a distance 4 centimeters"},{"Start":"01:39.030 ","End":"01:43.455","Text":"away from Source 1 and 2.8 centimeters away from Source 2."},{"Start":"01:43.455 ","End":"01:53.070","Text":"First of all, we remember our equation which said that the absolute value of S_2 to"},{"Start":"01:53.070 ","End":"02:03.090","Text":"whichever point where doing minus S_1 to that point is equal to n Lambda."},{"Start":"02:03.090 ","End":"02:07.305","Text":"This defines a maximum line."},{"Start":"02:07.305 ","End":"02:11.795","Text":"This is where we had complete constructive interference."},{"Start":"02:11.795 ","End":"02:18.490","Text":"Or alternatively, if we have that S_2P minus S_1P,"},{"Start":"02:18.490 ","End":"02:25.715","Text":"in absolute value is equal to n-1.5 of Lambda."},{"Start":"02:25.715 ","End":"02:29.930","Text":"Then we know that this defines a point on"},{"Start":"02:29.930 ","End":"02:35.675","Text":"the minimum line where we have complete destructive interference."},{"Start":"02:35.675 ","End":"02:38.980","Text":"Of course, we know that in both of these cases,"},{"Start":"02:38.980 ","End":"02:44.705","Text":"n represents of what order the line is."},{"Start":"02:44.705 ","End":"02:49.190","Text":"Let\u0027s look first of all at point a."},{"Start":"02:49.190 ","End":"02:53.255","Text":"Let\u0027s substitute this n. We have"},{"Start":"02:53.255 ","End":"02:59.990","Text":"that S_2 to our point a is given to us here,"},{"Start":"02:59.990 ","End":"03:05.675","Text":"2.8 centimeters minus the distance from S_1 to a,"},{"Start":"03:05.675 ","End":"03:09.645","Text":"which is 4 centimeters."},{"Start":"03:09.645 ","End":"03:13.120","Text":"This we can see,"},{"Start":"03:14.600 ","End":"03:21.300","Text":"is equal to 1.2 centimeters."},{"Start":"03:21.300 ","End":"03:25.740","Text":"1.2 centimeters, we saw in Question number 1"},{"Start":"03:25.740 ","End":"03:32.370","Text":"that the wavelength of the wave is 1.2 centimeters."},{"Start":"03:32.370 ","End":"03:40.350","Text":"We can see that this over here is equal to Lambda."},{"Start":"03:41.080 ","End":"03:45.690","Text":"This is equal to Lambda multiplied by 1."},{"Start":"03:46.460 ","End":"03:49.310","Text":"Here n=1."},{"Start":"03:49.310 ","End":"03:59.040","Text":"Therefore, point a is lying on a maximum line of,"},{"Start":"03:59.040 ","End":"04:02.800","Text":"we can see first order."},{"Start":"04:05.360 ","End":"04:08.130","Text":"This is the answer for a."},{"Start":"04:08.130 ","End":"04:13.350","Text":"Now let\u0027s answer b. Let\u0027s see."},{"Start":"04:13.350 ","End":"04:18.235","Text":"We have that S_2 to point b is"},{"Start":"04:18.235 ","End":"04:24.850","Text":"3.2 centimeters minus S_1 to be, which is 5."},{"Start":"04:24.850 ","End":"04:27.805","Text":"If we plug this into our calculator,"},{"Start":"04:27.805 ","End":"04:33.660","Text":"we get that this is equal to 1.8 centimeters."},{"Start":"04:33.660 ","End":"04:39.475","Text":"Now we can see that 1.8 centimeters"},{"Start":"04:39.475 ","End":"04:46.365","Text":"is exactly 1.5 multiplied by Lambda."},{"Start":"04:46.365 ","End":"04:53.810","Text":"If we try that, so we do 1.5 multiplied by Lambda,"},{"Start":"04:53.810 ","End":"05:00.460","Text":"which is 1.2, will get that this is equal to 1.8."},{"Start":"05:00.950 ","End":"05:04.940","Text":"First of all, we can see that if we have 1.5,"},{"Start":"05:04.940 ","End":"05:08.825","Text":"we can see that this isn\u0027t a whole number,"},{"Start":"05:08.825 ","End":"05:14.600","Text":"which means that we can\u0027t be working with the equation for a maximum line."},{"Start":"05:14.600 ","End":"05:19.025","Text":"We have to be working with an equation for a minimum line."},{"Start":"05:19.025 ","End":"05:22.295","Text":"Because n is always a whole number,"},{"Start":"05:22.295 ","End":"05:28.145","Text":"which means that if we see that we have some fraction or non whole number"},{"Start":"05:28.145 ","End":"05:35.730","Text":"multiplying our Lambda that can only come from a whole number minus a 1/2."},{"Start":"05:36.560 ","End":"05:40.815","Text":"Then we can plug this in to the equation."},{"Start":"05:40.815 ","End":"05:45.280","Text":"We know that S_2P minus S_1P=1.8."},{"Start":"05:46.760 ","End":"05:51.500","Text":"On the other hand, we know that we\u0027re going to be using the minimum."},{"Start":"05:51.500 ","End":"05:56.400","Text":"Let\u0027s just write here that this is using the minimum equation."},{"Start":"05:57.820 ","End":"06:03.890","Text":"1.8=n minus 1.5 multiplied by Lambda,"},{"Start":"06:03.890 ","End":"06:07.220","Text":"which is equal to 1.2."},{"Start":"06:07.220 ","End":"06:11.525","Text":"If we divide both sides by 1.2,"},{"Start":"06:11.525 ","End":"06:15.200","Text":"then we\u0027ll get 1.5."},{"Start":"06:15.200 ","End":"06:23.730","Text":"Then we have the 1.5=n minus 1/2 or minus 0.5."},{"Start":"06:23.730 ","End":"06:26.625","Text":"We add 0.5 onto both sides,"},{"Start":"06:26.625 ","End":"06:33.270","Text":"and therefore we get that n= 2."},{"Start":"06:33.270 ","End":"06:39.825","Text":"Therefore, we can see that b lies on the minimum lines."},{"Start":"06:39.825 ","End":"06:45.955","Text":"b is a nodal point and it is of second order."},{"Start":"06:45.955 ","End":"06:49.740","Text":"It\u0027s a second-order minimum line."},{"Start":"06:49.750 ","End":"06:52.985","Text":"Now let\u0027s take a look at"},{"Start":"06:52.985 ","End":"06:58.740","Text":"point C. point C"},{"Start":"06:58.740 ","End":"07:04.485","Text":"is 7 centimeters away from S_1 and 3.4 centimeters away from S_2."},{"Start":"07:04.485 ","End":"07:15.044","Text":"We have 3.4 minus 7=3.6."},{"Start":"07:15.044 ","End":"07:20.125","Text":"Now let\u0027s try this."},{"Start":"07:20.125 ","End":"07:28.195","Text":"So 3.6, we can maybe see that there\u0027s a whole number multiple for 1.2."},{"Start":"07:28.195 ","End":"07:31.120","Text":"But if you can\u0027t see this,"},{"Start":"07:31.120 ","End":"07:35.260","Text":"you can just try dividing 3.6 by 1.2."},{"Start":"07:35.260 ","End":"07:36.940","Text":"If you get a whole number,"},{"Start":"07:36.940 ","End":"07:39.535","Text":"that means it lies on a maximum line,"},{"Start":"07:39.535 ","End":"07:43.465","Text":"and if you get some decimal number,"},{"Start":"07:43.465 ","End":"07:46.285","Text":"then you know it lies on a minimum line."},{"Start":"07:46.285 ","End":"07:53.630","Text":"Let\u0027s just do that, so 3.6 divided by 1.2=3,"},{"Start":"07:55.200 ","End":"07:59.110","Text":"which means that we get a whole number."},{"Start":"07:59.110 ","End":"08:06.595","Text":"Therefore, we can say that point C lies on a maximum line of,"},{"Start":"08:06.595 ","End":"08:08.739","Text":"so we can see in this equation,"},{"Start":"08:08.739 ","End":"08:11.110","Text":"it\u0027s 3 times Lambda."},{"Start":"08:11.110 ","End":"08:13.960","Text":"Because if we do 3 multiplied by Lambda,"},{"Start":"08:13.960 ","End":"08:16.990","Text":"which is 1.2, then we get 3.6."},{"Start":"08:16.990 ","End":"08:18.985","Text":"That means n=3,"},{"Start":"08:18.985 ","End":"08:25.720","Text":"so that means that c lies on a maximum line of third order,"},{"Start":"08:25.720 ","End":"08:29.570","Text":"or third order maximum line."},{"Start":"08:29.730 ","End":"08:36.100","Text":"Now let\u0027s answer d. I\u0027m going to scroll down a little bit more."},{"Start":"08:36.100 ","End":"08:37.870","Text":"Here we have it,"},{"Start":"08:37.870 ","End":"08:40.420","Text":"so S_2 to a point,"},{"Start":"08:40.420 ","End":"08:43.420","Text":"or we\u0027re using 1 of these equations, we don\u0027t know yet."},{"Start":"08:43.420 ","End":"08:51.925","Text":"S_2 to our point d is 6.5 centimeters minus S_1 to our point d,"},{"Start":"08:51.925 ","End":"08:53.950","Text":"which is 8 centimeters."},{"Start":"08:53.950 ","End":"09:00.500","Text":"So 6.5 minus 8 is equal to 1.5."},{"Start":"09:02.790 ","End":"09:08.650","Text":"Let\u0027s test to see if this lies on a minimum or maximum line."},{"Start":"09:08.650 ","End":"09:16.390","Text":"We can do 1.5 divided by 1.2,"},{"Start":"09:16.390 ","End":"09:21.320","Text":"and this is equal to 1.25."},{"Start":"09:21.510 ","End":"09:25.465","Text":"First of all, 1.25 is not a whole number,"},{"Start":"09:25.465 ","End":"09:32.170","Text":"which means that our point d does not lie on the maximum line."},{"Start":"09:32.170 ","End":"09:35.280","Text":"But also what else can we notice?"},{"Start":"09:35.280 ","End":"09:38.100","Text":"1.25, so therefore,"},{"Start":"09:38.100 ","End":"09:41.040","Text":"we could say maybe it\u0027s on the minimum line."},{"Start":"09:41.040 ","End":"09:47.045","Text":"What we can see is that we know that n has to be a whole number,"},{"Start":"09:47.045 ","End":"09:52.825","Text":"and then we see that this whole section of the equation,"},{"Start":"09:52.825 ","End":"09:54.760","Text":"n minus 0.5,"},{"Start":"09:54.760 ","End":"10:00.835","Text":"multiplied by Lambda has to be equal to this 1.5."},{"Start":"10:00.835 ","End":"10:06.580","Text":"Lambda is equal to 1.2 and n minus a 1/2 as n is a whole number minus a 1/2,"},{"Start":"10:06.580 ","End":"10:10.165","Text":"so we\u0027re going to have some number, 0.5."},{"Start":"10:10.165 ","End":"10:11.530","Text":"If it\u0027s 1.5,"},{"Start":"10:11.530 ","End":"10:13.555","Text":"2.5, 3.5,"},{"Start":"10:13.555 ","End":"10:16.240","Text":"5.5, 10.5,"},{"Start":"10:16.240 ","End":"10:17.470","Text":"whatever it must be,"},{"Start":"10:17.470 ","End":"10:20.470","Text":"it\u0027s always going to be 0.5."},{"Start":"10:20.470 ","End":"10:26.815","Text":"But here we can see that we have a number 0.25."},{"Start":"10:26.815 ","End":"10:30.970","Text":"This isn\u0027t a multiple of 1,"},{"Start":"10:30.970 ","End":"10:33.790","Text":"as in this isn\u0027t a whole number, sorry."},{"Start":"10:33.790 ","End":"10:39.115","Text":"This isn\u0027t a multiple of a 1/2, of 0.5."},{"Start":"10:39.115 ","End":"10:43.660","Text":"Therefore, we can see that point d,"},{"Start":"10:43.660 ","End":"10:47.905","Text":"also doesn\u0027t lie on the minimum line,"},{"Start":"10:47.905 ","End":"10:53.570","Text":"so therefore d lies on neither."},{"Start":"10:59.970 ","End":"11:02.515","Text":"Before we go on to Question 3,"},{"Start":"11:02.515 ","End":"11:07.120","Text":"I wrote a quick note that is very important to remember."},{"Start":"11:07.120 ","End":"11:10.510","Text":"This is something to test."},{"Start":"11:10.510 ","End":"11:18.250","Text":"If S_2P minus S_1P divided by Lambda is equal to whole number,"},{"Start":"11:18.250 ","End":"11:21.700","Text":"then for sure it\u0027s on the max line."},{"Start":"11:21.700 ","End":"11:29.349","Text":"If however, S_2P minus S_1P divided by Lambda is some multiple of a 1/2."},{"Start":"11:29.349 ","End":"11:32.800","Text":"Now note, 1 is also a multiple of 1/2,"},{"Start":"11:32.800 ","End":"11:34.225","Text":"it\u0027s 1/2 and 1/2,"},{"Start":"11:34.225 ","End":"11:36.205","Text":"or 2 times 1/2 is 1."},{"Start":"11:36.205 ","End":"11:41.365","Text":"We\u0027re referring to all the multiples of 1/2 that are not whole numbers,"},{"Start":"11:41.365 ","End":"11:44.230","Text":"so that means 0.5,"},{"Start":"11:44.230 ","End":"11:46.120","Text":"1.5, 2.5,"},{"Start":"11:46.120 ","End":"11:48.250","Text":"3.5 and so on."},{"Start":"11:48.250 ","End":"11:53.455","Text":"Anything that\u0027s a multiple of 1/2 but is not a whole number,"},{"Start":"11:53.455 ","End":"11:55.315","Text":"I\u0027ll underline this,"},{"Start":"11:55.315 ","End":"11:58.015","Text":"that means therefore that it is a minimum line."},{"Start":"11:58.015 ","End":"12:00.910","Text":"If it is a whole number such as 1, 2,"},{"Start":"12:00.910 ","End":"12:02.530","Text":"3, 4, or 0, whatever,"},{"Start":"12:02.530 ","End":"12:04.420","Text":"then it\u0027s on the maximum line."},{"Start":"12:04.420 ","End":"12:10.210","Text":"If we do this equation,"},{"Start":"12:10.210 ","End":"12:12.805","Text":"S_2P minus n_1P divided by Lambda,"},{"Start":"12:12.805 ","End":"12:14.845","Text":"and we don\u0027t get a whole number,"},{"Start":"12:14.845 ","End":"12:18.745","Text":"and we also don\u0027t get a multiple of 1/2,"},{"Start":"12:18.745 ","End":"12:23.870","Text":"that means that the point lies on neither."},{"Start":"12:25.140 ","End":"12:30.475","Text":"Now let\u0027s answer question 3."},{"Start":"12:30.475 ","End":"12:37.460","Text":"Question 3, is how many maximum and minimum lines will be visible in the tank."},{"Start":"12:38.970 ","End":"12:45.640","Text":"You may remember from previous lesson that we derived 2 equations."},{"Start":"12:45.640 ","End":"12:55.105","Text":"The first was for the maximum order of maximum lines."},{"Start":"12:55.105 ","End":"12:59.140","Text":"The highest order of maximum lines,"},{"Start":"12:59.140 ","End":"13:02.290","Text":"so that\u0027s n_max highest order maximum lines."},{"Start":"13:02.290 ","End":"13:07.480","Text":"We said that this was equal to d,"},{"Start":"13:07.480 ","End":"13:14.000","Text":"where d over here is distance between sources."},{"Start":"13:15.090 ","End":"13:19.990","Text":"That means the distance between Source 1 and Source 2,"},{"Start":"13:19.990 ","End":"13:24.950","Text":"and we divided it by Lambda, the wavelength."},{"Start":"13:25.620 ","End":"13:30.550","Text":"Then the highest number,"},{"Start":"13:30.550 ","End":"13:35.750","Text":"or the highest order of minimum lines,"},{"Start":"13:36.750 ","End":"13:41.110","Text":"was smaller or equal to d,"},{"Start":"13:41.110 ","End":"13:45.025","Text":"which again is the distance between the sources, so not this point d,"},{"Start":"13:45.025 ","End":"13:47.035","Text":"the distance between the sources,"},{"Start":"13:47.035 ","End":"13:55.345","Text":"divided by Lambda plus 1/2 for the minimum."},{"Start":"13:55.345 ","End":"13:58.660","Text":"First of all, let\u0027s go back to the question,"},{"Start":"13:58.660 ","End":"14:01.270","Text":"and we can see that we have"},{"Start":"14:01.270 ","End":"14:06.440","Text":"2 wave generators a distance of 7 centimeters away from each other."},{"Start":"14:08.010 ","End":"14:13.540","Text":"Therefore, we know that d,"},{"Start":"14:13.540 ","End":"14:18.220","Text":"the distance between the 2 sources is 7 centimeters."},{"Start":"14:18.220 ","End":"14:26.830","Text":"Don\u0027t look at this. Let\u0027s see what\u0027s the highest order maximum line?"},{"Start":"14:26.830 ","End":"14:32.980","Text":"N_max, the highest order that we\u0027ll see is smaller or equal to d,"},{"Start":"14:32.980 ","End":"14:35.415","Text":"which is 7 divided by Lambda."},{"Start":"14:35.415 ","End":"14:41.035","Text":"The wavelength which we got was equal to 1.2 centimeters,"},{"Start":"14:41.035 ","End":"14:43.265","Text":"and once we plug this into the calculator,"},{"Start":"14:43.265 ","End":"14:47.585","Text":"we\u0027ll get that this is equal to 5.83."},{"Start":"14:47.585 ","End":"14:49.170","Text":"Now of course, n,"},{"Start":"14:49.170 ","End":"14:54.665","Text":"we know the order of the maximum line always has to be a whole number."},{"Start":"14:54.665 ","End":"14:58.970","Text":"As we said, it doesn\u0027t matter what our decimal over here is,"},{"Start":"14:58.970 ","End":"15:02.400","Text":"we always round down."},{"Start":"15:02.520 ","End":"15:14.455","Text":"Therefore, the highest order of maximum line that we\u0027ll get is going to be equal to 5."},{"Start":"15:14.455 ","End":"15:22.220","Text":"Then we said, in order to know how many maximum lines we\u0027ll see,"},{"Start":"15:22.650 ","End":"15:33.295","Text":"the number of lines visible for max lines is going to be equal to 2n plus 1,"},{"Start":"15:33.295 ","End":"15:36.590","Text":"and for minimum lines,"},{"Start":"15:36.590 ","End":"15:42.990","Text":"it is going to be just 2n."},{"Start":"15:53.340 ","End":"15:59.050","Text":"So 5 is the maximum order of maximum lines that will get,"},{"Start":"15:59.050 ","End":"16:00.955","Text":"so we go up to the firth order,"},{"Start":"16:00.955 ","End":"16:07.310","Text":"which means that number of maximum lines"},{"Start":"16:07.590 ","End":"16:16.305","Text":"visible in the tank is equal to 2n plus 1."},{"Start":"16:16.305 ","End":"16:18.165","Text":"That\u0027s 2 times n,"},{"Start":"16:18.165 ","End":"16:21.340","Text":"which is 5 plus 1,"},{"Start":"16:21.340 ","End":"16:24.295","Text":"which is equal to 11."},{"Start":"16:24.295 ","End":"16:29.810","Text":"In the tank we\u0027ll see 11 maximum lines,"},{"Start":"16:30.160 ","End":"16:34.945","Text":"so now what about the minimum lines?"},{"Start":"16:34.945 ","End":"16:41.330","Text":"N min, the highest order of minimum line that will get is smaller or equal to d,"},{"Start":"16:41.330 ","End":"16:43.160","Text":"which is 7 centimeters,"},{"Start":"16:43.160 ","End":"16:45.280","Text":"divided by our wavelength Lambda,"},{"Start":"16:45.280 ","End":"16:50.580","Text":"which is 1.2 centimeters plus 1/2."},{"Start":"16:50.580 ","End":"16:53.285","Text":"Once we plug this into the calculator,"},{"Start":"16:53.285 ","End":"17:00.310","Text":"we\u0027ll get that n_min is smaller or equal to 6.33."},{"Start":"17:00.310 ","End":"17:05.285","Text":"Again, n_min also has to be a whole number,"},{"Start":"17:05.285 ","End":"17:07.610","Text":"so we always round down."},{"Start":"17:07.610 ","End":"17:13.510","Text":"That means that n_min is equal to 6."},{"Start":"17:13.510 ","End":"17:17.815","Text":"We\u0027ll get up to the sixth order of minimum line,"},{"Start":"17:17.815 ","End":"17:24.020","Text":"and that therefore the number of minimum lines"},{"Start":"17:24.020 ","End":"17:33.635","Text":"visible in the tank is simply equal to for the minimum 2n."},{"Start":"17:33.635 ","End":"17:36.790","Text":"So 2 times 6,"},{"Start":"17:36.790 ","End":"17:38.665","Text":"which is equal to 12."},{"Start":"17:38.665 ","End":"17:43.680","Text":"We\u0027ll see a total of 12 minimum lines in the tank."},{"Start":"17:44.260 ","End":"17:48.060","Text":"That\u0027s the end of the lesson."}],"ID":12495},{"Watched":false,"Name":"Interference Pattern from Out of Phase Waves","Duration":"19m 56s","ChapterTopicVideoID":12028,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"Hello. In previous lessons,"},{"Start":"00:02.160 ","End":"00:05.100","Text":"we spoke about the interference pattern coming"},{"Start":"00:05.100 ","End":"00:09.525","Text":"from waves that were in phase with one another."},{"Start":"00:09.525 ","End":"00:13.725","Text":"In those examples, we had, for example,"},{"Start":"00:13.725 ","End":"00:19.200","Text":"a bar over here with 2 pins, like so."},{"Start":"00:19.200 ","End":"00:22.860","Text":"They were hitting the water,"},{"Start":"00:22.860 ","End":"00:28.935","Text":"like so, such that when the green wave,"},{"Start":"00:28.935 ","End":"00:32.520","Text":"the wave coming from the green source was at a peak,"},{"Start":"00:32.520 ","End":"00:36.475","Text":"the wave coming from a red source was also at a peak and"},{"Start":"00:36.475 ","End":"00:40.950","Text":"when the wave coming from the green source was at a trough,"},{"Start":"00:40.950 ","End":"00:45.605","Text":"the wave coming from the red source was also at a trough."},{"Start":"00:45.605 ","End":"00:50.015","Text":"Here, we were looking at what happens and"},{"Start":"00:50.015 ","End":"00:54.380","Text":"what the interference pattern looks like and here we had in phase."},{"Start":"00:54.380 ","End":"00:57.710","Text":"What we\u0027re going to do in this lesson is speak about"},{"Start":"00:57.710 ","End":"01:03.525","Text":"the interference pattern that comes from waves that are out of phase."},{"Start":"01:03.525 ","End":"01:13.040","Text":"What do we mean? If here we have our water level and here we\u0027ll have our bar where"},{"Start":"01:13.040 ","End":"01:23.310","Text":"attached to it on one end is the green pin and on the other end is the red pin."},{"Start":"01:23.360 ","End":"01:32.715","Text":"Just like before, but now we can see that when the green pin is down,"},{"Start":"01:32.715 ","End":"01:35.415","Text":"the red pin is up."},{"Start":"01:35.415 ","End":"01:37.005","Text":"What will that mean?"},{"Start":"01:37.005 ","End":"01:40.160","Text":"That means that when the wavefront,"},{"Start":"01:40.160 ","End":"01:43.390","Text":"the peak of the green wave is coming,"},{"Start":"01:43.390 ","End":"01:47.970","Text":"the trough of the red wave will be coming."},{"Start":"01:47.970 ","End":"01:49.700","Text":"Because these alternate first,"},{"Start":"01:49.700 ","End":"01:51.950","Text":"the green pin taps into the water,"},{"Start":"01:51.950 ","End":"01:56.040","Text":"then it goes up, and then the red pin taps into the water."},{"Start":"01:56.040 ","End":"02:01.445","Text":"We\u0027re going to have peaks with the green coming with troughs from the red,"},{"Start":"02:01.445 ","End":"02:08.805","Text":"or troughs from the green coming with peaks from the red."},{"Start":"02:08.805 ","End":"02:16.500","Text":"Here, we can see that these waves are going to be out of phase with each other."},{"Start":"02:16.500 ","End":"02:21.615","Text":"What we want to know is what will the interference pattern look like."},{"Start":"02:21.615 ","End":"02:25.250","Text":"Let\u0027s look at a diagram."},{"Start":"02:25.250 ","End":"02:32.635","Text":"Let\u0027s look over here at our pink source over here and this is called S_1."},{"Start":"02:32.635 ","End":"02:34.860","Text":"You can see that S_1,"},{"Start":"02:34.860 ","End":"02:36.365","Text":"the first ring around it,"},{"Start":"02:36.365 ","End":"02:37.955","Text":"is the solid ring,"},{"Start":"02:37.955 ","End":"02:41.525","Text":"indicating that this is a peak."},{"Start":"02:41.525 ","End":"02:45.358","Text":"But now, if we look at the green source over here,"},{"Start":"02:45.358 ","End":"02:47.030","Text":"this is still S_2,"},{"Start":"02:47.030 ","End":"02:52.115","Text":"we can see that the first ring about it is this dotted line,"},{"Start":"02:52.115 ","End":"02:54.185","Text":"which means that it\u0027s a trough."},{"Start":"02:54.185 ","End":"02:58.235","Text":"We can see that they\u0027re both out of phase and eventually"},{"Start":"02:58.235 ","End":"03:02.315","Text":"these 2 rings will interact with each other where we\u0027ll have,"},{"Start":"03:02.315 ","End":"03:09.185","Text":"let\u0027s say a peak from the green and a trough from the pink,"},{"Start":"03:09.185 ","End":"03:11.405","Text":"and then in total,"},{"Start":"03:11.405 ","End":"03:16.980","Text":"they\u0027ll cancel each other out and we\u0027ll have deconstructive interference."},{"Start":"03:17.300 ","End":"03:24.545","Text":"Let\u0027s wrap this all out and now let\u0027s start drawing our maximum lines."},{"Start":"03:24.545 ","End":"03:26.885","Text":"Let\u0027s draw this in blue."},{"Start":"03:26.885 ","End":"03:31.580","Text":"I\u0027m reminding, maximum lines are where we have constructive interference,"},{"Start":"03:31.580 ","End":"03:34.970","Text":"which means that this is all the points where either peaks"},{"Start":"03:34.970 ","End":"03:39.015","Text":"and peaks meet or troughs and troughs meet."},{"Start":"03:39.015 ","End":"03:41.445","Text":"Let\u0027s take a look."},{"Start":"03:41.445 ","End":"03:47.030","Text":"The first one that I can see is this point over here."},{"Start":"03:47.030 ","End":"03:51.640","Text":"Now I\u0027m just going to follow all of this in a line."},{"Start":"03:51.640 ","End":"03:53.420","Text":"We can see that we have trough and trough,"},{"Start":"03:53.420 ","End":"03:54.560","Text":"peak and peak, trough and trough,"},{"Start":"03:54.560 ","End":"03:55.850","Text":"peak and peak, trough and trough,"},{"Start":"03:55.850 ","End":"03:57.710","Text":"and of course from the other side,"},{"Start":"03:57.710 ","End":"04:02.420","Text":"trough and trough, peak and peak, troughs and peaks."},{"Start":"04:02.420 ","End":"04:09.645","Text":"Here, we have constructive interference."},{"Start":"04:09.645 ","End":"04:16.620","Text":"Now I\u0027m just going to join all the points like we saw in previous lessons."},{"Start":"04:16.620 ","End":"04:20.250","Text":"This is, of course, a maximum line and"},{"Start":"04:20.250 ","End":"04:24.555","Text":"it\u0027s the first one that I see so it\u0027s a first-order maximum line."},{"Start":"04:24.555 ","End":"04:29.350","Text":"We can see that it looks a little bit different to how we expected"},{"Start":"04:29.350 ","End":"04:36.440","Text":"our first maximum line to look in a diffraction pattern or in an interference pattern."},{"Start":"04:36.900 ","End":"04:41.815","Text":"Before, when both of the sources were in phase with one another,"},{"Start":"04:41.815 ","End":"04:46.840","Text":"our first maximum line went right through the middle and it was of order 0."},{"Start":"04:46.840 ","End":"04:50.000","Text":"Here we can see that because it\u0027s not going right through the middle,"},{"Start":"04:50.000 ","End":"04:54.550","Text":"it\u0027s not a void of 0 and it\u0027s actually of Order 1, a first order."},{"Start":"04:54.550 ","End":"04:56.673","Text":"Then of course, just like before,"},{"Start":"04:56.673 ","End":"04:58.565","Text":"we have the mirror case."},{"Start":"04:58.565 ","End":"05:00.035","Text":"Here we have a trough and a trough,"},{"Start":"05:00.035 ","End":"05:01.490","Text":"peak and peak, trough and trough."},{"Start":"05:01.490 ","End":"05:07.439","Text":"I\u0027m just filling out all of these points where we have constructive interference,"},{"Start":"05:07.439 ","End":"05:15.920","Text":"where a pink peak meets a green peak and a pink dotted line meets a green dotted line."},{"Start":"05:15.920 ","End":"05:23.920","Text":"Then I\u0027m going to join all of these together as well."},{"Start":"05:27.350 ","End":"05:33.910","Text":"This is also a maximum line of first-order as well."},{"Start":"05:33.910 ","End":"05:37.515","Text":"Now let\u0027s look at the minimum lines."},{"Start":"05:37.515 ","End":"05:44.280","Text":"We know that this is where in the minimum line we have complete destructive interference."},{"Start":"05:44.280 ","End":"05:48.575","Text":"That\u0027s where a pink peak meets a green trough,"},{"Start":"05:48.575 ","End":"05:52.775","Text":"or a green peak meets a pink trough."},{"Start":"05:52.775 ","End":"05:57.630","Text":"Then what we get is this 0 line over here."},{"Start":"05:57.630 ","End":"05:59.145","Text":"Let\u0027s draw these points."},{"Start":"05:59.145 ","End":"06:03.035","Text":"Here we see a pink solid line meeting a green dotted line,"},{"Start":"06:03.035 ","End":"06:06.080","Text":"a green solid line meeting a pink dotted line."},{"Start":"06:06.080 ","End":"06:08.810","Text":"I\u0027m just following where I have a solid line of"},{"Start":"06:08.810 ","End":"06:13.060","Text":"1 color meeting a dotted line of the other color."},{"Start":"06:13.060 ","End":"06:17.240","Text":"Then similarly, over here going down,"},{"Start":"06:17.240 ","End":"06:21.575","Text":"we can see that we have a green peak with the pink trough,"},{"Start":"06:21.575 ","End":"06:23.870","Text":"a green trough with the pink peak,"},{"Start":"06:23.870 ","End":"06:25.715","Text":"and here and here."},{"Start":"06:25.715 ","End":"06:29.925","Text":"Now, I\u0027m just going to connect these lines."},{"Start":"06:29.925 ","End":"06:36.295","Text":"We can see that this minimum line is equidistant from Source 1 and 2,"},{"Start":"06:36.295 ","End":"06:40.930","Text":"which means that this is a minimum line of a zeroth-order."},{"Start":"06:40.930 ","End":"06:46.880","Text":"We can see that the lines that we\u0027re getting their formation is different when"},{"Start":"06:46.880 ","End":"06:53.055","Text":"the sources are out of phase to what we expected when the sources were in phase."},{"Start":"06:53.055 ","End":"06:54.660","Text":"When the sources were in phase,"},{"Start":"06:54.660 ","End":"06:56.670","Text":"our maximum line was in the middle,"},{"Start":"06:56.670 ","End":"07:01.550","Text":"and then to either side of our maximum line of zeroth-order,"},{"Start":"07:01.550 ","End":"07:03.125","Text":"we had minimum lines."},{"Start":"07:03.125 ","End":"07:07.850","Text":"Here, it\u0027s the exact opposite because the 2 sources are out of phase."},{"Start":"07:07.850 ","End":"07:13.095","Text":"Now we have this minimum line of zeroth-order instead."},{"Start":"07:13.095 ","End":"07:14.985","Text":"Now let\u0027s carry on."},{"Start":"07:14.985 ","End":"07:18.485","Text":"Here we have the next minimum line over here"},{"Start":"07:18.485 ","End":"07:22.910","Text":"is where the pink trough meets the green peak and so on."},{"Start":"07:22.910 ","End":"07:30.990","Text":"I\u0027m just going to follow off again everywhere where a dotted line meets a solid line."},{"Start":"07:31.000 ","End":"07:38.460","Text":"Also of course, over here can see that we\u0027re"},{"Start":"07:38.460 ","End":"07:47.340","Text":"just going along all of these points and now we can connect all of these points."},{"Start":"07:47.690 ","End":"07:54.750","Text":"Here we have a minimum line of first-order."},{"Start":"07:54.750 ","End":"07:58.865","Text":"Similarly, we\u0027ll have over here a peak and a trough."},{"Start":"07:58.865 ","End":"08:02.340","Text":"We\u0027re just following all of this,"},{"Start":"08:03.320 ","End":"08:08.950","Text":"like so, and also down here."},{"Start":"08:09.500 ","End":"08:14.145","Text":"Just following this pattern."},{"Start":"08:14.145 ","End":"08:21.120","Text":"Now we can go like so and this is also a minimum line of first-order."},{"Start":"08:21.710 ","End":"08:28.485","Text":"Now let\u0027s find the next maximum line."},{"Start":"08:28.485 ","End":"08:33.465","Text":"Again, this is where peaks meet peaks and troughs meet troughs."},{"Start":"08:33.465 ","End":"08:34.895","Text":"I\u0027m just drawing this in."},{"Start":"08:34.895 ","End":"08:39.525","Text":"Now it\u0027s important to note that a computer generated this image,"},{"Start":"08:39.525 ","End":"08:46.695","Text":"not in the most ideal or exact way."},{"Start":"08:46.695 ","End":"08:52.760","Text":"We can see that some of these points aren\u0027t exactly in a line like this."},{"Start":"08:52.760 ","End":"08:57.380","Text":"We\u0027re just going to ignore that because in the lab you\u0027ll see this as"},{"Start":"08:57.380 ","End":"09:01.910","Text":"just a straight curve and not with little points jutting out."},{"Start":"09:01.910 ","End":"09:04.325","Text":"Let\u0027s just ignore that."},{"Start":"09:04.325 ","End":"09:07.670","Text":"This is the second-order maximum line and let\u0027s just pretend that"},{"Start":"09:07.670 ","End":"09:10.950","Text":"all of these points sit on all of the lines."},{"Start":"09:10.950 ","End":"09:17.700","Text":"Then also on this side of maximum order is going to be over here, here."},{"Start":"09:21.270 ","End":"09:24.505","Text":"All of these points, again,"},{"Start":"09:24.505 ","End":"09:34.045","Text":"ignoring the computer has drawn these waves not exactly on the same line."},{"Start":"09:34.045 ","End":"09:37.705","Text":"Then this is the mirror image."},{"Start":"09:37.705 ","End":"09:41.935","Text":"This is also maximum line of second-order."},{"Start":"09:41.935 ","End":"09:44.380","Text":"Then, we can see over here,"},{"Start":"09:44.380 ","End":"09:47.275","Text":"and this happens with sources that are out of phase."},{"Start":"09:47.275 ","End":"09:51.775","Text":"We can see that here we have a trough and a trough, peak and the peak."},{"Start":"09:51.775 ","End":"09:58.120","Text":"We can see it here, that here we have this straight line"},{"Start":"09:58.120 ","End":"10:04.885","Text":"that goes like so and this is a maximum line of third order."},{"Start":"10:04.885 ","End":"10:06.970","Text":"Also over here, we can see that we have"},{"Start":"10:06.970 ","End":"10:09.175","Text":"peaks and peaks and troughs and troughs together."},{"Start":"10:09.175 ","End":"10:10.540","Text":"Then, due to the computer,"},{"Start":"10:10.540 ","End":"10:14.320","Text":"we can see that they\u0027re not exactly connected,"},{"Start":"10:14.320 ","End":"10:15.865","Text":"but we know that they\u0027re meant to be."},{"Start":"10:15.865 ","End":"10:19.585","Text":"This is just the fault of the computer."},{"Start":"10:19.585 ","End":"10:23.860","Text":"This is also a maximum line of third order."},{"Start":"10:23.860 ","End":"10:26.530","Text":"Now let\u0027s go back to our minimum lines."},{"Start":"10:26.530 ","End":"10:29.410","Text":"We\u0027re looking for a minimum line over here,"},{"Start":"10:29.410 ","End":"10:31.090","Text":"so a peak and a trough."},{"Start":"10:31.090 ","End":"10:34.295","Text":"Any point that has a solid line of 1 color"},{"Start":"10:34.295 ","End":"10:39.060","Text":"crisscrossed or meeting a dotted line of the other color."},{"Start":"10:39.060 ","End":"10:42.840","Text":"Again, we can see that these points aren\u0027t sitting on the line,"},{"Start":"10:42.840 ","End":"10:45.910","Text":"but just imagine that they are."},{"Start":"10:46.260 ","End":"10:53.840","Text":"Then, this goes over here and I\u0027m just following these points."},{"Start":"10:54.600 ","End":"10:59.515","Text":"Then this connects over here."},{"Start":"10:59.515 ","End":"11:05.635","Text":"This is a minimum line of second-order because this was the first-order."},{"Start":"11:05.635 ","End":"11:10.310","Text":"Then we\u0027re also going to have a second-order over here."},{"Start":"11:11.910 ","End":"11:21.590","Text":"All of these points and over here."},{"Start":"11:28.050 ","End":"11:34.790","Text":"Everywhere where a peak meets a trough."},{"Start":"11:35.100 ","End":"11:43.600","Text":"Again, we\u0027re going to pretend that all of these dots stand on the line."},{"Start":"11:43.600 ","End":"11:48.295","Text":"This is also a minimum line of second-order."},{"Start":"11:48.295 ","End":"11:54.080","Text":"Then that\u0027s it. We can see that there aren\u0027t any more minimum lines."},{"Start":"11:55.560 ","End":"11:58.960","Text":"When our sources were in phase,"},{"Start":"11:58.960 ","End":"12:04.675","Text":"we saw that when a peak in green comes in,"},{"Start":"12:04.675 ","End":"12:08.380","Text":"then a peak in pink comes in."},{"Start":"12:08.380 ","End":"12:15.565","Text":"Then that means that we have a maximum line."},{"Start":"12:15.565 ","End":"12:21.085","Text":"Then, in this case,"},{"Start":"12:21.085 ","End":"12:23.590","Text":"our change in lambda our delta wavelength,"},{"Start":"12:23.590 ","End":"12:26.305","Text":"was some kind of multiple N,"},{"Start":"12:26.305 ","End":"12:30.625","Text":"a whole number of lambda."},{"Start":"12:30.625 ","End":"12:38.665","Text":"This means that when the sources were in phase that we had constructive interference."},{"Start":"12:38.665 ","End":"12:40.480","Text":"Whereas if we had,"},{"Start":"12:40.480 ","End":"12:42.445","Text":"when the sources were in phase,"},{"Start":"12:42.445 ","End":"12:47.005","Text":"destructive interference so that was a case like this."},{"Start":"12:47.005 ","End":"12:51.055","Text":"Then we know that we had a minimum line."},{"Start":"12:51.055 ","End":"12:53.110","Text":"When our sources were in phase,"},{"Start":"12:53.110 ","End":"12:59.410","Text":"the way that we define this minimum line was that the change in distance traveled,"},{"Start":"12:59.410 ","End":"13:00.625","Text":"let\u0027s actually do this."},{"Start":"13:00.625 ","End":"13:04.945","Text":"Delta d gave the change in distance traveled,"},{"Start":"13:04.945 ","End":"13:15.670","Text":"was equal to some kind of number multiplied by a 1/2 times the wavelength."},{"Start":"13:15.670 ","End":"13:19.150","Text":"It was a 1/2 a wavelength or 1.5 wavelengths,"},{"Start":"13:19.150 ","End":"13:23.260","Text":"so 8.5 wavelengths. But it was always a 0.5."},{"Start":"13:23.260 ","End":"13:28.850","Text":"Whereas in the maximum lines it was always a whole number."},{"Start":"13:33.390 ","End":"13:37.270","Text":"Now, when the sources are out of phase."},{"Start":"13:37.270 ","End":"13:43.275","Text":"What once defined the maximum line when the sources are in phase,"},{"Start":"13:43.275 ","End":"13:48.810","Text":"it now defines the minimum line when the sources are out of phase."},{"Start":"13:48.810 ","End":"13:50.940","Text":"When the sources are out of phase,"},{"Start":"13:50.940 ","End":"13:55.600","Text":"we\u0027ll have a green coming in as a peak and the pink,"},{"Start":"13:55.600 ","End":"13:58.315","Text":"let\u0027s say coming in as a trough."},{"Start":"13:58.315 ","End":"14:03.315","Text":"This we know is a minimum line,"},{"Start":"14:03.315 ","End":"14:12.500","Text":"but it is defined as the difference in distance traveled of the waves is equal to,"},{"Start":"14:12.500 ","End":"14:16.405","Text":"so here this same definition stands,"},{"Start":"14:16.405 ","End":"14:19.060","Text":"n multiplied by lambda."},{"Start":"14:19.060 ","End":"14:22.120","Text":"When we have a source out of phase."},{"Start":"14:22.120 ","End":"14:24.700","Text":"How it\u0027s defined is this n,"},{"Start":"14:24.700 ","End":"14:27.950","Text":"a whole number multiplied by lambda."},{"Start":"14:28.380 ","End":"14:32.335","Text":"Similarly, when our sources are out of phase,"},{"Start":"14:32.335 ","End":"14:36.220","Text":"once defined for sources in phase,"},{"Start":"14:36.220 ","End":"14:42.420","Text":"the minimum line, now defines the maximum line for sources in phase."},{"Start":"14:42.420 ","End":"14:44.310","Text":"If we have a green coming in as"},{"Start":"14:44.310 ","End":"14:48.495","Text":"a peak and the pink also coming in as a peak at obviously,"},{"Start":"14:48.495 ","End":"14:53.210","Text":"these are interchangeable where they\u0027re coming in as troughs."},{"Start":"14:53.210 ","End":"14:56.380","Text":"Then we know that this is constructive interference,"},{"Start":"14:56.380 ","End":"14:57.820","Text":"so it\u0027s a maximum line."},{"Start":"14:57.820 ","End":"15:00.175","Text":"But when the sources are out of phase,"},{"Start":"15:00.175 ","End":"15:04.195","Text":"what once defined the minimum line when the source in phase,"},{"Start":"15:04.195 ","End":"15:08.380","Text":"now defines the maximum line when the source is out of phase."},{"Start":"15:08.380 ","End":"15:15.490","Text":"The difference in distance traveled is some multiple of a 1/2 multiplied by lambda."},{"Start":"15:15.490 ","End":"15:21.230","Text":"1.5, 10.5, 1000.5, 8.5."},{"Start":"15:22.020 ","End":"15:28.015","Text":"Now we can see why our interference pattern has switched over."},{"Start":"15:28.015 ","End":"15:30.700","Text":"Where now we have the minimum line of zeroth-order in"},{"Start":"15:30.700 ","End":"15:35.300","Text":"the middle and the maximum on either side."},{"Start":"15:36.150 ","End":"15:39.445","Text":"Now I\u0027m going to rub out what we see over here"},{"Start":"15:39.445 ","End":"15:42.625","Text":"and I want to speak to you a little bit about"},{"Start":"15:42.625 ","End":"15:50.380","Text":"the condition that has to be met in order to get some kind of interference pattern."},{"Start":"15:50.380 ","End":"15:58.075","Text":"Interference patterns need to have a constant phase difference."},{"Start":"15:58.075 ","End":"16:01.280","Text":"Let\u0027s speak about this."},{"Start":"16:01.470 ","End":"16:06.909","Text":"Interference patterns need constant phase difference"},{"Start":"16:06.909 ","End":"16:10.765","Text":"in order to exist. What does that mean?"},{"Start":"16:10.765 ","End":"16:16.190","Text":"Let\u0027s say we have this green wave traveling like so."},{"Start":"16:18.540 ","End":"16:23.185","Text":"What\u0027s important, it doesn\u0027t matter if the pink wave is in phase with it."},{"Start":"16:23.185 ","End":"16:27.130","Text":"It doesn\u0027t matter if the pink wave has peaks exactly"},{"Start":"16:27.130 ","End":"16:31.765","Text":"where the green wave has peaks and troughs exactly where the green wave had troughs."},{"Start":"16:31.765 ","End":"16:35.065","Text":"It doesn\u0027t matter if the pink wave is completely out of phase."},{"Start":"16:35.065 ","End":"16:38.260","Text":"If it\u0027s the opposite like so,"},{"Start":"16:38.260 ","End":"16:43.240","Text":"where a peak in the pink is where a trough is in the green,"},{"Start":"16:43.240 ","End":"16:47.710","Text":"so none of this matters for an interference pattern."},{"Start":"16:47.710 ","End":"16:53.125","Text":"What does matter is that the phase difference is constant."},{"Start":"16:53.125 ","End":"16:59.335","Text":"If we have our pink wave traveling like so,"},{"Start":"16:59.335 ","End":"17:01.960","Text":"we can see it\u0027s very similar to the green wave,"},{"Start":"17:01.960 ","End":"17:04.090","Text":"just it\u0027s shifted a little bit."},{"Start":"17:04.090 ","End":"17:07.450","Text":"We can see that here the green has a peak,"},{"Start":"17:07.450 ","End":"17:09.565","Text":"but here the pink has a peak."},{"Start":"17:09.565 ","End":"17:14.739","Text":"The green wave arrives at a peak at a slightly different time than the pink wave,"},{"Start":"17:14.739 ","End":"17:18.010","Text":"so there\u0027s this delta t over here."},{"Start":"17:18.010 ","End":"17:23.905","Text":"As long as this delta t remains constant all the time."},{"Start":"17:23.905 ","End":"17:29.395","Text":"The time difference between the green being at a peak and the pink being at a peak,"},{"Start":"17:29.395 ","End":"17:32.710","Text":"so as long as this remains constant,"},{"Start":"17:32.710 ","End":"17:35.995","Text":"we will get some interference pattern."},{"Start":"17:35.995 ","End":"17:40.570","Text":"Of course, it will look different and it can look slightly irregular."},{"Start":"17:40.570 ","End":"17:44.290","Text":"I\u0027m just drawing, don\u0027t focus on what the picture looks like,"},{"Start":"17:44.290 ","End":"17:51.655","Text":"but just imagine that here we would have some kind of maximum line of zeroth-order."},{"Start":"17:51.655 ","End":"17:53.095","Text":"Let\u0027s imagine."},{"Start":"17:53.095 ","End":"17:56.335","Text":"Then we would have another 1 here."},{"Start":"17:56.335 ","End":"18:00.565","Text":"It would be quite irregular to what we were used to before."},{"Start":"18:00.565 ","End":"18:03.895","Text":"This would be our maximum of first-order."},{"Start":"18:03.895 ","End":"18:10.870","Text":"Then here, we would have something like this for a maximum of first-order on this side."},{"Start":"18:10.870 ","End":"18:19.345","Text":"We\u0027ll have a very irregular-looking interference pattern."},{"Start":"18:19.345 ","End":"18:24.730","Text":"However, we\u0027ll still get an interference pattern so long as"},{"Start":"18:24.730 ","End":"18:31.465","Text":"this time interval or this phase difference between the 2 waves is even."},{"Start":"18:31.465 ","End":"18:38.965","Text":"Then between every 2 maximum lines will have these minimum lines."},{"Start":"18:38.965 ","End":"18:42.230","Text":"Here would be the minimum line of first-order and here we would"},{"Start":"18:42.230 ","End":"18:46.590","Text":"have minimum of second-order."},{"Start":"18:46.590 ","End":"18:53.055","Text":"Then here we would have another minimum of first-order."},{"Start":"18:53.055 ","End":"18:57.550","Text":"Here, another minimum of second-order."},{"Start":"18:57.550 ","End":"19:02.808","Text":"We still get this diffraction pattern or"},{"Start":"19:02.808 ","End":"19:05.615","Text":"this interference pattern but it looks a bit weird"},{"Start":"19:05.615 ","End":"19:08.915","Text":"because the waves aren\u0027t completely in phase,"},{"Start":"19:08.915 ","End":"19:11.090","Text":"or the sources aren\u0027t completely in phase,"},{"Start":"19:11.090 ","End":"19:13.550","Text":"but they\u0027re also not completely out of phase."},{"Start":"19:13.550 ","End":"19:15.920","Text":"We\u0027ll get something. If however,"},{"Start":"19:15.920 ","End":"19:18.110","Text":"this delta t is changing."},{"Start":"19:18.110 ","End":"19:22.820","Text":"If we have a green wave coming in like so,"},{"Start":"19:22.820 ","End":"19:28.085","Text":"and then suddenly it starts coming in like this."},{"Start":"19:28.085 ","End":"19:33.800","Text":"Then we can see that we\u0027ll have"},{"Start":"19:33.800 ","End":"19:39.515","Text":"a very messy diffraction pattern because sometimes they\u0027re in phase,"},{"Start":"19:39.515 ","End":"19:42.745","Text":"sometimes they\u0027re completely out of phase."},{"Start":"19:42.745 ","End":"19:45.200","Text":"Sometimes they\u0027re in the middle and in this type of"},{"Start":"19:45.200 ","End":"19:48.965","Text":"case when there\u0027s no delta t that\u0027s constant,"},{"Start":"19:48.965 ","End":"19:52.865","Text":"we won\u0027t get a diffraction pattern at all."},{"Start":"19:52.865 ","End":"19:56.230","Text":"That\u0027s the end of this lesson."}],"ID":12496},{"Watched":false,"Name":"Generating Out of Phase Interference Patterns","Duration":"4m 42s","ChapterTopicVideoID":12029,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.845","Text":"Hello. In this lesson,"},{"Start":"00:01.845 ","End":"00:04.590","Text":"we\u0027re going to be speaking about how to form or how to"},{"Start":"00:04.590 ","End":"00:08.430","Text":"generate out-of-phase interference patterns."},{"Start":"00:08.430 ","End":"00:11.610","Text":"A few lessons ago, we spoke about generating"},{"Start":"00:11.610 ","End":"00:15.015","Text":"interference patterns with sources that are in phase,"},{"Start":"00:15.015 ","End":"00:18.420","Text":"so here we\u0027re speaking about sources that are out of phase."},{"Start":"00:18.420 ","End":"00:22.650","Text":"The first way that we can do this"},{"Start":"00:22.650 ","End":"00:29.160","Text":"is if we take our body of water over here and then,"},{"Start":"00:29.160 ","End":"00:34.950","Text":"where we had the in-phase interference pattern generator,"},{"Start":"00:34.950 ","End":"00:39.020","Text":"so we had a pin over here and the pen over here,"},{"Start":"00:39.020 ","End":"00:42.545","Text":"and both of them struck the water at the same time,"},{"Start":"00:42.545 ","End":"00:45.390","Text":"so this was in phase."},{"Start":"00:45.590 ","End":"00:48.470","Text":"When we\u0027re speaking about out-of-phase,"},{"Start":"00:48.470 ","End":"00:54.269","Text":"we have something in this shape where at 1 time,"},{"Start":"00:54.269 ","End":"01:00.045","Text":"the green pen hits the water and the red pen or the pink pen is up,"},{"Start":"01:00.045 ","End":"01:08.010","Text":"and half a wavelength later or half a period later, so here,"},{"Start":"01:08.010 ","End":"01:12.750","Text":"it\u0027s half a period later,"},{"Start":"01:12.750 ","End":"01:17.840","Text":"remember that period is represented by the letter T,"},{"Start":"01:17.840 ","End":"01:19.910","Text":"half a period later,"},{"Start":"01:19.910 ","End":"01:28.145","Text":"the pink pen will be in the water and the green pen will be up over her."},{"Start":"01:28.145 ","End":"01:36.065","Text":"This is the first way of forming out of phase interference patterns."},{"Start":"01:36.065 ","End":"01:42.470","Text":"The second way that we can do this is by using diffraction."},{"Start":"01:42.470 ","End":"01:45.470","Text":"Diffraction we spoke about in"},{"Start":"01:45.470 ","End":"01:49.205","Text":"the video dealing with forming in phase interference patterns,"},{"Start":"01:49.205 ","End":"01:51.890","Text":"so you can look a few lessons back,"},{"Start":"01:51.890 ","End":"01:57.585","Text":"but what we had when we were doing the in-phase."},{"Start":"01:57.585 ","End":"02:04.650","Text":"We had straight wavefronts that were traveling like so."},{"Start":"02:04.650 ","End":"02:10.110","Text":"Then we had some barrier that had"},{"Start":"02:10.110 ","End":"02:16.520","Text":"1 or 2 holes in it and then we saw that after the barrier,"},{"Start":"02:16.520 ","End":"02:22.130","Text":"we had circular wavefronts that were formed,"},{"Start":"02:22.130 ","End":"02:28.235","Text":"something like this and then we got the diffraction pattern."},{"Start":"02:28.235 ","End":"02:33.990","Text":"This was for in-phase as well,"},{"Start":"02:35.500 ","End":"02:39.905","Text":"I\u0027ve just highlighted, this is what we\u0027re looking at for the out-of-phase."},{"Start":"02:39.905 ","End":"02:45.980","Text":"Now let\u0027s speak about the how to use diffraction to form in phase."},{"Start":"02:45.980 ","End":"02:52.400","Text":"What we\u0027ll do is if we again have a barrier with 2 holes in it,"},{"Start":"02:52.400 ","End":"02:57.825","Text":"so what we\u0027ll do is we\u0027ll still send in the straight wavefronts, however,"},{"Start":"02:57.825 ","End":"03:03.990","Text":"we\u0027ll send them in at an angle."},{"Start":"03:03.990 ","End":"03:15.220","Text":"They\u0027ll be traveling like so and then we can see that over here,"},{"Start":"03:15.620 ","End":"03:19.650","Text":"a wavefront is hitting this barrier,"},{"Start":"03:19.650 ","End":"03:23.960","Text":"but the wavefront still hasn\u0027t hit this barrier so that"},{"Start":"03:23.960 ","End":"03:29.045","Text":"means that we\u0027re going to have these pink wavefronts,"},{"Start":"03:29.045 ","End":"03:32.010","Text":"circular wavefronts forming spreading out."},{"Start":"03:32.010 ","End":"03:39.640","Text":"Then a few moments later or rather half a wavelength later,"},{"Start":"03:39.700 ","End":"03:44.225","Text":"so imagine that this represents half a wavelength,"},{"Start":"03:44.225 ","End":"03:50.195","Text":"this distance over here so then the part of the wave is going to hit"},{"Start":"03:50.195 ","End":"03:53.270","Text":"this hole in the barrier and then we\u0027re going to"},{"Start":"03:53.270 ","End":"03:57.259","Text":"get these blue wavefronts that are of course,"},{"Start":"03:57.259 ","End":"04:02.130","Text":"going to interfere with 1 another."},{"Start":"04:02.590 ","End":"04:09.260","Text":"What\u0027s important to note is that the difference between the distance,"},{"Start":"04:09.260 ","End":"04:12.485","Text":"the distance difference between this wavefront in black"},{"Start":"04:12.485 ","End":"04:16.309","Text":"and this wavefront and black is this half a wavelength,"},{"Start":"04:16.309 ","End":"04:22.525","Text":"whereas in general, the distance between each wavefront is a wavelength."},{"Start":"04:22.525 ","End":"04:26.870","Text":"First we\u0027re going to get waves form a circular wave fronts"},{"Start":"04:26.870 ","End":"04:30.990","Text":"forming over here and half a wavelength later or out-of-phase,"},{"Start":"04:30.990 ","End":"04:34.280","Text":"we\u0027re going to get circular wave fronts forming over here and then"},{"Start":"04:34.280 ","End":"04:38.750","Text":"we have an out-of-phase interference pattern."},{"Start":"04:38.750 ","End":"04:42.030","Text":"That\u0027s the end of this lesson."}],"ID":12497},{"Watched":false,"Name":"Exercise - Interference 1","Duration":"12m 7s","ChapterTopicVideoID":12030,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"Hello. In this question,"},{"Start":"00:02.100 ","End":"00:06.300","Text":"we\u0027re being told that 2 wave sources are located at 0,0 and at"},{"Start":"00:06.300 ","End":"00:10.740","Text":"5,0 and that these values are given in centimeters."},{"Start":"00:10.740 ","End":"00:17.018","Text":"They both have the same phase and generate waves with a wavelength of 2 centimeters,"},{"Start":"00:17.018 ","End":"00:23.485","Text":"where on the y-axis will there be a first-order constructive interference."},{"Start":"00:23.485 ","End":"00:29.700","Text":"First of all, let\u0027s write out our equation for first-order constructive interference,"},{"Start":"00:29.700 ","End":"00:32.530","Text":"or for constructive interference in general,"},{"Start":"00:32.530 ","End":"00:42.935","Text":"we know that it\u0027s the absolute value of S_2P minus S_1P is equal to nLambda."},{"Start":"00:42.935 ","End":"00:48.030","Text":"Now here we\u0027re being told that n is equal to 1 because we\u0027re dealing"},{"Start":"00:48.030 ","End":"00:53.550","Text":"with first-order and that Lambda is equal to 2 because that\u0027s the wavelength."},{"Start":"00:53.550 ","End":"01:00.295","Text":"We\u0027re trying to find where S_2P minus S_1P is equal to 2."},{"Start":"01:00.295 ","End":"01:05.689","Text":"Now, let\u0027s draw out a graph with some axes."},{"Start":"01:05.689 ","End":"01:07.220","Text":"This is the x-axis,"},{"Start":"01:07.220 ","End":"01:08.825","Text":"this is the y-axis,"},{"Start":"01:08.825 ","End":"01:12.680","Text":"and then we have our source, S_1,"},{"Start":"01:12.680 ","End":"01:19.960","Text":"which is located at 0,0 and we have our second source,"},{"Start":"01:19.970 ","End":"01:24.600","Text":"S_2, which is located at 5,0."},{"Start":"01:24.600 ","End":"01:28.470","Text":"Now, we\u0027re trying to find where on the y-axis."},{"Start":"01:28.470 ","End":"01:30.890","Text":"Let\u0027s call this point P,"},{"Start":"01:30.890 ","End":"01:33.965","Text":"which is located at 0,y,"},{"Start":"01:33.965 ","End":"01:35.360","Text":"where we\u0027re trying to find this y."},{"Start":"01:35.360 ","End":"01:40.405","Text":"We\u0027re going to have this first-order constructive interference."},{"Start":"01:40.405 ","End":"01:43.835","Text":"Now, notice that 0, y,"},{"Start":"01:43.835 ","End":"01:47.960","Text":"point P can be in the positive section of the y-axis,"},{"Start":"01:47.960 ","End":"01:53.105","Text":"but it could also be on the negative section of the y-axis."},{"Start":"01:53.105 ","End":"02:02.140","Text":"Point P could also be at 0 with some value of y, which is negative."},{"Start":"02:02.140 ","End":"02:09.890","Text":"Let\u0027s just erase that and we can therefore say that where"},{"Start":"02:09.890 ","End":"02:20.075","Text":"y can be either less than 0 or y can be greater than 0,"},{"Start":"02:20.075 ","End":"02:22.340","Text":"or y can also be 0."},{"Start":"02:22.340 ","End":"02:26.330","Text":"It probably isn\u0027t, but you should always know that that option exists."},{"Start":"02:26.330 ","End":"02:30.650","Text":"What we\u0027re trying to find is what this value for y can be."},{"Start":"02:30.650 ","End":"02:32.410","Text":"If there\u0027s 1 value, 2 values,"},{"Start":"02:32.410 ","End":"02:35.930","Text":"or 3 values that are available."},{"Start":"02:35.930 ","End":"02:39.020","Text":"Let\u0027s do this."},{"Start":"02:39.020 ","End":"02:48.010","Text":"The distance from S_1 to P is equal to?"},{"Start":"02:48.010 ","End":"02:50.425","Text":"Let\u0027s do the full equation."},{"Start":"02:50.425 ","End":"02:53.290","Text":"We\u0027re finding this distance over here."},{"Start":"02:53.290 ","End":"03:00.800","Text":"It\u0027s going to be x of P minus x of source 1"},{"Start":"03:00.800 ","End":"03:09.265","Text":"squared plus y of p minus the y value of source 1 squared,"},{"Start":"03:09.265 ","End":"03:12.470","Text":"the square root of all of this."},{"Start":"03:13.070 ","End":"03:18.375","Text":"Now let\u0027s plug in our values."},{"Start":"03:18.375 ","End":"03:25.555","Text":"We have the square root of x of p is equal to 0 minus x of S_1,"},{"Start":"03:25.555 ","End":"03:30.380","Text":"which is also equal to 0 squared plus y of p,"},{"Start":"03:30.380 ","End":"03:34.430","Text":"some y-value that we don\u0027t know if it\u0027s positive or negative,"},{"Start":"03:34.430 ","End":"03:37.580","Text":"minus the y value of 1,"},{"Start":"03:37.580 ","End":"03:42.455","Text":"which is 0 squared and the square root of all of that."},{"Start":"03:42.455 ","End":"03:47.120","Text":"0 squared is 0 plus y minus 0 squared is"},{"Start":"03:47.120 ","End":"03:53.425","Text":"y^2 and then this is equal to the square root of y^2."},{"Start":"03:53.425 ","End":"03:57.770","Text":"Here is something that\u0027s important to note."},{"Start":"03:57.770 ","End":"04:03.005","Text":"Whenever you take the square root of a square number or the square of a square root,"},{"Start":"04:03.005 ","End":"04:08.870","Text":"know that this value could be either a positive or a negative,"},{"Start":"04:08.870 ","End":"04:14.420","Text":"so we can have either a positive or a negative y. Y is this."},{"Start":"04:14.420 ","End":"04:17.480","Text":"Let\u0027s just write a little note over here."},{"Start":"04:17.480 ","End":"04:21.860","Text":"If we take negative y and we square it,"},{"Start":"04:21.860 ","End":"04:24.365","Text":"this will be equal to y^2."},{"Start":"04:24.365 ","End":"04:28.265","Text":"If we take positive y and square it,"},{"Start":"04:28.265 ","End":"04:30.980","Text":"this is also equal to y-squared."},{"Start":"04:30.980 ","End":"04:34.160","Text":"Therefore, if you take the square root of this"},{"Start":"04:34.160 ","End":"04:42.050","Text":"y-squared you could either get that the square root of y^2 is positive y,"},{"Start":"04:42.050 ","End":"04:46.265","Text":"or you could get the square root of y^2 is negative y."},{"Start":"04:46.265 ","End":"04:49.370","Text":"Here, we\u0027re taking the square root of y^2,"},{"Start":"04:49.370 ","End":"04:50.829","Text":"this exact situation,"},{"Start":"04:50.829 ","End":"04:56.510","Text":"and we don\u0027t know if the y started off as a negative or a positive,"},{"Start":"04:56.510 ","End":"05:00.125","Text":"so we have to take into account both of these cases."},{"Start":"05:00.125 ","End":"05:04.185","Text":"Mathematically, this is correct as we saw over here."},{"Start":"05:04.185 ","End":"05:13.505","Text":"Therefore, we have that the distance of S_1 to P is positive, negative y."},{"Start":"05:13.505 ","End":"05:16.355","Text":"Now let\u0027s do the same for S_2P."},{"Start":"05:16.355 ","End":"05:20.625","Text":"The distance between S and P. Again,"},{"Start":"05:20.625 ","End":"05:29.230","Text":"this is x_p minus x_2 so the x value so x_2 squared plus the y value of"},{"Start":"05:29.230 ","End":"05:33.310","Text":"point P minus the y value of the second source"},{"Start":"05:33.310 ","End":"05:39.175","Text":"squared and the square root of all of this and now let\u0027s plug in all of our numbers."},{"Start":"05:39.175 ","End":"05:44.665","Text":"The x value of point P is 0 minus the x value of S_2,"},{"Start":"05:44.665 ","End":"05:47.125","Text":"which is 5 squared,"},{"Start":"05:47.125 ","End":"05:49.810","Text":"plus the y value of point P,"},{"Start":"05:49.810 ","End":"05:52.070","Text":"which is some y,"},{"Start":"05:52.070 ","End":"05:57.235","Text":"minus the y value of 2, which is 0."},{"Start":"05:57.235 ","End":"05:59.169","Text":"All of this is squared,"},{"Start":"05:59.169 ","End":"06:01.915","Text":"and we take the square root of all of this."},{"Start":"06:01.915 ","End":"06:05.100","Text":"First of all, here,"},{"Start":"06:05.100 ","End":"06:11.555","Text":"we can see that this is going to be the square root of 0 minus 5 is negative 5."},{"Start":"06:11.555 ","End":"06:14.240","Text":"Negative 5 squared is 25."},{"Start":"06:14.240 ","End":"06:19.040","Text":"Plus here, we have y minus 0 squared,"},{"Start":"06:19.040 ","End":"06:24.570","Text":"so y minus 0 is y but also if it\u0027s a negative y,"},{"Start":"06:24.570 ","End":"06:32.404","Text":"it would still be negative y-squared will still be y^2."},{"Start":"06:32.404 ","End":"06:33.920","Text":"Here, it doesn\u0027t make a difference."},{"Start":"06:33.920 ","End":"06:35.930","Text":"If inside this bracket it\u0027s negative y,"},{"Start":"06:35.930 ","End":"06:41.275","Text":"y^2, it\u0027s still y^2."},{"Start":"06:41.275 ","End":"06:46.335","Text":"Here we have what our S_2P is equal 2."},{"Start":"06:46.335 ","End":"06:50.660","Text":"Now, let\u0027s scroll down a little bit more to give us some more space."},{"Start":"06:50.660 ","End":"06:54.005","Text":"Now, let\u0027s plug in the values to this equation."},{"Start":"06:54.005 ","End":"06:58.190","Text":"S_2P is the square root of"},{"Start":"06:58.190 ","End":"07:07.240","Text":"25 plus y\u00262, minus S_1P."},{"Start":"07:07.240 ","End":"07:09.780","Text":"We have 2 equations."},{"Start":"07:09.780 ","End":"07:15.270","Text":"We have minus, let\u0027s use the positive y, so minus y."},{"Start":"07:15.290 ","End":"07:20.780","Text":"Because it\u0027s minus positive y is equal to 2."},{"Start":"07:20.780 ","End":"07:26.315","Text":"This is our first equation and our second equation is S_2P,"},{"Start":"07:26.315 ","End":"07:30.901","Text":"which is the square root of 25 plus y^2,"},{"Start":"07:30.901 ","End":"07:35.625","Text":"minus and this time we\u0027ll use this 1"},{"Start":"07:35.625 ","End":"07:42.510","Text":"minus negative y. K is equal to 2."},{"Start":"07:42.510 ","End":"07:48.305","Text":"Positive y and negative y fit S_1P."},{"Start":"07:48.305 ","End":"07:53.120","Text":"Let\u0027s now scroll down and let\u0027s solve each of these equations."},{"Start":"07:53.120 ","End":"07:56.190","Text":"First, let\u0027s solve equation number 1."},{"Start":"07:56.570 ","End":"07:59.300","Text":"Here, we have the square root of"},{"Start":"07:59.300 ","End":"08:08.810","Text":"25 plus y^2 minus positive y is just minus y is equal to 2."},{"Start":"08:08.810 ","End":"08:14.270","Text":"Now, let\u0027s add y to both sides so we\u0027ll get that the square root of"},{"Start":"08:14.270 ","End":"08:21.345","Text":"25 plus y^2 is equal to y plus 2."},{"Start":"08:21.345 ","End":"08:24.780","Text":"Now, let\u0027s square both sides."},{"Start":"08:24.780 ","End":"08:28.834","Text":"If we square both sides,"},{"Start":"08:28.834 ","End":"08:31.280","Text":"we\u0027re going to get here that the square root sign"},{"Start":"08:31.280 ","End":"08:34.790","Text":"cancels out so that\u0027s going to be 25 plus y^2"},{"Start":"08:34.790 ","End":"08:41.660","Text":"and that here we\u0027re doing y plus 2 squared so the right hand side,"},{"Start":"08:41.660 ","End":"08:46.345","Text":"which is simply equal to y plus 2 multiplied by y plus 2."},{"Start":"08:46.345 ","End":"08:49.830","Text":"Then we just open up the brackets."},{"Start":"08:49.830 ","End":"08:56.430","Text":"That\u0027s going to be equal to y^2 plus 4y plus 4."},{"Start":"08:56.430 ","End":"09:02.075","Text":"Now, what we\u0027re going to do is we\u0027re going to subtract y^2 from both sides."},{"Start":"09:02.075 ","End":"09:09.595","Text":"This is going to cancel out and we\u0027re going to subtract 4 from both sides."},{"Start":"09:09.595 ","End":"09:12.050","Text":"We\u0027ve done 2 things here."},{"Start":"09:12.050 ","End":"09:15.535","Text":"We\u0027ve subtracted y^2 and subtracted 4."},{"Start":"09:15.535 ","End":"09:20.510","Text":"We\u0027re going to have that 25 is equal to 4y."},{"Start":"09:20.510 ","End":"09:25.100","Text":"Therefore, if we isolate out our y because we\u0027re trying to find"},{"Start":"09:25.100 ","End":"09:31.445","Text":"where we\u0027ll have first order constructive interference in the y-axis."},{"Start":"09:31.445 ","End":"09:34.240","Text":"That means we have to isolate out y,"},{"Start":"09:34.240 ","End":"09:38.465","Text":"so I get that y is equal to 21 divided by 4."},{"Start":"09:38.465 ","End":"09:43.709","Text":"This is 1 answer."},{"Start":"09:43.900 ","End":"09:49.595","Text":"Now, let\u0027s do question or equation number 2 rather."},{"Start":"09:49.595 ","End":"09:57.470","Text":"Number 2 is we have that the square root of 25 plus y^2."},{"Start":"09:57.470 ","End":"10:01.620","Text":"Then we have negative, negative y."},{"Start":"10:01.620 ","End":"10:03.780","Text":"A negative and a negative is a positive."},{"Start":"10:03.780 ","End":"10:06.755","Text":"We just have plus y is equal to 2."},{"Start":"10:06.755 ","End":"10:09.740","Text":"Now again, we\u0027re going to,"},{"Start":"10:09.740 ","End":"10:12.050","Text":"instead of adding y to both sides,"},{"Start":"10:12.050 ","End":"10:14.645","Text":"we\u0027re going to subtract y from both sides."},{"Start":"10:14.645 ","End":"10:24.765","Text":"We\u0027re going to have that the square root of 25 plus y^2 is equal to 2 minus y."},{"Start":"10:24.765 ","End":"10:30.010","Text":"Now what we\u0027re going to do is we\u0027re going to square both sides."},{"Start":"10:30.010 ","End":"10:34.670","Text":"We\u0027re going to get that 25 plus y^2 is equal"},{"Start":"10:34.670 ","End":"10:41.485","Text":"to y^2 minus 4y plus 4."},{"Start":"10:41.485 ","End":"10:47.590","Text":"Now again, we\u0027re going to subtract y^2 from both sides"},{"Start":"10:47.590 ","End":"10:54.545","Text":"and we\u0027re going to subtract 4 from both sides just like before."},{"Start":"10:54.545 ","End":"11:01.685","Text":"What we\u0027re going to get is these will cancel out."},{"Start":"11:01.685 ","End":"11:09.535","Text":"Then we\u0027ll have that 21 is equal to negative 4y."},{"Start":"11:09.535 ","End":"11:19.090","Text":"Again, we want to isolate out our y. I get that y is equal to negative 21 divided by 4."},{"Start":"11:20.420 ","End":"11:25.220","Text":"These are the 2 possible points along the y-axis where we"},{"Start":"11:25.220 ","End":"11:29.585","Text":"can have constructive interference of the first-order."},{"Start":"11:29.585 ","End":"11:36.400","Text":"It\u0027s at y is it plus 21 divided by 4 or negative 21 divided by 4."},{"Start":"11:36.400 ","End":"11:41.254","Text":"If we scroll back up to our graph,"},{"Start":"11:41.254 ","End":"11:46.085","Text":"that means that we can get over here,"},{"Start":"11:46.085 ","End":"11:49.248","Text":"so at 0,"},{"Start":"11:49.248 ","End":"11:53.180","Text":"21 divided by 4 or we could get"},{"Start":"11:53.180 ","End":"11:56.195","Text":"constructive interference of first-order"},{"Start":"11:56.195 ","End":"11:59.960","Text":"in the mirror image and the negative y\u0027s section at 0,"},{"Start":"11:59.960 ","End":"12:03.370","Text":"negative 21 divided by 4."},{"Start":"12:03.370 ","End":"12:07.020","Text":"That\u0027s the end of this lesson."}],"ID":12498},{"Watched":false,"Name":"Exercise - Interference 2","Duration":"5m 45s","ChapterTopicVideoID":12031,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.085","Text":"Hello. In this question,"},{"Start":"00:02.085 ","End":"00:06.920","Text":"we\u0027re being given two identical wave sources that are placed at 0,"},{"Start":"00:06.920 ","End":"00:09.490","Text":"negative 5, and 0, 5."},{"Start":"00:09.710 ","End":"00:13.275","Text":"We\u0027re being told that at point 10, 10,"},{"Start":"00:13.275 ","End":"00:18.270","Text":"there is a first-order constructive interference."},{"Start":"00:18.270 ","End":"00:21.795","Text":"We\u0027re being asked to find the wavelength of the wave."},{"Start":"00:21.795 ","End":"00:29.730","Text":"Let\u0027s just sketch this out because it will make it easier to understand."},{"Start":"00:29.730 ","End":"00:33.735","Text":"We\u0027re being told that we have one source over here,"},{"Start":"00:33.735 ","End":"00:38.460","Text":"S_1, and this is at 0, 5."},{"Start":"00:38.460 ","End":"00:44.235","Text":"Then we have another source, S_2,"},{"Start":"00:44.235 ","End":"00:49.980","Text":"and this is at 0, negative 5."},{"Start":"00:49.980 ","End":"00:53.700","Text":"Then we\u0027re being told that at point 10, 10."},{"Start":"00:53.700 ","End":"00:57.200","Text":"Somewhere over here, let\u0027s say,"},{"Start":"00:57.200 ","End":"00:59.705","Text":"and let\u0027s call this point P,"},{"Start":"00:59.705 ","End":"01:02.840","Text":"which is at 10, 10."},{"Start":"01:02.840 ","End":"01:07.310","Text":"We have this first-order constructive interference."},{"Start":"01:07.310 ","End":"01:15.495","Text":"First of all, we know that our equation for constructive interference is S_2P minus S_1P."},{"Start":"01:15.495 ","End":"01:19.730","Text":"The distance between S_2 to the point minus the distance"},{"Start":"01:19.730 ","End":"01:24.240","Text":"between S_1 to the point is equal to n Lambda."},{"Start":"01:24.240 ","End":"01:29.435","Text":"Now n is equal to 1 because this is first-order constructive interference,"},{"Start":"01:29.435 ","End":"01:34.805","Text":"and Lambda this is what we\u0027re trying to find. This is what we don\u0027t know."},{"Start":"01:34.805 ","End":"01:39.270","Text":"Let\u0027s find out what S_1P is,"},{"Start":"01:39.270 ","End":"01:43.700","Text":"the distance between S_1 to P. The equation is"},{"Start":"01:43.700 ","End":"01:49.725","Text":"the square root of x_p minus x_1,"},{"Start":"01:49.725 ","End":"01:58.575","Text":"so the x-coordinate of source 1 plus y_p minus y_1."},{"Start":"01:58.575 ","End":"02:04.625","Text":"The y-coordinate of p minus the y-coordinate of source 1, squared."},{"Start":"02:04.625 ","End":"02:07.705","Text":"Then we take the square root of all of this."},{"Start":"02:07.705 ","End":"02:10.260","Text":"Now let\u0027s plug in the numbers,"},{"Start":"02:10.260 ","End":"02:16.080","Text":"x of point p is 10 minus x of S_1,"},{"Start":"02:16.080 ","End":"02:19.785","Text":"which is 0, the x coordinate."},{"Start":"02:19.785 ","End":"02:23.465","Text":"This squared plus the y coordinate of point P,"},{"Start":"02:23.465 ","End":"02:24.770","Text":"which is 10,"},{"Start":"02:24.770 ","End":"02:27.470","Text":"minus the y-coordinate of source 1,"},{"Start":"02:27.470 ","End":"02:30.395","Text":"which is 5, and then this squared,"},{"Start":"02:30.395 ","End":"02:33.060","Text":"and then we take the square root of that."},{"Start":"02:33.130 ","End":"02:37.715","Text":"This is equal to the square root of 10 minus 0 squared,"},{"Start":"02:37.715 ","End":"02:41.105","Text":"so 10^2, which is 100,"},{"Start":"02:41.105 ","End":"02:44.715","Text":"plus 10 and minus 5,"},{"Start":"02:44.715 ","End":"02:49.560","Text":"which is 5^2 which is 25."},{"Start":"02:49.560 ","End":"02:58.880","Text":"Then, the square root of 100 plus 25 is simply the square root of 125,"},{"Start":"02:58.880 ","End":"03:05.350","Text":"which is equal to 5 root 5."},{"Start":"03:06.110 ","End":"03:09.710","Text":"This, if we put this in decimal form,"},{"Start":"03:09.710 ","End":"03:16.285","Text":"is approximately equal to 11.18 centimeters."},{"Start":"03:16.285 ","End":"03:21.360","Text":"Now let\u0027s find the distance of S_2P."},{"Start":"03:21.360 ","End":"03:23.700","Text":"Again, we\u0027re doing the same thing."},{"Start":"03:23.700 ","End":"03:31.475","Text":"We have the square root of x_p minus the x coordinate of our 0.2"},{"Start":"03:31.475 ","End":"03:39.900","Text":"squared plus (y_p-y_2)^2 and square root of all of this."},{"Start":"03:39.900 ","End":"03:41.490","Text":"Let\u0027s plug in our numbers."},{"Start":"03:41.490 ","End":"03:45.885","Text":"We have the square root of x_p is 10,"},{"Start":"03:45.885 ","End":"03:53.870","Text":"and x coordinate of S_2 is 0 squared plus the y-coordinate of p,"},{"Start":"03:53.870 ","End":"03:57.230","Text":"which is 10 minus the y-coordinate of S_2,"},{"Start":"03:57.230 ","End":"04:00.235","Text":"which is minus 5,"},{"Start":"04:00.235 ","End":"04:05.045","Text":"and all of this is squared and the square root of this."},{"Start":"04:05.045 ","End":"04:10.640","Text":"This is equal to the square root of 10 minus 0 squared is"},{"Start":"04:10.640 ","End":"04:18.450","Text":"100 plus 10 minus 5 is 10 plus 5, which is 15."},{"Start":"04:18.450 ","End":"04:25.130","Text":"Then we have 15 squared is equal to 225,"},{"Start":"04:25.130 ","End":"04:30.550","Text":"so plus 225 is equal to this."},{"Start":"04:30.550 ","End":"04:37.190","Text":"Then, we have the square root of 100 plus 225,"},{"Start":"04:37.190 ","End":"04:40.730","Text":"which is the square root of 325,"},{"Start":"04:40.730 ","End":"04:46.565","Text":"which is equal to 5 root 13 or in decimals,"},{"Start":"04:46.565 ","End":"04:48.620","Text":"we can round this or not round this,"},{"Start":"04:48.620 ","End":"04:57.820","Text":"that\u0027s approximately equal to 18.03 to two decimal places, centimeters."},{"Start":"04:57.820 ","End":"05:00.555","Text":"Now we have S_1P and S_2P."},{"Start":"05:00.555 ","End":"05:02.760","Text":"We can plug this into the equation."},{"Start":"05:02.760 ","End":"05:04.500","Text":"We have S_2P,"},{"Start":"05:04.500 ","End":"05:10.355","Text":"which is equal to 18.03 minus S_1P,"},{"Start":"05:10.355 ","End":"05:18.745","Text":"so minus 11.18 is equal to Lambda."},{"Start":"05:18.745 ","End":"05:21.480","Text":"Now, let\u0027s do this calculation."},{"Start":"05:21.480 ","End":"05:28.025","Text":"We have 18.03 minus 11.18,"},{"Start":"05:28.025 ","End":"05:37.000","Text":"and this is equal to 6.85 centimeters."},{"Start":"05:37.550 ","End":"05:40.980","Text":"This is the wavelength of the wave."},{"Start":"05:40.980 ","End":"05:43.174","Text":"We\u0027ve answered our question,"},{"Start":"05:43.174 ","End":"05:46.380","Text":"and that is the end of the lesson."}],"ID":12499},{"Watched":false,"Name":"Exercise - Interference 3","Duration":"4m 25s","ChapterTopicVideoID":12032,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.890","Text":"Hello. In this question,"},{"Start":"00:01.890 ","End":"00:05.340","Text":"we\u0027re being told that we have 2 identical waves sources that are"},{"Start":"00:05.340 ","End":"00:09.030","Text":"placed at point (4,2) and (7,6)."},{"Start":"00:09.030 ","End":"00:15.615","Text":"We\u0027re being told that at (10,10) there is a third-order constructive interference."},{"Start":"00:15.615 ","End":"00:18.765","Text":"What is the wavelength of the wave?"},{"Start":"00:18.765 ","End":"00:20.645","Text":"I\u0027m not going to draw out the graph."},{"Start":"00:20.645 ","End":"00:22.730","Text":"If it\u0027s easier for you to visualize it,"},{"Start":"00:22.730 ","End":"00:26.990","Text":"then please draw out x and y axis and plot these points."},{"Start":"00:26.990 ","End":"00:32.330","Text":"But I\u0027m just going to write out the mathematical equations."},{"Start":"00:32.330 ","End":"00:36.650","Text":"We know that the equation for constructive interference"},{"Start":"00:36.650 ","End":"00:42.200","Text":"is the absolute value of S_2p minus S_1P."},{"Start":"00:42.200 ","End":"00:44.765","Text":"That this is equal to n Lambda,"},{"Start":"00:44.765 ","End":"00:46.295","Text":"where n over here,"},{"Start":"00:46.295 ","End":"00:48.440","Text":"we\u0027re being told is of third order,"},{"Start":"00:48.440 ","End":"00:50.840","Text":"so n is equal to 3,"},{"Start":"00:50.840 ","End":"00:53.720","Text":"and Lambda is what we\u0027re trying to calculate."},{"Start":"00:53.720 ","End":"00:58.090","Text":"Let\u0027s work out the distance between S_1 and"},{"Start":"00:58.090 ","End":"01:03.445","Text":"p. Let\u0027s say that"},{"Start":"01:03.445 ","End":"01:08.335","Text":"S_1 is this and that S_2 is this."},{"Start":"01:08.335 ","End":"01:09.820","Text":"We\u0027ve just given it a name."},{"Start":"01:09.820 ","End":"01:16.210","Text":"S_1P is equal to the square root of y_p minus y_1"},{"Start":"01:16.210 ","End":"01:22.225","Text":"squared plus x_p minus x_1 squared,"},{"Start":"01:22.225 ","End":"01:24.790","Text":"and the square root of all of this."},{"Start":"01:24.790 ","End":"01:27.990","Text":"Let\u0027s plug in the numbers now."},{"Start":"01:27.990 ","End":"01:32.520","Text":"So y_p, this is p. Ten"},{"Start":"01:32.520 ","End":"01:39.135","Text":"minus 2 squared plus x_p,"},{"Start":"01:39.135 ","End":"01:42.390","Text":"which is 10 minus x_1,"},{"Start":"01:42.390 ","End":"01:45.795","Text":"which is 4 squared."},{"Start":"01:45.795 ","End":"01:51.390","Text":"This is equal to 10 minus 2 is 8, 8 squared,"},{"Start":"01:51.390 ","End":"01:55.950","Text":"is 64, plus 10 minus 4 which is 6,"},{"Start":"01:55.950 ","End":"01:59.640","Text":"6 squared is 36."},{"Start":"01:59.640 ","End":"02:01.940","Text":"We\u0027re finding the square root of all of that."},{"Start":"02:01.940 ","End":"02:05.960","Text":"I\u0027m just going to plug all of this into my calculator."},{"Start":"02:05.960 ","End":"02:13.470","Text":"The square root of 64 plus 36 is equal to 10."},{"Start":"02:16.640 ","End":"02:25.884","Text":"That\u0027s the distance from S_1 to P. Now let\u0027s find the distance between S_2 and p. Again,"},{"Start":"02:25.884 ","End":"02:30.200","Text":"we have y_p minus y_2 squared"},{"Start":"02:30.200 ","End":"02:37.210","Text":"plus x_p minus x_2 squared and the square root of all of that."},{"Start":"02:37.210 ","End":"02:39.085","Text":"Let\u0027s plug in our values,"},{"Start":"02:39.085 ","End":"02:49.185","Text":"so y_p is equal to 10 minus y of source 2 is 6 squared plus x_p,"},{"Start":"02:49.185 ","End":"02:50.880","Text":"which is 10,"},{"Start":"02:50.880 ","End":"02:56.260","Text":"minus x_2, which is 7 squared."},{"Start":"02:57.740 ","End":"03:02.190","Text":"Square root of 10 minus 6 is 4,"},{"Start":"03:02.190 ","End":"03:06.075","Text":"4 squared is 16 plus 10 minus 7,"},{"Start":"03:06.075 ","End":"03:08.985","Text":"which is 3, 3 squared is 9,"},{"Start":"03:08.985 ","End":"03:11.870","Text":"so 16 plus 9 is 25 and of course,"},{"Start":"03:11.870 ","End":"03:14.665","Text":"the square root of 25 is 5."},{"Start":"03:14.665 ","End":"03:18.130","Text":"Now back to our equation, so S_2P."},{"Start":"03:18.130 ","End":"03:21.590","Text":"We\u0027re using the absolute value of S_2P,"},{"Start":"03:21.590 ","End":"03:24.860","Text":"which is 5 minus S_1P,"},{"Start":"03:24.860 ","End":"03:27.390","Text":"which is 10,"},{"Start":"03:27.390 ","End":"03:31.275","Text":"is equal to 3 Lambda."},{"Start":"03:31.275 ","End":"03:34.515","Text":"Five minus 10 is negative 5,"},{"Start":"03:34.515 ","End":"03:37.160","Text":"but because we have the absolute value,"},{"Start":"03:37.160 ","End":"03:41.105","Text":"we know that this will turn to a positive 5."},{"Start":"03:41.105 ","End":"03:45.590","Text":"Let\u0027s just write that out. The absolute value of negative 5 is"},{"Start":"03:45.590 ","End":"03:50.710","Text":"equal to simply positive 5 and this is equal to 3 Lambda."},{"Start":"03:50.710 ","End":"03:54.020","Text":"Therefore, we\u0027re going to isolate out our Lambda."},{"Start":"03:54.020 ","End":"03:56.405","Text":"We\u0027re going to divide both sides by 3."},{"Start":"03:56.405 ","End":"04:00.350","Text":"Lambda is equal to 5 divided by 3,"},{"Start":"04:00.350 ","End":"04:03.110","Text":"which is in decimals."},{"Start":"04:03.110 ","End":"04:09.820","Text":"This is equal to 1.67."},{"Start":"04:09.820 ","End":"04:12.050","Text":"Let\u0027s say that we\u0027re working,"},{"Start":"04:12.050 ","End":"04:16.770","Text":"everything is in meters, so 1.67 meters."},{"Start":"04:17.170 ","End":"04:22.100","Text":"This is the wavelength of the wave,"},{"Start":"04:22.100 ","End":"04:25.350","Text":"and that is the end of this lesson."}],"ID":12500},{"Watched":false,"Name":"Intro to Interference of Light","Duration":"30m 26s","ChapterTopicVideoID":12033,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.144","Text":"Hello. In this lesson,"},{"Start":"00:02.144 ","End":"00:05.190","Text":"we\u0027re moving on to a slightly different topic,"},{"Start":"00:05.190 ","End":"00:10.305","Text":"where here, we\u0027re going to be speaking about the interference of light."},{"Start":"00:10.305 ","End":"00:16.529","Text":"Light was proven to have wavelike qualities."},{"Start":"00:16.529 ","End":"00:22.845","Text":"We can say that lights are similar to waves."},{"Start":"00:22.845 ","End":"00:25.634","Text":"Just like we saw with,"},{"Start":"00:25.634 ","End":"00:28.290","Text":"let\u0027s say, water waves, 2-dimensional waves,"},{"Start":"00:28.290 ","End":"00:37.530","Text":"we saw that 2-dimensional waves exhibit refraction."},{"Start":"00:37.530 ","End":"00:39.630","Text":"What was refraction?"},{"Start":"00:39.630 ","End":"00:41.580","Text":"That was when, let\u0027s say,"},{"Start":"00:41.580 ","End":"00:46.640","Text":"the wave is passed through a different medium and so they change their angle."},{"Start":"00:46.640 ","End":"00:52.699","Text":"Similarly, light waves also exhibit refraction."},{"Start":"00:52.699 ","End":"00:57.349","Text":"You might see that if you put a pencil into a glass of water,"},{"Start":"00:57.349 ","End":"00:59.615","Text":"the pencil will look like it\u0027s bending,"},{"Start":"00:59.615 ","End":"01:04.530","Text":"and that\u0027s just due to refraction of light inside the glass."},{"Start":"01:05.300 ","End":"01:11.550","Text":"2-dimensional waves also experience dispersion."},{"Start":"01:12.290 ","End":"01:17.830","Text":"With dispersion, we had different wavelengths for our water waves."},{"Start":"01:17.830 ","End":"01:21.940","Text":"We saw that the angle at which the waves are"},{"Start":"01:21.940 ","End":"01:24.730","Text":"refracted when they reached the different medium"},{"Start":"01:24.730 ","End":"01:28.030","Text":"was different depending on the different wavelengths."},{"Start":"01:28.030 ","End":"01:30.669","Text":"Similarly, with light waves,"},{"Start":"01:30.669 ","End":"01:33.085","Text":"we also see dispersion."},{"Start":"01:33.085 ","End":"01:35.560","Text":"Where do we see dispersion in light waves?"},{"Start":"01:35.560 ","End":"01:40.014","Text":"Let\u0027s say if we send white light through a triangular prism,"},{"Start":"01:40.014 ","End":"01:42.505","Text":"then we\u0027ll see that all the different wavelengths,"},{"Start":"01:42.505 ","End":"01:44.529","Text":"so all the different colors,"},{"Start":"01:44.529 ","End":"01:49.920","Text":"red light and blue light and green light and so on and so forth,"},{"Start":"01:49.920 ","End":"01:52.240","Text":"they all have different wavelengths."},{"Start":"01:52.240 ","End":"01:55.090","Text":"That\u0027s what changes the color of the light that we see."},{"Start":"01:55.090 ","End":"01:57.880","Text":"They also disperse, because"},{"Start":"01:57.880 ","End":"02:05.914","Text":"the different wavelengths refract to different angles when they go through this prism,"},{"Start":"02:05.914 ","End":"02:09.590","Text":"just like the water waves when we change"},{"Start":"02:09.590 ","End":"02:14.274","Text":"the wavelengths also had a different refractive angle."},{"Start":"02:14.274 ","End":"02:19.189","Text":"Dispersion in light, you might remember from certain famous album"},{"Start":"02:19.189 ","End":"02:26.210","Text":"covers where we see this dispersion of white light through a triangular prism."},{"Start":"02:26.210 ","End":"02:29.824","Text":"In our water waves,"},{"Start":"02:29.824 ","End":"02:32.840","Text":"we saw that the water wave is reflected."},{"Start":"02:32.840 ","End":"02:37.600","Text":"The water wave, which was an example of a 2-dimensional wave,"},{"Start":"02:37.600 ","End":"02:41.495","Text":"and similarly, we have light reflection."},{"Start":"02:41.495 ","End":"02:44.795","Text":"When do we have light reflection?"},{"Start":"02:44.795 ","End":"02:46.984","Text":"Anytime we see something,"},{"Start":"02:46.984 ","End":"02:50.719","Text":"that means that light is being reflected back into our eye,"},{"Start":"02:50.719 ","End":"02:53.380","Text":"and specifically also in a mirror."},{"Start":"02:53.380 ","End":"02:58.595","Text":"We can see that light has all of these wave-like qualities."},{"Start":"02:58.595 ","End":"03:02.840","Text":"Just like we saw that water waves can"},{"Start":"03:02.840 ","End":"03:07.445","Text":"be interfered with and can have an interference pattern,"},{"Start":"03:07.445 ","End":"03:11.970","Text":"similarly so can light waves."},{"Start":"03:12.410 ","End":"03:19.970","Text":"What we\u0027re going to see now is that light also exhibits this phenomenon of interference,"},{"Start":"03:19.970 ","End":"03:24.359","Text":"and that\u0027s what we\u0027re going to learn in this lesson."},{"Start":"03:25.820 ","End":"03:30.125","Text":"Originally, what was done in order to check if"},{"Start":"03:30.125 ","End":"03:34.085","Text":"light exhibited this phenomenon of interference,"},{"Start":"03:34.085 ","End":"03:39.225","Text":"they took 2 monochromatic light sources."},{"Start":"03:39.225 ","End":"03:44.090","Text":"This was source 1 and this was source 2."},{"Start":"03:44.090 ","End":"03:46.069","Text":"What does monochromatic mean?"},{"Start":"03:46.069 ","End":"03:52.020","Text":"It means that both light sources have the same wavelength."},{"Start":"03:52.390 ","End":"03:57.194","Text":"The 2 sources had the same wavelength."},{"Start":"03:57.194 ","End":"04:01.025","Text":"That means that both had the same color of light,"},{"Start":"04:01.025 ","End":"04:05.090","Text":"be it red or green or blue light or whatever it is."},{"Start":"04:05.090 ","End":"04:09.184","Text":"Now what I\u0027m going to do is I\u0027m going to draw"},{"Start":"04:09.184 ","End":"04:14.945","Text":"the light coming out and I\u0027m going to draw this in different colors."},{"Start":"04:14.945 ","End":"04:18.886","Text":"Now, just imagine that the different colors are the same,"},{"Start":"04:18.886 ","End":"04:26.170","Text":"I\u0027m just using this to show that these wave fronts are coming from which light source."},{"Start":"04:26.170 ","End":"04:30.440","Text":"But imagine that the red and green have the same color,"},{"Start":"04:30.440 ","End":"04:35.339","Text":"in fact, and that they have the same wavelength."},{"Start":"04:35.710 ","End":"04:42.349","Text":"They shun these 2 monochromatic sources onto some screen and they"},{"Start":"04:42.349 ","End":"04:48.670","Text":"waited for the light waves to reach the screen and then be reflected back."},{"Start":"04:48.670 ","End":"04:52.840","Text":"Imagine that this is an eye looking. It\u0027s creepy."},{"Start":"04:52.840 ","End":"04:56.975","Text":"They waited for these waves to reflect back,"},{"Start":"04:56.975 ","End":"05:02.980","Text":"because that\u0027s the only way we can see light waves is from their reflection and to see"},{"Start":"05:02.980 ","End":"05:08.735","Text":"if we would get an interference pattern similar to what we saw in the wave tank."},{"Start":"05:08.735 ","End":"05:11.570","Text":"What they saw is when this happened,"},{"Start":"05:11.570 ","End":"05:13.759","Text":"they got no interference."},{"Start":"05:13.759 ","End":"05:22.044","Text":"They didn\u0027t see any interference whatsoever on the screen."},{"Start":"05:22.044 ","End":"05:23.809","Text":"What actually happened?"},{"Start":"05:23.809 ","End":"05:24.889","Text":"Why didn\u0027t this happen?"},{"Start":"05:24.889 ","End":"05:30.005","Text":"This is why I changed the colors between the 2 light sources."},{"Start":"05:30.005 ","End":"05:31.654","Text":"This will be clearer,"},{"Start":"05:31.654 ","End":"05:35.980","Text":"but imagine that they\u0027re both the same color because they have the same wavelength,"},{"Start":"05:35.980 ","End":"05:39.139","Text":"that\u0027s the definition of a monochromatic source."},{"Start":"05:39.139 ","End":"05:42.505","Text":"How do these torches work?"},{"Start":"05:42.505 ","End":"05:47.350","Text":"The touch will let out some packet of waves,"},{"Start":"05:47.350 ","End":"05:49.640","Text":"a few waves at the same time."},{"Start":"05:49.640 ","End":"05:51.020","Text":"They will all have,"},{"Start":"05:51.020 ","End":"05:53.330","Text":"of course, the same wavelength,"},{"Start":"05:53.330 ","End":"05:59.070","Text":"and it will shoot them out and they\u0027ll begin traveling towards the wall."},{"Start":"05:59.230 ","End":"06:05.420","Text":"Simultaneously or around about at the same time,"},{"Start":"06:05.420 ","End":"06:11.295","Text":"the light source number 2 will send out the same wave,"},{"Start":"06:11.295 ","End":"06:14.840","Text":"but rather it just has to have the same wavelength as"},{"Start":"06:14.840 ","End":"06:19.395","Text":"the first torch because that\u0027s the definition of monochromatic,"},{"Start":"06:19.395 ","End":"06:22.380","Text":"and it would send out this package."},{"Start":"06:22.380 ","End":"06:28.840","Text":"But it doesn\u0027t necessarily mean that these 2 waves are in phase."},{"Start":"06:28.840 ","End":"06:33.920","Text":"We can see that the red wave is at a peak here and the green wave is at a peak here,"},{"Start":"06:33.920 ","End":"06:38.080","Text":"and we can see that they have this difference in phase."},{"Start":"06:38.080 ","End":"06:41.960","Text":"Then a few moments later,"},{"Start":"06:41.960 ","End":"06:46.269","Text":"source number 1 in red would let"},{"Start":"06:46.269 ","End":"06:51.625","Text":"out another wave and it might not start at the same point, it\u0027s random."},{"Start":"06:51.625 ","End":"06:57.114","Text":"Here, we started from 0 amplitude and carried on here."},{"Start":"06:57.114 ","End":"07:01.404","Text":"The wave can be let out at the peak amplitude"},{"Start":"07:01.404 ","End":"07:06.525","Text":"and so on and so forth after some time period,"},{"Start":"07:06.525 ","End":"07:12.720","Text":"and then after another time periods source number 2 will let out its next wave."},{"Start":"07:12.720 ","End":"07:20.474","Text":"It might not be at the same time that the red torch or source number 1 let out the wave."},{"Start":"07:20.474 ","End":"07:28.114","Text":"Even though the wavelengths of both of these waves is Lambda,"},{"Start":"07:28.114 ","End":"07:31.319","Text":"even though this distance here is"},{"Start":"07:31.319 ","End":"07:37.260","Text":"Lambda and so is this over here Lambda and this is the same Lambda,"},{"Start":"07:37.260 ","End":"07:40.110","Text":"the same wavelength and so as this,"},{"Start":"07:40.110 ","End":"07:45.015","Text":"this difference over here,"},{"Start":"07:45.015 ","End":"07:47.085","Text":"let\u0027s call it Delta t,"},{"Start":"07:47.085 ","End":"07:48.963","Text":"the time difference between the red wave"},{"Start":"07:48.963 ","End":"07:51.455","Text":"being at a peak and the green wave being at a peak,"},{"Start":"07:51.455 ","End":"07:56.310","Text":"isn\u0027t necessarily constant as we can see here."},{"Start":"07:56.310 ","End":"08:00.855","Text":"This is Delta t_1 and here we have Delta t_2."},{"Start":"08:00.855 ","End":"08:05.719","Text":"What we could see is that we can\u0027t get a diffraction pattern like this"},{"Start":"08:05.719 ","End":"08:12.125","Text":"because the waves aren\u0027t always in the exact same formation as one another."},{"Start":"08:12.125 ","End":"08:15.380","Text":"The waves, although of the same wavelength,"},{"Start":"08:15.380 ","End":"08:19.190","Text":"are being shot out at random compared to the other ones,"},{"Start":"08:19.190 ","End":"08:24.840","Text":"so they\u0027re never going to be in phase with one another or out of phase with one another,"},{"Start":"08:24.840 ","End":"08:27.979","Text":"and it\u0027s always changing, and therefore,"},{"Start":"08:27.979 ","End":"08:32.754","Text":"we don\u0027t get an interference pattern on the screen."},{"Start":"08:32.754 ","End":"08:37.070","Text":"Now, we actually spoke about this a few lessons ago and we said that we"},{"Start":"08:37.070 ","End":"08:40.865","Text":"can always get an interference pattern so long"},{"Start":"08:40.865 ","End":"08:44.075","Text":"as the time interval between"},{"Start":"08:44.075 ","End":"08:50.389","Text":"both waves being at a peak is the same or is constant throughout."},{"Start":"08:50.389 ","End":"08:53.329","Text":"Then we can get an interference pattern."},{"Start":"08:53.329 ","End":"08:57.140","Text":"But if this time period is changing,"},{"Start":"08:57.140 ","End":"08:59.465","Text":"so at 1 stage it\u0027s Delta t_1,"},{"Start":"08:59.465 ","End":"09:03.680","Text":"but then next time it\u0027s Delta t_2 and then Delta t_3 and so on,"},{"Start":"09:03.680 ","End":"09:05.930","Text":"then we won\u0027t get an interference pattern."},{"Start":"09:05.930 ","End":"09:08.110","Text":"We spoke about this a few lessons ago,"},{"Start":"09:08.110 ","End":"09:11.239","Text":"and this was exactly the scientists\u0027 problem"},{"Start":"09:11.239 ","End":"09:15.750","Text":"when they did this experiment with the monochromatic sources."},{"Start":"09:16.010 ","End":"09:18.355","Text":"A few years later,"},{"Start":"09:18.355 ","End":"09:23.335","Text":"a scientist by the name of Young decided to try"},{"Start":"09:23.335 ","End":"09:30.025","Text":"a different experiment and to do this diffraction or interference experiment with light,"},{"Start":"09:30.025 ","End":"09:32.360","Text":"but in a slightly different way."},{"Start":"09:32.360 ","End":"09:37.434","Text":"How did he do it? First I\u0027m going to remind you about our water waves."},{"Start":"09:37.434 ","End":"09:43.000","Text":"Remember, when we had our water waves that were traveling like so."},{"Start":"09:43.000 ","End":"09:47.789","Text":"We had a straight wavefront and then we"},{"Start":"09:47.789 ","End":"09:54.045","Text":"had some barrier with a single slit inside."},{"Start":"09:54.045 ","End":"09:59.865","Text":"Then we said that once our water wave reaches this barrier,"},{"Start":"09:59.865 ","End":"10:05.660","Text":"only this part of the wave will travel through."},{"Start":"10:05.660 ","End":"10:11.274","Text":"Then this hole in the barrier will act like a source."},{"Start":"10:11.274 ","End":"10:19.320","Text":"Then the water wave will start spreading out with these circular wavefronts,"},{"Start":"10:20.150 ","End":"10:28.269","Text":"and of course, the wavelength over here is the same as the wavelength over here."},{"Start":"10:28.269 ","End":"10:30.750","Text":"This water wave will spread out."},{"Start":"10:30.750 ","End":"10:34.410","Text":"Similarly, we saw that if we had, again,"},{"Start":"10:34.410 ","End":"10:38.610","Text":"the straight wavefronts that hit a barrier that"},{"Start":"10:38.610 ","End":"10:43.949","Text":"has 2 holes or 2 slits inside, each slit,"},{"Start":"10:43.949 ","End":"10:47.635","Text":"just like before, will become a source,"},{"Start":"10:47.635 ","End":"10:51.124","Text":"and the water waves will begin spreading out like this"},{"Start":"10:51.124 ","End":"10:55.684","Text":"again with this circular wavefront."},{"Start":"10:55.684 ","End":"10:57.950","Text":"Now I\u0027m just drawing it in a different color,"},{"Start":"10:57.950 ","End":"10:59.450","Text":"but it has the same wavelength,"},{"Start":"10:59.450 ","End":"11:04.320","Text":"of course just to illustrate the second source."},{"Start":"11:06.070 ","End":"11:11.960","Text":"Then we would get another source of circular wavefronts."},{"Start":"11:11.960 ","End":"11:14.600","Text":"Then these would interfere with one another,"},{"Start":"11:14.600 ","End":"11:16.955","Text":"where of course the wavelength over here,"},{"Start":"11:16.955 ","End":"11:23.446","Text":"Lambda is the same as the wavelength for the waves coming out of the first slit,"},{"Start":"11:23.446 ","End":"11:26.990","Text":"and is the wavelength of the waves coming out of"},{"Start":"11:26.990 ","End":"11:30.979","Text":"the second slit and we would get interference."},{"Start":"11:30.979 ","End":"11:36.085","Text":"We would have superposition or constructive superposition over here,"},{"Start":"11:36.085 ","End":"11:40.565","Text":"constructive interference at all of these points,"},{"Start":"11:40.565 ","End":"11:45.303","Text":"and of course many others."},{"Start":"11:45.303 ","End":"11:52.449","Text":"Young decided to do this exact same experiment like what we did with the water waves,"},{"Start":"11:52.449 ","End":"11:55.834","Text":"however, to do it with light."},{"Start":"11:55.834 ","End":"11:57.900","Text":"Just a quick note,"},{"Start":"11:57.900 ","End":"11:59.744","Text":"why does this work over here?"},{"Start":"11:59.744 ","End":"12:03.509","Text":"But it\u0027s because of course this straight wavefront reaches"},{"Start":"12:03.509 ","End":"12:09.545","Text":"the barrier at the same time every single point on this straight wavefront."},{"Start":"12:09.545 ","End":"12:14.529","Text":"Which means that the part of the wavefront that reaches the top hole in"},{"Start":"12:14.529 ","End":"12:19.210","Text":"the barrier and the part of the wavefront that reaches the bottom hole"},{"Start":"12:19.210 ","End":"12:25.509","Text":"in the barrier reach at the same time which means that the secular wavefronts"},{"Start":"12:25.509 ","End":"12:32.905","Text":"coming from the 2 holes in the barrier or the 2 slits are going to be in-phase."},{"Start":"12:32.905 ","End":"12:39.412","Text":"Because they\u0027re in phase or the phase difference between them is constant throughout,"},{"Start":"12:39.412 ","End":"12:42.790","Text":"then we certainly get this diffraction pattern."},{"Start":"12:42.790 ","End":"12:45.325","Text":"What did Young do?"},{"Start":"12:45.325 ","End":"12:48.969","Text":"Young took 1 light source,"},{"Start":"12:48.969 ","End":"12:51.500","Text":"so I\u0027m going to draw it really big."},{"Start":"12:51.500 ","End":"12:59.864","Text":"Here we have a light source and this is some torch or a lamp,"},{"Start":"12:59.864 ","End":"13:06.599","Text":"then this sent out some beam of light,"},{"Start":"13:06.599 ","End":"13:09.099","Text":"where this beam of light,"},{"Start":"13:10.290 ","End":"13:24.864","Text":"its diameter is between 1 millimeter to 2 millimeters."},{"Start":"13:24.864 ","End":"13:26.559","Text":"That\u0027s pretty small."},{"Start":"13:26.559 ","End":"13:33.070","Text":"Then what he did is he formed some barrier so that\u0027s drawn in red where"},{"Start":"13:33.070 ","End":"13:37.435","Text":"this barrier had 2 little holes"},{"Start":"13:37.435 ","End":"13:42.234","Text":"in it just like in the water waves that had 2 little holes."},{"Start":"13:42.234 ","End":"13:48.475","Text":"The 2 little holes we can see are encompassed within this 1-2 millimeters."},{"Start":"13:48.475 ","End":"13:52.240","Text":"The 2 little holes are really small and they\u0027re close together."},{"Start":"13:52.240 ","End":"13:56.308","Text":"This is just to zoom in to make this clearer."},{"Start":"13:56.308 ","End":"13:59.859","Text":"Of course, the waves and of course,"},{"Start":"13:59.859 ","End":"14:04.640","Text":"this torch has monochromatic light,"},{"Start":"14:05.160 ","End":"14:08.680","Text":"light of a certain wavelength."},{"Start":"14:08.680 ","End":"14:11.815","Text":"That means just light of 1 color,"},{"Start":"14:11.815 ","End":"14:18.265","Text":"not white light but let\u0027s say red light or green light or whatever it might be."},{"Start":"14:18.265 ","End":"14:23.365","Text":"These waves are coming out just like we saw before."},{"Start":"14:23.365 ","End":"14:25.585","Text":"They come out at random intervals."},{"Start":"14:25.585 ","End":"14:26.964","Text":"Let\u0027s say it\u0027s green."},{"Start":"14:26.964 ","End":"14:31.270","Text":"The first time a wave like this will come out,"},{"Start":"14:31.270 ","End":"14:38.215","Text":"the second time it will come from a drop a few moments later like this."},{"Start":"14:38.215 ","End":"14:42.430","Text":"A few moments later it will come from the peak."},{"Start":"14:42.430 ","End":"14:47.829","Text":"We can see that it\u0027s coming out at random bursts and starting"},{"Start":"14:47.829 ","End":"14:53.695","Text":"from a peak or a trough or the 0 points whichever it might be."},{"Start":"14:53.695 ","End":"14:58.599","Text":"But this wave is traveling through here,"},{"Start":"14:58.599 ","End":"15:04.600","Text":"is going to reach these 2 slits in the barrier at the exact same time just"},{"Start":"15:04.600 ","End":"15:11.349","Text":"like our water waves hits the 2 slits in the barrier at the exact same time."},{"Start":"15:11.349 ","End":"15:13.847","Text":"It\u0027s the same thing."},{"Start":"15:13.847 ","End":"15:18.288","Text":"These 2 slits become light sources,"},{"Start":"15:18.288 ","End":"15:25.825","Text":"then we can see that we\u0027re going to have these circular wavefronts spreading out."},{"Start":"15:25.825 ","End":"15:31.179","Text":"I\u0027m just drawing this in different colors but of course they\u0027re meant to be of"},{"Start":"15:31.179 ","End":"15:34.119","Text":"the same wavelength so they\u0027re meant to be the same color as just"},{"Start":"15:34.119 ","End":"15:38.101","Text":"to illustrate the difference."},{"Start":"15:38.101 ","End":"15:43.900","Text":"What we have is that first of all, these 2 sources."},{"Start":"15:43.900 ","End":"15:48.370","Text":"New sources that were formed at the barrier are"},{"Start":"15:48.370 ","End":"15:52.765","Text":"monochromatic which means that they have the same wavelength which as we remember,"},{"Start":"15:52.765 ","End":"15:57.474","Text":"is imperative in order to get an interference pattern."},{"Start":"15:57.474 ","End":"16:02.725","Text":"We can see that because the 2 sources are formed"},{"Start":"16:02.725 ","End":"16:08.619","Text":"from 1 original source of monochromatic light,"},{"Start":"16:08.619 ","End":"16:12.535","Text":"it means that each wave-packets,"},{"Start":"16:12.535 ","End":"16:14.275","Text":"be it 1,"},{"Start":"16:14.275 ","End":"16:16.255","Text":"2, 3,"},{"Start":"16:16.255 ","End":"16:22.209","Text":"and so on is going to reach the barrier at the exact same time."},{"Start":"16:22.209 ","End":"16:30.340","Text":"Which means that these new waves formed at the barrier are always going to be in phase"},{"Start":"16:30.340 ","End":"16:34.420","Text":"with 1 another or maybe not in phase but they\u0027re"},{"Start":"16:34.420 ","End":"16:38.753","Text":"going to have a constant phase difference,"},{"Start":"16:38.753 ","End":"16:43.220","Text":"and therefore, we can get an interference pattern."},{"Start":"16:43.950 ","End":"16:50.545","Text":"These 2 ingredients are what we need in order to get a diffraction pattern."},{"Start":"16:50.545 ","End":"16:57.085","Text":"This experimental setup works and gives us these 2 ingredients."},{"Start":"16:57.085 ","End":"17:02.060","Text":"Therefore, we can get a diffraction pattern from light."},{"Start":"17:02.390 ","End":"17:10.425","Text":"Now, let\u0027s take a look at our whole system of the experiment but from a close-up."},{"Start":"17:10.425 ","End":"17:16.169","Text":"Here, we have our light beam."},{"Start":"17:16.169 ","End":"17:18.434","Text":"Remember all of this,"},{"Start":"17:18.434 ","End":"17:23.039","Text":"this is a massive close-up but this whole diameter of the light beam is"},{"Start":"17:23.039 ","End":"17:28.705","Text":"only between 2 millimeters or 1 millimeter."},{"Start":"17:28.705 ","End":"17:31.120","Text":"This is very small."},{"Start":"17:31.120 ","End":"17:40.779","Text":"Over here, we hit some barrier that has 2 holes in it."},{"Start":"17:40.779 ","End":"17:44.230","Text":"After this barrier,"},{"Start":"17:44.230 ","End":"17:47.870","Text":"we\u0027re going to have an interference pattern."},{"Start":"17:47.970 ","End":"17:56.650","Text":"Here we have our monochromatic light beam and here we have a barrier with 2 slits."},{"Start":"17:56.650 ","End":"18:05.305","Text":"We can see that we have this pink or purple source over here and we can call this S1."},{"Start":"18:05.305 ","End":"18:11.095","Text":"We have a second green source over here which we can call S2."},{"Start":"18:11.095 ","End":"18:15.069","Text":"Of course, the pink and green have the same wavelength and they\u0027re the same color,"},{"Start":"18:15.069 ","End":"18:21.142","Text":"they\u0027re just illustrated in different colors to make this clearer for us to understand."},{"Start":"18:21.142 ","End":"18:28.675","Text":"We can see that these wavelets or these new wavefronts start spreading out."},{"Start":"18:28.675 ","End":"18:33.250","Text":"We can say again that the solid lines are peaks and"},{"Start":"18:33.250 ","End":"18:38.289","Text":"that the dotted lines are the troughs of the wave."},{"Start":"18:38.289 ","End":"18:45.109","Text":"We can just see this and they\u0027re spreading out as we can see."},{"Start":"18:45.510 ","End":"18:48.069","Text":"Now, because this is light,"},{"Start":"18:48.069 ","End":"18:50.950","Text":"we won\u0027t be able to see all of this."},{"Start":"18:50.950 ","End":"18:53.500","Text":"This is just to illustrate what is happening but we can\u0027t"},{"Start":"18:53.500 ","End":"18:56.005","Text":"really see it because it\u0027s light."},{"Start":"18:56.005 ","End":"18:57.775","Text":"But what will we be able to see?"},{"Start":"18:57.775 ","End":"19:05.065","Text":"We\u0027ll be able to see all this light once it reflects off some screen and into our eyes."},{"Start":"19:05.065 ","End":"19:08.739","Text":"We\u0027ll be able to see this pattern or at least the last line of"},{"Start":"19:08.739 ","End":"19:14.380","Text":"the pattern once it\u0027s reflected from some screen."},{"Start":"19:14.380 ","End":"19:17.929","Text":"We place a screen at the end over here."},{"Start":"19:18.000 ","End":"19:25.810","Text":"Now, what\u0027s important to note is that because we\u0027re dealing with light or light waves,"},{"Start":"19:25.810 ","End":"19:29.935","Text":"so these are 3-dimensional waves."},{"Start":"19:29.935 ","End":"19:32.880","Text":"With the water waves we were dealing with 2D waves,"},{"Start":"19:32.880 ","End":"19:37.209","Text":"we had these circular wavefronts and water waves but because"},{"Start":"19:37.209 ","End":"19:43.315","Text":"here light is a 3-dimensional wave so these wavefronts are spherical."},{"Start":"19:43.315 ","End":"19:53.870","Text":"Therefore, we have spherical wavefronts."},{"Start":"19:53.870 ","End":"20:01.569","Text":"What we can see is that here we have a peak in pink with a peak in green."},{"Start":"20:01.569 ","End":"20:05.020","Text":"Here we have constructive interference and"},{"Start":"20:05.020 ","End":"20:09.339","Text":"similarly here we have a peak and a peak so it\u0027s constructive interference."},{"Start":"20:09.339 ","End":"20:11.515","Text":"Of course, just like before,"},{"Start":"20:11.515 ","End":"20:14.959","Text":"we also have here a trough and a trough."},{"Start":"20:14.959 ","End":"20:18.789","Text":"So here is also constructive interference."},{"Start":"20:18.789 ","End":"20:24.129","Text":"What do we can see is that there is this central line over"},{"Start":"20:24.129 ","End":"20:31.129","Text":"here where we have constructive interference throughout."},{"Start":"20:31.490 ","End":"20:34.784","Text":"Let\u0027s just mark them in red."},{"Start":"20:34.784 ","End":"20:38.410","Text":"Here we can see that we have the center."},{"Start":"20:38.410 ","End":"20:43.510","Text":"Here we have constructive interference and here and here and here and here."},{"Start":"20:43.510 ","End":"20:47.185","Text":"Peaks to peaks and troughs to troughs."},{"Start":"20:47.185 ","End":"20:52.899","Text":"What we can see is that we\u0027re going to have right over here through"},{"Start":"20:52.899 ","End":"20:58.870","Text":"the middle our 0th order maximum line."},{"Start":"20:58.870 ","End":"21:01.300","Text":"But in the water waves,"},{"Start":"21:01.300 ","End":"21:05.245","Text":"we saw this line also in the middle of the wave tank."},{"Start":"21:05.245 ","End":"21:08.140","Text":"However, because we\u0027re dealing with lights,"},{"Start":"21:08.140 ","End":"21:12.294","Text":"we won\u0027t see this line in the middle."},{"Start":"21:12.294 ","End":"21:17.305","Text":"But we will see it on the screen over here with"},{"Start":"21:17.305 ","End":"21:23.664","Text":"some line or a patch of light over here,"},{"Start":"21:23.664 ","End":"21:30.889","Text":"where we have our 0 orderth maximum line."},{"Start":"21:31.860 ","End":"21:35.170","Text":"Now similarly with the water waves,"},{"Start":"21:35.170 ","End":"21:39.249","Text":"if we had this maximum of 0th order."},{"Start":"21:39.249 ","End":"21:42.805","Text":"Now let\u0027s find the one of 1st order."},{"Start":"21:42.805 ","End":"21:51.670","Text":"I am just following this, peak-to-peak."},{"Start":"21:51.670 ","End":"22:01.194","Text":"That means that over here we\u0027ll have our first order maximum."},{"Start":"22:01.194 ","End":"22:03.699","Text":"Remember, we don\u0027t see any of"},{"Start":"22:03.699 ","End":"22:06.280","Text":"these pink and green lines in"},{"Start":"22:06.280 ","End":"22:09.175","Text":"the middle over here and we won\u0027t see this black dotted line."},{"Start":"22:09.175 ","End":"22:11.005","Text":"This is just to illustrate."},{"Start":"22:11.005 ","End":"22:15.864","Text":"All we will see is what is reflected to us from the screen."},{"Start":"22:15.864 ","End":"22:18.205","Text":"Then, similarly,"},{"Start":"22:18.205 ","End":"22:24.250","Text":"on the other side we\u0027re going to have this 1st order maximum line."},{"Start":"22:24.250 ","End":"22:27.069","Text":"We have to follow it through."},{"Start":"22:27.069 ","End":"22:31.539","Text":"It starts from here. Let\u0027s draw it in black."},{"Start":"22:31.539 ","End":"22:35.365","Text":"Trough to trough, peak to peak, trough, peak,"},{"Start":"22:35.365 ","End":"22:40.435","Text":"just like before. Then we get here."},{"Start":"22:40.435 ","End":"22:48.970","Text":"Here we have another maximum of 1st order and now let\u0027s find our maximum of 2nd order."},{"Start":"22:48.970 ","End":"22:52.510","Text":"Here, peak to peak, trough to trough."},{"Start":"22:52.510 ","End":"22:56.809","Text":"Just following this through."},{"Start":"22:58.140 ","End":"23:06.249","Text":"Here we have our maximum of 2nd order and similarly,"},{"Start":"23:06.249 ","End":"23:08.544","Text":"on the other side,"},{"Start":"23:08.544 ","End":"23:13.179","Text":"going to follow this out as well."},{"Start":"23:13.179 ","End":"23:22.015","Text":"Here we have therefore our 2nd maximum of 2nd order."},{"Start":"23:22.015 ","End":"23:26.230","Text":"Then, of course, we\u0027re going to have over here a maximum of 3rd order."},{"Start":"23:26.230 ","End":"23:30.040","Text":"However, that won\u0027t reach the screen."},{"Start":"23:30.040 ","End":"23:33.565","Text":"It won\u0027t be reflected back to us and so we won\u0027t see it."},{"Start":"23:33.565 ","End":"23:35.649","Text":"What we\u0027re going to see,"},{"Start":"23:35.649 ","End":"23:41.163","Text":"we can cross out this block and imagine that we don\u0027t see anything over here,"},{"Start":"23:41.163 ","End":"23:46.224","Text":"and all we\u0027ll see are these maximums where here we have patches of light."},{"Start":"23:46.224 ","End":"23:48.805","Text":"Of course, between each maximum,"},{"Start":"23:48.805 ","End":"23:54.470","Text":"we\u0027re going to have a minimum line where we have patches of darkness."},{"Start":"23:55.350 ","End":"23:58.405","Text":"Let\u0027s draw out our minimums."},{"Start":"23:58.405 ","End":"24:01.149","Text":"Here we have a peak and trough,"},{"Start":"24:01.149 ","End":"24:04.765","Text":"peak and trough, peak and trough."},{"Start":"24:04.765 ","End":"24:07.749","Text":"We just got to follow this out."},{"Start":"24:07.749 ","End":"24:16.180","Text":"It\u0027s between 1st and 2nd,"},{"Start":"24:16.180 ","End":"24:19.360","Text":"so it\u0027s a minimum of 2nd order."},{"Start":"24:19.360 ","End":"24:25.550","Text":"Here we have a peak and a trough and a peak and trough."},{"Start":"24:25.550 ","End":"24:29.965","Text":"We\u0027re going to follow this over here."},{"Start":"24:29.965 ","End":"24:34.989","Text":"Here we have our minimum between a maximum of 0 and 1st order."},{"Start":"24:34.989 ","End":"24:37.419","Text":"So this is a minimum of 1st order."},{"Start":"24:37.419 ","End":"24:45.580","Text":"Then, of course, we have the mirror image on the other side of the screen."},{"Start":"24:45.580 ","End":"24:50.050","Text":"Here we have our 2nd minimum of 1st order and"},{"Start":"24:50.050 ","End":"24:54.775","Text":"then here we have all of our peaks and troughs meeting."},{"Start":"24:54.775 ","End":"24:58.327","Text":"Here we have complete destructive interference,"},{"Start":"24:58.327 ","End":"25:03.679","Text":"of course, and here we have our 2nd minimum of 2nd order."},{"Start":"25:03.870 ","End":"25:07.404","Text":"Everywhere where we have a maximum,"},{"Start":"25:07.404 ","End":"25:12.339","Text":"we\u0027re going to have a bright patch of light and everywhere where we have a minimum,"},{"Start":"25:12.339 ","End":"25:15.800","Text":"we\u0027re going to have darkness."},{"Start":"25:16.200 ","End":"25:21.085","Text":"What this will look like is exactly where the line is"},{"Start":"25:21.085 ","End":"25:25.540","Text":"that I wrote maximum of 0 1st and 2nd order."},{"Start":"25:25.540 ","End":"25:28.419","Text":"So here we\u0027ll have very bright lights."},{"Start":"25:28.419 ","End":"25:30.279","Text":"However, on either side,"},{"Start":"25:30.279 ","End":"25:31.974","Text":"up until our minimum points,"},{"Start":"25:31.974 ","End":"25:35.439","Text":"we\u0027re still going to have light which is brightest at"},{"Start":"25:35.439 ","End":"25:40.569","Text":"the middle and then dim slightly towards the dark patch."},{"Start":"25:40.569 ","End":"25:46.555","Text":"A bright light in the middle and dimmer light at the edges and then a dark patch."},{"Start":"25:46.555 ","End":"25:48.219","Text":"Then, again, same for here,"},{"Start":"25:48.219 ","End":"25:53.620","Text":"bright light in the middle and then it dims towards the dark patch."},{"Start":"25:53.620 ","End":"25:55.899","Text":"Again, bright light in the middle for maximum of"},{"Start":"25:55.899 ","End":"25:59.590","Text":"2nd order and it dims towards the bright patch."},{"Start":"25:59.590 ","End":"26:01.464","Text":"The same over here."},{"Start":"26:01.464 ","End":"26:05.319","Text":"Bright light in the middle and dimming towards the dark patch and bright"},{"Start":"26:05.319 ","End":"26:10.089","Text":"light in the middle that dims towards the dark patch like that."},{"Start":"26:10.089 ","End":"26:14.140","Text":"If we move this a little bit more to the side,"},{"Start":"26:14.140 ","End":"26:17.004","Text":"what we\u0027ll have is right in the middle."},{"Start":"26:17.004 ","End":"26:21.519","Text":"Intense light here that dims, dark patch."},{"Start":"26:21.519 ","End":"26:28.900","Text":"Right here in the middle, intense light that dims towards the sides."},{"Start":"26:28.900 ","End":"26:34.900","Text":"Again, bright light in the middle that dims towards"},{"Start":"26:34.900 ","End":"26:41.095","Text":"the sides and the same on the other side."},{"Start":"26:41.095 ","End":"26:47.260","Text":"We\u0027re going to have this bright light in the middle that"},{"Start":"26:47.260 ","End":"26:49.450","Text":"dims towards where we have"},{"Start":"26:49.450 ","End":"26:53.335","Text":"our minimum points and then we have darkness at the minimum point."},{"Start":"26:53.335 ","End":"26:55.689","Text":"Again, all of this bright light,"},{"Start":"26:55.689 ","End":"26:59.530","Text":"which is brightest at the middle and dims towards"},{"Start":"26:59.530 ","End":"27:06.380","Text":"the edges and darkness in-between these red patches."},{"Start":"27:06.870 ","End":"27:15.130","Text":"Now let\u0027s scroll down and I\u0027m going to show you an image of this diffraction pattern."},{"Start":"27:15.130 ","End":"27:18.920","Text":"An image taken from an actual experiment."},{"Start":"27:18.930 ","End":"27:24.085","Text":"This is an image taken of this diffraction."},{"Start":"27:24.085 ","End":"27:28.615","Text":"We can really see where the maximum and minimums are."},{"Start":"27:28.615 ","End":"27:30.609","Text":"Over here, of course,"},{"Start":"27:30.609 ","End":"27:33.490","Text":"we have the 0th order maximum,"},{"Start":"27:33.490 ","End":"27:36.009","Text":"so that\u0027s a maximum when n=0."},{"Start":"27:36.009 ","End":"27:41.259","Text":"Here we have 1st order maximum, so when n=1."},{"Start":"27:41.259 ","End":"27:44.850","Text":"Here we also have 1st order maximum,"},{"Start":"27:44.850 ","End":"27:47.200","Text":"so, again, n=1."},{"Start":"27:47.200 ","End":"27:50.169","Text":"Here we have our 2nd order maximum,"},{"Start":"27:50.169 ","End":"27:54.474","Text":"so that\u0027s when n=2 and similarly over here,"},{"Start":"27:54.474 ","End":"27:57.895","Text":"our 2nd order maximum when n=2."},{"Start":"27:57.895 ","End":"28:00.954","Text":"Similarly, where we have our dark patches,"},{"Start":"28:00.954 ","End":"28:02.800","Text":"this is our minimum."},{"Start":"28:02.800 ","End":"28:07.569","Text":"This is our 1st order minimum, so when n=1."},{"Start":"28:07.569 ","End":"28:12.865","Text":"Here we have our 2nd order minimum, so when n=2."},{"Start":"28:12.865 ","End":"28:16.119","Text":"Here we have our other 1st order minimum,"},{"Start":"28:16.119 ","End":"28:18.344","Text":"so when n=1,"},{"Start":"28:18.344 ","End":"28:24.140","Text":"and the other 2nd order minimum, so when n=2."},{"Start":"28:25.410 ","End":"28:30.819","Text":"Another thing that I want you to notice here is that we can see that in"},{"Start":"28:30.819 ","End":"28:36.760","Text":"the middle we have this bright light all over here."},{"Start":"28:36.760 ","End":"28:39.729","Text":"We have all of this bright light."},{"Start":"28:39.729 ","End":"28:43.414","Text":"I\u0027ll draw it in gray."},{"Start":"28:43.414 ","End":"28:48.845","Text":"We can see that over here we have these gray lines on the edges."},{"Start":"28:48.845 ","End":"28:52.510","Text":"Or rather, let me draw it in green."},{"Start":"28:52.510 ","End":"28:55.824","Text":"We have these gray lines on the edges. Can you see that?"},{"Start":"28:55.824 ","End":"29:03.550","Text":"We have the bright in the middle and then on the edge it\u0027s gray on all of these."},{"Start":"29:03.550 ","End":"29:07.060","Text":"This is exactly what we were speaking about before,"},{"Start":"29:07.060 ","End":"29:11.109","Text":"where we were saying that we have this bright light right in the middle and that it"},{"Start":"29:11.109 ","End":"29:15.564","Text":"dims towards the edge and then it goes into a dark patch and then again,"},{"Start":"29:15.564 ","End":"29:22.010","Text":"bright light in the middle and it dims towards the edges where it meets the dark patches."},{"Start":"29:22.650 ","End":"29:28.089","Text":"This is the diffraction pattern of light"},{"Start":"29:28.089 ","End":"29:34.995","Text":"coming from a 2-slit barrier where the barrier has 2 holes in it."},{"Start":"29:34.995 ","End":"29:38.895","Text":"If we remember, we drew the barrier, let\u0027s say,"},{"Start":"29:38.895 ","End":"29:45.475","Text":"like so and then our light came in like this."},{"Start":"29:45.475 ","End":"29:49.210","Text":"This distance between the 2 slits,"},{"Start":"29:49.210 ","End":"29:52.379","Text":"this distance over here,"},{"Start":"29:52.379 ","End":"29:54.124","Text":"so between the 2 slits,"},{"Start":"29:54.124 ","End":"29:56.215","Text":"let\u0027s call it d,"},{"Start":"29:56.215 ","End":"30:02.785","Text":"has to be smaller or equal to 1 millimeter."},{"Start":"30:02.785 ","End":"30:05.650","Text":"Nowadays this is easy to do."},{"Start":"30:05.650 ","End":"30:09.264","Text":"But back in the day when Young came up with this experiment"},{"Start":"30:09.264 ","End":"30:13.389","Text":"and he had to make these two tiny slits in the barrier,"},{"Start":"30:13.389 ","End":"30:16.002","Text":"a distance of 1 millimeter apart,"},{"Start":"30:16.002 ","End":"30:18.670","Text":"that was very impressive."},{"Start":"30:19.910 ","End":"30:27.160","Text":"This is the end of the lesson on light interference and introduction."}],"ID":12501},{"Watched":false,"Name":"Deriving Interference Equation","Duration":"31m 53s","ChapterTopicVideoID":12034,"CourseChapterTopicPlaylistID":84748,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.965","Text":"Hello. In this lesson,"},{"Start":"00:01.965 ","End":"00:03.840","Text":"we\u0027re going to see how we can use"},{"Start":"00:03.840 ","End":"00:10.035","Text":"the light interference pattern in order to calculate the wavelength of light."},{"Start":"00:10.035 ","End":"00:14.145","Text":"This is how Young did it back in the day."},{"Start":"00:14.145 ","End":"00:17.760","Text":"This is how we discovered what the wavelength of light is."},{"Start":"00:17.760 ","End":"00:19.274","Text":"I\u0027m just warning you,"},{"Start":"00:19.274 ","End":"00:24.840","Text":"this is a very complicated video which is going to include lots of"},{"Start":"00:24.840 ","End":"00:31.770","Text":"diagrams and lots of calculations and we\u0027re going to derive the equation."},{"Start":"00:31.770 ","End":"00:37.290","Text":"Let\u0027s begin. Here we have our diagram."},{"Start":"00:37.290 ","End":"00:47.200","Text":"Now, here we have our screen and here we have our barrier."},{"Start":"00:47.870 ","End":"00:56.583","Text":"We shine a laser on to our barrier like so,"},{"Start":"00:56.583 ","End":"01:04.820","Text":"where the diameter of the laser beam is 1 millimeter."},{"Start":"01:04.820 ","End":"01:07.985","Text":"Here we have 2 slits."},{"Start":"01:07.985 ","End":"01:09.724","Text":"Here we have the first slit."},{"Start":"01:09.724 ","End":"01:13.035","Text":"So let\u0027s call that S_1 and here we have the second slit,"},{"Start":"01:13.035 ","End":"01:14.614","Text":"let\u0027s call that S_2,"},{"Start":"01:14.614 ","End":"01:18.137","Text":"and then here we have the distance between the 2 slits,"},{"Start":"01:18.137 ","End":"01:21.365","Text":"so the distance between S_1 and S_2,"},{"Start":"01:21.365 ","End":"01:31.495","Text":"and we\u0027ll call that distance d. Now we know that at the mid point over here,"},{"Start":"01:31.495 ","End":"01:35.032","Text":"so imagine that this is straight line,"},{"Start":"01:35.032 ","End":"01:37.047","Text":"so right over here,"},{"Start":"01:37.047 ","End":"01:45.790","Text":"we have the midpoint of our maximum line of zeroth order."},{"Start":"01:45.790 ","End":"01:47.815","Text":"We can color that in. Remember,"},{"Start":"01:47.815 ","End":"01:49.524","Text":"in the middle it\u0027s the brightest,"},{"Start":"01:49.524 ","End":"01:52.000","Text":"and then it dims towards the end."},{"Start":"01:52.000 ","End":"01:55.965","Text":"Then here we have another maximum,"},{"Start":"01:55.965 ","End":"01:59.184","Text":"our maximum of first-order,"},{"Start":"01:59.184 ","End":"02:05.129","Text":"and then we have our maximum over here of second-order."},{"Start":"02:05.129 ","End":"02:09.304","Text":"Then somewhere down the screen,"},{"Start":"02:09.304 ","End":"02:11.191","Text":"imagine the screen is a bit longer,"},{"Start":"02:11.191 ","End":"02:16.200","Text":"so somewhere down here,"},{"Start":"02:16.200 ","End":"02:18.745","Text":"we have our point P,"},{"Start":"02:18.745 ","End":"02:26.220","Text":"and our point P is located right in the middle of a maximum point."},{"Start":"02:26.220 ","End":"02:29.444","Text":"P is right in the center over here,"},{"Start":"02:29.444 ","End":"02:35.819","Text":"and this is a maximum point of nth order."},{"Start":"02:36.500 ","End":"02:42.960","Text":"P sits right in the center of this maximum point of nth order."},{"Start":"02:43.910 ","End":"02:48.560","Text":"Let\u0027s speak about our maximum zeroth order and also by the way,"},{"Start":"02:48.560 ","End":"02:49.610","Text":"our maximum of first,"},{"Start":"02:49.610 ","End":"02:54.480","Text":"second and so on order are also spreading up top here."},{"Start":"02:54.480 ","End":"02:59.295","Text":"Let\u0027s draw it in blue from our S_1."},{"Start":"02:59.295 ","End":"03:06.845","Text":"From S_1 we have a wave coming in like so,"},{"Start":"03:06.845 ","End":"03:12.484","Text":"traveling in this direction and let\u0027s say that it began at a peak in S_1,"},{"Start":"03:12.484 ","End":"03:16.914","Text":"and it arrives over here at Max 0,"},{"Start":"03:16.914 ","End":"03:18.779","Text":"also at a peak."},{"Start":"03:18.779 ","End":"03:20.359","Text":"Now in green,"},{"Start":"03:20.359 ","End":"03:26.390","Text":"let\u0027s see the direction of the wave propagation."},{"Start":"03:26.390 ","End":"03:28.894","Text":"It also goes here."},{"Start":"03:28.894 ","End":"03:34.580","Text":"We can see that Max 0 is equidistant from S_1 and S_2."},{"Start":"03:34.580 ","End":"03:37.084","Text":"That\u0027s actually what makes it Max 0."},{"Start":"03:37.084 ","End":"03:38.870","Text":"That\u0027s equidistant."},{"Start":"03:38.870 ","End":"03:43.219","Text":"If we said that the blue wave starts at a peak,"},{"Start":"03:43.219 ","End":"03:49.400","Text":"so we know that the green wave coming from S_2 also has to"},{"Start":"03:49.400 ","End":"03:52.310","Text":"start at a peak because they\u0027re"},{"Start":"03:52.310 ","End":"03:56.000","Text":"in phase with one another because they coming from the same source."},{"Start":"03:56.000 ","End":"03:58.490","Text":"If S_1 is at a trough,"},{"Start":"03:58.490 ","End":"04:00.035","Text":"then S_2 is at a trough."},{"Start":"04:00.035 ","End":"04:02.704","Text":"If it\u0027s at a peak, then S_2 is at a peak,"},{"Start":"04:02.704 ","End":"04:07.844","Text":"and so on for all the other examples and options,"},{"Start":"04:07.844 ","End":"04:09.675","Text":"so they\u0027re always in phase."},{"Start":"04:09.675 ","End":"04:16.066","Text":"Then the green, if S_1 arrived to the screen at a peak,"},{"Start":"04:16.066 ","End":"04:19.415","Text":"so they traveled the same exact distance,"},{"Start":"04:19.415 ","End":"04:21.703","Text":"and they started both in phase,"},{"Start":"04:21.703 ","End":"04:25.555","Text":"so that means the green will also arrive at a peak."},{"Start":"04:25.555 ","End":"04:28.189","Text":"Then we have constructive interference,"},{"Start":"04:28.189 ","End":"04:33.170","Text":"which means that we superimpose both of their amplitudes,"},{"Start":"04:33.170 ","End":"04:37.020","Text":"and we get very bright lights over here."},{"Start":"04:37.280 ","End":"04:42.029","Text":"This is for a maximum of zeroth order."},{"Start":"04:42.029 ","End":"04:44.663","Text":"I\u0027m going to rub this out."},{"Start":"04:44.663 ","End":"04:49.860","Text":"Now let\u0027s look at what is happening at point P over here."},{"Start":"04:50.390 ","End":"04:53.895","Text":"Let\u0027s use green for this one."},{"Start":"04:53.895 ","End":"04:59.962","Text":"We\u0027re going to have our wave coming in from S_1,"},{"Start":"04:59.962 ","End":"05:06.675","Text":"and we\u0027re going to have our wave coming in to point P from S_2."},{"Start":"05:06.675 ","End":"05:14.729","Text":"Because we know that P is located in the center of a maximum line of nth order,"},{"Start":"05:14.729 ","End":"05:19.295","Text":"we know that we\u0027re going to be using our equation."},{"Start":"05:19.295 ","End":"05:20.825","Text":"What was our equation?"},{"Start":"05:20.825 ","End":"05:26.060","Text":"It\u0027s the absolute value of S_1P minus S_2P."},{"Start":"05:26.060 ","End":"05:28.144","Text":"This is equal to,"},{"Start":"05:28.144 ","End":"05:30.428","Text":"because we\u0027re on a maximum line,"},{"Start":"05:30.428 ","End":"05:32.566","Text":"our point is on a maximum line,"},{"Start":"05:32.566 ","End":"05:37.355","Text":"so this is some multiple n multiplied by Lambda,"},{"Start":"05:37.355 ","End":"05:41.050","Text":"an n because it\u0027s the nth order."},{"Start":"05:41.050 ","End":"05:49.110","Text":"Now what I\u0027m going to do is I\u0027m going to label a point N along this line S_1P."},{"Start":"05:49.720 ","End":"05:54.484","Text":"Let\u0027s say that it\u0027s around about over here."},{"Start":"05:54.484 ","End":"06:02.370","Text":"This is N, such that when I connect S_2 to N over here,"},{"Start":"06:02.370 ","End":"06:06.044","Text":"then this length N to P,"},{"Start":"06:06.044 ","End":"06:09.930","Text":"and S_2 to P is the same length."},{"Start":"06:09.930 ","End":"06:11.670","Text":"Now I\u0027ve said that"},{"Start":"06:11.670 ","End":"06:19.529","Text":"the S_2PN triangle"},{"Start":"06:19.529 ","End":"06:23.050","Text":"is an isosceles triangle."},{"Start":"06:23.540 ","End":"06:31.635","Text":"If S_2PN is an isosceles triangle, therefore,"},{"Start":"06:31.635 ","End":"06:39.085","Text":"we can say that S_2P is equal to NP,"},{"Start":"06:39.085 ","End":"06:42.744","Text":"which was exactly why we placed N over here."},{"Start":"06:42.744 ","End":"06:46.940","Text":"We said that the length of S_2 to P is equal to"},{"Start":"06:46.940 ","End":"06:52.295","Text":"the length of N to P. That\u0027s exactly what we defined."},{"Start":"06:52.295 ","End":"06:55.040","Text":"Then if S_2P is equal to NP,"},{"Start":"06:55.040 ","End":"06:57.980","Text":"so when we plug this into our equation,"},{"Start":"06:57.980 ","End":"07:03.465","Text":"then we\u0027ll have that S_1P minus S_2P,"},{"Start":"07:03.465 ","End":"07:06.615","Text":"which is equal to NP,"},{"Start":"07:06.615 ","End":"07:10.740","Text":"is equal to n Lambda."},{"Start":"07:10.740 ","End":"07:14.340","Text":"What is S_1P minus NP?"},{"Start":"07:14.340 ","End":"07:18.140","Text":"S_1P is this whole green line over here,"},{"Start":"07:18.140 ","End":"07:23.399","Text":"and NP is from this point on the green line to the end."},{"Start":"07:24.790 ","End":"07:30.819","Text":"S_1P minus NP is"},{"Start":"07:30.819 ","End":"07:36.790","Text":"simply equal to this length over here, S_1N."},{"Start":"07:36.790 ","End":"07:44.960","Text":"Then we can say that S_1N is equal to n Lambda."},{"Start":"07:45.300 ","End":"07:52.670","Text":"S_1P minus NP just leaves us with this length over here."},{"Start":"07:52.820 ","End":"07:55.035","Text":"S_1 to N,"},{"Start":"07:55.035 ","End":"07:59.144","Text":"which is equal to n Lambda."},{"Start":"07:59.144 ","End":"08:03.899","Text":"Now the distance, of course,"},{"Start":"08:03.899 ","End":"08:06.630","Text":"this whole sketch isn\u0027t drawn to scale,"},{"Start":"08:06.630 ","End":"08:10.065","Text":"but just so you know that the distance between"},{"Start":"08:10.065 ","End":"08:15.014","Text":"our barrier to our screen is labeled with the letter L,"},{"Start":"08:15.014 ","End":"08:19.450","Text":"and L is of the size of a few meters."},{"Start":"08:19.610 ","End":"08:22.829","Text":"We\u0027re dealing with meters over here,"},{"Start":"08:22.829 ","End":"08:25.845","Text":"not millimeters or centimeters."},{"Start":"08:25.845 ","End":"08:30.270","Text":"So L, this distance between the bearing and the screen can be 1 meter,"},{"Start":"08:30.270 ","End":"08:31.425","Text":"2 meters, 3 meters,"},{"Start":"08:31.425 ","End":"08:33.089","Text":"4 meters, and so on."},{"Start":"08:33.089 ","End":"08:37.559","Text":"But we\u0027re always dealing with lengths that are in meters."},{"Start":"08:37.559 ","End":"08:44.415","Text":"However, we saw that the width of our laser beam is 1 millimeter."},{"Start":"08:44.415 ","End":"08:47.430","Text":"So it\u0027s a 1000th of a meter,"},{"Start":"08:47.430 ","End":"08:49.560","Text":"and we know that this length d,"},{"Start":"08:49.560 ","End":"08:51.840","Text":"the distance between the 2 slits,"},{"Start":"08:51.840 ","End":"08:57.670","Text":"S_1 and S_2, is even smaller than 1 millimeter."},{"Start":"08:58.670 ","End":"09:03.255","Text":"We can see that this distance S_2 to n,"},{"Start":"09:03.255 ","End":"09:06.870","Text":"so this distance over here,"},{"Start":"09:06.870 ","End":"09:10.275","Text":"is also approximately 1 millimeter,"},{"Start":"09:10.275 ","End":"09:14.140","Text":"or approximately less than 1 millimeter."},{"Start":"09:14.510 ","End":"09:19.455","Text":"S_2 to n is approximately a millimeter."},{"Start":"09:19.455 ","End":"09:22.530","Text":"Let\u0027s just draw this over here."},{"Start":"09:22.530 ","End":"09:26.628","Text":"Let\u0027s draw this triangle S_2,"},{"Start":"09:26.628 ","End":"09:30.600","Text":"P, N. So our triangle S_2 PN."},{"Start":"09:30.600 ","End":"09:33.359","Text":"We can see that N to P,"},{"Start":"09:33.359 ","End":"09:35.159","Text":"it\u0027s a few meters away,"},{"Start":"09:35.159 ","End":"09:39.165","Text":"and we can see that S_2 to P is also a few meters away."},{"Start":"09:39.165 ","End":"09:43.559","Text":"Because this distance L is a few meters if we\u0027re going straight,"},{"Start":"09:43.559 ","End":"09:45.059","Text":"and if we\u0027re going diagonal,"},{"Start":"09:45.059 ","End":"09:48.210","Text":"it\u0027s obviously going to be even a bit longer."},{"Start":"09:48.210 ","End":"09:50.294","Text":"So it\u0027s a few meters."},{"Start":"09:50.294 ","End":"09:56.339","Text":"That means that S_2 to P is a few meters away."},{"Start":"09:56.339 ","End":"09:58.124","Text":"This is S_2,"},{"Start":"09:58.124 ","End":"10:07.080","Text":"and this is P. Then we have approximately a millimeter in-between our N to P,"},{"Start":"10:07.080 ","End":"10:10.139","Text":"and our N to P is also a few meters away."},{"Start":"10:10.139 ","End":"10:12.375","Text":"Here we have a point N,"},{"Start":"10:12.375 ","End":"10:20.730","Text":"and it connects to point P. Now we can see the type of triangle that we\u0027re looking at,"},{"Start":"10:20.730 ","End":"10:22.920","Text":"and it\u0027s almost a straight line."},{"Start":"10:22.920 ","End":"10:25.289","Text":"The angle here at the head of the triangle,"},{"Start":"10:25.289 ","End":"10:26.970","Text":"which I can\u0027t even draw."},{"Start":"10:26.970 ","End":"10:29.535","Text":"Let\u0027s call it Theta."},{"Start":"10:29.535 ","End":"10:34.905","Text":"This angle is approaching 0 degrees, as we can see."},{"Start":"10:34.905 ","End":"10:39.615","Text":"The tip of this triangle is extremely, sharp and narrow."},{"Start":"10:39.615 ","End":"10:41.880","Text":"We have a very acute angle."},{"Start":"10:41.880 ","End":"10:44.370","Text":"Then all of the angles over here,"},{"Start":"10:44.370 ","End":"10:47.580","Text":"which soon I\u0027ll draw in the bigger triangle,"},{"Start":"10:47.580 ","End":"10:50.985","Text":"so let\u0027s call this angle over here,"},{"Start":"10:50.985 ","End":"10:53.456","Text":"I don\u0027t know, Beta."},{"Start":"10:53.456 ","End":"10:56.310","Text":"The angles over here, well, there\u0027s 2 angles,"},{"Start":"10:56.310 ","End":"11:01.125","Text":"so Alpha and Beta will be approaching 90 degrees,"},{"Start":"11:01.125 ","End":"11:04.184","Text":"and we\u0027re soon going to draw that over here."},{"Start":"11:04.184 ","End":"11:07.484","Text":"Let\u0027s call this angle Alpha."},{"Start":"11:07.484 ","End":"11:13.530","Text":"Then we can see that because this triangle obviously isn\u0027t to scale,"},{"Start":"11:13.530 ","End":"11:18.615","Text":"but because we saw that this angle over here is approaching 0 degrees,"},{"Start":"11:18.615 ","End":"11:24.914","Text":"so this angle over here is approximately 90 degrees,"},{"Start":"11:24.914 ","End":"11:28.860","Text":"and if this angle is 90 degrees in here,"},{"Start":"11:28.860 ","End":"11:32.730","Text":"we have an angle of 180 along the green line,"},{"Start":"11:32.730 ","End":"11:36.405","Text":"then this angle over here is also 90 degrees,"},{"Start":"11:36.405 ","End":"11:41.309","Text":"and this angle over here is 90 degrees."},{"Start":"11:41.309 ","End":"11:47.474","Text":"What we can see if I scroll down is that"},{"Start":"11:47.474 ","End":"11:54.660","Text":"our triangle S_1NS_2 triangle,"},{"Start":"11:54.660 ","End":"12:01.179","Text":"so this is a right angled triangle."},{"Start":"12:01.820 ","End":"12:04.634","Text":"If this is a right angle,"},{"Start":"12:04.634 ","End":"12:09.345","Text":"so therefore I can write that sine of"},{"Start":"12:09.345 ","End":"12:15.930","Text":"my angle Alpha over here is equal to the opposite,"},{"Start":"12:15.930 ","End":"12:17.595","Text":"so that\u0027s n Lambda,"},{"Start":"12:17.595 ","End":"12:19.440","Text":"the opposite side n Lambda,"},{"Start":"12:19.440 ","End":"12:26.385","Text":"divided by the hypotenuse."},{"Start":"12:26.385 ","End":"12:28.754","Text":"The hypotenuse is this side over here,"},{"Start":"12:28.754 ","End":"12:36.142","Text":"which is d. Now let\u0027s mark the midpoint between the 2 slits,"},{"Start":"12:36.142 ","End":"12:42.270","Text":"and call this point M. Now let\u0027s join between point"},{"Start":"12:42.270 ","End":"12:51.620","Text":"M. So we have this line going straight through to the maximum of zeroth-order."},{"Start":"12:51.620 ","End":"12:53.315","Text":"So we have this line."},{"Start":"12:53.315 ","End":"12:55.820","Text":"Then from the same point M,"},{"Start":"12:55.820 ","End":"13:03.945","Text":"we have this line going straight to the maximum of nth order,"},{"Start":"13:03.945 ","End":"13:05.715","Text":"or to our point P,"},{"Start":"13:05.715 ","End":"13:08.790","Text":"because P is at a maximum of nth order."},{"Start":"13:08.790 ","End":"13:15.010","Text":"Then, let\u0027s call this angle over here Theta."},{"Start":"13:16.520 ","End":"13:21.600","Text":"Now let\u0027s write in blue so that we don\u0027t get lost."},{"Start":"13:21.600 ","End":"13:26.565","Text":"Because we see that these are 90 degree angles over here,"},{"Start":"13:26.565 ","End":"13:31.845","Text":"what we can say is that S_2 to N,"},{"Start":"13:31.845 ","End":"13:36.525","Text":"so this length over here is approaching 0."},{"Start":"13:36.525 ","End":"13:42.269","Text":"Because it\u0027s less than 1 millimeter relative to this."},{"Start":"13:42.269 ","End":"13:46.470","Text":"The lengths between the slits and our point P or"},{"Start":"13:46.470 ","End":"13:51.270","Text":"the slits and any point along the screen which are a few meters away."},{"Start":"13:51.270 ","End":"13:55.199","Text":"We can say that this length is approaching 0."},{"Start":"13:55.199 ","End":"14:02.528","Text":"Then we can also say that, therefore,"},{"Start":"14:02.528 ","End":"14:10.965","Text":"S_1 to P is approximately parallel to S_2 to P,"},{"Start":"14:10.965 ","End":"14:18.509","Text":"and S_2 to P is approximately parallel to this point M,"},{"Start":"14:18.509 ","End":"14:22.200","Text":"the midpoint between the 2 slits to"},{"Start":"14:22.200 ","End":"14:28.200","Text":"P. We can really see that in this sketch, this sketch,"},{"Start":"14:28.200 ","End":"14:31.305","Text":"although of course also not to scale,"},{"Start":"14:31.305 ","End":"14:34.199","Text":"it illustrates this diagram slightly"},{"Start":"14:34.199 ","End":"14:38.235","Text":"more in the dimensions that are really being dealt with."},{"Start":"14:38.235 ","End":"14:41.520","Text":"Nothing here is drawn to scale."},{"Start":"14:41.520 ","End":"14:46.230","Text":"Because here the 1 millimeter is almost the same size as this L,"},{"Start":"14:46.230 ","End":"14:48.194","Text":"which is a few meters long."},{"Start":"14:48.194 ","End":"14:50.040","Text":"This looks a bit more like it,"},{"Start":"14:50.040 ","End":"14:53.069","Text":"and if I try and draw from over here as M,"},{"Start":"14:53.069 ","End":"14:57.180","Text":"and I try to draw this line M all the way to point P,"},{"Start":"14:57.180 ","End":"15:06.930","Text":"so here we have M. It is approximately parallel to S_1 to P,"},{"Start":"15:06.930 ","End":"15:08.925","Text":"which is just over here."},{"Start":"15:08.925 ","End":"15:11.640","Text":"It\u0027s just an extension of this N to P,"},{"Start":"15:11.640 ","End":"15:17.279","Text":"and S_2 to P. We can see that all 3 of the lines, the green,"},{"Start":"15:17.279 ","End":"15:18.599","Text":"the blue dotted line,"},{"Start":"15:18.599 ","End":"15:24.460","Text":"and the blue solid line are approximately parallel."},{"Start":"15:25.680 ","End":"15:32.499","Text":"Once we\u0027ve said that this line M all the way down to P is"},{"Start":"15:32.499 ","End":"15:39.144","Text":"parallel to S1 P and S2 P we can say that therefore,"},{"Start":"15:39.144 ","End":"15:43.825","Text":"this angle here is also 90 degrees."},{"Start":"15:43.825 ","End":"15:47.305","Text":"Then let\u0027s do some labeling over here."},{"Start":"15:47.305 ","End":"15:52.120","Text":"Let\u0027s call this point over here from M until"},{"Start":"15:52.120 ","End":"15:57.325","Text":"it reaches straight down to this line over here, S_2N."},{"Start":"15:57.325 ","End":"16:00.279","Text":"Let\u0027s call this point over here K,"},{"Start":"16:00.279 ","End":"16:03.065","Text":"and let\u0027s call this point over here O."},{"Start":"16:03.065 ","End":"16:08.175","Text":"Then because M to P is an extension of K to P,"},{"Start":"16:08.175 ","End":"16:10.604","Text":"so here is still straight line."},{"Start":"16:10.604 ","End":"16:16.025","Text":"We can say that this angle over here is also 90 degrees."},{"Start":"16:16.025 ","End":"16:19.700","Text":"We said that this angle over here was Alpha."},{"Start":"16:19.710 ","End":"16:22.420","Text":"Now what we\u0027re trying to find is,"},{"Start":"16:22.420 ","End":"16:26.215","Text":"what is this angle over here?"},{"Start":"16:26.215 ","End":"16:33.939","Text":"First of all, we can say that the triangle S_2 to M to K,"},{"Start":"16:33.939 ","End":"16:40.840","Text":"so S_2MK, is also right angled."},{"Start":"16:40.840 ","End":"16:46.430","Text":"It\u0027s also a right angle triangle."},{"Start":"16:48.150 ","End":"16:58.310","Text":"Therefore, this angle over here is called S_2MK."},{"Start":"16:58.530 ","End":"17:04.120","Text":"The angle here is going to be equal to,"},{"Start":"17:04.120 ","End":"17:05.875","Text":"let\u0027s write it over here,"},{"Start":"17:05.875 ","End":"17:11.830","Text":"so 180 is the angle in a triangle, total angle,"},{"Start":"17:11.830 ","End":"17:15.535","Text":"minus 90 for this angle,"},{"Start":"17:15.535 ","End":"17:21.640","Text":"MKS_2, which is 90, minus Alpha."},{"Start":"17:21.640 ","End":"17:26.710","Text":"That is simply equal to 90 minus Alpha."},{"Start":"17:26.710 ","End":"17:37.880","Text":"This angle S_2MK over here is equal to 90 minus Alpha."},{"Start":"17:37.950 ","End":"17:44.500","Text":"Now if we go from O to M to P,"},{"Start":"17:44.500 ","End":"17:48.160","Text":"so let\u0027s draw this in pink to not get confused,"},{"Start":"17:48.160 ","End":"17:49.705","Text":"let\u0027s do it over here."},{"Start":"17:49.705 ","End":"17:57.535","Text":"Now we\u0027re going from O to M to P. We\u0027re going O to M,"},{"Start":"17:57.535 ","End":"18:06.430","Text":"and then we\u0027re going to P. We said that this angle over here is equal to Theta."},{"Start":"18:06.430 ","End":"18:08.124","Text":"We\u0027ve already seen that."},{"Start":"18:08.124 ","End":"18:13.345","Text":"But also, this angle over here is 90 degrees."},{"Start":"18:13.345 ","End":"18:15.684","Text":"This angle over here,"},{"Start":"18:15.684 ","End":"18:20.889","Text":"which was MK to S_2 is 90 degrees."},{"Start":"18:20.889 ","End":"18:22.330","Text":"Remember we said that."},{"Start":"18:22.330 ","End":"18:25.989","Text":"Then we know that this angle over here at the head of"},{"Start":"18:25.989 ","End":"18:31.225","Text":"the triangle over here is 90 minus Alpha."},{"Start":"18:31.225 ","End":"18:36.910","Text":"We know that the angles in this triangle are going to be"},{"Start":"18:36.910 ","End":"18:43.810","Text":"180 minus this 90 over here,"},{"Start":"18:43.810 ","End":"18:46.360","Text":"the right angle, so minus 90,"},{"Start":"18:46.360 ","End":"18:52.855","Text":"minus this angle over here at the head of OMP."},{"Start":"18:52.855 ","End":"18:55.870","Text":"At the head of our pink triangle."},{"Start":"18:55.870 ","End":"19:03.769","Text":"The head of our pink triangle is 90 minus Alpha."},{"Start":"19:03.870 ","End":"19:07.659","Text":"180 minus 90 is 90."},{"Start":"19:07.659 ","End":"19:11.709","Text":"Then we have 90 minus 90,"},{"Start":"19:11.709 ","End":"19:15.115","Text":"and then minus and minus plus Alpha."},{"Start":"19:15.115 ","End":"19:17.950","Text":"Therefore, this is equal to Alpha,"},{"Start":"19:17.950 ","End":"19:21.980","Text":"this angle right over here."},{"Start":"19:22.620 ","End":"19:25.450","Text":"What is this angle over here?"},{"Start":"19:25.450 ","End":"19:28.090","Text":"It\u0027s the same as if I drew that angle over here,"},{"Start":"19:28.090 ","End":"19:32.470","Text":"or if I draw that angle over here."},{"Start":"19:32.470 ","End":"19:40.400","Text":"Therefore, we can see that Theta is equal to Alpha."},{"Start":"19:40.800 ","End":"19:43.569","Text":"This angle over here is Theta,"},{"Start":"19:43.569 ","End":"19:46.375","Text":"which is also equal to Alpha,"},{"Start":"19:46.375 ","End":"19:49.940","Text":"which was also this angle over here."},{"Start":"19:49.940 ","End":"19:54.599","Text":"Now let\u0027s label the distance between our O,"},{"Start":"19:54.599 ","End":"19:57.345","Text":"so our zeroth-order maximum,"},{"Start":"19:57.345 ","End":"20:00.675","Text":"2P, our nth order maximum."},{"Start":"20:00.675 ","End":"20:02.430","Text":"Let\u0027s call this distance X_n."},{"Start":"20:02.430 ","End":"20:10.915","Text":"X_n is the distance between our 0th order maximum and our nth order maximum."},{"Start":"20:10.915 ","End":"20:16.225","Text":"L, remember is this distance between the barrier and the screen."},{"Start":"20:16.225 ","End":"20:20.275","Text":"Let\u0027s scroll down a little bit."},{"Start":"20:20.275 ","End":"20:26.695","Text":"Then we can say that tangent of Theta,"},{"Start":"20:26.695 ","End":"20:33.174","Text":"which we\u0027ve already said that Theta and Alpha are the same angle, we proved that,"},{"Start":"20:33.174 ","End":"20:40.889","Text":"is equal to X_n divided by L. That\u0027s our first equation."},{"Start":"20:40.889 ","End":"20:44.085","Text":"Our second equation is this equation over here;"},{"Start":"20:44.085 ","End":"20:47.070","Text":"sine of Alpha, where Alpha is equal to Theta,"},{"Start":"20:47.070 ","End":"20:48.994","Text":"so we can just write sine of Theta,"},{"Start":"20:48.994 ","End":"20:58.330","Text":"is equal to n Lambda divided by d. Now I\u0027m going to write what X_n,"},{"Start":"20:58.330 ","End":"20:59.979","Text":"L, n Lambda,"},{"Start":"20:59.979 ","End":"21:02.200","Text":"and d are equal to."},{"Start":"21:02.200 ","End":"21:04.735","Text":"What they represent."},{"Start":"21:04.735 ","End":"21:09.850","Text":"X_n is the distance between the zeroth order maximum to the nth order maximum."},{"Start":"21:09.850 ","End":"21:13.000","Text":"L is the distance between the barrier and the screen,"},{"Start":"21:13.000 ","End":"21:16.420","Text":"Theta is the angle between the normal to the screen,"},{"Start":"21:16.420 ","End":"21:18.100","Text":"and the nth order maximum,"},{"Start":"21:18.100 ","End":"21:22.075","Text":"and d is the distance between the slits."},{"Start":"21:22.075 ","End":"21:25.705","Text":"Of course, Lambda is the wavelength."},{"Start":"21:25.705 ","End":"21:31.450","Text":"Then all you would have to do is measure out all of these distances."},{"Start":"21:31.450 ","End":"21:34.105","Text":"These are all things that you can measure in the lab."},{"Start":"21:34.105 ","End":"21:39.040","Text":"Of course, n, you just count how many maximums you got to."},{"Start":"21:39.040 ","End":"21:41.740","Text":"You\u0027d start, let\u0027s just scroll up again."},{"Start":"21:41.740 ","End":"21:45.160","Text":"You would start at 0,1,2,3,4,5,"},{"Start":"21:45.160 ","End"