1 00:00:24,000 --> 00:00:28,531 Today's a big day for the physics department and for MIT. 2 00:00:28,531 --> 00:00:32,819 Professor Frank Wilczek was sharing the Nobel Prize in 3 00:00:32,819 --> 00:00:37,512 physics for his seminal work when he was a graduate student 4 00:00:37,512 --> 00:00:42,124 on quantum chromo-dynamics. We are now ready to tackle the 5 00:00:42,124 --> 00:00:44,875 normal modes in continuous mediums. 6 00:00:44,875 --> 00:00:49,325 And I will do that starting first with a string which is 7 00:00:49,325 --> 00:00:53,290 fixed at both ends. One way you can do that is you 8 00:00:53,290 --> 00:00:57,659 can go back to the results of last lecture when we have 9 00:00:57,659 --> 00:01:00,815 capital N bits, and you can make N go to 10 00:01:00,815 --> 00:01:05,463 infinity. And the shapes that you get for 11 00:01:05,463 --> 00:01:10,072 your continuous media for the strings when they're fixed at 12 00:01:10,072 --> 00:01:14,841 both ends are sinusoidal motions would be the first harmonic, 13 00:01:14,841 --> 00:01:18,973 the lowest frequency, and then you get high harmonics 14 00:01:18,973 --> 00:01:21,913 with higher frequencies. The question, 15 00:01:21,913 --> 00:01:26,125 now, is one of those normal mode frequencies for those 16 00:01:26,125 --> 00:01:29,781 continued media? And I will derive these today. 17 00:01:29,781 --> 00:01:34,549 I will derive these normal mode frequencies using a different 18 00:01:34,549 --> 00:01:39,000 approach than the N go to infinity route. 19 00:01:39,000 --> 00:01:43,265 And, I find the same results, of course. 20 00:01:43,265 --> 00:01:49,281 So, imagine that I have a string here and the tension is 21 00:01:49,281 --> 00:01:53,328 T, and the mass per unit length is mu. 22 00:01:53,328 --> 00:02:00,000 And, I wiggle one side, and I generate in there a wave. 23 00:02:00,000 --> 00:02:04,899 So, I moved the [NOISE OBSCURES] angular frequency, 24 00:02:04,899 --> 00:02:08,916 omega, and I do that with an amplitude, A. 25 00:02:08,916 --> 00:02:14,305 I will then generate waves whereby this is what we call, 26 00:02:14,305 --> 00:02:19,792 in physics, the wavelength. That is the distance that the 27 00:02:19,792 --> 00:02:24,005 disturbance travels in one oscillation time. 28 00:02:24,005 --> 00:02:28,904 If I call this a positive direction, this wave will 29 00:02:28,904 --> 00:02:34,000 propagate with beat V in that direction. 30 00:02:34,000 --> 00:02:39,205 And we know what V is. V is the square root of T 31 00:02:39,205 --> 00:02:43,636 divided by mu. We derived that last time. 32 00:02:43,636 --> 00:02:49,174 And so, Y as a function of X and T, Y being in this 33 00:02:49,174 --> 00:02:55,598 direction, X being in this direction, is that an amplitude, 34 00:02:55,598 --> 00:03:00,471 A, times the sine, or if you wish the cosine, 35 00:03:00,471 --> 00:03:05,787 that's fine with me, times 2 pi divided by lambda 36 00:03:05,787 --> 00:03:11,488 times X minus VT. Let's first make T equals zero. 37 00:03:11,488 --> 00:03:16,063 Then you see what you have here, the sine 2 pi over lambda 38 00:03:16,063 --> 00:03:19,836 times X, if you take X zero, then you find zero. 39 00:03:19,836 --> 00:03:23,528 And, if you make X larger by an amount, lambda, 40 00:03:23,528 --> 00:03:27,541 you get zero again. So, that's why lambda is called 41 00:03:27,541 --> 00:03:30,752 the wavelength. The parent repeats over a 42 00:03:30,752 --> 00:03:35,090 distance, lambda. Clearly, this must be a 43 00:03:35,090 --> 00:03:40,000 solution to the wave equation because any single valued 44 00:03:40,000 --> 00:03:44,818 function I told you which is a function of X minus VT, 45 00:03:44,818 --> 00:03:49,727 satisfies the wave equation. And so, this one does too. 46 00:03:49,727 --> 00:03:53,545 Now, lambda is V, which is this V times the 47 00:03:53,545 --> 00:03:58,272 period of one oscillation, for which I will write a P 48 00:03:58,272 --> 00:04:02,454 today, not a T, because I don't want to confuse 49 00:04:02,454 --> 00:04:07,270 you. And, that P is obviously 2 pi 50 00:04:07,270 --> 00:04:12,458 divided by omega. This is my omega with which I 51 00:04:12,458 --> 00:04:15,278 am shaking. And so, omega, 52 00:04:15,278 --> 00:04:18,436 which is, then, my frequency, 53 00:04:18,436 --> 00:04:24,187 is 2 pi divided by lambda. I will introduce for 2 pi 54 00:04:24,187 --> 00:04:30,954 divide by lambda a new symbol, which I call K which is called 55 00:04:30,954 --> 00:04:35,428 the wave number. It has dimensions, 56 00:04:35,428 --> 00:04:38,920 meters minus one. This is done in most books. 57 00:04:38,920 --> 00:04:42,888 It is unfortunate that French calls K1 over lambda. 58 00:04:42,888 --> 00:04:47,650 It's not his fault because in the old days it was always done 59 00:04:47,650 --> 00:04:50,269 that way. I will always follow the 60 00:04:50,269 --> 00:04:53,444 convention of Beckafee and Barrett [SP?], 61 00:04:53,444 --> 00:04:56,698 and I will call K always 2 pi over lambda. 62 00:04:56,698 --> 00:05:00,507 If I use that new K, then I can rewrite this in a 63 00:05:00,507 --> 00:05:06,710 form that you see very often. But there's nothing wrong with 64 00:05:06,710 --> 00:05:10,234 that form. That would now be A times the 65 00:05:10,234 --> 00:05:14,933 sine of KX minus omega T. Notice the 2 pi lambda over 66 00:05:14,933 --> 00:05:17,825 lambda becomes K. So, that is KX. 67 00:05:17,825 --> 00:05:23,156 But, omega is always K times V. And so, this form is nice in 68 00:05:23,156 --> 00:05:27,945 the sense that if you have these two numbers, you know 69 00:05:27,945 --> 00:05:30,927 immediately what the frequency is. 70 00:05:30,927 --> 00:05:36,100 That's nonnegotiable. And you know immediately what 71 00:05:36,100 --> 00:05:39,214 the wavelength is. It's 2 pi divided by that 72 00:05:39,214 --> 00:05:41,676 number. And, you even know that the 73 00:05:41,676 --> 00:05:44,863 ratio of those two, omega divided by K is UV, 74 00:05:44,863 --> 00:05:48,267 which is that V. So, it is a nice way of writing 75 00:05:48,267 --> 00:05:51,019 things down. Now, I'm going to not only 76 00:05:51,019 --> 00:05:55,147 generate a wave on this side that moves in this direction. 77 00:05:55,147 --> 00:06:00,000 I'm going to generate one that goes in this direction. 78 00:06:00,000 --> 00:06:04,459 And then I want to see what they do together. 79 00:06:04,459 --> 00:06:10,033 And so, this is now the wave that goes in the positive X 80 00:06:10,033 --> 00:06:13,682 direction. And now, I'm going to have 81 00:06:13,682 --> 00:06:17,128 another one: Y2 XT, same amplitude, 82 00:06:17,128 --> 00:06:21,385 same frequency. Therefore, same wavelength, 83 00:06:21,385 --> 00:06:25,439 but now it's going not in this direction. 84 00:06:25,439 --> 00:06:30,000 But it's going in this direction. 85 00:06:30,000 --> 00:06:34,423 Notice there is a plus here, but there is a minus there. 86 00:06:34,423 --> 00:06:38,686 And so, I want to know, now, what some of these two is 87 00:06:38,686 --> 00:06:43,672 because one is going like this. Another one is going like this. 88 00:06:43,672 --> 00:06:48,418 So, I want to know what the superposition of those two waves 89 00:06:48,418 --> 00:06:50,911 do. And so, Y, which is then the 90 00:06:50,911 --> 00:06:53,485 total displacement is Y1 plus Y2. 91 00:06:53,485 --> 00:06:57,747 And so, I get the sine of alpha plus the sine of beta. 92 00:06:57,747 --> 00:07:02,955 That is twice. So that becomes 2A times half 93 00:07:02,955 --> 00:07:06,319 the sum, the sine of half the sum. 94 00:07:06,319 --> 00:07:11,517 Half the sum becomes KX times the cosine of half the 95 00:07:11,517 --> 00:07:15,493 difference. Half the difference would be 96 00:07:15,493 --> 00:07:19,977 minus omega T. But minus and plus for cosine, 97 00:07:19,977 --> 00:07:24,055 so I will just write down cosign omega T. 98 00:07:24,055 --> 00:07:27,826 And this, now, is a very unusual wave. 99 00:07:27,826 --> 00:07:31,393 All the spatial information is here. 100 00:07:31,393 --> 00:07:37,000 So, this is all the spatial information. 101 00:07:37,000 --> 00:07:40,154 And, all the time information is here. 102 00:07:40,154 --> 00:07:45,268 And so, whenever this sine is zero, if the X is such that the 103 00:07:45,268 --> 00:07:47,740 sine is zero, just tough luck. 104 00:07:47,740 --> 00:07:50,724 Then, that location, X, never moves. 105 00:07:50,724 --> 00:07:54,390 The cosine omega T could never make it move. 106 00:07:54,390 --> 00:07:58,055 It always stands still. We call those nodes, 107 00:07:58,055 --> 00:08:03,000 and I will show you, of course, some examples. 108 00:08:03,000 --> 00:08:06,845 This has a name, a very nice name. 109 00:08:06,845 --> 00:08:13,721 It's called a standing wave, whereas those that I have there 110 00:08:13,721 --> 00:08:19,432 we call traveling waves. Now I'm going to make the 111 00:08:19,432 --> 00:08:25,026 string, fixed at this end, and fixed at this end. 112 00:08:25,026 --> 00:08:28,755 I may shake it a little bit here. 113 00:08:28,755 --> 00:08:34,000 You'll see a demonstration of that. 114 00:08:34,000 --> 00:08:39,195 And now, at this X equals zero, and at this X equals L, 115 00:08:39,195 --> 00:08:44,005 I now have boundary conditions that I have to meet. 116 00:08:44,005 --> 00:08:49,201 Y cannot be anything but zero there because it's fixed. 117 00:08:49,201 --> 00:08:54,588 And so, I now demand that X equals zero and at X equals L 118 00:08:54,588 --> 00:08:59,784 that Y must become zero. And so, the only way that that 119 00:08:59,784 --> 00:09:05,268 can be done is that you only allow certain values for K to 120 00:09:05,268 --> 00:09:09,357 exist. And, those values for K, 121 00:09:09,357 --> 00:09:15,250 which I will now give an index, N, as in Nancy which is going 122 00:09:15,250 --> 00:09:18,687 to be the normal mode, N equals one, 123 00:09:18,687 --> 00:09:23,401 two, three, four, is now going to be N pi divided 124 00:09:23,401 --> 00:09:26,053 by L. Clearly, if X is zero. 125 00:09:26,053 --> 00:09:30,439 Oh sorry, we're here. If X equals zero, 126 00:09:30,439 --> 00:09:35,010 then surely we are here. Then the whole thing is always 127 00:09:35,010 --> 00:09:37,126 zero. That's not an issue. 128 00:09:37,126 --> 00:09:39,666 But you see now, if X equals L, 129 00:09:39,666 --> 00:09:42,883 then you get N times pi. And so, again, 130 00:09:42,883 --> 00:09:46,947 the sine becomes zero. So, I've met this boundary 131 00:09:46,947 --> 00:09:50,079 condition that this point cannot move. 132 00:09:50,079 --> 00:09:53,888 So, lambda of N, which is 2 pi divided by K of 133 00:09:53,888 --> 00:09:59,137 N, that's the way that I define K of N, then becomes 2L divided 134 00:09:59,137 --> 00:10:03,705 by N. And, being that one, 135 00:10:03,705 --> 00:10:07,898 two, three, four, five, omega N, 136 00:10:07,898 --> 00:10:16,014 which is always K of N times V, therefore becomes N pi V over 137 00:10:16,014 --> 00:10:22,642 L, and the frequency in hertz, if you prefer that, 138 00:10:22,642 --> 00:10:29,000 is 2 pi smaller would become NV over 2L. 139 00:10:29,000 --> 00:10:36,000 140 00:10:36,000 --> 00:10:42,223 And so, what are you going to see when you plot dysfunction, 141 00:10:42,223 --> 00:10:47,814 this standing wave function? Well, it depends on the N 142 00:10:47,814 --> 00:10:51,084 number. Let's take N equals one. 143 00:10:51,084 --> 00:10:56,886 And so, the length is L. If you plot that sinusoid for N 144 00:10:56,886 --> 00:11:01,000 equals one. It looks like this. 145 00:11:01,000 --> 00:11:04,948 And then, the cosine term will make you do this. 146 00:11:04,948 --> 00:11:07,888 That's the task of this cosine term. 147 00:11:07,888 --> 00:11:12,677 The sine term is just this thing, which would then have an 148 00:11:12,677 --> 00:11:15,533 amplitude. That could be different, 149 00:11:15,533 --> 00:11:18,390 of course, for the different modes. 150 00:11:18,390 --> 00:11:22,422 But it would be this amplitude that you see here. 151 00:11:22,422 --> 00:11:25,111 Recall this one, the fundamental. 152 00:11:25,111 --> 00:11:29,899 It's the lowest normal mode, but we also call it the first 153 00:11:29,899 --> 00:11:33,189 harmonic. I will do both. 154 00:11:33,189 --> 00:11:37,790 I will sometimes refer to this as a fundamental because I'm 155 00:11:37,790 --> 00:11:41,597 more used to that. But I will also refer to it as 156 00:11:41,597 --> 00:11:44,453 the first harmonic. Lambda one is 2L. 157 00:11:44,453 --> 00:11:47,229 That's just staring you in the face. 158 00:11:47,229 --> 00:11:51,671 In order to make a full wave out of this, you have to add 159 00:11:51,671 --> 00:11:53,733 this. So, lambda one is 2L. 160 00:11:53,733 --> 00:11:56,430 And, of course, when you look here, 161 00:11:56,430 --> 00:12:02,538 lambda one, 2L divided by one. That's exactly what we have 162 00:12:02,538 --> 00:12:05,829 there. So now we go to N equals two, 163 00:12:05,829 --> 00:12:09,119 which is called the second harmonic. 164 00:12:09,119 --> 00:12:13,726 So, let me write this down. So, this is called the 165 00:12:13,726 --> 00:12:18,615 fundamental, which is also called the first harmonic. 166 00:12:18,615 --> 00:12:23,692 And this is then called the second harmonic whereby the 167 00:12:23,692 --> 00:12:26,700 point in the middle stands still. 168 00:12:26,700 --> 00:12:33,000 And then, the cosine term will change shape like this. 169 00:12:33,000 --> 00:12:36,714 It is unfortunate that there are books that call this the 170 00:12:36,714 --> 00:12:39,169 fundamental and is the first harmonic. 171 00:12:39,169 --> 00:12:42,818 That's enormously confusing because now you had to start 172 00:12:42,818 --> 00:12:46,267 from zero which is the fundamental, and then N equals 173 00:12:46,267 --> 00:12:49,452 one is the second harmonic. I will never do that. 174 00:12:49,452 --> 00:12:51,906 This is always for the first harmonic. 175 00:12:51,906 --> 00:12:54,162 This is always the second harmonic. 176 00:12:54,162 --> 00:12:57,877 And so, for N equals two, you see immediately that lambda 177 00:12:57,877 --> 00:13:02,576 two is simply L. And that's exactly what you see 178 00:13:02,576 --> 00:13:05,318 here. And then, you can put in the 179 00:13:05,318 --> 00:13:09,889 third harmonic and I will demonstrate this very shortly. 180 00:13:09,889 --> 00:13:14,958 So here you have an omega one, and here you have an omega two. 181 00:13:14,958 --> 00:13:19,612 And, they follow this pattern. Omega two is exactly twice 182 00:13:19,612 --> 00:13:23,268 omega one because when you make N equals one, 183 00:13:23,268 --> 00:13:26,925 you have omega one. And when you make it two, 184 00:13:26,925 --> 00:13:31,767 you get twice that much. So, now you see that the 185 00:13:31,767 --> 00:13:34,787 ratios, omega one, omega two, omega three, 186 00:13:34,787 --> 00:13:37,145 relate as one, to three, to four, 187 00:13:37,145 --> 00:13:40,017 to five. And so, you can also write down 188 00:13:40,017 --> 00:13:43,185 here then that omega N is N times omega one, 189 00:13:43,185 --> 00:13:47,383 and therefore that F of N, which is the frequency in hertz 190 00:13:47,383 --> 00:13:50,993 is also N times F1. So, it's very easy to think in 191 00:13:50,993 --> 00:13:53,792 terms of the series of these harmonics. 192 00:13:53,792 --> 00:13:56,665 They're also sometimes called overtones. 193 00:13:56,665 --> 00:14:00,569 When we talk music next Thursday, I will often use the 194 00:14:00,569 --> 00:14:05,790 word overtone. So, if the lowest frequency in 195 00:14:05,790 --> 00:14:10,627 this mode were 100 Hz, then the second harmonic would 196 00:14:10,627 --> 00:14:14,441 be 200 Hz. The third harmonic would be 300 197 00:14:14,441 --> 00:14:18,162 Hz, and so on. Now, when I'm driving this 198 00:14:18,162 --> 00:14:22,720 string at one end, I will have a wave going in and 199 00:14:22,720 --> 00:14:27,651 have a wave coming back. That is exactly recipe that I 200 00:14:27,651 --> 00:14:32,401 need for a standing wave. I get reflection here, 201 00:14:32,401 --> 00:14:36,095 and so I have one wave going in, and I have one wave coming 202 00:14:36,095 --> 00:14:38,324 back. And, if I drive these at these 203 00:14:38,324 --> 00:14:42,210 discrete frequencies which are set by the boundary conditions, 204 00:14:42,210 --> 00:14:44,757 then the system will react very strongly. 205 00:14:44,757 --> 00:14:48,133 We'll go into resonance. You build up a huge amplitude 206 00:14:48,133 --> 00:14:50,235 because you keep feeding in waves. 207 00:14:50,235 --> 00:14:53,547 And, they keep coming back. So the whole thing starts 208 00:14:53,547 --> 00:14:55,840 building up. And, that is the idea of 209 00:14:55,840 --> 00:14:58,452 resonance. And, resonance and normal modes 210 00:14:58,452 --> 00:15:03,619 are one and the same. So, that's the normal mode of 211 00:15:03,619 --> 00:15:06,510 these strings fixed at both ends. 212 00:15:06,510 --> 00:15:10,395 We also refer to them, sometimes, as natural 213 00:15:10,395 --> 00:15:14,190 frequencies. These points here have a name. 214 00:15:14,190 --> 00:15:18,887 They are called nodes. So, this is a node and this is 215 00:15:18,887 --> 00:15:21,507 a node. And these points here, 216 00:15:21,507 --> 00:15:25,934 also here and here, the ones that have the largest 217 00:15:25,934 --> 00:15:32,698 amplitude are called anti-nodes. In Dutch, we call them tummies, 218 00:15:32,698 --> 00:15:35,745 very strange. We call the [barker?], 219 00:15:35,745 --> 00:15:39,314 barker is this. So, the boundary condition 220 00:15:39,314 --> 00:15:43,927 leads to discrete values of the resonance frequencies. 221 00:15:43,927 --> 00:15:48,715 And, if this were quantum mechanics, we would call these 222 00:15:48,715 --> 00:15:51,588 [igan?] solutions and igan states. 223 00:15:51,588 --> 00:15:56,898 So, if you want to write down, now, the situation for your Nth 224 00:15:56,898 --> 00:16:01,772 mode, in its most general form, then you would get Y as a 225 00:16:01,772 --> 00:16:06,386 function of X and T in the Nth mode would have its own 226 00:16:06,386 --> 00:16:11,000 amplitude, A of N, whatever that is. 227 00:16:11,000 --> 00:16:15,259 You can pick that for different values of N. 228 00:16:15,259 --> 00:16:21,004 And then, you have here the sine of N pi times X divided by 229 00:16:21,004 --> 00:16:24,075 L. And here, you have the cosine 230 00:16:24,075 --> 00:16:28,235 of omega NT. That, then, meets the boundary 231 00:16:28,235 --> 00:16:34,471 conditions for two fixed ends. And, any linear superposition, 232 00:16:34,471 --> 00:16:37,931 any combinations of various values Nancy N, 233 00:16:37,931 --> 00:16:42,379 and various values of A will satisfy the wave equation. 234 00:16:42,379 --> 00:16:46,334 So, this string can simultaneously oscillate in a 235 00:16:46,334 --> 00:16:49,135 whole series of these normal modes. 236 00:16:49,135 --> 00:16:53,254 And when we do music next Thursday you'll see that. 237 00:16:53,254 --> 00:16:58,279 I will demonstrate that to you. I want to show you now is that 238 00:16:58,279 --> 00:17:00,997 if I take a string, I need, again, 239 00:17:00,997 --> 00:17:06,022 even though it has spring-like qualities we will treat it as a 240 00:17:06,022 --> 00:17:10,245 string. Then I will show you that if I 241 00:17:10,245 --> 00:17:13,763 drive this at the end at the proper frequencies, 242 00:17:13,763 --> 00:17:16,308 that I can generate the resonances. 243 00:17:16,308 --> 00:17:19,302 And, I can make you see the normal modes. 244 00:17:19,302 --> 00:17:22,371 So, who is willing to assist me this time? 245 00:17:22,371 --> 00:17:25,065 You were dying last time also, right? 246 00:17:25,065 --> 00:17:29,182 But Nicole won the battle then. So, hold it in your hand 247 00:17:29,182 --> 00:17:33,000 firmly. Just walk back and do nothing. 248 00:17:33,000 --> 00:17:34,259 Just walk back, walk back. 249 00:17:34,259 --> 00:17:35,921 We need a little tension on there. 250 00:17:35,921 --> 00:17:38,138 OK, that is fine. So, I'm now going to wiggle 251 00:17:38,138 --> 00:17:39,750 this. This is really a fixed end. 252 00:17:39,750 --> 00:17:42,520 You will see when I hit resonance that my hand is hardly 253 00:17:42,520 --> 00:17:45,442 moving at all because I keep pumping waves in and they keep 254 00:17:45,442 --> 00:17:47,759 coming back at me. And the amplitude will build 255 00:17:47,759 --> 00:17:49,371 up. And then I have to search for 256 00:17:49,371 --> 00:17:51,184 these residences. And when I hit one, 257 00:17:51,184 --> 00:17:53,401 I know I hit one. You will see why I know it. 258 00:17:53,401 --> 00:17:55,718 I feel it in my stomach. I feel it in my tummy. 259 00:17:55,718 --> 00:17:58,287 I feel it in my hands, and my whole body knows I hit 260 00:17:58,287 --> 00:18:01,699 resonance. And you'll see that. 261 00:18:01,699 --> 00:18:05,099 OK, there we go. This is the lowest mode. 262 00:18:05,099 --> 00:18:09,093 This is N equals one. And notice that my hand is 263 00:18:09,093 --> 00:18:13,342 hardly moving at all. So my hand, for all practical 264 00:18:13,342 --> 00:18:18,186 reasons, is really a fixed end. So, you get that solution. 265 00:18:18,186 --> 00:18:22,096 So, I will now try to find the second harmonic, 266 00:18:22,096 --> 00:18:26,175 which would end up, then, as a node in the middle 267 00:18:26,175 --> 00:18:30,916 in addition to nodes at the end. Is that it? 268 00:18:30,916 --> 00:18:35,416 And again, when I hit that resonance, which is a normal 269 00:18:35,416 --> 00:18:38,916 mode, notice that my hand is hardly moving. 270 00:18:38,916 --> 00:18:43,000 And, boy, do I feel it. I really know that I am on 271 00:18:43,000 --> 00:18:45,833 resonance. I can try the third one, 272 00:18:45,833 --> 00:18:50,000 in which case you will see two nodes in the middle, 273 00:18:50,000 --> 00:18:53,416 and two at the end. Is this the third one? 274 00:18:53,416 --> 00:18:55,916 Well, I'm good today, am I not? 275 00:18:55,916 --> 00:19:01,000 You see: very clear. These points stand still. 276 00:19:01,000 --> 00:19:04,620 Normal modes: you're not supposed to do that. 277 00:19:04,620 --> 00:19:07,830 Oh man, you're ruining my demonstration. 278 00:19:07,830 --> 00:19:12,438 So, now I'll try to generate the highest frequency normal 279 00:19:12,438 --> 00:19:16,306 mode that I can. And so, you count the number of 280 00:19:16,306 --> 00:19:19,680 nodes in the middle. And if you find five, 281 00:19:19,680 --> 00:19:24,371 then that's the six harmonic. I think I can do better than 282 00:19:24,371 --> 00:19:27,169 that. But it's not so easy to get a 283 00:19:27,169 --> 00:19:30,955 very high frequency. So, you do nothing because 284 00:19:30,955 --> 00:19:35,623 you'd really ruin it. Yeah, yeah, yeah, 285 00:19:35,623 --> 00:19:38,967 yeah, yeah, yeah, I'm getting there. 286 00:19:38,967 --> 00:19:42,405 I'm getting there. I'm getting there. 287 00:19:42,405 --> 00:19:46,226 Oh, no, no, no. You shouldn't have moved, 288 00:19:46,226 --> 00:19:49,474 you see? No, it was not your fault. 289 00:19:49,474 --> 00:19:53,486 I'll try it again. I got another resonance. 290 00:19:53,486 --> 00:19:56,925 I got a resonance. I got a resonance. 291 00:19:56,925 --> 00:19:58,740 Count. Please count. 292 00:19:58,740 --> 00:20:01,223 How many did you see? Five? 293 00:20:01,223 --> 00:20:06,000 I counted 12 nodes in the middle. 294 00:20:06,000 --> 00:20:10,851 What is your name? All right, so that's just the 295 00:20:10,851 --> 00:20:16,219 way that it works but you seem normal mode solutions, 296 00:20:16,219 --> 00:20:20,038 which either result in standing waves. 297 00:20:20,038 --> 00:20:24,064 And the demonstration speaks for itself. 298 00:20:24,064 --> 00:20:29,948 I have here a rubber hose, which we are going to excite in 299 00:20:29,948 --> 00:20:34,490 the fifth harmonic, let's see how that works, 300 00:20:34,490 --> 00:20:39,910 whether we were close. If anyone touches the table, 301 00:20:39,910 --> 00:20:42,298 they are, then, the tangent changes. 302 00:20:42,298 --> 00:20:45,028 And immediately, of course, the resonance 303 00:20:45,028 --> 00:20:48,371 frequency changes because the resonance frequency, 304 00:20:48,371 --> 00:20:51,988 notice, has this V in it. So, the moment you change T, 305 00:20:51,988 --> 00:20:54,171 it's different. But it's not bad. 306 00:20:54,171 --> 00:20:58,334 And so, now I will strobe this one for you because you have no 307 00:20:58,334 --> 00:21:01,950 idea that what is happening that it's going like this. 308 00:21:01,950 --> 00:21:07,000 It's going too fast for you. And so, I make it easy on you. 309 00:21:07,000 --> 00:21:10,435 Could you turn the lights off? Thank you. 310 00:21:10,435 --> 00:21:13,355 So, I'm going to strobe it for you. 311 00:21:13,355 --> 00:21:17,907 And then I can more or less make the rope stand still. 312 00:21:17,907 --> 00:21:22,631 But I can also offset the frequency of my strobe light a 313 00:21:22,631 --> 00:21:27,871 little bit so that you actually see the rope move very slowly. 314 00:21:27,871 --> 00:21:32,251 So, I can purposely give the strobe light a slightly 315 00:21:32,251 --> 00:21:37,735 different frequency. So, that's what I'm doing now. 316 00:21:37,735 --> 00:21:42,750 You see, so you have no doubts anymore about the effect, 317 00:21:42,750 --> 00:21:46,670 no doubts that indeed if the middle goes up, 318 00:21:46,670 --> 00:21:50,773 the one on the side goes down, and vice versa. 319 00:21:50,773 --> 00:21:54,967 What I can also do, to turn this into a work of 320 00:21:54,967 --> 00:21:59,344 art, if I can turn this one off again, yes I can, 321 00:21:59,344 --> 00:22:04,450 I worked with an artist who actually liked these things a 322 00:22:04,450 --> 00:22:08,440 lot. And so, I can also strobe it 323 00:22:08,440 --> 00:22:11,578 twice as often. So, this is actually, 324 00:22:11,578 --> 00:22:14,978 the frequency of the rope is about 9 Hz. 325 00:22:14,978 --> 00:22:17,941 So, my red light was close to 9 Hz. 326 00:22:17,941 --> 00:22:21,951 The green light is 18 Hz, and a little bit off. 327 00:22:21,951 --> 00:22:25,438 So, now you see it twice per oscillation. 328 00:22:25,438 --> 00:22:30,668 Of course, to make it really wonderful, to make it very sexy, 329 00:22:30,668 --> 00:22:33,544 I can do them both simultaneously. 330 00:22:33,544 --> 00:22:38,251 So, now you see the red one doing his own thing and the 331 00:22:38,251 --> 00:22:44,153 green one doing its thing. And what I find interesting, 332 00:22:44,153 --> 00:22:47,615 it may not work for you, but there are moments 333 00:22:47,615 --> 00:22:52,000 occasionally that the red exactly overlaps with the green. 334 00:22:52,000 --> 00:22:54,769 When that happens, I see white light. 335 00:22:54,769 --> 00:22:58,307 Let's see whether we can have a case like that. 336 00:22:58,307 --> 00:23:01,692 Yeah, that was at the bottom. Did you see it? 337 00:23:01,692 --> 00:23:05,000 It's white. It's really amazing. 338 00:23:05,000 --> 00:23:09,216 There it is again at the top. All right. 339 00:23:09,216 --> 00:23:13,000 So, we can have some light again. 340 00:23:13,000 --> 00:23:20,000 341 00:23:20,000 --> 00:23:21,000 Thank you, Marcos. 342 00:23:21,000 --> 00:23:37,000 343 00:23:37,000 --> 00:23:40,575 I can now do the same thing for sound. 344 00:23:40,575 --> 00:23:45,890 These were transverse waves. But there was nothing wrong 345 00:23:45,890 --> 00:23:51,592 with doing the same thing for sound, in which case you would 346 00:23:51,592 --> 00:23:55,554 have a tube which has a certain length, L. 347 00:23:55,554 --> 00:24:00,000 Say it's closed on both sides for now. 348 00:24:00,000 --> 00:24:03,680 And, I can now generate in there pressure waves, 349 00:24:03,680 --> 00:24:08,221 make the air column oscillate, and the air particles in the 350 00:24:08,221 --> 00:24:13,154 lowest mode, all I have to do is what is here in the Y direction 351 00:24:13,154 --> 00:24:17,382 that is a transverse position. I now have to offset the 352 00:24:17,382 --> 00:24:20,436 molecules of the air in the X direction. 353 00:24:20,436 --> 00:24:24,742 So, this is the X direction. And, the direction in which 354 00:24:24,742 --> 00:24:29,440 they move is in the X direction. But, the displacement I will 355 00:24:29,440 --> 00:24:32,807 call psi [SP?], how much they displaced from 356 00:24:32,807 --> 00:24:36,529 equilibrium. So, in the lowest mode, 357 00:24:36,529 --> 00:24:39,921 that means in the fundamental in the first harmonic, 358 00:24:39,921 --> 00:24:44,045 in the middle the motion would be very large like the Y is very 359 00:24:44,045 --> 00:24:46,505 large. And then, it's a little smaller 360 00:24:46,505 --> 00:24:49,897 and a little smaller. And here it would stand still. 361 00:24:49,897 --> 00:24:53,156 And here it's a little smaller. Here it's smaller. 362 00:24:53,156 --> 00:24:56,016 Here it's smaller, and here it stands still. 363 00:24:56,016 --> 00:24:57,945 So therefore, in terms of psi, 364 00:24:57,945 --> 00:25:03,000 which is the position of the molecules, this would be a node. 365 00:25:03,000 --> 00:25:07,805 And this would be a node. And this would be an anti-node 366 00:25:07,805 --> 00:25:10,689 in the case of the first harmonic. 367 00:25:10,689 --> 00:25:15,669 And so, I could write down that psi in its end's mode as a 368 00:25:15,669 --> 00:25:20,213 function of X and T is, then, effectively the same as 369 00:25:20,213 --> 00:25:24,582 my Y that I had before, except I have to replace Y, 370 00:25:24,582 --> 00:25:28,339 now, by [xi?]. So, I give it some amplitude, 371 00:25:28,339 --> 00:25:33,048 A of N. Then I get the sine of N pi X 372 00:25:33,048 --> 00:25:39,250 over L, and I get the cosine omega NT through an equation is 373 00:25:39,250 --> 00:25:44,190 completely identical. However, keep in mind that 374 00:25:44,190 --> 00:25:49,656 omega N, which is this value, which is N pi V over L, 375 00:25:49,656 --> 00:25:53,440 that is no different. That is, again, 376 00:25:53,440 --> 00:25:57,644 N pi V over L. But, V is now the speed of 377 00:25:57,644 --> 00:26:00,272 sound. It's nonnegotiable. 378 00:26:00,272 --> 00:26:07,000 V is 300 m per second, and you are stuck with that. 379 00:26:07,000 --> 00:26:09,141 And that has major consequences, 380 00:26:09,141 --> 00:26:13,423 as I will show you Thursday for the design of wind instruments, 381 00:26:13,423 --> 00:26:16,876 whereas with string instruments, you can manipulate 382 00:26:16,876 --> 00:26:20,951 V, and you can change T and mu. You can make the tension and 383 00:26:20,951 --> 00:26:25,096 the strings larger or smaller. You do not have such an option 384 00:26:25,096 --> 00:26:28,480 with wind instruments. The only thing you can play 385 00:26:28,480 --> 00:26:32,278 with is L. So, this is the approximate 386 00:26:32,278 --> 00:26:35,654 speed of sound. More often than not do we 387 00:26:35,654 --> 00:26:40,465 express the standing wave in the case of sound in terms of 388 00:26:40,465 --> 00:26:43,588 overpressure. So, we don't look at the 389 00:26:43,588 --> 00:26:47,892 displacement of the air molecules, which I did here, 390 00:26:47,892 --> 00:26:51,437 which is psi. But we want to know where the 391 00:26:51,437 --> 00:26:55,826 pressure is larger than 1 atmosphere, and where it is 392 00:26:55,826 --> 00:26:59,539 lower than 1 atmosphere. It is the same idea, 393 00:26:59,539 --> 00:27:05,270 but it is very often done. Now, keep in mind that if these 394 00:27:05,270 --> 00:27:08,858 particles flow in this direction and pile up here, 395 00:27:08,858 --> 00:27:13,106 that the pressure here becomes higher and the pressure here 396 00:27:13,106 --> 00:27:16,987 becomes lower because the particles move away from it. 397 00:27:16,987 --> 00:27:21,454 So that means where you have a node in psi, you always have an 398 00:27:21,454 --> 00:27:25,335 anti-node in pressure. So these other locations by the 399 00:27:25,335 --> 00:27:29,070 pressure becomes high. And the overpressure over and 400 00:27:29,070 --> 00:27:34,191 above ambient is zero here. There was nothing that prevents 401 00:27:34,191 --> 00:27:38,222 the pressure from building up. These particles are free to 402 00:27:38,222 --> 00:27:40,767 move. And so, if you write it down in 403 00:27:40,767 --> 00:27:43,383 terms of P, which is now overpressure, 404 00:27:43,383 --> 00:27:47,272 it is not the total pressure. But it is what is over and 405 00:27:47,272 --> 00:27:50,878 above one atmosphere. Then you get some amplitude in 406 00:27:50,878 --> 00:27:53,282 pressure. I give you just a P of N. 407 00:27:53,282 --> 00:27:57,595 And now, I get the cosine of N pi X over L, and then I get the 408 00:27:57,595 --> 00:28:02,370 cosine of omega NT. And you understand now why you 409 00:28:02,370 --> 00:28:06,794 get the cosine because the anti-nodes are now here at the 410 00:28:06,794 --> 00:28:11,218 ends, which is the consequence of the boundary condition. 411 00:28:11,218 --> 00:28:15,168 If I drew a curve as I made a plot of the pressure, 412 00:28:15,168 --> 00:28:19,276 I will do it here if it doesn't become too cluttered. 413 00:28:19,276 --> 00:28:22,041 So, here is now zero, and here is L. 414 00:28:22,041 --> 00:28:26,307 If I put here the pressure, then in the first harmonic, 415 00:28:26,307 --> 00:28:29,941 the fundamental, here is a pressure node in the 416 00:28:29,941 --> 00:28:34,444 middle where there was an anti-node from psi that is now a 417 00:28:34,444 --> 00:28:39,994 pressure node in the middle. And then the pressure will 418 00:28:39,994 --> 00:28:43,677 change in this way. So, this is now N equals one. 419 00:28:43,677 --> 00:28:47,589 So, the pressure here at one moment in time is high. 420 00:28:47,589 --> 00:28:50,965 Then it's zero here, and then it is low here. 421 00:28:50,965 --> 00:28:55,492 It is below ambient pressure, and that of course is changing 422 00:28:55,492 --> 00:28:58,484 with time, with the cosine omega T term. 423 00:28:58,484 --> 00:29:02,243 And, again, you see that lambda one is, of course, 424 00:29:02,243 --> 00:29:04,621 2L. There is no difference there 425 00:29:04,621 --> 00:29:09,700 with the string. I can demonstrate this to you 426 00:29:09,700 --> 00:29:14,449 in a very, very nice way. I have here a two which is 427 00:29:14,449 --> 00:29:17,615 closed at both ends. So here it is. 428 00:29:17,615 --> 00:29:23,388 But at one end I have a piston. And so I can change the length, 429 00:29:23,388 --> 00:29:26,368 L. You will see in a minute why I 430 00:29:26,368 --> 00:29:30,000 like to change that length, L. 431 00:29:30,000 --> 00:29:34,509 We are going to drive the inside with a microphone. 432 00:29:34,509 --> 00:29:39,289 Here's the microphone. And we're going to do that with 433 00:29:39,289 --> 00:29:42,896 a frequency. We really wanted to do a 403 434 00:29:42,896 --> 00:29:46,684 Hz, at 8.03 Hz has a rather low wavelength. 435 00:29:46,684 --> 00:29:51,734 I wanted to have it shorter. So I do it three times 8.03. 436 00:29:51,734 --> 00:29:55,071 So, the frequency in [UNINTELLIGIBLE], 437 00:29:55,071 --> 00:29:58,769 which is 24.09 Hz. And so, the wavelength, 438 00:29:58,769 --> 00:30:04,000 lambda, which is V divided by the frequency. 439 00:30:04,000 --> 00:30:07,011 That's correct. The wavelength, 440 00:30:07,011 --> 00:30:13,135 lambda, equals V divided by the frequency is then very roughly 441 00:30:13,135 --> 00:30:16,347 14 cm. So, that gives you an idea 442 00:30:16,347 --> 00:30:20,864 about this much. In here, we have a microphone 443 00:30:20,864 --> 00:30:26,185 that we can move in and out, and a microphone measures 444 00:30:26,185 --> 00:30:30,000 pressure. It has a membrane. 445 00:30:30,000 --> 00:30:32,960 And so, it's sensitive for pressure. 446 00:30:32,960 --> 00:30:37,274 And this microphone is connected to a loudspeaker so 447 00:30:37,274 --> 00:30:40,574 you can hear it. It'll also show to you, 448 00:30:40,574 --> 00:30:44,719 the signal the microphone. It'll show to you as we 449 00:30:44,719 --> 00:30:49,709 recorded it on an oscilloscope. If I put the microphone at a 450 00:30:49,709 --> 00:30:52,416 pressure node, you hear no sound. 451 00:30:52,416 --> 00:30:58,000 And if I move it to an anti-node, you will hear sound. 452 00:30:58,000 --> 00:31:02,000 453 00:31:02,000 --> 00:31:05,602 I will create in here pressure nodes, nodes, 454 00:31:05,602 --> 00:31:07,781 nodes, nodes, nodes, nodes, 455 00:31:07,781 --> 00:31:11,132 which are always half a wavelength apart. 456 00:31:11,132 --> 00:31:15,656 Think about why that is, if you hear the nodes are half 457 00:31:15,656 --> 00:31:19,343 a wavelength apart. The same will be true for 458 00:31:19,343 --> 00:31:22,108 sound. I am then going to put this 459 00:31:22,108 --> 00:31:26,799 microphone somewhere at a node that we all agree you hear 460 00:31:26,799 --> 00:31:30,570 almost no sound. And we see no signal from the 461 00:31:30,570 --> 00:31:35,361 oscilloscope. And then, I'm going to move it 462 00:31:35,361 --> 00:31:39,060 back and search for one, two, three, four nodes. 463 00:31:39,060 --> 00:31:43,231 And then, I know that this distance is going to be two 464 00:31:43,231 --> 00:31:45,513 lambda. And I have, therefore, 465 00:31:45,513 --> 00:31:48,819 measured lambda. And, the amazing thing is, 466 00:31:48,819 --> 00:31:53,069 which actually surprised me, that you can do that to an 467 00:31:53,069 --> 00:31:57,555 accuracy of about one million. That is so enormously clear 468 00:31:57,555 --> 00:32:01,648 when you are at a node, if you move your mike by 1 mm 469 00:32:01,648 --> 00:32:07,000 you can really see that you're no longer at the node. 470 00:32:07,000 --> 00:32:10,570 So we'll know lambda probably to 1 mm accuracy. 471 00:32:10,570 --> 00:32:14,140 Once we know that, the speed of sound is lambda 472 00:32:14,140 --> 00:32:16,158 times F. We know what F is. 473 00:32:16,158 --> 00:32:20,737 That's known to one part in, we couldn't be off by more than 474 00:32:20,737 --> 00:32:23,376 1 Hz. So, that's a very well known. 475 00:32:23,376 --> 00:32:27,101 And so, we now have a measurement of the speed of 476 00:32:27,101 --> 00:32:29,585 sound, a high degree of accuracy. 477 00:32:29,585 --> 00:32:35,147 We measure the speed of sound. And so, we catch three birds 478 00:32:35,147 --> 00:32:37,295 with one stone. First of all, 479 00:32:37,295 --> 00:32:40,210 I can show you the nodes in anti-nodes. 480 00:32:40,210 --> 00:32:42,818 So you see how they build up there. 481 00:32:42,818 --> 00:32:47,420 And then, at the same time we can measure the speed of sound. 482 00:32:47,420 --> 00:32:51,715 The only reason why we make this end movable is that as I 483 00:32:51,715 --> 00:32:56,548 move it and bring the mike first at a node, I can make sure that 484 00:32:56,548 --> 00:33:03,011 the length is just at resonance. And so, that's just my 485 00:33:03,011 --> 00:33:11,361 beginning to make sure that the node is as close to zero as it 486 00:33:11,361 --> 00:33:17,794 possibly can be. And so, if we now get the image 487 00:33:17,794 --> 00:33:25,323 up there, and then I think we are going to get the light 488 00:33:25,323 --> 00:33:32,304 situation like this, and going to turn on the sound, 489 00:33:32,304 --> 00:33:38,094 here is the sound. I can turn up the volume 490 00:33:38,094 --> 00:33:41,672 seeking hear it better. Is this connected, 491 00:33:41,672 --> 00:33:43,941 Marcus? Oh yeah, of course, 492 00:33:43,941 --> 00:33:47,519 thank you. I didn't connect the microphone 493 00:33:47,519 --> 00:33:49,265 yet. It has a switch. 494 00:33:49,265 --> 00:33:53,018 OK, let me first use the, oh, this is awful. 495 00:33:53,018 --> 00:33:55,112 This 2409 Hz, by the way. 496 00:33:55,112 --> 00:34:00,000 I'll turn it down a little. Is that better? 497 00:34:00,000 --> 00:34:06,210 OK, let's first move it around. Let's first bring to a node. 498 00:34:06,210 --> 00:34:10,421 Boy, there's nothing left anymore, right? 499 00:34:10,421 --> 00:34:16,631 And now, I'm going to change L. OK, I cannot do much better. 500 00:34:16,631 --> 00:34:20,421 So, what L is, is not very important. 501 00:34:20,421 --> 00:34:26,631 I set L so that I think the system is at resonance in a node 502 00:34:26,631 --> 00:34:34,000 that you can calculate if you know how many nodes there are. 503 00:34:34,000 --> 00:34:37,500 So, I'm going to search for, well here, you see the 504 00:34:37,500 --> 00:34:40,019 anti-nodes, by the way. You see that? 505 00:34:40,019 --> 00:34:43,659 I'm moving it in now. I'm moving it further that way. 506 00:34:43,659 --> 00:34:45,969 And here, you get to another node. 507 00:34:45,969 --> 00:34:48,489 You see that? You see that wonderful, 508 00:34:48,489 --> 00:34:52,619 and I go again to an anti-node. And here comes another node. 509 00:34:52,619 --> 00:34:55,909 And this note I'm going to measure the position. 510 00:34:55,909 --> 00:34:59,409 I have a ruler in there, and the ruler gives me the 511 00:34:59,409 --> 00:35:04,038 position of the mike. And this one is 52.8. 512 00:35:04,038 --> 00:35:08,213 So, 52.8 cm is this position here somewhere. 513 00:35:08,213 --> 00:35:14,038 And, I can put a mark here for you, not that it will help you 514 00:35:14,038 --> 00:35:17,631 very much because my accuracy is 1 mm. 515 00:35:17,631 --> 00:35:23,262 And, this is a very crude way. So, now I'm going to pull it 516 00:35:23,262 --> 00:35:25,883 back. You're going to see an 517 00:35:25,883 --> 00:35:28,407 anti-node. You can hear it, 518 00:35:28,407 --> 00:35:36,979 a little bit more sound. And you're going to see a node. 519 00:35:36,979 --> 00:35:43,564 That's number one. Here's node number two, 520 00:35:43,564 --> 00:35:49,186 node number three, node number four. 521 00:35:49,186 --> 00:35:57,217 And I'm going to look; and I'm going to look now at 522 00:35:57,217 --> 00:36:03,000 the reading. And I read 24.3. 523 00:36:03,000 --> 00:36:06,900 So, I read 24.3 cm. So, I subtract them. 524 00:36:06,900 --> 00:36:09,000 That is five, 28.5 cm. 525 00:36:09,000 --> 00:36:13,900 And this is two lambda. And I know this really to, 526 00:36:13,900 --> 00:36:16,400 I would say, a millimeter. 527 00:36:16,400 --> 00:36:20,800 It's certainly no worse than two millimeters. 528 00:36:20,800 --> 00:36:25,400 And, there is no uncertainty for sure in the F, 529 00:36:25,400 --> 00:36:30,000 if you can give me the light back. 530 00:36:30,000 --> 00:36:35,426 So, I can now calculate what the velocity of sound is. 531 00:36:35,426 --> 00:36:39,317 So, that is 28.5. I divide that by two. 532 00:36:39,317 --> 00:36:44,334 It gives me 14.25 cm. This was just a rough number 533 00:36:44,334 --> 00:36:48,839 that I gave you. And now I multiplied that by 534 00:36:48,839 --> 00:36:52,832 the frequency 2409. And then I find 343. 535 00:36:52,832 --> 00:37:00,000 So, V is 343 I would say plus or minus maybe 2 m per second. 536 00:37:00,000 --> 00:37:05,561 So, we have measured the speed of sound to a high degree of 537 00:37:05,561 --> 00:37:07,863 accuracy. And, of course, 538 00:37:07,863 --> 00:37:13,712 at the same time you have seen this wonderful resonance normal 539 00:37:13,712 --> 00:37:19,082 mode behavior where you see the nodes and the anti-nodes, 540 00:37:19,082 --> 00:37:21,479 in this case, of pressure. 541 00:37:21,479 --> 00:37:25,027 Earlier you saw them [NOISE OBSCURES]. 542 00:37:25,027 --> 00:37:30,205 Now you have seen them in terms of longitudinal motion. 543 00:37:30,205 --> 00:37:35,000 There is energy in a traveling wave. 544 00:37:35,000 --> 00:37:42,316 If I have here a traveling wave, which is moving in this 545 00:37:42,316 --> 00:37:49,766 direction, has tension T, mu is the mass per unit length, 546 00:37:49,766 --> 00:37:56,151 then I can write down Y as a function of X and T. 547 00:37:56,151 --> 00:38:00,940 I can give it a certain amplitude, A, 548 00:38:00,940 --> 00:38:06,690 times the sine. And I can write this now in 549 00:38:06,690 --> 00:38:11,773 many different ways. But let's write it down in this 550 00:38:11,773 --> 00:38:16,257 form: P times X minus VT. And, we know this V, 551 00:38:16,257 --> 00:38:21,240 that V, let's call it V squared is T divided by mu. 552 00:38:21,240 --> 00:38:26,522 So, this is a traveling wave that goes into the plus X 553 00:38:26,522 --> 00:38:30,209 direction. Is any matter moving in the 554 00:38:30,209 --> 00:38:33,251 plus X direction? No. 555 00:38:33,251 --> 00:38:35,515 But is anything moving? Yes. 556 00:38:35,515 --> 00:38:38,114 It is moving in the Y direction. 557 00:38:38,114 --> 00:38:42,978 So, since these particles have mass, and they are moving in 558 00:38:42,978 --> 00:38:46,751 this direction, there is kinetic energy due to 559 00:38:46,751 --> 00:38:51,698 the motion in this direction, not due to any mass that moves 560 00:38:51,698 --> 00:38:54,717 along. No mass moves in the direction 561 00:38:54,717 --> 00:38:57,736 of here. Suppose I carve out here the 562 00:38:57,736 --> 00:39:03,322 section, DX. So this is the direction of X. 563 00:39:03,322 --> 00:39:06,755 And, I take a small section, DX. 564 00:39:06,755 --> 00:39:12,734 Then the kinetic energy, DE kinetic, tiny little bit of 565 00:39:12,734 --> 00:39:19,157 energy is one half times the mass, DM times the velocity in 566 00:39:19,157 --> 00:39:23,255 the Y direction squared. That is 8.01, 567 00:39:23,255 --> 00:39:27,130 right? Kinetic energy is one half MV 568 00:39:27,130 --> 00:39:31,338 squared. Never confuse this V with that 569 00:39:31,338 --> 00:39:35,343 V. That is the velocity of 570 00:39:35,343 --> 00:39:39,318 propagation. This is the velocity of the 571 00:39:39,318 --> 00:39:44,515 string in this direction. So, I can write this down. 572 00:39:44,515 --> 00:39:50,121 DM is obviously mu times DX. If I have a length DX here, 573 00:39:50,121 --> 00:39:56,031 and I have mu kg per meter, so I get here one half mu times 574 00:39:56,031 --> 00:40:02,248 DX, and for this I write down DY DT squared is the velocity in 575 00:40:02,248 --> 00:40:07,971 the Y direction. And I use partial derivatives 576 00:40:07,971 --> 00:40:12,172 because I do it at a given location for X. 577 00:40:12,172 --> 00:40:15,860 So, what is DY DT? Well, that's easy. 578 00:40:15,860 --> 00:40:20,368 That's a piece of cake. There is my function. 579 00:40:20,368 --> 00:40:24,672 So, I get an A. Then I get [NOISE OBSCURES] 580 00:40:24,672 --> 00:40:28,668 minus V. So, I get a minus and a K and a 581 00:40:28,668 --> 00:40:32,482 V. And then, the sine becomes a 582 00:40:32,482 --> 00:40:35,674 cosine. And so, I get K times X minus 583 00:40:35,674 --> 00:40:38,246 VT. I should have put brackets 584 00:40:38,246 --> 00:40:40,729 around there, too, but that's 585 00:40:40,729 --> 00:40:44,985 self-explanatory. And now I want to know what the 586 00:40:44,985 --> 00:40:49,152 square of this is. So, I'm going to square this. 587 00:40:49,152 --> 00:40:53,054 Now I can calculate what the total energy is, 588 00:40:53,054 --> 00:40:55,980 kinetic energy, in one wavelength. 589 00:40:55,980 --> 00:41:01,389 All I have to do is integrate, now, from zero to lambda to get 590 00:41:01,389 --> 00:41:06,000 the total energy in one wavelength. 591 00:41:06,000 --> 00:41:11,849 So, I'm going to do that. I'm going to write down here 592 00:41:11,849 --> 00:41:15,381 now E kinetic. Follow me closely. 593 00:41:15,381 --> 00:41:18,250 I have a half. I have a mu. 594 00:41:18,250 --> 00:41:21,892 And then, I get the DY DT squared. 595 00:41:21,892 --> 00:41:26,418 So, I get an A squared. I get a K squared. 596 00:41:26,418 --> 00:41:31,384 I get a V squared, and then I get the integral 597 00:41:31,384 --> 00:41:39,000 from zero to lambda of this function: cosine KX minus VT. 598 00:41:39,000 --> 00:41:42,208 And then I have my DX, which is this DX. 599 00:41:42,208 --> 00:41:46,240 That's what the integral is in the direction of X. 600 00:41:46,240 --> 00:41:48,791 Yeah? Yeah, thank you very much. 601 00:41:48,791 --> 00:41:52,246 Extra course credit. What is this integral? 602 00:41:52,246 --> 00:41:56,854 Maybe you don't remember, but I've done this often enough 603 00:41:56,854 --> 00:42:00,556 that I do know, the integral of cosine squared 604 00:42:00,556 --> 00:42:05,000 of that function is lambda divided by two. 605 00:42:05,000 --> 00:42:09,792 And I leave you with that. That's a very easy exercise. 606 00:42:09,792 --> 00:42:14,852 And I can also take the V squared out and write for that T 607 00:42:14,852 --> 00:42:17,514 over mu. So, [in kinetic?] now, 608 00:42:17,514 --> 00:42:22,840 and this is in one wavelength. This is my shorthand notation. 609 00:42:22,840 --> 00:42:27,633 So all in calculating for you now is how much is in one 610 00:42:27,633 --> 00:42:30,650 wavelength. So you're going to get, 611 00:42:30,650 --> 00:42:35,000 now, one half mu to get an A squared. 612 00:42:35,000 --> 00:42:40,185 The K squared I can write down as 4 pi squared divided by 613 00:42:40,185 --> 00:42:43,240 lambda squared. For the V squared, 614 00:42:43,240 --> 00:42:46,203 I can write down T divided by mu. 615 00:42:46,203 --> 00:42:49,444 And then, I have my lambda over two. 616 00:42:49,444 --> 00:42:54,629 Well, this mu kills this mu. This two kills this four and 617 00:42:54,629 --> 00:42:57,777 this two. And, one lambda kills one 618 00:42:57,777 --> 00:43:03,479 lambda here. So, now I have the final result 619 00:43:03,479 --> 00:43:09,877 that E kinetic in one wavelength equals A squared times pi 620 00:43:09,877 --> 00:43:13,581 squared times T divided by lambda. 621 00:43:13,581 --> 00:43:19,867 Now, if you look at this and you ask me, is that obvious? 622 00:43:19,867 --> 00:43:22,336 I would say, not to me. 623 00:43:22,336 --> 00:43:25,816 There is a T. There is a lambda. 624 00:43:25,816 --> 00:43:30,193 My goodness, and I can also cocktail the 625 00:43:30,193 --> 00:43:35,266 whole thing. I can get the T out and get a V 626 00:43:35,266 --> 00:43:38,483 back in again. I will admit that I don't have 627 00:43:38,483 --> 00:43:42,285 a very good feeling for this function except for one. 628 00:43:42,285 --> 00:43:46,160 I do know that always the energy in the wave is always 629 00:43:46,160 --> 00:43:48,865 proportional to the amplitude squared. 630 00:43:48,865 --> 00:43:53,179 You're going to see that when we do electromagnetic waves in 631 00:43:53,179 --> 00:43:55,665 8.03. It's always proportional to A 632 00:43:55,665 --> 00:43:58,224 squared. So, that's the only one for 633 00:43:58,224 --> 00:44:02,318 which I may not have a feeling. But I know it's got to be 634 00:44:02,318 --> 00:44:06,460 there. And the rest I will leave you 635 00:44:06,460 --> 00:44:11,381 with that to see whether perhaps you can talk yourself into 636 00:44:11,381 --> 00:44:15,793 understanding why you see the symbols where they are. 637 00:44:15,793 --> 00:44:19,442 Now, clearly, there is also potential energy 638 00:44:19,442 --> 00:44:24,787 because it takes energy to make that straight line into a curve. 639 00:44:24,787 --> 00:44:28,181 And, that means work that you have to do. 640 00:44:28,181 --> 00:44:32,000 You have to squeeze to stretch it. 641 00:44:32,000 --> 00:44:35,412 There is a tension, and you have to stretch that. 642 00:44:35,412 --> 00:44:38,683 And so, you have to work to just get the shape. 643 00:44:38,683 --> 00:44:42,736 And that is potential energy. And the potential energy per 644 00:44:42,736 --> 00:44:46,006 wavelength, which I want you to do on your own. 645 00:44:46,006 --> 00:44:48,282 It's worked out nicely in French. 646 00:44:48,282 --> 00:44:52,050 I will not do it today, happens to be exactly the same 647 00:44:52,050 --> 00:44:55,676 as the kinetic energy, which is by no means obvious. 648 00:44:55,676 --> 00:44:59,729 So, the potential energy per wavelength is the same as the 649 00:44:59,729 --> 00:45:06,172 kinetic energy per wavelength. And so, what that means, 650 00:45:06,172 --> 00:45:13,839 then, is that the total energy in a traveling wave is twice 651 00:45:13,839 --> 00:45:19,390 this: kinetic energy plus potential energy. 652 00:45:19,390 --> 00:45:26,000 That is the total energy in a traveling wave. 653 00:45:26,000 --> 00:45:35,000 654 00:45:35,000 --> 00:45:41,540 I can now make you see in a nice way an energy balance to 655 00:45:41,540 --> 00:45:46,678 compare traveling waves with a standing wave. 656 00:45:46,678 --> 00:45:53,102 If I have a standing wave, and the standing wave is like 657 00:45:53,102 --> 00:45:59,992 this, and I look at this picture at the moment that the wave 658 00:45:59,992 --> 00:46:03,379 stands still, that VY is zero, 659 00:46:03,379 --> 00:46:08,492 so it goes like this. That's a standing wave. 660 00:46:08,492 --> 00:46:11,088 You've seen that. I do it at this moment, 661 00:46:11,088 --> 00:46:15,048 and there's no kinetic energy. There is only potential energy. 662 00:46:15,048 --> 00:46:17,903 A little later, there is only kinetic energy, 663 00:46:17,903 --> 00:46:19,980 and there is no potential energy. 664 00:46:19,980 --> 00:46:23,161 And although later, there is only potential energy 665 00:46:23,161 --> 00:46:26,471 and no kinetic energy. If I pick that moment that it 666 00:46:26,471 --> 00:46:29,067 stands still, then I know that that's the 667 00:46:29,067 --> 00:46:34,000 total energy because it's only potential but it is the total. 668 00:46:34,000 --> 00:46:37,587 In other words, for a standing wave, 669 00:46:37,587 --> 00:46:42,404 if this has an amplitude, A, it must be the same 670 00:46:42,404 --> 00:46:48,759 potential energy as you have in the traveling wave because it's 671 00:46:48,759 --> 00:46:52,346 simply due to the fact of the shape. 672 00:46:52,346 --> 00:46:56,445 It has nothing to do anymore with motion. 673 00:46:56,445 --> 00:47:01,980 So, for a standing wave, the total must be that number, 674 00:47:01,980 --> 00:47:07,822 whereas for a traveling wave, you have kinetic energy plus 675 00:47:07,822 --> 00:47:13,299 potential energy. So, you have twice as much. 676 00:47:13,299 --> 00:47:17,728 Now, I can convince you. And that was my plan earlier 677 00:47:17,728 --> 00:47:22,241 today that if I have one traveling wave with amplitude 678 00:47:22,241 --> 00:47:27,096 one half, A, and we have another traveling wave again with 679 00:47:27,096 --> 00:47:31,013 amplitude one half, A, I know that I'm going to 680 00:47:31,013 --> 00:47:34,760 make a standing wave with amplitude A, right, 681 00:47:34,760 --> 00:47:39,274 because one half A and one half A, and we add them up, 682 00:47:39,274 --> 00:47:43,623 you got twice this. You've got these [two A's?]. 683 00:47:43,623 --> 00:47:46,927 But, there must be a certain amount of energy in that wave 684 00:47:46,927 --> 00:47:50,347 that comes in with one half A and a certain amount of energy 685 00:47:50,347 --> 00:47:52,840 in this one. And when you add those energies 686 00:47:52,840 --> 00:47:55,913 up, you must exactly get the energy in a standing wave 687 00:47:55,913 --> 00:47:59,628 because no energy was lost. So, therefore, 688 00:47:59,628 --> 00:48:05,795 I make the following statement now, which is testable, 689 00:48:05,795 --> 00:48:12,078 that to traveling waves each with amplitude one half A, 690 00:48:12,078 --> 00:48:19,059 I'm going to compare them with a standing wave with amplitude 691 00:48:19,059 --> 00:48:22,666 A. There must be the same energy 692 00:48:22,666 --> 00:48:28,833 in a standing wave with amplitude A as there is in two 693 00:48:28,833 --> 00:48:35,000 traveling waves with amplitude one half A. 694 00:48:35,000 --> 00:48:39,054 Did we follow that? Was that too difficult? 695 00:48:39,054 --> 00:48:44,362 Because, we know that we can make that wave the standing 696 00:48:44,362 --> 00:48:47,355 wave. OK, we know that the total 697 00:48:47,355 --> 00:48:50,637 energy in a traveling wave is this. 698 00:48:50,637 --> 00:48:55,559 But, we have two of them. So, in the traveling wave, 699 00:48:55,559 --> 00:49:00,000 in two traveling waves, we have two. 700 00:49:00,000 --> 00:49:03,665 Then we get that two there. That's nonnegotiable. 701 00:49:03,665 --> 00:49:06,491 Then we get pi squared. Then we get T, 702 00:49:06,491 --> 00:49:10,463 and we divided by lambda. But the amplitude A that we 703 00:49:10,463 --> 00:49:12,754 have there is now half A, yeah? 704 00:49:12,754 --> 00:49:16,572 Is that too difficult? Give it an amplitude half A. 705 00:49:16,572 --> 00:49:19,704 So now, I get one quarter times A squared. 706 00:49:19,704 --> 00:49:22,988 So, this is the energy into traveling waves. 707 00:49:22,988 --> 00:49:25,813 Do we agree? What is the energy in one 708 00:49:25,813 --> 00:49:28,334 standing wave with an amplitude A? 709 00:49:28,334 --> 00:49:33,498 Well, we have the answer here. In a standing wave, 710 00:49:33,498 --> 00:49:37,363 all the energy must be A, pi squared, T divided by 711 00:49:37,363 --> 00:49:38,783 lambda. Look at it. 712 00:49:38,783 --> 00:49:42,174 They are the same. These two, and these two, 713 00:49:42,174 --> 00:49:44,698 and these four eat each other up. 714 00:49:44,698 --> 00:49:48,011 Just a second. So, you see the consistency, 715 00:49:48,011 --> 00:49:52,270 I'll give you a chance, that two traveling waves making 716 00:49:52,270 --> 00:49:56,608 up one standing wave in the exercise I did earlier that, 717 00:49:56,608 --> 00:50:01,995 indeed, energy is conserved. The energy in the two traveling 718 00:50:01,995 --> 00:50:04,892 waves with half the amplitude gives you, then, 719 00:50:04,892 --> 00:50:06,630 a standing wave with A. Yes? 720 00:50:06,630 --> 00:50:09,721 A squared, thank you very much. Always A squared. 721 00:50:09,721 --> 00:50:13,133 Isn't that what I said? Amplitude is always A squared. 722 00:50:13,133 --> 00:50:15,000 Thank you very much. 723 00:50:15,000 --> 00:50:22,000 724 00:50:22,000 --> 00:50:27,033 If I generate traveling waves, I, Walter Lewin, 725 00:50:27,033 --> 00:50:33,052 start shaking strings or some other instrument is making 726 00:50:33,052 --> 00:50:38,196 traveling waves, then there is an energy flow in 727 00:50:38,196 --> 00:50:44,543 the direction of propagation because these wavelengths keep 728 00:50:44,543 --> 00:50:51,000 moving, these waves, and so there's an energy flow. 729 00:50:51,000 --> 00:50:55,744 There's no mass flowing. Oh, there's energy flowing. 730 00:50:55,744 --> 00:51:00,023 The mass is only doing this. And so, therefore, 731 00:51:00,023 --> 00:51:05,697 if I generate a certain amount of energy, then I need power to 732 00:51:05,697 --> 00:51:09,337 do that. Namely, power is energy per 733 00:51:09,337 --> 00:51:12,713 unit time. And so, I can now very easily 734 00:51:12,713 --> 00:51:17,474 calculate for you the power to generate a standing wave. 735 00:51:17,474 --> 00:51:22,841 That is utterly trivial because the energy in the standing wave 736 00:51:22,841 --> 00:51:26,737 is 2A squared, pi squared T divided by lambda. 737 00:51:26,737 --> 00:51:31,844 And all I have to do now is to divide it by the time that it 738 00:51:31,844 --> 00:51:36,000 takes me to generate one oscillation. 739 00:51:36,000 --> 00:51:38,891 So, that's the period of the oscillation. 740 00:51:38,891 --> 00:51:42,506 And the period of the oscillation, I try not to put 741 00:51:42,506 --> 00:51:45,831 too many P's in there. And I hate to call it T. 742 00:51:45,831 --> 00:51:49,518 So, the period of the oscillation is the same as one 743 00:51:49,518 --> 00:51:53,060 over the frequency. And, one over the frequency is 744 00:51:53,060 --> 00:51:55,445 V divided by lambda. So therefore, 745 00:51:55,445 --> 00:52:00,000 the power is simply the energy that I'm generating. 746 00:52:00,000 --> 00:52:05,450 And, V divided by lambda is the same, is one over the period of 747 00:52:05,450 --> 00:52:09,054 one oscillation, energy divided by period, 748 00:52:09,054 --> 00:52:13,890 the time that it takes to make one oscillation is [per?] 749 00:52:13,890 --> 00:52:18,109 definition power. And so, then you get the result 750 00:52:18,109 --> 00:52:21,186 here. You get a lambda squared here, 751 00:52:21,186 --> 00:52:24,000 and you get a V there. 752 00:52:24,000 --> 00:52:29,000 753 00:52:29,000 --> 00:52:34,262 This is a traveling wave. It's good that you asked that. 754 00:52:34,262 --> 00:52:37,228 I'm generating a traveling wave. 755 00:52:37,228 --> 00:52:42,012 I have to work for every wavelength that I produce. 756 00:52:42,012 --> 00:52:46,222 This is the amount of work that I have to do. 757 00:52:46,222 --> 00:52:49,475 Do we agree? There is energy in the 758 00:52:49,475 --> 00:52:54,546 potential energy because I have to give it that shape, 759 00:52:54,546 --> 00:53:00,000 and there is kinetic energy because it moves. 760 00:53:00,000 --> 00:53:03,751 If I generate this amount of energy, what is your name? 761 00:53:03,751 --> 00:53:07,503 James, if I generate this amount of energy and I divide 762 00:53:07,503 --> 00:53:11,741 that by the time that it takes for me to generate that energy, 763 00:53:11,741 --> 00:53:14,868 I get the power, the average power that I have 764 00:53:14,868 --> 00:53:17,369 to put in. It's really average power. 765 00:53:17,369 --> 00:53:21,607 And so, I have to divide this by the period of my oscillation, 766 00:53:21,607 --> 00:53:24,803 the time that it takes to make one oscillation. 767 00:53:24,803 --> 00:53:28,000 And that is lambda divided by V. 768 00:53:28,000 --> 00:53:34,146 That is the period and so I had to multiply this by V divided by 769 00:53:34,146 --> 00:53:37,756 lambda. So this is really what we call 770 00:53:37,756 --> 00:53:41,756 frequency. And so, that you get the power. 771 00:53:41,756 --> 00:53:47,902 And so you can move energy from one point to another without any 772 00:53:47,902 --> 00:53:51,707 math moving. All right, this is an ideal 773 00:53:51,707 --> 00:53:55,804 moment for a break and for a fatherly talk. 774 00:53:55,804 --> 00:54:01,560 I've had some very interesting e-mail exchanges with quite a 775 00:54:01,560 --> 00:54:05,789 few of you. And as a result of that, 776 00:54:05,789 --> 00:54:10,485 I have decided to lower the 10% of the mini-quizzes to 5% and to 777 00:54:10,485 --> 00:54:13,020 raise the homework from 10% to 15%. 778 00:54:13,020 --> 00:54:16,822 I think that will make probably most of you happier, 779 00:54:16,822 --> 00:54:19,953 though not all. Some of them thought it was 780 00:54:19,953 --> 00:54:23,308 fine, but I think most of you will be happier. 781 00:54:23,308 --> 00:54:27,557 And I will make that change this afternoon on the website. 782 00:54:27,557 --> 00:54:33,000 So you'll see that reflected when you look tomorrow morning. 783 00:54:33,000 --> 00:54:37,378 I will also be more careful that the people here in the 784 00:54:37,378 --> 00:54:41,837 front row don't have five minutes, and the people in the 785 00:54:41,837 --> 00:54:46,216 back maybe only one minute. And so, therefore I'll make 786 00:54:46,216 --> 00:54:51,081 sure each one of you from now on has the same amount of time. 787 00:54:51,081 --> 00:54:55,945 And so what I'm asking you now is not to start until everyone 788 00:54:55,945 --> 00:55:00,648 has, and then I will blow the whistle, and then I will give 789 00:55:00,648 --> 00:55:06,378 you five minutes. So, will you start handing this 790 00:55:06,378 --> 00:55:09,320 out? [SOUND OFF/THEN ON] OK. 791 00:55:09,320 --> 00:55:14,442 I'm now going to change the boundary conditions, 792 00:55:14,442 --> 00:55:20,435 which is something that you will need very much for your 793 00:55:20,435 --> 00:55:23,596 problem set. Here is a string, 794 00:55:23,596 --> 00:55:27,410 mu one, V one, and here is a string, 795 00:55:27,410 --> 00:55:32,205 mu two, V two. And obviously they're the same 796 00:55:32,205 --> 00:55:37,000 [tangent/tension?], one string. 797 00:55:37,000 --> 00:55:41,861 And so, V1 is the square root of T divided by mu one, 798 00:55:41,861 --> 00:55:46,630 and V two is the square root of T divided by mu two. 799 00:55:46,630 --> 00:55:50,650 I will call this point here, for simplicity, 800 00:55:50,650 --> 00:55:53,922 X equals zero. That is the junction. 801 00:55:53,922 --> 00:56:00,000 And this is the Y direction, and this is the X direction. 802 00:56:00,000 --> 00:56:04,612 And I'm going to have a wave coming in from the left, 803 00:56:04,612 --> 00:56:07,451 which I called the incident wave. 804 00:56:07,451 --> 00:56:11,532 I will give that an I moving in this direction. 805 00:56:11,532 --> 00:56:16,056 And then, there will be a reflection which I call R. 806 00:56:16,056 --> 00:56:21,645 Something may come back so that goes into this direction both in 807 00:56:21,645 --> 00:56:25,282 medium one. This is medium one and this is 808 00:56:25,282 --> 00:56:28,653 medium two. And then something is being 809 00:56:28,653 --> 00:56:33,000 transmitted into the second medium. 810 00:56:33,000 --> 00:56:38,515 I will give that subscript TR. And, that is going into medium 811 00:56:38,515 --> 00:56:41,181 two. And that's what I want to 812 00:56:41,181 --> 00:56:45,225 evaluate now. Now, I have boundary conditions 813 00:56:45,225 --> 00:56:48,534 at X equals zero. I told you earlier. 814 00:56:48,534 --> 00:56:53,774 All of 8.03 hangs together: the wave equation and boundary 815 00:56:53,774 --> 00:56:56,256 conditions. And that's true. 816 00:56:56,256 --> 00:57:00,576 The boundary condition is that at X equals zero, 817 00:57:00,576 --> 00:57:06,000 Y one must be Y two unless the string breaks. 818 00:57:06,000 --> 00:57:11,064 So, the Y right here must be the same as the Y right there. 819 00:57:11,064 --> 00:57:15,081 Otherwise there would be a break of the string. 820 00:57:15,081 --> 00:57:19,185 But not only that, DY1, DX, must also be DY2 DX. 821 00:57:19,185 --> 00:57:23,901 If that were not the case, there would be a kink in the 822 00:57:23,901 --> 00:57:28,180 rope at the junction, and the kink would mean that 823 00:57:28,180 --> 00:57:33,507 there is a tension here and that there is a tension here which 824 00:57:33,507 --> 00:57:38,788 would give a net force down. But the junction itself has 825 00:57:38,788 --> 00:57:41,172 zero mass. And so, that would give an 826 00:57:41,172 --> 00:57:44,682 infinite acceleration. So you can never have a kink in 827 00:57:44,682 --> 00:57:48,854 the string, not even when it is connected like this with another 828 00:57:48,854 --> 00:57:51,238 string. So, it must be something that 829 00:57:51,238 --> 00:57:54,284 is always fluid here, which could be like that, 830 00:57:54,284 --> 00:57:57,463 which could be like that. It cannot be like this. 831 00:57:57,463 --> 00:58:00,245 And so that's, then, the condition that the 832 00:58:00,245 --> 00:58:03,887 derivative, the spatial derivative from the left must be 833 00:58:03,887 --> 00:58:09,400 the same as on the right side. So, let us start with an 834 00:58:09,400 --> 00:58:13,199 incident wave that comes from the left. 835 00:58:13,199 --> 00:58:15,900 And so, Y1, YI, I, incident, 836 00:58:15,900 --> 00:58:21,300 has some amplitude A of I. That is the amplitude of the 837 00:58:21,300 --> 00:58:26,900 incident one times the sine. And, I'm going to write this 838 00:58:26,900 --> 00:58:29,699 now. I can write that in many 839 00:58:29,699 --> 00:58:34,000 different ways. But I'm going to write this, 840 00:58:34,000 --> 00:58:40,861 now, as omega 1T minus K1X. That's clearly a traveling wave 841 00:58:40,861 --> 00:58:46,769 that moves in the plus direction because the signs are different. 842 00:58:46,769 --> 00:58:51,661 And now I have a reflected wave, amplitude A reflected 843 00:58:51,661 --> 00:58:55,723 times the sine, obviously the same frequency, 844 00:58:55,723 --> 00:59:02,000 but now I get plus K1 times X. Notice the difference in sign. 845 00:59:02,000 --> 00:59:06,250 This one is going this way. This one is coming back. 846 00:59:06,250 --> 00:59:10,666 The amplitudes are different, but the K's are the same 847 00:59:10,666 --> 00:59:14,666 because the wavelength in medium one is the same. 848 00:59:14,666 --> 00:59:19,000 The K is 2 pi over lambda, and that wavelength is not 849 00:59:19,000 --> 00:59:23,333 going to change for those ways as it is for this one. 850 00:59:23,333 --> 00:59:27,583 And so, now I get to transmit it one, this amplitude 851 00:59:27,583 --> 00:59:32,500 transmitted times the sine of the same omega because if this 852 00:59:32,500 --> 00:59:36,750 junction shakes up and down with frequency of omega, 853 00:59:36,750 --> 00:59:42,330 [NOISE OBSCURES] given. You can't change the omega. 854 00:59:42,330 --> 00:59:47,510 You can say omega is different from the right side than it is 855 00:59:47,510 --> 00:59:50,446 from the left side. In other words, 856 00:59:50,446 --> 00:59:55,194 omega one is V1 times K1. But, that is also V2 times K2. 857 00:59:55,194 --> 00:59:59,251 It's the same omega. And so, I get here minus K2 858 00:59:59,251 --> 1:00:02,446 times X. So, this goes into the second 859 1:00:02,446 --> 1:00:06,811 medium. The minus sign indicates that 860 1:00:06,811 --> 1:00:12,627 it's going in this direction, but the K2 is different because 861 1:00:12,627 --> 1:00:17,862 the speed is different because V2 is different from V1. 862 1:00:17,862 --> 1:00:23,775 And so, with the same frequency I get a different value for K. 863 1:00:23,775 --> 1:00:29,494 So, now if I go to my boundary conditions, Y1 equals Y2 that 864 1:00:29,494 --> 1:00:33,566 you see here, then I get there that A of Y, 865 1:00:33,566 --> 1:00:38,801 A of I, so, X equals zero, at any moment in time I must 866 1:00:38,801 --> 1:00:45,979 meet that condition. So, A of I plus A of R must be 867 1:00:45,979 --> 1:00:50,357 ATR. If not, then the string would 868 1:00:50,357 --> 1:00:55,000 break. I can now take the derivative 869 1:00:55,000 --> 1:00:57,918 of my function, DY1 DX. 870 1:00:57,918 --> 1:01:05,081 And so, when I do that, I get the first one is going to 871 1:01:05,081 --> 1:01:11,130 be an A of I. And then the derivative against 872 1:01:11,130 --> 1:01:16,986 the X gives me a minus K1 here. And then, I get a cosine of 873 1:01:16,986 --> 1:01:22,237 omega 1T because X1 zero, remember, so I get a cosine 874 1:01:22,237 --> 1:01:27,387 omega 1T, but each term will have a cosine omega 1T. 875 1:01:27,387 --> 1:01:31,527 So, I will dump all the cosine omega 1T's. 876 1:01:31,527 --> 1:01:36,980 So, I go to my next one, which gives me now plus A of R 877 1:01:36,980 --> 1:01:40,235 times K1. And that now, 878 1:01:40,235 --> 1:01:44,321 so this is the left side, this is my DY1 DX, 879 1:01:44,321 --> 1:01:49,737 and my DY2 DX in my second medium is now this one gives me 880 1:01:49,737 --> 1:01:54,108 a minus K2 times A times minute times minus K2, 881 1:01:54,108 --> 1:01:58,289 and I dump the cosine. So, this is the second 882 1:01:58,289 --> 1:02:01,983 equation. And you could solve these two 883 1:02:01,983 --> 1:02:04,583 equations easily, except you have three 884 1:02:04,583 --> 1:02:08,140 questions, two equations, sorry, with three unknowns. 885 1:02:08,140 --> 1:02:10,466 You have AI, AR, and A transmitted. 886 1:02:10,466 --> 1:02:14,160 And of course you cannot solve two equations with three 887 1:02:14,160 --> 1:02:18,334 unknowns, but what you can find, that is the only goal I have, 888 1:02:18,334 --> 1:02:22,575 is the ratio of the amplitudes of what comes in over what comes 889 1:02:22,575 --> 1:02:24,354 back. That's my whole goal. 890 1:02:24,354 --> 1:02:28,390 If you calculate for this one K1 over K2, and you substitute 891 1:02:28,390 --> 1:02:31,263 that in here, then you can replace the case 892 1:02:31,263 --> 1:02:35,213 by the ratios of V's. You see that? 893 1:02:35,213 --> 1:02:40,533 So you take K1 divided by K2, and you get rid of your K's. 894 1:02:40,533 --> 1:02:45,013 So, when you do that, you get A of I minus A of R 895 1:02:45,013 --> 1:02:48,559 times V2 equals A transmitted times V1. 896 1:02:48,559 --> 1:02:52,106 So, that is simply combining these two. 897 1:02:52,106 --> 1:02:57,893 And so, now all you have to do is calculate the ratios of those 898 1:02:57,893 --> 1:03:02,746 amplitudes for this equation, one equation with three 899 1:03:02,746 --> 1:03:09,000 unknowns, and the second equation with three unknowns. 900 1:03:09,000 --> 1:03:14,438 And, what comes out of that, this is also worked out in 901 1:03:14,438 --> 1:03:20,582 French, what comes out of that is that the reflected amplitude 902 1:03:20,582 --> 1:03:26,625 over the incident amplitude is V2 minus V1 divided by V1 plus 903 1:03:26,625 --> 1:03:32,769 V2, and A transmitted divided by the amplitude or the incident 904 1:03:32,769 --> 1:03:37,000 one is 2V2 divided by V1 minus V2. 905 1:03:37,000 --> 1:03:40,371 Sometimes I call this with shorthand notation, 906 1:03:40,371 --> 1:03:43,742 R, reflectivity. It's the ratio of amplitudes. 907 1:03:43,742 --> 1:03:47,338 And, sometimes I call this shorthand notation TR, 908 1:03:47,338 --> 1:03:51,833 which is transmitidity (sic), the ratio of the amplitude that 909 1:03:51,833 --> 1:03:56,328 penetrates the second medium divided by the one that came in. 910 1:03:56,328 --> 1:04:00,000 So, that is just my shorthand notation. 911 1:04:00,000 --> 1:04:05,407 We can now do some interesting examples, and you will begin to 912 1:04:05,407 --> 1:04:08,332 understand what is happening here. 913 1:04:08,332 --> 1:04:10,903 Take an example, for instance, 914 1:04:10,903 --> 1:04:13,650 that mu two is infinitely large. 915 1:04:13,650 --> 1:04:16,576 That's a wall. So, in other words, 916 1:04:16,576 --> 1:04:21,451 that second string is a wall. If that second string as a 917 1:04:21,451 --> 1:04:26,326 wall, you better believe it that that point cannot move. 918 1:04:26,326 --> 1:04:32,000 The incoming wave couldn't possibly move the wall. 919 1:04:32,000 --> 1:04:36,548 So this is the same as what we earlier called, 920 1:04:36,548 --> 1:04:39,783 it's a fixed end. In other words, 921 1:04:39,783 --> 1:04:43,826 V2 is a zero, right, because if mu two is 922 1:04:43,826 --> 1:04:49,689 infinity [NOISE OBSCURES] zero. So, now, I'm going to go to 923 1:04:49,689 --> 1:04:52,519 these equations, and I'd say, 924 1:04:52,519 --> 1:04:55,046 what is now R? V2 is zero. 925 1:04:55,046 --> 1:05:00,000 Well, if V2 is zero, I get minus one. 926 1:05:00,000 --> 1:05:04,386 Hey, hey, minus one, that means when a mountain 927 1:05:04,386 --> 1:05:07,724 rolls in, what comes back? A valley. 928 1:05:07,724 --> 1:05:11,824 And when a valley rolls in, what comes back? 929 1:05:11,824 --> 1:05:14,971 A mountain. And we have seen that. 930 1:05:14,971 --> 1:05:19,072 We demonstrated that last time, a fixed end. 931 1:05:19,072 --> 1:05:22,791 And what do you think TR is going to be? 932 1:05:22,791 --> 1:05:26,033 What do you think it's going to be? 933 1:05:26,033 --> 1:05:28,512 Zero. Nothing goes through. 934 1:05:28,512 --> 1:05:31,850 And, look at it. If V2 goes to zero, 935 1:05:31,850 --> 1:05:37,000 ah, TR is zero. So, aren't we happy? 936 1:05:37,000 --> 1:05:42,087 Take now an example that V1 is smaller than V2. 937 1:05:42,087 --> 1:05:47,396 So, in other words, mu one is larger than mu two. 938 1:05:47,396 --> 1:05:54,032 When we have a case like that, notice that R is always larger 939 1:05:54,032 --> 1:05:57,903 than zero. If V1 is smaller than V2, 940 1:05:57,903 --> 1:06:02,835 this is a speed now. This is always larger than 941 1:06:02,835 --> 1:06:04,318 zero. What that means, 942 1:06:04,318 --> 1:06:06,931 just a second, I'll give you a chance. 943 1:06:06,931 --> 1:06:10,391 What that means is that a mountain comes back as a 944 1:06:10,391 --> 1:06:12,791 mountain. Now, the amplitude of the 945 1:06:12,791 --> 1:06:16,746 mountain will be changed, but it comes back as a mountain 946 1:06:16,746 --> 1:06:20,135 because that's the plus sign. And notice that the 947 1:06:20,135 --> 1:06:23,383 transmitivity (sic) is always larger than zero. 948 1:06:23,383 --> 1:06:27,479 It never gets a minus sign. Of course it cannot get a minus 949 1:06:27,479 --> 1:06:29,950 sign. That you can't explain to your 950 1:06:29,950 --> 1:06:33,269 kid brother because if a mountain comes in here, 951 1:06:33,269 --> 1:06:38,000 that point is going to move up no matter what. 952 1:06:38,000 --> 1:06:42,011 And if this point is going to move up, what goes into the 953 1:06:42,011 --> 1:06:45,952 second medium as a mountain. And if this point goes down 954 1:06:45,952 --> 1:06:49,749 because it is a valley, then what goes into the second 955 1:06:49,749 --> 1:06:53,259 medium is a valley. So, it is completely illogical 956 1:06:53,259 --> 1:06:56,125 that here you never can get a minus sign. 957 1:06:56,125 --> 1:06:58,776 But here you can. That was a question. 958 1:06:58,776 --> 1:07:02,000 Think you very much, my goodness. 959 1:07:02,000 --> 1:07:06,710 Yes, V1 plus V2, thank you very much. 960 1:07:06,710 --> 1:07:12,728 Thank you very much. So now, we take an example 961 1:07:12,728 --> 1:07:18,355 which is absolutely thrilling, V1 equals V2. 962 1:07:18,355 --> 1:07:22,411 That means mu one equals mu two. 963 1:07:22,411 --> 1:07:30,000 That's another way of saying there is no junction. 964 1:07:30,000 --> 1:07:33,352 [NOISE OBSCURES] anything will reflect. 965 1:07:33,352 --> 1:07:36,794 Of course not. So I predict without even 966 1:07:36,794 --> 1:07:42,352 looking at the result that R is zero, and that the transmitivity 967 1:07:42,352 --> 1:07:45,970 is going to be plus one. Let's check that. 968 1:07:45,970 --> 1:07:48,352 When V1 is V2, this is zero. 969 1:07:48,352 --> 1:07:52,411 And when V1 is V2, you just corrected me at the 970 1:07:52,411 --> 1:07:55,676 right time, by the way. When V1 is V2, 971 1:07:55,676 --> 1:08:00,000 you see that this goes to plus one. 972 1:08:00,000 --> 1:08:04,934 With a minus sign it would have gone to infinity. 973 1:08:04,934 --> 1:08:09,663 I would have caught it. But you got it earlier. 974 1:08:09,663 --> 1:08:15,626 OK, so now let's do a specific example whereby we give some 975 1:08:15,626 --> 1:08:20,663 numbers because that's what I plan to demonstrate. 976 1:08:20,663 --> 1:08:26,112 So, the example that I have in mind is that V1 is 2V2. 977 1:08:26,112 --> 1:08:29,093 So, V1 is 2V2. In other words, 978 1:08:29,093 --> 1:08:35,673 mu one is one half mu two. If I put that in that equation, 979 1:08:35,673 --> 1:08:38,778 I find that R equals minus one third. 980 1:08:38,778 --> 1:08:42,659 And, I find that the transmitivity is plus two 981 1:08:42,659 --> 1:08:45,850 thirds. It's just a matter of sticking 982 1:08:45,850 --> 1:08:48,610 those numbers in. And that means, 983 1:08:48,610 --> 1:08:52,664 then, the following, that if this is my incident 984 1:08:52,664 --> 1:08:55,855 wave, so this is now the incident one, 985 1:08:55,855 --> 1:08:59,909 and the incident one, say, has an amplitude one, 986 1:08:59,909 --> 1:09:04,221 and is moving in this direction, that the returning 987 1:09:04,221 --> 1:09:10,000 one has an amplitude which is three times smaller. 988 1:09:10,000 --> 1:09:14,591 And it is flipped over but it has the same length because it 989 1:09:14,591 --> 1:09:18,249 has the same speed. So suppose the returning one 990 1:09:18,249 --> 1:09:20,739 were here. Then you would see the 991 1:09:20,739 --> 1:09:24,630 returning one with only one third of its amplitude. 992 1:09:24,630 --> 1:09:28,443 So, that's about here. So, you would see something 993 1:09:28,443 --> 1:09:32,642 like this. And it would be moving in this 994 1:09:32,642 --> 1:09:35,726 direction. And this amplitude is one 995 1:09:35,726 --> 1:09:38,721 third. And now, what goes into that 996 1:09:38,721 --> 1:09:43,742 media number two is a pulse with two thirds the amplitude. 997 1:09:43,742 --> 1:09:46,473 So, this is the two thirds mark. 998 1:09:46,473 --> 1:09:49,645 So, the amplitude is only two thirds. 999 1:09:49,645 --> 1:09:53,609 But, since the speed is lower, because notice, 1000 1:09:53,609 --> 1:09:56,780 speed is lower. Therefore it shrinks. 1001 1:09:56,780 --> 1:10:02,066 And so, not only will amplitude be two thirds of the incident 1002 1:10:02,066 --> 1:10:07,000 one, but this length will only be half. 1003 1:10:07,000 --> 1:10:12,155 And so, you will see this. And this is now two thirds. 1004 1:10:12,155 --> 1:10:15,560 And this is moving with velocity V2. 1005 1:10:15,560 --> 1:10:18,575 This is moving with velocity V1. 1006 1:10:18,575 --> 1:10:23,342 And this is moving with velocity [UNINTELLIGIBLE]. 1007 1:10:23,342 --> 1:10:26,649 So, that's the meaning of R and TR. 1008 1:10:26,649 --> 1:10:32,000 There was a question here. Mu one, V1 is 2V2. 1009 1:10:32,000 --> 1:10:36,151 Then, mu one is one half mu two. 1010 1:10:36,151 --> 1:10:40,705 Is that one? You have a good point. 1011 1:10:40,705 --> 1:10:44,455 Let's just remove it. OK now? 1012 1:10:44,455 --> 1:10:48,741 Thank you. So, this now I want to 1013 1:10:48,741 --> 1:10:53,964 demonstrate. And the way we are going to 1014 1:10:53,964 --> 1:11:02,000 demonstrate it is we're going to use this machine. 1015 1:11:02,000 --> 1:11:06,150 And this machine allows me, this has a medium here whereby 1016 1:11:06,150 --> 1:11:09,354 the speed is twice as high as the speed here. 1017 1:11:09,354 --> 1:11:12,048 I'm going to generate here a mountain. 1018 1:11:12,048 --> 1:11:15,033 And the mountain will go into that medium. 1019 1:11:15,033 --> 1:11:19,402 And the first thing I want you to notice is that as it enters 1020 1:11:19,402 --> 1:11:23,844 this medium where the speed is lower, the speed is higher here 1021 1:11:23,844 --> 1:11:27,194 than there, I want you to see two things: that, 1022 1:11:27,194 --> 1:11:30,179 first of all, the mountain go through as a 1023 1:11:30,179 --> 1:11:33,608 mountain. Whatever goes through must 1024 1:11:33,608 --> 1:11:37,010 always have the same polarity. If the mountain comes in, 1025 1:11:37,010 --> 1:11:40,103 a mountain goes through. So look at the two things. 1026 1:11:40,103 --> 1:11:43,257 The mountain remains a mountain, and it will shrink. 1027 1:11:43,257 --> 1:11:46,103 And then, later, we'll overlook the reflection. 1028 1:11:46,103 --> 1:11:48,824 So, here, we have to meet it completely dark, 1029 1:11:48,824 --> 1:11:51,422 don't we, for this? And I think that's more 1030 1:11:51,422 --> 1:11:53,649 romantic. Yet, that's what they like, 1031 1:11:53,649 --> 1:11:55,690 yeah. OK, so I'm going to generate 1032 1:11:55,690 --> 1:11:57,670 here a mountain as fast as I can. 1033 1:11:57,670 --> 1:12:02,000 The speed here is larger than the speed over there. 1034 1:12:02,000 --> 1:12:06,090 And you will see that the pulse shrinks. 1035 1:12:06,090 --> 1:12:09,762 But the mountain remains a mountain. 1036 1:12:09,762 --> 1:12:14,797 Are you ready for that? You see that the mountain 1037 1:12:14,797 --> 1:12:19,727 remains a mountain? Can you see that it shrinks? 1038 1:12:19,727 --> 1:12:25,076 I need some light because I think I broke something. 1039 1:12:25,076 --> 1:12:26,860 No problem. Fixed. 1040 1:12:26,860 --> 1:12:32,000 OK, now we want to do something else. 1041 1:12:32,000 --> 1:12:36,214 Now I want you to see that when I drive in a mountain that a 1042 1:12:36,214 --> 1:12:39,642 mountain goes through, but the valley comes back. 1043 1:12:39,642 --> 1:12:43,714 And, the valley that comes back has the same length as the 1044 1:12:43,714 --> 1:12:46,714 incident one. That is, the minus one third, 1045 1:12:46,714 --> 1:12:49,571 remember? So, the valley is very shallow, 1046 1:12:49,571 --> 1:12:53,857 but the mountain will come back as a valley when it hits this 1047 1:12:53,857 --> 1:12:56,357 point. So, we have already seen that 1048 1:12:56,357 --> 1:12:59,285 the mountain goes through with a mountain. 1049 1:12:59,285 --> 1:13:04,000 You have already seen that the pulse gets shorter. 1050 1:13:04,000 --> 1:13:07,773 And now you are going to see that when it hits this point, 1051 1:13:07,773 --> 1:13:10,288 the reflection will be minus one third. 1052 1:13:10,288 --> 1:13:13,134 So, the mountain will come back as a valley. 1053 1:13:13,134 --> 1:13:15,914 Are you ready for this? And there it comes. 1054 1:13:15,914 --> 1:13:19,026 Could you see it, that it came back as a valley? 1055 1:13:19,026 --> 1:13:21,938 Shall I try again? You see, the speed is very 1056 1:13:21,938 --> 1:13:23,527 high. That's the problem. 1057 1:13:23,527 --> 1:13:26,241 It's very hard to see. I'll try once more. 1058 1:13:26,241 --> 1:13:29,881 Yeah, yeah, I could see that the mountain came back as a 1059 1:13:29,881 --> 1:13:33,825 valley effect. What would happen if I drive a 1060 1:13:33,825 --> 1:13:36,684 mountain in from this end? What would come back, 1061 1:13:36,684 --> 1:13:39,847 a mountain or a valley? So, if I do it from that end, 1062 1:13:39,847 --> 1:13:41,794 the mountain returns as a valley. 1063 1:13:41,794 --> 1:13:45,079 If I do it from this end, the mountain must return as a 1064 1:13:45,079 --> 1:13:48,000 mountain. Shall we take a look at that? 1065 1:13:48,000 --> 1:13:54,000 1066 1:13:54,000 --> 1:13:55,722 And it comes back as a mountain. 1067 1:13:55,722 --> 1:13:57,000 I'll do it once more. 1068 1:13:57,000 --> 1:14:02,000 1069 1:14:02,000 --> 1:14:06,046 I think I broke that one, too. 1070 1:14:06,046 --> 1:14:10,790 Yeah, boy, I only have one problem. 1071 1:14:10,790 --> 1:14:16,930 And that problem is going to be your problem. 1072 1:14:16,930 --> 1:14:24,883 And that is the following. I would like to explore now the 1073 1:14:24,883 --> 1:14:30,883 possibility that mu two is going to be zero. 1074 1:14:30,883 --> 1:14:38,000 That means V2 has become infinitely high. 1075 1:14:38,000 --> 1:14:40,976 If mu two is zero, there is nothing here. 1076 1:14:40,976 --> 1:14:43,729 It means there is tension on the rope. 1077 1:14:43,729 --> 1:14:46,482 But there is nothing here. It's empty. 1078 1:14:46,482 --> 1:14:50,500 We call that an open end. So, imagine in your head that 1079 1:14:50,500 --> 1:14:54,517 something is holding it tight. The end can move freely. 1080 1:14:54,517 --> 1:14:57,940 That's no problem. But, there is no mass there. 1081 1:14:57,940 --> 1:15:01,065 Remember, it was like the mass-less string, 1082 1:15:01,065 --> 1:15:04,805 and the rod. So, now, in this case, 1083 1:15:04,805 --> 1:15:09,564 which is really an open end, there's nothing in medium two, 1084 1:15:09,564 --> 1:15:13,256 I can now go to my equation and ask what R is. 1085 1:15:13,256 --> 1:15:17,194 And, I will be very pleased. When mu two is zero, 1086 1:15:17,194 --> 1:15:21,297 when V2 goes to infinity, when V2 goes to infinity, 1087 1:15:21,297 --> 1:15:25,071 there's this plus one. Mountain comes back as a 1088 1:15:25,071 --> 1:15:28,271 mountain. All of the predicted that last 1089 1:15:28,271 --> 1:15:31,800 time an open end. A mountain comes back as a 1090 1:15:31,800 --> 1:15:35,000 mountain. Physics works. 1091 1:15:35,000 --> 1:15:41,476 What do you think is going to be the transmitivity? 1092 1:15:41,476 --> 1:15:45,751 Zero, yeah, that's what you think. 1093 1:15:45,751 --> 1:15:49,766 Put in there V2 equals infinity. 1094 1:15:49,766 --> 1:15:54,430 That takes you by surprise, plus two. 1095 1:15:54,430 --> 1:16:00,000 Holy smokes! What is going on here? 1096 1:16:00,000 --> 1:16:04,116 If this were true, we would have solved the energy 1097 1:16:04,116 --> 1:16:08,905 crisis of the world because a pulse comes in and the whole 1098 1:16:08,905 --> 1:16:12,769 pulse comes back. Everything that went in comes 1099 1:16:12,769 --> 1:16:15,205 back. Mountain comes back as a 1100 1:16:15,205 --> 1:16:20,078 mountain, but there is something in addition that goes into 1101 1:16:20,078 --> 1:16:23,355 nothing. I want you to think about this, 1102 1:16:23,355 --> 1:16:26,715 and you may not be able to sleep tonight. 1103 1:16:26,715 --> 1:16:31,000 And if you can't, it is not my fault. 1104 1:16:31,000 --> 1:16:33,248 It's the fault of physics. See you next time.