1 00:00:02,550 --> 00:00:04,920 The following content is provided under a Creative 2 00:00:04,920 --> 00:00:06,310 Commons license. 3 00:00:06,310 --> 00:00:08,520 Your support will help MIT OpenCourseWare 4 00:00:08,520 --> 00:00:12,610 continue to offer high quality educational resources for free. 5 00:00:12,610 --> 00:00:15,150 To make a donation or to view additional materials 6 00:00:15,150 --> 00:00:19,110 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,110 --> 00:00:20,154 at ocw.mit.edu. 8 00:00:24,630 --> 00:00:29,520 YEN-JIE LEE: So welcome back again to 8.03. 9 00:00:29,520 --> 00:00:34,045 Today, my plan is to continue the discussion of the two 10 00:00:34,045 --> 00:00:38,100 string system, which we were working really hard last time 11 00:00:38,100 --> 00:00:40,210 but we sort of run out of time. 12 00:00:40,210 --> 00:00:43,830 So we didn't have time to enjoy what we have done, right? 13 00:00:43,830 --> 00:00:46,560 So today we are going to discuss all the outcome 14 00:00:46,560 --> 00:00:48,270 of that calculation. 15 00:00:48,270 --> 00:00:55,970 And so we will start to discuss more examples which can be 16 00:00:55,970 --> 00:00:58,180 described by the wave equation. 17 00:00:58,180 --> 00:01:01,950 Today we are going to talk about another example, which 18 00:01:01,950 --> 00:01:04,099 is sound waves. 19 00:01:04,099 --> 00:01:06,300 It's a very exciting topic. 20 00:01:06,300 --> 00:01:10,350 And afterwards, we will start the discussion 21 00:01:10,350 --> 00:01:11,965 about electromagnetic waves. 22 00:01:14,580 --> 00:01:17,640 So this is the wave equation which we have been using, 23 00:01:17,640 --> 00:01:19,970 and over the last few lectures, we 24 00:01:19,970 --> 00:01:24,150 have been discussing two specials kinds of solutions-- 25 00:01:24,150 --> 00:01:26,960 the normal modes, which is actually 26 00:01:26,960 --> 00:01:30,710 standing waves in the end, which we identified, 27 00:01:30,710 --> 00:01:32,930 and the progressing wave solution, which 28 00:01:32,930 --> 00:01:37,010 is very powerful in describing the phenomena which 29 00:01:37,010 --> 00:01:40,160 we are familiar with. 30 00:01:40,160 --> 00:01:42,650 Last time, in the end of the lecture, 31 00:01:42,650 --> 00:01:47,370 we were discussing about an interesting example, 32 00:01:47,370 --> 00:01:51,210 which involves two strings in the system. 33 00:01:51,210 --> 00:01:57,440 One essentially in the left-hand side have mass per unit length, 34 00:01:57,440 --> 00:02:02,720 rho l equals to rho 1, and right-hand side one is thicker, 35 00:02:02,720 --> 00:02:06,070 and therefore, the mass per unit length, rho, 36 00:02:06,070 --> 00:02:09,199 is larger, which is called rho 2. 37 00:02:09,199 --> 00:02:11,450 What we did last time is to assume 38 00:02:11,450 --> 00:02:15,260 that we have a progressing wave, which essentially 39 00:02:15,260 --> 00:02:16,220 going into this-- 40 00:02:19,540 --> 00:02:23,240 which essentially first initiated in the left-hand side 41 00:02:23,240 --> 00:02:29,510 string and it's going towards the boundary of the two 42 00:02:29,510 --> 00:02:31,040 systems-- 43 00:02:31,040 --> 00:02:35,220 the more massive one and the less massive one. 44 00:02:35,220 --> 00:02:37,590 And see what is going to happen. 45 00:02:37,590 --> 00:02:41,100 And what we actually tried to describe last time 46 00:02:41,100 --> 00:02:44,660 is that we actually define incident wave described 47 00:02:44,660 --> 00:02:51,130 by this fi function, transmitting wave, ft, 48 00:02:51,130 --> 00:02:54,290 and the refractive wave, fr. 49 00:02:54,290 --> 00:02:58,310 By using boundary conditions which we described last time 50 00:02:58,310 --> 00:03:00,860 we can conclude that there's a fixed 51 00:03:00,860 --> 00:03:09,270 relation between the three waves equation-- wave functions. 52 00:03:09,270 --> 00:03:11,090 What we actually concluded that last time 53 00:03:11,090 --> 00:03:18,240 is that fr is proportional to fi, which is the incident wave 54 00:03:18,240 --> 00:03:22,610 function, by some constant, which is called r. 55 00:03:22,610 --> 00:03:24,280 And what is r? 56 00:03:24,280 --> 00:03:29,900 As we solved last time, it's V2 minus V1 over V1 plus V2, 57 00:03:29,900 --> 00:03:34,250 which are the velocities of the first and second string. 58 00:03:34,250 --> 00:03:39,290 And the transmitted wave, ft, is actually also proportional 59 00:03:39,290 --> 00:03:43,950 to the incident wave. 60 00:03:43,950 --> 00:03:48,950 And the coefficient-- we call it tau-- 61 00:03:48,950 --> 00:03:54,050 to describe the amplitude of the transmitted wave. 62 00:03:54,050 --> 00:03:56,900 So what we can do is the variance. 63 00:03:56,900 --> 00:04:01,820 So we actually discussed two examples last time, 64 00:04:01,820 --> 00:04:04,820 plugging in V1 and V2. 65 00:04:04,820 --> 00:04:07,820 And there are two more examples we can actually 66 00:04:07,820 --> 00:04:12,110 make use of this equation we obtained last time to discuss 67 00:04:12,110 --> 00:04:14,540 what would be the physics outcome 68 00:04:14,540 --> 00:04:16,290 of this kind of situation. 69 00:04:16,290 --> 00:04:20,750 So the first example which we can actually discuss 70 00:04:20,750 --> 00:04:24,460 is that, OK, now I have this string. 71 00:04:24,460 --> 00:04:31,190 Assume, though, they're connected to a wall. 72 00:04:31,190 --> 00:04:33,530 So this time we can say, oh, wait, wait, wait, wait 73 00:04:33,530 --> 00:04:34,030 a second. 74 00:04:34,030 --> 00:04:35,960 There's only one string now, right? 75 00:04:35,960 --> 00:04:38,570 But last time we were solving two strings, right? 76 00:04:38,570 --> 00:04:42,310 But what I'm doing now is to treat the wall 77 00:04:42,310 --> 00:04:44,300 as if it's a string. 78 00:04:44,300 --> 00:04:47,720 But this wall is really massive. 79 00:04:47,720 --> 00:04:54,080 Therefore, the rho l, or the mass per unit length, 80 00:04:54,080 --> 00:04:55,410 is really large. 81 00:04:55,410 --> 00:04:55,910 OK. 82 00:04:55,910 --> 00:04:59,540 It goes to infinity. 83 00:04:59,540 --> 00:05:02,570 If that's the case, if you're set this idea-- 84 00:05:02,570 --> 00:05:06,990 this is still a two string system-- 85 00:05:06,990 --> 00:05:09,200 then I can now go ahead and calculate 86 00:05:09,200 --> 00:05:11,510 what will be the velocity. 87 00:05:11,510 --> 00:05:17,240 The velocity V2 goes to 0. 88 00:05:17,240 --> 00:05:21,340 Then I can go ahead and plug into my equations. 89 00:05:21,340 --> 00:05:25,610 So we spend a lot of time in the last lecture 90 00:05:25,610 --> 00:05:27,650 to obtain those equations. 91 00:05:27,650 --> 00:05:32,900 And we will find that if I have V2 goes to 0, 92 00:05:32,900 --> 00:05:36,560 I have r equal to minus 1. 93 00:05:36,560 --> 00:05:40,400 And the tau, which is actually related 94 00:05:40,400 --> 00:05:43,010 to the amplitude of the transmitted wave, 95 00:05:43,010 --> 00:05:47,070 is equal to 0, which you can see from this equation. 96 00:05:47,070 --> 00:05:48,590 What does that mean? 97 00:05:48,590 --> 00:05:51,890 That means once we solve that question, 98 00:05:51,890 --> 00:05:57,800 we also know what would be the outcome of this experiment, 99 00:05:57,800 --> 00:05:59,090 this physical situation. 100 00:05:59,090 --> 00:06:02,390 When we have a string attached to a wall 101 00:06:02,390 --> 00:06:06,920 and we have an incident wave, as a function of time, what 102 00:06:06,920 --> 00:06:10,370 is going to happen afterward is that this wave is 103 00:06:10,370 --> 00:06:13,930 going to propagate and hit the wall and get 104 00:06:13,930 --> 00:06:17,210 refracted completely. 105 00:06:17,210 --> 00:06:21,230 The amplitude ratio is minus 1. 106 00:06:21,230 --> 00:06:25,280 Therefore, all the energy is refracted 107 00:06:25,280 --> 00:06:30,800 by the wall in this highly idealized situation. 108 00:06:30,800 --> 00:06:33,150 So that's kind of interesting. 109 00:06:33,150 --> 00:06:37,440 The second example is also very interesting. 110 00:06:37,440 --> 00:06:42,740 So if you have a string attached to a massive ring, 111 00:06:42,740 --> 00:06:51,850 and this ring can go up and down freely without any friction, 112 00:06:51,850 --> 00:06:55,750 you can again say, no, no, no, this is again a single string 113 00:06:55,750 --> 00:06:57,040 system, right? 114 00:06:57,040 --> 00:07:00,490 But what I'm going to argue now is that, OK, 115 00:07:00,490 --> 00:07:05,680 there's another string which is so light mass per unit length 116 00:07:05,680 --> 00:07:09,340 is close to 0. 117 00:07:09,340 --> 00:07:13,000 So it is actually in the air. 118 00:07:13,000 --> 00:07:16,150 If I do that, what is going to happen? 119 00:07:16,150 --> 00:07:21,790 The rho l is going to go to the limit of 0, 120 00:07:21,790 --> 00:07:27,070 because the right-hand side you almost cannot see it. 121 00:07:27,070 --> 00:07:31,300 And the V2, which is the velocity 122 00:07:31,300 --> 00:07:36,130 of the transmitted wave, goes to infinity. 123 00:07:36,130 --> 00:07:38,300 If that's the case, you can then again, 124 00:07:38,300 --> 00:07:40,930 plugging into this equation we obtained, and then 125 00:07:40,930 --> 00:07:46,030 you conclude that r would be equal to 1 126 00:07:46,030 --> 00:07:50,270 and tau will be equal to 2. 127 00:07:50,270 --> 00:07:51,590 So what does that mean? 128 00:07:51,590 --> 00:07:55,600 This means that if you have this end, which 129 00:07:55,600 --> 00:07:59,230 is the open end, attached to a ring which 130 00:07:59,230 --> 00:08:04,740 can move up and down, then you get that refraction. 131 00:08:04,740 --> 00:08:07,280 The amplitude of the refractive wave 132 00:08:07,280 --> 00:08:11,950 doesn't change, because i is equal to 1. 133 00:08:11,950 --> 00:08:16,980 So it's still in the positive direction we defined. 134 00:08:16,980 --> 00:08:19,060 But now, it goes backward. 135 00:08:19,060 --> 00:08:22,700 Again, all the energy is actually refracted, 136 00:08:22,700 --> 00:08:25,000 as you can see from this equation. 137 00:08:25,000 --> 00:08:27,370 But the curious case is that-- 138 00:08:27,370 --> 00:08:33,350 the strangest thing is that the tau is equal to 2. 139 00:08:33,350 --> 00:08:35,159 That's kind of strange, right? 140 00:08:35,159 --> 00:08:37,347 Tau is equal to 2. 141 00:08:37,347 --> 00:08:38,180 What does that mean? 142 00:08:38,180 --> 00:08:43,840 That means you are going to predict a transmitted wave 143 00:08:43,840 --> 00:08:49,250 with amplitude exactly the two times the incident wave, 144 00:08:49,250 --> 00:08:52,390 and it's going to be propagating in the right-hand side 145 00:08:52,390 --> 00:08:55,180 and the speed goes to infinity. 146 00:08:57,970 --> 00:08:59,070 What does that mean? 147 00:08:59,070 --> 00:09:02,650 Does that mean the energy is not conserved? 148 00:09:02,650 --> 00:09:07,660 Have we found the cure of the energy crisis? 149 00:09:07,660 --> 00:09:12,530 Because now-- I can actually take all those energy. 150 00:09:12,530 --> 00:09:14,480 I can design this thing, and then this thing 151 00:09:14,480 --> 00:09:17,290 will bounce around all over the place. 152 00:09:17,290 --> 00:09:20,180 And that is going to emit energy. 153 00:09:20,180 --> 00:09:22,550 Oh my god, we solve all the problem. 154 00:09:22,550 --> 00:09:25,650 You should be really excited about it, right? 155 00:09:25,650 --> 00:09:28,430 No? 156 00:09:28,430 --> 00:09:33,430 But unfortunately, rho l goes to 0. 157 00:09:33,430 --> 00:09:38,540 So there's actually nothing oscillating out of this system. 158 00:09:38,540 --> 00:09:43,100 So therefore, there is no additional energy radiated out 159 00:09:43,100 --> 00:09:44,360 of this system. 160 00:09:44,360 --> 00:09:45,290 Too bad. 161 00:09:45,290 --> 00:09:48,530 Go back to work. 162 00:09:48,530 --> 00:09:49,520 All right. 163 00:09:49,520 --> 00:09:52,220 So that's actually what we discussed last time, 164 00:09:52,220 --> 00:09:54,920 and I hope that complete the loop. 165 00:09:54,920 --> 00:10:00,800 And today, before we actually move to sound wave, 166 00:10:00,800 --> 00:10:04,000 I would like to talk about, very briefly, 167 00:10:04,000 --> 00:10:08,060 harmonic progressing waves. 168 00:10:08,060 --> 00:10:13,580 So now, we can see that harmonic progressing wave looks really 169 00:10:13,580 --> 00:10:15,740 beautiful, as you can see here. 170 00:10:15,740 --> 00:10:23,820 And it can be described by a cosine kx minus omega t plus 5. 171 00:10:23,820 --> 00:10:25,600 5 is actually the face. 172 00:10:25,600 --> 00:10:30,620 And you can always write it in different forms. 173 00:10:30,620 --> 00:10:33,770 And since we have learned how to describe 174 00:10:33,770 --> 00:10:37,670 in general the progressing wave, this is just to remind you 175 00:10:37,670 --> 00:10:39,880 that, OK, there's no proper notion 176 00:10:39,880 --> 00:10:43,800 to describe a harmonic progressing wave. 177 00:10:47,450 --> 00:10:52,520 So since we have learned about waves, 178 00:10:52,520 --> 00:10:58,160 which involve oscillation in the transverse direction. 179 00:10:58,160 --> 00:11:00,540 So basically, we always say, OK, things 180 00:11:00,540 --> 00:11:05,000 are oscillating up and down in the case of string. 181 00:11:05,000 --> 00:11:07,990 Before I start, though, the sound wave, there's 182 00:11:07,990 --> 00:11:12,830 a different kind of wave which we can also see 183 00:11:12,830 --> 00:11:15,480 very often in the daily life. 184 00:11:15,480 --> 00:11:19,220 This is called longitudinal waves. 185 00:11:19,220 --> 00:11:25,100 For example, I can have a spring wave, and I can actually-- 186 00:11:25,100 --> 00:11:26,890 imagine I have a spring wave. 187 00:11:26,890 --> 00:11:28,220 And I can do this. 188 00:11:28,220 --> 00:11:31,250 I oscillate in the horizontal direction. 189 00:11:31,250 --> 00:11:37,100 Then that can produce displacement with respect 190 00:11:37,100 --> 00:11:39,200 to the equilibrium position. 191 00:11:39,200 --> 00:11:43,970 And this kind of behavior is like a density wave. 192 00:11:43,970 --> 00:11:46,253 We call it longitudinal waves. 193 00:11:49,040 --> 00:11:53,640 This is exactly what is happening with sound wave. 194 00:11:53,640 --> 00:11:55,180 So what is actually sound wave? 195 00:11:55,180 --> 00:12:00,830 Essentially, a collection or motion of air molecules. 196 00:12:00,830 --> 00:12:04,350 And they are actually oscillating back and forth. 197 00:12:04,350 --> 00:12:11,840 And we may use that to extend energy all over the place. 198 00:12:11,840 --> 00:12:15,560 And today we are going to discuss the sound wave. 199 00:12:15,560 --> 00:12:18,950 And by the way, just for simplicity, 200 00:12:18,950 --> 00:12:24,500 because drawing all those dots really take a lot of time. 201 00:12:24,500 --> 00:12:27,470 So what we sometimes do is that, OK, 202 00:12:27,470 --> 00:12:32,150 we can now draw the pressure, the amplitude of the pressure, 203 00:12:32,150 --> 00:12:35,260 or the amplitude of the displacement 204 00:12:35,260 --> 00:12:39,770 of individual molecules in the discussion 205 00:12:39,770 --> 00:12:40,800 as a function of time. 206 00:12:40,800 --> 00:12:45,190 So if we draw the amplitude as a function of time 207 00:12:45,190 --> 00:12:49,310 or as a function of location, then it 208 00:12:49,310 --> 00:12:52,830 looks exactly the same as what we discussed before 209 00:12:52,830 --> 00:12:56,600 for the transverse waves. 210 00:12:56,600 --> 00:12:58,550 So just some clarification. 211 00:12:58,550 --> 00:13:02,370 It's not like the molecules are going up and down. 212 00:13:02,370 --> 00:13:05,810 They are going back and forth, and it's just a matter 213 00:13:05,810 --> 00:13:08,270 presenting that these are. 214 00:13:08,270 --> 00:13:12,010 So this is actually an example of a travelling wave 215 00:13:12,010 --> 00:13:14,250 in the longitudinal direction. 216 00:13:14,250 --> 00:13:16,470 And you can see that it is actually 217 00:13:16,470 --> 00:13:20,970 the density which is actually changing as a function of time. 218 00:13:20,970 --> 00:13:23,450 And as you can see, it's actually 219 00:13:23,450 --> 00:13:29,410 traveling at a fixed speed and going in the right direction 220 00:13:29,410 --> 00:13:33,460 of the blackboard. 221 00:13:33,460 --> 00:13:38,110 So those being said, we can actually 222 00:13:38,110 --> 00:13:41,780 get started with a concrete example. 223 00:13:41,780 --> 00:13:46,400 So I would like to discuss with you now a system, 224 00:13:46,400 --> 00:13:53,676 which is like you have a tube, with cross section area, A. 225 00:13:53,676 --> 00:13:58,850 So A is actually the area of the cross section. 226 00:13:58,850 --> 00:14:01,850 And I can now wonder-- 227 00:14:01,850 --> 00:14:07,550 now the physics question I'm asking is, what would be the-- 228 00:14:07,550 --> 00:14:14,700 what would be the behavior of the air inside this tube? 229 00:14:14,700 --> 00:14:18,730 So before I go ahead and solve this problem, 230 00:14:18,730 --> 00:14:21,450 I need to define and give you some more information 231 00:14:21,450 --> 00:14:26,580 about this tube and also the condition or the environment 232 00:14:26,580 --> 00:14:29,050 this tube is living in. 233 00:14:29,050 --> 00:14:32,220 So the first information I would like to give you 234 00:14:32,220 --> 00:14:38,080 is that the pressure, the room pressure, 235 00:14:38,080 --> 00:14:42,100 is actually P0 in this example. 236 00:14:42,100 --> 00:14:44,970 So the P0 is actually the room pressure. 237 00:14:44,970 --> 00:14:51,600 And I can now define coordinates is the x direction is actually 238 00:14:51,600 --> 00:14:53,910 in the horizontal direction pointing 239 00:14:53,910 --> 00:14:56,280 to the right-hand side. 240 00:14:56,280 --> 00:15:03,630 And now, I can actually try to describe a small unit volume 241 00:15:03,630 --> 00:15:09,150 inside the tube by location x. 242 00:15:09,150 --> 00:15:15,180 And the width of this volume, I call it delta x. 243 00:15:15,180 --> 00:15:23,092 And if I go ahead and prepare this system and at time, t, 244 00:15:23,092 --> 00:15:25,890 something is happening to this system-- 245 00:15:25,890 --> 00:15:29,960 so now, you need length. 246 00:15:29,960 --> 00:15:32,210 You need volume, I was discussing. 247 00:15:32,210 --> 00:15:38,150 This get displaced with respect to the equilibrium position. 248 00:15:38,150 --> 00:15:40,730 So that means, assuming that something happened 249 00:15:40,730 --> 00:15:46,210 at the t equal to t, the left inside edge of the volume 250 00:15:46,210 --> 00:15:50,760 is shifted toward a positive direction, which is described 251 00:15:50,760 --> 00:15:54,540 by wave function psi x. 252 00:15:54,540 --> 00:15:57,510 And the position of the right-hand side 253 00:15:57,510 --> 00:16:06,240 edge of this volume is shifted to side x plus delta x. 254 00:16:06,240 --> 00:16:09,750 So something happened to this system. 255 00:16:09,750 --> 00:16:12,570 We can also say that-- we can also 256 00:16:12,570 --> 00:16:16,230 describe this system, the pressure of this system, 257 00:16:16,230 --> 00:16:18,030 by P function. 258 00:16:18,030 --> 00:16:22,230 P of x is actually equal to P0, which 259 00:16:22,230 --> 00:16:25,860 is actually room pressure-- 260 00:16:25,860 --> 00:16:29,190 P0 is the baseline room pressure-- 261 00:16:29,190 --> 00:16:39,500 plus some kind of displacement in pressure, psi P. 262 00:16:39,500 --> 00:16:45,010 So now we describe the pressure acting on the left inside edge 263 00:16:45,010 --> 00:16:49,470 of the small unit volume, and the right-hand side, 264 00:16:49,470 --> 00:16:52,000 you can also do the same thing. 265 00:16:52,000 --> 00:16:59,530 P of x plus delta x will be equal to P0 plus psi P, 266 00:16:59,530 --> 00:17:02,450 describing the displacement or how 267 00:17:02,450 --> 00:17:07,300 offset the pressure is as a function of x but now evaluated 268 00:17:07,300 --> 00:17:10,780 at x plus delta x. 269 00:17:10,780 --> 00:17:17,329 So once we have all of those elements defined-- 270 00:17:17,329 --> 00:17:19,000 these are essentially just a copy 271 00:17:19,000 --> 00:17:24,280 of what I have in the slide and those are a reminder here-- 272 00:17:24,280 --> 00:17:30,310 now, we can actually calculate the motion of all the molecules 273 00:17:30,310 --> 00:17:38,670 in this volume, because I have pressure, I have displacement. 274 00:17:38,670 --> 00:17:41,770 The displacement is described by psi, 275 00:17:41,770 --> 00:17:45,880 end position is described by psi, the change in pressure 276 00:17:45,880 --> 00:17:50,260 is described by psi P. And now I can go ahead and apply, 277 00:17:50,260 --> 00:17:52,270 for example, Newton's law. 278 00:17:52,270 --> 00:17:54,070 Then I can calculate what would be 279 00:17:54,070 --> 00:18:02,830 the acceleration for all the molecules inside this volume. 280 00:18:02,830 --> 00:18:05,320 But wait a second. 281 00:18:05,320 --> 00:18:09,260 That sounds all great, but I don't know yet 282 00:18:09,260 --> 00:18:13,040 how to relate pressure and the volume, 283 00:18:13,040 --> 00:18:16,790 because pressure is actually expressed by psi P 284 00:18:16,790 --> 00:18:20,510 and the volume is related to psi. 285 00:18:20,510 --> 00:18:25,070 I need to know is actually the relation between pressure 286 00:18:25,070 --> 00:18:28,190 and the displacement or pressure between psi 287 00:18:28,190 --> 00:18:30,650 so that I can make progress. 288 00:18:30,650 --> 00:18:33,830 So that this actually the main discussion which I would 289 00:18:33,830 --> 00:18:37,610 like to do in this lecture. 290 00:18:37,610 --> 00:18:41,970 So given those information, I can now 291 00:18:41,970 --> 00:18:47,540 calculate what is actually the change in this little volume. 292 00:18:47,540 --> 00:18:55,190 So I can calculate the change in volume, 293 00:18:55,190 --> 00:19:00,380 which is described by delta V. But delta V 294 00:19:00,380 --> 00:19:05,180 can be actually calculated by a, which is actually 295 00:19:05,180 --> 00:19:08,700 the area of the cross section, times 296 00:19:08,700 --> 00:19:19,970 psi x plus delta xt minus psi xt. 297 00:19:19,970 --> 00:19:27,620 So basically, just calculate how much the boundary is actually 298 00:19:27,620 --> 00:19:28,850 displaced. 299 00:19:28,850 --> 00:19:35,630 And if we always take very small amplitude approximation, 300 00:19:35,630 --> 00:19:39,500 then basically this expression is 301 00:19:39,500 --> 00:19:47,870 roughly equal to A partial psi partial x times delta x, 302 00:19:47,870 --> 00:19:50,990 where the delta x is really very small. 303 00:19:50,990 --> 00:19:55,280 So a very small volume I was talking about. 304 00:19:55,280 --> 00:19:59,165 And I can also calculate the pressure. 305 00:20:05,400 --> 00:20:06,990 What is the pressure difference? 306 00:20:06,990 --> 00:20:10,620 The pressure difference is between the pressure 307 00:20:10,620 --> 00:20:14,280 acting in the left-hand side edge and the pressure which 308 00:20:14,280 --> 00:20:17,380 is acting on the right-hand side edge. 309 00:20:17,380 --> 00:20:20,090 So I can now calculate pressure difference, 310 00:20:20,090 --> 00:20:32,500 delta P. Delta P would be minus psi P x plus delta x t plus psi 311 00:20:32,500 --> 00:20:36,230 P x t. 312 00:20:36,230 --> 00:20:40,740 So basically, one is essentially the pressure 313 00:20:40,740 --> 00:20:44,030 pushing the body in the right-hand side. 314 00:20:44,030 --> 00:20:45,530 The other one essentially pushing it 315 00:20:45,530 --> 00:20:47,950 in the left-hand side direction. 316 00:20:47,950 --> 00:20:52,504 Again, I can take very small delta x approximation. 317 00:20:52,504 --> 00:20:54,170 And basically, what you are going to get 318 00:20:54,170 --> 00:21:00,360 is minus partial psi P partial x delta x. 319 00:21:03,790 --> 00:21:06,910 So we have prepared all of those information 320 00:21:06,910 --> 00:21:11,020 about volume and the pressure. 321 00:21:11,020 --> 00:21:13,540 As I mentioned before, the big question 322 00:21:13,540 --> 00:21:17,380 which we would like to ask is, how do I 323 00:21:17,380 --> 00:21:23,110 relate pressure and the volume so that I can make progress? 324 00:21:23,110 --> 00:21:26,530 If I can relate pressure and volume, 325 00:21:26,530 --> 00:21:28,990 then I can know what is the relation 326 00:21:28,990 --> 00:21:32,290 between psi P and the psi. 327 00:21:32,290 --> 00:21:35,260 Then I can ask you to make use of Newton's Law. 328 00:21:35,260 --> 00:21:40,870 Then I can calculate the resulting equation of motion. 329 00:21:40,870 --> 00:21:46,900 So there's two possible interesting scenarios 330 00:21:46,900 --> 00:21:50,530 which we can relate temperature-- so sorry, 331 00:21:50,530 --> 00:21:53,380 relate pressure and the volume. 332 00:21:53,380 --> 00:21:58,870 The first one was proposed by Newton. 333 00:21:58,870 --> 00:22:05,060 Newton said that, OK, this is an interesting phenomena. 334 00:22:05,060 --> 00:22:09,820 In my opinion, although you actually displaced 335 00:22:09,820 --> 00:22:12,820 this volume-- 336 00:22:12,820 --> 00:22:19,780 make the displacement for those molecules in the tube-- 337 00:22:19,780 --> 00:22:23,620 but because the heat was conducted from one region 338 00:22:23,620 --> 00:22:25,510 to the other region, all those regions 339 00:22:25,510 --> 00:22:27,840 are connected to each other. 340 00:22:27,840 --> 00:22:34,460 And the speed of this heat transfer is so fast. 341 00:22:34,460 --> 00:22:37,400 It's really fast, like instant. 342 00:22:37,400 --> 00:22:42,670 This heat is actually transferred from one direction 343 00:22:42,670 --> 00:22:43,360 to the other-- 344 00:22:43,360 --> 00:22:46,530 one position to the other position. 345 00:22:46,530 --> 00:22:55,030 Therefore, over the course of this evolution, 346 00:22:55,030 --> 00:22:59,040 the temperature should be unchanged. 347 00:22:59,040 --> 00:23:03,610 No matter what you do to the air inside the tube, 348 00:23:03,610 --> 00:23:05,500 the temperature should be unchanged, 349 00:23:05,500 --> 00:23:11,860 because Newton thinks that heat should be-- 350 00:23:11,860 --> 00:23:13,725 the speed of the heat distribution 351 00:23:13,725 --> 00:23:15,880 is really, really fast. 352 00:23:15,880 --> 00:23:18,550 Much faster than all those vibration 353 00:23:18,550 --> 00:23:21,760 happening in the tube. 354 00:23:21,760 --> 00:23:25,270 If that is the case-- 355 00:23:25,270 --> 00:23:27,050 that is the case-- 356 00:23:27,050 --> 00:23:36,300 then that means we can use ideal gas law. 357 00:23:41,670 --> 00:23:44,825 P times V is equal to nRT. 358 00:23:50,480 --> 00:23:56,570 I hope that you have learned this before in 8.01 and 8.02. 359 00:23:56,570 --> 00:23:58,750 If that's the case, that means-- 360 00:23:58,750 --> 00:24:00,890 so all those things are constant, 361 00:24:00,890 --> 00:24:05,770 because we assume that temperature is unchanged. 362 00:24:12,900 --> 00:24:17,080 Therefore, the right-hand side is essentially a constant. 363 00:24:17,080 --> 00:24:23,290 Therefore, P times V would be some kind of constant. 364 00:24:23,290 --> 00:24:32,710 The V would be proportional to 1/P. 365 00:24:32,710 --> 00:24:35,320 So that essentially is the first idea, 366 00:24:35,320 --> 00:24:38,390 which is coming from Newton, in order 367 00:24:38,390 --> 00:24:43,450 to relate pressure and the volume. 368 00:24:47,240 --> 00:24:52,120 The second idea is coming from Laplace. 369 00:24:52,120 --> 00:24:56,350 Laplace says, OK, he has a different opinion 370 00:24:56,350 --> 00:24:58,210 on this matter. 371 00:24:58,210 --> 00:25:04,570 He think that this essentially is an adiabatic process. 372 00:25:04,570 --> 00:25:06,210 What does that mean? 373 00:25:06,210 --> 00:25:12,220 That means the heat flow from the compressed region 374 00:25:12,220 --> 00:25:17,430 to the other region is really negligible, 375 00:25:17,430 --> 00:25:20,190 because the oscillation is really fast 376 00:25:20,190 --> 00:25:24,980 and the speed of the transfer of the heat 377 00:25:24,980 --> 00:25:30,330 is really slow compared to the time scale of the oscillation. 378 00:25:30,330 --> 00:25:35,970 Therefore, in Laplace's opinion, he 379 00:25:35,970 --> 00:25:40,485 thinks that the whole process is adiabatic process. 380 00:25:50,790 --> 00:25:55,150 If that's the case, which I will show you later, 381 00:25:55,150 --> 00:26:01,490 that means you have this relation between pressure 382 00:26:01,490 --> 00:26:02,580 and the volume. 383 00:26:02,580 --> 00:26:05,890 P times V to the gamma. 384 00:26:05,890 --> 00:26:09,140 Gamma essentially related to the decrease 385 00:26:09,140 --> 00:26:13,650 of freedom of the molecule, which we will discuss later 386 00:26:13,650 --> 00:26:15,910 in the class. 387 00:26:15,910 --> 00:26:19,615 This would be equal to constant. 388 00:26:24,070 --> 00:26:27,490 So the very interesting thing of this lecture 389 00:26:27,490 --> 00:26:33,460 is that we are going to be able to test which one is correct. 390 00:26:33,460 --> 00:26:37,930 You will be able to see if Newton win or Laplace win. 391 00:26:40,690 --> 00:26:44,080 So as I mentioned before, one is assuming 392 00:26:44,080 --> 00:26:49,540 the heat transfer, the speed of the heat propagation, 393 00:26:49,540 --> 00:26:54,490 is really, really much larger than the speed of oscillation. 394 00:26:54,490 --> 00:26:57,010 The other viewpoint from Laplace is 395 00:26:57,010 --> 00:27:01,420 that the heat flow is actually really negligible 396 00:27:01,420 --> 00:27:05,350 compared to the oscillation we are talking about here. 397 00:27:05,350 --> 00:27:09,340 And now, as usual, I would like to have a vote now. 398 00:27:09,340 --> 00:27:15,910 How many of you support Newton's idea? 399 00:27:15,910 --> 00:27:22,716 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11-- 400 00:27:30,020 --> 00:27:34,604 so 15 of you vote for Newton. 401 00:27:38,510 --> 00:27:42,060 How many of you saying Laplace is correct? 402 00:28:01,990 --> 00:28:04,950 How about the others? 403 00:28:04,950 --> 00:28:06,390 OK, very good. 404 00:28:06,390 --> 00:28:12,510 So we have a majority of you support the idea of Laplace, 405 00:28:12,510 --> 00:28:15,550 and some of you actually support Newton. 406 00:28:15,550 --> 00:28:19,590 And we are going to see what is going to happen in the lecture 407 00:28:19,590 --> 00:28:21,510 today. 408 00:28:21,510 --> 00:28:26,160 So let's go ahead and apply these two ideas. 409 00:28:26,160 --> 00:28:29,730 So PV gamma equal to constant. 410 00:28:29,730 --> 00:28:34,760 So in the case if ideal gas law, gamma is equal to 1. 411 00:28:34,760 --> 00:28:37,927 Therefore, I just have to work on these function of form. 412 00:28:37,927 --> 00:28:39,510 And then also later we will figure out 413 00:28:39,510 --> 00:28:43,500 what is gamma all together. 414 00:28:43,500 --> 00:28:48,360 So that's consider only small vibration. 415 00:28:48,360 --> 00:28:56,940 Small vibration means-- small vibration means 416 00:28:56,940 --> 00:29:01,170 that I have psi P, which essentially, 417 00:29:01,170 --> 00:29:07,470 from this definition, psi P is the change 418 00:29:07,470 --> 00:29:12,840 in pressure with respect to the room pressure, P0. 419 00:29:12,840 --> 00:29:18,060 So psi P, assuming that's much, much smaller than P0. 420 00:29:21,210 --> 00:29:27,810 And also, I assume that the changing volume, delta V, which 421 00:29:27,810 --> 00:29:30,870 I calculated there, is much, much smaller 422 00:29:30,870 --> 00:29:36,600 than P0, which is essentially the original volume 423 00:29:36,600 --> 00:29:42,050 of this little area-- 424 00:29:42,050 --> 00:29:48,420 original volume of this area I was working here. 425 00:29:48,420 --> 00:29:49,260 All right. 426 00:29:49,260 --> 00:29:51,055 So that's the two assumption. 427 00:29:54,660 --> 00:30:00,920 Before I change the position of the boundary, which essentially 428 00:30:00,920 --> 00:30:09,340 is the upper graph, if I change anything, I have P0 V0 gamma. 429 00:30:09,340 --> 00:30:17,310 This is equal to some constant, C. And gamma can be equal to 1 430 00:30:17,310 --> 00:30:19,900 so that you have ideal gas law. 431 00:30:23,180 --> 00:30:28,530 After the vibration initially happened, 432 00:30:28,530 --> 00:30:34,470 after the wall's initially displaced from the equilibrium 433 00:30:34,470 --> 00:30:38,110 position, what I'm going to get is-- 434 00:30:38,110 --> 00:30:47,910 I will have P plus delta P times V0 plus delta V to the gamma. 435 00:30:47,910 --> 00:30:56,080 This is equal to C. 436 00:30:56,080 --> 00:31:01,120 So based on those, actually I have already 437 00:31:01,120 --> 00:31:14,650 calculated delta P and delta V. So in this case, this delta P 438 00:31:14,650 --> 00:31:17,190 should be-- 439 00:31:17,190 --> 00:31:19,040 OK, this is actually not the delta P 440 00:31:19,040 --> 00:31:23,020 I was talking about there, so I should change it to delta P. 441 00:31:23,020 --> 00:31:27,120 Because that's, essentially, the difference 442 00:31:27,120 --> 00:31:31,440 between the resulting pressure and the original pressure. 443 00:31:31,440 --> 00:31:34,416 It's not the difference between the left-hand side pressure 444 00:31:34,416 --> 00:31:35,790 and the right-hand side pressure, 445 00:31:35,790 --> 00:31:39,180 which essentially is showing there. 446 00:31:39,180 --> 00:31:43,960 So this delta P, which I have already defined, 447 00:31:43,960 --> 00:31:49,190 essentially called psi P. Therefore, I 448 00:31:49,190 --> 00:31:59,120 will derive this to be P plus psi P. 449 00:31:59,120 --> 00:32:04,130 And I can now also copy these. 450 00:32:04,130 --> 00:32:08,150 You are going to get V0 plus delta V to the gamma. 451 00:32:08,150 --> 00:32:12,160 And this is equal to C. 452 00:32:12,160 --> 00:32:17,470 So as I mentioned before, I'm considering small vibration. 453 00:32:17,470 --> 00:32:21,880 Delta V is much, much smaller than V0. 454 00:32:21,880 --> 00:32:27,875 Therefore, this expression can be written as P-- 455 00:32:31,540 --> 00:32:33,040 sorry, this should be P0. 456 00:32:33,040 --> 00:32:36,100 I'm making some mistake here. 457 00:32:36,100 --> 00:32:44,160 This expression should be written as P0 plus psi P. 458 00:32:44,160 --> 00:32:46,810 V0 to the gamma. 459 00:32:46,810 --> 00:32:58,620 1 plus gamma delta V over V0, because delta V is much, 460 00:32:58,620 --> 00:33:02,120 much smarter than V0. 461 00:33:02,120 --> 00:33:05,360 Can everybody follow? 462 00:33:05,360 --> 00:33:11,510 So here I will already take small vibration approximation. 463 00:33:11,510 --> 00:33:15,440 And basically, I can now rewrite this thing. 464 00:33:15,440 --> 00:33:17,570 I just expend all those terms. 465 00:33:17,570 --> 00:33:22,400 Basically, I call this equation number two. 466 00:33:22,400 --> 00:33:36,406 Equation number two will become P0 V0 gamma plus gamma delta V 467 00:33:36,406 --> 00:33:48,850 V0 gamma to the minus 1 P0 plus psi P V0 468 00:33:48,850 --> 00:34:04,130 to the gamma plus gamma delta V psi P V0 to the gamma minus 1. 469 00:34:04,130 --> 00:34:05,580 So there's no magic. 470 00:34:05,580 --> 00:34:08,870 Essentially, it's just expanding these terms. 471 00:34:08,870 --> 00:34:11,750 Then, basically, you are going to get four terms. 472 00:34:11,750 --> 00:34:18,020 And basically, if you do write down the equation number 473 00:34:18,020 --> 00:34:20,190 two especially, that is essentially 474 00:34:20,190 --> 00:34:21,320 what you are going to get. 475 00:34:24,199 --> 00:34:27,620 So we are making progress, and we 476 00:34:27,620 --> 00:34:30,139 would like to simplify things. 477 00:34:30,139 --> 00:34:34,090 And you can quickly identify the hardest term, 478 00:34:34,090 --> 00:34:36,230 P0 V0 to the gamma. 479 00:34:36,230 --> 00:34:38,389 I know what is the value of that, right? 480 00:34:38,389 --> 00:34:42,350 That essentially is the original situation, 481 00:34:42,350 --> 00:34:52,000 and that is essentially equal to C. 482 00:34:52,000 --> 00:34:56,409 Let's take a look at also this term. 483 00:34:56,409 --> 00:35:00,010 This term is proportional to what? 484 00:35:00,010 --> 00:35:05,590 Proportional to delta V, which is a very small quantity, 485 00:35:05,590 --> 00:35:12,400 and proportional to psi P, which is another very small quantity. 486 00:35:12,400 --> 00:35:17,540 Therefore, taking a small vibration approximation, 487 00:35:17,540 --> 00:35:19,960 I would just simply ignore this term. 488 00:35:22,590 --> 00:35:24,920 Is everybody following? 489 00:35:24,920 --> 00:35:26,270 All right. 490 00:35:26,270 --> 00:35:29,080 So this term, this original term, 491 00:35:29,080 --> 00:35:32,580 is equal to C based on this expression. 492 00:35:32,580 --> 00:35:38,540 So we start from the system before the vibration happened. 493 00:35:38,540 --> 00:35:42,380 After the vibration happened there's a change in pressure, 494 00:35:42,380 --> 00:35:49,880 there's a change in delta V. But if you multiply PV 495 00:35:49,880 --> 00:35:53,680 to the gamma, this expression is still 496 00:35:53,680 --> 00:35:56,240 equal to C, some kind of constant. 497 00:35:56,240 --> 00:36:02,060 And then, now I do small vibration approximation, 498 00:36:02,060 --> 00:36:05,990 and I drop the term which is actually proportional 499 00:36:05,990 --> 00:36:10,730 to delta V times psi P. And basically what I get 500 00:36:10,730 --> 00:36:17,760 is that C is equal to C plus gamma delta P V0 501 00:36:17,760 --> 00:36:23,510 gamma minus 1 P0 plus psi P V0 to the gamma. 502 00:36:28,480 --> 00:36:32,890 This term, these two constants cancel. 503 00:36:32,890 --> 00:36:36,940 And now, I can actually move one of the terms 504 00:36:36,940 --> 00:36:38,280 to the left-hand side. 505 00:36:38,280 --> 00:36:40,620 Then basically, what I am going to get 506 00:36:40,620 --> 00:36:48,700 is psi P V0 to the gamma would be equal to minus gamma delta V 507 00:36:48,700 --> 00:36:51,670 gamma 0 gamma to the-- 508 00:36:51,670 --> 00:36:59,260 V0 to the gamma minus 1 times P0. 509 00:36:59,260 --> 00:37:02,935 We can't immediately cancel V0 to the gamma. 510 00:37:06,660 --> 00:37:09,000 Therefore, what are we going to get? 511 00:37:09,000 --> 00:37:18,195 I'm getting psi P would be equal to minus gamma P0 over V0 512 00:37:18,195 --> 00:37:26,002 delta V. Everybody following? 513 00:37:33,410 --> 00:37:35,300 So this is essentially the expression. 514 00:37:35,300 --> 00:37:38,220 And we also know why essentially is 515 00:37:38,220 --> 00:37:42,470 delta V. Based on this expression, 516 00:37:42,470 --> 00:37:48,920 delta V is essentially A partial psi partial x delta x, 517 00:37:48,920 --> 00:37:52,310 if we look at the upper board, which we actually just 518 00:37:52,310 --> 00:37:54,620 derived a moment ago. 519 00:37:54,620 --> 00:38:00,580 Therefore, I can write, replace delta V by that expression. 520 00:38:00,580 --> 00:38:05,090 Then basically what I get is psi P would be equal 521 00:38:05,090 --> 00:38:15,396 to minus gamma P0 A delta x divided by V0 partial psi 522 00:38:15,396 --> 00:38:17,180 partial x. 523 00:38:17,180 --> 00:38:21,360 I'm just plugging in the expression for delta V 524 00:38:21,360 --> 00:38:22,930 to that equation. 525 00:38:22,930 --> 00:38:25,460 A lot of mathematics, but all of them 526 00:38:25,460 --> 00:38:27,740 should be pretty straightforward. 527 00:38:27,740 --> 00:38:29,890 You don't actually have to copy because all of them 528 00:38:29,890 --> 00:38:31,440 are in the lecture note. 529 00:38:31,440 --> 00:38:32,810 OK? 530 00:38:32,810 --> 00:38:34,010 All right. 531 00:38:34,010 --> 00:38:36,800 So here, you can see I can, again, 532 00:38:36,800 --> 00:38:39,620 simplify this expression. 533 00:38:39,620 --> 00:38:44,210 A is actually the cross section of the tube, 534 00:38:44,210 --> 00:38:48,470 and delta x is the width in the x direction. 535 00:38:48,470 --> 00:38:53,400 So A times delta x is just V0. 536 00:38:53,400 --> 00:38:55,070 Oh, very good! 537 00:38:55,070 --> 00:39:00,040 I get this very simple expression, minus gamma P0 538 00:39:00,040 --> 00:39:01,776 partial psi partial x. 539 00:39:04,760 --> 00:39:09,920 So we have achieved our goal to simplify the expression 540 00:39:09,920 --> 00:39:15,050 and to find the relation between psi P and the psi. 541 00:39:15,050 --> 00:39:19,660 That's actually what originally we were hoping to do, 542 00:39:19,660 --> 00:39:22,340 and we have achieved that. 543 00:39:22,340 --> 00:39:26,080 And I call it equation number three here. 544 00:39:26,080 --> 00:39:29,590 Don't forget what is psi and psi P. Psi 545 00:39:29,590 --> 00:39:34,270 P is the amount of change in pressure, 546 00:39:34,270 --> 00:39:39,400 and the psi is the amount of displacement of the wall-- 547 00:39:39,400 --> 00:39:42,120 of the molecule in the volume. 548 00:39:45,600 --> 00:39:50,460 So now, I'm really close to my solution, 549 00:39:50,460 --> 00:39:55,170 because now I can now calculate the force acting 550 00:39:55,170 --> 00:39:58,260 on this little volume, because now I 551 00:39:58,260 --> 00:40:01,860 know what is the pressure. 552 00:40:01,860 --> 00:40:05,250 So what is the F total? 553 00:40:05,250 --> 00:40:08,700 The F total is essentially delta P, 554 00:40:08,700 --> 00:40:11,760 which is the difference in pressure 555 00:40:11,760 --> 00:40:15,011 from the left-hand side end compared to the right-hand side 556 00:40:15,011 --> 00:40:15,510 end. 557 00:40:15,510 --> 00:40:17,705 So that's what we calculated before. 558 00:40:20,640 --> 00:40:30,240 Now this, A partial psi P partial x delta x. 559 00:40:30,240 --> 00:40:32,625 We also know what would be the mass. 560 00:40:35,820 --> 00:40:41,300 We know that the little mass in this volume, delta m, 561 00:40:41,300 --> 00:40:49,100 will be equal to rho, which is the density of the air, 562 00:40:49,100 --> 00:40:53,970 times A, which is the cross section, times delta x. 563 00:40:53,970 --> 00:40:59,130 That will give you the little area, V0. 564 00:40:59,130 --> 00:41:05,640 So rho times A times delta x will be your delta m. 565 00:41:05,640 --> 00:41:08,370 We are almost there. 566 00:41:08,370 --> 00:41:12,750 I have the m, I have the force, what kind of law 567 00:41:12,750 --> 00:41:15,782 do I need to use to get my equation of motion? 568 00:41:15,782 --> 00:41:16,740 AUDIENCE: Newton's law. 569 00:41:16,740 --> 00:41:18,360 YEN-JIE LEE: Newton's law, right? 570 00:41:18,360 --> 00:41:20,100 Newton's law. 571 00:41:20,100 --> 00:41:23,670 So F is equal to m times a, right? 572 00:41:23,670 --> 00:41:26,950 So therefore, I can now calculate 573 00:41:26,950 --> 00:41:33,400 and essentially I can now plug in rho times A times delta x. 574 00:41:33,400 --> 00:41:34,780 What is A? 575 00:41:34,780 --> 00:41:38,630 A is essentially psi double-dot. 576 00:41:38,630 --> 00:41:42,630 That's essentially describing the displacement with respect 577 00:41:42,630 --> 00:41:45,420 to the equilibrium position. 578 00:41:45,420 --> 00:41:53,400 And now I know this is equal to force, which is A times delta x 579 00:41:53,400 --> 00:41:57,056 partial psi P partial x. 580 00:42:02,620 --> 00:42:05,680 And you can see that both ends you have a delta x. 581 00:42:05,680 --> 00:42:07,960 So I can cancel that. 582 00:42:07,960 --> 00:42:11,140 Both ends you have an A, so now I can cancel that. 583 00:42:11,140 --> 00:42:15,130 Then basically you get rho psi double-prime. 584 00:42:15,130 --> 00:42:20,240 This is equal to partial psi P partial x. 585 00:42:23,580 --> 00:42:26,755 From the beginning we're talking about the relation between psi 586 00:42:26,755 --> 00:42:32,430 P, the displacement in pressure, and psi, how much 587 00:42:32,430 --> 00:42:35,430 the molecules are displaced. 588 00:42:35,430 --> 00:42:38,460 And then we have the solution here. 589 00:42:38,460 --> 00:42:41,550 If you assume the relation which was 590 00:42:41,550 --> 00:42:46,230 given by Newton or by Laplace, basically, you 591 00:42:46,230 --> 00:42:53,490 can conclude that this would be equal to gamma 592 00:42:53,490 --> 00:43:00,370 P0 partial square psi P partial x square. 593 00:43:10,560 --> 00:43:11,490 All right. 594 00:43:11,490 --> 00:43:14,220 And I can now put all the constants 595 00:43:14,220 --> 00:43:15,300 to the right-hand side. 596 00:43:18,880 --> 00:43:24,880 Basically, what you get is psi double-prime. 597 00:43:24,880 --> 00:43:27,640 So here it should be partial square psi partial x square, 598 00:43:27,640 --> 00:43:33,500 because I replaced psi P by psi. 599 00:43:33,500 --> 00:43:36,230 And I must miss one-- 600 00:43:36,230 --> 00:43:39,320 I must miss one negative sign somewhere. 601 00:43:39,320 --> 00:43:40,220 AUDIENCE: Over there. 602 00:43:40,220 --> 00:43:40,670 YEN-JIE LEE: Where? 603 00:43:40,670 --> 00:43:41,570 AUDIENCE: [INAUDIBLE] 604 00:43:41,570 --> 00:43:43,320 YEN-JIE LEE: Oh, this essentially-- 605 00:43:43,320 --> 00:43:44,736 there's a minus sign there, right? 606 00:43:44,736 --> 00:43:49,692 AUDIENCE: [INAUDIBLE] On the left side. 607 00:43:49,692 --> 00:43:51,150 YEN-JIE LEE: On the left-hand side. 608 00:43:51,150 --> 00:43:51,450 Yeah. 609 00:43:51,450 --> 00:43:51,790 That's right. 610 00:43:51,790 --> 00:43:52,498 AUDIENCE: Oh, no. 611 00:43:52,498 --> 00:43:55,020 There should be A. That's wrong. 612 00:43:55,020 --> 00:43:57,630 YEN-JIE LEE: Yes, you are right. 613 00:43:57,630 --> 00:44:00,010 And the minus sign should belong there. 614 00:44:00,010 --> 00:44:02,870 So that actually-- sorry for that. 615 00:44:02,870 --> 00:44:06,700 So there should be a minus sign here. 616 00:44:06,700 --> 00:44:08,840 And there should be a minus sign here. 617 00:44:08,840 --> 00:44:12,790 And after I plug in equation number three, 618 00:44:12,790 --> 00:44:16,900 then I get psi double-prime equal to gamma P0 619 00:44:16,900 --> 00:44:21,730 over rho partial square psi partial x square. 620 00:44:21,730 --> 00:44:24,620 Any other problems you find? 621 00:44:24,620 --> 00:44:26,080 Not yet? 622 00:44:26,080 --> 00:44:28,270 OK. 623 00:44:28,270 --> 00:44:30,700 So look at this equation. 624 00:44:30,700 --> 00:44:31,390 Oh my god! 625 00:44:31,390 --> 00:44:32,710 What is this equation? 626 00:44:32,710 --> 00:44:34,300 AUDIENCE: Wave. 627 00:44:34,300 --> 00:44:37,000 YEN-JIE LEE: Wave equation. 628 00:44:37,000 --> 00:44:38,500 Again. 629 00:44:38,500 --> 00:44:39,040 Again. 630 00:44:39,040 --> 00:44:42,010 Wave equation. 631 00:44:42,010 --> 00:44:43,810 You can say that, huh, I'm not surprised, 632 00:44:43,810 --> 00:44:48,930 because this system is used so many times. 633 00:44:48,930 --> 00:44:50,530 I have learned this so many times, 634 00:44:50,530 --> 00:44:53,200 but I am still surprised that this 635 00:44:53,200 --> 00:44:57,970 is so identical to the physics which 636 00:44:57,970 --> 00:45:02,160 we have been studying for the strings 637 00:45:02,160 --> 00:45:07,030 for over the few lectures. 638 00:45:07,030 --> 00:45:08,710 So that's very nice. 639 00:45:08,710 --> 00:45:12,550 And now, the question we have an answer 640 00:45:12,550 --> 00:45:17,230 is that, OK, what essentially is gamma? 641 00:45:17,230 --> 00:45:20,090 What is essentially gamma? 642 00:45:20,090 --> 00:45:26,960 So gamma, in the case of the adiabatic process, 643 00:45:26,960 --> 00:45:33,260 gamma is actually equals to alpha plus 1 minus alpha. 644 00:45:33,260 --> 00:45:39,620 And alpha is related to the number of degrees of freedom. 645 00:45:39,620 --> 00:45:43,190 So if you haven't done this before, 646 00:45:43,190 --> 00:45:47,870 I have a concrete proof of the adiabatic process. 647 00:45:47,870 --> 00:45:51,890 And this is actually coming from the first law 648 00:45:51,890 --> 00:45:54,660 of thermodynamics. 649 00:45:54,660 --> 00:45:57,650 And basically, you will be able to conclude 650 00:45:57,650 --> 00:46:02,645 that gamma will be equal to alpha plus 1 divided by alpha. 651 00:46:05,180 --> 00:46:07,220 The value of alpha-- 652 00:46:07,220 --> 00:46:11,090 the value you get for alpha is related 653 00:46:11,090 --> 00:46:15,680 to how many degrees of freedom you can actually 654 00:46:15,680 --> 00:46:17,930 have in this system. 655 00:46:17,930 --> 00:46:28,700 For example, if I have a system which is made of atomic gas-- 656 00:46:28,700 --> 00:46:33,140 so there's only one atom in a molecule-- 657 00:46:33,140 --> 00:46:36,530 and basically you have 3 degrees of freedom. 658 00:46:36,530 --> 00:46:40,160 So you can move this thing in the horizontal direction. 659 00:46:40,160 --> 00:46:45,130 You can move this atom upside down or back and forth. 660 00:46:45,130 --> 00:46:48,920 So there are three degrees of freedom. 661 00:46:48,920 --> 00:46:54,860 And if you calculate alpha, that will give you 3/2. 662 00:46:54,860 --> 00:46:59,240 And basically, if you calculate gamma according to the equation 663 00:46:59,240 --> 00:47:03,620 you are going to get 1.67. 664 00:47:03,620 --> 00:47:08,470 On the other hand, if you have atomic gas, 665 00:47:08,470 --> 00:47:12,740 that means you have more degrees of freedom. 666 00:47:12,740 --> 00:47:15,320 So basically, you have not only the translation 667 00:47:15,320 --> 00:47:16,940 of degrees of freedom-- 668 00:47:16,940 --> 00:47:23,270 the three ones which are identical to the atomic gas-- 669 00:47:23,270 --> 00:47:27,000 you can also have two rotational degrees of freedom. 670 00:47:27,000 --> 00:47:32,540 So you can have these two atoms rotating like this, 671 00:47:32,540 --> 00:47:37,030 you can have that rotating like that. 672 00:47:37,030 --> 00:47:41,330 The trickiest thing is this one vibration degree of freedom, 673 00:47:41,330 --> 00:47:46,460 as you learned from the couple equations before. 674 00:47:46,460 --> 00:47:52,070 But this one, very trickily, is not excited at all 675 00:47:52,070 --> 00:47:52,880 at low temperature. 676 00:47:52,880 --> 00:47:55,640 You have to go to really, really super high temperature 677 00:47:55,640 --> 00:48:03,470 so that this actually contribute to the overall degrees 678 00:48:03,470 --> 00:48:04,380 of freedom. 679 00:48:04,380 --> 00:48:08,410 So therefore, you have a total of available degrees 680 00:48:08,410 --> 00:48:10,010 of freedom of 5. 681 00:48:10,010 --> 00:48:17,140 And if you calculate the gamma, you basically will get 1.4. 682 00:48:17,140 --> 00:48:22,610 So let's now calculate what will be 683 00:48:22,610 --> 00:48:28,145 the resulting speed of light. 684 00:48:28,145 --> 00:48:29,010 Sorry, no. 685 00:48:29,010 --> 00:48:30,140 Not speed of light. 686 00:48:30,140 --> 00:48:38,890 The speed of sound So if the temperature remain unchanged, 687 00:48:38,890 --> 00:48:44,380 if you take this equation here, this is a wave equation. 688 00:48:44,380 --> 00:48:51,050 Therefore, I know how to calculate the speed of sound. 689 00:48:51,050 --> 00:48:54,340 The speed of sound will be equal to-- 690 00:48:54,340 --> 00:48:56,790 P will be equal to the square root 691 00:48:56,790 --> 00:49:02,200 of gamma P0 divided by rho. 692 00:49:02,200 --> 00:49:09,160 So I have figured out the rho and the room pressure for you. 693 00:49:09,160 --> 00:49:17,290 So the P0 will be 10 to 5 kilogram ms squared, 694 00:49:17,290 --> 00:49:20,950 and the rho will be equal to 1.2. 695 00:49:20,950 --> 00:49:24,480 Rho is actually the density of the air. 696 00:49:24,480 --> 00:49:29,890 It's essentially 1.2 kilogram per meter cubed. 697 00:49:29,890 --> 00:49:38,230 If I have a gamma equal to 1, which is the case for ideal gas 698 00:49:38,230 --> 00:49:43,300 law temperature unchanged, if I calculate the resulting 699 00:49:43,300 --> 00:49:46,090 speed of sound you are going to get something 700 00:49:46,090 --> 00:49:51,590 like 389 meters per second. 701 00:49:51,590 --> 00:49:55,060 So that's the prediction for Newton. 702 00:49:55,060 --> 00:50:02,960 And the second case, if we have-- 703 00:50:02,960 --> 00:50:06,100 if we are believing what Laplace actually said, 704 00:50:06,100 --> 00:50:08,980 the heat flow is really, really negligible 705 00:50:08,980 --> 00:50:12,340 compared with the speed of oscillation, 706 00:50:12,340 --> 00:50:17,410 then we have, as we discussed last slide, 707 00:50:17,410 --> 00:50:20,140 gamma would be equal to 1.4. 708 00:50:20,140 --> 00:50:23,710 Therefore, you would be able to calculate the resulting 709 00:50:23,710 --> 00:50:29,650 speed of sound, and that is actually 342. 710 00:50:29,650 --> 00:50:31,650 So those are the predictions. 711 00:50:31,650 --> 00:50:35,770 And what I'm going to do now is to really demonstrate 712 00:50:35,770 --> 00:50:43,420 that we can actually measure the speed of sound in front of you. 713 00:50:47,460 --> 00:50:49,210 So the first thing which I will need to do 714 00:50:49,210 --> 00:50:54,510 is to switch so that you can see the camera. 715 00:50:54,510 --> 00:50:59,860 And now, I have a set up here. 716 00:50:59,860 --> 00:51:03,980 Basically, this set up is like the following. 717 00:51:03,980 --> 00:51:08,000 So basically, very similar to the setup we have here. 718 00:51:08,000 --> 00:51:12,580 But at one end, we actually have a speaker 719 00:51:12,580 --> 00:51:15,440 which produced sound wave. 720 00:51:15,440 --> 00:51:17,200 So this is essentially what we have. 721 00:51:21,950 --> 00:51:29,280 This is the tube, and we have, one end, 722 00:51:29,280 --> 00:51:37,150 there's a speaker attached to here, produce a sound wave. 723 00:51:37,150 --> 00:51:41,190 And basically, the amplitude will look like this. 724 00:51:41,190 --> 00:51:47,350 So basically, this will create some kind of standing wave 725 00:51:47,350 --> 00:51:49,660 inside the tube. 726 00:51:49,660 --> 00:51:55,320 And I have another device, which is actually a microphone. 727 00:51:55,320 --> 00:52:00,280 A microphone is connected to this scope, which shows 728 00:52:00,280 --> 00:52:04,450 you the amplitude of the-- 729 00:52:04,450 --> 00:52:09,100 basically, the amplitude measured by this microphone. 730 00:52:09,100 --> 00:52:14,890 And you see that if I move this, as a function of position 731 00:52:14,890 --> 00:52:17,770 you see that the am is changing. 732 00:52:17,770 --> 00:52:25,360 It's getting smaller when it is actually hitting the note here, 733 00:52:25,360 --> 00:52:31,920 because here there's almost no oscillation in the air. 734 00:52:31,920 --> 00:52:37,250 Therefore, you will measure a very small signal 735 00:52:37,250 --> 00:52:39,590 at that position. 736 00:52:39,590 --> 00:52:42,100 And if you continue to move, then you 737 00:52:42,100 --> 00:52:45,740 can see that, aha, I move away from the note, 738 00:52:45,740 --> 00:52:48,890 therefore I see some kind of maxima. 739 00:52:48,890 --> 00:52:54,870 Then, I see that this amplitude is dropping again. 740 00:52:54,870 --> 00:52:59,520 If I continue-- if I continue, then say that, aha, again, 741 00:52:59,520 --> 00:53:06,700 this amplitude is increasing to a very large value. 742 00:53:06,700 --> 00:53:12,710 Then it decreases to a minima around the note. 743 00:53:12,710 --> 00:53:16,340 So what I going to do now is to measure 744 00:53:16,340 --> 00:53:20,040 the distance between notes. 745 00:53:20,040 --> 00:53:26,930 And since the sound waves, which I actually put into the system, 746 00:53:26,930 --> 00:53:31,580 have a frequency of-- 747 00:53:31,580 --> 00:53:32,720 let me see. 748 00:53:32,720 --> 00:53:35,010 The frequency I put in-- 749 00:53:35,010 --> 00:53:39,120 the frequency I put in is actually 1 kilohertz. 750 00:53:42,800 --> 00:53:48,070 With the location of the note, I can know what will be, what? 751 00:53:48,070 --> 00:53:52,820 What will be the wavelengths of the sound waves. 752 00:53:52,820 --> 00:53:56,120 So therefore, I can now measure the distance 753 00:53:56,120 --> 00:53:59,750 between those three notes. 754 00:53:59,750 --> 00:54:03,830 Then I would be able to measure the wavelengths. 755 00:54:03,830 --> 00:54:08,450 Then I would be able to know who is correct-- 756 00:54:08,450 --> 00:54:14,570 if Newton is correct or Laplace is correct. 757 00:54:14,570 --> 00:54:16,580 So let's do that. 758 00:54:16,580 --> 00:54:20,360 So let me find the first minima. 759 00:54:20,360 --> 00:54:25,920 So the first minima is around 64. 760 00:54:25,920 --> 00:54:29,690 64 centimeter. 761 00:54:29,690 --> 00:54:31,640 And you can see that. 762 00:54:31,640 --> 00:54:37,130 Say stop when you see that it's reaching the minima again. 763 00:54:46,040 --> 00:54:46,610 Stop. 764 00:54:46,610 --> 00:54:48,320 OK, very good. 765 00:54:48,320 --> 00:54:50,630 So this is actually the first note, the location 766 00:54:50,630 --> 00:54:53,210 of the first note, and I was trying 767 00:54:53,210 --> 00:54:55,760 to find the next note so that I can actually 768 00:54:55,760 --> 00:54:57,560 readout the wavelengths. 769 00:54:57,560 --> 00:55:00,770 Now, this will increase again. 770 00:55:06,910 --> 00:55:10,600 And reach-- stop? 771 00:55:10,600 --> 00:55:12,180 Is that a stop sign? 772 00:55:12,180 --> 00:55:14,090 OK, very good. 773 00:55:14,090 --> 00:55:15,290 All right. 774 00:55:15,290 --> 00:55:24,860 So I get the value, which is actually 30 centimeter. 775 00:55:24,860 --> 00:55:27,200 So now I can calculate what would be the lambda. 776 00:55:27,200 --> 00:55:31,880 The lambda is actually 64 minus 30. 777 00:55:31,880 --> 00:55:35,040 Then, what I get is 34 centimeters. 778 00:55:38,570 --> 00:55:46,070 And if I calculate the velocity of the sound wave, then 779 00:55:46,070 --> 00:55:51,530 basically I have F times lambda, and that will give you-- 780 00:55:51,530 --> 00:55:55,820 this is actually equal to 0.34 centimeter. 781 00:55:55,820 --> 00:56:00,920 So that would give you 340 meters per second. 782 00:56:00,920 --> 00:56:01,500 Oh my god. 783 00:56:01,500 --> 00:56:07,056 This is so close to the prediction of Laplace. 784 00:56:07,056 --> 00:56:08,960 First of all, this is amazing. 785 00:56:08,960 --> 00:56:09,560 Why? 786 00:56:09,560 --> 00:56:13,490 Because with this looks really crappy thing, 787 00:56:13,490 --> 00:56:19,360 I can measure speed of sound. 788 00:56:19,360 --> 00:56:23,980 Secondly, ooh, the measurement is really great. 789 00:56:23,980 --> 00:56:26,870 It match within 1%. 790 00:56:26,870 --> 00:56:30,290 You guys did a good job of stopping me. 791 00:56:30,290 --> 00:56:32,360 Very nice. 792 00:56:32,360 --> 00:56:38,860 And finally, very unfortunately, people who voted for Newton 793 00:56:38,860 --> 00:56:40,520 is wrong. 794 00:56:40,520 --> 00:56:43,220 So what happened is the following. 795 00:56:45,884 --> 00:56:49,850 What happened is that Newton actually 796 00:56:49,850 --> 00:56:54,680 assumed that the speed of propagation of the heat 797 00:56:54,680 --> 00:56:59,210 is really fast, but actually that's not true. 798 00:56:59,210 --> 00:57:03,130 Because, for example, I'm standing here and heating up 799 00:57:03,130 --> 00:57:05,600 the air. 800 00:57:05,600 --> 00:57:09,590 In the last few lecture, I even heat up the air 801 00:57:09,590 --> 00:57:13,412 by some kind of fire in front of you. 802 00:57:13,412 --> 00:57:14,870 But you don't feel the heat, right? 803 00:57:14,870 --> 00:57:18,320 So the heat propagation is really not really, really fast 804 00:57:18,320 --> 00:57:21,560 compared to the speed of vibration. 805 00:57:21,560 --> 00:57:24,440 The vibration is really quick, because it's really 806 00:57:24,440 --> 00:57:31,100 vibrating up and down 1,000 times per second. 807 00:57:31,100 --> 00:57:35,950 So that means what is actually much more reasonable 808 00:57:35,950 --> 00:57:43,160 is to describe this process is adiabatic process. 809 00:57:43,160 --> 00:57:45,080 So we will take a five minute break 810 00:57:45,080 --> 00:57:47,390 here so that we can take questions, 811 00:57:47,390 --> 00:57:50,730 and then we'll come back at the 39. 812 00:58:03,590 --> 00:58:06,800 So welcome back, everybody. 813 00:58:06,800 --> 00:58:12,170 So we can see that from the last-- 814 00:58:12,170 --> 00:58:15,980 so from the discussion we had before the break, 815 00:58:15,980 --> 00:58:22,070 we see that the sound wave can be described by something which 816 00:58:22,070 --> 00:58:24,230 we are now very familiar with-- 817 00:58:24,230 --> 00:58:26,430 the wave equation. 818 00:58:26,430 --> 00:58:32,600 And also, we know what is the speed of the sound, which, 819 00:58:32,600 --> 00:58:36,290 based on this wave equation, the speed of sound 820 00:58:36,290 --> 00:58:44,990 is actually equal to square root of gamma P0 over rho. 821 00:58:44,990 --> 00:58:51,840 So gamma is actually obtained from this discussion 822 00:58:51,840 --> 00:58:56,750 of how many degrees of freedom we have. 823 00:58:56,750 --> 00:58:59,150 So the first case we discussed is, 824 00:58:59,150 --> 00:59:04,060 if you have a single atom of which you make your air, 825 00:59:04,060 --> 00:59:10,790 then basically the gamma is actually higher. 826 00:59:10,790 --> 00:59:15,800 On the other hand, if you have diatomic gas, 827 00:59:15,800 --> 00:59:18,590 then the gamma is actually slightly smaller. 828 00:59:18,590 --> 00:59:22,520 It's 1.4. 829 00:59:22,520 --> 00:59:29,990 So what would happen if I change the air in my lung 830 00:59:29,990 --> 00:59:35,990 to monatomic atom? 831 00:59:35,990 --> 00:59:39,050 So what is going to happen is that the speed of sound 832 00:59:39,050 --> 00:59:41,870 is going to be increased. 833 00:59:41,870 --> 00:59:43,910 The speed of sound will increase. 834 00:59:43,910 --> 00:59:46,730 And I have a fixed sized of lung. 835 00:59:46,730 --> 00:59:49,250 I didn't increase the size of my lung. 836 00:59:49,250 --> 00:59:53,525 Therefore, the frequency of my sound will what? 837 00:59:53,525 --> 00:59:56,320 Will increase. 838 00:59:56,320 --> 00:59:59,880 So how about we do that experiment and see if it works? 839 00:59:59,880 --> 01:00:07,520 So here, I have a balloon here, which is full of helium. 840 01:00:07,520 --> 01:00:09,080 Let me see if it works. 841 01:00:09,080 --> 01:00:12,740 I'm not sure if it will work, but let's see. 842 01:00:12,740 --> 01:00:14,040 Fingers crossed. 843 01:00:14,040 --> 01:00:16,580 We'll see what happens. 844 01:00:16,580 --> 01:00:21,050 Now I'm going to do a measure of operation 845 01:00:21,050 --> 01:00:25,190 to replace all the air in my lung by this. 846 01:00:48,960 --> 01:00:51,126 Does my sound change? 847 01:00:51,126 --> 01:00:51,625 [LAUGHTER] 848 01:00:51,625 --> 01:00:53,450 No? 849 01:00:53,450 --> 01:00:55,171 Didn't work. 850 01:00:55,171 --> 01:00:56,270 Let me do that again. 851 01:01:06,750 --> 01:01:08,140 I speak more aggressive. 852 01:01:08,140 --> 01:01:10,230 [LAUGHTER] 853 01:01:11,275 --> 01:01:13,501 Did you hear any difference? 854 01:01:13,501 --> 01:01:14,000 No. 855 01:01:20,318 --> 01:01:21,786 (HIGH PITCHED) Any difference? 856 01:01:25,520 --> 01:01:26,120 Works? 857 01:01:26,120 --> 01:01:26,660 Works now? 858 01:01:29,270 --> 01:01:30,797 Very good. 859 01:01:30,797 --> 01:01:31,880 Maybe we should use some-- 860 01:01:36,260 --> 01:01:38,402 maybe we should use that sound to go over 861 01:01:38,402 --> 01:01:39,360 all the lecture, right? 862 01:01:42,800 --> 01:01:44,450 It's a very dangerous experiment, 863 01:01:44,450 --> 01:01:47,600 because you are replacing all the air in your lung. 864 01:01:47,600 --> 01:01:48,505 So you may choke. 865 01:01:51,290 --> 01:01:55,894 Fortunately, I survived this experiment and hope you enjoy. 866 01:01:55,894 --> 01:01:58,738 [APPLAUSE] 867 01:02:03,960 --> 01:02:06,240 What happened is the following. 868 01:02:06,240 --> 01:02:13,460 So basically-- basically, the gamma becomes large. 869 01:02:13,460 --> 01:02:17,400 Therefore, the speed of sound in my lung becomes large. 870 01:02:17,400 --> 01:02:21,110 Therefore, the frequency of my sound 871 01:02:21,110 --> 01:02:24,590 increased and you hear some really strange sound. 872 01:02:24,590 --> 01:02:25,640 OK, very good. 873 01:02:25,640 --> 01:02:29,870 So before the end, I would like-- 874 01:02:29,870 --> 01:02:31,750 poor Newton. 875 01:02:31,750 --> 01:02:35,870 Before the end, I would like to discuss with you something 876 01:02:35,870 --> 01:02:40,620 which I hope I would not see again in the exam 877 01:02:40,620 --> 01:02:42,650 but I saw before. 878 01:02:42,650 --> 01:02:49,640 So if I create a progressing wave in a single-- 879 01:02:49,640 --> 01:02:56,540 in a closed end, open end tube, this progressing wave 880 01:02:56,540 --> 01:03:00,068 is going to be propagating at the speed of what? 881 01:03:00,068 --> 01:03:01,490 The speed of what? 882 01:03:01,490 --> 01:03:02,170 AUDIENCE: Sound. 883 01:03:02,170 --> 01:03:03,170 YEN-JIE LEE: Sound, yes. 884 01:03:03,170 --> 01:03:06,380 It's going to be propagating at the speed of sound, 885 01:03:06,380 --> 01:03:08,525 and you would reach the boundary. 886 01:03:11,880 --> 01:03:16,356 The question is, will we see this? 887 01:03:16,356 --> 01:03:23,880 Would this progressing wave just simply leak out of the tube. 888 01:03:26,910 --> 01:03:28,965 How many of you think that's going to happen? 889 01:03:31,700 --> 01:03:35,330 I hope I will never see that again in the exam. 890 01:03:35,330 --> 01:03:36,415 This would never happen. 891 01:03:39,050 --> 01:03:40,620 Why? 892 01:03:40,620 --> 01:03:49,370 That means you will have a super narrow collimated progressing 893 01:03:49,370 --> 01:03:52,370 wave going straight out of a tube, 894 01:03:52,370 --> 01:03:58,110 and that will not actually match the boundary conditions 895 01:03:58,110 --> 01:04:00,500 at the end of the tube. 896 01:04:00,500 --> 01:04:03,740 So that means, basically, first of all, 897 01:04:03,740 --> 01:04:08,190 there was no refraction. 898 01:04:08,190 --> 01:04:12,470 That means that all the energy is transferred 899 01:04:12,470 --> 01:04:14,370 outside of the world. 900 01:04:14,370 --> 01:04:18,190 And according to what we discussed before, 901 01:04:18,190 --> 01:04:21,420 what you would expect is that, OK, now you suddenly 902 01:04:21,420 --> 01:04:29,040 change to environment which you have really very large volume. 903 01:04:29,040 --> 01:04:33,240 Therefore, you it will be very difficult to change 904 01:04:33,240 --> 01:04:36,410 the pressure outside of the tube. 905 01:04:36,410 --> 01:04:38,160 Because what you are actually connected to 906 01:04:38,160 --> 01:04:43,470 is a reserve of infinite number of molecules outside. 907 01:04:43,470 --> 01:04:48,320 It's going to be really hard to change the pressure. 908 01:04:48,320 --> 01:04:51,300 Therefore, apparently this behavior 909 01:04:51,300 --> 01:04:53,850 doesn't match the boundary condition. 910 01:04:53,850 --> 01:04:57,000 And therefore, what you should expect 911 01:04:57,000 --> 01:05:02,530 is something like this, which I can show you here. 912 01:05:02,530 --> 01:05:06,220 So in this case, you have both side opened. 913 01:05:06,220 --> 01:05:10,540 What is going to happen is that at the boundary-- actually, 914 01:05:10,540 --> 01:05:14,530 it's like the case of hitting a wall, 915 01:05:14,530 --> 01:05:19,000 because outside of the tube you have really, 916 01:05:19,000 --> 01:05:23,600 really large volume, huge amount of air out of it. 917 01:05:23,600 --> 01:05:26,680 Therefore, it's like hitting a wall. 918 01:05:26,680 --> 01:05:29,590 The amplitude of the progressing wave 919 01:05:29,590 --> 01:05:34,030 changes sine and goes back through the tube. 920 01:05:34,030 --> 01:05:37,070 And of course, this system is actually not perfect. 921 01:05:37,070 --> 01:05:41,590 Therefore, there can be some leaking out-- 922 01:05:41,590 --> 01:05:46,390 some energy leak out of the tube, which essentially 923 01:05:46,390 --> 01:05:50,320 must be happening, because we can actually roll the tube 924 01:05:50,320 --> 01:05:52,000 and we can hear the sound. 925 01:05:52,000 --> 01:05:54,210 That is because some of the sound wave 926 01:05:54,210 --> 01:05:56,110 actually leaks out of the tube. 927 01:05:56,110 --> 01:05:59,530 And this process will go over and over again. 928 01:05:59,530 --> 01:06:02,290 And this progressing wave is going 929 01:06:02,290 --> 01:06:06,850 to be going back and forth, like that. 930 01:06:06,850 --> 01:06:11,730 So I hope that after this demonstration everybody 931 01:06:11,730 --> 01:06:15,136 will expect that, OK, this will be not-- 932 01:06:15,136 --> 01:06:21,370 the result will be like this progressing wave 933 01:06:21,370 --> 01:06:25,900 is going to be reflected because of the boundary condition 934 01:06:25,900 --> 01:06:29,768 and also change sine in terms of amplitude. 935 01:06:35,450 --> 01:06:37,280 So what we have learned today-- 936 01:06:37,280 --> 01:06:39,930 so it's already close to the end. 937 01:06:39,930 --> 01:06:46,390 We have learned example of a longitudinal wave 938 01:06:46,390 --> 01:06:49,790 And basically, longitudinal wave is actually 939 01:06:49,790 --> 01:06:54,140 in the form of density wave in the example 940 01:06:54,140 --> 01:06:55,940 which we covered today. 941 01:06:55,940 --> 01:07:00,080 And the mathematical description of the sound wave 942 01:07:00,080 --> 01:07:03,350 is going to be almost identical to what 943 01:07:03,350 --> 01:07:09,170 we have learned from the string case, which we actually 944 01:07:09,170 --> 01:07:13,370 discussed last time. 945 01:07:13,370 --> 01:07:15,770 There are two boundary conditions 946 01:07:15,770 --> 01:07:20,330 which I would like to briefly discuss before we 947 01:07:20,330 --> 01:07:23,310 end the lecture today. 948 01:07:23,310 --> 01:07:32,380 So in the case of open end, as we discussed before, 949 01:07:32,380 --> 01:07:34,750 we can have a system which contains 950 01:07:34,750 --> 01:07:40,400 a closed end and an open end. 951 01:07:40,400 --> 01:07:43,900 What will be the boundary condition for a closed end? 952 01:07:43,900 --> 01:07:47,080 So the closed end have a wall here. 953 01:07:47,080 --> 01:07:52,438 Therefore, when you have your molecule, 954 01:07:52,438 --> 01:07:55,540 the air molecule oscillating back and forth, 955 01:07:55,540 --> 01:08:02,590 when they are actually close to the wall, they cannot vibrate. 956 01:08:02,590 --> 01:08:03,090 Why? 957 01:08:03,090 --> 01:08:04,790 Because it's hitting the wall. 958 01:08:04,790 --> 01:08:07,420 It cannot vibrate so that actually, 959 01:08:07,420 --> 01:08:11,010 the boundary condition at the closed end, 960 01:08:11,010 --> 01:08:17,279 where you have a wall closing the tube, is psi equal to 0. 961 01:08:20,790 --> 01:08:25,370 And on the other hand, if you have an open end-- 962 01:08:25,370 --> 01:08:32,520 if you have an open end, that means outside of the tube 963 01:08:32,520 --> 01:08:34,770 the pressure is equal to what? 964 01:08:34,770 --> 01:08:36,460 It's equal to P0. 965 01:08:36,460 --> 01:08:39,300 The room pressure. 966 01:08:39,300 --> 01:08:42,250 And you have so many stuff there. 967 01:08:42,250 --> 01:08:46,260 Therefore, it's not possible to actually change 968 01:08:46,260 --> 01:08:52,020 the pressure dramatically at the edge of the open end. 969 01:08:52,020 --> 01:08:54,050 Therefore, what will be the condition? 970 01:08:54,050 --> 01:08:59,910 Psi P. Psi P is again the displacement with respect 971 01:08:59,910 --> 01:09:00,840 to the room. 972 01:09:00,840 --> 01:09:05,189 Pressure will be equal to 0. 973 01:09:05,189 --> 01:09:09,810 Based on what we actually have learned 974 01:09:09,810 --> 01:09:16,060 from this expression-- sorry for that, my finger slipped. 975 01:09:16,060 --> 01:09:24,410 From this expression, psi P is equal to minus gamma P0 d psi 976 01:09:24,410 --> 01:09:32,800 dx So psi P is proportional to partial psi partial x. 977 01:09:32,800 --> 01:09:38,240 Therefore, this boundary condition 978 01:09:38,240 --> 01:09:45,000 actually translates to partial psi partial x equal to 0. 979 01:09:47,609 --> 01:09:52,260 So this issue looks really familiar to you, 980 01:09:52,260 --> 01:09:56,480 because in terms of psi, if you forget about this system, 981 01:09:56,480 --> 01:09:59,940 what those boundary conditions mean to you 982 01:09:59,940 --> 01:10:04,140 is exactly the same as you have some kind 983 01:10:04,140 --> 01:10:09,450 of a wall in the left-hand side and it's connected to a string, 984 01:10:09,450 --> 01:10:11,090 and the right-hand side of the string 985 01:10:11,090 --> 01:10:15,030 is connected to a massless ring which can actually 986 01:10:15,030 --> 01:10:17,140 move up and down. 987 01:10:17,140 --> 01:10:20,780 These two systems, if you actually 988 01:10:20,780 --> 01:10:23,430 don't look at the detail-- 989 01:10:23,430 --> 01:10:28,620 only look at the wave functions and the boundary conditions-- 990 01:10:28,620 --> 01:10:31,950 they are identical. 991 01:10:31,950 --> 01:10:35,250 So that's actually the first lesson we learn from here. 992 01:10:35,250 --> 01:10:39,330 So when I talk about sound wave or when 993 01:10:39,330 --> 01:10:41,910 you think about sound wave problem, 994 01:10:41,910 --> 01:10:44,760 there's nothing to be afraid of anymore, 995 01:10:44,760 --> 01:10:48,760 because that's actually the same as what we have learned 996 01:10:48,760 --> 01:10:52,050 with wall and a string system. 997 01:10:52,050 --> 01:10:54,090 That's the first thing we learned. 998 01:10:54,090 --> 01:11:00,660 Secondly, that only works when I write my wave function 999 01:11:00,660 --> 01:11:03,390 in the form of psi. 1000 01:11:03,390 --> 01:11:08,180 So now I can actually get the first normal mode 1001 01:11:08,180 --> 01:11:11,965 would be like this if I plot psi as a function of x. 1002 01:11:11,965 --> 01:11:13,570 The second normal mode-- 1003 01:11:13,570 --> 01:11:16,600 doesn't surprise you-- will look like this, 1004 01:11:16,600 --> 01:11:18,830 et cetera, et cetera. 1005 01:11:18,830 --> 01:11:22,330 If I plot psi-- 1006 01:11:22,330 --> 01:11:25,810 if I plot psi as a function of x. 1007 01:11:31,120 --> 01:11:35,110 On the other hand, we also know that psi P 1008 01:11:35,110 --> 01:11:41,950 is proportional to d psi partial psi partial x. 1009 01:11:41,950 --> 01:11:50,370 Therefore, you can also plot psi P as a function of x. 1010 01:11:50,370 --> 01:11:54,720 Then what you are going to get is something like this. 1011 01:11:54,720 --> 01:12:01,870 In the closed end, the psi P is actually reaching the maxima, 1012 01:12:01,870 --> 01:12:03,640 because it's got the wall. 1013 01:12:03,640 --> 01:12:06,980 Therefore, it can actually produce pressure 1014 01:12:06,980 --> 01:12:08,080 on top of the wall. 1015 01:12:08,080 --> 01:12:13,090 But you cannot move the position of all of those molecules 1016 01:12:13,090 --> 01:12:14,650 in front of the wall. 1017 01:12:14,650 --> 01:12:16,230 Therefore, that makes sense. 1018 01:12:16,230 --> 01:12:20,110 You will see exactly in the opposite direction, 1019 01:12:20,110 --> 01:12:25,360 if you plot the amplitude as a function of x, 1020 01:12:25,360 --> 01:12:28,870 you see a picture which is almost like flipped. 1021 01:12:28,870 --> 01:12:32,800 Of course, you can also do the same thing for the second one. 1022 01:12:32,800 --> 01:12:35,920 And basically, what you are going to get is something like. 1023 01:12:35,920 --> 01:12:39,880 The second normal mode, et cetera, et cetera. 1024 01:12:39,880 --> 01:12:46,930 So be careful about the matching between the boundary 1025 01:12:46,930 --> 01:12:53,020 condition obtained from the tube and string-wall system. 1026 01:12:53,020 --> 01:12:54,430 They are identical. 1027 01:12:54,430 --> 01:12:56,830 Open corresponds responds to open, 1028 01:12:56,830 --> 01:13:00,160 closed corresponds to closed when you express 1029 01:13:00,160 --> 01:13:04,060 your equation of motion in terms of psi. 1030 01:13:04,060 --> 01:13:07,390 On the other hand, if you change that to psi P, 1031 01:13:07,390 --> 01:13:10,200 then the relation is actually flipped. 1032 01:13:10,200 --> 01:13:13,450 Thank you very much, and I hope you enjoyed the lecture today. 1033 01:13:13,450 --> 01:13:16,920 And I will see you next week.