1 00:00:00,000 --> 00:00:30,000 2 00:00:30,000 --> 00:00:34,580 Today we are going to talk about wave guides and about 3 00:00:34,580 --> 00:00:38,728 resonance cavities. Last Thursday I discussed the 4 00:00:38,728 --> 00:00:43,049 boundary conditions of electromagnetic waves on the 5 00:00:43,049 --> 00:00:46,333 surface of an idea conductor in vacuum. 6 00:00:46,333 --> 00:00:51,000 And today we will see some of the amazing consequences. 7 00:00:51,000 --> 00:00:56,185 And I will return to something that I have discussed with you 8 00:00:56,185 --> 00:01:00,851 before but never fully explained, and that is the setup 9 00:01:00,851 --> 00:01:05,000 whereby we have met a metal plate. 10 00:01:05,000 --> 00:01:10,400 This is my coordinate system, x, y z. 11 00:01:10,400 --> 00:01:18,200 This is the z direction. And then I have here another 12 00:01:18,200 --> 00:01:25,549 plate, two parallel plates, and this will be the x 13 00:01:25,549 --> 00:01:32,000 direction. I call this x equals zero. 14 00:01:32,000 --> 00:01:34,425 And this x, zero, a. 15 00:01:34,425 --> 00:01:38,510 And this then is the y direction. 16 00:01:38,510 --> 00:01:42,978 That is the setup. You see it there. 17 00:01:42,978 --> 00:01:49,361 I am going to try to send electromagnetic radiation 18 00:01:49,361 --> 00:01:54,340 through this gap. And what holds always, 19 00:01:54,340 --> 00:02:01,234 so what now current holds always, that k is always kx x 20 00:02:01,234 --> 00:02:08,000 roof plus ky y roof plus kz times z roof. 21 00:02:08,000 --> 00:02:13,393 K is the direction then of propagation. 22 00:02:13,393 --> 00:02:19,212 Also lambda equals 2 pi divided by that k. 23 00:02:19,212 --> 00:02:27,870 And the magnitude of that k is the square root of kx2 plus ky2 24 00:02:27,870 --> 00:02:32,754 plus kz2. And omega equals kc if this is 25 00:02:32,754 --> 00:02:36,518 in vacuum. I am going to send through this 26 00:02:36,518 --> 00:02:41,199 gap linearly polarized radiation in the y direction. 27 00:02:41,199 --> 00:02:46,157 That is a choice that I make. I do not have to do that, 28 00:02:46,157 --> 00:02:49,095 but that is a choice that I make. 29 00:02:49,095 --> 00:02:53,777 And think about the y direction and the z direction, 30 00:02:53,777 --> 00:02:58,000 very, very large, infinitely large. 31 00:02:58,000 --> 00:03:01,278 Or, in practice, many, many, many times the 32 00:03:01,278 --> 00:03:05,024 wavelength of the radiation. To meet the boundary 33 00:03:05,024 --> 00:03:09,160 conditions, the electric field, which is only in the y 34 00:03:09,160 --> 00:03:13,765 direction, must becomes zero here and must become zero there 35 00:03:13,765 --> 00:03:17,668 because the electric field cannot be in this plane. 36 00:03:17,668 --> 00:03:20,790 Remember E of t, which was the tangential 37 00:03:20,790 --> 00:03:25,082 component of the surface of the conductor, must be zero. 38 00:03:25,082 --> 00:03:30,000 E of y must vanish here and it must vanish there. 39 00:03:30,000 --> 00:03:35,900 And so for this case whereby we only have radiation linearly 40 00:03:35,900 --> 00:03:41,800 polarized in the y direction, so E of y must become zero for 41 00:03:41,800 --> 00:03:45,000 x equals a and for x equals zero. 42 00:03:45,000 --> 00:03:51,199 Since there is no dependence of the E field in the y direction, 43 00:03:51,199 --> 00:03:56,300 k of y will be zero. And you will see that come back 44 00:03:56,300 --> 00:04:03,432 many times in this problem. Now I would like to take a look 45 00:04:03,432 --> 00:04:09,632 at this geometry from above looking down on the xz plane. 46 00:04:09,632 --> 00:04:15,389 Here is the z direction. And this is the x direction. 47 00:04:15,389 --> 00:04:21,921 And the y direction is coming straight out of the blackboard 48 00:04:21,921 --> 00:04:28,785 and pointing in your direction. I am looking down on that plane 49 00:04:28,785 --> 00:04:34,033 from above. Let the k vector be this. 50 00:04:34,033 --> 00:04:39,615 This is a vector. This k vector has a component 51 00:04:39,615 --> 00:04:43,861 in the x direction, which is k of x. 52 00:04:43,861 --> 00:04:50,899 And it has a component in the z direction, which is k of z. 53 00:04:50,899 --> 00:04:57,087 But it does not have a component in the y direction. 54 00:04:57,087 --> 00:05:00,000 K of y is zero. 55 00:05:00,000 --> 00:05:05,000 56 00:05:05,000 --> 00:05:09,140 Think of them as water waves, as far as I am concerned. 57 00:05:09,140 --> 00:05:12,744 This is the direction of propagation in the gap. 58 00:05:12,744 --> 00:05:16,884 The water waves would be perpendicular to the direction 59 00:05:16,884 --> 00:05:21,332 of propagation which are the planes with constant E fields. 60 00:05:21,332 --> 00:05:25,012 I am going to put on in here. This is 90 degrees. 61 00:05:25,012 --> 00:05:29,000 And I am going to put another one in here. 62 00:05:29,000 --> 00:05:33,959 And so this is now propagating with speed c in this direction. 63 00:05:33,959 --> 00:05:38,268 And the separation between these two crests is lambda. 64 00:05:38,268 --> 00:05:40,788 That is my definition of lambda. 65 00:05:40,788 --> 00:05:44,040 Think of this surface, which is a surface 66 00:05:44,040 --> 00:05:49,000 perpendicular to the blackboard, as having a constant phase of 67 00:05:49,000 --> 00:05:50,869 the wave. It means then, 68 00:05:50,869 --> 00:05:54,284 for instance, that the E vector is pointing 69 00:05:54,284 --> 00:05:59,000 in your direction and has reached a maximum. 70 00:05:59,000 --> 00:06:04,111 It would also have reached a maximum here then. 71 00:06:04,111 --> 00:06:10,555 We go through one complete cycle, 2pi radiance from here to 72 00:06:10,555 --> 00:06:13,888 here. That is the definition of 73 00:06:13,888 --> 00:06:18,222 wavelength. Now, in one period that this 74 00:06:18,222 --> 00:06:24,444 wave moves from here to here, in that same amount of time 75 00:06:24,444 --> 00:06:30,000 this wave here moves from here to there. 76 00:06:30,000 --> 00:06:37,404 And so I will give this here a name, as I have done before, 77 00:06:37,404 --> 00:06:43,404 and I call that L of z. And so it is immediately 78 00:06:43,404 --> 00:06:49,021 obvious, by the definition of phase velocity, 79 00:06:49,021 --> 00:06:56,680 that the phase velocity in the z direction must be Lz divided 80 00:06:56,680 --> 00:07:02,876 by lambda times c. Because it goes from here to 81 00:07:02,876 --> 00:07:08,424 here with the velocity z. But here it goes much further 82 00:07:08,424 --> 00:07:13,972 in the same amount of time, so now that must be times c 83 00:07:13,972 --> 00:07:16,849 knot. So it is larger than c. 84 00:07:16,849 --> 00:07:21,575 And this, by the way, and that follows home the 85 00:07:21,575 --> 00:07:26,198 geometry, is also k divided by k of z times c. 86 00:07:26,198 --> 00:07:32,496 And that is larger than c. The phase velocity is larger 87 00:07:32,496 --> 00:07:37,757 than c, and I have discussed that with you at length before. 88 00:07:37,757 --> 00:07:40,968 Now, look at what this wave is doing. 89 00:07:40,968 --> 00:07:46,229 These waves hit the wall and they then reflect off the wall. 90 00:07:46,229 --> 00:07:48,993 I will try to make you see that. 91 00:07:48,993 --> 00:07:53,719 They reflect off the wall, they come back this way and 92 00:07:53,719 --> 00:07:58,000 then they reflect off the wall again. 93 00:07:58,000 --> 00:08:03,666 And so these waves which zigzag slowly work their way through 94 00:08:03,666 --> 00:08:08,861 the gap in the z direction. And you can immediately see, 95 00:08:08,861 --> 00:08:14,055 therefore, that the speed with which they work their way 96 00:08:14,055 --> 00:08:19,155 through the gap in the z direction must be less than c. 97 00:08:19,155 --> 00:08:24,255 Because when they go from this point A to this point D, 98 00:08:24,255 --> 00:08:29,733 in that same amount of time in the z direction they only go 99 00:08:29,733 --> 00:08:35,715 from B to D. And so the group velocity in 100 00:08:35,715 --> 00:08:42,085 the z direction is then BD divided by AD times c. 101 00:08:42,085 --> 00:08:46,729 And that follows from this geometry. 102 00:08:46,729 --> 00:08:54,824 That equals k of z divided by k times c, and that is less than 103 00:08:54,824 --> 00:09:00,000 c. And it better be less than c. 104 00:09:00,000 --> 00:09:04,052 Because you can just see how difficult it is for this wave 105 00:09:04,052 --> 00:09:08,033 that reflects to work its way through in the z direction. 106 00:09:08,033 --> 00:09:11,658 We discussed earlier, and I explained that using the 107 00:09:11,658 --> 00:09:16,208 analogy with water waves hitting the shore, that a phase velocity 108 00:09:16,208 --> 00:09:18,696 larger than c has really no meaning. 109 00:09:18,696 --> 00:09:22,962 It is a geometrical effect of the consequences that the water 110 00:09:22,962 --> 00:09:26,872 slams on the wall everywhere at the same moment in time. 111 00:09:26,872 --> 00:09:31,691 However, the group velocity -- What is that? 112 00:09:31,691 --> 00:09:35,594 What did I do wrong? Excuse me? 113 00:09:35,594 --> 00:09:41,449 Thank you very much. This is larger than zero. 114 00:09:41,449 --> 00:09:47,174 Boy, it is not my day. Is this what you want? 115 00:09:47,174 --> 00:09:49,776 Thank you. All right. 116 00:09:49,776 --> 00:09:56,802 Notice that in this specific case, the product of group 117 00:09:56,802 --> 00:10:05,000 velocity and phase velocity happens to be c squared. 118 00:10:05,000 --> 00:10:08,857 It is a special case. It is not always the case. 119 00:10:08,857 --> 00:10:13,781 But what is important that you recognize, and this picture is 120 00:10:13,781 --> 00:10:18,705 so nice, you can really digest that picture and relate to it, 121 00:10:18,705 --> 00:10:23,136 that you immediately recognize that the phase velocity, 122 00:10:23,136 --> 00:10:27,732 which is the speed of this distance, is larger than c and 123 00:10:27,732 --> 00:10:32,000 that the group velocity is smaller than c. 124 00:10:32,000 --> 00:10:41,007 Now we can do the math, and I can write down the wave 125 00:10:41,007 --> 00:10:49,842 equation dell square E. There is only a component in 126 00:10:49,842 --> 00:11:00,062 the y direction because that is the radiation I have chosen. 127 00:11:00,062 --> 00:11:09,763 I get D2Ey/Dx2 plus D2Ey/Dy2 plus D2Ey/Dz2 equals epsilon 128 00:11:09,763 --> 00:11:20,987 zero mu zero times D2Ey/Dt2. And this term does not exist 129 00:11:20,987 --> 00:11:30,606 because there is no dependence of the E vector in the y 130 00:11:30,606 --> 00:11:37,731 direction. The solution for the wave that 131 00:11:37,731 --> 00:11:46,637 goes from here to here immediately presents itself, 132 00:11:46,637 --> 00:11:53,584 because we have a lot of experience now, 133 00:11:53,584 --> 00:12:03,737 as a standing wave in the x direction and is going to be a 134 00:12:03,737 --> 00:12:09,615 traveling wave in the z direction. 135 00:12:09,615 --> 00:12:13,000 11:44 136 00:12:13,000 --> 00:12:27,000 137 00:12:27,000 --> 00:12:30,223 And so you see here, it is staring you in the face 138 00:12:30,223 --> 00:12:33,250 because I knew, of course, the answer that this 139 00:12:33,250 --> 00:12:37,000 is a traveling wave and that this is a standing wave. 140 00:12:37,000 --> 00:12:41,072 And if I use that solution, which has to be correct, 141 00:12:41,072 --> 00:12:44,427 and I put it in that equation, which I can, 142 00:12:44,427 --> 00:12:48,260 it will meet the three-dimensional wave equation, 143 00:12:48,260 --> 00:12:52,812 you substitute that in here, you will find something that, 144 00:12:52,812 --> 00:12:56,805 of course, you also know already, namely that omega 145 00:12:56,805 --> 00:13:03,587 squared, which is k2 times c2. But k now only has a kx and has 146 00:13:03,587 --> 00:13:09,027 a kz, and so you get this kx2 plus kz2 times c2. 147 00:13:09,027 --> 00:13:15,972 That is what you will find if you substitute that in the wave 148 00:13:15,972 --> 00:13:19,907 equation. And you will find that c2 149 00:13:19,907 --> 00:13:25,000 equals one over epsilon zero mu zero. 150 00:13:25,000 --> 00:13:30,983 Now the boundary conditions, which we haven't met yet, 151 00:13:30,983 --> 00:13:36,177 require that the vector, which is only in the y 152 00:13:36,177 --> 00:13:42,161 direction, and once more to remind you, it is linearly 153 00:13:42,161 --> 00:13:48,709 polarized in the y direction, now must vanish when x equals 154 00:13:48,709 --> 00:13:54,806 zero and when x equals a. And so that now means that kx 155 00:13:54,806 --> 00:14:00,000 must now be n times pi divided by a. 156 00:14:00,000 --> 00:14:07,584 And n then can be one or two or three or four. 157 00:14:07,584 --> 00:14:16,853 And, if we look in a different direction, I make another 158 00:14:16,853 --> 00:14:23,932 cross-section, this is y and this now is x, 159 00:14:23,932 --> 00:14:33,347 so this is zero and this is a. And the z direction is now 160 00:14:33,347 --> 00:14:38,111 coming straight out of the blackboard. 161 00:14:38,111 --> 00:14:44,420 Then for n equals one kx is going to be pi over a, 162 00:14:44,420 --> 00:14:49,957 and that is this wave. That is the sinusoid. 163 00:14:49,957 --> 00:14:57,553 And so at a particular moment in time the E vectors would be 164 00:14:57,553 --> 00:15:02,448 like this. And you see they vanish here 165 00:15:02,448 --> 00:15:06,559 and they vanish here. There can be no tangential 166 00:15:06,559 --> 00:15:11,720 component in this plane here, and there can be no tangential 167 00:15:11,720 --> 00:15:16,705 component in that plane here. And so this is then n equals 168 00:15:16,705 --> 00:15:18,367 one. And, of course, 169 00:15:18,367 --> 00:15:22,390 it oscillates like this. It is a standard wave. 170 00:15:22,390 --> 00:15:26,326 For n equals two there is another possibility, 171 00:15:26,326 --> 00:15:33,066 kx would be 2pi divided by a. And I can put that in a 172 00:15:33,066 --> 00:15:38,399 different color. That would be like this. 173 00:15:38,399 --> 00:15:45,333 And so now the E vector has right here a nodal plane. 174 00:15:45,333 --> 00:15:52,799 And now it goes like this. And so those are possibilities 175 00:15:52,799 --> 00:16:00,000 which are only restrictions in the x direction. 176 00:16:00,000 --> 00:16:05,294 We can write down now the k vector. 177 00:16:05,294 --> 00:16:14,792 The k vector now is kx times x, but kx is n pi divided by a in 178 00:16:14,792 --> 00:16:21,799 the x direction. And then I have a kz in the z 179 00:16:21,799 --> 00:16:26,314 direction. And there is no ky. 180 00:16:26,314 --> 00:16:31,608 Ky is zero. And omega, which is kc, 181 00:16:31,608 --> 00:16:41,418 is therefore c times the square root of n pi over a squared plus 182 00:16:41,418 --> 00:16:47,583 kz squared. And this equation is one that I 183 00:16:47,583 --> 00:16:52,583 had on the blackboard earlier before we understood why it was 184 00:16:52,583 --> 00:16:56,333 that way when I discussed with you dispersion. 185 00:16:56,333 --> 00:17:00,000 This is called the dispersion relation. 186 00:17:00,000 --> 00:17:08,648 This, by the way, under the square root is simply 187 00:17:08,648 --> 00:17:14,594 k. And so now you can ask yourself 188 00:17:14,594 --> 00:17:23,423 what is now the phase velocity in the z direction. 189 00:17:23,423 --> 00:17:31,171 We already know what it is. It must be that. 190 00:17:31,171 --> 00:17:39,279 We already reasoned that purely from geometry, 191 00:17:39,279 --> 00:17:44,504 but it must come out now, too. 192 00:17:44,504 --> 00:17:55,675 Well, the phase velocity in the z direction is omega divided by 193 00:17:55,675 --> 00:18:00,000 kz. 17:29 194 00:18:00,000 --> 00:18:04,774 Now you can also calculate what the group velocity is. 195 00:18:04,774 --> 00:18:09,459 The group velocity in the z direction is d omega dkz. 196 00:18:09,459 --> 00:18:14,774 And, since you have this term here, you should be able to do 197 00:18:14,774 --> 00:18:17,477 that. That is an 18.01 problem. 198 00:18:17,477 --> 00:18:22,072 If you take d omega dkz, you will find now that that 199 00:18:22,072 --> 00:18:27,027 becomes, and I will leave you with that, that becomes kz 200 00:18:27,027 --> 00:18:32,743 divided by k times c. And that is always less than c. 201 00:18:32,743 --> 00:18:35,663 This is what we also knew already. 202 00:18:35,663 --> 00:18:40,619 I had already predicted that based on geometry arguments. 203 00:18:40,619 --> 00:18:45,398 That comes out now if you use this dispersion relation. 204 00:18:45,398 --> 00:18:50,530 And this dispersion relation then has the absurdity that if 205 00:18:50,530 --> 00:18:55,752 kz goes to zero that the phase velocity goes to infinity but 206 00:18:55,752 --> 00:19:00,000 that the group velocity goes to zero. 207 00:19:00,000 --> 00:19:04,730 And so that now I want to address further. 208 00:19:04,730 --> 00:19:11,423 And the best way that we do that is to make what we call an 209 00:19:11,423 --> 00:19:16,961 omega kz diagram. I am going to plot for you this 210 00:19:16,961 --> 00:19:20,538 curve. It is extremely useful to 211 00:19:20,538 --> 00:19:25,500 always think of this as an omega kz diagram. 212 00:19:25,500 --> 00:19:32,619 Here is kz and here is omega. And this straight line, 213 00:19:32,619 --> 00:19:39,338 which would be a non-dispersive medium, would be omega is kz 214 00:19:39,338 --> 00:19:43,551 times c. But that is not what we have. 215 00:19:43,551 --> 00:19:49,587 What we have is for n equals one there is a particular 216 00:19:49,587 --> 00:19:54,256 frequency omega c, we call that the cutoff 217 00:19:54,256 --> 00:20:00,291 frequency, below which no radiation can go through the 218 00:20:00,291 --> 00:20:04,947 gap. And that cutoff frequency is 219 00:20:04,947 --> 00:20:09,368 when n equals one and when kz becomes zero. 220 00:20:09,368 --> 00:20:12,736 And so that is c times pi over a. 221 00:20:12,736 --> 00:20:16,631 And this curve then, for n equals one, 222 00:20:16,631 --> 00:20:20,421 goes like this. This is n equals one. 223 00:20:20,421 --> 00:20:26,631 If I radiate electromagnetic radiation linearly polarized in 224 00:20:26,631 --> 00:20:32,000 this direction and I know the frequency. 225 00:20:32,000 --> 00:20:36,231 Let us say this is the frequency that I radiate, 226 00:20:36,231 --> 00:20:40,372 that is a given, then radiation will go through 227 00:20:40,372 --> 00:20:44,244 beautifully. And this will then be the value 228 00:20:44,244 --> 00:20:46,405 of kz. The value of kx is 229 00:20:46,405 --> 00:20:50,276 nonnegotiable. That value for kx must be the 230 00:20:50,276 --> 00:20:53,607 value that I had earlier. Where is it? 231 00:20:53,607 --> 00:21:00,000 That must be pi divided by a. Let me write that down again. 232 00:21:00,000 --> 00:21:04,910 For n equals one k of x must be pi divided by a. 233 00:21:04,910 --> 00:21:10,343 But it is kz that adjusts itself so that it meets the 234 00:21:10,343 --> 00:21:15,149 boundary conditions. Kx does not adjust itself. 235 00:21:15,149 --> 00:21:19,537 Kx has no choice. Kx must meet the boundary 236 00:21:19,537 --> 00:21:26,014 conditions so that the E vector vanishes at x equals zero and x 237 00:21:26,014 --> 00:21:30,208 equals a. And so any solution must lie on 238 00:21:30,208 --> 00:21:33,058 this curve. And so it is kz that pays the 239 00:21:33,058 --> 00:21:37,048 price and that adjusts itself. Now, keep in mind that all 240 00:21:37,048 --> 00:21:41,323 radiation with frequency above this value here can go through 241 00:21:41,323 --> 00:21:43,104 this gap. There is often a 242 00:21:43,104 --> 00:21:47,165 misunderstanding among students. They think that there are 243 00:21:47,165 --> 00:21:50,300 resonance frequencies. There are no resonance 244 00:21:50,300 --> 00:21:53,292 frequencies. Any frequency above that value 245 00:21:53,292 --> 00:22:00,688 can go through the gap. You can immediately see here 246 00:22:00,688 --> 00:22:10,231 that the phase velocity is larger than c because the phase 247 00:22:10,231 --> 00:22:20,277 velocity is omega divided by kz. And so when I draw this line 248 00:22:20,277 --> 00:22:27,142 then this slope here, omega divided by kz, 249 00:22:27,142 --> 00:22:35,848 is larger than this slope. And that corresponded to a 250 00:22:35,848 --> 00:22:39,699 speed c. And so you see, 251 00:22:39,699 --> 00:22:49,075 since this slope is steeper, the phase velocity is larger 252 00:22:49,075 --> 00:22:55,438 than c. And you see that when you reach 253 00:22:55,438 --> 00:23:05,149 the situation that kz becomes zero that then you have this. 254 00:23:05,149 --> 00:23:09,000 22:30 255 00:23:09,000 --> 00:23:15,000 256 00:23:15,000 --> 00:23:21,555 I now want to write down for you something that may help you 257 00:23:21,555 --> 00:23:26,222 later if you want to understand your notes. 258 00:23:26,222 --> 00:23:32,022 I will put this back up again. I am going to do the following 259 00:23:32,022 --> 00:23:34,314 experience now with you in my head. 260 00:23:34,314 --> 00:23:36,876 I start with a certain frequency omega, 261 00:23:36,876 --> 00:23:39,438 and I am going to lower that frequency. 262 00:23:39,438 --> 00:23:42,471 I do it in my head. It is a Duncan experiment. 263 00:23:42,471 --> 00:23:46,584 I am going to write down step by step what is going to happen. 264 00:23:46,584 --> 00:23:50,629 This point will slowly go down. And so we will come down this 265 00:23:50,629 --> 00:23:53,056 line here. And finally we reach here. 266 00:23:53,056 --> 00:23:56,831 I probably can write it here, it is nicer for you because 267 00:23:56,831 --> 00:24:02,028 then you have it all together. I start with omega, 268 00:24:02,028 --> 00:24:07,259 which is larger than omega c. That is how I start. 269 00:24:07,259 --> 00:24:11,637 It is a given. And now I am going to lower 270 00:24:11,637 --> 00:24:15,587 omega. The next thing I am going to do 271 00:24:15,587 --> 00:24:20,711 is to lower omega. When you lower omega kx cannot 272 00:24:20,711 --> 00:24:24,021 change. It must remain pi over a 273 00:24:24,021 --> 00:24:30,000 because it must meet the boundary conditions. 274 00:24:30,000 --> 00:24:36,338 Therefore, kz gets smaller. You are going down this line 275 00:24:36,338 --> 00:24:40,486 and the kz gets smaller. Omega is kz. 276 00:24:40,486 --> 00:24:46,940 Omega is always k times z. Clearly, if omega comes down k 277 00:24:46,940 --> 00:24:52,241 must come down. And that means kz comes down in 278 00:24:52,241 --> 00:24:57,312 this case because kx is not going to give up. 279 00:24:57,312 --> 00:25:05,826 And so k gets smaller. And if k gets smaller then, 280 00:25:05,826 --> 00:25:15,835 of course, lambda increases because lambda is 2pi over k. 281 00:25:15,835 --> 00:25:23,700 Until disaster strikes omega becomes omega c. 282 00:25:23,700 --> 00:25:33,888 And when you reach that point kz has become zero and kx is 283 00:25:33,888 --> 00:25:41,622 still pi over a. And so now k itself is kx. 284 00:25:41,622 --> 00:25:48,590 And for lambda now is 2pi divided by that value kx. 285 00:25:48,590 --> 00:25:55,000 And that means 2pi divided by kx equals 2a. 286 00:25:55,000 --> 00:26:00,000 287 00:26:00,000 --> 00:26:04,792 It means if I have a given frequency, which we do have here 288 00:26:04,792 --> 00:26:08,179 that I cannot change, I have 10 gigahertz, 289 00:26:08,179 --> 00:26:12,558 and 10 gigahertz has a wavelength of three centimeters 290 00:26:12,558 --> 00:26:15,945 in vacuum. It means the moment that I make 291 00:26:15,945 --> 00:26:19,663 a any smaller than 1.5 centimeters there is no 292 00:26:19,663 --> 00:26:24,455 propagation anymore in that direction, because now I can no 293 00:26:24,455 --> 00:26:29,000 longer meet the boundary conditions for kx. 294 00:26:29,000 --> 00:26:32,351 And I will demonstrate that, as I did before. 295 00:26:32,351 --> 00:26:35,702 This is not the first time you will see that. 296 00:26:35,702 --> 00:26:39,282 Before I do that, I want to discuss with you the 297 00:26:39,282 --> 00:26:43,243 meaning now of n equals two. What does that mean now? 298 00:26:43,243 --> 00:26:46,594 In other words, we had here the n equals one. 299 00:26:46,594 --> 00:26:51,240 And we also have n equals two. And so I am going to make a new 300 00:26:51,240 --> 00:26:54,820 plot because it becomes otherwise too cluttered. 301 00:26:54,820 --> 00:26:58,095 I will leave this on. Make sure you get that 302 00:26:58,095 --> 00:27:02,395 sequence. That is very important, 303 00:27:02,395 --> 00:27:05,729 and I will put it up later again. 304 00:27:05,729 --> 00:27:11,458 I am going to make a new plot. Otherwise, it becomes too 305 00:27:11,458 --> 00:27:14,479 confusing, too many lines. kz. 306 00:27:14,479 --> 00:27:17,500 Omega. Here is your line known 307 00:27:17,500 --> 00:27:21,145 dispersion. Omega is c times k of c. 308 00:27:21,145 --> 00:27:27,291 And here now is your n equals one, omega c equals pi divided 309 00:27:27,291 --> 00:27:33,083 by a times c. And here, exactly twice as 310 00:27:33,083 --> 00:27:39,367 high, is n equals two. Whereby, another omega c is 2pi 311 00:27:39,367 --> 00:27:45,533 times c divided by a. And so I have one line here and 312 00:27:45,533 --> 00:27:51,818 I have another line here. This is now the plot of this 313 00:27:51,818 --> 00:27:58,458 equation for n equals one, and independently for n equals 314 00:27:58,458 --> 00:28:02,803 two. This is n equals one and this 315 00:28:02,803 --> 00:28:07,057 is n equals two. Imagine now that this is our 316 00:28:07,057 --> 00:28:09,184 omega. That is a given. 317 00:28:09,184 --> 00:28:12,858 That is the one that we happen to have. 318 00:28:12,858 --> 00:28:17,401 This frequency can obviously go through the gap. 319 00:28:17,401 --> 00:28:22,042 But now it has a choice out of two possibilities. 320 00:28:22,042 --> 00:28:27,456 It can either go through this way with this value for kz, 321 00:28:27,456 --> 00:28:32,000 which corresponds to n equals two. 322 00:28:32,000 --> 00:28:34,378 That is the mote in green there. 323 00:28:34,378 --> 00:28:38,980 Or, it can go through in this mote with this value for kz for 324 00:28:38,980 --> 00:28:41,435 n equals one. Or, it can do both. 325 00:28:41,435 --> 00:28:45,271 It can do any linear combination between these two, 326 00:28:45,271 --> 00:28:49,336 because each one is a solution to Maxwell's equations. 327 00:28:49,336 --> 00:28:53,863 Each one obeys the boundary condition and each one meets the 328 00:28:53,863 --> 00:28:57,161 wave equation. And so there are now two ways 329 00:28:57,161 --> 00:29:02,074 that it can go through. If it decides to choose this 330 00:29:02,074 --> 00:29:04,888 route then kx is a given, is pi over a. 331 00:29:04,888 --> 00:29:08,592 If it decides to go this way then kx is 2pi over a. 332 00:29:08,592 --> 00:29:12,296 The only difference between these two is that kx is 333 00:29:12,296 --> 00:29:15,111 different. K is not different because I 334 00:29:15,111 --> 00:29:18,148 have the same omega. It is the same omega, 335 00:29:18,148 --> 00:29:20,962 it has the same k, the same wavelength, 336 00:29:20,962 --> 00:29:24,370 but this one has a different k of x, therefore, 337 00:29:24,370 --> 00:29:27,037 a different k of z. And you see that. 338 00:29:27,037 --> 00:29:30,444 And this one has a different k of x, therefore, 339 00:29:30,444 --> 00:29:35,876 a different value for k of z. I cannot be more specific. 340 00:29:35,876 --> 00:29:39,773 Before I want to demonstrate to you, which is a repeat, 341 00:29:39,773 --> 00:29:43,020 that if I have here electromagnetic radiation, 342 00:29:43,020 --> 00:29:46,845 three centimeter waves polarized in this direction and 343 00:29:46,845 --> 00:29:51,175 I am going to send it to this receiver, which can receive it, 344 00:29:51,175 --> 00:29:54,927 before I am going to demonstrate that when I make the 345 00:29:54,927 --> 00:29:59,257 gap smaller than 1.5 centimeters that it will all of a sudden 346 00:29:59,257 --> 00:30:03,461 disappear, the radiation. Before I do that, 347 00:30:03,461 --> 00:30:08,446 I want to ask you a question which is important so can test 348 00:30:08,446 --> 00:30:11,455 whether you understand this concept. 349 00:30:11,455 --> 00:30:15,925 My radiation is linearly polarized in this direction. 350 00:30:15,925 --> 00:30:18,676 Out of that came all this misery. 351 00:30:18,676 --> 00:30:23,404 Is there a way that I can send electromagnetic radiation 352 00:30:23,404 --> 00:30:28,389 through that gap when the gap spacing for my transmitter is 353 00:30:28,389 --> 00:30:34,256 smaller than 1.5 centimeters? I know I cannot do it when it 354 00:30:34,256 --> 00:30:38,846 is polarized in this direction. It will immediate disappear. 355 00:30:38,846 --> 00:30:42,736 You will hear that. Is there a way that I can still 356 00:30:42,736 --> 00:30:47,327 get radiation through when the opening of the gap is smaller 357 00:30:47,327 --> 00:30:51,295 than 1.5 centimeters? I realize that it may take you 358 00:30:51,295 --> 00:30:54,018 a little bit of time to think of it. 359 00:30:54,018 --> 00:30:58,686 And I helped you by saying all this misery is the consequence 360 00:30:58,686 --> 00:31:03,354 of the fact that I chose the direction of linear polarization 361 00:31:03,354 --> 00:31:06,000 in the y direction. 362 00:31:06,000 --> 00:31:09,000 363 00:31:09,000 --> 00:31:10,872 Exactly. I will do that, 364 00:31:10,872 --> 00:31:14,127 believe me. If I rotate my transmitter by 365 00:31:14,127 --> 00:31:18,523 90 degrees and I send in radiation whereby the E vector 366 00:31:18,523 --> 00:31:23,488 is linearly polarized in the x direction there is no misery at 367 00:31:23,488 --> 00:31:28,046 all because all boundary conditions will always be met by 368 00:31:28,046 --> 00:31:32,233 Mother Nature. Because all that Mother Nature 369 00:31:32,233 --> 00:31:36,476 has to do is to make sure that the normal component at the 370 00:31:36,476 --> 00:31:39,528 conductor is sigma S divided epsilon zero. 371 00:31:39,528 --> 00:31:43,697 All it has to do is rearrange the charges on the surface, 372 00:31:43,697 --> 00:31:46,228 but nothing has ever to go to zero. 373 00:31:46,228 --> 00:31:50,173 The moment that I make the radiation polarized in this 374 00:31:50,173 --> 00:31:52,630 direction, there is no dispersion. 375 00:31:52,630 --> 00:31:56,203 The phase velocity is c. The group velocity is c, 376 00:31:56,203 --> 00:32:00,000 and this is the line that holds now. 377 00:32:00,000 --> 00:32:04,650 It is only the component in the y direction that gives you this 378 00:32:04,650 --> 00:32:07,049 misery. That is very interesting, 379 00:32:07,049 --> 00:32:10,049 isn't it? Because what that means is that 380 00:32:10,049 --> 00:32:14,700 if the radiation that you send in had a component both in the y 381 00:32:14,700 --> 00:32:17,474 and in the x direction, I can do that. 382 00:32:17,474 --> 00:32:21,900 I can send radiation through that is linearly polarized like 383 00:32:21,900 --> 00:32:24,525 this. Then something unbelievable is 384 00:32:24,525 --> 00:32:27,375 going to happen. The component in the x 385 00:32:27,375 --> 00:32:33,000 direction will go straight through with the speed of light. 386 00:32:33,000 --> 00:32:40,659 And the component in the y direction has to suffer like 387 00:32:40,659 --> 00:32:44,489 this. And so they completely 388 00:32:44,489 --> 00:32:48,460 decouple. Your x direction is 389 00:32:48,460 --> 00:32:53,283 nondispersive, group velocity is c, 390 00:32:53,283 --> 00:33:00,375 phase velocity is c, and it is the component in the 391 00:33:00,375 --> 00:33:08,460 y direction that decouples. And that can go so slowly that 392 00:33:08,460 --> 00:33:14,276 it can even have a group velocity of zero. 393 00:33:14,276 --> 00:33:21,936 When I show you that I can change the size of that gap, 394 00:33:21,936 --> 00:33:29,595 what that means in terms of this line is the following. 395 00:33:29,595 --> 00:33:36,184 33:05 Because, when I make a smaller, 396 00:33:36,184 --> 00:33:39,986 it is this point that omega c goes up. 397 00:33:39,986 --> 00:33:45,739 And so I make omega c go up until it hits my 10 to the 10 398 00:33:45,739 --> 00:33:48,513 hertz. In our thinking here, 399 00:33:48,513 --> 00:33:53,342 we lowered omega. And I showed you what happens. 400 00:33:53,342 --> 00:33:59,198 Now, to do the experiment, I lower a, and I move this line 401 00:33:59,198 --> 00:34:04,928 up until I hit that point. I think we are ready for that 402 00:34:04,928 --> 00:34:07,055 now. You will be able to hear it. 403 00:34:07,055 --> 00:34:09,316 I would like someone to witness it. 404 00:34:09,316 --> 00:34:13,305 Could you come and witness it? Because I want you to see that 405 00:34:13,305 --> 00:34:16,829 the separation between the gap is now two centimeters. 406 00:34:16,829 --> 00:34:20,686 You stand a little bit on the side here and no in the beam. 407 00:34:20,686 --> 00:34:22,547 Look here. What is your name? 408 00:34:22,547 --> 00:34:25,539 Chrystally, wonderful name. My name is Walter, 409 00:34:25,539 --> 00:34:28,000 by the way. [LAUGHTER] 410 00:34:28,000 --> 00:34:31,212 This is 2.0 centimeters. You see that? 411 00:34:31,212 --> 00:34:35,380 This is a transmitter. It is polarized like this. 412 00:34:35,380 --> 00:34:40,417 Here is the receiver which receives the polarized radiation 413 00:34:40,417 --> 00:34:43,803 like this. We modulate it with 550 hertz 414 00:34:43,803 --> 00:34:48,319 so that you can hear it. You cannot hear 10 to the 10 415 00:34:48,319 --> 00:34:51,098 hertz. That is not the way nature 416 00:34:51,098 --> 00:34:53,789 designed you. Nor can I hear it. 417 00:34:53,789 --> 00:34:59,000 This is 550 hertz. The gap is 2.0 centimeters. 418 00:34:59,000 --> 00:35:03,344 I am going to make the gap smaller with my hand. 419 00:35:03,344 --> 00:35:07,781 And you watch it when it gets to 1.5 centimeters. 420 00:35:07,781 --> 00:35:11,571 It is gone. All of a sudden it disappears. 421 00:35:11,571 --> 00:35:17,117 That is the consequence of the fact that this point hits here 422 00:35:17,117 --> 00:35:20,168 now. But now it is 1.5 centimeters 423 00:35:20,168 --> 00:35:22,941 apart. I can now rotate this 90 424 00:35:22,941 --> 00:35:28,210 degrees and rotate this 90 degrees so that now we have the 425 00:35:28,210 --> 00:35:34,005 E vector in this direction. There are no longer any 426 00:35:34,005 --> 00:35:37,755 restrictions. And the radiation goes through 427 00:35:37,755 --> 00:35:40,633 because now I have this red stuff. 428 00:35:40,633 --> 00:35:43,598 And now I can make the gap smaller. 429 00:35:43,598 --> 00:35:48,831 And what do you think when I make the gap smaller and smaller 430 00:35:48,831 --> 00:35:52,581 and smaller? Well, very slowly you will hear 431 00:35:52,581 --> 00:35:55,546 nothing. Because, if the gap has no 432 00:35:55,546 --> 00:36:02,000 cross-sectional opening anymore, no radiation can go through. 433 00:36:02,000 --> 00:36:05,264 Don't expect miracles. If I now make the gap smaller 434 00:36:05,264 --> 00:36:08,657 and smaller and smaller, nature has no problem sending 435 00:36:08,657 --> 00:36:11,794 radiation through, but nature is stuck through the 436 00:36:11,794 --> 00:36:14,547 cross-sectional opening. That is, of course, 437 00:36:14,547 --> 00:36:18,260 you make this opening narrower and narrower and narrow then 438 00:36:18,260 --> 00:36:21,333 less and less and less radiation will go through. 439 00:36:21,333 --> 00:36:23,766 And so I will now make the gap smaller. 440 00:36:23,766 --> 00:36:26,390 She is my witness. I will put this on top. 441 00:36:26,390 --> 00:36:28,375 It is now about 1.5 centimeters. 442 00:36:28,375 --> 00:36:34,569 I will make it smaller. It is now about 1.0 centimeter. 443 00:36:34,569 --> 00:36:39,060 Still going well. About 0.7 centimeters. 444 00:36:39,060 --> 00:36:44,012 About 0.5 centimeter. About 3.0 millimeters. 445 00:36:44,012 --> 00:36:48,387 And obviously we slowly lose intensity. 446 00:36:48,387 --> 00:36:52,533 Thank you very much for assisting me. 447 00:36:52,533 --> 00:36:54,951 My pleasure, actually. 448 00:36:54,951 --> 00:37:01,284 And so you see how bizarre it is that the behavior is so 449 00:37:01,284 --> 00:37:09,000 fundamentally different for these two different modes. 450 00:37:09,000 --> 00:37:13,508 And then the idea of the decoupling, if you oscillate in 451 00:37:13,508 --> 00:37:18,508 a direction that has an x and a y component that the radiation 452 00:37:18,508 --> 00:37:21,786 decouples. I think this is a nice natural 453 00:37:21,786 --> 00:37:26,131 moment to have our break. And I was just wondering why 454 00:37:26,131 --> 00:37:31,634 the attendance was so high. But then I was reminded of the 455 00:37:31,634 --> 00:37:36,201 fact that it is Tuesday. I am very flattered that you 456 00:37:36,201 --> 00:37:39,274 all came. And we will hand them out. 457 00:37:39,274 --> 00:37:44,016 And then we will not start yet until the whistle blows. 458 00:37:44,016 --> 00:37:48,495 I will raise all the blackboards so that you can see 459 00:37:48,495 --> 00:37:52,096 everything. Can you help handing this out? 460 00:37:52,096 --> 00:37:56,135 Can you also help? Could you also hand this out 461 00:37:56,135 --> 00:38:00,000 and make sure everyone has one? 462 00:38:00,000 --> 00:38:10,000 463 00:38:10,000 --> 00:38:18,358 Now I am going to change the situation very dramatically. 464 00:38:18,358 --> 00:38:26,268 And now I am going to give you a closed box conductor. 465 00:38:26,268 --> 00:38:32,560 There is a conductor. Our coordinate system is the 466 00:38:32,560 --> 00:38:36,128 same, x, y, z. And I simply call this a, 467 00:38:36,128 --> 00:38:38,963 I call this b and I call this c. 468 00:38:38,963 --> 00:38:43,079 That is the length. And that is the coordinate 469 00:38:43,079 --> 00:38:46,189 system. And now comes the question, 470 00:38:46,189 --> 00:38:50,853 if I now have electromagnetic radiation in this box, 471 00:38:50,853 --> 00:38:55,884 what is going to happen? Now there is no way that it can 472 00:38:55,884 --> 00:39:02,387 get in or out if it is in there. Clearly, you now have a 473 00:39:02,387 --> 00:39:08,825 situation that you have a cavity with very discrete resonances. 474 00:39:08,825 --> 00:39:13,704 It is totally different from what we did before. 475 00:39:13,704 --> 00:39:19,415 You only get standing waves with very discrete resonance 476 00:39:19,415 --> 00:39:23,464 frequencies. Well, I will write down for 477 00:39:23,464 --> 00:39:28,240 you, because I want to ram it down your throat, 478 00:39:28,240 --> 00:39:34,158 but I could have left it here, k equals kx times x plus ky 479 00:39:34,158 --> 00:39:42,850 times y plus kz times z. K equals the square root kx2 480 00:39:42,850 --> 00:39:49,952 plus ky2 plus kz2. Omega equals k times c. 481 00:39:49,952 --> 00:40:00,000 Lambda is 2pi divided by k. None of this is negotiable. 482 00:40:00,000 --> 00:40:03,750 Now, in this box, I can have an E vector which 483 00:40:03,750 --> 00:40:08,333 has an x, y and z component. I have an E vector which is 484 00:40:08,333 --> 00:40:12,583 something in the x direction plus something in the y 485 00:40:12,583 --> 00:40:16,166 direction plus something in the z direction. 486 00:40:16,166 --> 00:40:21,083 And our task is now to find the resonance frequency for this 487 00:40:21,083 --> 00:40:24,250 cavity. Let us only look for now at the 488 00:40:24,250 --> 00:40:29,000 component of the E vector in the x direction. 489 00:40:29,000 --> 00:40:34,354 Everything that follows now is only true for the x direction. 490 00:40:34,354 --> 00:40:39,083 E of x, which is the component of the E field indirect 491 00:40:39,083 --> 00:40:43,992 direction, must be zero. E of x is this component in the 492 00:40:43,992 --> 00:40:47,472 x direction. It must be zero at y equals 493 00:40:47,472 --> 00:40:52,291 zero and at y equals b. If it were not zero there would 494 00:40:52,291 --> 00:40:56,574 be a tangential electric field in that top plate. 495 00:40:56,574 --> 00:41:03,000 It must be zero when y equals zero and also when y equals b. 496 00:41:03,000 --> 00:41:12,407 But it must also be zero when z is zero and when z is c. 497 00:41:12,407 --> 00:41:22,157 Because, if the E vector is like this, it must be zero not 498 00:41:22,157 --> 00:41:29,855 only on the bottom plate and on the top plate, 499 00:41:29,855 --> 00:41:40,460 but it must also be zero in the front plate which is this plate 500 00:41:40,460 --> 00:41:51,407 and in the back plate because it cannot have any E vector in that 501 00:41:51,407 --> 00:41:59,105 surface either. The E of x alone must meet now 502 00:41:59,105 --> 00:42:04,578 four special boundary conditions. 503 00:42:04,578 --> 00:42:10,454 41:35 And, to remind you that it is 504 00:42:10,454 --> 00:42:13,501 in this direction, I put the x there. 505 00:42:13,501 --> 00:42:16,887 What you have now are two standing waves, 506 00:42:16,887 --> 00:42:21,711 one standing wave in the y direction and one standing wave 507 00:42:21,711 --> 00:42:25,435 in the z direction. And, to meet the boundary 508 00:42:25,435 --> 00:42:30,514 conditions which I wrote down here, it is immediately obvious 509 00:42:30,514 --> 00:42:35,000 that k of y must now be m pi divided by b. 510 00:42:35,000 --> 00:42:39,347 And k of z must be n pi divided by c. 511 00:42:39,347 --> 00:42:44,420 And then m equals, in this case it is sine, 512 00:42:44,420 --> 00:42:46,956 so it is 1, 2, 3, etc. 513 00:42:46,956 --> 00:42:49,975 And n equals 1, 2, 3, etc. 514 00:42:49,975 --> 00:42:56,980 And, as long as I meet this condition, the E of x component 515 00:42:56,980 --> 00:43:06,949 is quite happy. It means that k now only has a 516 00:43:06,949 --> 00:43:16,447 component in the z and in the y direction. 517 00:43:16,447 --> 00:43:24,092 K of x is zero for this component. 518 00:43:24,092 --> 00:43:35,212 And so we get k of y times y plus k of z times z. 519 00:43:35,212 --> 00:43:49,343 The wave equation dell squared E, if I apply the wave equation 520 00:43:49,343 --> 00:44:00,000 now only to the x component -- 43:32 521 00:44:00,000 --> 00:44:04,575 And this term is not there for that x component. 522 00:44:04,575 --> 00:44:09,637 But the other terms are there. And if I substitute my 523 00:44:09,637 --> 00:44:15,185 solution, which I know deep in my belly has to be correct, 524 00:44:15,185 --> 00:44:20,539 if I substitute that in this wave equation then what you 525 00:44:20,539 --> 00:44:24,530 find, of course, is no surprise that omega 526 00:44:24,530 --> 00:44:30,079 squared equals k2 times c2. And so that is going to be ky2 527 00:44:30,079 --> 00:44:36,059 plus kz2 times c2. But my ky and my kz are 528 00:44:36,059 --> 00:44:44,305 quantized because they are now only allowed for certain values. 529 00:44:44,305 --> 00:44:51,620 And so I can write then now that omega squared is then c 530 00:44:51,620 --> 00:45:00,000 squared time m pi over b squared plus n pi over c squared. 531 00:45:00,000 --> 00:45:05,000 532 00:45:05,000 --> 00:45:10,829 B is for the y direction and c is for the z direction. 533 00:45:10,829 --> 00:45:14,789 What is the problem? Only one person, 534 00:45:14,789 --> 00:45:16,000 please. 535 00:45:16,000 --> 00:45:25,000 536 00:45:25,000 --> 00:45:28,000 The z direction is c, the x direction is a and the y 537 00:45:28,000 --> 00:45:29,000 direction is b. 538 00:45:29,000 --> 00:45:35,000 539 00:45:35,000 --> 00:45:40,365 B is in the y direction. We have a boundary condition in 540 00:45:40,365 --> 00:45:46,121 the y direction and a boundary condition in the z direction. 541 00:45:46,121 --> 00:45:51,682 Why don't you come here and just make the change because I 542 00:45:51,682 --> 00:45:55,000 missed something. [LAUGHTER] 543 00:45:55,000 --> 00:46:00,000 544 00:46:00,000 --> 00:46:03,141 Oh, I leave it c. I cannot change that. 545 00:46:03,141 --> 00:46:07,440 Because, if I change that, then it becomes a terrible 546 00:46:07,440 --> 00:46:09,506 chain reaction. That is c. 547 00:46:09,506 --> 00:46:12,813 Oh, I am very sorry that I called that c. 548 00:46:12,813 --> 00:46:16,450 Thank you very much. Can I get my chalk back, 549 00:46:16,450 --> 00:46:19,757 by the way? [LAUGHTER] We agree there was 550 00:46:19,757 --> 00:46:24,717 no mistake on the blackboard, but there is confusion about c. 551 00:46:24,717 --> 00:46:28,354 My apologies. If I had called it d that it is 552 00:46:28,354 --> 00:46:32,856 another problem. Dx, dy and then you have 553 00:46:32,856 --> 00:46:35,717 another d. Indeed I have followed this 554 00:46:35,717 --> 00:46:37,573 convention. This was a c. 555 00:46:37,573 --> 00:46:40,357 This is a b. You see the same c here? 556 00:46:40,357 --> 00:46:42,754 That is that same c. Z equals c. 557 00:46:42,754 --> 00:46:46,157 It has nothing to do with the speed of light. 558 00:46:46,157 --> 00:46:49,018 It is one of those things that happen. 559 00:46:49,018 --> 00:46:52,112 M stands for Mary and n stands for Nancy. 560 00:46:52,112 --> 00:46:54,973 It doesn't stand for normal this time. 561 00:46:54,973 --> 00:46:57,525 It stands for Nancy. My apologies. 562 00:46:57,525 --> 00:47:01,670 Can we go on? I have your permission? 563 00:47:01,670 --> 00:47:02,251 Yeah. OK. 564 00:47:02,251 --> 00:47:06,392 Thank you for trying because, indeed, it could have been a 565 00:47:06,392 --> 00:47:09,079 mistake. But it wasn't meant that way. 566 00:47:09,079 --> 00:47:13,147 What you see there now is that you have a whole family of 567 00:47:13,147 --> 00:47:17,578 frequencies which meet all the conditions for the x direction. 568 00:47:17,578 --> 00:47:22,009 And that whole family then you can give Mary and Nancy here as 569 00:47:22,009 --> 00:47:24,188 subscript. There is an omega 1, 570 00:47:24,188 --> 00:47:26,803 1 there is an omega 1, 2, an omega 2, 571 00:47:26,803 --> 00:47:30,000 1, an omega 2, 2 and so on. 572 00:47:30,000 --> 00:47:34,120 I tried to see what actually this system was doing. 573 00:47:34,120 --> 00:47:38,241 It was awfully difficult, but I think I can make an 574 00:47:38,241 --> 00:47:41,373 attempt. Suppose we look at this system 575 00:47:41,373 --> 00:47:45,412 in this direction. We are looking at the yz plane. 576 00:47:45,412 --> 00:47:49,285 Here is the z direction which has this length c. 577 00:47:49,285 --> 00:47:53,571 Sorry, I will repeat it once more to avoid confusion. 578 00:47:53,571 --> 00:47:58,021 This is b so this is the y direction, and this is the z 579 00:47:58,021 --> 00:48:02,126 direction. And so x is coming straight out 580 00:48:02,126 --> 00:48:05,212 of the blackboard. We are looking now from the 581 00:48:05,212 --> 00:48:08,230 side like this. That means that the component 582 00:48:08,230 --> 00:48:11,384 of the E vector in the x direction must be zero 583 00:48:11,384 --> 00:48:14,059 everywhere. It must be zero here because 584 00:48:14,059 --> 00:48:16,734 this is a plane. And there cannot be any 585 00:48:16,734 --> 00:48:20,780 tangential component in that plane, but this is also a plane 586 00:48:20,780 --> 00:48:24,278 and there cannot be any tangential component in that 587 00:48:24,278 --> 00:48:26,404 plane. And this is also a plane, 588 00:48:26,404 --> 00:48:31,000 and there cannot be any tangential component here. 589 00:48:31,000 --> 00:48:35,017 Everywhere here must be the E vector and the x direction must 590 00:48:35,017 --> 00:48:37,227 be zero. And so what does it mean, 591 00:48:37,227 --> 00:48:40,174 the 1, 1 mode? It means that the whole thing, 592 00:48:40,174 --> 00:48:43,857 think of it as a membrane, that the whole membrane comes 593 00:48:43,857 --> 00:48:47,607 to you, the E vector is like this, it goes away from you, 594 00:48:47,607 --> 00:48:50,955 E vector is like this, it goes towards you and goes 595 00:48:50,955 --> 00:48:53,165 away from you. That is what the 1, 596 00:48:53,165 --> 00:48:55,575 1 mode is. Think of it as a membrane. 597 00:48:55,575 --> 00:48:59,125 But, keep in mind that, of course, it extends in the x 598 00:48:59,125 --> 00:49:03,548 direction. And it is everywhere the same 599 00:49:03,548 --> 00:49:07,239 in the x direction over the whole length a. 600 00:49:07,239 --> 00:49:11,721 I cannot change that and I cannot make you see that. 601 00:49:11,721 --> 00:49:15,324 For instance, the 2, 1 mode would then be, 602 00:49:15,324 --> 00:49:19,718 if I make m two and n one then, in the y direction, 603 00:49:19,718 --> 00:49:24,990 I would have here a complete nodal surface which runs all the 604 00:49:24,990 --> 00:49:30,000 way through the box, entirely through the box. 605 00:49:30,000 --> 00:49:34,336 No x component of the E field. And this one would then be an E 606 00:49:34,336 --> 00:49:38,530 field in your direction and this would be an E field in this 607 00:49:38,530 --> 00:49:41,018 direction and it would go like this. 608 00:49:41,018 --> 00:49:44,644 And then, if you have the 2, 2 mode, so x is in this 609 00:49:44,644 --> 00:49:47,061 direction, then you would get this. 610 00:49:47,061 --> 00:49:50,047 You would get one whole nodal surface here, 611 00:49:50,047 --> 00:49:53,317 no E component, no x component of the E vector, 612 00:49:53,317 --> 00:49:57,298 no x component of the E vector throughout the entire box, 613 00:49:57,298 --> 00:50:01,946 throughout complete x. And then this would come to 614 00:50:01,946 --> 00:50:06,199 you, this would come to you, this would go away from you and 615 00:50:06,199 --> 00:50:09,444 there would be this standing oscillating wave. 616 00:50:09,444 --> 00:50:12,472 Now, independently, I can now look at the y 617 00:50:12,472 --> 00:50:16,509 component of the E field. Of course, I only looked at the 618 00:50:16,509 --> 00:50:19,248 x component. So let's now look in the y 619 00:50:19,248 --> 00:50:23,502 direction and let's now ask what is the problem with E of y? 620 00:50:23,502 --> 00:50:26,674 Well, E of y also has to become zero when it, 621 00:50:26,674 --> 00:50:31,000 in a plane, becomes the tangential component. 622 00:50:31,000 --> 00:50:34,808 And, therefore, E of y must be zero when x 623 00:50:34,808 --> 00:50:40,568 equals zero and when x equals a and when z equals zero and when 624 00:50:40,568 --> 00:50:44,098 z equals c. If E of y in this direction 625 00:50:44,098 --> 00:50:49,579 must vanish for x equals zero, it must vanish for x equals a 626 00:50:49,579 --> 00:50:55,431 and it must also vanish in this front plate z equals zero and in 627 00:50:55,431 --> 00:50:59,426 the back plate z equals c. Now, by parallel, 628 00:50:59,426 --> 00:51:05,000 you can immediately come up with a relationship. 629 00:51:05,000 --> 00:51:09,269 I will call it now l as in lion for x. 630 00:51:09,269 --> 00:51:14,461 And then, for z, I will keep the n as in Nancy 631 00:51:14,461 --> 00:51:19,884 so that we don't get confused. So that is now c. 632 00:51:19,884 --> 00:51:23,576 I will do it in terms of squares. 633 00:51:23,576 --> 00:51:29,923 And then I will get l pi over a squared plus n pi over c 634 00:51:29,923 --> 00:51:33,234 squared. And that is it. 635 00:51:33,234 --> 00:51:37,927 Now we have another infinite family of values from l as in 636 00:51:37,927 --> 00:51:41,878 lion, n as in Nancy, which gives me the resonance 637 00:51:41,878 --> 00:51:46,242 condition in the y direction. And so if I had linearly 638 00:51:46,242 --> 00:51:50,440 polarized radiation, I could have in the x direction 639 00:51:50,440 --> 00:51:55,133 linearly polarized radiation which would then have to obey 640 00:51:55,133 --> 00:51:58,096 this. I could have linearly polarized 641 00:51:58,096 --> 00:52:02,789 radiation in the y direction which would have to obey this 642 00:52:02,789 --> 00:52:08,257 sequence of frequencies. And I could do one in the z 643 00:52:08,257 --> 00:52:12,104 direction which, of course, you can make up for 644 00:52:12,104 --> 00:52:15,282 yourself. And any linear combination of 645 00:52:15,282 --> 00:52:18,209 those would be fine. But, of course, 646 00:52:18,209 --> 00:52:22,724 it is also possible that I have an E vector which has a 647 00:52:22,724 --> 00:52:26,738 component x, y and z. And that simultaneously all 648 00:52:26,738 --> 00:52:33,212 boundary conditions are met. I don't just radiate linearly 649 00:52:33,212 --> 00:52:39,858 polarized radiation only in the x direction and only in the z 650 00:52:39,858 --> 00:52:46,615 direction, but now I say ah-ha. If now I make omega squared l, 651 00:52:46,615 --> 00:52:53,483 m, n, if that now is c squared times l pi over a squared plus m 652 00:52:53,483 --> 00:53:00,350 pi over b squared plus n pi over c squared then I can have an E 653 00:53:00,350 --> 00:53:06,000 vector, which is in some random direction. 654 00:53:06,000 --> 00:53:10,012 Well, maybe not too random because I have to meet this 655 00:53:10,012 --> 00:53:12,965 condition. And now I have a whole family 656 00:53:12,965 --> 00:53:16,447 of an infinite number of resonance frequencies, 657 00:53:16,447 --> 00:53:20,384 which are not only E vectors in xy or in z direction. 658 00:53:20,384 --> 00:53:23,261 Did I make a slip? Thank you very much. 659 00:53:23,261 --> 00:53:25,987 I have a c squared downstairs, right? 660 00:53:25,987 --> 00:53:30,000 Isn't that one over epsilon zero mu zero? 661 00:53:30,000 --> 00:53:32,411 Thank you. I appreciate it. 662 00:53:32,411 --> 00:53:35,750 Really, I do. Because it is so nasty, 663 00:53:35,750 --> 00:53:38,440 these slips. We are happy now? 664 00:53:38,440 --> 00:53:40,295 Yes, I am. All right. 665 00:53:40,295 --> 00:53:45,860 Now I am going to change gears because I wanted to find a way 666 00:53:45,860 --> 00:53:51,147 to demonstrate this to you. I am going to change gears and 667 00:53:51,147 --> 00:53:56,527 I am going to make this a box. And I am going to ask myself 668 00:53:56,527 --> 00:54:02,000 what are the resonance frequencies for sound? 669 00:54:02,000 --> 00:54:06,791 Sound is a longitudinal wave, so sound has no linear 670 00:54:06,791 --> 00:54:10,644 polarization. And so now I can immediately 671 00:54:10,644 --> 00:54:14,778 write down the function of the over pressure, 672 00:54:14,778 --> 00:54:18,912 p, over and above one atmosphere or below one 673 00:54:18,912 --> 00:54:24,738 atmosphere, because now I know that at all surfaces I must have 674 00:54:24,738 --> 00:54:30,000 pressure antinodes. The particles cannot move. 675 00:54:30,000 --> 00:54:32,354 They are stuck against the wall. 676 00:54:32,354 --> 00:54:35,924 The pressure can build up, pressure can get low, 677 00:54:35,924 --> 00:54:39,797 pressure can build up, but the particles at the wall 678 00:54:39,797 --> 00:54:42,759 themselves cannot move through the wall. 679 00:54:42,759 --> 00:54:46,556 And so now we have pressure antinodes at the walls. 680 00:54:46,556 --> 00:54:50,962 And so it is easy now for me, without any further thinking, 681 00:54:50,962 --> 00:54:55,291 to write down the general solution for the pressure in the 682 00:54:55,291 --> 00:55:00,000 box as a function of the dimensions of that box. 683 00:55:00,000 --> 00:55:04,393 And so it is p, which is either a little bit 684 00:55:04,393 --> 00:55:10,216 above one atmosphere or a little bit below one atmosphere, 685 00:55:10,216 --> 00:55:15,938 it is some amplitude p zero. And now we get the cosine kx 686 00:55:15,938 --> 00:55:21,761 times x times the cosine ky times y times the cosine of kz 687 00:55:21,761 --> 00:55:25,643 times z. And then the whole thing times 688 00:55:25,643 --> 00:55:30,139 cosine omega t, which is now the frequency of 689 00:55:30,139 --> 00:55:36,464 my sound source. And so now, in complete analogy 690 00:55:36,464 --> 00:55:41,047 with what I wrote on the blackboard there, 691 00:55:41,047 --> 00:55:47,976 I must have pressure antinodes at x equals zero and at x equals 692 00:55:47,976 --> 00:55:55,129 a, at y equals zero and y equals b, at k equals zero and k equals 693 00:55:55,129 --> 00:55:58,147 c. And so now I have that my 694 00:55:58,147 --> 00:56:03,735 boundary conditions require that kx is l pi over a, 695 00:56:03,735 --> 00:56:11,000 that ky equals m pi over b and that kz is n pi over c. 696 00:56:11,000 --> 00:56:15,724 Then I am happy. Then at all surfaces I have met 697 00:56:15,724 --> 00:56:21,354 the boundary conditions that the pressures are a maximum. 698 00:56:21,354 --> 00:56:27,084 And so what always holds is that omega squared is always k 699 00:56:27,084 --> 00:56:33,254 squared time v squared. But this v is now the speed of 700 00:56:33,254 --> 00:56:36,942 sound. And the wavelength lambda is 701 00:56:36,942 --> 00:56:41,606 2pi divided by k. I am going to rewrite that 702 00:56:41,606 --> 00:56:47,681 equation on the blackboard in terms of frequency in hertz 703 00:56:47,681 --> 00:56:53,972 because it is going to play a key role in the demonstration 704 00:56:53,972 --> 00:56:58,854 that is coming. It is exactly this same result 705 00:56:58,854 --> 00:57:05,682 that you see here. But I am going to write it now 706 00:57:05,682 --> 00:57:11,451 as f as a function of l, m, n, which is omega l, 707 00:57:11,451 --> 00:57:16,483 m, n divided by 2pi, which is therefore v, 708 00:57:16,483 --> 00:57:21,761 which is the speed of sound, divided by 2pi. 709 00:57:21,761 --> 00:57:29,371 And then I get here the square root of l pi over a squared plus 710 00:57:29,371 --> 00:57:36,000 m pi over b squared plus n pi over c squared. 711 00:57:36,000 --> 00:57:40,987 Notice that you lose your pi. This pi here eats up this pi, 712 00:57:40,987 --> 00:57:43,738 eats up this pi, eats up that pi, 713 00:57:43,738 --> 00:57:48,554 and this is now in hertz. And so now you have an equation 714 00:57:48,554 --> 00:57:53,799 that if you make a box and you know the dimensions of that box 715 00:57:53,799 --> 00:57:58,184 that you know a is the dimension in the x direction, 716 00:57:58,184 --> 00:58:03,000 b in the y direction and c in the z direction. 717 00:58:03,000 --> 00:58:08,954 You can now predict at what sound frequencies this box will 718 00:58:08,954 --> 00:58:11,315 resonate. Markos Hankin, 719 00:58:11,315 --> 00:58:16,448 who is sitting there, and I have spent the last two 720 00:58:16,448 --> 00:58:20,863 weeks of our lives, he more time than I did, 721 00:58:20,863 --> 00:58:24,969 to build such a box. And the box is here. 722 00:58:24,969 --> 00:58:30,000 It is a marvel. He even gave it colors. 723 00:58:30,000 --> 00:58:34,471 He loved it so much, he could hardly sleep at night 724 00:58:34,471 --> 00:58:37,333 he told me. And there is the box. 725 00:58:37,333 --> 00:58:40,731 And here is a loud speaker on the side. 726 00:58:40,731 --> 00:58:45,024 On this side he has a whole in there of one inch. 727 00:58:45,024 --> 00:58:49,943 And so the sound comes in there, and inside the box is a 728 00:58:49,943 --> 00:58:53,520 microphone. And this microphone will tell 729 00:58:53,520 --> 00:58:58,796 us whether we hit resonance, because then we will see a very 730 00:58:58,796 --> 00:59:03,000 high signal from this microphone. 731 00:59:03,000 --> 00:59:06,596 You have no idea how much fun we have had this with. 732 00:59:06,596 --> 00:59:10,544 And I will show you here. You have plenty of time to copy 733 00:59:10,544 --> 00:59:14,211 it because this is going to take at least 15 minutes, 734 00:59:14,211 --> 00:59:17,666 this demonstration. You see here the dimensions of 735 00:59:17,666 --> 00:59:20,346 the box. They were meant to be 30 by 40 736 00:59:20,346 --> 00:59:23,237 by 50 centimeters, but the accuracy of our 737 00:59:23,237 --> 00:59:26,692 measurements became so stunningly high that Markos 738 00:59:26,692 --> 00:59:30,993 said, well, maybe it is not 40, maybe it is 39.9 because there 739 00:59:30,993 --> 00:59:36,000 was one particular frequency that was a little bit off. 740 00:59:36,000 --> 00:59:38,045 And he was right. We measured it, 741 00:59:38,045 --> 00:59:41,113 and it was a little bit less than 40 centimeters. 742 00:59:41,113 --> 00:59:44,501 You see the sizes there and you see the uncertainties. 743 00:59:44,501 --> 00:59:48,527 The accuracy of our measurement was so enormous that we began to 744 00:59:48,527 --> 00:59:51,020 worry about what the speed of sound was. 745 00:59:51,020 --> 00:59:54,472 And so that means we have to know the room temperature. 746 00:59:54,472 --> 00:59:58,115 Markos and I agreed that we would just accept for now that 747 00:59:58,115 --> 1:00:01,311 it is probably very close to 344 meters per second, 748 1:00:01,311 --> 1:00:05,146 which is the speed of sound at room temperature of 20 degrees 749 1:00:05,146 --> 1:00:08,418 centigrade. But, keep in mind, 750 1:00:08,418 --> 1:00:12,672 it is linearly proportional with the square root of the 751 1:00:12,672 --> 1:00:16,690 temperature in degrees Kelvin. That means one degree 752 1:00:16,690 --> 1:00:20,157 centigrade difference will make a difference. 753 1:00:20,157 --> 1:00:24,412 Two degrees centigrade will be a noticeable difference. 754 1:00:24,412 --> 1:00:28,115 We would be surprised, though, if it is any more 755 1:00:28,115 --> 1:00:31,739 different than 1%, but it could be different by 756 1:00:31,739 --> 1:00:35,513 1%. Here you see the motes that 757 1:00:35,513 --> 1:00:39,982 Markos computerized and predicted with increasing 758 1:00:39,982 --> 1:00:43,706 frequency. Notice that when n equals one, 759 1:00:43,706 --> 1:00:47,896 that is when c, which is the largest dimension 760 1:00:47,896 --> 1:00:52,458 in the z direction, when that is the largest value 761 1:00:52,458 --> 1:00:55,717 that gives you the lowest frequency. 762 1:00:55,717 --> 1:01:00,000 That should be immediately obvious. 763 1:01:00,000 --> 1:01:03,703 Because if you make l zero and m zero and n is one, 764 1:01:03,703 --> 1:01:07,555 if this one is 50 centimeters, which is half a meter, 765 1:01:07,555 --> 1:01:10,444 you will get the largest value possible. 766 1:01:10,444 --> 1:01:13,333 Oh, we are here now. If c is 0.5 meters, 767 1:01:13,333 --> 1:01:17,185 and you multiply that by two, then you get one meter. 768 1:01:17,185 --> 1:01:21,407 This is zero and this is zero. And so it is 344 divided by 769 1:01:21,407 --> 1:01:24,962 one, which is 344. And so that is why this one is 770 1:01:24,962 --> 1:01:30,000 344, because one dimension happens to be half a meter. 771 1:01:30,000 --> 1:01:34,411 And then from that moment on it becomes way more difficult to 772 1:01:34,411 --> 1:01:36,764 see what is happening in the box. 773 1:01:36,764 --> 1:01:39,632 The next frequency that we predict is 0, 774 1:01:39,632 --> 1:01:42,132 1, 0. And we went all the way up to 775 1:01:42,132 --> 1:01:44,632 812. In fact, Markos went even past 776 1:01:44,632 --> 1:01:48,676 4,000, but they become so enormously close together that 777 1:01:48,676 --> 1:01:51,911 it is no longer fun. Here they are quite well 778 1:01:51,911 --> 1:01:54,705 separated. We are going to make you see 779 1:01:54,705 --> 1:02:00,000 the result and we are going to make you hear the result. 780 1:02:00,000 --> 1:02:05,820 Hearing is easy because we turn on the sound as we receive it 781 1:02:05,820 --> 1:02:09,119 from the microphone inside the box. 782 1:02:09,119 --> 1:02:13,000 And I am going to turn that one now. 783 1:02:13,000 --> 1:02:21,000 784 1:02:21,000 --> 1:02:25,588 Now you, in principle, should be able to hear the 785 1:02:25,588 --> 1:02:29,602 frequency of the microphone inside the box. 786 1:02:29,602 --> 1:02:34,000 We are also going to show you the results. 787 1:02:34,000 --> 1:02:48,000 788 1:02:48,000 --> 1:02:49,721 One at the bottom is the speaker. 789 1:02:49,721 --> 1:02:52,089 That is the driving frequency of the speaker. 790 1:02:52,089 --> 1:02:54,026 The frequency, as I have said it now, 791 1:02:54,026 --> 1:02:56,771 you can also see here. It is very important that you 792 1:02:56,771 --> 1:02:58,439 see that. That frequency is now, 793 1:02:58,439 --> 1:03:02,252 I think, 325 hertz. You will see it come up. 794 1:03:02,252 --> 1:03:06,757 Way off resonance. The one on top is the one that 795 1:03:06,757 --> 1:03:10,417 the microphone records. And, if I speak, 796 1:03:10,417 --> 1:03:15,673 you see nonsense at the top because the microphone inside 797 1:03:15,673 --> 1:03:19,333 can hear me. It is important that we are 798 1:03:19,333 --> 1:03:23,744 relatively quiet. And I am going to increase the 799 1:03:23,744 --> 1:03:29,000 frequency and search for the first resonance. 800 1:03:29,000 --> 1:03:32,893 And you can see how accurate it becomes because I will go over 801 1:03:32,893 --> 1:03:36,021 and under the resonance, and you can see there the 802 1:03:36,021 --> 1:03:38,000 response of the microphone. 803 1:03:38,000 --> 1:03:45,000 804 1:03:45,000 --> 1:03:48,695 That is it. Did you notice that I went over 805 1:03:48,695 --> 1:03:53,184 and I went under it? Look at what incredible signal. 806 1:03:53,184 --> 1:03:56,000 I shouldn't be talking. 344. 807 1:03:56,000 --> 1:04:04,000 808 1:04:04,000 --> 1:04:06,000 I am going to find the next one. 809 1:04:06,000 --> 1:04:31,000 810 1:04:31,000 --> 1:04:38,853 434. Less than 0.5% off. 811 1:04:38,853 --> 1:04:45,000 Next one is 552. 812 1:04:45,000 --> 1:05:01,000 813 1:05:01,000 --> 1:05:08,000 548. Less than 1% off. Next one. 814 1:05:08,000 --> 1:05:21,000 815 1:05:21,000 --> 1:05:24,473 575. Amazing what Mother Nature is 816 1:05:24,473 --> 1:05:30,052 doing all the time making sure that there are pressure 817 1:05:30,052 --> 1:05:35,842 antinodes of all the volts. And only then do you see the 818 1:05:35,842 --> 1:05:39,000 resonance. The next one. 819 1:05:39,000 --> 1:05:50,000 820 1:05:50,000 --> 1:05:52,266 670. The next one. 821 1:05:52,266 --> 1:05:57,733 You may get bored, but Markos and I don't. 822 1:05:57,733 --> 1:06:00,000 We love this. 823 1:06:00,000 --> 1:06:09,000 824 1:06:09,000 --> 1:06:11,867 691. Unbelievable. 825 1:06:11,867 --> 1:06:20,975 If you say physics works, that is the understatement of 826 1:06:20,975 --> 1:06:23,000 the day. 827 1:06:23,000 --> 1:06:36,000 828 1:06:36,000 --> 1:06:43,000 721. Next one. 829 1:06:43,000 --> 1:06:57,000 830 1:06:57,000 --> 1:07:00,692 792. Only about 0.5% off. 831 1:07:00,692 --> 1:07:07,000 And now I will go only to the next one. 832 1:07:07,000 --> 1:07:13,000 833 1:07:13,000 --> 1:07:15,000 812. 834 1:07:15,000 --> 1:07:22,000 835 1:07:22,000 --> 1:07:29,484 What Markos also did, he scanned very slowly from 300 836 1:07:29,484 --> 1:07:38,696 hertz to 850 extremely slowly so you don't see the wave structure 837 1:07:38,696 --> 1:07:47,045 anymore, and he recorded only the amplitude recorded by the 838 1:07:47,045 --> 1:07:52,659 microphone. That means the largest value 839 1:07:52,659 --> 1:08:00,000 then of the sinusoid there, the peak value. 840 1:08:00,000 --> 1:08:04,024 And he made a plot of them. And you see that plot here, 841 1:08:04,024 --> 1:08:07,675 which will blow your mind. Horizontal frequencies. 842 1:08:07,675 --> 1:08:10,730 This is the experiment you have just seen. 843 1:08:10,730 --> 1:08:13,338 It was presented in a different way. 844 1:08:13,338 --> 1:08:16,468 Here you see, as we scan with the frequency 845 1:08:16,468 --> 1:08:20,269 very slowly over it, which we did but now you see it 846 1:08:20,269 --> 1:08:23,995 all in one picture, when you hit the 344 hertz that 847 1:08:23,995 --> 1:08:27,423 is the amplitude that is recorded inside by the 848 1:08:27,423 --> 1:08:32,339 microphone due to the resonance. And you see the 434. 849 1:08:32,339 --> 1:08:35,688 These are the values that we measured this morning. 850 1:08:35,688 --> 1:08:39,438 547, OK, it was one hertz off. 574, one hertz difference. 851 1:08:39,438 --> 1:08:41,247 696. You see them all there. 852 1:08:41,247 --> 1:08:43,993 And you also get a feeling for the height. 853 1:08:43,993 --> 1:08:47,408 And we tried to understand the difference in height, 854 1:08:47,408 --> 1:08:49,618 but we are not acoustic engineers. 855 1:08:49,618 --> 1:08:53,503 We contacted some people at MIT who are acoustic engineers, 856 1:08:53,503 --> 1:08:55,847 and they did not have a clue either. 857 1:08:55,847 --> 1:09:00,000 We are not alone. It is very, very difficult. 858 1:09:00,000 --> 1:09:04,209 Because now you have to understand where your microphone 859 1:09:04,209 --> 1:09:07,959 is located in the box. This is already so amazing. 860 1:09:07,959 --> 1:09:12,551 I mean absolutely so amazing. Then one afternoon when we knew 861 1:09:12,551 --> 1:09:16,913 this was going to work and when both of us were very proud 862 1:09:16,913 --> 1:09:19,438 Markos called me. He said, Walter, 863 1:09:19,438 --> 1:09:22,806 you have to come down. I am going to show you 864 1:09:22,806 --> 1:09:25,255 something that you won't believe. 865 1:09:25,255 --> 1:09:30,000 And I said, well, you have got to tell me first. 866 1:09:30,000 --> 1:09:35,743 He said we can show the student transient behavior in a way that 867 1:09:35,743 --> 1:09:39,116 no one has ever demonstrated in class. 868 1:09:39,116 --> 1:09:43,584 What did Markos do? He said when we turn on all of 869 1:09:43,584 --> 1:09:49,054 a sudden the sound then it is like starting to drive all of a 870 1:09:49,054 --> 1:09:54,433 sudden a coupled oscillator which has resonance frequencies. 871 1:09:54,433 --> 1:09:59,173 What happens when all of a sudden you start driving a 872 1:09:59,173 --> 1:10:04,476 coupled oscillator? Well, then you get steady-state 873 1:10:04,476 --> 1:10:07,192 solution plus a transient solution. 874 1:10:07,192 --> 1:10:11,347 The system is going to oscillate in its normal modes. 875 1:10:11,347 --> 1:10:15,741 And then, ultimately the transients will die out because 876 1:10:15,741 --> 1:10:19,815 the driver, of course, is the only one that survives 877 1:10:19,815 --> 1:10:24,528 because that is the frequency that you impose on the system. 878 1:10:24,528 --> 1:10:28,523 And so what Markos then did, as a function of time, 879 1:10:28,523 --> 1:10:34,121 chose a frequency. We will show shortly what he 880 1:10:34,121 --> 1:10:39,472 has chosen, but it is somewhere around 580 hertz. 881 1:10:39,472 --> 1:10:46,273 That is the driving frequency. And here he shows the amplitude 882 1:10:46,273 --> 1:10:51,067 of the microphone during a 1.5 second pulse, 883 1:10:51,067 --> 1:10:57,756 and he shows only the amplitude when it is at its highest and 884 1:10:57,756 --> 1:11:04,000 when it is at its lowest value of the sinusoid. 885 1:11:04,000 --> 1:11:08,311 Imagine that at this moment in time this is the zero value, 886 1:11:08,311 --> 1:11:12,326 this is the highest value and this is the lowest value. 887 1:11:12,326 --> 1:11:15,968 This is a huge sinusoid, highest and lowest value. 888 1:11:15,968 --> 1:11:19,685 But a little later in time because of the transient 889 1:11:19,685 --> 1:11:23,774 phenomenon, which is going to interfere with the driver, 890 1:11:23,774 --> 1:11:26,376 often you get even beat frequencies. 891 1:11:26,376 --> 1:11:30,762 A little later it may be that amplitude of the microphone is 892 1:11:30,762 --> 1:11:33,513 lower here and, therefore, lower here, 893 1:11:33,513 --> 1:11:36,545 too. And then ultimately, 894 1:11:36,545 --> 1:11:40,178 given enough time, this one and this one will no 895 1:11:40,178 --> 1:11:44,429 longer change because this now the stead-state solution. 896 1:11:44,429 --> 1:11:48,603 Now we have the amplitude of the steady-state solution. 897 1:11:48,603 --> 1:11:52,932 But in the beginning you will see the interference of the 898 1:11:52,932 --> 1:11:56,024 transient with the steady-state solution. 899 1:11:56,024 --> 1:12:00,661 And so we were wondering what the transient frequencies would 900 1:12:00,661 --> 1:12:05,376 be that would start up all of a sudden when we start with this 901 1:12:05,376 --> 1:12:08,476 pulse. And we had no clue. 902 1:12:08,476 --> 1:12:12,207 We had no prediction. We had acoustic experts who 903 1:12:12,207 --> 1:12:15,393 said, well, just try and see what happens. 904 1:12:15,393 --> 1:12:18,580 And we are going to show you what happens. 905 1:12:18,580 --> 1:12:23,398 And we have now all reasons to believe that the moment you turn 906 1:12:23,398 --> 1:12:27,906 on the 580 hertz that this one, which is only 7 hertz away, 907 1:12:27,906 --> 1:12:31,637 apparently becomes very potent, not so clear why, 908 1:12:31,637 --> 1:12:37,000 and that one is going to set up a transient oscillation. 909 1:12:37,000 --> 1:12:44,850 But that transient oscillation, which ultimately has to die 910 1:12:44,850 --> 1:12:52,566 out, is going to mix now with our 780, and it will cause a 911 1:12:52,566 --> 1:12:58,116 beat phenomenon. The two beat against each 912 1:12:58,116 --> 1:13:03,395 other, and so you will see this do this. 913 1:13:03,395 --> 1:13:10,298 Of course, it goes in time. You will see it do this, 914 1:13:10,298 --> 1:13:17,607 this and then it will die out. You will be able to see, 915 1:13:17,607 --> 1:13:23,292 for the first time, a transient phenomenon. 916 1:13:23,292 --> 1:13:31,000 73:03 [APPLAUSE] 917 1:13:31,000 --> 1:13:35,448 Markos, I don't think we need that any more. 918 1:13:35,448 --> 1:13:40,000 Should I turn that off? You do your thing. 919 1:13:40,000 --> 1:13:52,000 920 1:13:52,000 --> 1:13:56,000 Here is the frequency that he is going to use as the driver. 921 1:13:56,000 --> 1:14:06,000 922 1:14:06,000 --> 1:14:09,888 You are going to do roughly near 580 or 581 or whatever. 923 1:14:09,888 --> 1:14:13,353 It is very important, again, that none of you talk 924 1:14:13,353 --> 1:14:16,040 and that I don't talk, because talking, 925 1:14:16,040 --> 1:14:20,000 all the nonsense you see now is due to my talking. 926 1:14:20,000 --> 1:14:35,000 927 1:14:35,000 --> 1:14:41,103 Can you just freeze it? Now, from here to here is 1.5 928 1:14:41,103 --> 1:14:44,741 seconds. He has 581 as a driver. 929 1:14:44,741 --> 1:14:49,319 And here you see a transient phenomenon. 930 1:14:49,319 --> 1:14:53,192 From here to there we measured it. 931 1:14:53,192 --> 1:15:00,000 It is exactly what you would expect from the beat. 932 1:15:00,000 --> 1:15:03,583 In other words, if he has 581 and we measure 933 1:15:03,583 --> 1:15:08,000 here 575, which is this resonance, then the difference 934 1:15:08,000 --> 1:15:11,583 is six hertz. And we measured that from here 935 1:15:11,583 --> 1:15:14,833 to here is indeed one-sixth of a second. 936 1:15:14,833 --> 1:15:17,750 One-sixth of second means six hertz. 937 1:15:17,750 --> 1:15:22,583 The oscillation that you see right at the beginning when we 938 1:15:22,583 --> 1:15:26,500 start the driver, that oscillation is six hertz. 939 1:15:26,500 --> 1:15:31,666 That means it must be the beat frequency between the 581 driver 940 1:15:31,666 --> 1:15:36,500 and a normal mote frequency at which the system is going to 941 1:15:36,500 --> 1:15:42,827 fight back, so to speak. That must be the 575 hertz. 942 1:15:42,827 --> 1:15:49,109 And so the 575 hertz and the 581 give you a beat frequency of 943 1:15:49,109 --> 1:15:54,240 six hertz, but the 575, of course, cannot survive. 944 1:15:54,240 --> 1:15:58,324 It dies out. And the 581 is the one that 945 1:15:58,324 --> 1:16:01,866 survives. Now there is something else, 946 1:16:01,866 --> 1:16:05,133 which is an extra bonus. And that is something you 947 1:16:05,133 --> 1:16:08,333 haven't missed here. At the end, we turn all of a 948 1:16:08,333 --> 1:16:11,866 sudden the driver off. The moment that the driver goes 949 1:16:11,866 --> 1:16:15,533 off, the system can only oscillate in a superposition of 950 1:16:15,533 --> 1:16:18,533 its normal modes. Now, which normal modes I do 951 1:16:18,533 --> 1:16:21,000 not know. And the experts may not even 952 1:16:21,000 --> 1:16:23,266 know. But we can look at the moment 953 1:16:23,266 --> 1:16:26,266 that we turn it off. And what do we see there? 954 1:16:26,266 --> 1:16:30,000 We see a signal that is about 27 hertz. 955 1:16:30,000 --> 1:16:34,073 We have measured that it is very close to 27 hertz. 956 1:16:34,073 --> 1:16:37,412 I, therefore, conclude that it must be the 957 1:16:37,412 --> 1:16:42,219 beat frequency between the 575 normal mode and the 548 hertz 958 1:16:42,219 --> 1:16:45,477 normal mode. That difference is 27 hertz. 959 1:16:45,477 --> 1:16:49,713 And you see that beat phenomenon obviously dying out. 960 1:16:49,713 --> 1:16:53,705 It is a transient. Tell your parents that you have 961 1:16:53,705 --> 1:16:57,044 seen this. Tell your friends that you have 962 1:16:57,044 --> 1:17:00,730 seen this. Have a good vacation. 963 1:17:00,730 --> 1:17:04,107 I am leaving this afternoon for Amsterdam. 964 1:17:04,107 I will be back Sunday morning.