1 00:00:30,000 --> 00:00:34,772 When the speed of propagation of a wave and a medium is 2 00:00:34,772 --> 00:00:39,633 independent of the frequency, thereby independent of the 3 00:00:39,633 --> 00:00:43,876 wavelength, we call that a non-dispersive medium. 4 00:00:43,876 --> 00:00:48,295 When the speed of propagation depends on frequency, 5 00:00:48,295 --> 00:00:51,123 we call that a dispersive medium. 6 00:00:51,123 --> 00:00:56,603 And that is on our plate today. I want to revisit the case that 7 00:00:56,603 --> 00:00:59,078 we have N beats, each mass M, 8 00:00:59,078 --> 00:01:02,790 and they are connected with little strings, 9 00:01:02,790 --> 00:01:06,440 little L. And total length, 10 00:01:06,440 --> 00:01:09,745 say, is capital L. Perhaps you remember, 11 00:01:09,745 --> 00:01:13,813 you even had a problem whereby we had five beats. 12 00:01:13,813 --> 00:01:17,627 It was also demonstrated by me in my lectures. 13 00:01:17,627 --> 00:01:21,186 This end was fixed. And this end was fixed. 14 00:01:21,186 --> 00:01:25,423 So this was a very special case whereby N was five. 15 00:01:25,423 --> 00:01:30,000 And we derived the normal mode frequencies. 16 00:01:30,000 --> 00:01:36,129 The general result is that the normal mode frequencies in terms 17 00:01:36,129 --> 00:01:39,590 of N, N being one, two, three, etc., 18 00:01:39,590 --> 00:01:45,720 equals two omega zero times the sine of N pi divided by 2N plus 19 00:01:45,720 --> 00:01:48,884 one. I don't want you to remember 20 00:01:48,884 --> 00:01:52,245 that. I certainly don't remember it 21 00:01:52,245 --> 00:01:56,793 when we removed this. I certainly don't want to 22 00:01:56,793 --> 00:02:01,836 remember it, but this is something that today I will 23 00:02:01,836 --> 00:02:05,000 need, 2N plus one. 24 00:02:05,000 --> 00:02:11,000 25 00:02:11,000 --> 00:02:17,602 Omega zero was the square root of T divided by ML. 26 00:02:17,602 --> 00:02:23,666 So, this little m, let me make it a capital M. 27 00:02:23,666 --> 00:02:30,000 Each one of those beats has Mass capital M. 28 00:02:30,000 --> 00:02:36,180 You can see that the maximum possible frequency is two omega 29 00:02:36,180 --> 00:02:42,047 zero because that's when the sine of this function equals 30 00:02:42,047 --> 00:02:44,980 one. The boundary conditions, 31 00:02:44,980 --> 00:02:51,266 that it is fixed here and that it is fixed there demands that 32 00:02:51,266 --> 00:02:57,761 there are certain values of K of N which are allowed which is N 33 00:02:57,761 --> 00:03:03,000 pi divided by L if the length here is L. 34 00:03:03,000 --> 00:03:09,000 35 00:03:09,000 --> 00:03:11,887 Now, omega N, as you see here, 36 00:03:11,887 --> 00:03:17,861 does not increase linearly with little m because it increases 37 00:03:17,861 --> 00:03:22,740 with the sine of N. But, the speed of propagation, 38 00:03:22,740 --> 00:03:27,419 V, is of course the ratio of omega divided by K. 39 00:03:27,419 --> 00:03:32,000 So, that is omega N divided by K of N. 40 00:03:32,000 --> 00:03:35,449 And, since K of N goes up linearly with N, 41 00:03:35,449 --> 00:03:40,413 little N, but omega N does not go up linearly with little N, 42 00:03:40,413 --> 00:03:45,461 it is clear that the speed of propagation is lower for higher 43 00:03:45,461 --> 00:03:50,509 frequencies because when you go to higher values of little n, 44 00:03:50,509 --> 00:03:54,127 K goes linearly up, but omega N goes slower. 45 00:03:54,127 --> 00:03:57,913 And so, you see here an example of dispersion, 46 00:03:57,913 --> 00:04:02,540 namely that the speed of propagation is lower for higher 47 00:04:02,540 --> 00:04:07,000 frequencies than for lower frequencies. 48 00:04:07,000 --> 00:04:11,400 And therefore, it also depends on wavelength. 49 00:04:11,400 --> 00:04:16,199 A very nice way of seeing that is to make a plot. 50 00:04:16,199 --> 00:04:22,300 And, you will see many of these plots today of omega versus K. 51 00:04:22,300 --> 00:04:27,800 Let me make the plot here. So, this vertical is going to 52 00:04:27,800 --> 00:04:30,100 be omega. And this is K. 53 00:04:30,100 --> 00:04:34,000 And this is K. This is omega. 54 00:04:34,000 --> 00:04:37,638 So this is K1. This is K2, K3, 55 00:04:37,638 --> 00:04:40,775 K4, K5. Let me move the K. 56 00:04:40,775 --> 00:04:48,178 And then, here is the end of the string for this value here. 57 00:04:48,178 --> 00:04:52,444 It's K5, and this value here is K2. 58 00:04:52,444 --> 00:04:57,087 It was [those?] normal mode solutions. 59 00:04:57,087 --> 00:05:04,866 And so, the maximum value that omega can have is then two omega 60 00:05:04,866 --> 00:05:08,810 zero. And if I connect, 61 00:05:08,810 --> 00:05:13,496 if I draw this curve now, it looks like this. 62 00:05:13,496 --> 00:05:18,607 It's a sine curve. So, for K1, this is omega one. 63 00:05:18,607 --> 00:05:22,441 For K2, this is omega two, and so on. 64 00:05:22,441 --> 00:05:28,724 So, you can immediately see that the speed of propagation in 65 00:05:28,724 --> 00:05:35,007 the fifth mode is lower than the speed of propagation at the 66 00:05:35,007 --> 00:05:41,610 lower modes because this point here, which is K5 connected with 67 00:05:41,610 --> 00:05:48,000 omega five, I can connect that with the origin. 68 00:05:48,000 --> 00:05:52,338 And then, this angle is a measure for the speed of 69 00:05:52,338 --> 00:05:57,916 propagation because the speed of propagation is omega divided by 70 00:05:57,916 --> 00:06:00,130 K. So, omega divided by K, 71 00:06:00,130 --> 00:06:05,000 the slope is a measure for the propagation speed. 72 00:06:05,000 --> 00:06:08,709 And you see that if I go to this point here, 73 00:06:08,709 --> 00:06:12,591 and I draw this line that the slope is larger. 74 00:06:12,591 --> 00:06:17,680 And so, the speed is larger. So, you see here in a graphical 75 00:06:17,680 --> 00:06:22,424 way, which is always very nice, to use omega K diagrams. 76 00:06:22,424 --> 00:06:27,169 What I already argued with you here, that the higher the 77 00:06:27,169 --> 00:06:32,000 frequency, the lower the speed of propagation. 78 00:06:32,000 --> 00:06:34,686 And, that is what we call dispersion. 79 00:06:34,686 --> 00:06:37,746 Now, dispersion is very common in physics. 80 00:06:37,746 --> 00:06:41,253 As you will see today, water waves under certain 81 00:06:41,253 --> 00:06:44,835 conditions are dispersive. Electromagnetic waves, 82 00:06:44,835 --> 00:06:49,313 which we will begin to think about today, and next lecture we 83 00:06:49,313 --> 00:06:53,194 will go further into that, radio waves and lights and 84 00:06:53,194 --> 00:06:55,731 infrared are electromagnetic waves. 85 00:06:55,731 --> 00:06:58,716 They can be dispersive and certain media, 86 00:06:58,716 --> 00:07:02,000 not in vacuum, but in media. 87 00:07:02,000 --> 00:07:05,555 Waves on a string, as I will discuss very shortly, 88 00:07:05,555 --> 00:07:08,893 can also be dispersive, are in fact dispersive. 89 00:07:08,893 --> 00:07:12,666 The consequences of dispersion are very nonintuitive. 90 00:07:12,666 --> 00:07:16,222 And this is what I want to discuss with you first. 91 00:07:16,222 --> 00:07:19,777 I will start with two waves which have a different 92 00:07:19,777 --> 00:07:24,276 wavelength, and I will give them purposely a different speed of 93 00:07:24,276 --> 00:07:27,106 propagation. Think of this as being on a 94 00:07:27,106 --> 00:07:31,387 string because that's always very easy to see because that's 95 00:07:31,387 --> 00:07:35,560 a transverse motion. So, I have Y1, 96 00:07:35,560 --> 00:07:41,357 which is a certain amplitude. And I'm going to make a 97 00:07:41,357 --> 00:07:46,039 traveling wave. So, we get the sine of K1 X 98 00:07:46,039 --> 00:07:51,167 minus omega 1T, and so the speed of propagation 99 00:07:51,167 --> 00:07:57,521 is omega one divided by K1. And then, I have a second one, 100 00:07:57,521 --> 00:08:04,209 which for simplicity I give the same amplitude the sine of K2 101 00:08:04,209 --> 00:08:09,406 times X minus omega 2T. And this is V1. 102 00:08:09,406 --> 00:08:12,406 And V2 is omega T2 divided by K2. 103 00:08:12,406 --> 00:08:17,187 And, the two are not the same. If they are the same, 104 00:08:17,187 --> 00:08:20,281 then it's a non-dispersive medium. 105 00:08:20,281 --> 00:08:25,718 But, they are not the same. And so, now I want to add them. 106 00:08:25,718 --> 00:08:32,000 So, I want to know what Y is, which is the sum of the two. 107 00:08:32,000 --> 00:08:40,250 So, Y is Y1 plus Y2. So, that becomes 2A times the 108 00:08:40,250 --> 00:08:50,352 sine of half the sum times the cosine of half the difference. 109 00:08:50,352 --> 00:09:00,117 And so, that gives me the sine of K1 plus K2 divided by two 110 00:09:00,117 --> 00:09:07,845 times X plus this term. So, that becomes minus omega 111 00:09:07,845 --> 00:09:12,012 one plus omega two divided by two times T. 112 00:09:12,012 --> 00:09:15,670 So, that is the sine of half the sum. 113 00:09:15,670 --> 00:09:21,565 I should put brackets here so the minus sine holds for here 114 00:09:21,565 --> 00:09:26,036 and for there. And, I have to multiply by the 115 00:09:26,036 --> 00:09:33,454 cosine of half the difference. So, I get K1 minus K2 divided 116 00:09:33,454 --> 00:09:38,815 by two times X. And, now I get minus omega one 117 00:09:38,815 --> 00:09:43,342 minus omega two divided by two times T. 118 00:09:43,342 --> 00:09:47,870 So, look closely what I have done here. 119 00:09:47,870 --> 00:09:54,184 I've added two sine curves. I get twice the amplitude, 120 00:09:54,184 --> 00:10:00,379 the sine of half the sum times the cosine of half the 121 00:10:00,379 --> 00:10:04,347 difference. Now, let us take, 122 00:10:04,347 --> 00:10:07,836 as an example whereby K1 is approximately K2. 123 00:10:07,836 --> 00:10:11,086 And, omega one is approximately omega two. 124 00:10:11,086 --> 00:10:14,416 So, we take the frequencies close together. 125 00:10:14,416 --> 00:10:18,698 You would agree with me, then, that K1 plus K2 over two 126 00:10:18,698 --> 00:10:23,296 is then effectively K to the sum of the two divided by two. 127 00:10:23,296 --> 00:10:27,895 You would also agree we've made it omega one plus omega two 128 00:10:27,895 --> 00:10:32,176 divided by two is omega, sort of the mean value between 129 00:10:32,176 --> 00:10:37,585 the two. So, I can rewrite this now that 130 00:10:37,585 --> 00:10:41,748 Y is approximately 2A times the sine. 131 00:10:41,748 --> 00:10:46,721 And, now I will just write KX minus omega T, 132 00:10:46,721 --> 00:10:51,578 K being now the mean value between the two, 133 00:10:51,578 --> 00:10:58,285 omega being the mean value between the two times the cosine 134 00:10:58,285 --> 00:11:05,340 of delta K divided by two times X minus delta omega divided by 135 00:11:05,340 --> 00:11:10,873 two times T. And now, look at what we have 136 00:11:10,873 --> 00:11:13,647 here. This all by itself is a 137 00:11:13,647 --> 00:11:18,006 traveling wave. The sine alone is a traveling 138 00:11:18,006 --> 00:11:21,077 wave. And, that sine alone has a 139 00:11:21,077 --> 00:11:24,941 velocity. This wave is a traveling wave, 140 00:11:24,941 --> 00:11:31,083 which has a velocity which I'm going to call the phase velocity 141 00:11:31,083 --> 00:11:34,947 with a P. That's our definition of phase 142 00:11:34,947 --> 00:11:40,000 velocity, which is omega divided by K. 143 00:11:40,000 --> 00:11:45,375 So, when you see that wave moving, it will move with that 144 00:11:45,375 --> 00:11:48,831 speed. But the cosine term has a very 145 00:11:48,831 --> 00:11:53,440 different velocity. It has a velocity delta omega 146 00:11:53,440 --> 00:11:58,144 divided by delta K. And, that is the same as omega 147 00:11:58,144 --> 00:12:04,000 divided by K if the waves travel with the same speed. 148 00:12:04,000 --> 00:12:07,654 So, that V, which we call the group velocity, 149 00:12:07,654 --> 00:12:10,395 is delta omega divided by delta K. 150 00:12:10,395 --> 00:12:14,963 And, if the two are the same, then it's a non-dispersive 151 00:12:14,963 --> 00:12:18,036 medium. But, if the two are different, 152 00:12:18,036 --> 00:12:23,102 then you're going to see that this wave moves with a different 153 00:12:23,102 --> 00:12:27,338 speed than that wave. The consequences are even more 154 00:12:27,338 --> 00:12:31,794 bizarre. The wavelength of the sine 155 00:12:31,794 --> 00:12:36,847 curve is, of course, given by 2 pi divided by K. 156 00:12:36,847 --> 00:12:40,610 That's the definition of wavelength. 157 00:12:40,610 --> 00:12:45,985 And, if you think of the wavelength, the repetition 158 00:12:45,985 --> 00:12:52,649 pattern of the cosine function, that is 2 pi divided by this K. 159 00:12:52,649 --> 00:12:56,412 So, that is 4 pi divided by delta K. 160 00:12:56,412 --> 00:13:01,894 And, if delta K is very small, then this wavelength, 161 00:13:01,894 --> 00:13:08,452 the repetition of the cosine is way longer than the wavelength 162 00:13:08,452 --> 00:13:15,448 of the sine curve. If I made a curve of the sine 163 00:13:15,448 --> 00:13:21,989 curve, you would see this. And, that moves with the face 164 00:13:21,989 --> 00:13:26,745 velocity, say, in the positive direction. 165 00:13:26,745 --> 00:13:34,000 If I make a plot of the cosine, that would be like this. 166 00:13:34,000 --> 00:13:36,532 And, that would move with the group velocity. 167 00:13:36,532 --> 00:13:39,697 And, if the two are the same, then this overall pattern, 168 00:13:39,697 --> 00:13:42,633 which is the product of the two, which I will put in 169 00:13:42,633 --> 00:13:44,589 shortly, moves with the same speed. 170 00:13:44,589 --> 00:13:47,697 And if the two are the same, then this overall pattern, 171 00:13:47,697 --> 00:13:50,633 which is the product of the two, which I will put in 172 00:13:50,633 --> 00:13:52,589 shortly, moves with the same speed. 173 00:13:52,589 --> 00:13:56,043 If I multiply this with this, and I will try to do that here, 174 00:13:56,043 --> 00:13:58,000 you will get this. 175 00:13:58,000 --> 00:14:06,000 176 00:14:06,000 --> 00:14:11,152 So, what you are looking here at is exactly this function. 177 00:14:11,152 --> 00:14:15,310 The sines are in here, and the cosine is there, 178 00:14:15,310 --> 00:14:19,016 some kind of a beat phenomenon, of course. 179 00:14:19,016 --> 00:14:24,440 And, it is this envelop that moves with the group velocity as 180 00:14:24,440 --> 00:14:29,050 I already have here. It is the individual sines that 181 00:14:29,050 --> 00:14:34,293 move with the phase velocity. And, if the two are the same, 182 00:14:34,293 --> 00:14:38,753 it's a non-dispersive medium. If, however, 183 00:14:38,753 --> 00:14:41,895 the two are not the same, and that's what I wanted, 184 00:14:41,895 --> 00:14:45,036 I wanted the two to be different, then you will see 185 00:14:45,036 --> 00:14:48,178 that the group velocity is different from the phase 186 00:14:48,178 --> 00:14:50,502 velocity. Group velocity can be larger 187 00:14:50,502 --> 00:14:53,832 than the phase velocity, and so then this red envelope 188 00:14:53,832 --> 00:14:56,219 moves faster than the individual sines. 189 00:14:56,219 --> 00:14:59,109 It can also be smaller than the phase velocity. 190 00:14:59,109 --> 00:15:03,739 I will demonstrate that to you. When you look at omega K 191 00:15:03,739 --> 00:15:07,000 diagrams, in general you get the full story. 192 00:15:07,000 --> 00:15:14,000 193 00:15:14,000 --> 00:15:20,730 I'll make you an omega T diagram, omega K diagram of a 194 00:15:20,730 --> 00:15:26,063 nondispersive medium, which is very boring. 195 00:15:26,063 --> 00:15:32,470 So, this is omega versus K. And, this is a nondispersive 196 00:15:32,470 --> 00:15:35,911 medium, straight line. It's clear that V 197 00:15:35,911 --> 00:15:40,058 [UNINTELLIGIBLE], which is omega divided by K is 198 00:15:40,058 --> 00:15:45,176 this indicated by this angle by the slope, and is the same. 199 00:15:45,176 --> 00:15:49,235 Independent of omega, it's everywhere the same. 200 00:15:49,235 --> 00:15:54,264 That's a nondispersive medium. That is the definition of a 201 00:15:54,264 --> 00:15:59,294 nondispersive medium that always, regardless of frequency, 202 00:15:59,294 --> 00:16:05,368 have the same phase velocity. And so, you will see since the 203 00:16:05,368 --> 00:16:10,026 group velocity is the omega DK, which is the tension of this 204 00:16:10,026 --> 00:16:14,131 line, you'll see that the group velocity is the same, 205 00:16:14,131 --> 00:16:18,473 of course, because if you take the tension of this line, 206 00:16:18,473 --> 00:16:22,421 no matter where you take it that is the same slope. 207 00:16:22,421 --> 00:16:25,105 So this is also the group velocity. 208 00:16:25,105 --> 00:16:28,105 And so, this is a nondispersive medium. 209 00:16:28,105 --> 00:16:32,289 When we dealt with strings, so far, we always made the 210 00:16:32,289 --> 00:16:36,868 assumption that the speed of propagation was independent of 211 00:16:36,868 --> 00:16:42,514 frequency. And so we always treated string 212 00:16:42,514 --> 00:16:49,089 as if they were nondispersive. And, we derive that from the 213 00:16:49,089 --> 00:16:53,396 wave equation. I can show you another K 214 00:16:53,396 --> 00:16:56,797 relationship now. For instance, 215 00:16:56,797 --> 00:17:01,672 the omega K relationship could go like this. 216 00:17:01,672 --> 00:17:07,000 That would be a dispersive medium now. 217 00:17:07,000 --> 00:17:11,583 You should be able to tell me now whether the speed is higher 218 00:17:11,583 --> 00:17:16,090 or lower at high frequencies. Well, the speed of propagation 219 00:17:16,090 --> 00:17:20,291 is clearly lower at high frequency because it's dictated 220 00:17:20,291 --> 00:17:23,652 by this angle. And this angle is smaller here 221 00:17:23,652 --> 00:17:27,243 than it is there. But the group velocity is also 222 00:17:27,243 --> 00:17:29,381 dependent, now, on frequency. 223 00:17:29,381 --> 00:17:32,590 For instance, right at this frequency here, 224 00:17:32,590 --> 00:17:36,944 so right at this frequency, the group velocity is given by 225 00:17:36,944 --> 00:17:41,756 the slope, the omega DK. And that group velocity, 226 00:17:41,756 --> 00:17:44,975 which is the omega DK, has a different angle, 227 00:17:44,975 --> 00:17:49,000 the slope, than this slope. And so, you see that for all 228 00:17:49,000 --> 00:17:52,585 these points here, the group velocity is different 229 00:17:52,585 --> 00:17:56,317 from the face velocity. And, the exact values of the 230 00:17:56,317 --> 00:17:59,390 phase velocity, and the exact values of the 231 00:17:59,390 --> 00:18:04,000 group velocity follows from the shape of that curve. 232 00:18:04,000 --> 00:18:06,384 So, you have to know that curve. 233 00:18:06,384 --> 00:18:10,307 And that curve is called, in general, the dispersion 234 00:18:10,307 --> 00:18:13,307 relation. So, any relation between omega 235 00:18:13,307 --> 00:18:17,769 and K gives the whole show away. Suppose you had an omega K 236 00:18:17,769 --> 00:18:20,230 relationship that goes like this. 237 00:18:20,230 --> 00:18:23,615 And, you will see today that that's possible. 238 00:18:23,615 --> 00:18:27,615 So, now, you have enough knowledge to tell me that at 239 00:18:27,615 --> 00:18:30,923 high frequency, the phase velocity is higher 240 00:18:30,923 --> 00:18:36,000 than at low frequency because the slope is higher. 241 00:18:36,000 --> 00:18:40,262 You also can see that the group velocity at high frequency, 242 00:18:40,262 --> 00:18:43,569 which is the tangent, is higher than the phase 243 00:18:43,569 --> 00:18:45,700 velocity. Pick any point here, 244 00:18:45,700 --> 00:18:50,330 and you'll see that the tangent has a higher slope than the line 245 00:18:50,330 --> 00:18:54,078 straight through here. So, the group velocity at any 246 00:18:54,078 --> 00:18:56,944 point is larger than the phase velocity. 247 00:18:56,944 --> 00:19:00,839 It is even possible in physics, but it is a rare case, 248 00:19:00,839 --> 00:19:06,531 that you have this situation. And I will demonstrate it you, 249 00:19:06,531 --> 00:19:11,764 but it's a computer simulation. So, it's always easy to do that 250 00:19:11,764 --> 00:19:15,478 on a computer. It is even possible that there 251 00:19:15,478 --> 00:19:19,782 are parts of the omega K diagram which go like this. 252 00:19:19,782 --> 00:19:23,411 They may go up here, but they go down again. 253 00:19:23,411 --> 00:19:27,294 If that's the case, that is remarkable what you 254 00:19:27,294 --> 00:19:30,163 have. This slope would indicate the 255 00:19:30,163 --> 00:19:33,370 phase velocity. But the slope would now 256 00:19:33,370 --> 00:19:39,180 indicate the group velocity. And that group velocity now is 257 00:19:39,180 --> 00:19:42,595 in the opposite direction of the phase velocity. 258 00:19:42,595 --> 00:19:44,921 So, the sine wave goes like this. 259 00:19:44,921 --> 00:19:48,264 And the group goes back. Even that is possible. 260 00:19:48,264 --> 00:19:51,026 That's a rare case, but it does happen. 261 00:19:51,026 --> 00:19:54,223 So what you have here is that the wavelength, 262 00:19:54,223 --> 00:19:58,584 remember wavelength increases in this direction because K and 263 00:19:58,584 --> 00:20:02,000 lambda are inversely proportional. 264 00:20:02,000 --> 00:20:06,586 So, the wavelength increases now with increasing frequency. 265 00:20:06,586 --> 00:20:10,619 If my frequency increases, the wavelength increases, 266 00:20:10,619 --> 00:20:14,811 which is something absurd. We always think that if the 267 00:20:14,811 --> 00:20:18,607 frequency increases, that the wavelength will get 268 00:20:18,607 --> 00:20:21,533 shorter. Well, there are media whereby 269 00:20:21,533 --> 00:20:25,566 that is not the case. That is what dispersion is all 270 00:20:25,566 --> 00:20:28,176 about. I can create a pattern like 271 00:20:28,176 --> 00:20:33,000 this with two way this review, which I will do. 272 00:20:33,000 --> 00:20:37,956 And I do that in a graphical sense, that I have two 273 00:20:37,956 --> 00:20:42,516 transparencies. One transparency has black bars 274 00:20:42,516 --> 00:20:45,886 on it. Many, as you'll see shortly, 275 00:20:45,886 --> 00:20:48,760 and they have a separation, D. 276 00:20:48,760 --> 00:20:52,230 And, I have many of them, maybe 100. 277 00:20:52,230 --> 00:20:55,997 I didn't count them. And I have another 278 00:20:55,997 --> 00:20:59,962 transparency, which has bars which are 5% 279 00:20:59,962 --> 00:21:03,629 thicker. And also, the opening between 280 00:21:03,629 --> 00:21:09,941 them is 5% larger. So here, we have 1.05 D. 281 00:21:09,941 --> 00:21:16,191 And, I line these up. So, this is the very first one 282 00:21:16,191 --> 00:21:19,990 on my sheet. This is number one, 283 00:21:19,990 --> 00:21:24,892 and this is number one. So, they line up. 284 00:21:24,892 --> 00:21:32,000 This one, the opening, the separation is 5% larger. 285 00:21:32,000 --> 00:21:37,396 So that means when I reach here on my sheet number 20, 286 00:21:37,396 --> 00:21:41,876 then I reach here at number 19. But they are, 287 00:21:41,876 --> 00:21:47,781 again, on top of each other. They are in phase so to speak. 288 00:21:47,781 --> 00:21:54,094 So, my number 20 coincides with number 19 here because this one 289 00:21:54,094 --> 00:22:00,000 is 5% larger spacing. The wavelength is 5% larger. 290 00:22:00,000 --> 00:22:05,083 That means halfway in between, the black lines from this sheet 291 00:22:05,083 --> 00:22:09,000 will exactly eclipse the open spots of this one. 292 00:22:09,000 --> 00:22:13,083 And so, these black ones here will all [occurlt?], 293 00:22:13,083 --> 00:22:17,416 to use the astronomical phrase, occurlt the openings. 294 00:22:17,416 --> 00:22:21,583 And so this whole central portion will [do?] black. 295 00:22:21,583 --> 00:22:26,333 Black will be on top of light, and light will be on top of 296 00:22:26,333 --> 00:22:30,000 black. And black will top light. 297 00:22:30,000 --> 00:22:33,283 Here, however, it's back to what was here. 298 00:22:33,283 --> 00:22:37,207 So, you are beginning to see this kind of pattern. 299 00:22:37,207 --> 00:22:40,170 Think of this, for now, as being dark. 300 00:22:40,170 --> 00:22:43,054 And think of this, now, as the light. 301 00:22:43,054 --> 00:22:47,859 This kind of pattern is what you're going to see if I put the 302 00:22:47,859 --> 00:22:52,344 two transparencies on top of each other as I will do very 303 00:22:52,344 --> 00:22:55,387 shortly. Now comes the wonderful thing. 304 00:22:55,387 --> 00:23:00,112 Not only will you see such a pattern, but imagine now that I 305 00:23:00,112 --> 00:23:03,716 move one of these sheets, say the upper sheet, 306 00:23:03,716 --> 00:23:09,258 over a distance half D. So I move it only over this 307 00:23:09,258 --> 00:23:12,436 distance. Remember, here the black ones 308 00:23:12,436 --> 00:23:17,538 were occurlting the open ones. But, if I shift that by half D, 309 00:23:17,538 --> 00:23:21,636 it will be reversed. The black ones will no longer 310 00:23:21,636 --> 00:23:25,818 occult the open ones. And so, this whole black area 311 00:23:25,818 --> 00:23:30,000 will instantaneously turn to light again. 312 00:23:30,000 --> 00:23:34,594 And, that means that this overall pattern moves 20 times 313 00:23:34,594 --> 00:23:38,354 faster than the motion of one of these sheets. 314 00:23:38,354 --> 00:23:41,779 If I move this sheet over a full distance, 315 00:23:41,779 --> 00:23:46,792 D, this entire pattern that you see here, which I think of as 316 00:23:46,792 --> 00:23:50,718 being the group velocity, moves 20 times faster. 317 00:23:50,718 --> 00:23:53,475 And that so I'm going to show you. 318 00:23:53,475 --> 00:23:57,235 And, the reason why I get 20 times, of course, 319 00:23:57,235 --> 00:24:03,000 because I have chosen the wavelength is to be 5% apart. 320 00:24:03,000 --> 00:24:06,887 And I am going to give them a different phase velocity. 321 00:24:06,887 --> 00:24:11,208 The first thing that I want to do is give them the same phase 322 00:24:11,208 --> 00:24:13,728 velocity. So here, you see those two 323 00:24:13,728 --> 00:24:17,327 sheets on top of each other. I've each marked them, 324 00:24:17,327 --> 00:24:20,496 one with a red mark and one with a blue mark. 325 00:24:20,496 --> 00:24:24,599 I think a blue one is the one that has 5% larger spacings. 326 00:24:24,599 --> 00:24:27,192 And you see exactly what I predicted. 327 00:24:27,192 --> 00:24:32,400 You see those dark bends. So these are the areas here 328 00:24:32,400 --> 00:24:36,133 where, these are the areas, the dark areas, 329 00:24:36,133 --> 00:24:41,022 where the black bend of one occurlts the openings of the 330 00:24:41,022 --> 00:24:43,422 other. And then, in between, 331 00:24:43,422 --> 00:24:46,355 you see light here and light here. 332 00:24:46,355 --> 00:24:50,977 And you see light there. If I can manage to move them 333 00:24:50,977 --> 00:24:55,599 both with the same speed, so I take them firmly in my 334 00:24:55,599 --> 00:24:58,444 hand. And, I'm going to move them 335 00:24:58,444 --> 00:25:03,766 both with the same speed. Then, notice that the bars, 336 00:25:03,766 --> 00:25:07,562 the individual bars move with exactly the same speed as the 337 00:25:07,562 --> 00:25:09,983 whole pattern. So that means the group 338 00:25:09,983 --> 00:25:12,601 velocity and phase velocity are the same. 339 00:25:12,601 --> 00:25:14,957 I'll try that once more. There we go. 340 00:25:14,957 --> 00:25:17,051 I have it firmly in my hands now. 341 00:25:17,051 --> 00:25:20,977 So, you see the group velocity and the phase velocity are the 342 00:25:20,977 --> 00:25:23,267 same. But now, I'm going to move one 343 00:25:23,267 --> 00:25:25,623 relative to the other. Watch closely. 344 00:25:25,623 --> 00:25:28,699 Now I'm going to move one relative to the other. 345 00:25:28,699 --> 00:25:34,000 And you can tell by preparing the red spot with the blue spot. 346 00:25:34,000 --> 00:25:37,741 Compare the red spot with the blue spot, and you can see the 347 00:25:37,741 --> 00:25:40,024 relative motion. I'm moving them now. 348 00:25:40,024 --> 00:25:43,512 And, look how fast the group velocity is compared to the 349 00:25:43,512 --> 00:25:46,682 motion between the two. You can actually hardly see 350 00:25:46,682 --> 00:25:49,917 that the red one is moving relative to the blue one. 351 00:25:49,917 --> 00:25:53,151 You can hardly see that. But look how fast the group 352 00:25:53,151 --> 00:25:54,736 velocity is. You see that? 353 00:25:54,736 --> 00:25:57,209 20 times faster, you see how the red one 354 00:25:57,209 --> 00:26:01,525 separates from the blue one? It's a very low-speed. 355 00:26:01,525 --> 00:26:05,548 That is the phase velocity. It's the difference between the 356 00:26:05,548 --> 00:26:08,460 phase velocity. That's really what I should 357 00:26:08,460 --> 00:26:11,304 have said. It's the difference between the 358 00:26:11,304 --> 00:26:14,217 phase velocity. That's really what I should 359 00:26:14,217 --> 00:26:17,060 have said. It's the difference between the 360 00:26:17,060 --> 00:26:20,250 phase velocity. And so, then the group velocity 361 00:26:20,250 --> 00:26:23,856 goes 20 times faster. So now, I want to return to our 362 00:26:23,856 --> 00:26:27,185 continuous strings. When we dealt with continuous 363 00:26:27,185 --> 00:26:31,000 strings, we derived the wave equation. 364 00:26:31,000 --> 00:26:34,320 And I still remember how we derived it. 365 00:26:34,320 --> 00:26:39,475 We took into account that the tension is responsible for the 366 00:26:39,475 --> 00:26:43,320 restoring force. So, we made the [wire?] like 367 00:26:43,320 --> 00:26:48,300 this, and then we had tension. And then, we worked out the 368 00:26:48,300 --> 00:26:51,446 math. And then, we found that in that 369 00:26:51,446 --> 00:26:56,252 case, D2Y DX squared was one over V squared times D2Y DT 370 00:26:56,252 --> 00:26:58,000 squared. 371 00:26:58,000 --> 00:27:03,000 372 00:27:03,000 --> 00:27:07,278 And we found by substituting the solution into the wave 373 00:27:07,278 --> 00:27:11,795 equation that the speed of propagation was the square root 374 00:27:11,795 --> 00:27:14,885 of T divided by mu, T being the tension. 375 00:27:14,885 --> 00:27:17,341 Mu was the mass per unit length. 376 00:27:17,341 --> 00:27:20,510 In this speed is no K. There is no omega. 377 00:27:20,510 --> 00:27:22,967 So, it's a nondispersive medium. 378 00:27:22,967 --> 00:27:26,057 It says that, you tell me what omega is, 379 00:27:26,057 --> 00:27:28,672 and this is going to be the speed. 380 00:27:28,672 --> 00:27:34,614 You tell me what omega is. That's going to be the speed. 381 00:27:34,614 --> 00:27:39,844 It's independent of frequency. It's a nondispersive medium. 382 00:27:39,844 --> 00:27:44,442 And the reason for that is because we only took into 383 00:27:44,442 --> 00:27:49,942 account that the tension in the string was responsible for the 384 00:27:49,942 --> 00:27:52,827 restoring force. So that gave us, 385 00:27:52,827 --> 00:27:58,147 then, that omega was VK that came out of this wave equation. 386 00:27:58,147 --> 00:28:03,106 And so, we found that omega squared is V squared times K 387 00:28:03,106 --> 00:28:06,923 squared. That is the result of the 388 00:28:06,923 --> 00:28:11,692 calculation when the tension is exclusively responsible for the 389 00:28:11,692 --> 00:28:15,000 restoring force. And you get a nondispersive 390 00:28:15,000 --> 00:28:17,615 medium. However, there is something 391 00:28:17,615 --> 00:28:20,153 that we did not take into account. 392 00:28:20,153 --> 00:28:23,000 And that is the stiffness of the wire. 393 00:28:23,000 --> 00:28:26,076 Imagine for now that the tension is zero. 394 00:28:26,076 --> 00:28:30,307 So you take a piano string or you take the string from a 395 00:28:30,307 --> 00:28:34,053 violin. And assume there is no tension 396 00:28:34,053 --> 00:28:36,518 at all. And you take it in your hands 397 00:28:36,518 --> 00:28:38,914 and you bend it. There's no tension. 398 00:28:38,914 --> 00:28:41,926 You just bend it. It wants to straighten out. 399 00:28:41,926 --> 00:28:46,102 That's the result of stiffness. And it is due to the stiffness 400 00:28:46,102 --> 00:28:49,799 now that you get an extra restoring force which we have 401 00:28:49,799 --> 00:28:51,990 ignored. We didn't take that into 402 00:28:51,990 --> 00:28:54,249 account. And, the restoring force, 403 00:28:54,249 --> 00:28:58,083 due to stiffness turns out to be proportional to K to the 404 00:28:58,083 --> 00:29:02,397 power four. It's inversely proportional to 405 00:29:02,397 --> 00:29:05,355 the wave lengths to the power of four. 406 00:29:05,355 --> 00:29:08,312 It's an approximation. In other words, 407 00:29:08,312 --> 00:29:13,107 our wave equation that we had is no longer valid because this 408 00:29:13,107 --> 00:29:16,944 wave equation only took into account the tension. 409 00:29:16,944 --> 00:29:20,700 If now you go through the whole procedure again, 410 00:29:20,700 --> 00:29:24,697 which is slightly more complicated, you'll find now 411 00:29:24,697 --> 00:29:29,013 that you get a different relationship between omega and 412 00:29:29,013 --> 00:29:31,171 K. And now, you see that the 413 00:29:31,171 --> 00:29:35,646 medium becomes dispersive. Now you get that omega squared 414 00:29:35,646 --> 00:29:40,682 equals V squared times K squared plus alpha times K to the power 415 00:29:40,682 --> 00:29:44,895 of four. And this is the result of that 416 00:29:44,895 --> 00:29:47,423 stiffness. Beckafee and Barrett call this 417 00:29:47,423 --> 00:29:49,193 alpha A squared. That's fine, 418 00:29:49,193 --> 00:29:51,278 of course. That's just a matter of 419 00:29:51,278 --> 00:29:53,427 definition. A is a positive number. 420 00:29:53,427 --> 00:29:57,029 And, what this tells you is that the higher the frequency, 421 00:29:57,029 --> 00:30:00,000 the higher the speed of propagation. 422 00:30:00,000 --> 00:30:04,172 I can make for you an omega K diagram. 423 00:30:04,172 --> 00:30:09,586 So, let's have here omega. And let's have here K. 424 00:30:09,586 --> 00:30:16,015 So, what we have before when we only took the tension into 425 00:30:16,015 --> 00:30:21,541 account, we had this. Omega is linearly [with?] K, 426 00:30:21,541 --> 00:30:25,601 and that slope is the phase velocity. 427 00:30:25,601 --> 00:30:32,277 It's also the group velocity. All wavelengths have the same 428 00:30:32,277 --> 00:30:35,104 speed. But now, because of this term, 429 00:30:35,104 --> 00:30:38,874 it's going to curve up because Alpha is positive. 430 00:30:38,874 --> 00:30:41,387 And now you're going to get this. 431 00:30:41,387 --> 00:30:45,785 And now you see in front of your own eyes that the higher 432 00:30:45,785 --> 00:30:48,219 the frequency, when you're here, 433 00:30:48,219 --> 00:30:52,460 the phase velocity here is higher than at low-frequency 434 00:30:52,460 --> 00:30:55,130 here. This slope is lower than that 435 00:30:55,130 --> 00:30:57,251 slope. And so, now we have a 436 00:30:57,251 --> 00:31:02,174 dispersive medium. And also, the group velocity 437 00:31:02,174 --> 00:31:06,910 right here is even higher than the phase velocity because this 438 00:31:06,910 --> 00:31:11,802 slope here which is the tangent is higher than the connection to 439 00:31:11,802 --> 00:31:13,821 zero. So, a piano string is 440 00:31:13,821 --> 00:31:16,150 dispersive. And, that has major 441 00:31:16,150 --> 00:31:19,178 consequences. Now, the values for alpha, 442 00:31:19,178 --> 00:31:22,051 as you can imagine, depend, of course, 443 00:31:22,051 --> 00:31:25,778 entirely on your string. If you have a very thick 444 00:31:25,778 --> 00:31:29,894 string, then it is very stiff. It's very hard to bend. 445 00:31:29,894 --> 00:31:35,349 So, alpha will be high. Also, if Young's modulus of the 446 00:31:35,349 --> 00:31:38,987 wire is very high, then it is extremely difficult 447 00:31:38,987 --> 00:31:42,094 to deform it. So, alpha will also be high. 448 00:31:42,094 --> 00:31:46,717 So, there is no such thing as one value for alpha for all your 449 00:31:46,717 --> 00:31:50,052 piano strings. They will all be different for 450 00:31:50,052 --> 00:31:52,932 all the strings. But, I do want to do a 451 00:31:52,932 --> 00:31:57,555 calculation to give you an order of magnitude idea what effect 452 00:31:57,555 --> 00:32:01,875 this would have on a piano. So, I choose a value for alpha 453 00:32:01,875 --> 00:32:06,271 which is not entirely absurd, although way higher values of 454 00:32:06,271 --> 00:32:11,148 alpha are possible. And then, I want to calculate 455 00:32:11,148 --> 00:32:15,592 with you quantitatively what difference that would make for a 456 00:32:15,592 --> 00:32:19,962 particular string in the piano. And so, the case that I have 457 00:32:19,962 --> 00:32:23,222 chosen, I take alpha is ten to the minus two. 458 00:32:23,222 --> 00:32:26,851 I take the tension, which is very common for piano 459 00:32:26,851 --> 00:32:30,703 strings, which is 250 N. I take the length of a piano 460 00:32:30,703 --> 00:32:35,000 spring which is one meter, just for simplicity. 461 00:32:35,000 --> 00:32:42,375 And I give it a mass per unit length of 10 g per meter. 462 00:32:42,375 --> 00:32:49,751 So, that would be ten to the minus two in our SI units, 463 00:32:49,751 --> 00:32:57,673 ten to the -2 kg per meter. And, I want to explore with you 464 00:32:57,673 --> 00:33:03,000 the 10th harmonic of this string. 465 00:33:03,000 --> 00:33:08,543 Well, T divided by mu, by the way, this can be 466 00:33:08,543 --> 00:33:16,057 written, I will write it down again in, this is also T divided 467 00:33:16,057 --> 00:33:21,971 by mu times K squared plus alpha K to the fourth. 468 00:33:21,971 --> 00:33:26,528 This V, remember, was T divided by mu. 469 00:33:26,528 --> 00:33:31,333 So, T divided by mu, that's easy enough. 470 00:33:31,333 --> 00:33:37,000 That is 2.5 times ten to the fourth. 471 00:33:37,000 --> 00:33:41,544 I have to know what K is. Well, lambda ten, 472 00:33:41,544 --> 00:33:47,062 lambda one is two meters, right, fixed at both ends. 473 00:33:47,062 --> 00:33:51,606 If it's 1 m, the length, then lambda one is 474 00:33:51,606 --> 00:33:54,960 2 m. So, lambda ten is ten times 475 00:33:54,960 --> 00:33:59,180 smaller. So, that is two divided by ten. 476 00:33:59,180 --> 00:34:04,049 So, K ten, which is, of course, by definition, 477 00:34:04,049 --> 00:34:10,000 2 pi divided by lambda ten, is then ten pi. 478 00:34:10,000 --> 00:34:14,223 And so, now I have all the ingredients to compare this 479 00:34:14,223 --> 00:34:17,808 omega squared, this term with that term to see 480 00:34:17,808 --> 00:34:22,191 by how much the frequency changes over the case where we 481 00:34:22,191 --> 00:34:25,139 had no dispersion. So this first term, 482 00:34:25,139 --> 00:34:30,000 now, and you can check that, of course, for yourself. 483 00:34:30,000 --> 00:34:33,450 So I will write down omega squared, again, 484 00:34:33,450 --> 00:34:36,143 which is that term, is T over mu. 485 00:34:36,143 --> 00:34:39,846 And then, I have to multiply it by K squared. 486 00:34:39,846 --> 00:34:43,297 And then, I find 2.5 times ten to the 7th. 487 00:34:43,297 --> 00:34:47,252 And then, I can calculate alpha K to the fourth, 488 00:34:47,252 --> 00:34:51,292 and I have all the ingredients on the blackboard. 489 00:34:51,292 --> 00:34:55,668 And, you'll find that this term is ten to the fourth. 490 00:34:55,668 --> 00:35:00,549 If I calculate roughly what omega is, I'm not interested in 491 00:35:00,549 --> 00:35:05,957 the exact value of omega. You'll see shortly why. 492 00:35:05,957 --> 00:35:10,723 You will find that omega is about 5,000 rad per second in 493 00:35:10,723 --> 00:35:15,829 this case, which translates into a frequency of about 800 Hz. 494 00:35:15,829 --> 00:35:18,893 The value itself is not so important. 495 00:35:18,893 --> 00:35:23,659 But what is important now, that as a result of this extra 496 00:35:23,659 --> 00:35:28,000 number, the frequency will go up by 0.02%. 497 00:35:28,000 --> 00:35:31,888 And this, of course, you can check immediately for 498 00:35:31,888 --> 00:35:36,333 yourself with your calculator. And, 0.02% increase means, 499 00:35:36,333 --> 00:35:38,714 in this case, 1/6 of the hertz. 500 00:35:38,714 --> 00:35:43,000 So at the 10th harmonic if you expected exactly 800 Hz, 501 00:35:43,000 --> 00:35:47,523 this is of course not exact, but I take around off number, 502 00:35:47,523 --> 00:35:51,809 then it would be 1/6 of the hertz higher because of the 503 00:35:51,809 --> 00:35:55,698 dispersion relation. But, if you make alpha ten to 504 00:35:55,698 --> 00:35:58,396 the minus one, which is by no means 505 00:35:58,396 --> 00:36:03,000 impossible, then it would go up by 1.6 Hz. 506 00:36:03,000 --> 00:36:08,068 And so the bottom line, now, is that for a real piano 507 00:36:08,068 --> 00:36:12,552 string, omega N is no longer N times omega one. 508 00:36:12,552 --> 00:36:17,620 And, that is the reason. That is why pianos go sharp, 509 00:36:17,620 --> 00:36:21,617 as musicians say. That means at the higher 510 00:36:21,617 --> 00:36:27,075 harmonics, the frequency is a little higher than linearly 511 00:36:27,075 --> 00:36:32,133 proportional with N. And that's what's called, 512 00:36:32,133 --> 00:36:36,655 makes the piano go sharp. Professor Wyslouch developed 513 00:36:36,655 --> 00:36:40,496 several years ago, when he was lecturing 8.03, 514 00:36:40,496 --> 00:36:44,079 a toy model, which he uses on the computer. 515 00:36:44,079 --> 00:36:48,773 And the toy model has the following dispersion relation. 516 00:36:48,773 --> 00:36:52,442 It has no direct connection with the string, 517 00:36:52,442 --> 00:36:56,197 no direct connection with any physical thing. 518 00:36:56,197 --> 00:37:02,000 But it is simply used for the purpose of demonstration. 519 00:37:02,000 --> 00:37:07,731 And that is that omega squared equals V squared times K squared 520 00:37:07,731 --> 00:37:13,092 times one plus alpha K squared. And, if you're not careful, 521 00:37:13,092 --> 00:37:16,882 you may think that it is the same as this. 522 00:37:16,882 --> 00:37:19,932 But it's not. It's very different. 523 00:37:19,932 --> 00:37:25,386 The difference is that this term here has a V squared in it. 524 00:37:25,386 --> 00:37:29,638 And this term here doesn't have this V squared. 525 00:37:29,638 --> 00:37:35,000 So, totally different dispersion relationship. 526 00:37:35,000 --> 00:37:40,354 In this program you are going to see, [Bolech?] Wyslouch shows 527 00:37:40,354 --> 00:37:43,427 you six waves. All six have the same 528 00:37:43,427 --> 00:37:45,797 amplitude. But, they differ. 529 00:37:45,797 --> 00:37:50,889 The shortest wavelengths and the longest wavelengths differ 530 00:37:50,889 --> 00:37:54,225 by 12%. So, two neighboring wavelengths 531 00:37:54,225 --> 00:37:58,000 are two and a half percent apart. 532 00:37:58,000 --> 00:38:01,257 And, the velocities of the individual waves, 533 00:38:01,257 --> 00:38:04,515 the phase velocity, is then dictated by this 534 00:38:04,515 --> 00:38:08,075 dispersion relation. That is omega divided by K. 535 00:38:08,075 --> 00:38:12,318 So, these velocities are not the same if you put in alpha 536 00:38:12,318 --> 00:38:15,500 nonzero, of course. If you make alpha zero, 537 00:38:15,500 --> 00:38:18,000 then they are the same. 538 00:38:18,000 --> 00:38:26,000 539 00:38:26,000 --> 00:38:29,130 First, we have to understand what you're going to see when 540 00:38:29,130 --> 00:38:31,822 the waves are not moving. If I take six waves with 541 00:38:31,822 --> 00:38:34,513 different wavelengths, there I showed you only two 542 00:38:34,513 --> 00:38:36,271 waves with different wavelengths. 543 00:38:36,271 --> 00:38:38,084 And you saw these beautiful beats. 544 00:38:38,084 --> 00:38:40,885 If you do that with six waves and you lined them up, 545 00:38:40,885 --> 00:38:44,016 say, at the center of your screen, that the amplitudes all 546 00:38:44,016 --> 00:38:46,762 line up with each other, you will see six times the 547 00:38:46,762 --> 00:38:48,794 amplitude at the center of the screen. 548 00:38:48,794 --> 00:38:52,090 But when you go away from the center, you do not see anything 549 00:38:52,090 --> 00:38:54,067 like this. What you are going to see, 550 00:38:54,067 --> 00:38:56,759 the computer will show you. It's not so intuitive. 551 00:38:56,759 --> 00:39:00,000 But you'll see something like this. 552 00:39:00,000 --> 00:39:03,246 And then, it bulges a little and then bulges again. 553 00:39:03,246 --> 00:39:05,714 And that's simply a matter of geometry. 554 00:39:05,714 --> 00:39:08,181 That has nothing to do with dispersion. 555 00:39:08,181 --> 00:39:10,389 We haven't even touched dispersion. 556 00:39:10,389 --> 00:39:13,896 We simply take six waves and we draw six waves all with 557 00:39:13,896 --> 00:39:17,337 different wavelengths, to a half percent apart between 558 00:39:17,337 --> 00:39:20,389 adjacent wavelengths. We put them on top of each 559 00:39:20,389 --> 00:39:23,311 other like I did there, and you see a pattern. 560 00:39:23,311 --> 00:39:26,818 And that pattern you'll see. And, to make sure that you 561 00:39:26,818 --> 00:39:29,805 remember that pattern for the next ten minutes, 562 00:39:29,805 --> 00:39:35,000 I'm first going to show you what happens when alpha is zero. 563 00:39:35,000 --> 00:39:39,194 That means all the waves will move with exactly the same 564 00:39:39,194 --> 00:39:41,025 speed. And, the envelope, 565 00:39:41,025 --> 00:39:45,372 crazy as the envelope will be, is moving with exactly that 566 00:39:45,372 --> 00:39:49,415 same speed because the envelope is the group velocity. 567 00:39:49,415 --> 00:39:53,762 And then, when you have really remembered, when you really 568 00:39:53,762 --> 00:39:57,881 remember that envelope, then I'm going to change alpha. 569 00:39:57,881 --> 00:40:02,000 If I make alpha positive, it will turn up. 570 00:40:02,000 --> 00:40:05,466 That's the general idea [of your plot?] goes like this. 571 00:40:05,466 --> 00:40:08,932 If you make alpha negative, which is not the case for a 572 00:40:08,932 --> 00:40:12,655 string, it's just in this toy model, I can put alpha minus, 573 00:40:12,655 --> 00:40:14,516 right? If you put alpha minus, 574 00:40:14,516 --> 00:40:18,175 then you're going to get this. And, if you go to very high 575 00:40:18,175 --> 00:40:21,962 values of K, it will even turn over because if alpha becomes 576 00:40:21,962 --> 00:40:25,364 [high enough negative?], ultimately it will turn over. 577 00:40:25,364 --> 00:40:28,638 And, if I can do that, if I can choose a value for K 578 00:40:28,638 --> 00:40:32,618 very high with a negative alpha, this is the situation that was 579 00:40:32,618 --> 00:40:35,827 required, remember, for the phase velocity to be in 580 00:40:35,827 --> 00:40:40,000 the opposite direction as the group velocity. 581 00:40:40,000 --> 00:40:44,626 So, I can also show you that, which is really cute. 582 00:40:44,626 --> 00:40:49,992 So you see, the envelope move in a different direction than 583 00:40:49,992 --> 00:40:54,526 the individual waves. So, Marcos, did we decide on 584 00:40:54,526 --> 00:40:57,857 five? So, the first thing that I want 585 00:40:57,857 --> 00:41:01,836 to run with you is, then, alpha equals zero. 586 00:41:01,836 --> 00:41:06,000 You are going to see the six waves. 587 00:41:06,000 --> 00:41:11,000 588 00:41:11,000 --> 00:41:13,526 There it comes. Now, look very closely. 589 00:41:13,526 --> 00:41:16,519 The red one is the superposition of six waves. 590 00:41:16,519 --> 00:41:20,309 The blue ones are the waves. The blue ones all travel with 591 00:41:20,309 --> 00:41:22,636 the same speed. Take my word for it. 592 00:41:22,636 --> 00:41:25,296 I put alpha zero. And so, this red curve, 593 00:41:25,296 --> 00:41:28,953 which is the superposition, which has nothing to do with 594 00:41:28,953 --> 00:41:32,877 dispersion, it has to do with geometry, that red curve moves 595 00:41:32,877 --> 00:41:37,000 with exactly the same velocity as the blue ones. 596 00:41:37,000 --> 00:41:40,492 The group velocity is the same as the phase velocity. 597 00:41:40,492 --> 00:41:44,589 And this crazy shape of the red one, [UNINTELLIGIBLE] geometry 598 00:41:44,589 --> 00:41:48,552 to the overlay of the six waves, that shape is not changing. 599 00:41:48,552 --> 00:41:51,507 So, hold onto it. Make sure you remember that 600 00:41:51,507 --> 00:41:54,798 five minutes from now. That shape is not changing. 601 00:41:54,798 --> 00:41:58,626 And, the reason why it's not changing is because the waves 602 00:41:58,626 --> 00:42:01,111 move with the same speed, all of them. 603 00:42:01,111 --> 00:42:05,008 So, that shape cannot change. All right. 604 00:42:05,008 --> 00:42:08,637 So now I'm going to make alpha 1,000. 605 00:42:08,637 --> 00:42:13,980 And I make it positive. And, I also want you to see it 606 00:42:13,980 --> 00:42:20,231 a little longer in time because I want you to see that now that 607 00:42:20,231 --> 00:42:25,474 the phase velocities are different for the individual 608 00:42:25,474 --> 00:42:28,700 waves. I want you to see that the 609 00:42:28,700 --> 00:42:34,850 overall envelope is not going to change, which is no surprise, 610 00:42:34,850 --> 00:42:38,378 of course. That is now the result of 611 00:42:38,378 --> 00:42:43,579 dispersion. So, this overall envelope will 612 00:42:43,579 --> 00:42:46,760 not change. Because now you change the 613 00:42:46,760 --> 00:42:50,542 geometry of how they line up with each other. 614 00:42:50,542 --> 00:42:55,615 So, what you are going to see here now is an example whereby 615 00:42:55,615 --> 00:43:00,000 the six waves all have different velocities. 616 00:43:00,000 --> 00:43:04,334 The group velocity is larger than the phase velocity because 617 00:43:04,334 --> 00:43:08,081 I made alpha positive, and the shape of the envelope 618 00:43:08,081 --> 00:43:11,093 is slowly going to change. You cannot see, 619 00:43:11,093 --> 00:43:14,400 of course, with your naked eye. Neither can I, 620 00:43:14,400 --> 00:43:18,000 that the six waves don't move with the same speed. 621 00:43:18,000 --> 00:43:22,114 It's impossible to see that. Neither can you see that the 622 00:43:22,114 --> 00:43:25,861 group velocity is different from the phase velocity. 623 00:43:25,861 --> 00:43:29,240 You just don't have the resolution in her eyes. 624 00:43:29,240 --> 00:43:32,546 But notice, now, that the overall [red?] shape 625 00:43:32,546 --> 00:43:37,836 already has changed somewhat. It's no longer as crispy. 626 00:43:37,836 --> 00:43:41,508 It's no longer as sharp. That's why I'm giving it a 627 00:43:41,508 --> 00:43:45,841 little more time than the first one so that you can actually 628 00:43:45,841 --> 00:43:48,926 begin to see that the shape, the red shape, 629 00:43:48,926 --> 00:43:52,231 is changing in time. And that is the result of 630 00:43:52,231 --> 00:43:53,847 dispersion. Here, look. 631 00:43:53,847 --> 00:43:56,491 Look when this red one is coming out. 632 00:43:56,491 --> 00:44:00,090 It's way more drawn out now. That is the result of 633 00:44:00,090 --> 00:44:02,000 dispersion. 634 00:44:02,000 --> 00:44:07,000 635 00:44:07,000 --> 00:44:11,890 Alpha, by the way, is only 1,000 in this toy 636 00:44:11,890 --> 00:44:14,620 model. Hey, look at that. 637 00:44:14,620 --> 00:44:18,488 You see that? Very, very different. 638 00:44:18,488 --> 00:44:24,061 And now, I want you, oh, let's look at the omega K 639 00:44:24,061 --> 00:44:28,270 plane. This program also shows you the 640 00:44:28,270 --> 00:44:32,625 omega K plane. Now, this is not omega zero 641 00:44:32,625 --> 00:44:35,395 here, so be very careful when you look at this. 642 00:44:35,395 --> 00:44:38,467 You see only a very short part of the omega K plane. 643 00:44:38,467 --> 00:44:41,177 Something here, you see, a very high value for 644 00:44:41,177 --> 00:44:43,104 K. You see a short section there. 645 00:44:43,104 --> 00:44:46,054 So you see omega vertically plotted, and you see K 646 00:44:46,054 --> 00:44:48,885 horizontally plotted. This end, the fact that it 647 00:44:48,885 --> 00:44:52,136 looks like a straight line there doesn't mean that it's 648 00:44:52,136 --> 00:44:55,328 nondispersive because it looks that so many very small 649 00:44:55,328 --> 00:44:57,616 sections. So this looks like a straight 650 00:44:57,616 --> 00:44:59,122 line. But the whole thing, 651 00:44:59,122 --> 00:45:01,771 of course, is on a curve. And that's why it's 652 00:45:01,771 --> 00:45:06,645 nondispersive. And so, now I'm going to show 653 00:45:06,645 --> 00:45:10,203 you a value for alpha which is negative. 654 00:45:10,203 --> 00:45:14,582 And I'm going to give it the same amount of time. 655 00:45:14,582 --> 00:45:19,873 So, we start with the same original shape that you are used 656 00:45:19,873 --> 00:45:23,066 to, now, which is that crispy shape. 657 00:45:23,066 --> 00:45:27,354 And now if you look very closely at the red one, 658 00:45:27,354 --> 00:45:31,914 look very closely. I will keep pointing at the top 659 00:45:31,914 --> 00:45:34,446 of the red one. You see it's moving in the 660 00:45:34,446 --> 00:45:36,422 opposite direction. You see that? 661 00:45:36,422 --> 00:45:38,274 You see that? It's moving back. 662 00:45:38,274 --> 00:45:41,980 So the group velocity is now moving in the opposite direction 663 00:45:41,980 --> 00:45:44,944 as the phase velocity. And the shape is changing. 664 00:45:44,944 --> 00:45:46,858 Look. Look how fast the shape is 665 00:45:46,858 --> 00:45:49,884 changing, much faster even than in the first case. 666 00:45:49,884 --> 00:45:52,169 So, we are now in our omega K diagram. 667 00:45:52,169 --> 00:45:54,886 We are now here. So, we have a linear portion 668 00:45:54,886 --> 00:45:57,171 there, what it looks like linear here. 669 00:45:57,171 --> 00:46:01,000 But it is, of course, part of this curve. 670 00:46:01,000 --> 00:46:06,196 Look how dispersive this is. So, the change of the envelope 671 00:46:06,196 --> 00:46:11,124 is the result of dispersion, and the fact that the group 672 00:46:11,124 --> 00:46:16,141 velocity is opposite the direction of the phase velocity. 673 00:46:16,141 --> 00:46:19,815 Boy, this whole envelope is falling apart, 674 00:46:19,815 --> 00:46:22,771 right? It's totally falling apart. 675 00:46:22,771 --> 00:46:27,699 And, that's only due to the fact that these six waves no 676 00:46:27,699 --> 00:46:32,000 longer have the same phase velocity. 677 00:46:32,000 --> 00:46:40,000 678 00:46:40,000 --> 00:46:44,108 And so, we are going to look at the omega K plane when this is 679 00:46:44,108 --> 00:46:47,072 coming to a halt. I expect it's very closely, 680 00:46:47,072 --> 00:46:49,227 now, hey, boy, yeah, there we go. 681 00:46:49,227 --> 00:46:53,134 So, here's your omega K plane. Again, it's a little section 682 00:46:53,134 --> 00:46:55,155 here. You can see it's slightly 683 00:46:55,155 --> 00:46:57,512 curved. But, even if it looks like a 684 00:46:57,512 --> 00:47:00,274 straight line, it's because of this curve. 685 00:47:00,274 --> 00:47:03,507 So, that's where [UNINTELLIGIBLE] very high value 686 00:47:03,507 --> 00:47:08,881 for K that he picked. And it was the negative value 687 00:47:08,881 --> 00:47:13,155 for alpha, then, that gives you that result. 688 00:47:13,155 --> 00:47:17,726 So, as we just saw, now, the fact that there is 689 00:47:17,726 --> 00:47:23,689 dispersion changes the shape if you have the superposition of 690 00:47:23,689 --> 00:47:28,360 many different waves. Now, suppose you plucked a 691 00:47:28,360 --> 00:47:32,732 string, or as we did during the last lecture, 692 00:47:32,732 --> 00:47:38,000 you have a square wave on top of a string. 693 00:47:38,000 --> 00:47:42,761 And you do a Fourier analysis of this square wave. 694 00:47:42,761 --> 00:47:46,551 And you say to the Fourier analysis, OK, 695 00:47:46,551 --> 00:47:51,410 show me what you're going to do if we look in time, 696 00:47:51,410 --> 00:47:57,435 that each one of those Poirier components is going to oscillate 697 00:47:57,435 --> 00:48:01,808 with its own frequency. And the cause that was 698 00:48:01,808 --> 00:48:06,084 nondispersive, we always use that omega N was 699 00:48:06,084 --> 00:48:10,038 N times omega one. And therefore, 700 00:48:10,038 --> 00:48:13,748 if we waited one full period of the first harmonic, 701 00:48:13,748 --> 00:48:18,051 the shape, whatever it was, was back to what it was before. 702 00:48:18,051 --> 00:48:22,429 It never changed because after one full oscillation of Nancy 703 00:48:22,429 --> 00:48:25,396 one, Nancy two has made two oscillations. 704 00:48:25,396 --> 00:48:28,290 Nancy three had made three oscillations. 705 00:48:28,290 --> 00:48:32,000 So, you're exactly back where you were. 706 00:48:32,000 --> 00:48:36,222 That is no longer the case when you have dispersion because now 707 00:48:36,222 --> 00:48:40,172 one wave takes a little longer or a little shorter than the 708 00:48:40,172 --> 00:48:43,032 other wave. So, they no longer meet up with 709 00:48:43,032 --> 00:48:45,688 each other. Another way of putting it is 710 00:48:45,688 --> 00:48:49,365 that the traveling waves, of course, standing waves are 711 00:48:49,365 --> 00:48:53,043 the result of traveling waves, the traveling waves have 712 00:48:53,043 --> 00:48:56,175 different velocities because of the dispersion. 713 00:48:56,175 --> 00:48:59,240 And that, now, will make any sharp features in 714 00:48:59,240 --> 00:49:02,236 a pulse disappear because the sharp features, 715 00:49:02,236 --> 00:49:05,641 if you have a rectangular pulse, the sharp features 716 00:49:05,641 --> 00:49:10,000 require a high frequency to make that steep. 717 00:49:10,000 --> 00:49:13,569 And there will be a large difference there between the 718 00:49:13,569 --> 00:49:16,868 phase velocity of the high frequencies and the low 719 00:49:16,868 --> 00:49:19,494 frequencies. And so, you're going to see 720 00:49:19,494 --> 00:49:23,265 your sharp pulse fall apart. And that's what I'm going to 721 00:49:23,265 --> 00:49:26,565 demonstrate to you. This toy model will be used to 722 00:49:26,565 --> 00:49:30,000 do the Fourier analysis of a square wave. 723 00:49:30,000 --> 00:49:35,172 I will first show you the square way of the way I showed 724 00:49:35,172 --> 00:49:39,310 it last Tuesday when there was no dispersion. 725 00:49:39,310 --> 00:49:44,576 You will see the evolution in time, and I waited half the 726 00:49:44,576 --> 00:49:50,313 period of the fundamental when the square wave came out upside 727 00:49:50,313 --> 00:49:53,605 down. I will show you that to remind 728 00:49:53,605 --> 00:49:57,931 you of what that was like. And then I will add, 729 00:49:57,931 --> 00:50:04,378 all right. I think I have to find that 730 00:50:04,378 --> 00:50:11,513 program moving triangle. OK, this is the one. 731 00:50:11,513 --> 00:50:17,837 This is it, moving square at the center. 732 00:50:17,837 --> 00:50:26,756 So, are we ready for this? So, this is what I showed you 733 00:50:26,756 --> 00:50:31,987 last Tuesday. The square is half the length 734 00:50:31,987 --> 00:50:34,453 of the string. We Fourier analyze it. 735 00:50:34,453 --> 00:50:38,496 All these Fourier components are going to oscillate in their 736 00:50:38,496 --> 00:50:42,676 own way, which is like saying we have two pulses moving in the 737 00:50:42,676 --> 00:50:45,623 opposite direction. And they are reflecting, 738 00:50:45,623 --> 00:50:48,706 and they come back. I have 100 terms here this 739 00:50:48,706 --> 00:50:51,104 time. Last time I only had 50 terms. 740 00:50:51,104 --> 00:50:53,297 I have 100. And only the odd ones 741 00:50:53,297 --> 00:50:56,518 contribute, remember? So, we effectively have 50 742 00:50:56,518 --> 00:50:59,122 terms: Mary one, Mary three, Mary five, 743 00:50:59,122 --> 00:51:02,857 all the way to Mary 99. And so, look. 744 00:51:02,857 --> 00:51:05,627 This is one half period of the fundamental. 745 00:51:05,627 --> 00:51:08,331 And your shape is back, well, upside down. 746 00:51:08,331 --> 00:51:11,365 But that's because I only ran it half a period. 747 00:51:11,365 --> 00:51:13,674 The shape is back from where it was. 748 00:51:13,674 --> 00:51:16,774 And the reason is, it is a nondispersive medium. 749 00:51:16,774 --> 00:51:20,797 And so, the reason is that the Fourier components oscillate in 750 00:51:20,797 --> 00:51:23,634 this fashion. But now, I'm going to use [one 751 00:51:23,634 --> 00:51:26,668 Bolex?] toy model. And, I am going to do in the 752 00:51:26,668 --> 00:51:29,900 these are the Fourier components, which are not so 753 00:51:29,900 --> 00:51:35,395 important for you right now. And so, now I'm going to add 754 00:51:35,395 --> 00:51:38,646 dispersion. And the dispersion that I'm 755 00:51:38,646 --> 00:51:41,896 adding is alpha is 0.01, one hundredth, 756 00:51:41,896 --> 00:51:46,687 and we do the same pulse. We get exactly the same Fourier 757 00:51:46,687 --> 00:51:50,023 analysis. The blue curves are identical, 758 00:51:50,023 --> 00:51:55,156 except they oscillate no longer according to omega N equals N 759 00:51:55,156 --> 00:51:57,979 omega one. But, they now oscillate 760 00:51:57,979 --> 00:52:02,000 according to our new toy relation. 761 00:52:02,000 --> 00:52:05,103 And now look what's going to happen. 762 00:52:05,103 --> 00:52:10,247 So, the higher frequencies, now, the traveling speed of the 763 00:52:10,247 --> 00:52:15,746 higher frequencies is different from the traveling speed of the 764 00:52:15,746 --> 00:52:19,914 lower frequencies. And if now you wait one half. 765 00:52:19,914 --> 00:52:24,171 Of the fundamental, you are going to see when the 766 00:52:24,171 --> 00:52:28,605 thing is upside down, it really doesn't look like a 767 00:52:28,605 --> 00:52:34,263 square anymore. You can sort of still recognize 768 00:52:34,263 --> 00:52:37,947 it a little. The Fourier components, 769 00:52:37,947 --> 00:52:41,842 the blue components, have not changed. 770 00:52:41,842 --> 00:52:45,842 They are exactly the same as they were. 771 00:52:45,842 --> 00:52:52,157 And this is now what you have. So, that is the consequence of 772 00:52:52,157 --> 00:52:56,157 dispersion. So, I think this is a great 773 00:52:56,157 --> 00:53:00,789 moment for a break. We will reconvene in five 774 00:53:00,789 --> 00:53:04,651 minutes. [SOUND OFF/THEN ON] So I 775 00:53:04,651 --> 00:53:07,953 summarize for you here, which is really something 776 00:53:07,953 --> 00:53:11,530 you'll have to remember. And there is really not much 777 00:53:11,530 --> 00:53:14,764 more to it than that, that the phase velocity is 778 00:53:14,764 --> 00:53:18,135 omega divided by K, and that the group velocity is 779 00:53:18,135 --> 00:53:20,955 D omega DK. And so, there is no such thing 780 00:53:20,955 --> 00:53:24,257 as one phase velocity for a nondispersive medium. 781 00:53:24,257 --> 00:53:27,285 There is no such thing as one group velocity. 782 00:53:27,285 --> 00:53:31,000 It depends on the curve omega versus K. 783 00:53:31,000 --> 00:53:36,919 So, you have to know what we call the dispersion relation in 784 00:53:36,919 --> 00:53:42,737 order to say what the phase velocity is for what frequency. 785 00:53:42,737 --> 00:53:47,453 And, dispersion is extremely common and physics. 786 00:53:47,453 --> 00:53:53,171 If we take deep water waves, and I would imagine that what 787 00:53:53,171 --> 00:53:59,090 is meant by that is that the water is deeper than the length 788 00:53:59,090 --> 00:54:04,189 of your wavelength. That is probably the 789 00:54:04,189 --> 00:54:09,007 operational definition. It's very dispersive, 790 00:54:09,007 --> 00:54:14,262 the dispersion relation, which I will not derive. 791 00:54:14,262 --> 00:54:18,642 I didn't derive it either for the string. 792 00:54:18,642 --> 00:54:24,226 Omega squared equals GK. G is just the gravitational 793 00:54:24,226 --> 00:54:32,000 acceleration plus S divided by rho times K to the power three. 794 00:54:32,000 --> 00:54:38,099 And, S is the surface tension, which is for water about 0.72 N 795 00:54:38,099 --> 00:54:41,800 per meter. I don't know whether you've 796 00:54:41,800 --> 00:54:45,199 dealt with surface tension in 8.01. 797 00:54:45,199 --> 00:54:49,000 But if not, you just can accept this S. 798 00:54:49,000 --> 00:54:54,199 And, the density of water, rho, is about 1,000 kg per 799 00:54:54,199 --> 00:54:57,000 cubic meter. That, of course, 800 00:54:57,000 --> 00:55:02,400 is a little different for seawater, but that's the nice 801 00:55:02,400 --> 00:55:08,713 number sort of to use. Now, this S term is only 802 00:55:08,713 --> 00:55:12,895 important for a very short wavelength. 803 00:55:12,895 --> 00:55:17,417 If you take the case that lambda is much, 804 00:55:17,417 --> 00:55:23,182 much larger than one centimeter, then I can show you 805 00:55:23,182 --> 00:55:29,852 that the S term is unimportant. So, let's take lambda equals 806 00:55:29,852 --> 00:55:34,411 one meter. If lambda is one meter, 807 00:55:34,411 --> 00:55:39,865 and K is 2 pi divided by lambda, so you can calculate 808 00:55:39,865 --> 00:55:45,528 what GK is, it's about 62 in SI units because G is 9.8. 809 00:55:45,528 --> 00:55:52,029 And, you can also calculate now S divided by rho times K to the 810 00:55:52,029 --> 00:55:55,805 third. Remember, K is 2 pi divided by 811 00:55:55,805 --> 00:55:58,007 lambda. And this, now, 812 00:55:58,007 --> 00:56:02,700 is 0.02. So, it's insignificant. 813 00:56:02,700 --> 00:56:07,162 It's very small. And so, for this case, 814 00:56:07,162 --> 00:56:12,681 you might as well, for first order approximation 815 00:56:12,681 --> 00:56:18,200 forget that term. And so, you can write down now 816 00:56:18,200 --> 00:56:24,306 that omega divided by K, which is the phase velocity, 817 00:56:24,306 --> 00:56:28,886 in this case, comes out to be 1.25 m per 818 00:56:28,886 --> 00:56:32,812 second. And what you also find, 819 00:56:32,812 --> 00:56:37,540 and I will show you that very shortly, but the group velocity 820 00:56:37,540 --> 00:56:41,875 is half the phase velocity. And so it's half this value. 821 00:56:41,875 --> 00:56:46,524 And if you want to know what frequency, the water is hopping 822 00:56:46,524 --> 00:56:48,573 up and down. The frequency, 823 00:56:48,573 --> 00:56:52,513 of course, is the phase velocity divided by lambda. 824 00:56:52,513 --> 00:56:56,847 So, that is about 1.25 Hz. So, it's a little faster than 825 00:56:56,847 --> 00:57:00,000 one oscillation per second. 826 00:57:00,000 --> 00:57:07,000 827 00:57:07,000 --> 00:57:11,679 If we have the case that lambda is much larger than 1 cm, 828 00:57:11,679 --> 00:57:15,021 I will continue here. So, lambda is much, 829 00:57:15,021 --> 00:57:18,865 much larger than 1 cm. I can use the dispersion 830 00:57:18,865 --> 00:57:23,878 relation, and I can calculate now what the phase velocity and 831 00:57:23,878 --> 00:57:27,555 the group velocity is in these general terms. 832 00:57:27,555 --> 00:57:31,315 So, I know now that omega, to a high degree of 833 00:57:31,315 --> 00:57:34,490 approximation, is the square root of GK 834 00:57:34,490 --> 00:57:40,892 because I forget the S term. And so, omega divided by K, 835 00:57:40,892 --> 00:57:47,214 which is the phase velocity, is there for the square root of 836 00:57:47,214 --> 00:57:50,107 G divided by K. Do we agree? 837 00:57:50,107 --> 00:57:56,000 And, notice that this is, therefore, proportional to the 838 00:57:56,000 --> 00:58:01,357 square root of the wavelength because one over K is 839 00:58:01,357 --> 00:58:07,369 proportional with lambda. So, what it means is that the 840 00:58:07,369 --> 00:58:11,617 larger the wavelength, the higher the phase velocity. 841 00:58:11,617 --> 00:58:14,313 If lambda is larger, K is smaller, 842 00:58:14,313 --> 00:58:19,297 the higher the phase velocity. What now is the group velocity? 843 00:58:19,297 --> 00:58:22,565 Well, the group velocity is the omega DK. 844 00:58:22,565 --> 00:58:27,058 So, you take this one and you take the derivative of the 845 00:58:27,058 --> 00:58:32,441 square root of G times K. So, you first get your one 846 00:58:32,441 --> 00:58:35,633 half. And then, you get your square 847 00:58:35,633 --> 00:58:41,267 root of G, but now we have to divide by the square root of K. 848 00:58:41,267 --> 00:58:45,492 So, you get times the square root of G over K. 849 00:58:45,492 --> 00:58:50,657 But, the square root of G over K was the phase velocity. 850 00:58:50,657 --> 00:58:53,661 So, it's half the phase velocity. 851 00:58:53,661 --> 00:58:57,699 And, that's what I already anticipated here. 852 00:58:57,699 --> 00:59:02,206 So, for water waves, which are much larger than 1 853 00:59:02,206 --> 00:59:07,370 cm, you see that the higher the velocity, the larger the 854 00:59:07,370 --> 00:59:12,471 wavelength. And you also see that the group 855 00:59:12,471 --> 00:59:15,260 velocity is half the phase velocity. 856 00:59:15,260 --> 00:59:18,847 So, let's now take a situation that lambda is, 857 00:59:18,847 --> 00:59:23,471 say, much smaller than 1 cm. It doesn't even have to be all 858 00:59:23,471 --> 00:59:27,057 that much smaller, but just smaller than 1 cm. 859 00:59:27,057 --> 00:59:32,000 Then, it's the second term that completely dominates. 860 00:59:32,000 --> 00:59:37,775 So, now you get that omega squared, to a very good 861 00:59:37,775 --> 00:59:44,139 approximation is S divided by rho times K to the third. 862 00:59:44,139 --> 00:59:50,974 So, now, the phase velocity, which is omega divided by K is 863 00:59:50,974 --> 00:59:56,514 now the square root of S divided by rho times K, 864 00:59:56,514 --> 1:00:02,996 which now is proportional to the square root of one over 865 1:00:02,996 --> 1:00:06,813 lambda. So, now you have to reverse. 866 1:00:06,813 --> 1:00:09,727 So now you see that the larger the wavelength, 867 1:00:09,727 --> 1:00:12,577 the smaller the phase velocity. None of this, 868 1:00:12,577 --> 1:00:16,010 of course, is intuitive. You have no feeling for that. 869 1:00:16,010 --> 1:00:19,702 I have no feeling for that. What I'm asking you in problem 870 1:00:19,702 --> 1:00:22,551 set number six is to prove that in this case, 871 1:00:22,551 --> 1:00:25,919 the group velocity is 1 1/2 times the phase velocity, 872 1:00:25,919 --> 1:00:30,000 not at all obvious, but that's what will come out. 873 1:00:30,000 --> 1:00:33,956 Here, the group velocity was half the phase velocity. 874 1:00:33,956 --> 1:00:38,293 And, the phase velocity was larger for larger wavelengths. 875 1:00:38,293 --> 1:00:42,250 Here, it is reversed. And the group velocity is 1 1/2 876 1:00:42,250 --> 1:00:45,673 times the phase velocity. Shallow water waves, 877 1:00:45,673 --> 1:00:50,010 which is also part of your problem set, are nondispersive, 878 1:00:50,010 --> 1:00:53,434 not so intuitive. So, if you have very shallow 879 1:00:53,434 --> 1:00:57,619 water waves, which means, then, that the amount of water 880 1:00:57,619 --> 1:01:00,663 that you have, the height of the water is 881 1:01:00,663 --> 1:01:05,000 substantially lower than the wavelengths. 882 1:01:05,000 --> 1:01:10,019 Then, you get a nondispersive relation, which is not at all 883 1:01:10,019 --> 1:01:14,000 obvious either of course. Sound in this room is 884 1:01:14,000 --> 1:01:16,509 nondispersive. Thank goodness. 885 1:01:16,509 --> 1:01:21,788 Imagine for a minute that the high-frequency would travel with 886 1:01:21,788 --> 1:01:26,115 substantially higher speed than my low frequencies. 887 1:01:26,115 --> 1:01:32,000 Then only the people in front here knew what I was saying. 888 1:01:32,000 --> 1:01:35,521 And the one in back had no clue because the higher frequencies 889 1:01:35,521 --> 1:01:38,523 might arrive much earlier than the lower frequencies. 890 1:01:38,523 --> 1:01:41,236 And you could not make sense of what I'm saying. 891 1:01:41,236 --> 1:01:44,584 Imagine you go to a concert. If sound were very dispersive, 892 1:01:44,584 --> 1:01:47,990 then you would have to pay a very high price for the tickets 893 1:01:47,990 --> 1:01:50,299 in the front row. And, the tickets in the 894 1:01:50,299 --> 1:01:53,763 balcony, they would give them away for free because you would 895 1:01:53,763 --> 1:01:56,996 have no clue what you are listening at anyhow because the 896 1:01:56,996 --> 1:02:00,286 different frequencies of the violins and of the bass would 897 1:02:00,286 --> 1:02:04,976 reach you at different times. So, we are fortunate that 898 1:02:04,976 --> 1:02:08,454 sounds in this lecture hall is nondispersive. 899 1:02:08,454 --> 1:02:11,932 The next lecture, we will enter the domain of 900 1:02:11,932 --> 1:02:14,857 electromagnetic waves. In other words, 901 1:02:14,857 --> 1:02:19,047 you're going to love it. You're going to see Maxwell's 902 1:02:19,047 --> 1:02:22,130 equations. And I know you love Maxwell's 903 1:02:22,130 --> 1:02:24,343 equations. You remember 8.02, 904 1:02:24,343 --> 1:02:27,664 the good old days. And we will derive using 905 1:02:27,664 --> 1:02:31,458 Maxwell's equations that there is such a thing as 906 1:02:31,458 --> 1:02:35,569 electromagnetic waves, and that they move in a vacuum 907 1:02:35,569 --> 1:02:40,470 with a speed which we call [C?] in physics, which is 300,000 km 908 1:02:40,470 --> 1:02:46,540 per second. That can actually be derived. 909 1:02:46,540 --> 1:02:51,302 So, in free space, that means in vacuum, 910 1:02:51,302 --> 1:02:56,063 electromagnetic waves are nondispersive. 911 1:02:56,063 --> 1:03:01,558 Omega equals C times K. That's a nondispersive 912 1:03:01,558 --> 1:03:06,125 relationship. So, the frequency of 913 1:03:06,125 --> 1:03:11,012 electromagnetic waves is C, the speed of light, 914 1:03:11,012 --> 1:03:16,006 divided by lambda. And so we will encounter with 915 1:03:16,006 --> 1:03:20,787 8.03 radio waves. And we will actually produce 916 1:03:20,787 --> 1:03:23,762 them. And let's say we take a 917 1:03:23,762 --> 1:03:28,862 frequency of about 1 MHz, ten to the sixth hertz, 918 1:03:28,862 --> 1:03:33,856 that would give you a wavelength of about 300 m. 919 1:03:33,856 --> 1:03:39,670 You can use this equation. If we go to radar, 920 1:03:39,670 --> 1:03:44,869 which we have right here in front of us, the frequency is 921 1:03:44,869 --> 1:03:48,210 about 10 GHz, ten to the tenth hertz, 922 1:03:48,210 --> 1:03:52,109 and so you have a wavelength of about 3 cm. 923 1:03:52,109 --> 1:03:57,400 And if you go to infrared radiation, you have a wavelength 924 1:03:57,400 --> 1:04:02,647 of tens of microns. And if you go to visible light, 925 1:04:02,647 --> 1:04:07,599 you get frequencies up to five times ten to the 14th hertz. 926 1:04:07,599 --> 1:04:11,698 And, you get a wavelength of about half a micron. 927 1:04:11,698 --> 1:04:16,480 And when you go to ultraviolet and X-rays and gamma rays, 928 1:04:16,480 --> 1:04:19,555 the frequencies go higher and higher. 929 1:04:19,555 --> 1:04:23,226 And the wavelengths get shorter and shorter. 930 1:04:23,226 --> 1:04:27,923 Nondispersive [in vacuum?], they all travel with exactly 931 1:04:27,923 --> 1:04:30,485 the same speed. Light in glass, 932 1:04:30,485 --> 1:04:32,876 or in water, for that matter, 933 1:04:32,876 --> 1:04:38,000 is not nondispersive. It's very dispersive. 934 1:04:38,000 --> 1:04:43,280 As we will derive in 8.03, the speed of electromagnetic 935 1:04:43,280 --> 1:04:48,951 radiation is C divided by the square root of the dielectric 936 1:04:48,951 --> 1:04:53,644 constant, kappa E, and the relative permeability, 937 1:04:53,644 --> 1:05:00,000 kappa N, which you may or may not recognize from 8.02 days. 938 1:05:00,000 --> 1:05:00,658 And in vacuum, this is one, 939 1:05:00,658 --> 1:05:01,746 and this is one. But that's not the case for 940 1:05:01,746 --> 1:05:02,759 water, and that's not the case for glass. 941 1:05:02,759 --> 1:05:03,670 And in fact, this kappa E is strongly 942 1:05:03,670 --> 1:05:04,784 frequency dependent. So if I take water as an 943 1:05:04,784 --> 1:05:06,303 example for its kappa M is very close to one because water is 944 1:05:06,303 --> 1:05:07,746 not ferromagnetic material, and all materials that are not 945 1:05:07,746 --> 1:05:09,341 ferromagnetic have a kappa M or relative permeability very close 946 1:05:09,341 --> 1:05:14,485 to 1.0000. So, forget kappa M now. 947 1:05:14,485 --> 1:05:26,382 But I'll show you the enormous frequency dependency of kappa E 948 1:05:26,382 --> 1:05:35,549 of the dielectric constant. It's huge for water. 949 1:05:35,549 --> 1:05:47,251 If you look in the frequency range, F is about zero to ten to 950 1:05:47,251 --> 1:05:56,418 the tenth hertz. So, that is our 10th gigahertz. 951 1:05:56,418 --> 1:06:05,000 That is radar. Kappa E is about 78. 952 1:06:05,000 --> 1:06:10,030 So, that means the speed of radar in water is one ninth, 953 1:06:10,030 --> 1:06:12,957 roughly, of C. So, it's only 10%, 954 1:06:12,957 --> 1:06:15,701 roughly. It's ten times slower. 955 1:06:15,701 --> 1:06:19,908 If we go to visible light, so this is the rate, 956 1:06:19,908 --> 1:06:23,201 and this is, of course, an even lower 957 1:06:23,201 --> 1:06:26,128 frequency that goes to the radio. 958 1:06:26,128 --> 1:06:31,067 If you look at visible light, you may say that light by 959 1:06:31,067 --> 1:06:37,089 definition has to be visible. Well, we physicists make a 960 1:06:37,089 --> 1:06:41,026 distinction between light in general, which is all 961 1:06:41,026 --> 1:06:45,125 electromagnetic radiation, and a certain part of the 962 1:06:45,125 --> 1:06:49,464 spectrum that we can see. And that, we call the visible 963 1:06:49,464 --> 1:06:51,794 part. So that's visible light, 964 1:06:51,794 --> 1:06:56,455 which has a frequency of very roughly five times ten to the 965 1:06:56,455 --> 1:06:58,142 14th hertz. It's huge. 966 1:06:58,142 --> 1:07:02,000 And, kappa E there is roughly 1.77. 967 1:07:02,000 --> 1:07:04,606 Look at the enormous difference. 968 1:07:04,606 --> 1:07:09,399 At that very high frequency, they make the electric dipole 969 1:07:09,399 --> 1:07:13,687 of water cannot follow the changing electric fields. 970 1:07:13,687 --> 1:07:17,303 And therefore, it just sits there and it has 971 1:07:17,303 --> 1:07:21,339 almost no impact. And that's why the number is so 972 1:07:21,339 --> 1:07:24,114 low. So, that means that in water, 973 1:07:24,114 --> 1:07:30,000 visible light has a speed which is about 30% lower than C. 974 1:07:30,000 --> 1:07:33,608 That is dispersion. And, later in the course, 975 1:07:33,608 --> 1:07:38,529 we will see how a prison made of water or made of glass makes 976 1:07:38,529 --> 1:07:42,548 it possible to decompose white lights into colors. 977 1:07:42,548 --> 1:07:45,911 That is only possible cause of dispersion. 978 1:07:45,911 --> 1:07:49,684 It is only possible because the speed of light, 979 1:07:49,684 --> 1:07:53,375 for blue light, is different from the speed of 980 1:07:53,375 --> 1:07:56,000 light for red light. 981 1:07:56,000 --> 1:08:01,000 982 1:08:01,000 --> 1:08:04,712 We will also explore in 8.03 the transports of 983 1:08:04,712 --> 1:08:08,094 electromagnetic waves through metal pipes. 984 1:08:08,094 --> 1:08:12,219 We call them waveguides. Here, I have a metal pipe. 985 1:08:12,219 --> 1:08:16,345 The opening here is about 1 1/2 centimeters across. 986 1:08:16,345 --> 1:08:19,727 Light has no difficulties getting through. 987 1:08:19,727 --> 1:08:22,614 I can look at you and I can see you. 988 1:08:22,614 --> 1:08:26,244 And I can see you. You may be able to see me. 989 1:08:26,244 --> 1:08:28,885 I don't know. I have no problems. 990 1:08:28,885 --> 1:08:34,000 Lights has no difficulty getting through here. 991 1:08:34,000 --> 1:08:40,109 But, radar, with a wavelength of 3 cm would have great 992 1:08:40,109 --> 1:08:43,912 difficulties getting through here. 993 1:08:43,912 --> 1:08:50,713 I will derive in the future what happens with radar when you 994 1:08:50,713 --> 1:08:57,629 send it through metal plates. You will see a dramatic example 995 1:08:57,629 --> 1:09:02,585 of dispersion. And this is the result of the 996 1:09:02,585 --> 1:09:09,040 boundary conditions of the electromagnetic waves with the 997 1:09:09,040 --> 1:09:12,869 pipe. But I want to tell you the 998 1:09:12,869 --> 1:09:16,392 results because it is such a remarkable example of 999 1:09:16,392 --> 1:09:19,411 dispersion. And I want to demonstrate it to 1000 1:09:19,411 --> 1:09:22,718 you because it's very relevant in this lecture. 1001 1:09:22,718 --> 1:09:25,522 Because it all comes down to dispersion. 1002 1:09:25,522 --> 1:09:29,692 We have there two aluminum metal plates that you see there. 1003 1:09:29,692 --> 1:09:33,000 Here is one, and here is the other. 1004 1:09:33,000 --> 1:09:36,152 Let's call this the direction of Z. 1005 1:09:36,152 --> 1:09:40,046 And, let's say they are separation A apart. 1006 1:09:40,046 --> 1:09:43,569 So, that's the separation between them. 1007 1:09:43,569 --> 1:09:48,854 We have a 10 GHz transmitter, which is a radar transmitter 1008 1:09:48,854 --> 1:09:51,913 here. And here we have a receiver, 1009 1:09:51,913 --> 1:09:57,384 which can receive the 10 GHz. And so, we're going to send it 1010 1:09:57,384 --> 1:10:02,604 from here to the other end. The wavelengths, 1011 1:10:02,604 --> 1:10:07,017 lambda, is 3 cm. So, this is ten to the tenth 1012 1:10:07,017 --> 1:10:09,524 hertz. So, lambda is 3 cm. 1013 1:10:09,524 --> 1:10:14,538 When A is less than 1.5 cm, radar cannot go through 1014 1:10:14,538 --> 1:10:17,747 anymore. It's larger than 1.5 cm. 1015 1:10:17,747 --> 1:10:21,257 It can, and I will demonstrate that. 1016 1:10:21,257 --> 1:10:26,873 [UNINTELLIGIBLE] behind that is, and that's really what I 1017 1:10:26,873 --> 1:10:32,489 want to get across is to dispersion relation which I will 1018 1:10:32,489 --> 1:10:38,538 derive, but not now. But I will derive it a few 1019 1:10:38,538 --> 1:10:42,346 weeks from now when we reach that point. 1020 1:10:42,346 --> 1:10:47,717 But I want to show you what that dispersion relationship 1021 1:10:47,717 --> 1:10:52,111 will be if we have done our homework: omega N, 1022 1:10:52,111 --> 1:10:55,821 whereby N equals one, two, three, four, 1023 1:10:55,821 --> 1:11:01,191 five equals C times the square root of N pi divided by A 1024 1:11:01,191 --> 1:11:07,036 squared plus K of Z squared. Let's first look at this term 1025 1:11:07,036 --> 1:11:10,828 even though you don't see directly where it comes from. 1026 1:11:10,828 --> 1:11:14,761 You can connect to it already. It has to do with boundary 1027 1:11:14,761 --> 1:11:17,640 conditions. Remember, when we had a string 1028 1:11:17,640 --> 1:11:21,151 that is fixed at both ends, we got terms like this. 1029 1:11:21,151 --> 1:11:23,328 Remember, we had sound in a box. 1030 1:11:23,328 --> 1:11:27,752 We had N pi divided by L in the X direction, and pi divided by L 1031 1:11:27,752 --> 1:11:32,873 in the Y direction. So, you recognize here that it 1032 1:11:32,873 --> 1:11:38,523 has to do with the boundary condition of the electromagnetic 1033 1:11:38,523 --> 1:11:43,887 radiation without exactly understanding why it is the way 1034 1:11:43,887 --> 1:11:47,047 it is. And so, the phase velocity, 1035 1:11:47,047 --> 1:11:51,932 V phase, in the Z direction, is omega divided by KZ. 1036 1:11:51,932 --> 1:11:57,391 And, the group velocity in the Z direction is D omega DKZ. 1037 1:11:57,391 --> 1:12:04,000 I will concentrate now only on the case that N equals one. 1038 1:12:04,000 --> 1:12:09,748 So I make this a one because that gives me the lowest 1039 1:12:09,748 --> 1:12:14,834 frequency possible in this dispersion relation. 1040 1:12:14,834 --> 1:12:20,693 And I'm now going to graph that thing, omega versus K. 1041 1:12:20,693 --> 1:12:26,000 This is nondispersion. [SOUND OFF/THEN ON] 1042 1:12:26,000 --> 1:12:38,000 1043 1:12:38,000 --> 1:12:42,233 I can write here omega equals KZ times C. 1044 1:12:42,233 --> 1:12:47,948 That is the case when K is huge, very short wavelength, 1045 1:12:47,948 --> 1:12:52,500 way shorter than the dimension of the [pi?]. 1046 1:12:52,500 --> 1:12:57,686 Then, that system acts like a nondispersive media. 1047 1:12:57,686 --> 1:13:02,660 You're somewhere here. But, if you go to the low 1048 1:13:02,660 --> 1:13:08,743 frequencies, this is the curve. And what you see here, 1049 1:13:08,743 --> 1:13:13,260 there is a value which we call the cutoff frequency omega. 1050 1:13:13,260 --> 1:13:17,856 It's normally called omega C. Below it, no radiation can go 1051 1:13:17,856 --> 1:13:21,581 through the pipe. This is not a solution anymore 1052 1:13:21,581 --> 1:13:25,543 to my wave equation. You are stuck to this line for 1053 1:13:25,543 --> 1:13:28,000 the mode N equals one. 1054 1:13:28,000 --> 1:13:32,000 1055 1:13:32,000 --> 1:13:36,000 So, what is required: that the frequency for this 1056 1:13:36,000 --> 1:13:39,500 radiation, well, first let me give you what 1057 1:13:39,500 --> 1:13:43,416 omega minimum is. Omega minimum is when you make 1058 1:13:43,416 --> 1:13:47,166 in the extreme case KZ zero, and N equals one. 1059 1:13:47,166 --> 1:13:52,083 So, you get C pi divided by A. That is the minimum value for 1060 1:13:52,083 --> 1:13:56,166 omega that can propagate through these two plates. 1061 1:13:56,166 --> 1:14:01,333 And so, a required frequency is that F is larger than C divided 1062 1:14:01,333 --> 1:14:06,032 by 2A. That means A has to be larger 1063 1:14:06,032 --> 1:14:10,320 than 1.5 cm. And, that is so nonintuitive. 1064 1:14:10,320 --> 1:14:15,026 And this is what I want to demonstrate to you. 1065 1:14:15,026 --> 1:14:18,477 So we have there those two plates. 1066 1:14:18,477 --> 1:14:23,496 And the plates are now something like 2 cm apart. 1067 1:14:23,496 --> 1:14:27,261 You can check this after the lecture. 1068 1:14:27,261 --> 1:14:33,326 We modulate the 10 GHz with a nasty triangular sound of 550 1069 1:14:33,326 --> 1:14:36,851 Hz. And the reason why that is 1070 1:14:36,851 --> 1:14:40,698 nasty: because you now understand that anything that is 1071 1:14:40,698 --> 1:14:44,971 modulated triangular requires high harmonics because think in 1072 1:14:44,971 --> 1:14:48,105 terms of Fourier space, and higher harmonics, 1073 1:14:48,105 --> 1:14:51,524 not so nice in your ears. So, if this were a nice 1074 1:14:51,524 --> 1:14:54,943 sinusoid, you would hear a beautiful 550 Hz tone. 1075 1:14:54,943 --> 1:15:00,000 But because we modulated with a triangle, it's a nasty tone. 1076 1:15:00,000 --> 1:15:04,803 But the only reason why we wanted you to hear something is 1077 1:15:04,803 --> 1:15:08,932 that when this receiver receives the radar signal, 1078 1:15:08,932 --> 1:15:12,050 we want to hear this modulated signal. 1079 1:15:12,050 --> 1:15:15,674 Unfortunately, you and I cannot hear 10 GHz. 1080 1:15:15,674 --> 1:15:19,803 Our ears are not designed to follow a frequency of 1081 1:15:19,803 --> 1:15:23,174 10,000,000,000 Hz. It cuts off at 20 kHz, 1082 1:15:23,174 --> 1:15:26,882 as you remember. So, we are going to show you 1083 1:15:26,882 --> 1:15:32,016 this transmitter first. The transmitter is the signal 1084 1:15:32,016 --> 1:15:34,519 upstairs. So, it is 10 to your hertz, 1085 1:15:34,519 --> 1:15:37,926 but it is a triangular modulation which is 550 Hz. 1086 1:15:37,926 --> 1:15:41,263 Here's the receiver. When I turn on the receiver, 1087 1:15:41,263 --> 1:15:43,975 you'll hear that signal coming out here. 1088 1:15:43,975 --> 1:15:47,521 [SOUND PLAYS] And at the bottom, you see this signal 1089 1:15:47,521 --> 1:15:50,997 received by the receiver. And the fact that it is a 1090 1:15:50,997 --> 1:15:55,100 little larger than the upstairs is simply a matter of how we 1091 1:15:55,100 --> 1:15:58,646 adjust our amplifiers. Nicole, can you be my witness 1092 1:15:58,646 --> 1:16:04,000 and look here so that you can tell the class what I'm doing? 1093 1:16:04,000 --> 1:16:07,053 I'm going to take these two plates now. 1094 1:16:07,053 --> 1:16:10,187 Please come up here and just watch here. 1095 1:16:10,187 --> 1:16:14,526 And tell them that what I'm doing is completely honest. 1096 1:16:14,526 --> 1:16:18,383 You see the opening here now of these two plates? 1097 1:16:18,383 --> 1:16:22,482 And these plates are about 1.8 cm apart, almost two. 1098 1:16:22,482 --> 1:16:27,062 And now, I'm going to make it a little less than 1 1/2 cm. 1099 1:16:27,062 --> 1:16:31,000 Watch there and listen. [SOUND PLAYS] 1100 1:16:31,000 --> 1:16:33,830 It's gone. It's completely gone, 1101 1:16:33,830 --> 1:16:38,942 and you tell the class that the opening is still sizable, 1102 1:16:38,942 --> 1:16:42,046 right? The opening is just a little 1103 1:16:42,046 --> 1:16:46,246 less than 1 1/2 cm. And it instantaneously goes 1104 1:16:46,246 --> 1:16:49,076 away. [SOUND PLAYS] There it is. 1105 1:16:49,076 --> 1:16:51,997 Why don't you squeeze it, Nicole? 1106 1:16:51,997 --> 1:16:55,467 And it's gone. [SOUND PLAYS] Thank you, 1107 1:16:55,467 --> 1:16:57,749 Nicole. You are an expert. 1108 1:16:57,749 --> 1:17:01,857 Now, there's one more thing that is important. 1109 1:17:01,857 --> 1:17:06,696 And that has to do with your sleepless nights because, 1110 1:17:06,696 --> 1:17:14,000 you know, I'm very concerned about you getting enough sleep. 1111 1:17:14,000 --> 1:17:17,775 Look at that dispersion relation. 1112 1:17:17,775 --> 1:17:23,438 Look at this curve. This line indicates the speed 1113 1:17:23,438 --> 1:17:28,275 of light, C. That line indicates that when 1114 1:17:28,275 --> 1:17:35,000 omega divided by K is C, that's the speed of light. 1115 1:17:35,000 --> 1:17:39,919 So, what now is the phase velocity, for instance, 1116 1:17:39,919 --> 1:17:44,941 at this frequency so that it has this value for K? 1117 1:17:44,941 --> 1:17:49,554 Well, it's this. This flow is larger than this 1118 1:17:49,554 --> 1:17:52,936 one. It's higher than the speed of 1119 1:17:52,936 --> 1:17:56,626 light. Not only is it higher than the 1120 1:17:56,626 --> 1:18:00,110 speed of light, the phase velocity, 1121 1:18:00,110 --> 1:18:06,157 that by the time I reach this point here, the phase velocity 1122 1:18:06,157 --> 1:18:12,000 goes to infinity because look at the slope. 1123 1:18:12,000 --> 1:18:16,023 It's 90°. So, we now have a case that the 1124 1:18:16,023 --> 1:18:19,544 phase velocity is larger than C. No. 1125 1:18:19,544 --> 1:18:24,372 The phase velocity can be much, much, much, much, 1126 1:18:24,372 --> 1:18:29,000 much larger than C approaching infinity. 1127 1:18:29,000 --> 1:18:34,218 And what now is the group velocity when we reach this 1128 1:18:34,218 --> 1:18:39,235 point of the cutoff frequency? That is the tangent. 1129 1:18:39,235 --> 1:18:43,951 And so, this is the slope of the group velocity. 1130 1:18:43,951 --> 1:18:49,570 So, the group velocity goes to zero in that extreme case. 1131 1:18:49,570 --> 1:18:54,186 I sure as hell hope that you can sleep tonight. 1132 1:18:54,186 See you next Tuesday.