1 00:00:02,450 --> 00:00:04,820 The following content is provided under a Creative 2 00:00:04,820 --> 00:00:06,210 Commons license. 3 00:00:06,210 --> 00:00:08,420 Your support will help MIT OpenCourseWare 4 00:00:08,420 --> 00:00:12,510 continue to offer high quality educational resources for free. 5 00:00:12,510 --> 00:00:15,050 To make a donation, or to view additional materials 6 00:00:15,050 --> 00:00:19,010 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,010 --> 00:00:20,316 at ocw.mit.edu. 8 00:00:24,390 --> 00:00:27,660 YEN-JIE LEE: OK, so welcome back to 803. 9 00:00:27,660 --> 00:00:30,280 Happy to see you again. 10 00:00:30,280 --> 00:00:32,564 So today, we are going to continue our discussion 11 00:00:32,564 --> 00:00:34,770 of dispersive medium. 12 00:00:34,770 --> 00:00:36,900 And there are two questions we are 13 00:00:36,900 --> 00:00:39,420 going to ask in this lecture, and we 14 00:00:39,420 --> 00:00:44,190 will answer also these two questions in this lecture. 15 00:00:44,190 --> 00:00:47,440 So just to warn you in advance, this lecture 16 00:00:47,440 --> 00:00:53,640 will have a lot of mathematics, so fasten the seatbelt 17 00:00:53,640 --> 00:00:57,380 and follow me. 18 00:00:57,380 --> 00:01:01,320 And stop me any time you don't feel like you 19 00:01:01,320 --> 00:01:04,560 know you understand something. 20 00:01:04,560 --> 00:01:06,840 So let's get started. 21 00:01:06,840 --> 00:01:11,790 OK, so today, we are going to talk about phenomena 22 00:01:11,790 --> 00:01:13,890 related to dispersion. 23 00:01:13,890 --> 00:01:16,740 And last time, we started a discussion 24 00:01:16,740 --> 00:01:19,980 about how to send information from one 25 00:01:19,980 --> 00:01:21,600 place to the other place right. 26 00:01:21,600 --> 00:01:26,550 So what we should be before was to send square pulse. 27 00:01:26,550 --> 00:01:33,210 So if I do have a machine which can produce a square pulse, 28 00:01:33,210 --> 00:01:36,770 then I can define something like this. 29 00:01:36,770 --> 00:01:40,350 So over some ratio, which I set, I 30 00:01:40,350 --> 00:01:43,470 can actually separate 0 and 1. 31 00:01:43,470 --> 00:01:45,700 So if I have a pulse which is actually 32 00:01:45,700 --> 00:01:49,110 having an amplitude greater than some threshold and I say, 33 00:01:49,110 --> 00:01:52,120 OK, I've got 1, and if it's actually 34 00:01:52,120 --> 00:01:55,230 below some straight line, say, OK, I've got a 0. 35 00:01:55,230 --> 00:01:58,350 And with that way, we actually can send information 36 00:01:58,350 --> 00:02:00,570 from one place to the other place. 37 00:02:00,570 --> 00:02:03,770 So that sounds really nice. 38 00:02:03,770 --> 00:02:09,130 However, if we work on a dispersive medium, 39 00:02:09,130 --> 00:02:11,580 which is really very common-- 40 00:02:11,580 --> 00:02:15,980 for example, light and gas is actually-- 41 00:02:15,980 --> 00:02:21,810 not all the lights with different wavelengths are 42 00:02:21,810 --> 00:02:26,340 traveling at the same speed, and also, 43 00:02:26,340 --> 00:02:30,906 as you've seen before in the p-set, deep water, 44 00:02:30,906 --> 00:02:36,765 and also the strings, considering a realistic string 45 00:02:36,765 --> 00:02:40,380 with stiffness, et cetera, et cetera-- 46 00:02:40,380 --> 00:02:47,490 to the wavelength of the input wave is going to affect 47 00:02:47,490 --> 00:02:50,340 the speed of this travelling wave. 48 00:02:50,340 --> 00:02:54,540 So in short, the speed of the wave propagation 49 00:02:54,540 --> 00:02:59,230 in a dispersive medium will depend on the wavelengths 50 00:02:59,230 --> 00:03:01,410 of this wave. 51 00:03:01,410 --> 00:03:03,780 So that brings us a lot of trouble 52 00:03:03,780 --> 00:03:08,310 because, for example, here we are trying to send a Gaussian 53 00:03:08,310 --> 00:03:13,260 pulse through the medium, but after a while 54 00:03:13,260 --> 00:03:17,130 this pulse actually becomes wider and wider because 55 00:03:17,130 --> 00:03:18,240 of the dispersion. 56 00:03:18,240 --> 00:03:21,810 Because all the components with different wavelengths 57 00:03:21,810 --> 00:03:26,140 which actually construct this narrow pulse, 58 00:03:26,140 --> 00:03:29,920 actually are traveling at different speeds. 59 00:03:29,920 --> 00:03:32,670 Therefore, if you wait long enough, 60 00:03:32,670 --> 00:03:40,300 all the different frequencies, or all the different frequency 61 00:03:40,300 --> 00:03:43,290 harmonic waves are travel at different speeds, 62 00:03:43,290 --> 00:03:46,080 therefore you get the, dispersion, 63 00:03:46,080 --> 00:03:51,300 which results in a much wider pulse in the end. 64 00:03:51,300 --> 00:03:52,950 And at some point, this pulse is going 65 00:03:52,950 --> 00:03:56,670 to be really wide, such that it's actually 66 00:03:56,670 --> 00:04:02,460 going to be very difficult to separate 0 from 1. 67 00:04:02,460 --> 00:04:04,110 So that's the problem. 68 00:04:04,110 --> 00:04:07,260 And we also did some simulations with computer. 69 00:04:07,260 --> 00:04:12,860 We do see this behavior also in our computer simulation. 70 00:04:12,860 --> 00:04:16,839 If I put in triangular pulse and allow it to evolve, 71 00:04:16,839 --> 00:04:19,680 and like what we did before, we assume 72 00:04:19,680 --> 00:04:24,070 that there's a stiffness in this string system. 73 00:04:24,070 --> 00:04:27,510 And you will see that, OK, as a function of time, 74 00:04:27,510 --> 00:04:31,290 this part is now longer a triangular shape, 75 00:04:31,290 --> 00:04:35,170 but you have a very complicated structure. 76 00:04:35,170 --> 00:04:37,710 So that is actually a problem we are 77 00:04:37,710 --> 00:04:40,920 going to try to solve today. 78 00:04:40,920 --> 00:04:44,250 And during that discussion last time, in the lecture, 79 00:04:44,250 --> 00:04:49,360 we also introduced dispersion relation omega k and also tried 80 00:04:49,360 --> 00:04:55,920 to overlap two travelling waves with similar wavelengths. 81 00:04:55,920 --> 00:04:59,520 And that would give you beat phenomenon. 82 00:04:59,520 --> 00:05:02,852 That probably doesn't surprise you any more. 83 00:05:02,852 --> 00:05:05,820 As you can see from this example, 84 00:05:05,820 --> 00:05:07,770 you have the beat phenomenon, and you 85 00:05:07,770 --> 00:05:12,640 can see the amplitude is actually variating slowly, 86 00:05:12,640 --> 00:05:15,210 the function of position. 87 00:05:15,210 --> 00:05:18,400 And if you follow the red point, which is actually 88 00:05:18,400 --> 00:05:23,220 associated with one of the peak, in the structure called 89 00:05:23,220 --> 00:05:26,690 carrier, OK, it's actually moving 90 00:05:26,690 --> 00:05:32,390 at the phase velocity, the P, which we introduced last time. 91 00:05:32,390 --> 00:05:37,540 The formula for BP, which is actually the speed of harmonic, 92 00:05:37,540 --> 00:05:42,935 oscillating travelling wave is actually defined as vp equal 93 00:05:42,935 --> 00:05:48,350 to omega over k, and the green point, 94 00:05:48,350 --> 00:05:53,540 which actually always at the minima of the distribution, 95 00:05:53,540 --> 00:05:58,730 which is actually associated with the speed of the envelope. 96 00:05:58,730 --> 00:06:01,520 You can see that, indeed, it actually 97 00:06:01,520 --> 00:06:04,310 can move at different speeds. 98 00:06:04,310 --> 00:06:07,460 It depends on the dispersion relation omega 99 00:06:07,460 --> 00:06:10,250 as a function of k you have in this system. 100 00:06:10,250 --> 00:06:16,100 And we call the speed of these envelope, 101 00:06:16,100 --> 00:06:20,120 which we construct from these two overlapping travelling 102 00:06:20,120 --> 00:06:21,630 waves to be-- 103 00:06:21,630 --> 00:06:25,130 we call it group velocity. 104 00:06:25,130 --> 00:06:27,130 And the definition of group velocity 105 00:06:27,130 --> 00:06:31,310 is vg equal to d omega dk. 106 00:06:31,310 --> 00:06:35,480 So that's what we have learned last time. 107 00:06:35,480 --> 00:06:41,720 OK, you may ask, OK, what do I mean by group velocity? 108 00:06:41,720 --> 00:06:45,500 And can I use it beyond what we have 109 00:06:45,500 --> 00:06:48,830 done for the beat phenomena. 110 00:06:48,830 --> 00:06:52,240 But what do I mean by group velocity? 111 00:06:52,240 --> 00:06:54,270 Is that really useful, and it's actually 112 00:06:54,270 --> 00:06:57,650 which part of the structure I was talking about. 113 00:06:57,650 --> 00:07:02,780 So in that case of two overlapping progressing waves 114 00:07:02,780 --> 00:07:06,410 with similar length or similar frequency, 115 00:07:06,410 --> 00:07:08,360 when we see that the group velocity actually 116 00:07:08,360 --> 00:07:12,080 present the speed of the envelope, right? 117 00:07:12,080 --> 00:07:14,070 Can we actually learn something more 118 00:07:14,070 --> 00:07:17,570 general about group velocity? 119 00:07:17,570 --> 00:07:19,430 The second question which we are asking 120 00:07:19,430 --> 00:07:24,800 is, OK, now we have this problem of dispersion. 121 00:07:24,800 --> 00:07:29,690 This square pulse is going to be something which is really 122 00:07:29,690 --> 00:07:33,070 wide after some period of time. 123 00:07:33,070 --> 00:07:37,820 So that's clearly a problem, and how do we actually 124 00:07:37,820 --> 00:07:40,650 solve this problem, and how do we actually send information 125 00:07:40,650 --> 00:07:45,620 like, for example, music over a large distance 126 00:07:45,620 --> 00:07:47,660 from one place to another place. 127 00:07:47,660 --> 00:07:49,610 So that's essentially what we're going 128 00:07:49,610 --> 00:07:53,360 to try to understand today. 129 00:07:53,360 --> 00:07:59,150 So let's start with an infinitely long string. 130 00:08:02,000 --> 00:08:05,780 And this string is actually very long, and this began from here, 131 00:08:05,780 --> 00:08:10,220 and it goes to some place which is really, really far away. 132 00:08:10,220 --> 00:08:12,920 And, of course, as usually, I can actually 133 00:08:12,920 --> 00:08:17,580 hold one end of this string and shake it a bit, 134 00:08:17,580 --> 00:08:21,420 then I can actually create some kind of pulse 135 00:08:21,420 --> 00:08:25,420 which is going to travel along this string 136 00:08:25,420 --> 00:08:28,340 towards a positive x direction. 137 00:08:28,340 --> 00:08:31,150 In this case, I defined the x coordinate 138 00:08:31,150 --> 00:08:33,360 will be pointing to a right hand side, and thus 139 00:08:33,360 --> 00:08:35,450 the positive direction. 140 00:08:35,450 --> 00:08:38,690 So of course I can hold this string, and I just shake it, 141 00:08:38,690 --> 00:08:43,700 and I would prepare a pulse on this medium, which is 142 00:08:43,700 --> 00:08:46,520 a string with constant tension. 143 00:08:46,520 --> 00:08:50,750 So I can describe the motion-- 144 00:08:50,750 --> 00:08:55,060 you can describe the motion of Yen-Jie's hand by a function. 145 00:08:55,060 --> 00:09:00,210 So you can say, OK, Yen-Jie is somehow doing a really nice job 146 00:09:00,210 --> 00:09:05,510 and oscillating at constant frequency. 147 00:09:05,510 --> 00:09:09,620 Like I can say, OK, Yen-Jie is shaking this thing 148 00:09:09,620 --> 00:09:12,120 to produce a harmonic wave, for example. 149 00:09:12,120 --> 00:09:15,890 And that, I can actually describe the motion of the hand 150 00:09:15,890 --> 00:09:17,440 by f of t. 151 00:09:17,440 --> 00:09:19,400 That's very good. 152 00:09:19,400 --> 00:09:24,690 And from what we have learned in the last lecture, 153 00:09:24,690 --> 00:09:27,725 we've found that, basically, waves, 154 00:09:27,725 --> 00:09:31,850 harmonic waves with different frequency, 155 00:09:31,850 --> 00:09:33,670 or with different wavelengths, are 156 00:09:33,670 --> 00:09:36,950 traveling at different speeds. 157 00:09:36,950 --> 00:09:43,420 Therefore, we would like to actually decompose the motion 158 00:09:43,420 --> 00:09:47,780 of Yen-Jie's hand into many, many harmonic waves-- 159 00:09:47,780 --> 00:09:51,740 then attack them one by one, to follow them one by one, 160 00:09:51,740 --> 00:09:54,020 then I can solve this problem. 161 00:09:54,020 --> 00:09:56,540 So that's actually what we are going to do. 162 00:09:56,540 --> 00:10:01,400 And that will involve some math, which we would follow 163 00:10:01,400 --> 00:10:03,830 from the math department. 164 00:10:03,830 --> 00:10:07,430 And before that, I would like to introduce the imitation first. 165 00:10:07,430 --> 00:10:10,970 As I said f of t is actually the displacement 166 00:10:10,970 --> 00:10:14,540 as a function of time as x is equal to 0. 167 00:10:14,540 --> 00:10:16,504 So basically, I'm holding this string, 168 00:10:16,504 --> 00:10:19,820 and I move things up and down, so that, actually, I 169 00:10:19,820 --> 00:10:23,490 move this string away from the equilibrium positive, which 170 00:10:23,490 --> 00:10:27,230 is actually y equal to 0. 171 00:10:27,230 --> 00:10:28,690 Then what is going to happen? 172 00:10:28,690 --> 00:10:30,230 What is going to happen is that I'm 173 00:10:30,230 --> 00:10:35,270 going to produce some kind of pulse, 174 00:10:35,270 --> 00:10:38,030 and this pulse, I can actually describe it 175 00:10:38,030 --> 00:10:42,470 by a function, which is psi x and p, 176 00:10:42,470 --> 00:10:46,460 this psi is actually describing the displacement 177 00:10:46,460 --> 00:10:50,030 as a function of x, and as a function of t. 178 00:10:50,030 --> 00:10:52,940 Apparently, if you put x equals to zero, 179 00:10:52,940 --> 00:10:56,220 then you go back to f of t, right? 180 00:10:56,220 --> 00:10:58,640 Basically that's the idea. 181 00:10:58,640 --> 00:11:00,410 OK. 182 00:11:00,410 --> 00:11:06,770 So what we have learned before we introduce 183 00:11:06,770 --> 00:11:20,890 dispersive medium is that, if I have a non-dispersive medium, 184 00:11:20,890 --> 00:11:24,080 OK, if I have a non-dispersive medium, 185 00:11:24,080 --> 00:11:29,300 then things are pretty simple because omega over K 186 00:11:29,300 --> 00:11:33,870 is actually a constant, which is the phase velocity, vp. 187 00:11:33,870 --> 00:11:39,355 And omega is actually just equal to vp times k. 188 00:11:42,610 --> 00:11:46,390 That means, no matter what kind of wavelength 189 00:11:46,390 --> 00:11:49,600 we are talking about, no matter what kind of angular frequency 190 00:11:49,600 --> 00:11:55,030 we are talking about, harmonic progressing wave 191 00:11:55,030 --> 00:11:58,900 is going to travel at the speed of Vp. 192 00:11:58,900 --> 00:12:02,590 No matter what's the frequency, or what's the wavelength. 193 00:12:02,590 --> 00:12:05,140 So that makes our life much simpler 194 00:12:05,140 --> 00:12:07,870 when we work on non-dispersive medium. 195 00:12:07,870 --> 00:12:11,820 In this case, if I have a non-dispersive medium, 196 00:12:11,820 --> 00:12:15,670 then psi would be equal to-- 197 00:12:15,670 --> 00:12:18,730 maybe I write it here-- 198 00:12:18,730 --> 00:12:21,670 if I have non-dispersive medium where, 199 00:12:21,670 --> 00:12:25,090 no matter what kind of frequency, 200 00:12:25,090 --> 00:12:27,690 the speed of the harmonic traveling wave 201 00:12:27,690 --> 00:12:31,210 is a constant, which is actually Vp, I can write down psi 202 00:12:31,210 --> 00:12:43,420 x t to could be equal to f of t minus x over v. 203 00:12:43,420 --> 00:12:45,640 Just remember f is actually describing 204 00:12:45,640 --> 00:12:49,240 how I shake one end of the string, 205 00:12:49,240 --> 00:12:51,850 and, basically you can see that ha! 206 00:12:51,850 --> 00:12:54,800 What is happening is that my hand is actually 207 00:12:54,800 --> 00:13:00,380 generating the shape of the pulse as a function of x, 208 00:13:00,380 --> 00:13:02,650 as a function of time, and it can 209 00:13:02,650 --> 00:13:08,410 be described by a really simple formula here. 210 00:13:08,410 --> 00:13:11,680 So this is actually really nice for non-dispersive. 211 00:13:11,680 --> 00:13:17,050 As I introduced before, when we talk about dispersive medium, 212 00:13:17,050 --> 00:13:27,970 then, if I go to dispersive, omega 213 00:13:27,970 --> 00:13:34,840 is actually a function of k, and can be a non-linear function. 214 00:13:34,840 --> 00:13:36,060 So what does that mean? 215 00:13:36,060 --> 00:13:41,020 That means, if I evaluate vp, which is actually 216 00:13:41,020 --> 00:13:45,100 the phase velocity, which is the formula there, 217 00:13:45,100 --> 00:13:51,800 this is going to be omega of k divided by k. 218 00:13:51,800 --> 00:13:57,010 That means BP is going to be a function of k, 219 00:13:57,010 --> 00:13:59,220 the wavelength-- wave number. 220 00:13:59,220 --> 00:14:02,390 It's not going to be a constant in general-- 221 00:14:02,390 --> 00:14:07,680 unless omega is actually equal to vp times k, 222 00:14:07,680 --> 00:14:13,300 in general, vp can actually be some quantity 223 00:14:13,300 --> 00:14:15,900 which is variating as a function of k. 224 00:14:15,900 --> 00:14:16,540 OK? 225 00:14:16,540 --> 00:14:20,770 Then we have trouble because that means, 226 00:14:20,770 --> 00:14:24,880 when I produce progressing wave from the left hand side end, 227 00:14:24,880 --> 00:14:29,560 it's actually made of many, many harmonic waves, 228 00:14:29,560 --> 00:14:32,410 right, with different angular frequency. 229 00:14:32,410 --> 00:14:37,240 So I can shake this like [MAKES NOISE],, different speed. 230 00:14:37,240 --> 00:14:41,740 And I can always decompose the motion of Yen-Jie's hand 231 00:14:41,740 --> 00:14:44,230 into many, many harmonic waves. 232 00:14:44,230 --> 00:14:49,150 The problem is, all those harmonic waves are going to be 233 00:14:49,150 --> 00:14:52,690 travelling at different speed. 234 00:14:52,690 --> 00:14:56,190 How do we actually describe this? 235 00:14:56,190 --> 00:14:58,110 So that's the trouble. 236 00:14:58,110 --> 00:15:02,647 And I was really frustrated when I think about this problem, 237 00:15:02,647 --> 00:15:04,230 and my friend from the math department 238 00:15:04,230 --> 00:15:07,440 said, hey, we have solved this problem a long time ago. 239 00:15:07,440 --> 00:15:10,320 [LAUGHTER] 240 00:15:10,320 --> 00:15:11,820 So this is not the problem anymore. 241 00:15:11,820 --> 00:15:16,360 And I say, oh, what is the idea you're talking about? 242 00:15:16,360 --> 00:15:18,600 And they actually told me that you 243 00:15:18,600 --> 00:15:22,960 should use Fourier transform to attack this problem. 244 00:15:22,960 --> 00:15:23,990 OK? 245 00:15:23,990 --> 00:15:25,840 This is the idea. 246 00:15:25,840 --> 00:15:30,180 The idea is that I can now write down 247 00:15:30,180 --> 00:15:35,610 f of t, which is motion of Yen-Jie's hand, 248 00:15:35,610 --> 00:15:41,550 and this can even returned as a superposition 249 00:15:41,550 --> 00:15:46,890 of infinite number of waves. 250 00:15:46,890 --> 00:15:49,350 I can integrate from minus infinity 251 00:15:49,350 --> 00:15:55,270 to infinity, t omega, which is the angular frequency. 252 00:15:55,270 --> 00:16:03,450 And each contributing wave has an amplitutde associated 253 00:16:03,450 --> 00:16:08,340 with it, which is, as you see, is a function omega. 254 00:16:08,340 --> 00:16:11,900 And the actual wave is actually written in terms 255 00:16:11,900 --> 00:16:16,900 of exponential minus i omega t. 256 00:16:16,900 --> 00:16:21,750 So now, let's actually you look at this thing really carefully. 257 00:16:21,750 --> 00:16:23,340 What am I doing? 258 00:16:23,340 --> 00:16:29,670 I am saying that I, now, can shake one end of the string up 259 00:16:29,670 --> 00:16:33,950 and down according to my will. 260 00:16:33,950 --> 00:16:36,840 And, if I do this for a long time, 261 00:16:36,840 --> 00:16:41,640 I can actually describe the motion of my hand 262 00:16:41,640 --> 00:16:47,070 by infinite number of harmonic waves, which is actually 263 00:16:47,070 --> 00:16:50,160 kind of like exponential i omega t 264 00:16:50,160 --> 00:16:55,380 describing the frequency of these waves, and each of them 265 00:16:55,380 --> 00:17:00,020 got associated amplitude. 266 00:17:00,020 --> 00:17:02,230 And you may ask, OK, wait a second, 267 00:17:02,230 --> 00:17:07,240 you call this Fourier transform, and I have learned that before, 268 00:17:07,240 --> 00:17:09,060 but I learned a different version. 269 00:17:09,060 --> 00:17:12,450 I learned a version of cosine and sine? 270 00:17:12,450 --> 00:17:15,329 And what is going on? 271 00:17:15,329 --> 00:17:17,336 Actually, they are all the same. 272 00:17:17,336 --> 00:17:18,960 No matter what you do, you can actually 273 00:17:18,960 --> 00:17:21,780 also do that with cosine and sine, 274 00:17:21,780 --> 00:17:25,710 but what I actually found is that it's actually easier 275 00:17:25,710 --> 00:17:29,610 to deal with exponential functional form 276 00:17:29,610 --> 00:17:33,720 You can always write exponential i omega 277 00:17:33,720 --> 00:17:40,470 t in terms of sine and cosine and absorb the i 278 00:17:40,470 --> 00:17:42,890 into a c omega. 279 00:17:42,890 --> 00:17:45,520 Basically, these things are identical between these two 280 00:17:45,520 --> 00:17:47,130 forms of this answer. 281 00:17:47,130 --> 00:17:49,350 So therefore, in this lecture, I'm 282 00:17:49,350 --> 00:17:52,790 going to stick with this functional form. 283 00:17:52,790 --> 00:17:54,232 OK, any questions? 284 00:17:54,232 --> 00:17:59,900 STUDENT: We don't include dx in the [INAUDIBLE]?? 285 00:17:59,900 --> 00:18:01,000 YEN-JIE LEE: Not yet. 286 00:18:01,000 --> 00:18:02,590 We are going to include that. 287 00:18:02,590 --> 00:18:07,590 Because, for that, in order to actually-- 288 00:18:07,590 --> 00:18:11,250 OK, so now I actually decompose the motion of my hand 289 00:18:11,250 --> 00:18:14,020 into many, many waves-- 290 00:18:14,020 --> 00:18:17,020 which should be or is say it many, many oscillation 291 00:18:17,020 --> 00:18:19,860 with different frequencies. 292 00:18:19,860 --> 00:18:23,590 So I actually describe the motion 293 00:18:23,590 --> 00:18:26,500 of my hand infinite number of oscillation 294 00:18:26,500 --> 00:18:28,300 with different frequency. 295 00:18:28,300 --> 00:18:34,900 And the trouble we are facing is that all those oscillations are 296 00:18:34,900 --> 00:18:37,960 going to be charged travelling at different speeds 297 00:18:37,960 --> 00:18:42,160 because of the dispersion relation. 298 00:18:42,160 --> 00:18:45,760 Therefore, what I am going to do afterwards 299 00:18:45,760 --> 00:18:50,170 is to show you that, OK, I can write down the functional form 300 00:18:50,170 --> 00:18:54,130 for psi in this general case. 301 00:18:54,130 --> 00:18:58,100 So for that, that's actually what I'm going to do now. 302 00:18:58,100 --> 00:19:03,730 So now, I would like to know what would be the psi xt, which 303 00:19:03,730 --> 00:19:06,970 actually the position of the string as a function of x, 304 00:19:06,970 --> 00:19:11,020 and at some specific time equal to t. 305 00:19:11,020 --> 00:19:16,690 And that can be written as, I do the an integration from minus 306 00:19:16,690 --> 00:19:23,380 infinity to infinity over frequency omega, 307 00:19:23,380 --> 00:19:27,310 and I have the usual amplitude associated 308 00:19:27,310 --> 00:19:29,770 with the angular frequency omega, 309 00:19:29,770 --> 00:19:35,810 and the exponential i omega t-- 310 00:19:35,810 --> 00:19:38,890 minus i omega t because that's the convention I'm using here-- 311 00:19:38,890 --> 00:19:46,020 and I say, OK, plus ik, which is a function omega-- 312 00:19:46,020 --> 00:19:46,960 x. 313 00:19:46,960 --> 00:19:49,510 So now you can to see that what I'm 314 00:19:49,510 --> 00:19:53,250 doing here is that I am now progressing, 315 00:19:53,250 --> 00:19:56,680 I am making infinite number of progressing waves. 316 00:19:56,680 --> 00:19:59,320 Each of these exponential functions 317 00:19:59,320 --> 00:20:04,630 is a progressing wave with angular frequency omega. 318 00:20:04,630 --> 00:20:10,110 And why do I write k as a function of omega x here? 319 00:20:10,110 --> 00:20:12,572 It's because they are going to be travelling at the speed 320 00:20:12,572 --> 00:20:15,160 of omega over k. 321 00:20:15,160 --> 00:20:19,630 Therefore, I need to actually put k here, 322 00:20:19,630 --> 00:20:22,660 and this k is actually-- 323 00:20:22,660 --> 00:20:27,040 this k is actually not the independent parameter. 324 00:20:27,040 --> 00:20:29,800 It's actually a function of omega. 325 00:20:29,800 --> 00:20:33,730 So we can see that, here, we do an integration over omega 326 00:20:33,730 --> 00:20:36,960 from minus infinity to infinity-- 327 00:20:36,960 --> 00:20:42,040 for each omega you can actually find the corresponding k, 328 00:20:42,040 --> 00:20:42,760 right? 329 00:20:42,760 --> 00:20:45,370 Because of the dispersion relation. 330 00:20:45,370 --> 00:20:47,770 Because omega is a function of k, 331 00:20:47,770 --> 00:20:52,420 therefore you can always solve the corresponding k, right? 332 00:20:52,420 --> 00:20:54,130 Then you put it there? 333 00:20:54,130 --> 00:20:58,610 Because you are now trying to propagate 334 00:20:58,610 --> 00:21:01,660 how many waves with different angular 335 00:21:01,660 --> 00:21:04,130 frequency at different speed-- 336 00:21:04,130 --> 00:21:05,307 then we are done. 337 00:21:08,080 --> 00:21:10,690 That looks like a wonderful solution, 338 00:21:10,690 --> 00:21:17,150 and we can actually see how it works for our purpose. 339 00:21:17,150 --> 00:21:18,310 Any questions? 340 00:21:20,950 --> 00:21:21,850 All right. 341 00:21:21,850 --> 00:21:23,090 So that's really nice. 342 00:21:23,090 --> 00:21:26,650 And I can now do a really simple test 343 00:21:26,650 --> 00:21:29,640 to see if this really works. 344 00:21:29,640 --> 00:21:32,790 Let me try a very simple case. 345 00:21:32,790 --> 00:21:34,358 OK, a spatial case. 346 00:21:37,230 --> 00:21:41,970 If I now go back to use this description 347 00:21:41,970 --> 00:21:46,080 to describe non-dispersive medium and see what 348 00:21:46,080 --> 00:21:47,340 will happen. 349 00:21:47,340 --> 00:21:52,210 Now my k as a function omega is actually rather simple. 350 00:21:52,210 --> 00:21:53,740 It's actually omega over vp-- 351 00:21:58,200 --> 00:22:02,760 according to the dispersion relation here. 352 00:22:02,760 --> 00:22:05,820 I can solve k, as I was mentioning, 353 00:22:05,820 --> 00:22:08,610 with these dispersion relation formula. 354 00:22:08,610 --> 00:22:11,250 And then I can conclude k as a function 355 00:22:11,250 --> 00:22:15,920 omega is omega over vp. 356 00:22:15,920 --> 00:22:18,680 Then I can now put that into this equation, 357 00:22:18,680 --> 00:22:22,380 and I'm going to get psi x of t-- 358 00:22:22,380 --> 00:22:24,930 this would be equal to minus infinity 359 00:22:24,930 --> 00:22:33,270 to infinity d omega, c omega, exponential minus i omega 360 00:22:33,270 --> 00:22:37,140 t minus omega over vx. 361 00:22:41,640 --> 00:22:47,160 And we can actually take omega out of this, minus infinity 362 00:22:47,160 --> 00:22:54,060 to infinity d omega c omega exponential minus i omega 363 00:22:54,060 --> 00:23:02,850 t minus x divided by v. And you can see that, huh, indeed, 364 00:23:02,850 --> 00:23:09,990 this is actually ft minus x over v. OK. 365 00:23:09,990 --> 00:23:11,670 I'm dropping the vp here. 366 00:23:11,670 --> 00:23:16,030 This should be vp all over the place. 367 00:23:16,030 --> 00:23:21,450 So you can see that, now, if I have solved the k as a function 368 00:23:21,450 --> 00:23:25,380 omega, and I plug it in in this special case, 369 00:23:25,380 --> 00:23:31,590 which is non-dispersive medium, omega over k equal to vp, 370 00:23:31,590 --> 00:23:36,310 then I really calculate this integral, 371 00:23:36,310 --> 00:23:38,450 then I can quickly identify that-- 372 00:23:38,450 --> 00:23:43,750 huh, I can write the functional form in this way. 373 00:23:43,750 --> 00:23:48,300 And this is actually really familiar to me 374 00:23:48,300 --> 00:23:52,580 because that's actually using this definition, f is actually 375 00:23:52,580 --> 00:23:56,550 equal to integration minus infinity to infinity, d omega, 376 00:23:56,550 --> 00:23:59,670 c omega, exponential minus i omega t. 377 00:23:59,670 --> 00:24:06,280 If I replace t, by t minus x over vp, then I'm done. 378 00:24:06,280 --> 00:24:10,500 So I have evaluated this integration, which is actually 379 00:24:10,500 --> 00:24:14,670 just f t minus x over vp. 380 00:24:14,670 --> 00:24:19,740 So that's exactly what guessed from the beginning. 381 00:24:19,740 --> 00:24:22,170 So if I have a non-dispersive medium, 382 00:24:22,170 --> 00:24:26,490 then psi xt will be equal to this function. 383 00:24:26,490 --> 00:24:28,440 So that gives us some kind of confidence that, 384 00:24:28,440 --> 00:24:33,600 OK, at the easy case, it works. 385 00:24:33,600 --> 00:24:39,510 All right, so that's very nice, all sounds very good. 386 00:24:39,510 --> 00:24:41,190 But wait a second. 387 00:24:41,190 --> 00:24:43,470 How do I actually extract this c, 388 00:24:43,470 --> 00:24:45,690 which is a function of omega? 389 00:24:45,690 --> 00:24:50,580 I'm troubled because this is an infinite integral from 390 00:24:50,580 --> 00:24:52,480 minus infinity to infinity. 391 00:24:52,480 --> 00:24:55,560 And that means I have infinite number of constants, which I 392 00:24:55,560 --> 00:24:58,830 have to determine the c omega. 393 00:24:58,830 --> 00:24:59,970 How do I actually do this? 394 00:25:02,670 --> 00:25:06,180 So that is another point which I would 395 00:25:06,180 --> 00:25:09,180 like to discuss before we actually go ahead 396 00:25:09,180 --> 00:25:15,360 and really use this function for the dispersive medium case. 397 00:25:15,360 --> 00:25:19,680 So how to we actually extract c as a function of omega? 398 00:25:22,230 --> 00:25:27,630 So for that, we really need to employ a few uses 399 00:25:27,630 --> 00:25:31,860 for formula, which are actually documented here. 400 00:25:31,860 --> 00:25:36,030 How many of you actually have not heard about delta function 401 00:25:36,030 --> 00:25:37,520 before? 402 00:25:37,520 --> 00:25:41,560 OK, a few of you actually have not heard about delta function. 403 00:25:41,560 --> 00:25:44,730 So what is actually your delta function? 404 00:25:44,730 --> 00:25:47,910 This is a delta function. 405 00:25:47,910 --> 00:25:52,155 So a delta function is actually a notation 406 00:25:52,155 --> 00:25:58,280 which actually shows you a function, which 407 00:25:58,280 --> 00:26:03,180 is should only be non-zero, at x equal to zero. 408 00:26:03,180 --> 00:26:07,200 And the x equal to zero, the size of this function 409 00:26:07,200 --> 00:26:11,290 as you're going to infinity, and on the other hand, 410 00:26:11,290 --> 00:26:16,520 all the other points at x not equal to zero, 411 00:26:16,520 --> 00:26:20,980 the delta function is equal to zero. 412 00:26:20,980 --> 00:26:23,820 So that's actually the kind of function I was talking about. 413 00:26:23,820 --> 00:26:28,010 And the area of this function, if you're doing the equation 414 00:26:28,010 --> 00:26:32,040 over minus infinity infinity over x, 415 00:26:32,040 --> 00:26:38,220 the integration of these delta m the area is actually 1. 416 00:26:38,220 --> 00:26:40,420 So that is actually the kind of function. 417 00:26:40,420 --> 00:26:45,870 So essentially, it's a really, really narrow function, OK, 418 00:26:45,870 --> 00:26:50,040 very narrow, very narrow, very narrow. 419 00:26:50,040 --> 00:26:54,870 But the area is finite, which is why. 420 00:26:54,870 --> 00:26:57,870 So you can have a square. 421 00:26:57,870 --> 00:27:01,320 You can actually start with a square pulse, 422 00:27:01,320 --> 00:27:03,390 or square function, and you can actually 423 00:27:03,390 --> 00:27:07,670 make the width of the square narrower, smaller and smaller 424 00:27:07,670 --> 00:27:09,610 and smaller, go to 0. 425 00:27:09,610 --> 00:27:11,680 Then what you are going to get is essentially 426 00:27:11,680 --> 00:27:13,620 the delta function. 427 00:27:13,620 --> 00:27:18,990 That's actually how we understand this delta function. 428 00:27:18,990 --> 00:27:20,900 All right, really quickly. 429 00:27:20,900 --> 00:27:24,930 And also, we would like to use a few formula which 430 00:27:24,930 --> 00:27:26,800 are documented here. 431 00:27:26,800 --> 00:27:32,040 So if I do an integration from minus infinity to infinity, 432 00:27:32,040 --> 00:27:35,490 exponential i omega minus omega prime, 433 00:27:35,490 --> 00:27:40,830 t over the t which is integrating over time, t, here. 434 00:27:40,830 --> 00:27:45,770 And then divide the whole formula by 1 and over 2pi. 435 00:27:45,770 --> 00:27:49,030 What I'm going to get is a delta function, 436 00:27:49,030 --> 00:27:53,670 which is a delta function which is omega minus omega prime. 437 00:27:53,670 --> 00:27:58,590 So that means when this delta function formula tells us 438 00:27:58,590 --> 00:28:02,610 that omega is equal to omega prime, 439 00:28:02,610 --> 00:28:06,900 then this function is actually going to infinity. 440 00:28:06,900 --> 00:28:10,620 And only when omega equal to omega prime, 441 00:28:10,620 --> 00:28:12,700 this function is not zero. 442 00:28:12,700 --> 00:28:16,750 Any other place, this function is always zero. 443 00:28:19,290 --> 00:28:22,230 And this strange integration should give you 444 00:28:22,230 --> 00:28:23,530 this delta function. 445 00:28:23,530 --> 00:28:26,140 So that's the first thing which we will use, 446 00:28:26,140 --> 00:28:28,350 was one useful formula. 447 00:28:28,350 --> 00:28:32,130 The second thing which what just I talked about, 448 00:28:32,130 --> 00:28:35,560 if I do an integration over minus infinity to infinity, 449 00:28:35,560 --> 00:28:40,410 delta x dx, then basically you get 1. 450 00:28:40,410 --> 00:28:43,290 The third one is actually kind of interesting. 451 00:28:43,290 --> 00:28:44,400 Let's take a look. 452 00:28:44,400 --> 00:28:48,510 So if I do an integration over from minus infinity 453 00:28:48,510 --> 00:28:54,630 to infinity, delta function x minus alpha. 454 00:28:54,630 --> 00:28:57,060 Let's look at this delta function first. 455 00:28:57,060 --> 00:29:04,596 This function is only non-zero when x is equal to what? 456 00:29:04,596 --> 00:29:05,262 AUDIENCE: Alpha. 457 00:29:05,262 --> 00:29:06,012 YEN-JIE LEE: Yeah. 458 00:29:06,012 --> 00:29:10,310 When x is equal to alpha, only when that happen, 459 00:29:10,310 --> 00:29:12,410 this is actually non-zero. 460 00:29:12,410 --> 00:29:18,050 If you multiply this delta function to some function which 461 00:29:18,050 --> 00:29:22,610 is f of alpha, and integrate over alpha from minus 462 00:29:22,610 --> 00:29:25,040 infinity to infinity. 463 00:29:25,040 --> 00:29:27,620 And that means that when alpha is 464 00:29:27,620 --> 00:29:32,000 equal to x, or x equal to alpha, this integration 465 00:29:32,000 --> 00:29:35,750 give you non-zero result. All the other ways, 466 00:29:35,750 --> 00:29:38,420 you will get zero. 467 00:29:38,420 --> 00:29:41,470 The interesting thing is that if you do this integration, what 468 00:29:41,470 --> 00:29:47,316 you are going to get is that OK, when I integrate over alpha, 469 00:29:47,316 --> 00:29:51,260 only when alpha is equal to x this thing is non-zero. 470 00:29:51,260 --> 00:29:52,730 Therefore what you are going to get 471 00:29:52,730 --> 00:29:56,780 is, you get only one point of the width, which is actually 472 00:29:56,780 --> 00:29:59,180 f of x. 473 00:29:59,180 --> 00:30:02,282 So that's the intuition about this formula. 474 00:30:02,282 --> 00:30:02,990 That's just fine. 475 00:30:02,990 --> 00:30:05,731 Any questions related to those formulas? 476 00:30:05,731 --> 00:30:08,677 AUDIENCE: [INAUDIBLE]? 477 00:30:08,677 --> 00:30:10,150 YEN-JIE LEE: Hm? 478 00:30:10,150 --> 00:30:11,270 AUDIENCE: [INAUDIBLE]? 479 00:30:13,599 --> 00:30:15,890 YEN-JIE LEE: Yeah, this is actually pretty complicated, 480 00:30:15,890 --> 00:30:21,300 so it would take a few 10, 20 minutes to explain that. 481 00:30:21,300 --> 00:30:23,790 But let's just take the words from the math department-- 482 00:30:23,790 --> 00:30:24,802 we trust them. 483 00:30:28,100 --> 00:30:31,040 All right, so once I have those formula, 484 00:30:31,040 --> 00:30:35,610 I can now demonstrate you how I can actually 485 00:30:35,610 --> 00:30:38,750 track C as a function omega. 486 00:30:38,750 --> 00:30:41,320 So this is actually the goal, right? 487 00:30:41,320 --> 00:30:43,480 So don't forget why we are doing what we are doing, 488 00:30:43,480 --> 00:30:47,810 is to try to extract what is actually the C omega, 489 00:30:47,810 --> 00:30:54,410 so that we can actually finish this formula. 490 00:30:54,410 --> 00:30:56,450 So how do we do that? 491 00:30:56,450 --> 00:31:00,170 So suppose, if I evaluate this. 492 00:31:05,120 --> 00:31:12,910 This function, 1 over 2pi, minus infinity 493 00:31:12,910 --> 00:31:23,610 to infinity dt, ft, exponential i omega t. 494 00:31:23,610 --> 00:31:26,480 If I evaluate this function. 495 00:31:26,480 --> 00:31:28,860 This is coming out of nowhere, right? 496 00:31:28,860 --> 00:31:34,090 So coming out of Yen-Jie's hand, maybe, I don't know. 497 00:31:34,090 --> 00:31:38,490 Suppose if I evaluate this function, 498 00:31:38,490 --> 00:31:40,980 and now I have ft here, right? 499 00:31:40,980 --> 00:31:47,790 I can replace ft by these interesting formula. 500 00:31:47,790 --> 00:31:54,360 If I do that, then basically I get 1 over 2pi minus infinity 501 00:31:54,360 --> 00:32:04,010 to infinity dt, minus infinity infinity C omega 502 00:32:04,010 --> 00:32:10,600 prime, exponential minus i omega prime t. 503 00:32:10,600 --> 00:32:15,330 And the last is actually integrating over d omega prime. 504 00:32:17,880 --> 00:32:21,480 So this is actually the f of t. 505 00:32:21,480 --> 00:32:24,490 This is actually f of t. 506 00:32:24,490 --> 00:32:28,380 I'm just replacing that formula into this integral. 507 00:32:28,380 --> 00:32:33,140 And then I have the rest, which is exponential i omega t. 508 00:32:36,770 --> 00:32:39,660 And of course, I can continue and collect 509 00:32:39,660 --> 00:32:42,810 all the relevant terms together. 510 00:32:42,810 --> 00:32:51,930 This is actually equal to 1 over 2i, minus infinity to infinity. 511 00:32:51,930 --> 00:32:57,510 I collect all the terms related to omega prime to the left hand 512 00:32:57,510 --> 00:32:58,690 side. 513 00:32:58,690 --> 00:33:04,940 Basically what I get is C omega prime d omega prime. 514 00:33:04,940 --> 00:33:10,920 This is actually coming from here, except-- 515 00:33:10,920 --> 00:33:15,420 yeah, OK, it is actually coming from here. 516 00:33:15,420 --> 00:33:20,830 And I have another integral which is from minus infinity 517 00:33:20,830 --> 00:33:26,770 to infinity, this time integrating over delta dt. 518 00:33:26,770 --> 00:33:36,480 And I have dt here, exponential i omega minus omega prime t. 519 00:33:36,480 --> 00:33:44,650 So basically I'm collecting these two terms together. 520 00:33:44,650 --> 00:33:48,660 They now become exponential i omega minus omega prime t. 521 00:33:51,390 --> 00:33:55,980 So basically, no magic happened, but I'm just re-writing things 522 00:33:55,980 --> 00:34:01,680 and we are arranging things from this formula to that formula. 523 00:34:01,680 --> 00:34:08,130 Then if we look at this formula, this formula here, 524 00:34:08,130 --> 00:34:10,889 and the formula sheet we have. 525 00:34:10,889 --> 00:34:14,840 1 over 2pi minus infinity to infinity 526 00:34:14,840 --> 00:34:17,920 to this integration over t, exponential i omega 527 00:34:17,920 --> 00:34:19,940 minus omega prime t. 528 00:34:19,940 --> 00:34:22,510 That will give you delta function, 529 00:34:22,510 --> 00:34:25,130 which is delta omega minus omega prime. 530 00:34:27,929 --> 00:34:31,750 Therefore, I can continue this calculation here. 531 00:34:34,600 --> 00:34:45,800 Thus it's going to give you minus infinity to infinity. 532 00:34:45,800 --> 00:34:53,870 I identify this part, this part, and this part, 533 00:34:53,870 --> 00:34:55,440 to be the delta function. 534 00:34:58,330 --> 00:35:03,340 Therefore, what I get is minus infinity to infinity, C 535 00:35:03,340 --> 00:35:14,850 omega prime, delta omega minus omega prime, d omega prime. 536 00:35:14,850 --> 00:35:17,450 Am I going too fast? 537 00:35:17,450 --> 00:35:20,637 Everybody's following? 538 00:35:20,637 --> 00:35:22,470 So you can see that what we have been doing, 539 00:35:22,470 --> 00:35:26,490 I use this formula coming out of nowhere. 540 00:35:26,490 --> 00:35:31,980 I replace f by the formula I was writing there. 541 00:35:31,980 --> 00:35:34,920 And then I collect the terms I like together. 542 00:35:34,920 --> 00:35:36,910 That's all I did. 543 00:35:36,910 --> 00:35:38,790 And then I found, aha! 544 00:35:38,790 --> 00:35:44,240 One part of the formula is actually the delta function. 545 00:35:44,240 --> 00:35:46,420 Then I put the delta a function here. 546 00:35:46,420 --> 00:35:50,200 And then finally, I use the third formula here, 547 00:35:50,200 --> 00:35:54,230 which I have related to delta function, and I found, aha! 548 00:35:54,230 --> 00:35:56,250 If I do this integration, I know how 549 00:35:56,250 --> 00:35:58,740 to do this integration even without knowing 550 00:35:58,740 --> 00:36:02,610 the structure of C. This is actually just changing 551 00:36:02,610 --> 00:36:06,320 the omega prime to omega. 552 00:36:06,320 --> 00:36:10,650 So that's actually what this integration actually does. 553 00:36:10,650 --> 00:36:16,500 Therefore, I get C omega. 554 00:36:16,500 --> 00:36:20,190 Look at what we have done. 555 00:36:20,190 --> 00:36:23,350 What we have done is that, we have proof 556 00:36:23,350 --> 00:36:27,240 that this formula coming out of nowhere, 557 00:36:27,240 --> 00:36:32,080 to be a continuous version of mode picker. 558 00:36:32,080 --> 00:36:35,130 You remember the fourth year decomposition before? 559 00:36:35,130 --> 00:36:39,840 You were using the orthogonality of the sine function, 560 00:36:39,840 --> 00:36:42,840 and I can do some kind of fancy integration 561 00:36:42,840 --> 00:36:47,700 to actually extract a m from one of the-- 562 00:36:47,700 --> 00:36:50,440 which is associated with one of the normal mode, right? 563 00:36:50,440 --> 00:36:53,460 What we are doing here is actually a continuous version. 564 00:36:53,460 --> 00:36:55,740 Now omega is actually continuous. 565 00:36:55,740 --> 00:37:02,220 And I'm now using the orthogonality 566 00:37:02,220 --> 00:37:05,220 of the exponential function. 567 00:37:05,220 --> 00:37:08,130 If I do this integration, that will only 568 00:37:08,130 --> 00:37:14,020 give you non-zero value when omega is equal to omega prime. 569 00:37:14,020 --> 00:37:16,210 It's exactly the same thing, right? 570 00:37:16,210 --> 00:37:21,220 Then I can construct an integration like this. 571 00:37:21,220 --> 00:37:24,360 And now will give you the redoubting C 572 00:37:24,360 --> 00:37:29,142 as a function omega, which is like the amplitude of one 573 00:37:29,142 --> 00:37:34,470 of the associated harmonics exponential i omega t. 574 00:37:34,470 --> 00:37:37,240 So in short, from this exercise, we 575 00:37:37,240 --> 00:37:40,330 have shown you that C of omega can 576 00:37:40,330 --> 00:37:46,360 be extracted using this formula 1 over 2pi, minus infinity 577 00:37:46,360 --> 00:37:54,790 to infinity dt, f of t, exponential i omega t. 578 00:37:54,790 --> 00:37:58,660 That's actually how we actually can determine 579 00:37:58,660 --> 00:38:06,100 all the amplitude associated to a specific exponential 580 00:38:06,100 --> 00:38:07,000 function. 581 00:38:07,000 --> 00:38:08,040 Any questions so far? 582 00:38:11,280 --> 00:38:17,760 OK, so if no question, then we can actually continue. 583 00:38:17,760 --> 00:38:22,040 So let's actually go back to the original question, 584 00:38:22,040 --> 00:38:24,710 which we were posting. 585 00:38:24,710 --> 00:38:29,540 So we have a problem related to the transmission 586 00:38:29,540 --> 00:38:31,830 of information. 587 00:38:31,830 --> 00:38:35,570 So this is actually where we got started. 588 00:38:35,570 --> 00:38:42,320 If I send a square pulse on a dispersive median, 589 00:38:42,320 --> 00:38:45,170 then I have some trouble, which is 590 00:38:45,170 --> 00:38:49,190 that this pulse is going to disperse and become 591 00:38:49,190 --> 00:38:50,700 wider and wider. 592 00:38:50,700 --> 00:38:53,000 It's changing as a function of time, 593 00:38:53,000 --> 00:38:55,280 as a function of distance it travel. 594 00:38:55,280 --> 00:38:57,680 That's not cool. 595 00:38:57,680 --> 00:39:03,130 All right, so therefore what I am going to do is this. 596 00:39:03,130 --> 00:39:06,330 There was a very smart idea which 597 00:39:06,330 --> 00:39:10,970 were discovered long time ago, during maybe World War I, 598 00:39:10,970 --> 00:39:17,665 and widely used in World War II, which is the AM radio. 599 00:39:17,665 --> 00:39:18,640 What is AM? 600 00:39:18,640 --> 00:39:25,120 Is actually amplitude modulation radio. 601 00:39:25,120 --> 00:39:28,850 This smart idea is the following. 602 00:39:28,850 --> 00:39:32,840 I will describe it before we take a break. 603 00:39:32,840 --> 00:39:41,840 So this smart idea, AM radio is the following. 604 00:39:41,840 --> 00:39:47,270 If I have some kind of information which is fs t. 605 00:39:47,270 --> 00:39:50,180 s here means signal. 606 00:39:50,180 --> 00:39:51,890 If I have some kind of information 607 00:39:51,890 --> 00:39:53,870 I would like to send, I can send it 608 00:39:53,870 --> 00:39:59,300 by oscillating one end of the string. 609 00:39:59,300 --> 00:40:03,200 And this is what I want to send. 610 00:40:03,200 --> 00:40:08,430 And there are two ways you can send this fs function. 611 00:40:08,430 --> 00:40:14,050 The first one is actually what did before, I send it directly. 612 00:40:14,050 --> 00:40:16,940 I just said OK, if I want to send this function, 613 00:40:16,940 --> 00:40:19,850 then I just oscillate the string according 614 00:40:19,850 --> 00:40:22,180 to the functional form. 615 00:40:22,180 --> 00:40:25,130 Yen-Jie just have to be really careful, right? 616 00:40:25,130 --> 00:40:27,650 So that you can send this function. 617 00:40:27,650 --> 00:40:31,680 And that fails miserably. 618 00:40:31,680 --> 00:40:32,720 Why? 619 00:40:32,720 --> 00:40:38,580 Because all the components which actually produce the fs, 620 00:40:38,580 --> 00:40:43,690 in this case the square pulse, all those components are 621 00:40:43,690 --> 00:40:46,200 travelling at different speed. 622 00:40:46,200 --> 00:40:49,020 Therefore, the information will never get there, 623 00:40:49,020 --> 00:40:53,460 because of the dispersion. 624 00:40:53,460 --> 00:40:56,840 So now what should we do instead? 625 00:40:56,840 --> 00:41:02,440 Instead of doing sending fs as a function of t directly, 626 00:41:02,440 --> 00:41:05,040 what you could do is that I can now 627 00:41:05,040 --> 00:41:15,940 send f of t, which is equal to f of s t cosine omega0t. 628 00:41:19,630 --> 00:41:26,610 Where omega0 is a very, very large number. 629 00:41:26,610 --> 00:41:29,310 And basically, look at what we have been doing. 630 00:41:29,310 --> 00:41:34,350 So that means I, instead of sending fs directly, 631 00:41:34,350 --> 00:41:39,960 I send fs, but modulated by a really high frequency 632 00:41:39,960 --> 00:41:43,410 function, cosine omega0t. 633 00:41:43,410 --> 00:41:45,690 And this will work. 634 00:41:45,690 --> 00:41:48,480 And you will only know that after we come back 635 00:41:48,480 --> 00:41:53,040 from the break, which is twenty first. 636 00:41:53,040 --> 00:41:54,900 Let's take five minute break. 637 00:41:54,900 --> 00:41:58,380 And if you have any questions, you can actually ask me here. 638 00:42:06,000 --> 00:42:12,750 So we will continue the discussion about AM radio. 639 00:42:12,750 --> 00:42:16,890 So before the break, actually we introduced this one 640 00:42:16,890 --> 00:42:23,220 possible solution to solve this dispersive median problem, 641 00:42:23,220 --> 00:42:29,450 is that I can now actually send instead of fs as a function 642 00:42:29,450 --> 00:42:33,200 t, which is actually the signal I want to send, 643 00:42:33,200 --> 00:42:38,460 I could send fs, but multiplied by cosine omega0t. 644 00:42:41,960 --> 00:42:48,660 If I assume that fs is some really slow function, 645 00:42:48,660 --> 00:42:52,290 slowly varying as a function of time, 646 00:42:52,290 --> 00:42:55,870 compared to cosine omega0t. 647 00:42:55,870 --> 00:43:00,600 Cosine omega0t is a really fast function, 648 00:43:00,600 --> 00:43:03,930 oscillating up and down like crazy, really fast. 649 00:43:03,930 --> 00:43:08,190 If I multiply fs by this function, 650 00:43:08,190 --> 00:43:10,490 what is going to happen? 651 00:43:10,490 --> 00:43:13,030 We are going to show you that actually that 652 00:43:13,030 --> 00:43:19,410 means I am going to only have non-zero C 653 00:43:19,410 --> 00:43:23,150 function, or a large contribution of C, 654 00:43:23,150 --> 00:43:25,610 in a very thin middle range of omega. 655 00:43:28,440 --> 00:43:29,470 So we'll show that. 656 00:43:29,470 --> 00:43:35,700 So in a typical case, fs is really slow, which is like, 657 00:43:35,700 --> 00:43:38,620 for example, my sound, et cetera, 658 00:43:38,620 --> 00:43:41,460 in the label of one kilohertz. 659 00:43:41,460 --> 00:43:43,710 And you can actually design a system 660 00:43:43,710 --> 00:43:51,570 which will actually multiply this fs by cosine omega0t. 661 00:43:51,570 --> 00:43:57,960 Omega0 can be as fast as 1.1 to 30 megahertz. 662 00:43:57,960 --> 00:44:01,470 If you do this calculation, then you 663 00:44:01,470 --> 00:44:20,210 will find that OK, the range of omega, with sizable C omega 664 00:44:20,210 --> 00:44:22,500 is small. 665 00:44:26,690 --> 00:44:34,550 It's roughly equal to omega0 minus omega s, 666 00:44:34,550 --> 00:44:41,470 to omega0 plus omega s, where omega s is 667 00:44:41,470 --> 00:44:47,780 the typical frequency in your signal. 668 00:44:47,780 --> 00:44:52,210 And the omega0 is the typical frequency of-- 669 00:44:52,210 --> 00:44:55,820 the frequency of your cosine omega t 670 00:44:55,820 --> 00:44:58,120 term, which is actually, later, you 671 00:44:58,120 --> 00:44:59,765 will recognize this as carrier. 672 00:45:03,280 --> 00:45:07,080 So what I want to say is that if I do this trick, 673 00:45:07,080 --> 00:45:08,870 what is going to happen is that the range 674 00:45:08,870 --> 00:45:14,450 of omega, which you have sizable contribution from C omega-- 675 00:45:14,450 --> 00:45:17,670 C omega is the associated amplitude, 676 00:45:17,670 --> 00:45:20,090 associated amplitude. 677 00:45:20,090 --> 00:45:23,450 It's going to be confined to a really small region 678 00:45:23,450 --> 00:45:28,590 from omega0 minus omega s, to omega0 plus omega s. 679 00:45:28,590 --> 00:45:33,670 So that's the trick which actually makes this problem 680 00:45:33,670 --> 00:45:34,170 solvable. 681 00:45:36,710 --> 00:45:40,940 How do we know this? 682 00:45:40,940 --> 00:45:43,760 That is because, if I now, for example, I 683 00:45:43,760 --> 00:45:51,610 send fs equal to cosine omega st. 684 00:45:51,610 --> 00:45:54,710 If this is actually the signal which I would like to send, 685 00:45:54,710 --> 00:45:58,430 just a harmonic wave, then what is going to happen 686 00:45:58,430 --> 00:46:07,730 is that I'm going to get ft is equal to cosine omega st 687 00:46:07,730 --> 00:46:09,390 cosine-- 688 00:46:09,390 --> 00:46:13,520 so this is actually multiplied by cosine omega0t. 689 00:46:13,520 --> 00:46:17,960 So I have cosine omega0t here. 690 00:46:17,960 --> 00:46:21,800 I have cosine multiplied by cosine. 691 00:46:21,800 --> 00:46:24,620 Therefore I have the question which 692 00:46:24,620 --> 00:46:28,360 I prepare here, the formula, of cosine alpha times 693 00:46:28,360 --> 00:46:32,540 cosine beta will be equal to the functional form. 694 00:46:32,540 --> 00:46:35,240 There's a remainder, therefore I can now 695 00:46:35,240 --> 00:46:42,130 write it as 1 over 2 cosine omega s 696 00:46:42,130 --> 00:46:53,770 minus omega0 t plus cosine omega s plus omega0. 697 00:46:57,540 --> 00:47:03,640 You can see that when I actually multiply two cosine functions 698 00:47:03,640 --> 00:47:10,480 together, then what I get is actually the omega0 minus omega 699 00:47:10,480 --> 00:47:11,580 s. 700 00:47:11,580 --> 00:47:15,420 You can actually put the minus sign there, it didn't matter. 701 00:47:15,420 --> 00:47:21,930 And cosine omega0 plus omega s times t. 702 00:47:21,930 --> 00:47:24,660 So therefore, you can see that the frequency, there are only 703 00:47:24,660 --> 00:47:29,250 two frequencies which contribute to this C of omega, which is 704 00:47:29,250 --> 00:47:31,500 actually these two frequencies. 705 00:47:31,500 --> 00:47:36,280 So that is actually why, if you do this trick, 706 00:47:36,280 --> 00:47:41,100 you actually try to modulate your slow signal function 707 00:47:41,100 --> 00:47:45,000 by a fast carrier frequency. 708 00:47:45,000 --> 00:47:49,950 Then you are going to confine the effective range of omega 709 00:47:49,950 --> 00:47:53,460 into a very small range. 710 00:47:53,460 --> 00:47:56,660 Why is that useful? 711 00:47:56,660 --> 00:48:00,240 That's actually what I want to answer to you. 712 00:48:00,240 --> 00:48:06,060 Suppose I have this crazy dispersion relation, which 713 00:48:06,060 --> 00:48:10,940 is omega as a function of K. You can graph it, 714 00:48:10,940 --> 00:48:14,350 and suppose it looks really crazy like this. 715 00:48:18,660 --> 00:48:26,010 And if I set my carrier oscillation frequency 716 00:48:26,010 --> 00:48:35,130 to be omega0, and that will give you a corresponding wave 717 00:48:35,130 --> 00:48:39,240 number which is K0. 718 00:48:39,240 --> 00:48:40,290 I hope you can see it. 719 00:48:43,060 --> 00:48:47,130 That's the corresponding K0. 720 00:48:47,130 --> 00:48:54,990 Before we actually multiply this function, it's a slow function. 721 00:48:54,990 --> 00:48:57,220 It's not exactly one cosine function. 722 00:48:57,220 --> 00:49:00,270 So if you just have a cosine function harmonic wave, 723 00:49:00,270 --> 00:49:02,086 then you don't really need this trick, 724 00:49:02,086 --> 00:49:03,460 because it's actually going to be 725 00:49:03,460 --> 00:49:07,620 traveling at a speed of some constant speed. 726 00:49:07,620 --> 00:49:09,690 It's a harmonic traveling wave. 727 00:49:09,690 --> 00:49:14,100 But if this is actually a slow function, but not 728 00:49:14,100 --> 00:49:18,780 really a single harmonic wave, then what is going to happen 729 00:49:18,780 --> 00:49:23,700 is that you are going to need a wide range of K value or omega 730 00:49:23,700 --> 00:49:28,470 value to describe fs. 731 00:49:28,470 --> 00:49:31,360 Then you are in trouble because now, 732 00:49:31,360 --> 00:49:35,260 all the waves with different wavelengths are going to be 733 00:49:35,260 --> 00:49:37,290 travelling at different speed. 734 00:49:37,290 --> 00:49:40,900 Then you have this dispersion problem. 735 00:49:40,900 --> 00:49:44,460 On the other hand, if I multiply this function, 736 00:49:44,460 --> 00:49:48,940 this slow function, by a fast oscillating function, 737 00:49:48,940 --> 00:49:55,420 I am confining the effective range of omega 738 00:49:55,420 --> 00:49:58,690 into this small box, which is actually 739 00:49:58,690 --> 00:50:06,100 between omega0 minus or plus omega s. 740 00:50:06,100 --> 00:50:09,870 This is actually omega0 plus minus omega s. 741 00:50:09,870 --> 00:50:13,830 This is actually the range of the possible omega, 742 00:50:13,830 --> 00:50:17,650 which contribute to this resulting f of t. 743 00:50:17,650 --> 00:50:21,870 Therefore, the behavior of this function 744 00:50:21,870 --> 00:50:26,470 is actually much easier to understand. 745 00:50:26,470 --> 00:50:33,370 So with that given there, suppose now I have 746 00:50:33,370 --> 00:50:37,330 a large difference between-- 747 00:50:37,330 --> 00:50:40,945 suppose I have a large difference between omega s 748 00:50:40,945 --> 00:50:42,490 and omega0. 749 00:50:42,490 --> 00:50:46,780 Then I can actually focus on a very small range 750 00:50:46,780 --> 00:50:53,530 in this dispersion relation diagram. 751 00:50:53,530 --> 00:50:56,830 Then I can write omega as a function of K, 752 00:50:56,830 --> 00:51:03,790 the dispersion relation equal to omega0 plus K 753 00:51:03,790 --> 00:51:10,400 minus K0, partial omega, partial K. Evaluate 754 00:51:10,400 --> 00:51:15,700 it at a equal to K0, plus higher order term. 755 00:51:15,700 --> 00:51:20,470 Basically I can do this Taylor expansion. 756 00:51:20,470 --> 00:51:26,640 And maybe it surprised you, you can immediately identify, ha! 757 00:51:26,640 --> 00:51:35,060 This is delta d omega dk, is the group velocity. 758 00:51:35,060 --> 00:51:39,210 Suddenly it show up in the Taylor expansion 759 00:51:39,210 --> 00:51:42,550 of the dispersion relation. 760 00:51:42,550 --> 00:51:48,760 so if I focus on the region which is around omega0, 761 00:51:48,760 --> 00:51:53,750 then I can actually re-write omega in this functional form. 762 00:51:53,750 --> 00:51:59,500 Omega is actually equal to omega0 plus K 763 00:51:59,500 --> 00:52:04,810 minus K0 times Vg, which is the group velocity. 764 00:52:09,770 --> 00:52:14,570 Suppose this is happening, then now I can actually 765 00:52:14,570 --> 00:52:23,540 go ahead and really calculate the functional form for f of t. 766 00:52:23,540 --> 00:52:29,200 So suppose I have this definition of t. 767 00:52:29,200 --> 00:52:41,330 The definition of t is equal to fs t times cosine omega0t. 768 00:52:41,330 --> 00:52:45,230 Or say I can actually write it in a complex form, 769 00:52:45,230 --> 00:52:51,230 instead of writing it in a cosine omega0t functional form, 770 00:52:51,230 --> 00:52:54,540 I can write it in exponential functional form. 771 00:52:54,540 --> 00:52:59,750 Exponential minus i omega0t, which is actually more 772 00:52:59,750 --> 00:53:04,130 convenient for the discussion. 773 00:53:04,130 --> 00:53:09,190 So what is going to happen if I actually do this calculation? 774 00:53:09,190 --> 00:53:12,080 Then basically that's one example signal 775 00:53:12,080 --> 00:53:14,990 which I would like to send on the slides. 776 00:53:14,990 --> 00:53:21,680 So if I am trying to send a progressing harmonic wave, 777 00:53:21,680 --> 00:53:27,740 then after multiplying by this exponential i omega0t 778 00:53:27,740 --> 00:53:30,380 function, or a cosine function, basically 779 00:53:30,380 --> 00:53:32,990 you get something which is actually oscillating really 780 00:53:32,990 --> 00:53:37,100 fast, which is actually the AM signal we are trying 781 00:53:37,100 --> 00:53:41,030 to send through this media. 782 00:53:41,030 --> 00:53:43,640 So we can actually identify, this 783 00:53:43,640 --> 00:53:47,300 is actually the structure of this, actually the carrier. 784 00:53:50,360 --> 00:53:55,700 And this signal become the analogue, 785 00:53:55,700 --> 00:54:00,630 in analogy to what we actually have discussed for that beat 786 00:54:00,630 --> 00:54:01,710 phenomenon case. 787 00:54:04,650 --> 00:54:14,810 So now, if the omega range is really small, 788 00:54:14,810 --> 00:54:18,700 then I can actually write this down. 789 00:54:18,700 --> 00:54:23,060 Write a functional form of omega in this functional form. 790 00:54:23,060 --> 00:54:29,090 Or I can actually take out the kVg term, 791 00:54:29,090 --> 00:54:31,910 and the rest is actually going to be something 792 00:54:31,910 --> 00:54:39,650 like some constant a, where a is actually equal to omega0 793 00:54:39,650 --> 00:54:43,220 minus Vg times K0. 794 00:54:43,220 --> 00:54:47,960 So basically I'm just taking out this K term here, 795 00:54:47,960 --> 00:54:51,620 and this become this term. 796 00:54:51,620 --> 00:54:54,290 With this formula, I can solve what 797 00:54:54,290 --> 00:54:58,350 would be the functional form for K, 798 00:54:58,350 --> 00:55:01,860 as a preparation for what I'm going to do later. 799 00:55:01,860 --> 00:55:08,330 So K can be also expressed as omega over Vg. 800 00:55:08,330 --> 00:55:12,910 So basically I just solve the K plus b. 801 00:55:12,910 --> 00:55:16,030 b is actually just some constant. 802 00:55:16,030 --> 00:55:17,810 Just do it for convenience, I can 803 00:55:17,810 --> 00:55:25,520 write b equal to K0 minus omega0 divided by Vg. 804 00:55:25,520 --> 00:55:31,010 What we learned here is that if the range of effective omega 805 00:55:31,010 --> 00:55:35,710 is really small around omega zero, 806 00:55:35,710 --> 00:55:39,590 then the relation between omega and the K 807 00:55:39,590 --> 00:55:42,170 becomes a linear function. 808 00:55:42,170 --> 00:55:47,910 Of course, it's still not like the case for the non dispersive 809 00:55:47,910 --> 00:55:53,700 median, where omega over K is a constant. 810 00:55:53,700 --> 00:55:56,922 But at least it becomes a linear function, 811 00:55:56,922 --> 00:55:58,130 which is actually much nicer. 812 00:56:01,070 --> 00:56:05,780 So finally, with all those preparation we have done, 813 00:56:05,780 --> 00:56:10,100 we would like to show one important consequence. 814 00:56:10,100 --> 00:56:17,300 So what we are trying to do is to show that psi xt. 815 00:56:17,300 --> 00:56:22,130 Now I send, I oscillate the median, 816 00:56:22,130 --> 00:56:27,450 the string, by this f of t, which I designed there. 817 00:56:27,450 --> 00:56:33,930 ft is actually fs times exponential i omega0t. 818 00:56:33,930 --> 00:56:37,460 That's actually designed there. 819 00:56:37,460 --> 00:56:41,810 I would like to show that the resulting amplitude will 820 00:56:41,810 --> 00:56:56,130 be equal to fs t minus x divided by Vg, exponential minus i 821 00:56:56,130 --> 00:57:00,460 omega0t minus K0x. 822 00:57:00,460 --> 00:57:03,636 Of course I need to take the real part of this 823 00:57:03,636 --> 00:57:07,730 in, to go back to the real axis. 824 00:57:07,730 --> 00:57:14,990 Basically I dropped the i sine omega t contribution. 825 00:57:14,990 --> 00:57:18,500 So this is actually what I want to show. 826 00:57:18,500 --> 00:57:21,540 Before I go through all those math, 827 00:57:21,540 --> 00:57:25,070 let's do get the conclusion which we would like to draw, 828 00:57:25,070 --> 00:57:28,120 before we actually really go through the math. 829 00:57:28,120 --> 00:57:31,730 The conclusion which I would like to draw is that, OK, 830 00:57:31,730 --> 00:57:35,540 this is actually my analogue. 831 00:57:35,540 --> 00:57:41,480 My analogue is going to be travelling at the speed of Vg, 832 00:57:41,480 --> 00:57:43,504 which is the group velocity. 833 00:57:43,504 --> 00:57:45,170 That's the conclusion which I would like 834 00:57:45,170 --> 00:57:47,660 to draw from this exercise. 835 00:57:47,660 --> 00:57:54,350 And this thing is actually cosine omega0t minus K0x. 836 00:57:54,350 --> 00:57:57,830 Therefore, this is actually a harmonic wave. 837 00:57:57,830 --> 00:58:02,540 The carrier is a harmonic wave travelling at Vp equal 838 00:58:02,540 --> 00:58:06,590 to omega0 divided by K0. 839 00:58:06,590 --> 00:58:08,660 That's the kind of conclusion which I would like 840 00:58:08,660 --> 00:58:13,050 to draw from this exercise. 841 00:58:13,050 --> 00:58:18,110 Any questions about what we have discussed so far? 842 00:58:18,110 --> 00:58:23,750 OK, then really you have to hold tight and follow me really, 843 00:58:23,750 --> 00:58:26,350 100% focus, because this is actually 844 00:58:26,350 --> 00:58:29,010 a complicated calculation. 845 00:58:29,010 --> 00:58:35,150 So now what I can do is, now I need to express my fs in terms 846 00:58:35,150 --> 00:58:41,360 of C. So I do integration from minus infinity 847 00:58:41,360 --> 00:58:50,150 to infinity, d omega, C omega, exponential minus i omega t. 848 00:58:50,150 --> 00:58:53,870 So basically I can write my f of s 849 00:58:53,870 --> 00:58:59,690 in a functional form, which we introduced before. 850 00:58:59,690 --> 00:59:07,640 Then my f function is actually equal to fs times exponential i 851 00:59:07,640 --> 00:59:10,910 omega minus i omega0t. 852 00:59:10,910 --> 00:59:14,480 So that's actually what we defined there. 853 00:59:14,480 --> 00:59:19,270 And this would be equal to minus infinity to infinity. 854 00:59:19,270 --> 00:59:27,090 I do this integration number, omega C omega exponential minus 855 00:59:27,090 --> 00:59:33,800 i omega plus omega0 times t. 856 00:59:33,800 --> 00:59:38,200 So there's nothing special, I just take my expression for fs, 857 00:59:38,200 --> 00:59:42,877 multiply that by exponential minus i omega0t. 858 00:59:42,877 --> 00:59:44,210 Then that's actually what I get. 859 00:59:54,740 --> 00:59:58,320 So since this is actually integration over omega 860 00:59:58,320 --> 01:00:02,700 from minus infinity to infinity, therefore I 861 01:00:02,700 --> 01:00:07,410 can always have the freedom to shift the origin. 862 01:00:07,410 --> 01:00:15,600 So that means f of t can be returned as minus infinity 863 01:00:15,600 --> 01:00:24,210 to infinity d omega C omega minus omega0 exponential minus 864 01:00:24,210 --> 01:00:27,440 i omega t. 865 01:00:27,440 --> 01:00:31,950 Then we can see that is fix a relation between C 866 01:00:31,950 --> 01:00:38,760 of the f function, and the C of the fs function. 867 01:00:38,760 --> 01:00:41,420 So so far, everything is exact. 868 01:00:41,420 --> 01:00:46,110 I haven't made any approximation so far. 869 01:00:46,110 --> 01:00:54,660 So now, I can take this function and propagate that to all x. 870 01:00:54,660 --> 01:00:58,800 In other words, I can now take this ft, 871 01:00:58,800 --> 01:01:03,570 and write down the psi as a function of x and t. 872 01:01:03,570 --> 01:01:06,690 So that means all the different components are 873 01:01:06,690 --> 01:01:10,860 traveling at different speeds. 874 01:01:10,860 --> 01:01:12,570 So basically, I can write it down 875 01:01:12,570 --> 01:01:20,160 like d omega C omega minus omega0, exponential minus i 876 01:01:20,160 --> 01:01:24,170 omega t, exponential ikx. 877 01:01:24,170 --> 01:01:28,650 Kx k is actually a function of omega. 878 01:01:31,180 --> 01:01:32,410 Any questions? 879 01:01:32,410 --> 01:01:35,550 So that's actually just identical to what we actually 880 01:01:35,550 --> 01:01:36,900 have done before. 881 01:01:36,900 --> 01:01:45,250 So now I can go from f of t to sine, if you are following me. 882 01:01:45,250 --> 01:01:48,270 So until here, everything exact. 883 01:01:48,270 --> 01:01:50,890 You have all the problems you have, 884 01:01:50,890 --> 01:01:53,240 like you know this dispersion essentially, 885 01:01:53,240 --> 01:01:54,730 because all the little components, 886 01:01:54,730 --> 01:01:59,820 as you can see here, can be travelling at different speeds. 887 01:01:59,820 --> 01:02:04,430 So now, what I could do is that if I 888 01:02:04,430 --> 01:02:11,820 assume that C omega is only sizable at the small range 889 01:02:11,820 --> 01:02:18,130 around, it's only sizable around omega0. 890 01:02:18,130 --> 01:02:23,040 If now I take this assumption and propagate 891 01:02:23,040 --> 01:02:27,360 into this formula, then I can write this psi 892 01:02:27,360 --> 01:02:32,820 function roughly like minus infinity 893 01:02:32,820 --> 01:02:40,780 to infinity d omega C omega minus omega0 exponential minus 894 01:02:40,780 --> 01:02:45,880 i omega t exponential i. 895 01:02:45,880 --> 01:02:50,800 Now I can take the formula which I actually 896 01:02:50,800 --> 01:02:56,040 did an approximation, around omega0. 897 01:02:56,040 --> 01:03:03,450 Around omega0, K can be returned us omega over Vg plus b. 898 01:03:03,450 --> 01:03:05,910 This is actually where I take the approximation. 899 01:03:05,910 --> 01:03:10,320 Only consider the first order in the Taylor expansion. 900 01:03:10,320 --> 01:03:13,650 So you can see now here, it's not exact anymore. 901 01:03:13,650 --> 01:03:19,110 But now I write approximate function of form for K omega. 902 01:03:19,110 --> 01:03:25,180 So what I'm going to get is omega over Vg 903 01:03:25,180 --> 01:03:28,170 plus b, multiplied by x. 904 01:03:32,980 --> 01:03:33,931 Any questions? 905 01:03:36,510 --> 01:03:38,350 Now I have the approximation. 906 01:03:38,350 --> 01:03:42,760 And of course now I can gather all the terms related to omega 907 01:03:42,760 --> 01:03:44,530 together. 908 01:03:44,530 --> 01:03:47,650 I'm getting minus infinity to infinity 909 01:03:47,650 --> 01:03:56,830 d omega C omega minus omega0 exponential minus i omega t 910 01:03:56,830 --> 01:04:05,560 minus x over Vg, exponential ibx. 911 01:04:05,560 --> 01:04:09,550 So basically, I am merging this term and that term. 912 01:04:09,550 --> 01:04:14,040 This term and that term will give you this term. 913 01:04:14,040 --> 01:04:18,160 And what is essentially the rest is the exponential ibx. 914 01:04:21,520 --> 01:04:22,750 We are almost there. 915 01:04:27,530 --> 01:04:32,570 So now I would like to use this board, so I need to erase that. 916 01:04:41,050 --> 01:04:48,160 So now I continue from here, and I can now again, 917 01:04:48,160 --> 01:04:52,750 I can again change the origin of this infinite integral 918 01:04:52,750 --> 01:04:58,720 so that this can be written as minus infinity 919 01:04:58,720 --> 01:05:04,400 to infinity, d omega C function of omega, 920 01:05:04,400 --> 01:05:16,960 exponential minus i omega plus omega0, t minus x divided 921 01:05:16,960 --> 01:05:22,320 by Vg, and exponential ibx. 922 01:05:25,100 --> 01:05:29,320 So what I come from this board to that formula, 923 01:05:29,320 --> 01:05:32,260 if you are following me we are almost there, 924 01:05:32,260 --> 01:05:34,900 because I am changing the origin again, 925 01:05:34,900 --> 01:05:38,200 so that omega minus omega0 becomes 926 01:05:38,200 --> 01:05:40,510 omega, become a new omega. 927 01:05:40,510 --> 01:05:43,840 Is everybody accepting this fact? 928 01:05:43,840 --> 01:05:46,100 And that means the original omega 929 01:05:46,100 --> 01:05:52,360 will become omega plus omega0. 930 01:05:52,360 --> 01:05:58,120 I'm trying to go really slow, so that everybody can follow. 931 01:05:58,120 --> 01:05:59,980 I hope you are following. 932 01:05:59,980 --> 01:06:02,470 All right, then now I can actually 933 01:06:02,470 --> 01:06:05,200 redistribute, arrange all those terms 934 01:06:05,200 --> 01:06:07,930 and the magic will happen. 935 01:06:07,930 --> 01:06:10,680 So that means rearrange all those terms, 936 01:06:10,680 --> 01:06:16,480 minus infinity to infinity d omega C omega 937 01:06:16,480 --> 01:06:22,450 exponential minus i omega t minus x divided 938 01:06:22,450 --> 01:06:33,760 by Vg, exponential minus i omega0t, exponential i omega0 939 01:06:33,760 --> 01:06:39,310 over Vg plus b x. 940 01:06:39,310 --> 01:06:42,070 So basically, there's really no magic. 941 01:06:42,070 --> 01:06:45,940 What I'm doing is really to rearrange all those terms, 942 01:06:45,940 --> 01:06:49,010 so that this term is actually rearranged 943 01:06:49,010 --> 01:06:53,150 so that it's now omega times t minus x over Vg. 944 01:06:53,150 --> 01:06:55,090 It's an independent exponential term. 945 01:06:58,080 --> 01:07:06,310 And I actually extract this term times t to be returned here. 946 01:07:06,310 --> 01:07:08,850 I'm just rearranging things, OK? 947 01:07:08,850 --> 01:07:12,220 I'm not changing anything. 948 01:07:12,220 --> 01:07:17,800 And finally, I can merge this term and that term, 949 01:07:17,800 --> 01:07:20,900 and become this function field. 950 01:07:23,740 --> 01:07:30,420 I can immediately recognize that after this rearrangement, 951 01:07:30,420 --> 01:07:32,800 this is just re-writing the formula, 952 01:07:32,800 --> 01:07:36,460 putting all those terms in different place. 953 01:07:36,460 --> 01:07:38,400 Of course, you can actually review 954 01:07:38,400 --> 01:07:42,160 this part of the lecture in the lecture notes later. 955 01:07:42,160 --> 01:07:46,000 But basically, we're not doing anything fancy but rearranging 956 01:07:46,000 --> 01:07:50,050 things over in different place. 957 01:07:50,050 --> 01:07:52,840 Then I can actually quickly identify 958 01:07:52,840 --> 01:07:55,750 what I am trying to integrate. 959 01:07:55,750 --> 01:07:59,590 So this integration is over omega. 960 01:07:59,590 --> 01:08:03,730 Therefore all those terms are now related to omega. 961 01:08:03,730 --> 01:08:07,240 Therefore, they are just some terms which are sitting there, 962 01:08:07,240 --> 01:08:08,840 they don't participate. 963 01:08:08,840 --> 01:08:16,479 And if you focus on this part, what is this? 964 01:08:16,479 --> 01:08:21,880 If you compare that to the original equation of which 965 01:08:21,880 --> 01:08:23,500 I have here. 966 01:08:23,500 --> 01:08:27,399 If you compare that to the original fs equation here, 967 01:08:27,399 --> 01:08:29,710 you can't immediately identify that actually that's 968 01:08:29,710 --> 01:08:31,580 a function of fs. 969 01:08:31,580 --> 01:08:37,490 Originally this function fs is a function of t. 970 01:08:37,490 --> 01:08:40,149 And I'm going to that board now. 971 01:08:40,149 --> 01:08:48,290 This is actually fs with t minus x over Vg. 972 01:08:48,290 --> 01:08:49,960 Surprisingly simple. 973 01:08:53,380 --> 01:08:58,510 Now let's look at the right hand side, this mass here. 974 01:08:58,510 --> 01:09:05,899 This is actually K0, which actually you cannot see 975 01:09:05,899 --> 01:09:06,399 anymore. 976 01:09:06,399 --> 01:09:10,700 It's in the back of this board. 977 01:09:10,700 --> 01:09:13,189 And then if you combine these two terms, 978 01:09:13,189 --> 01:09:15,810 basically what you get is exponential 979 01:09:15,810 --> 01:09:19,720 minus i omega0t minus K0x. 980 01:09:22,880 --> 01:09:24,229 So look at what we have done. 981 01:09:26,760 --> 01:09:33,660 I got started with this Fourier transform functional 982 01:09:33,660 --> 01:09:36,810 form of fs. 983 01:09:36,810 --> 01:09:40,800 I multiplied fs by cosine omega0t 984 01:09:40,800 --> 01:09:43,210 and go to the complex notation. 985 01:09:43,210 --> 01:09:47,660 It becomes exponential minus i omega0t. 986 01:09:47,660 --> 01:09:53,250 If I multiplied that, I get my f function, which is like this. 987 01:09:53,250 --> 01:09:55,910 You get additional term there. 988 01:09:55,910 --> 01:10:00,030 I rearrange things and change the origin, 989 01:10:00,030 --> 01:10:03,750 and I can rewrite ft in this functional form. 990 01:10:03,750 --> 01:10:08,700 And I can have a relation between the C related to fs 991 01:10:08,700 --> 01:10:12,930 to the C related to f of t. 992 01:10:12,930 --> 01:10:18,120 I propagate ft over the full space, 993 01:10:18,120 --> 01:10:21,690 and attain my sine, which is the amplitude as a function 994 01:10:21,690 --> 01:10:25,910 of place and the time. 995 01:10:25,910 --> 01:10:29,060 Until here, everything is exact. 996 01:10:29,060 --> 01:10:31,750 Then I have introduced assumption, 997 01:10:31,750 --> 01:10:36,850 which is C of omega is only sizable, only contributing, 998 01:10:36,850 --> 01:10:41,570 around omega zero, therefore I can do approximation form 999 01:10:41,570 --> 01:10:46,520 for the K function, which is this functional form. 1000 01:10:46,520 --> 01:10:48,770 Then I just do the integration. 1001 01:10:48,770 --> 01:10:50,690 Then I found that, interesting! 1002 01:10:55,450 --> 01:10:58,020 This side is-- you should be taking 1003 01:10:58,020 --> 01:11:01,130 the real part of this thing. 1004 01:11:01,130 --> 01:11:03,860 This side have two components. 1005 01:11:03,860 --> 01:11:06,770 The first component is fs, which is 1006 01:11:06,770 --> 01:11:12,630 the original signal you put in, the signal you want to send. 1007 01:11:12,630 --> 01:11:18,080 It's actually progressing at the speed of group velocity. 1008 01:11:18,080 --> 01:11:22,440 So now you understand what this group velocity means. 1009 01:11:22,440 --> 01:11:25,190 That's the speed of the signal you 1010 01:11:25,190 --> 01:11:29,510 want to send in the AM radio. 1011 01:11:29,510 --> 01:11:32,460 And this thing is actually modulated 1012 01:11:32,460 --> 01:11:36,270 by exponential function, which is actually 1013 01:11:36,270 --> 01:11:44,720 the propagating at the speed of Vp, equal to omega0 over K0. 1014 01:11:44,720 --> 01:11:48,620 So the carrier still, after you actually 1015 01:11:48,620 --> 01:11:53,750 include many, many terms contracting 1016 01:11:53,750 --> 01:11:59,900 the f function, the sine which is the amplitude, 1017 01:11:59,900 --> 01:12:05,162 the trick is that only the omega value around omega0 1018 01:12:05,162 --> 01:12:06,460 contributes. 1019 01:12:06,460 --> 01:12:10,550 If that happen, then you can see that there 1020 01:12:10,550 --> 01:12:13,610 are two structures actually propagating 1021 01:12:13,610 --> 01:12:16,520 at different speeds, and that you can actually understand 1022 01:12:16,520 --> 01:12:20,620 the structure independently. 1023 01:12:20,620 --> 01:12:25,610 That means your signal will not be distorted 1024 01:12:25,610 --> 01:12:27,620 if you're sending it this way. 1025 01:12:27,620 --> 01:12:30,020 But the difference is that the speed 1026 01:12:30,020 --> 01:12:34,350 of the signal you are sending is actually 1027 01:12:34,350 --> 01:12:37,860 at the speed of group velocity. 1028 01:12:37,860 --> 01:12:40,250 That is actually the amazing fact 1029 01:12:40,250 --> 01:12:44,230 which actually enables us to send signal 1030 01:12:44,230 --> 01:12:49,640 over thousands and thousands of miles away from the source. 1031 01:12:49,640 --> 01:12:53,470 So what is actually done is actually that, 1032 01:12:53,470 --> 01:12:56,810 suppose you have some kind of radio station. 1033 01:12:56,810 --> 01:13:01,860 You can send the radio, and the radio will go over the place, 1034 01:13:01,860 --> 01:13:05,450 and got refracted by atmosphere-- 1035 01:13:05,450 --> 01:13:08,490 the atmosphere on Earth. 1036 01:13:08,490 --> 01:13:13,350 Got refracted, and the receiver from some place which is really 1037 01:13:13,350 --> 01:13:15,710 distant from the source can still 1038 01:13:15,710 --> 01:13:21,170 see it without any dispersion, as we show here. 1039 01:13:21,170 --> 01:13:23,330 And it's actually going to be propagating 1040 01:13:23,330 --> 01:13:27,870 at the speed of group velocity. 1041 01:13:27,870 --> 01:13:33,620 So you may not actually believe that. 1042 01:13:33,620 --> 01:13:37,360 How about we do a simulation like what 1043 01:13:37,360 --> 01:13:39,950 we did before with MIT wave? 1044 01:13:39,950 --> 01:13:43,730 So this is actually the example which we did last time. 1045 01:13:43,730 --> 01:13:45,450 We have an nit wave. 1046 01:13:45,450 --> 01:13:48,950 We can compose that into many, many pieces. 1047 01:13:48,950 --> 01:13:52,730 And then see how it evolved as a function of time. 1048 01:13:52,730 --> 01:13:54,590 This is actually without dispersion, 1049 01:13:54,590 --> 01:13:57,770 therefore everything is perfect. 1050 01:13:57,770 --> 01:14:00,760 So now I would like to introduce some excitement there. 1051 01:14:03,820 --> 01:14:10,440 If I have dispersion, like 0.1, alpha is equal to 0.1, 1052 01:14:10,440 --> 01:14:11,860 and see will happen. 1053 01:14:11,860 --> 01:14:16,690 Then just a reminder that things will not go super well. 1054 01:14:16,690 --> 01:14:19,140 Wait a second, what am I doing? 1055 01:14:19,140 --> 01:14:23,080 This is actually still without dispersion. 1056 01:14:23,080 --> 01:14:24,230 Sorry for that. 1057 01:14:28,840 --> 01:14:31,730 It should be-- 1058 01:14:31,730 --> 01:14:35,180 OK, so let's take a look at the triangular case. 1059 01:14:35,180 --> 01:14:39,550 This is now with dispersion. 1060 01:14:39,550 --> 01:14:43,390 And you can see that as a reminder as a function of time, 1061 01:14:43,390 --> 01:14:46,190 the shape of the signal which you would like to send 1062 01:14:46,190 --> 01:14:49,160 is actually changing as a function of time. 1063 01:14:49,160 --> 01:14:53,020 And after a few thousands of miles, 1064 01:14:53,020 --> 01:14:57,310 you will not even recognize the original structure we put in. 1065 01:14:57,310 --> 01:15:03,600 So that's the trouble we are actually facing. 1066 01:15:03,600 --> 01:15:07,790 You can see that it's getting wider and wider et cetera. 1067 01:15:07,790 --> 01:15:18,250 So now what will happen if I send this kind of signal. 1068 01:15:18,250 --> 01:15:20,820 This is a signal which you have some kind of shape. 1069 01:15:20,820 --> 01:15:23,880 You can imagine that there's sounds kind of analogue. 1070 01:15:23,880 --> 01:15:27,540 And I am now doing the calculation to actually map 1071 01:15:27,540 --> 01:15:29,870 all the individual components. 1072 01:15:29,870 --> 01:15:32,820 And now I'm going to propagate through the median. 1073 01:15:32,820 --> 01:15:39,060 And the blue is the original non-dispersive median 1074 01:15:39,060 --> 01:15:40,350 situation. 1075 01:15:40,350 --> 01:15:44,360 And the red is actually the propagation 1076 01:15:44,360 --> 01:15:45,910 in a dispersive median. 1077 01:15:45,910 --> 01:15:48,770 You can see the propagation in a dispersive median 1078 01:15:48,770 --> 01:15:53,160 is faster, because alpha is actually larger than one-- 1079 01:15:53,160 --> 01:15:54,270 larger than zero. 1080 01:15:54,270 --> 01:15:56,310 So it's actually, in this case it's 0.1. 1081 01:15:56,310 --> 01:16:03,975 And you can see that the red cosine omega0t modulated signal 1082 01:16:03,975 --> 01:16:12,480 is progressing, and the shape of the analogue is not changing. 1083 01:16:12,480 --> 01:16:13,770 You can see that, right? 1084 01:16:13,770 --> 01:16:17,380 So it's very different from what we actually see before 1085 01:16:17,380 --> 01:16:20,130 with a single triangular pulse. 1086 01:16:20,130 --> 01:16:21,420 Now you can see that, ha! 1087 01:16:21,420 --> 01:16:24,360 Only when n gets very large, I start 1088 01:16:24,360 --> 01:16:27,150 to be able to feed all those little structures. 1089 01:16:27,150 --> 01:16:31,740 That means the end value, or say the omega value, which I need 1090 01:16:31,740 --> 01:16:34,429 is you really narrow, a very narrow range, 1091 01:16:34,429 --> 01:16:36,720 which will actually match with what we have been doing. 1092 01:16:36,720 --> 01:16:39,460 And now I start to propagate all those things. 1093 01:16:39,460 --> 01:16:43,530 And you can see that the red is actually traveling faster 1094 01:16:43,530 --> 01:16:46,470 than the blue, which is what we expect. 1095 01:16:46,470 --> 01:16:48,730 And you can see now, in the instance 1096 01:16:48,730 --> 01:16:51,120 they actually overlap each other, 1097 01:16:51,120 --> 01:16:54,420 you can see that envelope, the shape of the envelope, 1098 01:16:54,420 --> 01:16:55,230 is still the same. 1099 01:16:55,230 --> 01:16:57,970 It's exactly what we actually printed. 1100 01:16:57,970 --> 01:17:01,800 And that actually brings me to the end of my lecture. 1101 01:17:01,800 --> 01:17:06,960 We have on understood how the AM radio actually works. 1102 01:17:06,960 --> 01:17:08,820 And next time, we are going to talk 1103 01:17:08,820 --> 01:17:11,170 about uncertainty principles. 1104 01:17:11,170 --> 01:17:12,150 What the hell? 1105 01:17:12,150 --> 01:17:14,520 What happened? 1106 01:17:14,520 --> 01:17:17,460 And believe me, they are actually 1107 01:17:17,460 --> 01:17:19,272 connected to each other. 1108 01:17:19,272 --> 01:17:21,970 Uncertainty principle is actually highly related 1109 01:17:21,970 --> 01:17:24,640 to wave and the vibrations. 1110 01:17:24,640 --> 01:17:29,510 Thank you very much, and let me know if you have any questions.