1 00:00:27,000 --> 00:00:30,823 Today, I'm going to couple a huge number of oscillators. 2 00:00:30,823 --> 00:00:34,925 And that ultimately we'll make that number infinitely large. 3 00:00:34,925 --> 00:00:39,096 And that's the way that you can couple oscillators is through 4 00:00:39,096 --> 00:00:42,224 springs, as we did last time on the air track. 5 00:00:42,224 --> 00:00:44,866 And we can let the cars move like this. 6 00:00:44,866 --> 00:00:47,229 We call that a longitudinal motion. 7 00:00:47,229 --> 00:00:51,331 That means that the motion is along the line of oscillators. 8 00:00:51,331 --> 00:00:54,807 And then, we have ways that we can oscillate things 9 00:00:54,807 --> 00:00:58,213 perpendicular to the direction of the oscillators. 10 00:00:58,213 --> 00:01:00,716 And we call that a transverse motion. 11 00:01:00,716 --> 00:01:06,000 So that goes like this on the oscillators are like this. 12 00:01:06,000 --> 00:01:11,086 And the algebra is identical. That is easier to use a 13 00:01:11,086 --> 00:01:15,489 transverse motion when you do the derivations. 14 00:01:15,489 --> 00:01:19,010 And so, I'll use a transverse motion. 15 00:01:19,010 --> 00:01:24,489 Suppose I have a string and I put on this string [beats?] 16 00:01:24,489 --> 00:01:26,836 masses. I have a tangent, 17 00:01:26,836 --> 00:01:28,500 T. Each mass is M, 18 00:01:28,500 --> 00:01:33,000 and the length between these is L. 19 00:01:33,000 --> 00:01:37,913 And, this is a fixed end. And this and it is also fixed. 20 00:01:37,913 --> 00:01:41,665 It cannot move. And so, this is number one. 21 00:01:41,665 --> 00:01:45,239 This is number two. This is number three. 22 00:01:45,239 --> 00:01:49,527 And then here is the last one, which is number N. 23 00:01:49,527 --> 00:01:53,815 I'm going to have N of these beats on the string. 24 00:01:53,815 --> 00:01:57,478 So, this point here, I can call that zero. 25 00:01:57,478 --> 00:02:02,570 I think of this direction, X, and this is the direction of 26 00:02:02,570 --> 00:02:06,534 Y. And so, at zero there is no 27 00:02:06,534 --> 00:02:09,537 beat. And here, at the position N 28 00:02:09,537 --> 00:02:12,729 plus one, there is no beat, either. 29 00:02:12,729 --> 00:02:17,234 And so, the question, now is, what are the normal 30 00:02:17,234 --> 00:02:22,866 mode solutions to that system? There must be capital N normal 31 00:02:22,866 --> 00:02:25,776 modes. And, if N goes to 10,000, 32 00:02:25,776 --> 00:02:30,000 there must be 10,000 normal modes. 33 00:02:30,000 --> 00:02:33,664 At a particular moment in time, you can imagine, 34 00:02:33,664 --> 00:02:36,393 for instance, that this one is here. 35 00:02:36,393 --> 00:02:40,448 And maybe this one is here. And this one may be here. 36 00:02:40,448 --> 00:02:44,892 And maybe this one is here. The only thing we have to keep 37 00:02:44,892 --> 00:02:49,337 an eye on that this is always zero, and the Y here is also 38 00:02:49,337 --> 00:02:52,066 always zero. That's what we call the 39 00:02:52,066 --> 00:02:55,964 boundary condition. Now, if you make the amplitudes 40 00:02:55,964 --> 00:03:00,097 modest, then it's easy to demonstrate that the tangent 41 00:03:00,097 --> 00:03:04,385 will remain constant in these pieces, modest amplitudes, 42 00:03:04,385 --> 00:03:10,000 and that there is no motion in the X direction either. 43 00:03:10,000 --> 00:03:14,654 At least you can make it negligibly small. 44 00:03:14,654 --> 00:03:20,556 And so, we will only concentrate on the forces in the 45 00:03:20,556 --> 00:03:25,324 Y direction. And so now I'm going to make a 46 00:03:25,324 --> 00:03:31,000 [loner?] version here for particle number P. 47 00:03:31,000 --> 00:03:36,021 So, this is particle P, and here's the location of P 48 00:03:36,021 --> 00:03:39,566 minus one. And here's the location of 49 00:03:39,566 --> 00:03:44,095 particle P plus one. And at a certain moment in 50 00:03:44,095 --> 00:03:48,624 time, let's assume that that particle is there, 51 00:03:48,624 --> 00:03:53,547 that little mass N. And, let's assume that this one 52 00:03:53,547 --> 00:03:57,289 is here. And let's assume that this one 53 00:03:57,289 --> 00:04:01,030 is here. So, this vertical displacement 54 00:04:01,030 --> 00:04:04,956 is YP. This is YP plus one. 55 00:04:04,956 --> 00:04:07,989 And this, then, is YP minus one. 56 00:04:07,989 --> 00:04:12,195 But the strings are just attached like this. 57 00:04:12,195 --> 00:04:18,065 And so, the fact that strings get longer: we will ignore that 58 00:04:18,065 --> 00:04:21,293 because they are small amplitudes. 59 00:04:21,293 --> 00:04:25,304 So the [tangent/tension?] will not change. 60 00:04:25,304 --> 00:04:30,000 We draw this line. We draw this line. 61 00:04:30,000 --> 00:04:36,290 I will call this angle alpha P. And I will call this angle 62 00:04:36,290 --> 00:04:41,588 alpha P minus one. So, the tension is going to be 63 00:04:41,588 --> 00:04:47,879 on that point P on this little mass, M, in this direction. 64 00:04:47,879 --> 00:04:52,294 And there is a tangent in this direction. 65 00:04:52,294 --> 00:04:57,040 And, we will assume that these tangents are, 66 00:04:57,040 --> 00:05:03,000 then, the same for reasons that I mentioned. 67 00:05:03,000 --> 00:05:08,723 So I can write down, now, for that mass for that 68 00:05:08,723 --> 00:05:13,228 location, P. I can write down Newton's 69 00:05:13,228 --> 00:05:16,516 Second Law. And so, I get M, 70 00:05:16,516 --> 00:05:23,457 Y of P, second derivative, D2YDT squared of that position, 71 00:05:23,457 --> 00:05:26,501 P. And, you see I have one 72 00:05:26,501 --> 00:05:31,129 vertical component that drives it down. 73 00:05:31,129 --> 00:05:39,724 That is due to this tangent. And then I have one that drives 74 00:05:39,724 --> 00:05:46,451 it away from equilibrium. And so I'm going to get minus T 75 00:05:46,451 --> 00:05:50,655 times the sine of alpha T minus one. 76 00:05:50,655 --> 00:05:55,701 And I get plus T times the sine of alpha P. 77 00:05:55,701 --> 00:06:01,227 But, of course, I know what the sine of alpha P 78 00:06:01,227 --> 00:06:08,915 minus one is because the sine of alpha P minus one is YP minus YP 79 00:06:08,915 --> 00:06:18,095 minus one divided by L. And so, I can write here Y of P 80 00:06:18,095 --> 00:06:26,479 minus YP minus one divided by L. And here, I have T. 81 00:06:26,479 --> 00:06:37,000 The sine of alpha P is YP plus one minus Y of P divided by L. 82 00:06:37,000 --> 00:06:44,000 83 00:06:44,000 --> 00:06:47,221 I'm going to introduce a shorthand notation. 84 00:06:47,221 --> 00:06:51,491 I'm going to introduce the omega zero squared is T divided 85 00:06:51,491 --> 00:06:53,140 by ML. As time goes on, 86 00:06:53,140 --> 00:06:56,885 you will get more insight as to why we choose that. 87 00:06:56,885 --> 00:07:00,482 At least convince yourself, if you have the time, 88 00:07:00,482 --> 00:07:02,579 that this is, as I mentioned, 89 00:07:02,579 --> 00:07:07,000 one over second squared is the right dimension. 90 00:07:07,000 --> 00:07:11,368 T, by the way, whenever I use this throughout 91 00:07:11,368 --> 00:07:16,531 the lecture, is never a period. It is always tension. 92 00:07:16,531 --> 00:07:22,191 So, I will stay away from the capital T when we're dealing 93 00:07:22,191 --> 00:07:25,865 with a period. So, I can divide M out, 94 00:07:25,865 --> 00:07:30,432 and I can rewrite this to get the M downstairs. 95 00:07:30,432 --> 00:07:35,000 And so we now get Y of P double dot. 96 00:07:35,000 --> 00:07:39,545 And what I'm going to do now, I'm going to take the YP here 97 00:07:39,545 --> 00:07:43,228 and the YP there. And, I bring them to the left. 98 00:07:43,228 --> 00:07:46,598 I have a minus here and I have a minus here. 99 00:07:46,598 --> 00:07:49,811 So, I get plus omega squared times Y of P, 100 00:07:49,811 --> 00:07:53,416 and then I'm going to bring the P minus one in. 101 00:07:53,416 --> 00:07:56,630 And, I'm going to bring the P plus one in. 102 00:07:56,630 --> 00:08:02,950 Notice I have a plus sign here. So, I've got minus omega zero 103 00:08:02,950 --> 00:08:08,038 squared times YP plus one. And here, I have a plus. 104 00:08:08,038 --> 00:08:13,126 Minus-minus is a plus, so when I bring that back to 105 00:08:13,126 --> 00:08:17,298 get a minus. But I already have the minus, 106 00:08:17,298 --> 00:08:22,182 so I get YP minus one. And, that now equals zero. 107 00:08:22,182 --> 00:08:26,863 But that is Newton's Second Law for particle P. 108 00:08:26,863 --> 00:08:29,000 Excuse me? 109 00:08:29,000 --> 00:08:35,000 110 00:08:35,000 --> 00:08:39,470 What is your problem? Yes, thank you very much. 111 00:08:39,470 --> 00:08:45,009 Very good, I appreciate it. This is a two because you have 112 00:08:45,009 --> 00:08:48,702 two P's. You have one here and you have 113 00:08:48,702 --> 00:08:51,520 one here. Thank you very much. 114 00:08:51,520 --> 00:08:56,379 OK, very attentive. So, now we have to do these for 115 00:08:56,379 --> 00:09:00,363 every single object. So, we have capital N 116 00:09:00,363 --> 00:09:06,000 differential equations, one for each particle. 117 00:09:06,000 --> 00:09:11,422 And the only thing that we have to keep in mind now when we 118 00:09:11,422 --> 00:09:16,189 solve it, that Y zero, which refers to that location 119 00:09:16,189 --> 00:09:21,705 zero there must always be zero, and that YN plus one is also 120 00:09:21,705 --> 00:09:24,977 zero. So, if I made a sketch sort of 121 00:09:24,977 --> 00:09:30,118 to warm you up to the idea, and I made a sketch for only 122 00:09:30,118 --> 00:09:34,232 two particles, so this is number one and this 123 00:09:34,232 --> 00:09:39,000 is number two, and this is a fixed end. 124 00:09:39,000 --> 00:09:43,756 And this is a fixed end. Then you can sort of see that 125 00:09:43,756 --> 00:09:49,051 in the lowest normal mode you're going to see something like 126 00:09:49,051 --> 00:09:51,653 this. So, this is omega minus, 127 00:09:51,653 --> 00:09:54,974 and it's going to oscillate like this. 128 00:09:54,974 --> 00:09:59,282 But you can also imagine that in the second mode, 129 00:09:59,282 --> 00:10:04,217 which is the highest one, that number one is up and then 130 00:10:04,217 --> 00:10:09,146 number two is down. So, you get this situation. 131 00:10:09,146 --> 00:10:11,611 And so they oscillate like this. 132 00:10:11,611 --> 00:10:15,587 Just to make you see, I called this little n equals 133 00:10:15,587 --> 00:10:17,892 one. And, I call this little n 134 00:10:17,892 --> 00:10:21,073 equals 2n. They're referring to the mode: 135 00:10:21,073 --> 00:10:23,935 mode one. I will use this n later on, 136 00:10:23,935 --> 00:10:26,639 and this is, then, mode number two. 137 00:10:26,639 --> 00:10:31,012 And, there are only two modes because there are only two 138 00:10:31,012 --> 00:10:35,501 particles. So, let us now proceed with the 139 00:10:35,501 --> 00:10:39,536 equation that we have. And it let us write down for 140 00:10:39,536 --> 00:10:42,925 this system the two differential equations. 141 00:10:42,925 --> 00:10:47,282 And so, look at this one. We're going to substitute for 142 00:10:47,282 --> 00:10:50,429 P, first, number one, which is this one. 143 00:10:50,429 --> 00:10:53,011 That's one differential equation. 144 00:10:53,011 --> 00:10:56,884 And then, we're going to put in for P number two. 145 00:10:56,884 --> 00:11:01,000 We get a second differential equation. 146 00:11:01,000 --> 00:11:06,590 So, if you're ready, then we're going to get Y1 147 00:11:06,590 --> 00:11:13,638 double dot plus two omega zero squared times Y1 minus omega 148 00:11:13,638 --> 00:11:17,649 zero squared. And then, we get Y2. 149 00:11:17,649 --> 00:11:23,847 That's this particle. And then, we get plus YP minus 150 00:11:23,847 --> 00:11:30,045 one, which is plus Y zero, which happens to be zero, 151 00:11:30,045 --> 00:11:36,000 by the way because Y zero here is zero. 152 00:11:36,000 --> 00:11:39,899 So now, we go to [D?], and this is zero. 153 00:11:39,899 --> 00:11:43,100 Now we go to the second particle. 154 00:11:43,100 --> 00:11:48,799 And so we get Y2 double dot plus two omega zero squared Y2 155 00:11:48,799 --> 00:11:54,100 minus omega zero squared. And, now we get first P plus 156 00:11:54,100 --> 00:11:58,000 one, which is number three, which is Y3, 157 00:11:58,000 --> 00:12:04,000 which happens to be zero because Y3 is this point. 158 00:12:04,000 --> 00:12:09,214 And that's zero in this specific case with only two 159 00:12:09,214 --> 00:12:11,717 objects. And then we get, 160 00:12:11,717 --> 00:12:16,828 here, plus YP minus one. And, that is Y1 plus one. 161 00:12:16,828 --> 00:12:19,852 That is Y1. And, that is zero. 162 00:12:19,852 --> 00:12:24,858 So, these two coupled oscillators will have to be 163 00:12:24,858 --> 00:12:27,883 solved. And in the normal mode 164 00:12:27,883 --> 00:12:33,932 situation, we are clearly going to put this in as our trial 165 00:12:33,932 --> 00:12:39,770 function, cosine omega T. They must oscillate with the 166 00:12:39,770 --> 00:12:42,678 same frequency, omega, otherwise we wouldn't be 167 00:12:42,678 --> 00:12:46,028 dealing with normal modes. And so, these are our trial 168 00:12:46,028 --> 00:12:49,316 functions that we're going to put in these equations. 169 00:12:49,316 --> 00:12:51,528 And we will put them in, number one. 170 00:12:51,528 --> 00:12:53,931 We are going to put them in number two. 171 00:12:53,931 --> 00:12:56,839 And then we will put them in particle number P. 172 00:12:56,839 --> 00:13:00,000 And then, we can all the way go to N. 173 00:13:00,000 --> 00:13:03,780 And if N is 10,000, we have to write down 10,000 174 00:13:03,780 --> 00:13:06,997 differential equations on the blackboard. 175 00:13:06,997 --> 00:13:09,812 That will take the rest of the hour. 176 00:13:09,812 --> 00:13:12,627 So, I'm going to do number one here. 177 00:13:12,627 --> 00:13:15,120 So I get a Y1 here, which is A1. 178 00:13:15,120 --> 00:13:19,865 The second derivative always gives me a minus omega squared. 179 00:13:19,865 --> 00:13:23,083 So, you get minus omega squared times A1. 180 00:13:23,083 --> 00:13:27,506 I ditch the cosine omega T because each term will have a 181 00:13:27,506 --> 00:13:33,026 cosine omega T. Then I get plus two omega zero 182 00:13:33,026 --> 00:13:37,869 squared times A1, and then I get minus omega zero 183 00:13:37,869 --> 00:13:42,712 squared times A2, right, because now I get a two, 184 00:13:42,712 --> 00:13:46,445 plus A zero, which happens to be zero. 185 00:13:46,445 --> 00:13:51,994 But I just put it there; you will see shortly why I want 186 00:13:51,994 --> 00:13:56,534 to keep it there. I go to particle number two. 187 00:13:56,534 --> 00:13:59,964 I get minus omega squared times A2. 188 00:13:59,964 --> 00:14:07,026 It becomes a little boring. Two omega zero squared times A2 189 00:14:07,026 --> 00:14:11,616 minus omega zero squared. Now, I have a Y3 here. 190 00:14:11,616 --> 00:14:15,619 So, I get an A3, which happens to be zero. 191 00:14:15,619 --> 00:14:19,134 But that's not so relevant right now. 192 00:14:19,134 --> 00:14:24,212 And then I get plus A1. And, the reason why I started 193 00:14:24,212 --> 00:14:29,289 off with one and two: that now you can see how we can 194 00:14:29,289 --> 00:14:36,960 put it in the P's particle. So, the P's particle now is 195 00:14:36,960 --> 00:14:44,146 going to be rather easy. Maybe I should do that in 196 00:14:44,146 --> 00:14:52,799 color, minus omega squared AP plus two omega zero squared AP 197 00:14:52,799 --> 00:15:01,013 minus omega zero squared. And, now we are going to get AP 198 00:15:01,013 --> 00:15:07,968 plus one plus AP minus one. And that equals zero. 199 00:15:07,968 --> 00:15:12,562 Also, this equals zero. And also this equals zero. 200 00:15:12,562 --> 00:15:17,062 And so, now you see the differential equation for 201 00:15:17,062 --> 00:15:21,375 particle number P. And so, you can go on now to 202 00:15:21,375 --> 00:15:25,687 particle capital N, and now you have to solve N 203 00:15:25,687 --> 00:15:30,000 differential equations. That's a zoo. 204 00:15:30,000 --> 00:15:35,073 That's a terrible thing. Now, we will take a shortcut, 205 00:15:35,073 --> 00:15:40,241 which is not very rigid. But, it really with save a lot 206 00:15:40,241 --> 00:15:43,687 of math. And, that is we will use our 207 00:15:43,687 --> 00:15:49,143 intuition, something that we know sort of from experience, 208 00:15:49,143 --> 00:15:54,408 if you have a lot of beats on here, fixed here and fixed 209 00:15:54,408 --> 00:15:59,768 there, and you ask yourself what's going to happen in the 210 00:15:59,768 --> 00:16:04,745 lowest possible mode, then you just know that you get 211 00:16:04,745 --> 00:16:09,656 something like this. It goes like this, 212 00:16:09,656 --> 00:16:12,233 and like this, and like this. 213 00:16:12,233 --> 00:16:16,282 And, you know that on the second normal mode, 214 00:16:16,282 --> 00:16:20,699 the one that follows that has a higher frequency, 215 00:16:20,699 --> 00:16:23,644 you expect that the side goes up. 216 00:16:23,644 --> 00:16:27,877 This goes down. And then it will oscillate like 217 00:16:27,877 --> 00:16:30,822 this. So, we use that experience, 218 00:16:30,822 --> 00:16:35,791 which is not very rigid in order to decide on our trial 219 00:16:35,791 --> 00:16:41,859 function. This would be mode one, 220 00:16:41,859 --> 00:16:51,422 and this would be mode two. And so, not going to put in as 221 00:16:51,422 --> 00:17:01,154 a trial solution A for particle P, which is in mode N as in 222 00:17:01,154 --> 00:17:04,599 Nancy. This is the mode. 223 00:17:04,599 --> 00:17:10,149 I want a sinusoid in there that is always zero here and zero 224 00:17:10,149 --> 00:17:13,442 there. And it can have an amplitude, 225 00:17:13,442 --> 00:17:16,829 of course, which I can freely choose. 226 00:17:16,829 --> 00:17:21,439 So, this C of N is the amplitude of this sinusoid. 227 00:17:21,439 --> 00:17:25,578 This is C1, and this value is going to be C2. 228 00:17:25,578 --> 00:17:30,000 Each one can have its own amplitude. 229 00:17:30,000 --> 00:17:35,766 And then I get the sine of PN pi over N plus one. 230 00:17:35,766 --> 00:17:43,214 So, let's look together at this equation so that we have a full 231 00:17:43,214 --> 00:17:48,980 understanding what we are trying to put in there. 232 00:17:48,980 --> 00:17:55,467 Notice that P equals zero that the sine is always zero. 233 00:17:55,467 --> 00:18:02,194 That's obvious because we wanted that because this point, 234 00:18:02,194 --> 00:18:07,000 the zero point, is not moving. 235 00:18:07,000 --> 00:18:11,079 Notice, also, that if you put in P equals N 236 00:18:11,079 --> 00:18:15,546 plus one, that AN plus one is also always zero. 237 00:18:15,546 --> 00:18:21,276 It's [put in?] P N plus one. The sine of a multiple times pi 238 00:18:21,276 --> 00:18:25,161 is always zero because N is now our mode. 239 00:18:25,161 --> 00:18:28,755 It's going to be one, two, three, etc. 240 00:18:28,755 --> 00:18:34,000 These are the modes that we are looking for. 241 00:18:34,000 --> 00:18:37,000 For instance, if we take N equals one -- 242 00:18:37,000 --> 00:18:44,000 243 00:18:44,000 --> 00:18:50,077 Let's take N equals one. So, we have particles and they 244 00:18:50,077 --> 00:18:55,705 are all in mode one. So then we get that AP in mode 245 00:18:55,705 --> 00:19:01,108 one, we have a C1, which is the amplitude of that 246 00:19:01,108 --> 00:19:07,861 sinusoid, and then you would have the sine of N pi divided by 247 00:19:07,861 --> 00:19:11,175 N plus one. And you can, 248 00:19:11,175 --> 00:19:16,955 indeed, convince yourself that that is exactly this sinusoid 249 00:19:16,955 --> 00:19:21,559 with an amplitude C1. And, you can also convince 250 00:19:21,559 --> 00:19:26,653 yourself that P zero, A zero, and particle N plus one 251 00:19:26,653 --> 00:19:29,200 is, again, zero, of course. 252 00:19:29,200 --> 00:19:34,000 And if you take N equals two, excuse me? 253 00:19:34,000 --> 00:19:37,247 Yeah, boy, yeah, yeah, yeah, yeah, 254 00:19:37,247 --> 00:19:39,117 the N is a P, right? 255 00:19:39,117 --> 00:19:43,938 Thank you very much. Is that what you were saying? 256 00:19:43,938 --> 00:19:48,465 Yeah, thank you very much. Today is not my day. 257 00:19:48,465 --> 00:19:52,696 Yeah, there is a P. So, for particle number, 258 00:19:52,696 --> 00:19:56,632 when P is zero you see this goes to zero. 259 00:19:56,632 --> 00:20:01,553 And, when P is N plus one, this again goes to zero. 260 00:20:01,553 --> 00:20:07,260 So, you see here that AP2 is now C2 times the sine of 2P pi 261 00:20:07,260 --> 00:20:12,907 divided by N plus one. And, that exactly is this 262 00:20:12,907 --> 00:20:15,425 curve. Notice that if you put in P 263 00:20:15,425 --> 00:20:19,698 equals zero, you get a zero. If you put in P is capital N 264 00:20:19,698 --> 00:20:23,361 plus one, you get a zero. But you will also find, 265 00:20:23,361 --> 00:20:27,863 now, a zero precisely in the middle when P is capital N plus 266 00:20:27,863 --> 00:20:33,110 one divided by two. And so, that is the consequence 267 00:20:33,110 --> 00:20:36,722 of the introduction of this function. 268 00:20:36,722 --> 00:20:41,036 And, of course, the ratios of the individual 269 00:20:41,036 --> 00:20:45,551 P's for a particular mode state, for instance, 270 00:20:45,551 --> 00:20:50,167 mode number one, when the system is oscillating 271 00:20:50,167 --> 00:20:56,187 like this, the ratios of the amplitudes of the individual P's 272 00:20:56,187 --> 00:21:02,041 is then given by this sign. P1 has its own amplitude. 273 00:21:02,041 --> 00:21:06,125 P2 has its own amplitude. P3 has its own amplitude. 274 00:21:06,125 --> 00:21:09,147 So, P1 will be here. P2 will be there. 275 00:21:09,147 --> 00:21:11,924 P3 will be there. P4 will be there, 276 00:21:11,924 --> 00:21:14,865 and so on. And so, then the amplitude 277 00:21:14,865 --> 00:21:18,214 first goes up. And then the amplitude goes 278 00:21:18,214 --> 00:21:21,317 down again. But now comes the question, 279 00:21:21,317 --> 00:21:25,075 what is omega N? What is the frequency omega at 280 00:21:25,075 --> 00:21:28,750 these normal modes N equals one, N equals two, 281 00:21:28,750 --> 00:21:31,119 N equals three, N equals four, 282 00:21:31,119 --> 00:21:37,000 and N cannot go all the way down, then, to capital N. 283 00:21:37,000 --> 00:21:42,230 And for that, we have to return to this 284 00:21:42,230 --> 00:21:47,461 equation. And I'm going to write it now 285 00:21:47,461 --> 00:21:54,619 slightly differently. I'm going to take the [ace?] to 286 00:21:54,619 --> 00:21:59,574 one side. So, I'm going to write down 287 00:21:59,574 --> 00:22:07,971 here AP plus one plus AP minus one, and divide that by AP just 288 00:22:07,971 --> 00:22:14,384 rearranging. And then you will find that it 289 00:22:14,384 --> 00:22:21,153 is minus omega squared plus two omega zero squared divided by 290 00:22:21,153 --> 00:22:24,989 omega squared. Take a look at this, 291 00:22:24,989 --> 00:22:31,758 and convince yourself that this equation is identical to this 292 00:22:31,758 --> 00:22:35,598 one. If you look in French, 293 00:22:35,598 --> 00:22:41,149 French will take you from here one step further, 294 00:22:41,149 --> 00:22:47,645 which is pure trigonometry. And I decided not to go that 295 00:22:47,645 --> 00:22:51,897 route. But you can use the values for 296 00:22:51,897 --> 00:22:56,622 AP that we have defined, namely this one. 297 00:22:56,622 --> 00:23:03,000 And so, you can now massage the trigonometry. 298 00:23:03,000 --> 00:23:07,552 And, you can find that this ratio is very simple. 299 00:23:07,552 --> 00:23:12,390 It's twice the cosine. So, this whole thing is twice 300 00:23:12,390 --> 00:23:15,615 the cosine of N pi over N plus one. 301 00:23:15,615 --> 00:23:19,124 That is correct. N pi over N plus one, 302 00:23:19,124 --> 00:23:24,056 so, there's no physics there. It's purely a matter of 303 00:23:24,056 --> 00:23:27,661 trigonometry. We can now put an N here, 304 00:23:27,661 --> 00:23:33,636 Nancy, because we know now that we are going to get solutions as 305 00:23:33,636 --> 00:23:40,454 a function of the mode number N. And then, with a little bit 306 00:23:40,454 --> 00:23:44,909 more trigonometry, and you really want to check up 307 00:23:44,909 --> 00:23:48,000 on French there, which is page 141, 308 00:23:48,000 --> 00:23:52,454 he then comes up with the normal mode frequencies, 309 00:23:52,454 --> 00:23:57,272 which equals our goal. So, I will give you the result, 310 00:23:57,272 --> 00:24:02,000 but it really is implicit, already in here. 311 00:24:02,000 --> 00:24:08,336 You will get two omega zero. The result is by no means 312 00:24:08,336 --> 00:24:15,989 intuitive times the sign of N pi divided by two times N plus one. 313 00:24:15,989 --> 00:24:21,369 And, of course, I'm going to look through that 314 00:24:21,369 --> 00:24:26,271 result with you. It now looks very opaque. 315 00:24:26,271 --> 00:24:31,293 So, this, then, is the solution to omega N. 316 00:24:31,293 --> 00:24:37,409 This is the solution. I will take one color to show 317 00:24:37,409 --> 00:24:41,230 you how these link. This is the solution for A. 318 00:24:41,230 --> 00:24:45,467 If you know the mode, and you know which particle it 319 00:24:45,467 --> 00:24:50,203 is, and you have specified the amplitude of the sinusoids. 320 00:24:50,203 --> 00:24:54,772 Then this tells you each particle what the amplitude is. 321 00:24:54,772 --> 00:24:58,926 If you know the mode, N, then you know that this is 322 00:24:58,926 --> 00:25:04,658 going to be the frequency. And so, you can write down now 323 00:25:04,658 --> 00:25:09,000 that Y, which is the displacement as a function of 324 00:25:09,000 --> 00:25:12,455 time, A is amplitude. Y is displacement. 325 00:25:12,455 --> 00:25:16,531 Particle P in mode N, and now we can put in the 326 00:25:16,531 --> 00:25:20,784 amplitude that we know. That is the PN amplitude, 327 00:25:20,784 --> 00:25:23,886 APN. And now it's going to oscillate 328 00:25:23,886 --> 00:25:27,696 with cosine omega N times T. And, of course, 329 00:25:27,696 --> 00:25:32,835 you can always add a phase angle depending upon at T equals 330 00:25:32,835 --> 00:25:39,168 zero what the particle is doing. And so, this one gives you the 331 00:25:39,168 --> 00:25:41,481 amplitude. This one gives you the 332 00:25:41,481 --> 00:25:43,289 frequency. And this, then, 333 00:25:43,289 --> 00:25:47,626 is the time dependence of the displacement of particle number 334 00:25:47,626 --> 00:25:49,000 P in mode N. 335 00:25:49,000 --> 00:25:59,000 336 00:25:59,000 --> 00:26:02,967 What I want to do now is to take a specific example which I 337 00:26:02,967 --> 00:26:07,003 also will try to demonstrate, which will give you tremendous 338 00:26:07,003 --> 00:26:09,534 insight. You can actually do it on the 339 00:26:09,534 --> 00:26:12,475 left here because these are all very opaque. 340 00:26:12,475 --> 00:26:16,511 But when you see an example worked out, and you see actually 341 00:26:16,511 --> 00:26:20,000 how it oscillates, then it comes to life. 342 00:26:20,000 --> 00:26:26,000 343 00:26:26,000 --> 00:26:30,695 I'm going to have five beats on a string: one, 344 00:26:30,695 --> 00:26:34,243 two, three, four, five, fixed here, 345 00:26:34,243 --> 00:26:37,686 fixed here, one, two, three, four, 346 00:26:37,686 --> 00:26:39,982 five. The tangent is T. 347 00:26:39,982 --> 00:26:45,930 The mass of each one is N, and the separation between them 348 00:26:45,930 --> 00:26:48,330 is L. And so, N is five. 349 00:26:48,330 --> 00:26:52,400 So, keep in mind that N plus one is six. 350 00:26:52,400 --> 00:26:58,243 The reason why I write that down: because you're going to 351 00:26:58,243 --> 00:27:05,159 need the N plus one. And then, omega zero is the 352 00:27:05,159 --> 00:27:12,180 square root of T divided by ML. So, I'm interested in knowing 353 00:27:12,180 --> 00:27:17,914 what the frequency is in the lowest possible mode, 354 00:27:17,914 --> 00:27:24,000 which is going to resemble something like this. 355 00:27:24,000 --> 00:27:29,226 And so, that frequency, omega one, is then two omega 356 00:27:29,226 --> 00:27:32,607 zero times the sine, N equals one, 357 00:27:32,607 --> 00:27:37,936 capital N plus one is six, 180° divided by 12 is 15°. 358 00:27:37,936 --> 00:27:43,367 So, there's the sine of 15°. I write it now in degrees 359 00:27:43,367 --> 00:27:49,310 because I have a better feeling for degrees than I have for 360 00:27:49,310 --> 00:27:53,000 radians. And that is 0.51. 361 00:27:53,000 --> 00:27:58,000 362 00:27:58,000 --> 00:28:03,217 I now go to omega two. And, I get the same thing, 363 00:28:03,217 --> 00:28:06,586 except I get 30°. So, I get 0.5. 364 00:28:06,586 --> 00:28:11,913 Sorry, I get exactly omega one, omega zero, right, 365 00:28:11,913 --> 00:28:15,391 because the sine 30° is one half. 366 00:28:15,391 --> 00:28:20,826 And, that eats up this one. And, I get omega three. 367 00:28:20,826 --> 00:28:27,021 And, now I get the sine of 45°. And, that is approximately 368 00:28:27,021 --> 00:28:32,239 1.41, not exactly, but as the square root of two, 369 00:28:32,239 --> 00:28:38,000 so that is about 1.41 times omega zero. 370 00:28:38,000 --> 00:28:44,696 Then, I go to omega four. And so, I get 60°. 371 00:28:44,696 --> 00:28:52,795 And that can only be approximated again by about 1.73 372 00:28:52,795 --> 00:28:58,557 omega zero. And then I have omega five 373 00:28:58,557 --> 00:29:06,032 which is the last one. So I get this sign of 75°. 374 00:29:06,032 --> 00:29:10,393 And that, then, becomes 3.73, 375 00:29:10,393 --> 00:29:18,408 no, 1.93 times omega zero. What I want to concentrate on, 376 00:29:18,408 --> 00:29:21,620 because that's part of the demonstration, 377 00:29:21,620 --> 00:29:25,072 is not so much on the meaning of omega zero. 378 00:29:25,072 --> 00:29:29,890 It's just some arbitrary thing that I have called omega zero. 379 00:29:29,890 --> 00:29:34,868 But, I want to concentrate with you on the ratios of the higher 380 00:29:34,868 --> 00:29:41,111 frequencies to the lowest one. And so, I call the lowest one 381 00:29:41,111 --> 00:29:45,296 omega one. I simply call this omega one. 382 00:29:45,296 --> 00:29:51,304 If that one is omega one, then the next one is 1.93 omega 383 00:29:51,304 --> 00:29:55,703 one, again, not exactly but approximately. 384 00:29:55,703 --> 00:30:03,000 It is this one divided by 0.51, the ratio of the frequencies. 385 00:30:03,000 --> 00:30:08,954 And if I take this one and divide it by that one, 386 00:30:08,954 --> 00:30:15,281 then I get 2.73 omega one. Then I take the next one. 387 00:30:15,281 --> 00:30:18,755 I get 3.35. And the last one, 388 00:30:18,755 --> 00:30:24,338 then, is 3.73. So, the bottom line is that the 389 00:30:24,338 --> 00:30:31,657 ratio of these frequencies are not at all very nice numbers, 390 00:30:31,657 --> 00:30:38,047 as you may have expected. But the ratios are quite 391 00:30:38,047 --> 00:30:41,972 bizarre: 1.93 times higher than the lowest one, 392 00:30:41,972 --> 00:30:45,044 2.73 times higher, 3.35 times higher, 393 00:30:45,044 --> 00:30:48,457 and 3.73. And so, the general solution to 394 00:30:48,457 --> 00:30:52,296 that system is, then, the linear superposition 395 00:30:52,296 --> 00:30:56,904 of all these normal modes. That's the general solution. 396 00:30:56,904 --> 00:31:01,000 You give them very modest amplitudes. 397 00:31:01,000 --> 00:31:05,366 And, you can choose the amplitude of each one of them. 398 00:31:05,366 --> 00:31:09,321 That's effective, like, say you were choosing C1, 399 00:31:09,321 --> 00:31:12,781 C2, C3, C4, C5. And, you also can give them 400 00:31:12,781 --> 00:31:15,582 initial velocities, if you want to. 401 00:31:15,582 --> 00:31:19,701 So, at T equals zero, they do not all have to stand 402 00:31:19,701 --> 00:31:22,255 still. You have that choice too, 403 00:31:22,255 --> 00:31:25,633 of course. So, you can change the relative 404 00:31:25,633 --> 00:31:30,000 phase between the five different modes. 405 00:31:30,000 --> 00:31:33,203 What I will do is I want to make life simple. 406 00:31:33,203 --> 00:31:36,479 I will generate all five normal modes for you. 407 00:31:36,479 --> 00:31:40,702 And I will start them off all at zero speed when I show you 408 00:31:40,702 --> 00:31:43,760 the simulation. And, the first one that I'm 409 00:31:43,760 --> 00:31:46,526 going to show you, then, is number one. 410 00:31:46,526 --> 00:31:49,584 And then, I'm going to show you number two. 411 00:31:49,584 --> 00:31:53,079 I want you to appreciate that if I showed you the 412 00:31:53,079 --> 00:31:57,301 superposition of one and two, so I let it oscillate in this 413 00:31:57,301 --> 00:32:00,650 mode and in this mode, I start off at a certain 414 00:32:00,650 --> 00:32:06,478 position of these particles. And they start to oscillate in 415 00:32:06,478 --> 00:32:10,940 this mode in that mode, that the shape that I have will 416 00:32:10,940 --> 00:32:14,988 never ever become the same as it was at time zero. 417 00:32:14,988 --> 00:32:18,871 And, why is that? So, you just let it oscillate. 418 00:32:18,871 --> 00:32:23,829 You can wait 100 billion years, and you'll never see the same 419 00:32:23,829 --> 00:32:26,390 shape. If I start with a certain 420 00:32:26,390 --> 00:32:30,934 shape, and it never ever, ever comes back to that shape, 421 00:32:30,934 --> 00:32:34,500 why is that? Yeah, that's right, 422 00:32:34,500 --> 00:32:37,184 that the sine of 15° is the killer. 423 00:32:37,184 --> 00:32:40,105 That is not the ratio of two integers. 424 00:32:40,105 --> 00:32:44,526 Therefore, you'll never get it back to the same position. 425 00:32:44,526 --> 00:32:48,157 Maybe approximately, but you never get it back. 426 00:32:48,157 --> 00:32:52,578 This demonstration is going to be a cocktail between very 427 00:32:52,578 --> 00:32:57,157 low-tech and very high-tech. And, I will start with myself, 428 00:32:57,157 --> 00:33:01,501 which is very low-tech. And that is this. 429 00:33:01,501 --> 00:33:05,652 I was sitting in my office and I said to myself, 430 00:33:05,652 --> 00:33:10,156 gee, what will I see? So, I took a pencil and a just 431 00:33:10,156 --> 00:33:14,131 sketched very roughly the sinusoid right here. 432 00:33:14,131 --> 00:33:19,430 And, I know according to these solutions, if you accept them, 433 00:33:19,430 --> 00:33:23,316 that these beats, these particles must lie on 434 00:33:23,316 --> 00:33:27,025 that sinusoid. And C1 is then the choice of 435 00:33:27,025 --> 00:33:32,632 that amplitude of that sinusoid. In the second mode, 436 00:33:32,632 --> 00:33:36,500 you pick another value for the amplitude, say, 437 00:33:36,500 --> 00:33:39,421 C2. And then, the beats have to lie 438 00:33:39,421 --> 00:33:41,742 on that sinusoid, and so on. 439 00:33:41,742 --> 00:33:45,695 What you will see, however, is that these beats 440 00:33:45,695 --> 00:33:48,531 are connected with straight wires. 441 00:33:48,531 --> 00:33:51,625 So, you will not see those nice arcs. 442 00:33:51,625 --> 00:33:54,804 What you will see is, of course, this. 443 00:33:54,804 --> 00:33:57,898 The red lines are the actual strings. 444 00:33:57,898 --> 00:34:02,195 And so, for instance, if you go to the second mode, 445 00:34:02,195 --> 00:34:05,890 we call that a second harmonic, if you like, 446 00:34:05,890 --> 00:34:10,789 then notice that this point here and this point here never 447 00:34:10,789 --> 00:34:15,000 reached the amplitude, C2. 448 00:34:15,000 --> 00:34:18,148 The sinusoid does, and the C2 in that equation 449 00:34:18,148 --> 00:34:22,135 does, but those points will never reach that because their 450 00:34:22,135 --> 00:34:25,913 location is such that they never make it to that point, 451 00:34:25,913 --> 00:34:28,082 here. This [won't?] stand still, 452 00:34:28,082 --> 00:34:29,761 then. That was intuitive. 453 00:34:29,761 --> 00:34:33,841 So, we have that here. And if you go to higher 454 00:34:33,841 --> 00:34:37,525 frequencies, particularly the very highest one, 455 00:34:37,525 --> 00:34:42,090 neighboring beats always are out of phase with each other. 456 00:34:42,090 --> 00:34:44,173 You see that: up, down, up, 457 00:34:44,173 --> 00:34:47,376 down, up. And again, this little particle 458 00:34:47,376 --> 00:34:50,019 will never reach the amplitude C5. 459 00:34:50,019 --> 00:34:52,582 This one does. This one does not. 460 00:34:52,582 --> 00:34:55,465 This one does not. This one does not. 461 00:34:55,465 --> 00:34:59,149 But this one does. And so, as I'm going to show 462 00:34:59,149 --> 00:35:03,233 you this simulation, we will keep this going because 463 00:35:03,233 --> 00:35:09,000 it will be great to anticipate what we may be seeing. 464 00:35:09,000 --> 00:35:13,896 So, the first thing that I'm going to show you is one 465 00:35:13,896 --> 00:35:18,605 complete oscillation in the normal mode number one, 466 00:35:18,605 --> 00:35:23,219 which I have set to be 15 seconds with the help of 467 00:35:23,219 --> 00:35:27,457 [UNINTELLIGIBLE], who has guided me greatly in 468 00:35:27,457 --> 00:35:32,807 this demonstration. So, let me first of all give us 469 00:35:32,807 --> 00:35:37,971 the right light conditions. And, now I will start the last 470 00:35:37,971 --> 00:35:41,594 15 seconds. I have given amplitude a two, 471 00:35:41,594 --> 00:35:44,764 which is very large. And, of course, 472 00:35:44,764 --> 00:35:48,478 that's unrealistic, these high amplitudes. 473 00:35:48,478 --> 00:35:53,097 But I want you to see the relative position of these 474 00:35:53,097 --> 00:35:55,000 particles. 475 00:35:55,000 --> 00:36:01,000 476 00:36:01,000 --> 00:36:03,351 There we go. Now, you will see, 477 00:36:03,351 --> 00:36:06,408 they will make one complete oscillation. 478 00:36:06,408 --> 00:36:09,152 And that, of course, is no surprise. 479 00:36:09,152 --> 00:36:12,287 If I clicked only once, it will stop now. 480 00:36:12,287 --> 00:36:15,971 And that will be. If I click twice it will start 481 00:36:15,971 --> 00:36:18,558 again. No, thank goodness only did 482 00:36:18,558 --> 00:36:21,301 once. Now, I'm going to show you the 483 00:36:21,301 --> 00:36:23,888 second one. And I give it the same 484 00:36:23,888 --> 00:36:26,396 amplitude. So, C2, I give it two. 485 00:36:26,396 --> 00:36:30,629 And, I want you to count how many oscillations it makes 486 00:36:30,629 --> 00:36:35,218 before comes to a stop. It will, again, 487 00:36:35,218 --> 00:36:39,362 be exactly 15 seconds. And so, we will have to agree 488 00:36:39,362 --> 00:36:43,912 that the number of oscillations that it makes is now 1.9. 489 00:36:43,912 --> 00:36:47,650 It misses the two. It will knock it back to the 490 00:36:47,650 --> 00:36:52,281 two complete oscillations. And so, you're going to look at 491 00:36:52,281 --> 00:36:55,287 this mode. So these two particles will 492 00:36:55,287 --> 00:36:57,400 never reach the value, two. 493 00:36:57,400 --> 00:37:02,631 Two is marked here. And it will go down, 494 00:37:02,631 --> 00:37:10,263 up, down, and it stops very just short of two oscillations. 495 00:37:10,263 --> 00:37:16,578 So we'll do that now. So, I make amplitude of the 496 00:37:16,578 --> 00:37:22,631 first one zero. And now we get the amplitude of 497 00:37:22,631 --> 00:37:26,842 number two. And, I make that two. 498 00:37:26,842 --> 00:37:30,526 And, there we go. Now, count. 499 00:37:30,526 --> 00:37:37,400 That's one oscillation. And it will stop just short of 500 00:37:37,400 --> 00:37:39,320 two. You can't tell that, 501 00:37:39,320 --> 00:37:40,440 of course. Why? 502 00:37:40,440 --> 00:37:43,480 Because you don't have that resolution. 503 00:37:43,480 --> 00:37:46,039 But it stopped just short of two. 504 00:37:46,039 --> 00:37:50,440 Now we're going to this one, again, give it amplitude of 505 00:37:50,440 --> 00:37:53,079 two. And now we're going to count. 506 00:37:53,079 --> 00:37:58,039 And you will definitely be able to see that it just misses 2.75 507 00:37:58,039 --> 00:38:03,000 because 2.75 is something that you can eyeball. 508 00:38:03,000 --> 00:38:07,410 So, we're going to number three now. 509 00:38:07,410 --> 00:38:14,719 Number three already has two interesting points which don't 510 00:38:14,719 --> 00:38:20,012 move at all. This point won't move and that 511 00:38:20,012 --> 00:38:25,682 point won't move. They will reach the plus two 512 00:38:25,682 --> 00:38:30,723 and the plus two. Zero, and we get a two. 513 00:38:30,723 --> 00:38:35,246 And there we go. That's one. 514 00:38:35,246 --> 00:38:38,987 That's two. That's two and a half. 515 00:38:38,987 --> 00:38:43,862 And that's 2.73. You see, it's just short of 516 00:38:43,862 --> 00:38:47,149 2.75. So, that is this number. 517 00:38:47,149 --> 00:38:52,024 Now we're going to number four. Number four, 518 00:38:52,024 --> 00:38:57,919 this one will stand still. And others do not have the 519 00:38:57,919 --> 00:39:02,000 maximum amplitude of two. 520 00:39:02,000 --> 00:39:09,000 521 00:39:09,000 --> 00:39:19,388 So, again, count. There we go. 522 00:39:19,388 --> 00:39:33,000 One, two, three, and it stops there. 523 00:39:33,000 --> 00:39:36,251 It stops at 3.35. So now, the last one before 524 00:39:36,251 --> 00:39:39,724 we're going to cocktail them, there is this one. 525 00:39:39,724 --> 00:39:42,532 So, again, this point will be plus two. 526 00:39:42,532 --> 00:39:44,970 So this one will never reach that. 527 00:39:44,970 --> 00:39:48,000 This one will reach the plus two. 528 00:39:48,000 --> 00:39:57,000 529 00:39:57,000 --> 00:39:58,000 OK. 530 00:39:58,000 --> 00:40:04,000 531 00:40:04,000 --> 00:40:08,175 One, see how it doesn't make it to two? 532 00:40:08,175 --> 00:40:14,109 It doesn't get that high. Two, three, three and a half, 533 00:40:14,109 --> 00:40:17,626 and it stops. Just under the 3.5, 534 00:40:17,626 --> 00:40:21,362 it stops. Sorry, just over the 3.5, 535 00:40:21,362 --> 00:40:25,868 it stops short. This would have been 3.75. 536 00:40:25,868 --> 00:40:31,692 It's just under there. Now I will cocktail for you two 537 00:40:31,692 --> 00:40:35,862 and four. If I cocktail two and four, 538 00:40:35,862 --> 00:40:38,827 then this point would stand still, and this, 539 00:40:38,827 --> 00:40:41,310 of course is going to be starting up. 540 00:40:41,310 --> 00:40:45,172 And, this will start up because I give them both the same 541 00:40:45,172 --> 00:40:47,931 amplitude. I will not go to plus two now, 542 00:40:47,931 --> 00:40:51,793 but I will make it plus one. Otherwise, you get two large 543 00:40:51,793 --> 00:40:56,000 values and it becomes a little bit unrealistic in what you see 544 00:40:56,000 --> 00:40:58,344 there. So, I'm going to give number 545 00:40:58,344 --> 00:41:00,206 two a one. That's amplitude, 546 00:41:00,206 --> 00:41:04,275 and I'm going to give number five zero, and number four also 547 00:41:04,275 --> 00:41:07,784 a one. And now, already you're going 548 00:41:07,784 --> 00:41:11,661 to begin to see that the motion that you're going to see becomes 549 00:41:11,661 --> 00:41:14,984 already sort of a little bit chaotic, a little erratic. 550 00:41:14,984 --> 00:41:17,938 So, it's a superposition now of two normal modes. 551 00:41:17,938 --> 00:41:21,753 This one, which I start off at a one here, and this one which I 552 00:41:21,753 --> 00:41:24,707 start off as a one here, and at T equals zero are 553 00:41:24,707 --> 00:41:26,676 released in both with zero speed. 554 00:41:26,676 --> 00:41:30,000 So too I have the one in there? Yes I do. 555 00:41:30,000 --> 00:41:34,012 There we go. This motion is already not so 556 00:41:34,012 --> 00:41:37,339 predictable. But it's still sort of 557 00:41:37,339 --> 00:41:43,113 symmetric for obvious reasons because this one stands still. 558 00:41:43,113 --> 00:41:47,321 And now I want to do is start them all five. 559 00:41:47,321 --> 00:41:52,018 Now, if we start all five, the start will be very 560 00:41:52,018 --> 00:41:54,366 asymmetric because, look. 561 00:41:54,366 --> 00:41:59,357 Particle number one: positive because I set them all 562 00:41:59,357 --> 00:42:03,190 off positive. Positive, positive, 563 00:42:03,190 --> 00:42:04,944 positive, positive, positive. 564 00:42:04,944 --> 00:42:08,639 So it will start up very high. Now, look at the number five. 565 00:42:08,639 --> 00:42:11,020 Positive, negative, positive, negative, 566 00:42:11,020 --> 00:42:13,525 positive. So, number five will start very 567 00:42:13,525 --> 00:42:16,031 low, and number one will start very high. 568 00:42:16,031 --> 00:42:19,476 And then, when it starts to oscillate, it will take more 569 00:42:19,476 --> 00:42:23,234 than the age of the universe to come back to that same shape. 570 00:42:23,234 --> 00:42:26,742 But it is extremely erratic. You and I, no one can really 571 00:42:26,742 --> 00:42:30,000 relate any more to what's going on. 572 00:42:30,000 --> 00:42:35,008 And it is even possible to imagine that the motion in a way 573 00:42:35,008 --> 00:42:38,548 is very simple, namely, a superposition of 574 00:42:38,548 --> 00:42:42,520 five, very well behaving normal mode solutions. 575 00:42:42,520 --> 00:42:47,615 It is a linear superposition of five very simple normal mode 576 00:42:47,615 --> 00:42:50,810 solutions. But the net result is total 577 00:42:50,810 --> 00:42:55,387 utter chaos, at least that's the way it appears to us. 578 00:42:55,387 --> 00:43:01,000 But, it can be dissected into five very simple modes. 579 00:43:01,000 --> 00:43:12,000 580 00:43:12,000 --> 00:43:15,193 So, these were transverse motions. 581 00:43:15,193 --> 00:43:19,838 And the same idea holds for longitudinal motions. 582 00:43:19,838 --> 00:43:25,258 So, you can have five beats with six springs and then the 583 00:43:25,258 --> 00:43:28,354 oscillation is in this direction. 584 00:43:28,354 --> 00:43:33,000 We call that a longitudinal oscillation. 585 00:43:33,000 --> 00:43:35,406 In this case, the displacement is 586 00:43:35,406 --> 00:43:37,812 perpendicular to the oscillators. 587 00:43:37,812 --> 00:43:40,819 We call that transverse. But the algebra, 588 00:43:40,819 --> 00:43:43,225 as you can imagine, is identical, 589 00:43:43,225 --> 00:43:47,736 except that the displacement are, then, not in this direction 590 00:43:47,736 --> 00:43:51,270 for the longitudinal one, but in this direction. 591 00:43:51,270 --> 00:43:55,556 We will shortly answer the domain of waves to make you see 592 00:43:55,556 --> 00:43:58,263 the idea. The big difference would be 593 00:43:58,263 --> 00:44:01,195 transverse waves and longitudinal waves, 594 00:44:01,195 --> 00:44:04,525 sound. This is a pressure wave. 595 00:44:04,525 --> 00:44:07,638 My direction to you, the air, the pressure wave is 596 00:44:07,638 --> 00:44:10,180 doing this. So, the air is oscillating in 597 00:44:10,180 --> 00:44:12,214 the same direction that it moves. 598 00:44:12,214 --> 00:44:15,709 That is a longitudinal wave. But, this is a nice moment. 599 00:44:15,709 --> 00:44:19,458 Will break for five minutes and will start exactly 5 minutes 600 00:44:19,458 --> 00:44:22,000 from now. [SOUND OFF/THEN ON] 601 00:44:22,000 --> 00:44:47,000 602 00:44:47,000 --> 00:44:50,594 All right. Thank you very much for the 603 00:44:50,594 --> 00:44:53,800 performance. That was prearranged, 604 00:44:53,800 --> 00:44:57,588 by the way. So, we are now ready to make 605 00:44:57,588 --> 00:45:04,000 the step to continue this medium whereby N goes to infinity. 606 00:45:04,000 --> 00:45:08,188 Well, you can argue that it goes to as many items as we can 607 00:45:08,188 --> 00:45:11,294 line up on a string. It's close to infinity. 608 00:45:11,294 --> 00:45:14,761 And, it should not come as a surprise, of course, 609 00:45:14,761 --> 00:45:18,516 that now you are going to get that the entire string, 610 00:45:18,516 --> 00:45:22,127 which is now continuous mass, so you no longer have 611 00:45:22,127 --> 00:45:26,750 individual beats that the entire string is now going to oscillate 612 00:45:26,750 --> 00:45:30,000 as a sinusoid in its lowest mode. 613 00:45:30,000 --> 00:45:34,533 So this is N equals one. And then it's going to 614 00:45:34,533 --> 00:45:38,180 oscillate like this. For N equals two, 615 00:45:38,180 --> 00:45:41,334 N equals two. And N equals three, 616 00:45:41,334 --> 00:45:45,572 going up like this, and this goes like this. 617 00:45:45,572 --> 00:45:48,725 It should not come as a surprise. 618 00:45:48,725 --> 00:45:54,639 I will not pursue that today. We will get back to that later. 619 00:45:54,639 --> 00:45:59,862 What I want to mention, though, what is interesting is 620 00:45:59,862 --> 00:46:05,579 that the ratio of these normal mode frequencies will now be 621 00:46:05,579 --> 00:46:11,000 one, two, three, four, five, and so on. 622 00:46:11,000 --> 00:46:15,259 So, now you get that second mode is twice the frequency of 623 00:46:15,259 --> 00:46:18,697 the first, which is what we didn't have there.. 624 00:46:18,697 --> 00:46:22,060 That's the big difference between number of N, 625 00:46:22,060 --> 00:46:25,572 which is finite, and an infinity number of these 626 00:46:25,572 --> 00:46:28,562 oscillators. What I want to pursue today, 627 00:46:28,562 --> 00:46:32,000 will get back to this in the future. 628 00:46:32,000 --> 00:46:37,151 What I want to do today is to generate a disturbance in a 629 00:46:37,151 --> 00:46:42,578 medium, which has an infinite number of coupled oscillators, 630 00:46:42,578 --> 00:46:46,074 which is a string to generate in there. 631 00:46:46,074 --> 00:46:49,753 So, I take a string and I wiggle the end. 632 00:46:49,753 --> 00:46:55,272 And then, I want to evaluate with you what's going to happen. 633 00:46:55,272 --> 00:46:58,952 So, for this, I need some assistance from 634 00:46:58,952 --> 00:47:03,000 someone. Nicole, would you mind? 635 00:47:03,000 --> 00:47:07,394 Hold those firmly in your hand. Now, most of you may think that 636 00:47:07,394 --> 00:47:10,017 this is a spring with a P as in Peter. 637 00:47:10,017 --> 00:47:12,994 But, no, it is a string with a T as in Tom. 638 00:47:12,994 --> 00:47:16,113 You will see that. I'm going to use this as a 639 00:47:16,113 --> 00:47:18,807 string. I'm going to put tension on it, 640 00:47:18,807 --> 00:47:22,281 T, which is what we needed, also for the N coupled 641 00:47:22,281 --> 00:47:26,392 oscillators, and the amount of mass that we have we express 642 00:47:26,392 --> 00:47:31,000 that normally in terms of the mass per unit length. 643 00:47:31,000 --> 00:47:35,185 Remember, in the other case, we had little M divided by L. 644 00:47:35,185 --> 00:47:37,534 Well, we call that now [gnu/mu?]. 645 00:47:37,534 --> 00:47:41,059 So, that's how much mass per unit length we have. 646 00:47:41,059 --> 00:47:44,510 And what I want to do now is just shake my hand. 647 00:47:44,510 --> 00:47:46,933 And then you tell me what you see. 648 00:47:46,933 --> 00:47:48,181 Ready? Here we go. 649 00:47:48,181 --> 00:47:49,723 Are you ready, Nicole? 650 00:47:49,723 --> 00:47:52,000 What did you see? 651 00:47:52,000 --> 00:47:59,000 652 00:47:59,000 --> 00:48:03,417 Just some in what you see right after I do this. 653 00:48:03,417 --> 00:48:07,083 What did you see right after I did this? 654 00:48:07,083 --> 00:48:11,971 The disturbance moved. That's number one that we have 655 00:48:11,971 --> 00:48:14,885 to understand. Why does it move? 656 00:48:14,885 --> 00:48:18,550 Now, look what happens at Nicole's side. 657 00:48:18,550 --> 00:48:21,934 I generate a pose which is like this. 658 00:48:21,934 --> 00:48:25,318 I would call that a mountain for now. 659 00:48:25,318 --> 00:48:30,958 And, only look at the moment that the mountain reaches her in 660 00:48:30,958 --> 00:48:37,851 something comes back to me. And then, stop looking because 661 00:48:37,851 --> 00:48:41,368 things begin to wander back and forth. 662 00:48:41,368 --> 00:48:44,505 And tell me what comes back at me. 663 00:48:44,505 --> 00:48:48,117 I'm going to send a mountain to Nicole. 664 00:48:48,117 --> 00:48:50,969 What came back at me? A valley. 665 00:48:50,969 --> 00:48:54,771 Now I'm going to send a valley to Nicole. 666 00:48:54,771 --> 00:48:57,813 What do you think is coming back? 667 00:48:57,813 --> 00:49:02,031 Very good. I don't know why it is. 668 00:49:02,031 --> 00:49:05,122 It's very hard to generate a valley. 669 00:49:05,122 --> 00:49:09,184 Let me do a mountain again. This is a mountain. 670 00:49:09,184 --> 00:49:13,246 That comes back as a valley. I'll try a valley. 671 00:49:13,246 --> 00:49:17,485 OK, I'll try to do a valley. So I go down and up. 672 00:49:17,485 --> 00:49:22,519 Yeah, that was a good one. Well, because if you it worked. 673 00:49:22,519 --> 00:49:26,051 Thank you very much. You did a great job. 674 00:49:26,051 --> 00:49:30,732 So, now we have to understand two things, and that is, 675 00:49:30,732 --> 00:49:37,157 why does it propagate? And why does a mountain come 676 00:49:37,157 --> 00:49:42,571 back as a valley, and why does a valley come back 677 00:49:42,571 --> 00:49:46,180 as a mountain? Continuous medium: 678 00:49:46,180 --> 00:49:50,466 infinite number of coupled oscillators. 679 00:49:50,466 --> 00:49:54,751 I start here with a piece of that rope. 680 00:49:54,751 --> 00:49:58,135 That's called this position, X. 681 00:49:58,135 --> 00:50:04,000 And, I call this position X plus delta X. 682 00:50:04,000 --> 00:50:09,710 I call this Y. I call this angle theta plus 683 00:50:09,710 --> 00:50:15,285 delta theta. And, I call this angle theta. 684 00:50:15,285 --> 00:50:19,771 We have a tension, T, on the line, 685 00:50:19,771 --> 00:50:24,394 and mu is the mass per unit length. 686 00:50:24,394 --> 00:50:30,649 So, you tell me what the mass per one meter is, 687 00:50:30,649 --> 00:50:36,344 and I know what mu is. If the length, 688 00:50:36,344 --> 00:50:41,625 the mass per unit length. Well, if our displacements are 689 00:50:41,625 --> 00:50:45,754 not absurdly high, then we can make the same 690 00:50:45,754 --> 00:50:51,516 assumption that we made with the beated (sic) string that the 691 00:50:51,516 --> 00:50:54,685 tension is the same on both sides. 692 00:50:54,685 --> 00:50:59,390 It's an approximation. But, for modest amplitudes, 693 00:50:59,390 --> 00:51:04,000 it's a very reasonable approximation. 694 00:51:04,000 --> 00:51:07,393 So, we have a T here. And, we have a T there. 695 00:51:07,393 --> 00:51:10,092 And they are, then, to a decent to a 696 00:51:10,092 --> 00:51:12,714 reasonable approximation, the same. 697 00:51:12,714 --> 00:51:16,262 Just like with the beats for modest amplitudes, 698 00:51:16,262 --> 00:51:20,426 we don't have to worry about motion in the X direction. 699 00:51:20,426 --> 00:51:25,053 The only thing that matters is the motion in the Y direction. 700 00:51:25,053 --> 00:51:28,910 So, I will concentrate exclusively on the motion in 701 00:51:28,910 --> 00:51:31,840 this direction, which drives it back to 702 00:51:31,840 --> 00:51:37,529 equilibrium. And so, F of Y on this segment 703 00:51:37,529 --> 00:51:44,470 is, then, minus T sine theta because this component is down, 704 00:51:44,470 --> 00:51:51,529 minus T sine theta plus T sine theta plus delta theta because 705 00:51:51,529 --> 00:51:57,764 this component in the Y direction is driving away from 706 00:51:57,764 --> 00:52:01,740 equilibrium. But for small angles, 707 00:52:01,740 --> 00:52:05,553 and we have to have small angles, otherwise our 708 00:52:05,553 --> 00:52:09,283 assumptions are wrong, these are not the same. 709 00:52:09,283 --> 00:52:13,096 For small angles, the sine of theta is the same 710 00:52:13,096 --> 00:52:17,157 as theta in radians. And so, this becomes a theta. 711 00:52:17,157 --> 00:52:22,213 This becomes theta plus delta theta, and so this thing becomes 712 00:52:22,213 --> 00:52:25,612 T delta theta. That's an approximation for 713 00:52:25,612 --> 00:52:28,844 small angles. Now, I will apply Newton's 714 00:52:28,844 --> 00:52:33,770 Second Law. The amount of mass that is in 715 00:52:33,770 --> 00:52:37,592 here is DM. And, I will calculate shortly 716 00:52:37,592 --> 00:52:41,127 what DM is. It's a little bit of mass. 717 00:52:41,127 --> 00:52:46,382 We're going to make DX go to zero, infinitesimally small 718 00:52:46,382 --> 00:52:50,777 amount of mass. And so that mass times Y double 719 00:52:50,777 --> 00:52:55,554 dot must now be this force that we just calculated. 720 00:52:55,554 --> 00:53:01,000 So, it must be T delta theta. But, what is DM? 721 00:53:01,000 --> 00:53:07,319 Well, we know that the length of the string is delta X. 722 00:53:07,319 --> 00:53:14,691 So, DM must be delta X times mu because mu is the amount of mass 723 00:53:14,691 --> 00:53:19,957 per unit length. And, if my length is delta X, 724 00:53:19,957 --> 00:53:25,691 then DM is mu delta X. So, I can write this now as 725 00:53:25,691 --> 00:53:34,000 delta X times mu times Y double dot equals T times delta theta. 726 00:53:34,000 --> 00:53:41,393 We are getting there. Now, since we are in a limiting 727 00:53:41,393 --> 00:53:46,938 case, we are going to make delta X zero. 728 00:53:46,938 --> 00:53:52,625 The tangent of theta, so that's becoming, 729 00:53:52,625 --> 00:53:59,876 then, this direction, the tangent of theta is DY DX, 730 00:53:59,876 --> 00:54:04,000 right? That is DY DX. 731 00:54:04,000 --> 00:54:08,623 And the reason why I use partial derivatives is that I 732 00:54:08,623 --> 00:54:11,763 think of it as the time not changing. 733 00:54:11,763 --> 00:54:14,641 Any moment in time, this is DY DX. 734 00:54:14,641 --> 00:54:19,613 That's the only justification for the partial derivatives. 735 00:54:19,613 --> 00:54:24,498 I take the derivatives on the side and on this side in X. 736 00:54:24,498 --> 00:54:28,336 So, the left side, I take D tangent theta DX, 737 00:54:28,336 --> 00:54:34,154 and I do it on the right side. Now, the derivatives of the 738 00:54:34,154 --> 00:54:39,102 tangent of theta of the function is one over the cosine squared 739 00:54:39,102 --> 00:54:42,134 of theta. That can't take you more than 740 00:54:42,134 --> 00:54:46,124 20 seconds to confirm that. You can do that in many 741 00:54:46,124 --> 00:54:49,635 different ways. So, this is the derivative of 742 00:54:49,635 --> 00:54:52,827 the function itself. And then, of course, 743 00:54:52,827 --> 00:54:57,057 I have to multiply it by D theta DX because I take the 744 00:54:57,057 --> 00:55:01,127 whole function derivative in DX. And so, here I get, 745 00:55:01,127 --> 00:55:06,551 then, D2Y DX squared. But, for small angle 746 00:55:06,551 --> 00:55:12,292 approximation, cosine squared of theta is one. 747 00:55:12,292 --> 00:55:19,181 And so, I'm going to substitute now this result into my 748 00:55:19,181 --> 00:55:25,432 differential equation. I read this as delta theta, 749 00:55:25,432 --> 00:55:30,917 which is here. And I read this in my mind as 750 00:55:30,917 --> 00:55:37,463 delta X, which is here. Now, mathematicians would 751 00:55:37,463 --> 00:55:42,686 probably never do that, but physicists have no problem 752 00:55:42,686 --> 00:55:46,233 with that. So I'm going to write now, 753 00:55:46,233 --> 00:55:50,667 here, mu times delta X. And here, I write D2Y. 754 00:55:50,667 --> 00:55:54,215 DT squared. I use partial derivatives 755 00:55:54,215 --> 00:55:59,635 because I'm not changing X. That's the justification for 756 00:55:59,635 --> 00:56:03,746 the partials. And now, I get T. 757 00:56:03,746 --> 00:56:09,397 And now, this delta theta, I'm going to write [whether?] 758 00:56:09,397 --> 00:56:14,328 this times delta X. So, now you get delta X times 759 00:56:14,328 --> 00:56:18,746 D2Y DX squared. And, now I'm doing something 760 00:56:18,746 --> 00:56:22,239 that mathematicians would never do. 761 00:56:22,239 --> 00:56:25,424 I'm going to divide out delta X. 762 00:56:25,424 --> 00:56:32,000 Don't tell your 18.0 whatever people that I did that. 763 00:56:32,000 --> 00:56:38,460 So, now what you have is that mu divided by T times D2Y DT 764 00:56:38,460 --> 00:56:44,126 squared, constant value of X is now D2Y DX squared. 765 00:56:44,126 --> 00:56:49,793 And, believe it or not, this is a big moment in our 766 00:56:49,793 --> 00:56:53,873 life. You have here with differential 767 00:56:53,873 --> 00:56:58,973 equations of Y, which is a function of X and T 768 00:56:58,973 --> 00:57:06,000 whereby here you take the double derivative in time. 769 00:57:06,000 --> 00:57:12,281 And here you take the double derivative in space in location. 770 00:57:12,281 --> 00:57:18,249 What is a possible solution to this differential equation? 771 00:57:18,249 --> 00:57:22,018 You can just see it by looking at it. 772 00:57:22,018 --> 00:57:26,729 You immediately see what the solution must be. 773 00:57:26,729 --> 00:57:30,916 Any function, any single valued function, 774 00:57:30,916 --> 00:57:37,317 you can come up with any one. I don't care with which one, 775 00:57:37,317 --> 00:57:41,953 any single valued function of X, plus or minus a constant 776 00:57:41,953 --> 00:57:45,844 times T will satisfy this differential equation. 777 00:57:45,844 --> 00:57:49,735 Just look at it. You can see immediately that it 778 00:57:49,735 --> 00:57:52,632 works. Take the second derivative in 779 00:57:52,632 --> 00:57:54,950 time. You get a C square out, 780 00:57:54,950 --> 00:58:00,000 and you get the second derivative of the function. 781 00:58:00,000 --> 00:58:05,455 Take the second derivative in X; you would only get the second 782 00:58:05,455 --> 00:58:09,211 derivative of the function, and that's all. 783 00:58:09,211 --> 00:58:14,577 So, all it requires is that C is the square root of T divided 784 00:58:14,577 --> 00:58:17,617 by mu. Then, I'll bet you a month's 785 00:58:17,617 --> 00:58:22,536 salary that any single valued function will satisfy this 786 00:58:22,536 --> 00:58:27,097 differential equation. What is the dimension of that 787 00:58:27,097 --> 00:58:29,869 C? What is the dimension of that 788 00:58:29,869 --> 00:58:36,146 C, meters per second? It's a velocity because if I 789 00:58:36,146 --> 00:58:41,426 have apples here, I must also have apples there. 790 00:58:41,426 --> 00:58:47,943 And so, this can only be an apple if C has the dimension of 791 00:58:47,943 --> 00:58:50,752 a velocity. So, therefore, 792 00:58:50,752 --> 00:58:56,146 you might as well write this as plus or minus VT. 793 00:58:56,146 --> 00:59:03,000 And, you might as well write V for here, a velocity. 794 00:59:03,000 --> 00:59:09,722 And we might as well change now this differential equation in a 795 00:59:09,722 --> 00:59:15,144 way more uniform way, which is what I'm going to do 796 00:59:15,144 --> 00:59:21,975 now, which is one over V squared times D2Y DT squared equals D2Y 797 00:59:21,975 --> 00:59:26,313 DX squared. And, this equation is what is 798 00:59:26,313 --> 00:59:30,325 generally called the [rate?] equation. 799 00:59:30,325 --> 00:59:36,831 It will be with us until the end of the course until death do 800 00:59:36,831 --> 00:59:41,430 us part. It is really a big moment 801 00:59:41,430 --> 00:59:47,263 because you're going to see this equation many times for many 802 00:59:47,263 --> 00:59:51,930 different systems. But now you have seen it being 803 00:59:51,930 --> 00:59:55,333 derived for this very specific case. 804 00:59:55,333 --> 1:00:00,000 Let's now evaluate the meaning of that V. 805 1:00:00,000 --> 1:00:03,666 Well, if I have, here, X, and here, 806 1:00:03,666 --> 1:00:07,225 Y, and I picture [us?] a function. 807 1:00:07,225 --> 1:00:11,539 It could be a sine. It could be a cosine. 808 1:00:11,539 --> 1:00:15,745 I pick one that is way more imaginative. 809 1:00:15,745 --> 1:00:20,705 I pick this one. [WHISTLES] That's my function. 810 1:00:20,705 --> 1:00:24,264 It has to be single value, though. 811 1:00:24,264 --> 1:00:29,441 You have to be careful. It must be single valued. 812 1:00:29,441 --> 1:00:34,727 You cannot go back. That's my function. 813 1:00:34,727 --> 1:00:39,454 And so, that's my function, F, at time T equals zero. 814 1:00:39,454 --> 1:00:44,636 Let us take for V always a positive number for simplicity. 815 1:00:44,636 --> 1:00:47,454 I'm going to call it even speed. 816 1:00:47,454 --> 1:00:50,272 Speed is always positive, right? 817 1:00:50,272 --> 1:00:54,909 And, I want to know now, if I look a little later in 818 1:00:54,909 --> 1:01:00,363 time when there is a minus sign here, what the function looks 819 1:01:00,363 --> 1:01:03,836 like. So, at T equals zero, 820 1:01:03,836 --> 1:01:07,508 I gave it to you. What would it look like a 821 1:01:07,508 --> 1:01:12,316 little bit later in time if there is a minus sign there? 822 1:01:12,316 --> 1:01:15,901 Any suggestions? The function has shifted? 823 1:01:15,901 --> 1:01:18,786 In what direction? Use your hands. 824 1:01:18,786 --> 1:01:21,672 Who thinks it's in this direction? 825 1:01:21,672 --> 1:01:24,557 Who thinks it's in this direction? 826 1:01:24,557 --> 1:01:27,442 Very good. It's in this direction. 827 1:01:27,442 --> 1:01:34,000 So, you'll see a little later in time, you will see it here. 828 1:01:34,000 --> 1:01:38,428 And, what is it doing? It is moving with speed V in 829 1:01:38,428 --> 1:01:42,325 that direction. Now, we are going to evaluate 830 1:01:42,325 --> 1:01:45,071 the plus sign. What will happen? 831 1:01:45,071 --> 1:01:49,500 We now look at the function a little later in time. 832 1:01:49,500 --> 1:01:54,282 A little later in time, it has moved in this direction. 833 1:01:54,282 --> 1:01:57,737 And it's moving [UNINTELLIGIBLE] in this 834 1:01:57,737 --> 1:02:01,191 direction. So, now, you can look through 835 1:02:01,191 --> 1:02:06,833 the meaning of this equation. You now understand why, 836 1:02:06,833 --> 1:02:10,739 when I wiggled here, why the string had no choice. 837 1:02:10,739 --> 1:02:14,565 It must propagate that function that I generated. 838 1:02:14,565 --> 1:02:19,188 And it must propagate that with the speed, square root of T 839 1:02:19,188 --> 1:02:22,137 divided by mu. We derived the speed of 840 1:02:22,137 --> 1:02:26,840 propagation for that string. Mu is the mass per unit length. 841 1:02:26,840 --> 1:02:29,391 T is the tension. If I asked you, 842 1:02:29,391 --> 1:02:33,775 is it obvious that the higher tension gives you a higher 843 1:02:33,775 --> 1:02:38,000 speed, is it completely obvious to me? 844 1:02:38,000 --> 1:02:40,892 Sort of, not quite. But I accept that. 845 1:02:40,892 --> 1:02:45,114 Is it obvious if I made mu large, that I make it a very 846 1:02:45,114 --> 1:02:49,257 thick, very heavy per meter, that propagation speed is 847 1:02:49,257 --> 1:02:50,664 lower? Yeah, maybe. 848 1:02:50,664 --> 1:02:54,104 Now that I know the answer I would say, yeah, 849 1:02:54,104 --> 1:02:57,622 that's quite obvious. But it's not so trivial. 850 1:02:57,622 --> 1:03:02,000 So, in any case, we have derived two things. 851 1:03:02,000 --> 1:03:05,689 We have derived that there is such a thing as speed. 852 1:03:05,689 --> 1:03:08,656 But we even have derived the speed itself, 853 1:03:08,656 --> 1:03:12,346 the square root of T over mu. So, if we had done the 854 1:03:12,346 --> 1:03:16,832 experiment again with a higher tension, then the [pulse?] would 855 1:03:16,832 --> 1:03:20,087 have moved faster. But now, there is something 856 1:03:20,087 --> 1:03:23,994 else that we had to explain. Why on Earth is a mountain 857 1:03:23,994 --> 1:03:27,974 coming back as a valley? And why is a valley coming back 858 1:03:27,974 --> 1:03:31,478 as a mountain? And that, now, 859 1:03:31,478 --> 1:03:35,280 is the result of boundary conditions. 860 1:03:35,280 --> 1:03:40,772 Some people who have lectured 8.03 make a very simple 861 1:03:40,772 --> 1:03:45,102 statement. They say 8.03 is only about two 862 1:03:45,102 --> 1:03:49,854 things: this equation and boundary conditions. 863 1:03:49,854 --> 1:03:54,607 And all the rest follows. It's quite accurate. 864 1:03:54,607 --> 1:04:02,000 So, we have here the string that Nicole and I were holding. 865 1:04:02,000 --> 1:04:08,054 And, here is the answer. That's where Nicole was. 866 1:04:08,054 --> 1:04:11,963 I hope I spelled that correctly. 867 1:04:11,963 --> 1:04:18,774 And we know that that end must stay fixed, cannot move. 868 1:04:18,774 --> 1:04:22,810 I'll put the line a little lower. 869 1:04:22,810 --> 1:04:26,972 I'll put it here. This is the end. 870 1:04:26,972 --> 1:04:32,018 And, my pulse came in. This is the pulse. 871 1:04:32,018 --> 1:04:39,585 And, let us evaluate the moment in time that this part of the 872 1:04:39,585 --> 1:04:46,000 pulse reaches Nicole. Are you ready for that? 873 1:04:46,000 --> 1:04:51,000 874 1:04:51,000 --> 1:04:56,000 So, this part is here. 875 1:04:56,000 --> 1:05:01,000 876 1:05:01,000 --> 1:05:05,673 And the part that, yeah, maybe it's in heaven, 877 1:05:05,673 --> 1:05:09,516 is here. I have to make these a little 878 1:05:09,516 --> 1:05:15,229 steeper to make it look alive, make it a little steeper. 879 1:05:15,229 --> 1:05:19,695 And this, yeah, who knows what happened with 880 1:05:19,695 --> 1:05:23,434 that? But, Nicole knew very well that 881 1:05:23,434 --> 1:05:28,835 this point cannot move. Therefore, she very sneakily, 882 1:05:28,835 --> 1:05:34,859 without telling you and me generated a pulse that came back 883 1:05:34,859 --> 1:05:41,091 to me, which made sure that at all moments in time this point 884 1:05:41,091 --> 1:05:45,785 stood still. So, at this very moment in 885 1:05:45,785 --> 1:05:49,356 time, she must've generated the pulse, which had this 886 1:05:49,356 --> 1:05:52,171 displacement, so that this part is exactly 887 1:05:52,171 --> 1:05:55,467 the same as this. And so, [UNINTELLIGIBLE] stands 888 1:05:55,467 --> 1:05:57,939 still. But she must have done that at 889 1:05:57,939 --> 1:06:02,270 every moment in time. She must have done that when 890 1:06:02,270 --> 1:06:05,594 this part arrived, when this part arrived, 891 1:06:05,594 --> 1:06:09,324 when this part arrived, when that part arrived. 892 1:06:09,324 --> 1:06:13,702 So, that means he must've generated a pulse on her side 893 1:06:13,702 --> 1:06:17,027 that is a valley that now looks like this. 894 1:06:17,027 --> 1:06:21,000 So, this part is here. And at this moment in time, 895 1:06:21,000 --> 1:06:24,243 all she has to do is generate this pulse. 896 1:06:24,243 --> 1:06:29,270 And so, the net result is that if you took a photograph of this 897 1:06:29,270 --> 1:06:33,972 string at this moment in time, you would see something very 898 1:06:33,972 --> 1:06:38,355 bizarre. It is the sum of this with 899 1:06:38,355 --> 1:06:41,526 this. And, you try to draw what that 900 1:06:41,526 --> 1:06:43,790 looks like. For one thing, 901 1:06:43,790 --> 1:06:47,414 this point will be here. That's for sure. 902 1:06:47,414 --> 1:06:52,487 And then, whatever you see here, well, you try to add the 903 1:06:52,487 --> 1:06:55,839 two up. And this thing is moving in my 904 1:06:55,839 --> 1:07:02,000 direction with peak V because she is generating a valley. 905 1:07:02,000 --> 1:07:06,322 And so, the consequence of the boundary condition is since this 906 1:07:06,322 --> 1:07:09,459 point is fixed, a mountain must come back as a 907 1:07:09,459 --> 1:07:13,014 valley, and the valley must come back as a mountain. 908 1:07:13,014 --> 1:07:17,336 And given a little bit of time, when this point here has passed 909 1:07:17,336 --> 1:07:19,567 Nicole completely, then there is, 910 1:07:19,567 --> 1:07:23,331 of course, a very nice, healthy post on the way back to 911 1:07:23,331 --> 1:07:25,841 me, which is mirrored, now, this way. 912 1:07:25,841 --> 1:07:29,605 The mountain is a valley, but it also has mirrored this 913 1:07:29,605 --> 1:07:33,181 way. That's why I made the pulse 914 1:07:33,181 --> 1:07:37,300 purposely asymmetric. So, that is what is happening. 915 1:07:37,300 --> 1:07:41,662 So, now I want to do this experiment again with Nicole, 916 1:07:41,662 --> 1:07:44,974 of course, because she knows how to do it. 917 1:07:44,974 --> 1:07:49,579 And you're going to look at this with completely different 918 1:07:49,579 --> 1:07:52,325 eyes. Your eyes were closed when we 919 1:07:52,325 --> 1:07:55,233 did it the first time. You are blind. 920 1:07:55,233 --> 1:08:00,000 Let's face it. But now, you've seen the light. 921 1:08:00,000 --> 1:08:03,913 This is a big moment in your life because you now know, 922 1:08:03,913 --> 1:08:06,159 first of all, why it propagates. 923 1:08:06,159 --> 1:08:10,217 And now when it arrives there, you know that the mountain 924 1:08:10,217 --> 1:08:13,188 becomes a valley. So, I'm just going to do 925 1:08:13,188 --> 1:08:17,101 exactly the same thing, only to allow you to look at it 926 1:08:17,101 --> 1:08:21,086 now through different eyes. And that's what education is 927 1:08:21,086 --> 1:08:25,434 all about regardless of whether it's physics or whether it is 928 1:08:25,434 --> 1:08:27,028 art. Without education, 929 1:08:27,028 --> 1:08:31,107 you cannot appreciate it. Now you can. 930 1:08:31,107 --> 1:08:33,690 Watch it. You ready for this, 931 1:08:33,690 --> 1:08:35,996 Nicole? You see, it moves. 932 1:08:35,996 --> 1:08:40,332 It has no choice. And the mountain comes back as 933 1:08:40,332 --> 1:08:43,837 a valley. I will do that once more very 934 1:08:43,837 --> 1:08:46,697 clear. You deserve an A for this 935 1:08:46,697 --> 1:08:48,542 course. That's clear. 936 1:08:48,542 --> 1:08:52,693 Oh, what, you don't want it? I'll make it a B. 937 1:08:52,693 --> 1:08:55,000 [LAUGHTER] 938 1:08:55,000 --> 1:09:02,000 939 1:09:02,000 --> 1:09:06,264 I can change the boundary conditions. 940 1:09:06,264 --> 1:09:10,648 I don't have to keep this point fixed. 941 1:09:10,648 --> 1:09:15,268 And, I can do that in the following way. 942 1:09:15,268 --> 1:09:20,125 Here's my string. I have here a metal rod. 943 1:09:20,125 --> 1:09:27,351 We put oil and grease on it so it's completely frictionless in 944 1:09:27,351 --> 1:09:32,682 this direction. And, we mount here a mass-less 945 1:09:32,682 --> 1:09:37,556 ring, mass-less. But the tension, 946 1:09:37,556 --> 1:09:42,225 of course, is there. And, mass per unit length is 947 1:09:42,225 --> 1:09:44,560 mu. None of that changes. 948 1:09:44,560 --> 1:09:50,007 But here is a mass-less ring, and this is a rod with zero 949 1:09:50,007 --> 1:09:53,217 friction. Those are very different 950 1:09:53,217 --> 1:09:58,081 boundary conditions. This point can now move up and 951 1:09:58,081 --> 1:10:01,000 down. And, it will. 952 1:10:01,000 --> 1:10:06,416 However, the shape of that string right here is now very 953 1:10:06,416 --> 1:10:09,469 special. At all moments in time, 954 1:10:09,469 --> 1:10:15,181 what will the shape of this string be when we photograph it 955 1:10:15,181 --> 1:10:18,333 no matter when you photograph it? 956 1:10:18,333 --> 1:10:22,666 You can photograph it before the poses there, 957 1:10:22,666 --> 1:10:27,000 after the poses there, at any moment in time. 958 1:10:27,000 --> 1:10:31,924 What will this point look like? It comes in at 90°. 959 1:10:31,924 --> 1:10:36,848 DY DX, if this is Y, this is X, at that location DY 960 1:10:36,848 --> 1:10:41,229 DX must be zero. If it were not zero, 961 1:10:41,229 --> 1:10:43,564 this would be zero. This would be zero. 962 1:10:43,564 --> 1:10:45,162 This is zero. This is zero. 963 1:10:45,162 --> 1:10:47,128 This is zero. That is whole zero. 964 1:10:47,128 --> 1:10:49,217 If it weren't zero, if it was this, 965 1:10:49,217 --> 1:10:52,843 then there would be a force on this ring because the tension 966 1:10:52,843 --> 1:10:56,039 would be in this direction. But the ring has no mass. 967 1:10:56,039 --> 1:10:59,787 And so, the acceleration of the ring would be infinitely high, 968 1:10:59,787 --> 1:11:03,015 which we don't allow. So, therefore, 969 1:11:03,015 --> 1:11:06,860 in the extreme case that you can go to this situation, 970 1:11:06,860 --> 1:11:09,834 you will now see something very different. 971 1:11:09,834 --> 1:11:13,606 You will see that the string, at all moments in time, 972 1:11:13,606 --> 1:11:16,000 will have to be like this. 973 1:11:16,000 --> 1:11:21,000 974 1:11:21,000 --> 1:11:25,243 If now I sent in a mountain, what you think will come back? 975 1:11:25,243 --> 1:11:28,243 The mountain comes back. Mountain goes in, 976 1:11:28,243 --> 1:11:32,038 mountain comes back. Who thinks it a mountain goes 977 1:11:32,038 --> 1:11:34,569 in, a mountain comes back? Very good. 978 1:11:34,569 --> 1:11:38,575 And that's the consequence of the fact that it is open now 979 1:11:38,575 --> 1:11:42,653 because the only reason why a mountain goes in and a valley 980 1:11:42,653 --> 1:11:45,183 came back, there was only one reason. 981 1:11:45,183 --> 1:11:48,276 The end could not move. Now the end can move, 982 1:11:48,276 --> 1:11:50,526 and I will demonstrate it to you. 983 1:11:50,526 --> 1:11:54,040 And now, the mountain will come back as a mountain. 984 1:11:54,040 --> 1:11:57,274 We referred to this in physics as a closed end, 985 1:11:57,274 --> 1:12:01,000 and we refer to this as an open end. 986 1:12:01,000 --> 1:12:06,864 And when you have an open end, and this is the pulse that 987 1:12:06,864 --> 1:12:13,358 comes in, say it has amplitude, A, then what comes back at some 988 1:12:13,358 --> 1:12:19,222 point in time is again a mountain going in this direction 989 1:12:19,222 --> 1:12:23,412 with speed V. This comes in with speed V. 990 1:12:23,412 --> 1:12:27,496 We call that the incident pulse in this. 991 1:12:27,496 --> 1:12:32,000 We call that the reflected pulse. 992 1:12:32,000 --> 1:12:33,675 This has amplitude, A. 993 1:12:33,675 --> 1:12:37,025 And this has very interesting consequences, 994 1:12:37,025 --> 1:12:41,333 namely, at the moment that this point here reaches that 995 1:12:41,333 --> 1:12:44,843 mass-less ring. The mass-less ring must go up 996 1:12:44,843 --> 1:12:47,794 by an amount, 2A because it generates, 997 1:12:47,794 --> 1:12:52,740 that ring generates this pulse. And so, the ring generates this 998 1:12:52,740 --> 1:12:55,133 pulse. But this was also there. 999 1:12:55,133 --> 1:12:58,005 And remember, you have to add the two 1000 1:12:58,005 --> 1:13:02,261 together. Like, this one was added to 1001 1:13:02,261 --> 1:13:04,272 this. That gives me zero. 1002 1:13:04,272 --> 1:13:07,287 Now you have to add this A to that A. 1003 1:13:07,287 --> 1:13:11,727 And so, what you'll see is that if [E?] was your ring, 1004 1:13:11,727 --> 1:13:15,329 it will go up to 2A. So, it will make a huge 1005 1:13:15,329 --> 1:13:19,434 excursion, goes twice as high as the incoming one, 1006 1:13:19,434 --> 1:13:22,115 and then it will go back to zero. 1007 1:13:22,115 --> 1:13:24,795 And then the mountain rolls back. 1008 1:13:24,795 --> 1:13:29,151 And needless to say that we, of course, would like to 1009 1:13:29,151 --> 1:13:33,809 demonstrate that. Now, to make a rod, 1010 1:13:33,809 --> 1:13:38,000 which is nearly frictionless, it's difficult, 1011 1:13:38,000 --> 1:13:42,571 but we can use a lot of oil, and a lot of grease, 1012 1:13:42,571 --> 1:13:46,761 and a lot of soap. So, that was not our major 1013 1:13:46,761 --> 1:13:49,904 hang-up. But when we looked at the 1014 1:13:49,904 --> 1:13:54,571 Amazon.com, and we wanted to buy a mass-less ring, 1015 1:13:54,571 --> 1:14:00,000 Marcos and I really tried, but it didn't work. 1016 1:14:00,000 --> 1:14:02,666 We couldn't buy a mass-less ring. 1017 1:14:02,666 --> 1:14:05,833 And so, therefore, it is not so easy to 1018 1:14:05,833 --> 1:14:09,416 demonstrate this in a way that I have there. 1019 1:14:09,416 --> 1:14:13,416 So, we will demonstrate it to you in another way. 1020 1:14:13,416 --> 1:14:16,083 And that is with this instrument. 1021 1:14:16,083 --> 1:14:19,833 I will first explain it. This is not a string. 1022 1:14:19,833 --> 1:14:22,750 These are rods, all the same length. 1023 1:14:22,750 --> 1:14:25,916 They're connected here with some metal. 1024 1:14:25,916 --> 1:14:30,500 And so, you can move these. And then, the pulse that you 1025 1:14:30,500 --> 1:14:35,511 generate will propagate. So, they are coupled. 1026 1:14:35,511 --> 1:14:37,741 I don't know how many there are. 1027 1:14:37,741 --> 1:14:41,771 Do you know how many there are? OK, let's say I count 40. 1028 1:14:41,771 --> 1:14:44,074 Then it's 40 coupled oscillators. 1029 1:14:44,074 --> 1:14:48,320 And now, I have the option with this machine that I can hold 1030 1:14:48,320 --> 1:14:51,342 this one fixed, which is then a closed end. 1031 1:14:51,342 --> 1:14:55,588 But I can also let this one open, and then it's an open end. 1032 1:14:55,588 --> 1:14:59,833 So, if I hope this one close, and I sent in a mountain here, 1033 1:14:59,833 --> 1:15:04,717 then a valley will come back. But if I keep it open, 1034 1:15:04,717 --> 1:15:09,090 and I send in a mountain, then a mountain will come back. 1035 1:15:09,090 --> 1:15:13,228 And you should be able to see that the end gets a huge 1036 1:15:13,228 --> 1:15:17,210 amplitude at the moment that it reaches the maximum. 1037 1:15:17,210 --> 1:15:20,333 And, so that is what is on our plate now. 1038 1:15:20,333 --> 1:15:24,003 And, we will make it extremely romantic for you, 1039 1:15:24,003 --> 1:15:26,969 believe me. We're going to do this very 1040 1:15:26,969 --> 1:15:29,000 romantic way. 1041 1:15:29,000 --> 1:15:35,000 1042 1:15:35,000 --> 1:15:36,000 I told you. 1043 1:15:36,000 --> 1:15:44,000 1044 1:15:44,000 --> 1:15:50,758 So here, I have a clip here. So I will first lock this in 1045 1:15:50,758 --> 1:15:54,620 place so that this N cannot move. 1046 1:15:54,620 --> 1:16:00,413 That's what I will do first. And, from this side, 1047 1:16:00,413 --> 1:16:05,000 I will then generate a mountain. 1048 1:16:05,000 --> 1:16:08,969 The speed at which it propagates is actually quite 1049 1:16:08,969 --> 1:16:12,452 decent, not as fast as it was with a string. 1050 1:16:12,452 --> 1:16:17,312 And so, I want you to see that first of all it propagates and 1051 1:16:17,312 --> 1:16:22,011 then comes back as a valley. So, the end here is now fixed. 1052 1:16:22,011 --> 1:16:23,874 It's a fixed end. Ready? 1053 1:16:23,874 --> 1:16:26,385 Mountain, and now it's a valley. 1054 1:16:26,385 --> 1:16:29,382 You see it? OK, now it's always a pain 1055 1:16:29,382 --> 1:16:34,000 because the system is a very high Q system. 1056 1:16:34,000 --> 1:16:36,012 So, it doesn't want to damp out. 1057 1:16:36,012 --> 1:16:37,699 I can try to send in. Yeah. 1058 1:16:37,699 --> 1:16:39,971 I know exactly what you're thinking. 1059 1:16:39,971 --> 1:16:43,021 We are aware of this. If they try to com calm it 1060 1:16:43,021 --> 1:16:45,163 down, it makes it worse sometimes. 1061 1:16:45,163 --> 1:16:47,564 I will now try to generate the valley. 1062 1:16:47,564 --> 1:16:50,420 It is a little harder. I don't know why this, 1063 1:16:50,420 --> 1:16:53,341 why it's a little harder. I have to talk to my 1064 1:16:53,341 --> 1:16:56,651 psychiatrist about it. It's easier to go up and down 1065 1:16:56,651 --> 1:17:01,000 than to go down and up. I don't know why that is. 1066 1:17:01,000 --> 1:17:03,938 So, I'll go down and up, make a valley, 1067 1:17:03,938 --> 1:17:07,341 and then when it comes back, it's a mountain. 1068 1:17:07,341 --> 1:17:10,125 There it goes. And it comes back as a 1069 1:17:10,125 --> 1:17:11,981 mountain. But you see it? 1070 1:17:11,981 --> 1:17:13,682 Did you? If you didn't, 1071 1:17:13,682 --> 1:17:16,775 just say so. We can do it once more but I 1072 1:17:16,775 --> 1:17:20,410 don't think we have to. Now comes the big thing. 1073 1:17:20,410 --> 1:17:23,890 Now I'm going to make this, and freely moving. 1074 1:17:23,890 --> 1:17:27,525 So now, it's an open end. And, I will generate a 1075 1:17:27,525 --> 1:17:32,036 mountain, now. And I want you to not only 1076 1:17:32,036 --> 1:17:35,404 appreciate that it comes back as a mountain. 1077 1:17:35,404 --> 1:17:38,616 But above all, the end will have twice the 1078 1:17:38,616 --> 1:17:43,237 amplitude at one moment in time when the top of the mountain 1079 1:17:43,237 --> 1:17:46,135 reaches that end. And then, of course, 1080 1:17:46,135 --> 1:17:50,287 it will go back to zero. And the regular mountain will 1081 1:17:50,287 --> 1:17:53,812 grow back to me. So, if you're ready for this, 1082 1:17:53,812 --> 1:17:58,041 there goes the mountain. Sweep high, and it comes back. 1083 1:17:58,041 --> 1:18:02,084 [LAUGHTER] What is so funny about that? 1084 1:18:02,084 --> 1:18:04,392 Did you see that huge amplitude? 1085 1:18:04,392 --> 1:18:08,858 I'll do it once more because I don't want you to forget that. 1086 1:18:08,858 --> 1:18:13,176 Let's give it just ten seconds. Oh boy, it's like an ocean. 1087 1:18:13,176 --> 1:18:17,047 I will do it once more. I'm not going to try a valley 1088 1:18:17,047 --> 1:18:20,024 because that's what the problem comes in. 1089 1:18:20,024 --> 1:18:23,598 I will simply go up, and then let's look again at 1090 1:18:23,598 --> 1:18:28,213 the end and see whether we can see that double amplitude of the 1091 1:18:28,213 --> 1:18:31,128 mountain. Mountain: whoa, 1092 1:18:31,128 --> 1:18:32,333 biggie, oh, man. Whoa, OK, have a good weekend.