1 00:00:00 --> 00:00:00,499 2 00:00:00,499 --> 00:00:06,527 So today, I will start with a general discussion on waves, 3 00:00:06,527 --> 00:00:11,075 as an introduction to electromagnetic waves, 4 00:00:11,075 --> 00:00:14,353 which we will discuss next week. 5 00:00:14,353 --> 00:00:19,218 We'll start with a very down-to-earth equation, 6 00:00:19,218 --> 00:00:24,612 Y equals one-third X. And I'm going to plot that for 7 00:00:24,612 --> 00:00:30,746 you, so here is Y and here is X, and that's a straight line 8 00:00:30,746 --> 00:00:36,458 through the origin, Y equals one-third X. 9 00:00:36,458 --> 00:00:40,13 Suppose, now, I want this line to move. 10 00:00:40,13 --> 00:00:46,025 I want this line to move with a speed of six meters per second 11 00:00:46,025 --> 00:00:51,341 in the plus X direction. All I will have to do now is to 12 00:00:51,341 --> 00:00:55,496 replace X in that equation by X minus six T. 13 00:00:55,496 --> 00:00:59,169 Notice the minus sign. I will go, then, 14 00:00:59,169 --> 00:01:04,097 in the plus X direction. The equation then becomes Y 15 00:01:04,097 --> 00:01:08,93 equals one-third times X minus six T. 16 00:01:08,93 --> 00:01:14,189 So look at it at T equals 1. At T equals zero, 17 00:01:14,189 --> 00:01:19,097 you already have the line. At T equals one, 18 00:01:19,097 --> 00:01:24,122 you now have Y equals one-third X minus two. 19 00:01:24,122 --> 00:01:29,615 That means, here it will intersect at minus two, 20 00:01:29,615 --> 00:01:36,744 and there it will intersect at plus six, and the line parallel 21 00:01:36,744 --> 00:01:42,839 to the first one, this line is now T equals one, 22 00:01:42,839 --> 00:01:47,543 and this is T equals zero. And it has moved in this 23 00:01:47,543 --> 00:01:52,058 direction, with a speed of six meters per second. 24 00:01:52,058 --> 00:01:57,797 And so what this is telling us, that if we ever want something 25 00:01:57,797 --> 00:02:02,125 to move with a speed V in the plus X direction, 26 00:02:02,125 --> 00:02:07,393 then all we have to do in our equations to replace X by X 27 00:02:07,393 --> 00:02:14,354 minus V T, and if we want it to move in the minus X direction, 28 00:02:14,354 --> 00:02:18,099 then we replace X by X plus V T. 29 00:02:18,099 --> 00:02:24,742 That's all we have to do. So now, I'm going to change to 30 00:02:24,742 --> 00:02:31,265 something that is a real wave. I now have Y equals two, 31 00:02:31,265 --> 00:02:36,096 times the sine of three X. That's a wave. 32 00:02:36,096 --> 00:02:38,995 It's not moving, not yet. 33 00:02:38,995 --> 00:02:44,31 So I can make a plot of Y as a function of X, 34 00:02:44,31 --> 00:02:49,746 and that plot will be like this. 35 00:02:49,746 --> 00:02:54,029 This is zero, so when the sine is zero, 36 00:02:54,029 --> 00:03:00,117 this is pi divided by three, and this is hundred eighty 37 00:03:00,117 --> 00:03:06,204 degrees, and it's again zero, this is two pi divided by 38 00:03:06,204 --> 00:03:12,066 three, it's again zero. And lambda, which we call the 39 00:03:12,066 --> 00:03:15,674 wavelength, lambda, in this case, 40 00:03:15,674 --> 00:03:20,86 is from here to here, that is two pi divided by 41 00:03:20,86 --> 00:03:25,537 three, this goes also from here to 42 00:03:25,537 --> 00:03:28,636 there. I will introduce a symbol K 43 00:03:28,636 --> 00:03:33,613 that you will often see, we call that the wave number, 44 00:03:33,613 --> 00:03:38,402 and K is simply defined as two pi divided by lambda. 45 00:03:38,402 --> 00:03:41,688 So in our specific case, K is three. 46 00:03:41,688 --> 00:03:45,35 This here is K. If you know this number, 47 00:03:45,35 --> 00:03:49,764 you can immediately tell what the wavelength is. 48 00:03:49,764 --> 00:03:52,957 Now, I want to have this wave move. 49 00:03:52,957 --> 00:03:57,365 I want to have a traveling wave. 50 00:03:57,365 --> 00:04:02,843 And I want to have it move with six meters per second in the 51 00:04:02,843 --> 00:04:06,743 plus X direction. So the recipe is now very 52 00:04:06,743 --> 00:04:11,943 simple, all I have to do replace this X by X minus six T. 53 00:04:11,943 --> 00:04:16,865 So now I get Y equals two times the sine, times three, 54 00:04:16,865 --> 00:04:21,229 times X minus six T. And if you now look at this 55 00:04:21,229 --> 00:04:25,779 curve, this equation, and you plot it a little bit 56 00:04:25,779 --> 00:04:31,918 later in time than T zero -- this is already T zero -- a 57 00:04:31,918 --> 00:04:35,191 little later in time, you will see that, 58 00:04:35,191 --> 00:04:38,884 indeed, it has moved in the plus X direction. 59 00:04:38,884 --> 00:04:43,331 And it's moving with a speed of six meters per second. 60 00:04:43,331 --> 00:04:46,436 So this equation, when you look at it, 61 00:04:46,436 --> 00:04:50,465 holds all the characteristics of the oscillation. 62 00:04:50,465 --> 00:04:54,577 It holds the amplitude. This two is the amplitude. 63 00:04:54,577 --> 00:04:59,108 This is minus two, that's the amplitude. 64 00:04:59,108 --> 00:05:02,928 This information, K, holds the information on the 65 00:05:02,928 --> 00:05:07,702 wavelength, and this information tells you what the speed is. 66 00:05:07,702 --> 00:05:10,726 And the minus sign, which is important, 67 00:05:10,726 --> 00:05:14,624 tells you that it's going in the plus X direction, 68 00:05:14,624 --> 00:05:17,171 and not in the minus X direction. 69 00:05:17,171 --> 00:05:19,796 Can we make such a traveling wave? 70 00:05:19,796 --> 00:05:23,218 Yes, we can do that, actually, quite easily. 71 00:05:23,218 --> 00:05:27,833 Suppose I have here a rotating wheel -- rotate with angular 72 00:05:27,833 --> 00:05:32,848 frequency omega, and let this has a radius R, 73 00:05:32,848 --> 00:05:36,775 and I get the two units, so that I get the same 74 00:05:36,775 --> 00:05:41,64 amplitude that I have here. And I attach to this a string, 75 00:05:41,64 --> 00:05:46,761 and I put some tension on the string, so that I create a wave 76 00:05:46,761 --> 00:05:50,26 as I rotate it, and the string is attached 77 00:05:50,26 --> 00:05:54,869 here, and as it rotates, the wave is going to propagate 78 00:05:54,869 --> 00:05:58,71 into the string with a velocity, let's say, V. 79 00:05:58,71 --> 00:06:02,79 So I can generate a traveling wave. 80 00:06:02,79 --> 00:06:07,3 The period of one oscillation -- if you were here on the 81 00:06:07,3 --> 00:06:10,743 string, you're going up, you're going down, 82 00:06:10,743 --> 00:06:13,531 you're going up, you're going down, 83 00:06:13,531 --> 00:06:17,958 that's all you're doing, when the wave passes by -- the 84 00:06:17,958 --> 00:06:22,959 period of one whole oscillation is obviously two pi divided by 85 00:06:22,959 --> 00:06:26,321 this omega. The wavelength lambda that you 86 00:06:26,321 --> 00:06:30,502 are creating -- from here to here is lambda -- well, 87 00:06:30,502 --> 00:06:35,831 if you know the speed with which it is traveling, 88 00:06:35,831 --> 00:06:40,727 and you know it has been traveling capital T seconds, 89 00:06:40,727 --> 00:06:44,587 one oscillation, that's a distance lambda. 90 00:06:44,587 --> 00:06:49,295 So this is V times T. But this is also V divided by 91 00:06:49,295 --> 00:06:52,402 F, if F is the frequency in Hertz. 92 00:06:52,402 --> 00:06:58,145 And so the frequency F is then also given by the speed divided 93 00:06:58,145 --> 00:07:01,723 by lambda. And so I can write down this 94 00:07:01,723 --> 00:07:05,583 equation now in a somewhat different form, 95 00:07:05,583 --> 00:07:12,139 Y equals two times the sine, and now I bring the three 96 00:07:12,139 --> 00:07:16,252 inside, so I get three X minus eighteen T. 97 00:07:16,252 --> 00:07:19,361 This eighteen is now that omega. 98 00:07:19,361 --> 00:07:23,474 This is omega T. In here is all the timing 99 00:07:23,474 --> 00:07:26,683 information. Omega, the period T, 100 00:07:26,683 --> 00:07:31,197 everything is in here. Here is all the spatial 101 00:07:31,197 --> 00:07:33,404 information. This is K. 102 00:07:33,404 --> 00:07:37,316 In here is the information about lambda. 103 00:07:37,316 --> 00:07:42,204 And so if I know omega, and I know K, 104 00:07:42,204 --> 00:07:46,411 then I can also find the velocity, which is omega divided 105 00:07:46,411 --> 00:07:48,665 by K. So everything is in here, 106 00:07:48,665 --> 00:07:53,248 omega divided by three gives me back my six meters per second. 107 00:07:53,248 --> 00:07:57,455 So once you have the equation, I can ask you any question 108 00:07:57,455 --> 00:08:00,384 about that wave, and you should be able, 109 00:08:00,384 --> 00:08:02,488 then, to answer. Wavelengths, 110 00:08:02,488 --> 00:08:05,643 frequency, in Hertz, in radians per second, 111 00:08:05,643 --> 00:08:08,423 speed, everything. You may ask me now, 112 00:08:08,423 --> 00:08:10,827 "Why do you discuss this with us? 113 00:08:10,827 --> 00:08:15,824 Well, we are coming up to electromagnetic waves next 114 00:08:15,824 --> 00:08:19,553 week, and electromagnetic waves, you're going to see lambdas, 115 00:08:19,553 --> 00:08:23,158 you're going to see omegas, you're going to see capital Ts, 116 00:08:23,158 --> 00:08:26,452 you're going to see frequency, you're going to see Ks, 117 00:08:26,452 --> 00:08:29,809 everything you see there you're going to see next week. 118 00:08:29,809 --> 00:08:32,357 One exception, that Y, the displacement Y, 119 00:08:32,357 --> 00:08:35,714 will not be in centimeters or meters, but it will be an 120 00:08:35,714 --> 00:08:38,324 electric field, a traveling electric field, 121 00:08:38,324 --> 00:08:41,183 volts per meter. Or a traveling magnetic field, 122 00:08:41,183 --> 00:08:42,799 tesla. But other than that, 123 00:08:42,799 --> 00:08:47,586 all these quantities will return in exactly the same way. 124 00:08:47,586 --> 00:08:52,951 Now I want to discuss with you a standing wave first, 125 00:08:52,951 --> 00:08:57,903 because standing waves are going to be important. 126 00:08:57,903 --> 00:09:03,371 This is a traveling wave. And now comes something even 127 00:09:03,371 --> 00:09:07,601 more intriguing, which is a standing wave. 128 00:09:07,601 --> 00:09:12,657 Suppose I have a wave traveling in this direction, 129 00:09:12,657 --> 00:09:17,815 and I call that Y one, and Y zero is the amplitude, 130 00:09:17,815 --> 00:09:22,526 sine K X minus omega T. And notice now, 131 00:09:22,526 --> 00:09:25,714 I have all the symbols that we are familiar with. 132 00:09:25,714 --> 00:09:28,504 We have the K here, we have the omega here, 133 00:09:28,504 --> 00:09:32,423 and we have the amplitude here. And the minus sign tells me, 134 00:09:32,423 --> 00:09:35,346 [wssshhht], it's going in the plus direction. 135 00:09:35,346 --> 00:09:38,468 But I have another wave. And the wave is exactly 136 00:09:38,468 --> 00:09:41,391 identical, in terms of amplitude, in terms of 137 00:09:41,391 --> 00:09:44,313 wavelength, in terms of frequency, identical, 138 00:09:44,313 --> 00:09:46,705 but it's traveling in this direction. 139 00:09:46,705 --> 00:09:50,092 And so this is Y two, which is Y zero times the sine 140 00:09:50,092 --> 00:09:53,001 of K X plus omega T. 141 00:09:53,001 --> 00:09:57,557 This plus sign tells me it's going in this direction. 142 00:09:57,557 --> 00:10:02,376 And so if this is a string, the net result is the sum of 143 00:10:02,376 --> 00:10:05,267 the two. So I have to add them up. 144 00:10:05,267 --> 00:10:09,998 So Y is Y one plus Y two. So I have to do some trigono- 145 00:10:09,998 --> 00:10:14,73 trigonometric manipulation, and this is what I leave -- 146 00:10:14,73 --> 00:10:19,636 I'll leave you with that, that's high school stuff -- you 147 00:10:19,636 --> 00:10:24,806 add the two up and you'll find two by zero -- 148 00:10:24,806 --> 00:10:29,568 notice that the amplitude has doubled -- times the sine of K X 149 00:10:29,568 --> 00:10:34,019 times the cosine of omega T. That's the some of those two. 150 00:10:34,019 --> 00:10:37,767 And this is very, very different from a traveling 151 00:10:37,767 --> 00:10:40,499 wave. Nowhere will you see K X minus 152 00:10:40,499 --> 00:10:42,764 omega T any more. K X is here, 153 00:10:42,764 --> 00:10:46,902 separate under the sine, and omega T is separate under 154 00:10:46,902 --> 00:10:50,025 the cosine. All the timing information is 155 00:10:50,025 --> 00:10:54,476 now separate from the spatial information. 156 00:10:54,476 --> 00:10:59,884 And so what does a standing wave like this look like? 157 00:10:59,884 --> 00:11:05,916 Well, let's a bracket here. Let's make a drawing of such a 158 00:11:05,916 --> 00:11:09,244 standing wave. So here we have Y, 159 00:11:09,244 --> 00:11:14,444 and here we have X. Let's only look at the sine K X 160 00:11:14,444 --> 00:11:16,628 for now. If X is zero, 161 00:11:16,628 --> 00:11:24,221 the sine is always zero, so this point will never move. 162 00:11:24,221 --> 00:11:28,65 But if K X is hundred eighty degrees, it's also zero, 163 00:11:28,65 --> 00:11:31,716 always. So lambda over two will never 164 00:11:31,716 --> 00:11:33,164 move. X is lambda, 165 00:11:33,164 --> 00:11:36,571 when this is three hundred sixty degrees, 166 00:11:36,571 --> 00:11:40,063 it will never move. Minus lambda over two, 167 00:11:40,063 --> 00:11:43,64 will never move. So what will it look like? 168 00:11:43,64 --> 00:11:47,473 Well, you're going to see something like this, 169 00:11:47,473 --> 00:11:52,158 let's take the moment when T equals zero, so when cosine 170 00:11:52,158 --> 00:11:55,268 omega T is plus one. 171 00:11:55,268 --> 00:12:00,933 So we're going to have a curve like this, so this goes up to 172 00:12:00,933 --> 00:12:06,695 two Y zero like this -- and this here is then my two Y zero. 173 00:12:06,695 --> 00:12:12,265 These points will never move, they will always stand still. 174 00:12:12,265 --> 00:12:15,818 There's nothing like a traveling wave. 175 00:12:15,818 --> 00:12:21,099 If it's a traveling wave, these points will see the wave 176 00:12:21,099 --> 00:12:27,45 go by, they will go up and down, they never do that. 177 00:12:27,45 --> 00:12:29,897 They sit still. They have a name. 178 00:12:29,897 --> 00:12:33,721 We call them nodes. Let's now look at little later. 179 00:12:33,721 --> 00:12:37,24 Let's look at T equals one quarter of a period. 180 00:12:37,24 --> 00:12:41,293 Now, the cosine is zero. So there's not a single point 181 00:12:41,293 --> 00:12:45,959 on the string that is not zero. So the string looks like this. 182 00:12:45,959 --> 00:12:50,395 If you took a picture of the string, you wouldn't even know 183 00:12:50,395 --> 00:12:53,76 it's oscillating. It would be just a straight 184 00:12:53,76 --> 00:12:56,208 line. And now, if we do -- look a 185 00:12:56,208 --> 00:13:00,094 little later, and we look at T equals 186 00:13:00,094 --> 00:13:03,442 one-half the period, then the cosine is minus one. 187 00:13:03,442 --> 00:13:05,902 So now the curve will look like this. 188 00:13:05,902 --> 00:13:09,456 And so what does it mean? If we just look what's here 189 00:13:09,456 --> 00:13:12,258 happening, this is what's going to happen. 190 00:13:12,258 --> 00:13:16,427 The string is just doing this, and there are points that stand 191 00:13:16,427 --> 00:13:18,682 still. Nothing is going like this, 192 00:13:18,682 --> 00:13:22,645 nothing is going like this. You see this point going up and 193 00:13:22,645 --> 00:13:24,695 down, up and down, up and down, 194 00:13:24,695 --> 00:13:28,044 and this will do the same, and these nodes will do 195 00:13:28,044 --> 00:13:32,381 nothing. So that is what a standing wave 196 00:13:32,381 --> 00:13:35,82 will look like, and I think the name standing 197 00:13:35,82 --> 00:13:39,651 wave is a very appropriate name, very descriptive, 198 00:13:39,651 --> 00:13:44,107 because it's really standing, it's not -- it's not moving. 199 00:13:44,107 --> 00:13:47,624 At least, not traveling along the X direction. 200 00:13:47,624 --> 00:13:50,751 Can we make a standing wave? Yes, we can, 201 00:13:50,751 --> 00:13:55,051 and I will do that today. A standing wave can be made by 202 00:13:55,051 --> 00:13:59,038 shaking -- or rotating, in that fashion -- a string. 203 00:13:59,038 --> 00:14:04,237 So here I have a string, I -- say I attach the string to 204 00:14:04,237 --> 00:14:07,312 the wall there, and I move it up and down here. 205 00:14:07,312 --> 00:14:11,389 So a wave goes in -- I do just this, like the rotating disc -- 206 00:14:11,389 --> 00:14:14,264 the wave travels, but the wave is reflected, 207 00:14:14,264 --> 00:14:18,208 and so I have a wave going in and I have a wave coming back, 208 00:14:18,208 --> 00:14:21,417 so I have now two waves going through each other. 209 00:14:21,417 --> 00:14:25,294 And if the conditions are just right, then these reflective 210 00:14:25,294 --> 00:14:28,77 waves -- this one will reflect, when it arrives here, 211 00:14:28,77 --> 00:14:31,51 it will reflect again, it goes back again, 212 00:14:31,51 --> 00:14:34,986 and it will continue to reflect -- 213 00:14:34,986 --> 00:14:39,149 so if the conditions are just right, then these reflective 214 00:14:39,149 --> 00:14:43,531 waves will support each other, and they will generate a large 215 00:14:43,531 --> 00:14:47,475 amplitude -- as I will demonstrate to you -- but that's 216 00:14:47,475 --> 00:14:51,93 only the case for very specific frequencies, and we call those 217 00:14:51,93 --> 00:14:55,581 resonant frequencies. The lowest possible frequency 218 00:14:55,581 --> 00:15:00,109 for which this happens -- which we call the fundamental -- will 219 00:15:00,109 --> 00:15:02,519 make the string vibrate like this. 220 00:15:02,519 --> 00:15:06,244 So the whole thing goes. [wssshhht], 221 00:15:06,244 --> 00:15:10,154 [wssshhht], [wssshhht], and we call that the 222 00:15:10,154 --> 00:15:13,702 fundamental. We call that also the first 223 00:15:13,702 --> 00:15:16,43 harmonic. If now I increase the 224 00:15:16,43 --> 00:15:21,069 frequencies, then I get a second resonant frequency, 225 00:15:21,069 --> 00:15:26,708 and a node jumps in the middle -- there is already a node here, 226 00:15:26,708 --> 00:15:31,71 and there is a node here, because this motion of my hand 227 00:15:31,71 --> 00:15:37,35 here is very small, as I will demonstrate to you, 228 00:15:37,35 --> 00:15:41,902 for all practical purpose you can think of this being a node 229 00:15:41,902 --> 00:15:46,609 -- and so now the string in the second harmonic will oscillate 230 00:15:46,609 --> 00:15:49,155 like this. [Wssshhht], [wssshhht], 231 00:15:49,155 --> 00:15:53,322 [wssshhht], [wssshhht], so this is the second harmonic. 232 00:15:53,322 --> 00:15:58,106 And if we go up in frequencies, then -- this should be right in 233 00:15:58,106 --> 00:16:02,813 the middle, by the way -- and if I go up in frequency one step 234 00:16:02,813 --> 00:16:06,749 more, then I get another resonance whereby we get an 235 00:16:06,749 --> 00:16:11,379 extra node, and so we get the third harmonic, 236 00:16:11,379 --> 00:16:14,3 and you just can go on like that. 237 00:16:14,3 --> 00:16:18,59 You get a whole series of resonance frequencies. 238 00:16:18,59 --> 00:16:23,976 And so, for the fundamental, lambda one -- the one refers to 239 00:16:23,976 --> 00:16:28,997 the first harmonic -- is two L, if L is the length of my 240 00:16:28,997 --> 00:16:30,549 string. This is L. 241 00:16:30,549 --> 00:16:35,204 You only have half a wavelength here, so L is two L. 242 00:16:35,204 --> 00:16:41,96 But we know that the frequency is the velocity divided by the 243 00:16:41,96 --> 00:16:46,418 wavelength -- we see that there, frequency is velocity 244 00:16:46,418 --> 00:16:51,212 divided by the wavelength -- so the frequency F one is the 245 00:16:51,212 --> 00:16:56,091 velocity divided by lambda one, so that's divided by two L. 246 00:16:56,091 --> 00:17:00,802 So that's the frequency in the fundamental for which this 247 00:17:00,802 --> 00:17:05,176 resonance phenomenon occurs. For the second harmonic, 248 00:17:05,176 --> 00:17:07,951 lambda two equals L. You can tell, 249 00:17:07,951 --> 00:17:10,811 you see a complete wavelength here. 250 00:17:10,811 --> 00:17:16,615 And F two, that frequency, is going to be twice F one. 251 00:17:16,615 --> 00:17:21,353 And F three is going to be three times F one. 252 00:17:21,353 --> 00:17:26,198 And if you want to know, for the Nth harmonic, 253 00:17:26,198 --> 00:17:31,043 N being Nancy, then lambda of N equals N two L 254 00:17:31,043 --> 00:17:34,597 divided by N. Substitute in N one, 255 00:17:34,597 --> 00:17:39,98 and you find the wavelength for the first harmonic. 256 00:17:39,98 --> 00:17:45,579 Substitute for N two, and you find the wavelength for 257 00:17:45,579 --> 00:17:48,735 the second harmonic. 258 00:17:48,735 --> 00:17:51,677 And so on. And the frequency for the Nth 259 00:17:51,677 --> 00:17:56,205 harmonic -- N stands for Nancy -- is N times V divided by two 260 00:17:56,205 --> 00:17:58,318 L. So here you see the entire 261 00:17:58,318 --> 00:18:02,392 series of frequencies and wavelengths for which we have 262 00:18:02,392 --> 00:18:05,335 resonance. Unlike in our LRC system that 263 00:18:05,335 --> 00:18:09,108 we discussed last time, where you had one resonance 264 00:18:09,108 --> 00:18:13,182 frequency, now you have an infinite number of resonance 265 00:18:13,182 --> 00:18:18,163 frequencies, and they are at very discrete values, 266 00:18:18,163 --> 00:18:21,11 equally spaced. I want to demonstrate this to 267 00:18:21,11 --> 00:18:24,527 you with a violin string, it's a very special violin 268 00:18:24,527 --> 00:18:27,542 string, it's here on the floor, it's a biggie, 269 00:18:27,542 --> 00:18:29,752 and I need some help from someone. 270 00:18:29,752 --> 00:18:32,834 You helped me before, would you mind helping me 271 00:18:32,834 --> 00:18:34,241 again? So here is, uh, 272 00:18:34,241 --> 00:18:37,524 one end of the string, which you're going to hold, 273 00:18:37,524 --> 00:18:40,471 you're going to be a node, believe it or not. 274 00:18:40,471 --> 00:18:42,548 Hold it better, two hands -- no, 275 00:18:42,548 --> 00:18:44,759 much better. You will see shortly, 276 00:18:44,759 --> 00:18:48,242 why -- no, no, no, much better. 277 00:18:48,242 --> 00:18:50,377 That's it. And walk back a little, 278 00:18:50,377 --> 00:18:52,318 walk further. Yes, that's good, 279 00:18:52,318 --> 00:18:54,647 hold it. I will put on a white glove, 280 00:18:54,647 --> 00:18:58,529 and there is a reason for that, because I want you to be able 281 00:18:58,529 --> 00:19:01,57 to see my hand when we're going to make it dark, 282 00:19:01,57 --> 00:19:04,61 so that you will convince yourself that my hand, 283 00:19:04,61 --> 00:19:08,039 which is generating the wave, is hardly moving at all. 284 00:19:08,039 --> 00:19:10,304 For practical purposes, it's a node, 285 00:19:10,304 --> 00:19:13,538 and yet we get this wonderful resonance phenomenon. 286 00:19:13,538 --> 00:19:17,42 So I'm going to make it very dark so that the U V will do its 287 00:19:17,42 --> 00:19:21,234 job, and you can see the string better, 288 00:19:21,234 --> 00:19:24,784 that's the only way we can make you see the string well. 289 00:19:24,784 --> 00:19:26,268 All right. Don't let go, 290 00:19:26,268 --> 00:19:29,753 er- under any circumstances, you will hurt me if you do 291 00:19:29,753 --> 00:19:31,947 that. Of course, if I let go first, 292 00:19:31,947 --> 00:19:33,754 then [pfft], I will hurt you, 293 00:19:33,754 --> 00:19:37,239 but that's not my plan. OK, so let's try to go a little 294 00:19:37,239 --> 00:19:39,175 bit further back. Let's try to, 295 00:19:39,175 --> 00:19:41,628 uh, find, first the -- the fundamental. 296 00:19:41,628 --> 00:19:44,79 And I'll try to find it by exciting just the right 297 00:19:44,79 --> 00:19:47,049 frequency with my hand. There it is. 298 00:19:47,049 --> 00:19:51,115 I think I got it. That's the fundamental. 299 00:19:51,115 --> 00:19:54,087 And look how little my hand is moving here. 300 00:19:54,087 --> 00:19:57,838 And you will see a very large amplitude in the middle. 301 00:19:57,838 --> 00:20:01,023 And so these reflected waves, one runs to him, 302 00:20:01,023 --> 00:20:03,783 it runs back at me, it runs back at him, 303 00:20:03,783 --> 00:20:07,747 keeps reflecting many times, they support each other in a 304 00:20:07,747 --> 00:20:10,932 constructive way, that's what resonance is all 305 00:20:10,932 --> 00:20:13,338 about. And now I'll try to find the 306 00:20:13,338 --> 00:20:17,656 second harmonic -- so you'll see another note coming in at the 307 00:20:17,656 --> 00:20:20,345 middle. It's easier for you to see than 308 00:20:20,345 --> 00:20:24,663 for me, actually. And it's not always easy 309 00:20:24,663 --> 00:20:28,353 to find the -- no, no, no, I'm too low frequency, 310 00:20:28,353 --> 00:20:31,197 I have to go up. I think I got it now. 311 00:20:31,197 --> 00:20:34,042 Is this it? Yes, one extra node in the 312 00:20:34,042 --> 00:20:35,579 middle? Speak out up, 313 00:20:35,579 --> 00:20:38,039 please. [chorus of agreement] Ah, 314 00:20:38,039 --> 00:20:40,576 that's better. Now I can hear you, 315 00:20:40,576 --> 00:20:43,651 thank you. Um, there are three nodes now. 316 00:20:43,651 --> 00:20:47,88 My friend there is a node, I'm a node, and then there is 317 00:20:47,88 --> 00:20:50,801 one in the middle. If you subtract one, 318 00:20:50,801 --> 00:20:56,721 the three minus one is two, then it's the second harmonic. 319 00:20:56,721 --> 00:21:01,685 And so now I will try to generate a very high frequency, 320 00:21:01,685 --> 00:21:05,476 in resonance, and then you count the number 321 00:21:05,476 --> 00:21:09,628 of nodes, subtract one, and then you know which 322 00:21:09,628 --> 00:21:14,954 harmonic I was able to generate. But I will try to -- not so 323 00:21:14,954 --> 00:21:17,842 easy to get a resonance in there. 324 00:21:17,842 --> 00:21:20,099 No, I'm off resonance. No. 325 00:21:20,099 --> 0. 326 0. --> 00:21:20,64 Yeah! 327 00:21:20,64 --> 00:21:21,543 Yeah! Yeah! 328 00:21:21,543 --> 00:21:22,446 Yeah! Yeah! 329 00:21:22,446 --> 00:21:23,348 Yeah! Yeah! 330 00:21:23,348 --> 00:21:28,73 [laughter] Got it, got it, got it! 331 00:21:28,73 --> 00:21:30,778 Got it! You keep counting. 332 00:21:30,778 --> 00:21:33,81 Keep counting. Oh, that's a super-high 333 00:21:33,81 --> 00:21:36,924 harmonic! [crowd responds] How many did 334 00:21:36,924 --> 00:21:39,464 you count? [crowd responds] Ten, 335 00:21:39,464 --> 00:21:42,742 do I hear ten? [crowd responds] Do I hear 336 00:21:42,742 --> 00:21:44,954 twenty? [laughter] Actually, 337 00:21:44,954 --> 00:21:48,478 I counted about twenty-seven, but that's OK. 338 00:21:48,478 --> 00:21:51,919 [laughter]. All right, thank you very much, 339 00:21:51,919 --> 00:21:54,459 it was great that you helped me. 340 00:21:54,459 --> 00:21:58,155 [applause]. So that's, uh, 341 00:21:58,155 --> 00:22:01,292 standing waves. And you see the shapes, 342 00:22:01,292 --> 00:22:04,428 and you saw the, the mode of operation, 343 00:22:04,428 --> 00:22:07,564 very characteristic for standing waves. 344 00:22:07,564 --> 00:22:10,371 When I pluck a string, of a violin, 345 00:22:10,371 --> 00:22:13,92 or I strike it with a bow, or with a hammer, 346 00:22:13,92 --> 00:22:18,625 on a piano, just a hammer comes down, that is exposing the 347 00:22:18,625 --> 00:22:21,596 string to a whole set of frequencies. 348 00:22:21,596 --> 00:22:25,558 And so the string, now, decided which frequencies 349 00:22:25,558 --> 00:22:30,629 it like to oscillate in. And so it selects these 350 00:22:30,629 --> 00:22:33,978 resonance frequencies. And so if the string has a 351 00:22:33,978 --> 00:22:37,885 fundamental of four hundred Hertz, then it would start to 352 00:22:37,885 --> 00:22:40,955 resonate at four hundred, but simultaneously, 353 00:22:40,955 --> 00:22:44,165 it will be very happy with eight hundred Hertz, 354 00:22:44,165 --> 00:22:47,793 and with twelve hundred Hertz. And so the string will 355 00:22:47,793 --> 00:22:50,863 simultaneously -- if I bang it, or strike it, 356 00:22:50,863 --> 00:22:54,91 or pluck it -- simultaneously oscillate, often at more than 357 00:22:54,91 --> 00:22:56,724 one frequency. Fundamental, 358 00:22:56,724 --> 00:23:00,491 and several of the higher harmonics. 359 00:23:00,491 --> 00:23:04,577 And all the others that are present, all the other 360 00:23:04,577 --> 00:23:08,079 frequencies in this striking with the bow, 361 00:23:08,079 --> 00:23:10,664 I ignore, they're off resonance. 362 00:23:10,664 --> 00:23:15,333 So if you design a string instrument, then this is really 363 00:23:15,333 --> 00:23:18,584 a key equation. If you want a particular 364 00:23:18,584 --> 00:23:23,087 fundamental -- say your fundamental is four hundred and 365 00:23:23,087 --> 00:23:25,838 forty, and this is a given number. 366 00:23:25,838 --> 00:23:31,174 And so N is one. You can now manipulate V, 367 00:23:31,174 --> 00:23:34,741 because V depends on the tension on the string, 368 00:23:34,741 --> 00:23:38,307 and it depends on what kind of string you have. 369 00:23:38,307 --> 00:23:43,114 The speed in the string is the square root of the tension -- if 370 00:23:43,114 --> 00:23:47,223 you take eight oh three, you will even see a proof for 371 00:23:47,223 --> 00:23:51,797 that -- divided by the mass of the string per unit length of 372 00:23:51,797 --> 00:23:54,976 the string. So you take four strings for a 373 00:23:54,976 --> 00:23:59,472 violin -- six for a guitar -- and you make them out of very 374 00:23:59,472 --> 00:24:04,061 different material -- different mass per unit length 375 00:24:04,061 --> 00:24:07,356 -- and so that gives you, then, difference velocities -- 376 00:24:07,356 --> 00:24:10,891 you can also fool around with the tension -- and so the four 377 00:24:10,891 --> 00:24:13,108 strings, then, have all four different 378 00:24:13,108 --> 00:24:14,666 fundamental. In the violin, 379 00:24:14,666 --> 00:24:18,081 they may have the same length. What are going to do now to 380 00:24:18,081 --> 00:24:20,657 play the violin? All that is left over is L, 381 00:24:20,657 --> 00:24:23,892 that's the only thing you can change, and that's what a 382 00:24:23,892 --> 00:24:26,289 violinist is doing. Goes with the finger, 383 00:24:26,289 --> 00:24:29,285 back and forth over the strings, make them shorter, 384 00:24:29,285 --> 00:24:31,202 pitch goes up, frequency goes up, 385 00:24:31,202 --> 00:24:34,78 makes them longer, frequency goes down. 386 00:24:34,78 --> 00:24:38,206 And you do the same with a guitar, and you do the same with 387 00:24:38,206 --> 00:24:40,924 the bass and the cello. So when you're playing, 388 00:24:40,924 --> 00:24:44,409 what you're doing all the time is changing L so that you get 389 00:24:44,409 --> 00:24:47,127 all these frequencies that you want to produce. 390 00:24:47,127 --> 00:24:49,785 If you take your instrument out of the closet, 391 00:24:49,785 --> 00:24:53,153 you may have noticed that it's really not in tune anymore, 392 00:24:53,153 --> 00:24:56,047 it's slightly off-tune. Well, what you can do now, 393 00:24:56,047 --> 00:24:59,651 you can change V a little bit -- uh, these little knobs -- and 394 00:24:59,651 --> 00:25:03,195 you can change the tension in the strings. 395 00:25:03,195 --> 00:25:06,251 And that's what violinists do when they tune their violin, 396 00:25:06,251 --> 00:25:09,361 they change the tension on the string to get just the right 397 00:25:09,361 --> 00:25:11,237 frequency. But the playing means you 398 00:25:11,237 --> 00:25:12,845 change L. A piano is different, 399 00:25:12,845 --> 00:25:15,686 that's really a luxury. A piano has eighty eight keys, 400 00:25:15,686 --> 00:25:18,152 and the length of each set of strings is fixed, 401 00:25:18,152 --> 00:25:20,136 so you don't have to worry about that. 402 00:25:20,136 --> 00:25:21,959 It's a great luxury, you may think, 403 00:25:21,959 --> 00:25:25,068 therefore, that it is much easier to play the piano that to 404 00:25:25,068 --> 00:25:27,32 play a violin, because you don't have to do 405 00:25:27,32 --> 00:25:29,732 this all the time, and be exactly at the right 406 00:25:29,732 --> 00:25:31,126 length. Well, that is true, 407 00:25:31,126 --> 00:25:35,146 of course, but given the fact that you have eighty 408 00:25:35,146 --> 00:25:39,111 eight keys, you can imagine you can hit occasionally the wrong 409 00:25:39,111 --> 00:25:41,256 key, and that's not what you want. 410 00:25:41,256 --> 00:25:44,506 If a string is vibrating, it is pushing on the air, 411 00:25:44,506 --> 00:25:48,34 and it's pulling on the air, and it's producing thereby what 412 00:25:48,34 --> 00:25:51,33 we call pressure waves. If I have a string that 413 00:25:51,33 --> 00:25:54,97 oscillates four hundred Hertz, it makes pressure waves -- 414 00:25:54,97 --> 00:25:57,049 pressure goes up, down, up, down, 415 00:25:57,049 --> 00:26:00,429 up, down -- four hundred times per second it goes up, 416 00:26:00,429 --> 00:26:03,679 it reaches your eardrum, and your eardrum starts to 417 00:26:03,679 --> 00:26:07,253 shake four hundred times per second, 418 00:26:07,253 --> 00:26:09,781 it goes back and forth, and your brain say, 419 00:26:09,781 --> 00:26:12,068 "I hear sound. That's the way it works. 420 00:26:12,068 --> 00:26:14,535 So it is the string, then you get the air, 421 00:26:14,535 --> 00:26:17,243 pressure waves, and then you get your eardrum, 422 00:26:17,243 --> 00:26:19,891 and then you get to brains, if there are any. 423 00:26:19,891 --> 00:26:23,442 Now, I want to discuss with you before I demonstrate some of 424 00:26:23,442 --> 00:26:27,052 this, I want to discuss with you instruments which don't have 425 00:26:27,052 --> 00:26:30,302 strings, and I will call them all woodwind instruments, 426 00:26:30,302 --> 00:26:33,732 although that's perhaps not an appropriate name for all of 427 00:26:33,732 --> 00:26:35,417 them. But I'll just call them 428 00:26:35,417 --> 00:26:37,778 woodwinds for now. 429 00:26:37,778 --> 00:26:40,975 And suppose I have, here, a box which is filled 430 00:26:40,975 --> 00:26:43,13 with air. Completely closed box, 431 00:26:43,13 --> 00:26:45,979 and it has a length L. And I put in here a 432 00:26:45,979 --> 00:26:50,079 loudspeaker, and I generate a particular frequency of sound. 433 00:26:50,079 --> 00:26:54,249 Then pressure waves are going to run, they're going to bounce 434 00:26:54,249 --> 00:26:57,933 off, and they come back, and I get reflected traveling 435 00:26:57,933 --> 00:27:00,295 waves. And what I get inside the -- 436 00:27:00,295 --> 00:27:03,145 the box, now, I get standing waves of air. 437 00:27:03,145 --> 00:27:06,759 It's not the box that goes into a standing resonance, 438 00:27:06,759 --> 00:27:10,373 but it's the air itself. And the 439 00:27:10,373 --> 00:27:14,213 frequencies that are produced, at which the system is in 440 00:27:14,213 --> 00:27:17,564 resonance, is given exactly by the same equation. 441 00:27:17,564 --> 00:27:20,427 Except, now, that V is non-negotiable -- V 442 00:27:20,427 --> 00:27:24,127 is now the speed of sound, which, at room temperature, 443 00:27:24,127 --> 00:27:27,618 is about three hundred and forty meters per second. 444 00:27:27,618 --> 00:27:31,598 So whenever you design an -- wi- we- woodwind instruments, 445 00:27:31,598 --> 00:27:34,6 that is non-negotiable. You cannot change V, 446 00:27:34,6 --> 00:27:38,301 which you can do when you are an instrument builder of 447 00:27:38,301 --> 00:27:40,942 strings. You will say, 448 00:27:40,942 --> 00:27:44,938 "Gee, if I have an instrument whereby the sound is inside a 449 00:27:44,938 --> 00:27:48,589 closed box, you're not going to hear very much." Well, 450 00:27:48,589 --> 00:27:51,414 that's true. You must let the sound go out 451 00:27:51,414 --> 00:27:53,549 somehow. And what is surprising, 452 00:27:53,549 --> 00:27:57,683 that if you take this end out -- off -- and you take this end 453 00:27:57,683 --> 00:28:00,714 off, that this box, which is now open on both 454 00:28:00,714 --> 00:28:04,503 sides, will still resonate at exactly those frequencies. 455 00:28:04,503 --> 00:28:08,637 And you've got to take eight oh three to get to the bottom of 456 00:28:08,637 --> 00:28:11,914 this. There are also resonant 457 00:28:11,914 --> 00:28:16,328 frequencies in case that the sound cavity -- if I call this a 458 00:28:16,328 --> 00:28:20,448 sound cavity -- is closed in one end and open on one end. 459 00:28:20,448 --> 00:28:24,2 The series of resonant frequencies is different from 460 00:28:24,2 --> 00:28:27,069 this one, though. It's not so important, 461 00:28:27,069 --> 00:28:30,821 but it is different. But you also get a whole series 462 00:28:30,821 --> 00:28:34,941 of resonance frequencies. The velocity of sound in a gas, 463 00:28:34,941 --> 00:28:40,385 V, is the square root of the temperature -- so it's a little 464 00:28:40,385 --> 00:28:43,511 temperature-dependent -- divided by the molecular weight. 465 00:28:43,511 --> 00:28:46,86 Well, you can't do much about the temperature in a room -- in 466 00:28:46,86 --> 00:28:49,428 general, it's room temperature -- and with air, 467 00:28:49,428 --> 00:28:51,605 you are stuck with the molecular weight, 468 00:28:51,605 --> 00:28:54,786 oxygen and nitrogen is about thirty, there is not much you 469 00:28:54,786 --> 00:28:57,466 can do about that. But every one of you who plays 470 00:28:57,466 --> 00:29:00,871 woodwind instruments know that if you go from a cold room to a 471 00:29:00,871 --> 00:29:03,717 warm room that your instrument is no longer in tune. 472 00:29:03,717 --> 00:29:06,508 And that's because of this, the temperature change. 473 00:29:06,508 --> 00:29:08,797 So V changes, so your fundamentals change. 474 00:29:08,797 --> 00:29:13,201 And what do you do know? These people know what they do. 475 00:29:13,201 --> 00:29:16,68 They have a way of making the cavity a little shorter or a 476 00:29:16,68 --> 00:29:18,694 little longer. It's not very much, 477 00:29:18,694 --> 00:29:21,075 but they have a little bit to play with. 478 00:29:21,075 --> 00:29:24,127 And when they do that, so they compensate L for the 479 00:29:24,127 --> 00:29:27,606 slight difference in V to get back to the same fundamental 480 00:29:27,606 --> 00:29:30,109 that they need. So now you have a woodwind 481 00:29:30,109 --> 00:29:32,123 instrument. Low-frequency woodwind 482 00:29:32,123 --> 00:29:35,236 instruments will be big. And high-frequency woodwind 483 00:29:35,236 --> 00:29:38,594 instruments will be small, because in L lies the secret, 484 00:29:38,594 --> 00:29:42,575 you can't fool around with V, V is a God-given. 485 00:29:42,575 --> 00:29:44,906 And so how do you play an instrument now? 486 00:29:44,906 --> 00:29:47,644 Well, you have to change, that's all you can do. 487 00:29:47,644 --> 00:29:50,266 And if you have a trombone, you're doing this, 488 00:29:50,266 --> 00:29:53,82 it's clear that you're changing L, you make the cavity shorter 489 00:29:53,82 --> 00:29:55,335 and longer. So that's easy. 490 00:29:55,335 --> 00:29:58,131 If you have a flute, you have holes in the flute. 491 00:29:58,131 --> 00:30:01,336 And if all the holes are closed, the flute is this long. 492 00:30:01,336 --> 00:30:03,783 But if you take your fingers off the holes, 493 00:30:03,783 --> 00:30:06,405 it gets shorter. And so when you take all your 494 00:30:06,405 --> 00:30:09,435 fingers off the holes, then you have a high frequency 495 00:30:09,435 --> 00:30:14,526 -- flute is only this long, if you put all your fingers on 496 00:30:14,526 --> 00:30:18,18 it, it's this long, and so the frequency is lower. 497 00:30:18,18 --> 00:30:22,803 And a trumpet is the same idea. You have valves that open holes 498 00:30:22,803 --> 00:30:25,562 and close holes. If you blow air in an 499 00:30:25,562 --> 00:30:30,11 instrument, it is like plucking a string, it's like exciting a 500 00:30:30,11 --> 00:30:33,167 string with a bow, you are dumping a whole 501 00:30:33,167 --> 00:30:36,448 spectrum of frequencies onto that air cavity. 502 00:30:36,448 --> 00:30:40,176 And you let the air cavity decide where it wants to 503 00:30:40,176 --> 00:30:43,925 resonate. And it will pick out the ones 504 00:30:43,925 --> 00:30:46,134 that it likes, it will pick out the 505 00:30:46,134 --> 00:30:49,772 fundamental, and maybe the second and the third harmonic. 506 00:30:49,772 --> 00:30:52,891 So in that sense, blowing air is like striking it 507 00:30:52,891 --> 00:30:55,879 with a bow, in the case of a string instrument. 508 00:30:55,879 --> 00:30:59,778 But blowing air is not always as easy as you may think it is. 509 00:30:59,778 --> 00:31:02,766 Have you ever tried to blow air into a trumpet? 510 00:31:02,766 --> 00:31:04,715 Nothing happens. You just blow, 511 00:31:04,715 --> 00:31:07,964 [pffff], and you hear nothing. You have to do this. 512 00:31:07,964 --> 00:31:09,913 [ppppppp]. Something like that. 513 00:31:09,913 --> 00:31:13,356 A bizarre sound you have to make, 514 00:31:13,356 --> 00:31:17,62 you have to know how to spit in the instrument just the right 515 00:31:17,62 --> 00:31:21,458 way to get a sound out of it. I've tried it many times, 516 00:31:21,458 --> 00:31:25,082 it's really not easy. So blowing air is just said in 517 00:31:25,082 --> 00:31:28,138 a simple way, but in order to get it exc- to 518 00:31:28,138 --> 00:31:32,188 resonate, you've got to really know how to hold your lips, 519 00:31:32,188 --> 00:31:35,955 and how to excite that cavity. I can show you the easy 520 00:31:35,955 --> 00:31:39,934 relation between the frequency of the fundamental and the 521 00:31:39,934 --> 00:31:44,34 length of woodwind instruments. That's a one-to-one correlation 522 00:31:44,34 --> 00:31:46,825 -- this is on the web, 523 00:31:46,825 --> 00:31:49,81 you can download it, you don't have to copy this -- 524 00:31:49,81 --> 00:31:52,915 and you see there that, uh, this is only for an open, 525 00:31:52,915 --> 00:31:55,363 open system, this is not for a closed open 526 00:31:55,363 --> 00:31:57,572 system. The number would be different. 527 00:31:57,572 --> 00:32:00,498 So if you are interested in very high frequencies, 528 00:32:00,498 --> 00:32:03,901 then an open open system which is only one centimeter long 529 00:32:03,901 --> 00:32:07,245 would give you a fundamental of seventeen thousand Hertz, 530 00:32:07,245 --> 00:32:10,469 which most of you can hear, because you're still young, 531 00:32:10,469 --> 00:32:13,156 you can hear up to twenty kiloHertz, probably. 532 00:32:13,156 --> 00:32:17,327 The second harmonic you would not be able to hear, 533 00:32:17,327 --> 00:32:20,473 that is too high for you. An instrument which is ten 534 00:32:20,473 --> 00:32:23,311 centimeters long, open open, you would here the 535 00:32:23,311 --> 00:32:25,964 fundamental easily, seventeen hundred Hertz, 536 00:32:25,964 --> 00:32:28,555 second harmonic, thirty four hundred Hertz, 537 00:32:28,555 --> 00:32:31,207 no problem, third harmonic, fourth harmonic, 538 00:32:31,207 --> 00:32:33,551 no problem. And then when you go to the 539 00:32:33,551 --> 00:32:36,574 very low frequencies, uh, organ pipes that produce 540 00:32:36,574 --> 00:32:39,597 fundamentals in the range twenty and thirty Hertz, 541 00:32:39,597 --> 00:32:42,002 are huge in size. And you -- in general, 542 00:32:42,002 --> 00:32:44,161 that holds. When you have a woodwind 543 00:32:44,161 --> 00:32:47,647 instrument which is tunes for low 544 00:32:47,647 --> 00:32:50,877 frequencies, it's big. And for high frequencies, 545 00:32:50,877 --> 00:32:52,527 like a flute, it's small. 546 00:32:52,527 --> 00:32:55,894 In a way, that's also true for string instruments. 547 00:32:55,894 --> 00:32:59,261 A bass which generates low frequencies, it's a big 548 00:32:59,261 --> 00:33:01,048 instrument. But the violin, 549 00:33:01,048 --> 00:33:04,484 which generates high frequencies, is a much shorter 550 00:33:04,484 --> 00:33:06,408 instrument. So in that sense, 551 00:33:06,408 --> 00:33:09,088 the reason it, they have both an L here. 552 00:33:09,088 --> 00:33:12,112 And it's the L, of course, that is crucial in 553 00:33:12,112 --> 00:33:16,647 terms of the fundamental. So I can now ge- uh, 554 00:33:16,647 --> 00:33:24,131 demonstrate to you the basic idea of a flute -- this is a 555 00:33:24,131 --> 00:33:28,675 flute. Now, the flute is this long. 556 00:33:28,675 --> 00:33:32,417 [plays flute]. Low frequency. 557 00:33:32,417 --> 00:33:36,56 [plays flute]. Higher frequency, 558 00:33:36,56 --> 00:33:41,238 because it's shorter. [plays flute]. 559 00:33:41,238 --> 00:33:46,984 Even higher frequency, because it's shorter. 560 00:33:46,984 --> 00:33:55,154 [plays flute]. That's all it takes. 561 00:33:55,154 --> 00:33:59,715 [applause]. Trombone. 562 00:33:59,715 --> 00:34:06,328 That speaks for itself, right? 563 00:34:06,328 --> 00:34:14,31 Make it longer, you make it shorter. 564 00:34:14,31 --> 00:34:23,204 I'll try it. [plays trombone] [applause] 565 00:34:23,204 --> 00:34:27,765 Trombone. [applause]. 566 00:34:27,765 --> 00:34:34,376 Wind organ. Eighty centimeters long. 567 00:34:34,376 --> 00:34:35,969 Open and open, on both sides. 568 00:34:35,969 --> 00:34:38,757 Eighty centimeters, it would give me a fundamental 569 00:34:38,757 --> 00:34:41,146 a little higher than a hundred and seventy. 570 00:34:41,146 --> 00:34:43,536 And then it will give me a second harmonic, 571 00:34:43,536 --> 00:34:46,437 and a third harmonic, it all depends on how fast the 572 00:34:46,437 --> 00:34:49,168 air is flowing by, and there will be moments that 573 00:34:49,168 --> 00:34:51,216 you will hear more than one harmonic. 574 00:34:51,216 --> 00:34:54,573 I'll try to swing it around. It's not easy for me to hit the 575 00:34:54,573 --> 00:34:56,564 fundamental, but I'll try that, too. 576 00:34:56,564 --> 00:34:59,238 This is the second harmonic. [plays wind organ]. 577 00:34:59,238 --> 00:35:01,741 Third harmonic. [wind organ] Fourth harmonic. 578 00:35:01,741 --> 00:35:07,346 [wind organ] Fifth harmonic. [wind organ] Fourth. 579 00:35:07,346 --> 00:35:13,035 [wind organ] Furdamen-, this is fundamental. 580 00:35:13,035 --> 00:35:20,313 [wind organ] This is the fundamental, two hundred twelve 581 00:35:20,313 --> 00:35:24,944 Hertz. [wind organ] Four twenty-five 582 00:35:24,944 --> 00:35:29,707 Hertz. [wind organ] Six thirty seven, 583 00:35:29,707 --> 00:35:35 six thirty seven. [wind organ] 584 00:35:35 --> 00:35:38,762 [applause] Thank you, thank you, thank you. 585 00:35:38,762 --> 00:35:43,599 If you bang on a tuning fork, or you pluck on a string, 586 00:35:43,599 --> 00:35:46,287 in isolation, you hear nothing, 587 00:35:46,287 --> 00:35:49,959 almost nothing. I have here a tuning fork, 588 00:35:49,959 --> 00:35:53,274 and if I bang on it, you hear nothing, 589 00:35:53,274 --> 00:35:56,499 and I hear nothing -- almost nothing. 590 00:35:56,499 --> 00:36:00,351 Unless I heel- hold it very close to my ear. 591 00:36:00,351 --> 00:36:05,277 What we do now, with string instruments, 592 00:36:05,277 --> 00:36:08,023 we mount the strings on a box with air. 593 00:36:08,023 --> 00:36:10,696 A sound cavity. Sound- sounding board, 594 00:36:10,696 --> 00:36:13,658 it's called. And now the air inside can s- 595 00:36:13,658 --> 00:36:18,065 oscillate with it -- it doesn't always have to be precisely at 596 00:36:18,065 --> 00:36:21,749 resonance -- and also, the surface itself of the box 597 00:36:21,749 --> 00:36:25,433 can start to vibrate. So you're displacing more air, 598 00:36:25,433 --> 00:36:28,034 and the sound becomes loud and clear. 599 00:36:28,034 --> 00:36:31,863 You don't gain energy, but you drain the energy out of 600 00:36:31,863 --> 00:36:37,856 the oscillating string faster, and so for that short amount of 601 00:36:37,856 --> 00:36:42,464 time, you get louder sound. And I will demonstrate that, 602 00:36:42,464 --> 00:36:46,82 first, with the tuning fork. I hear it now very well, 603 00:36:46,82 --> 00:36:50,59 it's harder for you because it's farther away. 604 00:36:50,59 --> 00:36:54,025 Now you hear nothing. And now you hear it. 605 00:36:54,025 --> 00:36:58,968 It can actually be much better demonstrated with this little 606 00:36:58,968 --> 00:37:02,99 music box that I bought years ago in Switzerland. 607 00:37:02,99 --> 00:37:08,772 If I rotate this music box, it has a very romantic tune, 608 00:37:08,772 --> 00:37:11,875 you hear nothing. I hear a little bit. 609 00:37:11,875 --> 00:37:15,48 And now I put it on this box , unmistakable. 610 00:37:15,48 --> 00:37:18,582 So that's the idea of sounding boards. 611 00:37:18,582 --> 00:37:22,439 You have them violins, you have them on pianos, 612 00:37:22,439 --> 00:37:26,045 and, of course, the design of these sounding 613 00:37:26,045 --> 00:37:30,237 boards is top-secret, the manufacturer is not going 614 00:37:30,237 --> 00:37:35,1 to tell you how they built them, because the quality of the 615 00:37:35,1 --> 00:37:39,88 sound, of course, is partly in the 616 00:37:39,88 --> 00:37:44,178 design of the sounding board. I can make you hear, 617 00:37:44,178 --> 00:37:49,266 and I can make you see sound. And my goal for the remaining 618 00:37:49,266 --> 00:37:53,565 time is to make you see and hear at the same time. 619 00:37:53,565 --> 00:37:58,126 I have here a microphone, which is like your eardrum, 620 00:37:58,126 --> 00:38:02,6 and suppose I generate four hundred and forty Hertz, 621 00:38:02,6 --> 00:38:05,934 and I can do that with the tuning fork. 622 00:38:05,934 --> 00:38:12,514 So here is the amplitude of the oscillation of the membrane in 623 00:38:12,514 --> 00:38:15,328 the microphone, which is your eardrum, 624 00:38:15,328 --> 00:38:18,523 say, we amplify that, and we show you on an 625 00:38:18,523 --> 00:38:20,805 oscilloscope, the current after 626 00:38:20,805 --> 00:38:24,001 amplification. And so you're going to see a 627 00:38:24,001 --> 00:38:27,652 signal like this. And if this is four hundred and 628 00:38:27,652 --> 00:38:32,216 forty Hertz, so this is time, and this is the displacement of 629 00:38:32,216 --> 00:38:35,716 your eardrum -- in our case, it's a microphone, 630 00:38:35,716 --> 00:38:40,128 it's really a current after amplification -- and if this is 631 00:38:40,128 --> 00:38:45,619 four hundred and forty Hertz, then this time T will be about 632 00:38:45,619 --> 00:38:49,602 two point three milliseconds. One divided by four hundred and 633 00:38:49,602 --> 00:38:51,593 forty. That's no problem for an 634 00:38:51,593 --> 00:38:54,182 oscilloscope. We can do much better than 635 00:38:54,182 --> 00:38:56,572 that. So the time resolution is not a 636 00:38:56,572 --> 00:38:58,962 problem. And so I will show you there 637 00:38:58,962 --> 00:39:02,879 the output of our microphone, I will show you this signal as 638 00:39:02,879 --> 00:39:05,866 a function of time. For four hundred and forty 639 00:39:05,866 --> 00:39:09,915 Hertz, you see a boring signal. And I can make a boring signal 640 00:39:09,915 --> 00:39:12,902 with a tuning fork, it's almost a pure sine of 641 00:39:12,902 --> 00:39:16,554 sorts. But now, it just so happens, 642 00:39:16,554 --> 00:39:20,129 we have in our audie- in our audience, someone who can play 643 00:39:20,129 --> 00:39:22,471 the violin. And that person is going to 644 00:39:22,471 --> 00:39:24,814 produce a four hundred and forty Hertz. 645 00:39:24,814 --> 00:39:27,958 But at the same time, he's going to produce a second 646 00:39:27,958 --> 00:39:31,348 harmonic, and maybe a third harmonic, and maybe a fourth 647 00:39:31,348 --> 00:39:33,136 harmonic. And so, imagine now, 648 00:39:33,136 --> 00:39:36,465 that at th- simultaneously, your eardrum is going to do 649 00:39:36,465 --> 00:39:39,793 this, but at the same time, your eardrum is going to do 650 00:39:39,793 --> 00:39:42,382 this, because this is some higher harmonic. 651 00:39:42,382 --> 00:39:45,711 Then the net result is that your eardrum is going to do 652 00:39:45,711 --> 00:39:47,807 this. And that is what I'm going to 653 00:39:47,807 --> 00:39:49,731 show you. 654 00:39:49,731 --> 00:39:53,75 And so when you see the various instruments, you will recognize 655 00:39:53,75 --> 00:39:57,25 that on top of the fundamental, you will see these very 656 00:39:57,25 --> 00:40:00,816 characteristic harmonics, each instrument having its own 657 00:40:00,816 --> 00:40:04,511 cocktail, it's own unique cocktail, and when you hear that 658 00:40:04,511 --> 00:40:07,557 cocktail, you say, "Oh, yes, that's a saxophone. 659 00:40:07,557 --> 00:40:09,632 Or you say, "No, that's a violin. 660 00:40:09,632 --> 00:40:12,808 You would never mistaken a saxophone for a violin. 661 00:40:12,808 --> 00:40:16,049 And that's because of the combination of the higher 662 00:40:16,049 --> 00:40:19,614 harmonics. And so we are so fortunate 663 00:40:19,614 --> 00:40:23,285 that we have four musicians in our audience. 664 00:40:23,285 --> 00:40:26,785 Tom, who is the violinist -- where is Tom? 665 00:40:26,785 --> 00:40:28,748 There's Tom. [applause]. 666 00:40:28,748 --> 00:40:32,077 I hope you brought your violin [laughs]. 667 00:40:32,077 --> 00:40:36,516 Oh, you got it there. And then we have Emily -- I saw 668 00:40:36,516 --> 00:40:41,382 her already, with the clarinet -- so if you come this way, 669 00:40:41,382 --> 00:40:45,394 Emily [applause]. And we have Aaron -- Aaron has 670 00:40:45,394 --> 00:40:50,088 a bassoon. You may never have a bassoon. 671 00:40:50,088 --> 00:40:56,364 A bassoon is an instrument that produces a very low tone. 672 00:40:56,364 --> 00:41:00,959 So the instrument is going to be very big. 673 00:41:00,959 --> 00:41:04,545 Bigger than -- bigger than Aaron. 674 00:41:04,545 --> 00:41:10,036 Just wait and see. Beauty, it's a really beautiful 675 00:41:10,036 --> 00:41:14,071 instrument. A flute is only this big. 676 00:41:14,071 --> 00:41:17,657 Ah, look at that, beautiful, big, 677 00:41:17,657 --> 00:41:22,277 bassoon. And then we have Fabian, 678 00:41:22,277 --> 00:41:24,922 with a saxophone. [applause]. 679 00:41:24,922 --> 00:41:29,834 So if you stand here, then I will first do the boring 680 00:41:29,834 --> 00:41:33,896 part, and what I will do is I will show you, 681 00:41:33,896 --> 00:41:39,468 then, what a four-hundred and forty Hertz signal looks like, 682 00:41:39,468 --> 00:41:44,475 produced with a tuning fork -- and we'll see it there, 683 00:41:44,475 --> 00:41:51,086 and so I have to change the light situation substantially. 684 00:41:51,086 --> 00:41:56,572 [unintelligible] the musicians will get a little bit into the 685 00:41:56,572 --> 00:42:00,595 dark, but you will still be able to see them. 686 00:42:00,595 --> 00:42:04,984 And so I'm going to turn on, now, the microphone, 687 00:42:04,984 --> 00:42:09,281 and that's where you're going to see the signal, 688 00:42:09,281 --> 00:42:14,035 and when you make noise, you can hear -- hear and see 689 00:42:14,035 --> 00:42:15,863 yourself. Four forty. 690 00:42:15,863 --> 00:42:19,521 Boring, and no signs of higher harmonics. 691 00:42:19,521 --> 00:42:26,531 Now Tom will try to produce four-forty in his violin -- or 692 00:42:26,531 --> 00:42:32,978 close to four-forty -- and then look for the higher harmonics, 693 00:42:32,978 --> 00:42:36,888 which makes the violin characteristic. 694 00:42:36,888 --> 00:42:40,904 [plays violin]. Notice that the average 695 00:42:40,904 --> 00:42:46,612 spacing, the repetition, is, indeed, the same as it was 696 00:42:46,612 --> 00:42:51,579 with the four forty, but you saw this incredible 697 00:42:51,579 --> 00:42:57,263 richness of harmonics. 698 00:42:57,263 --> 00:43:09,074 Tom happens to be, also, an excellent violin 699 00:43:09,074 --> 00:43:23,082 player, and so he insisted that he demonstrate that. 700 00:43:23,082 --> 00:43:29,949 [laughter] All right, Tom? 701 00:43:29,949 --> 00:43:35,992 Student: All right. OK. 702 00:43:35,992 --> 00:43:42,584 Go ahead. [plays violin]. 703 00:43:42,584 --> 00:43:53,021 Terrific, Tom. [applause]. 704 00:43:53,021 --> 00:43:58,408 Emily, would you mind producing something that comes close to 705 00:43:58,408 --> 00:44:03,435 four hundred and forty Hertz? Come a little closer to the 706 00:44:03,435 --> 00:44:05,949 microphone. [plays clarinet]. 707 00:44:05,949 --> 00:44:09,63 Notice the big difference with the violin. 708 00:44:09,63 --> 00:44:13,041 Violin has many, many higher harmonics. 709 00:44:13,041 --> 00:44:15,734 Her instrument, maybe only one, 710 00:44:15,734 --> 00:44:20,223 maybe only the fundamental and the second harmonic. 711 00:44:20,223 --> 00:44:23,365 Can you try again? [plays clarinet]. 712 00:44:23,365 --> 00:44:26,418 Now we have more, now we have more. 713 00:44:26,418 --> 00:44:33,558 [plays clarinet]. Now, Emily did not insist that 714 00:44:33,558 --> 00:44:37,335 she wanted to play, but I did. 715 00:44:37,335 --> 00:44:43,067 So Emily, would you please? [plays clarinet]. 716 00:44:43,067 --> 00:44:45,933 Impressive. [applause]. 717 00:44:45,933 --> 00:44:52,576 Aaron, with his bassoon. He ordered a special chair, 718 00:44:52,576 --> 00:44:58,308 because he says, "Look, with an instrument so 719 00:44:58,308 --> 00:45:01,594 big, L is so large, 720 00:45:01,594 --> 00:45:05,629 it's heavy." Clearly, a bass, which produces low 721 00:45:05,629 --> 00:45:10,607 frequencies, is heavier that a violin, and the same is true 722 00:45:10,607 --> 00:45:15,414 with woodwind instruments. Aaron, could you try something 723 00:45:15,414 --> 00:45:18,505 close to four-forty? [plays bassoon]. 724 00:45:18,505 --> 00:45:21,252 Bizarre instrument, isn't it, eh? 725 00:45:21,252 --> 00:45:26,488 You see a weird combination of probably fundamental and second 726 00:45:26,488 --> 00:45:29,75 harmonic. Aaron, would you mind showing 727 00:45:29,75 --> 00:45:44,667 some of your expertise? This is a wonderful instrument. 728 00:45:44,667 --> 00:45:58,103 You don't see them too often, do you? 729 00:45:58,103 --> 00:46:07,433 [plays bassoon]. Terrific. 730 00:46:07,433 --> 00:46:18,63 [applause]. Last, but -- 731 00:46:18,63 --> 00:46:20,771 but not least, we have a saxophone, 732 00:46:20,771 --> 00:46:22,976 Fabian. Now, you may have to stand a 733 00:46:22,976 --> 00:46:26,001 long distance from this microphone, because these 734 00:46:26,001 --> 00:46:29,34 instruments make a hell of a lot of noise, don't they? 735 00:46:29,34 --> 00:46:31,67 So give it a shot, and try four-forty, 736 00:46:31,67 --> 00:46:34,82 or come close to that. It doesn't have to be exact. 737 00:46:34,82 --> 00:46:36,71 [plays saxophone]. Interesting. 738 00:46:36,71 --> 00:46:40,427 Also, you see several higher harmonics, it's hard to s- hard 739 00:46:40,427 --> 00:46:42,255 to tell which. Would you mind, 740 00:46:42,255 --> 00:46:45,972 uh, playing something real hot? [laughter] [plays saxophone] 741 00:46:45,972 --> 00:46:49,5 [unintelligible] [laughter] [noise] 742 00:46:49,5 --> 00:46:57,08 If you think we're interested in hearing it, 743 00:46:57,08 --> 00:47:02,546 you're wrong, we want to see it! 744 00:47:02,546 --> 00:47:12,419 [laughter] [plays saxophone]. [unintelligible] [laughter] 745 00:47:12,419 --> 00:47:20 [unintelligible]. [plays saxophone]. 746 00:47:20 --> 00:47:22,78 [applause]. Thank all of them. 747 00:47:22,78 --> 00:47:28,533 [applause] Thank you very much. [applause] So during the last 748 00:47:28,533 --> 00:47:32,56 three minutes, I would like to discuss with 749 00:47:32,56 --> 00:47:37,354 you the speed of sound in a little bit more detail. 750 00:47:37,354 --> 00:47:43,107 Uh, you notice that the speed of sound is the -- proportional 751 00:47:43,107 --> 00:47:47,422 with the temperature and the molecular weight. 752 00:47:47,422 --> 00:47:51,833 [unintelligible] few other things 753 00:47:51,833 --> 00:47:55,205 upstairs here. But these are the -- these are 754 00:47:55,205 --> 00:47:59,038 the major contributors. So I should really say it's 755 00:47:59,038 --> 00:48:01,874 proportional. If you take air -- as we 756 00:48:01,874 --> 00:48:05,324 discussed earlier, molecular weight is thirty, 757 00:48:05,324 --> 00:48:08,85 that's a God-given, there's not much you can do 758 00:48:08,85 --> 00:48:11,839 about it. I would like to demonstrate to 759 00:48:11,839 --> 00:48:14,829 you, the dependence on molecular weight. 760 00:48:14,829 --> 00:48:19,351 And one way I could do that, I could take all the air out of 761 00:48:19,351 --> 00:48:23,184 twenty six one hundred, and replace it with helium. 762 00:48:23,184 --> 00:48:26,44 And then I would ask the same 763 00:48:26,44 --> 00:48:28,888 musicians to come, and I would ask them, 764 00:48:28,888 --> 00:48:31,148 then, to play their wind instruments. 765 00:48:31,148 --> 00:48:33,848 The Ls are fixed, there's nothing you can do 766 00:48:33,848 --> 00:48:36,359 about it. So their instruments don't know 767 00:48:36,359 --> 00:48:39,874 that I put helium in the audience, so the only thing that 768 00:48:39,874 --> 00:48:42,636 changes is V. The speed would go up by almost 769 00:48:42,636 --> 00:48:45,712 a factor of three, and so the fundamental would be 770 00:48:45,712 --> 00:48:49,04 free -- three times higher. And the harmonics would be 771 00:48:49,04 --> 00:48:52,053 three times higher. So you would hear much higher 772 00:48:52,053 --> 00:48:56,07 frequencies. And you wouldn't even recognize 773 00:48:56,07 --> 00:48:59,46 these instruments. This wouldn't be very 774 00:48:59,46 --> 00:49:02,762 practical. I cannot take the air out of 775 00:49:02,762 --> 00:49:07,107 twenty six one hundred, and replace it with helium. 776 00:49:07,107 --> 00:49:11,366 But what I can do, as I have done so often here in 777 00:49:11,366 --> 00:49:15,798 twenty six one hundred, I can suffer myself -- I can 778 00:49:15,798 --> 00:49:18,839 suffer, and put helium in my system. 779 00:49:18,839 --> 00:49:23,098 I have, here -- I have, here, my own sound cavity. 780 00:49:23,098 --> 00:49:27,791 I am, in a way, like a wind instrument. 781 00:49:27,791 --> 00:49:32,022 And, um, if I swallow helium, my sound cavity doesn't know 782 00:49:32,022 --> 00:49:35,215 that I'm producing helium. And you will say, 783 00:49:35,215 --> 00:49:38,333 when I talk to you as I do right now, "Yes, 784 00:49:38,333 --> 00:49:41,303 that's typical, that's Walter Lewin." You 785 00:49:41,303 --> 00:49:45,238 recognize my fundamentals, you recognize my harmonics, 786 00:49:45,238 --> 00:49:49,544 and it's unique for my voice. And so you will recognize me. 787 00:49:49,544 --> 00:49:53,108 But the moment that I fill my system with helium, 788 00:49:53,108 --> 00:49:56,449 nothing is changing in my system except for V. 789 00:49:56,449 --> 00:50:00,086 And so the frequency will go up. 790 00:50:00,086 --> 00:50:03,183 And that will be noticeable. And, in fact, 791 00:50:03,183 --> 00:50:07,563 change are that you will say, "Hm, that's really not Walter 792 00:50:07,563 --> 00:50:11,565 Lewin any more." There's only one problem with helium. 793 00:50:11,565 --> 00:50:14,586 And that is there is no oxygen in helium. 794 00:50:14,586 --> 00:50:17,531 And that is also very noticeable for me. 795 00:50:17,531 --> 00:50:21,912 And yet, I really have to fill my lungs with helium all the 796 00:50:21,912 --> 00:50:24,479 way, and so I will be, for a while, 797 00:50:24,479 --> 00:50:27,802 without oxygen, and, um, so you may catch two 798 00:50:27,802 --> 00:50:33,088 birds with one stone. You may [unintelligible] hear a 799 00:50:33,088 --> 00:50:36,609 strange frequency, and you see me on the floor. 800 00:50:36,609 --> 00:50:39,899 So I'll really try not to fall on the floor, 801 00:50:39,899 --> 00:50:41,506 then. OK, there we go. 802 00:50:41,506 --> 00:50:44,414 [laughter]. And it really doesn't sound 803 00:50:44,414 --> 00:50:47,092 like Walter Lewin any more, does it? 804 00:50:47,092 --> 00:50:49,77 I will see you Friday. All the best. 805 00:50:49,77 --> 50:55 [applause]. Whew.