1 00:00:28,000 --> 00:00:33,241 I will summarize what we have learned about the normal modes 2 00:00:33,241 --> 00:00:38,128 of the string fixed at both ends, which is very relevant 3 00:00:38,128 --> 00:00:43,192 today for musical instruments. So, suppose I have a string 4 00:00:43,192 --> 00:00:46,568 with length L, mass per unit length mu, 5 00:00:46,568 --> 00:00:50,388 and tension T, then I know that the speed of 6 00:00:50,388 --> 00:00:54,741 propagation is the square root of T divided by mu. 7 00:00:54,741 --> 00:01:00,161 And, the wavelength is always this P to propagation divided by 8 00:01:00,161 --> 00:01:04,070 the frequency, speed of propagation times the 9 00:01:04,070 --> 00:01:11,000 period of one oscillation that I avoid capital T's for period. 10 00:01:11,000 --> 00:01:15,137 So I will just write it down as frequency. 11 00:01:15,137 --> 00:01:20,688 And so, in its lowest mode, recall that the fundamental, 12 00:01:20,688 --> 00:01:24,422 first harmonic, you will get then this 13 00:01:24,422 --> 00:01:29,871 situation, N equals one, and lambda one is then clearly 14 00:01:29,871 --> 00:01:36,440 2L if this length is L. And, the frequency is then V 15 00:01:36,440 --> 00:01:41,601 divided by 2L. So, that's the frequency for 16 00:01:41,601 --> 00:01:47,008 the lowest possible mode, the first harmonic. 17 00:01:47,008 --> 00:01:52,783 If we go to the second harmonic, and the picture 18 00:01:52,783 --> 00:01:57,207 changes. We get a node in the middle. 19 00:01:57,207 --> 00:02:05,000 And we have here N equals two. So, lambda two is now L. 20 00:02:05,000 --> 00:02:10,125 And so, F2 is now, again, V divided by the 21 00:02:10,125 --> 00:02:13,500 wavelength. So that becomes, 22 00:02:13,500 --> 00:02:19,250 now, V divided by L. And so, we can write down, 23 00:02:19,250 --> 00:02:24,500 now, the general equation for the N's mode, 24 00:02:24,500 --> 00:02:30,000 N being Nancy. Lambda of L is then 2L divided 25 00:02:30,000 --> 00:02:34,599 by N. In the frequency in the Nth 26 00:02:34,599 --> 00:02:39,990 mode, that is the Nth harmonic, is then NV divided by 2L. 27 00:02:39,990 --> 00:02:44,708 And, this V is given there. So, what you see here, 28 00:02:44,708 --> 00:02:49,232 that if the fundamental here were, for instance, 29 00:02:49,232 --> 00:02:53,854 100 Hz, then the second harmonic would be 200 Hz, 30 00:02:53,854 --> 00:02:58,667 the third harmonic 300 Hz. They come in series one, 31 00:02:58,667 --> 00:03:03,000 two, three, four, five, and so on. 32 00:03:03,000 --> 00:03:06,410 Now, clearly, there is the possibility that I 33 00:03:06,410 --> 00:03:09,356 would have one end of that string open. 34 00:03:09,356 --> 00:03:11,759 We call this then closed-closed, 35 00:03:11,759 --> 00:03:15,015 or fixed-fixed. But suppose I have the same 36 00:03:15,015 --> 00:03:19,046 length, L, but here I have this infamous rod which is 37 00:03:19,046 --> 00:03:23,232 frictionless with the infamous ring, which has no mass. 38 00:03:23,232 --> 00:03:27,263 And it is fixed here, and now I want to know what the 39 00:03:27,263 --> 00:03:30,364 normal modes are. And in the lowest mode, 40 00:03:30,364 --> 00:03:34,739 you will get this. The angle is 90°. 41 00:03:34,739 --> 00:03:37,297 DY DX must be zero, there. 42 00:03:37,297 --> 00:03:42,207 And then, it oscillates like this back and forth. 43 00:03:42,207 --> 00:03:47,834 And so, now you have that N equals one, which is now the 44 00:03:47,834 --> 00:03:53,359 fundamental first harmonic. Lambda one is now 4L lambda 45 00:03:53,359 --> 00:03:55,201 one. And therefore, 46 00:03:55,201 --> 00:04:01,033 the frequency that you now generate is twice as low is now 47 00:04:01,033 --> 00:04:05,874 V divided by 4L. So, you can go one step 48 00:04:05,874 --> 00:04:07,504 further. You can ask, 49 00:04:07,504 --> 00:04:11,090 now, for the second harmonic. So here, again, 50 00:04:11,090 --> 00:04:14,269 is that rod. And now I introduce another 51 00:04:14,269 --> 00:04:17,366 node here. And so, now the string would 52 00:04:17,366 --> 00:04:19,811 look like this. This is, again, 53 00:04:19,811 --> 00:04:21,849 90°. And as it oscillates, 54 00:04:21,849 --> 00:04:26,087 it would move like this. And so now, this is N equals 55 00:04:26,087 --> 00:04:30,000 two. This is the second harmonic. 56 00:04:30,000 --> 00:04:37,475 You can just look at it. Lambda two is four thirds times 57 00:04:37,475 --> 00:04:42,368 L, right? You need to make it longer. 58 00:04:42,368 --> 00:04:48,213 And therefore, F2 is now three divided by 3V 59 00:04:48,213 --> 00:04:54,058 divided by 4L. And so, now we can write down 60 00:04:54,058 --> 00:05:00,854 the general recipe for the Nth mode, N being Nancy. 61 00:05:00,854 --> 00:05:07,106 So, we will find, then, that lambda N is now 4L 62 00:05:07,106 --> 00:05:13,202 divided by 2N minus one. And that, now, 63 00:05:13,202 --> 00:05:17,666 changes the picture quite dramatically because if you 64 00:05:17,666 --> 00:05:22,217 follow here that you go from lambda one to lambda two, 65 00:05:22,217 --> 00:05:25,308 the change is not by a factor of two. 66 00:05:25,308 --> 00:05:29,000 But it changes by a factor of three. 67 00:05:29,000 --> 00:05:34,739 And so the frequency in the Nth mode, which is the velocity 68 00:05:34,739 --> 00:05:40,776 divided by lambda N is 2N minus one times the velocity divided 69 00:05:40,776 --> 00:05:43,843 by 4L. And so, what you see now: 70 00:05:43,843 --> 00:05:48,989 if we take both systems, and suppose L were the same, 71 00:05:48,989 --> 00:05:52,651 mu were the same, and T were the same, 72 00:05:52,651 --> 00:05:57,401 then F1 in this mode, this instrument is half the 73 00:05:57,401 --> 00:06:03,338 frequency of F1 there because there you have downstairs a 2L, 74 00:06:03,338 --> 00:06:08,424 and here you have a 4L. So, for instance, 75 00:06:08,424 --> 00:06:12,530 if that were 100 Hz, with the same length here you 76 00:06:12,530 --> 00:06:16,553 would get 50 Hz if all other things are the same. 77 00:06:16,553 --> 00:06:21,245 And then the second harmonic would be 150 Hz because if N 78 00:06:21,245 --> 00:06:24,346 becomes one, you get a three upstairs. 79 00:06:24,346 --> 00:06:27,782 If N becomes two, you get a five upstairs. 80 00:06:27,782 --> 00:06:30,798 So now the ratio is one, three, five, 81 00:06:30,798 --> 00:06:34,905 seven, whereas there the ratio is one, two, three, 82 00:06:34,905 --> 00:06:38,774 four, etc. Now, if I take a string in 83 00:06:38,774 --> 00:06:42,186 isolation and I oscillated, I almost hear no sound. 84 00:06:42,186 --> 00:06:46,145 There is not enough to air that is displaced by the string. 85 00:06:46,145 --> 00:06:50,103 And so, what we do is we have to mount it on a surface that 86 00:06:50,103 --> 00:06:53,994 starts to vibrate with it. And you see that on all musical 87 00:06:53,994 --> 00:06:56,860 instruments. I will demonstrate that to you 88 00:06:56,860 --> 00:07:00,000 in a simple way with a tuning fork. 89 00:07:00,000 --> 00:07:03,959 I have here a tuning fork. And if I just excite this 90 00:07:03,959 --> 00:07:07,920 tuning fork, it is 440 Hz. You'll you're practically 91 00:07:07,920 --> 00:07:09,550 nothing. I hit it now, 92 00:07:09,550 --> 00:07:11,802 I can hear it. I'm very close. 93 00:07:11,802 --> 00:07:16,616 But most of you cannot hear it. However, the moment I put it on 94 00:07:16,616 --> 00:07:20,498 the surface, the surface starts to vibrate with it. 95 00:07:20,498 --> 00:07:24,225 So, I drain energy faster out of the tuning fork, 96 00:07:24,225 --> 00:07:26,943 but you get more volume, more sound. 97 00:07:26,943 --> 00:07:30,903 A larger surface oscillates, so the pressure wave is 98 00:07:30,903 --> 00:07:34,000 stronger. So I'll demonstrate that. 99 00:07:34,000 --> 00:07:39,000 100 00:07:39,000 --> 00:07:43,356 Do you hear it now? Do it here, big difference, 101 00:07:43,356 --> 00:07:46,670 right? Huge difference when I put it 102 00:07:46,670 --> 00:07:51,026 on the surface. I have here a music box which I 103 00:07:51,026 --> 00:07:54,056 bought many years ago in Austria. 104 00:07:54,056 --> 00:07:58,507 And, it has these prongs. And, when I rotate it, 105 00:07:58,507 --> 00:08:04,000 I may be able to hear it. But you won't hear it. 106 00:08:04,000 --> 00:08:08,200 Maybe some of you are very close. 107 00:08:08,200 --> 00:08:12,007 Can you hear it? Good for you. 108 00:08:12,007 --> 00:08:17,258 Now listen. [MUSIC PLAYS] Big difference. 109 00:08:17,258 --> 00:08:21,459 Now the whole surface oscillates. 110 00:08:21,459 --> 00:08:26,841 Again, you drain energy faster, of course. 111 00:08:26,841 --> 00:08:32,617 But you get more sound. I can put it on here. 112 00:08:32,617 --> 00:08:40,865 [MUSIC PLAYS] Big difference. And of course you will see that 113 00:08:40,865 --> 00:08:44,563 in the design of all musical instruments. 114 00:08:44,563 --> 00:08:48,445 Needless to say, the design of the sounding 115 00:08:48,445 --> 00:08:52,605 boards which are connected to the strings are, 116 00:08:52,605 --> 00:08:58,058 of course, secrets that the company are very fond of and not 117 00:08:58,058 --> 00:09:02,029 telling you. I am not aware of musical 118 00:09:02,029 --> 00:09:06,495 instruments, of stringed instruments, whereby one end of 119 00:09:06,495 --> 00:09:10,068 the string is attached to a frictionless rod, 120 00:09:10,068 --> 00:09:14,290 and whereby that end is attached to a mass-less ring. 121 00:09:14,290 --> 00:09:19,162 So, I will restrict myself in discussing musical instruments, 122 00:09:19,162 --> 00:09:23,141 string instruments, to the one's whereby both ends 123 00:09:23,141 --> 00:09:25,414 are fixed. And so, therefore, 124 00:09:25,414 --> 00:09:28,905 the frequency, F1, that is the one that I am 125 00:09:28,905 --> 00:09:32,316 interested in, is the speed of propagation, 126 00:09:32,316 --> 00:09:38,000 which is T divided by mu, and then one divided by 2L. 127 00:09:38,000 --> 00:09:41,206 So, these are the key players in the design of instruments. 128 00:09:41,206 --> 00:09:43,582 If you make L longer, you get a lower pitch. 129 00:09:43,582 --> 00:09:46,070 If you make L shorter, you get a higher pitch. 130 00:09:46,070 --> 00:09:48,502 And, we all know that. If you make L shorter, 131 00:09:48,502 --> 00:09:51,487 you get a higher pitch. If you make the tension higher, 132 00:09:51,487 --> 00:09:54,361 you get a higher pitch. If you make the mass per unit 133 00:09:54,361 --> 00:09:56,351 length lower, you get a higher pitch. 134 00:09:56,351 --> 00:10:00,000 And so, that's key in designing musical instruments. 135 00:10:00,000 --> 00:10:02,786 If you take a piano, it is 88 keys. 136 00:10:02,786 --> 00:10:07,213 The lowest frequency is 28 Hz. The highest is 4,000 Hz. 137 00:10:07,213 --> 00:10:11,065 It covers seven octaves. And when you hit a key, 138 00:10:11,065 --> 00:10:14,672 a hammer actually comes down onto the string. 139 00:10:14,672 --> 00:10:19,180 And it excites the string in the accommodation of normal 140 00:10:19,180 --> 00:10:21,557 modes. In fact, in many cases, 141 00:10:21,557 --> 00:10:25,245 when you hit one key without you realizing it, 142 00:10:25,245 --> 00:10:30,000 you hit more than one string simultaneously. 143 00:10:30,000 --> 00:10:33,508 But that's a detail now which I will not further expand on. 144 00:10:33,508 --> 00:10:37,016 If you take a Steinway grand piano, even though it has only 145 00:10:37,016 --> 00:10:40,524 88 keys, it has 216 strings. And so, the idea of the piano, 146 00:10:40,524 --> 00:10:43,245 then, is you change the length of the strings. 147 00:10:43,245 --> 00:10:45,846 So that's a player. The shorter the lengths, 148 00:10:45,846 --> 00:10:48,689 the higher the frequency. And you can change mu. 149 00:10:48,689 --> 00:10:52,016 And, when you open a can oh, and you look at the various 150 00:10:52,016 --> 00:10:55,524 strings, you immediately see that some strings are as thick 151 00:10:55,524 --> 00:10:57,399 as my pinky. You have a huge mu, 152 00:10:57,399 --> 00:11:01,951 and others are very thin. They have a very low value. 153 00:11:01,951 --> 00:11:04,987 And that will give you, then, higher pitch. 154 00:11:04,987 --> 00:11:09,325 The tension of the strings in the piano are approximately all 155 00:11:09,325 --> 00:11:11,421 the same. They are quite high. 156 00:11:11,421 --> 00:11:14,963 They are near 200 N per string. So, a grand piano, 157 00:11:14,963 --> 00:11:18,433 then the total force, all the strings together is 158 00:11:18,433 --> 00:11:22,048 something like 45,000 N, which is an immense force, 159 00:11:22,048 --> 00:11:25,879 when you think about it. Now, if you go to violins and 160 00:11:25,879 --> 00:11:30,000 cellos and a bass, they have four strings. 161 00:11:30,000 --> 00:11:33,549 Then the length is a given. But if you look at the various 162 00:11:33,549 --> 00:11:36,040 strings, and we will see a violent later, 163 00:11:36,040 --> 00:11:39,715 you will see that one string is much thicker than the other. 164 00:11:39,715 --> 00:11:42,144 So, you play with mu. You change the mu. 165 00:11:42,144 --> 00:11:44,635 So, you only have four fundamentals then. 166 00:11:44,635 --> 00:11:47,375 If you go to a Museum of Fine Arts in Boston, 167 00:11:47,375 --> 00:11:49,991 they have a wonderful collection of musical 168 00:11:49,991 --> 00:11:52,606 instruments. You'll see musical instruments 169 00:11:52,606 --> 00:11:55,782 with just two strings, and you will also see musical 170 00:11:55,782 --> 00:11:59,145 instruments with one string. But I have never seen one, 171 00:11:59,145 --> 00:12:02,072 as I said earlier, whereby one end of the string 172 00:12:02,072 --> 00:12:06,610 [can/can't?] freely move. But maybe they do exist. 173 00:12:06,610 --> 00:12:09,562 So, now comes the issue, if you have built an 174 00:12:09,562 --> 00:12:13,655 instrument, how do you tune it? Well, with the piano you ask a 175 00:12:13,655 --> 00:12:17,950 piano tuner maybe once a year or once every year to come into the 176 00:12:17,950 --> 00:12:20,835 piano for you. And what the candidates in or 177 00:12:20,835 --> 00:12:24,056 does it simply changes the tension in the string, 178 00:12:24,056 --> 00:12:27,209 which is a major job, of course, if you have 216 179 00:12:27,209 --> 00:12:28,753 strings. But the violin, 180 00:12:28,753 --> 00:12:31,974 and guitar, and cello, and bass, it is the player 181 00:12:31,974 --> 00:12:36,000 herself or himself who is doing the tuning. 182 00:12:36,000 --> 00:12:39,782 We will see that later today. They will actually change the 183 00:12:39,782 --> 00:12:43,239 tension in the string. And then, they listen carefully 184 00:12:43,239 --> 00:12:46,891 to get just the right tone. And, they do that before they 185 00:12:46,891 --> 00:12:49,239 start playing. If you play the piano, 186 00:12:49,239 --> 00:12:53,217 with all due respect for piano players, all they have to do is 187 00:12:53,217 --> 00:12:55,826 hit the right keys in the right sequence. 188 00:12:55,826 --> 00:13:00,000 That's all there is to it when you are piano player. 189 00:13:00,000 --> 00:13:03,000 Now, think about the violinist, or a guitar, 190 00:13:03,000 --> 00:13:05,441 or a cello. They cannot just hit one 191 00:13:05,441 --> 00:13:08,023 string. They have to change the length 192 00:13:08,023 --> 00:13:11,651 of the string all the time to change the fundamental. 193 00:13:11,651 --> 00:13:15,627 And that's the whole idea when you have the violin and you 194 00:13:15,627 --> 00:13:18,418 strike it with a bow. You rub the string, 195 00:13:18,418 --> 00:13:22,534 and the string will start to oscillate in the fundamental in 196 00:13:22,534 --> 00:13:26,093 the higher harmonic. By making the string no shorter 197 00:13:26,093 --> 00:13:30,000 with your hands, you increase the pitch. 198 00:13:30,000 --> 00:13:33,783 And so, playing there is that you as a player continuously 199 00:13:33,783 --> 00:13:37,700 have to move the length of the string and just hit the right 200 00:13:37,700 --> 00:13:40,089 length. It goes beyond my imagination 201 00:13:40,089 --> 00:13:42,280 that anyone can do that. I cannot. 202 00:13:42,280 --> 00:13:45,599 I tried it when I was young. I took violin lessons. 203 00:13:45,599 --> 00:13:48,387 I was a total disaster. If you take a harp, 204 00:13:48,387 --> 00:13:51,839 it is in a way like a piano except there's no hammer. 205 00:13:51,839 --> 00:13:55,556 But you pluck the string. And now there is something very 206 00:13:55,556 --> 00:13:59,074 special about the harp. You can decide where you pluck 207 00:13:59,074 --> 00:14:02,995 the string. And that makes a difference in 208 00:14:02,995 --> 00:14:06,322 the percentage of higher harmonics, in which higher 209 00:14:06,322 --> 00:14:09,649 harmonics you excite. And we will deal with that in 210 00:14:09,649 --> 00:14:11,778 8.03 when we do Fourier analysis. 211 00:14:11,778 --> 00:14:14,306 You will see, then, that if you pluck a 212 00:14:14,306 --> 00:14:17,633 string in the middle, do you get a different series 213 00:14:17,633 --> 00:14:20,427 of higher harmonics than when you pluck at, 214 00:14:20,427 --> 00:14:23,421 say, a certain distance of 20 cm from the end. 215 00:14:23,421 --> 00:14:27,346 It's a very interesting part. In fact, the piano is designed 216 00:14:27,346 --> 00:14:31,405 in such a way that the hammer hits the string about 1/7 of its 217 00:14:31,405 --> 00:14:36,031 length from one end. And that is done to suppress 218 00:14:36,031 --> 00:14:39,658 the seventh harmonic. Now, why the seventh harmonic 219 00:14:39,658 --> 00:14:43,139 has to be depressed? That is up to you to decide. 220 00:14:43,139 --> 00:14:45,823 Maybe we don't like seventh harmonics. 221 00:14:45,823 --> 00:14:47,637 I don't know. It beats me. 222 00:14:47,637 --> 00:14:51,046 But in any case, it's purposely done to suppress 223 00:14:51,046 --> 00:14:54,673 the seventh harmonic. There is a musical instrument 224 00:14:54,673 --> 00:14:58,735 that has only one string. One was designed by my daughter 225 00:14:58,735 --> 00:15:03,922 when she was a nursery school. That's a long time ago. 226 00:15:03,922 --> 00:15:07,182 And I have it here. I'm very fond of it. 227 00:15:07,182 --> 00:15:11,862 I demonstrated it many times. It's called a washtub bass, 228 00:15:11,862 --> 00:15:14,537 I think. So, you see here the one 229 00:15:14,537 --> 00:15:16,543 string. And you see here, 230 00:15:16,543 --> 00:15:21,140 then, the surface that is necessary to make you hear the 231 00:15:21,140 --> 00:15:22,811 string. In this case, 232 00:15:22,811 --> 00:15:26,155 it is, I think, a box from Kentucky Fried 233 00:15:26,155 --> 00:15:31,681 Chicken or something like that. So, I need three hands to play 234 00:15:31,681 --> 00:15:33,699 this. So the only thing that I can do 235 00:15:33,699 --> 00:15:35,717 now, I can't change mu. It's a given. 236 00:15:35,717 --> 00:15:38,015 I can't change the length. That's a given. 237 00:15:38,015 --> 00:15:39,977 So, what is the only thing I can do? 238 00:15:39,977 --> 00:15:43,116 I can change the tension, and that's the way you play it. 239 00:15:43,116 --> 00:15:44,910 And for that, I need three hands. 240 00:15:44,910 --> 00:15:46,984 But I will try to do it with my mouth. 241 00:15:46,984 --> 00:15:49,730 And then, you listen. When I increase the tension, 242 00:15:49,730 --> 00:15:51,580 you will hear the frequency go up. 243 00:15:51,580 --> 00:15:55,000 When I decrease the tension, it goes down. 244 00:15:55,000 --> 00:16:08,000 245 00:16:08,000 --> 00:16:09,627 You could hear it, right? 246 00:16:09,627 --> 00:16:12,000 Clearly different in frequencies. 247 00:16:12,000 --> 00:16:16,000 248 00:16:16,000 --> 00:16:19,252 Pythagoras, who lives in the sixth century B.C. 249 00:16:19,252 --> 00:16:23,141 discovered that musical notes are very pleasing when the 250 00:16:23,141 --> 00:16:26,181 length of the strings come in simple ratios. 251 00:16:26,181 --> 00:16:30,000 This is quite remarkable when you think of that. 252 00:16:30,000 --> 00:16:34,744 Two-two-one give you an octave. Three-two-two give you a fifth, 253 00:16:34,744 --> 00:16:37,576 and four-two-three gives you a fourth. 254 00:16:37,576 --> 00:16:41,556 And, the evolution of Western music is based on that. 255 00:16:41,556 --> 00:16:45,153 When you take a piano, the piano has the octave: 256 00:16:45,153 --> 00:16:47,448 one-two-two. It has the fourth, 257 00:16:47,448 --> 00:16:50,204 four-two-three, and it has the fifth, 258 00:16:50,204 --> 00:16:53,265 three-two-two. And somehow these are very 259 00:16:53,265 --> 00:16:57,780 pleasant intervals when you play them in combination for our 260 00:16:57,780 --> 00:17:02,492 Western ears. It doesn't mean that it is also 261 00:17:02,492 --> 00:17:06,834 true for other cultures. The ancient Greek astronomers, 262 00:17:06,834 --> 00:17:11,658 even all the way up to Kepler in the 17th century belief that 263 00:17:11,658 --> 00:17:16,482 the musical instruments could also explain orbits of planets. 264 00:17:16,482 --> 00:17:20,180 And this was known as the music of the spheres. 265 00:17:20,180 --> 00:17:24,603 It was believed that the movement of the planet produces 266 00:17:24,603 --> 00:17:29,427 music, but our ears were just not sensitive enough to be able 267 00:17:29,427 --> 00:17:34,592 to hear that music. And of course it shouldn't 268 00:17:34,592 --> 00:17:39,876 surprise you that all this was mixed up with religion. 269 00:17:39,876 --> 00:17:45,460 The fact that it was pleasing to hear music that comes in 270 00:17:45,460 --> 00:17:51,642 simple ratios must show the hand of God, if you want to believe 271 00:17:51,642 --> 00:17:55,231 that. I now want to turn towards wind 272 00:17:55,231 --> 00:17:59,319 instruments. And we're going to put a wind 273 00:17:59,319 --> 00:18:03,706 instrument here, which is nothing but a tube, 274 00:18:03,706 --> 00:18:08,161 length, L. Let's start with one that is 275 00:18:08,161 --> 00:18:10,476 closed on both sides, length L. 276 00:18:10,476 --> 00:18:15,107 And, if it is closed on both sides, then the pressure here on 277 00:18:15,107 --> 00:18:19,506 this side and this side can build up, as we discussed last 278 00:18:19,506 --> 00:18:21,667 time. So, there's going to be 279 00:18:21,667 --> 00:18:25,218 pressure anti-nodes. And, there are going to be 280 00:18:25,218 --> 00:18:29,000 nodes for the motion of the molecules. 281 00:18:29,000 --> 00:18:33,606 If I have here a sound cavity which is open on both sides and 282 00:18:33,606 --> 00:18:37,751 I have here a pressure node, and I have here a pressure 283 00:18:37,751 --> 00:18:42,588 node, the pressure can never be higher than the ambient pressure 284 00:18:42,588 --> 00:18:45,659 because it's connected with the universe. 285 00:18:45,659 --> 00:18:49,574 When I say pressure, I always mean overpressure over 286 00:18:49,574 --> 00:18:53,643 and above 1 atmosphere. And so, it's immediately clear 287 00:18:53,643 --> 00:18:57,175 that the solutions that would work out here for 288 00:18:57,175 --> 00:19:01,320 closed-closed-system must be exactly identical to this. 289 00:19:01,320 --> 00:19:04,852 So, you do get precisely here that lambda N is, 290 00:19:04,852 --> 00:19:10,317 again, 2L divided by N. There is no difference. 291 00:19:10,317 --> 00:19:13,440 And, F of N is NV divided by 2L. 292 00:19:13,440 --> 00:19:17,873 But, of course, the enormous differences that 293 00:19:17,873 --> 00:19:22,809 the V is God-given, that is, the speed of sound in 294 00:19:22,809 --> 00:19:26,738 air, the air column begins to oscillate. 295 00:19:26,738 --> 00:19:30,768 And so V, now, is approximately 340 m per 296 00:19:30,768 --> 00:19:37,318 second, which is nonnegotiable. So, there is no way that you 297 00:19:37,318 --> 00:19:41,872 can manipulate mu NT to change the speed of propagation. 298 00:19:41,872 --> 00:19:46,095 The speed of sound in a gas, something that I really 299 00:19:46,095 --> 00:19:50,898 wouldn't want you to remember, I will return to that at the 300 00:19:50,898 --> 00:19:54,541 end of my lecture, is given by this equation. 301 00:19:54,541 --> 00:19:58,267 But you may forget it as far as I'm concerned. 302 00:19:58,267 --> 00:20:02,962 R is the gas constant. T is the temperature of the gas 303 00:20:02,962 --> 00:20:05,622 in degrees Kelvin. And gamma, if any of you 304 00:20:05,622 --> 00:20:08,978 remember that from 8.01, is the ratio of specific heat 305 00:20:08,978 --> 00:20:12,524 at constant pressure divided by specific heat at constant 306 00:20:12,524 --> 00:20:14,866 volume. But, if you've never had that, 307 00:20:14,866 --> 00:20:15,879 that's fine, too. 308 00:20:15,879 --> 00:20:19,235 Just take my word for it. I will give you some numbers 309 00:20:19,235 --> 00:20:22,021 for gamma later. And then, M is the molecular 310 00:20:22,021 --> 00:20:24,364 weight. So, this is the speed of sound 311 00:20:24,364 --> 00:20:26,517 in a gas. So, it's a given for air. 312 00:20:26,517 --> 00:20:30,000 There's nothing you can do about it. 313 00:20:30,000 --> 00:20:31,955 Room temperature, it's given. 314 00:20:31,955 --> 00:20:35,376 It changes with temperature, which is interesting. 315 00:20:35,376 --> 00:20:39,705 So, the way you get the system going, you must somehow blow air 316 00:20:39,705 --> 00:20:43,546 past maybe in this direction, or so, and then the column 317 00:20:43,546 --> 00:20:46,827 starts to excite. It's very hard to see why that 318 00:20:46,827 --> 00:20:50,389 happens, but then the column gets into normal modes. 319 00:20:50,389 --> 00:20:53,950 And you may get a whole collection of various normal 320 00:20:53,950 --> 00:20:58,000 modes, a superposition of many normal modes. 321 00:20:58,000 --> 00:21:00,969 Now, there is, in the case of sound cavities, 322 00:21:00,969 --> 00:21:04,748 what is very easy to make a system which is closed at one 323 00:21:04,748 --> 00:21:07,042 end and which is open at the other. 324 00:21:07,042 --> 00:21:09,944 That is very easy, which you cannot do here. 325 00:21:09,944 --> 00:21:12,576 You can do that here. And so, therefore, 326 00:21:12,576 --> 00:21:15,478 in this case, that one side of the cavity is 327 00:21:15,478 --> 00:21:18,852 closed and the other one is open, there are musical 328 00:21:18,852 --> 00:21:21,619 instruments. The clarinet comes very close 329 00:21:21,619 --> 00:21:25,196 to be almost closed at one side and open at the other. 330 00:21:25,196 --> 00:21:28,907 But there are many musical instruments which are open on 331 00:21:28,907 --> 00:21:33,374 both sides. So, then you would have here 332 00:21:33,374 --> 00:21:37,868 that F of M equals 2M minus one times V divided by 2L. 333 00:21:37,868 --> 00:21:41,514 So then you have, indeed, this case which we 334 00:21:41,514 --> 00:21:44,397 dismissed for stringed instruments. 335 00:21:44,397 --> 00:21:48,383 [That's?] divided by 4L. And, the easiest way to 336 00:21:48,383 --> 00:21:51,690 demonstrate that to you, to take my pen, 337 00:21:51,690 --> 00:21:56,269 this is the cover of my pen. It is closed on this side, 338 00:21:56,269 --> 00:22:00,000 believe me. And it's open here. 339 00:22:00,000 --> 00:22:04,725 And it is about 3 cm long. You may not like the frequency. 340 00:22:04,725 --> 00:22:07,544 It's a very high-pitched frequency. 341 00:22:07,544 --> 00:22:11,357 And it may be a cocktail of more than one mode. 342 00:22:11,357 --> 00:22:14,839 That is a musical instrument. Not the best, 343 00:22:14,839 --> 00:22:19,896 but it is a musical instrument: closed and open on one end and 344 00:22:19,896 --> 00:22:23,129 the other. So, now, it is important that 345 00:22:23,129 --> 00:22:28,020 we get some feeling for the frequencies that you can produce 346 00:22:28,020 --> 00:22:32,000 with the various wind instruments. 347 00:22:32,000 --> 00:22:34,134 And so, here, I have listed for you the 348 00:22:34,134 --> 00:22:35,369 length, L. And this is, 349 00:22:35,369 --> 00:22:38,514 then, an open-open system. Of course, it's the same for a 350 00:22:38,514 --> 00:22:41,153 closed-closed system. But a closed-closed system 351 00:22:41,153 --> 00:22:44,692 doesn't make a very nice musical instrument because the sound is 352 00:22:44,692 --> 00:22:46,826 not coming out. So that's not the idea. 353 00:22:46,826 --> 00:22:48,454 So, therefore, it's open-open. 354 00:22:48,454 --> 00:22:51,262 Or, it is closed-open. And you see here the length. 355 00:22:51,262 --> 00:22:53,789 And, of course, the speed of sound is a given. 356 00:22:53,789 --> 00:22:55,811 So, I can't change that. And this is, 357 00:22:55,811 --> 00:22:58,114 then, the fundamental, the first harmonic, 358 00:22:58,114 --> 00:23:01,483 covering the entire range of your hearing all the way from 17 359 00:23:01,483 --> 00:23:05,783 Hz. So, you would take a 10 m long 360 00:23:05,783 --> 00:23:11,257 tube open on both sides to get in the fundamental the 17 Hz. 361 00:23:11,257 --> 00:23:16,360 And for a closed-open system, that would then be easier. 362 00:23:16,360 --> 00:23:21,742 So you would get away with half the length to get the same. 363 00:23:21,742 --> 00:23:24,711 You would get half the frequency. 364 00:23:24,711 --> 00:23:30,000 That's basically what a closed-open system does. 365 00:23:30,000 --> 00:23:35,231 And so, we'll keep that on so that we can understand when we 366 00:23:35,231 --> 00:23:40,108 hear the various frequencies, why the frequencies are as 367 00:23:40,108 --> 00:23:45,162 higher as low as they are. Now here, we have a tuning fork 368 00:23:45,162 --> 00:23:48,975 which is 256 Hz, which is mounted on a sound 369 00:23:48,975 --> 00:23:53,940 cavity which is closed on one side and open on the other. 370 00:23:53,940 --> 00:23:59,261 And that is done in order to get perfect matching between the 371 00:23:59,261 --> 00:24:03,428 resonance frequency, the fundamental of the box, 372 00:24:03,428 --> 00:24:07,539 and the 256 Hz. You can do that, 373 00:24:07,539 --> 00:24:12,639 of course, if you're only interested in one particular 374 00:24:12,639 --> 00:24:15,333 frequency. So, for this case, 375 00:24:15,333 --> 00:24:19,374 then, you would have that the frequency, F, 376 00:24:19,374 --> 00:24:23,415 is the velocity. So we go to this case now, 377 00:24:23,415 --> 00:24:27,938 to the case of this equation. So, the frequency, 378 00:24:27,938 --> 00:24:34,000 this is the first harmonic, now, is V divided by 4L. 379 00:24:34,000 --> 00:24:36,666 So, that is 256. So, that is 340, 380 00:24:36,666 --> 00:24:39,000 approximately, divided by 4L. 381 00:24:39,000 --> 00:24:42,250 And then, L comes out to be about 33 cm. 382 00:24:42,250 --> 00:24:45,666 And, indeed, if you measure the lengths of 383 00:24:45,666 --> 00:24:49,333 this, you will find that it is roughly 33 cm. 384 00:24:49,333 --> 00:24:54,250 And so, now you have a case where you have perfect matching. 385 00:24:54,250 --> 00:24:58,833 So, when I hear this tuning fork, then the box starts to 386 00:24:58,833 --> 00:25:03,000 oscillate exactly at that frequency. 387 00:25:03,000 --> 00:25:07,296 So you get a very large sound. So you drain the energy very 388 00:25:07,296 --> 00:25:10,037 quickly. You do see often tuning forks 389 00:25:10,037 --> 00:25:12,703 which are mounted, instead, this way. 390 00:25:12,703 --> 00:25:16,777 You can also have a box like this, open and open on both 391 00:25:16,777 --> 00:25:19,148 sides. Then it has to be twice as 392 00:25:19,148 --> 00:25:21,518 long. So, for very high frequency 393 00:25:21,518 --> 00:25:24,481 tuning forks, you often see it open-open. 394 00:25:24,481 --> 00:25:28,777 If we have to do it for this one, it would have to be twice 395 00:25:28,777 --> 00:25:32,925 as long, and that is just unpractical, and not necessary, 396 00:25:32,925 --> 00:25:37,114 of course. Now, with musical instruments, 397 00:25:37,114 --> 00:25:39,864 you produce an infinite number of tones. 398 00:25:39,864 --> 00:25:42,825 So there is no way that you can, of course, 399 00:25:42,825 --> 00:25:47,125 design your sounding board that it resonates with every single 400 00:25:47,125 --> 00:25:49,592 frequency. And so, this is where the 401 00:25:49,592 --> 00:25:52,130 secrecy of the manufacturer comes in. 402 00:25:52,130 --> 00:25:55,655 And they're not going to tell you how they do that. 403 00:25:55,655 --> 00:25:58,545 And therefore, some instruments are better 404 00:25:58,545 --> 00:26:02,000 than others. And you pay for that. 405 00:26:02,000 --> 00:26:06,196 If you have a barrel filled with liquid, I remember that in 406 00:26:06,196 --> 00:26:10,320 France they did this technique that I'm telling you about, 407 00:26:10,320 --> 00:26:14,444 and the wine was up to here, say, and they wanted to know, 408 00:26:14,444 --> 00:26:16,832 you couldn't look into the barrel. 409 00:26:16,832 --> 00:26:20,160 You wanted to know where the level of wine was. 410 00:26:20,160 --> 00:26:23,922 They would knock on this. And the sound here would be 411 00:26:23,922 --> 00:26:27,974 very different from the sound here because here's air and 412 00:26:27,974 --> 00:26:31,931 here's liquid. So it's obvious that where you 413 00:26:31,931 --> 00:26:35,343 have liquid the residences are very different from the 414 00:26:35,343 --> 00:26:38,369 residences here. So, that's one way that you can 415 00:26:38,369 --> 00:26:40,557 tell what the level is of the wine. 416 00:26:40,557 --> 00:26:43,776 When you have a cold, which I happen to have today, 417 00:26:43,776 --> 00:26:47,639 I remember as a kid I would go to the doctor and he would ask 418 00:26:47,639 --> 00:26:50,343 me to inhale. And then he would knock on my 419 00:26:50,343 --> 00:26:52,596 chest. And I would exhale and inhale 420 00:26:52,596 --> 00:26:55,815 and knock on my chest. And then he would be able to 421 00:26:55,815 --> 00:27:00,000 tell whether there is fluid in my lungs are not. 422 00:27:00,000 --> 00:27:02,812 I don't know how sensitive that is. 423 00:27:02,812 --> 00:27:06,864 It's not my field, but that's the way it was done. 424 00:27:06,864 --> 00:27:10,586 And maybe you have experienced the same thing, 425 00:27:10,586 --> 00:27:13,646 that sometimes also, I don't know why, 426 00:27:13,646 --> 00:27:17,864 they knock on your back even. Now, I have developed, 427 00:27:17,864 --> 00:27:22,413 in fact I can even say invented a method to test for the 428 00:27:22,413 --> 00:27:26,466 presence of brains. And, the way I do that is that 429 00:27:26,466 --> 00:27:31,263 I strike a tuning fork and put the tuning fork on someone's 430 00:27:31,263 --> 00:27:35,344 head. And you can imagine if it's 431 00:27:35,344 --> 00:27:40,553 empty here, then you get the same effect as you have what the 432 00:27:40,553 --> 00:27:44,025 wine barrel. And then you hear very clear 433 00:27:44,025 --> 00:27:47,064 sound. But if there are brains here, 434 00:27:47,064 --> 00:27:49,842 like the liquid, you get nothing. 435 00:27:49,842 --> 00:27:54,617 And I have patented that. So you can't try that at home. 436 00:27:54,617 --> 00:27:57,308 But I can demonstrate it to you. 437 00:27:57,308 --> 00:28:00,000 You hear much? Oh. 438 00:28:00,000 --> 00:28:04,113 You hear much? Maybe I shouldn't be lecturing 439 00:28:04,113 --> 00:28:06,543 8.03, then. Now, of course, 440 00:28:06,543 --> 00:28:11,311 in my case, I think I passed the test within reason. 441 00:28:11,311 --> 00:28:15,237 It is not so clear, of course, that all the 442 00:28:15,237 --> 00:28:20,566 students would pass that test. And I was wondering whether 443 00:28:20,566 --> 00:28:24,492 anyone has the courage that I may try that. 444 00:28:24,492 --> 00:28:27,203 Anyone of you? You are afraid, 445 00:28:27,203 --> 00:28:30,662 right? You're worried that it will all 446 00:28:30,662 --> 00:28:35,470 come out. Do you mind? 447 00:28:35,470 --> 00:28:42,676 You don't mind. You're a strong man. 448 00:28:42,676 --> 00:28:52,352 You have to be quiet, otherwise I can't tell the 449 00:28:52,352 --> 00:29:01,000 difference. [LAUGHTER] I say no more. 450 00:29:01,000 --> 00:29:05,000 I think it's better that I teach at 8.03 than you do. 451 00:29:05,000 --> 00:29:10,000 452 00:29:10,000 --> 00:29:14,135 So, when you design a wood instrument, the only thing you 453 00:29:14,135 --> 00:29:18,270 have to play with is L because T is no longer negotiable. 454 00:29:18,270 --> 00:29:22,406 And so, with organ pipes, when you go to churches and you 455 00:29:22,406 --> 00:29:25,655 see organ pipes, you see a whole zoo of these 456 00:29:25,655 --> 00:29:28,461 organ pipes: open-open and closed-open. 457 00:29:28,461 --> 00:29:31,784 And every fundamental that you want to excite, 458 00:29:31,784 --> 00:29:36,114 you need one pipe. So, it's a huge number of 459 00:29:36,114 --> 00:29:37,891 types. It's, in a way, 460 00:29:37,891 --> 00:29:40,851 like the piano. If you take a flute, 461 00:29:40,851 --> 00:29:45,842 then you make it longer and shorter simply by drilling holes 462 00:29:45,842 --> 00:29:48,803 in it. And if you hold your hands on 463 00:29:48,803 --> 00:29:53,794 both holes, then this is the length of the flute which gives 464 00:29:53,794 --> 00:29:58,785 you, then, a lower frequency than when you take one hand off 465 00:29:58,785 --> 00:30:04,854 because now it's shorter. This is connected with the 466 00:30:04,854 --> 00:30:09,575 universe, so here, no pressure can build up. 467 00:30:09,575 --> 00:30:12,979 So this becomes a pressure node. 468 00:30:12,979 --> 00:30:17,919 So it's shorter. And if you take this one off, 469 00:30:17,919 --> 00:30:23,080 it's even shorter, and so the pitch will go even 470 00:30:23,080 --> 00:30:26,703 up. That's the basic idea behind a 471 00:30:26,703 --> 00:30:30,655 flute. And, I can demonstrate that to 472 00:30:30,655 --> 00:30:34,278 you. [MUSIC PLAYS] Do you hear the 473 00:30:34,278 --> 00:30:39,000 pitch go down? [MUSIC PLAYS] 474 00:30:39,000 --> 00:30:43,344 I have here a flute-like instrument which is open on this 475 00:30:43,344 --> 00:30:47,922 side, and it's also open on here where we passed the air by. 476 00:30:47,922 --> 00:30:52,732 So, you can consider this to a very good approximation is open, 477 00:30:52,732 --> 00:30:54,206 open. It is 16.6 cm, 478 00:30:54,206 --> 00:30:58,086 and so if you want to know what the fundamental is, 479 00:30:58,086 --> 00:31:01,810 well, then you apply this equation, N equals one. 480 00:31:01,810 --> 00:31:06,000 And then, you get something like 1,024 Hz. 481 00:31:06,000 --> 00:31:11,727 But, if you may get closed at one end, and you apply this 482 00:31:11,727 --> 00:31:15,613 equation, you would only get half that. 483 00:31:15,613 --> 00:31:20,215 So you only get 512 Hz. It's a big difference, 484 00:31:20,215 --> 00:31:23,488 factors of two. So this is, then, 485 00:31:23,488 --> 00:31:29,318 the fundamental for open-open. [MUSIC PLAYS] And now comes 486 00:31:29,318 --> 00:31:33,000 open-closed. [MUSIC PLAYS] 487 00:31:33,000 --> 00:31:36,477 Factor of two difference. I have, here, 488 00:31:36,477 --> 00:31:40,962 the very special tube. It is open and open on both 489 00:31:40,962 --> 00:31:46,088 sides, and if you don't believe me, Annie, I can see you. 490 00:31:46,088 --> 00:31:48,193 Can you see me? Can you? 491 00:31:48,193 --> 00:31:52,037 Can you see me open and open on both sides? 492 00:31:52,037 --> 00:31:56,064 And it is corrugated, which of course is very 493 00:31:56,064 --> 00:32:02,431 important why it works so well. And this has a length of 77 cm. 494 00:32:02,431 --> 00:32:06,431 And so, you can calculate, now, using this equation, 495 00:32:06,431 --> 00:32:09,333 N equals one, what the fundamental is. 496 00:32:09,333 --> 00:32:13,725 And, the fundamental is about 220 Hz, the first harmonic. 497 00:32:13,725 --> 00:32:18,352 And, since it's open-open on both sides, the second harmonic 498 00:32:18,352 --> 00:32:21,803 will be 440 Hz. In the third harmonic is 660. 499 00:32:21,803 --> 00:32:25,568 And, by twirling this around, with a little luck, 500 00:32:25,568 --> 00:32:29,882 you can get a wind flow past here which only excites the 501 00:32:29,882 --> 00:32:34,313 lowest mode. Sometimes you hear the lowest 502 00:32:34,313 --> 00:32:38,222 one and the second. But, when you twirl it faster, 503 00:32:38,222 --> 00:32:42,609 you get higher harmonics. And I want to demonstrate this 504 00:32:42,609 --> 00:32:45,561 to you. I will first try to excite the 505 00:32:45,561 --> 00:32:50,108 220, which is the lowest mode possible, which is the first 506 00:32:50,108 --> 00:32:54,655 harmonic, the fundamental. And, I'll try to make it clean, 507 00:32:54,655 --> 00:32:57,527 just only the fundamental. That's it. 508 00:32:57,527 --> 00:33:07,500 It's about 220 Hz. [MUSIC PLAYS] 440, 509 00:33:07,500 --> 00:33:24,583 660, 880, 660, 440, I can't get above 880. 510 00:33:24,583 --> 00:33:31,842 [APPLAUSE] There are ways that you can 511 00:33:31,842 --> 00:33:34,434 change the length of a wind instrument. 512 00:33:34,434 --> 00:33:38,051 One is by making holes in it. Another one is by really 513 00:33:38,051 --> 00:33:42,076 physically changing the length. And there was one instrument 514 00:33:42,076 --> 00:33:45,420 which is well-known for that, which is a trombone. 515 00:33:45,420 --> 00:33:49,172 So, you actually changed, this is a system which is open 516 00:33:49,172 --> 00:33:51,901 and it is closed at the end in this case. 517 00:33:51,901 --> 00:33:54,358 You can see that. It's like a piston, 518 00:33:54,358 --> 00:33:57,223 so it's closed. And so, now by changing the 519 00:33:57,223 --> 00:34:02,000 length of the cavity, you can change the fundamental. 520 00:34:02,000 --> 00:34:30,000 So, we get it this way. [MUSIC PLAYS] [APPLAUSE] 521 00:34:30,000 --> 00:34:33,773 We recognize instruments by the cocktail of the harmonics that 522 00:34:33,773 --> 00:34:36,247 they generate, and depending upon how you 523 00:34:36,247 --> 00:34:38,412 excite them. I mentioned already the 524 00:34:38,412 --> 00:34:40,762 plucking of the harp, but also the way, 525 00:34:40,762 --> 00:34:42,742 for instance, you blow on musical 526 00:34:42,742 --> 00:34:46,020 instruments, I don't know whether any of you have ever 527 00:34:46,020 --> 00:34:49,546 tried to play on the trumpet, but if I gave you a trumpet, 528 00:34:49,546 --> 00:34:53,010 chances are that you would get no sound out of it at all. 529 00:34:53,010 --> 00:34:55,670 You have to hold your lips in a special way. 530 00:34:55,670 --> 00:35:00,000 You have to know how to spit in there the right way. 531 00:35:00,000 --> 00:35:03,373 It goes like [MAKES NOISE], something like that. 532 00:35:03,373 --> 00:35:07,823 So, that is also a way that you can excite certain harmonics in 533 00:35:07,823 --> 00:35:10,910 relation to the fundamental. And that, then, 534 00:35:10,910 --> 00:35:14,283 determines the sound quality of your instrument. 535 00:35:14,283 --> 00:35:18,518 And, I'd like to demonstrate this to you now in various ways 536 00:35:18,518 --> 00:35:22,753 that the different instruments have different sound quality. 537 00:35:22,753 --> 00:35:26,916 What it comes down to is that if you ask each instrument to 538 00:35:26,916 --> 00:35:29,141 play, for instance, 440 Hz tone, 539 00:35:29,141 --> 00:35:33,376 you will see that the violin cannot just simply produce 440, 540 00:35:33,376 --> 00:35:36,822 but it will automatically also, at the same time, 541 00:35:36,822 --> 00:35:39,693 generate 880, and made that as the second 542 00:35:39,693 --> 00:35:44,000 harmonic, and maybe a higher harmonic. 543 00:35:44,000 --> 00:35:47,467 And that's different for different instruments. 544 00:35:47,467 --> 00:35:50,934 And that's, of course, the idea behind the tone 545 00:35:50,934 --> 00:35:53,874 quality. And the way that we're going to 546 00:35:53,874 --> 00:35:56,437 demonstrate that you is as follows. 547 00:35:56,437 --> 00:36:00,932 We have here a microphone. And I will first make you 548 00:36:00,932 --> 00:36:03,791 listen to a tuning fork, for instance, 549 00:36:03,791 --> 00:36:08,274 440 Hz, which will give you, then, this signal [times?] the 550 00:36:08,274 --> 00:36:11,597 amplitude of the membrane of the microphone. 551 00:36:11,597 --> 00:36:14,843 If, however, an instrument were to generate 552 00:36:14,843 --> 00:36:19,557 440 in addition higher harmonic, you would get the sum of this 553 00:36:19,557 --> 00:36:22,880 signal and, of course, the higher harmonics. 554 00:36:22,880 --> 00:36:26,435 And so, you will see no longer a nice sinusoid. 555 00:36:26,435 --> 00:36:32,000 But, you see on top of the sinusoid the higher harmonics. 556 00:36:32,000 --> 00:36:36,382 So, let me now, first, show you the 440 Hz. 557 00:36:36,382 --> 00:36:40,869 Here's the microphone. [SOUND PLAYS] Boring: 558 00:36:40,869 --> 00:36:45,147 just one frequency. Nothing rich about it. 559 00:36:45,147 --> 00:36:50,052 256 [SOUND PLAYS], there's no indication for any 560 00:36:50,052 --> 00:36:54,539 higher harmonics. That's the way that tuning 561 00:36:54,539 --> 00:36:58,713 forks are designed. If I take this flute, 562 00:36:58,713 --> 00:37:04,452 [SOUND PLAYS] as far as I can tell only the fundamental, 563 00:37:04,452 --> 00:37:10,473 no higher harmonics. Now, maybe if you blow in a 564 00:37:10,473 --> 00:37:14,320 very special way, maybe I can excite higher 565 00:37:14,320 --> 00:37:16,702 harmonics. But this is now, 566 00:37:16,702 --> 00:37:22,106 oh, I want to show you 4,000 Hz so that you can easily hear. 567 00:37:22,106 --> 00:37:25,496 This is 4,000 Hz. [SOUND PLAYS] Right, 568 00:37:25,496 --> 00:37:29,801 so you get the idea, so you know now what you're 569 00:37:29,801 --> 00:37:34,434 going to see. But you don't know yet what 570 00:37:34,434 --> 00:37:37,500 you're going to hear. Neither do I. 571 00:37:37,500 --> 00:37:41,016 We have six students who are very brave. 572 00:37:41,016 --> 00:37:44,713 I would like all of them to come down now. 573 00:37:44,713 --> 00:37:48,860 And, they're going to demonstrate their musical 574 00:37:48,860 --> 00:37:52,196 instruments. So, all of you come down, 575 00:37:52,196 --> 00:37:55,352 please. Yes, you got your instrument 576 00:37:55,352 --> 00:37:57,245 there. Just come here, 577 00:37:57,245 --> 00:37:59,409 that's fine. Don't worry. 578 00:37:59,409 --> 00:38:02,385 Come here. All right, where is the 579 00:38:02,385 --> 00:38:07,360 violinist? There's the violinist. 580 00:38:07,360 --> 00:38:09,720 Yeah, oh, boy. Oh, boy. 581 00:38:09,720 --> 00:38:15,942 [LAUGHTER] The first person who is going to demonstrate the 582 00:38:15,942 --> 00:38:19,804 violin is going to be Mark [Avarro?]. 583 00:38:19,804 --> 00:38:25,704 And, I'm going to ask Mark first to only produce what he 584 00:38:25,704 --> 00:38:28,172 thinks, then, is 440 Hz. 585 00:38:28,172 --> 00:38:33,000 We have, for you, a special chair. 586 00:38:33,000 --> 00:38:37,212 We will bring the chair very shortly. 587 00:38:37,212 --> 00:38:41,659 So let me remind you, then, of the 440. 588 00:38:41,659 --> 00:38:45,872 This is 440. Come a little closer and 589 00:38:45,872 --> 00:38:48,446 produce 440. Now, look. 590 00:38:48,446 --> 00:38:54,765 Did you see that you could really see the 440 in there? 591 00:38:54,765 --> 00:39:01,904 It was the same spacing as my tuning fork, but there were many 592 00:39:01,904 --> 00:39:08,580 higher harmonics there. So it [wants?] more. 593 00:39:08,580 --> 00:39:14,000 And they, then, give you all these wiggles. 594 00:39:14,000 --> 00:39:21,483 Now, the audience is yours. And now, I'll give you 20 or 30 595 00:39:21,483 --> 00:39:27,935 seconds, anything you want. Play anything you love. 596 00:39:27,935 --> 00:40:00,000 Go ahead. [MUSIC PLAYS] 597 00:40:00,000 --> 00:40:03,857 You are fantastic. [APPLAUSE] Is Sharon here? 598 00:40:03,857 --> 00:40:08,942 Sharon, so, Sharon is going to demonstrate to us the flute. 599 00:40:08,942 --> 00:40:12,361 And, maybe you can show it to the class. 600 00:40:12,361 --> 00:40:15,780 See, her flute has more holes than mine. 601 00:40:15,780 --> 00:40:19,726 I only have two, and she has quite a few more. 602 00:40:19,726 --> 00:40:23,320 And she opens and closes them with valves. 603 00:40:23,320 --> 00:40:28,493 Sharon, you may not be able to exactly get the 440 with this 604 00:40:28,493 --> 00:40:36,009 instrument, but that's OK. Come close and show us your 605 00:40:36,009 --> 00:40:38,089 440. Excellent. 606 00:40:38,089 --> 00:40:44,178 Did you see? You could really see the 440, 607 00:40:44,178 --> 00:40:48,930 but you could see more than that. 608 00:40:48,930 --> 00:40:55,168 It separates it from the violin, of course. 609 00:40:55,168 --> 00:41:02,000 The audience is yours. Make us happy. 610 00:41:02,000 --> 00:41:17,125 [MUSIC PLAYS] Terrific. 611 00:41:17,125 --> 00:41:22,625 [APPLAUSE] Now, we have a very special guest, 612 00:41:22,625 --> 00:41:29,250 which is Shawn [Augin?], who was so kind to go through 613 00:41:29,250 --> 00:41:33,375 the trouble of bringing the cello. 614 00:41:33,375 --> 00:41:39,000 Now, you think, Shawn, that you can come close 615 00:41:39,000 --> 00:41:41,000 to 440? Try that. 616 00:41:41,000 --> 00:41:46,250 And if you can do it just in a fundamental. 617 00:41:46,250 --> 00:41:53,750 You may have to tell us whether you had to shorten the string 618 00:41:53,750 --> 00:41:58,750 with your finger. You do have to do that? 619 00:41:58,750 --> 00:42:05,612 Go ahead. [MUSIC PLAYS] Try it once more. 620 00:42:05,612 --> 00:42:12,488 I can see the underlying 440. And, you shortened the string? 621 00:42:12,488 --> 00:42:16,800 Oh, there was no finger on the string? 622 00:42:16,800 --> 00:42:23,793 So, it is the fundamental of that string, the middle A of the 623 00:42:23,793 --> 00:42:26,124 piano. Show us yours. 624 00:42:26,124 --> 00:42:30,203 You practiced yours? Oh, you didn't, 625 00:42:30,203 --> 00:42:35,673 good for you. So, you prefer not to play? 626 00:42:35,673 --> 00:42:38,841 That's fine. Thank you very much. 627 00:42:38,841 --> 00:42:43,792 Now we have a Chinese clarinet from [Nyugen Choo?], 628 00:42:43,792 --> 00:42:49,732 and she told me that the nice thing about it is that it's not 629 00:42:49,732 --> 00:42:53,792 really a clarinet. So, can you make a 440? 630 00:42:53,792 --> 00:42:59,732 Well, since it's on a clarinet, I think you have good reasons 631 00:42:59,732 --> 00:43:03,000 to be off. OK, try it. 632 00:43:03,000 --> 00:43:12,763 [Use your shoulder clamp?]. See all these holes in here? 633 00:43:12,763 --> 00:43:20,396 One, two, three, four, it even says F there. 634 00:43:20,396 --> 00:43:28,029 [MUSIC PLAYS] Very nice. I do see overtones. 635 00:43:28,029 --> 00:43:33,000 The audience is yours. 636 00:43:33,000 --> 00:43:46,000 637 00:43:46,000 --> 00:43:56,548 [MUSIC PLAYS] It's all right. 638 00:43:56,548 --> 00:44:05,537 I also did very poorly on the trombone. 639 00:44:05,537 --> 00:44:15,000 That's the way it goes. [MUSIC PLAYS] 640 00:44:15,000 --> 00:44:57,000 You're doing great. [MUSIC PLAYS] 641 00:44:57,000 --> 00:45:09,114 Wonderful. [APPLAUSE] Where is the saxophone? 642 00:45:09,114 --> 00:45:14,125 There is the saxophone. Colin Johnson. 643 00:45:14,125 --> 00:45:18,052 Colin, 440. [MUSIC PLAYS] Man. 644 00:45:18,052 --> 00:45:22,791 You got all the way down there? 440. 645 00:45:22,791 --> 00:45:28,479 [MUSIC PLAYS] You can't see it, but we can. 646 00:45:28,479 --> 00:45:41,818 That's very nice. If you stay there, 647 00:45:41,818 --> 00:46:02,000 the audience is yours. [MUSIC PLAYS] 648 00:46:02,000 --> 00:46:04,975 Terrific. [APPLAUSE] Now, 649 00:46:04,975 --> 00:46:07,826 you get the hero, Aston. 650 00:46:07,826 --> 00:46:14,768 Aston is going to play the 440 Hz tone on his percussion. 651 00:46:14,768 --> 00:46:19,727 Show us yours. But, we had email exchange 652 00:46:19,727 --> 00:46:25,553 about it, and you said you could produce 440 Hz. 653 00:46:25,553 --> 00:46:32,000 Can you show the percussion for one thing? 654 00:46:32,000 --> 00:46:36,821 Oh, no, we're going to do percussions, right? 655 00:46:36,821 --> 00:46:40,000 Go ahead. Small percussion. 656 00:46:40,000 --> 00:46:50,000 657 00:46:50,000 --> 00:46:56,500 [SOUND PLAYS] I think it's nowhere near 440. 658 00:46:56,500 --> 00:47:01,516 What do you think? Oh, it has 440 overtones. 659 00:47:01,516 --> 00:47:05,250 It can never have a 440 overtone. 660 00:47:05,250 --> 00:47:08,866 The overtones are always higher. 661 00:47:08,866 --> 00:47:12,133 That maybe an undertone, 440. 662 00:47:12,133 --> 00:47:16,333 Go ahead. [SOUND PLAYS] Let's call it 663 00:47:16,333 --> 00:47:20,183 440. Well, that's why you're here, 664 00:47:20,183 --> 00:47:23,099 right? Man, we can't wait. 665 00:47:23,099 --> 00:47:42,000 20 seconds, it's your show. [MUSIC PLAYS] 666 00:47:42,000 --> 00:47:49,916 That's closer to 440 than what I've seen. 667 00:47:49,916 --> 00:47:57,635 You could see it. We need one more bang. 668 00:47:57,635 --> 00:48:02,462 [SOUND PLAYS] Thank you all. 669 00:48:02,462 --> 00:48:05,699 Great. [APPLAUSE] If there's any 670 00:48:05,699 --> 00:48:11,129 singer in the audience who wants to try, be my guest. 671 00:48:11,129 --> 00:48:15,724 I think this is a great moment for the break. 672 00:48:15,724 --> 00:48:21,050 While we have a break, feel free to play around with 673 00:48:21,050 --> 00:48:25,958 the wind organs, and also with the tuning forks, 674 00:48:25,958 --> 00:48:31,075 be a little careful. You may also try the trombone 675 00:48:31,075 --> 00:48:35,058 and the flutes. And then, five minutes from 676 00:48:35,058 --> 00:48:36,000 now, we will reconvene. 677 00:48:36,000 --> 00:52:41,000 678 00:52:41,000 --> 00:52:49,571 So, what we have been looking at our one-dimensional standing 679 00:52:49,571 --> 00:52:58,000 waves, normal mode solutions to the famous wave equation. 680 00:52:58,000 --> 00:53:02,947 In the case of transverse motion, we have D2Y, 681 00:53:02,947 --> 00:53:08,994 DX squared is one over V squared times D2Y [G?] squared. 682 00:53:08,994 --> 00:53:14,931 So, this is the wave equation, Y now in this direction. 683 00:53:14,931 --> 00:53:19,000 So, for instance, if we have a string, 684 00:53:19,000 --> 00:53:25,267 and we have a string with length L and it is fixed on both 685 00:53:25,267 --> 00:53:31,204 ends than the boundary conditions determine the precise 686 00:53:31,204 --> 00:53:36,481 allowed solutions. We call them [igan?] states in 687 00:53:36,481 --> 00:53:42,258 quantum mechanics. And I can write down, 688 00:53:42,258 --> 00:53:47,207 now, the displacement, Y, in the N'th mode as a 689 00:53:47,207 --> 00:53:52,478 function of X and T. And, we give the amplitude of 690 00:53:52,478 --> 00:53:56,996 the N'th mode, say, a value that can differ 691 00:53:56,996 --> 00:54:00,868 from mode to mode. So, we get A of N, 692 00:54:00,868 --> 00:54:07,000 and then we get the sine of N pi X divided by L. 693 00:54:07,000 --> 00:54:10,956 And then, we get the cosine of omega N times T. 694 00:54:10,956 --> 00:54:15,946 So, this is the time domain, which makes it shake with that 695 00:54:15,946 --> 00:54:19,301 frequency. And this is now the part that 696 00:54:19,301 --> 00:54:22,741 makes sure that the N's are always fixed. 697 00:54:22,741 --> 00:54:25,838 If you make X equals zero, Y is zero. 698 00:54:25,838 --> 00:54:29,021 And if you make X equals L, Y is zero. 699 00:54:29,021 --> 00:54:33,236 And, that gives you, then, a whole zoo of possible 700 00:54:33,236 --> 00:54:39,000 frequencies because omega N is always V times K of N. 701 00:54:39,000 --> 00:54:44,666 And so, that then becomes V times N pi divided by L. 702 00:54:44,666 --> 00:54:49,444 And so, you see the ratios, one, two, three, 703 00:54:49,444 --> 00:54:53,222 four, answer one NVL. That's right. 704 00:54:53,222 --> 00:55:00,000 Now if we go the [SOUND OFF/THEN ON] of longitudinal -- 705 00:55:00,000 --> 00:55:09,000 706 00:55:09,000 --> 00:55:13,057 -- then if we have a system which is open-open like we've 707 00:55:13,057 --> 00:55:15,739 discussed before, and it has length L, 708 00:55:15,739 --> 00:55:19,869 then you get exactly the same solutions, provided that you 709 00:55:19,869 --> 00:55:24,144 replace what we have Y there, you have to replace that by P, 710 00:55:24,144 --> 00:55:27,695 P being the overpressure because at the ends of an 711 00:55:27,695 --> 00:55:31,028 open-open system, the pressure can never become 712 00:55:31,028 --> 00:55:36,063 higher or lower than ambient. And so, you have pressure 713 00:55:36,063 --> 00:55:38,787 nodes. And so, you get exactly the 714 00:55:38,787 --> 00:55:43,657 same solution that you have there, provided that you replace 715 00:55:43,657 --> 00:55:48,196 this by P, and this you replace maybe by some capital P, 716 00:55:48,196 --> 00:55:52,984 which is then the amplitude. But, this and this is the same 717 00:55:52,984 --> 00:55:56,203 except, of course, that omega, this V is 718 00:55:56,203 --> 00:56:00,000 nonnegotiable in the case of sound. 719 00:56:00,000 --> 00:56:04,954 If you have a closed system, so now you close one end. 720 00:56:04,954 --> 00:56:10,189 Then you want the boundary conditions at P is zero at the 721 00:56:10,189 --> 00:56:15,331 open end, and P has to be an anti-node at the closed end 722 00:56:15,331 --> 00:56:19,725 where the pressure can build up. That gives you, 723 00:56:19,725 --> 00:56:23,277 then, different values for this K of N. 724 00:56:23,277 --> 00:56:28,699 The sine may even have to be changed to a cosine it depends 725 00:56:28,699 --> 00:56:33,000 on where you define X equals zero. 726 00:56:33,000 --> 00:56:38,303 That's not so important now. But what is important now that 727 00:56:38,303 --> 00:56:43,973 K of N in that equation now gets the form 2N minus one times pi 728 00:56:43,973 --> 00:56:47,539 divided by 2L. So, everything comes from 729 00:56:47,539 --> 00:56:52,203 boundary conditions. And now I want to make the step 730 00:56:52,203 --> 00:56:55,769 from one-dimensional to two-dimensional. 731 00:56:55,769 --> 00:56:59,884 If we go two-dimensional, which is easy to do, 732 00:56:59,884 --> 00:57:04,000 and we can even demonstrate that. 733 00:57:04,000 --> 00:57:07,486 So, this has length LX, and this has length LY. 734 00:57:07,486 --> 00:57:10,669 It could be a [frame?] with a [soap film?], 735 00:57:10,669 --> 00:57:13,019 a membrane, which can oscillate. 736 00:57:13,019 --> 00:57:17,339 And, we'll assume that the membrane is attached everywhere 737 00:57:17,339 --> 00:57:20,144 to the frame. So, that is the boundary 738 00:57:20,144 --> 00:57:24,616 conditions, everywhere attached. And so, we want to know now 739 00:57:24,616 --> 00:57:27,723 what the normal modes are for this system. 740 00:57:27,723 --> 00:57:31,513 So now, our one-dimensional wave equation becomes a 741 00:57:31,513 --> 00:57:37,678 two-dimensional wave equation. And so, if this is my 742 00:57:37,678 --> 00:57:43,673 coordinate system, XYZ, then now we get that D2Z 743 00:57:43,673 --> 00:57:51,454 DX squared plus D2Z DY squared is now one over V squared times 744 00:57:51,454 --> 00:57:57,321 D2Z DT squared. That, now, is a two dimensional 745 00:57:57,321 --> 00:58:02,175 wave equation. And when you put in your 746 00:58:02,175 --> 00:58:05,711 boundary conditions, which in this case, 747 00:58:05,711 --> 00:58:10,063 suppose we call this zero, so, for X equals zero, 748 00:58:10,063 --> 00:58:14,325 N for X equals LX, you want Z to be zero because 749 00:58:14,325 --> 00:58:19,311 the membrane cannot move. And, for Y equals zero and for 750 00:58:19,311 --> 00:58:22,847 Y equals LY, you also want Z to be zero. 751 00:58:22,847 --> 00:58:26,384 Then, you can now, in analogy with that, 752 00:58:26,384 --> 00:58:31,552 you can write down now the solution for Z as a function of 753 00:58:31,552 --> 00:58:36,728 X, Y, and T. Now you had to introduce to 754 00:58:36,728 --> 00:58:42,386 quantum numbers if I call N Nancy and M a quantum number. 755 00:58:42,386 --> 00:58:47,439 So, now, you can get Z, and now I have two integers 756 00:58:47,439 --> 00:58:50,975 that can change. So, M now goes one, 757 00:58:50,975 --> 00:58:53,905 two, three, etc. Nancy is one, 758 00:58:53,905 --> 00:58:58,452 two, three, etc. And this is now a function of 759 00:58:58,452 --> 00:59:03,000 X, Y, and T. It's two-dimensional. 760 00:59:03,000 --> 00:59:08,461 So, I have to introduce some kind of an amplitude, 761 00:59:08,461 --> 00:59:13,589 which can be different for the different modes, 762 00:59:13,589 --> 00:59:17,378 Mary Nancy. And now I get something 763 00:59:17,378 --> 00:59:22,840 extremely similar to this, except I get two sines. 764 00:59:22,840 --> 00:59:29,528 So, I get here sine of M pi X divided by LX times the sine of 765 00:59:29,528 --> 00:59:35,770 Nancy pi Y divided by L of Y times the cosine of omega MN 766 00:59:35,770 --> 00:59:40,117 times T. And you see that it meets the 767 00:59:40,117 --> 00:59:43,769 boundary conditions. If you put in X equals zero or 768 00:59:43,769 --> 00:59:46,033 X equals L of X, you get a zero. 769 00:59:46,033 --> 00:59:49,174 If you put Y equals zero or Y equals L of Y, 770 00:59:49,174 --> 00:59:53,645 you get your zero. And, if you substitute this 771 00:59:53,645 --> 00:59:57,957 solution, which we clearly, intuitively know, 772 00:59:57,957 --> 1:00:02,954 it has to be correct, if you substitute that in that 773 1:00:02,954 --> 1:00:06,972 wave equation, then you get the connection 774 1:00:06,972 --> 1:00:12,067 between the K values, N omega, and then you will find 775 1:00:12,067 --> 1:00:16,183 that omega MN, which is always the velocity 776 1:00:16,183 --> 1:00:22,356 times K value of M of N becomes, now, V times the square root of 777 1:00:22,356 --> 1:00:27,746 Mary pi divided by LX squared plus Nancy I divided by LY 778 1:00:27,746 --> 1:00:31,989 squared. This, now, sets the entire 779 1:00:31,989 --> 1:00:35,433 range of all possible normal mode frequencies. 780 1:00:35,433 --> 1:00:40,178 So, now you can play with Mary M and you can play with Nancy N, 781 1:00:40,178 --> 1:00:44,540 and you can cocktail that in all combination that you want 782 1:00:44,540 --> 1:00:46,147 to. And, for instance, 783 1:00:46,147 --> 1:00:50,204 if we ask this membrane to oscillate in M equals one N 784 1:00:50,204 --> 1:00:53,647 equals one mode, then it means that the entire 785 1:00:53,647 --> 1:00:56,403 membrane comes to you, and goes back, 786 1:00:56,403 --> 1:01:00,000 and comes to you, and goes back. 787 1:01:00,000 --> 1:01:02,719 That's the lowest possible mode. 788 1:01:02,719 --> 1:01:07,105 All points are in phase, and they all have the same 789 1:01:07,105 --> 1:01:10,701 frequency. That's the characteristic for a 790 1:01:10,701 --> 1:01:13,245 normal mode. If, for instance, 791 1:01:13,245 --> 1:01:17,105 we had Mary equals two, and Nancy equals one, 792 1:01:17,105 --> 1:01:20,877 then we would have in the direction of Mary, 793 1:01:20,877 --> 1:01:25,877 we would have plus and minus. This part would come to you, 794 1:01:25,877 --> 1:01:30,000 but this part goes away from you. 795 1:01:30,000 --> 1:01:34,632 So, a plus and a minus, and this now is a nodal line. 796 1:01:34,632 --> 1:01:39,265 So, we get a nodal line. So, it's not anymore a nodal 797 1:01:39,265 --> 1:01:43,630 point, which you get in one-dimensional solutions, 798 1:01:43,630 --> 1:01:47,461 but you get a nodal line. This stands still. 799 1:01:47,461 --> 1:01:51,826 And then, the membrane oscillates like this and is 800 1:01:51,826 --> 1:01:55,567 attached to the edge. And if, for instance, 801 1:01:55,567 --> 1:02:00,378 you wanted to know the Mary equals two Nancy equals two 802 1:02:00,378 --> 1:02:03,496 mode, so M equals two, N equals two, 803 1:02:03,496 --> 1:02:10,000 then you would get two nodal lines plus-minus minus-plus. 804 1:02:10,000 --> 1:02:13,348 If this comes out of the blackboard, this comes out of 805 1:02:13,348 --> 1:02:16,192 the blackboard, this goes into the blackboard, 806 1:02:16,192 --> 1:02:19,035 this goes into the blackboard. And, in French, 807 1:02:19,035 --> 1:02:22,574 you will see some remarkable, wonderful pictures of a guy 808 1:02:22,574 --> 1:02:26,618 with beautiful hair from the 50s whereby you see the oscillations 809 1:02:26,618 --> 1:02:29,082 of the soap film in these various modes. 810 1:02:29,082 --> 1:02:31,041 It's really very, very exciting, 811 1:02:31,041 --> 1:02:35,450 very interesting. And I want to demonstrate this 812 1:02:35,450 --> 1:02:39,371 to you in a way that I cannot possibly calculate. 813 1:02:39,371 --> 1:02:43,047 I doubt whether there are many people who can, 814 1:02:43,047 --> 1:02:46,887 although maybe some civil engineers can do that, 815 1:02:46,887 --> 1:02:51,788 because this system is way more complicated than the membrane 816 1:02:51,788 --> 1:02:55,628 that we have here that is attached to the frame. 817 1:02:55,628 --> 1:03:00,366 What we have is a square plate, and it is only fixed in the 818 1:03:00,366 --> 1:03:03,494 middle. That's where it's held. 819 1:03:03,494 --> 1:03:06,092 That's the only point that is not moving. 820 1:03:06,092 --> 1:03:07,976 That is here. But, what we do, 821 1:03:07,976 --> 1:03:10,965 we drive the thing. Actually, there is a piston 822 1:03:10,965 --> 1:03:13,628 below here which drives that middle point. 823 1:03:13,628 --> 1:03:16,941 And, it oscillates it. And then, you get an infinite 824 1:03:16,941 --> 1:03:20,580 number of frequencies whereby the system goes into normal 825 1:03:20,580 --> 1:03:22,723 mode. We can change the frequency. 826 1:03:22,723 --> 1:03:26,751 And the way we can make you see these frequencies is by putting 827 1:03:26,751 --> 1:03:30,000 a powder on it, which we will do. 828 1:03:30,000 --> 1:03:33,199 And then, the places where there are nodal lines, 829 1:03:33,199 --> 1:03:36,400 the powder stays. And, the places where there are 830 1:03:36,400 --> 1:03:39,400 anti-nodes, the power is literally thrown off. 831 1:03:39,400 --> 1:03:41,533 And, it accumulates at the nodes. 832 1:03:41,533 --> 1:03:44,599 And, that's the idea of the next demonstration. 833 1:03:44,599 --> 1:03:46,733 We call these the Chladni plates. 834 1:03:46,733 --> 1:03:50,199 And after the lecture, I would invite you to actually 835 1:03:50,199 --> 1:03:53,866 play here with this violin bow. So, here we have similar 836 1:03:53,866 --> 1:03:57,533 Chladni plates which have different boundary conditions. 837 1:03:57,533 --> 1:04:02,000 This is a square, triangle, and this is a circle. 838 1:04:02,000 --> 1:04:03,855 And, you can put stuff on there. 839 1:04:03,855 --> 1:04:07,386 And then, you can try to hit these resonant frequencies with 840 1:04:07,386 --> 1:04:09,181 the bow. And you get remarkable 841 1:04:09,181 --> 1:04:11,216 patterns. So, let's first go to our 842 1:04:11,216 --> 1:04:14,328 system, which is driven. So, that is very controlled. 843 1:04:14,328 --> 1:04:17,619 We can adjust the frequencies almost any way we want to. 844 1:04:17,619 --> 1:04:20,013 And, I will start with the low frequency. 845 1:04:20,013 --> 1:04:22,167 Sometimes the sound may be very high. 846 1:04:22,167 --> 1:04:24,980 I may have to adjust the sound. You can hear it. 847 1:04:24,980 --> 1:04:28,032 We'll start somewhere I think in the range of about, 848 1:04:28,032 --> 1:04:31,084 we are close to 300 Hz. And then, we'll increase the 849 1:04:31,084 --> 1:04:35,864 frequency. And I will add powder as 850 1:04:35,864 --> 1:04:40,637 necessary. And so, what you're looking at 851 1:04:40,637 --> 1:04:47,201 now a two-dimensional normal modes, which give you nodal 852 1:04:47,201 --> 1:04:50,065 lines. [SOUND PLAYS] Wow, 853 1:04:50,065 --> 1:04:55,078 I'm right on resonance. That's an accident. 854 1:04:55,078 --> 1:05:02,000 We didn't try that. That's a beautiful resonance. 855 1:05:02,000 --> 1:05:08,580 Who on earth could have predicted these nodal lines? 856 1:05:08,580 --> 1:05:12,580 I will go to a higher frequency. 857 1:05:12,580 --> 1:05:16,709 I will lower the volume a little. 858 1:05:16,709 --> 1:05:18,000 Wow. 859 1:05:18,000 --> 1:05:25,000 860 1:05:25,000 --> 1:05:27,958 Look at that. You see where the anti-nodes 861 1:05:27,958 --> 1:05:29,402 are. It's thrown off. 862 1:05:29,402 --> 1:05:32,000 And it accumulates at the nodes. 863 1:05:32,000 --> 1:05:41,000 864 1:05:41,000 --> 1:05:43,901 Hey, that's a weird one coming up. 865 1:05:43,901 --> 1:05:47,065 My goodness, I've done it many times. 866 1:05:47,065 --> 1:05:50,582 Every time I do it, I see different ones. 867 1:05:50,582 --> 1:05:56,120 No, the normal mode frequencies are often so close together that 868 1:05:56,120 --> 1:05:59,989 you can very easily skip one and go over one. 869 1:05:59,989 --> 1:06:05,000 Isn't that amazing? You look at the lines here. 870 1:06:05,000 --> 1:06:08,000 It's completely ridiculous. [LAUGHTER] 871 1:06:08,000 --> 1:06:22,000 872 1:06:22,000 --> 1:06:24,343 This is a different one, right? 873 1:06:24,343 --> 1:06:27,000 Different from the previous one. 874 1:06:27,000 --> 1:06:33,000 875 1:06:33,000 --> 1:06:39,016 I'll increase the volume. When you are coming up to a 876 1:06:39,016 --> 1:06:45,380 resonance, you can even hear it because the sound volume 877 1:06:45,380 --> 1:06:47,000 increases. 878 1:06:47,000 --> 1:06:55,000 879 1:06:55,000 --> 1:06:57,484 Holy smokes! Isn't that amazing? 880 1:06:57,484 --> 1:07:01,331 You try to calculate that. I would like to think, 881 1:07:01,331 --> 1:07:05,658 though, that aero and astro people who design airplanes 882 1:07:05,658 --> 1:07:09,585 [would have rings?], and people who build bridges, 883 1:07:09,585 --> 1:07:13,672 that they should be able to calculate these resonant 884 1:07:13,672 --> 1:07:18,000 frequencies because you may get destruction. 885 1:07:18,000 --> 1:07:33,000 886 1:07:33,000 --> 1:07:38,182 Look at this one. I've never seen this one in my 887 1:07:38,182 --> 1:07:42,152 course. [LAUGHTER] This is completely 888 1:07:42,152 --> 1:07:46,452 bizarre. All right, so you get the idea. 889 1:07:46,452 --> 1:07:53,178 I really would like you to play with that, with the ones there 890 1:07:53,178 --> 1:07:59,794 at the end of the lecture to see whether you can excite them. 891 1:07:59,794 --> 1:08:04,846 [APPLAUSE] So, now, I want to make the 892 1:08:04,846 --> 1:08:09,064 step to three dimensional wave equations. 893 1:08:09,064 --> 1:08:13,598 This was a problem that Professor Mavalvala, 894 1:08:13,598 --> 1:08:19,503 who is sitting in the back, put on the final exam of 8.03 895 1:08:19,503 --> 1:08:24,037 this spring this year. If you don't like it, 896 1:08:24,037 --> 1:08:27,200 there she is. So, this problem, 897 1:08:27,200 --> 1:08:33,000 who has suggested that problems you, Nergis? 898 1:08:33,000 --> 1:08:38,210 Don't tell them. Here we have a sound cavity in 899 1:08:38,210 --> 1:08:42,742 three dimensions: A, B, C are the length. 900 1:08:42,742 --> 1:08:47,273 And, this is our coordinate system, X, Y, 901 1:08:47,273 --> 1:08:50,105 Z. And, it is open in this 902 1:08:50,105 --> 1:08:53,730 direction. And, it is open there. 903 1:08:53,730 --> 1:08:59,507 So, it is a box which is open on both sides in the Z 904 1:08:59,507 --> 1:09:05,959 direction. And, the rest is all closed. 905 1:09:05,959 --> 1:09:12,040 So, now the three dimensional wave equation, 906 1:09:12,040 --> 1:09:17,272 D2P DX squared, P is the overpressure, 907 1:09:17,272 --> 1:09:24,909 plus D2P DY squared is D2P DZ squared equals one over V 908 1:09:24,909 --> 1:09:32,193 squared times D2P DT squared. That is the three dimensional 909 1:09:32,193 --> 1:09:35,522 wave equation. And so, it should really be no 910 1:09:35,522 --> 1:09:40,212 problem for you or me to write down immediately the solution of 911 1:09:40,212 --> 1:09:43,541 the normal modes. You must make sure that you 912 1:09:43,541 --> 1:09:47,702 have pressure nodes in the Z direction at Z equals zero, 913 1:09:47,702 --> 1:09:51,182 and at Z equals C. And, you must make sure that 914 1:09:51,182 --> 1:09:55,494 you have anti-nodes in the X and the Y direction where the 915 1:09:55,494 --> 1:09:59,352 pressure can build up. That's the boundary condition 916 1:09:59,352 --> 1:10:03,826 for a closed system. And so, the solution, 917 1:10:03,826 --> 1:10:09,043 then, for the pressure in terms now of three quantum numbers, 918 1:10:09,043 --> 1:10:11,304 L as in Lion, M as in Mary, 919 1:10:11,304 --> 1:10:16,347 N as in Nancy is now some kind of an amplitude which can be 920 1:10:16,347 --> 1:10:20,000 chosen differently for the different modes. 921 1:10:20,000 --> 1:10:24,000 So, I give them, then, the index LMN. 922 1:10:24,000 --> 1:10:31,831 And then, you get the cosine of pi times L times X divided by A, 923 1:10:31,831 --> 1:10:37,798 which is my L of X. And then, I get the cosine of 924 1:10:37,798 --> 1:10:42,024 pi times Mary times Y divided by B. 925 1:10:42,024 --> 1:10:49,483 And then, I get the sine of pi times Nancy times Z divided by 926 1:10:49,483 --> 1:10:53,337 C. And then, I get the cosine of 927 1:10:53,337 --> 1:10:59,179 omega LNN times T. And so, you see when you look 928 1:10:59,179 --> 1:11:05,892 there, convince yourself that you meet all the boundary 929 1:11:05,892 --> 1:11:10,561 conditions. This is the one that is the 930 1:11:10,561 --> 1:11:12,818 open end. That means if you substitute Z 931 1:11:12,818 --> 1:11:15,074 equals zero, this one will make it zero. 932 1:11:15,074 --> 1:11:17,909 That's where you have the pressure node where it's 933 1:11:17,909 --> 1:11:20,743 connected with universe. No pressure can build up. 934 1:11:20,743 --> 1:11:24,157 But if you substitute in here Z equals C, you also get zero. 935 1:11:24,157 --> 1:11:26,008 And, you get the anti-nodes here. 936 1:11:26,008 --> 1:11:30,000 So, this is a solution that immediately presents itself. 937 1:11:30,000 --> 1:11:34,670 If you have followed what we did 1D and what we did 2D, 938 1:11:34,670 --> 1:11:39,081 this is the obvious solution. And so, the system can 939 1:11:39,081 --> 1:11:43,578 oscillate in a combination of all these normal modes. 940 1:11:43,578 --> 1:11:47,816 But what is interesting now is that L can be zero, 941 1:11:47,816 --> 1:11:51,708 one, two, three, because there's nothing wrong 942 1:11:51,708 --> 1:11:55,772 with making L zero. The cosine remains one then. 943 1:11:55,772 --> 1:11:59,405 Nothing goes to zero. And Mary can go zero, 944 1:11:59,405 --> 1:12:03,948 one, two, three. It's only Nancy that cannot be 945 1:12:03,948 --> 1:12:07,522 zero because if Nancy is zero, then there is no pressure 946 1:12:07,522 --> 1:12:09,341 everywhere. There is no wave. 947 1:12:09,341 --> 1:12:12,199 So, you see here that you now allow for zero, 948 1:12:12,199 --> 1:12:14,993 zero, one modes, which will give you certain 949 1:12:14,993 --> 1:12:17,656 frequencies. And, the frequencies that you 950 1:12:17,656 --> 1:12:21,359 will get, you will find by substituting this solution back 951 1:12:21,359 --> 1:12:24,802 into the differential equation into the wave equation, 952 1:12:24,802 --> 1:12:27,596 which gives you, then, always the connection 953 1:12:27,596 --> 1:12:34,086 between omega and K. And so, that connection is then 954 1:12:34,086 --> 1:12:40,942 that omega LMN becomes V, which is the speed of sound 955 1:12:40,942 --> 1:12:46,610 times the square root. If you are ready now, 956 1:12:46,610 --> 1:12:53,729 we get pi L divided by A squared plus pi N divided by B 957 1:12:53,729 --> 1:12:59,266 squared plus pi Nancy divided by C squared. 958 1:12:59,266 --> 1:13:07,045 And so, the lowest possible frequency for this system is the 959 1:13:07,045 --> 1:13:13,307 zero-zero-one mode. If I assume that C is the 960 1:13:13,307 --> 1:13:18,199 largest dimension in the system, so that's the length, 961 1:13:18,199 --> 1:13:23,369 L, I didn't put the length in there, but C is sort of the 962 1:13:23,369 --> 1:13:27,800 length of my musical instrument, so you will get, 963 1:13:27,800 --> 1:13:31,769 then, that zero-zero-one mode would give me, 964 1:13:31,769 --> 1:13:36,826 then, omega zero-zero-one. So, this is not there. 965 1:13:36,826 --> 1:13:40,727 This is not there. I call this now L to make the 966 1:13:40,727 --> 1:13:43,881 connection with my musical instruments. 967 1:13:43,881 --> 1:13:47,616 And so, I get V times N times pi divided by L. 968 1:13:47,616 --> 1:13:52,098 And, that is precisely what we had before for a musical 969 1:13:52,098 --> 1:13:54,588 instrument. For a sound cavity, 970 1:13:54,588 --> 1:13:59,486 which is open and open at both sides, or which is closed and 971 1:13:59,486 --> 1:14:03,802 closed on both sides, and so what you see now is that 972 1:14:03,802 --> 1:14:08,035 if you have a musical instrument, even though it has 973 1:14:08,035 --> 1:14:12,185 dimensions, A and B, that the lowest frequency that 974 1:14:12,185 --> 1:14:17,000 you get is always dictated, then, by L. 975 1:14:17,000 --> 1:14:20,123 But not only that, if A and B are much, 976 1:14:20,123 --> 1:14:23,986 much smaller than C, then the second harmonic is 977 1:14:23,986 --> 1:14:26,616 almost certainly going to be 002. 978 1:14:26,616 --> 1:14:31,301 So, by making this one go to two, you are still at a lower 979 1:14:31,301 --> 1:14:35,000 frequency than by making M go to one. 980 1:14:35,000 --> 1:14:38,769 And, the third harmonic is very likely to be here. 981 1:14:38,769 --> 1:14:42,769 And then there comes a time, of course, that the next 982 1:14:42,769 --> 1:14:46,769 frequency comes in when you change the L and the M's. 983 1:14:46,769 --> 1:14:50,076 And, you can get a whole zoo of frequencies. 984 1:14:50,076 --> 1:14:53,384 And that is what that problem was all about. 985 1:14:53,384 --> 1:14:55,769 In spring 2004, it's on the Web, 986 1:14:55,769 --> 1:14:58,615 and the solutions are also on the Web. 987 1:14:58,615 --> 1:15:03,000 The final exam was one of several problems. 988 1:15:03,000 --> 1:15:07,307 If I return, now, to the speed of sound in 989 1:15:07,307 --> 1:15:11,089 gas, R is the universal gas constant. 990 1:15:11,089 --> 1:15:17,498 You may have had that in 8.01. T is the temperature in degrees 991 1:15:17,498 --> 1:15:20,649 Kelvin. What I want to mention, 992 1:15:20,649 --> 1:15:26,533 though, what is interesting: a friend of mine was a flute 993 1:15:26,533 --> 1:15:31,707 player in an orchestra. And I asked him whether they 994 1:15:31,707 --> 1:15:35,365 have to take into account the change in temperature when they 995 1:15:35,365 --> 1:15:37,195 start playing. And he said yes, 996 1:15:37,195 --> 1:15:40,000 they actually do. They tune their instrument to 997 1:15:40,000 --> 1:15:43,719 adjust it to the temperature by making their estimate a longer 998 1:15:43,719 --> 1:15:46,341 or little shorter. So, the temperature which 999 1:15:46,341 --> 1:15:49,085 changes the velocity of sound must, of course, 1000 1:15:49,085 --> 1:15:52,012 be taken into account if you are in an orchestra, 1001 1:15:52,012 --> 1:15:54,756 and you have the flutes, and you have the wind 1002 1:15:54,756 --> 1:15:56,951 instruments together with the violin. 1003 1:15:56,951 --> 1:16:00,304 So, the violence can change the tension and the strings, 1004 1:16:00,304 --> 1:16:05,000 but the wind instruments also adjust to the temperature. 1005 1:16:05,000 --> 1:16:11,798 Now, if we have air and gamma, air is a diatomic molecule. 1006 1:16:11,798 --> 1:16:17,880 Nitrogen and oxygen, for those of you who have taken 1007 1:16:17,880 --> 1:16:23,844 8.01, and when this was covered for a diatomic gas, 1008 1:16:23,844 --> 1:16:29,568 gamma is about 7/5, which would make it roughly a 1009 1:16:29,568 --> 1:16:33,502 1.4. And, the molecular weight, 1010 1:16:33,502 --> 1:16:37,834 [while?] oxygen two is 32, nitrogen two is 28. 1011 1:16:37,834 --> 1:16:42,165 You have 80% nitrogen and you have 20% oxygen. 1012 1:16:42,165 --> 1:16:47,171 So, anything you take, you take 2930 is fine with me. 1013 1:16:47,171 --> 1:16:51,213 So, that is, then, the molecular weight for 1014 1:16:51,213 --> 1:16:54,582 air. So, now, suppose we do the same 1015 1:16:54,582 --> 1:16:58,336 for helium. Now, helium is a mono-atomic 1016 1:16:58,336 --> 1:17:00,935 gas. And, a mono-atomic gas, 1017 1:17:00,935 --> 1:17:07,000 gamma is about five thirds, which is about 1.66. 1018 1:17:07,000 --> 1:17:12,047 But the molecular weight, which in this case is atomic 1019 1:17:12,047 --> 1:17:16,714 weight, and helium is just a single atom, is four. 1020 1:17:16,714 --> 1:17:21,190 And, consequently, the speed of sound and helium 1021 1:17:21,190 --> 1:17:25,761 is roughly three times the speed of sound in air. 1022 1:17:25,761 --> 1:17:31,000 You can just substitute those numbers in there. 1023 1:17:31,000 --> 1:17:35,293 And you'll find that it's about three times higher. 1024 1:17:35,293 --> 1:17:39,587 Now, my voice is some kind of a musical instrument, 1025 1:17:39,587 --> 1:17:42,764 not unlike the violin. It has strings, 1026 1:17:42,764 --> 1:17:46,371 which are my vocal chords, which oscillate, 1027 1:17:46,371 --> 1:17:51,695 and these strings are connected to a sounding board which is my 1028 1:17:51,695 --> 1:17:56,590 larynx, which is my throat. And that creates a voice which 1029 1:17:56,590 --> 1:17:58,994 you recognize. You say, yeah, 1030 1:17:58,994 --> 1:18:03,401 that is Walter Lewin. My sounding board, 1031 1:18:03,401 --> 1:18:08,152 just like a sounding board of the violin, resonates at certain 1032 1:18:08,152 --> 1:18:10,722 frequencies better than at others. 1033 1:18:10,722 --> 1:18:14,460 And that gives you, then, the characteristic tone 1034 1:18:14,460 --> 1:18:18,276 of that particular violin, that particular person. 1035 1:18:18,276 --> 1:18:22,637 However, when I inhale helium, my poor throat has no idea 1036 1:18:22,637 --> 1:18:25,907 about that. And so it starts to resonate at 1037 1:18:25,907 --> 1:18:28,711 different frequencies. And therefore, 1038 1:18:28,711 --> 1:18:33,306 my voice will be very different because of the difference in 1039 1:18:33,306 --> 1:18:37,823 speed of sound because remember, the resonance frequency of 1040 1:18:37,823 --> 1:18:40,237 sound cavities, their resonance, 1041 1:18:40,237 --> 1:18:43,274 their frequency, depends directly on the 1042 1:18:43,274 --> 1:18:47,766 velocity, V. I have hear omega, 1043 1:18:47,766 --> 1:18:51,398 but the same is true for F, of course. 1044 1:18:51,398 --> 1:18:56,601 Now, there's only one problem, and the problem for me, 1045 1:18:56,601 --> 1:19:00,625 not for you, is that there is no oxygen in 1046 1:19:00,625 --> 1:19:04,139 helium. And if I take in too much 1047 1:19:04,139 --> 1:19:08,673 helium, I'll be on the floor. If I take in too little, 1048 1:19:08,673 --> 1:19:12,438 it won't work. And so, there is always a very 1049 1:19:12,438 --> 1:19:15,604 fine line. But I'm not joking about it 1050 1:19:15,604 --> 1:19:19,796 that this is really an experiment that can make me 1051 1:19:19,796 --> 1:19:24,930 extremely dizzy because will be without oxygen for some time. 1052 1:19:24,930 --> 1:19:29,978 And, I'm actually unaware of it because you just inhale that 1053 1:19:29,978 --> 1:19:31,860 helium. It feels great. 1054 1:19:31,860 --> 1:19:34,000 [LAUGHTER] 1055 1:19:34,000 --> 1:19:39,000 1056 1:19:39,000 --> 1:19:43,815 The person who's going to talk to you next you'll think is not 1057 1:19:43,815 --> 1:19:45,000 Walter Lewin. 1058 1:19:45,000 --> 1:20:16,000 1059 1:20:16,000 --> 1:21:48,597 I wasn't joking, really. 1060 1:21:48,597 --> 1:24:37,688 I hope you're going to have a good weekend. 1061 1:24:37,688 See you.