1 00:00:02,520 --> 00:00:04,860 The following content is provided under a Creative 2 00:00:04,860 --> 00:00:06,280 Commons license. 3 00:00:06,280 --> 00:00:08,490 Your support will help MIT OpenCourseWare 4 00:00:08,490 --> 00:00:12,580 continue to offer high quality educational resources for free. 5 00:00:12,580 --> 00:00:15,120 To make a donation or to view additional materials 6 00:00:15,120 --> 00:00:19,080 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,080 --> 00:00:20,114 at ocw.mit.edu. 8 00:00:23,560 --> 00:00:25,880 YEN-JIE LEE: So let's get started. 9 00:00:25,880 --> 00:00:29,180 This is our goal for 8.03. 10 00:00:29,180 --> 00:00:32,299 So you can see, during the exam number one, 11 00:00:32,299 --> 00:00:35,120 we have covered the first half of the goal, 12 00:00:35,120 --> 00:00:37,640 and we are actually making progress 13 00:00:37,640 --> 00:00:40,520 to learn about boundary conditions 14 00:00:40,520 --> 00:00:42,860 in one dimensional system and also 15 00:00:42,860 --> 00:00:44,930 in two dimensional system today. 16 00:00:44,930 --> 00:00:48,200 And we will actually talk about phenomena related 17 00:00:48,200 --> 00:00:51,350 to electromagnetic waves and optics 18 00:00:51,350 --> 00:00:54,800 today, which we will be able to learn 19 00:00:54,800 --> 00:00:58,090 two very important fundamental laws related 20 00:00:58,090 --> 00:01:05,370 to geometrical optics. 21 00:01:05,370 --> 00:01:08,620 OK, so that's the excitement. 22 00:01:08,620 --> 00:01:12,830 And then we started a discussion of two dimensional or three 23 00:01:12,830 --> 00:01:15,500 dimensional wave last time. 24 00:01:15,500 --> 00:01:19,760 And just in case you haven't realized that, 25 00:01:19,760 --> 00:01:23,780 there are two ways to go to higher dimension. 26 00:01:23,780 --> 00:01:28,900 So the first way is to increase the number of objects 27 00:01:28,900 --> 00:01:31,650 and place that in two dimensional or three 28 00:01:31,650 --> 00:01:33,230 dimensional space. 29 00:01:33,230 --> 00:01:38,190 And that is the kind of things which you will discuss today. 30 00:01:38,190 --> 00:01:40,340 So for example, I can have particles 31 00:01:40,340 --> 00:01:44,770 arranged in two dimensions which form membranes. 32 00:01:44,770 --> 00:01:47,750 And then we can also, on the other hand, 33 00:01:47,750 --> 00:01:51,830 change the direction of the electromagnetic wave, 34 00:01:51,830 --> 00:01:54,360 for example, as a function of time, 35 00:01:54,360 --> 00:01:59,270 and that's another way to go to a higher dimension. 36 00:01:59,270 --> 00:02:01,580 And today, as I mentioned before, we 37 00:02:01,580 --> 00:02:05,420 are going to talk about the first case, and on Thursday 38 00:02:05,420 --> 00:02:07,730 we are going to talk about the second way 39 00:02:07,730 --> 00:02:11,280 to go to higher dimension, which is related to polarization, 40 00:02:11,280 --> 00:02:11,820 et cetera. 41 00:02:14,510 --> 00:02:22,390 In general, higher order dimensions are hopeless. 42 00:02:22,390 --> 00:02:24,930 They are super complicated. 43 00:02:24,930 --> 00:02:27,490 And, in general, we don't really know how 44 00:02:27,490 --> 00:02:30,680 to solve this kind of system. 45 00:02:30,680 --> 00:02:34,810 Fortunately, in 8.03, what we have been doing 46 00:02:34,810 --> 00:02:41,020 is focusing on a small subset of questions of which are actually 47 00:02:41,020 --> 00:02:43,240 highly symmetric. 48 00:02:43,240 --> 00:02:45,860 Therefore, we can actually solve it analytically. 49 00:02:45,860 --> 00:02:49,630 So that will be the focus of 8.03, 50 00:02:49,630 --> 00:02:52,360 so that we can actually learn some physics intuition out 51 00:02:52,360 --> 00:02:56,310 of this kind highly idealized system. 52 00:02:56,310 --> 00:03:00,750 And the system which we are going to focus on today 53 00:03:00,750 --> 00:03:02,200 is shown here. 54 00:03:02,200 --> 00:03:04,200 It's a two dimensional system, which 55 00:03:04,200 --> 00:03:12,190 you have array of masses placing the x and y, x, y plan. 56 00:03:12,190 --> 00:03:17,860 And that is the system we are going to solve today. 57 00:03:17,860 --> 00:03:21,720 And we will learn a lot of interesting phenomena coming 58 00:03:21,720 --> 00:03:24,870 from the solution of this kind of system. 59 00:03:24,870 --> 00:03:28,020 Before we start a discussion of two dimensional system, 60 00:03:28,020 --> 00:03:30,780 I would like to remind you of what we have already 61 00:03:30,780 --> 00:03:32,970 learned from lecture eight. 62 00:03:32,970 --> 00:03:37,950 So that was about a system which consists of infinite number 63 00:03:37,950 --> 00:03:42,330 of mass and the infinite number of strings, and each string 64 00:03:42,330 --> 00:03:46,020 have string tension T. And all the mass, when 65 00:03:46,020 --> 00:03:49,920 they are in equilibrium position, 66 00:03:49,920 --> 00:03:53,690 the distance between all those mass in the x direction 67 00:03:53,690 --> 00:03:55,320 is a, OK? 68 00:03:55,320 --> 00:04:00,600 So we have solved this system before with space translation 69 00:04:00,600 --> 00:04:02,210 symmetry. 70 00:04:02,210 --> 00:04:06,900 And this is just a reminder that the dispersion relation, which 71 00:04:06,900 --> 00:04:11,160 we got a lot time, omega, as a function of k, 72 00:04:11,160 --> 00:04:15,510 is t over ma sine Ka over 2. 73 00:04:15,510 --> 00:04:18,000 So that was just a reminder of what we have 74 00:04:18,000 --> 00:04:20,040 learned from lecture eight. 75 00:04:20,040 --> 00:04:24,750 So by now you should realize that, OK, 76 00:04:24,750 --> 00:04:28,860 dispersion relation is unusual. 77 00:04:28,860 --> 00:04:31,170 This is actually telling you that this 78 00:04:31,170 --> 00:04:34,320 is a dispersive media, right? 79 00:04:34,320 --> 00:04:37,710 Because if you calculate the ratio of omega and k, 80 00:04:37,710 --> 00:04:40,260 you'll see that this is actually not a constant. 81 00:04:40,260 --> 00:04:43,150 So after all the discussion from previous lecture, 82 00:04:43,150 --> 00:04:46,800 you should be able to immediately realize that. 83 00:04:46,800 --> 00:04:52,190 And any wave propagating on this kind of system, 84 00:04:52,190 --> 00:04:56,220 there will be a dispersion phenomena happening 85 00:04:56,220 --> 00:04:58,350 in this kind of system. 86 00:04:58,350 --> 00:04:59,510 OK? 87 00:04:59,510 --> 00:05:03,150 And also from the previous lecture, 88 00:05:03,150 --> 00:05:07,220 we'll have learned that the eigenvectors based 89 00:05:07,220 --> 00:05:09,620 on space translation symmetry, it's 90 00:05:09,620 --> 00:05:17,150 exponential of ikx, where x is defined as j times a, where 91 00:05:17,150 --> 00:05:21,050 it's a is a label to tell you what which mass 92 00:05:21,050 --> 00:05:23,870 I was talking about. 93 00:05:23,870 --> 00:05:28,190 Now today, we are going to extend this 94 00:05:28,190 --> 00:05:31,160 to a two dimensional system. 95 00:05:31,160 --> 00:05:34,460 So instead of a one dimensional system 96 00:05:34,460 --> 00:05:37,980 we have a two dimensional array. 97 00:05:37,980 --> 00:05:42,950 So all the little mass all have mass equal to n, 98 00:05:42,950 --> 00:05:46,370 and they are placing xy plan. 99 00:05:46,370 --> 00:05:50,010 The coordinate system, which I defined is here. 100 00:05:50,010 --> 00:05:53,840 x is horizontal, and the y is vertical, 101 00:05:53,840 --> 00:05:57,090 and the z is actually pointing to you. 102 00:05:57,090 --> 00:05:59,690 And all those little mass can only 103 00:05:59,690 --> 00:06:05,010 oscillate toward you or going away from you, 104 00:06:05,010 --> 00:06:07,070 so in the z direction. 105 00:06:07,070 --> 00:06:12,470 It can only oscillate up and down in the z direction. 106 00:06:12,470 --> 00:06:16,250 And in this system we have the length scale, which 107 00:06:16,250 --> 00:06:21,310 is the horizontal distance between mass, is called aH. 108 00:06:21,310 --> 00:06:24,350 And in the vertical direction, the scale 109 00:06:24,350 --> 00:06:29,220 of the distance between mass is av. 110 00:06:29,220 --> 00:06:31,500 Also, we have string tension-- 111 00:06:31,500 --> 00:06:33,380 two different kinds of string tension 112 00:06:33,380 --> 00:06:36,080 for the vertical and horizontal direction. 113 00:06:36,080 --> 00:06:39,230 The vertical direction, you have string tension Tv, 114 00:06:39,230 --> 00:06:44,040 and in the horizontal direction you have string tension Th. 115 00:06:44,040 --> 00:06:44,810 OK 116 00:06:44,810 --> 00:06:48,680 So how do we actually describe this kind of system, right? 117 00:06:48,680 --> 00:06:52,520 The first thing, as we did before, 118 00:06:52,520 --> 00:06:57,770 is to label those little mass by my label. 119 00:06:57,770 --> 00:07:01,770 And my label is called Jx and Jy, 120 00:07:01,770 --> 00:07:09,960 which tells you which mass I was talking about in this system. 121 00:07:09,960 --> 00:07:13,580 Once I have defined that, the labels, 122 00:07:13,580 --> 00:07:19,220 I will be able to write the position of all those mass, 123 00:07:19,220 --> 00:07:22,490 the x direction position and y direction position, 124 00:07:22,490 --> 00:07:26,540 in terms of J and the A. So for example, 125 00:07:26,540 --> 00:07:32,510 x position of 1 over the mass will be written as Jx times ah. 126 00:07:32,510 --> 00:07:35,630 And y position of a specific mass, 127 00:07:35,630 --> 00:07:39,820 you can write it down in terms of a Jy times av. 128 00:07:39,820 --> 00:07:44,420 So all those things should be pretty straight forward. 129 00:07:44,420 --> 00:07:47,000 The interesting part is that, as we 130 00:07:47,000 --> 00:07:50,720 identified in the last lecture, this system 131 00:07:50,720 --> 00:07:52,790 is highly symmetric. 132 00:07:52,790 --> 00:07:56,220 It has space translation symmetry, right? 133 00:07:56,220 --> 00:08:00,200 Therefore, we can actually immediately figure out 134 00:08:00,200 --> 00:08:04,430 what will be the eigenvectors for this system. 135 00:08:04,430 --> 00:08:08,150 So the eigenvector-- very similar to what 136 00:08:08,150 --> 00:08:10,550 has been discussed here, where you have a one 137 00:08:10,550 --> 00:08:14,480 dimensional space translation symmetric system. 138 00:08:14,480 --> 00:08:18,170 Exponential of ikx was the eigenvector. 139 00:08:18,170 --> 00:08:22,100 Now you have eigenvector which is in two dimension, 140 00:08:22,100 --> 00:08:24,410 because you would like to describe 141 00:08:24,410 --> 00:08:27,690 not only the x direction but also y direction. 142 00:08:27,690 --> 00:08:31,820 And the eigenvector have exactly the same functional form 143 00:08:31,820 --> 00:08:34,270 because of space translational symmetry, 144 00:08:34,270 --> 00:08:40,460 and it is like exponentional of ixk times x. 145 00:08:40,460 --> 00:08:43,250 Multiply that by another exponential function-- 146 00:08:43,250 --> 00:08:47,470 exponential iky times y. 147 00:08:47,470 --> 00:08:54,770 So I think, until now, nothing should surprise you 148 00:08:54,770 --> 00:08:58,130 because this is what we have learned from the one 149 00:08:58,130 --> 00:09:02,800 dimensional system analysis. 150 00:09:02,800 --> 00:09:05,180 Based on what we have learned before, 151 00:09:05,180 --> 00:09:07,750 we can also immediately write down 152 00:09:07,750 --> 00:09:09,820 what would be the dispersion relation. 153 00:09:09,820 --> 00:09:14,570 Since we are always considering very small vibration, 154 00:09:14,570 --> 00:09:18,940 and this formula is still applies, 155 00:09:18,940 --> 00:09:23,080 therefore you can actually write down the dispersion relation-- 156 00:09:23,080 --> 00:09:42,260 omega squared k will be equal to 4 Th over Mah sine square Kx 157 00:09:42,260 --> 00:09:56,940 Ah, divided by 2 plus 4 Tv divided by Mav sine square Ky 158 00:09:56,940 --> 00:10:02,040 Av divided by 2. 159 00:10:02,040 --> 00:10:08,160 So this is actually pretty straightforward. 160 00:10:08,160 --> 00:10:18,440 And you can see that omega is a function of both Kx and the Ky. 161 00:10:18,440 --> 00:10:22,580 From the eigenvector we can also write down 162 00:10:22,580 --> 00:10:26,930 what would be the possible Psi xy. 163 00:10:26,930 --> 00:10:29,490 Now the Psi is actually the displacement 164 00:10:29,490 --> 00:10:33,560 in the z direction, with respect to the equilibrium position. 165 00:10:33,560 --> 00:10:36,980 And that is actually proportional 166 00:10:36,980 --> 00:10:38,810 to the eigenvector. 167 00:10:38,810 --> 00:10:45,635 So basically it's going to be a exponential ikx times x 168 00:10:45,635 --> 00:10:50,690 exponential iky times y. 169 00:10:50,690 --> 00:10:54,830 And of course I can write these two terms together, right? 170 00:10:54,830 --> 00:11:00,140 So basically what I would get is a exponential i, 171 00:11:00,140 --> 00:11:05,720 k is a vector times r, which is a vector. 172 00:11:05,720 --> 00:11:10,700 So basically k contains two components, Kx and Ky, 173 00:11:10,700 --> 00:11:14,110 and r have also two components, which is x and y. 174 00:11:17,820 --> 00:11:21,580 Again, we see that this is actually a non dispersive 175 00:11:21,580 --> 00:11:23,700 medium. 176 00:11:23,700 --> 00:11:28,440 And what we are going to do is to make linear combination 177 00:11:28,440 --> 00:11:32,190 of all those eigenvectors and figure out 178 00:11:32,190 --> 00:11:36,900 what would be the behavior when this system is oscillating 179 00:11:36,900 --> 00:11:42,840 at a specific frequency omega, and that is actually 180 00:11:42,840 --> 00:11:47,160 the corresponding normal mode at the angular frequency omega. 181 00:11:47,160 --> 00:11:51,000 So that is actually pretty similar to what we have done 182 00:11:51,000 --> 00:11:52,980 for one dimensional system. 183 00:11:56,940 --> 00:11:59,400 So this is a two dimensional system. 184 00:11:59,400 --> 00:12:05,310 Just a reminder about one dimensional system for a while. 185 00:12:05,310 --> 00:12:09,810 So there are two eigenvectors which 186 00:12:09,810 --> 00:12:12,540 have identical omega, right? 187 00:12:12,540 --> 00:12:16,860 So the first one is exponential ikx, 188 00:12:16,860 --> 00:12:20,550 and the second one is exponential of minus ikx. 189 00:12:23,190 --> 00:12:25,980 What we have done before is to do 190 00:12:25,980 --> 00:12:29,970 a linear combination of the two exponential functions, right? 191 00:12:29,970 --> 00:12:32,400 So what we can do is that-- 192 00:12:32,400 --> 00:12:36,540 OK, now I can create cosine Kx. 193 00:12:36,540 --> 00:12:41,510 This is actually 1 over 2 exponential ikx 194 00:12:41,510 --> 00:12:43,500 plus exponential minus ikx. 195 00:12:46,560 --> 00:12:49,440 Or I can also create sine Kx. 196 00:12:49,440 --> 00:12:54,900 And this is actually 1 over 2i, exponential ikx 197 00:12:54,900 --> 00:13:01,090 minus exponential minus ikx, for this one dimensional system. 198 00:13:01,090 --> 00:13:06,180 So that's how we figure out when the system is 199 00:13:06,180 --> 00:13:09,890 doing one of the normal mode. 200 00:13:09,890 --> 00:13:13,780 The shape of the system is like a cosine or a sine. 201 00:13:13,780 --> 00:13:17,490 Or in general, you can add these two together, 202 00:13:17,490 --> 00:13:19,470 and in general it can be something 203 00:13:19,470 --> 00:13:24,440 like cosine Kx plus 5, but 5 is actually 204 00:13:24,440 --> 00:13:30,630 some phase angle, which you can figure out 205 00:13:30,630 --> 00:13:32,880 by boundary condition. 206 00:13:32,880 --> 00:13:36,250 But before we introduce any boundary condition, 207 00:13:36,250 --> 00:13:40,000 all the k values, all the five values are allowed. 208 00:13:40,000 --> 00:13:44,040 Just a reminder about what we have learned before. 209 00:13:44,040 --> 00:13:46,630 So the situation is pretty simple. 210 00:13:46,630 --> 00:13:49,350 You have just plus and minus k, and then 211 00:13:49,350 --> 00:13:51,720 you make linear combination of these two, 212 00:13:51,720 --> 00:13:57,350 then you know what will be the shape of the normal mode. 213 00:13:57,350 --> 00:14:01,190 On the other hand, we are now talking 214 00:14:01,190 --> 00:14:04,430 about two dimensional case. 215 00:14:04,430 --> 00:14:08,330 So let's take a look at this dispersion relation. 216 00:14:08,330 --> 00:14:11,540 The dispersion relation we will have here, omega 217 00:14:11,540 --> 00:14:15,890 is a function of Kx, is a function of Ky as well. 218 00:14:15,890 --> 00:14:18,360 OK, what does that mean? 219 00:14:18,360 --> 00:14:25,850 That means I can have multiple choice of Kx and Ky, 220 00:14:25,850 --> 00:14:30,460 which they all produce the same omega value. 221 00:14:33,000 --> 00:14:35,490 So it's not as simple as this one 222 00:14:35,490 --> 00:14:38,070 any more, as you can see, right? 223 00:14:38,070 --> 00:14:43,920 Because when I slightly increase Kx, what I could do is-- 224 00:14:43,920 --> 00:14:49,530 OK, I can slightly reduce Ky to compensate the difference. 225 00:14:49,530 --> 00:14:52,840 Therefore, I can still keep omega, 226 00:14:52,840 --> 00:14:57,640 which is the angular frequency of the oscillation the same. 227 00:14:57,640 --> 00:15:01,260 OK, so that can be seen from this demonstration 228 00:15:01,260 --> 00:15:02,580 on the slide. 229 00:15:02,580 --> 00:15:07,290 You can see that this is actually one example dispersion 230 00:15:07,290 --> 00:15:08,980 relation. 231 00:15:08,980 --> 00:15:12,150 This is actually the formula which we have on the board. 232 00:15:12,150 --> 00:15:16,320 And what about if I set all those parameters and get, 233 00:15:16,320 --> 00:15:21,030 example, omega squared equal to 5 sine squared Kx, 234 00:15:21,030 --> 00:15:26,160 and the plus 5 sine squared Ky? 235 00:15:26,160 --> 00:15:28,960 OK, so what will happen? 236 00:15:28,960 --> 00:15:35,190 If I set my aht and the m value, so that I have this example. 237 00:15:35,190 --> 00:15:38,520 What will happen if I have that dispersion relation? 238 00:15:38,520 --> 00:15:43,260 So if I go ahead, and then plot allowed Kx and Ky 239 00:15:43,260 --> 00:15:49,090 value, which gives 1, you see a very beautiful pattern, right, 240 00:15:49,090 --> 00:15:49,960 on this. 241 00:15:49,960 --> 00:15:56,040 So you can see that ha, all those things on the circle 242 00:15:56,040 --> 00:16:00,180 can produce angular frequency omega equal to 1. 243 00:16:00,180 --> 00:16:02,715 And 1 will be their-- 244 00:16:02,715 --> 00:16:04,970 1 will be the-- normal mode will be 245 00:16:04,970 --> 00:16:09,570 all possible linear combination of all those possible Kx, Ky 246 00:16:09,570 --> 00:16:10,260 pairs. 247 00:16:10,260 --> 00:16:11,330 You have a question? 248 00:16:11,330 --> 00:16:14,830 AUDIENCE: [INAUDIBLE] 249 00:16:23,460 --> 00:16:25,540 YEN-JIE LEE: In general, I think this-- 250 00:16:25,540 --> 00:16:27,120 you mean a circular shape? 251 00:16:27,120 --> 00:16:29,610 AUDIENCE: [INAUDIBLE] 252 00:16:29,610 --> 00:16:31,470 YEN-JIE LEE: I think, in general, yes, 253 00:16:31,470 --> 00:16:34,480 you do have determinacy because you can-- 254 00:16:34,480 --> 00:16:39,450 but the shape will not be circular, for example. 255 00:16:39,450 --> 00:16:41,970 OK, so it can be a general function 256 00:16:41,970 --> 00:16:47,640 which is like the formula above, but this argument still 257 00:16:47,640 --> 00:16:48,330 applies. 258 00:16:48,330 --> 00:16:51,390 So if you have some intermediate omega value, 259 00:16:51,390 --> 00:16:56,010 you can always slightly increase Kx and slightly decrease Ky, 260 00:16:56,010 --> 00:17:00,030 and that will still satisfy the same omega value. 261 00:17:00,030 --> 00:17:02,085 Therefore, all the normal modes, we 262 00:17:02,085 --> 00:17:05,339 set specific omega will be a linear combination of all 263 00:17:05,339 --> 00:17:07,990 those possible normal mode. 264 00:17:07,990 --> 00:17:11,880 All those possible Kx and Ky pairs if you 265 00:17:11,880 --> 00:17:15,720 have an infinitely long system. 266 00:17:15,720 --> 00:17:19,710 And the law also applies to the other example, when 267 00:17:19,710 --> 00:17:22,290 I have omega equal to 5, then you have 268 00:17:22,290 --> 00:17:23,579 slightly different behavior. 269 00:17:23,579 --> 00:17:25,710 But the take home message is that there 270 00:17:25,710 --> 00:17:28,109 are many, many pairs of Kx and Ky, 271 00:17:28,109 --> 00:17:34,890 which can create the same amount of omega. 272 00:17:34,890 --> 00:17:37,670 So that makes things pretty complicated. 273 00:17:37,670 --> 00:17:42,050 Potentially, we can always still try 274 00:17:42,050 --> 00:17:44,790 to understand this by investigating 275 00:17:44,790 --> 00:17:48,910 all the possible k pairs of Kx and Ky. 276 00:17:48,910 --> 00:17:50,140 On the other hand-- 277 00:17:53,080 --> 00:17:55,350 this is what we have discussed before. 278 00:17:55,350 --> 00:17:58,840 Before you introduce boundary condition 279 00:17:58,840 --> 00:18:02,290 for the one dimensional system, there 280 00:18:02,290 --> 00:18:05,720 are infinite number of possible k value, right? 281 00:18:05,720 --> 00:18:08,530 All the possible k values are allowed. 282 00:18:08,530 --> 00:18:13,550 But after you add boundary condition-- for example, 283 00:18:13,550 --> 00:18:20,045 I add walls around this system, so that I basically have 284 00:18:20,045 --> 00:18:22,480 a fixed boundary condition. 285 00:18:22,480 --> 00:18:24,730 So basically, the boundary condition 286 00:18:24,730 --> 00:18:30,370 is that the amplitude at x equal to 0, y equal to 0, or x 287 00:18:30,370 --> 00:18:38,550 equal to 5ah or y equal to 4av. 288 00:18:38,550 --> 00:18:43,960 At the boundary, the amplitude has to be equal to 0, 289 00:18:43,960 --> 00:18:46,960 because it is attached to a wall. 290 00:18:46,960 --> 00:18:53,510 OK, when this happens this means that we 291 00:18:53,510 --> 00:18:59,750 will have have four wall which will 292 00:18:59,750 --> 00:19:02,430 have a corresponding boundary condition. 293 00:19:02,430 --> 00:19:06,700 So that means I have to satisfy this four boundary 294 00:19:06,700 --> 00:19:13,580 consideration of side 0,y evaluated at any time will be 295 00:19:13,580 --> 00:19:25,790 equal to Psi Lh, y, t, will be equal to Psi x,0,t, 296 00:19:25,790 --> 00:19:29,555 will be equal to Psi x,Lv,t. 297 00:19:32,750 --> 00:19:36,590 And this is all equal to 0. 298 00:19:36,590 --> 00:19:40,610 So those are not very difficult to understand. 299 00:19:40,610 --> 00:19:47,170 Those are just the four walls around the system. 300 00:19:47,170 --> 00:19:51,350 Once you have all those conditions-- 301 00:19:51,350 --> 00:19:57,530 and of course I define Lh will be equal to 5 times Ah, 302 00:19:57,530 --> 00:20:05,150 because there are 5 strings between the two walls 303 00:20:05,150 --> 00:20:07,160 in the horizontal direction. 304 00:20:07,160 --> 00:20:10,110 And of course I also have defined here, 305 00:20:10,110 --> 00:20:12,610 Lv will be 4 times equal to av. 306 00:20:16,810 --> 00:20:20,550 So once I have all those four boundary conditions in place 307 00:20:20,550 --> 00:20:25,890 that means I cannot arbitrarily choose k value and the fact, 308 00:20:25,890 --> 00:20:26,640 right? 309 00:20:26,640 --> 00:20:31,050 Otherwise, I will not be able to satisfy these four boundary 310 00:20:31,050 --> 00:20:33,870 conditions. 311 00:20:33,870 --> 00:20:39,300 So now we actually will be able to figure out 312 00:20:39,300 --> 00:20:44,490 that there will be only four modes in this two 313 00:20:44,490 --> 00:20:48,890 dimensional problem, which will give the same omega. 314 00:20:48,890 --> 00:20:52,140 What are the four possible nodes-- 315 00:20:52,140 --> 00:20:56,250 what are the four possible eigenvectors? 316 00:20:56,250 --> 00:21:02,520 Those are a exponential plus or minus ikx 317 00:21:02,520 --> 00:21:15,150 times x, exponential plus minus iky times y, where the Kx-- 318 00:21:15,150 --> 00:21:16,930 because of the boundary condition, 319 00:21:16,930 --> 00:21:20,730 which we have solved in the one dimensional system-- 320 00:21:20,730 --> 00:21:26,690 Kx will be equal to Nx pi divided by Lh-- 321 00:21:34,000 --> 00:21:40,720 in order to match the boundary condition, add x equal to 0 322 00:21:40,720 --> 00:21:45,130 and x equal to Lh. 323 00:21:45,130 --> 00:21:53,950 And Ky will be equal to y times pi divided by Lv. 324 00:21:53,950 --> 00:21:57,490 That's actually designed to match 325 00:21:57,490 --> 00:22:02,410 the boundary condition at y equal to 0 326 00:22:02,410 --> 00:22:05,990 and then y equal to Lv. 327 00:22:05,990 --> 00:22:09,040 So you can see that, like when we've 328 00:22:09,040 --> 00:22:14,090 seen before with one dimensional system, after you introduced 329 00:22:14,090 --> 00:22:18,460 the boundary condition it's not an infinity long system anymore 330 00:22:18,460 --> 00:22:24,400 that allowed k value, which is the length number in the x 331 00:22:24,400 --> 00:22:25,830 and y direction. 332 00:22:25,830 --> 00:22:31,150 For example, in this case, it's also become limited, 333 00:22:31,150 --> 00:22:37,960 and only a limited number of possible values are allowed. 334 00:22:37,960 --> 00:22:45,970 In this case, Nx is allowed to be equal to 1, 2 until 4, 335 00:22:45,970 --> 00:22:56,496 and Ny is equal 1,2,3 in this system we are talking about. 336 00:22:56,496 --> 00:22:57,370 Any questions so far? 337 00:23:00,021 --> 00:23:00,520 Yep? 338 00:23:00,520 --> 00:23:05,018 AUDIENCE: I think you mentioned that Kx and Ky are 339 00:23:05,018 --> 00:23:06,976 directly related rather than inversely related, 340 00:23:06,976 --> 00:23:08,928 but I'm sort of confused as to why that is. 341 00:23:08,928 --> 00:23:11,368 Because if you want to maintain the frequency, 342 00:23:11,368 --> 00:23:13,320 it increases the wave numer and [INAUDIBLE].. 343 00:23:17,430 --> 00:23:19,960 YEN-JIE LEE: Yeah, so I was talking 344 00:23:19,960 --> 00:23:26,240 about when I choose Kx and Ky in the infinitely long system. 345 00:23:26,240 --> 00:23:29,270 OK, all of the possible values of Kx and Ky 346 00:23:29,270 --> 00:23:32,830 are allowed because I have a infinitely long system 347 00:23:32,830 --> 00:23:34,730 with no boundary condition. 348 00:23:34,730 --> 00:23:40,450 And in that case, going back to this dispersion relation, 349 00:23:40,450 --> 00:23:42,870 I have the freedom to-- 350 00:23:42,870 --> 00:23:46,150 OK, so when I increase a little bit, Kx, 351 00:23:46,150 --> 00:23:49,720 I can always decrease a little bit, the Ky. 352 00:23:49,720 --> 00:23:54,520 OK, so the question is why that's not the case, right, 353 00:23:54,520 --> 00:23:56,130 for the discrete case. 354 00:23:56,130 --> 00:24:01,690 As you can see from here, after we introduced the boundary 355 00:24:01,690 --> 00:24:04,590 condition, the four boundary conditions 356 00:24:04,590 --> 00:24:11,620 especially describe the boundary of the four walls. 357 00:24:11,620 --> 00:24:15,040 And what is going to happen is that you will also see that 358 00:24:15,040 --> 00:24:19,810 the allowed Kx value is becoming limited, 359 00:24:19,810 --> 00:24:23,830 because you cannot arbitrarily choose with lengths, right, 360 00:24:23,830 --> 00:24:27,340 if you choose a side along the wavelengths like what we have 361 00:24:27,340 --> 00:24:29,840 been trying to do for the infinitely long system-- 362 00:24:29,840 --> 00:24:33,460 not that it matched the boundary condition. 363 00:24:33,460 --> 00:24:36,850 Therefore, you don't have this degree of freedom 364 00:24:36,850 --> 00:24:41,880 to choose slightly higher or slightly lower 365 00:24:41,880 --> 00:24:46,380 Ky when I change a Kx. 366 00:24:46,380 --> 00:24:49,900 So you can see that the allowed value are discrete. 367 00:24:49,900 --> 00:24:53,380 Therefore, the number of possible combinations 368 00:24:53,380 --> 00:24:55,870 of Kx and Ky is also limited. 369 00:24:55,870 --> 00:24:59,650 And in this case, it's actually very likely 370 00:24:59,650 --> 00:25:02,920 to be limited to be only four pairs, which is actually 371 00:25:02,920 --> 00:25:05,330 plus, minus Kx and the plus, minus Ky. 372 00:25:07,730 --> 00:25:08,230 All right. 373 00:25:11,420 --> 00:25:12,880 Thank you for the question. 374 00:25:12,880 --> 00:25:15,800 OK, then once I have those I can do 375 00:25:15,800 --> 00:25:21,560 a linear combination of these four possible eigenvectors. 376 00:25:21,560 --> 00:25:23,360 And also, at the same time, I will 377 00:25:23,360 --> 00:25:26,870 try to match the boundary condition. 378 00:25:26,870 --> 00:25:30,250 So if I jump forward, basically what you can conclude 379 00:25:30,250 --> 00:25:38,570 is that Psi Nx and y, so that's with an Nx value y 380 00:25:38,570 --> 00:25:43,910 value chosen for the determination of Kx and Ky. 381 00:25:43,910 --> 00:25:50,020 And is this actually a function of x and y and of course also 382 00:25:50,020 --> 00:25:52,560 time, when I also make it oscillate 383 00:25:52,560 --> 00:25:54,060 as a function of time. 384 00:25:54,060 --> 00:25:57,920 This will be equal to some arbitrary constant, A 385 00:25:57,920 --> 00:26:08,710 of amplitude Nx, Ny sine Nx pi x, 386 00:26:08,710 --> 00:26:22,470 divided by Lh sine Ny times y divided by Lv. 387 00:26:22,470 --> 00:26:27,240 And of course, you can see that this is actually sine, right? 388 00:26:27,240 --> 00:26:29,670 It's actually, the same as what we have done for the one 389 00:26:29,670 --> 00:26:31,030 dimensional system, right? 390 00:26:31,030 --> 00:26:34,185 So if you have two boundary conditions that said, 391 00:26:34,185 --> 00:26:38,200 look, the beginning and the end, therefor, 392 00:26:38,200 --> 00:26:41,890 the corresponding normal mode is always a sine function. 393 00:26:41,890 --> 00:26:45,360 So that's what we have learned from the one 394 00:26:45,360 --> 00:26:46,410 dimensional system. 395 00:26:46,410 --> 00:26:51,390 And this is also the case for the two dimensional system. 396 00:26:51,390 --> 00:26:57,030 And of course, don't forget this wave function 397 00:26:57,030 --> 00:26:59,730 is changing as a function of time 398 00:26:59,730 --> 00:27:03,420 oscillating up and down harmonically. 399 00:27:03,420 --> 00:27:14,200 Therefore, you have sine omega Nx, Ny times T plus Theta, 400 00:27:14,200 --> 00:27:18,520 which is a phase to be determined 401 00:27:18,520 --> 00:27:21,610 by initial conditions. 402 00:27:21,610 --> 00:27:27,730 And you can see that the whole equation, a sine sine 403 00:27:27,730 --> 00:27:31,620 is multiplied by a sine omega T plus 5 because 404 00:27:31,620 --> 00:27:33,340 of beta function, right? 405 00:27:33,340 --> 00:27:36,960 So that means the shape is actually 406 00:27:36,960 --> 00:27:39,820 going up and down harmonically. 407 00:27:39,820 --> 00:27:43,240 So the shape is fixed, which is sine times sine, 408 00:27:43,240 --> 00:27:45,960 and the whole thing is oscillating 409 00:27:45,960 --> 00:27:48,790 at the same frequency at the same phase, which 410 00:27:48,790 --> 00:27:51,410 is the definition of normal mode, right? 411 00:27:51,410 --> 00:27:52,570 Just a reminder. 412 00:27:52,570 --> 00:27:56,740 And how do we actually imagine what is actually happening? 413 00:27:56,740 --> 00:27:59,980 That brings me to the demonstration, 414 00:27:59,980 --> 00:28:05,980 so we can really visualize how this kind of system 415 00:28:05,980 --> 00:28:11,470 will look like by a little simulation. 416 00:28:11,470 --> 00:28:21,340 So, suppose I choose Nx and Ny equal to 1 417 00:28:21,340 --> 00:28:23,250 and see what will happen. 418 00:28:23,250 --> 00:28:28,550 This is the kind of oscillation you will expect, right? 419 00:28:28,550 --> 00:28:33,820 So if you choose Nx equal to 1, Ny equal to 1, 420 00:28:33,820 --> 00:28:36,070 then this is a system. 421 00:28:36,070 --> 00:28:40,360 Basically you have sine function with no node 422 00:28:40,360 --> 00:28:43,690 in x and y direction. 423 00:28:43,690 --> 00:28:49,840 Therefore, if you do get this simulation, 424 00:28:49,840 --> 00:28:55,830 you can see that there will be no node in the x,y plane, 425 00:28:55,830 --> 00:29:02,770 and all those particles are either going toward you 426 00:29:02,770 --> 00:29:05,620 or going away from you. 427 00:29:05,620 --> 00:29:13,190 They only oscillate in the z direction in this simulation. 428 00:29:13,190 --> 00:29:16,120 And also, you can see that now I can increase, for example-- 429 00:29:19,930 --> 00:29:24,980 I can increase the Kx by setting Nx to be 2 and see 430 00:29:24,980 --> 00:29:26,190 what will happen. 431 00:29:26,190 --> 00:29:30,200 So what is going to happen is that if I have higher 432 00:29:30,200 --> 00:29:36,110 Kx in the x direction-- so the next possible normal mode is 433 00:29:36,110 --> 00:29:43,510 that you have a full sine wave in the x direction, then 434 00:29:43,510 --> 00:29:46,610 you are going to see two components 435 00:29:46,610 --> 00:29:49,730 in this demonstration. 436 00:29:49,730 --> 00:29:54,980 And one part of the system is actually moving toward you 437 00:29:54,980 --> 00:29:58,780 while the other half part of the system 438 00:29:58,780 --> 00:30:00,680 is actually moving away from you. 439 00:30:00,680 --> 00:30:05,290 And you can actually see the node, or nodal line 440 00:30:05,290 --> 00:30:06,890 in this case, because we are talking 441 00:30:06,890 --> 00:30:10,700 about a two dimensional system in the middle 442 00:30:10,700 --> 00:30:13,730 of the distribution. 443 00:30:13,730 --> 00:30:16,730 Of course, we can always go crazy, right? 444 00:30:16,730 --> 00:30:20,640 I can set this to a really high value. 445 00:30:20,640 --> 00:30:22,310 So in this case, the highest value 446 00:30:22,310 --> 00:30:27,210 I can set these is 3 and 4, and see what happens. 447 00:30:27,210 --> 00:30:30,290 And this is actually a beautiful shape 448 00:30:30,290 --> 00:30:35,060 which is actually complicated but understandable, as you 449 00:30:35,060 --> 00:30:39,380 see in this demonstration. 450 00:30:39,380 --> 00:30:43,760 And all those little particles in this system 451 00:30:43,760 --> 00:30:48,650 are oscillating up and down at the same angular frequency 452 00:30:48,650 --> 00:30:50,540 and also at the same phase. 453 00:30:53,450 --> 00:30:56,600 Any questions? 454 00:30:56,600 --> 00:31:02,100 OK, so now we have done the discrete case, right? 455 00:31:02,100 --> 00:31:06,910 And of course we can also go to the continuous case. 456 00:31:06,910 --> 00:31:10,800 So if we go to a continuous limit, 457 00:31:10,800 --> 00:31:17,590 now I can assume that there is a symmetry 458 00:31:17,590 --> 00:31:22,720 between a horizontal direction and the vertical direction. 459 00:31:22,720 --> 00:31:29,460 I assume that Th is equal to Tv is equal to T. 460 00:31:29,460 --> 00:31:33,690 And also I assume that the length scale in the x direction 461 00:31:33,690 --> 00:31:37,030 and the y directions is equal, and the length scale 462 00:31:37,030 --> 00:31:43,450 is A. In order to make the whole system continuous, 463 00:31:43,450 --> 00:31:47,740 I need to increase the number of objects in the system, 464 00:31:47,740 --> 00:31:52,570 and at the same time I also need to decrease the distance 465 00:31:52,570 --> 00:31:54,860 between all those objects. 466 00:31:54,860 --> 00:31:56,970 So therefore, I need to have-- 467 00:31:56,970 --> 00:32:00,790 this length scale goes to 0. 468 00:32:00,790 --> 00:32:04,690 And what is going to happen is that if I rewrite 469 00:32:04,690 --> 00:32:11,050 my omega, which is a dispersion relation, what I am going 470 00:32:11,050 --> 00:32:19,360 to get is 4T divided by Na Kx squared, A squared, 471 00:32:19,360 --> 00:32:29,060 divided by 4, plus 4T over Na, Ky squared a squared over 4. 472 00:32:32,520 --> 00:32:37,290 This is issue because I am taking-- 473 00:32:37,290 --> 00:32:41,880 Ah and V need to be equal to-- and also having 474 00:32:41,880 --> 00:32:44,290 to be a very, very small value. 475 00:32:44,290 --> 00:32:50,300 Therefore, sine theta is roughly theta, right? 476 00:32:50,300 --> 00:32:54,860 Therefore, I can immediately write down this expression. 477 00:32:54,860 --> 00:33:03,080 And this will be equal to Ta divided by N, Kx squared 478 00:33:03,080 --> 00:33:04,100 plus Ky squared. 479 00:33:07,780 --> 00:33:11,590 So we are facing exactly the same situation. 480 00:33:11,590 --> 00:33:17,200 When I decrease A, I am going to add more objects 481 00:33:17,200 --> 00:33:19,390 into the system, but I don't want 482 00:33:19,390 --> 00:33:22,750 to have an infinitely large mass. 483 00:33:22,750 --> 00:33:25,300 Therefore, I also need to ensure the fix 484 00:33:25,300 --> 00:33:32,980 the ratio of m and a, so that when I actually increase 485 00:33:32,980 --> 00:33:37,510 the number of objects, I don't actually make the total mass 486 00:33:37,510 --> 00:33:40,330 go to infinity. 487 00:33:40,330 --> 00:33:45,770 So what I could do is I can define Rho S is actually 488 00:33:45,770 --> 00:33:49,120 the surface mass density. 489 00:33:49,120 --> 00:33:51,460 So the surface mass density is defined 490 00:33:51,460 --> 00:33:55,360 as m divided by a squared. 491 00:33:55,360 --> 00:33:59,770 And I can also define a surface tension. 492 00:33:59,770 --> 00:34:04,060 Surface tension Ts will be equal to T over a. 493 00:34:06,830 --> 00:34:12,050 And in this case, basically, I will be able to control, 494 00:34:12,050 --> 00:34:16,159 so that when I increase the number of objects, 495 00:34:16,159 --> 00:34:18,560 mass doesn't go to infinity, and I 496 00:34:18,560 --> 00:34:24,560 have constant surface tension and constant surface mass 497 00:34:24,560 --> 00:34:26,239 density. 498 00:34:26,239 --> 00:34:29,449 If I have defined this to quantity 499 00:34:29,449 --> 00:34:38,900 then this will become Ts divided by Rho S Kx squared 500 00:34:38,900 --> 00:34:43,010 plus Ky squared, and this will be 501 00:34:43,010 --> 00:34:51,830 equal to Ts divided by Rho S, k vector squared. 502 00:34:51,830 --> 00:34:57,230 And this k vector is a two dimensional vector. 503 00:35:01,280 --> 00:35:06,020 So we are actually almost there to make it continuous. 504 00:35:06,020 --> 00:35:10,360 So now I can make a goes to a very small value. 505 00:35:10,360 --> 00:35:14,120 We fixed the Ts and the row S. Very 506 00:35:14,120 --> 00:35:17,790 similar to what we have learned from the one dimensional case. 507 00:35:17,790 --> 00:35:20,030 Basically what we actually found is 508 00:35:20,030 --> 00:35:21,845 that time in the one dimensional case 509 00:35:21,845 --> 00:35:26,890 is that M minus 1 K metrics become minus T over Rho 510 00:35:26,890 --> 00:35:29,710 L, partial squared, partial x squared 511 00:35:29,710 --> 00:35:33,200 in the one dimensional case. 512 00:35:33,200 --> 00:35:37,040 And in the two dimensional case, without working 513 00:35:37,040 --> 00:35:40,160 through all the detail of mass, basically 514 00:35:40,160 --> 00:35:43,340 what we are going to get is partial square partial T 515 00:35:43,340 --> 00:35:48,280 square Psi xy-- 516 00:35:48,280 --> 00:35:52,150 It's actually a function of x and y and the time, 517 00:35:52,150 --> 00:35:55,730 right, because this is actually a two dimensional system. 518 00:35:55,730 --> 00:36:02,660 And this will be equal to V squared, partial squared, 519 00:36:02,660 --> 00:36:08,630 partial x squared plus partial squared, partial y squared, 520 00:36:08,630 --> 00:36:12,680 Psi xy and T-- 521 00:36:12,680 --> 00:36:14,840 very similar to what we have done 522 00:36:14,840 --> 00:36:17,570 for the one dimensional system. 523 00:36:17,570 --> 00:36:22,130 And of course I can, as you define this, as del squared. 524 00:36:22,130 --> 00:36:25,796 And basically what you are going to get is V squared, 525 00:36:25,796 --> 00:36:28,730 del squared, Psi x,y,t. 526 00:36:34,130 --> 00:36:37,430 So basically we again see this wave equation, 527 00:36:37,430 --> 00:36:41,990 but this wave equation is now a two dimensional wave equation. 528 00:36:41,990 --> 00:36:46,370 And we can also figure out what will be the V value, right, 529 00:36:46,370 --> 00:36:48,440 so what will be the velocity? 530 00:36:48,440 --> 00:36:55,330 The velocity which is going to be square root of Ts 531 00:36:55,330 --> 00:37:01,490 over Rho S. This is very similar what we have done 532 00:37:01,490 --> 00:37:05,905 for the continuous case, and in this case, what replaced 533 00:37:05,905 --> 00:37:10,970 T over Rho L is Ts over Rho S. Therefore, 534 00:37:10,970 --> 00:37:15,890 what we actually see that if I increase the surface tension, 535 00:37:15,890 --> 00:37:18,380 then the velocity will increase. 536 00:37:18,380 --> 00:37:24,620 If I decrease the mass per unit area, 537 00:37:24,620 --> 00:37:31,070 Rho S, then I will be able to have a much faster traveling 538 00:37:31,070 --> 00:37:34,790 wave from this kind of media. 539 00:37:34,790 --> 00:37:38,130 And what we can actually immediately also write down 540 00:37:38,130 --> 00:37:45,876 is that the Psi will be proportional to A sine Kx 541 00:37:45,876 --> 00:37:54,530 times x, sine Ky times y, and Psi omega T plus 5, 542 00:37:54,530 --> 00:37:59,930 where omega is calculated from the input Kx and Ky 543 00:37:59,930 --> 00:38:04,805 for this standing wave solution. 544 00:38:08,430 --> 00:38:13,080 And very similarly, I can also argue that-- 545 00:38:13,080 --> 00:38:16,290 in the three dimensional case I can actually 546 00:38:16,290 --> 00:38:18,890 follow exactly the same argument. 547 00:38:18,890 --> 00:38:20,750 Basically, in the three dimensional case, 548 00:38:20,750 --> 00:38:25,510 as well, we already see in the electromagnetic wave 549 00:38:25,510 --> 00:38:29,640 discussion, the three dimensional wave equation can 550 00:38:29,640 --> 00:38:32,100 be written as partial squared, partial T squared 551 00:38:32,100 --> 00:38:37,020 Psi is a function of x, y, and z and T. 552 00:38:37,020 --> 00:38:42,700 And this will be equal to V squared, partial squared, 553 00:38:42,700 --> 00:38:46,920 partial x squared plus partial squared, partial y squared 554 00:38:46,920 --> 00:38:51,300 plus partial squared, partial V squared-- 555 00:38:51,300 --> 00:39:03,141 Psi is a function of x,y,z, and T. Any questions so far? 556 00:39:03,141 --> 00:39:03,640 Nope? 557 00:39:07,320 --> 00:39:12,640 OK, so everything is crystal clear, right? 558 00:39:12,640 --> 00:39:15,570 OK, so this is actually the animation, which 559 00:39:15,570 --> 00:39:17,280 I showed you before already. 560 00:39:17,280 --> 00:39:21,670 So this is actually the two dimensional vibration 561 00:39:21,670 --> 00:39:23,070 of membranous. 562 00:39:23,070 --> 00:39:26,490 So basically the first one is what I have shown you 563 00:39:26,490 --> 00:39:31,770 when I choose a very small K value, which 564 00:39:31,770 --> 00:39:35,430 only make half of the sign and which 565 00:39:35,430 --> 00:39:38,010 match the boundary condition. 566 00:39:38,010 --> 00:39:41,340 Basically you see that there are oscillation, 567 00:39:41,340 --> 00:39:44,460 which you have the middle part, which 568 00:39:44,460 --> 00:39:48,480 is either going toward you or going away from you 569 00:39:48,480 --> 00:39:49,930 in this continuous system. 570 00:39:49,930 --> 00:39:55,030 So basically the solution is actually remarkably the same 571 00:39:55,030 --> 00:39:57,840 as what we have seen in the discrete system. 572 00:39:57,840 --> 00:39:59,810 OK, that's actually what I wanted to say. 573 00:39:59,810 --> 00:40:04,590 And also, of course, you can increase the k value, 574 00:40:04,590 --> 00:40:10,290 so that you go to the higher frequency normal mode. 575 00:40:10,290 --> 00:40:14,240 And you can see that if you have more and more nodal 576 00:40:14,240 --> 00:40:21,171 lines, which is actually the lines describing the-- 577 00:40:21,171 --> 00:40:24,870 the lines which you actually have no oscillation 578 00:40:24,870 --> 00:40:26,400 at all on the surface. 579 00:40:26,400 --> 00:40:29,040 For example, in this case, the nodal lines, 580 00:40:29,040 --> 00:40:32,070 as you're passing through the middle of this figure-- 581 00:40:32,070 --> 00:40:36,570 because all those little mass, all the other high particles 582 00:40:36,570 --> 00:40:38,730 are vibrating like crazy. 583 00:40:38,730 --> 00:40:43,840 But all the particles on this line, the nodal line, 584 00:40:43,840 --> 00:40:48,470 they're not at all moving, because that's actually 585 00:40:48,470 --> 00:40:49,920 at this position-- 586 00:40:49,920 --> 00:40:53,880 which is having one of the sine function equal to 0. 587 00:40:53,880 --> 00:40:57,090 Therefore, no matter what you do as a function of time, 588 00:40:57,090 --> 00:41:00,850 how you evolve the system, all those particle at that line 589 00:41:00,850 --> 00:41:04,230 will not move at all. 590 00:41:04,230 --> 00:41:10,470 And this was demonstrated from this table here. 591 00:41:10,470 --> 00:41:13,890 It's actually Chladni figures. 592 00:41:13,890 --> 00:41:17,610 You can see that in a two dimensional case 593 00:41:17,610 --> 00:41:21,600 the figures can look very complicated. 594 00:41:21,600 --> 00:41:23,670 So basically what it's showing here 595 00:41:23,670 --> 00:41:27,150 is that you have a square plate and it's 596 00:41:27,150 --> 00:41:34,080 attached to a vibrator, and basically this vibrator 597 00:41:34,080 --> 00:41:35,460 can be controlled. 598 00:41:35,460 --> 00:41:39,390 I can change the frequency of that vibration. 599 00:41:39,390 --> 00:41:47,940 When I reach resonance, which excites one of the normal mode, 600 00:41:47,940 --> 00:41:52,620 then this plate will be oscillating 601 00:41:52,620 --> 00:41:56,440 in a specific pattern. 602 00:41:56,440 --> 00:41:58,470 And those lines are actually showing you 603 00:41:58,470 --> 00:42:02,670 that the plates, which you have no oscillation at all 604 00:42:02,670 --> 00:42:04,050 as a function of time. 605 00:42:04,050 --> 00:42:09,820 Because if I, for example, turn on this demo again, 606 00:42:09,820 --> 00:42:12,483 you can see that if I turn on this demo-- 607 00:42:17,320 --> 00:42:21,430 you can see that all the sand on the plates 608 00:42:21,430 --> 00:42:25,430 are vibrating because now I am oscillating 609 00:42:25,430 --> 00:42:32,430 this plate by the vibration generator and the button-- 610 00:42:32,430 --> 00:42:33,720 by the motor and the button. 611 00:42:33,720 --> 00:42:38,050 And if I change the oscillation frequency so 612 00:42:38,050 --> 00:42:43,020 you can see that this frequency doesn't match with one 613 00:42:43,020 --> 00:42:46,860 of the normal mode frequency. 614 00:42:46,860 --> 00:42:50,160 Therefore, they will not be a lot of activity. 615 00:42:50,160 --> 00:42:56,590 But if I now change the frequency, 616 00:42:56,590 --> 00:43:02,370 so that it matches with one of the oscillation frequency 617 00:43:02,370 --> 00:43:05,280 for one of the normal mode of this system, 618 00:43:05,280 --> 00:43:11,450 you can see the, oh, some really cryptic pattern is formed! 619 00:43:11,450 --> 00:43:16,080 You can see that, oh, it have a very complicated pattern. 620 00:43:16,080 --> 00:43:19,810 And if I put my finger in one of the lines 621 00:43:19,810 --> 00:43:23,480 here I don't feel the vibration, but on the other hand, 622 00:43:23,480 --> 00:43:25,980 if I put my finger here, I can actually 623 00:43:25,980 --> 00:43:30,920 feel that there's lot of vibration at that point. 624 00:43:30,920 --> 00:43:32,960 I can always change the frequency 625 00:43:32,960 --> 00:43:36,000 and see what will happen. 626 00:43:36,000 --> 00:43:40,110 And then you can see that now I increase the frequency, 627 00:43:40,110 --> 00:43:45,509 and now I am actually trying to excite another mode. 628 00:43:45,509 --> 00:43:50,440 Now I need some more sand. 629 00:43:50,440 --> 00:43:55,740 You can see that I randomly throw sand on this plate, 630 00:43:55,740 --> 00:43:58,880 and then you can see that those centered as you bounce it 631 00:43:58,880 --> 00:44:06,060 around until it sits on the nodal line, which 632 00:44:06,060 --> 00:44:09,000 no vibration actually happens. 633 00:44:09,000 --> 00:44:15,576 OK, so let's go back to one of the lower frequency mode, which 634 00:44:15,576 --> 00:44:17,052 we showed you before. 635 00:44:21,000 --> 00:44:24,720 Now the question is, OK, you can see this complicated pattern 636 00:44:24,720 --> 00:44:32,490 almost look ridiculous, can we actually reproduce this pattern 637 00:44:32,490 --> 00:44:33,720 by our calculation? 638 00:44:36,650 --> 00:44:42,740 So we have seen that, OK, I can conclude that the normal mode 639 00:44:42,740 --> 00:44:44,090 looks like this, right? 640 00:44:44,090 --> 00:44:51,009 So therefore, I must be able to explain 641 00:44:51,009 --> 00:44:52,550 all those patterns, which is actually 642 00:44:52,550 --> 00:44:56,780 shown in this experiment. 643 00:44:56,780 --> 00:45:00,555 So that's actually what I am going to do to give you a try. 644 00:45:04,070 --> 00:45:07,650 So this is a little demonstration 645 00:45:07,650 --> 00:45:12,110 which I actually wrote. 646 00:45:12,110 --> 00:45:17,210 This demonstration actually has the solution to this two 647 00:45:17,210 --> 00:45:19,460 dimensional problem. 648 00:45:19,460 --> 00:45:23,300 And also the boundary condition is that-- 649 00:45:23,300 --> 00:45:26,600 or say, the condition which, as you can see on my solution, 650 00:45:26,600 --> 00:45:32,000 is that I require the center of the plate to be driven, 651 00:45:32,000 --> 00:45:35,840 because that's where I start to vibrate this plate. 652 00:45:35,840 --> 00:45:39,770 And I drive this plate up and down to see 653 00:45:39,770 --> 00:45:42,020 what is going to happen. 654 00:45:42,020 --> 00:45:44,670 So from this analytical calculation 655 00:45:44,670 --> 00:45:49,770 you can see that you expect a circle in the middle and also 656 00:45:49,770 --> 00:45:55,170 four lines which actually cover this circle. 657 00:45:55,170 --> 00:45:57,560 And also there are some strange structure 658 00:45:57,560 --> 00:46:02,360 at the edge of the plate. 659 00:46:02,360 --> 00:46:05,780 And you can actually compare this calculation 660 00:46:05,780 --> 00:46:10,430 to this result. It doesn't really match perfectly. 661 00:46:10,430 --> 00:46:12,840 So you can see that there is some imperfection, 662 00:46:12,840 --> 00:46:16,310 but you get that ring in the middle, 663 00:46:16,310 --> 00:46:20,078 and you do see these 1-2-3-4-5-6-7-8, 664 00:46:20,078 --> 00:46:24,170 8 lines produced in this experiment. 665 00:46:24,170 --> 00:46:28,250 So this experiment is not perfect because there's a, 666 00:46:28,250 --> 00:46:30,060 you know, stiffness of this thing 667 00:46:30,060 --> 00:46:34,640 and also some energy dissipation, et cetera. 668 00:46:34,640 --> 00:46:36,590 But you can see that, sort of, we 669 00:46:36,590 --> 00:46:41,030 can actually use our calculation to explain this pattern! 670 00:46:41,030 --> 00:46:42,920 That's really cool, right? 671 00:46:42,920 --> 00:46:49,250 And the advantage is that now I have this wonderful simulation. 672 00:46:49,250 --> 00:46:53,000 I can put in all the crazy numbers, 673 00:46:53,000 --> 00:46:56,760 and you see that, huh, if I increase the K value, 674 00:46:56,760 --> 00:47:02,240 I can really make all kinds of ridiculous patterns out 675 00:47:02,240 --> 00:47:03,020 of this. 676 00:47:03,020 --> 00:47:08,640 And all these things can be kind of realized by this experiment. 677 00:47:08,640 --> 00:47:10,050 So you can see that, for example, 678 00:47:10,050 --> 00:47:17,960 I can now also turn on this, and I can actually 679 00:47:17,960 --> 00:47:25,720 increase the frequency to a very high frequency, for example. 680 00:47:25,720 --> 00:47:28,690 Then I can see that, oh, the pattern really 681 00:47:28,690 --> 00:47:30,690 becomes much more complicated. 682 00:47:38,498 --> 00:47:42,330 Now I have a circle and there are many, many more structures 683 00:47:42,330 --> 00:47:44,740 which you're seeing in the surrounding area. 684 00:47:44,740 --> 00:47:48,380 And of course I can again increase, increase, 685 00:47:48,380 --> 00:47:51,180 and see what happens. 686 00:47:51,180 --> 00:47:54,390 I don't know what is going to happen because every time I 687 00:47:54,390 --> 00:47:57,270 do this experiment I get a different pattern. 688 00:48:01,450 --> 00:48:07,800 OK, now this seems to be a very nice frequency. 689 00:48:07,800 --> 00:48:11,790 It's getting harder and harder. 690 00:48:11,790 --> 00:48:16,100 You can see that this is really crazy. 691 00:48:16,100 --> 00:48:18,510 Holy mackerel, right? 692 00:48:18,510 --> 00:48:21,010 What the hell is this? 693 00:48:21,010 --> 00:48:23,880 So you can see that all those crazy patterns can be created. 694 00:48:26,440 --> 00:48:28,410 And of course, during the break, you 695 00:48:28,410 --> 00:48:33,130 are welcome to come forward and play with this. 696 00:48:33,130 --> 00:48:36,180 So you can see that we can actually 697 00:48:36,180 --> 00:48:41,710 understand, sort of, the pattern produced from this experiment. 698 00:48:41,710 --> 00:48:43,960 That's actually very exciting, because that's actually 699 00:48:43,960 --> 00:48:45,290 why we are physicists, right? 700 00:48:45,290 --> 00:48:48,100 We would like to know why those patterns are formed, 701 00:48:48,100 --> 00:48:49,660 and now you know why. 702 00:48:49,660 --> 00:48:51,670 Those patterns are formed because there 703 00:48:51,670 --> 00:48:56,830 are nodal lines in this two dimensional normal mode modes. 704 00:48:56,830 --> 00:48:59,440 And the little sands really love to sit there, 705 00:48:59,440 --> 00:49:02,050 because you want to sit in a place which you 706 00:49:02,050 --> 00:49:04,600 don't have a lot of vibration. 707 00:49:04,600 --> 00:49:06,420 It's not very comfortable, right? 708 00:49:06,420 --> 00:49:10,030 So you sit in the place, which, hm, vibrates? 709 00:49:10,030 --> 00:49:10,922 Your problem. 710 00:49:10,922 --> 00:49:11,880 Vibrate's your problem. 711 00:49:11,880 --> 00:49:15,740 I sit here where there is no vibration. 712 00:49:15,740 --> 00:49:20,260 So that's basically how we explain these strange figures 713 00:49:20,260 --> 00:49:21,480 which we can see. 714 00:49:21,480 --> 00:49:25,270 And just for fun you can see that I can also generate 715 00:49:25,270 --> 00:49:27,420 all kinds of craziness. 716 00:49:27,420 --> 00:49:31,480 You can input all kinds of different Nx and Ny values, 717 00:49:31,480 --> 00:49:35,110 and you get all those wonderful figures for free. 718 00:49:35,110 --> 00:49:37,600 Maybe we can actually make some T-shirts 719 00:49:37,600 --> 00:49:41,260 with all those figures on the T-shirt, right? 720 00:49:41,260 --> 00:49:46,790 OK, so we had a lot of fun with this two dimensional plate. 721 00:49:46,790 --> 00:49:52,480 How about what will happen if I have a circular plate? 722 00:49:52,480 --> 00:49:54,620 What does it do? 723 00:49:54,620 --> 00:49:58,390 Unfortunately, I would not be able to solve the two 724 00:49:58,390 --> 00:50:01,600 dimensional plate problem in front of you 725 00:50:01,600 --> 00:50:05,920 because that will give you a Bessel function, which is not 726 00:50:05,920 --> 00:50:09,410 the end of the world, but that's actually kind of complicated. 727 00:50:09,410 --> 00:50:12,790 If I put it in mid-term exam, that's actually not 728 00:50:12,790 --> 00:50:15,520 very encouraging, right? 729 00:50:15,520 --> 00:50:17,980 But I can actually tell you what will be the solution. 730 00:50:17,980 --> 00:50:19,780 The solution will be a Bessel function. 731 00:50:19,780 --> 00:50:24,400 Basically you will have a lot of ring-like structures 732 00:50:24,400 --> 00:50:27,190 if I have a circular plate. 733 00:50:27,190 --> 00:50:30,460 And I can actually do an experiment which actually shows 734 00:50:30,460 --> 00:50:35,740 you the behavior of the circular boundary condition 735 00:50:35,740 --> 00:50:39,370 and see what kind of pattern can we see. 736 00:50:39,370 --> 00:50:45,790 So here I have a kind of complicated experiment. 737 00:50:45,790 --> 00:50:48,760 So here I have this ring, which I 738 00:50:48,760 --> 00:50:56,580 would like to produce some film on this ring. 739 00:50:56,580 --> 00:50:59,140 So see if I am successful. 740 00:50:59,140 --> 00:51:00,770 Kind of. 741 00:51:00,770 --> 00:51:01,620 OK. 742 00:51:01,620 --> 00:51:12,100 Now I can put this a soft film in front of the speaker. 743 00:51:12,100 --> 00:51:16,460 I can actually oscillate this thing-- 744 00:51:16,460 --> 00:51:18,527 membranes by the speaker. 745 00:51:18,527 --> 00:51:20,360 Oops, don't want to destroy everything here. 746 00:51:24,510 --> 00:51:28,910 All right, so now I can turn on this, so that we have light. 747 00:51:28,910 --> 00:51:32,990 And of course I will turn on the signal generator, 748 00:51:32,990 --> 00:51:34,580 so that I can hear-- 749 00:51:34,580 --> 00:51:39,020 I can actually start to vibrate the membranes. 750 00:51:39,020 --> 00:51:47,185 Before I do that, I have to turn everything off, hide images. 751 00:51:47,185 --> 00:51:49,870 All right, I hope you can see something. 752 00:51:49,870 --> 00:51:50,370 Can you? 753 00:51:53,690 --> 00:51:55,420 Can you see something on the-- 754 00:51:58,210 --> 00:52:05,030 it's kind of difficult to see it, but that should be there. 755 00:52:05,030 --> 00:52:10,140 OK, now I can turn on this speaker, 756 00:52:10,140 --> 00:52:13,040 and you can see that there are some patterns which 757 00:52:13,040 --> 00:52:16,160 it's probably difficult to see. 758 00:52:16,160 --> 00:52:18,360 Kind of see, right? 759 00:52:18,360 --> 00:52:20,560 There are rings. 760 00:52:20,560 --> 00:52:23,336 You can see it on the speaker. 761 00:52:23,336 --> 00:52:27,710 So you can see that now I have one, two, three-- three rings, 762 00:52:27,710 --> 00:52:28,850 right? 763 00:52:28,850 --> 00:52:33,950 Because I couldn't turn off the light, which is actually 764 00:52:33,950 --> 00:52:37,220 emitted from the sun, right? 765 00:52:37,220 --> 00:52:39,080 So I cannot turn off sun. 766 00:52:39,080 --> 00:52:42,130 Therefore, you can barely see this figure. 767 00:52:42,130 --> 00:52:45,470 So we shall explain the result of this experiment. 768 00:52:45,470 --> 00:52:51,540 And you can see that if I increase the frequency, 769 00:52:51,540 --> 00:52:54,270 according to the solution from Bessel function, 770 00:52:54,270 --> 00:52:58,312 you will see more rings got excited. 771 00:52:58,312 --> 00:53:00,270 So you can see that now I have one, two, three, 772 00:53:00,270 --> 00:53:01,740 four-- four rings. 773 00:53:01,740 --> 00:53:04,542 And of course I can continue to increase, increase, 774 00:53:04,542 --> 00:53:06,102 and increase. 775 00:53:06,102 --> 00:53:10,470 And you will see that there are even more rings produced. 776 00:53:10,470 --> 00:53:13,280 Essentially what I'm doing is actually really 777 00:53:13,280 --> 00:53:18,500 trying to vibrate and excite one of the normal mode 778 00:53:18,500 --> 00:53:21,530 by this loud speaker. 779 00:53:21,530 --> 00:53:23,420 And you can actually kind of see-- 780 00:53:23,420 --> 00:53:26,270 I hope you can kind of see it. 781 00:53:26,270 --> 00:53:28,950 If you can still see it, that means 782 00:53:28,950 --> 00:53:32,145 you need to check your eye because the membranes is 783 00:53:32,145 --> 00:53:32,645 broken. 784 00:53:36,560 --> 00:53:39,800 OK, so I think you sort of get this idea, 785 00:53:39,800 --> 00:53:45,470 and I'm going to turn off this wonderful machine 786 00:53:45,470 --> 00:53:50,250 and go back to the lecture. 787 00:53:50,250 --> 00:53:56,570 So this experiment is kind of hard to reproduce in your study 788 00:53:56,570 --> 00:53:58,010 room, right? 789 00:53:58,010 --> 00:54:00,200 I think everybody will agree. 790 00:54:00,200 --> 00:54:03,170 And there's another one which is actually 791 00:54:03,170 --> 00:54:05,300 kind of easy to reproduce, which I 792 00:54:05,300 --> 00:54:07,730 will encourage you to try it-- 793 00:54:07,730 --> 00:54:08,990 so if you have time. 794 00:54:08,990 --> 00:54:11,500 So this is from Jake. 795 00:54:11,500 --> 00:54:14,510 He sent me this wonderful video when 796 00:54:14,510 --> 00:54:16,700 I was teaching the 8.03 class. 797 00:54:16,700 --> 00:54:20,780 They found that they could excite two dimensional waves 798 00:54:20,780 --> 00:54:23,470 in this way. 799 00:54:23,470 --> 00:54:25,400 Can you see it? 800 00:54:25,400 --> 00:54:27,410 It's wonderful. 801 00:54:27,410 --> 00:54:33,620 You can see there are very high frequency oscillation, which 802 00:54:33,620 --> 00:54:37,640 actually excite these two dimensional wave. 803 00:54:37,640 --> 00:54:41,210 And you can see that lots, and lots, and lots of rings 804 00:54:41,210 --> 00:54:43,280 are excited. 805 00:54:43,280 --> 00:54:44,990 And then you can see very clearly 806 00:54:44,990 --> 00:54:50,240 from this simple experiment, what you really 807 00:54:50,240 --> 00:54:53,420 need is a cup of water. 808 00:54:53,420 --> 00:54:57,740 And you rub it against the surface of a table, then 809 00:54:57,740 --> 00:55:01,610 you'll be able to excite all the crazy patterns, which 810 00:55:01,610 --> 00:55:07,510 you can actually see from this two dimensional system 811 00:55:07,510 --> 00:55:11,180 and with two dimensional boundary conditions. 812 00:55:11,180 --> 00:55:13,580 OK, so we will take a five minute 813 00:55:13,580 --> 00:55:18,560 break before we enter the next part of the discussion. 814 00:55:18,560 --> 00:55:20,280 And we come back at 35. 815 00:55:26,940 --> 00:55:29,250 OK, welcome back, everybody. 816 00:55:29,250 --> 00:55:33,180 So what I'm going to do now is to continue the discussion, 817 00:55:33,180 --> 00:55:37,550 the one we actually got started, of the two dimensional 818 00:55:37,550 --> 00:55:39,600 and three dimensional system. 819 00:55:39,600 --> 00:55:45,030 And we have actually studied the behavior 820 00:55:45,030 --> 00:55:48,520 of standing wave, or normal mode, 821 00:55:48,520 --> 00:55:51,220 for this two dimensional system. 822 00:55:51,220 --> 00:55:55,080 And what I am going to do is discuss with you, a two 823 00:55:55,080 --> 00:55:56,900 dimensional progressive wave. 824 00:56:06,470 --> 00:56:11,120 So I will stick to a really simple example, 825 00:56:11,120 --> 00:56:12,480 which are plane waves. 826 00:56:18,720 --> 00:56:24,010 OK, so in the case of plane waves, which we discussed when 827 00:56:24,010 --> 00:56:27,520 we actually discussed the EM waves, 828 00:56:27,520 --> 00:56:31,100 you have the following functional form. 829 00:56:31,100 --> 00:56:34,600 Psi is a function of r and the t. 830 00:56:34,600 --> 00:56:41,160 And this will be equal to A exponential i. 831 00:56:41,160 --> 00:56:44,770 The k is a vector now, and it's pointing 832 00:56:44,770 --> 00:56:50,570 to the direction of the propagation of this plan wave. 833 00:56:50,570 --> 00:56:57,080 And this k is dot with r vector minus omega T, which 834 00:56:57,080 --> 00:57:01,310 is the oscillation frequency-- angular frequency. 835 00:57:01,310 --> 00:57:04,670 And evaluated at a specific time. 836 00:57:04,670 --> 00:57:09,590 And this is expression actually describes a plane wave 837 00:57:09,590 --> 00:57:14,540 where the direction of propagation 838 00:57:14,540 --> 00:57:18,110 is described by this k vector. 839 00:57:18,110 --> 00:57:22,820 And of course you can actually have the wave front, 840 00:57:22,820 --> 00:57:27,320 which is actually the peak position of this plain wave. 841 00:57:27,320 --> 00:57:31,220 And the distance between the peak position-- 842 00:57:31,220 --> 00:57:33,710 so if you can imagine that this is like this. 843 00:57:44,030 --> 00:57:48,530 So if you look at the distance between peak position 844 00:57:48,530 --> 00:57:51,920 that will give you the wavelengths, right? 845 00:57:51,920 --> 00:57:54,530 The wavelengths, now that will be 846 00:57:54,530 --> 00:57:58,310 equal to 2 pi divided by k, right? 847 00:57:58,310 --> 00:58:03,330 In this case, it's the length of this k vector. 848 00:58:03,330 --> 00:58:06,560 Just a reminder about what we introduced 849 00:58:06,560 --> 00:58:09,170 in the previous lecture. 850 00:58:09,170 --> 00:58:13,520 And we were using this to describe electromagnetic wave 851 00:58:13,520 --> 00:58:17,570 and such a kind of expression can be also 852 00:58:17,570 --> 00:58:22,880 be used to describe sound waves and also 853 00:58:22,880 --> 00:58:28,010 vibration on the membranes, et cetera, progressive waves. 854 00:58:28,010 --> 00:58:33,230 So if there are no other medium like what we actually 855 00:58:33,230 --> 00:58:35,450 have in this slide-- 856 00:58:35,450 --> 00:58:37,460 so we have nothing else. 857 00:58:37,460 --> 00:58:42,410 I have a membrane with a surface tension Ts, 858 00:58:42,410 --> 00:58:47,360 and Rho S is the mass per unit area. 859 00:58:47,360 --> 00:58:50,080 Then basically, this progressing wave 860 00:58:50,080 --> 00:58:53,540 is going to be traveling at the speed of v, 861 00:58:53,540 --> 00:58:57,830 which is equal to square root of Ts over Rho S, 862 00:58:57,830 --> 00:59:02,600 and I can actually define that to be some constant c divided 863 00:59:02,600 --> 00:59:03,860 by n. 864 00:59:03,860 --> 00:59:13,100 So c is some constant, and m is another constant 865 00:59:13,100 --> 00:59:16,610 which actually, the ratio c and n is equal to v. 866 00:59:16,610 --> 00:59:20,800 And I will need that expression later, only later, not now. 867 00:59:20,800 --> 00:59:26,030 If I have nothing else and that this system actually 868 00:59:26,030 --> 00:59:28,850 filled the whole universe, then what is going to happen 869 00:59:28,850 --> 00:59:31,160 is that this progressing wave is going 870 00:59:31,160 --> 00:59:34,600 to be propagating, propagating, propagating, propagating. 871 00:59:34,600 --> 00:59:39,680 Nothing will change until the edge of the universe. 872 00:59:39,680 --> 00:59:43,700 It doesn't actually introduce any excitement. 873 00:59:43,700 --> 00:59:45,890 So that's what we have already learned 874 00:59:45,890 --> 00:59:52,010 from when we have discussed electromagnetic interaction, 875 00:59:52,010 --> 00:59:54,290 and now the same expression can also 876 00:59:54,290 --> 01:00:00,080 be used for the description of the membranes. 877 01:00:00,080 --> 01:00:04,850 And then now to make this problem more exciting, 878 01:00:04,850 --> 01:00:08,410 what I'm going to do is to introduce a boundary. 879 01:00:08,410 --> 01:00:12,340 So the boundary is in the middle of this slide. 880 01:00:12,340 --> 01:00:17,690 And I will assume that the horizontal direction 881 01:00:17,690 --> 01:00:22,070 to be x equal to 0-- 882 01:00:22,070 --> 01:00:27,350 the horizontal direction to be in x direction and the boundary 883 01:00:27,350 --> 01:00:29,210 is at x equal to 0. 884 01:00:32,330 --> 01:00:39,130 And when you pass this boundary, there's 885 01:00:39,130 --> 01:00:44,500 another kind of material with surface tension Ts prime 886 01:00:44,500 --> 01:00:50,920 and slightly different mass per unit area, your s prime. 887 01:00:50,920 --> 01:00:54,360 Based on the expression we got for the velocity 888 01:00:54,360 --> 01:00:56,780 we will be able to conclude that v prime will 889 01:00:56,780 --> 01:01:02,650 be equal to square root of T prime S divided by Rho S prime. 890 01:01:02,650 --> 01:01:06,580 And that will be equal to c over n prime. 891 01:01:06,580 --> 01:01:10,630 And c is the same constant which I used for the left hand side 892 01:01:10,630 --> 01:01:12,260 system. 893 01:01:12,260 --> 01:01:16,900 And n, later, you will realize that that's a refraction 894 01:01:16,900 --> 01:01:20,710 index in a discussion. 895 01:01:20,710 --> 01:01:24,580 So the question which I would like to ask 896 01:01:24,580 --> 01:01:29,820 is, OK, now I have a prime wave propagating 897 01:01:29,820 --> 01:01:31,200 in the first system. 898 01:01:31,200 --> 01:01:35,020 And it met a boundary, and the question 899 01:01:35,020 --> 01:01:41,020 is what will happen when I have the incident wave coming 900 01:01:41,020 --> 01:01:42,160 into the system? 901 01:01:44,830 --> 01:01:50,920 So before that, I also need to write down the dispersion 902 01:01:50,920 --> 01:01:51,940 relation, right. 903 01:01:51,940 --> 01:01:55,090 So dispersion relation can be attempted 904 01:01:55,090 --> 01:02:01,660 by plugging in a normal mode Psi function 905 01:02:01,660 --> 01:02:03,730 into the wave equation. 906 01:02:03,730 --> 01:02:06,700 So what I can immediately obtain is 907 01:02:06,700 --> 01:02:10,020 that the dispersion relation, omega squared 908 01:02:10,020 --> 01:02:16,990 is equal to V squared, Kx squared, plus Ky squared. 909 01:02:16,990 --> 01:02:19,390 You can actually check this expression 910 01:02:19,390 --> 01:02:24,430 by plugging in this function into the two dimensional wave 911 01:02:24,430 --> 01:02:29,170 equation, and you will get that expression, OK? 912 01:02:29,170 --> 01:02:33,160 And that means omega cannot be arbitrary number. 913 01:02:33,160 --> 01:02:36,640 It's as you decided by Kx and the Ky. 914 01:02:36,640 --> 01:02:41,410 Or say, if omega is the side and one of the k is the side, 915 01:02:41,410 --> 01:02:45,490 then the third number, for example in this case, Kx, 916 01:02:45,490 --> 01:02:48,940 is as you decided by the dispersion relation 917 01:02:48,940 --> 01:02:51,790 which we have here. 918 01:02:51,790 --> 01:02:56,160 So, coming back to the original problem we are posting, 919 01:02:56,160 --> 01:03:01,860 I have, now, the incident wave coming into this system. 920 01:03:01,860 --> 01:03:06,390 I would like to know what will happen at the boundary 921 01:03:06,390 --> 01:03:12,070 when I have two systems with a left hand side propagating at-- 922 01:03:12,070 --> 01:03:14,280 the speed of the propagation is v, 923 01:03:14,280 --> 01:03:17,400 and right hand side's speed of propagation is v prime. 924 01:03:17,400 --> 01:03:18,450 What is going to happen? 925 01:03:21,990 --> 01:03:28,810 Assume my guess that I am going to get a refractive wave 926 01:03:28,810 --> 01:03:31,210 and a transmitted wave. 927 01:03:31,210 --> 01:03:34,330 So that's based on what we have learned from the one 928 01:03:34,330 --> 01:03:36,370 dimensional system. 929 01:03:36,370 --> 01:03:40,580 If I call this the left hand side, and call the right hand 930 01:03:40,580 --> 01:03:43,710 side system right hand-- 931 01:03:43,710 --> 01:03:44,992 the right hand system, r. 932 01:03:47,860 --> 01:03:56,150 So I can write down the wave function Psi L describing 933 01:03:56,150 --> 01:03:57,680 the left hand side. 934 01:03:57,680 --> 01:04:05,140 This will be equal to A exponential of ik dot r 935 01:04:05,140 --> 01:04:10,070 minus omega T. This is actually the incident wave-- 936 01:04:19,750 --> 01:04:22,500 describing this incident wave. 937 01:04:22,500 --> 01:04:25,380 And as you might guess, there should be 938 01:04:25,380 --> 01:04:27,660 some kind of refraction, right? 939 01:04:27,660 --> 01:04:31,710 So once this wave actually passed 940 01:04:31,710 --> 01:04:35,070 through the boundary, or touch the boundary, 941 01:04:35,070 --> 01:04:37,470 there should be some kind of refraction, right? 942 01:04:37,470 --> 01:04:39,970 So the refraction, I can actually write it down 943 01:04:39,970 --> 01:04:44,490 in this form as sum over alpha, r alpha 944 01:04:44,490 --> 01:04:48,660 is actually the coefficient over amplitude as function 945 01:04:48,660 --> 01:04:51,440 of the normal modes-- 946 01:04:51,440 --> 01:04:58,150 as a function of the progressing wave number, 947 01:04:58,150 --> 01:04:59,590 which I have shown. 948 01:04:59,590 --> 01:05:01,650 And I can actually sum over all kinds 949 01:05:01,650 --> 01:05:05,110 of progressing wave numbers. 950 01:05:05,110 --> 01:05:12,990 Exponential ik alpha times r minus omega T. 951 01:05:12,990 --> 01:05:18,805 So this is a general form of refracting wave. 952 01:05:18,805 --> 01:05:23,820 k alpha is describing the direction 953 01:05:23,820 --> 01:05:25,950 of the individual refractive wave, 954 01:05:25,950 --> 01:05:30,899 and alpha is labeling the individual refractive wave. 955 01:05:30,899 --> 01:05:32,940 But I don't know what will be the functional form 956 01:05:32,940 --> 01:05:35,230 for the k alpha for the moment. 957 01:05:35,230 --> 01:05:40,380 So therefore, I try to sum over all the possible alpha. 958 01:05:40,380 --> 01:05:43,590 And I would like to figure out what 959 01:05:43,590 --> 01:05:47,700 will be the allowed alpha by matching the boundary 960 01:05:47,700 --> 01:05:48,850 condition. 961 01:05:48,850 --> 01:05:52,460 So in short, the right hand side turn 962 01:05:52,460 --> 01:05:54,972 essentially is actually describing the refractive wave. 963 01:06:00,540 --> 01:06:04,710 And finally, passing through this boundary condition, 964 01:06:04,710 --> 01:06:07,500 let's look at the right hand side. 965 01:06:07,500 --> 01:06:12,990 Right hand side, Psi r, is going to be 966 01:06:12,990 --> 01:06:20,880 sum over beta on the transmission coefficients tau 967 01:06:20,880 --> 01:06:28,570 beta, which is the original amplitude, exponential of i, 968 01:06:28,570 --> 01:06:36,790 k beta times r minus omega T. So again I 969 01:06:36,790 --> 01:06:39,880 don't know what will be the behavior 970 01:06:39,880 --> 01:06:41,080 of the transmitted wave. 971 01:06:41,080 --> 01:06:44,380 Therefore, I have summed over all the possible values. 972 01:06:44,380 --> 01:06:46,547 And this is actually the functional form 973 01:06:46,547 --> 01:06:47,588 for the transmitted wave. 974 01:06:56,750 --> 01:07:04,190 I also know that k alpha vector squared will 975 01:07:04,190 --> 01:07:10,550 be equal to omega squared Rho s over Ts, 976 01:07:10,550 --> 01:07:15,890 and this will be equal to omega squared v squared, 977 01:07:15,890 --> 01:07:19,760 because of the dispersion relation in the left hand side. 978 01:07:19,760 --> 01:07:24,110 So basically, if you look at the left hand side dispersion 979 01:07:24,110 --> 01:07:31,680 relation, the length squared of this k vector 980 01:07:31,680 --> 01:07:34,850 will be equal to omega squared times v squared, right? 981 01:07:34,850 --> 01:07:37,660 This is just a dispersion relation 982 01:07:37,660 --> 01:07:41,070 of a non-dispersive medium. 983 01:07:41,070 --> 01:07:45,720 And also, I can actually figure out what will be the-- 984 01:07:45,720 --> 01:07:49,200 allowed length for the k theta. 985 01:07:49,200 --> 01:07:54,810 So the k theta squared will be equal to omega squared 986 01:07:54,810 --> 01:08:00,110 v prime square, because this progressing wave is actually 987 01:08:00,110 --> 01:08:02,580 the transmitted wave, is actually 988 01:08:02,580 --> 01:08:06,680 traveling in a second medium. 989 01:08:10,620 --> 01:08:13,180 So look at what we have done here. 990 01:08:13,180 --> 01:08:16,270 So we have an incident wave. 991 01:08:16,270 --> 01:08:18,729 We will wonder, then, what is going to happen. 992 01:08:18,729 --> 01:08:21,569 Our physics intuition tells me that, you 993 01:08:21,569 --> 01:08:25,000 must get a refracting wave, oscillation frequency 994 01:08:25,000 --> 01:08:25,979 should be the same. 995 01:08:25,979 --> 01:08:27,600 Otherwise, as a function over time, 996 01:08:27,600 --> 01:08:30,930 you cannot match the left hand and right hand side. 997 01:08:30,930 --> 01:08:35,170 And you also get a transmitted wave. 998 01:08:35,170 --> 01:08:39,180 But I'm now in trouble because I have so many turns. 999 01:08:39,180 --> 01:08:42,090 I'm summing over alpha infinite number of turns, 1000 01:08:42,090 --> 01:08:46,220 and I don't know what will be the coefficient for the r alpha 1001 01:08:46,220 --> 01:08:48,770 and the tau beta, which are the transmission 1002 01:08:48,770 --> 01:08:52,380 coefficient and then refraction coefficients. 1003 01:08:52,380 --> 01:08:56,550 So what I need to do, as you might guess, 1004 01:08:56,550 --> 01:09:00,380 is to use the boundary condition. 1005 01:09:00,380 --> 01:09:02,770 So now I am writing down, already, 1006 01:09:02,770 --> 01:09:04,170 the general expression. 1007 01:09:04,170 --> 01:09:06,600 Now I'm going to use the boundary condition 1008 01:09:06,600 --> 01:09:11,910 to actually limit the choice of the possible k alpha and the k 1009 01:09:11,910 --> 01:09:12,810 beta. 1010 01:09:12,810 --> 01:09:14,710 What is actually the boundary condition? 1011 01:09:21,580 --> 01:09:28,729 The boundary conditions are that at x equal to 0-- 1012 01:09:28,729 --> 01:09:32,180 that's actually at the position of this line-- 1013 01:09:32,180 --> 01:09:34,335 the membranes doesn't break. 1014 01:09:37,750 --> 01:09:42,850 Otherwise, suddenly the membranes break, 1015 01:09:42,850 --> 01:09:46,485 and this is the end of the discussion, right? 1016 01:09:46,485 --> 01:09:48,430 Like, what we have done before, right? 1017 01:09:48,430 --> 01:09:50,229 So the membranes doesn't break, so 1018 01:09:50,229 --> 01:09:53,620 that the propagation can continue. 1019 01:09:53,620 --> 01:09:56,420 So what does that mean? 1020 01:09:56,420 --> 01:10:02,620 This means that if I evaluate Psi L and Psi 1021 01:10:02,620 --> 01:10:08,650 r at x equal to 0, Psi 0, y, t. 1022 01:10:11,500 --> 01:10:17,890 The left hand side will be equal to A exponential i, Ky times y 1023 01:10:17,890 --> 01:10:26,360 minus omega T, plus summing over all possible alpha, r alpha, 1024 01:10:26,360 --> 01:10:37,780 A exponential i, K alpha y times y, minus omega T. This 1025 01:10:37,780 --> 01:10:42,400 is the incident wave transmitted wave evaluated at the left hand 1026 01:10:42,400 --> 01:10:46,230 side, which is the upper formula. 1027 01:10:46,230 --> 01:10:48,990 And that will be equal to the right hand 1028 01:10:48,990 --> 01:10:52,320 side, which is containing only the transmitted wave. 1029 01:10:52,320 --> 01:10:55,150 So basically you have summing over 1030 01:10:55,150 --> 01:11:02,790 beta, tau beta, A exponential i, k beta, 1031 01:11:02,790 --> 01:11:08,440 y times r minus omega T. 1032 01:11:08,440 --> 01:11:13,650 And this expression, this boundary condition, 1033 01:11:13,650 --> 01:11:17,600 should hold true for all the possible y, 1034 01:11:17,600 --> 01:11:19,920 right, because the boundary condition 1035 01:11:19,920 --> 01:11:22,530 is valid at x equal to 0. 1036 01:11:22,530 --> 01:11:26,670 I didn't specify the value of y. 1037 01:11:26,670 --> 01:11:30,090 So therefore I can actually put in all the possible-- oh, 1038 01:11:30,090 --> 01:11:31,830 this should be y. 1039 01:11:31,830 --> 01:11:32,700 Sorry for that. 1040 01:11:32,700 --> 01:11:37,590 I can actually vary the y, and I will figure out that, ah, 1041 01:11:37,590 --> 01:11:42,680 if I have Ky not equal to k alpha y, that 1042 01:11:42,680 --> 01:11:46,920 means the wavelengths of the refractive wave 1043 01:11:46,920 --> 01:11:49,590 and the incident wave will be different. 1044 01:11:49,590 --> 01:11:54,750 If I have Ky not equal to beta y that means 1045 01:11:54,750 --> 01:11:57,150 the transmitted wavelengths is going 1046 01:11:57,150 --> 01:12:00,330 to be different from the incident wave. 1047 01:12:00,330 --> 01:12:04,110 That means, no matter what I do as a boundary of y, 1048 01:12:04,110 --> 01:12:05,455 the membranes will break. 1049 01:12:08,290 --> 01:12:13,840 Therefore, in order to make this equation valid, 1050 01:12:13,840 --> 01:12:18,810 the only choice is that when k alpha y 1051 01:12:18,810 --> 01:12:23,840 will be equal to k beta y and equal to Ky. 1052 01:12:23,840 --> 01:12:28,000 So that means the wavelengths projected in the y direction 1053 01:12:28,000 --> 01:12:33,550 should be equal for the incident wave, transmitted wave, 1054 01:12:33,550 --> 01:12:34,870 and the refractive wave. 1055 01:12:34,870 --> 01:12:39,390 Otherwise, as you always move a little bit in the y direction, 1056 01:12:39,390 --> 01:12:42,250 the membranes will break. 1057 01:12:42,250 --> 01:12:47,770 So that's actually the condition which you can actually get. 1058 01:12:47,770 --> 01:12:53,810 And the interesting thing is that, based on this expression, 1059 01:12:53,810 --> 01:12:57,130 k alpha, the length of the k alpha, 1060 01:12:57,130 --> 01:13:01,960 and the length of the k beta is fixed. 1061 01:13:01,960 --> 01:13:07,850 And I also know what will be the component for the y direction. 1062 01:13:07,850 --> 01:13:15,820 Therefore, that means the x direction Psi's for that k 1063 01:13:15,820 --> 01:13:21,310 alpha x and the k beta x are also fixed because 1064 01:13:21,310 --> 01:13:24,750 of the dispersion relation. 1065 01:13:24,750 --> 01:13:29,680 So that immediately brings me to this conclusion 1066 01:13:29,680 --> 01:13:37,660 that basically k alpha x will be equal to minus omega squared 1067 01:13:37,660 --> 01:13:41,230 over v squared minus Ky squared, and that 1068 01:13:41,230 --> 01:13:45,280 will be equal to minus Kx. 1069 01:13:45,280 --> 01:13:48,730 So this is the x component of the refractive wave. 1070 01:13:48,730 --> 01:13:51,520 And the transmitting wave, k beta x, 1071 01:13:51,520 --> 01:13:56,290 will be equal to square root of omega squared over v squared 1072 01:13:56,290 --> 01:13:59,470 minus Ky squared. 1073 01:13:59,470 --> 01:14:04,450 If I draw, visualize the relative direction 1074 01:14:04,450 --> 01:14:07,875 of all the three components, basically, this 1075 01:14:07,875 --> 01:14:13,840 is essentially the direction of the incident wave, k, 1076 01:14:13,840 --> 01:14:17,370 and the incident angle is theta. 1077 01:14:17,370 --> 01:14:20,410 And from this expression, you see that the Ky 1078 01:14:20,410 --> 01:14:23,380 is equal to k alpha y. 1079 01:14:23,380 --> 01:14:26,180 Therefore, you have a refractive wave. 1080 01:14:26,180 --> 01:14:28,910 But actually only the x direction has changed sides. 1081 01:14:28,910 --> 01:14:31,430 Therefore, you have a refractive wave 1082 01:14:31,430 --> 01:14:37,060 with exactly the same angle as the incident angle theta. 1083 01:14:37,060 --> 01:14:41,420 The refraction angle will be zeta as well. 1084 01:14:41,420 --> 01:14:44,380 And that there will be a transmitted wave 1085 01:14:44,380 --> 01:14:46,330 with theta prime. 1086 01:14:46,330 --> 01:14:51,430 And this is essentially the direction of the k prime. 1087 01:14:51,430 --> 01:14:57,580 And the interesting thing is that the projection 1088 01:14:57,580 --> 01:15:01,210 toward the y direction, that k prime y, 1089 01:15:01,210 --> 01:15:05,200 has to be equal to the progression 1090 01:15:05,200 --> 01:15:12,460 of the original incident wave in the y direction. 1091 01:15:12,460 --> 01:15:17,710 So that means I will be able to conclude that-- 1092 01:15:17,710 --> 01:15:19,400 the y components are the same. 1093 01:15:19,400 --> 01:15:23,350 Therefore, I can conclude that k sine 1094 01:15:23,350 --> 01:15:28,860 theta will be equal to k prime sine theta prime. 1095 01:15:31,610 --> 01:15:34,000 I'm kind of running out of time. 1096 01:15:34,000 --> 01:15:38,980 And if I define, already as I defined here, 1097 01:15:38,980 --> 01:15:41,980 velocity is equal to c over n, and the v 1098 01:15:41,980 --> 01:15:46,840 prime is equal to c over n prime, what I can immediately 1099 01:15:46,840 --> 01:15:49,870 conclude is that-- 1100 01:15:49,870 --> 01:15:51,910 give me one more minute-- 1101 01:15:51,910 --> 01:15:57,760 is that if I have n equal to c over v and the n 1102 01:15:57,760 --> 01:16:00,370 prime is equal to c over v prime, 1103 01:16:00,370 --> 01:16:03,940 I can conclude that sine theta will 1104 01:16:03,940 --> 01:16:08,280 be equal to n prime sine theta prime. 1105 01:16:08,280 --> 01:16:12,520 Does this look familiar to you? 1106 01:16:12,520 --> 01:16:15,610 This is essentially Snell's law. 1107 01:16:15,610 --> 01:16:19,810 How many of you haven't heard about Snell's law. 1108 01:16:19,810 --> 01:16:21,250 There were a few before. 1109 01:16:21,250 --> 01:16:22,590 Yeah, OK. 1110 01:16:22,590 --> 01:16:23,900 No problem at all. 1111 01:16:23,900 --> 01:16:25,370 Then you learned it. 1112 01:16:25,370 --> 01:16:31,310 So that means if I have two kinds of systems in my hand, 1113 01:16:31,310 --> 01:16:38,000 and I will be able to relate the transmitted wave according 1114 01:16:38,000 --> 01:16:40,370 to what I have in the incident wave. 1115 01:16:40,370 --> 01:16:43,430 And you can see that Snell's law-- 1116 01:16:43,430 --> 01:16:49,100 which were famous for the discussion of optics-- 1117 01:16:49,100 --> 01:16:53,270 and here, I have no knowledge about optics 1118 01:16:53,270 --> 01:16:55,520 or electromagnetic waves. 1119 01:16:55,520 --> 01:16:59,810 So in short, what I want to tell you is that, we have just 1120 01:16:59,810 --> 01:17:07,730 proved two of the most important laws of the geometrical optics, 1121 01:17:07,730 --> 01:17:12,680 the refraction angle is equal to incident angle and the Snell's 1122 01:17:12,680 --> 01:17:16,520 law without using any information about the dynamics. 1123 01:17:16,520 --> 01:17:21,350 That means all those laws are coming from purely boundary 1124 01:17:21,350 --> 01:17:23,850 condition and the waves. 1125 01:17:23,850 --> 01:17:26,700 Therefore, you will expect that this 1126 01:17:26,700 --> 01:17:31,370 will work for water wave, sound wave, electromagnetic wave, et 1127 01:17:31,370 --> 01:17:31,870 cetera. 1128 01:17:31,870 --> 01:17:34,900 O So we will continue the discussion next time. 1129 01:17:34,900 --> 01:17:37,860 Thanks for the attention. 1130 01:17:37,860 --> 01:17:40,496 And if you have any questions, let me know. 1131 01:17:40,496 --> 01:17:41,120 I will be here. 1132 01:17:46,940 --> 01:17:48,020 Hello, everybody. 1133 01:17:48,020 --> 01:17:50,570 We are going to show you a demonstration, a really 1134 01:17:50,570 --> 01:17:52,790 nice one. 1135 01:17:52,790 --> 01:17:56,040 It consists of the following setup. 1136 01:17:56,040 --> 01:17:59,920 So basically I'm going to place some film here. 1137 01:17:59,920 --> 01:18:04,190 And then behind that there's a loud speaker, 1138 01:18:04,190 --> 01:18:08,540 which I use as a signal generator and to actually 1139 01:18:08,540 --> 01:18:09,770 produce sound wave. 1140 01:18:09,770 --> 01:18:13,850 And this sound wave is going to oscillate the soft film, 1141 01:18:13,850 --> 01:18:17,490 and then you are going to see the oscillation, 1142 01:18:17,490 --> 01:18:21,470 or the normal mode's pattern, on the screen. 1143 01:18:21,470 --> 01:18:25,130 OK, so that is actually the setup 1144 01:18:25,130 --> 01:18:27,650 which we can actually demonstrate to you two 1145 01:18:27,650 --> 01:18:31,020 dimensional normal modes. 1146 01:18:31,020 --> 01:18:33,170 So the first thing which I am going to do 1147 01:18:33,170 --> 01:18:35,676 is to produce a soft film. 1148 01:18:38,430 --> 01:18:43,560 Now I am going to put it back into this setup here. 1149 01:18:43,560 --> 01:18:46,218 You should be able to see the pattern on the screen. 1150 01:18:50,800 --> 01:18:55,470 Then I am going to turn on the sound wave generator. 1151 01:18:55,470 --> 01:19:01,110 You can see, immediately, that the pattern on the screen 1152 01:19:01,110 --> 01:19:03,960 changed because of the sound wave trying 1153 01:19:03,960 --> 01:19:07,080 to oscillate the soft film. 1154 01:19:07,080 --> 01:19:09,660 You can see it directly from here, 1155 01:19:09,660 --> 01:19:16,230 but it actually looks much more prominent on the screen. 1156 01:19:16,230 --> 01:19:22,080 And now what am I going to do is change the frequency 1157 01:19:22,080 --> 01:19:23,400 of the sound wave. 1158 01:19:23,400 --> 01:19:25,350 And you can see that I'm changing it 1159 01:19:25,350 --> 01:19:26,730 to a higher frequency. 1160 01:19:29,460 --> 01:19:32,940 And you can see that there is a more and more complicated 1161 01:19:32,940 --> 01:19:36,080 pattern formed on the screen. 1162 01:19:36,080 --> 01:19:42,560 That is because I'm now exciting higher and higher frequency 1163 01:19:42,560 --> 01:19:43,480 normal modes. 1164 01:19:46,930 --> 01:19:51,160 And you can see that now I can actually continue and increase 1165 01:19:51,160 --> 01:19:56,424 the frequency. 1166 01:19:56,424 --> 01:19:57,340 And you can see that-- 1167 01:20:01,480 --> 01:20:04,620 now we can see that the pattern becomes really, 1168 01:20:04,620 --> 01:20:07,680 really infinitely complicated. 1169 01:20:07,680 --> 01:20:10,620 You can see this grid developing. 1170 01:20:10,620 --> 01:20:14,460 And then you can see that eventually that's 1171 01:20:14,460 --> 01:20:18,330 basically two sine functions multiply each other. 1172 01:20:18,330 --> 01:20:20,405 One sine function is in the x direction. 1173 01:20:20,405 --> 01:20:23,580 The other one is in the y direction-- horizontal 1174 01:20:23,580 --> 01:20:25,300 and the vertical direction. 1175 01:20:25,300 --> 01:20:28,230 And you can see a really beautiful pattern 1176 01:20:28,230 --> 01:20:34,320 forming due to the solution we derived during the lecture. 1177 01:20:34,320 --> 01:20:40,580 And the higher frequency I go, I can see more and more 1178 01:20:40,580 --> 01:20:46,800 complicating patterns, many more lines 1179 01:20:46,800 --> 01:20:53,690 developing on the screen due to oscillation of the soft film. 1180 01:20:53,690 --> 01:20:55,810 You can see now we have even more lines. 1181 01:20:59,136 --> 01:21:00,760 And it's actually getting more and more 1182 01:21:00,760 --> 01:21:04,075 difficult to see the pattern because now the lines are 1183 01:21:04,075 --> 01:21:05,200 really close to each other. 1184 01:21:05,200 --> 01:21:09,846 The nodal line, we can see clearly on the screen. 1185 01:21:12,714 --> 01:21:17,270 Now I am going back down to a lower frequency, 1186 01:21:17,270 --> 01:21:20,660 and you can see that at low frequency oscillation, 1187 01:21:20,660 --> 01:21:24,650 the number of lines is actually smaller, 1188 01:21:24,650 --> 01:21:29,270 and that is because of the smaller oscillation 1189 01:21:29,270 --> 01:21:33,640 frequency and the longer wavelengths 1190 01:21:33,640 --> 01:21:36,090 of the normal modes.