1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,850 Commons license. 3 00:00:03,850 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,650 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,650 --> 00:00:18,012 at ocw.mit.edu. 8 00:00:21,363 --> 00:00:23,280 GEORGE BARBASTATHIS: OK, so let's get started. 9 00:00:28,870 --> 00:00:30,600 I'm just trying to wire myself up. 10 00:00:30,600 --> 00:00:31,600 OK. 11 00:00:31,600 --> 00:00:39,600 So before we get into the PowerPoint here. 12 00:00:39,600 --> 00:00:42,620 So basically, what I'm doing is I'm picking up from the point 13 00:00:42,620 --> 00:00:47,570 where we left last time, where we're discussing waves. 14 00:00:47,570 --> 00:00:53,900 And what I would like to do now is derive a wave equation 15 00:00:53,900 --> 00:00:58,760 basically, for a mathematical form 16 00:00:58,760 --> 00:01:02,930 that should allow us to predict different kinds of waves. 17 00:01:02,930 --> 00:01:05,459 Of waves, not the waves. 18 00:01:05,459 --> 00:01:08,930 All right, so what I start with here. 19 00:01:08,930 --> 00:01:15,060 I start with a general function, psi, of an argument, eta. 20 00:01:15,060 --> 00:01:24,080 So for those of you who are not fluent in Greek, this is psi, 21 00:01:24,080 --> 00:01:25,840 and this is eta. 22 00:01:25,840 --> 00:01:26,340 OK. 23 00:01:30,850 --> 00:01:32,680 At the moment, these don't necessarily 24 00:01:32,680 --> 00:01:35,290 have a physical meaning. 25 00:01:35,290 --> 00:01:39,190 But in order to produce a wave out 26 00:01:39,190 --> 00:01:45,310 of this function, what I can do is 27 00:01:45,310 --> 00:01:56,640 I can write it as psi, or f of z comma t, 28 00:01:56,640 --> 00:02:02,070 equals psi of z minus ct. 29 00:02:02,070 --> 00:02:05,910 OK, now why is that a wave? 30 00:02:05,910 --> 00:02:11,220 First of all, z now has become a spacial coordinate. 31 00:02:11,220 --> 00:02:13,980 t, as usual, is time. 32 00:02:13,980 --> 00:02:16,650 So if you actually hide time. 33 00:02:16,650 --> 00:02:18,660 So now time is zero. 34 00:02:18,660 --> 00:02:20,760 Time is a constant or something like that. 35 00:02:20,760 --> 00:02:24,330 Then you actually have basically the same argument. 36 00:02:24,330 --> 00:02:27,060 But if you allow time to increase, 37 00:02:27,060 --> 00:02:30,360 then you can see that the time increase will 38 00:02:30,360 --> 00:02:35,060 have the effect of sliding the same shape 39 00:02:35,060 --> 00:02:37,350 along the spatial axis. 40 00:02:37,350 --> 00:02:43,560 So for example, let's take this as our reference point here. 41 00:02:43,560 --> 00:02:46,440 The first peak after you advance time 42 00:02:46,440 --> 00:02:50,330 a little further than this point has moved fast, 43 00:02:50,330 --> 00:02:51,970 has moved to the right. 44 00:02:51,970 --> 00:02:55,780 And how much is this distance? 45 00:02:55,780 --> 00:02:58,440 Well, anybody? 46 00:02:58,440 --> 00:02:59,700 This is easy. 47 00:02:59,700 --> 00:03:03,210 I'm just trying to see if the people in Boston have woken up. 48 00:03:11,950 --> 00:03:12,450 Yes? 49 00:03:15,800 --> 00:03:18,020 Nobody's awake in Boston. 50 00:03:18,020 --> 00:03:20,630 So this distance equals ct. 51 00:03:20,630 --> 00:03:25,910 Because if I set the argument of this function 52 00:03:25,910 --> 00:03:29,250 to zero, then basically I recover my original function. 53 00:03:29,250 --> 00:03:32,510 So basically as I advance time, the fact 54 00:03:32,510 --> 00:03:35,010 is that this waveform moves to the right. 55 00:03:41,000 --> 00:03:43,770 And at the moment, there's no obvious motivation 56 00:03:43,770 --> 00:03:47,620 for why I'm going to do what I'm about to do. 57 00:03:47,620 --> 00:03:50,730 But please take my word for it for a while. 58 00:03:50,730 --> 00:03:52,620 I'm going to take two partial derivatives. 59 00:03:52,620 --> 00:04:01,560 I'm going to take df dz, and df dt. 60 00:04:01,560 --> 00:04:09,420 OK, df dz equals basically c prime, where c prime-- 61 00:04:09,420 --> 00:04:14,800 I use c prime simply to denote the derivative of c 62 00:04:14,800 --> 00:04:18,459 with respect to its mysterious argument. 63 00:04:18,459 --> 00:04:20,200 So that's very easy. 64 00:04:20,200 --> 00:04:21,579 What about the affinity? 65 00:04:31,058 --> 00:04:31,600 That's right. 66 00:04:31,600 --> 00:04:33,870 I have to apply the chain rule, because there 67 00:04:33,870 --> 00:04:36,390 is a multiplicative argument inside. 68 00:04:36,390 --> 00:04:42,360 So this is going to be minus c times c prime. 69 00:04:42,360 --> 00:04:47,880 So because c prime appears in both places over here, 70 00:04:47,880 --> 00:04:56,460 I can combine these equations and say, df over dz 71 00:04:56,460 --> 00:05:02,700 equals minus 1 over c, df dt. 72 00:05:05,610 --> 00:05:07,760 So that is a partial differential equation. 73 00:05:07,760 --> 00:05:10,720 And you might say, well, what I did is a bit stupid, right? 74 00:05:10,720 --> 00:05:13,540 Because I had a relatively simple explanation 75 00:05:13,540 --> 00:05:14,290 for the wave. 76 00:05:14,290 --> 00:05:17,110 I actually had the wave just in front of me. 77 00:05:17,110 --> 00:05:20,200 And what I did is I wrote a partial differential equation 78 00:05:20,200 --> 00:05:21,907 which is not very intuitive. 79 00:05:21,907 --> 00:05:23,740 But the benefit of doing that is, of course, 80 00:05:23,740 --> 00:05:31,060 that the wave that I wrote here is a very special, very limited 81 00:05:31,060 --> 00:05:34,420 kind of wave that propagates without changing shape, 82 00:05:34,420 --> 00:05:36,940 without anything fancy happening. 83 00:05:36,940 --> 00:05:39,220 We know from experience, for example, in real life, 84 00:05:39,220 --> 00:05:41,780 if you have ever seen water waves. 85 00:05:41,780 --> 00:05:44,290 Water waves, they don't quite propagate. 86 00:05:44,290 --> 00:05:48,700 They change shape in all kinds of complicated ways. 87 00:05:48,700 --> 00:05:51,820 So even though I derive this equation 88 00:05:51,820 --> 00:05:53,355 from that simple shape. 89 00:05:53,355 --> 00:05:54,730 What we'll see a little bit later 90 00:05:54,730 --> 00:05:57,100 that this equation actually can describe 91 00:05:57,100 --> 00:05:59,713 much more general waves. 92 00:05:59,713 --> 00:06:01,630 We're not quite done yet, because the question 93 00:06:01,630 --> 00:06:07,940 that I derived here applies for a wave moving to the right. 94 00:06:07,940 --> 00:06:10,420 I can equally well contribute a wave 95 00:06:10,420 --> 00:06:12,150 that is moving to the left. 96 00:06:12,150 --> 00:06:14,410 And in this case, what I would have to do 97 00:06:14,410 --> 00:06:19,890 is I would have to write f of z comma t equals psi 98 00:06:19,890 --> 00:06:21,825 or z plus ct. 99 00:06:21,825 --> 00:06:23,700 A little bit of thought will convince you now 100 00:06:23,700 --> 00:06:26,280 that if you advance time, then the wave 101 00:06:26,280 --> 00:06:29,460 will have actually to move to the left, towards negative z 102 00:06:29,460 --> 00:06:33,170 in order to maintain its argument. 103 00:06:33,170 --> 00:06:36,300 And if I do the same game now. 104 00:06:36,300 --> 00:06:40,420 Again, I can write df dz equals c prime. 105 00:06:40,420 --> 00:06:41,790 That did not change. 106 00:06:41,790 --> 00:06:47,160 This time, df dt equals plus c prime. 107 00:06:47,160 --> 00:06:51,420 Now, I have another partial differential equation, 108 00:06:51,420 --> 00:06:59,160 df dz equals 1 over c, df dt. 109 00:06:59,160 --> 00:07:01,540 OK, let me put back the equation. 110 00:07:01,540 --> 00:07:11,200 So this is a wave going to the left, backwards. 111 00:07:11,200 --> 00:07:17,870 And I had another equation, df dz equals minus 1 112 00:07:17,870 --> 00:07:20,790 over c, df dt. 113 00:07:20,790 --> 00:07:23,210 And that is a way of going to the right. 114 00:07:28,230 --> 00:07:29,260 OK. 115 00:07:29,260 --> 00:07:32,980 You can see very easily that if you take a second derivative 116 00:07:32,980 --> 00:07:35,350 of these equations. 117 00:07:35,350 --> 00:07:39,000 Let's go back to the first one. 118 00:07:39,000 --> 00:07:49,530 If you do the second derivative, d squared f dz square. 119 00:07:49,530 --> 00:07:53,140 You will pick up an extra minus sign from the same equation. 120 00:07:53,140 --> 00:07:57,520 And you will find that it is 1 over c square, d square f, 121 00:07:57,520 --> 00:08:00,980 dt square. 122 00:08:00,980 --> 00:08:03,275 You can do the same in this equation. 123 00:08:08,530 --> 00:08:10,610 There's no sign here to begin with. 124 00:08:10,610 --> 00:08:12,770 But there is one over here, c square, 125 00:08:12,770 --> 00:08:15,150 d squared f, dt squared. 126 00:08:15,150 --> 00:08:16,640 So basically, this single equation. 127 00:08:28,630 --> 00:08:32,530 This single equation can describe both forward 128 00:08:32,530 --> 00:08:37,210 and backward propagating waves. 129 00:08:37,210 --> 00:08:39,730 And this equation. 130 00:08:39,730 --> 00:08:41,679 Actually, it's a three dimensional version. 131 00:08:41,679 --> 00:08:44,770 Not this one, but it's three dimensional version. 132 00:08:44,770 --> 00:08:52,415 It has the name Helmholtz wave equation. 133 00:08:56,690 --> 00:09:00,430 And because of Helmholtz, it's a little bit difficult 134 00:09:00,430 --> 00:09:02,560 to pronounce. 135 00:09:02,560 --> 00:09:04,240 Actually, that's not the real one. 136 00:09:04,240 --> 00:09:07,630 Anyway, it's a very commonly encountered wave equation. 137 00:09:07,630 --> 00:09:10,690 So typically we simply omit Helmholtz, 138 00:09:10,690 --> 00:09:12,710 and we say wave equation. 139 00:09:12,710 --> 00:09:17,400 Now, why you might want to distinguish this as opposed to 140 00:09:17,400 --> 00:09:19,210 are there other wave equations? 141 00:09:19,210 --> 00:09:22,210 Yes, there is the Schrodinger equation. 142 00:09:22,210 --> 00:09:26,410 There is the Klein-Gordon equation. 143 00:09:26,410 --> 00:09:28,150 For those of you who do fluid mechanics, 144 00:09:28,150 --> 00:09:32,110 there is a Korteweg-de Vries equation, also known as KDV. 145 00:09:32,110 --> 00:09:34,960 There's a bunch of different wave equations. 146 00:09:34,960 --> 00:09:38,170 So we need some way to discriminate. 147 00:09:38,170 --> 00:09:40,563 Nevertheless, the other wave equations, 148 00:09:40,563 --> 00:09:41,980 they are referred with their name. 149 00:09:41,980 --> 00:09:44,500 So when you want to refer to Schrodinger's equation, 150 00:09:44,500 --> 00:09:46,860 you say Schrodinger's equation. 151 00:09:46,860 --> 00:09:49,150 By convention, when you say wave equation, 152 00:09:49,150 --> 00:09:51,610 you mean this one, so the Helmholtz. 153 00:09:51,610 --> 00:09:53,950 OK. 154 00:09:53,950 --> 00:09:56,930 So this is all very nice, but it's actually 155 00:09:56,930 --> 00:09:58,460 down on your slide. 156 00:09:58,460 --> 00:10:04,670 So there's no reason to agonize over what I did over here. 157 00:10:08,070 --> 00:10:10,480 What I would like to remind you before I 158 00:10:10,480 --> 00:10:15,430 move on is for the simple case of a-- 159 00:10:15,430 --> 00:10:20,070 first of all, you can imagine that this 160 00:10:20,070 --> 00:10:22,530 is a very general kind of solution. 161 00:10:22,530 --> 00:10:23,790 Because what is c? 162 00:10:23,790 --> 00:10:29,250 After all, I put up this strange shape over here. 163 00:10:29,250 --> 00:10:30,670 I didn't say where it came from. 164 00:10:30,670 --> 00:10:33,930 So for that matter, if someone gives you the wave question, 165 00:10:33,930 --> 00:10:36,300 how can you go ahead and solve it? 166 00:10:36,300 --> 00:10:40,080 Well, we're not going to do this in great detail here. 167 00:10:40,080 --> 00:10:42,750 This is a little bit beside the scope of the class. 168 00:10:42,750 --> 00:10:46,500 But generally, if you need to solve a partial differential 169 00:10:46,500 --> 00:10:48,900 like this one, you need additional information. 170 00:10:48,900 --> 00:10:50,970 You need the initial conditions. 171 00:10:50,970 --> 00:10:54,030 How did the wave start at that time equals zero? 172 00:10:54,030 --> 00:10:55,860 And you also need the boundary conditions. 173 00:10:55,860 --> 00:10:59,130 So what happens to the wave as it expands? 174 00:10:59,130 --> 00:11:01,200 Sometimes, the boundary condition is at infinity. 175 00:11:01,200 --> 00:11:02,760 Maybe the wave is not bounded. 176 00:11:02,760 --> 00:11:05,705 It can propagate outwards all the way. 177 00:11:05,705 --> 00:11:07,080 And that's a good example of what 178 00:11:07,080 --> 00:11:13,600 happens if you are flying with a helicopter on top of the ocean. 179 00:11:13,600 --> 00:11:15,048 And you drop a-- 180 00:11:15,048 --> 00:11:15,590 I don't know. 181 00:11:15,590 --> 00:11:18,240 You drop a stone into the surface of the ocean. 182 00:11:18,240 --> 00:11:23,800 Then the wave that you generate, the wave that you generate. 183 00:11:23,800 --> 00:11:26,360 To a good approximation, you can use infinite boundary 184 00:11:26,360 --> 00:11:27,110 conditions. 185 00:11:27,110 --> 00:11:31,990 Because the ocean is not quite infinite, but very large. 186 00:11:31,990 --> 00:11:37,130 If you generate in a wave in a small channel, 187 00:11:37,130 --> 00:11:41,060 then you might have to include boundary conditions. 188 00:11:41,060 --> 00:11:42,770 Because the wave will be confined 189 00:11:42,770 --> 00:11:44,450 by the walls of the channel. 190 00:11:44,450 --> 00:11:46,490 And we will see later that this is actually 191 00:11:46,490 --> 00:11:48,320 quite commonly the case. 192 00:11:48,320 --> 00:11:50,040 In optics, we don't call them channels. 193 00:11:50,040 --> 00:11:52,220 We'll call them wave guides. 194 00:11:52,220 --> 00:11:56,810 But the boundary conditions can actually significantly alter 195 00:11:56,810 --> 00:12:00,400 the shape of the wave. 196 00:12:00,400 --> 00:12:04,230 So what I did here is just to give you an idea, 197 00:12:04,230 --> 00:12:08,070 I quote unquote, "solved." 198 00:12:08,070 --> 00:12:09,660 Nothing interesting happens here, 199 00:12:09,660 --> 00:12:12,720 but this is actually solution of the wave 200 00:12:12,720 --> 00:12:17,850 equation with a particular initial condition, where 201 00:12:17,850 --> 00:12:19,560 time equals zero. 202 00:12:19,560 --> 00:12:23,250 The wave is a sinusoid. 203 00:12:23,250 --> 00:12:26,320 I said that this is my equation. 204 00:12:26,320 --> 00:12:31,200 And I say that f of z at t equals 0, 205 00:12:31,200 --> 00:12:39,750 equals some sinusoid amplitude a, and wave number k. 206 00:12:39,750 --> 00:12:43,670 If that is the case, then by inspection. 207 00:12:43,670 --> 00:12:45,000 I don't have to do anything. 208 00:12:45,000 --> 00:12:46,450 All very important. 209 00:12:46,450 --> 00:12:50,290 And what I did not write down, but it is implied, 210 00:12:50,290 --> 00:12:52,480 is that the boundary conditions are free. 211 00:12:52,480 --> 00:12:54,790 So this wave is free to go as far 212 00:12:54,790 --> 00:13:00,350 as it pleases without any special constraints. 213 00:13:00,350 --> 00:13:05,410 So if that is the case, then you can write the solution almost 214 00:13:05,410 --> 00:13:06,460 by inspection. 215 00:13:06,460 --> 00:13:15,250 Half of z comma t equals a cosine kz minus ct. 216 00:13:15,250 --> 00:13:17,940 And you can see by inspection here that because of the way 217 00:13:17,940 --> 00:13:20,230 I wrote it, this actually satisfies 218 00:13:20,230 --> 00:13:23,740 both the partial differential equation, the wave equation, 219 00:13:23,740 --> 00:13:25,250 and the boundary condition. 220 00:13:25,250 --> 00:13:30,560 So therefore, this is a proper solution of the wave equation. 221 00:13:30,560 --> 00:13:32,740 In fact, I only wrote half of it. 222 00:13:32,740 --> 00:13:38,170 The wave, remember, it can go forwards, and can go backwards. 223 00:13:38,170 --> 00:13:44,540 To cover both cases, I actually write the solution this way. 224 00:13:44,540 --> 00:13:46,470 With the data that I have here, actually, I 225 00:13:46,470 --> 00:13:48,780 don't have enough to specify. 226 00:13:48,780 --> 00:13:51,880 Or actually, in principle, the wave can go both ways. 227 00:13:51,880 --> 00:13:54,900 I would have to apply to give you additional information 228 00:13:54,900 --> 00:13:58,920 to know which actually happened. 229 00:13:58,920 --> 00:14:01,200 The other thing that I wanted to point out, 230 00:14:01,200 --> 00:14:05,190 which is not really difficult, but it takes some getting used 231 00:14:05,190 --> 00:14:08,190 to, is the various terms that appear 232 00:14:08,190 --> 00:14:11,740 inside this equation over here. 233 00:14:11,740 --> 00:14:14,850 So the term that we have been very familiar with 234 00:14:14,850 --> 00:14:16,670 is the wavelength. 235 00:14:16,670 --> 00:14:19,620 There's another of those crazy Greek symbols. 236 00:14:19,620 --> 00:14:21,420 It's called lambda. 237 00:14:21,420 --> 00:14:24,960 And the wavelength is related to the quantity 238 00:14:24,960 --> 00:14:32,430 that we have over here, as 2 pi over lambda equals k. 239 00:14:32,430 --> 00:14:34,980 So lambda is the wavelength, as I said. 240 00:14:34,980 --> 00:14:39,280 K is known as the wave number. 241 00:14:39,280 --> 00:14:42,960 So the relationship that they have 242 00:14:42,960 --> 00:14:45,510 is if you think of the wavelength as a period-- 243 00:14:45,510 --> 00:14:48,570 which it is, the period of the wave in the space domain-- 244 00:14:48,570 --> 00:14:55,620 then k would actually be the angular frequency. 245 00:14:55,620 --> 00:14:59,550 But we'll call it wave number to keep them separated. 246 00:14:59,550 --> 00:15:00,570 OK. 247 00:15:00,570 --> 00:15:05,010 Equivalently, you can define the actual angular frequency 248 00:15:05,010 --> 00:15:07,760 of the wave. 249 00:15:07,760 --> 00:15:11,300 Usually this goes by the symbol omega. 250 00:15:11,300 --> 00:15:13,550 And from this equation over here, 251 00:15:13,550 --> 00:15:16,550 very simply you can see that if you wanted to write it 252 00:15:16,550 --> 00:15:25,210 in the form a cosine kz minus omega t, then clearly, 253 00:15:25,210 --> 00:15:27,058 omega is the angular frequency. 254 00:15:27,058 --> 00:15:28,600 And when comparing the two equations, 255 00:15:28,600 --> 00:15:31,420 you can see that omega equals kc. 256 00:15:31,420 --> 00:15:34,330 So the angular frequency, then, is 257 00:15:34,330 --> 00:15:45,060 related to the spatial frequency, the wave number, 258 00:15:45,060 --> 00:15:46,960 and the speed of the wave. 259 00:15:46,960 --> 00:15:48,460 And of course, this is nothing new. 260 00:15:51,430 --> 00:15:54,407 We've seen this equation before, but we saw it in disguise. 261 00:15:54,407 --> 00:15:55,990 And that's why I'm going through this. 262 00:15:55,990 --> 00:15:58,780 Because all the symbols, they can be played around 263 00:15:58,780 --> 00:16:00,220 with quite a bit. 264 00:16:00,220 --> 00:16:03,040 So this equation, I can rewrite. 265 00:16:03,040 --> 00:16:04,690 And the [INAUDIBLE] is like this. 266 00:16:04,690 --> 00:16:07,360 So omega is the angular frequency. 267 00:16:07,360 --> 00:16:11,110 I also have the plane, so to speak. 268 00:16:11,110 --> 00:16:16,830 Frequency nu, which is related as omega over 2 pi. 269 00:16:16,830 --> 00:16:20,910 One of them goes [INAUDIBLE] radians. 270 00:16:20,910 --> 00:16:25,060 I'm sorry, it's measured in full radians per second. 271 00:16:25,060 --> 00:16:26,040 This is omega. 272 00:16:26,040 --> 00:16:29,920 Nu is measured in hertz, simply inverse seconds. 273 00:16:29,920 --> 00:16:32,600 Bless you. 274 00:16:32,600 --> 00:16:34,350 So I can substitute in this equation. 275 00:16:34,350 --> 00:16:37,510 Omega is 2 pi times nu. 276 00:16:37,510 --> 00:16:41,820 k, according to my equation over here, is 2 pi up on lambda. 277 00:16:41,820 --> 00:16:43,560 And c is c. 278 00:16:43,560 --> 00:16:45,360 So the two pis cancel here. 279 00:16:49,120 --> 00:16:51,830 The two pis cancel here, and I end up with equation c 280 00:16:51,830 --> 00:16:54,190 equals lambda nu, which I think we've already 281 00:16:54,190 --> 00:16:55,760 seen a couple of times before. 282 00:16:55,760 --> 00:16:59,120 Who call it the dispersion relation of the wave. 283 00:16:59,120 --> 00:17:01,550 So this is how all this came about. 284 00:17:01,550 --> 00:17:05,829 I haven't said anything new about this. 285 00:17:05,829 --> 00:17:09,680 I'm just trying to point out the different ways 286 00:17:09,680 --> 00:17:13,460 that this equation can manifest itself. 287 00:17:17,680 --> 00:17:21,310 And the other two terms that we encountered last time, 288 00:17:21,310 --> 00:17:22,829 and I wanted to remind you. 289 00:17:22,829 --> 00:17:25,910 One is the amplitude of the wave. 290 00:17:25,910 --> 00:17:29,710 This is this constant a over here. 291 00:17:29,710 --> 00:17:33,590 For some reason, in the slide, they decided to call it a0. 292 00:17:33,590 --> 00:17:35,590 Anyway, it is the same thing. 293 00:17:35,590 --> 00:17:40,360 And also, so we said last time that in general, the wave 294 00:17:40,360 --> 00:17:42,160 can also have a phase delay. 295 00:17:42,160 --> 00:17:46,180 So in here, I can not cite the constant plus 296 00:17:46,180 --> 00:17:50,060 fee, which does not really do anything by itself. 297 00:17:50,060 --> 00:17:53,140 All it does if you wish equivalently, 298 00:17:53,140 --> 00:17:56,840 it shifts the time origin. 299 00:17:56,840 --> 00:17:58,580 So that's a relatively trivial thing 300 00:17:58,580 --> 00:18:01,990 because after all, I am free to start my clock whenever I want. 301 00:18:01,990 --> 00:18:04,330 If I have a wave that never changes, 302 00:18:04,330 --> 00:18:06,048 I can start my clock whenever I want. 303 00:18:06,048 --> 00:18:07,840 So it doesn't really make a big difference. 304 00:18:07,840 --> 00:18:11,590 But we saw last time that if you have two waves that are, 305 00:18:11,590 --> 00:18:13,960 as we said, interference-- or they're simultaneously 306 00:18:13,960 --> 00:18:15,430 happening, so to speak-- 307 00:18:15,430 --> 00:18:19,750 then the relative phase delay between the two 308 00:18:19,750 --> 00:18:21,520 can actually be quite significant. 309 00:18:21,520 --> 00:18:24,010 And I will play this movie again. 310 00:18:24,010 --> 00:18:26,575 In this case, the two waves are actually-- 311 00:18:26,575 --> 00:18:29,930 they have the same phase between them. 312 00:18:29,930 --> 00:18:32,160 In other words, if you look at the center, 313 00:18:32,160 --> 00:18:36,010 they're both simultaneously bright or simultaneously dark. 314 00:18:36,010 --> 00:18:38,690 Whereas in the other case, it's the other way around. 315 00:18:38,690 --> 00:18:40,070 They're off by pi. 316 00:18:40,070 --> 00:18:42,908 So when one is maximum, the other is minimum. 317 00:18:42,908 --> 00:18:44,450 I think it is worth playing this one. 318 00:18:48,990 --> 00:18:50,700 This also has the interesting effect 319 00:18:50,700 --> 00:18:52,527 that this can hypnotize you. 320 00:18:52,527 --> 00:18:55,110 So for those of you who are in Boston and not fully awake yet. 321 00:18:55,110 --> 00:18:55,890 Yes. 322 00:18:55,890 --> 00:18:58,390 Oh, so when you ask a question, you need to push the button. 323 00:19:04,835 --> 00:19:06,210 AUDIENCE: So for them for now are 324 00:19:06,210 --> 00:19:08,970 two waves travel in the opposite direction. 325 00:19:08,970 --> 00:19:12,910 They have same amplitude, about perfect cancel each other. 326 00:19:12,910 --> 00:19:15,825 So in this case, where does the energy go? 327 00:19:15,825 --> 00:19:18,450 GEORGE BARBASTATHIS: They don't quite cancel each other, right? 328 00:19:18,450 --> 00:19:21,390 So as we will see I will actually do this in a second. 329 00:19:21,390 --> 00:19:22,935 It's called a standing wave. 330 00:19:22,935 --> 00:19:24,810 So you will see that they don't quite cancel. 331 00:19:24,810 --> 00:19:28,720 They just produce a, well, what is called a standing wave. 332 00:19:28,720 --> 00:19:31,493 So I will get to your question. 333 00:19:31,493 --> 00:19:33,035 I think it's actually coming up next. 334 00:19:38,370 --> 00:19:41,060 Yeah, so give me a second and I will get to your question. 335 00:19:46,920 --> 00:19:48,330 So that is a standing wave. 336 00:19:48,330 --> 00:19:51,560 So like your colleague asked over here. 337 00:19:51,560 --> 00:19:55,800 Suppose that you have a superposition of two waves. 338 00:19:55,800 --> 00:19:58,440 Now, both are solutions to the wave equation. 339 00:19:58,440 --> 00:20:01,150 Now, by truce. 340 00:20:01,150 --> 00:20:02,820 I don't want to do the proof here. 341 00:20:02,820 --> 00:20:04,290 It takes three lines. 342 00:20:04,290 --> 00:20:09,130 But by its looks, this is a linear differential equation. 343 00:20:09,130 --> 00:20:12,990 So basically, it means that if you know one solution, 344 00:20:12,990 --> 00:20:15,090 you know another solution. 345 00:20:15,090 --> 00:20:17,340 And then you add these two solutions 346 00:20:17,340 --> 00:20:20,190 after multiplying by arbitrary constants. 347 00:20:20,190 --> 00:20:22,650 The result from the addition is still 348 00:20:22,650 --> 00:20:24,510 a solution to this equation. 349 00:20:24,510 --> 00:20:28,740 A very fundamental property that we use again and again later 350 00:20:28,740 --> 00:20:30,330 in the class. 351 00:20:30,330 --> 00:20:35,310 So that said, suppose that I pick two solutions 352 00:20:35,310 --> 00:20:36,290 that I already saw. 353 00:20:36,290 --> 00:20:38,400 One of them is a forward propagating wave, 354 00:20:38,400 --> 00:20:40,680 the other is a backward propagating wave. 355 00:20:40,680 --> 00:20:41,810 So here they are. 356 00:20:41,810 --> 00:20:44,040 The one with the minus is propagating forward. 357 00:20:44,040 --> 00:20:46,680 The one with the plus is propagating backward. 358 00:20:46,680 --> 00:20:48,330 So your colleague here asked, what will 359 00:20:48,330 --> 00:20:50,568 happen if I superimpose them? 360 00:20:50,568 --> 00:20:52,110 So one of them is going to the right. 361 00:20:52,110 --> 00:20:52,988 One is to the left. 362 00:20:52,988 --> 00:20:54,030 Are they going to cancel? 363 00:20:54,030 --> 00:20:55,870 What are they going to do? 364 00:20:55,870 --> 00:20:58,280 OK, so this is a computational. 365 00:20:58,280 --> 00:21:00,100 So perhaps you should not trust it. 366 00:21:00,100 --> 00:21:02,650 After all, what is this? 367 00:21:02,650 --> 00:21:05,020 But anyway, this is what I got in Matlab 368 00:21:05,020 --> 00:21:07,460 when I simulated this. 369 00:21:07,460 --> 00:21:08,750 OK, let me play this again. 370 00:21:08,750 --> 00:21:12,960 So this is the result of the two waves. 371 00:21:12,960 --> 00:21:14,497 OK. 372 00:21:14,497 --> 00:21:15,330 I pushed the button. 373 00:21:15,330 --> 00:21:16,622 I did not play the movie again. 374 00:21:16,622 --> 00:21:19,640 I revealed the derivation, but that's quite all right. 375 00:21:22,520 --> 00:21:26,570 Actually, I meant to do this on the free hand here. 376 00:21:26,570 --> 00:21:27,630 OK, never mind. 377 00:21:27,630 --> 00:21:29,610 Ignore what you see on the slide, 378 00:21:29,610 --> 00:21:33,300 and just watch the whiteboard. 379 00:21:33,300 --> 00:21:35,050 I guess I can call it whiteboard, can't I? 380 00:21:35,050 --> 00:21:38,610 I mean, it is a whiteboard. 381 00:21:38,610 --> 00:21:41,570 OK, so here are the two waves. 382 00:21:41,570 --> 00:21:43,060 So I actually omitted. 383 00:21:43,060 --> 00:21:47,120 I should have said that not only are they counterpropagating, 384 00:21:47,120 --> 00:21:51,500 but also, I pick them both to be harmonic waves. 385 00:21:51,500 --> 00:21:55,140 So that we can do the math easily. 386 00:21:55,140 --> 00:21:56,330 OK, so let's add this out. 387 00:21:56,330 --> 00:22:03,140 So we picked the identical amplitude. 388 00:22:03,140 --> 00:22:05,530 So one of them will be written as-- did I use a, 389 00:22:05,530 --> 00:22:06,740 or was it-- a, yeah, OK. 390 00:22:14,500 --> 00:22:16,630 OK, now, for generality. 391 00:22:16,630 --> 00:22:19,630 I don't really have to, but for generality, I also 392 00:22:19,630 --> 00:22:24,250 added the phase, a phase delay field to the first wave. 393 00:22:24,250 --> 00:22:27,680 So this one is forward. 394 00:22:27,680 --> 00:22:29,180 And then I have the other one, which 395 00:22:29,180 --> 00:22:33,320 looks very similar, except for this devious minus 396 00:22:33,320 --> 00:22:38,370 sign that goes backward. 397 00:22:38,370 --> 00:22:39,640 OK. 398 00:22:39,640 --> 00:22:41,560 And the superposition principle says 399 00:22:41,560 --> 00:22:44,530 that I can simply add the two waves, 400 00:22:44,530 --> 00:22:48,680 and the summation is still a solution to the wave equation, 401 00:22:48,680 --> 00:22:52,400 therefore, still a valid wave. 402 00:22:52,400 --> 00:22:55,260 OK, so what is the result? 403 00:22:55,260 --> 00:23:02,070 So now, I don't know how to tell you to remember this. 404 00:23:02,070 --> 00:23:06,690 Most people look it up in books of trigonometry. 405 00:23:06,690 --> 00:23:09,750 I have a second mnemonic for it, but I will not tell you. 406 00:23:09,750 --> 00:23:17,950 So there's a mnemonic that says if you have 407 00:23:17,950 --> 00:23:21,538 the sum of two cosines equals-- 408 00:23:21,538 --> 00:23:23,080 now, this is a mathematical property, 409 00:23:23,080 --> 00:23:25,450 so there's no intuition to be derived here. 410 00:23:25,450 --> 00:23:29,530 This is just obtained by trigonometric manipulation. 411 00:23:29,530 --> 00:23:36,850 Equals twice cosine, a plus b over 2, cosine a minus b 412 00:23:36,850 --> 00:23:37,510 over 2. 413 00:23:40,782 --> 00:23:41,990 So I never remember this one. 414 00:23:41,990 --> 00:23:44,480 I will tell you actually my mnemonic. 415 00:23:44,480 --> 00:23:47,570 It's nothing strange. 416 00:23:47,570 --> 00:23:54,275 All I remember is that if you have cosine of two things, 417 00:23:54,275 --> 00:23:55,040 it equals-- 418 00:24:00,860 --> 00:24:04,380 let me write it cleanly on a clean piece of paper. 419 00:24:04,380 --> 00:24:06,515 So cosine of two things. 420 00:24:12,810 --> 00:24:14,040 It has a minus sign here. 421 00:24:18,180 --> 00:24:21,475 And sine of two things. 422 00:24:33,570 --> 00:24:35,260 OK, this, you have memorized. 423 00:24:35,260 --> 00:24:37,260 And you can see very easily that from these two, 424 00:24:37,260 --> 00:24:39,840 you can obtain the equation we said before. 425 00:24:39,840 --> 00:24:49,380 Because, for example, you can write cosine alpha plus beta, 426 00:24:49,380 --> 00:24:54,150 plus minus beta equals cosine alpha cosine beta minus 427 00:24:54,150 --> 00:24:57,660 plus sine alpha sine beta. 428 00:24:57,660 --> 00:25:00,150 OK, so if you add the term with a plus, 429 00:25:00,150 --> 00:25:01,650 and the term with a minus. 430 00:25:01,650 --> 00:25:06,120 The sine terms evaporate, and you end up with the identity 431 00:25:06,120 --> 00:25:07,120 that they had before. 432 00:25:07,120 --> 00:25:09,600 So this is my mnemonic for remembering this one. 433 00:25:09,600 --> 00:25:13,530 We will not have to do this too often because last time, 434 00:25:13,530 --> 00:25:16,170 if you remember, we introduced phasors. 435 00:25:16,170 --> 00:25:19,430 We introduced a complex notation for waves, 436 00:25:19,430 --> 00:25:22,800 so we don't have to deal with this trigonometry very often. 437 00:25:22,800 --> 00:25:24,353 In this case, there's no way out. 438 00:25:24,353 --> 00:25:25,770 Well, actually, there's a way out, 439 00:25:25,770 --> 00:25:29,700 but it is simpler to do it this way. 440 00:25:29,700 --> 00:25:36,030 So now, I have basically a equals kz 441 00:25:36,030 --> 00:25:39,090 minus omega t plus phi. 442 00:25:39,090 --> 00:25:43,960 b equals plus omega t plus phi. 443 00:25:43,960 --> 00:25:46,290 So if you apply my formula here, you 444 00:25:46,290 --> 00:25:49,710 will find that the wave actually-- 445 00:25:49,710 --> 00:25:51,870 OK, the amplitude is the same. 446 00:25:51,870 --> 00:25:54,830 I pick up a factor of two. 447 00:25:54,830 --> 00:25:57,960 So it is two. 448 00:25:57,960 --> 00:25:59,850 And then I have the cosine of the sum. 449 00:25:59,850 --> 00:26:05,350 So if I sum the two, the time term will disappear. 450 00:26:05,350 --> 00:26:08,880 So we'll have cosine. 451 00:26:08,880 --> 00:26:11,470 Well, you get twice of it, but there's a factor of two 452 00:26:11,470 --> 00:26:12,700 here, so they cancel. 453 00:26:12,700 --> 00:26:16,530 It'll be cosine kz plus phi. 454 00:26:16,530 --> 00:26:18,890 And if I subtract the two, then naturally, 455 00:26:18,890 --> 00:26:21,940 the spatial terms and the phase will disappear. 456 00:26:21,940 --> 00:26:25,080 And I will get the cosine omega t. 457 00:26:25,080 --> 00:26:27,737 OK, now do I believe Matlab? 458 00:26:27,737 --> 00:26:29,820 I think I should believe Matlab, because let's see 459 00:26:29,820 --> 00:26:31,400 what Matlab saw. 460 00:26:31,400 --> 00:26:33,270 Here, time is frozen. 461 00:26:33,270 --> 00:26:35,220 So you'll see basically the same term 462 00:26:35,220 --> 00:26:38,810 where this has been replaced by constant, because I froze time. 463 00:26:38,810 --> 00:26:44,370 So in there, this looks like a sinusoid of the same frequency 464 00:26:44,370 --> 00:26:46,170 as we had before. 465 00:26:46,170 --> 00:26:48,900 And actually, of course, you have no way of knowing that. 466 00:26:48,900 --> 00:26:51,570 But the waves that I entered had an amplitude 467 00:26:51,570 --> 00:26:52,830 that was half of this. 468 00:26:52,830 --> 00:26:54,800 So basically, the factor of two was also 469 00:26:54,800 --> 00:26:56,325 reproduced by my simulation. 470 00:26:56,325 --> 00:26:57,930 OK, and if I play the movie again, 471 00:26:57,930 --> 00:27:00,750 it is basically like unfreezing time. 472 00:27:07,760 --> 00:27:09,360 OK, so here, I have unfrozen time. 473 00:27:09,360 --> 00:27:13,220 And you see that I get a sinusoidal variation 474 00:27:13,220 --> 00:27:14,760 in the time domain. 475 00:27:14,760 --> 00:27:18,480 So therefore, that's my term cosine omega t here. 476 00:27:18,480 --> 00:27:20,680 So basically, the Matlab simulation is correct. 477 00:27:20,680 --> 00:27:21,690 So we're very happy. 478 00:27:21,690 --> 00:27:22,910 OK. 479 00:27:22,910 --> 00:27:24,960 So now, let me come to your question. 480 00:27:24,960 --> 00:27:29,690 What happens to the wave when it vanishes? 481 00:27:29,690 --> 00:27:31,315 First of all, it doesn't really vanish. 482 00:27:35,140 --> 00:27:36,373 I don't know how to go back. 483 00:27:44,563 --> 00:27:45,980 So it does look like at some point 484 00:27:45,980 --> 00:27:49,388 the wave vanishes, and then it reappears. 485 00:27:49,388 --> 00:27:51,430 OK, it is a little bit annoying that it vanishes. 486 00:27:51,430 --> 00:27:54,130 But in electromagnetic waves, and usually 487 00:27:54,130 --> 00:27:56,380 in other kinds of waves, we don't worry 488 00:27:56,380 --> 00:27:58,465 about instantaneous energy. 489 00:27:58,465 --> 00:28:01,000 We worry about the average energy. 490 00:28:01,000 --> 00:28:06,130 So what you can see here is that the average energy is actually 491 00:28:06,130 --> 00:28:06,640 conserved. 492 00:28:06,640 --> 00:28:10,600 Because as the wave oscillates from a maximum to a minimum, 493 00:28:10,600 --> 00:28:13,250 it's average energy remains the same. 494 00:28:13,250 --> 00:28:16,010 Now, you might ask, well, what happens in between? 495 00:28:16,010 --> 00:28:24,055 I mean, what happens when this wave has gone to zero? 496 00:28:24,055 --> 00:28:26,430 Again, I will [INAUDIBLE] kinds of electromagnetic waves. 497 00:28:31,510 --> 00:28:33,610 What you have done here is you have basically 498 00:28:33,610 --> 00:28:39,110 launched two waves that are moving in opposite directions. 499 00:28:39,110 --> 00:28:41,530 So who is providing the energy? 500 00:28:41,530 --> 00:28:44,560 Well, if you are providing the energy on one end, 501 00:28:44,560 --> 00:28:47,100 and someone else is providing the energy on the other end. 502 00:28:58,680 --> 00:29:00,840 What happens when the two waves cancel 503 00:29:00,840 --> 00:29:04,230 is that the energy is actually stored 504 00:29:04,230 --> 00:29:06,870 in another place, which does not show up in this equation. 505 00:29:06,870 --> 00:29:09,690 That's one way to think about it. 506 00:29:09,690 --> 00:29:12,193 So if you remember when I saw-- actually, 507 00:29:12,193 --> 00:29:13,860 you don't remember, because you were not 508 00:29:13,860 --> 00:29:16,020 in the class one week ago. 509 00:29:16,020 --> 00:29:19,450 But I showed an example of a simple harmonic oscillator. 510 00:29:19,450 --> 00:29:24,090 So simple harmonic oscillator goes from zero velocity 511 00:29:24,090 --> 00:29:31,660 maximum potential energy to the edge, where the velocity now 512 00:29:31,660 --> 00:29:32,160 is-- 513 00:29:32,160 --> 00:29:35,490 I'm sorry, to the rest position, where the velocity's maximum 514 00:29:35,490 --> 00:29:38,670 and the potential energy has become zero 515 00:29:38,670 --> 00:29:39,780 at the rest position. 516 00:29:39,780 --> 00:29:41,430 So it's the same kind of thing. 517 00:29:41,430 --> 00:29:42,908 The kinetic energy disappears. 518 00:29:42,908 --> 00:29:43,950 Well, what happens to it? 519 00:29:43,950 --> 00:29:45,500 It has become potential. 520 00:29:45,500 --> 00:29:47,130 So it is very similar to this. 521 00:29:47,130 --> 00:29:49,380 The energy has become potential energy, 522 00:29:49,380 --> 00:29:52,730 and you don't see it here. 523 00:29:52,730 --> 00:29:55,010 AUDIENCE: OK, maybe a follow up on the question. 524 00:29:55,010 --> 00:29:56,950 So now, let's assume the two wave 525 00:29:56,950 --> 00:29:59,610 are traveling along the same direction, 526 00:29:59,610 --> 00:30:02,470 and they are out of phase by pi. 527 00:30:02,470 --> 00:30:05,950 In this case, they'd be a perfect cancelling each other. 528 00:30:05,950 --> 00:30:07,510 So where does the energy go? 529 00:30:07,510 --> 00:30:10,300 Just for your information, the previous answer, again. 530 00:30:10,300 --> 00:30:13,180 Such thing will not happen in the first place, 531 00:30:13,180 --> 00:30:15,410 but just not comfortable with this answer. 532 00:30:15,410 --> 00:30:17,962 So I would like to know your comments. 533 00:30:17,962 --> 00:30:19,670 GEORGE BARBASTATHIS: [INAUDIBLE] the wave 534 00:30:19,670 --> 00:30:22,690 start at the pi phase shift. 535 00:30:22,690 --> 00:30:28,240 It's equivalent to not having launched any wave at all. 536 00:30:28,240 --> 00:30:31,750 Whoever launched the two waves basically 537 00:30:31,750 --> 00:30:36,730 canceled his own action. 538 00:30:36,730 --> 00:30:40,110 So if you think about it physically, what happened? 539 00:30:40,110 --> 00:30:43,910 First of all, have the two waves been launched at infinity, 540 00:30:43,910 --> 00:30:44,870 or at the finite time? 541 00:30:47,480 --> 00:30:50,930 AUDIENCE: So for the [INAUDIBLE] have, and as that equal zero. 542 00:30:50,930 --> 00:30:55,360 And this wave traveling to the positive direction 543 00:30:55,360 --> 00:30:56,850 and continuously. 544 00:30:56,850 --> 00:31:00,530 Now, I'm not sure another wave added them [INAUDIBLE] that 545 00:31:00,530 --> 00:31:01,880 equals minus 10. 546 00:31:01,880 --> 00:31:04,040 But this show is that we've only lasted 547 00:31:04,040 --> 00:31:06,390 for three period for the input. 548 00:31:06,390 --> 00:31:10,100 So as we are traveling into a positive direction, 549 00:31:10,100 --> 00:31:12,890 it is going to cancel three periods out the previous wave, 550 00:31:12,890 --> 00:31:13,390 right? 551 00:31:15,942 --> 00:31:18,400 GEORGE BARBASTATHIS: Now you're making it a little bit more 552 00:31:18,400 --> 00:31:18,940 complicated. 553 00:31:18,940 --> 00:31:20,990 Because what happens to the edges of the wave? 554 00:31:20,990 --> 00:31:25,150 Are you multiplying by a boxcar function? 555 00:31:25,150 --> 00:31:26,020 What happens later? 556 00:31:36,260 --> 00:31:38,090 You're not given enough information 557 00:31:38,090 --> 00:31:39,920 to answer the question. 558 00:31:39,920 --> 00:31:42,650 Because then we had three periods cancel, but then 559 00:31:42,650 --> 00:31:43,700 what happens later? 560 00:31:43,700 --> 00:31:44,960 Did you just stop the wave? 561 00:31:48,287 --> 00:31:49,870 AUDIENCE: I'm still not so sure, yeah. 562 00:31:49,870 --> 00:31:51,480 That's why I check. 563 00:31:51,480 --> 00:31:52,370 Yeah, thanks. 564 00:31:52,370 --> 00:31:53,690 GEORGE BARBASTATHIS: The question's a bit ill defined, 565 00:31:53,690 --> 00:31:53,960 I guess. 566 00:31:53,960 --> 00:31:54,920 So I can not answer it. 567 00:31:54,920 --> 00:31:58,420 But I can tell you this. 568 00:31:58,420 --> 00:32:03,510 Suppose that I launch a wave at some time, and then 569 00:32:03,510 --> 00:32:07,250 a certain time later, I launch the same wave 570 00:32:07,250 --> 00:32:09,943 with biphasic trait. 571 00:32:09,943 --> 00:32:12,110 So there's no doubt that the two waves at this point 572 00:32:12,110 --> 00:32:15,050 will cancel. 573 00:32:15,050 --> 00:32:17,660 So what happened to the energy? 574 00:32:17,660 --> 00:32:20,320 Well, it is equivalent to thinking that I did not really 575 00:32:20,320 --> 00:32:24,460 launch a second wave with a pi phase shift. 576 00:32:24,460 --> 00:32:27,423 But I actually turned off the swings 577 00:32:27,423 --> 00:32:29,590 that was producing my first wave in the first place. 578 00:32:33,090 --> 00:32:35,090 There might be a tangent or something like that, 579 00:32:35,090 --> 00:32:37,020 but in fact, I've just canceled the wave. 580 00:32:37,020 --> 00:32:38,630 So simply stop the flow of energy. 581 00:32:45,160 --> 00:32:46,590 Everybody happy with this answer, 582 00:32:46,590 --> 00:32:48,440 or should we discuss this? 583 00:33:05,380 --> 00:33:07,755 And I think to realize that actually this is interesting, 584 00:33:07,755 --> 00:33:10,050 that we brought up energy. 585 00:33:10,050 --> 00:33:12,960 In the standing wave, the energy is actually not going anywhere. 586 00:33:18,315 --> 00:33:19,940 Well, that's one way to think about it. 587 00:33:19,940 --> 00:33:21,230 It's not going anywhere. 588 00:33:21,230 --> 00:33:22,730 Or you can think about it in the way 589 00:33:22,730 --> 00:33:24,397 that the energy is going back and forth. 590 00:33:42,273 --> 00:33:43,440 I did not realize, actually. 591 00:33:43,440 --> 00:33:46,965 I did not the phasors last time, did I? 592 00:33:46,965 --> 00:33:47,465 Did I? 593 00:33:47,465 --> 00:33:48,455 I guess I didn't. 594 00:33:58,490 --> 00:34:00,460 So I guess I pulled a fast one. 595 00:34:00,460 --> 00:34:03,250 A few minutes ago, I was talking about the difficulty 596 00:34:03,250 --> 00:34:04,710 of trigonometry. 597 00:34:04,710 --> 00:34:07,240 And I said that's why I introduced phasors last time. 598 00:34:07,240 --> 00:34:08,230 I was mistaken. 599 00:34:08,230 --> 00:34:10,929 I had not introduced any phasors last time. 600 00:34:10,929 --> 00:34:13,835 I'm introducing phasors now. 601 00:34:13,835 --> 00:34:16,210 And the reason, of course, that we're introducing phasors 602 00:34:16,210 --> 00:34:19,690 is because we don't want to have to remember 603 00:34:19,690 --> 00:34:22,810 complicated identities like this one. 604 00:34:27,020 --> 00:34:29,159 By the way, another point that I would 605 00:34:29,159 --> 00:34:32,429 like to bring up before I move on to phasors 606 00:34:32,429 --> 00:34:37,110 is that the wave that I'm showing here 607 00:34:37,110 --> 00:34:41,199 is actually very different than the traveling wave that 608 00:34:41,199 --> 00:34:42,822 led us to the wave equation. 609 00:34:42,822 --> 00:34:43,780 You'll agree with that. 610 00:34:43,780 --> 00:34:46,360 That looks very different, both in the way 611 00:34:46,360 --> 00:34:50,650 it behaves with time and space, and the way 612 00:34:50,650 --> 00:34:52,639 it looks like an equation. 613 00:34:52,639 --> 00:34:56,600 Yet, it still originated from the same wave equation. 614 00:34:56,600 --> 00:35:00,190 So this is actually the power of partial differential equations 615 00:35:00,190 --> 00:35:01,870 and superposition. 616 00:35:01,870 --> 00:35:05,770 That I took two solutions with my partial differential 617 00:35:05,770 --> 00:35:08,005 equation that were not particularly interesting, 618 00:35:08,005 --> 00:35:10,560 I superimposed them, and then all of a sudden, 619 00:35:10,560 --> 00:35:13,390 I created another solution, which 620 00:35:13,390 --> 00:35:16,180 is still a very valid wave, but we have totally different. 621 00:35:16,180 --> 00:35:18,040 It is a stationary wave. 622 00:35:18,040 --> 00:35:21,230 It has this oscillation, and so on and so forth. 623 00:35:21,230 --> 00:35:25,150 So that is why we bother to write this partial differential 624 00:35:25,150 --> 00:35:28,840 equation, because it allows us to generate 625 00:35:28,840 --> 00:35:33,117 a much richer set of waves that are, 626 00:35:33,117 --> 00:35:34,700 of course, very important in practice. 627 00:35:34,700 --> 00:35:38,382 Standing waves happen all the time. 628 00:35:38,382 --> 00:35:39,715 OK, so now, let's go to phasors. 629 00:35:44,970 --> 00:35:49,750 OK, so just like before, I wrote two waves 630 00:35:49,750 --> 00:35:52,667 that were propagating in opposite directions. 631 00:35:52,667 --> 00:35:54,250 Now, what I'm going to do is I'm going 632 00:35:54,250 --> 00:35:59,110 to write two waves that are phase shifted by pi over two. 633 00:35:59,110 --> 00:36:05,465 So one of them is cosine kz minus omega t plus phi. 634 00:36:09,140 --> 00:36:10,920 And the other is phase shifted by pi 635 00:36:10,920 --> 00:36:13,550 over 2, which means instead of a cosine, it is a sine. 636 00:36:22,150 --> 00:36:26,570 OK, both are valid solutions to the wave equation 637 00:36:26,570 --> 00:36:28,410 with different initial conditions, 638 00:36:28,410 --> 00:36:29,830 but that doesn't bother us. 639 00:36:29,830 --> 00:36:32,300 They're still valid solutions. 640 00:36:32,300 --> 00:36:34,955 Therefore, their superposition is also a solution. 641 00:36:37,820 --> 00:36:40,610 And I said before that when you do a linear superposition, 642 00:36:40,610 --> 00:36:45,440 you can also multiply the waves by arbitrary constants. 643 00:36:45,440 --> 00:36:48,210 I said this a little bit fast, but I hope 644 00:36:48,210 --> 00:36:50,670 that remember that I said it. 645 00:36:50,670 --> 00:36:53,250 In fact, in this class, we have evidence of what I said 646 00:36:53,250 --> 00:36:54,320 and what I did not say. 647 00:36:54,320 --> 00:36:58,590 Then go back to the video and see if I said it, and I did. 648 00:36:58,590 --> 00:37:00,820 So I will pick a very particular constant, 649 00:37:00,820 --> 00:37:03,890 which may sound like a crazy choice, but I will pick it. 650 00:37:03,890 --> 00:37:07,690 And that is the imaginary constant, i, also known 651 00:37:07,690 --> 00:37:12,900 as one of the two square root of minus 1. 652 00:37:12,900 --> 00:37:15,360 OK, what is this now? 653 00:37:15,360 --> 00:37:17,580 So the first confession that I will make 654 00:37:17,580 --> 00:37:20,210 is that it is not physical anymore. 655 00:37:20,210 --> 00:37:23,280 There's no such thing as a complex wave. 656 00:37:23,280 --> 00:37:25,290 The waves that we see around us are real. 657 00:37:25,290 --> 00:37:28,290 They can be measured by instruments, 658 00:37:28,290 --> 00:37:32,030 such as oscilloscope, scanners, I don't know what else. 659 00:37:32,030 --> 00:37:34,110 They're real. 660 00:37:34,110 --> 00:37:37,410 As far as I know, nobody has ever measured anything complex. 661 00:37:37,410 --> 00:37:41,190 However, what I'm doing here is mathematically correct, 662 00:37:41,190 --> 00:37:43,560 even if it has no physical meaning. 663 00:37:43,560 --> 00:37:47,040 And as it turns out, it provides a huge mathematical 664 00:37:47,040 --> 00:37:48,090 convenience. 665 00:37:48,090 --> 00:37:52,380 So I'm justified in doing it as long as I remember what 666 00:37:52,380 --> 00:37:54,450 is the physics that is behind. 667 00:37:54,450 --> 00:37:56,040 And the physics that is behind it 668 00:37:56,040 --> 00:37:59,240 is that every time I see a wave like this, 669 00:37:59,240 --> 00:38:02,490 I have to remember that the physical quantity is 670 00:38:02,490 --> 00:38:04,720 the real part. 671 00:38:04,720 --> 00:38:06,700 So the physical wave-- 672 00:38:06,700 --> 00:38:10,480 the one that I can actually measure, observe, 673 00:38:10,480 --> 00:38:14,170 generate with natural means-- 674 00:38:14,170 --> 00:38:17,550 is the real part. 675 00:38:17,550 --> 00:38:19,890 The imaginary part is something that I added 676 00:38:19,890 --> 00:38:23,190 for mathematical convenience. 677 00:38:23,190 --> 00:38:26,520 And it isn't reality because I forget whose formula. 678 00:38:26,520 --> 00:38:28,840 It is called the De Moivre. 679 00:38:34,010 --> 00:38:35,760 Anyway, one of those. 680 00:38:40,500 --> 00:38:41,000 D-E? 681 00:38:43,930 --> 00:38:44,620 Moivre, yeah. 682 00:38:44,620 --> 00:38:45,100 AUDIENCE: De Moivre. 683 00:38:45,100 --> 00:38:46,642 GEORGE BARBASTATHIS: De Moivre, yeah. 684 00:38:46,642 --> 00:38:48,520 OK, I got it right. 685 00:38:48,520 --> 00:38:50,430 So De Moivre, then, if I remember correctly, 686 00:38:50,430 --> 00:38:51,430 that's how you spell it. 687 00:38:54,520 --> 00:39:03,180 De Moivre says that this thing here is a complex exponential. 688 00:39:10,180 --> 00:39:14,060 So that's very nice because for example, 689 00:39:14,060 --> 00:39:18,660 if you have to multiply two cosines, it's a big pain. 690 00:39:18,660 --> 00:39:21,660 You have to remember the trig to multiply two exponential. 691 00:39:21,660 --> 00:39:22,340 It is trivial. 692 00:39:22,340 --> 00:39:26,850 You just add their phases, I mean, their exponents. 693 00:39:26,850 --> 00:39:31,960 So a lot of inconveniences to be to be expected. 694 00:39:31,960 --> 00:39:35,940 So this representation, it comes with different names, 695 00:39:35,940 --> 00:39:39,090 which are sometimes used confusingly 696 00:39:39,090 --> 00:39:41,460 in an interchangeable way. 697 00:39:41,460 --> 00:39:45,240 So some people call this a phasor notation. 698 00:39:45,240 --> 00:39:47,760 I prefer to call this the complex representation 699 00:39:47,760 --> 00:39:49,820 of the wave. 700 00:39:49,820 --> 00:39:53,170 It becomes a phasor when we drop-- 701 00:39:53,170 --> 00:39:55,060 I'm going to do something nasty here. 702 00:39:55,060 --> 00:39:57,400 OK, remember that I can write this as a project. 703 00:39:57,400 --> 00:40:03,540 So I can write this as e to the ikz, e to the minus i omega t, 704 00:40:03,540 --> 00:40:06,020 e to the i phi. 705 00:40:06,020 --> 00:40:07,840 OK. 706 00:40:07,840 --> 00:40:12,660 If you recall during the beginnings of the class, 707 00:40:12,660 --> 00:40:18,820 we saw that the wavelength in the middle changes. 708 00:40:18,820 --> 00:40:22,896 But there is a property of the wave that does not change. 709 00:40:22,896 --> 00:40:24,550 And that property, if you recall, 710 00:40:24,550 --> 00:40:28,200 is the temporal frequency, omega, or nu, 711 00:40:28,200 --> 00:40:29,580 which are related by two pi. 712 00:40:32,540 --> 00:40:34,280 That is true in linear media. 713 00:40:34,280 --> 00:40:36,500 Of course, there is a class of optics 714 00:40:36,500 --> 00:40:39,390 called non-linear, where actually, the frequency can 715 00:40:39,390 --> 00:40:39,890 change. 716 00:40:39,890 --> 00:40:42,890 It can double, or it can half, or [INAUDIBLE] of things 717 00:40:42,890 --> 00:40:43,655 can happen to it. 718 00:40:43,655 --> 00:40:46,280 But in this class, we don't deal with these complicated things. 719 00:40:46,280 --> 00:40:48,620 We just deal with linear optics. 720 00:40:48,620 --> 00:40:51,260 So in linear optics, the temporal frequency 721 00:40:51,260 --> 00:40:53,730 of the light, no matter what happens to the light. 722 00:40:53,730 --> 00:40:57,050 If it goes through glass and reflects, 723 00:40:57,050 --> 00:40:58,220 and all kinds of things. 724 00:40:58,220 --> 00:41:00,690 The temporal frequency remains the same. 725 00:41:00,690 --> 00:41:04,970 So when we deal with optical waves of a single frequency-- 726 00:41:04,970 --> 00:41:10,340 that is the quality's fixed and the medium is linear-- 727 00:41:10,340 --> 00:41:12,860 then this term is actually superfluous. 728 00:41:12,860 --> 00:41:16,880 We don't have to remind ourselves that the light goes 729 00:41:16,880 --> 00:41:18,920 like e to the minus omega t. 730 00:41:18,920 --> 00:41:21,410 It does, but we don't have to write it. 731 00:41:21,410 --> 00:41:25,010 Because if we keep writing it, it clutters our equations. 732 00:41:25,010 --> 00:41:26,510 So we simply drop it. 733 00:41:29,750 --> 00:41:30,590 We drop it. 734 00:41:30,590 --> 00:41:31,770 Very important to remember. 735 00:41:31,770 --> 00:41:33,830 We are allowed to drop it if we know 736 00:41:33,830 --> 00:41:37,340 that we have a single color in our system, a single frequency. 737 00:41:37,340 --> 00:41:38,780 If there is multiple frequencies, 738 00:41:38,780 --> 00:41:40,040 we can not do that anymore. 739 00:41:40,040 --> 00:41:43,790 Because then these frequencies have to adapt in a certain way, 740 00:41:43,790 --> 00:41:47,000 and you have to keep track of their additive phases. 741 00:41:47,000 --> 00:41:49,880 So we can only drop this term if we-- 742 00:41:49,880 --> 00:41:50,570 two conditions. 743 00:41:50,570 --> 00:41:52,820 If we know that we have a linear medium, 744 00:41:52,820 --> 00:41:57,530 and a single frequency light propagating in that medium. 745 00:41:57,530 --> 00:42:03,090 So in my terminology, at least, after we drop this term, 746 00:42:03,090 --> 00:42:04,740 then it becomes a phasor. 747 00:42:04,740 --> 00:42:05,750 So what is left? 748 00:42:05,750 --> 00:42:08,830 e to the ikz plus 5. 749 00:42:08,830 --> 00:42:10,450 That, I call a phasor. 750 00:42:10,450 --> 00:42:12,040 And you notice what happened there 751 00:42:12,040 --> 00:42:16,990 is I only left the spatial dependency of the wave 752 00:42:16,990 --> 00:42:18,800 plus the phase delay. 753 00:42:18,800 --> 00:42:21,220 The temporal variation is-- 754 00:42:21,220 --> 00:42:21,760 it's hidden. 755 00:42:21,760 --> 00:42:22,450 It is not gone. 756 00:42:22,450 --> 00:42:25,710 It is just hidden. 757 00:42:25,710 --> 00:42:29,440 OK, now let's see this in action. 758 00:42:29,440 --> 00:42:32,860 So what I'm going to do is I'm going to rederive the standing 759 00:42:32,860 --> 00:42:34,750 wave, but using the phasor. 760 00:42:34,750 --> 00:42:36,970 So you can see how nice and simple it comes out, 761 00:42:36,970 --> 00:42:39,260 and we don't have to agonize over it. 762 00:42:39,260 --> 00:42:43,380 But before I do it, I actually have to solve a little problem. 763 00:42:43,380 --> 00:42:45,970 And the problem is that I did this very nice derivation 764 00:42:45,970 --> 00:42:49,630 for the phasor of the forward propagating wave. 765 00:42:49,630 --> 00:42:54,940 What about the phasor for the backward propagating wave? 766 00:42:54,940 --> 00:42:59,530 The backward propagating wave will be something of this sort. 767 00:42:59,530 --> 00:43:01,110 So this will always forward, right? 768 00:43:07,260 --> 00:43:08,788 Let me write the backward. 769 00:43:08,788 --> 00:43:10,830 I think I have enough space here, so I can do it. 770 00:43:16,670 --> 00:43:19,240 OK, so the backward would be something 771 00:43:19,240 --> 00:43:26,770 of the sort of the form of cosine, az plus omega t 772 00:43:26,770 --> 00:43:29,303 plus phi. 773 00:43:29,303 --> 00:43:30,470 And it is very inconvenient. 774 00:43:30,470 --> 00:43:32,960 Because the way I got the phasor is 775 00:43:32,960 --> 00:43:38,980 I got rid of a term of the forward to the minus omega t. 776 00:43:38,980 --> 00:43:44,200 If I add the imaginary part here, 777 00:43:44,200 --> 00:43:49,000 I don't have a term e to the minus omega t, so I'm stuck. 778 00:43:49,000 --> 00:43:51,850 But I can do some mathematical trickery. 779 00:43:51,850 --> 00:43:55,138 Now, this is pure trickery, but it works. 780 00:43:55,138 --> 00:43:56,430 The trick, it is the following. 781 00:43:56,430 --> 00:44:00,270 We know that the cosine is an even function. 782 00:44:00,270 --> 00:44:04,360 So this is actually identical. 783 00:44:04,360 --> 00:44:06,200 I can replace it, in other words, 784 00:44:06,200 --> 00:44:08,080 with the cosine of the negative element. 785 00:44:14,600 --> 00:44:15,440 No harm done. 786 00:44:15,440 --> 00:44:17,210 It is still a backward propagating wave. 787 00:44:17,210 --> 00:44:18,140 Nothing has changed. 788 00:44:18,140 --> 00:44:20,290 It is the same equation. 789 00:44:20,290 --> 00:44:24,540 Now, according to my good principle here, I add. 790 00:44:31,360 --> 00:44:34,310 That is not the same as before. 791 00:44:34,310 --> 00:44:37,430 Because the sign is an odd function. 792 00:44:37,430 --> 00:44:39,180 But this is a non-physical term, anyway. 793 00:44:39,180 --> 00:44:41,222 I just put it there for mathematical convenience, 794 00:44:41,222 --> 00:44:42,910 so I don't particularly care. 795 00:44:42,910 --> 00:44:45,320 I'm very happy, because the real part 796 00:44:45,320 --> 00:44:48,250 is the same as I started with. 797 00:44:48,250 --> 00:44:50,870 What the imaginary part does-- 798 00:44:50,870 --> 00:44:52,660 as long as it's mathematically convenient, 799 00:44:52,660 --> 00:44:54,360 I couldn't care less. 800 00:44:54,360 --> 00:44:56,780 OK, so that gives me the license to write this. 801 00:44:56,780 --> 00:45:02,890 Now, this, of course, equals e to the minus 802 00:45:02,890 --> 00:45:09,440 ikz minus omega t minus phi. 803 00:45:09,440 --> 00:45:11,780 Now very happy, because I did get a term e 804 00:45:11,780 --> 00:45:16,700 to the minus i omega t in the phase of this exponential. 805 00:45:16,700 --> 00:45:19,490 And therefore, I can drop it to produce my phasor. 806 00:45:19,490 --> 00:45:21,400 So the last one, then, is that the phasor 807 00:45:21,400 --> 00:45:29,960 for the backward wave is e to the minus ikz minus i phi. 808 00:45:29,960 --> 00:45:32,720 In other words, if I give you a forward propagating wave, 809 00:45:32,720 --> 00:45:34,475 and you compute its phasor. 810 00:45:34,475 --> 00:45:37,100 And then I asked you, well, what is the phasor for the backward 811 00:45:37,100 --> 00:45:38,630 propagating wave? 812 00:45:38,630 --> 00:45:41,000 Well, all you do is you flip the sign and the exponent. 813 00:45:48,030 --> 00:45:48,730 Whoa. 814 00:45:48,730 --> 00:45:49,870 Where did this come from? 815 00:45:49,870 --> 00:45:55,790 I think this is an old paper from some previous lecture. 816 00:45:55,790 --> 00:45:57,820 But anyway, ignore this. 817 00:46:01,010 --> 00:46:04,750 I have to remember to ask for a new paper next time. 818 00:46:04,750 --> 00:46:06,330 OK, so what I will do now. 819 00:46:06,330 --> 00:46:08,470 Let me reproduce the standing wave. 820 00:46:08,470 --> 00:46:12,930 So the standing wave is using the phasors now. 821 00:46:12,930 --> 00:46:17,460 It's e to the ikz plus phi. 822 00:46:17,460 --> 00:46:24,630 That is the forward wave, plus e to the minus ikz minus i phi. 823 00:46:28,680 --> 00:46:32,215 That is the backward wave. 824 00:46:32,215 --> 00:46:33,090 Can I deal with that? 825 00:46:33,090 --> 00:46:33,923 Well, yes, actually. 826 00:46:33,923 --> 00:46:36,350 Because, again, De Moivre says that if you add 827 00:46:36,350 --> 00:46:38,240 two exponential of this form-- 828 00:46:38,240 --> 00:46:42,660 notice they have the same phase within a minus sign-- 829 00:46:42,660 --> 00:46:47,980 I will simply get two i's cosine az plus phi. 830 00:46:47,980 --> 00:46:53,960 So I've got my thing here. 831 00:46:53,960 --> 00:46:56,900 Not quite, though, because the standing wave. 832 00:46:56,900 --> 00:46:58,960 It had the cosine omega t term. 833 00:46:58,960 --> 00:47:00,460 What happened to the cosine omega t? 834 00:47:05,850 --> 00:47:06,940 Oh, come on. 835 00:47:06,940 --> 00:47:07,710 OK. 836 00:47:07,710 --> 00:47:09,600 What happened to the cosine omega t? 837 00:47:09,600 --> 00:47:15,943 Well, remember, we neglected an e to the minus omega t. 838 00:47:15,943 --> 00:47:17,110 We didn't really neglect it. 839 00:47:17,110 --> 00:47:18,985 We hit it, because we didn't want to carry it 840 00:47:18,985 --> 00:47:21,660 when we write the equation. 841 00:47:21,660 --> 00:47:26,580 First of all, if I don't care about the temporal variation, 842 00:47:26,580 --> 00:47:28,470 I don't really have to do anything. 843 00:47:28,470 --> 00:47:29,350 I'm done. 844 00:47:29,350 --> 00:47:32,070 But if I want to put a temporal variation back in, 845 00:47:32,070 --> 00:47:36,030 what I have to do is I have to put back what I took 846 00:47:36,030 --> 00:47:37,510 when it did not belong to me. 847 00:47:37,510 --> 00:47:43,520 And what did not belong to me is this e to the minus i omega t. 848 00:47:43,520 --> 00:47:44,970 OK, I put it back. 849 00:47:44,970 --> 00:47:46,770 Now, that is still not a physical wave 850 00:47:46,770 --> 00:47:47,910 because it is complex. 851 00:47:47,910 --> 00:47:52,270 How can I find the actual, physical wave? 852 00:47:52,270 --> 00:47:54,030 Well, I have to take its real part. 853 00:47:54,030 --> 00:47:57,870 And the real part is easy here, because the only complex thing 854 00:47:57,870 --> 00:47:59,350 is this exponential. 855 00:47:59,350 --> 00:48:07,230 So if I take the real part of this, of all of this now. 856 00:48:07,230 --> 00:48:13,930 It would simply be 2 cosine kz plus phi cosine omega t. 857 00:48:13,930 --> 00:48:14,730 And I'm done. 858 00:48:14,730 --> 00:48:16,670 I've actually produced my standing wave. 859 00:48:23,800 --> 00:48:25,231 Any questions? 860 00:48:39,000 --> 00:48:40,540 We're almost done here. 861 00:48:40,540 --> 00:48:42,400 So I have about three minutes. 862 00:48:42,400 --> 00:48:44,730 So I think I'll use them to finish up this lecture, 863 00:48:44,730 --> 00:48:48,480 so we can move on to bigger and better things. 864 00:48:48,480 --> 00:48:51,160 The question that I wrote before is the one dimensional wave 865 00:48:51,160 --> 00:48:51,660 equation. 866 00:48:51,660 --> 00:48:54,930 Because the waves that I've written. 867 00:48:54,930 --> 00:49:00,140 They can only go down one spatial axis, z. 868 00:49:00,140 --> 00:49:03,780 The more generalized, of course, situation 869 00:49:03,780 --> 00:49:05,820 is three dimensional waves. 870 00:49:05,820 --> 00:49:08,880 And what you see here is the generalized three 871 00:49:08,880 --> 00:49:11,280 dimensional wave equation, which, of course, 872 00:49:11,280 --> 00:49:12,240 I will not derive. 873 00:49:12,240 --> 00:49:15,070 But it can be pretty easily derived 874 00:49:15,070 --> 00:49:17,230 using the same arguments that we did before, 875 00:49:17,230 --> 00:49:20,850 except you have to generalize in the third dimension. 876 00:49:20,850 --> 00:49:22,800 And this produces things that we've 877 00:49:22,800 --> 00:49:25,200 seen before in geometrical optics, 878 00:49:25,200 --> 00:49:27,540 for example, the plain wave. 879 00:49:27,540 --> 00:49:29,615 Except now, you see the wave description, 880 00:49:29,615 --> 00:49:36,530 it is you have alternating positive and negative-- 881 00:49:36,530 --> 00:49:39,360 how do you call those-- peaks and troughs, 882 00:49:39,360 --> 00:49:42,010 which are propagating along an arbitrary angle. 883 00:49:42,010 --> 00:49:45,900 Now, they can go in any angle in 3D space. 884 00:49:45,900 --> 00:49:48,510 So you just see a cross-section here. 885 00:49:48,510 --> 00:49:53,730 Or you can have a spherical wave, which we saw it before. 886 00:49:53,730 --> 00:50:00,075 It starts at a given point, and then radiates outwards. 887 00:50:00,075 --> 00:50:01,200 So that's a spherical wave. 888 00:50:06,170 --> 00:50:07,740 Well, we're not quite finished, yet. 889 00:50:07,740 --> 00:50:09,670 But I think we're almost out of time. 890 00:50:09,670 --> 00:50:13,220 So I think I'll stop here, and we can still 891 00:50:13,220 --> 00:50:15,090 continue next time. 892 00:50:15,090 --> 00:50:16,940 Are there any questions? 893 00:50:25,258 --> 00:50:26,800 There's no homework due on Wednesday, 894 00:50:26,800 --> 00:50:29,088 so what I would like to-- you have a question? 895 00:50:29,088 --> 00:50:30,130 Yeah, there's a question. 896 00:50:30,130 --> 00:50:30,630 Go ahead. 897 00:50:37,180 --> 00:50:39,380 AUDIENCE: How is the wave equation 898 00:50:39,380 --> 00:50:43,300 intuitively different from simple harmonic oscillator? 899 00:50:43,300 --> 00:50:48,200 The variation in space depends on the relation in time. 900 00:50:48,200 --> 00:50:49,060 So that's coupling? 901 00:50:51,697 --> 00:50:54,280 GEORGE BARBASTATHIS: Yeah, the mathematical answer, of course, 902 00:50:54,280 --> 00:50:54,580 is very easy. 903 00:50:54,580 --> 00:50:56,038 They're simply harmonic oscillators 904 00:50:56,038 --> 00:50:57,703 in an ordinary differential equation. 905 00:50:57,703 --> 00:50:59,370 This is a partial differential equation. 906 00:50:59,370 --> 00:51:02,770 But let me think how to formulate a physical answer. 907 00:51:02,770 --> 00:51:07,950 OK, so the physical answer is that the harmonic oscillator 908 00:51:07,950 --> 00:51:13,960 is a singly particle whose position you 909 00:51:13,960 --> 00:51:15,300 track as a function of time. 910 00:51:18,190 --> 00:51:21,690 A wave is a distributed physical system. 911 00:51:21,690 --> 00:51:23,730 So you can think of a wave-- 912 00:51:23,730 --> 00:51:25,410 in this case, for example. 913 00:51:25,410 --> 00:51:27,180 You can interpret the physical meaning 914 00:51:27,180 --> 00:51:31,590 of these black and white stripes as particles which are moving. 915 00:51:31,590 --> 00:51:34,920 For example, the particle that is on the white stripe 916 00:51:34,920 --> 00:51:36,120 is moving up. 917 00:51:36,120 --> 00:51:39,940 The particle in the black stripe is moving down. 918 00:51:39,940 --> 00:51:41,940 If I play the wave again, what you 919 00:51:41,940 --> 00:51:44,970 will see that the particles are kind of executing 920 00:51:44,970 --> 00:51:46,680 a coordinated motion. 921 00:51:46,680 --> 00:51:50,140 Because you have particles that were down, then they go up, 922 00:51:50,140 --> 00:51:51,480 and then they go down again. 923 00:51:51,480 --> 00:51:54,120 But they go out in a coordinated motion 924 00:51:54,120 --> 00:51:56,040 throughout the entire plane. 925 00:51:56,040 --> 00:51:56,750 How do they know? 926 00:51:56,750 --> 00:51:59,000 Well, there is a physical system associated with this. 927 00:51:59,000 --> 00:52:02,330 For example, this could be a membrane with the sound wave 928 00:52:02,330 --> 00:52:03,910 propagating on it. 929 00:52:03,910 --> 00:52:06,240 And now the reason that particles are connected, 930 00:52:06,240 --> 00:52:08,550 that they execute a coordinated motion 931 00:52:08,550 --> 00:52:10,950 is because of the interatomic forces 932 00:52:10,950 --> 00:52:13,320 between the membrane particles. 933 00:52:13,320 --> 00:52:15,180 If it is an electromagnetic wave, 934 00:52:15,180 --> 00:52:19,050 then it is a field that is distributed. 935 00:52:19,050 --> 00:52:21,210 And, of course, it knows to be connected 936 00:52:21,210 --> 00:52:23,060 because of Maxim's equations. 937 00:52:23,060 --> 00:52:25,350 There's things like charge conservation and so on that 938 00:52:25,350 --> 00:52:27,880 force it to act this way. 939 00:52:27,880 --> 00:52:29,790 So a wave is actually a distributed system, 940 00:52:29,790 --> 00:52:37,230 where the physics force an extended systems-- 941 00:52:37,230 --> 00:52:40,470 particles, or fields, or whatever the case may be-- 942 00:52:40,470 --> 00:52:42,878 to execute a coordinated motion. 943 00:52:42,878 --> 00:52:45,420 That's the difference between that simple harmonic oscillator 944 00:52:45,420 --> 00:52:46,795 where you just have one particle. 945 00:52:46,795 --> 00:52:48,378 Now, you can go from one to the other. 946 00:52:48,378 --> 00:52:50,370 Imagine you take a single harmonic oscillator. 947 00:52:50,370 --> 00:52:51,120 That's a good one. 948 00:52:51,120 --> 00:52:53,280 So let me draw it here. 949 00:52:53,280 --> 00:52:56,730 Wow, we're actually going backwards now in the knowledge 950 00:52:56,730 --> 00:52:57,900 that they used in the past. 951 00:52:57,900 --> 00:52:59,640 So you take a simple harmonic oscillator. 952 00:52:59,640 --> 00:53:01,950 Here it is, a pendulum. 953 00:53:01,950 --> 00:53:05,550 This is a simple harmonic oscillator. 954 00:53:05,550 --> 00:53:07,742 Then you couple it with a separate pendulum. 955 00:53:07,742 --> 00:53:08,700 And how do you couple-- 956 00:53:08,700 --> 00:53:10,117 they're not coupled at the moment, 957 00:53:10,117 --> 00:53:12,540 but you can couple them, for example, 958 00:53:12,540 --> 00:53:16,110 by connecting the spring in between. 959 00:53:16,110 --> 00:53:17,520 But that's not quite a wave. 960 00:53:17,520 --> 00:53:19,590 But you can see now that because I coupled them, 961 00:53:19,590 --> 00:53:22,110 if I kick one of these, it will also 962 00:53:22,110 --> 00:53:25,560 cause the other one to move, so they become coordinated. 963 00:53:25,560 --> 00:53:26,830 Well, now, let me generalize. 964 00:53:26,830 --> 00:53:32,080 Imagine that I have a bunch of harmonic oscillators. 965 00:53:32,080 --> 00:53:34,780 And, in fact, I can make them infinite, and I couple them. 966 00:53:34,780 --> 00:53:38,470 Now, in fact, the system will become wave like, in the sense 967 00:53:38,470 --> 00:53:42,440 that if you kick it at one end. 968 00:53:42,440 --> 00:53:43,810 Let's say not at infinity. 969 00:53:43,810 --> 00:53:45,550 Let's say it starts actually here, 970 00:53:45,550 --> 00:53:46,950 and then it goes to infinity. 971 00:53:46,950 --> 00:53:49,960 So if you kick it at one end, then the disturbance 972 00:53:49,960 --> 00:53:52,810 will actually propagate very similar to a wave. 973 00:53:52,810 --> 00:53:55,560 It's a discrete wave, but it's still a wave. 974 00:53:55,560 --> 00:53:59,800 So you can go from one to the other by introducing kind 975 00:53:59,800 --> 00:54:01,110 of a coordination mechanism. 976 00:54:01,110 --> 00:54:04,044 So it is the coupling here that does the coordination. 977 00:54:11,927 --> 00:54:12,760 Any other questions? 978 00:54:12,760 --> 00:54:13,570 We can stay here, actually. 979 00:54:13,570 --> 00:54:15,653 Nobody claims the classrooms either here or there. 980 00:54:15,653 --> 00:54:17,590 So if you have more questions, I'll 981 00:54:17,590 --> 00:54:19,180 be happy to stay and answer them. 982 00:54:28,860 --> 00:54:32,750 But about the phasors, is it clear? 983 00:54:32,750 --> 00:54:35,960 When I was your age, when I was a student, 984 00:54:35,960 --> 00:54:37,460 I was also very upset by phasors. 985 00:54:37,460 --> 00:54:40,740 Because they look very much like mathematical trickery. 986 00:54:40,740 --> 00:54:43,748 I mean, I'm trying to find my-- 987 00:54:43,748 --> 00:54:46,040 I guess that's one benefit of this over the whiteboard, 988 00:54:46,040 --> 00:54:48,190 that I have a history of what I did. 989 00:54:48,190 --> 00:54:48,910 But yeah. 990 00:54:48,910 --> 00:54:50,410 But the phasors, they look very much 991 00:54:50,410 --> 00:54:51,740 like mathematical trickery. 992 00:54:51,740 --> 00:54:56,280 I mean, this looks like someone pulled it out of a hat. 993 00:54:56,280 --> 00:55:00,200 But the only reason to justify it in your mind 994 00:55:00,200 --> 00:55:03,908 is that it provides huge mathematical convenience. 995 00:55:03,908 --> 00:55:05,450 Now, this looks like a trivial thing. 996 00:55:05,450 --> 00:55:07,400 I mean, the derivation of the standing wave 997 00:55:07,400 --> 00:55:10,670 was maybe four lines with trigonometry, two lines 998 00:55:10,670 --> 00:55:13,020 with the complex exponential. 999 00:55:13,020 --> 00:55:16,220 So OK, why mess with complex numbers? 1000 00:55:16,220 --> 00:55:18,530 Well, try superposition. 1001 00:55:18,530 --> 00:55:23,490 If you have several waves superimposed, 1002 00:55:23,490 --> 00:55:26,030 or if you have an integral, a continuum 1003 00:55:26,030 --> 00:55:28,940 of superimposed waves, which we will see very soon. 1004 00:55:28,940 --> 00:55:30,720 Like Fourier transforms and so on. 1005 00:55:30,720 --> 00:55:34,010 Fourier transforms are extremely inconvenient if you write them 1006 00:55:34,010 --> 00:55:35,960 as cosine transforms. 1007 00:55:35,960 --> 00:55:37,060 Big mess. 1008 00:55:37,060 --> 00:55:39,200 If you write them as complex exponentials, 1009 00:55:39,200 --> 00:55:43,370 it's actually very simple to do the math. 1010 00:55:43,370 --> 00:55:50,930 It is a very, very time saving tool, basically. 1011 00:55:50,930 --> 00:55:53,270 Strangely enough, it leads to some insights 1012 00:55:53,270 --> 00:55:55,180 that real numbers do not give you. 1013 00:55:55,180 --> 00:55:58,010 So that's an additional convenience. 1014 00:55:58,010 --> 00:56:00,120 We will appreciate that later. 1015 00:56:00,120 --> 00:56:02,690 For now, it is simply the mathematical convenience 1016 00:56:02,690 --> 00:56:06,155 that drives this mathematical wizardry. 1017 00:56:12,950 --> 00:56:14,670 AUDIENCE: Yeah. 1018 00:56:14,670 --> 00:56:16,830 You're saying [INAUDIBLE] angular frequency 1019 00:56:16,830 --> 00:56:21,850 and the temporal frequency are different, or it's the same? 1020 00:56:21,850 --> 00:56:27,342 Because you initially defined omega as angular frequency. 1021 00:56:27,342 --> 00:56:29,550 GEORGE BARBASTATHIS: I confuse those myself, usually, 1022 00:56:29,550 --> 00:56:30,570 the terms. 1023 00:56:30,570 --> 00:56:31,530 Usually nu. 1024 00:56:34,320 --> 00:56:39,620 Usually nu is called the frequency. 1025 00:56:39,620 --> 00:56:43,120 And it is measured in hertz. 1026 00:56:43,120 --> 00:56:49,290 2 pi nu is measured in radians per second. 1027 00:56:52,313 --> 00:56:53,480 Angular frequency, isn't it? 1028 00:56:57,336 --> 00:56:57,836 Yeah. 1029 00:57:06,480 --> 00:57:10,013 As opposed to what else? 1030 00:57:10,013 --> 00:57:11,680 I guess I don't understand the question. 1031 00:57:22,760 --> 00:57:28,190 Yeah, what I mean is that, for example, 1032 00:57:28,190 --> 00:57:29,890 take the propagating wave. 1033 00:57:33,410 --> 00:57:39,955 The propagating wave is e to the i-- 1034 00:57:39,955 --> 00:57:44,230 let's say forward-- kz minus omega t. 1035 00:57:44,230 --> 00:57:52,490 The standing wave is cosine kz e to the minus i omega t. 1036 00:57:56,127 --> 00:57:58,460 I'll step ahead a little bit, and write something awful. 1037 00:58:30,310 --> 00:58:32,230 This is actually a wave. 1038 00:58:32,230 --> 00:58:35,390 What we'll see is called a diffraction integral. 1039 00:58:35,390 --> 00:58:37,870 And you can see it is nothing-- 1040 00:58:37,870 --> 00:58:39,850 even though it looks really complicated, 1041 00:58:39,850 --> 00:58:41,170 it is just a superposition. 1042 00:58:41,170 --> 00:58:43,000 Because the integral is a summation. 1043 00:58:43,000 --> 00:58:45,250 And then what I have here is a phasor, 1044 00:58:45,250 --> 00:58:47,560 and I put a lot of these phasors together. 1045 00:58:47,560 --> 00:58:51,970 What they all share is this term, e to the i minus I 1046 00:58:51,970 --> 00:58:52,480 omega t. 1047 00:58:55,600 --> 00:58:57,147 Actually, to be strictly correct, 1048 00:58:57,147 --> 00:58:58,480 I should really carry this term. 1049 00:58:58,480 --> 00:59:00,220 I'm not really allowed to drop it. 1050 00:59:00,220 --> 00:59:03,040 But because it is the same always, 1051 00:59:03,040 --> 00:59:05,440 I might as well just as well not write it. 1052 00:59:05,440 --> 00:59:08,903 So it basically saves me writing. 1053 00:59:08,903 --> 00:59:10,570 And also confusion, because as you know, 1054 00:59:10,570 --> 00:59:12,320 the more things you write, the more likely 1055 00:59:12,320 --> 00:59:15,350 you are to make mistakes. 1056 00:59:15,350 --> 00:59:17,710 It is typical in these derivations. 1057 00:59:17,710 --> 00:59:19,960 If you don't need something, you drop it. 1058 00:59:19,960 --> 00:59:23,578 Because if you start carrying it around, it involves mistakes. 1059 00:59:23,578 --> 00:59:25,120 So that's why I would drop this term. 1060 00:59:29,020 --> 00:59:34,790 So I can just as well say that if I have a propagating wave, 1061 00:59:34,790 --> 00:59:37,290 I can just write it as e to the ikz. 1062 00:59:37,290 --> 00:59:39,290 That's a propagating wave. 1063 00:59:39,290 --> 00:59:40,400 I have a standing wave. 1064 00:59:40,400 --> 00:59:43,800 I can just as well write it cosine kz. 1065 00:59:43,800 --> 00:59:47,140 OK, the e to the i minus i omega t is implicit, right? 1066 00:59:47,140 --> 00:59:48,640 If I have this diffraction integral, 1067 00:59:48,640 --> 00:59:51,987 I can just write it without this term. 1068 00:59:51,987 --> 00:59:53,320 But again, the term is implicit. 1069 00:59:53,320 --> 00:59:57,640 If I have to figure out what is the temporal variation 1070 00:59:57,640 --> 01:00:00,450 of the wave, I have to put it back in. 1071 01:00:00,450 --> 01:00:04,290 And again, I cannot do it if I have two waves of different 1072 01:00:04,290 --> 01:00:04,810 frequencies. 1073 01:00:04,810 --> 01:00:09,110 If I have e to the i omega. 1074 01:00:09,110 --> 01:00:11,300 Let me write another superposition. e 1075 01:00:11,300 --> 01:00:16,850 to the i k1 z minus omega one t, plus e 1076 01:00:16,850 --> 01:00:22,230 to the i k2 z minus omega 2t. 1077 01:00:22,230 --> 01:00:25,440 That's still a valid superposition, right? 1078 01:00:25,440 --> 01:00:31,740 Provided that the ratios of these, k1 over omega 1 1079 01:00:31,740 --> 01:00:33,220 equals c. 1080 01:00:33,220 --> 01:00:35,600 k2 over omega 2 equal c. 1081 01:00:35,600 --> 01:00:36,950 Still a valid superposition. 1082 01:00:36,950 --> 01:00:39,100 I'm not violating anything. 1083 01:00:39,100 --> 01:00:41,770 But I cannot do the phasor trick anymore. 1084 01:00:41,770 --> 01:00:45,480 I have to keep the e to the i omega times, 1085 01:00:45,480 --> 01:00:47,200 or I will make a mistake. 1086 01:00:47,200 --> 01:00:49,630 There's one more case when if you 1087 01:00:49,630 --> 01:00:52,215 produce any kind of nonlinearity, 1088 01:00:52,215 --> 01:00:55,650 where well we'll see later. 1089 01:00:55,650 --> 01:00:58,930 And again, we're going a little bit ahead. 1090 01:00:58,930 --> 01:01:01,360 I'm in trouble. 1091 01:01:01,360 --> 01:01:03,840 These are from the quiz. 1092 01:01:03,840 --> 01:01:06,060 We'll a little bit later. 1093 01:01:06,060 --> 01:01:10,220 When we try to compute the flux of electromagnetic energy, 1094 01:01:10,220 --> 01:01:12,630 there's something called the pointing vector, which 1095 01:01:12,630 --> 01:01:19,810 is defined as the electric field cross with the magnetic field. 1096 01:01:19,810 --> 01:01:22,690 If you tried to write the expression for the pointing 1097 01:01:22,690 --> 01:01:28,200 vector, you cannot just multiply the phasors of the electric 1098 01:01:28,200 --> 01:01:31,310 and the magnetic field, because it's a product. 1099 01:01:31,310 --> 01:01:34,330 So you can use phasors only when you have linear operations, 1100 01:01:34,330 --> 01:01:36,660 such as addition or integrals. 1101 01:01:36,660 --> 01:01:37,660 In this case, you can't. 1102 01:01:37,660 --> 01:01:39,202 We will see what you do in this case. 1103 01:01:39,202 --> 01:01:42,620 You do a temporal average. 1104 01:01:42,620 --> 01:01:46,090 There's a few cases where one has 1105 01:01:46,090 --> 01:01:48,510 to be careful with this phasor.