WEBVTT 00:00:02.520 --> 00:00:04.860 The following content is provided under a Creative 00:00:04.860 --> 00:00:06.280 Commons license. 00:00:06.280 --> 00:00:08.490 Your support will help MIT OpenCourseWare 00:00:08.490 --> 00:00:12.580 continue to offer high quality educational resources for free. 00:00:12.580 --> 00:00:15.120 To make a donation or to view additional materials 00:00:15.120 --> 00:00:19.080 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.080 --> 00:00:20.114 at ocw.mit.edu. 00:00:23.560 --> 00:00:25.880 YEN-JIE LEE: So let's get started. 00:00:25.880 --> 00:00:29.180 This is our goal for 8.03. 00:00:29.180 --> 00:00:32.299 So you can see, during the exam number one, 00:00:32.299 --> 00:00:35.120 we have covered the first half of the goal, 00:00:35.120 --> 00:00:37.640 and we are actually making progress 00:00:37.640 --> 00:00:40.520 to learn about boundary conditions 00:00:40.520 --> 00:00:42.860 in one dimensional system and also 00:00:42.860 --> 00:00:44.930 in two dimensional system today. 00:00:44.930 --> 00:00:48.200 And we will actually talk about phenomena related 00:00:48.200 --> 00:00:51.350 to electromagnetic waves and optics 00:00:51.350 --> 00:00:54.800 today, which we will be able to learn 00:00:54.800 --> 00:00:58.090 two very important fundamental laws related 00:00:58.090 --> 00:01:05.370 to geometrical optics. 00:01:05.370 --> 00:01:08.620 OK, so that's the excitement. 00:01:08.620 --> 00:01:12.830 And then we started a discussion of two dimensional or three 00:01:12.830 --> 00:01:15.500 dimensional wave last time. 00:01:15.500 --> 00:01:19.760 And just in case you haven't realized that, 00:01:19.760 --> 00:01:23.780 there are two ways to go to higher dimension. 00:01:23.780 --> 00:01:28.900 So the first way is to increase the number of objects 00:01:28.900 --> 00:01:31.650 and place that in two dimensional or three 00:01:31.650 --> 00:01:33.230 dimensional space. 00:01:33.230 --> 00:01:38.190 And that is the kind of things which you will discuss today. 00:01:38.190 --> 00:01:40.340 So for example, I can have particles 00:01:40.340 --> 00:01:44.770 arranged in two dimensions which form membranes. 00:01:44.770 --> 00:01:47.750 And then we can also, on the other hand, 00:01:47.750 --> 00:01:51.830 change the direction of the electromagnetic wave, 00:01:51.830 --> 00:01:54.360 for example, as a function of time, 00:01:54.360 --> 00:01:59.270 and that's another way to go to a higher dimension. 00:01:59.270 --> 00:02:01.580 And today, as I mentioned before, we 00:02:01.580 --> 00:02:05.420 are going to talk about the first case, and on Thursday 00:02:05.420 --> 00:02:07.730 we are going to talk about the second way 00:02:07.730 --> 00:02:11.280 to go to higher dimension, which is related to polarization, 00:02:11.280 --> 00:02:11.820 et cetera. 00:02:14.510 --> 00:02:22.390 In general, higher order dimensions are hopeless. 00:02:22.390 --> 00:02:24.930 They are super complicated. 00:02:24.930 --> 00:02:27.490 And, in general, we don't really know how 00:02:27.490 --> 00:02:30.680 to solve this kind of system. 00:02:30.680 --> 00:02:34.810 Fortunately, in 8.03, what we have been doing 00:02:34.810 --> 00:02:41.020 is focusing on a small subset of questions of which are actually 00:02:41.020 --> 00:02:43.240 highly symmetric. 00:02:43.240 --> 00:02:45.860 Therefore, we can actually solve it analytically. 00:02:45.860 --> 00:02:49.630 So that will be the focus of 8.03, 00:02:49.630 --> 00:02:52.360 so that we can actually learn some physics intuition out 00:02:52.360 --> 00:02:56.310 of this kind highly idealized system. 00:02:56.310 --> 00:03:00.750 And the system which we are going to focus on today 00:03:00.750 --> 00:03:02.200 is shown here. 00:03:02.200 --> 00:03:04.200 It's a two dimensional system, which 00:03:04.200 --> 00:03:12.190 you have array of masses placing the x and y, x, y plan. 00:03:12.190 --> 00:03:17.860 And that is the system we are going to solve today. 00:03:17.860 --> 00:03:21.720 And we will learn a lot of interesting phenomena coming 00:03:21.720 --> 00:03:24.870 from the solution of this kind of system. 00:03:24.870 --> 00:03:28.020 Before we start a discussion of two dimensional system, 00:03:28.020 --> 00:03:30.780 I would like to remind you of what we have already 00:03:30.780 --> 00:03:32.970 learned from lecture eight. 00:03:32.970 --> 00:03:37.950 So that was about a system which consists of infinite number 00:03:37.950 --> 00:03:42.330 of mass and the infinite number of strings, and each string 00:03:42.330 --> 00:03:46.020 have string tension T. And all the mass, when 00:03:46.020 --> 00:03:49.920 they are in equilibrium position, 00:03:49.920 --> 00:03:53.690 the distance between all those mass in the x direction 00:03:53.690 --> 00:03:55.320 is a, OK? 00:03:55.320 --> 00:04:00.600 So we have solved this system before with space translation 00:04:00.600 --> 00:04:02.210 symmetry. 00:04:02.210 --> 00:04:06.900 And this is just a reminder that the dispersion relation, which 00:04:06.900 --> 00:04:11.160 we got a lot time, omega, as a function of k, 00:04:11.160 --> 00:04:15.510 is t over ma sine Ka over 2. 00:04:15.510 --> 00:04:18.000 So that was just a reminder of what we have 00:04:18.000 --> 00:04:20.040 learned from lecture eight. 00:04:20.040 --> 00:04:24.750 So by now you should realize that, OK, 00:04:24.750 --> 00:04:28.860 dispersion relation is unusual. 00:04:28.860 --> 00:04:31.170 This is actually telling you that this 00:04:31.170 --> 00:04:34.320 is a dispersive media, right? 00:04:34.320 --> 00:04:37.710 Because if you calculate the ratio of omega and k, 00:04:37.710 --> 00:04:40.260 you'll see that this is actually not a constant. 00:04:40.260 --> 00:04:43.150 So after all the discussion from previous lecture, 00:04:43.150 --> 00:04:46.800 you should be able to immediately realize that. 00:04:46.800 --> 00:04:52.190 And any wave propagating on this kind of system, 00:04:52.190 --> 00:04:56.220 there will be a dispersion phenomena happening 00:04:56.220 --> 00:04:58.350 in this kind of system. 00:04:58.350 --> 00:04:59.510 OK? 00:04:59.510 --> 00:05:03.150 And also from the previous lecture, 00:05:03.150 --> 00:05:07.220 we'll have learned that the eigenvectors based 00:05:07.220 --> 00:05:09.620 on space translation symmetry, it's 00:05:09.620 --> 00:05:17.150 exponential of ikx, where x is defined as j times a, where 00:05:17.150 --> 00:05:21.050 it's a is a label to tell you what which mass 00:05:21.050 --> 00:05:23.870 I was talking about. 00:05:23.870 --> 00:05:28.190 Now today, we are going to extend this 00:05:28.190 --> 00:05:31.160 to a two dimensional system. 00:05:31.160 --> 00:05:34.460 So instead of a one dimensional system 00:05:34.460 --> 00:05:37.980 we have a two dimensional array. 00:05:37.980 --> 00:05:42.950 So all the little mass all have mass equal to n, 00:05:42.950 --> 00:05:46.370 and they are placing xy plan. 00:05:46.370 --> 00:05:50.010 The coordinate system, which I defined is here. 00:05:50.010 --> 00:05:53.840 x is horizontal, and the y is vertical, 00:05:53.840 --> 00:05:57.090 and the z is actually pointing to you. 00:05:57.090 --> 00:05:59.690 And all those little mass can only 00:05:59.690 --> 00:06:05.010 oscillate toward you or going away from you, 00:06:05.010 --> 00:06:07.070 so in the z direction. 00:06:07.070 --> 00:06:12.470 It can only oscillate up and down in the z direction. 00:06:12.470 --> 00:06:16.250 And in this system we have the length scale, which 00:06:16.250 --> 00:06:21.310 is the horizontal distance between mass, is called aH. 00:06:21.310 --> 00:06:24.350 And in the vertical direction, the scale 00:06:24.350 --> 00:06:29.220 of the distance between mass is av. 00:06:29.220 --> 00:06:31.500 Also, we have string tension-- 00:06:31.500 --> 00:06:33.380 two different kinds of string tension 00:06:33.380 --> 00:06:36.080 for the vertical and horizontal direction. 00:06:36.080 --> 00:06:39.230 The vertical direction, you have string tension Tv, 00:06:39.230 --> 00:06:44.040 and in the horizontal direction you have string tension Th. 00:06:44.040 --> 00:06:44.810 OK 00:06:44.810 --> 00:06:48.680 So how do we actually describe this kind of system, right? 00:06:48.680 --> 00:06:52.520 The first thing, as we did before, 00:06:52.520 --> 00:06:57.770 is to label those little mass by my label. 00:06:57.770 --> 00:07:01.770 And my label is called Jx and Jy, 00:07:01.770 --> 00:07:09.960 which tells you which mass I was talking about in this system. 00:07:09.960 --> 00:07:13.580 Once I have defined that, the labels, 00:07:13.580 --> 00:07:19.220 I will be able to write the position of all those mass, 00:07:19.220 --> 00:07:22.490 the x direction position and y direction position, 00:07:22.490 --> 00:07:26.540 in terms of J and the A. So for example, 00:07:26.540 --> 00:07:32.510 x position of 1 over the mass will be written as Jx times ah. 00:07:32.510 --> 00:07:35.630 And y position of a specific mass, 00:07:35.630 --> 00:07:39.820 you can write it down in terms of a Jy times av. 00:07:39.820 --> 00:07:44.420 So all those things should be pretty straight forward. 00:07:44.420 --> 00:07:47.000 The interesting part is that, as we 00:07:47.000 --> 00:07:50.720 identified in the last lecture, this system 00:07:50.720 --> 00:07:52.790 is highly symmetric. 00:07:52.790 --> 00:07:56.220 It has space translation symmetry, right? 00:07:56.220 --> 00:08:00.200 Therefore, we can actually immediately figure out 00:08:00.200 --> 00:08:04.430 what will be the eigenvectors for this system. 00:08:04.430 --> 00:08:08.150 So the eigenvector-- very similar to what 00:08:08.150 --> 00:08:10.550 has been discussed here, where you have a one 00:08:10.550 --> 00:08:14.480 dimensional space translation symmetric system. 00:08:14.480 --> 00:08:18.170 Exponential of ikx was the eigenvector. 00:08:18.170 --> 00:08:22.100 Now you have eigenvector which is in two dimension, 00:08:22.100 --> 00:08:24.410 because you would like to describe 00:08:24.410 --> 00:08:27.690 not only the x direction but also y direction. 00:08:27.690 --> 00:08:31.820 And the eigenvector have exactly the same functional form 00:08:31.820 --> 00:08:34.270 because of space translational symmetry, 00:08:34.270 --> 00:08:40.460 and it is like exponentional of ixk times x. 00:08:40.460 --> 00:08:43.250 Multiply that by another exponential function-- 00:08:43.250 --> 00:08:47.470 exponential iky times y. 00:08:47.470 --> 00:08:54.770 So I think, until now, nothing should surprise you 00:08:54.770 --> 00:08:58.130 because this is what we have learned from the one 00:08:58.130 --> 00:09:02.800 dimensional system analysis. 00:09:02.800 --> 00:09:05.180 Based on what we have learned before, 00:09:05.180 --> 00:09:07.750 we can also immediately write down 00:09:07.750 --> 00:09:09.820 what would be the dispersion relation. 00:09:09.820 --> 00:09:14.570 Since we are always considering very small vibration, 00:09:14.570 --> 00:09:18.940 and this formula is still applies, 00:09:18.940 --> 00:09:23.080 therefore you can actually write down the dispersion relation-- 00:09:23.080 --> 00:09:42.260 omega squared k will be equal to 4 Th over Mah sine square Kx 00:09:42.260 --> 00:09:56.940 Ah, divided by 2 plus 4 Tv divided by Mav sine square Ky 00:09:56.940 --> 00:10:02.040 Av divided by 2. 00:10:02.040 --> 00:10:08.160 So this is actually pretty straightforward. 00:10:08.160 --> 00:10:18.440 And you can see that omega is a function of both Kx and the Ky. 00:10:18.440 --> 00:10:22.580 From the eigenvector we can also write down 00:10:22.580 --> 00:10:26.930 what would be the possible Psi xy. 00:10:26.930 --> 00:10:29.490 Now the Psi is actually the displacement 00:10:29.490 --> 00:10:33.560 in the z direction, with respect to the equilibrium position. 00:10:33.560 --> 00:10:36.980 And that is actually proportional 00:10:36.980 --> 00:10:38.810 to the eigenvector. 00:10:38.810 --> 00:10:45.635 So basically it's going to be a exponential ikx times x 00:10:45.635 --> 00:10:50.690 exponential iky times y. 00:10:50.690 --> 00:10:54.830 And of course I can write these two terms together, right? 00:10:54.830 --> 00:11:00.140 So basically what I would get is a exponential i, 00:11:00.140 --> 00:11:05.720 k is a vector times r, which is a vector. 00:11:05.720 --> 00:11:10.700 So basically k contains two components, Kx and Ky, 00:11:10.700 --> 00:11:14.110 and r have also two components, which is x and y. 00:11:17.820 --> 00:11:21.580 Again, we see that this is actually a non dispersive 00:11:21.580 --> 00:11:23.700 medium. 00:11:23.700 --> 00:11:28.440 And what we are going to do is to make linear combination 00:11:28.440 --> 00:11:32.190 of all those eigenvectors and figure out 00:11:32.190 --> 00:11:36.900 what would be the behavior when this system is oscillating 00:11:36.900 --> 00:11:42.840 at a specific frequency omega, and that is actually 00:11:42.840 --> 00:11:47.160 the corresponding normal mode at the angular frequency omega. 00:11:47.160 --> 00:11:51.000 So that is actually pretty similar to what we have done 00:11:51.000 --> 00:11:52.980 for one dimensional system. 00:11:56.940 --> 00:11:59.400 So this is a two dimensional system. 00:11:59.400 --> 00:12:05.310 Just a reminder about one dimensional system for a while. 00:12:05.310 --> 00:12:09.810 So there are two eigenvectors which 00:12:09.810 --> 00:12:12.540 have identical omega, right? 00:12:12.540 --> 00:12:16.860 So the first one is exponential ikx, 00:12:16.860 --> 00:12:20.550 and the second one is exponential of minus ikx. 00:12:23.190 --> 00:12:25.980 What we have done before is to do 00:12:25.980 --> 00:12:29.970 a linear combination of the two exponential functions, right? 00:12:29.970 --> 00:12:32.400 So what we can do is that-- 00:12:32.400 --> 00:12:36.540 OK, now I can create cosine Kx. 00:12:36.540 --> 00:12:41.510 This is actually 1 over 2 exponential ikx 00:12:41.510 --> 00:12:43.500 plus exponential minus ikx. 00:12:46.560 --> 00:12:49.440 Or I can also create sine Kx. 00:12:49.440 --> 00:12:54.900 And this is actually 1 over 2i, exponential ikx 00:12:54.900 --> 00:13:01.090 minus exponential minus ikx, for this one dimensional system. 00:13:01.090 --> 00:13:06.180 So that's how we figure out when the system is 00:13:06.180 --> 00:13:09.890 doing one of the normal mode. 00:13:09.890 --> 00:13:13.780 The shape of the system is like a cosine or a sine. 00:13:13.780 --> 00:13:17.490 Or in general, you can add these two together, 00:13:17.490 --> 00:13:19.470 and in general it can be something 00:13:19.470 --> 00:13:24.440 like cosine Kx plus 5, but 5 is actually 00:13:24.440 --> 00:13:30.630 some phase angle, which you can figure out 00:13:30.630 --> 00:13:32.880 by boundary condition. 00:13:32.880 --> 00:13:36.250 But before we introduce any boundary condition, 00:13:36.250 --> 00:13:40.000 all the k values, all the five values are allowed. 00:13:40.000 --> 00:13:44.040 Just a reminder about what we have learned before. 00:13:44.040 --> 00:13:46.630 So the situation is pretty simple. 00:13:46.630 --> 00:13:49.350 You have just plus and minus k, and then 00:13:49.350 --> 00:13:51.720 you make linear combination of these two, 00:13:51.720 --> 00:13:57.350 then you know what will be the shape of the normal mode. 00:13:57.350 --> 00:14:01.190 On the other hand, we are now talking 00:14:01.190 --> 00:14:04.430 about two dimensional case. 00:14:04.430 --> 00:14:08.330 So let's take a look at this dispersion relation. 00:14:08.330 --> 00:14:11.540 The dispersion relation we will have here, omega 00:14:11.540 --> 00:14:15.890 is a function of Kx, is a function of Ky as well. 00:14:15.890 --> 00:14:18.360 OK, what does that mean? 00:14:18.360 --> 00:14:25.850 That means I can have multiple choice of Kx and Ky, 00:14:25.850 --> 00:14:30.460 which they all produce the same omega value. 00:14:33.000 --> 00:14:35.490 So it's not as simple as this one 00:14:35.490 --> 00:14:38.070 any more, as you can see, right? 00:14:38.070 --> 00:14:43.920 Because when I slightly increase Kx, what I could do is-- 00:14:43.920 --> 00:14:49.530 OK, I can slightly reduce Ky to compensate the difference. 00:14:49.530 --> 00:14:52.840 Therefore, I can still keep omega, 00:14:52.840 --> 00:14:57.640 which is the angular frequency of the oscillation the same. 00:14:57.640 --> 00:15:01.260 OK, so that can be seen from this demonstration 00:15:01.260 --> 00:15:02.580 on the slide. 00:15:02.580 --> 00:15:07.290 You can see that this is actually one example dispersion 00:15:07.290 --> 00:15:08.980 relation. 00:15:08.980 --> 00:15:12.150 This is actually the formula which we have on the board. 00:15:12.150 --> 00:15:16.320 And what about if I set all those parameters and get, 00:15:16.320 --> 00:15:21.030 example, omega squared equal to 5 sine squared Kx, 00:15:21.030 --> 00:15:26.160 and the plus 5 sine squared Ky? 00:15:26.160 --> 00:15:28.960 OK, so what will happen? 00:15:28.960 --> 00:15:35.190 If I set my aht and the m value, so that I have this example. 00:15:35.190 --> 00:15:38.520 What will happen if I have that dispersion relation? 00:15:38.520 --> 00:15:43.260 So if I go ahead, and then plot allowed Kx and Ky 00:15:43.260 --> 00:15:49.090 value, which gives 1, you see a very beautiful pattern, right, 00:15:49.090 --> 00:15:49.960 on this. 00:15:49.960 --> 00:15:56.040 So you can see that ha, all those things on the circle 00:15:56.040 --> 00:16:00.180 can produce angular frequency omega equal to 1. 00:16:00.180 --> 00:16:02.715 And 1 will be their-- 00:16:02.715 --> 00:16:04.970 1 will be the-- normal mode will be 00:16:04.970 --> 00:16:09.570 all possible linear combination of all those possible Kx, Ky 00:16:09.570 --> 00:16:10.260 pairs. 00:16:10.260 --> 00:16:11.330 You have a question? 00:16:11.330 --> 00:16:14.830 AUDIENCE: [INAUDIBLE] 00:16:23.460 --> 00:16:25.540 YEN-JIE LEE: In general, I think this-- 00:16:25.540 --> 00:16:27.120 you mean a circular shape? 00:16:27.120 --> 00:16:29.610 AUDIENCE: [INAUDIBLE] 00:16:29.610 --> 00:16:31.470 YEN-JIE LEE: I think, in general, yes, 00:16:31.470 --> 00:16:34.480 you do have determinacy because you can-- 00:16:34.480 --> 00:16:39.450 but the shape will not be circular, for example. 00:16:39.450 --> 00:16:41.970 OK, so it can be a general function 00:16:41.970 --> 00:16:47.640 which is like the formula above, but this argument still 00:16:47.640 --> 00:16:48.330 applies. 00:16:48.330 --> 00:16:51.390 So if you have some intermediate omega value, 00:16:51.390 --> 00:16:56.010 you can always slightly increase Kx and slightly decrease Ky, 00:16:56.010 --> 00:17:00.030 and that will still satisfy the same omega value. 00:17:00.030 --> 00:17:02.085 Therefore, all the normal modes, we 00:17:02.085 --> 00:17:05.339 set specific omega will be a linear combination of all 00:17:05.339 --> 00:17:07.990 those possible normal mode. 00:17:07.990 --> 00:17:11.880 All those possible Kx and Ky pairs if you 00:17:11.880 --> 00:17:15.720 have an infinitely long system. 00:17:15.720 --> 00:17:19.710 And the law also applies to the other example, when 00:17:19.710 --> 00:17:22.290 I have omega equal to 5, then you have 00:17:22.290 --> 00:17:23.579 slightly different behavior. 00:17:23.579 --> 00:17:25.710 But the take home message is that there 00:17:25.710 --> 00:17:28.109 are many, many pairs of Kx and Ky, 00:17:28.109 --> 00:17:34.890 which can create the same amount of omega. 00:17:34.890 --> 00:17:37.670 So that makes things pretty complicated. 00:17:37.670 --> 00:17:42.050 Potentially, we can always still try 00:17:42.050 --> 00:17:44.790 to understand this by investigating 00:17:44.790 --> 00:17:48.910 all the possible k pairs of Kx and Ky. 00:17:48.910 --> 00:17:50.140 On the other hand-- 00:17:53.080 --> 00:17:55.350 this is what we have discussed before. 00:17:55.350 --> 00:17:58.840 Before you introduce boundary condition 00:17:58.840 --> 00:18:02.290 for the one dimensional system, there 00:18:02.290 --> 00:18:05.720 are infinite number of possible k value, right? 00:18:05.720 --> 00:18:08.530 All the possible k values are allowed. 00:18:08.530 --> 00:18:13.550 But after you add boundary condition-- for example, 00:18:13.550 --> 00:18:20.045 I add walls around this system, so that I basically have 00:18:20.045 --> 00:18:22.480 a fixed boundary condition. 00:18:22.480 --> 00:18:24.730 So basically, the boundary condition 00:18:24.730 --> 00:18:30.370 is that the amplitude at x equal to 0, y equal to 0, or x 00:18:30.370 --> 00:18:38.550 equal to 5ah or y equal to 4av. 00:18:38.550 --> 00:18:43.960 At the boundary, the amplitude has to be equal to 0, 00:18:43.960 --> 00:18:46.960 because it is attached to a wall. 00:18:46.960 --> 00:18:53.510 OK, when this happens this means that we 00:18:53.510 --> 00:18:59.750 will have have four wall which will 00:18:59.750 --> 00:19:02.430 have a corresponding boundary condition. 00:19:02.430 --> 00:19:06.700 So that means I have to satisfy this four boundary 00:19:06.700 --> 00:19:13.580 consideration of side 0,y evaluated at any time will be 00:19:13.580 --> 00:19:25.790 equal to Psi Lh, y, t, will be equal to Psi x,0,t, 00:19:25.790 --> 00:19:29.555 will be equal to Psi x,Lv,t. 00:19:32.750 --> 00:19:36.590 And this is all equal to 0. 00:19:36.590 --> 00:19:40.610 So those are not very difficult to understand. 00:19:40.610 --> 00:19:47.170 Those are just the four walls around the system. 00:19:47.170 --> 00:19:51.350 Once you have all those conditions-- 00:19:51.350 --> 00:19:57.530 and of course I define Lh will be equal to 5 times Ah, 00:19:57.530 --> 00:20:05.150 because there are 5 strings between the two walls 00:20:05.150 --> 00:20:07.160 in the horizontal direction. 00:20:07.160 --> 00:20:10.110 And of course I also have defined here, 00:20:10.110 --> 00:20:12.610 Lv will be 4 times equal to av. 00:20:16.810 --> 00:20:20.550 So once I have all those four boundary conditions in place 00:20:20.550 --> 00:20:25.890 that means I cannot arbitrarily choose k value and the fact, 00:20:25.890 --> 00:20:26.640 right? 00:20:26.640 --> 00:20:31.050 Otherwise, I will not be able to satisfy these four boundary 00:20:31.050 --> 00:20:33.870 conditions. 00:20:33.870 --> 00:20:39.300 So now we actually will be able to figure out 00:20:39.300 --> 00:20:44.490 that there will be only four modes in this two 00:20:44.490 --> 00:20:48.890 dimensional problem, which will give the same omega. 00:20:48.890 --> 00:20:52.140 What are the four possible nodes-- 00:20:52.140 --> 00:20:56.250 what are the four possible eigenvectors? 00:20:56.250 --> 00:21:02.520 Those are a exponential plus or minus ikx 00:21:02.520 --> 00:21:15.150 times x, exponential plus minus iky times y, where the Kx-- 00:21:15.150 --> 00:21:16.930 because of the boundary condition, 00:21:16.930 --> 00:21:20.730 which we have solved in the one dimensional system-- 00:21:20.730 --> 00:21:26.690 Kx will be equal to Nx pi divided by Lh-- 00:21:34.000 --> 00:21:40.720 in order to match the boundary condition, add x equal to 0 00:21:40.720 --> 00:21:45.130 and x equal to Lh. 00:21:45.130 --> 00:21:53.950 And Ky will be equal to y times pi divided by Lv. 00:21:53.950 --> 00:21:57.490 That's actually designed to match 00:21:57.490 --> 00:22:02.410 the boundary condition at y equal to 0 00:22:02.410 --> 00:22:05.990 and then y equal to Lv. 00:22:05.990 --> 00:22:09.040 So you can see that, like when we've 00:22:09.040 --> 00:22:14.090 seen before with one dimensional system, after you introduced 00:22:14.090 --> 00:22:18.460 the boundary condition it's not an infinity long system anymore 00:22:18.460 --> 00:22:24.400 that allowed k value, which is the length number in the x 00:22:24.400 --> 00:22:25.830 and y direction. 00:22:25.830 --> 00:22:31.150 For example, in this case, it's also become limited, 00:22:31.150 --> 00:22:37.960 and only a limited number of possible values are allowed. 00:22:37.960 --> 00:22:45.970 In this case, Nx is allowed to be equal to 1, 2 until 4, 00:22:45.970 --> 00:22:56.496 and Ny is equal 1,2,3 in this system we are talking about. 00:22:56.496 --> 00:22:57.370 Any questions so far? 00:23:00.021 --> 00:23:00.520 Yep? 00:23:00.520 --> 00:23:05.018 AUDIENCE: I think you mentioned that Kx and Ky are 00:23:05.018 --> 00:23:06.976 directly related rather than inversely related, 00:23:06.976 --> 00:23:08.928 but I'm sort of confused as to why that is. 00:23:08.928 --> 00:23:11.368 Because if you want to maintain the frequency, 00:23:11.368 --> 00:23:13.320 it increases the wave numer and [INAUDIBLE].. 00:23:17.430 --> 00:23:19.960 YEN-JIE LEE: Yeah, so I was talking 00:23:19.960 --> 00:23:26.240 about when I choose Kx and Ky in the infinitely long system. 00:23:26.240 --> 00:23:29.270 OK, all of the possible values of Kx and Ky 00:23:29.270 --> 00:23:32.830 are allowed because I have a infinitely long system 00:23:32.830 --> 00:23:34.730 with no boundary condition. 00:23:34.730 --> 00:23:40.450 And in that case, going back to this dispersion relation, 00:23:40.450 --> 00:23:42.870 I have the freedom to-- 00:23:42.870 --> 00:23:46.150 OK, so when I increase a little bit, Kx, 00:23:46.150 --> 00:23:49.720 I can always decrease a little bit, the Ky. 00:23:49.720 --> 00:23:54.520 OK, so the question is why that's not the case, right, 00:23:54.520 --> 00:23:56.130 for the discrete case. 00:23:56.130 --> 00:24:01.690 As you can see from here, after we introduced the boundary 00:24:01.690 --> 00:24:04.590 condition, the four boundary conditions 00:24:04.590 --> 00:24:11.620 especially describe the boundary of the four walls. 00:24:11.620 --> 00:24:15.040 And what is going to happen is that you will also see that 00:24:15.040 --> 00:24:19.810 the allowed Kx value is becoming limited, 00:24:19.810 --> 00:24:23.830 because you cannot arbitrarily choose with lengths, right, 00:24:23.830 --> 00:24:27.340 if you choose a side along the wavelengths like what we have 00:24:27.340 --> 00:24:29.840 been trying to do for the infinitely long system-- 00:24:29.840 --> 00:24:33.460 not that it matched the boundary condition. 00:24:33.460 --> 00:24:36.850 Therefore, you don't have this degree of freedom 00:24:36.850 --> 00:24:41.880 to choose slightly higher or slightly lower 00:24:41.880 --> 00:24:46.380 Ky when I change a Kx. 00:24:46.380 --> 00:24:49.900 So you can see that the allowed value are discrete. 00:24:49.900 --> 00:24:53.380 Therefore, the number of possible combinations 00:24:53.380 --> 00:24:55.870 of Kx and Ky is also limited. 00:24:55.870 --> 00:24:59.650 And in this case, it's actually very likely 00:24:59.650 --> 00:25:02.920 to be limited to be only four pairs, which is actually 00:25:02.920 --> 00:25:05.330 plus, minus Kx and the plus, minus Ky. 00:25:07.730 --> 00:25:08.230 All right. 00:25:11.420 --> 00:25:12.880 Thank you for the question. 00:25:12.880 --> 00:25:15.800 OK, then once I have those I can do 00:25:15.800 --> 00:25:21.560 a linear combination of these four possible eigenvectors. 00:25:21.560 --> 00:25:23.360 And also, at the same time, I will 00:25:23.360 --> 00:25:26.870 try to match the boundary condition. 00:25:26.870 --> 00:25:30.250 So if I jump forward, basically what you can conclude 00:25:30.250 --> 00:25:38.570 is that Psi Nx and y, so that's with an Nx value y 00:25:38.570 --> 00:25:43.910 value chosen for the determination of Kx and Ky. 00:25:43.910 --> 00:25:50.020 And is this actually a function of x and y and of course also 00:25:50.020 --> 00:25:52.560 time, when I also make it oscillate 00:25:52.560 --> 00:25:54.060 as a function of time. 00:25:54.060 --> 00:25:57.920 This will be equal to some arbitrary constant, A 00:25:57.920 --> 00:26:08.710 of amplitude Nx, Ny sine Nx pi x, 00:26:08.710 --> 00:26:22.470 divided by Lh sine Ny times y divided by Lv. 00:26:22.470 --> 00:26:27.240 And of course, you can see that this is actually sine, right? 00:26:27.240 --> 00:26:29.670 It's actually, the same as what we have done for the one 00:26:29.670 --> 00:26:31.030 dimensional system, right? 00:26:31.030 --> 00:26:34.185 So if you have two boundary conditions that said, 00:26:34.185 --> 00:26:38.200 look, the beginning and the end, therefor, 00:26:38.200 --> 00:26:41.890 the corresponding normal mode is always a sine function. 00:26:41.890 --> 00:26:45.360 So that's what we have learned from the one 00:26:45.360 --> 00:26:46.410 dimensional system. 00:26:46.410 --> 00:26:51.390 And this is also the case for the two dimensional system. 00:26:51.390 --> 00:26:57.030 And of course, don't forget this wave function 00:26:57.030 --> 00:26:59.730 is changing as a function of time 00:26:59.730 --> 00:27:03.420 oscillating up and down harmonically. 00:27:03.420 --> 00:27:14.200 Therefore, you have sine omega Nx, Ny times T plus Theta, 00:27:14.200 --> 00:27:18.520 which is a phase to be determined 00:27:18.520 --> 00:27:21.610 by initial conditions. 00:27:21.610 --> 00:27:27.730 And you can see that the whole equation, a sine sine 00:27:27.730 --> 00:27:31.620 is multiplied by a sine omega T plus 5 because 00:27:31.620 --> 00:27:33.340 of beta function, right? 00:27:33.340 --> 00:27:36.960 So that means the shape is actually 00:27:36.960 --> 00:27:39.820 going up and down harmonically. 00:27:39.820 --> 00:27:43.240 So the shape is fixed, which is sine times sine, 00:27:43.240 --> 00:27:45.960 and the whole thing is oscillating 00:27:45.960 --> 00:27:48.790 at the same frequency at the same phase, which 00:27:48.790 --> 00:27:51.410 is the definition of normal mode, right? 00:27:51.410 --> 00:27:52.570 Just a reminder. 00:27:52.570 --> 00:27:56.740 And how do we actually imagine what is actually happening? 00:27:56.740 --> 00:27:59.980 That brings me to the demonstration, 00:27:59.980 --> 00:28:05.980 so we can really visualize how this kind of system 00:28:05.980 --> 00:28:11.470 will look like by a little simulation. 00:28:11.470 --> 00:28:21.340 So, suppose I choose Nx and Ny equal to 1 00:28:21.340 --> 00:28:23.250 and see what will happen. 00:28:23.250 --> 00:28:28.550 This is the kind of oscillation you will expect, right? 00:28:28.550 --> 00:28:33.820 So if you choose Nx equal to 1, Ny equal to 1, 00:28:33.820 --> 00:28:36.070 then this is a system. 00:28:36.070 --> 00:28:40.360 Basically you have sine function with no node 00:28:40.360 --> 00:28:43.690 in x and y direction. 00:28:43.690 --> 00:28:49.840 Therefore, if you do get this simulation, 00:28:49.840 --> 00:28:55.830 you can see that there will be no node in the x,y plane, 00:28:55.830 --> 00:29:02.770 and all those particles are either going toward you 00:29:02.770 --> 00:29:05.620 or going away from you. 00:29:05.620 --> 00:29:13.190 They only oscillate in the z direction in this simulation. 00:29:13.190 --> 00:29:16.120 And also, you can see that now I can increase, for example-- 00:29:19.930 --> 00:29:24.980 I can increase the Kx by setting Nx to be 2 and see 00:29:24.980 --> 00:29:26.190 what will happen. 00:29:26.190 --> 00:29:30.200 So what is going to happen is that if I have higher 00:29:30.200 --> 00:29:36.110 Kx in the x direction-- so the next possible normal mode is 00:29:36.110 --> 00:29:43.510 that you have a full sine wave in the x direction, then 00:29:43.510 --> 00:29:46.610 you are going to see two components 00:29:46.610 --> 00:29:49.730 in this demonstration. 00:29:49.730 --> 00:29:54.980 And one part of the system is actually moving toward you 00:29:54.980 --> 00:29:58.780 while the other half part of the system 00:29:58.780 --> 00:30:00.680 is actually moving away from you. 00:30:00.680 --> 00:30:05.290 And you can actually see the node, or nodal line 00:30:05.290 --> 00:30:06.890 in this case, because we are talking 00:30:06.890 --> 00:30:10.700 about a two dimensional system in the middle 00:30:10.700 --> 00:30:13.730 of the distribution. 00:30:13.730 --> 00:30:16.730 Of course, we can always go crazy, right? 00:30:16.730 --> 00:30:20.640 I can set this to a really high value. 00:30:20.640 --> 00:30:22.310 So in this case, the highest value 00:30:22.310 --> 00:30:27.210 I can set these is 3 and 4, and see what happens. 00:30:27.210 --> 00:30:30.290 And this is actually a beautiful shape 00:30:30.290 --> 00:30:35.060 which is actually complicated but understandable, as you 00:30:35.060 --> 00:30:39.380 see in this demonstration. 00:30:39.380 --> 00:30:43.760 And all those little particles in this system 00:30:43.760 --> 00:30:48.650 are oscillating up and down at the same angular frequency 00:30:48.650 --> 00:30:50.540 and also at the same phase. 00:30:53.450 --> 00:30:56.600 Any questions? 00:30:56.600 --> 00:31:02.100 OK, so now we have done the discrete case, right? 00:31:02.100 --> 00:31:06.910 And of course we can also go to the continuous case. 00:31:06.910 --> 00:31:10.800 So if we go to a continuous limit, 00:31:10.800 --> 00:31:17.590 now I can assume that there is a symmetry 00:31:17.590 --> 00:31:22.720 between a horizontal direction and the vertical direction. 00:31:22.720 --> 00:31:29.460 I assume that Th is equal to Tv is equal to T. 00:31:29.460 --> 00:31:33.690 And also I assume that the length scale in the x direction 00:31:33.690 --> 00:31:37.030 and the y directions is equal, and the length scale 00:31:37.030 --> 00:31:43.450 is A. In order to make the whole system continuous, 00:31:43.450 --> 00:31:47.740 I need to increase the number of objects in the system, 00:31:47.740 --> 00:31:52.570 and at the same time I also need to decrease the distance 00:31:52.570 --> 00:31:54.860 between all those objects. 00:31:54.860 --> 00:31:56.970 So therefore, I need to have-- 00:31:56.970 --> 00:32:00.790 this length scale goes to 0. 00:32:00.790 --> 00:32:04.690 And what is going to happen is that if I rewrite 00:32:04.690 --> 00:32:11.050 my omega, which is a dispersion relation, what I am going 00:32:11.050 --> 00:32:19.360 to get is 4T divided by Na Kx squared, A squared, 00:32:19.360 --> 00:32:29.060 divided by 4, plus 4T over Na, Ky squared a squared over 4. 00:32:32.520 --> 00:32:37.290 This is issue because I am taking-- 00:32:37.290 --> 00:32:41.880 Ah and V need to be equal to-- and also having 00:32:41.880 --> 00:32:44.290 to be a very, very small value. 00:32:44.290 --> 00:32:50.300 Therefore, sine theta is roughly theta, right? 00:32:50.300 --> 00:32:54.860 Therefore, I can immediately write down this expression. 00:32:54.860 --> 00:33:03.080 And this will be equal to Ta divided by N, Kx squared 00:33:03.080 --> 00:33:04.100 plus Ky squared. 00:33:07.780 --> 00:33:11.590 So we are facing exactly the same situation. 00:33:11.590 --> 00:33:17.200 When I decrease A, I am going to add more objects 00:33:17.200 --> 00:33:19.390 into the system, but I don't want 00:33:19.390 --> 00:33:22.750 to have an infinitely large mass. 00:33:22.750 --> 00:33:25.300 Therefore, I also need to ensure the fix 00:33:25.300 --> 00:33:32.980 the ratio of m and a, so that when I actually increase 00:33:32.980 --> 00:33:37.510 the number of objects, I don't actually make the total mass 00:33:37.510 --> 00:33:40.330 go to infinity. 00:33:40.330 --> 00:33:45.770 So what I could do is I can define Rho S is actually 00:33:45.770 --> 00:33:49.120 the surface mass density. 00:33:49.120 --> 00:33:51.460 So the surface mass density is defined 00:33:51.460 --> 00:33:55.360 as m divided by a squared. 00:33:55.360 --> 00:33:59.770 And I can also define a surface tension. 00:33:59.770 --> 00:34:04.060 Surface tension Ts will be equal to T over a. 00:34:06.830 --> 00:34:12.050 And in this case, basically, I will be able to control, 00:34:12.050 --> 00:34:16.159 so that when I increase the number of objects, 00:34:16.159 --> 00:34:18.560 mass doesn't go to infinity, and I 00:34:18.560 --> 00:34:24.560 have constant surface tension and constant surface mass 00:34:24.560 --> 00:34:26.239 density. 00:34:26.239 --> 00:34:29.449 If I have defined this to quantity 00:34:29.449 --> 00:34:38.900 then this will become Ts divided by Rho S Kx squared 00:34:38.900 --> 00:34:43.010 plus Ky squared, and this will be 00:34:43.010 --> 00:34:51.830 equal to Ts divided by Rho S, k vector squared. 00:34:51.830 --> 00:34:57.230 And this k vector is a two dimensional vector. 00:35:01.280 --> 00:35:06.020 So we are actually almost there to make it continuous. 00:35:06.020 --> 00:35:10.360 So now I can make a goes to a very small value. 00:35:10.360 --> 00:35:14.120 We fixed the Ts and the row S. Very 00:35:14.120 --> 00:35:17.790 similar to what we have learned from the one dimensional case. 00:35:17.790 --> 00:35:20.030 Basically what we actually found is 00:35:20.030 --> 00:35:21.845 that time in the one dimensional case 00:35:21.845 --> 00:35:26.890 is that M minus 1 K metrics become minus T over Rho 00:35:26.890 --> 00:35:29.710 L, partial squared, partial x squared 00:35:29.710 --> 00:35:33.200 in the one dimensional case. 00:35:33.200 --> 00:35:37.040 And in the two dimensional case, without working 00:35:37.040 --> 00:35:40.160 through all the detail of mass, basically 00:35:40.160 --> 00:35:43.340 what we are going to get is partial square partial T 00:35:43.340 --> 00:35:48.280 square Psi xy-- 00:35:48.280 --> 00:35:52.150 It's actually a function of x and y and the time, 00:35:52.150 --> 00:35:55.730 right, because this is actually a two dimensional system. 00:35:55.730 --> 00:36:02.660 And this will be equal to V squared, partial squared, 00:36:02.660 --> 00:36:08.630 partial x squared plus partial squared, partial y squared, 00:36:08.630 --> 00:36:12.680 Psi xy and T-- 00:36:12.680 --> 00:36:14.840 very similar to what we have done 00:36:14.840 --> 00:36:17.570 for the one dimensional system. 00:36:17.570 --> 00:36:22.130 And of course I can, as you define this, as del squared. 00:36:22.130 --> 00:36:25.796 And basically what you are going to get is V squared, 00:36:25.796 --> 00:36:28.730 del squared, Psi x,y,t. 00:36:34.130 --> 00:36:37.430 So basically we again see this wave equation, 00:36:37.430 --> 00:36:41.990 but this wave equation is now a two dimensional wave equation. 00:36:41.990 --> 00:36:46.370 And we can also figure out what will be the V value, right, 00:36:46.370 --> 00:36:48.440 so what will be the velocity? 00:36:48.440 --> 00:36:55.330 The velocity which is going to be square root of Ts 00:36:55.330 --> 00:37:01.490 over Rho S. This is very similar what we have done 00:37:01.490 --> 00:37:05.905 for the continuous case, and in this case, what replaced 00:37:05.905 --> 00:37:10.970 T over Rho L is Ts over Rho S. Therefore, 00:37:10.970 --> 00:37:15.890 what we actually see that if I increase the surface tension, 00:37:15.890 --> 00:37:18.380 then the velocity will increase. 00:37:18.380 --> 00:37:24.620 If I decrease the mass per unit area, 00:37:24.620 --> 00:37:31.070 Rho S, then I will be able to have a much faster traveling 00:37:31.070 --> 00:37:34.790 wave from this kind of media. 00:37:34.790 --> 00:37:38.130 And what we can actually immediately also write down 00:37:38.130 --> 00:37:45.876 is that the Psi will be proportional to A sine Kx 00:37:45.876 --> 00:37:54.530 times x, sine Ky times y, and Psi omega T plus 5, 00:37:54.530 --> 00:37:59.930 where omega is calculated from the input Kx and Ky 00:37:59.930 --> 00:38:04.805 for this standing wave solution. 00:38:08.430 --> 00:38:13.080 And very similarly, I can also argue that-- 00:38:13.080 --> 00:38:16.290 in the three dimensional case I can actually 00:38:16.290 --> 00:38:18.890 follow exactly the same argument. 00:38:18.890 --> 00:38:20.750 Basically, in the three dimensional case, 00:38:20.750 --> 00:38:25.510 as well, we already see in the electromagnetic wave 00:38:25.510 --> 00:38:29.640 discussion, the three dimensional wave equation can 00:38:29.640 --> 00:38:32.100 be written as partial squared, partial T squared 00:38:32.100 --> 00:38:37.020 Psi is a function of x, y, and z and T. 00:38:37.020 --> 00:38:42.700 And this will be equal to V squared, partial squared, 00:38:42.700 --> 00:38:46.920 partial x squared plus partial squared, partial y squared 00:38:46.920 --> 00:38:51.300 plus partial squared, partial V squared-- 00:38:51.300 --> 00:39:03.141 Psi is a function of x,y,z, and T. Any questions so far? 00:39:03.141 --> 00:39:03.640 Nope? 00:39:07.320 --> 00:39:12.640 OK, so everything is crystal clear, right? 00:39:12.640 --> 00:39:15.570 OK, so this is actually the animation, which 00:39:15.570 --> 00:39:17.280 I showed you before already. 00:39:17.280 --> 00:39:21.670 So this is actually the two dimensional vibration 00:39:21.670 --> 00:39:23.070 of membranous. 00:39:23.070 --> 00:39:26.490 So basically the first one is what I have shown you 00:39:26.490 --> 00:39:31.770 when I choose a very small K value, which 00:39:31.770 --> 00:39:35.430 only make half of the sign and which 00:39:35.430 --> 00:39:38.010 match the boundary condition. 00:39:38.010 --> 00:39:41.340 Basically you see that there are oscillation, 00:39:41.340 --> 00:39:44.460 which you have the middle part, which 00:39:44.460 --> 00:39:48.480 is either going toward you or going away from you 00:39:48.480 --> 00:39:49.930 in this continuous system. 00:39:49.930 --> 00:39:55.030 So basically the solution is actually remarkably the same 00:39:55.030 --> 00:39:57.840 as what we have seen in the discrete system. 00:39:57.840 --> 00:39:59.810 OK, that's actually what I wanted to say. 00:39:59.810 --> 00:40:04.590 And also, of course, you can increase the k value, 00:40:04.590 --> 00:40:10.290 so that you go to the higher frequency normal mode. 00:40:10.290 --> 00:40:14.240 And you can see that if you have more and more nodal 00:40:14.240 --> 00:40:21.171 lines, which is actually the lines describing the-- 00:40:21.171 --> 00:40:24.870 the lines which you actually have no oscillation 00:40:24.870 --> 00:40:26.400 at all on the surface. 00:40:26.400 --> 00:40:29.040 For example, in this case, the nodal lines, 00:40:29.040 --> 00:40:32.070 as you're passing through the middle of this figure-- 00:40:32.070 --> 00:40:36.570 because all those little mass, all the other high particles 00:40:36.570 --> 00:40:38.730 are vibrating like crazy. 00:40:38.730 --> 00:40:43.840 But all the particles on this line, the nodal line, 00:40:43.840 --> 00:40:48.470 they're not at all moving, because that's actually 00:40:48.470 --> 00:40:49.920 at this position-- 00:40:49.920 --> 00:40:53.880 which is having one of the sine function equal to 0. 00:40:53.880 --> 00:40:57.090 Therefore, no matter what you do as a function of time, 00:40:57.090 --> 00:41:00.850 how you evolve the system, all those particle at that line 00:41:00.850 --> 00:41:04.230 will not move at all. 00:41:04.230 --> 00:41:10.470 And this was demonstrated from this table here. 00:41:10.470 --> 00:41:13.890 It's actually Chladni figures. 00:41:13.890 --> 00:41:17.610 You can see that in a two dimensional case 00:41:17.610 --> 00:41:21.600 the figures can look very complicated. 00:41:21.600 --> 00:41:23.670 So basically what it's showing here 00:41:23.670 --> 00:41:27.150 is that you have a square plate and it's 00:41:27.150 --> 00:41:34.080 attached to a vibrator, and basically this vibrator 00:41:34.080 --> 00:41:35.460 can be controlled. 00:41:35.460 --> 00:41:39.390 I can change the frequency of that vibration. 00:41:39.390 --> 00:41:47.940 When I reach resonance, which excites one of the normal mode, 00:41:47.940 --> 00:41:52.620 then this plate will be oscillating 00:41:52.620 --> 00:41:56.440 in a specific pattern. 00:41:56.440 --> 00:41:58.470 And those lines are actually showing you 00:41:58.470 --> 00:42:02.670 that the plates, which you have no oscillation at all 00:42:02.670 --> 00:42:04.050 as a function of time. 00:42:04.050 --> 00:42:09.820 Because if I, for example, turn on this demo again, 00:42:09.820 --> 00:42:12.483 you can see that if I turn on this demo-- 00:42:17.320 --> 00:42:21.430 you can see that all the sand on the plates 00:42:21.430 --> 00:42:25.430 are vibrating because now I am oscillating 00:42:25.430 --> 00:42:32.430 this plate by the vibration generator and the button-- 00:42:32.430 --> 00:42:33.720 by the motor and the button. 00:42:33.720 --> 00:42:38.050 And if I change the oscillation frequency so 00:42:38.050 --> 00:42:43.020 you can see that this frequency doesn't match with one 00:42:43.020 --> 00:42:46.860 of the normal mode frequency. 00:42:46.860 --> 00:42:50.160 Therefore, they will not be a lot of activity. 00:42:50.160 --> 00:42:56.590 But if I now change the frequency, 00:42:56.590 --> 00:43:02.370 so that it matches with one of the oscillation frequency 00:43:02.370 --> 00:43:05.280 for one of the normal mode of this system, 00:43:05.280 --> 00:43:11.450 you can see the, oh, some really cryptic pattern is formed! 00:43:11.450 --> 00:43:16.080 You can see that, oh, it have a very complicated pattern. 00:43:16.080 --> 00:43:19.810 And if I put my finger in one of the lines 00:43:19.810 --> 00:43:23.480 here I don't feel the vibration, but on the other hand, 00:43:23.480 --> 00:43:25.980 if I put my finger here, I can actually 00:43:25.980 --> 00:43:30.920 feel that there's lot of vibration at that point. 00:43:30.920 --> 00:43:32.960 I can always change the frequency 00:43:32.960 --> 00:43:36.000 and see what will happen. 00:43:36.000 --> 00:43:40.110 And then you can see that now I increase the frequency, 00:43:40.110 --> 00:43:45.509 and now I am actually trying to excite another mode. 00:43:45.509 --> 00:43:50.440 Now I need some more sand. 00:43:50.440 --> 00:43:55.740 You can see that I randomly throw sand on this plate, 00:43:55.740 --> 00:43:58.880 and then you can see that those centered as you bounce it 00:43:58.880 --> 00:44:06.060 around until it sits on the nodal line, which 00:44:06.060 --> 00:44:09.000 no vibration actually happens. 00:44:09.000 --> 00:44:15.576 OK, so let's go back to one of the lower frequency mode, which 00:44:15.576 --> 00:44:17.052 we showed you before. 00:44:21.000 --> 00:44:24.720 Now the question is, OK, you can see this complicated pattern 00:44:24.720 --> 00:44:32.490 almost look ridiculous, can we actually reproduce this pattern 00:44:32.490 --> 00:44:33.720 by our calculation? 00:44:36.650 --> 00:44:42.740 So we have seen that, OK, I can conclude that the normal mode 00:44:42.740 --> 00:44:44.090 looks like this, right? 00:44:44.090 --> 00:44:51.009 So therefore, I must be able to explain 00:44:51.009 --> 00:44:52.550 all those patterns, which is actually 00:44:52.550 --> 00:44:56.780 shown in this experiment. 00:44:56.780 --> 00:45:00.555 So that's actually what I am going to do to give you a try. 00:45:04.070 --> 00:45:07.650 So this is a little demonstration 00:45:07.650 --> 00:45:12.110 which I actually wrote. 00:45:12.110 --> 00:45:17.210 This demonstration actually has the solution to this two 00:45:17.210 --> 00:45:19.460 dimensional problem. 00:45:19.460 --> 00:45:23.300 And also the boundary condition is that-- 00:45:23.300 --> 00:45:26.600 or say, the condition which, as you can see on my solution, 00:45:26.600 --> 00:45:32.000 is that I require the center of the plate to be driven, 00:45:32.000 --> 00:45:35.840 because that's where I start to vibrate this plate. 00:45:35.840 --> 00:45:39.770 And I drive this plate up and down to see 00:45:39.770 --> 00:45:42.020 what is going to happen. 00:45:42.020 --> 00:45:44.670 So from this analytical calculation 00:45:44.670 --> 00:45:49.770 you can see that you expect a circle in the middle and also 00:45:49.770 --> 00:45:55.170 four lines which actually cover this circle. 00:45:55.170 --> 00:45:57.560 And also there are some strange structure 00:45:57.560 --> 00:46:02.360 at the edge of the plate. 00:46:02.360 --> 00:46:05.780 And you can actually compare this calculation 00:46:05.780 --> 00:46:10.430 to this result. It doesn't really match perfectly. 00:46:10.430 --> 00:46:12.840 So you can see that there is some imperfection, 00:46:12.840 --> 00:46:16.310 but you get that ring in the middle, 00:46:16.310 --> 00:46:20.078 and you do see these 1-2-3-4-5-6-7-8, 00:46:20.078 --> 00:46:24.170 8 lines produced in this experiment. 00:46:24.170 --> 00:46:28.250 So this experiment is not perfect because there's a, 00:46:28.250 --> 00:46:30.060 you know, stiffness of this thing 00:46:30.060 --> 00:46:34.640 and also some energy dissipation, et cetera. 00:46:34.640 --> 00:46:36.590 But you can see that, sort of, we 00:46:36.590 --> 00:46:41.030 can actually use our calculation to explain this pattern! 00:46:41.030 --> 00:46:42.920 That's really cool, right? 00:46:42.920 --> 00:46:49.250 And the advantage is that now I have this wonderful simulation. 00:46:49.250 --> 00:46:53.000 I can put in all the crazy numbers, 00:46:53.000 --> 00:46:56.760 and you see that, huh, if I increase the K value, 00:46:56.760 --> 00:47:02.240 I can really make all kinds of ridiculous patterns out 00:47:02.240 --> 00:47:03.020 of this. 00:47:03.020 --> 00:47:08.640 And all these things can be kind of realized by this experiment. 00:47:08.640 --> 00:47:10.050 So you can see that, for example, 00:47:10.050 --> 00:47:17.960 I can now also turn on this, and I can actually 00:47:17.960 --> 00:47:25.720 increase the frequency to a very high frequency, for example. 00:47:25.720 --> 00:47:28.690 Then I can see that, oh, the pattern really 00:47:28.690 --> 00:47:30.690 becomes much more complicated. 00:47:38.498 --> 00:47:42.330 Now I have a circle and there are many, many more structures 00:47:42.330 --> 00:47:44.740 which you're seeing in the surrounding area. 00:47:44.740 --> 00:47:48.380 And of course I can again increase, increase, 00:47:48.380 --> 00:47:51.180 and see what happens. 00:47:51.180 --> 00:47:54.390 I don't know what is going to happen because every time I 00:47:54.390 --> 00:47:57.270 do this experiment I get a different pattern. 00:48:01.450 --> 00:48:07.800 OK, now this seems to be a very nice frequency. 00:48:07.800 --> 00:48:11.790 It's getting harder and harder. 00:48:11.790 --> 00:48:16.100 You can see that this is really crazy. 00:48:16.100 --> 00:48:18.510 Holy mackerel, right? 00:48:18.510 --> 00:48:21.010 What the hell is this? 00:48:21.010 --> 00:48:23.880 So you can see that all those crazy patterns can be created. 00:48:26.440 --> 00:48:28.410 And of course, during the break, you 00:48:28.410 --> 00:48:33.130 are welcome to come forward and play with this. 00:48:33.130 --> 00:48:36.180 So you can see that we can actually 00:48:36.180 --> 00:48:41.710 understand, sort of, the pattern produced from this experiment. 00:48:41.710 --> 00:48:43.960 That's actually very exciting, because that's actually 00:48:43.960 --> 00:48:45.290 why we are physicists, right? 00:48:45.290 --> 00:48:48.100 We would like to know why those patterns are formed, 00:48:48.100 --> 00:48:49.660 and now you know why. 00:48:49.660 --> 00:48:51.670 Those patterns are formed because there 00:48:51.670 --> 00:48:56.830 are nodal lines in this two dimensional normal mode modes. 00:48:56.830 --> 00:48:59.440 And the little sands really love to sit there, 00:48:59.440 --> 00:49:02.050 because you want to sit in a place which you 00:49:02.050 --> 00:49:04.600 don't have a lot of vibration. 00:49:04.600 --> 00:49:06.420 It's not very comfortable, right? 00:49:06.420 --> 00:49:10.030 So you sit in the place, which, hm, vibrates? 00:49:10.030 --> 00:49:10.922 Your problem. 00:49:10.922 --> 00:49:11.880 Vibrate's your problem. 00:49:11.880 --> 00:49:15.740 I sit here where there is no vibration. 00:49:15.740 --> 00:49:20.260 So that's basically how we explain these strange figures 00:49:20.260 --> 00:49:21.480 which we can see. 00:49:21.480 --> 00:49:25.270 And just for fun you can see that I can also generate 00:49:25.270 --> 00:49:27.420 all kinds of craziness. 00:49:27.420 --> 00:49:31.480 You can input all kinds of different Nx and Ny values, 00:49:31.480 --> 00:49:35.110 and you get all those wonderful figures for free. 00:49:35.110 --> 00:49:37.600 Maybe we can actually make some T-shirts 00:49:37.600 --> 00:49:41.260 with all those figures on the T-shirt, right? 00:49:41.260 --> 00:49:46.790 OK, so we had a lot of fun with this two dimensional plate. 00:49:46.790 --> 00:49:52.480 How about what will happen if I have a circular plate? 00:49:52.480 --> 00:49:54.620 What does it do? 00:49:54.620 --> 00:49:58.390 Unfortunately, I would not be able to solve the two 00:49:58.390 --> 00:50:01.600 dimensional plate problem in front of you 00:50:01.600 --> 00:50:05.920 because that will give you a Bessel function, which is not 00:50:05.920 --> 00:50:09.410 the end of the world, but that's actually kind of complicated. 00:50:09.410 --> 00:50:12.790 If I put it in mid-term exam, that's actually not 00:50:12.790 --> 00:50:15.520 very encouraging, right? 00:50:15.520 --> 00:50:17.980 But I can actually tell you what will be the solution. 00:50:17.980 --> 00:50:19.780 The solution will be a Bessel function. 00:50:19.780 --> 00:50:24.400 Basically you will have a lot of ring-like structures 00:50:24.400 --> 00:50:27.190 if I have a circular plate. 00:50:27.190 --> 00:50:30.460 And I can actually do an experiment which actually shows 00:50:30.460 --> 00:50:35.740 you the behavior of the circular boundary condition 00:50:35.740 --> 00:50:39.370 and see what kind of pattern can we see. 00:50:39.370 --> 00:50:45.790 So here I have a kind of complicated experiment. 00:50:45.790 --> 00:50:48.760 So here I have this ring, which I 00:50:48.760 --> 00:50:56.580 would like to produce some film on this ring. 00:50:56.580 --> 00:50:59.140 So see if I am successful. 00:50:59.140 --> 00:51:00.770 Kind of. 00:51:00.770 --> 00:51:01.620 OK. 00:51:01.620 --> 00:51:12.100 Now I can put this a soft film in front of the speaker. 00:51:12.100 --> 00:51:16.460 I can actually oscillate this thing-- 00:51:16.460 --> 00:51:18.527 membranes by the speaker. 00:51:18.527 --> 00:51:20.360 Oops, don't want to destroy everything here. 00:51:24.510 --> 00:51:28.910 All right, so now I can turn on this, so that we have light. 00:51:28.910 --> 00:51:32.990 And of course I will turn on the signal generator, 00:51:32.990 --> 00:51:34.580 so that I can hear-- 00:51:34.580 --> 00:51:39.020 I can actually start to vibrate the membranes. 00:51:39.020 --> 00:51:47.185 Before I do that, I have to turn everything off, hide images. 00:51:47.185 --> 00:51:49.870 All right, I hope you can see something. 00:51:49.870 --> 00:51:50.370 Can you? 00:51:53.690 --> 00:51:55.420 Can you see something on the-- 00:51:58.210 --> 00:52:05.030 it's kind of difficult to see it, but that should be there. 00:52:05.030 --> 00:52:10.140 OK, now I can turn on this speaker, 00:52:10.140 --> 00:52:13.040 and you can see that there are some patterns which 00:52:13.040 --> 00:52:16.160 it's probably difficult to see. 00:52:16.160 --> 00:52:18.360 Kind of see, right? 00:52:18.360 --> 00:52:20.560 There are rings. 00:52:20.560 --> 00:52:23.336 You can see it on the speaker. 00:52:23.336 --> 00:52:27.710 So you can see that now I have one, two, three-- three rings, 00:52:27.710 --> 00:52:28.850 right? 00:52:28.850 --> 00:52:33.950 Because I couldn't turn off the light, which is actually 00:52:33.950 --> 00:52:37.220 emitted from the sun, right? 00:52:37.220 --> 00:52:39.080 So I cannot turn off sun. 00:52:39.080 --> 00:52:42.130 Therefore, you can barely see this figure. 00:52:42.130 --> 00:52:45.470 So we shall explain the result of this experiment. 00:52:45.470 --> 00:52:51.540 And you can see that if I increase the frequency, 00:52:51.540 --> 00:52:54.270 according to the solution from Bessel function, 00:52:54.270 --> 00:52:58.312 you will see more rings got excited. 00:52:58.312 --> 00:53:00.270 So you can see that now I have one, two, three, 00:53:00.270 --> 00:53:01.740 four-- four rings. 00:53:01.740 --> 00:53:04.542 And of course I can continue to increase, increase, 00:53:04.542 --> 00:53:06.102 and increase. 00:53:06.102 --> 00:53:10.470 And you will see that there are even more rings produced. 00:53:10.470 --> 00:53:13.280 Essentially what I'm doing is actually really 00:53:13.280 --> 00:53:18.500 trying to vibrate and excite one of the normal mode 00:53:18.500 --> 00:53:21.530 by this loud speaker. 00:53:21.530 --> 00:53:23.420 And you can actually kind of see-- 00:53:23.420 --> 00:53:26.270 I hope you can kind of see it. 00:53:26.270 --> 00:53:28.950 If you can still see it, that means 00:53:28.950 --> 00:53:32.145 you need to check your eye because the membranes is 00:53:32.145 --> 00:53:32.645 broken. 00:53:36.560 --> 00:53:39.800 OK, so I think you sort of get this idea, 00:53:39.800 --> 00:53:45.470 and I'm going to turn off this wonderful machine 00:53:45.470 --> 00:53:50.250 and go back to the lecture. 00:53:50.250 --> 00:53:56.570 So this experiment is kind of hard to reproduce in your study 00:53:56.570 --> 00:53:58.010 room, right? 00:53:58.010 --> 00:54:00.200 I think everybody will agree. 00:54:00.200 --> 00:54:03.170 And there's another one which is actually 00:54:03.170 --> 00:54:05.300 kind of easy to reproduce, which I 00:54:05.300 --> 00:54:07.730 will encourage you to try it-- 00:54:07.730 --> 00:54:08.990 so if you have time. 00:54:08.990 --> 00:54:11.500 So this is from Jake. 00:54:11.500 --> 00:54:14.510 He sent me this wonderful video when 00:54:14.510 --> 00:54:16.700 I was teaching the 8.03 class. 00:54:16.700 --> 00:54:20.780 They found that they could excite two dimensional waves 00:54:20.780 --> 00:54:23.470 in this way. 00:54:23.470 --> 00:54:25.400 Can you see it? 00:54:25.400 --> 00:54:27.410 It's wonderful. 00:54:27.410 --> 00:54:33.620 You can see there are very high frequency oscillation, which 00:54:33.620 --> 00:54:37.640 actually excite these two dimensional wave. 00:54:37.640 --> 00:54:41.210 And you can see that lots, and lots, and lots of rings 00:54:41.210 --> 00:54:43.280 are excited. 00:54:43.280 --> 00:54:44.990 And then you can see very clearly 00:54:44.990 --> 00:54:50.240 from this simple experiment, what you really 00:54:50.240 --> 00:54:53.420 need is a cup of water. 00:54:53.420 --> 00:54:57.740 And you rub it against the surface of a table, then 00:54:57.740 --> 00:55:01.610 you'll be able to excite all the crazy patterns, which 00:55:01.610 --> 00:55:07.510 you can actually see from this two dimensional system 00:55:07.510 --> 00:55:11.180 and with two dimensional boundary conditions. 00:55:11.180 --> 00:55:13.580 OK, so we will take a five minute 00:55:13.580 --> 00:55:18.560 break before we enter the next part of the discussion. 00:55:18.560 --> 00:55:20.280 And we come back at 35. 00:55:26.940 --> 00:55:29.250 OK, welcome back, everybody. 00:55:29.250 --> 00:55:33.180 So what I'm going to do now is to continue the discussion, 00:55:33.180 --> 00:55:37.550 the one we actually got started, of the two dimensional 00:55:37.550 --> 00:55:39.600 and three dimensional system. 00:55:39.600 --> 00:55:45.030 And we have actually studied the behavior 00:55:45.030 --> 00:55:48.520 of standing wave, or normal mode, 00:55:48.520 --> 00:55:51.220 for this two dimensional system. 00:55:51.220 --> 00:55:55.080 And what I am going to do is discuss with you, a two 00:55:55.080 --> 00:55:56.900 dimensional progressive wave. 00:56:06.470 --> 00:56:11.120 So I will stick to a really simple example, 00:56:11.120 --> 00:56:12.480 which are plane waves. 00:56:18.720 --> 00:56:24.010 OK, so in the case of plane waves, which we discussed when 00:56:24.010 --> 00:56:27.520 we actually discussed the EM waves, 00:56:27.520 --> 00:56:31.100 you have the following functional form. 00:56:31.100 --> 00:56:34.600 Psi is a function of r and the t. 00:56:34.600 --> 00:56:41.160 And this will be equal to A exponential i. 00:56:41.160 --> 00:56:44.770 The k is a vector now, and it's pointing 00:56:44.770 --> 00:56:50.570 to the direction of the propagation of this plan wave. 00:56:50.570 --> 00:56:57.080 And this k is dot with r vector minus omega T, which 00:56:57.080 --> 00:57:01.310 is the oscillation frequency-- angular frequency. 00:57:01.310 --> 00:57:04.670 And evaluated at a specific time. 00:57:04.670 --> 00:57:09.590 And this is expression actually describes a plane wave 00:57:09.590 --> 00:57:14.540 where the direction of propagation 00:57:14.540 --> 00:57:18.110 is described by this k vector. 00:57:18.110 --> 00:57:22.820 And of course you can actually have the wave front, 00:57:22.820 --> 00:57:27.320 which is actually the peak position of this plain wave. 00:57:27.320 --> 00:57:31.220 And the distance between the peak position-- 00:57:31.220 --> 00:57:33.710 so if you can imagine that this is like this. 00:57:44.030 --> 00:57:48.530 So if you look at the distance between peak position 00:57:48.530 --> 00:57:51.920 that will give you the wavelengths, right? 00:57:51.920 --> 00:57:54.530 The wavelengths, now that will be 00:57:54.530 --> 00:57:58.310 equal to 2 pi divided by k, right? 00:57:58.310 --> 00:58:03.330 In this case, it's the length of this k vector. 00:58:03.330 --> 00:58:06.560 Just a reminder about what we introduced 00:58:06.560 --> 00:58:09.170 in the previous lecture. 00:58:09.170 --> 00:58:13.520 And we were using this to describe electromagnetic wave 00:58:13.520 --> 00:58:17.570 and such a kind of expression can be also 00:58:17.570 --> 00:58:22.880 be used to describe sound waves and also 00:58:22.880 --> 00:58:28.010 vibration on the membranes, et cetera, progressive waves. 00:58:28.010 --> 00:58:33.230 So if there are no other medium like what we actually 00:58:33.230 --> 00:58:35.450 have in this slide-- 00:58:35.450 --> 00:58:37.460 so we have nothing else. 00:58:37.460 --> 00:58:42.410 I have a membrane with a surface tension Ts, 00:58:42.410 --> 00:58:47.360 and Rho S is the mass per unit area. 00:58:47.360 --> 00:58:50.080 Then basically, this progressing wave 00:58:50.080 --> 00:58:53.540 is going to be traveling at the speed of v, 00:58:53.540 --> 00:58:57.830 which is equal to square root of Ts over Rho S, 00:58:57.830 --> 00:59:02.600 and I can actually define that to be some constant c divided 00:59:02.600 --> 00:59:03.860 by n. 00:59:03.860 --> 00:59:13.100 So c is some constant, and m is another constant 00:59:13.100 --> 00:59:16.610 which actually, the ratio c and n is equal to v. 00:59:16.610 --> 00:59:20.800 And I will need that expression later, only later, not now. 00:59:20.800 --> 00:59:26.030 If I have nothing else and that this system actually 00:59:26.030 --> 00:59:28.850 filled the whole universe, then what is going to happen 00:59:28.850 --> 00:59:31.160 is that this progressing wave is going 00:59:31.160 --> 00:59:34.600 to be propagating, propagating, propagating, propagating. 00:59:34.600 --> 00:59:39.680 Nothing will change until the edge of the universe. 00:59:39.680 --> 00:59:43.700 It doesn't actually introduce any excitement. 00:59:43.700 --> 00:59:45.890 So that's what we have already learned 00:59:45.890 --> 00:59:52.010 from when we have discussed electromagnetic interaction, 00:59:52.010 --> 00:59:54.290 and now the same expression can also 00:59:54.290 --> 01:00:00.080 be used for the description of the membranes. 01:00:00.080 --> 01:00:04.850 And then now to make this problem more exciting, 01:00:04.850 --> 01:00:08.410 what I'm going to do is to introduce a boundary. 01:00:08.410 --> 01:00:12.340 So the boundary is in the middle of this slide. 01:00:12.340 --> 01:00:17.690 And I will assume that the horizontal direction 01:00:17.690 --> 01:00:22.070 to be x equal to 0-- 01:00:22.070 --> 01:00:27.350 the horizontal direction to be in x direction and the boundary 01:00:27.350 --> 01:00:29.210 is at x equal to 0. 01:00:32.330 --> 01:00:39.130 And when you pass this boundary, there's 01:00:39.130 --> 01:00:44.500 another kind of material with surface tension Ts prime 01:00:44.500 --> 01:00:50.920 and slightly different mass per unit area, your s prime. 01:00:50.920 --> 01:00:54.360 Based on the expression we got for the velocity 01:00:54.360 --> 01:00:56.780 we will be able to conclude that v prime will 01:00:56.780 --> 01:01:02.650 be equal to square root of T prime S divided by Rho S prime. 01:01:02.650 --> 01:01:06.580 And that will be equal to c over n prime. 01:01:06.580 --> 01:01:10.630 And c is the same constant which I used for the left hand side 01:01:10.630 --> 01:01:12.260 system. 01:01:12.260 --> 01:01:16.900 And n, later, you will realize that that's a refraction 01:01:16.900 --> 01:01:20.710 index in a discussion. 01:01:20.710 --> 01:01:24.580 So the question which I would like to ask 01:01:24.580 --> 01:01:29.820 is, OK, now I have a prime wave propagating 01:01:29.820 --> 01:01:31.200 in the first system. 01:01:31.200 --> 01:01:35.020 And it met a boundary, and the question 01:01:35.020 --> 01:01:41.020 is what will happen when I have the incident wave coming 01:01:41.020 --> 01:01:42.160 into the system? 01:01:44.830 --> 01:01:50.920 So before that, I also need to write down the dispersion 01:01:50.920 --> 01:01:51.940 relation, right. 01:01:51.940 --> 01:01:55.090 So dispersion relation can be attempted 01:01:55.090 --> 01:02:01.660 by plugging in a normal mode Psi function 01:02:01.660 --> 01:02:03.730 into the wave equation. 01:02:03.730 --> 01:02:06.700 So what I can immediately obtain is 01:02:06.700 --> 01:02:10.020 that the dispersion relation, omega squared 01:02:10.020 --> 01:02:16.990 is equal to V squared, Kx squared, plus Ky squared. 01:02:16.990 --> 01:02:19.390 You can actually check this expression 01:02:19.390 --> 01:02:24.430 by plugging in this function into the two dimensional wave 01:02:24.430 --> 01:02:29.170 equation, and you will get that expression, OK? 01:02:29.170 --> 01:02:33.160 And that means omega cannot be arbitrary number. 01:02:33.160 --> 01:02:36.640 It's as you decided by Kx and the Ky. 01:02:36.640 --> 01:02:41.410 Or say, if omega is the side and one of the k is the side, 01:02:41.410 --> 01:02:45.490 then the third number, for example in this case, Kx, 01:02:45.490 --> 01:02:48.940 is as you decided by the dispersion relation 01:02:48.940 --> 01:02:51.790 which we have here. 01:02:51.790 --> 01:02:56.160 So, coming back to the original problem we are posting, 01:02:56.160 --> 01:03:01.860 I have, now, the incident wave coming into this system. 01:03:01.860 --> 01:03:06.390 I would like to know what will happen at the boundary 01:03:06.390 --> 01:03:12.070 when I have two systems with a left hand side propagating at-- 01:03:12.070 --> 01:03:14.280 the speed of the propagation is v, 01:03:14.280 --> 01:03:17.400 and right hand side's speed of propagation is v prime. 01:03:17.400 --> 01:03:18.450 What is going to happen? 01:03:21.990 --> 01:03:28.810 Assume my guess that I am going to get a refractive wave 01:03:28.810 --> 01:03:31.210 and a transmitted wave. 01:03:31.210 --> 01:03:34.330 So that's based on what we have learned from the one 01:03:34.330 --> 01:03:36.370 dimensional system. 01:03:36.370 --> 01:03:40.580 If I call this the left hand side, and call the right hand 01:03:40.580 --> 01:03:43.710 side system right hand-- 01:03:43.710 --> 01:03:44.992 the right hand system, r. 01:03:47.860 --> 01:03:56.150 So I can write down the wave function Psi L describing 01:03:56.150 --> 01:03:57.680 the left hand side. 01:03:57.680 --> 01:04:05.140 This will be equal to A exponential of ik dot r 01:04:05.140 --> 01:04:10.070 minus omega T. This is actually the incident wave-- 01:04:19.750 --> 01:04:22.500 describing this incident wave. 01:04:22.500 --> 01:04:25.380 And as you might guess, there should be 01:04:25.380 --> 01:04:27.660 some kind of refraction, right? 01:04:27.660 --> 01:04:31.710 So once this wave actually passed 01:04:31.710 --> 01:04:35.070 through the boundary, or touch the boundary, 01:04:35.070 --> 01:04:37.470 there should be some kind of refraction, right? 01:04:37.470 --> 01:04:39.970 So the refraction, I can actually write it down 01:04:39.970 --> 01:04:44.490 in this form as sum over alpha, r alpha 01:04:44.490 --> 01:04:48.660 is actually the coefficient over amplitude as function 01:04:48.660 --> 01:04:51.440 of the normal modes-- 01:04:51.440 --> 01:04:58.150 as a function of the progressing wave number, 01:04:58.150 --> 01:04:59.590 which I have shown. 01:04:59.590 --> 01:05:01.650 And I can actually sum over all kinds 01:05:01.650 --> 01:05:05.110 of progressing wave numbers. 01:05:05.110 --> 01:05:12.990 Exponential ik alpha times r minus omega T. 01:05:12.990 --> 01:05:18.805 So this is a general form of refracting wave. 01:05:18.805 --> 01:05:23.820 k alpha is describing the direction 01:05:23.820 --> 01:05:25.950 of the individual refractive wave, 01:05:25.950 --> 01:05:30.899 and alpha is labeling the individual refractive wave. 01:05:30.899 --> 01:05:32.940 But I don't know what will be the functional form 01:05:32.940 --> 01:05:35.230 for the k alpha for the moment. 01:05:35.230 --> 01:05:40.380 So therefore, I try to sum over all the possible alpha. 01:05:40.380 --> 01:05:43.590 And I would like to figure out what 01:05:43.590 --> 01:05:47.700 will be the allowed alpha by matching the boundary 01:05:47.700 --> 01:05:48.850 condition. 01:05:48.850 --> 01:05:52.460 So in short, the right hand side turn 01:05:52.460 --> 01:05:54.972 essentially is actually describing the refractive wave. 01:06:00.540 --> 01:06:04.710 And finally, passing through this boundary condition, 01:06:04.710 --> 01:06:07.500 let's look at the right hand side. 01:06:07.500 --> 01:06:12.990 Right hand side, Psi r, is going to be 01:06:12.990 --> 01:06:20.880 sum over beta on the transmission coefficients tau 01:06:20.880 --> 01:06:28.570 beta, which is the original amplitude, exponential of i, 01:06:28.570 --> 01:06:36.790 k beta times r minus omega T. So again I 01:06:36.790 --> 01:06:39.880 don't know what will be the behavior 01:06:39.880 --> 01:06:41.080 of the transmitted wave. 01:06:41.080 --> 01:06:44.380 Therefore, I have summed over all the possible values. 01:06:44.380 --> 01:06:46.547 And this is actually the functional form 01:06:46.547 --> 01:06:47.588 for the transmitted wave. 01:06:56.750 --> 01:07:04.190 I also know that k alpha vector squared will 01:07:04.190 --> 01:07:10.550 be equal to omega squared Rho s over Ts, 01:07:10.550 --> 01:07:15.890 and this will be equal to omega squared v squared, 01:07:15.890 --> 01:07:19.760 because of the dispersion relation in the left hand side. 01:07:19.760 --> 01:07:24.110 So basically, if you look at the left hand side dispersion 01:07:24.110 --> 01:07:31.680 relation, the length squared of this k vector 01:07:31.680 --> 01:07:34.850 will be equal to omega squared times v squared, right? 01:07:34.850 --> 01:07:37.660 This is just a dispersion relation 01:07:37.660 --> 01:07:41.070 of a non-dispersive medium. 01:07:41.070 --> 01:07:45.720 And also, I can actually figure out what will be the-- 01:07:45.720 --> 01:07:49.200 allowed length for the k theta. 01:07:49.200 --> 01:07:54.810 So the k theta squared will be equal to omega squared 01:07:54.810 --> 01:08:00.110 v prime square, because this progressing wave is actually 01:08:00.110 --> 01:08:02.580 the transmitted wave, is actually 01:08:02.580 --> 01:08:06.680 traveling in a second medium. 01:08:10.620 --> 01:08:13.180 So look at what we have done here. 01:08:13.180 --> 01:08:16.270 So we have an incident wave. 01:08:16.270 --> 01:08:18.729 We will wonder, then, what is going to happen. 01:08:18.729 --> 01:08:21.569 Our physics intuition tells me that, you 01:08:21.569 --> 01:08:25.000 must get a refracting wave, oscillation frequency 01:08:25.000 --> 01:08:25.979 should be the same. 01:08:25.979 --> 01:08:27.600 Otherwise, as a function over time, 01:08:27.600 --> 01:08:30.930 you cannot match the left hand and right hand side. 01:08:30.930 --> 01:08:35.170 And you also get a transmitted wave. 01:08:35.170 --> 01:08:39.180 But I'm now in trouble because I have so many turns. 01:08:39.180 --> 01:08:42.090 I'm summing over alpha infinite number of turns, 01:08:42.090 --> 01:08:46.220 and I don't know what will be the coefficient for the r alpha 01:08:46.220 --> 01:08:48.770 and the tau beta, which are the transmission 01:08:48.770 --> 01:08:52.380 coefficient and then refraction coefficients. 01:08:52.380 --> 01:08:56.550 So what I need to do, as you might guess, 01:08:56.550 --> 01:09:00.380 is to use the boundary condition. 01:09:00.380 --> 01:09:02.770 So now I am writing down, already, 01:09:02.770 --> 01:09:04.170 the general expression. 01:09:04.170 --> 01:09:06.600 Now I'm going to use the boundary condition 01:09:06.600 --> 01:09:11.910 to actually limit the choice of the possible k alpha and the k 01:09:11.910 --> 01:09:12.810 beta. 01:09:12.810 --> 01:09:14.710 What is actually the boundary condition? 01:09:21.580 --> 01:09:28.729 The boundary conditions are that at x equal to 0-- 01:09:28.729 --> 01:09:32.180 that's actually at the position of this line-- 01:09:32.180 --> 01:09:34.335 the membranes doesn't break. 01:09:37.750 --> 01:09:42.850 Otherwise, suddenly the membranes break, 01:09:42.850 --> 01:09:46.485 and this is the end of the discussion, right? 01:09:46.485 --> 01:09:48.430 Like, what we have done before, right? 01:09:48.430 --> 01:09:50.229 So the membranes doesn't break, so 01:09:50.229 --> 01:09:53.620 that the propagation can continue. 01:09:53.620 --> 01:09:56.420 So what does that mean? 01:09:56.420 --> 01:10:02.620 This means that if I evaluate Psi L and Psi 01:10:02.620 --> 01:10:08.650 r at x equal to 0, Psi 0, y, t. 01:10:11.500 --> 01:10:17.890 The left hand side will be equal to A exponential i, Ky times y 01:10:17.890 --> 01:10:26.360 minus omega T, plus summing over all possible alpha, r alpha, 01:10:26.360 --> 01:10:37.780 A exponential i, K alpha y times y, minus omega T. This 01:10:37.780 --> 01:10:42.400 is the incident wave transmitted wave evaluated at the left hand 01:10:42.400 --> 01:10:46.230 side, which is the upper formula. 01:10:46.230 --> 01:10:48.990 And that will be equal to the right hand 01:10:48.990 --> 01:10:52.320 side, which is containing only the transmitted wave. 01:10:52.320 --> 01:10:55.150 So basically you have summing over 01:10:55.150 --> 01:11:02.790 beta, tau beta, A exponential i, k beta, 01:11:02.790 --> 01:11:08.440 y times r minus omega T. 01:11:08.440 --> 01:11:13.650 And this expression, this boundary condition, 01:11:13.650 --> 01:11:17.600 should hold true for all the possible y, 01:11:17.600 --> 01:11:19.920 right, because the boundary condition 01:11:19.920 --> 01:11:22.530 is valid at x equal to 0. 01:11:22.530 --> 01:11:26.670 I didn't specify the value of y. 01:11:26.670 --> 01:11:30.090 So therefore I can actually put in all the possible-- oh, 01:11:30.090 --> 01:11:31.830 this should be y. 01:11:31.830 --> 01:11:32.700 Sorry for that. 01:11:32.700 --> 01:11:37.590 I can actually vary the y, and I will figure out that, ah, 01:11:37.590 --> 01:11:42.680 if I have Ky not equal to k alpha y, that 01:11:42.680 --> 01:11:46.920 means the wavelengths of the refractive wave 01:11:46.920 --> 01:11:49.590 and the incident wave will be different. 01:11:49.590 --> 01:11:54.750 If I have Ky not equal to beta y that means 01:11:54.750 --> 01:11:57.150 the transmitted wavelengths is going 01:11:57.150 --> 01:12:00.330 to be different from the incident wave. 01:12:00.330 --> 01:12:04.110 That means, no matter what I do as a boundary of y, 01:12:04.110 --> 01:12:05.455 the membranes will break. 01:12:08.290 --> 01:12:13.840 Therefore, in order to make this equation valid, 01:12:13.840 --> 01:12:18.810 the only choice is that when k alpha y 01:12:18.810 --> 01:12:23.840 will be equal to k beta y and equal to Ky. 01:12:23.840 --> 01:12:28.000 So that means the wavelengths projected in the y direction 01:12:28.000 --> 01:12:33.550 should be equal for the incident wave, transmitted wave, 01:12:33.550 --> 01:12:34.870 and the refractive wave. 01:12:34.870 --> 01:12:39.390 Otherwise, as you always move a little bit in the y direction, 01:12:39.390 --> 01:12:42.250 the membranes will break. 01:12:42.250 --> 01:12:47.770 So that's actually the condition which you can actually get. 01:12:47.770 --> 01:12:53.810 And the interesting thing is that, based on this expression, 01:12:53.810 --> 01:12:57.130 k alpha, the length of the k alpha, 01:12:57.130 --> 01:13:01.960 and the length of the k beta is fixed. 01:13:01.960 --> 01:13:07.850 And I also know what will be the component for the y direction. 01:13:07.850 --> 01:13:15.820 Therefore, that means the x direction Psi's for that k 01:13:15.820 --> 01:13:21.310 alpha x and the k beta x are also fixed because 01:13:21.310 --> 01:13:24.750 of the dispersion relation. 01:13:24.750 --> 01:13:29.680 So that immediately brings me to this conclusion 01:13:29.680 --> 01:13:37.660 that basically k alpha x will be equal to minus omega squared 01:13:37.660 --> 01:13:41.230 over v squared minus Ky squared, and that 01:13:41.230 --> 01:13:45.280 will be equal to minus Kx. 01:13:45.280 --> 01:13:48.730 So this is the x component of the refractive wave. 01:13:48.730 --> 01:13:51.520 And the transmitting wave, k beta x, 01:13:51.520 --> 01:13:56.290 will be equal to square root of omega squared over v squared 01:13:56.290 --> 01:13:59.470 minus Ky squared. 01:13:59.470 --> 01:14:04.450 If I draw, visualize the relative direction 01:14:04.450 --> 01:14:07.875 of all the three components, basically, this 01:14:07.875 --> 01:14:13.840 is essentially the direction of the incident wave, k, 01:14:13.840 --> 01:14:17.370 and the incident angle is theta. 01:14:17.370 --> 01:14:20.410 And from this expression, you see that the Ky 01:14:20.410 --> 01:14:23.380 is equal to k alpha y. 01:14:23.380 --> 01:14:26.180 Therefore, you have a refractive wave. 01:14:26.180 --> 01:14:28.910 But actually only the x direction has changed sides. 01:14:28.910 --> 01:14:31.430 Therefore, you have a refractive wave 01:14:31.430 --> 01:14:37.060 with exactly the same angle as the incident angle theta. 01:14:37.060 --> 01:14:41.420 The refraction angle will be zeta as well. 01:14:41.420 --> 01:14:44.380 And that there will be a transmitted wave 01:14:44.380 --> 01:14:46.330 with theta prime. 01:14:46.330 --> 01:14:51.430 And this is essentially the direction of the k prime. 01:14:51.430 --> 01:14:57.580 And the interesting thing is that the projection 01:14:57.580 --> 01:15:01.210 toward the y direction, that k prime y, 01:15:01.210 --> 01:15:05.200 has to be equal to the progression 01:15:05.200 --> 01:15:12.460 of the original incident wave in the y direction. 01:15:12.460 --> 01:15:17.710 So that means I will be able to conclude that-- 01:15:17.710 --> 01:15:19.400 the y components are the same. 01:15:19.400 --> 01:15:23.350 Therefore, I can conclude that k sine 01:15:23.350 --> 01:15:28.860 theta will be equal to k prime sine theta prime. 01:15:31.610 --> 01:15:34.000 I'm kind of running out of time. 01:15:34.000 --> 01:15:38.980 And if I define, already as I defined here, 01:15:38.980 --> 01:15:41.980 velocity is equal to c over n, and the v 01:15:41.980 --> 01:15:46.840 prime is equal to c over n prime, what I can immediately 01:15:46.840 --> 01:15:49.870 conclude is that-- 01:15:49.870 --> 01:15:51.910 give me one more minute-- 01:15:51.910 --> 01:15:57.760 is that if I have n equal to c over v and the n 01:15:57.760 --> 01:16:00.370 prime is equal to c over v prime, 01:16:00.370 --> 01:16:03.940 I can conclude that sine theta will 01:16:03.940 --> 01:16:08.280 be equal to n prime sine theta prime. 01:16:08.280 --> 01:16:12.520 Does this look familiar to you? 01:16:12.520 --> 01:16:15.610 This is essentially Snell's law. 01:16:15.610 --> 01:16:19.810 How many of you haven't heard about Snell's law. 01:16:19.810 --> 01:16:21.250 There were a few before. 01:16:21.250 --> 01:16:22.590 Yeah, OK. 01:16:22.590 --> 01:16:23.900 No problem at all. 01:16:23.900 --> 01:16:25.370 Then you learned it. 01:16:25.370 --> 01:16:31.310 So that means if I have two kinds of systems in my hand, 01:16:31.310 --> 01:16:38.000 and I will be able to relate the transmitted wave according 01:16:38.000 --> 01:16:40.370 to what I have in the incident wave. 01:16:40.370 --> 01:16:43.430 And you can see that Snell's law-- 01:16:43.430 --> 01:16:49.100 which were famous for the discussion of optics-- 01:16:49.100 --> 01:16:53.270 and here, I have no knowledge about optics 01:16:53.270 --> 01:16:55.520 or electromagnetic waves. 01:16:55.520 --> 01:16:59.810 So in short, what I want to tell you is that, we have just 01:16:59.810 --> 01:17:07.730 proved two of the most important laws of the geometrical optics, 01:17:07.730 --> 01:17:12.680 the refraction angle is equal to incident angle and the Snell's 01:17:12.680 --> 01:17:16.520 law without using any information about the dynamics. 01:17:16.520 --> 01:17:21.350 That means all those laws are coming from purely boundary 01:17:21.350 --> 01:17:23.850 condition and the waves. 01:17:23.850 --> 01:17:26.700 Therefore, you will expect that this 01:17:26.700 --> 01:17:31.370 will work for water wave, sound wave, electromagnetic wave, et 01:17:31.370 --> 01:17:31.870 cetera. 01:17:31.870 --> 01:17:34.900 O So we will continue the discussion next time. 01:17:34.900 --> 01:17:37.860 Thanks for the attention. 01:17:37.860 --> 01:17:40.496 And if you have any questions, let me know. 01:17:40.496 --> 01:17:41.120 I will be here. 01:17:46.940 --> 01:17:48.020 Hello, everybody. 01:17:48.020 --> 01:17:50.570 We are going to show you a demonstration, a really 01:17:50.570 --> 01:17:52.790 nice one. 01:17:52.790 --> 01:17:56.040 It consists of the following setup. 01:17:56.040 --> 01:17:59.920 So basically I'm going to place some film here. 01:17:59.920 --> 01:18:04.190 And then behind that there's a loud speaker, 01:18:04.190 --> 01:18:08.540 which I use as a signal generator and to actually 01:18:08.540 --> 01:18:09.770 produce sound wave. 01:18:09.770 --> 01:18:13.850 And this sound wave is going to oscillate the soft film, 01:18:13.850 --> 01:18:17.490 and then you are going to see the oscillation, 01:18:17.490 --> 01:18:21.470 or the normal mode's pattern, on the screen. 01:18:21.470 --> 01:18:25.130 OK, so that is actually the setup 01:18:25.130 --> 01:18:27.650 which we can actually demonstrate to you two 01:18:27.650 --> 01:18:31.020 dimensional normal modes. 01:18:31.020 --> 01:18:33.170 So the first thing which I am going to do 01:18:33.170 --> 01:18:35.676 is to produce a soft film. 01:18:38.430 --> 01:18:43.560 Now I am going to put it back into this setup here. 01:18:43.560 --> 01:18:46.218 You should be able to see the pattern on the screen. 01:18:50.800 --> 01:18:55.470 Then I am going to turn on the sound wave generator. 01:18:55.470 --> 01:19:01.110 You can see, immediately, that the pattern on the screen 01:19:01.110 --> 01:19:03.960 changed because of the sound wave trying 01:19:03.960 --> 01:19:07.080 to oscillate the soft film. 01:19:07.080 --> 01:19:09.660 You can see it directly from here, 01:19:09.660 --> 01:19:16.230 but it actually looks much more prominent on the screen. 01:19:16.230 --> 01:19:22.080 And now what am I going to do is change the frequency 01:19:22.080 --> 01:19:23.400 of the sound wave. 01:19:23.400 --> 01:19:25.350 And you can see that I'm changing it 01:19:25.350 --> 01:19:26.730 to a higher frequency. 01:19:29.460 --> 01:19:32.940 And you can see that there is a more and more complicated 01:19:32.940 --> 01:19:36.080 pattern formed on the screen. 01:19:36.080 --> 01:19:42.560 That is because I'm now exciting higher and higher frequency 01:19:42.560 --> 01:19:43.480 normal modes. 01:19:46.930 --> 01:19:51.160 And you can see that now I can actually continue and increase 01:19:51.160 --> 01:19:56.424 the frequency. 01:19:56.424 --> 01:19:57.340 And you can see that-- 01:20:01.480 --> 01:20:04.620 now we can see that the pattern becomes really, 01:20:04.620 --> 01:20:07.680 really infinitely complicated. 01:20:07.680 --> 01:20:10.620 You can see this grid developing. 01:20:10.620 --> 01:20:14.460 And then you can see that eventually that's 01:20:14.460 --> 01:20:18.330 basically two sine functions multiply each other. 01:20:18.330 --> 01:20:20.405 One sine function is in the x direction. 01:20:20.405 --> 01:20:23.580 The other one is in the y direction-- horizontal 01:20:23.580 --> 01:20:25.300 and the vertical direction. 01:20:25.300 --> 01:20:28.230 And you can see a really beautiful pattern 01:20:28.230 --> 01:20:34.320 forming due to the solution we derived during the lecture. 01:20:34.320 --> 01:20:40.580 And the higher frequency I go, I can see more and more 01:20:40.580 --> 01:20:46.800 complicating patterns, many more lines 01:20:46.800 --> 01:20:53.690 developing on the screen due to oscillation of the soft film. 01:20:53.690 --> 01:20:55.810 You can see now we have even more lines. 01:20:59.136 --> 01:21:00.760 And it's actually getting more and more 01:21:00.760 --> 01:21:04.075 difficult to see the pattern because now the lines are 01:21:04.075 --> 01:21:05.200 really close to each other. 01:21:05.200 --> 01:21:09.846 The nodal line, we can see clearly on the screen. 01:21:12.714 --> 01:21:17.270 Now I am going back down to a lower frequency, 01:21:17.270 --> 01:21:20.660 and you can see that at low frequency oscillation, 01:21:20.660 --> 01:21:24.650 the number of lines is actually smaller, 01:21:24.650 --> 01:21:29.270 and that is because of the smaller oscillation 01:21:29.270 --> 01:21:33.640 frequency and the longer wavelengths 01:21:33.640 --> 01:21:36.090 of the normal modes.