1 00:00:02,620 --> 00:00:04,960 The following content is provided under a Creative 2 00:00:04,960 --> 00:00:06,380 Commons license. 3 00:00:06,380 --> 00:00:08,590 Your support will help MIT OpenCourseWare 4 00:00:08,590 --> 00:00:12,680 continue to offer high-quality educational resources for free. 5 00:00:12,680 --> 00:00:15,220 To make a donation or to view additional materials 6 00:00:15,220 --> 00:00:19,180 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,180 --> 00:00:20,214 at ocw.mit.edu. 8 00:00:23,681 --> 00:00:24,930 YEN-JIE LEE: Hello, everybody. 9 00:00:24,930 --> 00:00:27,610 Welcome back to 8.03. 10 00:00:27,610 --> 00:00:31,070 Today we are going to continue the discussion of waves. 11 00:00:31,070 --> 00:00:33,380 We will discuss a very interesting phenomenon 12 00:00:33,380 --> 00:00:36,130 not today, which is dispersion. 13 00:00:36,130 --> 00:00:38,720 And before that, we will discuss a bit, 14 00:00:38,720 --> 00:00:41,530 just to give you some reminder, about what we 15 00:00:41,530 --> 00:00:42,950 have learned so far. 16 00:00:42,950 --> 00:00:46,690 So we discovered this wave equation, 17 00:00:46,690 --> 00:00:49,280 which is showing here, in the class, 18 00:00:49,280 --> 00:00:52,200 and then we also show you that it described 19 00:00:52,200 --> 00:00:54,850 three different kinds of systems, which 20 00:00:54,850 --> 00:00:57,120 we included in the lecture-- 21 00:00:57,120 --> 00:01:00,380 the massive strings, which are the strings 22 00:01:00,380 --> 00:01:05,680 can actually oscillate up and down in a wide direction. 23 00:01:05,680 --> 00:01:10,030 And also we discussed about sound waves. 24 00:01:10,030 --> 00:01:13,450 This is also discussed in a previous lecture. 25 00:01:13,450 --> 00:01:18,440 And sound waves can be described by wave equation. 26 00:01:18,440 --> 00:01:21,490 And finally, the last time we discussed 27 00:01:21,490 --> 00:01:23,110 electromagnetic waves. 28 00:01:23,110 --> 00:01:25,870 It's a special kind of wave involving 29 00:01:25,870 --> 00:01:28,060 two oscillating fields. 30 00:01:28,060 --> 00:01:30,580 One is actually the electric field, 31 00:01:30,580 --> 00:01:33,670 the other one is magnetic field. 32 00:01:33,670 --> 00:01:36,304 So that's kind of interesting, because this is actually 33 00:01:36,304 --> 00:01:37,720 slightly different from what we've 34 00:01:37,720 --> 00:01:41,020 discussed before in the previous two cases. 35 00:01:41,020 --> 00:01:45,010 This is actually a three dimensional wave, 36 00:01:45,010 --> 00:01:48,290 and also involving two different components. 37 00:01:48,290 --> 00:01:50,740 And we also discussed the solution, 38 00:01:50,740 --> 00:01:54,220 the traveling wave solution of the electromagnetic waves. 39 00:01:54,220 --> 00:01:57,940 As you can see from here, the electric field 40 00:01:57,940 --> 00:02:00,970 is showing us the red, and the magnetic field 41 00:02:00,970 --> 00:02:02,470 is showing us the blue. 42 00:02:02,470 --> 00:02:04,910 And you can see that in case of traveling wave, 43 00:02:04,910 --> 00:02:06,430 they are in phase. 44 00:02:06,430 --> 00:02:10,090 And the magnitude reach maxima simultaneously 45 00:02:10,090 --> 00:02:13,620 for electric field and the magnetic field. 46 00:02:13,620 --> 00:02:16,680 And while in the case of standing wave, 47 00:02:16,680 --> 00:02:20,150 there's a phase difference. so they don't reach maxima 48 00:02:20,150 --> 00:02:25,080 simultaneously in the standing electromagnetic field case. 49 00:02:25,080 --> 00:02:29,170 OK, so what are we going to discuss today? 50 00:02:29,170 --> 00:02:35,270 We would like to discuss the strategy to send information 51 00:02:35,270 --> 00:02:36,670 using waves. 52 00:02:36,670 --> 00:02:40,780 How do we actually send information using waves? 53 00:02:40,780 --> 00:02:44,440 So you can say, OK, maybe I can just 54 00:02:44,440 --> 00:02:47,620 send a harmonic oscillation. 55 00:02:47,620 --> 00:02:50,870 So If I do this harmonic oscillation, 56 00:02:50,870 --> 00:02:54,710 I can basically produce harmonic waves. 57 00:02:54,710 --> 00:02:58,800 They are moving up and down, and is actually 58 00:02:58,800 --> 00:03:05,650 always constant angular momentum and angular frequency. 59 00:03:05,650 --> 00:03:09,650 And maybe that's a way to send the information. 60 00:03:09,650 --> 00:03:14,630 But this kind of wave is, in reality, not super helpful, 61 00:03:14,630 --> 00:03:19,690 because if you fill the whole space with harmonic waves, then 62 00:03:19,690 --> 00:03:22,840 you don't know when did you actually send the signal. 63 00:03:22,840 --> 00:03:24,790 Because it's always oscillating up and down, 64 00:03:24,790 --> 00:03:28,570 so you don't know the starting time of the signal. 65 00:03:28,570 --> 00:03:34,090 So in reality, these kind of simple harmonic oscillating 66 00:03:34,090 --> 00:03:37,010 traveling wave is not super helpful. 67 00:03:37,010 --> 00:03:39,740 So what is actually helpful? 68 00:03:39,740 --> 00:03:41,000 That's the question. 69 00:03:41,000 --> 00:03:44,720 So what is actually helpful is to produce square pulse, 70 00:03:44,720 --> 00:03:45,800 for example. 71 00:03:45,800 --> 00:03:50,910 We can create square pulse, for example, in this case, 72 00:03:50,910 --> 00:03:52,860 I can create a square pulse here. 73 00:03:52,860 --> 00:03:56,800 And in the next time interval, I don't create a square pulse. 74 00:03:56,800 --> 00:03:59,470 In the next time interval, I don't do anything. 75 00:03:59,470 --> 00:04:02,040 And I create another square pulse here, et cetera, 76 00:04:02,040 --> 00:04:03,980 et cetera, OK. 77 00:04:03,980 --> 00:04:07,310 If you use this kind of strategy what we can do 78 00:04:07,310 --> 00:04:12,170 is to have some kind of receiver here to actually measure 79 00:04:12,170 --> 00:04:18,170 the magnitude of the pulse. 80 00:04:18,170 --> 00:04:21,079 And then we can actually interpret this data. 81 00:04:21,079 --> 00:04:25,920 So this wave is going to where the positive x direction 82 00:04:25,920 --> 00:04:28,070 or going to the right-hand side of the board. 83 00:04:28,070 --> 00:04:32,360 And the receiver will be able to interpret this data 84 00:04:32,360 --> 00:04:36,050 by appraising this ratio on the energy 85 00:04:36,050 --> 00:04:38,210 or on the measure of the amplitude. 86 00:04:38,210 --> 00:04:41,920 Then I can say, oh, now I'd receive a 0, 87 00:04:41,920 --> 00:04:43,900 and then the next signal I'm receiving 88 00:04:43,900 --> 00:04:48,020 is 1, and this one is 0, and 0, and 1, and 0. 89 00:04:48,020 --> 00:04:51,830 In this way, I can actually send information 90 00:04:51,830 --> 00:04:55,550 and that this information can be verified as a function of time. 91 00:04:55,550 --> 00:04:58,220 So in short, what would be useful 92 00:04:58,220 --> 00:05:02,750 is probably to use a narrow square pulse, 93 00:05:02,750 --> 00:05:08,750 and that would be very helpful in transmitting information. 94 00:05:08,750 --> 00:05:15,350 So if we consider an ideal string case-- 95 00:05:15,350 --> 00:05:24,810 if I have an ideal string, as we learned before, 96 00:05:24,810 --> 00:05:30,070 the behavior of the string is described by the wave equation. 97 00:05:30,070 --> 00:05:32,620 Partial squared psi partial t squared, 98 00:05:32,620 --> 00:05:36,210 and this is equal to v squared partial squared 99 00:05:36,210 --> 00:05:40,180 psi partial t squared. 100 00:05:40,180 --> 00:05:44,530 And this v is actually related to the speed 101 00:05:44,530 --> 00:05:46,990 of the progressing wave, as we discussed 102 00:05:46,990 --> 00:05:50,260 before-- the progressing wave solution. 103 00:05:50,260 --> 00:05:54,940 And if I have this idealized string, 104 00:05:54,940 --> 00:05:58,240 and it obey the wave equation, the simple version of wave 105 00:05:58,240 --> 00:06:03,100 equation, then I would be able to divide the dispersion 106 00:06:03,100 --> 00:06:03,790 relation. 107 00:06:03,790 --> 00:06:08,200 So I can now write down my harmonic progressing wave 108 00:06:08,200 --> 00:06:12,730 in the form of sine kx minus omega t. 109 00:06:12,730 --> 00:06:17,080 If I have a harmonic oscillating wave propagating 110 00:06:17,080 --> 00:06:21,880 toward the positive x direction at the speed of v. 111 00:06:21,880 --> 00:06:25,630 I can write it down in this functional form, where 112 00:06:25,630 --> 00:06:29,080 k, as a reminder, is the wavenumber, 113 00:06:29,080 --> 00:06:35,260 and the omega is actually the angular frequency. 114 00:06:35,260 --> 00:06:39,310 And therefore, if I plug in this solution, and of course, 115 00:06:39,310 --> 00:06:41,890 it can have arbitrary amplitude. 116 00:06:41,890 --> 00:06:45,580 If I plug in this solution to this equation, 117 00:06:45,580 --> 00:06:47,770 then what I'm going to get is, as we 118 00:06:47,770 --> 00:06:50,290 did in the last few lectures, there 119 00:06:50,290 --> 00:06:53,440 would be a fixed relation between k, 120 00:06:53,440 --> 00:06:57,590 which is the wavenumber, and the omega, the angular frequency. 121 00:06:57,590 --> 00:07:02,570 So the fixed relation is actually omega over k 122 00:07:02,570 --> 00:07:05,980 would be equal to v, which is actually 123 00:07:05,980 --> 00:07:09,610 the velocity in this wave equation. 124 00:07:09,610 --> 00:07:11,540 And from the previous discussion, 125 00:07:11,540 --> 00:07:15,490 we know this is actually equal to a squared root of T 126 00:07:15,490 --> 00:07:19,400 over rho L, where T is actually the tension, 127 00:07:19,400 --> 00:07:23,410 the constant tension, which we apply on this string, 128 00:07:23,410 --> 00:07:29,740 and the rho L is actually the mass per unit as a reminder. 129 00:07:29,740 --> 00:07:31,550 So what does this mean? 130 00:07:31,550 --> 00:07:33,490 What does this equation mean? 131 00:07:33,490 --> 00:07:37,270 We call it dispersion relation a lot of time, right? 132 00:07:37,270 --> 00:07:40,720 But we actually didn't explain why do I do that. 133 00:07:40,720 --> 00:07:43,760 So we are going to learn why this is actually called 134 00:07:43,760 --> 00:07:45,190 this dispersion relation. 135 00:07:45,190 --> 00:07:47,260 Omega is a function of k. 136 00:07:47,260 --> 00:07:53,260 And in this case, in this very simplified idealized case, 137 00:07:53,260 --> 00:07:57,610 omega over k is ratio. 138 00:07:57,610 --> 00:08:02,200 we know this is related to the speed of propagation 139 00:08:02,200 --> 00:08:09,040 of the harmonic wave is equal to v. v is a constant that 140 00:08:09,040 --> 00:08:12,010 is independent of k. 141 00:08:12,010 --> 00:08:14,240 This ratio is independent of k. 142 00:08:14,240 --> 00:08:15,160 What does that mean? 143 00:08:15,160 --> 00:08:23,410 That means if I prepare waves with different wavenumber, 144 00:08:23,410 --> 00:08:28,450 or in other words, waves with different wavelengths, 145 00:08:28,450 --> 00:08:31,375 they are going to propagate at the same speed. 146 00:08:34,000 --> 00:08:41,669 So the speed of the harmonic progressing wave 147 00:08:41,669 --> 00:08:45,110 is independent of the wavelength. 148 00:08:45,110 --> 00:08:46,780 That's actually very good, because 149 00:08:46,780 --> 00:08:51,280 in this case, if I prepared the square pulse, 150 00:08:51,280 --> 00:08:56,250 as we learned before, this square pulse is actually 151 00:08:56,250 --> 00:08:58,030 a very complicated object. 152 00:08:58,030 --> 00:09:00,390 Square pulse is really very complicated. 153 00:09:00,390 --> 00:09:04,790 You can do a Fourier decomposition as we did before. 154 00:09:04,790 --> 00:09:07,715 And we need infinite number of turns 155 00:09:07,715 --> 00:09:11,260 of harmonic oscillating waves. 156 00:09:11,260 --> 00:09:14,170 We actually add them together so that I 157 00:09:14,170 --> 00:09:17,740 can produce a square pulse. 158 00:09:17,740 --> 00:09:22,630 And as I mentioned here, if the dispersion relation, omega 159 00:09:22,630 --> 00:09:30,450 over k, is is our constant, v. That means all the 160 00:09:30,450 --> 00:09:37,390 whatever wavelengths pulse, which should be added together 161 00:09:37,390 --> 00:09:39,790 and produce the square pulse, are 162 00:09:39,790 --> 00:09:42,490 going to be traveling at the that speed. 163 00:09:42,490 --> 00:09:48,300 Therefore, if I have this square pulse in the beginning, 164 00:09:48,300 --> 00:09:54,210 after some time, t, what I'm going to get 165 00:09:54,210 --> 00:09:58,050 is that this is the original position of the square pulse, 166 00:09:58,050 --> 00:10:01,220 and after some time, t, this square pulse 167 00:10:01,220 --> 00:10:06,550 will move by v times t in the horizontal direction. 168 00:10:06,550 --> 00:10:08,940 And the shape of the pulse is not 169 00:10:08,940 --> 00:10:14,100 going to be changed, because no matter what kind of wavelengths 170 00:10:14,100 --> 00:10:16,530 which produce the square pulse, all 171 00:10:16,530 --> 00:10:18,535 the components in the square pulse 172 00:10:18,535 --> 00:10:24,600 are propagating at the same speed. 173 00:10:24,600 --> 00:10:30,600 So this kind of system, which has satisfied 174 00:10:30,600 --> 00:10:38,430 this kind of dispersion relation is called nondispersive media. 175 00:10:38,430 --> 00:10:42,240 no dispersion was happening in this case, 176 00:10:42,240 --> 00:10:44,370 in this highly idealized case. 177 00:10:47,130 --> 00:10:51,780 We also know that in case of the string, 178 00:10:51,780 --> 00:10:55,380 we are actually making it too idealized. 179 00:10:55,380 --> 00:11:00,870 So if we consider a more realistic string, 180 00:11:00,870 --> 00:11:06,240 then I have to consider an important phenomenon, which 181 00:11:06,240 --> 00:11:07,170 is-- 182 00:11:07,170 --> 00:11:10,620 or is a important property of the string, for example-- 183 00:11:10,620 --> 00:11:13,590 stiffness 184 00:11:13,590 --> 00:11:15,240 What do I mean by stiffness? 185 00:11:15,240 --> 00:11:20,250 So for example, if I take a string from a piano, 186 00:11:20,250 --> 00:11:26,670 a piano string, even if I don't apply any tension 187 00:11:26,670 --> 00:11:30,060 to the string, if I bend the string, 188 00:11:30,060 --> 00:11:31,590 it don't like it, all right? 189 00:11:31,590 --> 00:11:37,240 It's going to bounce back and restore to its original shape. 190 00:11:37,240 --> 00:11:39,520 So that's what I call stiffness. 191 00:11:39,520 --> 00:11:44,400 It's a different contribution compared to the string tension. 192 00:11:44,400 --> 00:11:49,410 So what we have been discussing so far that this distorting 193 00:11:49,410 --> 00:11:54,000 force is actually coming from the string tension, t. 194 00:11:54,000 --> 00:11:54,810 OK? 195 00:11:54,810 --> 00:12:00,750 What will happen if I introduce additional contribution 196 00:12:00,750 --> 00:12:03,500 from the stiffness? 197 00:12:03,500 --> 00:12:07,250 The stiffness is actually not completely related 198 00:12:07,250 --> 00:12:09,650 to the string tension, and that also 199 00:12:09,650 --> 00:12:12,620 wants to restore the shape of the string. 200 00:12:12,620 --> 00:12:13,790 OK? 201 00:12:13,790 --> 00:12:16,760 Before we go to the modeling, I would 202 00:12:16,760 --> 00:12:19,220 like to take some votes to predict 203 00:12:19,220 --> 00:12:20,880 what is going to happen. 204 00:12:20,880 --> 00:12:26,240 How many of you were predict that if I introduce and include 205 00:12:26,240 --> 00:12:30,680 the stiffness of the string into my equation, 206 00:12:30,680 --> 00:12:33,710 will the speed of propagation increase? 207 00:12:33,710 --> 00:12:36,260 How many of you think it's going to happen? 208 00:12:36,260 --> 00:12:39,780 1, 2, 3, 4, 5. 209 00:12:39,780 --> 00:12:40,280 OK. 210 00:12:44,600 --> 00:12:51,280 So some of you predict the speed of propagation will increase. 211 00:12:51,280 --> 00:12:53,760 How many of you predict that the speed 212 00:12:53,760 --> 00:12:57,840 of propagation of the harmonic wave will stay the same? 213 00:12:57,840 --> 00:13:00,250 How many of you? 214 00:13:00,250 --> 00:13:02,640 One? 215 00:13:02,640 --> 00:13:03,590 OK, only one. 216 00:13:09,310 --> 00:13:11,720 OK, how many of you actually predict 217 00:13:11,720 --> 00:13:25,750 that the speed of propagation would decrease OK 218 00:13:25,750 --> 00:13:29,320 so all the other students don't have opinion. 219 00:13:29,320 --> 00:13:32,350 OK, want to wait for the answer. 220 00:13:32,350 --> 00:13:32,980 All right. 221 00:13:32,980 --> 00:13:36,610 So you can see that it is actually not completely obvious 222 00:13:36,610 --> 00:13:38,710 before we solve this question. 223 00:13:38,710 --> 00:13:43,890 And we are going to solve it with a simple model, which 224 00:13:43,890 --> 00:13:52,690 actually slightly modifies the idealized wave equation. 225 00:13:52,690 --> 00:13:58,990 So now, one semi-realistic model which I can introduce 226 00:13:58,990 --> 00:14:03,800 is to add a term additional term to the wave equation. 227 00:14:03,800 --> 00:14:07,990 So I can now rewrite my wave equation 228 00:14:07,990 --> 00:14:13,630 to include the effect that to describe a realistic string, 229 00:14:13,630 --> 00:14:16,850 and now this is your partial squared psi partial t squared. 230 00:14:16,850 --> 00:14:20,860 This will be equal to v squared partial squared 231 00:14:20,860 --> 00:14:24,400 psi partial t squared. 232 00:14:24,400 --> 00:14:28,560 And the additional term, which I put into this, again, 233 00:14:28,560 --> 00:14:36,080 is minus alpha partial to the 4 psi partial x to the 4. 234 00:14:36,080 --> 00:14:39,350 And this is actually the contribution 235 00:14:39,350 --> 00:14:42,310 from the stiffness. 236 00:14:42,310 --> 00:14:45,160 This is stiffness. 237 00:14:45,160 --> 00:14:51,050 OK, so you can see that the wave equation is now modified. 238 00:14:51,050 --> 00:14:56,300 And what I could do in order to get the relation between omega 239 00:14:56,300 --> 00:14:57,470 and the k-- 240 00:14:57,470 --> 00:15:00,260 what I could do is that I can now 241 00:15:00,260 --> 00:15:06,320 start with this harmonic wave solution progressing wave 242 00:15:06,320 --> 00:15:11,450 solution, plug that in to this equation, this modified 243 00:15:11,450 --> 00:15:14,090 equation, and see what will happen. 244 00:15:14,090 --> 00:15:17,330 If I plug this equation into it that 245 00:15:17,330 --> 00:15:20,960 modified wave equation, what I am going to get 246 00:15:20,960 --> 00:15:21,840 is the following. 247 00:15:21,840 --> 00:15:27,560 So basically the left-hand, side you're going to get minus omega 248 00:15:27,560 --> 00:15:29,540 squared. 249 00:15:29,540 --> 00:15:31,160 And then the right-hand side, you 250 00:15:31,160 --> 00:15:43,190 get v squared minus k squared and plus alpha k to the 4 251 00:15:43,190 --> 00:15:44,560 in the right-hand side. 252 00:15:44,560 --> 00:15:48,890 OK, so of course, I can now cancel this minus sign. 253 00:15:48,890 --> 00:15:52,490 This will become plus and this will become minus. 254 00:15:52,490 --> 00:15:56,390 And then you can see that the relation between omega 255 00:15:56,390 --> 00:16:02,330 and the k is now different after I introduce this term, which 256 00:16:02,330 --> 00:16:03,610 is proportional to alpha. 257 00:16:03,610 --> 00:16:11,980 Alpha is actually describing how stiff this string is. 258 00:16:11,980 --> 00:16:14,490 Of course, now I can calculate omega 259 00:16:14,490 --> 00:16:18,280 over k, which is actually, as we learned 260 00:16:18,280 --> 00:16:22,440 before, right is the speed of the propagation 261 00:16:22,440 --> 00:16:26,920 of a harmonic wave. 262 00:16:26,920 --> 00:16:28,900 So basically, if I calculate omega 263 00:16:28,900 --> 00:16:32,980 over k from this equation, then basically what you get 264 00:16:32,980 --> 00:16:38,230 is v square root of 1 plus alpha k squared. 265 00:16:41,680 --> 00:16:45,340 So if you look at this equation, the first reaction 266 00:16:45,340 --> 00:16:51,610 is, oh, now this omega and the k ratio 267 00:16:51,610 --> 00:16:57,780 is not a constant anymore as a function of k. 268 00:16:57,780 --> 00:16:59,310 What does that mean? 269 00:16:59,310 --> 00:17:04,040 That means if I prepare progressing waves 270 00:17:04,040 --> 00:17:08,849 with different wavelengths for wavenumber k, 271 00:17:08,849 --> 00:17:13,920 it's going to be propagating at different speed, OK? 272 00:17:13,920 --> 00:17:19,260 Before we introduce this into the model, 273 00:17:19,260 --> 00:17:25,440 the ratio omega and k is a constant v, independent of k. 274 00:17:25,440 --> 00:17:31,170 Now, once you introduce this model into the equation, 275 00:17:31,170 --> 00:17:35,550 and you plug in the progressing wave solution 276 00:17:35,550 --> 00:17:39,270 to actually check the dispersion relation obtained 277 00:17:39,270 --> 00:17:46,550 from this equation, you'll find that the speed of progressing 278 00:17:46,550 --> 00:17:56,110 wave depends on how distorted this progressing wave is, OK? 279 00:17:56,110 --> 00:18:01,750 So let me compare this to situation 280 00:18:01,750 --> 00:18:06,010 in this graph, omega versus k. 281 00:18:06,010 --> 00:18:10,820 So we will see this dispersion relation graph pretty often 282 00:18:10,820 --> 00:18:12,250 in the class today. 283 00:18:12,250 --> 00:18:17,170 The y-axis is actually the omega, angular frequency, 284 00:18:17,170 --> 00:18:21,990 and the k is the wavenumber, two pi over lambda. 285 00:18:21,990 --> 00:18:23,410 OK. 286 00:18:23,410 --> 00:18:27,100 In the original case, in the case 287 00:18:27,100 --> 00:18:31,180 I have this idealized string, obey 288 00:18:31,180 --> 00:18:33,580 the wave equation which we introduced 289 00:18:33,580 --> 00:18:35,920 in the previous lectures. 290 00:18:35,920 --> 00:18:39,940 If I plug omega as a function of k, what I'm getting 291 00:18:39,940 --> 00:18:40,840 is a straight line. 292 00:18:43,890 --> 00:18:44,628 question. 293 00:18:44,628 --> 00:18:49,020 AUDIENCE: Why are you [INAUDIBLE] 294 00:18:49,020 --> 00:18:50,972 minus alpha [INAUDIBLE]. 295 00:18:53,362 --> 00:18:54,570 YEN-JIE LEE: This one, right? 296 00:18:54,570 --> 00:18:58,680 AUDIENCE: [INAUDIBLE] 297 00:18:58,680 --> 00:19:00,980 YEN-JIE LEE: Oh, maybe I made some mistake here. 298 00:19:00,980 --> 00:19:03,830 So this should be also plus here, right. 299 00:19:12,810 --> 00:19:14,060 So you have this-- 300 00:19:14,060 --> 00:19:21,190 OK, so this is omega squared, and I shouldn't have 301 00:19:21,190 --> 00:19:22,890 this minus sign here, right? 302 00:19:22,890 --> 00:19:26,970 So this should be minus, and this should be-- 303 00:19:26,970 --> 00:19:29,790 OK, let's go back to the original equation, OK. 304 00:19:29,790 --> 00:19:31,820 So basically, you get-- 305 00:19:31,820 --> 00:19:36,180 so if I plug in this equation to this equation, 306 00:19:36,180 --> 00:19:39,540 so basically I get minus omega squared out of it. 307 00:19:39,540 --> 00:19:43,270 And I get minus k squared out of this. 308 00:19:43,270 --> 00:19:47,190 And I'm going to get plus k to the 4 out 309 00:19:47,190 --> 00:19:52,560 of this partial square to the 4 psi partial x to the 4. 310 00:19:52,560 --> 00:19:53,910 Therefore, this would be minus. 311 00:19:53,910 --> 00:19:55,200 OK, maybe I made a mistake. 312 00:19:55,200 --> 00:19:57,360 Thank you very much for spotting that. 313 00:19:57,360 --> 00:20:03,660 AUDIENCE: [INAUDIBLE] 314 00:20:03,660 --> 00:20:05,800 YEN-JIE LEE: Oh, yeah, I'm sorry. 315 00:20:05,800 --> 00:20:07,980 Not my best day today. 316 00:20:07,980 --> 00:20:09,590 AUDIENCE: [INAUDIBLE] 317 00:20:09,590 --> 00:20:10,500 YEN-JIE LEE: Yeah. 318 00:20:10,500 --> 00:20:12,020 Well, then I do it. 319 00:20:12,020 --> 00:20:14,150 OK. 320 00:20:14,150 --> 00:20:15,600 I must be drunk today. 321 00:20:15,600 --> 00:20:17,420 [LAUGHTER] 322 00:20:19,700 --> 00:20:20,860 Thank you very much. 323 00:20:20,860 --> 00:20:23,220 Anymore mistake? 324 00:20:23,220 --> 00:20:23,990 OK. 325 00:20:23,990 --> 00:20:25,730 Fortunately not, right? 326 00:20:25,730 --> 00:20:26,840 OK. 327 00:20:26,840 --> 00:20:27,350 Very good. 328 00:20:27,350 --> 00:20:31,020 So let me do this again. 329 00:20:31,020 --> 00:20:35,610 So now I can modify my wave equation, right? 330 00:20:35,610 --> 00:20:38,110 Originally, the wave equation is partial squared 331 00:20:38,110 --> 00:20:41,810 psi partial t squared equal to v squared partial squared psi 332 00:20:41,810 --> 00:20:43,640 partial x squared. 333 00:20:43,640 --> 00:20:48,470 And now I add additional term, which is actually 334 00:20:48,470 --> 00:20:55,100 proportional to the partial to the 4 psi partial x to the 4. 335 00:20:55,100 --> 00:20:57,620 OK, if I add this term into again. 336 00:20:57,620 --> 00:21:03,560 And now I plug in the wave equation, the progressing wave 337 00:21:03,560 --> 00:21:07,040 solution, into this equation, and I 338 00:21:07,040 --> 00:21:09,020 would get this formula, OK? 339 00:21:09,020 --> 00:21:11,960 So now everything should be correct, 340 00:21:11,960 --> 00:21:15,380 and I have clear evidence that everybody's following. 341 00:21:15,380 --> 00:21:17,030 So that is very good. 342 00:21:17,030 --> 00:21:21,080 And now, I can now cancel all the minus sign, 343 00:21:21,080 --> 00:21:23,090 right and then it's become plus. 344 00:21:23,090 --> 00:21:25,100 And now I can actually calculate what 345 00:21:25,100 --> 00:21:28,700 would be the speed of propagation 346 00:21:28,700 --> 00:21:32,470 for this specific harmonic progressing wave 347 00:21:32,470 --> 00:21:37,040 and omega over k will be equal to v square root of 1 348 00:21:37,040 --> 00:21:39,260 plus alpha k squared. 349 00:21:39,260 --> 00:21:40,910 OK? 350 00:21:40,910 --> 00:21:42,810 Thank you very much for the contribution. 351 00:21:42,810 --> 00:21:49,310 And then now we see that here this ratio depends on k. 352 00:21:49,310 --> 00:21:55,280 So if I plug this on top of the previous curve, which 353 00:21:55,280 --> 00:22:01,310 is actually obtained from here, then what I'm going to get 354 00:22:01,310 --> 00:22:03,340 is something like this. 355 00:22:03,340 --> 00:22:05,942 In the beginning it's pretty close 356 00:22:05,942 --> 00:22:08,900 to the nondispersive case. 357 00:22:08,900 --> 00:22:14,720 And it goes up, because of this alpha contribution. 358 00:22:14,720 --> 00:22:18,740 Alpha is actually a positive number in my model. 359 00:22:18,740 --> 00:22:23,780 And the k is actually the wavenumber. 360 00:22:23,780 --> 00:22:28,170 So what is going to happen is that basically after you 361 00:22:28,170 --> 00:22:32,920 include stiffness, the slope of this curve 362 00:22:32,920 --> 00:22:35,630 is changing as a function of k. 363 00:22:35,630 --> 00:22:36,130 OK? 364 00:22:39,000 --> 00:22:42,010 What do I learn from this exercise 365 00:22:42,010 --> 00:22:49,010 is that if I increase k, if I have a very large k-- 366 00:22:49,010 --> 00:22:55,620 that means I have a very small lambda, because k is actually 367 00:22:55,620 --> 00:22:57,200 2 pi over lambda. 368 00:22:57,200 --> 00:22:57,860 OK? 369 00:22:57,860 --> 00:23:02,060 So that means I'm looking at something really distorted 370 00:23:02,060 --> 00:23:02,630 like this. 371 00:23:05,572 --> 00:23:09,830 Both string tension and the stiffness 372 00:23:09,830 --> 00:23:16,020 wants to restore the string back to normal. 373 00:23:16,020 --> 00:23:18,500 Therefore, what is happening is that you 374 00:23:18,500 --> 00:23:23,840 are going to get additional restoring force. 375 00:23:23,840 --> 00:23:27,350 Therefore, as we actually calculate to here 376 00:23:27,350 --> 00:23:33,260 if alpha is actually positive, then the velocity actually 377 00:23:33,260 --> 00:23:38,770 increased with respect to what we actually 378 00:23:38,770 --> 00:23:42,530 get before we actually had this into a model. 379 00:23:42,530 --> 00:23:45,710 So I think that makes sense, because the stiffness also 380 00:23:45,710 --> 00:23:49,850 wants to restore the distortion. 381 00:23:49,850 --> 00:23:55,520 Therefore, you have larger and larger restoring force. 382 00:23:55,520 --> 00:23:57,920 Therefore, the speed of propagation 383 00:23:57,920 --> 00:24:02,090 of this harmonic wave will increase. 384 00:24:02,090 --> 00:24:04,790 so that's pretty nice. 385 00:24:04,790 --> 00:24:09,670 But what does that mean to our project? 386 00:24:09,670 --> 00:24:13,100 OK, our project is to send information 387 00:24:13,100 --> 00:24:15,630 from one place to the other place, right? 388 00:24:15,630 --> 00:24:19,460 So what we just discussed is that we can actually 389 00:24:19,460 --> 00:24:24,440 send a square pulse and let it propagate. 390 00:24:24,440 --> 00:24:29,360 A square pulse can be decomposed into many, many pieces-- 391 00:24:29,360 --> 00:24:31,155 many, many harmonic waves. 392 00:24:31,155 --> 00:24:32,630 OK? 393 00:24:32,630 --> 00:24:39,050 Before the square pulse works, because all the waves 394 00:24:39,050 --> 00:24:44,150 with different wavelengths should 395 00:24:44,150 --> 00:24:47,775 be moving at this constant speed, independent 396 00:24:47,775 --> 00:24:49,190 of the wavelengths. 397 00:24:49,190 --> 00:24:51,650 Now we are in trouble. 398 00:24:51,650 --> 00:24:58,040 As you can see here, now the speed, which is omega over k 399 00:24:58,040 --> 00:25:03,200 depends on the wavenumber or wavelength. 400 00:25:03,200 --> 00:25:07,460 Therefore, different components, which 401 00:25:07,460 --> 00:25:12,110 actually are needed to create a square pulse, 402 00:25:12,110 --> 00:25:15,000 are going to be propagating at different speed. 403 00:25:15,000 --> 00:25:17,550 You can say, oh, come on, this is actually mathematics, 404 00:25:17,550 --> 00:25:19,340 so I don't believe you. 405 00:25:19,340 --> 00:25:21,440 A square pulse is a square pulse, 406 00:25:21,440 --> 00:25:23,910 and that's mathematics, that's math department. 407 00:25:23,910 --> 00:25:27,850 But we can actually really see that in the experiment. 408 00:25:27,850 --> 00:25:29,060 OK? 409 00:25:29,060 --> 00:25:33,050 So that's-- take a look at this demonstration. 410 00:25:33,050 --> 00:25:34,940 Maybe you didn't notice that before, 411 00:25:34,940 --> 00:25:39,830 but we have seen this effect from the previous lectures. 412 00:25:39,830 --> 00:25:43,685 OK, so I can now create a square-- 413 00:25:43,685 --> 00:25:46,900 not really a square pulse, but actually some kind of pulse. 414 00:25:46,900 --> 00:25:51,500 OK I can create some kind of pulse like this. 415 00:25:51,500 --> 00:25:52,340 OK? 416 00:25:52,340 --> 00:25:56,240 And as we learned before, when this pulse pass 417 00:25:56,240 --> 00:26:00,230 through an open end, it's going to be bounced back. 418 00:26:00,230 --> 00:26:02,780 so therefore, I can have-- 419 00:26:02,780 --> 00:26:07,140 I can actually show you this demo in a limited set-up. 420 00:26:07,140 --> 00:26:11,420 But this pulse is going to be going back and forth, because I 421 00:26:11,420 --> 00:26:14,870 have open end, as we've discussed before. 422 00:26:14,870 --> 00:26:17,810 What is going to happen is that since we have 423 00:26:17,810 --> 00:26:21,950 a realistic system, what is going to happen 424 00:26:21,950 --> 00:26:27,420 is that this pulse will become wider and wider, right? 425 00:26:27,420 --> 00:26:31,370 That's the prediction coming from this equation. 426 00:26:31,370 --> 00:26:34,840 Different component with different wavelengths 427 00:26:34,840 --> 00:26:38,000 is going to be propagating at different speed. 428 00:26:38,000 --> 00:26:41,600 Therefore this pulse is going to become wider, 429 00:26:41,600 --> 00:26:42,680 and we can see that. 430 00:26:42,680 --> 00:26:47,950 OK, so let me quickly produce a pulse and see what will happen. 431 00:26:47,950 --> 00:26:49,170 OK. 432 00:26:49,170 --> 00:26:52,020 Originally, it's actually really sharp. 433 00:26:52,020 --> 00:26:57,030 And you can see that really the width of the pulse 434 00:26:57,030 --> 00:26:59,040 become wider and wider. 435 00:26:59,040 --> 00:27:02,430 And at some point, it disappear. 436 00:27:02,430 --> 00:27:05,400 If I have a very long set-up, what you are going to see 437 00:27:05,400 --> 00:27:08,550 is that it's going to be propagating 438 00:27:08,550 --> 00:27:11,100 toward the same direction. 439 00:27:11,100 --> 00:27:14,190 And the width of the pulse is actually 440 00:27:14,190 --> 00:27:17,460 going to be increasing as a function of time. 441 00:27:17,460 --> 00:27:20,400 Let's take a look at this again. 442 00:27:20,400 --> 00:27:23,100 Now, this time we have a negative pulse. 443 00:27:23,100 --> 00:27:24,540 You sort of see-- 444 00:27:24,540 --> 00:27:25,432 very similar, see. 445 00:27:25,432 --> 00:27:26,890 And also you can see that there are 446 00:27:26,890 --> 00:27:33,420 some strange vibration actually left behind the main pulse. 447 00:27:33,420 --> 00:27:41,520 So that means harmonic waves with different wavelengths 448 00:27:41,520 --> 00:27:47,080 really propagating at different speed. 449 00:27:47,080 --> 00:27:49,900 And for that, to demonstrate this effect, 450 00:27:49,900 --> 00:27:52,870 I also prepared some demonstration, 451 00:27:52,870 --> 00:27:58,000 which actually are based on our calculation, OK. 452 00:27:58,000 --> 00:28:04,160 So you can say that, OK, now I'm convinced I can see dispersion 453 00:28:04,160 --> 00:28:05,170 in the experiment. 454 00:28:05,170 --> 00:28:08,830 How do I know this calculation actually match 455 00:28:08,830 --> 00:28:10,360 with the experimental data, right? 456 00:28:10,360 --> 00:28:14,955 How about we really run a simulation and see what 457 00:28:14,955 --> 00:28:16,180 would happen. 458 00:28:16,180 --> 00:28:22,920 So what this example actually do is, in the beginning, 459 00:28:22,920 --> 00:28:26,500 you would do integration like crazy 460 00:28:26,500 --> 00:28:30,040 in order to get all the components calculated. 461 00:28:30,040 --> 00:28:33,640 Then it's going to propagate all those pulses-- 462 00:28:33,640 --> 00:28:37,840 all those pulse with different components through the medium, 463 00:28:37,840 --> 00:28:38,620 OK? 464 00:28:38,620 --> 00:28:41,080 And then there will be two different colors, 465 00:28:41,080 --> 00:28:44,840 one is actually blue, which is the original shape. 466 00:28:44,840 --> 00:28:47,110 The other one is actually the one 467 00:28:47,110 --> 00:28:50,110 which is stiffness turned up. 468 00:28:50,110 --> 00:28:54,220 So now, in the beginning I can set the alpha value 469 00:28:54,220 --> 00:28:57,200 to be 0.02 and see what will happen. 470 00:28:57,200 --> 00:29:01,870 And I will put produce triangular pulse. 471 00:29:01,870 --> 00:29:02,860 You can see that now. 472 00:29:02,860 --> 00:29:05,380 The program is really working very hard 473 00:29:05,380 --> 00:29:10,930 to capture all the components from 1 to 199 and equal to 1 474 00:29:10,930 --> 00:29:12,460 until 99. 475 00:29:12,460 --> 00:29:16,810 And then now, these individual components 476 00:29:16,810 --> 00:29:19,580 are propagating through the medium. 477 00:29:19,580 --> 00:29:23,490 And you can see that originally the shape is like-- 478 00:29:23,490 --> 00:29:27,050 the blue shape-- triangular shape. 479 00:29:27,050 --> 00:29:30,470 And you can see that is a function of time. 480 00:29:30,470 --> 00:29:34,640 The pulse become wider and wider, OK? 481 00:29:34,640 --> 00:29:39,200 Now, of course, I can increase the alpha to 0.02 482 00:29:39,200 --> 00:29:40,910 and see what happen-- 483 00:29:40,910 --> 00:29:43,550 from 0.02 to 0.2 and see what will happen. 484 00:29:43,550 --> 00:29:48,050 You should expect a much larger dispersion. 485 00:29:48,050 --> 00:29:50,450 And you can see that now in the beginning, 486 00:29:50,450 --> 00:29:54,560 it's doing the integration. 487 00:29:54,560 --> 00:29:56,890 And you can see that this time because the alpha 488 00:29:56,890 --> 00:29:58,640 is actually larger. 489 00:29:58,640 --> 00:30:03,110 Therefore, you see that this effect, this broadening, 490 00:30:03,110 --> 00:30:05,350 is actually happening earlier, and it 491 00:30:05,350 --> 00:30:07,640 become broader and broader, and that there 492 00:30:07,640 --> 00:30:11,120 are a lot of strange structures, as you can see also 493 00:30:11,120 --> 00:30:15,250 from the demo, produce because different components are 494 00:30:15,250 --> 00:30:18,070 actually propagating at different speeds. 495 00:30:21,200 --> 00:30:26,560 So of course, we are MIT, so in this course we have MIT-- 496 00:30:26,560 --> 00:30:28,290 MIT waves. 497 00:30:28,290 --> 00:30:31,460 So let's take a look at the MIT wave and see what will happen. 498 00:30:31,460 --> 00:30:35,240 Now you see that there are very sharp edge, which 499 00:30:35,240 --> 00:30:39,680 actually require really a lot of effort to reproduce that. 500 00:30:39,680 --> 00:30:44,320 And you can see that MIT is kind of distorted 501 00:30:44,320 --> 00:30:46,140 as a function of time. 502 00:30:46,140 --> 00:30:48,775 We can kind of still identify the peak, 503 00:30:48,775 --> 00:30:51,270 but it's actually now displaced. 504 00:30:51,270 --> 00:30:54,900 And in the end of the simulation, 505 00:30:54,900 --> 00:30:57,356 you can not even recognize that's actually 506 00:30:57,356 --> 00:31:03,540 originally MIT signal, which was sent from your source. 507 00:31:03,540 --> 00:31:08,790 So what I want to say is that this effect, this dispersion 508 00:31:08,790 --> 00:31:13,980 effect, is really an enemy, which 509 00:31:13,980 --> 00:31:15,570 is actually very dangerous. 510 00:31:15,570 --> 00:31:17,790 And that actually will prevent us 511 00:31:17,790 --> 00:31:22,390 from sending high quality signals. 512 00:31:22,390 --> 00:31:25,890 OK, any questions about all those demos? 513 00:31:28,650 --> 00:31:30,108 Yes. 514 00:31:30,108 --> 00:31:32,052 AUDIENCE: Why do we model the [INAUDIBLE]?? 515 00:31:36,510 --> 00:31:40,972 YEN-JIE LEE: So this is because the stiffness is actually 516 00:31:40,972 --> 00:31:41,680 symmetric, right. 517 00:31:41,680 --> 00:31:47,535 So if you bend the string, then there 518 00:31:47,535 --> 00:31:52,820 are contribution from the positive and negative part, OK? 519 00:31:52,820 --> 00:31:57,520 If you have partial to the 3, partial to the x to the 3 520 00:31:57,520 --> 00:32:00,710 component, then it's going to be a symmetric and so actually 521 00:32:00,710 --> 00:32:03,110 against our physics intuition. 522 00:32:03,110 --> 00:32:05,960 And also, in this modeling, you also 523 00:32:05,960 --> 00:32:09,490 match with our experimental data pretty well. 524 00:32:09,490 --> 00:32:10,270 OK. 525 00:32:10,270 --> 00:32:11,990 Very good question. 526 00:32:11,990 --> 00:32:16,760 And on the other hand, we now consider then the stiffness. 527 00:32:16,760 --> 00:32:20,270 you can also go back to the infinite number coupled 528 00:32:20,270 --> 00:32:21,740 oscillator case. 529 00:32:21,740 --> 00:32:28,190 If you instead take an example which is actually not 530 00:32:28,190 --> 00:32:32,600 super small displacement approximation, 531 00:32:32,600 --> 00:32:34,580 you take the next to leading order term. 532 00:32:37,470 --> 00:32:40,580 Then you will see that the partial to the 3 partial x 533 00:32:40,580 --> 00:32:43,540 to the 3 term as you cancel because it's symmetric, 534 00:32:43,540 --> 00:32:45,200 or so I argued. 535 00:32:45,200 --> 00:32:48,860 And then you will be able to also obtain this tern when 536 00:32:48,860 --> 00:32:53,390 you have slightly larger displacement with respect 537 00:32:53,390 --> 00:32:56,590 to the equilibrium position. 538 00:32:56,590 --> 00:32:58,190 So I hope that answers your question. 539 00:32:58,190 --> 00:32:58,981 Any other question? 540 00:32:58,981 --> 00:33:00,098 Yes? 541 00:33:00,098 --> 00:33:03,930 AUDIENCE: If you were looking at [INAUDIBLE],, for example, what 542 00:33:03,930 --> 00:33:08,037 would be [INAUDIBLE]? 543 00:33:08,037 --> 00:33:09,953 YEN-JIE LEE: When you pass through the medium. 544 00:33:09,953 --> 00:33:12,510 AUDIENCE: So [INAUDIBLE] 545 00:33:12,510 --> 00:33:14,010 YEN-JIE LEE: A molecule can actually 546 00:33:14,010 --> 00:33:18,270 change the speed of different wavelengths, 547 00:33:18,270 --> 00:33:19,980 actually, differently, right? 548 00:33:19,980 --> 00:33:21,450 Very good question. 549 00:33:21,450 --> 00:33:23,260 OK, so very good. 550 00:33:23,260 --> 00:33:26,580 We got two questions, and we can see that if I now 551 00:33:26,580 --> 00:33:32,260 turn on the alpha and make the alpha value large, 552 00:33:32,260 --> 00:33:35,940 then you can see that the information is distorted. 553 00:33:35,940 --> 00:33:41,030 And this involve infinite number of terms. 554 00:33:41,030 --> 00:33:44,610 And in this case, in this new demo which I show here, 555 00:33:44,610 --> 00:33:47,200 I have alpha value equal to 0.2. 556 00:33:47,200 --> 00:33:52,260 Therefore, the effect of dispersion 557 00:33:52,260 --> 00:33:55,170 is actually much larger than what you showed before. 558 00:33:55,170 --> 00:33:57,690 And then you can see that this MIT wave quickly 559 00:33:57,690 --> 00:34:02,230 become something like a Gaussian-like wave, right? 560 00:34:02,230 --> 00:34:03,630 OK, so very good. 561 00:34:03,630 --> 00:34:11,370 So you can say, OK, you are making an example-- 562 00:34:11,370 --> 00:34:16,090 it's a very interesting example, but it involve too many terms. 563 00:34:16,090 --> 00:34:21,120 You have infinite number of progressing waves 564 00:34:21,120 --> 00:34:22,739 in this example. 565 00:34:22,739 --> 00:34:25,590 It's very difficult to understand. 566 00:34:25,590 --> 00:34:31,989 How about we go back to a much simpler example, OK? 567 00:34:31,989 --> 00:34:37,290 What we can do is that instead of going 568 00:34:37,290 --> 00:34:42,570 through infinite number of harmonic waves, now 569 00:34:42,570 --> 00:34:46,750 we just consider two waves, and overlap these two 570 00:34:46,750 --> 00:34:50,520 waves together and see what will happen. 571 00:34:50,520 --> 00:34:52,290 And let's see what we can learn from it, 572 00:34:52,290 --> 00:34:58,360 because the required number of harmonic wave to describe such 573 00:34:58,360 --> 00:35:01,170 a pulse is too complicated. 574 00:35:01,170 --> 00:35:04,980 So you can say that, OK, now let's just consider two waves 575 00:35:04,980 --> 00:35:06,600 and see what we can learn from this. 576 00:35:06,600 --> 00:35:10,570 And this is actually what I am going to do now. 577 00:35:10,570 --> 00:35:13,050 So from Bolek's lecture I hope that he 578 00:35:13,050 --> 00:35:15,850 covered the beat phenomenon. 579 00:35:15,850 --> 00:35:17,230 So basically, what is it? 580 00:35:17,230 --> 00:35:18,590 A beat phenomenon? 581 00:35:18,590 --> 00:35:22,380 Beat phenomenon happens when you overlap 582 00:35:22,380 --> 00:35:26,100 two waves, two harmonic waves. 583 00:35:26,100 --> 00:35:31,450 They have pretty close wavelengths. 584 00:35:31,450 --> 00:35:33,800 OK, but they're not the same. 585 00:35:33,800 --> 00:35:36,980 And now, if you add two waves together, 586 00:35:36,980 --> 00:35:38,730 that's actually what you are going to get. 587 00:35:38,730 --> 00:35:42,880 You are going to get something which is oscillating really, 588 00:35:42,880 --> 00:35:46,740 really fast, which is basically called the carrier. 589 00:35:46,740 --> 00:35:51,570 And also you can see that the magnitude of the oscillation 590 00:35:51,570 --> 00:35:54,910 is actually changing as a function of position, 591 00:35:54,910 --> 00:35:58,020 and that we call envelope. 592 00:35:58,020 --> 00:35:59,940 So that's essentially the beat phenomenon, 593 00:35:59,940 --> 00:36:02,740 which you learned from previous lectures. 594 00:36:02,740 --> 00:36:13,070 So in this example, I'm going to add two waves together. 595 00:36:13,070 --> 00:36:15,920 So the first wave is described by-- 596 00:36:15,920 --> 00:36:18,230 OK, is denoted by side one. 597 00:36:18,230 --> 00:36:23,020 It's a function of x and t, and it has a function of form 598 00:36:23,020 --> 00:36:25,740 A is the amplitude. 599 00:36:25,740 --> 00:36:32,580 And the sine k1 x minus omega1 t. 600 00:36:32,580 --> 00:36:35,050 This is actually a progressing wave 601 00:36:35,050 --> 00:36:37,360 propagating toward the right-hand side 602 00:36:37,360 --> 00:36:42,330 of the board, the positive direction of the x-axis 603 00:36:42,330 --> 00:36:44,500 in my coordinate system. 604 00:36:44,500 --> 00:36:50,850 And it has a wavenumber of k1 and angular frequency omega1 605 00:36:50,850 --> 00:36:53,230 And I can also write down my second wave, 606 00:36:53,230 --> 00:36:56,950 which I would like to overlap with the first wave. 607 00:36:56,950 --> 00:37:00,830 So this is actually having exactly the same amplitude, 608 00:37:00,830 --> 00:37:06,030 which is A. And it is described by a sine function, 609 00:37:06,030 --> 00:37:14,110 and you have a wavenumber k2 x minus omega2 t, angular 610 00:37:14,110 --> 00:37:17,620 frequency omega2. 611 00:37:17,620 --> 00:37:22,260 With these two equations, we can calculate 612 00:37:22,260 --> 00:37:25,520 the speed of propagation for the individual waves, right? 613 00:37:25,520 --> 00:37:28,160 So the first one, I can calculate 614 00:37:28,160 --> 00:37:33,350 the speed of propagation v1 would be equal to omega1 615 00:37:33,350 --> 00:37:36,420 over k1. 616 00:37:36,420 --> 00:37:38,600 Very similarly, you can also calculate 617 00:37:38,600 --> 00:37:41,100 the speed of propagation for the second wave, 618 00:37:41,100 --> 00:37:45,260 which is omega2 over k2. 619 00:37:45,260 --> 00:37:48,515 So now what I'm going to do is to calculate 620 00:37:48,515 --> 00:37:50,870 a sum of these two waves. 621 00:37:50,870 --> 00:37:56,900 So I have the total, which is psi is equal to psi1 plus psi2. 622 00:37:59,500 --> 00:38:02,530 So what I'm going to do is to overlap these two waves 623 00:38:02,530 --> 00:38:03,980 and see what will happen. 624 00:38:03,980 --> 00:38:06,310 And for that, I need this formula, 625 00:38:06,310 --> 00:38:11,560 which is a sine A plus sine B. And this 626 00:38:11,560 --> 00:38:21,670 would be equal to 2 times sine A plus B over 2 and sine-- 627 00:38:21,670 --> 00:38:24,200 it would become cosine here-- 628 00:38:24,200 --> 00:38:30,990 cosine A minus B over 2. 629 00:38:30,990 --> 00:38:35,660 So if I use that formula, basically what I'm going to get 630 00:38:35,660 --> 00:38:36,410 is-- 631 00:38:36,410 --> 00:38:38,780 we have two times from the formula. 632 00:38:38,780 --> 00:38:50,430 So if you have 2A sine k1 plus k2 over 2x 633 00:38:50,430 --> 00:38:56,850 minus omega1 plus omega2 over 2. 634 00:38:56,850 --> 00:39:00,540 So basically, the first term is the sine function. 635 00:39:00,540 --> 00:39:03,150 The sine function and the content 636 00:39:03,150 --> 00:39:07,840 is actually A plus B. Therefore, you add these two together, 637 00:39:07,840 --> 00:39:10,170 divide it by two, then basically this 638 00:39:10,170 --> 00:39:13,260 is as actually what you obtain. 639 00:39:13,260 --> 00:39:16,530 The second term is a cosine term. 640 00:39:16,530 --> 00:39:18,370 You get a cosine here. 641 00:39:18,370 --> 00:39:21,450 But now you calculate A minus B, which 642 00:39:21,450 --> 00:39:25,520 is this term minus that term divided by 2. 643 00:39:25,520 --> 00:39:29,850 Then basically what you get is k1 minus k2 divided 644 00:39:29,850 --> 00:39:36,570 by 2 times x minus omega1 minus omega2 over 2 t. 645 00:39:39,480 --> 00:39:41,540 OK, so now this actually-- 646 00:39:41,540 --> 00:39:46,140 what would happen if you add these two waves together? 647 00:39:49,000 --> 00:39:52,540 Until now, everything is exact. 648 00:39:52,540 --> 00:39:56,770 And I would like to add additional conditions 649 00:39:56,770 --> 00:40:01,360 or additional assumptions when I discuss this solution. 650 00:40:01,360 --> 00:40:02,200 OK? 651 00:40:02,200 --> 00:40:06,670 So how about in order to produce the beat phenomenon, 652 00:40:06,670 --> 00:40:11,110 I need to make the wavelengths very, very 653 00:40:11,110 --> 00:40:14,140 similar between the two waves. 654 00:40:14,140 --> 00:40:16,060 So therefore, what I am going to do 655 00:40:16,060 --> 00:40:22,090 is that I'm going to assume k1 is 656 00:40:22,090 --> 00:40:25,770 very close to k2 is roughly k. 657 00:40:29,090 --> 00:40:33,220 And because of this, since I have a continuous function, 658 00:40:33,220 --> 00:40:36,270 if k1 is really close to k2, that 659 00:40:36,270 --> 00:40:41,680 means omega1 is going to be also very close to omega2, right? 660 00:40:41,680 --> 00:40:45,010 So what I'm going to get is omega1 is 661 00:40:45,010 --> 00:40:48,820 going to be also very similar to omega2, 662 00:40:48,820 --> 00:40:50,760 and I will call it omega. 663 00:40:54,100 --> 00:41:02,230 So if I do this, when I have very similar k1 and k2, 664 00:41:02,230 --> 00:41:03,760 what is going to happen? 665 00:41:03,760 --> 00:41:11,050 What is going to happen is that k1 minus k2 will be very small. 666 00:41:11,050 --> 00:41:18,250 So this very small k means larger wavelengths. 667 00:41:18,250 --> 00:41:24,870 Therefore, this cosine term will become the envelope, 668 00:41:24,870 --> 00:41:29,560 because it's actually a slowly variating amplitude 669 00:41:29,560 --> 00:41:34,240 as a function of position, because the k is very small. 670 00:41:34,240 --> 00:41:37,070 K is small means lambda large. 671 00:41:37,070 --> 00:41:39,910 Therefore, the amplitude is going 672 00:41:39,910 --> 00:41:43,420 to be having this modulation, which is actually 673 00:41:43,420 --> 00:41:48,977 like the envelope, that the oscillation of this envelope 674 00:41:48,977 --> 00:41:50,560 is actually controlled by the k, okay? 675 00:41:53,950 --> 00:41:56,380 Let's look at the left-hand side term. 676 00:41:56,380 --> 00:42:00,280 k1 plus k2 over 2 is kind of like calculating 677 00:42:00,280 --> 00:42:08,320 the average of the wavenumber of the first and second wave. 678 00:42:08,320 --> 00:42:13,100 So if you calculate our average, you can be still pretty large. 679 00:42:13,100 --> 00:42:19,180 Therefore, you have small lambda compared to the difference. 680 00:42:19,180 --> 00:42:20,970 Therefore, you see that that actually 681 00:42:20,970 --> 00:42:25,115 contribute to those little structures in this graph, 682 00:42:25,115 --> 00:42:26,380 and it's called carrier. 683 00:42:26,380 --> 00:42:27,111 Yes? 684 00:42:27,111 --> 00:42:28,073 AUDIENCE: [INAUDIBLE]? 685 00:42:30,959 --> 00:42:34,326 If k1 were a lot bigger than k2, then [INAUDIBLE].. 686 00:42:39,140 --> 00:42:41,910 YEN-JIE LEE: So they can be different. 687 00:42:41,910 --> 00:42:43,920 Yeah, so you are absolutely right. 688 00:42:43,920 --> 00:42:49,130 So you can produce something like a carrier 689 00:42:49,130 --> 00:42:51,480 even when k1 is not equal to k2, right? 690 00:42:51,480 --> 00:42:53,470 Its just a average. 691 00:42:53,470 --> 00:42:54,050 You're right. 692 00:42:54,050 --> 00:43:00,130 But then on the other hand, the difference, k1 and k2 693 00:43:00,130 --> 00:43:02,040 will be also large. 694 00:43:02,040 --> 00:43:05,570 Therefore, it's not as easy as what 695 00:43:05,570 --> 00:43:09,370 we have been doing here to identify who is the carrier 696 00:43:09,370 --> 00:43:11,220 and who is the envelope. 697 00:43:11,220 --> 00:43:14,630 But you do get some kind of graph, 698 00:43:14,630 --> 00:43:16,370 which is oscillating really fast, 699 00:43:16,370 --> 00:43:20,550 but the envelope is going to be also oscillating very fast. 700 00:43:20,550 --> 00:43:23,660 That is harder to see all the structure. 701 00:43:23,660 --> 00:43:26,210 But you're absolutely right, yes. 702 00:43:26,210 --> 00:43:27,740 Very good question. 703 00:43:27,740 --> 00:43:30,320 So now I have this set-up. 704 00:43:30,320 --> 00:43:33,180 I assume that they are very close to each other. 705 00:43:33,180 --> 00:43:37,320 So now I can define phase velocity. 706 00:43:37,320 --> 00:43:41,660 Finally, we define what is actually the phase velocity. 707 00:43:41,660 --> 00:43:44,650 In The phase velocity-- 708 00:43:44,650 --> 00:43:46,570 I call it vp-- 709 00:43:46,570 --> 00:43:49,220 you can see that before I already 710 00:43:49,220 --> 00:43:52,400 have been using phase velocity vp 711 00:43:52,400 --> 00:43:55,060 for the previous discussions. 712 00:43:55,060 --> 00:43:58,280 In the case of nondispersive medium, 713 00:43:58,280 --> 00:44:01,070 the phase velocity is just a vp, which is 714 00:44:01,070 --> 00:44:04,170 the velocity in the equation. 715 00:44:04,170 --> 00:44:09,140 And in this case, vp will be equal to omega over k, 716 00:44:09,140 --> 00:44:10,254 as we discussed before. 717 00:44:10,254 --> 00:44:11,670 And that's actually the definition 718 00:44:11,670 --> 00:44:15,320 of this phase velocity. 719 00:44:15,320 --> 00:44:18,350 And I can now also define the group velocity. 720 00:44:23,800 --> 00:44:31,340 The group velocity is actually the velocity of the envelope. 721 00:44:31,340 --> 00:44:36,140 I can calculate the velocity of the envelope. 722 00:44:36,140 --> 00:44:38,030 in the case of phase velocity, I'm 723 00:44:38,030 --> 00:44:42,590 calculating the velocity of the carrier. 724 00:44:42,590 --> 00:44:44,790 I'm taking a ratio of the average, 725 00:44:44,790 --> 00:44:48,800 and actually the average is so close to k and omega, 726 00:44:48,800 --> 00:44:51,620 therefore the phase velocity vp would 727 00:44:51,620 --> 00:44:55,710 be just the speed of the propagation of the carrier, 728 00:44:55,710 --> 00:44:57,750 which is actually omega over k. 729 00:44:57,750 --> 00:44:59,060 I call it vp. 730 00:44:59,060 --> 00:45:04,040 And in case of group velocity, I call it vg. 731 00:45:04,040 --> 00:45:08,210 vg is describing the speed of propagation of the envelope. 732 00:45:08,210 --> 00:45:16,220 Therefore, what I am getting is omega1 minus omega2 733 00:45:16,220 --> 00:45:19,770 divided by k1 minus k2. 734 00:45:19,770 --> 00:45:23,220 Both of them have effect of 1 over 2, 735 00:45:23,220 --> 00:45:25,940 which we say is canceled. 736 00:45:25,940 --> 00:45:29,220 And when they are really so close to each other, 737 00:45:29,220 --> 00:45:33,161 this is actually roughly like d omega dk. 738 00:45:37,840 --> 00:45:38,770 Any questions so far? 739 00:45:42,280 --> 00:45:47,140 So we have derived two different kinds of speed. 740 00:45:47,140 --> 00:45:51,710 One is actually related to the phase velocity, which is-- 741 00:45:51,710 --> 00:45:54,790 one is actually called the phase velocity. 742 00:45:54,790 --> 00:45:59,260 It's related to the speed of the carrier. 743 00:45:59,260 --> 00:46:02,620 The other one is group velocity, which is actually related 744 00:46:02,620 --> 00:46:06,340 to the speed of the envelope. 745 00:46:06,340 --> 00:46:12,080 So let me describe you a few interesting examples. 746 00:46:12,080 --> 00:46:17,470 And let's see what we can actually learn from this. 747 00:46:17,470 --> 00:46:21,820 In the first example, I'm working on a non dispersive 748 00:46:21,820 --> 00:46:23,320 medium, OK? 749 00:46:23,320 --> 00:46:33,430 If I have a nondispersive medium, 750 00:46:33,430 --> 00:46:34,960 then basically what I'm going to get 751 00:46:34,960 --> 00:46:40,970 is that omega will be proportional to k. 752 00:46:40,970 --> 00:46:45,410 If I plot omega versus k, it's a straight line. 753 00:46:45,410 --> 00:46:49,310 Now, if I have omega-- 754 00:46:49,310 --> 00:46:53,780 I choose the omega of the two, omega1, omega2, 755 00:46:53,780 --> 00:46:54,830 of the two waves-- 756 00:46:54,830 --> 00:47:03,250 to be roughly equal to omega 0, I can now evaluate the vp. 757 00:47:03,250 --> 00:47:11,290 The vp will be the slope of this point 758 00:47:11,290 --> 00:47:13,390 on the slope of a line connecting 759 00:47:13,390 --> 00:47:17,570 the 0 to that point, which is actually the omega over k, 760 00:47:17,570 --> 00:47:18,070 right? 761 00:47:18,070 --> 00:47:19,445 So that's actually the definition 762 00:47:19,445 --> 00:47:21,430 of the phase velocity. 763 00:47:21,430 --> 00:47:24,450 I would get this slope. 764 00:47:24,450 --> 00:47:28,480 The slope of this line is actually 765 00:47:28,480 --> 00:47:32,830 related to the phase velocity. 766 00:47:32,830 --> 00:47:37,740 I can also calculate the slope of a line cuts 767 00:47:37,740 --> 00:47:39,770 through this point. 768 00:47:39,770 --> 00:47:43,610 But as it cuts through this curve, and in this case, 769 00:47:43,610 --> 00:47:50,080 I'm also going to get a line overlapping with phase 770 00:47:50,080 --> 00:47:54,040 velocity, because in this case, omega over k 771 00:47:54,040 --> 00:47:58,360 is a constant, which is v. Therefore, no matter 772 00:47:58,360 --> 00:48:02,290 what you calculate, if you calculate vp 773 00:48:02,290 --> 00:48:07,210 as a ratio of omega and a k, where you calculate vg, which 774 00:48:07,210 --> 00:48:09,910 is actually the slope of the line cutting 775 00:48:09,910 --> 00:48:15,820 through that point, is you always get actually v. 776 00:48:15,820 --> 00:48:17,950 Therefore, what we learned from here 777 00:48:17,950 --> 00:48:23,980 is that for a nondispersive medium, vp will be equal to vg. 778 00:48:28,200 --> 00:48:32,850 That means both of these two curves, 779 00:48:32,850 --> 00:48:37,120 both of the curve of envelope, describing the envelope 780 00:48:37,120 --> 00:48:38,950 and then describing the carrier, is 781 00:48:38,950 --> 00:48:41,850 going to be propagating at the same speed. 782 00:48:41,850 --> 00:48:43,580 OK, any questions? 783 00:48:43,580 --> 00:48:51,230 So the whole thing is going to be moving at constant speed. 784 00:48:51,230 --> 00:48:59,560 For that, I can now show you some example, which I prepared, 785 00:48:59,560 --> 00:49:01,460 some simulation which I prepared. 786 00:49:14,780 --> 00:49:19,060 So what it does is that it really-- 787 00:49:19,060 --> 00:49:20,410 oh, wait a second. 788 00:49:20,410 --> 00:49:25,510 This is 0. 789 00:49:25,510 --> 00:49:31,540 OK, so this is the case when I have a nondispersive medium. 790 00:49:31,540 --> 00:49:34,810 if I have a nondispersive medium, what is going to happen 791 00:49:34,810 --> 00:49:40,380 is that both the carrier, which is the speed of all 792 00:49:40,380 --> 00:49:43,000 those little structures, and the envelope 793 00:49:43,000 --> 00:49:45,210 is going to be propagating at the same speed. 794 00:49:45,210 --> 00:49:48,250 So you can see the high is like a fixed pattern. 795 00:49:48,250 --> 00:49:51,700 It's propagating toward the right-hand side. 796 00:49:51,700 --> 00:49:55,330 And the relative motion between the defined structure 797 00:49:55,330 --> 00:49:58,000 and the envelope is actually 0. 798 00:49:58,000 --> 00:50:00,750 So basically you have exactly the same pattern 799 00:50:00,750 --> 00:50:03,820 as a function of time. 800 00:50:03,820 --> 00:50:09,070 So now I'm going to move away from the nondispersive medium. 801 00:50:09,070 --> 00:50:11,620 How about we discuss what would happen 802 00:50:11,620 --> 00:50:16,600 if we have considered the stiffness of the string 803 00:50:16,600 --> 00:50:19,730 and see what we get from there. 804 00:50:19,730 --> 00:50:27,760 So if I plugged omega as a function of k, 805 00:50:27,760 --> 00:50:31,450 and consider alpha to be non-zero. 806 00:50:31,450 --> 00:50:32,990 It's a positive value. 807 00:50:32,990 --> 00:50:36,930 So if I have alpha to be a positive value, 808 00:50:36,930 --> 00:50:41,560 non-zero, in this case, I'm going to get a curve like this. 809 00:50:44,370 --> 00:50:50,970 The slope is actually changing and it's curving up 810 00:50:50,970 --> 00:50:54,840 because if you have k large, then 811 00:50:54,840 --> 00:51:00,150 you would see that the ratio of omega and k actually increase. 812 00:51:00,150 --> 00:51:01,870 So this is actually the kind of curve 813 00:51:01,870 --> 00:51:06,690 which we would get if I set the omega 814 00:51:06,690 --> 00:51:12,150 of the first and second wave of interest 815 00:51:12,150 --> 00:51:16,080 in this study to be omega 0. 816 00:51:16,080 --> 00:51:18,660 Then basically, what you are going to get is that-- 817 00:51:18,660 --> 00:51:23,340 OK, now I have this point here on the curve. 818 00:51:23,340 --> 00:51:27,492 If I calculate the phase velocity-- 819 00:51:27,492 --> 00:51:29,700 the phase velocity, how do I calculate that? 820 00:51:29,700 --> 00:51:37,010 I can now connect 0 and the point by a line. 821 00:51:37,010 --> 00:51:40,220 And I can now calculate the slope of this line, 822 00:51:40,220 --> 00:51:43,510 and I can get the phase velocity, vp. 823 00:51:46,820 --> 00:51:51,500 On the other hand, I can also calculate 824 00:51:51,500 --> 00:51:55,550 the slope of a line cutting through, tangential 825 00:51:55,550 --> 00:51:59,030 to the point of interest. 826 00:51:59,030 --> 00:52:04,560 And that is going to give me the group velocity. 827 00:52:04,560 --> 00:52:09,100 As you can see from here, which slope is actually larger? 828 00:52:09,100 --> 00:52:10,140 Anybody know? 829 00:52:10,140 --> 00:52:12,670 Can point it out? 830 00:52:12,670 --> 00:52:15,450 Group velocity's larger, right? 831 00:52:15,450 --> 00:52:18,750 So in this case, if I turn on alpha greater 832 00:52:18,750 --> 00:52:22,410 than 0, what is going to happen is that, since the group 833 00:52:22,410 --> 00:52:27,060 velocity is larger than the phase velocity, that means, 834 00:52:27,060 --> 00:52:31,310 if I go back to that picture, the envelope 835 00:52:31,310 --> 00:52:35,850 is going to be moving faster than the fine structure 836 00:52:35,850 --> 00:52:37,440 inside the envelope. 837 00:52:37,440 --> 00:52:40,170 How about we take a five-minute break from here? 838 00:52:40,170 --> 00:52:42,816 And then we continue the discussion after the break. 839 00:52:42,816 --> 00:52:44,190 It's a good time to take a break. 840 00:52:50,250 --> 00:52:52,140 Welcome back, everybody. 841 00:52:52,140 --> 00:52:54,180 So we will continue the discussion 842 00:52:54,180 --> 00:52:57,310 of the beat phenomenon. 843 00:52:57,310 --> 00:53:02,431 So what we have shown you is that, based on those curves, 844 00:53:02,431 --> 00:53:03,930 actually can actually determine what 845 00:53:03,930 --> 00:53:06,030 will be the relative velocity-- 846 00:53:09,810 --> 00:53:15,220 what would be the velocity of the carrier, which is actually 847 00:53:15,220 --> 00:53:19,470 denoted by vp, and the what would 848 00:53:19,470 --> 00:53:23,390 be the velocity of the envelope, which is actually 849 00:53:23,390 --> 00:53:28,170 denoted by our group velocity. 850 00:53:28,170 --> 00:53:33,480 And in this case, what I'm actually plotting here 851 00:53:33,480 --> 00:53:37,950 is that, in this case, because alpha is actually 852 00:53:37,950 --> 00:53:43,620 greater than 0, therefore, this curve is actually curving up. 853 00:53:43,620 --> 00:53:47,850 Therefore, you have larger group velocity 854 00:53:47,850 --> 00:53:51,010 compared to the phase velocity. 855 00:53:51,010 --> 00:53:54,360 So what you would expect is that the envelope 856 00:53:54,360 --> 00:53:57,110 is going to be actually progressing 857 00:53:57,110 --> 00:54:03,930 at a speed higher than the speed of the carrier. 858 00:54:03,930 --> 00:54:08,520 On the other hand, if magically I 859 00:54:08,520 --> 00:54:11,310 can construct some kind of medium which 860 00:54:11,310 --> 00:54:14,720 can be described in this situation, 861 00:54:14,720 --> 00:54:19,270 alpha smaller than 0, what is going to happen? 862 00:54:19,270 --> 00:54:26,800 So if I plot a situation with alpha smaller than 0, 863 00:54:26,800 --> 00:54:30,690 so now I plot omega was a function of k. 864 00:54:34,210 --> 00:54:36,890 What is going to happen is that this-- so basically, 865 00:54:36,890 --> 00:54:39,760 you have something which is actually curving downward. 866 00:54:43,670 --> 00:54:51,750 So if I now, again, work on some point of interest here, 867 00:54:51,750 --> 00:54:57,050 you can see that the slope of the phase velocity 868 00:54:57,050 --> 00:55:08,570 is now actually larger than the slope, which is actually 869 00:55:08,570 --> 00:55:11,210 from the line cutting through the-- 870 00:55:11,210 --> 00:55:15,350 tangential to the curve, which is actually 871 00:55:15,350 --> 00:55:20,510 getting you the group velocity. 872 00:55:20,510 --> 00:55:23,890 So in the case of alpha's more than 0, 873 00:55:23,890 --> 00:55:25,940 which is some strange medium I can 874 00:55:25,940 --> 00:55:28,290 which I can create from whatever, plasma, 875 00:55:28,290 --> 00:55:35,060 or some really strange new kind of material of interest. 876 00:55:35,060 --> 00:55:39,410 If that happens, then that means your group velocity 877 00:55:39,410 --> 00:55:43,478 will be smoother than the phase velocity. 878 00:55:48,340 --> 00:55:51,560 And if you look at this point here, 879 00:55:51,560 --> 00:55:57,790 you can see that this curve actually reach a maxima here. 880 00:55:57,790 --> 00:56:02,140 And if you actually are operating at this point, what 881 00:56:02,140 --> 00:56:03,910 is going to happen? 882 00:56:03,910 --> 00:56:08,650 What is going to happen is that if you calculate the group 883 00:56:08,650 --> 00:56:11,610 velocity, what will be the value? 884 00:56:11,610 --> 00:56:13,367 It will be 0. 885 00:56:13,367 --> 00:56:14,200 What does that mean? 886 00:56:14,200 --> 00:56:19,280 That means the envelope will not be moving a lot, 887 00:56:19,280 --> 00:56:23,910 but the carriers are still moving. 888 00:56:23,910 --> 00:56:28,190 So at this point, you are going to get 889 00:56:28,190 --> 00:56:31,130 group velocity equal to 0. 890 00:56:34,700 --> 00:56:41,750 And finally, the if you actually going to a very large k 891 00:56:41,750 --> 00:56:45,520 value in this scenario, alpha smaller than 0, 892 00:56:45,520 --> 00:56:49,020 you will see that even you can have 893 00:56:49,020 --> 00:56:56,460 phase velocity, vp, positive, because it's actually 894 00:56:56,460 --> 00:56:58,400 a positive slope. 895 00:56:58,400 --> 00:57:06,520 And that the group velocity actually is negative. 896 00:57:06,520 --> 00:57:07,920 What does that mean? 897 00:57:07,920 --> 00:57:13,660 That means you are going to see a situation that the carriers 898 00:57:13,660 --> 00:57:17,630 are progressing in the positive direction, 899 00:57:17,630 --> 00:57:24,910 and the envelope is going to be progressing 900 00:57:24,910 --> 00:57:29,440 in the negative direction, probably 901 00:57:29,440 --> 00:57:33,490 progressing to the left-hand side of the board. 902 00:57:33,490 --> 00:57:34,910 So what does that mean? 903 00:57:34,910 --> 00:57:40,820 That means this wave is doing what Michael Jackson's doing. 904 00:57:40,820 --> 00:57:42,875 It's actually doing the moonwalk. 905 00:57:42,875 --> 00:57:45,310 [LAUGHTER] 906 00:57:45,310 --> 00:57:48,880 So this is actually the kind of thing 907 00:57:48,880 --> 00:57:53,110 which could have happened, that it looks like 908 00:57:53,110 --> 00:57:57,910 and that you are doing-- 909 00:57:57,910 --> 00:58:03,280 going forward, because all the carriers 910 00:58:03,280 --> 00:58:05,460 are moving in a positive direction. 911 00:58:05,460 --> 00:58:10,660 But the body is actually going toward negative direction. 912 00:58:10,660 --> 00:58:13,882 maybe I can also learn moonwalk at some point. 913 00:58:13,882 --> 00:58:15,090 [LAUGHTER] 914 00:58:15,090 --> 00:58:16,630 OK. 915 00:58:16,630 --> 00:58:19,490 So let's go back to the demonstration 916 00:58:19,490 --> 00:58:24,160 which I got started, and somehow I got messed up. 917 00:58:24,160 --> 00:58:28,210 So let's take a look at the demo again so 918 00:58:28,210 --> 00:58:33,820 let's look at all the different situation at once. 919 00:58:33,820 --> 00:58:36,880 So in this case, as we discussed before, 920 00:58:36,880 --> 00:58:41,680 this is actually happening in the nondispersive situation. 921 00:58:41,680 --> 00:58:44,680 In this situation, you have a straight line, 922 00:58:44,680 --> 00:58:47,590 nondispersed medium actually give you 923 00:58:47,590 --> 00:58:53,590 always the group velocity equal to phase velocity. 924 00:58:53,590 --> 00:58:58,870 So that means the carrier and the envelope 925 00:58:58,870 --> 00:59:02,650 is going to be moving in the same direction 926 00:59:02,650 --> 00:59:04,540 at the same speed. 927 00:59:08,470 --> 00:59:11,590 On the other hand, in this case, we 928 00:59:11,590 --> 00:59:16,690 can actually have a situation that the phase velocity 929 00:59:16,690 --> 00:59:20,890 is actually faster than the group velocity. 930 00:59:20,890 --> 00:59:23,770 So what I mean is actually the situation here. 931 00:59:23,770 --> 00:59:28,120 The phase velocity calculated from a line connecting from 0 932 00:59:28,120 --> 00:59:32,500 to that point is actually having a larger slope compared 933 00:59:32,500 --> 00:59:35,070 to the tangential line. 934 00:59:35,070 --> 00:59:37,770 And you see this situation. 935 00:59:37,770 --> 00:59:41,310 So basically, you see that inside the envelope 936 00:59:41,310 --> 00:59:44,230 all those carriers are actually moving faster 937 00:59:44,230 --> 00:59:45,778 than the envelope. 938 00:59:50,170 --> 00:59:54,950 Now I can have a dispersive medium where the group 939 00:59:54,950 --> 00:59:56,650 velocity is equal to 0. 940 00:59:56,650 --> 01:00:02,230 So what is going to happen is that really the envelope 941 01:00:02,230 --> 01:00:03,400 is actually not moving. 942 01:00:03,400 --> 01:00:05,340 It's not like this. 943 01:00:05,340 --> 01:00:07,060 The body is not moving. 944 01:00:07,060 --> 01:00:12,245 So you have some carriers inside this structure 945 01:00:12,245 --> 01:00:14,600 is actually moving forward. 946 01:00:14,600 --> 01:00:17,148 But the envelope is actually not moving. 947 01:00:24,510 --> 01:00:29,950 So, finally the last situation is really interesting. 948 01:00:29,950 --> 01:00:33,800 So in this situation, this is actually 949 01:00:33,800 --> 01:00:35,840 having the group velocity-- 950 01:00:39,090 --> 01:00:41,860 the group velocity is actually having difference sine 951 01:00:41,860 --> 01:00:44,060 compared to the phase velocity. 952 01:00:44,060 --> 01:00:48,610 So you can see that the whole structure of the envelope 953 01:00:48,610 --> 01:00:51,910 is actually moving backwards. 954 01:00:51,910 --> 01:00:54,790 But the carrier is actually moving 955 01:00:54,790 --> 01:01:01,380 in the positive direction in this example. 956 01:01:04,190 --> 01:01:08,950 So this is actually what we have learned from this beat 957 01:01:08,950 --> 01:01:11,390 phenomenon, and then we have covered 958 01:01:11,390 --> 01:01:15,020 the idea of phase velocity and the group velocity. 959 01:01:15,020 --> 01:01:19,460 So how about bound system how do we 960 01:01:19,460 --> 01:01:23,380 understand when we have a bound system? 961 01:01:23,380 --> 01:01:27,600 And how does that evolve as a function of time? 962 01:01:27,600 --> 01:01:33,530 So if I have a system of two walls and one string, 963 01:01:33,530 --> 01:01:36,350 and of course, I give you the density 964 01:01:36,350 --> 01:01:41,660 for the unit length and the string tension, and also 965 01:01:41,660 --> 01:01:44,330 the alpha, which is actually telling you 966 01:01:44,330 --> 01:01:47,780 about the stiffness of the system. 967 01:01:47,780 --> 01:01:53,390 Again, I can write down psi xt to be 968 01:01:53,390 --> 01:01:59,530 the sum of all the normal mode from one to infinity, 969 01:01:59,530 --> 01:02:11,196 A m sine km x plus alpha m sine omega mt plus beta m. 970 01:02:15,390 --> 01:02:19,650 And then what we can do is that we can first 971 01:02:19,650 --> 01:02:23,280 get the initial conditions of this system, 972 01:02:23,280 --> 01:02:26,040 and those are the boundary conditions of this system. 973 01:02:26,040 --> 01:02:29,480 That we actually just follow exactly the same procedure 974 01:02:29,480 --> 01:02:32,000 to obtain all the unknown coefficients 975 01:02:32,000 --> 01:02:34,845 that we would be able to evolve this system as a function 976 01:02:34,845 --> 01:02:38,340 of time, as I have demonstrated to you in the beginning 977 01:02:38,340 --> 01:02:40,080 of the lecture. 978 01:02:40,080 --> 01:02:44,070 So in this case, you can have two boundary conditions. 979 01:02:44,070 --> 01:02:47,490 One is actually say at x equal to 0. 980 01:02:47,490 --> 01:02:51,810 And the other one is actually at x equal to L. 981 01:02:51,810 --> 01:02:54,070 In those boundaries, as we actually 982 01:02:54,070 --> 01:02:57,750 learned before, because the endpoints are 983 01:02:57,750 --> 01:02:59,910 fixed on the wall. 984 01:02:59,910 --> 01:03:07,680 Therefore, psi of 0 at that time, t, 985 01:03:07,680 --> 01:03:11,670 will be always equal to 0 for the left-hand side boundary 986 01:03:11,670 --> 01:03:13,370 condition. 987 01:03:13,370 --> 01:03:16,640 And very similarly, as we discussed before, 988 01:03:16,640 --> 01:03:22,440 psi of L t will be equal to 0 if you look at the right-hand side 989 01:03:22,440 --> 01:03:23,650 of the wall-- 990 01:03:23,650 --> 01:03:25,670 of the system. 991 01:03:25,670 --> 01:03:28,940 So I don't want to repeat this, because this is actually 992 01:03:28,940 --> 01:03:33,440 exactly the same calculation which we have done before. 993 01:03:33,440 --> 01:03:36,740 So with these two boundary conditions, 994 01:03:36,740 --> 01:03:44,660 we can actually conclude that k m will be equal to m pi over L, 995 01:03:44,660 --> 01:03:50,090 and alpha m will be equal to 0. 996 01:03:50,090 --> 01:03:55,690 So you can actually go back and check this out. 997 01:03:55,690 --> 01:03:58,350 So what I'm going to say is that until now, 998 01:03:58,350 --> 01:04:02,640 what we have been doing is identical to what 999 01:04:02,640 --> 01:04:06,120 we have been doing for the nondispersive media. 1000 01:04:06,120 --> 01:04:10,380 What I'm to say is that the shape of the normal mode 1001 01:04:10,380 --> 01:04:13,940 is actually set by the boundary condition. 1002 01:04:13,940 --> 01:04:16,710 It's determined by the boundary condition, 1003 01:04:16,710 --> 01:04:19,500 and it has actually, so far, nothing 1004 01:04:19,500 --> 01:04:24,950 to do with the dispersion relation omega 1005 01:04:24,950 --> 01:04:27,810 as a function of k. 1006 01:04:27,810 --> 01:04:31,350 So in short, boundary condition can give you 1007 01:04:31,350 --> 01:04:33,690 the shape of the normal mode, and that we 1008 01:04:33,690 --> 01:04:36,975 know that the first normal mode, second normal mode, 1009 01:04:36,975 --> 01:04:40,380 et cetera, et cetera, is actually 1010 01:04:40,380 --> 01:04:46,440 going to be identical to the case of nondispersive medium. 1011 01:04:46,440 --> 01:04:49,890 so that's actually the first thing which we learned. 1012 01:04:49,890 --> 01:04:54,270 The second thing we learned is that OK, now what we see 1013 01:04:54,270 --> 01:04:58,860 is that once the boundary condition is given, 1014 01:04:58,860 --> 01:05:02,610 then the k m is actually also given. 1015 01:05:02,610 --> 01:05:07,020 Therefore, since I have the dispersion relation omega 1016 01:05:07,020 --> 01:05:11,055 as a function of k, as shown there. 1017 01:05:11,055 --> 01:05:14,610 Omega over k is equal to v times square root of 1 1018 01:05:14,610 --> 01:05:17,130 plus alpha k squared. 1019 01:05:17,130 --> 01:05:24,480 Therefore, once k m is given, omega m is also given. 1020 01:05:27,550 --> 01:05:32,370 So you can see that that's actually where the dispersion 1021 01:05:32,370 --> 01:05:33,970 relation come into play. 1022 01:05:36,650 --> 01:05:42,290 The omega m will be different if you compare the dispersive case 1023 01:05:42,290 --> 01:05:45,560 and nondispersive case. 1024 01:05:45,560 --> 01:05:49,470 So that is actually what I want to say. 1025 01:05:49,470 --> 01:05:52,520 The k m, which is the shape of the normal mode, 1026 01:05:52,520 --> 01:05:57,690 doesn't depend on the dispersion relation. 1027 01:05:57,690 --> 01:06:00,480 On the other hand, the speed of the oscillation, 1028 01:06:00,480 --> 01:06:04,020 the angular frequency, omega, depends 1029 01:06:04,020 --> 01:06:06,690 on the dispersion relation, which is actually 1030 01:06:06,690 --> 01:06:11,260 what we obtained from there. 1031 01:06:11,260 --> 01:06:20,795 If I start to plot omega m as a function of k m-- 1032 01:06:23,980 --> 01:06:27,240 so in the case of nondispersive medium, 1033 01:06:27,240 --> 01:06:31,240 so what am I going to get is actually discrete points 1034 01:06:31,240 --> 01:06:35,790 along a straight line. 1035 01:06:35,790 --> 01:06:42,460 This is actually k1, k2, k3, k4, et cetera. 1036 01:06:42,460 --> 01:06:50,430 They are actually all sitting on a common straight line. 1037 01:06:50,430 --> 01:06:56,640 If you look at the relative difference between k1, k2, 1038 01:06:56,640 --> 01:07:01,860 and k3, they are constant according to this formula. 1039 01:07:01,860 --> 01:07:05,180 The difference between k1 and k2 is pi over 2. 1040 01:07:05,180 --> 01:07:08,190 k2 and k3 is actually also pi over 2-- 1041 01:07:08,190 --> 01:07:12,300 pi over L. It's always a fixed number. 1042 01:07:12,300 --> 01:07:16,620 And since omega is actually proportional to k. 1043 01:07:16,620 --> 01:07:24,640 Therefore, the spacing between omega 1, omega 2, omega 3, 1044 01:07:24,640 --> 01:07:25,900 is also constant. 1045 01:07:28,510 --> 01:07:34,080 In short, omega 2, omega 3, and omega 4, 1046 01:07:34,080 --> 01:07:38,080 et cetera is always multiple times 1047 01:07:38,080 --> 01:07:43,240 what you get from omega 1, according to this graph 1048 01:07:43,240 --> 01:07:48,200 and in the case of nondispersive medium. 1049 01:07:48,200 --> 01:07:50,280 So what does that mean? 1050 01:07:50,280 --> 01:07:56,690 That means OK, now if I have a very complicated 1051 01:07:56,690 --> 01:07:59,490 initial condition-- 1052 01:07:59,490 --> 01:08:02,030 this is actually what I have, an initial condition-- very 1053 01:08:02,030 --> 01:08:04,130 complicated. 1054 01:08:04,130 --> 01:08:06,030 I just need to wait. 1055 01:08:06,030 --> 01:08:08,930 If this is actually nondispersive medium, 1056 01:08:08,930 --> 01:08:16,100 I just have to wait until p equal to 2 pi over omega 1. 1057 01:08:16,100 --> 01:08:21,439 Then the system would restore to its original shape. 1058 01:08:21,439 --> 01:08:23,450 That's actually what I can learn from here, 1059 01:08:23,450 --> 01:08:30,700 because omega 2, omega 3, and any higher order normal modes, 1060 01:08:30,700 --> 01:08:32,809 the angular frequency is actually multiple times 1061 01:08:32,809 --> 01:08:34,910 of what I get from omega 1. 1062 01:08:38,430 --> 01:08:43,200 On the other hand, if I consider a situation 1063 01:08:43,200 --> 01:08:46,510 of dispersive medium-- 1064 01:08:46,510 --> 01:08:55,210 you can see that now the difference between omega m 1065 01:08:55,210 --> 01:08:58,149 is now the constant. 1066 01:08:58,149 --> 01:09:01,250 So what you would predict is that it would take much, 1067 01:09:01,250 --> 01:09:03,880 much longer for this system to go back 1068 01:09:03,880 --> 01:09:10,840 to the original shape compared to nondispersive media. 1069 01:09:10,840 --> 01:09:13,380 So that actually you can actually see. 1070 01:09:13,380 --> 01:09:20,500 In a real-life experiment, I can distort this equipment 1071 01:09:20,500 --> 01:09:23,680 in this boundless system, and it's actually 1072 01:09:23,680 --> 01:09:27,640 going to take forever or impossible to come back 1073 01:09:27,640 --> 01:09:30,850 to the original shape, because of that dispersion. 1074 01:09:30,850 --> 01:09:33,399 On the other hand, if I have a really highly 1075 01:09:33,399 --> 01:09:37,510 idealized situation, if I have both ends bound, 1076 01:09:37,510 --> 01:09:43,040 and I just have to wait until t equal to 2 pi over omega 1. 1077 01:09:43,040 --> 01:09:48,399 Then this system will go back to the original shape. 1078 01:09:48,399 --> 01:09:56,100 Before I end the lecture today, I 1079 01:09:56,100 --> 01:10:01,050 would like to discuss with you two interesting issues. 1080 01:10:01,050 --> 01:10:06,360 So many of you have seen water waves, 1081 01:10:06,360 --> 01:10:09,120 and Feynman actually told us in his lecture 1082 01:10:09,120 --> 01:10:14,460 that water waves are really easily seen by everybody, 1083 01:10:14,460 --> 01:10:17,520 but it's actually the worst possible example. 1084 01:10:17,520 --> 01:10:20,400 That's the bad news-- the worst possible example 1085 01:10:20,400 --> 01:10:24,960 because it has all the possible complications 1086 01:10:24,960 --> 01:10:26,760 that waves can have. 1087 01:10:26,760 --> 01:10:28,060 That's the bad news. 1088 01:10:28,060 --> 01:10:31,176 The good news is that you are going to do that in your P set. 1089 01:10:31,176 --> 01:10:32,880 [LAUGHTER] 1090 01:10:32,880 --> 01:10:37,080 So we will be able to understand the behavior of the water 1091 01:10:37,080 --> 01:10:38,280 waves. 1092 01:10:38,280 --> 01:10:41,230 So that's the good news. 1093 01:10:41,230 --> 01:10:43,530 The second thing which I would like to talk about 1094 01:10:43,530 --> 01:10:45,300 is phase velocity. 1095 01:10:48,270 --> 01:10:52,170 You can say, OK, you say that phase velocity or harmonic 1096 01:10:52,170 --> 01:10:54,980 waves doesn't send information, right? 1097 01:10:54,980 --> 01:10:58,830 And how do I actually know that? 1098 01:10:58,830 --> 01:10:59,430 Right? 1099 01:10:59,430 --> 01:11:00,660 So what does that mean? 1100 01:11:00,660 --> 01:11:04,930 OK, so let's take this horrible example of water wave. 1101 01:11:04,930 --> 01:11:09,520 OK, so the black line is actually the beach, 1102 01:11:09,520 --> 01:11:14,550 and there is a water wave from the ocean approaching 1103 01:11:14,550 --> 01:11:15,420 the beach. 1104 01:11:15,420 --> 01:11:17,820 And you can see that you can have 1105 01:11:17,820 --> 01:11:20,960 some kind of angle between the insert of water wave 1106 01:11:20,960 --> 01:11:22,980 and the line of the beach. 1107 01:11:25,530 --> 01:11:28,950 What I can actually do is that I can now 1108 01:11:28,950 --> 01:11:37,860 measure the shape of the water wave at the edge of the beach. 1109 01:11:37,860 --> 01:11:42,430 And I would see that, huh, now the phase velocity 1110 01:11:42,430 --> 01:11:45,660 which I observe there is actually 1111 01:11:45,660 --> 01:11:50,940 faster than the speed of propagation of the water wave, 1112 01:11:50,940 --> 01:11:56,040 because of this inserted angle, OK? 1113 01:11:56,040 --> 01:12:01,830 I can actually make it very, very fast. 1114 01:12:01,830 --> 01:12:05,460 I can make the speed actually even faster 1115 01:12:05,460 --> 01:12:07,550 than the speed of light. 1116 01:12:07,550 --> 01:12:09,041 Right? 1117 01:12:09,041 --> 01:12:15,570 I can now decrease the theta to 0. 1118 01:12:15,570 --> 01:12:19,420 Then you will have a phase velocity 1119 01:12:19,420 --> 01:12:21,560 which is faster than the speed of light. 1120 01:12:21,560 --> 01:12:23,740 It goes to infinity. 1121 01:12:23,740 --> 01:12:27,110 But does that mean anything? 1122 01:12:27,110 --> 01:12:28,880 Actually, that doesn't mean anything, 1123 01:12:28,880 --> 01:12:35,480 because I don't really move the water from a specific point 1124 01:12:35,480 --> 01:12:38,570 to another point infinitely fast. 1125 01:12:38,570 --> 01:12:40,880 Therefore, what I want to say is that, OK, 1126 01:12:40,880 --> 01:12:46,280 you can do whatever you want to make a fancy phase velocity. 1127 01:12:46,280 --> 01:12:49,610 But that will not help you with sending 1128 01:12:49,610 --> 01:12:52,540 things close to the speed of light or greater 1129 01:12:52,540 --> 01:12:54,230 than the speed of light. 1130 01:12:54,230 --> 01:12:56,630 So as you can see from this example, 1131 01:12:56,630 --> 01:12:59,610 I can easily construct a simple example, 1132 01:12:59,610 --> 01:13:03,050 which you see that is actually really not sending anything 1133 01:13:03,050 --> 01:13:04,850 from one place to the other. 1134 01:13:04,850 --> 01:13:09,540 But you still have really, really fast phase velocity. 1135 01:13:09,540 --> 01:13:11,580 OK, thank you very much, everybody, 1136 01:13:11,580 --> 01:13:14,990 for the attention and hope you enjoyed the lecture. 1137 01:13:14,990 --> 01:13:19,650 And if you have any questions, please let me know.