WEBVTT 00:00:02.620 --> 00:00:04.960 The following content is provided under a Creative 00:00:04.960 --> 00:00:06.380 Commons license. 00:00:06.380 --> 00:00:08.590 Your support will help MIT OpenCourseWare 00:00:08.590 --> 00:00:12.680 continue to offer high-quality educational resources for free. 00:00:12.680 --> 00:00:15.220 To make a donation or to view additional materials 00:00:15.220 --> 00:00:19.180 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.180 --> 00:00:20.214 at ocw.mit.edu. 00:00:23.681 --> 00:00:24.930 YEN-JIE LEE: Hello, everybody. 00:00:24.930 --> 00:00:27.610 Welcome back to 8.03. 00:00:27.610 --> 00:00:31.070 Today we are going to continue the discussion of waves. 00:00:31.070 --> 00:00:33.380 We will discuss a very interesting phenomenon 00:00:33.380 --> 00:00:36.130 not today, which is dispersion. 00:00:36.130 --> 00:00:38.720 And before that, we will discuss a bit, 00:00:38.720 --> 00:00:41.530 just to give you some reminder, about what we 00:00:41.530 --> 00:00:42.950 have learned so far. 00:00:42.950 --> 00:00:46.690 So we discovered this wave equation, 00:00:46.690 --> 00:00:49.280 which is showing here, in the class, 00:00:49.280 --> 00:00:52.200 and then we also show you that it described 00:00:52.200 --> 00:00:54.850 three different kinds of systems, which 00:00:54.850 --> 00:00:57.120 we included in the lecture-- 00:00:57.120 --> 00:01:00.380 the massive strings, which are the strings 00:01:00.380 --> 00:01:05.680 can actually oscillate up and down in a wide direction. 00:01:05.680 --> 00:01:10.030 And also we discussed about sound waves. 00:01:10.030 --> 00:01:13.450 This is also discussed in a previous lecture. 00:01:13.450 --> 00:01:18.440 And sound waves can be described by wave equation. 00:01:18.440 --> 00:01:21.490 And finally, the last time we discussed 00:01:21.490 --> 00:01:23.110 electromagnetic waves. 00:01:23.110 --> 00:01:25.870 It's a special kind of wave involving 00:01:25.870 --> 00:01:28.060 two oscillating fields. 00:01:28.060 --> 00:01:30.580 One is actually the electric field, 00:01:30.580 --> 00:01:33.670 the other one is magnetic field. 00:01:33.670 --> 00:01:36.304 So that's kind of interesting, because this is actually 00:01:36.304 --> 00:01:37.720 slightly different from what we've 00:01:37.720 --> 00:01:41.020 discussed before in the previous two cases. 00:01:41.020 --> 00:01:45.010 This is actually a three dimensional wave, 00:01:45.010 --> 00:01:48.290 and also involving two different components. 00:01:48.290 --> 00:01:50.740 And we also discussed the solution, 00:01:50.740 --> 00:01:54.220 the traveling wave solution of the electromagnetic waves. 00:01:54.220 --> 00:01:57.940 As you can see from here, the electric field 00:01:57.940 --> 00:02:00.970 is showing us the red, and the magnetic field 00:02:00.970 --> 00:02:02.470 is showing us the blue. 00:02:02.470 --> 00:02:04.910 And you can see that in case of traveling wave, 00:02:04.910 --> 00:02:06.430 they are in phase. 00:02:06.430 --> 00:02:10.090 And the magnitude reach maxima simultaneously 00:02:10.090 --> 00:02:13.620 for electric field and the magnetic field. 00:02:13.620 --> 00:02:16.680 And while in the case of standing wave, 00:02:16.680 --> 00:02:20.150 there's a phase difference. so they don't reach maxima 00:02:20.150 --> 00:02:25.080 simultaneously in the standing electromagnetic field case. 00:02:25.080 --> 00:02:29.170 OK, so what are we going to discuss today? 00:02:29.170 --> 00:02:35.270 We would like to discuss the strategy to send information 00:02:35.270 --> 00:02:36.670 using waves. 00:02:36.670 --> 00:02:40.780 How do we actually send information using waves? 00:02:40.780 --> 00:02:44.440 So you can say, OK, maybe I can just 00:02:44.440 --> 00:02:47.620 send a harmonic oscillation. 00:02:47.620 --> 00:02:50.870 So If I do this harmonic oscillation, 00:02:50.870 --> 00:02:54.710 I can basically produce harmonic waves. 00:02:54.710 --> 00:02:58.800 They are moving up and down, and is actually 00:02:58.800 --> 00:03:05.650 always constant angular momentum and angular frequency. 00:03:05.650 --> 00:03:09.650 And maybe that's a way to send the information. 00:03:09.650 --> 00:03:14.630 But this kind of wave is, in reality, not super helpful, 00:03:14.630 --> 00:03:19.690 because if you fill the whole space with harmonic waves, then 00:03:19.690 --> 00:03:22.840 you don't know when did you actually send the signal. 00:03:22.840 --> 00:03:24.790 Because it's always oscillating up and down, 00:03:24.790 --> 00:03:28.570 so you don't know the starting time of the signal. 00:03:28.570 --> 00:03:34.090 So in reality, these kind of simple harmonic oscillating 00:03:34.090 --> 00:03:37.010 traveling wave is not super helpful. 00:03:37.010 --> 00:03:39.740 So what is actually helpful? 00:03:39.740 --> 00:03:41.000 That's the question. 00:03:41.000 --> 00:03:44.720 So what is actually helpful is to produce square pulse, 00:03:44.720 --> 00:03:45.800 for example. 00:03:45.800 --> 00:03:50.910 We can create square pulse, for example, in this case, 00:03:50.910 --> 00:03:52.860 I can create a square pulse here. 00:03:52.860 --> 00:03:56.800 And in the next time interval, I don't create a square pulse. 00:03:56.800 --> 00:03:59.470 In the next time interval, I don't do anything. 00:03:59.470 --> 00:04:02.040 And I create another square pulse here, et cetera, 00:04:02.040 --> 00:04:03.980 et cetera, OK. 00:04:03.980 --> 00:04:07.310 If you use this kind of strategy what we can do 00:04:07.310 --> 00:04:12.170 is to have some kind of receiver here to actually measure 00:04:12.170 --> 00:04:18.170 the magnitude of the pulse. 00:04:18.170 --> 00:04:21.079 And then we can actually interpret this data. 00:04:21.079 --> 00:04:25.920 So this wave is going to where the positive x direction 00:04:25.920 --> 00:04:28.070 or going to the right-hand side of the board. 00:04:28.070 --> 00:04:32.360 And the receiver will be able to interpret this data 00:04:32.360 --> 00:04:36.050 by appraising this ratio on the energy 00:04:36.050 --> 00:04:38.210 or on the measure of the amplitude. 00:04:38.210 --> 00:04:41.920 Then I can say, oh, now I'd receive a 0, 00:04:41.920 --> 00:04:43.900 and then the next signal I'm receiving 00:04:43.900 --> 00:04:48.020 is 1, and this one is 0, and 0, and 1, and 0. 00:04:48.020 --> 00:04:51.830 In this way, I can actually send information 00:04:51.830 --> 00:04:55.550 and that this information can be verified as a function of time. 00:04:55.550 --> 00:04:58.220 So in short, what would be useful 00:04:58.220 --> 00:05:02.750 is probably to use a narrow square pulse, 00:05:02.750 --> 00:05:08.750 and that would be very helpful in transmitting information. 00:05:08.750 --> 00:05:15.350 So if we consider an ideal string case-- 00:05:15.350 --> 00:05:24.810 if I have an ideal string, as we learned before, 00:05:24.810 --> 00:05:30.070 the behavior of the string is described by the wave equation. 00:05:30.070 --> 00:05:32.620 Partial squared psi partial t squared, 00:05:32.620 --> 00:05:36.210 and this is equal to v squared partial squared 00:05:36.210 --> 00:05:40.180 psi partial t squared. 00:05:40.180 --> 00:05:44.530 And this v is actually related to the speed 00:05:44.530 --> 00:05:46.990 of the progressing wave, as we discussed 00:05:46.990 --> 00:05:50.260 before-- the progressing wave solution. 00:05:50.260 --> 00:05:54.940 And if I have this idealized string, 00:05:54.940 --> 00:05:58.240 and it obey the wave equation, the simple version of wave 00:05:58.240 --> 00:06:03.100 equation, then I would be able to divide the dispersion 00:06:03.100 --> 00:06:03.790 relation. 00:06:03.790 --> 00:06:08.200 So I can now write down my harmonic progressing wave 00:06:08.200 --> 00:06:12.730 in the form of sine kx minus omega t. 00:06:12.730 --> 00:06:17.080 If I have a harmonic oscillating wave propagating 00:06:17.080 --> 00:06:21.880 toward the positive x direction at the speed of v. 00:06:21.880 --> 00:06:25.630 I can write it down in this functional form, where 00:06:25.630 --> 00:06:29.080 k, as a reminder, is the wavenumber, 00:06:29.080 --> 00:06:35.260 and the omega is actually the angular frequency. 00:06:35.260 --> 00:06:39.310 And therefore, if I plug in this solution, and of course, 00:06:39.310 --> 00:06:41.890 it can have arbitrary amplitude. 00:06:41.890 --> 00:06:45.580 If I plug in this solution to this equation, 00:06:45.580 --> 00:06:47.770 then what I'm going to get is, as we 00:06:47.770 --> 00:06:50.290 did in the last few lectures, there 00:06:50.290 --> 00:06:53.440 would be a fixed relation between k, 00:06:53.440 --> 00:06:57.590 which is the wavenumber, and the omega, the angular frequency. 00:06:57.590 --> 00:07:02.570 So the fixed relation is actually omega over k 00:07:02.570 --> 00:07:05.980 would be equal to v, which is actually 00:07:05.980 --> 00:07:09.610 the velocity in this wave equation. 00:07:09.610 --> 00:07:11.540 And from the previous discussion, 00:07:11.540 --> 00:07:15.490 we know this is actually equal to a squared root of T 00:07:15.490 --> 00:07:19.400 over rho L, where T is actually the tension, 00:07:19.400 --> 00:07:23.410 the constant tension, which we apply on this string, 00:07:23.410 --> 00:07:29.740 and the rho L is actually the mass per unit as a reminder. 00:07:29.740 --> 00:07:31.550 So what does this mean? 00:07:31.550 --> 00:07:33.490 What does this equation mean? 00:07:33.490 --> 00:07:37.270 We call it dispersion relation a lot of time, right? 00:07:37.270 --> 00:07:40.720 But we actually didn't explain why do I do that. 00:07:40.720 --> 00:07:43.760 So we are going to learn why this is actually called 00:07:43.760 --> 00:07:45.190 this dispersion relation. 00:07:45.190 --> 00:07:47.260 Omega is a function of k. 00:07:47.260 --> 00:07:53.260 And in this case, in this very simplified idealized case, 00:07:53.260 --> 00:07:57.610 omega over k is ratio. 00:07:57.610 --> 00:08:02.200 we know this is related to the speed of propagation 00:08:02.200 --> 00:08:09.040 of the harmonic wave is equal to v. v is a constant that 00:08:09.040 --> 00:08:12.010 is independent of k. 00:08:12.010 --> 00:08:14.240 This ratio is independent of k. 00:08:14.240 --> 00:08:15.160 What does that mean? 00:08:15.160 --> 00:08:23.410 That means if I prepare waves with different wavenumber, 00:08:23.410 --> 00:08:28.450 or in other words, waves with different wavelengths, 00:08:28.450 --> 00:08:31.375 they are going to propagate at the same speed. 00:08:34.000 --> 00:08:41.669 So the speed of the harmonic progressing wave 00:08:41.669 --> 00:08:45.110 is independent of the wavelength. 00:08:45.110 --> 00:08:46.780 That's actually very good, because 00:08:46.780 --> 00:08:51.280 in this case, if I prepared the square pulse, 00:08:51.280 --> 00:08:56.250 as we learned before, this square pulse is actually 00:08:56.250 --> 00:08:58.030 a very complicated object. 00:08:58.030 --> 00:09:00.390 Square pulse is really very complicated. 00:09:00.390 --> 00:09:04.790 You can do a Fourier decomposition as we did before. 00:09:04.790 --> 00:09:07.715 And we need infinite number of turns 00:09:07.715 --> 00:09:11.260 of harmonic oscillating waves. 00:09:11.260 --> 00:09:14.170 We actually add them together so that I 00:09:14.170 --> 00:09:17.740 can produce a square pulse. 00:09:17.740 --> 00:09:22.630 And as I mentioned here, if the dispersion relation, omega 00:09:22.630 --> 00:09:30.450 over k, is is our constant, v. That means all the 00:09:30.450 --> 00:09:37.390 whatever wavelengths pulse, which should be added together 00:09:37.390 --> 00:09:39.790 and produce the square pulse, are 00:09:39.790 --> 00:09:42.490 going to be traveling at the that speed. 00:09:42.490 --> 00:09:48.300 Therefore, if I have this square pulse in the beginning, 00:09:48.300 --> 00:09:54.210 after some time, t, what I'm going to get 00:09:54.210 --> 00:09:58.050 is that this is the original position of the square pulse, 00:09:58.050 --> 00:10:01.220 and after some time, t, this square pulse 00:10:01.220 --> 00:10:06.550 will move by v times t in the horizontal direction. 00:10:06.550 --> 00:10:08.940 And the shape of the pulse is not 00:10:08.940 --> 00:10:14.100 going to be changed, because no matter what kind of wavelengths 00:10:14.100 --> 00:10:16.530 which produce the square pulse, all 00:10:16.530 --> 00:10:18.535 the components in the square pulse 00:10:18.535 --> 00:10:24.600 are propagating at the same speed. 00:10:24.600 --> 00:10:30.600 So this kind of system, which has satisfied 00:10:30.600 --> 00:10:38.430 this kind of dispersion relation is called nondispersive media. 00:10:38.430 --> 00:10:42.240 no dispersion was happening in this case, 00:10:42.240 --> 00:10:44.370 in this highly idealized case. 00:10:47.130 --> 00:10:51.780 We also know that in case of the string, 00:10:51.780 --> 00:10:55.380 we are actually making it too idealized. 00:10:55.380 --> 00:11:00.870 So if we consider a more realistic string, 00:11:00.870 --> 00:11:06.240 then I have to consider an important phenomenon, which 00:11:06.240 --> 00:11:07.170 is-- 00:11:07.170 --> 00:11:10.620 or is a important property of the string, for example-- 00:11:10.620 --> 00:11:13.590 stiffness 00:11:13.590 --> 00:11:15.240 What do I mean by stiffness? 00:11:15.240 --> 00:11:20.250 So for example, if I take a string from a piano, 00:11:20.250 --> 00:11:26.670 a piano string, even if I don't apply any tension 00:11:26.670 --> 00:11:30.060 to the string, if I bend the string, 00:11:30.060 --> 00:11:31.590 it don't like it, all right? 00:11:31.590 --> 00:11:37.240 It's going to bounce back and restore to its original shape. 00:11:37.240 --> 00:11:39.520 So that's what I call stiffness. 00:11:39.520 --> 00:11:44.400 It's a different contribution compared to the string tension. 00:11:44.400 --> 00:11:49.410 So what we have been discussing so far that this distorting 00:11:49.410 --> 00:11:54.000 force is actually coming from the string tension, t. 00:11:54.000 --> 00:11:54.810 OK? 00:11:54.810 --> 00:12:00.750 What will happen if I introduce additional contribution 00:12:00.750 --> 00:12:03.500 from the stiffness? 00:12:03.500 --> 00:12:07.250 The stiffness is actually not completely related 00:12:07.250 --> 00:12:09.650 to the string tension, and that also 00:12:09.650 --> 00:12:12.620 wants to restore the shape of the string. 00:12:12.620 --> 00:12:13.790 OK? 00:12:13.790 --> 00:12:16.760 Before we go to the modeling, I would 00:12:16.760 --> 00:12:19.220 like to take some votes to predict 00:12:19.220 --> 00:12:20.880 what is going to happen. 00:12:20.880 --> 00:12:26.240 How many of you were predict that if I introduce and include 00:12:26.240 --> 00:12:30.680 the stiffness of the string into my equation, 00:12:30.680 --> 00:12:33.710 will the speed of propagation increase? 00:12:33.710 --> 00:12:36.260 How many of you think it's going to happen? 00:12:36.260 --> 00:12:39.780 1, 2, 3, 4, 5. 00:12:39.780 --> 00:12:40.280 OK. 00:12:44.600 --> 00:12:51.280 So some of you predict the speed of propagation will increase. 00:12:51.280 --> 00:12:53.760 How many of you predict that the speed 00:12:53.760 --> 00:12:57.840 of propagation of the harmonic wave will stay the same? 00:12:57.840 --> 00:13:00.250 How many of you? 00:13:00.250 --> 00:13:02.640 One? 00:13:02.640 --> 00:13:03.590 OK, only one. 00:13:09.310 --> 00:13:11.720 OK, how many of you actually predict 00:13:11.720 --> 00:13:25.750 that the speed of propagation would decrease OK 00:13:25.750 --> 00:13:29.320 so all the other students don't have opinion. 00:13:29.320 --> 00:13:32.350 OK, want to wait for the answer. 00:13:32.350 --> 00:13:32.980 All right. 00:13:32.980 --> 00:13:36.610 So you can see that it is actually not completely obvious 00:13:36.610 --> 00:13:38.710 before we solve this question. 00:13:38.710 --> 00:13:43.890 And we are going to solve it with a simple model, which 00:13:43.890 --> 00:13:52.690 actually slightly modifies the idealized wave equation. 00:13:52.690 --> 00:13:58.990 So now, one semi-realistic model which I can introduce 00:13:58.990 --> 00:14:03.800 is to add a term additional term to the wave equation. 00:14:03.800 --> 00:14:07.990 So I can now rewrite my wave equation 00:14:07.990 --> 00:14:13.630 to include the effect that to describe a realistic string, 00:14:13.630 --> 00:14:16.850 and now this is your partial squared psi partial t squared. 00:14:16.850 --> 00:14:20.860 This will be equal to v squared partial squared 00:14:20.860 --> 00:14:24.400 psi partial t squared. 00:14:24.400 --> 00:14:28.560 And the additional term, which I put into this, again, 00:14:28.560 --> 00:14:36.080 is minus alpha partial to the 4 psi partial x to the 4. 00:14:36.080 --> 00:14:39.350 And this is actually the contribution 00:14:39.350 --> 00:14:42.310 from the stiffness. 00:14:42.310 --> 00:14:45.160 This is stiffness. 00:14:45.160 --> 00:14:51.050 OK, so you can see that the wave equation is now modified. 00:14:51.050 --> 00:14:56.300 And what I could do in order to get the relation between omega 00:14:56.300 --> 00:14:57.470 and the k-- 00:14:57.470 --> 00:15:00.260 what I could do is that I can now 00:15:00.260 --> 00:15:06.320 start with this harmonic wave solution progressing wave 00:15:06.320 --> 00:15:11.450 solution, plug that in to this equation, this modified 00:15:11.450 --> 00:15:14.090 equation, and see what will happen. 00:15:14.090 --> 00:15:17.330 If I plug this equation into it that 00:15:17.330 --> 00:15:20.960 modified wave equation, what I am going to get 00:15:20.960 --> 00:15:21.840 is the following. 00:15:21.840 --> 00:15:27.560 So basically the left-hand, side you're going to get minus omega 00:15:27.560 --> 00:15:29.540 squared. 00:15:29.540 --> 00:15:31.160 And then the right-hand side, you 00:15:31.160 --> 00:15:43.190 get v squared minus k squared and plus alpha k to the 4 00:15:43.190 --> 00:15:44.560 in the right-hand side. 00:15:44.560 --> 00:15:48.890 OK, so of course, I can now cancel this minus sign. 00:15:48.890 --> 00:15:52.490 This will become plus and this will become minus. 00:15:52.490 --> 00:15:56.390 And then you can see that the relation between omega 00:15:56.390 --> 00:16:02.330 and the k is now different after I introduce this term, which 00:16:02.330 --> 00:16:03.610 is proportional to alpha. 00:16:03.610 --> 00:16:11.980 Alpha is actually describing how stiff this string is. 00:16:11.980 --> 00:16:14.490 Of course, now I can calculate omega 00:16:14.490 --> 00:16:18.280 over k, which is actually, as we learned 00:16:18.280 --> 00:16:22.440 before, right is the speed of the propagation 00:16:22.440 --> 00:16:26.920 of a harmonic wave. 00:16:26.920 --> 00:16:28.900 So basically, if I calculate omega 00:16:28.900 --> 00:16:32.980 over k from this equation, then basically what you get 00:16:32.980 --> 00:16:38.230 is v square root of 1 plus alpha k squared. 00:16:41.680 --> 00:16:45.340 So if you look at this equation, the first reaction 00:16:45.340 --> 00:16:51.610 is, oh, now this omega and the k ratio 00:16:51.610 --> 00:16:57.780 is not a constant anymore as a function of k. 00:16:57.780 --> 00:16:59.310 What does that mean? 00:16:59.310 --> 00:17:04.040 That means if I prepare progressing waves 00:17:04.040 --> 00:17:08.849 with different wavelengths for wavenumber k, 00:17:08.849 --> 00:17:13.920 it's going to be propagating at different speed, OK? 00:17:13.920 --> 00:17:19.260 Before we introduce this into the model, 00:17:19.260 --> 00:17:25.440 the ratio omega and k is a constant v, independent of k. 00:17:25.440 --> 00:17:31.170 Now, once you introduce this model into the equation, 00:17:31.170 --> 00:17:35.550 and you plug in the progressing wave solution 00:17:35.550 --> 00:17:39.270 to actually check the dispersion relation obtained 00:17:39.270 --> 00:17:46.550 from this equation, you'll find that the speed of progressing 00:17:46.550 --> 00:17:56.110 wave depends on how distorted this progressing wave is, OK? 00:17:56.110 --> 00:18:01.750 So let me compare this to situation 00:18:01.750 --> 00:18:06.010 in this graph, omega versus k. 00:18:06.010 --> 00:18:10.820 So we will see this dispersion relation graph pretty often 00:18:10.820 --> 00:18:12.250 in the class today. 00:18:12.250 --> 00:18:17.170 The y-axis is actually the omega, angular frequency, 00:18:17.170 --> 00:18:21.990 and the k is the wavenumber, two pi over lambda. 00:18:21.990 --> 00:18:23.410 OK. 00:18:23.410 --> 00:18:27.100 In the original case, in the case 00:18:27.100 --> 00:18:31.180 I have this idealized string, obey 00:18:31.180 --> 00:18:33.580 the wave equation which we introduced 00:18:33.580 --> 00:18:35.920 in the previous lectures. 00:18:35.920 --> 00:18:39.940 If I plug omega as a function of k, what I'm getting 00:18:39.940 --> 00:18:40.840 is a straight line. 00:18:43.890 --> 00:18:44.628 question. 00:18:44.628 --> 00:18:49.020 AUDIENCE: Why are you [INAUDIBLE] 00:18:49.020 --> 00:18:50.972 minus alpha [INAUDIBLE]. 00:18:53.362 --> 00:18:54.570 YEN-JIE LEE: This one, right? 00:18:54.570 --> 00:18:58.680 AUDIENCE: [INAUDIBLE] 00:18:58.680 --> 00:19:00.980 YEN-JIE LEE: Oh, maybe I made some mistake here. 00:19:00.980 --> 00:19:03.830 So this should be also plus here, right. 00:19:12.810 --> 00:19:14.060 So you have this-- 00:19:14.060 --> 00:19:21.190 OK, so this is omega squared, and I shouldn't have 00:19:21.190 --> 00:19:22.890 this minus sign here, right? 00:19:22.890 --> 00:19:26.970 So this should be minus, and this should be-- 00:19:26.970 --> 00:19:29.790 OK, let's go back to the original equation, OK. 00:19:29.790 --> 00:19:31.820 So basically, you get-- 00:19:31.820 --> 00:19:36.180 so if I plug in this equation to this equation, 00:19:36.180 --> 00:19:39.540 so basically I get minus omega squared out of it. 00:19:39.540 --> 00:19:43.270 And I get minus k squared out of this. 00:19:43.270 --> 00:19:47.190 And I'm going to get plus k to the 4 out 00:19:47.190 --> 00:19:52.560 of this partial square to the 4 psi partial x to the 4. 00:19:52.560 --> 00:19:53.910 Therefore, this would be minus. 00:19:53.910 --> 00:19:55.200 OK, maybe I made a mistake. 00:19:55.200 --> 00:19:57.360 Thank you very much for spotting that. 00:19:57.360 --> 00:20:03.660 AUDIENCE: [INAUDIBLE] 00:20:03.660 --> 00:20:05.800 YEN-JIE LEE: Oh, yeah, I'm sorry. 00:20:05.800 --> 00:20:07.980 Not my best day today. 00:20:07.980 --> 00:20:09.590 AUDIENCE: [INAUDIBLE] 00:20:09.590 --> 00:20:10.500 YEN-JIE LEE: Yeah. 00:20:10.500 --> 00:20:12.020 Well, then I do it. 00:20:12.020 --> 00:20:14.150 OK. 00:20:14.150 --> 00:20:15.600 I must be drunk today. 00:20:15.600 --> 00:20:17.420 [LAUGHTER] 00:20:19.700 --> 00:20:20.860 Thank you very much. 00:20:20.860 --> 00:20:23.220 Anymore mistake? 00:20:23.220 --> 00:20:23.990 OK. 00:20:23.990 --> 00:20:25.730 Fortunately not, right? 00:20:25.730 --> 00:20:26.840 OK. 00:20:26.840 --> 00:20:27.350 Very good. 00:20:27.350 --> 00:20:31.020 So let me do this again. 00:20:31.020 --> 00:20:35.610 So now I can modify my wave equation, right? 00:20:35.610 --> 00:20:38.110 Originally, the wave equation is partial squared 00:20:38.110 --> 00:20:41.810 psi partial t squared equal to v squared partial squared psi 00:20:41.810 --> 00:20:43.640 partial x squared. 00:20:43.640 --> 00:20:48.470 And now I add additional term, which is actually 00:20:48.470 --> 00:20:55.100 proportional to the partial to the 4 psi partial x to the 4. 00:20:55.100 --> 00:20:57.620 OK, if I add this term into again. 00:20:57.620 --> 00:21:03.560 And now I plug in the wave equation, the progressing wave 00:21:03.560 --> 00:21:07.040 solution, into this equation, and I 00:21:07.040 --> 00:21:09.020 would get this formula, OK? 00:21:09.020 --> 00:21:11.960 So now everything should be correct, 00:21:11.960 --> 00:21:15.380 and I have clear evidence that everybody's following. 00:21:15.380 --> 00:21:17.030 So that is very good. 00:21:17.030 --> 00:21:21.080 And now, I can now cancel all the minus sign, 00:21:21.080 --> 00:21:23.090 right and then it's become plus. 00:21:23.090 --> 00:21:25.100 And now I can actually calculate what 00:21:25.100 --> 00:21:28.700 would be the speed of propagation 00:21:28.700 --> 00:21:32.470 for this specific harmonic progressing wave 00:21:32.470 --> 00:21:37.040 and omega over k will be equal to v square root of 1 00:21:37.040 --> 00:21:39.260 plus alpha k squared. 00:21:39.260 --> 00:21:40.910 OK? 00:21:40.910 --> 00:21:42.810 Thank you very much for the contribution. 00:21:42.810 --> 00:21:49.310 And then now we see that here this ratio depends on k. 00:21:49.310 --> 00:21:55.280 So if I plug this on top of the previous curve, which 00:21:55.280 --> 00:22:01.310 is actually obtained from here, then what I'm going to get 00:22:01.310 --> 00:22:03.340 is something like this. 00:22:03.340 --> 00:22:05.942 In the beginning it's pretty close 00:22:05.942 --> 00:22:08.900 to the nondispersive case. 00:22:08.900 --> 00:22:14.720 And it goes up, because of this alpha contribution. 00:22:14.720 --> 00:22:18.740 Alpha is actually a positive number in my model. 00:22:18.740 --> 00:22:23.780 And the k is actually the wavenumber. 00:22:23.780 --> 00:22:28.170 So what is going to happen is that basically after you 00:22:28.170 --> 00:22:32.920 include stiffness, the slope of this curve 00:22:32.920 --> 00:22:35.630 is changing as a function of k. 00:22:35.630 --> 00:22:36.130 OK? 00:22:39.000 --> 00:22:42.010 What do I learn from this exercise 00:22:42.010 --> 00:22:49.010 is that if I increase k, if I have a very large k-- 00:22:49.010 --> 00:22:55.620 that means I have a very small lambda, because k is actually 00:22:55.620 --> 00:22:57.200 2 pi over lambda. 00:22:57.200 --> 00:22:57.860 OK? 00:22:57.860 --> 00:23:02.060 So that means I'm looking at something really distorted 00:23:02.060 --> 00:23:02.630 like this. 00:23:05.572 --> 00:23:09.830 Both string tension and the stiffness 00:23:09.830 --> 00:23:16.020 wants to restore the string back to normal. 00:23:16.020 --> 00:23:18.500 Therefore, what is happening is that you 00:23:18.500 --> 00:23:23.840 are going to get additional restoring force. 00:23:23.840 --> 00:23:27.350 Therefore, as we actually calculate to here 00:23:27.350 --> 00:23:33.260 if alpha is actually positive, then the velocity actually 00:23:33.260 --> 00:23:38.770 increased with respect to what we actually 00:23:38.770 --> 00:23:42.530 get before we actually had this into a model. 00:23:42.530 --> 00:23:45.710 So I think that makes sense, because the stiffness also 00:23:45.710 --> 00:23:49.850 wants to restore the distortion. 00:23:49.850 --> 00:23:55.520 Therefore, you have larger and larger restoring force. 00:23:55.520 --> 00:23:57.920 Therefore, the speed of propagation 00:23:57.920 --> 00:24:02.090 of this harmonic wave will increase. 00:24:02.090 --> 00:24:04.790 so that's pretty nice. 00:24:04.790 --> 00:24:09.670 But what does that mean to our project? 00:24:09.670 --> 00:24:13.100 OK, our project is to send information 00:24:13.100 --> 00:24:15.630 from one place to the other place, right? 00:24:15.630 --> 00:24:19.460 So what we just discussed is that we can actually 00:24:19.460 --> 00:24:24.440 send a square pulse and let it propagate. 00:24:24.440 --> 00:24:29.360 A square pulse can be decomposed into many, many pieces-- 00:24:29.360 --> 00:24:31.155 many, many harmonic waves. 00:24:31.155 --> 00:24:32.630 OK? 00:24:32.630 --> 00:24:39.050 Before the square pulse works, because all the waves 00:24:39.050 --> 00:24:44.150 with different wavelengths should 00:24:44.150 --> 00:24:47.775 be moving at this constant speed, independent 00:24:47.775 --> 00:24:49.190 of the wavelengths. 00:24:49.190 --> 00:24:51.650 Now we are in trouble. 00:24:51.650 --> 00:24:58.040 As you can see here, now the speed, which is omega over k 00:24:58.040 --> 00:25:03.200 depends on the wavenumber or wavelength. 00:25:03.200 --> 00:25:07.460 Therefore, different components, which 00:25:07.460 --> 00:25:12.110 actually are needed to create a square pulse, 00:25:12.110 --> 00:25:15.000 are going to be propagating at different speed. 00:25:15.000 --> 00:25:17.550 You can say, oh, come on, this is actually mathematics, 00:25:17.550 --> 00:25:19.340 so I don't believe you. 00:25:19.340 --> 00:25:21.440 A square pulse is a square pulse, 00:25:21.440 --> 00:25:23.910 and that's mathematics, that's math department. 00:25:23.910 --> 00:25:27.850 But we can actually really see that in the experiment. 00:25:27.850 --> 00:25:29.060 OK? 00:25:29.060 --> 00:25:33.050 So that's-- take a look at this demonstration. 00:25:33.050 --> 00:25:34.940 Maybe you didn't notice that before, 00:25:34.940 --> 00:25:39.830 but we have seen this effect from the previous lectures. 00:25:39.830 --> 00:25:43.685 OK, so I can now create a square-- 00:25:43.685 --> 00:25:46.900 not really a square pulse, but actually some kind of pulse. 00:25:46.900 --> 00:25:51.500 OK I can create some kind of pulse like this. 00:25:51.500 --> 00:25:52.340 OK? 00:25:52.340 --> 00:25:56.240 And as we learned before, when this pulse pass 00:25:56.240 --> 00:26:00.230 through an open end, it's going to be bounced back. 00:26:00.230 --> 00:26:02.780 so therefore, I can have-- 00:26:02.780 --> 00:26:07.140 I can actually show you this demo in a limited set-up. 00:26:07.140 --> 00:26:11.420 But this pulse is going to be going back and forth, because I 00:26:11.420 --> 00:26:14.870 have open end, as we've discussed before. 00:26:14.870 --> 00:26:17.810 What is going to happen is that since we have 00:26:17.810 --> 00:26:21.950 a realistic system, what is going to happen 00:26:21.950 --> 00:26:27.420 is that this pulse will become wider and wider, right? 00:26:27.420 --> 00:26:31.370 That's the prediction coming from this equation. 00:26:31.370 --> 00:26:34.840 Different component with different wavelengths 00:26:34.840 --> 00:26:38.000 is going to be propagating at different speed. 00:26:38.000 --> 00:26:41.600 Therefore this pulse is going to become wider, 00:26:41.600 --> 00:26:42.680 and we can see that. 00:26:42.680 --> 00:26:47.950 OK, so let me quickly produce a pulse and see what will happen. 00:26:47.950 --> 00:26:49.170 OK. 00:26:49.170 --> 00:26:52.020 Originally, it's actually really sharp. 00:26:52.020 --> 00:26:57.030 And you can see that really the width of the pulse 00:26:57.030 --> 00:26:59.040 become wider and wider. 00:26:59.040 --> 00:27:02.430 And at some point, it disappear. 00:27:02.430 --> 00:27:05.400 If I have a very long set-up, what you are going to see 00:27:05.400 --> 00:27:08.550 is that it's going to be propagating 00:27:08.550 --> 00:27:11.100 toward the same direction. 00:27:11.100 --> 00:27:14.190 And the width of the pulse is actually 00:27:14.190 --> 00:27:17.460 going to be increasing as a function of time. 00:27:17.460 --> 00:27:20.400 Let's take a look at this again. 00:27:20.400 --> 00:27:23.100 Now, this time we have a negative pulse. 00:27:23.100 --> 00:27:24.540 You sort of see-- 00:27:24.540 --> 00:27:25.432 very similar, see. 00:27:25.432 --> 00:27:26.890 And also you can see that there are 00:27:26.890 --> 00:27:33.420 some strange vibration actually left behind the main pulse. 00:27:33.420 --> 00:27:41.520 So that means harmonic waves with different wavelengths 00:27:41.520 --> 00:27:47.080 really propagating at different speed. 00:27:47.080 --> 00:27:49.900 And for that, to demonstrate this effect, 00:27:49.900 --> 00:27:52.870 I also prepared some demonstration, 00:27:52.870 --> 00:27:58.000 which actually are based on our calculation, OK. 00:27:58.000 --> 00:28:04.160 So you can say that, OK, now I'm convinced I can see dispersion 00:28:04.160 --> 00:28:05.170 in the experiment. 00:28:05.170 --> 00:28:08.830 How do I know this calculation actually match 00:28:08.830 --> 00:28:10.360 with the experimental data, right? 00:28:10.360 --> 00:28:14.955 How about we really run a simulation and see what 00:28:14.955 --> 00:28:16.180 would happen. 00:28:16.180 --> 00:28:22.920 So what this example actually do is, in the beginning, 00:28:22.920 --> 00:28:26.500 you would do integration like crazy 00:28:26.500 --> 00:28:30.040 in order to get all the components calculated. 00:28:30.040 --> 00:28:33.640 Then it's going to propagate all those pulses-- 00:28:33.640 --> 00:28:37.840 all those pulse with different components through the medium, 00:28:37.840 --> 00:28:38.620 OK? 00:28:38.620 --> 00:28:41.080 And then there will be two different colors, 00:28:41.080 --> 00:28:44.840 one is actually blue, which is the original shape. 00:28:44.840 --> 00:28:47.110 The other one is actually the one 00:28:47.110 --> 00:28:50.110 which is stiffness turned up. 00:28:50.110 --> 00:28:54.220 So now, in the beginning I can set the alpha value 00:28:54.220 --> 00:28:57.200 to be 0.02 and see what will happen. 00:28:57.200 --> 00:29:01.870 And I will put produce triangular pulse. 00:29:01.870 --> 00:29:02.860 You can see that now. 00:29:02.860 --> 00:29:05.380 The program is really working very hard 00:29:05.380 --> 00:29:10.930 to capture all the components from 1 to 199 and equal to 1 00:29:10.930 --> 00:29:12.460 until 99. 00:29:12.460 --> 00:29:16.810 And then now, these individual components 00:29:16.810 --> 00:29:19.580 are propagating through the medium. 00:29:19.580 --> 00:29:23.490 And you can see that originally the shape is like-- 00:29:23.490 --> 00:29:27.050 the blue shape-- triangular shape. 00:29:27.050 --> 00:29:30.470 And you can see that is a function of time. 00:29:30.470 --> 00:29:34.640 The pulse become wider and wider, OK? 00:29:34.640 --> 00:29:39.200 Now, of course, I can increase the alpha to 0.02 00:29:39.200 --> 00:29:40.910 and see what happen-- 00:29:40.910 --> 00:29:43.550 from 0.02 to 0.2 and see what will happen. 00:29:43.550 --> 00:29:48.050 You should expect a much larger dispersion. 00:29:48.050 --> 00:29:50.450 And you can see that now in the beginning, 00:29:50.450 --> 00:29:54.560 it's doing the integration. 00:29:54.560 --> 00:29:56.890 And you can see that this time because the alpha 00:29:56.890 --> 00:29:58.640 is actually larger. 00:29:58.640 --> 00:30:03.110 Therefore, you see that this effect, this broadening, 00:30:03.110 --> 00:30:05.350 is actually happening earlier, and it 00:30:05.350 --> 00:30:07.640 become broader and broader, and that there 00:30:07.640 --> 00:30:11.120 are a lot of strange structures, as you can see also 00:30:11.120 --> 00:30:15.250 from the demo, produce because different components are 00:30:15.250 --> 00:30:18.070 actually propagating at different speeds. 00:30:21.200 --> 00:30:26.560 So of course, we are MIT, so in this course we have MIT-- 00:30:26.560 --> 00:30:28.290 MIT waves. 00:30:28.290 --> 00:30:31.460 So let's take a look at the MIT wave and see what will happen. 00:30:31.460 --> 00:30:35.240 Now you see that there are very sharp edge, which 00:30:35.240 --> 00:30:39.680 actually require really a lot of effort to reproduce that. 00:30:39.680 --> 00:30:44.320 And you can see that MIT is kind of distorted 00:30:44.320 --> 00:30:46.140 as a function of time. 00:30:46.140 --> 00:30:48.775 We can kind of still identify the peak, 00:30:48.775 --> 00:30:51.270 but it's actually now displaced. 00:30:51.270 --> 00:30:54.900 And in the end of the simulation, 00:30:54.900 --> 00:30:57.356 you can not even recognize that's actually 00:30:57.356 --> 00:31:03.540 originally MIT signal, which was sent from your source. 00:31:03.540 --> 00:31:08.790 So what I want to say is that this effect, this dispersion 00:31:08.790 --> 00:31:13.980 effect, is really an enemy, which 00:31:13.980 --> 00:31:15.570 is actually very dangerous. 00:31:15.570 --> 00:31:17.790 And that actually will prevent us 00:31:17.790 --> 00:31:22.390 from sending high quality signals. 00:31:22.390 --> 00:31:25.890 OK, any questions about all those demos? 00:31:28.650 --> 00:31:30.108 Yes. 00:31:30.108 --> 00:31:32.052 AUDIENCE: Why do we model the [INAUDIBLE]?? 00:31:36.510 --> 00:31:40.972 YEN-JIE LEE: So this is because the stiffness is actually 00:31:40.972 --> 00:31:41.680 symmetric, right. 00:31:41.680 --> 00:31:47.535 So if you bend the string, then there 00:31:47.535 --> 00:31:52.820 are contribution from the positive and negative part, OK? 00:31:52.820 --> 00:31:57.520 If you have partial to the 3, partial to the x to the 3 00:31:57.520 --> 00:32:00.710 component, then it's going to be a symmetric and so actually 00:32:00.710 --> 00:32:03.110 against our physics intuition. 00:32:03.110 --> 00:32:05.960 And also, in this modeling, you also 00:32:05.960 --> 00:32:09.490 match with our experimental data pretty well. 00:32:09.490 --> 00:32:10.270 OK. 00:32:10.270 --> 00:32:11.990 Very good question. 00:32:11.990 --> 00:32:16.760 And on the other hand, we now consider then the stiffness. 00:32:16.760 --> 00:32:20.270 you can also go back to the infinite number coupled 00:32:20.270 --> 00:32:21.740 oscillator case. 00:32:21.740 --> 00:32:28.190 If you instead take an example which is actually not 00:32:28.190 --> 00:32:32.600 super small displacement approximation, 00:32:32.600 --> 00:32:34.580 you take the next to leading order term. 00:32:37.470 --> 00:32:40.580 Then you will see that the partial to the 3 partial x 00:32:40.580 --> 00:32:43.540 to the 3 term as you cancel because it's symmetric, 00:32:43.540 --> 00:32:45.200 or so I argued. 00:32:45.200 --> 00:32:48.860 And then you will be able to also obtain this tern when 00:32:48.860 --> 00:32:53.390 you have slightly larger displacement with respect 00:32:53.390 --> 00:32:56.590 to the equilibrium position. 00:32:56.590 --> 00:32:58.190 So I hope that answers your question. 00:32:58.190 --> 00:32:58.981 Any other question? 00:32:58.981 --> 00:33:00.098 Yes? 00:33:00.098 --> 00:33:03.930 AUDIENCE: If you were looking at [INAUDIBLE],, for example, what 00:33:03.930 --> 00:33:08.037 would be [INAUDIBLE]? 00:33:08.037 --> 00:33:09.953 YEN-JIE LEE: When you pass through the medium. 00:33:09.953 --> 00:33:12.510 AUDIENCE: So [INAUDIBLE] 00:33:12.510 --> 00:33:14.010 YEN-JIE LEE: A molecule can actually 00:33:14.010 --> 00:33:18.270 change the speed of different wavelengths, 00:33:18.270 --> 00:33:19.980 actually, differently, right? 00:33:19.980 --> 00:33:21.450 Very good question. 00:33:21.450 --> 00:33:23.260 OK, so very good. 00:33:23.260 --> 00:33:26.580 We got two questions, and we can see that if I now 00:33:26.580 --> 00:33:32.260 turn on the alpha and make the alpha value large, 00:33:32.260 --> 00:33:35.940 then you can see that the information is distorted. 00:33:35.940 --> 00:33:41.030 And this involve infinite number of terms. 00:33:41.030 --> 00:33:44.610 And in this case, in this new demo which I show here, 00:33:44.610 --> 00:33:47.200 I have alpha value equal to 0.2. 00:33:47.200 --> 00:33:52.260 Therefore, the effect of dispersion 00:33:52.260 --> 00:33:55.170 is actually much larger than what you showed before. 00:33:55.170 --> 00:33:57.690 And then you can see that this MIT wave quickly 00:33:57.690 --> 00:34:02.230 become something like a Gaussian-like wave, right? 00:34:02.230 --> 00:34:03.630 OK, so very good. 00:34:03.630 --> 00:34:11.370 So you can say, OK, you are making an example-- 00:34:11.370 --> 00:34:16.090 it's a very interesting example, but it involve too many terms. 00:34:16.090 --> 00:34:21.120 You have infinite number of progressing waves 00:34:21.120 --> 00:34:22.739 in this example. 00:34:22.739 --> 00:34:25.590 It's very difficult to understand. 00:34:25.590 --> 00:34:31.989 How about we go back to a much simpler example, OK? 00:34:31.989 --> 00:34:37.290 What we can do is that instead of going 00:34:37.290 --> 00:34:42.570 through infinite number of harmonic waves, now 00:34:42.570 --> 00:34:46.750 we just consider two waves, and overlap these two 00:34:46.750 --> 00:34:50.520 waves together and see what will happen. 00:34:50.520 --> 00:34:52.290 And let's see what we can learn from it, 00:34:52.290 --> 00:34:58.360 because the required number of harmonic wave to describe such 00:34:58.360 --> 00:35:01.170 a pulse is too complicated. 00:35:01.170 --> 00:35:04.980 So you can say that, OK, now let's just consider two waves 00:35:04.980 --> 00:35:06.600 and see what we can learn from this. 00:35:06.600 --> 00:35:10.570 And this is actually what I am going to do now. 00:35:10.570 --> 00:35:13.050 So from Bolek's lecture I hope that he 00:35:13.050 --> 00:35:15.850 covered the beat phenomenon. 00:35:15.850 --> 00:35:17.230 So basically, what is it? 00:35:17.230 --> 00:35:18.590 A beat phenomenon? 00:35:18.590 --> 00:35:22.380 Beat phenomenon happens when you overlap 00:35:22.380 --> 00:35:26.100 two waves, two harmonic waves. 00:35:26.100 --> 00:35:31.450 They have pretty close wavelengths. 00:35:31.450 --> 00:35:33.800 OK, but they're not the same. 00:35:33.800 --> 00:35:36.980 And now, if you add two waves together, 00:35:36.980 --> 00:35:38.730 that's actually what you are going to get. 00:35:38.730 --> 00:35:42.880 You are going to get something which is oscillating really, 00:35:42.880 --> 00:35:46.740 really fast, which is basically called the carrier. 00:35:46.740 --> 00:35:51.570 And also you can see that the magnitude of the oscillation 00:35:51.570 --> 00:35:54.910 is actually changing as a function of position, 00:35:54.910 --> 00:35:58.020 and that we call envelope. 00:35:58.020 --> 00:35:59.940 So that's essentially the beat phenomenon, 00:35:59.940 --> 00:36:02.740 which you learned from previous lectures. 00:36:02.740 --> 00:36:13.070 So in this example, I'm going to add two waves together. 00:36:13.070 --> 00:36:15.920 So the first wave is described by-- 00:36:15.920 --> 00:36:18.230 OK, is denoted by side one. 00:36:18.230 --> 00:36:23.020 It's a function of x and t, and it has a function of form 00:36:23.020 --> 00:36:25.740 A is the amplitude. 00:36:25.740 --> 00:36:32.580 And the sine k1 x minus omega1 t. 00:36:32.580 --> 00:36:35.050 This is actually a progressing wave 00:36:35.050 --> 00:36:37.360 propagating toward the right-hand side 00:36:37.360 --> 00:36:42.330 of the board, the positive direction of the x-axis 00:36:42.330 --> 00:36:44.500 in my coordinate system. 00:36:44.500 --> 00:36:50.850 And it has a wavenumber of k1 and angular frequency omega1 00:36:50.850 --> 00:36:53.230 And I can also write down my second wave, 00:36:53.230 --> 00:36:56.950 which I would like to overlap with the first wave. 00:36:56.950 --> 00:37:00.830 So this is actually having exactly the same amplitude, 00:37:00.830 --> 00:37:06.030 which is A. And it is described by a sine function, 00:37:06.030 --> 00:37:14.110 and you have a wavenumber k2 x minus omega2 t, angular 00:37:14.110 --> 00:37:17.620 frequency omega2. 00:37:17.620 --> 00:37:22.260 With these two equations, we can calculate 00:37:22.260 --> 00:37:25.520 the speed of propagation for the individual waves, right? 00:37:25.520 --> 00:37:28.160 So the first one, I can calculate 00:37:28.160 --> 00:37:33.350 the speed of propagation v1 would be equal to omega1 00:37:33.350 --> 00:37:36.420 over k1. 00:37:36.420 --> 00:37:38.600 Very similarly, you can also calculate 00:37:38.600 --> 00:37:41.100 the speed of propagation for the second wave, 00:37:41.100 --> 00:37:45.260 which is omega2 over k2. 00:37:45.260 --> 00:37:48.515 So now what I'm going to do is to calculate 00:37:48.515 --> 00:37:50.870 a sum of these two waves. 00:37:50.870 --> 00:37:56.900 So I have the total, which is psi is equal to psi1 plus psi2. 00:37:59.500 --> 00:38:02.530 So what I'm going to do is to overlap these two waves 00:38:02.530 --> 00:38:03.980 and see what will happen. 00:38:03.980 --> 00:38:06.310 And for that, I need this formula, 00:38:06.310 --> 00:38:11.560 which is a sine A plus sine B. And this 00:38:11.560 --> 00:38:21.670 would be equal to 2 times sine A plus B over 2 and sine-- 00:38:21.670 --> 00:38:24.200 it would become cosine here-- 00:38:24.200 --> 00:38:30.990 cosine A minus B over 2. 00:38:30.990 --> 00:38:35.660 So if I use that formula, basically what I'm going to get 00:38:35.660 --> 00:38:36.410 is-- 00:38:36.410 --> 00:38:38.780 we have two times from the formula. 00:38:38.780 --> 00:38:50.430 So if you have 2A sine k1 plus k2 over 2x 00:38:50.430 --> 00:38:56.850 minus omega1 plus omega2 over 2. 00:38:56.850 --> 00:39:00.540 So basically, the first term is the sine function. 00:39:00.540 --> 00:39:03.150 The sine function and the content 00:39:03.150 --> 00:39:07.840 is actually A plus B. Therefore, you add these two together, 00:39:07.840 --> 00:39:10.170 divide it by two, then basically this 00:39:10.170 --> 00:39:13.260 is as actually what you obtain. 00:39:13.260 --> 00:39:16.530 The second term is a cosine term. 00:39:16.530 --> 00:39:18.370 You get a cosine here. 00:39:18.370 --> 00:39:21.450 But now you calculate A minus B, which 00:39:21.450 --> 00:39:25.520 is this term minus that term divided by 2. 00:39:25.520 --> 00:39:29.850 Then basically what you get is k1 minus k2 divided 00:39:29.850 --> 00:39:36.570 by 2 times x minus omega1 minus omega2 over 2 t. 00:39:39.480 --> 00:39:41.540 OK, so now this actually-- 00:39:41.540 --> 00:39:46.140 what would happen if you add these two waves together? 00:39:49.000 --> 00:39:52.540 Until now, everything is exact. 00:39:52.540 --> 00:39:56.770 And I would like to add additional conditions 00:39:56.770 --> 00:40:01.360 or additional assumptions when I discuss this solution. 00:40:01.360 --> 00:40:02.200 OK? 00:40:02.200 --> 00:40:06.670 So how about in order to produce the beat phenomenon, 00:40:06.670 --> 00:40:11.110 I need to make the wavelengths very, very 00:40:11.110 --> 00:40:14.140 similar between the two waves. 00:40:14.140 --> 00:40:16.060 So therefore, what I am going to do 00:40:16.060 --> 00:40:22.090 is that I'm going to assume k1 is 00:40:22.090 --> 00:40:25.770 very close to k2 is roughly k. 00:40:29.090 --> 00:40:33.220 And because of this, since I have a continuous function, 00:40:33.220 --> 00:40:36.270 if k1 is really close to k2, that 00:40:36.270 --> 00:40:41.680 means omega1 is going to be also very close to omega2, right? 00:40:41.680 --> 00:40:45.010 So what I'm going to get is omega1 is 00:40:45.010 --> 00:40:48.820 going to be also very similar to omega2, 00:40:48.820 --> 00:40:50.760 and I will call it omega. 00:40:54.100 --> 00:41:02.230 So if I do this, when I have very similar k1 and k2, 00:41:02.230 --> 00:41:03.760 what is going to happen? 00:41:03.760 --> 00:41:11.050 What is going to happen is that k1 minus k2 will be very small. 00:41:11.050 --> 00:41:18.250 So this very small k means larger wavelengths. 00:41:18.250 --> 00:41:24.870 Therefore, this cosine term will become the envelope, 00:41:24.870 --> 00:41:29.560 because it's actually a slowly variating amplitude 00:41:29.560 --> 00:41:34.240 as a function of position, because the k is very small. 00:41:34.240 --> 00:41:37.070 K is small means lambda large. 00:41:37.070 --> 00:41:39.910 Therefore, the amplitude is going 00:41:39.910 --> 00:41:43.420 to be having this modulation, which is actually 00:41:43.420 --> 00:41:48.977 like the envelope, that the oscillation of this envelope 00:41:48.977 --> 00:41:50.560 is actually controlled by the k, okay? 00:41:53.950 --> 00:41:56.380 Let's look at the left-hand side term. 00:41:56.380 --> 00:42:00.280 k1 plus k2 over 2 is kind of like calculating 00:42:00.280 --> 00:42:08.320 the average of the wavenumber of the first and second wave. 00:42:08.320 --> 00:42:13.100 So if you calculate our average, you can be still pretty large. 00:42:13.100 --> 00:42:19.180 Therefore, you have small lambda compared to the difference. 00:42:19.180 --> 00:42:20.970 Therefore, you see that that actually 00:42:20.970 --> 00:42:25.115 contribute to those little structures in this graph, 00:42:25.115 --> 00:42:26.380 and it's called carrier. 00:42:26.380 --> 00:42:27.111 Yes? 00:42:27.111 --> 00:42:28.073 AUDIENCE: [INAUDIBLE]? 00:42:30.959 --> 00:42:34.326 If k1 were a lot bigger than k2, then [INAUDIBLE].. 00:42:39.140 --> 00:42:41.910 YEN-JIE LEE: So they can be different. 00:42:41.910 --> 00:42:43.920 Yeah, so you are absolutely right. 00:42:43.920 --> 00:42:49.130 So you can produce something like a carrier 00:42:49.130 --> 00:42:51.480 even when k1 is not equal to k2, right? 00:42:51.480 --> 00:42:53.470 Its just a average. 00:42:53.470 --> 00:42:54.050 You're right. 00:42:54.050 --> 00:43:00.130 But then on the other hand, the difference, k1 and k2 00:43:00.130 --> 00:43:02.040 will be also large. 00:43:02.040 --> 00:43:05.570 Therefore, it's not as easy as what 00:43:05.570 --> 00:43:09.370 we have been doing here to identify who is the carrier 00:43:09.370 --> 00:43:11.220 and who is the envelope. 00:43:11.220 --> 00:43:14.630 But you do get some kind of graph, 00:43:14.630 --> 00:43:16.370 which is oscillating really fast, 00:43:16.370 --> 00:43:20.550 but the envelope is going to be also oscillating very fast. 00:43:20.550 --> 00:43:23.660 That is harder to see all the structure. 00:43:23.660 --> 00:43:26.210 But you're absolutely right, yes. 00:43:26.210 --> 00:43:27.740 Very good question. 00:43:27.740 --> 00:43:30.320 So now I have this set-up. 00:43:30.320 --> 00:43:33.180 I assume that they are very close to each other. 00:43:33.180 --> 00:43:37.320 So now I can define phase velocity. 00:43:37.320 --> 00:43:41.660 Finally, we define what is actually the phase velocity. 00:43:41.660 --> 00:43:44.650 In The phase velocity-- 00:43:44.650 --> 00:43:46.570 I call it vp-- 00:43:46.570 --> 00:43:49.220 you can see that before I already 00:43:49.220 --> 00:43:52.400 have been using phase velocity vp 00:43:52.400 --> 00:43:55.060 for the previous discussions. 00:43:55.060 --> 00:43:58.280 In the case of nondispersive medium, 00:43:58.280 --> 00:44:01.070 the phase velocity is just a vp, which is 00:44:01.070 --> 00:44:04.170 the velocity in the equation. 00:44:04.170 --> 00:44:09.140 And in this case, vp will be equal to omega over k, 00:44:09.140 --> 00:44:10.254 as we discussed before. 00:44:10.254 --> 00:44:11.670 And that's actually the definition 00:44:11.670 --> 00:44:15.320 of this phase velocity. 00:44:15.320 --> 00:44:18.350 And I can now also define the group velocity. 00:44:23.800 --> 00:44:31.340 The group velocity is actually the velocity of the envelope. 00:44:31.340 --> 00:44:36.140 I can calculate the velocity of the envelope. 00:44:36.140 --> 00:44:38.030 in the case of phase velocity, I'm 00:44:38.030 --> 00:44:42.590 calculating the velocity of the carrier. 00:44:42.590 --> 00:44:44.790 I'm taking a ratio of the average, 00:44:44.790 --> 00:44:48.800 and actually the average is so close to k and omega, 00:44:48.800 --> 00:44:51.620 therefore the phase velocity vp would 00:44:51.620 --> 00:44:55.710 be just the speed of the propagation of the carrier, 00:44:55.710 --> 00:44:57.750 which is actually omega over k. 00:44:57.750 --> 00:44:59.060 I call it vp. 00:44:59.060 --> 00:45:04.040 And in case of group velocity, I call it vg. 00:45:04.040 --> 00:45:08.210 vg is describing the speed of propagation of the envelope. 00:45:08.210 --> 00:45:16.220 Therefore, what I am getting is omega1 minus omega2 00:45:16.220 --> 00:45:19.770 divided by k1 minus k2. 00:45:19.770 --> 00:45:23.220 Both of them have effect of 1 over 2, 00:45:23.220 --> 00:45:25.940 which we say is canceled. 00:45:25.940 --> 00:45:29.220 And when they are really so close to each other, 00:45:29.220 --> 00:45:33.161 this is actually roughly like d omega dk. 00:45:37.840 --> 00:45:38.770 Any questions so far? 00:45:42.280 --> 00:45:47.140 So we have derived two different kinds of speed. 00:45:47.140 --> 00:45:51.710 One is actually related to the phase velocity, which is-- 00:45:51.710 --> 00:45:54.790 one is actually called the phase velocity. 00:45:54.790 --> 00:45:59.260 It's related to the speed of the carrier. 00:45:59.260 --> 00:46:02.620 The other one is group velocity, which is actually related 00:46:02.620 --> 00:46:06.340 to the speed of the envelope. 00:46:06.340 --> 00:46:12.080 So let me describe you a few interesting examples. 00:46:12.080 --> 00:46:17.470 And let's see what we can actually learn from this. 00:46:17.470 --> 00:46:21.820 In the first example, I'm working on a non dispersive 00:46:21.820 --> 00:46:23.320 medium, OK? 00:46:23.320 --> 00:46:33.430 If I have a nondispersive medium, 00:46:33.430 --> 00:46:34.960 then basically what I'm going to get 00:46:34.960 --> 00:46:40.970 is that omega will be proportional to k. 00:46:40.970 --> 00:46:45.410 If I plot omega versus k, it's a straight line. 00:46:45.410 --> 00:46:49.310 Now, if I have omega-- 00:46:49.310 --> 00:46:53.780 I choose the omega of the two, omega1, omega2, 00:46:53.780 --> 00:46:54.830 of the two waves-- 00:46:54.830 --> 00:47:03.250 to be roughly equal to omega 0, I can now evaluate the vp. 00:47:03.250 --> 00:47:11.290 The vp will be the slope of this point 00:47:11.290 --> 00:47:13.390 on the slope of a line connecting 00:47:13.390 --> 00:47:17.570 the 0 to that point, which is actually the omega over k, 00:47:17.570 --> 00:47:18.070 right? 00:47:18.070 --> 00:47:19.445 So that's actually the definition 00:47:19.445 --> 00:47:21.430 of the phase velocity. 00:47:21.430 --> 00:47:24.450 I would get this slope. 00:47:24.450 --> 00:47:28.480 The slope of this line is actually 00:47:28.480 --> 00:47:32.830 related to the phase velocity. 00:47:32.830 --> 00:47:37.740 I can also calculate the slope of a line cuts 00:47:37.740 --> 00:47:39.770 through this point. 00:47:39.770 --> 00:47:43.610 But as it cuts through this curve, and in this case, 00:47:43.610 --> 00:47:50.080 I'm also going to get a line overlapping with phase 00:47:50.080 --> 00:47:54.040 velocity, because in this case, omega over k 00:47:54.040 --> 00:47:58.360 is a constant, which is v. Therefore, no matter 00:47:58.360 --> 00:48:02.290 what you calculate, if you calculate vp 00:48:02.290 --> 00:48:07.210 as a ratio of omega and a k, where you calculate vg, which 00:48:07.210 --> 00:48:09.910 is actually the slope of the line cutting 00:48:09.910 --> 00:48:15.820 through that point, is you always get actually v. 00:48:15.820 --> 00:48:17.950 Therefore, what we learned from here 00:48:17.950 --> 00:48:23.980 is that for a nondispersive medium, vp will be equal to vg. 00:48:28.200 --> 00:48:32.850 That means both of these two curves, 00:48:32.850 --> 00:48:37.120 both of the curve of envelope, describing the envelope 00:48:37.120 --> 00:48:38.950 and then describing the carrier, is 00:48:38.950 --> 00:48:41.850 going to be propagating at the same speed. 00:48:41.850 --> 00:48:43.580 OK, any questions? 00:48:43.580 --> 00:48:51.230 So the whole thing is going to be moving at constant speed. 00:48:51.230 --> 00:48:59.560 For that, I can now show you some example, which I prepared, 00:48:59.560 --> 00:49:01.460 some simulation which I prepared. 00:49:14.780 --> 00:49:19.060 So what it does is that it really-- 00:49:19.060 --> 00:49:20.410 oh, wait a second. 00:49:20.410 --> 00:49:25.510 This is 0. 00:49:25.510 --> 00:49:31.540 OK, so this is the case when I have a nondispersive medium. 00:49:31.540 --> 00:49:34.810 if I have a nondispersive medium, what is going to happen 00:49:34.810 --> 00:49:40.380 is that both the carrier, which is the speed of all 00:49:40.380 --> 00:49:43.000 those little structures, and the envelope 00:49:43.000 --> 00:49:45.210 is going to be propagating at the same speed. 00:49:45.210 --> 00:49:48.250 So you can see the high is like a fixed pattern. 00:49:48.250 --> 00:49:51.700 It's propagating toward the right-hand side. 00:49:51.700 --> 00:49:55.330 And the relative motion between the defined structure 00:49:55.330 --> 00:49:58.000 and the envelope is actually 0. 00:49:58.000 --> 00:50:00.750 So basically you have exactly the same pattern 00:50:00.750 --> 00:50:03.820 as a function of time. 00:50:03.820 --> 00:50:09.070 So now I'm going to move away from the nondispersive medium. 00:50:09.070 --> 00:50:11.620 How about we discuss what would happen 00:50:11.620 --> 00:50:16.600 if we have considered the stiffness of the string 00:50:16.600 --> 00:50:19.730 and see what we get from there. 00:50:19.730 --> 00:50:27.760 So if I plugged omega as a function of k, 00:50:27.760 --> 00:50:31.450 and consider alpha to be non-zero. 00:50:31.450 --> 00:50:32.990 It's a positive value. 00:50:32.990 --> 00:50:36.930 So if I have alpha to be a positive value, 00:50:36.930 --> 00:50:41.560 non-zero, in this case, I'm going to get a curve like this. 00:50:44.370 --> 00:50:50.970 The slope is actually changing and it's curving up 00:50:50.970 --> 00:50:54.840 because if you have k large, then 00:50:54.840 --> 00:51:00.150 you would see that the ratio of omega and k actually increase. 00:51:00.150 --> 00:51:01.870 So this is actually the kind of curve 00:51:01.870 --> 00:51:06.690 which we would get if I set the omega 00:51:06.690 --> 00:51:12.150 of the first and second wave of interest 00:51:12.150 --> 00:51:16.080 in this study to be omega 0. 00:51:16.080 --> 00:51:18.660 Then basically, what you are going to get is that-- 00:51:18.660 --> 00:51:23.340 OK, now I have this point here on the curve. 00:51:23.340 --> 00:51:27.492 If I calculate the phase velocity-- 00:51:27.492 --> 00:51:29.700 the phase velocity, how do I calculate that? 00:51:29.700 --> 00:51:37.010 I can now connect 0 and the point by a line. 00:51:37.010 --> 00:51:40.220 And I can now calculate the slope of this line, 00:51:40.220 --> 00:51:43.510 and I can get the phase velocity, vp. 00:51:46.820 --> 00:51:51.500 On the other hand, I can also calculate 00:51:51.500 --> 00:51:55.550 the slope of a line cutting through, tangential 00:51:55.550 --> 00:51:59.030 to the point of interest. 00:51:59.030 --> 00:52:04.560 And that is going to give me the group velocity. 00:52:04.560 --> 00:52:09.100 As you can see from here, which slope is actually larger? 00:52:09.100 --> 00:52:10.140 Anybody know? 00:52:10.140 --> 00:52:12.670 Can point it out? 00:52:12.670 --> 00:52:15.450 Group velocity's larger, right? 00:52:15.450 --> 00:52:18.750 So in this case, if I turn on alpha greater 00:52:18.750 --> 00:52:22.410 than 0, what is going to happen is that, since the group 00:52:22.410 --> 00:52:27.060 velocity is larger than the phase velocity, that means, 00:52:27.060 --> 00:52:31.310 if I go back to that picture, the envelope 00:52:31.310 --> 00:52:35.850 is going to be moving faster than the fine structure 00:52:35.850 --> 00:52:37.440 inside the envelope. 00:52:37.440 --> 00:52:40.170 How about we take a five-minute break from here? 00:52:40.170 --> 00:52:42.816 And then we continue the discussion after the break. 00:52:42.816 --> 00:52:44.190 It's a good time to take a break. 00:52:50.250 --> 00:52:52.140 Welcome back, everybody. 00:52:52.140 --> 00:52:54.180 So we will continue the discussion 00:52:54.180 --> 00:52:57.310 of the beat phenomenon. 00:52:57.310 --> 00:53:02.431 So what we have shown you is that, based on those curves, 00:53:02.431 --> 00:53:03.930 actually can actually determine what 00:53:03.930 --> 00:53:06.030 will be the relative velocity-- 00:53:09.810 --> 00:53:15.220 what would be the velocity of the carrier, which is actually 00:53:15.220 --> 00:53:19.470 denoted by vp, and the what would 00:53:19.470 --> 00:53:23.390 be the velocity of the envelope, which is actually 00:53:23.390 --> 00:53:28.170 denoted by our group velocity. 00:53:28.170 --> 00:53:33.480 And in this case, what I'm actually plotting here 00:53:33.480 --> 00:53:37.950 is that, in this case, because alpha is actually 00:53:37.950 --> 00:53:43.620 greater than 0, therefore, this curve is actually curving up. 00:53:43.620 --> 00:53:47.850 Therefore, you have larger group velocity 00:53:47.850 --> 00:53:51.010 compared to the phase velocity. 00:53:51.010 --> 00:53:54.360 So what you would expect is that the envelope 00:53:54.360 --> 00:53:57.110 is going to be actually progressing 00:53:57.110 --> 00:54:03.930 at a speed higher than the speed of the carrier. 00:54:03.930 --> 00:54:08.520 On the other hand, if magically I 00:54:08.520 --> 00:54:11.310 can construct some kind of medium which 00:54:11.310 --> 00:54:14.720 can be described in this situation, 00:54:14.720 --> 00:54:19.270 alpha smaller than 0, what is going to happen? 00:54:19.270 --> 00:54:26.800 So if I plot a situation with alpha smaller than 0, 00:54:26.800 --> 00:54:30.690 so now I plot omega was a function of k. 00:54:34.210 --> 00:54:36.890 What is going to happen is that this-- so basically, 00:54:36.890 --> 00:54:39.760 you have something which is actually curving downward. 00:54:43.670 --> 00:54:51.750 So if I now, again, work on some point of interest here, 00:54:51.750 --> 00:54:57.050 you can see that the slope of the phase velocity 00:54:57.050 --> 00:55:08.570 is now actually larger than the slope, which is actually 00:55:08.570 --> 00:55:11.210 from the line cutting through the-- 00:55:11.210 --> 00:55:15.350 tangential to the curve, which is actually 00:55:15.350 --> 00:55:20.510 getting you the group velocity. 00:55:20.510 --> 00:55:23.890 So in the case of alpha's more than 0, 00:55:23.890 --> 00:55:25.940 which is some strange medium I can 00:55:25.940 --> 00:55:28.290 which I can create from whatever, plasma, 00:55:28.290 --> 00:55:35.060 or some really strange new kind of material of interest. 00:55:35.060 --> 00:55:39.410 If that happens, then that means your group velocity 00:55:39.410 --> 00:55:43.478 will be smoother than the phase velocity. 00:55:48.340 --> 00:55:51.560 And if you look at this point here, 00:55:51.560 --> 00:55:57.790 you can see that this curve actually reach a maxima here. 00:55:57.790 --> 00:56:02.140 And if you actually are operating at this point, what 00:56:02.140 --> 00:56:03.910 is going to happen? 00:56:03.910 --> 00:56:08.650 What is going to happen is that if you calculate the group 00:56:08.650 --> 00:56:11.610 velocity, what will be the value? 00:56:11.610 --> 00:56:13.367 It will be 0. 00:56:13.367 --> 00:56:14.200 What does that mean? 00:56:14.200 --> 00:56:19.280 That means the envelope will not be moving a lot, 00:56:19.280 --> 00:56:23.910 but the carriers are still moving. 00:56:23.910 --> 00:56:28.190 So at this point, you are going to get 00:56:28.190 --> 00:56:31.130 group velocity equal to 0. 00:56:34.700 --> 00:56:41.750 And finally, the if you actually going to a very large k 00:56:41.750 --> 00:56:45.520 value in this scenario, alpha smaller than 0, 00:56:45.520 --> 00:56:49.020 you will see that even you can have 00:56:49.020 --> 00:56:56.460 phase velocity, vp, positive, because it's actually 00:56:56.460 --> 00:56:58.400 a positive slope. 00:56:58.400 --> 00:57:06.520 And that the group velocity actually is negative. 00:57:06.520 --> 00:57:07.920 What does that mean? 00:57:07.920 --> 00:57:13.660 That means you are going to see a situation that the carriers 00:57:13.660 --> 00:57:17.630 are progressing in the positive direction, 00:57:17.630 --> 00:57:24.910 and the envelope is going to be progressing 00:57:24.910 --> 00:57:29.440 in the negative direction, probably 00:57:29.440 --> 00:57:33.490 progressing to the left-hand side of the board. 00:57:33.490 --> 00:57:34.910 So what does that mean? 00:57:34.910 --> 00:57:40.820 That means this wave is doing what Michael Jackson's doing. 00:57:40.820 --> 00:57:42.875 It's actually doing the moonwalk. 00:57:42.875 --> 00:57:45.310 [LAUGHTER] 00:57:45.310 --> 00:57:48.880 So this is actually the kind of thing 00:57:48.880 --> 00:57:53.110 which could have happened, that it looks like 00:57:53.110 --> 00:57:57.910 and that you are doing-- 00:57:57.910 --> 00:58:03.280 going forward, because all the carriers 00:58:03.280 --> 00:58:05.460 are moving in a positive direction. 00:58:05.460 --> 00:58:10.660 But the body is actually going toward negative direction. 00:58:10.660 --> 00:58:13.882 maybe I can also learn moonwalk at some point. 00:58:13.882 --> 00:58:15.090 [LAUGHTER] 00:58:15.090 --> 00:58:16.630 OK. 00:58:16.630 --> 00:58:19.490 So let's go back to the demonstration 00:58:19.490 --> 00:58:24.160 which I got started, and somehow I got messed up. 00:58:24.160 --> 00:58:28.210 So let's take a look at the demo again so 00:58:28.210 --> 00:58:33.820 let's look at all the different situation at once. 00:58:33.820 --> 00:58:36.880 So in this case, as we discussed before, 00:58:36.880 --> 00:58:41.680 this is actually happening in the nondispersive situation. 00:58:41.680 --> 00:58:44.680 In this situation, you have a straight line, 00:58:44.680 --> 00:58:47.590 nondispersed medium actually give you 00:58:47.590 --> 00:58:53.590 always the group velocity equal to phase velocity. 00:58:53.590 --> 00:58:58.870 So that means the carrier and the envelope 00:58:58.870 --> 00:59:02.650 is going to be moving in the same direction 00:59:02.650 --> 00:59:04.540 at the same speed. 00:59:08.470 --> 00:59:11.590 On the other hand, in this case, we 00:59:11.590 --> 00:59:16.690 can actually have a situation that the phase velocity 00:59:16.690 --> 00:59:20.890 is actually faster than the group velocity. 00:59:20.890 --> 00:59:23.770 So what I mean is actually the situation here. 00:59:23.770 --> 00:59:28.120 The phase velocity calculated from a line connecting from 0 00:59:28.120 --> 00:59:32.500 to that point is actually having a larger slope compared 00:59:32.500 --> 00:59:35.070 to the tangential line. 00:59:35.070 --> 00:59:37.770 And you see this situation. 00:59:37.770 --> 00:59:41.310 So basically, you see that inside the envelope 00:59:41.310 --> 00:59:44.230 all those carriers are actually moving faster 00:59:44.230 --> 00:59:45.778 than the envelope. 00:59:50.170 --> 00:59:54.950 Now I can have a dispersive medium where the group 00:59:54.950 --> 00:59:56.650 velocity is equal to 0. 00:59:56.650 --> 01:00:02.230 So what is going to happen is that really the envelope 01:00:02.230 --> 01:00:03.400 is actually not moving. 01:00:03.400 --> 01:00:05.340 It's not like this. 01:00:05.340 --> 01:00:07.060 The body is not moving. 01:00:07.060 --> 01:00:12.245 So you have some carriers inside this structure 01:00:12.245 --> 01:00:14.600 is actually moving forward. 01:00:14.600 --> 01:00:17.148 But the envelope is actually not moving. 01:00:24.510 --> 01:00:29.950 So, finally the last situation is really interesting. 01:00:29.950 --> 01:00:33.800 So in this situation, this is actually 01:00:33.800 --> 01:00:35.840 having the group velocity-- 01:00:39.090 --> 01:00:41.860 the group velocity is actually having difference sine 01:00:41.860 --> 01:00:44.060 compared to the phase velocity. 01:00:44.060 --> 01:00:48.610 So you can see that the whole structure of the envelope 01:00:48.610 --> 01:00:51.910 is actually moving backwards. 01:00:51.910 --> 01:00:54.790 But the carrier is actually moving 01:00:54.790 --> 01:01:01.380 in the positive direction in this example. 01:01:04.190 --> 01:01:08.950 So this is actually what we have learned from this beat 01:01:08.950 --> 01:01:11.390 phenomenon, and then we have covered 01:01:11.390 --> 01:01:15.020 the idea of phase velocity and the group velocity. 01:01:15.020 --> 01:01:19.460 So how about bound system how do we 01:01:19.460 --> 01:01:23.380 understand when we have a bound system? 01:01:23.380 --> 01:01:27.600 And how does that evolve as a function of time? 01:01:27.600 --> 01:01:33.530 So if I have a system of two walls and one string, 01:01:33.530 --> 01:01:36.350 and of course, I give you the density 01:01:36.350 --> 01:01:41.660 for the unit length and the string tension, and also 01:01:41.660 --> 01:01:44.330 the alpha, which is actually telling you 01:01:44.330 --> 01:01:47.780 about the stiffness of the system. 01:01:47.780 --> 01:01:53.390 Again, I can write down psi xt to be 01:01:53.390 --> 01:01:59.530 the sum of all the normal mode from one to infinity, 01:01:59.530 --> 01:02:11.196 A m sine km x plus alpha m sine omega mt plus beta m. 01:02:15.390 --> 01:02:19.650 And then what we can do is that we can first 01:02:19.650 --> 01:02:23.280 get the initial conditions of this system, 01:02:23.280 --> 01:02:26.040 and those are the boundary conditions of this system. 01:02:26.040 --> 01:02:29.480 That we actually just follow exactly the same procedure 01:02:29.480 --> 01:02:32.000 to obtain all the unknown coefficients 01:02:32.000 --> 01:02:34.845 that we would be able to evolve this system as a function 01:02:34.845 --> 01:02:38.340 of time, as I have demonstrated to you in the beginning 01:02:38.340 --> 01:02:40.080 of the lecture. 01:02:40.080 --> 01:02:44.070 So in this case, you can have two boundary conditions. 01:02:44.070 --> 01:02:47.490 One is actually say at x equal to 0. 01:02:47.490 --> 01:02:51.810 And the other one is actually at x equal to L. 01:02:51.810 --> 01:02:54.070 In those boundaries, as we actually 01:02:54.070 --> 01:02:57.750 learned before, because the endpoints are 01:02:57.750 --> 01:02:59.910 fixed on the wall. 01:02:59.910 --> 01:03:07.680 Therefore, psi of 0 at that time, t, 01:03:07.680 --> 01:03:11.670 will be always equal to 0 for the left-hand side boundary 01:03:11.670 --> 01:03:13.370 condition. 01:03:13.370 --> 01:03:16.640 And very similarly, as we discussed before, 01:03:16.640 --> 01:03:22.440 psi of L t will be equal to 0 if you look at the right-hand side 01:03:22.440 --> 01:03:23.650 of the wall-- 01:03:23.650 --> 01:03:25.670 of the system. 01:03:25.670 --> 01:03:28.940 So I don't want to repeat this, because this is actually 01:03:28.940 --> 01:03:33.440 exactly the same calculation which we have done before. 01:03:33.440 --> 01:03:36.740 So with these two boundary conditions, 01:03:36.740 --> 01:03:44.660 we can actually conclude that k m will be equal to m pi over L, 01:03:44.660 --> 01:03:50.090 and alpha m will be equal to 0. 01:03:50.090 --> 01:03:55.690 So you can actually go back and check this out. 01:03:55.690 --> 01:03:58.350 So what I'm going to say is that until now, 01:03:58.350 --> 01:04:02.640 what we have been doing is identical to what 01:04:02.640 --> 01:04:06.120 we have been doing for the nondispersive media. 01:04:06.120 --> 01:04:10.380 What I'm to say is that the shape of the normal mode 01:04:10.380 --> 01:04:13.940 is actually set by the boundary condition. 01:04:13.940 --> 01:04:16.710 It's determined by the boundary condition, 01:04:16.710 --> 01:04:19.500 and it has actually, so far, nothing 01:04:19.500 --> 01:04:24.950 to do with the dispersion relation omega 01:04:24.950 --> 01:04:27.810 as a function of k. 01:04:27.810 --> 01:04:31.350 So in short, boundary condition can give you 01:04:31.350 --> 01:04:33.690 the shape of the normal mode, and that we 01:04:33.690 --> 01:04:36.975 know that the first normal mode, second normal mode, 01:04:36.975 --> 01:04:40.380 et cetera, et cetera, is actually 01:04:40.380 --> 01:04:46.440 going to be identical to the case of nondispersive medium. 01:04:46.440 --> 01:04:49.890 so that's actually the first thing which we learned. 01:04:49.890 --> 01:04:54.270 The second thing we learned is that OK, now what we see 01:04:54.270 --> 01:04:58.860 is that once the boundary condition is given, 01:04:58.860 --> 01:05:02.610 then the k m is actually also given. 01:05:02.610 --> 01:05:07.020 Therefore, since I have the dispersion relation omega 01:05:07.020 --> 01:05:11.055 as a function of k, as shown there. 01:05:11.055 --> 01:05:14.610 Omega over k is equal to v times square root of 1 01:05:14.610 --> 01:05:17.130 plus alpha k squared. 01:05:17.130 --> 01:05:24.480 Therefore, once k m is given, omega m is also given. 01:05:27.550 --> 01:05:32.370 So you can see that that's actually where the dispersion 01:05:32.370 --> 01:05:33.970 relation come into play. 01:05:36.650 --> 01:05:42.290 The omega m will be different if you compare the dispersive case 01:05:42.290 --> 01:05:45.560 and nondispersive case. 01:05:45.560 --> 01:05:49.470 So that is actually what I want to say. 01:05:49.470 --> 01:05:52.520 The k m, which is the shape of the normal mode, 01:05:52.520 --> 01:05:57.690 doesn't depend on the dispersion relation. 01:05:57.690 --> 01:06:00.480 On the other hand, the speed of the oscillation, 01:06:00.480 --> 01:06:04.020 the angular frequency, omega, depends 01:06:04.020 --> 01:06:06.690 on the dispersion relation, which is actually 01:06:06.690 --> 01:06:11.260 what we obtained from there. 01:06:11.260 --> 01:06:20.795 If I start to plot omega m as a function of k m-- 01:06:23.980 --> 01:06:27.240 so in the case of nondispersive medium, 01:06:27.240 --> 01:06:31.240 so what am I going to get is actually discrete points 01:06:31.240 --> 01:06:35.790 along a straight line. 01:06:35.790 --> 01:06:42.460 This is actually k1, k2, k3, k4, et cetera. 01:06:42.460 --> 01:06:50.430 They are actually all sitting on a common straight line. 01:06:50.430 --> 01:06:56.640 If you look at the relative difference between k1, k2, 01:06:56.640 --> 01:07:01.860 and k3, they are constant according to this formula. 01:07:01.860 --> 01:07:05.180 The difference between k1 and k2 is pi over 2. 01:07:05.180 --> 01:07:08.190 k2 and k3 is actually also pi over 2-- 01:07:08.190 --> 01:07:12.300 pi over L. It's always a fixed number. 01:07:12.300 --> 01:07:16.620 And since omega is actually proportional to k. 01:07:16.620 --> 01:07:24.640 Therefore, the spacing between omega 1, omega 2, omega 3, 01:07:24.640 --> 01:07:25.900 is also constant. 01:07:28.510 --> 01:07:34.080 In short, omega 2, omega 3, and omega 4, 01:07:34.080 --> 01:07:38.080 et cetera is always multiple times 01:07:38.080 --> 01:07:43.240 what you get from omega 1, according to this graph 01:07:43.240 --> 01:07:48.200 and in the case of nondispersive medium. 01:07:48.200 --> 01:07:50.280 So what does that mean? 01:07:50.280 --> 01:07:56.690 That means OK, now if I have a very complicated 01:07:56.690 --> 01:07:59.490 initial condition-- 01:07:59.490 --> 01:08:02.030 this is actually what I have, an initial condition-- very 01:08:02.030 --> 01:08:04.130 complicated. 01:08:04.130 --> 01:08:06.030 I just need to wait. 01:08:06.030 --> 01:08:08.930 If this is actually nondispersive medium, 01:08:08.930 --> 01:08:16.100 I just have to wait until p equal to 2 pi over omega 1. 01:08:16.100 --> 01:08:21.439 Then the system would restore to its original shape. 01:08:21.439 --> 01:08:23.450 That's actually what I can learn from here, 01:08:23.450 --> 01:08:30.700 because omega 2, omega 3, and any higher order normal modes, 01:08:30.700 --> 01:08:32.809 the angular frequency is actually multiple times 01:08:32.809 --> 01:08:34.910 of what I get from omega 1. 01:08:38.430 --> 01:08:43.200 On the other hand, if I consider a situation 01:08:43.200 --> 01:08:46.510 of dispersive medium-- 01:08:46.510 --> 01:08:55.210 you can see that now the difference between omega m 01:08:55.210 --> 01:08:58.149 is now the constant. 01:08:58.149 --> 01:09:01.250 So what you would predict is that it would take much, 01:09:01.250 --> 01:09:03.880 much longer for this system to go back 01:09:03.880 --> 01:09:10.840 to the original shape compared to nondispersive media. 01:09:10.840 --> 01:09:13.380 So that actually you can actually see. 01:09:13.380 --> 01:09:20.500 In a real-life experiment, I can distort this equipment 01:09:20.500 --> 01:09:23.680 in this boundless system, and it's actually 01:09:23.680 --> 01:09:27.640 going to take forever or impossible to come back 01:09:27.640 --> 01:09:30.850 to the original shape, because of that dispersion. 01:09:30.850 --> 01:09:33.399 On the other hand, if I have a really highly 01:09:33.399 --> 01:09:37.510 idealized situation, if I have both ends bound, 01:09:37.510 --> 01:09:43.040 and I just have to wait until t equal to 2 pi over omega 1. 01:09:43.040 --> 01:09:48.399 Then this system will go back to the original shape. 01:09:48.399 --> 01:09:56.100 Before I end the lecture today, I 01:09:56.100 --> 01:10:01.050 would like to discuss with you two interesting issues. 01:10:01.050 --> 01:10:06.360 So many of you have seen water waves, 01:10:06.360 --> 01:10:09.120 and Feynman actually told us in his lecture 01:10:09.120 --> 01:10:14.460 that water waves are really easily seen by everybody, 01:10:14.460 --> 01:10:17.520 but it's actually the worst possible example. 01:10:17.520 --> 01:10:20.400 That's the bad news-- the worst possible example 01:10:20.400 --> 01:10:24.960 because it has all the possible complications 01:10:24.960 --> 01:10:26.760 that waves can have. 01:10:26.760 --> 01:10:28.060 That's the bad news. 01:10:28.060 --> 01:10:31.176 The good news is that you are going to do that in your P set. 01:10:31.176 --> 01:10:32.880 [LAUGHTER] 01:10:32.880 --> 01:10:37.080 So we will be able to understand the behavior of the water 01:10:37.080 --> 01:10:38.280 waves. 01:10:38.280 --> 01:10:41.230 So that's the good news. 01:10:41.230 --> 01:10:43.530 The second thing which I would like to talk about 01:10:43.530 --> 01:10:45.300 is phase velocity. 01:10:48.270 --> 01:10:52.170 You can say, OK, you say that phase velocity or harmonic 01:10:52.170 --> 01:10:54.980 waves doesn't send information, right? 01:10:54.980 --> 01:10:58.830 And how do I actually know that? 01:10:58.830 --> 01:10:59.430 Right? 01:10:59.430 --> 01:11:00.660 So what does that mean? 01:11:00.660 --> 01:11:04.930 OK, so let's take this horrible example of water wave. 01:11:04.930 --> 01:11:09.520 OK, so the black line is actually the beach, 01:11:09.520 --> 01:11:14.550 and there is a water wave from the ocean approaching 01:11:14.550 --> 01:11:15.420 the beach. 01:11:15.420 --> 01:11:17.820 And you can see that you can have 01:11:17.820 --> 01:11:20.960 some kind of angle between the insert of water wave 01:11:20.960 --> 01:11:22.980 and the line of the beach. 01:11:25.530 --> 01:11:28.950 What I can actually do is that I can now 01:11:28.950 --> 01:11:37.860 measure the shape of the water wave at the edge of the beach. 01:11:37.860 --> 01:11:42.430 And I would see that, huh, now the phase velocity 01:11:42.430 --> 01:11:45.660 which I observe there is actually 01:11:45.660 --> 01:11:50.940 faster than the speed of propagation of the water wave, 01:11:50.940 --> 01:11:56.040 because of this inserted angle, OK? 01:11:56.040 --> 01:12:01.830 I can actually make it very, very fast. 01:12:01.830 --> 01:12:05.460 I can make the speed actually even faster 01:12:05.460 --> 01:12:07.550 than the speed of light. 01:12:07.550 --> 01:12:09.041 Right? 01:12:09.041 --> 01:12:15.570 I can now decrease the theta to 0. 01:12:15.570 --> 01:12:19.420 Then you will have a phase velocity 01:12:19.420 --> 01:12:21.560 which is faster than the speed of light. 01:12:21.560 --> 01:12:23.740 It goes to infinity. 01:12:23.740 --> 01:12:27.110 But does that mean anything? 01:12:27.110 --> 01:12:28.880 Actually, that doesn't mean anything, 01:12:28.880 --> 01:12:35.480 because I don't really move the water from a specific point 01:12:35.480 --> 01:12:38.570 to another point infinitely fast. 01:12:38.570 --> 01:12:40.880 Therefore, what I want to say is that, OK, 01:12:40.880 --> 01:12:46.280 you can do whatever you want to make a fancy phase velocity. 01:12:46.280 --> 01:12:49.610 But that will not help you with sending 01:12:49.610 --> 01:12:52.540 things close to the speed of light or greater 01:12:52.540 --> 01:12:54.230 than the speed of light. 01:12:54.230 --> 01:12:56.630 So as you can see from this example, 01:12:56.630 --> 01:12:59.610 I can easily construct a simple example, 01:12:59.610 --> 01:13:03.050 which you see that is actually really not sending anything 01:13:03.050 --> 01:13:04.850 from one place to the other. 01:13:04.850 --> 01:13:09.540 But you still have really, really fast phase velocity. 01:13:09.540 --> 01:13:11.580 OK, thank you very much, everybody, 01:13:11.580 --> 01:13:14.990 for the attention and hope you enjoyed the lecture. 01:13:14.990 --> 01:13:19.650 And if you have any questions, please let me know.