WEBVTT 00:00:02.450 --> 00:00:04.820 The following content is provided under a Creative 00:00:04.820 --> 00:00:06.210 Commons license. 00:00:06.210 --> 00:00:08.420 Your support will help MIT OpenCourseWare 00:00:08.420 --> 00:00:12.510 continue to offer high quality educational resources for free. 00:00:12.510 --> 00:00:15.050 To make a donation, or to view additional materials 00:00:15.050 --> 00:00:19.010 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.010 --> 00:00:20.316 at ocw.mit.edu. 00:00:24.390 --> 00:00:27.660 YEN-JIE LEE: OK, so welcome back to 803. 00:00:27.660 --> 00:00:30.280 Happy to see you again. 00:00:30.280 --> 00:00:32.564 So today, we are going to continue our discussion 00:00:32.564 --> 00:00:34.770 of dispersive medium. 00:00:34.770 --> 00:00:36.900 And there are two questions we are 00:00:36.900 --> 00:00:39.420 going to ask in this lecture, and we 00:00:39.420 --> 00:00:44.190 will answer also these two questions in this lecture. 00:00:44.190 --> 00:00:47.440 So just to warn you in advance, this lecture 00:00:47.440 --> 00:00:53.640 will have a lot of mathematics, so fasten the seatbelt 00:00:53.640 --> 00:00:57.380 and follow me. 00:00:57.380 --> 00:01:01.320 And stop me any time you don't feel like you 00:01:01.320 --> 00:01:04.560 know you understand something. 00:01:04.560 --> 00:01:06.840 So let's get started. 00:01:06.840 --> 00:01:11.790 OK, so today, we are going to talk about phenomena 00:01:11.790 --> 00:01:13.890 related to dispersion. 00:01:13.890 --> 00:01:16.740 And last time, we started a discussion 00:01:16.740 --> 00:01:19.980 about how to send information from one 00:01:19.980 --> 00:01:21.600 place to the other place right. 00:01:21.600 --> 00:01:26.550 So what we should be before was to send square pulse. 00:01:26.550 --> 00:01:33.210 So if I do have a machine which can produce a square pulse, 00:01:33.210 --> 00:01:36.770 then I can define something like this. 00:01:36.770 --> 00:01:40.350 So over some ratio, which I set, I 00:01:40.350 --> 00:01:43.470 can actually separate 0 and 1. 00:01:43.470 --> 00:01:45.700 So if I have a pulse which is actually 00:01:45.700 --> 00:01:49.110 having an amplitude greater than some threshold and I say, 00:01:49.110 --> 00:01:52.120 OK, I've got 1, and if it's actually 00:01:52.120 --> 00:01:55.230 below some straight line, say, OK, I've got a 0. 00:01:55.230 --> 00:01:58.350 And with that way, we actually can send information 00:01:58.350 --> 00:02:00.570 from one place to the other place. 00:02:00.570 --> 00:02:03.770 So that sounds really nice. 00:02:03.770 --> 00:02:09.130 However, if we work on a dispersive medium, 00:02:09.130 --> 00:02:11.580 which is really very common-- 00:02:11.580 --> 00:02:15.980 for example, light and gas is actually-- 00:02:15.980 --> 00:02:21.810 not all the lights with different wavelengths are 00:02:21.810 --> 00:02:26.340 traveling at the same speed, and also, 00:02:26.340 --> 00:02:30.906 as you've seen before in the p-set, deep water, 00:02:30.906 --> 00:02:36.765 and also the strings, considering a realistic string 00:02:36.765 --> 00:02:40.380 with stiffness, et cetera, et cetera-- 00:02:40.380 --> 00:02:47.490 to the wavelength of the input wave is going to affect 00:02:47.490 --> 00:02:50.340 the speed of this travelling wave. 00:02:50.340 --> 00:02:54.540 So in short, the speed of the wave propagation 00:02:54.540 --> 00:02:59.230 in a dispersive medium will depend on the wavelengths 00:02:59.230 --> 00:03:01.410 of this wave. 00:03:01.410 --> 00:03:03.780 So that brings us a lot of trouble 00:03:03.780 --> 00:03:08.310 because, for example, here we are trying to send a Gaussian 00:03:08.310 --> 00:03:13.260 pulse through the medium, but after a while 00:03:13.260 --> 00:03:17.130 this pulse actually becomes wider and wider because 00:03:17.130 --> 00:03:18.240 of the dispersion. 00:03:18.240 --> 00:03:21.810 Because all the components with different wavelengths 00:03:21.810 --> 00:03:26.140 which actually construct this narrow pulse, 00:03:26.140 --> 00:03:29.920 actually are traveling at different speeds. 00:03:29.920 --> 00:03:32.670 Therefore, if you wait long enough, 00:03:32.670 --> 00:03:40.300 all the different frequencies, or all the different frequency 00:03:40.300 --> 00:03:43.290 harmonic waves are travel at different speeds, 00:03:43.290 --> 00:03:46.080 therefore you get the, dispersion, 00:03:46.080 --> 00:03:51.300 which results in a much wider pulse in the end. 00:03:51.300 --> 00:03:52.950 And at some point, this pulse is going 00:03:52.950 --> 00:03:56.670 to be really wide, such that it's actually 00:03:56.670 --> 00:04:02.460 going to be very difficult to separate 0 from 1. 00:04:02.460 --> 00:04:04.110 So that's the problem. 00:04:04.110 --> 00:04:07.260 And we also did some simulations with computer. 00:04:07.260 --> 00:04:12.860 We do see this behavior also in our computer simulation. 00:04:12.860 --> 00:04:16.839 If I put in triangular pulse and allow it to evolve, 00:04:16.839 --> 00:04:19.680 and like what we did before, we assume 00:04:19.680 --> 00:04:24.070 that there's a stiffness in this string system. 00:04:24.070 --> 00:04:27.510 And you will see that, OK, as a function of time, 00:04:27.510 --> 00:04:31.290 this part is now longer a triangular shape, 00:04:31.290 --> 00:04:35.170 but you have a very complicated structure. 00:04:35.170 --> 00:04:37.710 So that is actually a problem we are 00:04:37.710 --> 00:04:40.920 going to try to solve today. 00:04:40.920 --> 00:04:44.250 And during that discussion last time, in the lecture, 00:04:44.250 --> 00:04:49.360 we also introduced dispersion relation omega k and also tried 00:04:49.360 --> 00:04:55.920 to overlap two travelling waves with similar wavelengths. 00:04:55.920 --> 00:04:59.520 And that would give you beat phenomenon. 00:04:59.520 --> 00:05:02.852 That probably doesn't surprise you any more. 00:05:02.852 --> 00:05:05.820 As you can see from this example, 00:05:05.820 --> 00:05:07.770 you have the beat phenomenon, and you 00:05:07.770 --> 00:05:12.640 can see the amplitude is actually variating slowly, 00:05:12.640 --> 00:05:15.210 the function of position. 00:05:15.210 --> 00:05:18.400 And if you follow the red point, which is actually 00:05:18.400 --> 00:05:23.220 associated with one of the peak, in the structure called 00:05:23.220 --> 00:05:26.690 carrier, OK, it's actually moving 00:05:26.690 --> 00:05:32.390 at the phase velocity, the P, which we introduced last time. 00:05:32.390 --> 00:05:37.540 The formula for BP, which is actually the speed of harmonic, 00:05:37.540 --> 00:05:42.935 oscillating travelling wave is actually defined as vp equal 00:05:42.935 --> 00:05:48.350 to omega over k, and the green point, 00:05:48.350 --> 00:05:53.540 which actually always at the minima of the distribution, 00:05:53.540 --> 00:05:58.730 which is actually associated with the speed of the envelope. 00:05:58.730 --> 00:06:01.520 You can see that, indeed, it actually 00:06:01.520 --> 00:06:04.310 can move at different speeds. 00:06:04.310 --> 00:06:07.460 It depends on the dispersion relation omega 00:06:07.460 --> 00:06:10.250 as a function of k you have in this system. 00:06:10.250 --> 00:06:16.100 And we call the speed of these envelope, 00:06:16.100 --> 00:06:20.120 which we construct from these two overlapping travelling 00:06:20.120 --> 00:06:21.630 waves to be-- 00:06:21.630 --> 00:06:25.130 we call it group velocity. 00:06:25.130 --> 00:06:27.130 And the definition of group velocity 00:06:27.130 --> 00:06:31.310 is vg equal to d omega dk. 00:06:31.310 --> 00:06:35.480 So that's what we have learned last time. 00:06:35.480 --> 00:06:41.720 OK, you may ask, OK, what do I mean by group velocity? 00:06:41.720 --> 00:06:45.500 And can I use it beyond what we have 00:06:45.500 --> 00:06:48.830 done for the beat phenomena. 00:06:48.830 --> 00:06:52.240 But what do I mean by group velocity? 00:06:52.240 --> 00:06:54.270 Is that really useful, and it's actually 00:06:54.270 --> 00:06:57.650 which part of the structure I was talking about. 00:06:57.650 --> 00:07:02.780 So in that case of two overlapping progressing waves 00:07:02.780 --> 00:07:06.410 with similar length or similar frequency, 00:07:06.410 --> 00:07:08.360 when we see that the group velocity actually 00:07:08.360 --> 00:07:12.080 present the speed of the envelope, right? 00:07:12.080 --> 00:07:14.070 Can we actually learn something more 00:07:14.070 --> 00:07:17.570 general about group velocity? 00:07:17.570 --> 00:07:19.430 The second question which we are asking 00:07:19.430 --> 00:07:24.800 is, OK, now we have this problem of dispersion. 00:07:24.800 --> 00:07:29.690 This square pulse is going to be something which is really 00:07:29.690 --> 00:07:33.070 wide after some period of time. 00:07:33.070 --> 00:07:37.820 So that's clearly a problem, and how do we actually 00:07:37.820 --> 00:07:40.650 solve this problem, and how do we actually send information 00:07:40.650 --> 00:07:45.620 like, for example, music over a large distance 00:07:45.620 --> 00:07:47.660 from one place to another place. 00:07:47.660 --> 00:07:49.610 So that's essentially what we're going 00:07:49.610 --> 00:07:53.360 to try to understand today. 00:07:53.360 --> 00:07:59.150 So let's start with an infinitely long string. 00:08:02.000 --> 00:08:05.780 And this string is actually very long, and this began from here, 00:08:05.780 --> 00:08:10.220 and it goes to some place which is really, really far away. 00:08:10.220 --> 00:08:12.920 And, of course, as usually, I can actually 00:08:12.920 --> 00:08:17.580 hold one end of this string and shake it a bit, 00:08:17.580 --> 00:08:21.420 then I can actually create some kind of pulse 00:08:21.420 --> 00:08:25.420 which is going to travel along this string 00:08:25.420 --> 00:08:28.340 towards a positive x direction. 00:08:28.340 --> 00:08:31.150 In this case, I defined the x coordinate 00:08:31.150 --> 00:08:33.360 will be pointing to a right hand side, and thus 00:08:33.360 --> 00:08:35.450 the positive direction. 00:08:35.450 --> 00:08:38.690 So of course I can hold this string, and I just shake it, 00:08:38.690 --> 00:08:43.700 and I would prepare a pulse on this medium, which is 00:08:43.700 --> 00:08:46.520 a string with constant tension. 00:08:46.520 --> 00:08:50.750 So I can describe the motion-- 00:08:50.750 --> 00:08:55.060 you can describe the motion of Yen-Jie's hand by a function. 00:08:55.060 --> 00:09:00.210 So you can say, OK, Yen-Jie is somehow doing a really nice job 00:09:00.210 --> 00:09:05.510 and oscillating at constant frequency. 00:09:05.510 --> 00:09:09.620 Like I can say, OK, Yen-Jie is shaking this thing 00:09:09.620 --> 00:09:12.120 to produce a harmonic wave, for example. 00:09:12.120 --> 00:09:15.890 And that, I can actually describe the motion of the hand 00:09:15.890 --> 00:09:17.440 by f of t. 00:09:17.440 --> 00:09:19.400 That's very good. 00:09:19.400 --> 00:09:24.690 And from what we have learned in the last lecture, 00:09:24.690 --> 00:09:27.725 we've found that, basically, waves, 00:09:27.725 --> 00:09:31.850 harmonic waves with different frequency, 00:09:31.850 --> 00:09:33.670 or with different wavelengths, are 00:09:33.670 --> 00:09:36.950 traveling at different speeds. 00:09:36.950 --> 00:09:43.420 Therefore, we would like to actually decompose the motion 00:09:43.420 --> 00:09:47.780 of Yen-Jie's hand into many, many harmonic waves-- 00:09:47.780 --> 00:09:51.740 then attack them one by one, to follow them one by one, 00:09:51.740 --> 00:09:54.020 then I can solve this problem. 00:09:54.020 --> 00:09:56.540 So that's actually what we are going to do. 00:09:56.540 --> 00:10:01.400 And that will involve some math, which we would follow 00:10:01.400 --> 00:10:03.830 from the math department. 00:10:03.830 --> 00:10:07.430 And before that, I would like to introduce the imitation first. 00:10:07.430 --> 00:10:10.970 As I said f of t is actually the displacement 00:10:10.970 --> 00:10:14.540 as a function of time as x is equal to 0. 00:10:14.540 --> 00:10:16.504 So basically, I'm holding this string, 00:10:16.504 --> 00:10:19.820 and I move things up and down, so that, actually, I 00:10:19.820 --> 00:10:23.490 move this string away from the equilibrium positive, which 00:10:23.490 --> 00:10:27.230 is actually y equal to 0. 00:10:27.230 --> 00:10:28.690 Then what is going to happen? 00:10:28.690 --> 00:10:30.230 What is going to happen is that I'm 00:10:30.230 --> 00:10:35.270 going to produce some kind of pulse, 00:10:35.270 --> 00:10:38.030 and this pulse, I can actually describe it 00:10:38.030 --> 00:10:42.470 by a function, which is psi x and p, 00:10:42.470 --> 00:10:46.460 this psi is actually describing the displacement 00:10:46.460 --> 00:10:50.030 as a function of x, and as a function of t. 00:10:50.030 --> 00:10:52.940 Apparently, if you put x equals to zero, 00:10:52.940 --> 00:10:56.220 then you go back to f of t, right? 00:10:56.220 --> 00:10:58.640 Basically that's the idea. 00:10:58.640 --> 00:11:00.410 OK. 00:11:00.410 --> 00:11:06.770 So what we have learned before we introduce 00:11:06.770 --> 00:11:20.890 dispersive medium is that, if I have a non-dispersive medium, 00:11:20.890 --> 00:11:24.080 OK, if I have a non-dispersive medium, 00:11:24.080 --> 00:11:29.300 then things are pretty simple because omega over K 00:11:29.300 --> 00:11:33.870 is actually a constant, which is the phase velocity, vp. 00:11:33.870 --> 00:11:39.355 And omega is actually just equal to vp times k. 00:11:42.610 --> 00:11:46.390 That means, no matter what kind of wavelength 00:11:46.390 --> 00:11:49.600 we are talking about, no matter what kind of angular frequency 00:11:49.600 --> 00:11:55.030 we are talking about, harmonic progressing wave 00:11:55.030 --> 00:11:58.900 is going to travel at the speed of Vp. 00:11:58.900 --> 00:12:02.590 No matter what's the frequency, or what's the wavelength. 00:12:02.590 --> 00:12:05.140 So that makes our life much simpler 00:12:05.140 --> 00:12:07.870 when we work on non-dispersive medium. 00:12:07.870 --> 00:12:11.820 In this case, if I have a non-dispersive medium, 00:12:11.820 --> 00:12:15.670 then psi would be equal to-- 00:12:15.670 --> 00:12:18.730 maybe I write it here-- 00:12:18.730 --> 00:12:21.670 if I have non-dispersive medium where, 00:12:21.670 --> 00:12:25.090 no matter what kind of frequency, 00:12:25.090 --> 00:12:27.690 the speed of the harmonic traveling wave 00:12:27.690 --> 00:12:31.210 is a constant, which is actually Vp, I can write down psi 00:12:31.210 --> 00:12:43.420 x t to could be equal to f of t minus x over v. 00:12:43.420 --> 00:12:45.640 Just remember f is actually describing 00:12:45.640 --> 00:12:49.240 how I shake one end of the string, 00:12:49.240 --> 00:12:51.850 and, basically you can see that ha! 00:12:51.850 --> 00:12:54.800 What is happening is that my hand is actually 00:12:54.800 --> 00:13:00.380 generating the shape of the pulse as a function of x, 00:13:00.380 --> 00:13:02.650 as a function of time, and it can 00:13:02.650 --> 00:13:08.410 be described by a really simple formula here. 00:13:08.410 --> 00:13:11.680 So this is actually really nice for non-dispersive. 00:13:11.680 --> 00:13:17.050 As I introduced before, when we talk about dispersive medium, 00:13:17.050 --> 00:13:27.970 then, if I go to dispersive, omega 00:13:27.970 --> 00:13:34.840 is actually a function of k, and can be a non-linear function. 00:13:34.840 --> 00:13:36.060 So what does that mean? 00:13:36.060 --> 00:13:41.020 That means, if I evaluate vp, which is actually 00:13:41.020 --> 00:13:45.100 the phase velocity, which is the formula there, 00:13:45.100 --> 00:13:51.800 this is going to be omega of k divided by k. 00:13:51.800 --> 00:13:57.010 That means BP is going to be a function of k, 00:13:57.010 --> 00:13:59.220 the wavelength-- wave number. 00:13:59.220 --> 00:14:02.390 It's not going to be a constant in general-- 00:14:02.390 --> 00:14:07.680 unless omega is actually equal to vp times k, 00:14:07.680 --> 00:14:13.300 in general, vp can actually be some quantity 00:14:13.300 --> 00:14:15.900 which is variating as a function of k. 00:14:15.900 --> 00:14:16.540 OK? 00:14:16.540 --> 00:14:20.770 Then we have trouble because that means, 00:14:20.770 --> 00:14:24.880 when I produce progressing wave from the left hand side end, 00:14:24.880 --> 00:14:29.560 it's actually made of many, many harmonic waves, 00:14:29.560 --> 00:14:32.410 right, with different angular frequency. 00:14:32.410 --> 00:14:37.240 So I can shake this like [MAKES NOISE],, different speed. 00:14:37.240 --> 00:14:41.740 And I can always decompose the motion of Yen-Jie's hand 00:14:41.740 --> 00:14:44.230 into many, many harmonic waves. 00:14:44.230 --> 00:14:49.150 The problem is, all those harmonic waves are going to be 00:14:49.150 --> 00:14:52.690 travelling at different speed. 00:14:52.690 --> 00:14:56.190 How do we actually describe this? 00:14:56.190 --> 00:14:58.110 So that's the trouble. 00:14:58.110 --> 00:15:02.647 And I was really frustrated when I think about this problem, 00:15:02.647 --> 00:15:04.230 and my friend from the math department 00:15:04.230 --> 00:15:07.440 said, hey, we have solved this problem a long time ago. 00:15:07.440 --> 00:15:10.320 [LAUGHTER] 00:15:10.320 --> 00:15:11.820 So this is not the problem anymore. 00:15:11.820 --> 00:15:16.360 And I say, oh, what is the idea you're talking about? 00:15:16.360 --> 00:15:18.600 And they actually told me that you 00:15:18.600 --> 00:15:22.960 should use Fourier transform to attack this problem. 00:15:22.960 --> 00:15:23.990 OK? 00:15:23.990 --> 00:15:25.840 This is the idea. 00:15:25.840 --> 00:15:30.180 The idea is that I can now write down 00:15:30.180 --> 00:15:35.610 f of t, which is motion of Yen-Jie's hand, 00:15:35.610 --> 00:15:41.550 and this can even returned as a superposition 00:15:41.550 --> 00:15:46.890 of infinite number of waves. 00:15:46.890 --> 00:15:49.350 I can integrate from minus infinity 00:15:49.350 --> 00:15:55.270 to infinity, t omega, which is the angular frequency. 00:15:55.270 --> 00:16:03.450 And each contributing wave has an amplitutde associated 00:16:03.450 --> 00:16:08.340 with it, which is, as you see, is a function omega. 00:16:08.340 --> 00:16:11.900 And the actual wave is actually written in terms 00:16:11.900 --> 00:16:16.900 of exponential minus i omega t. 00:16:16.900 --> 00:16:21.750 So now, let's actually you look at this thing really carefully. 00:16:21.750 --> 00:16:23.340 What am I doing? 00:16:23.340 --> 00:16:29.670 I am saying that I, now, can shake one end of the string up 00:16:29.670 --> 00:16:33.950 and down according to my will. 00:16:33.950 --> 00:16:36.840 And, if I do this for a long time, 00:16:36.840 --> 00:16:41.640 I can actually describe the motion of my hand 00:16:41.640 --> 00:16:47.070 by infinite number of harmonic waves, which is actually 00:16:47.070 --> 00:16:50.160 kind of like exponential i omega t 00:16:50.160 --> 00:16:55.380 describing the frequency of these waves, and each of them 00:16:55.380 --> 00:17:00.020 got associated amplitude. 00:17:00.020 --> 00:17:02.230 And you may ask, OK, wait a second, 00:17:02.230 --> 00:17:07.240 you call this Fourier transform, and I have learned that before, 00:17:07.240 --> 00:17:09.060 but I learned a different version. 00:17:09.060 --> 00:17:12.450 I learned a version of cosine and sine? 00:17:12.450 --> 00:17:15.329 And what is going on? 00:17:15.329 --> 00:17:17.336 Actually, they are all the same. 00:17:17.336 --> 00:17:18.960 No matter what you do, you can actually 00:17:18.960 --> 00:17:21.780 also do that with cosine and sine, 00:17:21.780 --> 00:17:25.710 but what I actually found is that it's actually easier 00:17:25.710 --> 00:17:29.610 to deal with exponential functional form 00:17:29.610 --> 00:17:33.720 You can always write exponential i omega 00:17:33.720 --> 00:17:40.470 t in terms of sine and cosine and absorb the i 00:17:40.470 --> 00:17:42.890 into a c omega. 00:17:42.890 --> 00:17:45.520 Basically, these things are identical between these two 00:17:45.520 --> 00:17:47.130 forms of this answer. 00:17:47.130 --> 00:17:49.350 So therefore, in this lecture, I'm 00:17:49.350 --> 00:17:52.790 going to stick with this functional form. 00:17:52.790 --> 00:17:54.232 OK, any questions? 00:17:54.232 --> 00:17:59.900 STUDENT: We don't include dx in the [INAUDIBLE]?? 00:17:59.900 --> 00:18:01.000 YEN-JIE LEE: Not yet. 00:18:01.000 --> 00:18:02.590 We are going to include that. 00:18:02.590 --> 00:18:07.590 Because, for that, in order to actually-- 00:18:07.590 --> 00:18:11.250 OK, so now I actually decompose the motion of my hand 00:18:11.250 --> 00:18:14.020 into many, many waves-- 00:18:14.020 --> 00:18:17.020 which should be or is say it many, many oscillation 00:18:17.020 --> 00:18:19.860 with different frequencies. 00:18:19.860 --> 00:18:23.590 So I actually describe the motion 00:18:23.590 --> 00:18:26.500 of my hand infinite number of oscillation 00:18:26.500 --> 00:18:28.300 with different frequency. 00:18:28.300 --> 00:18:34.900 And the trouble we are facing is that all those oscillations are 00:18:34.900 --> 00:18:37.960 going to be charged travelling at different speeds 00:18:37.960 --> 00:18:42.160 because of the dispersion relation. 00:18:42.160 --> 00:18:45.760 Therefore, what I am going to do afterwards 00:18:45.760 --> 00:18:50.170 is to show you that, OK, I can write down the functional form 00:18:50.170 --> 00:18:54.130 for psi in this general case. 00:18:54.130 --> 00:18:58.100 So for that, that's actually what I'm going to do now. 00:18:58.100 --> 00:19:03.730 So now, I would like to know what would be the psi xt, which 00:19:03.730 --> 00:19:06.970 actually the position of the string as a function of x, 00:19:06.970 --> 00:19:11.020 and at some specific time equal to t. 00:19:11.020 --> 00:19:16.690 And that can be written as, I do the an integration from minus 00:19:16.690 --> 00:19:23.380 infinity to infinity over frequency omega, 00:19:23.380 --> 00:19:27.310 and I have the usual amplitude associated 00:19:27.310 --> 00:19:29.770 with the angular frequency omega, 00:19:29.770 --> 00:19:35.810 and the exponential i omega t-- 00:19:35.810 --> 00:19:38.890 minus i omega t because that's the convention I'm using here-- 00:19:38.890 --> 00:19:46.020 and I say, OK, plus ik, which is a function omega-- 00:19:46.020 --> 00:19:46.960 x. 00:19:46.960 --> 00:19:49.510 So now you can to see that what I'm 00:19:49.510 --> 00:19:53.250 doing here is that I am now progressing, 00:19:53.250 --> 00:19:56.680 I am making infinite number of progressing waves. 00:19:56.680 --> 00:19:59.320 Each of these exponential functions 00:19:59.320 --> 00:20:04.630 is a progressing wave with angular frequency omega. 00:20:04.630 --> 00:20:10.110 And why do I write k as a function of omega x here? 00:20:10.110 --> 00:20:12.572 It's because they are going to be travelling at the speed 00:20:12.572 --> 00:20:15.160 of omega over k. 00:20:15.160 --> 00:20:19.630 Therefore, I need to actually put k here, 00:20:19.630 --> 00:20:22.660 and this k is actually-- 00:20:22.660 --> 00:20:27.040 this k is actually not the independent parameter. 00:20:27.040 --> 00:20:29.800 It's actually a function of omega. 00:20:29.800 --> 00:20:33.730 So we can see that, here, we do an integration over omega 00:20:33.730 --> 00:20:36.960 from minus infinity to infinity-- 00:20:36.960 --> 00:20:42.040 for each omega you can actually find the corresponding k, 00:20:42.040 --> 00:20:42.760 right? 00:20:42.760 --> 00:20:45.370 Because of the dispersion relation. 00:20:45.370 --> 00:20:47.770 Because omega is a function of k, 00:20:47.770 --> 00:20:52.420 therefore you can always solve the corresponding k, right? 00:20:52.420 --> 00:20:54.130 Then you put it there? 00:20:54.130 --> 00:20:58.610 Because you are now trying to propagate 00:20:58.610 --> 00:21:01.660 how many waves with different angular 00:21:01.660 --> 00:21:04.130 frequency at different speed-- 00:21:04.130 --> 00:21:05.307 then we are done. 00:21:08.080 --> 00:21:10.690 That looks like a wonderful solution, 00:21:10.690 --> 00:21:17.150 and we can actually see how it works for our purpose. 00:21:17.150 --> 00:21:18.310 Any questions? 00:21:20.950 --> 00:21:21.850 All right. 00:21:21.850 --> 00:21:23.090 So that's really nice. 00:21:23.090 --> 00:21:26.650 And I can now do a really simple test 00:21:26.650 --> 00:21:29.640 to see if this really works. 00:21:29.640 --> 00:21:32.790 Let me try a very simple case. 00:21:32.790 --> 00:21:34.358 OK, a spatial case. 00:21:37.230 --> 00:21:41.970 If I now go back to use this description 00:21:41.970 --> 00:21:46.080 to describe non-dispersive medium and see what 00:21:46.080 --> 00:21:47.340 will happen. 00:21:47.340 --> 00:21:52.210 Now my k as a function omega is actually rather simple. 00:21:52.210 --> 00:21:53.740 It's actually omega over vp-- 00:21:58.200 --> 00:22:02.760 according to the dispersion relation here. 00:22:02.760 --> 00:22:05.820 I can solve k, as I was mentioning, 00:22:05.820 --> 00:22:08.610 with these dispersion relation formula. 00:22:08.610 --> 00:22:11.250 And then I can conclude k as a function 00:22:11.250 --> 00:22:15.920 omega is omega over vp. 00:22:15.920 --> 00:22:18.680 Then I can now put that into this equation, 00:22:18.680 --> 00:22:22.380 and I'm going to get psi x of t-- 00:22:22.380 --> 00:22:24.930 this would be equal to minus infinity 00:22:24.930 --> 00:22:33.270 to infinity d omega, c omega, exponential minus i omega 00:22:33.270 --> 00:22:37.140 t minus omega over vx. 00:22:41.640 --> 00:22:47.160 And we can actually take omega out of this, minus infinity 00:22:47.160 --> 00:22:54.060 to infinity d omega c omega exponential minus i omega 00:22:54.060 --> 00:23:02.850 t minus x divided by v. And you can see that, huh, indeed, 00:23:02.850 --> 00:23:09.990 this is actually ft minus x over v. OK. 00:23:09.990 --> 00:23:11.670 I'm dropping the vp here. 00:23:11.670 --> 00:23:16.030 This should be vp all over the place. 00:23:16.030 --> 00:23:21.450 So you can see that, now, if I have solved the k as a function 00:23:21.450 --> 00:23:25.380 omega, and I plug it in in this special case, 00:23:25.380 --> 00:23:31.590 which is non-dispersive medium, omega over k equal to vp, 00:23:31.590 --> 00:23:36.310 then I really calculate this integral, 00:23:36.310 --> 00:23:38.450 then I can quickly identify that-- 00:23:38.450 --> 00:23:43.750 huh, I can write the functional form in this way. 00:23:43.750 --> 00:23:48.300 And this is actually really familiar to me 00:23:48.300 --> 00:23:52.580 because that's actually using this definition, f is actually 00:23:52.580 --> 00:23:56.550 equal to integration minus infinity to infinity, d omega, 00:23:56.550 --> 00:23:59.670 c omega, exponential minus i omega t. 00:23:59.670 --> 00:24:06.280 If I replace t, by t minus x over vp, then I'm done. 00:24:06.280 --> 00:24:10.500 So I have evaluated this integration, which is actually 00:24:10.500 --> 00:24:14.670 just f t minus x over vp. 00:24:14.670 --> 00:24:19.740 So that's exactly what guessed from the beginning. 00:24:19.740 --> 00:24:22.170 So if I have a non-dispersive medium, 00:24:22.170 --> 00:24:26.490 then psi xt will be equal to this function. 00:24:26.490 --> 00:24:28.440 So that gives us some kind of confidence that, 00:24:28.440 --> 00:24:33.600 OK, at the easy case, it works. 00:24:33.600 --> 00:24:39.510 All right, so that's very nice, all sounds very good. 00:24:39.510 --> 00:24:41.190 But wait a second. 00:24:41.190 --> 00:24:43.470 How do I actually extract this c, 00:24:43.470 --> 00:24:45.690 which is a function of omega? 00:24:45.690 --> 00:24:50.580 I'm troubled because this is an infinite integral from 00:24:50.580 --> 00:24:52.480 minus infinity to infinity. 00:24:52.480 --> 00:24:55.560 And that means I have infinite number of constants, which I 00:24:55.560 --> 00:24:58.830 have to determine the c omega. 00:24:58.830 --> 00:24:59.970 How do I actually do this? 00:25:02.670 --> 00:25:06.180 So that is another point which I would 00:25:06.180 --> 00:25:09.180 like to discuss before we actually go ahead 00:25:09.180 --> 00:25:15.360 and really use this function for the dispersive medium case. 00:25:15.360 --> 00:25:19.680 So how to we actually extract c as a function of omega? 00:25:22.230 --> 00:25:27.630 So for that, we really need to employ a few uses 00:25:27.630 --> 00:25:31.860 for formula, which are actually documented here. 00:25:31.860 --> 00:25:36.030 How many of you actually have not heard about delta function 00:25:36.030 --> 00:25:37.520 before? 00:25:37.520 --> 00:25:41.560 OK, a few of you actually have not heard about delta function. 00:25:41.560 --> 00:25:44.730 So what is actually your delta function? 00:25:44.730 --> 00:25:47.910 This is a delta function. 00:25:47.910 --> 00:25:52.155 So a delta function is actually a notation 00:25:52.155 --> 00:25:58.280 which actually shows you a function, which 00:25:58.280 --> 00:26:03.180 is should only be non-zero, at x equal to zero. 00:26:03.180 --> 00:26:07.200 And the x equal to zero, the size of this function 00:26:07.200 --> 00:26:11.290 as you're going to infinity, and on the other hand, 00:26:11.290 --> 00:26:16.520 all the other points at x not equal to zero, 00:26:16.520 --> 00:26:20.980 the delta function is equal to zero. 00:26:20.980 --> 00:26:23.820 So that's actually the kind of function I was talking about. 00:26:23.820 --> 00:26:28.010 And the area of this function, if you're doing the equation 00:26:28.010 --> 00:26:32.040 over minus infinity infinity over x, 00:26:32.040 --> 00:26:38.220 the integration of these delta m the area is actually 1. 00:26:38.220 --> 00:26:40.420 So that is actually the kind of function. 00:26:40.420 --> 00:26:45.870 So essentially, it's a really, really narrow function, OK, 00:26:45.870 --> 00:26:50.040 very narrow, very narrow, very narrow. 00:26:50.040 --> 00:26:54.870 But the area is finite, which is why. 00:26:54.870 --> 00:26:57.870 So you can have a square. 00:26:57.870 --> 00:27:01.320 You can actually start with a square pulse, 00:27:01.320 --> 00:27:03.390 or square function, and you can actually 00:27:03.390 --> 00:27:07.670 make the width of the square narrower, smaller and smaller 00:27:07.670 --> 00:27:09.610 and smaller, go to 0. 00:27:09.610 --> 00:27:11.680 Then what you are going to get is essentially 00:27:11.680 --> 00:27:13.620 the delta function. 00:27:13.620 --> 00:27:18.990 That's actually how we understand this delta function. 00:27:18.990 --> 00:27:20.900 All right, really quickly. 00:27:20.900 --> 00:27:24.930 And also, we would like to use a few formula which 00:27:24.930 --> 00:27:26.800 are documented here. 00:27:26.800 --> 00:27:32.040 So if I do an integration from minus infinity to infinity, 00:27:32.040 --> 00:27:35.490 exponential i omega minus omega prime, 00:27:35.490 --> 00:27:40.830 t over the t which is integrating over time, t, here. 00:27:40.830 --> 00:27:45.770 And then divide the whole formula by 1 and over 2pi. 00:27:45.770 --> 00:27:49.030 What I'm going to get is a delta function, 00:27:49.030 --> 00:27:53.670 which is a delta function which is omega minus omega prime. 00:27:53.670 --> 00:27:58.590 So that means when this delta function formula tells us 00:27:58.590 --> 00:28:02.610 that omega is equal to omega prime, 00:28:02.610 --> 00:28:06.900 then this function is actually going to infinity. 00:28:06.900 --> 00:28:10.620 And only when omega equal to omega prime, 00:28:10.620 --> 00:28:12.700 this function is not zero. 00:28:12.700 --> 00:28:16.750 Any other place, this function is always zero. 00:28:19.290 --> 00:28:22.230 And this strange integration should give you 00:28:22.230 --> 00:28:23.530 this delta function. 00:28:23.530 --> 00:28:26.140 So that's the first thing which we will use, 00:28:26.140 --> 00:28:28.350 was one useful formula. 00:28:28.350 --> 00:28:32.130 The second thing which what just I talked about, 00:28:32.130 --> 00:28:35.560 if I do an integration over minus infinity to infinity, 00:28:35.560 --> 00:28:40.410 delta x dx, then basically you get 1. 00:28:40.410 --> 00:28:43.290 The third one is actually kind of interesting. 00:28:43.290 --> 00:28:44.400 Let's take a look. 00:28:44.400 --> 00:28:48.510 So if I do an integration over from minus infinity 00:28:48.510 --> 00:28:54.630 to infinity, delta function x minus alpha. 00:28:54.630 --> 00:28:57.060 Let's look at this delta function first. 00:28:57.060 --> 00:29:04.596 This function is only non-zero when x is equal to what? 00:29:04.596 --> 00:29:05.262 AUDIENCE: Alpha. 00:29:05.262 --> 00:29:06.012 YEN-JIE LEE: Yeah. 00:29:06.012 --> 00:29:10.310 When x is equal to alpha, only when that happen, 00:29:10.310 --> 00:29:12.410 this is actually non-zero. 00:29:12.410 --> 00:29:18.050 If you multiply this delta function to some function which 00:29:18.050 --> 00:29:22.610 is f of alpha, and integrate over alpha from minus 00:29:22.610 --> 00:29:25.040 infinity to infinity. 00:29:25.040 --> 00:29:27.620 And that means that when alpha is 00:29:27.620 --> 00:29:32.000 equal to x, or x equal to alpha, this integration 00:29:32.000 --> 00:29:35.750 give you non-zero result. All the other ways, 00:29:35.750 --> 00:29:38.420 you will get zero. 00:29:38.420 --> 00:29:41.470 The interesting thing is that if you do this integration, what 00:29:41.470 --> 00:29:47.316 you are going to get is that OK, when I integrate over alpha, 00:29:47.316 --> 00:29:51.260 only when alpha is equal to x this thing is non-zero. 00:29:51.260 --> 00:29:52.730 Therefore what you are going to get 00:29:52.730 --> 00:29:56.780 is, you get only one point of the width, which is actually 00:29:56.780 --> 00:29:59.180 f of x. 00:29:59.180 --> 00:30:02.282 So that's the intuition about this formula. 00:30:02.282 --> 00:30:02.990 That's just fine. 00:30:02.990 --> 00:30:05.731 Any questions related to those formulas? 00:30:05.731 --> 00:30:08.677 AUDIENCE: [INAUDIBLE]? 00:30:08.677 --> 00:30:10.150 YEN-JIE LEE: Hm? 00:30:10.150 --> 00:30:11.270 AUDIENCE: [INAUDIBLE]? 00:30:13.599 --> 00:30:15.890 YEN-JIE LEE: Yeah, this is actually pretty complicated, 00:30:15.890 --> 00:30:21.300 so it would take a few 10, 20 minutes to explain that. 00:30:21.300 --> 00:30:23.790 But let's just take the words from the math department-- 00:30:23.790 --> 00:30:24.802 we trust them. 00:30:28.100 --> 00:30:31.040 All right, so once I have those formula, 00:30:31.040 --> 00:30:35.610 I can now demonstrate you how I can actually 00:30:35.610 --> 00:30:38.750 track C as a function omega. 00:30:38.750 --> 00:30:41.320 So this is actually the goal, right? 00:30:41.320 --> 00:30:43.480 So don't forget why we are doing what we are doing, 00:30:43.480 --> 00:30:47.810 is to try to extract what is actually the C omega, 00:30:47.810 --> 00:30:54.410 so that we can actually finish this formula. 00:30:54.410 --> 00:30:56.450 So how do we do that? 00:30:56.450 --> 00:31:00.170 So suppose, if I evaluate this. 00:31:05.120 --> 00:31:12.910 This function, 1 over 2pi, minus infinity 00:31:12.910 --> 00:31:23.610 to infinity dt, ft, exponential i omega t. 00:31:23.610 --> 00:31:26.480 If I evaluate this function. 00:31:26.480 --> 00:31:28.860 This is coming out of nowhere, right? 00:31:28.860 --> 00:31:34.090 So coming out of Yen-Jie's hand, maybe, I don't know. 00:31:34.090 --> 00:31:38.490 Suppose if I evaluate this function, 00:31:38.490 --> 00:31:40.980 and now I have ft here, right? 00:31:40.980 --> 00:31:47.790 I can replace ft by these interesting formula. 00:31:47.790 --> 00:31:54.360 If I do that, then basically I get 1 over 2pi minus infinity 00:31:54.360 --> 00:32:04.010 to infinity dt, minus infinity infinity C omega 00:32:04.010 --> 00:32:10.600 prime, exponential minus i omega prime t. 00:32:10.600 --> 00:32:15.330 And the last is actually integrating over d omega prime. 00:32:17.880 --> 00:32:21.480 So this is actually the f of t. 00:32:21.480 --> 00:32:24.490 This is actually f of t. 00:32:24.490 --> 00:32:28.380 I'm just replacing that formula into this integral. 00:32:28.380 --> 00:32:33.140 And then I have the rest, which is exponential i omega t. 00:32:36.770 --> 00:32:39.660 And of course, I can continue and collect 00:32:39.660 --> 00:32:42.810 all the relevant terms together. 00:32:42.810 --> 00:32:51.930 This is actually equal to 1 over 2i, minus infinity to infinity. 00:32:51.930 --> 00:32:57.510 I collect all the terms related to omega prime to the left hand 00:32:57.510 --> 00:32:58.690 side. 00:32:58.690 --> 00:33:04.940 Basically what I get is C omega prime d omega prime. 00:33:04.940 --> 00:33:10.920 This is actually coming from here, except-- 00:33:10.920 --> 00:33:15.420 yeah, OK, it is actually coming from here. 00:33:15.420 --> 00:33:20.830 And I have another integral which is from minus infinity 00:33:20.830 --> 00:33:26.770 to infinity, this time integrating over delta dt. 00:33:26.770 --> 00:33:36.480 And I have dt here, exponential i omega minus omega prime t. 00:33:36.480 --> 00:33:44.650 So basically I'm collecting these two terms together. 00:33:44.650 --> 00:33:48.660 They now become exponential i omega minus omega prime t. 00:33:51.390 --> 00:33:55.980 So basically, no magic happened, but I'm just re-writing things 00:33:55.980 --> 00:34:01.680 and we are arranging things from this formula to that formula. 00:34:01.680 --> 00:34:08.130 Then if we look at this formula, this formula here, 00:34:08.130 --> 00:34:10.889 and the formula sheet we have. 00:34:10.889 --> 00:34:14.840 1 over 2pi minus infinity to infinity 00:34:14.840 --> 00:34:17.920 to this integration over t, exponential i omega 00:34:17.920 --> 00:34:19.940 minus omega prime t. 00:34:19.940 --> 00:34:22.510 That will give you delta function, 00:34:22.510 --> 00:34:25.130 which is delta omega minus omega prime. 00:34:27.929 --> 00:34:31.750 Therefore, I can continue this calculation here. 00:34:34.600 --> 00:34:45.800 Thus it's going to give you minus infinity to infinity. 00:34:45.800 --> 00:34:53.870 I identify this part, this part, and this part, 00:34:53.870 --> 00:34:55.440 to be the delta function. 00:34:58.330 --> 00:35:03.340 Therefore, what I get is minus infinity to infinity, C 00:35:03.340 --> 00:35:14.850 omega prime, delta omega minus omega prime, d omega prime. 00:35:14.850 --> 00:35:17.450 Am I going too fast? 00:35:17.450 --> 00:35:20.637 Everybody's following? 00:35:20.637 --> 00:35:22.470 So you can see that what we have been doing, 00:35:22.470 --> 00:35:26.490 I use this formula coming out of nowhere. 00:35:26.490 --> 00:35:31.980 I replace f by the formula I was writing there. 00:35:31.980 --> 00:35:34.920 And then I collect the terms I like together. 00:35:34.920 --> 00:35:36.910 That's all I did. 00:35:36.910 --> 00:35:38.790 And then I found, aha! 00:35:38.790 --> 00:35:44.240 One part of the formula is actually the delta function. 00:35:44.240 --> 00:35:46.420 Then I put the delta a function here. 00:35:46.420 --> 00:35:50.200 And then finally, I use the third formula here, 00:35:50.200 --> 00:35:54.230 which I have related to delta function, and I found, aha! 00:35:54.230 --> 00:35:56.250 If I do this integration, I know how 00:35:56.250 --> 00:35:58.740 to do this integration even without knowing 00:35:58.740 --> 00:36:02.610 the structure of C. This is actually just changing 00:36:02.610 --> 00:36:06.320 the omega prime to omega. 00:36:06.320 --> 00:36:10.650 So that's actually what this integration actually does. 00:36:10.650 --> 00:36:16.500 Therefore, I get C omega. 00:36:16.500 --> 00:36:20.190 Look at what we have done. 00:36:20.190 --> 00:36:23.350 What we have done is that, we have proof 00:36:23.350 --> 00:36:27.240 that this formula coming out of nowhere, 00:36:27.240 --> 00:36:32.080 to be a continuous version of mode picker. 00:36:32.080 --> 00:36:35.130 You remember the fourth year decomposition before? 00:36:35.130 --> 00:36:39.840 You were using the orthogonality of the sine function, 00:36:39.840 --> 00:36:42.840 and I can do some kind of fancy integration 00:36:42.840 --> 00:36:47.700 to actually extract a m from one of the-- 00:36:47.700 --> 00:36:50.440 which is associated with one of the normal mode, right? 00:36:50.440 --> 00:36:53.460 What we are doing here is actually a continuous version. 00:36:53.460 --> 00:36:55.740 Now omega is actually continuous. 00:36:55.740 --> 00:37:02.220 And I'm now using the orthogonality 00:37:02.220 --> 00:37:05.220 of the exponential function. 00:37:05.220 --> 00:37:08.130 If I do this integration, that will only 00:37:08.130 --> 00:37:14.020 give you non-zero value when omega is equal to omega prime. 00:37:14.020 --> 00:37:16.210 It's exactly the same thing, right? 00:37:16.210 --> 00:37:21.220 Then I can construct an integration like this. 00:37:21.220 --> 00:37:24.360 And now will give you the redoubting C 00:37:24.360 --> 00:37:29.142 as a function omega, which is like the amplitude of one 00:37:29.142 --> 00:37:34.470 of the associated harmonics exponential i omega t. 00:37:34.470 --> 00:37:37.240 So in short, from this exercise, we 00:37:37.240 --> 00:37:40.330 have shown you that C of omega can 00:37:40.330 --> 00:37:46.360 be extracted using this formula 1 over 2pi, minus infinity 00:37:46.360 --> 00:37:54.790 to infinity dt, f of t, exponential i omega t. 00:37:54.790 --> 00:37:58.660 That's actually how we actually can determine 00:37:58.660 --> 00:38:06.100 all the amplitude associated to a specific exponential 00:38:06.100 --> 00:38:07.000 function. 00:38:07.000 --> 00:38:08.040 Any questions so far? 00:38:11.280 --> 00:38:17.760 OK, so if no question, then we can actually continue. 00:38:17.760 --> 00:38:22.040 So let's actually go back to the original question, 00:38:22.040 --> 00:38:24.710 which we were posting. 00:38:24.710 --> 00:38:29.540 So we have a problem related to the transmission 00:38:29.540 --> 00:38:31.830 of information. 00:38:31.830 --> 00:38:35.570 So this is actually where we got started. 00:38:35.570 --> 00:38:42.320 If I send a square pulse on a dispersive median, 00:38:42.320 --> 00:38:45.170 then I have some trouble, which is 00:38:45.170 --> 00:38:49.190 that this pulse is going to disperse and become 00:38:49.190 --> 00:38:50.700 wider and wider. 00:38:50.700 --> 00:38:53.000 It's changing as a function of time, 00:38:53.000 --> 00:38:55.280 as a function of distance it travel. 00:38:55.280 --> 00:38:57.680 That's not cool. 00:38:57.680 --> 00:39:03.130 All right, so therefore what I am going to do is this. 00:39:03.130 --> 00:39:06.330 There was a very smart idea which 00:39:06.330 --> 00:39:10.970 were discovered long time ago, during maybe World War I, 00:39:10.970 --> 00:39:17.665 and widely used in World War II, which is the AM radio. 00:39:17.665 --> 00:39:18.640 What is AM? 00:39:18.640 --> 00:39:25.120 Is actually amplitude modulation radio. 00:39:25.120 --> 00:39:28.850 This smart idea is the following. 00:39:28.850 --> 00:39:32.840 I will describe it before we take a break. 00:39:32.840 --> 00:39:41.840 So this smart idea, AM radio is the following. 00:39:41.840 --> 00:39:47.270 If I have some kind of information which is fs t. 00:39:47.270 --> 00:39:50.180 s here means signal. 00:39:50.180 --> 00:39:51.890 If I have some kind of information 00:39:51.890 --> 00:39:53.870 I would like to send, I can send it 00:39:53.870 --> 00:39:59.300 by oscillating one end of the string. 00:39:59.300 --> 00:40:03.200 And this is what I want to send. 00:40:03.200 --> 00:40:08.430 And there are two ways you can send this fs function. 00:40:08.430 --> 00:40:14.050 The first one is actually what did before, I send it directly. 00:40:14.050 --> 00:40:16.940 I just said OK, if I want to send this function, 00:40:16.940 --> 00:40:19.850 then I just oscillate the string according 00:40:19.850 --> 00:40:22.180 to the functional form. 00:40:22.180 --> 00:40:25.130 Yen-Jie just have to be really careful, right? 00:40:25.130 --> 00:40:27.650 So that you can send this function. 00:40:27.650 --> 00:40:31.680 And that fails miserably. 00:40:31.680 --> 00:40:32.720 Why? 00:40:32.720 --> 00:40:38.580 Because all the components which actually produce the fs, 00:40:38.580 --> 00:40:43.690 in this case the square pulse, all those components are 00:40:43.690 --> 00:40:46.200 travelling at different speed. 00:40:46.200 --> 00:40:49.020 Therefore, the information will never get there, 00:40:49.020 --> 00:40:53.460 because of the dispersion. 00:40:53.460 --> 00:40:56.840 So now what should we do instead? 00:40:56.840 --> 00:41:02.440 Instead of doing sending fs as a function of t directly, 00:41:02.440 --> 00:41:05.040 what you could do is that I can now 00:41:05.040 --> 00:41:15.940 send f of t, which is equal to f of s t cosine omega0t. 00:41:19.630 --> 00:41:26.610 Where omega0 is a very, very large number. 00:41:26.610 --> 00:41:29.310 And basically, look at what we have been doing. 00:41:29.310 --> 00:41:34.350 So that means I, instead of sending fs directly, 00:41:34.350 --> 00:41:39.960 I send fs, but modulated by a really high frequency 00:41:39.960 --> 00:41:43.410 function, cosine omega0t. 00:41:43.410 --> 00:41:45.690 And this will work. 00:41:45.690 --> 00:41:48.480 And you will only know that after we come back 00:41:48.480 --> 00:41:53.040 from the break, which is twenty first. 00:41:53.040 --> 00:41:54.900 Let's take five minute break. 00:41:54.900 --> 00:41:58.380 And if you have any questions, you can actually ask me here. 00:42:06.000 --> 00:42:12.750 So we will continue the discussion about AM radio. 00:42:12.750 --> 00:42:16.890 So before the break, actually we introduced this one 00:42:16.890 --> 00:42:23.220 possible solution to solve this dispersive median problem, 00:42:23.220 --> 00:42:29.450 is that I can now actually send instead of fs as a function 00:42:29.450 --> 00:42:33.200 t, which is actually the signal I want to send, 00:42:33.200 --> 00:42:38.460 I could send fs, but multiplied by cosine omega0t. 00:42:41.960 --> 00:42:48.660 If I assume that fs is some really slow function, 00:42:48.660 --> 00:42:52.290 slowly varying as a function of time, 00:42:52.290 --> 00:42:55.870 compared to cosine omega0t. 00:42:55.870 --> 00:43:00.600 Cosine omega0t is a really fast function, 00:43:00.600 --> 00:43:03.930 oscillating up and down like crazy, really fast. 00:43:03.930 --> 00:43:08.190 If I multiply fs by this function, 00:43:08.190 --> 00:43:10.490 what is going to happen? 00:43:10.490 --> 00:43:13.030 We are going to show you that actually that 00:43:13.030 --> 00:43:19.410 means I am going to only have non-zero C 00:43:19.410 --> 00:43:23.150 function, or a large contribution of C, 00:43:23.150 --> 00:43:25.610 in a very thin middle range of omega. 00:43:28.440 --> 00:43:29.470 So we'll show that. 00:43:29.470 --> 00:43:35.700 So in a typical case, fs is really slow, which is like, 00:43:35.700 --> 00:43:38.620 for example, my sound, et cetera, 00:43:38.620 --> 00:43:41.460 in the label of one kilohertz. 00:43:41.460 --> 00:43:43.710 And you can actually design a system 00:43:43.710 --> 00:43:51.570 which will actually multiply this fs by cosine omega0t. 00:43:51.570 --> 00:43:57.960 Omega0 can be as fast as 1.1 to 30 megahertz. 00:43:57.960 --> 00:44:01.470 If you do this calculation, then you 00:44:01.470 --> 00:44:20.210 will find that OK, the range of omega, with sizable C omega 00:44:20.210 --> 00:44:22.500 is small. 00:44:26.690 --> 00:44:34.550 It's roughly equal to omega0 minus omega s, 00:44:34.550 --> 00:44:41.470 to omega0 plus omega s, where omega s is 00:44:41.470 --> 00:44:47.780 the typical frequency in your signal. 00:44:47.780 --> 00:44:52.210 And the omega0 is the typical frequency of-- 00:44:52.210 --> 00:44:55.820 the frequency of your cosine omega t 00:44:55.820 --> 00:44:58.120 term, which is actually, later, you 00:44:58.120 --> 00:44:59.765 will recognize this as carrier. 00:45:03.280 --> 00:45:07.080 So what I want to say is that if I do this trick, 00:45:07.080 --> 00:45:08.870 what is going to happen is that the range 00:45:08.870 --> 00:45:14.450 of omega, which you have sizable contribution from C omega-- 00:45:14.450 --> 00:45:17.670 C omega is the associated amplitude, 00:45:17.670 --> 00:45:20.090 associated amplitude. 00:45:20.090 --> 00:45:23.450 It's going to be confined to a really small region 00:45:23.450 --> 00:45:28.590 from omega0 minus omega s, to omega0 plus omega s. 00:45:28.590 --> 00:45:33.670 So that's the trick which actually makes this problem 00:45:33.670 --> 00:45:34.170 solvable. 00:45:36.710 --> 00:45:40.940 How do we know this? 00:45:40.940 --> 00:45:43.760 That is because, if I now, for example, I 00:45:43.760 --> 00:45:51.610 send fs equal to cosine omega st. 00:45:51.610 --> 00:45:54.710 If this is actually the signal which I would like to send, 00:45:54.710 --> 00:45:58.430 just a harmonic wave, then what is going to happen 00:45:58.430 --> 00:46:07.730 is that I'm going to get ft is equal to cosine omega st 00:46:07.730 --> 00:46:09.390 cosine-- 00:46:09.390 --> 00:46:13.520 so this is actually multiplied by cosine omega0t. 00:46:13.520 --> 00:46:17.960 So I have cosine omega0t here. 00:46:17.960 --> 00:46:21.800 I have cosine multiplied by cosine. 00:46:21.800 --> 00:46:24.620 Therefore I have the question which 00:46:24.620 --> 00:46:28.360 I prepare here, the formula, of cosine alpha times 00:46:28.360 --> 00:46:32.540 cosine beta will be equal to the functional form. 00:46:32.540 --> 00:46:35.240 There's a remainder, therefore I can now 00:46:35.240 --> 00:46:42.130 write it as 1 over 2 cosine omega s 00:46:42.130 --> 00:46:53.770 minus omega0 t plus cosine omega s plus omega0. 00:46:57.540 --> 00:47:03.640 You can see that when I actually multiply two cosine functions 00:47:03.640 --> 00:47:10.480 together, then what I get is actually the omega0 minus omega 00:47:10.480 --> 00:47:11.580 s. 00:47:11.580 --> 00:47:15.420 You can actually put the minus sign there, it didn't matter. 00:47:15.420 --> 00:47:21.930 And cosine omega0 plus omega s times t. 00:47:21.930 --> 00:47:24.660 So therefore, you can see that the frequency, there are only 00:47:24.660 --> 00:47:29.250 two frequencies which contribute to this C of omega, which is 00:47:29.250 --> 00:47:31.500 actually these two frequencies. 00:47:31.500 --> 00:47:36.280 So that is actually why, if you do this trick, 00:47:36.280 --> 00:47:41.100 you actually try to modulate your slow signal function 00:47:41.100 --> 00:47:45.000 by a fast carrier frequency. 00:47:45.000 --> 00:47:49.950 Then you are going to confine the effective range of omega 00:47:49.950 --> 00:47:53.460 into a very small range. 00:47:53.460 --> 00:47:56.660 Why is that useful? 00:47:56.660 --> 00:48:00.240 That's actually what I want to answer to you. 00:48:00.240 --> 00:48:06.060 Suppose I have this crazy dispersion relation, which 00:48:06.060 --> 00:48:10.940 is omega as a function of K. You can graph it, 00:48:10.940 --> 00:48:14.350 and suppose it looks really crazy like this. 00:48:18.660 --> 00:48:26.010 And if I set my carrier oscillation frequency 00:48:26.010 --> 00:48:35.130 to be omega0, and that will give you a corresponding wave 00:48:35.130 --> 00:48:39.240 number which is K0. 00:48:39.240 --> 00:48:40.290 I hope you can see it. 00:48:43.060 --> 00:48:47.130 That's the corresponding K0. 00:48:47.130 --> 00:48:54.990 Before we actually multiply this function, it's a slow function. 00:48:54.990 --> 00:48:57.220 It's not exactly one cosine function. 00:48:57.220 --> 00:49:00.270 So if you just have a cosine function harmonic wave, 00:49:00.270 --> 00:49:02.086 then you don't really need this trick, 00:49:02.086 --> 00:49:03.460 because it's actually going to be 00:49:03.460 --> 00:49:07.620 traveling at a speed of some constant speed. 00:49:07.620 --> 00:49:09.690 It's a harmonic traveling wave. 00:49:09.690 --> 00:49:14.100 But if this is actually a slow function, but not 00:49:14.100 --> 00:49:18.780 really a single harmonic wave, then what is going to happen 00:49:18.780 --> 00:49:23.700 is that you are going to need a wide range of K value or omega 00:49:23.700 --> 00:49:28.470 value to describe fs. 00:49:28.470 --> 00:49:31.360 Then you are in trouble because now, 00:49:31.360 --> 00:49:35.260 all the waves with different wavelengths are going to be 00:49:35.260 --> 00:49:37.290 travelling at different speed. 00:49:37.290 --> 00:49:40.900 Then you have this dispersion problem. 00:49:40.900 --> 00:49:44.460 On the other hand, if I multiply this function, 00:49:44.460 --> 00:49:48.940 this slow function, by a fast oscillating function, 00:49:48.940 --> 00:49:55.420 I am confining the effective range of omega 00:49:55.420 --> 00:49:58.690 into this small box, which is actually 00:49:58.690 --> 00:50:06.100 between omega0 minus or plus omega s. 00:50:06.100 --> 00:50:09.870 This is actually omega0 plus minus omega s. 00:50:09.870 --> 00:50:13.830 This is actually the range of the possible omega, 00:50:13.830 --> 00:50:17.650 which contribute to this resulting f of t. 00:50:17.650 --> 00:50:21.870 Therefore, the behavior of this function 00:50:21.870 --> 00:50:26.470 is actually much easier to understand. 00:50:26.470 --> 00:50:33.370 So with that given there, suppose now I have 00:50:33.370 --> 00:50:37.330 a large difference between-- 00:50:37.330 --> 00:50:40.945 suppose I have a large difference between omega s 00:50:40.945 --> 00:50:42.490 and omega0. 00:50:42.490 --> 00:50:46.780 Then I can actually focus on a very small range 00:50:46.780 --> 00:50:53.530 in this dispersion relation diagram. 00:50:53.530 --> 00:50:56.830 Then I can write omega as a function of K, 00:50:56.830 --> 00:51:03.790 the dispersion relation equal to omega0 plus K 00:51:03.790 --> 00:51:10.400 minus K0, partial omega, partial K. Evaluate 00:51:10.400 --> 00:51:15.700 it at a equal to K0, plus higher order term. 00:51:15.700 --> 00:51:20.470 Basically I can do this Taylor expansion. 00:51:20.470 --> 00:51:26.640 And maybe it surprised you, you can immediately identify, ha! 00:51:26.640 --> 00:51:35.060 This is delta d omega dk, is the group velocity. 00:51:35.060 --> 00:51:39.210 Suddenly it show up in the Taylor expansion 00:51:39.210 --> 00:51:42.550 of the dispersion relation. 00:51:42.550 --> 00:51:48.760 so if I focus on the region which is around omega0, 00:51:48.760 --> 00:51:53.750 then I can actually re-write omega in this functional form. 00:51:53.750 --> 00:51:59.500 Omega is actually equal to omega0 plus K 00:51:59.500 --> 00:52:04.810 minus K0 times Vg, which is the group velocity. 00:52:09.770 --> 00:52:14.570 Suppose this is happening, then now I can actually 00:52:14.570 --> 00:52:23.540 go ahead and really calculate the functional form for f of t. 00:52:23.540 --> 00:52:29.200 So suppose I have this definition of t. 00:52:29.200 --> 00:52:41.330 The definition of t is equal to fs t times cosine omega0t. 00:52:41.330 --> 00:52:45.230 Or say I can actually write it in a complex form, 00:52:45.230 --> 00:52:51.230 instead of writing it in a cosine omega0t functional form, 00:52:51.230 --> 00:52:54.540 I can write it in exponential functional form. 00:52:54.540 --> 00:52:59.750 Exponential minus i omega0t, which is actually more 00:52:59.750 --> 00:53:04.130 convenient for the discussion. 00:53:04.130 --> 00:53:09.190 So what is going to happen if I actually do this calculation? 00:53:09.190 --> 00:53:12.080 Then basically that's one example signal 00:53:12.080 --> 00:53:14.990 which I would like to send on the slides. 00:53:14.990 --> 00:53:21.680 So if I am trying to send a progressing harmonic wave, 00:53:21.680 --> 00:53:27.740 then after multiplying by this exponential i omega0t 00:53:27.740 --> 00:53:30.380 function, or a cosine function, basically 00:53:30.380 --> 00:53:32.990 you get something which is actually oscillating really 00:53:32.990 --> 00:53:37.100 fast, which is actually the AM signal we are trying 00:53:37.100 --> 00:53:41.030 to send through this media. 00:53:41.030 --> 00:53:43.640 So we can actually identify, this 00:53:43.640 --> 00:53:47.300 is actually the structure of this, actually the carrier. 00:53:50.360 --> 00:53:55.700 And this signal become the analogue, 00:53:55.700 --> 00:54:00.630 in analogy to what we actually have discussed for that beat 00:54:00.630 --> 00:54:01.710 phenomenon case. 00:54:04.650 --> 00:54:14.810 So now, if the omega range is really small, 00:54:14.810 --> 00:54:18.700 then I can actually write this down. 00:54:18.700 --> 00:54:23.060 Write a functional form of omega in this functional form. 00:54:23.060 --> 00:54:29.090 Or I can actually take out the kVg term, 00:54:29.090 --> 00:54:31.910 and the rest is actually going to be something 00:54:31.910 --> 00:54:39.650 like some constant a, where a is actually equal to omega0 00:54:39.650 --> 00:54:43.220 minus Vg times K0. 00:54:43.220 --> 00:54:47.960 So basically I'm just taking out this K term here, 00:54:47.960 --> 00:54:51.620 and this become this term. 00:54:51.620 --> 00:54:54.290 With this formula, I can solve what 00:54:54.290 --> 00:54:58.350 would be the functional form for K, 00:54:58.350 --> 00:55:01.860 as a preparation for what I'm going to do later. 00:55:01.860 --> 00:55:08.330 So K can be also expressed as omega over Vg. 00:55:08.330 --> 00:55:12.910 So basically I just solve the K plus b. 00:55:12.910 --> 00:55:16.030 b is actually just some constant. 00:55:16.030 --> 00:55:17.810 Just do it for convenience, I can 00:55:17.810 --> 00:55:25.520 write b equal to K0 minus omega0 divided by Vg. 00:55:25.520 --> 00:55:31.010 What we learned here is that if the range of effective omega 00:55:31.010 --> 00:55:35.710 is really small around omega zero, 00:55:35.710 --> 00:55:39.590 then the relation between omega and the K 00:55:39.590 --> 00:55:42.170 becomes a linear function. 00:55:42.170 --> 00:55:47.910 Of course, it's still not like the case for the non dispersive 00:55:47.910 --> 00:55:53.700 median, where omega over K is a constant. 00:55:53.700 --> 00:55:56.922 But at least it becomes a linear function, 00:55:56.922 --> 00:55:58.130 which is actually much nicer. 00:56:01.070 --> 00:56:05.780 So finally, with all those preparation we have done, 00:56:05.780 --> 00:56:10.100 we would like to show one important consequence. 00:56:10.100 --> 00:56:17.300 So what we are trying to do is to show that psi xt. 00:56:17.300 --> 00:56:22.130 Now I send, I oscillate the median, 00:56:22.130 --> 00:56:27.450 the string, by this f of t, which I designed there. 00:56:27.450 --> 00:56:33.930 ft is actually fs times exponential i omega0t. 00:56:33.930 --> 00:56:37.460 That's actually designed there. 00:56:37.460 --> 00:56:41.810 I would like to show that the resulting amplitude will 00:56:41.810 --> 00:56:56.130 be equal to fs t minus x divided by Vg, exponential minus i 00:56:56.130 --> 00:57:00.460 omega0t minus K0x. 00:57:00.460 --> 00:57:03.636 Of course I need to take the real part of this 00:57:03.636 --> 00:57:07.730 in, to go back to the real axis. 00:57:07.730 --> 00:57:14.990 Basically I dropped the i sine omega t contribution. 00:57:14.990 --> 00:57:18.500 So this is actually what I want to show. 00:57:18.500 --> 00:57:21.540 Before I go through all those math, 00:57:21.540 --> 00:57:25.070 let's do get the conclusion which we would like to draw, 00:57:25.070 --> 00:57:28.120 before we actually really go through the math. 00:57:28.120 --> 00:57:31.730 The conclusion which I would like to draw is that, OK, 00:57:31.730 --> 00:57:35.540 this is actually my analogue. 00:57:35.540 --> 00:57:41.480 My analogue is going to be travelling at the speed of Vg, 00:57:41.480 --> 00:57:43.504 which is the group velocity. 00:57:43.504 --> 00:57:45.170 That's the conclusion which I would like 00:57:45.170 --> 00:57:47.660 to draw from this exercise. 00:57:47.660 --> 00:57:54.350 And this thing is actually cosine omega0t minus K0x. 00:57:54.350 --> 00:57:57.830 Therefore, this is actually a harmonic wave. 00:57:57.830 --> 00:58:02.540 The carrier is a harmonic wave travelling at Vp equal 00:58:02.540 --> 00:58:06.590 to omega0 divided by K0. 00:58:06.590 --> 00:58:08.660 That's the kind of conclusion which I would like 00:58:08.660 --> 00:58:13.050 to draw from this exercise. 00:58:13.050 --> 00:58:18.110 Any questions about what we have discussed so far? 00:58:18.110 --> 00:58:23.750 OK, then really you have to hold tight and follow me really, 00:58:23.750 --> 00:58:26.350 100% focus, because this is actually 00:58:26.350 --> 00:58:29.010 a complicated calculation. 00:58:29.010 --> 00:58:35.150 So now what I can do is, now I need to express my fs in terms 00:58:35.150 --> 00:58:41.360 of C. So I do integration from minus infinity 00:58:41.360 --> 00:58:50.150 to infinity, d omega, C omega, exponential minus i omega t. 00:58:50.150 --> 00:58:53.870 So basically I can write my f of s 00:58:53.870 --> 00:58:59.690 in a functional form, which we introduced before. 00:58:59.690 --> 00:59:07.640 Then my f function is actually equal to fs times exponential i 00:59:07.640 --> 00:59:10.910 omega minus i omega0t. 00:59:10.910 --> 00:59:14.480 So that's actually what we defined there. 00:59:14.480 --> 00:59:19.270 And this would be equal to minus infinity to infinity. 00:59:19.270 --> 00:59:27.090 I do this integration number, omega C omega exponential minus 00:59:27.090 --> 00:59:33.800 i omega plus omega0 times t. 00:59:33.800 --> 00:59:38.200 So there's nothing special, I just take my expression for fs, 00:59:38.200 --> 00:59:42.877 multiply that by exponential minus i omega0t. 00:59:42.877 --> 00:59:44.210 Then that's actually what I get. 00:59:54.740 --> 00:59:58.320 So since this is actually integration over omega 00:59:58.320 --> 01:00:02.700 from minus infinity to infinity, therefore I 01:00:02.700 --> 01:00:07.410 can always have the freedom to shift the origin. 01:00:07.410 --> 01:00:15.600 So that means f of t can be returned as minus infinity 01:00:15.600 --> 01:00:24.210 to infinity d omega C omega minus omega0 exponential minus 01:00:24.210 --> 01:00:27.440 i omega t. 01:00:27.440 --> 01:00:31.950 Then we can see that is fix a relation between C 01:00:31.950 --> 01:00:38.760 of the f function, and the C of the fs function. 01:00:38.760 --> 01:00:41.420 So so far, everything is exact. 01:00:41.420 --> 01:00:46.110 I haven't made any approximation so far. 01:00:46.110 --> 01:00:54.660 So now, I can take this function and propagate that to all x. 01:00:54.660 --> 01:00:58.800 In other words, I can now take this ft, 01:00:58.800 --> 01:01:03.570 and write down the psi as a function of x and t. 01:01:03.570 --> 01:01:06.690 So that means all the different components are 01:01:06.690 --> 01:01:10.860 traveling at different speeds. 01:01:10.860 --> 01:01:12.570 So basically, I can write it down 01:01:12.570 --> 01:01:20.160 like d omega C omega minus omega0, exponential minus i 01:01:20.160 --> 01:01:24.170 omega t, exponential ikx. 01:01:24.170 --> 01:01:28.650 Kx k is actually a function of omega. 01:01:31.180 --> 01:01:32.410 Any questions? 01:01:32.410 --> 01:01:35.550 So that's actually just identical to what we actually 01:01:35.550 --> 01:01:36.900 have done before. 01:01:36.900 --> 01:01:45.250 So now I can go from f of t to sine, if you are following me. 01:01:45.250 --> 01:01:48.270 So until here, everything exact. 01:01:48.270 --> 01:01:50.890 You have all the problems you have, 01:01:50.890 --> 01:01:53.240 like you know this dispersion essentially, 01:01:53.240 --> 01:01:54.730 because all the little components, 01:01:54.730 --> 01:01:59.820 as you can see here, can be travelling at different speeds. 01:01:59.820 --> 01:02:04.430 So now, what I could do is that if I 01:02:04.430 --> 01:02:11.820 assume that C omega is only sizable at the small range 01:02:11.820 --> 01:02:18.130 around, it's only sizable around omega0. 01:02:18.130 --> 01:02:23.040 If now I take this assumption and propagate 01:02:23.040 --> 01:02:27.360 into this formula, then I can write this psi 01:02:27.360 --> 01:02:32.820 function roughly like minus infinity 01:02:32.820 --> 01:02:40.780 to infinity d omega C omega minus omega0 exponential minus 01:02:40.780 --> 01:02:45.880 i omega t exponential i. 01:02:45.880 --> 01:02:50.800 Now I can take the formula which I actually 01:02:50.800 --> 01:02:56.040 did an approximation, around omega0. 01:02:56.040 --> 01:03:03.450 Around omega0, K can be returned us omega over Vg plus b. 01:03:03.450 --> 01:03:05.910 This is actually where I take the approximation. 01:03:05.910 --> 01:03:10.320 Only consider the first order in the Taylor expansion. 01:03:10.320 --> 01:03:13.650 So you can see now here, it's not exact anymore. 01:03:13.650 --> 01:03:19.110 But now I write approximate function of form for K omega. 01:03:19.110 --> 01:03:25.180 So what I'm going to get is omega over Vg 01:03:25.180 --> 01:03:28.170 plus b, multiplied by x. 01:03:32.980 --> 01:03:33.931 Any questions? 01:03:36.510 --> 01:03:38.350 Now I have the approximation. 01:03:38.350 --> 01:03:42.760 And of course now I can gather all the terms related to omega 01:03:42.760 --> 01:03:44.530 together. 01:03:44.530 --> 01:03:47.650 I'm getting minus infinity to infinity 01:03:47.650 --> 01:03:56.830 d omega C omega minus omega0 exponential minus i omega t 01:03:56.830 --> 01:04:05.560 minus x over Vg, exponential ibx. 01:04:05.560 --> 01:04:09.550 So basically, I am merging this term and that term. 01:04:09.550 --> 01:04:14.040 This term and that term will give you this term. 01:04:14.040 --> 01:04:18.160 And what is essentially the rest is the exponential ibx. 01:04:21.520 --> 01:04:22.750 We are almost there. 01:04:27.530 --> 01:04:32.570 So now I would like to use this board, so I need to erase that. 01:04:41.050 --> 01:04:48.160 So now I continue from here, and I can now again, 01:04:48.160 --> 01:04:52.750 I can again change the origin of this infinite integral 01:04:52.750 --> 01:04:58.720 so that this can be written as minus infinity 01:04:58.720 --> 01:05:04.400 to infinity, d omega C function of omega, 01:05:04.400 --> 01:05:16.960 exponential minus i omega plus omega0, t minus x divided 01:05:16.960 --> 01:05:22.320 by Vg, and exponential ibx. 01:05:25.100 --> 01:05:29.320 So what I come from this board to that formula, 01:05:29.320 --> 01:05:32.260 if you are following me we are almost there, 01:05:32.260 --> 01:05:34.900 because I am changing the origin again, 01:05:34.900 --> 01:05:38.200 so that omega minus omega0 becomes 01:05:38.200 --> 01:05:40.510 omega, become a new omega. 01:05:40.510 --> 01:05:43.840 Is everybody accepting this fact? 01:05:43.840 --> 01:05:46.100 And that means the original omega 01:05:46.100 --> 01:05:52.360 will become omega plus omega0. 01:05:52.360 --> 01:05:58.120 I'm trying to go really slow, so that everybody can follow. 01:05:58.120 --> 01:05:59.980 I hope you are following. 01:05:59.980 --> 01:06:02.470 All right, then now I can actually 01:06:02.470 --> 01:06:05.200 redistribute, arrange all those terms 01:06:05.200 --> 01:06:07.930 and the magic will happen. 01:06:07.930 --> 01:06:10.680 So that means rearrange all those terms, 01:06:10.680 --> 01:06:16.480 minus infinity to infinity d omega C omega 01:06:16.480 --> 01:06:22.450 exponential minus i omega t minus x divided 01:06:22.450 --> 01:06:33.760 by Vg, exponential minus i omega0t, exponential i omega0 01:06:33.760 --> 01:06:39.310 over Vg plus b x. 01:06:39.310 --> 01:06:42.070 So basically, there's really no magic. 01:06:42.070 --> 01:06:45.940 What I'm doing is really to rearrange all those terms, 01:06:45.940 --> 01:06:49.010 so that this term is actually rearranged 01:06:49.010 --> 01:06:53.150 so that it's now omega times t minus x over Vg. 01:06:53.150 --> 01:06:55.090 It's an independent exponential term. 01:06:58.080 --> 01:07:06.310 And I actually extract this term times t to be returned here. 01:07:06.310 --> 01:07:08.850 I'm just rearranging things, OK? 01:07:08.850 --> 01:07:12.220 I'm not changing anything. 01:07:12.220 --> 01:07:17.800 And finally, I can merge this term and that term, 01:07:17.800 --> 01:07:20.900 and become this function field. 01:07:23.740 --> 01:07:30.420 I can immediately recognize that after this rearrangement, 01:07:30.420 --> 01:07:32.800 this is just re-writing the formula, 01:07:32.800 --> 01:07:36.460 putting all those terms in different place. 01:07:36.460 --> 01:07:38.400 Of course, you can actually review 01:07:38.400 --> 01:07:42.160 this part of the lecture in the lecture notes later. 01:07:42.160 --> 01:07:46.000 But basically, we're not doing anything fancy but rearranging 01:07:46.000 --> 01:07:50.050 things over in different place. 01:07:50.050 --> 01:07:52.840 Then I can actually quickly identify 01:07:52.840 --> 01:07:55.750 what I am trying to integrate. 01:07:55.750 --> 01:07:59.590 So this integration is over omega. 01:07:59.590 --> 01:08:03.730 Therefore all those terms are now related to omega. 01:08:03.730 --> 01:08:07.240 Therefore, they are just some terms which are sitting there, 01:08:07.240 --> 01:08:08.840 they don't participate. 01:08:08.840 --> 01:08:16.479 And if you focus on this part, what is this? 01:08:16.479 --> 01:08:21.880 If you compare that to the original equation of which 01:08:21.880 --> 01:08:23.500 I have here. 01:08:23.500 --> 01:08:27.399 If you compare that to the original fs equation here, 01:08:27.399 --> 01:08:29.710 you can't immediately identify that actually that's 01:08:29.710 --> 01:08:31.580 a function of fs. 01:08:31.580 --> 01:08:37.490 Originally this function fs is a function of t. 01:08:37.490 --> 01:08:40.149 And I'm going to that board now. 01:08:40.149 --> 01:08:48.290 This is actually fs with t minus x over Vg. 01:08:48.290 --> 01:08:49.960 Surprisingly simple. 01:08:53.380 --> 01:08:58.510 Now let's look at the right hand side, this mass here. 01:08:58.510 --> 01:09:05.899 This is actually K0, which actually you cannot see 01:09:05.899 --> 01:09:06.399 anymore. 01:09:06.399 --> 01:09:10.700 It's in the back of this board. 01:09:10.700 --> 01:09:13.189 And then if you combine these two terms, 01:09:13.189 --> 01:09:15.810 basically what you get is exponential 01:09:15.810 --> 01:09:19.720 minus i omega0t minus K0x. 01:09:22.880 --> 01:09:24.229 So look at what we have done. 01:09:26.760 --> 01:09:33.660 I got started with this Fourier transform functional 01:09:33.660 --> 01:09:36.810 form of fs. 01:09:36.810 --> 01:09:40.800 I multiplied fs by cosine omega0t 01:09:40.800 --> 01:09:43.210 and go to the complex notation. 01:09:43.210 --> 01:09:47.660 It becomes exponential minus i omega0t. 01:09:47.660 --> 01:09:53.250 If I multiplied that, I get my f function, which is like this. 01:09:53.250 --> 01:09:55.910 You get additional term there. 01:09:55.910 --> 01:10:00.030 I rearrange things and change the origin, 01:10:00.030 --> 01:10:03.750 and I can rewrite ft in this functional form. 01:10:03.750 --> 01:10:08.700 And I can have a relation between the C related to fs 01:10:08.700 --> 01:10:12.930 to the C related to f of t. 01:10:12.930 --> 01:10:18.120 I propagate ft over the full space, 01:10:18.120 --> 01:10:21.690 and attain my sine, which is the amplitude as a function 01:10:21.690 --> 01:10:25.910 of place and the time. 01:10:25.910 --> 01:10:29.060 Until here, everything is exact. 01:10:29.060 --> 01:10:31.750 Then I have introduced assumption, 01:10:31.750 --> 01:10:36.850 which is C of omega is only sizable, only contributing, 01:10:36.850 --> 01:10:41.570 around omega zero, therefore I can do approximation form 01:10:41.570 --> 01:10:46.520 for the K function, which is this functional form. 01:10:46.520 --> 01:10:48.770 Then I just do the integration. 01:10:48.770 --> 01:10:50.690 Then I found that, interesting! 01:10:55.450 --> 01:10:58.020 This side is-- you should be taking 01:10:58.020 --> 01:11:01.130 the real part of this thing. 01:11:01.130 --> 01:11:03.860 This side have two components. 01:11:03.860 --> 01:11:06.770 The first component is fs, which is 01:11:06.770 --> 01:11:12.630 the original signal you put in, the signal you want to send. 01:11:12.630 --> 01:11:18.080 It's actually progressing at the speed of group velocity. 01:11:18.080 --> 01:11:22.440 So now you understand what this group velocity means. 01:11:22.440 --> 01:11:25.190 That's the speed of the signal you 01:11:25.190 --> 01:11:29.510 want to send in the AM radio. 01:11:29.510 --> 01:11:32.460 And this thing is actually modulated 01:11:32.460 --> 01:11:36.270 by exponential function, which is actually 01:11:36.270 --> 01:11:44.720 the propagating at the speed of Vp, equal to omega0 over K0. 01:11:44.720 --> 01:11:48.620 So the carrier still, after you actually 01:11:48.620 --> 01:11:53.750 include many, many terms contracting 01:11:53.750 --> 01:11:59.900 the f function, the sine which is the amplitude, 01:11:59.900 --> 01:12:05.162 the trick is that only the omega value around omega0 01:12:05.162 --> 01:12:06.460 contributes. 01:12:06.460 --> 01:12:10.550 If that happen, then you can see that there 01:12:10.550 --> 01:12:13.610 are two structures actually propagating 01:12:13.610 --> 01:12:16.520 at different speeds, and that you can actually understand 01:12:16.520 --> 01:12:20.620 the structure independently. 01:12:20.620 --> 01:12:25.610 That means your signal will not be distorted 01:12:25.610 --> 01:12:27.620 if you're sending it this way. 01:12:27.620 --> 01:12:30.020 But the difference is that the speed 01:12:30.020 --> 01:12:34.350 of the signal you are sending is actually 01:12:34.350 --> 01:12:37.860 at the speed of group velocity. 01:12:37.860 --> 01:12:40.250 That is actually the amazing fact 01:12:40.250 --> 01:12:44.230 which actually enables us to send signal 01:12:44.230 --> 01:12:49.640 over thousands and thousands of miles away from the source. 01:12:49.640 --> 01:12:53.470 So what is actually done is actually that, 01:12:53.470 --> 01:12:56.810 suppose you have some kind of radio station. 01:12:56.810 --> 01:13:01.860 You can send the radio, and the radio will go over the place, 01:13:01.860 --> 01:13:05.450 and got refracted by atmosphere-- 01:13:05.450 --> 01:13:08.490 the atmosphere on Earth. 01:13:08.490 --> 01:13:13.350 Got refracted, and the receiver from some place which is really 01:13:13.350 --> 01:13:15.710 distant from the source can still 01:13:15.710 --> 01:13:21.170 see it without any dispersion, as we show here. 01:13:21.170 --> 01:13:23.330 And it's actually going to be propagating 01:13:23.330 --> 01:13:27.870 at the speed of group velocity. 01:13:27.870 --> 01:13:33.620 So you may not actually believe that. 01:13:33.620 --> 01:13:37.360 How about we do a simulation like what 01:13:37.360 --> 01:13:39.950 we did before with MIT wave? 01:13:39.950 --> 01:13:43.730 So this is actually the example which we did last time. 01:13:43.730 --> 01:13:45.450 We have an nit wave. 01:13:45.450 --> 01:13:48.950 We can compose that into many, many pieces. 01:13:48.950 --> 01:13:52.730 And then see how it evolved as a function of time. 01:13:52.730 --> 01:13:54.590 This is actually without dispersion, 01:13:54.590 --> 01:13:57.770 therefore everything is perfect. 01:13:57.770 --> 01:14:00.760 So now I would like to introduce some excitement there. 01:14:03.820 --> 01:14:10.440 If I have dispersion, like 0.1, alpha is equal to 0.1, 01:14:10.440 --> 01:14:11.860 and see will happen. 01:14:11.860 --> 01:14:16.690 Then just a reminder that things will not go super well. 01:14:16.690 --> 01:14:19.140 Wait a second, what am I doing? 01:14:19.140 --> 01:14:23.080 This is actually still without dispersion. 01:14:23.080 --> 01:14:24.230 Sorry for that. 01:14:28.840 --> 01:14:31.730 It should be-- 01:14:31.730 --> 01:14:35.180 OK, so let's take a look at the triangular case. 01:14:35.180 --> 01:14:39.550 This is now with dispersion. 01:14:39.550 --> 01:14:43.390 And you can see that as a reminder as a function of time, 01:14:43.390 --> 01:14:46.190 the shape of the signal which you would like to send 01:14:46.190 --> 01:14:49.160 is actually changing as a function of time. 01:14:49.160 --> 01:14:53.020 And after a few thousands of miles, 01:14:53.020 --> 01:14:57.310 you will not even recognize the original structure we put in. 01:14:57.310 --> 01:15:03.600 So that's the trouble we are actually facing. 01:15:03.600 --> 01:15:07.790 You can see that it's getting wider and wider et cetera. 01:15:07.790 --> 01:15:18.250 So now what will happen if I send this kind of signal. 01:15:18.250 --> 01:15:20.820 This is a signal which you have some kind of shape. 01:15:20.820 --> 01:15:23.880 You can imagine that there's sounds kind of analogue. 01:15:23.880 --> 01:15:27.540 And I am now doing the calculation to actually map 01:15:27.540 --> 01:15:29.870 all the individual components. 01:15:29.870 --> 01:15:32.820 And now I'm going to propagate through the median. 01:15:32.820 --> 01:15:39.060 And the blue is the original non-dispersive median 01:15:39.060 --> 01:15:40.350 situation. 01:15:40.350 --> 01:15:44.360 And the red is actually the propagation 01:15:44.360 --> 01:15:45.910 in a dispersive median. 01:15:45.910 --> 01:15:48.770 You can see the propagation in a dispersive median 01:15:48.770 --> 01:15:53.160 is faster, because alpha is actually larger than one-- 01:15:53.160 --> 01:15:54.270 larger than zero. 01:15:54.270 --> 01:15:56.310 So it's actually, in this case it's 0.1. 01:15:56.310 --> 01:16:03.975 And you can see that the red cosine omega0t modulated signal 01:16:03.975 --> 01:16:12.480 is progressing, and the shape of the analogue is not changing. 01:16:12.480 --> 01:16:13.770 You can see that, right? 01:16:13.770 --> 01:16:17.380 So it's very different from what we actually see before 01:16:17.380 --> 01:16:20.130 with a single triangular pulse. 01:16:20.130 --> 01:16:21.420 Now you can see that, ha! 01:16:21.420 --> 01:16:24.360 Only when n gets very large, I start 01:16:24.360 --> 01:16:27.150 to be able to feed all those little structures. 01:16:27.150 --> 01:16:31.740 That means the end value, or say the omega value, which I need 01:16:31.740 --> 01:16:34.429 is you really narrow, a very narrow range, 01:16:34.429 --> 01:16:36.720 which will actually match with what we have been doing. 01:16:36.720 --> 01:16:39.460 And now I start to propagate all those things. 01:16:39.460 --> 01:16:43.530 And you can see that the red is actually traveling faster 01:16:43.530 --> 01:16:46.470 than the blue, which is what we expect. 01:16:46.470 --> 01:16:48.730 And you can see now, in the instance 01:16:48.730 --> 01:16:51.120 they actually overlap each other, 01:16:51.120 --> 01:16:54.420 you can see that envelope, the shape of the envelope, 01:16:54.420 --> 01:16:55.230 is still the same. 01:16:55.230 --> 01:16:57.970 It's exactly what we actually printed. 01:16:57.970 --> 01:17:01.800 And that actually brings me to the end of my lecture. 01:17:01.800 --> 01:17:06.960 We have on understood how the AM radio actually works. 01:17:06.960 --> 01:17:08.820 And next time, we are going to talk 01:17:08.820 --> 01:17:11.170 about uncertainty principles. 01:17:11.170 --> 01:17:12.150 What the hell? 01:17:12.150 --> 01:17:14.520 What happened? 01:17:14.520 --> 01:17:17.460 And believe me, they are actually 01:17:17.460 --> 01:17:19.272 connected to each other. 01:17:19.272 --> 01:17:21.970 Uncertainty principle is actually highly related 01:17:21.970 --> 01:17:24.640 to wave and the vibrations. 01:17:24.640 --> 01:17:29.510 Thank you very much, and let me know if you have any questions.