WEBVTT 00:00:02.550 --> 00:00:04.920 The following content is provided under a Creative 00:00:04.920 --> 00:00:06.310 Commons license. 00:00:06.310 --> 00:00:08.520 Your support will help MIT OpenCourseWare 00:00:08.520 --> 00:00:12.610 continue to offer high quality educational resources for free. 00:00:12.610 --> 00:00:15.150 To make a donation or to view additional materials 00:00:15.150 --> 00:00:19.110 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.110 --> 00:00:20.154 at ocw.mit.edu. 00:00:24.630 --> 00:00:29.520 YEN-JIE LEE: So welcome back again to 8.03. 00:00:29.520 --> 00:00:34.045 Today, my plan is to continue the discussion of the two 00:00:34.045 --> 00:00:38.100 string system, which we were working really hard last time 00:00:38.100 --> 00:00:40.210 but we sort of run out of time. 00:00:40.210 --> 00:00:43.830 So we didn't have time to enjoy what we have done, right? 00:00:43.830 --> 00:00:46.560 So today we are going to discuss all the outcome 00:00:46.560 --> 00:00:48.270 of that calculation. 00:00:48.270 --> 00:00:55.970 And so we will start to discuss more examples which can be 00:00:55.970 --> 00:00:58.180 described by the wave equation. 00:00:58.180 --> 00:01:01.950 Today we are going to talk about another example, which 00:01:01.950 --> 00:01:04.099 is sound waves. 00:01:04.099 --> 00:01:06.300 It's a very exciting topic. 00:01:06.300 --> 00:01:10.350 And afterwards, we will start the discussion 00:01:10.350 --> 00:01:11.965 about electromagnetic waves. 00:01:14.580 --> 00:01:17.640 So this is the wave equation which we have been using, 00:01:17.640 --> 00:01:19.970 and over the last few lectures, we 00:01:19.970 --> 00:01:24.150 have been discussing two specials kinds of solutions-- 00:01:24.150 --> 00:01:26.960 the normal modes, which is actually 00:01:26.960 --> 00:01:30.710 standing waves in the end, which we identified, 00:01:30.710 --> 00:01:32.930 and the progressing wave solution, which 00:01:32.930 --> 00:01:37.010 is very powerful in describing the phenomena which 00:01:37.010 --> 00:01:40.160 we are familiar with. 00:01:40.160 --> 00:01:42.650 Last time, in the end of the lecture, 00:01:42.650 --> 00:01:47.370 we were discussing about an interesting example, 00:01:47.370 --> 00:01:51.210 which involves two strings in the system. 00:01:51.210 --> 00:01:57.440 One essentially in the left-hand side have mass per unit length, 00:01:57.440 --> 00:02:02.720 rho l equals to rho 1, and right-hand side one is thicker, 00:02:02.720 --> 00:02:06.070 and therefore, the mass per unit length, rho, 00:02:06.070 --> 00:02:09.199 is larger, which is called rho 2. 00:02:09.199 --> 00:02:11.450 What we did last time is to assume 00:02:11.450 --> 00:02:15.260 that we have a progressing wave, which essentially 00:02:15.260 --> 00:02:16.220 going into this-- 00:02:19.540 --> 00:02:23.240 which essentially first initiated in the left-hand side 00:02:23.240 --> 00:02:29.510 string and it's going towards the boundary of the two 00:02:29.510 --> 00:02:31.040 systems-- 00:02:31.040 --> 00:02:35.220 the more massive one and the less massive one. 00:02:35.220 --> 00:02:37.590 And see what is going to happen. 00:02:37.590 --> 00:02:41.100 And what we actually tried to describe last time 00:02:41.100 --> 00:02:44.660 is that we actually define incident wave described 00:02:44.660 --> 00:02:51.130 by this fi function, transmitting wave, ft, 00:02:51.130 --> 00:02:54.290 and the refractive wave, fr. 00:02:54.290 --> 00:02:58.310 By using boundary conditions which we described last time 00:02:58.310 --> 00:03:00.860 we can conclude that there's a fixed 00:03:00.860 --> 00:03:09.270 relation between the three waves equation-- wave functions. 00:03:09.270 --> 00:03:11.090 What we actually concluded that last time 00:03:11.090 --> 00:03:18.240 is that fr is proportional to fi, which is the incident wave 00:03:18.240 --> 00:03:22.610 function, by some constant, which is called r. 00:03:22.610 --> 00:03:24.280 And what is r? 00:03:24.280 --> 00:03:29.900 As we solved last time, it's V2 minus V1 over V1 plus V2, 00:03:29.900 --> 00:03:34.250 which are the velocities of the first and second string. 00:03:34.250 --> 00:03:39.290 And the transmitted wave, ft, is actually also proportional 00:03:39.290 --> 00:03:43.950 to the incident wave. 00:03:43.950 --> 00:03:48.950 And the coefficient-- we call it tau-- 00:03:48.950 --> 00:03:54.050 to describe the amplitude of the transmitted wave. 00:03:54.050 --> 00:03:56.900 So what we can do is the variance. 00:03:56.900 --> 00:04:01.820 So we actually discussed two examples last time, 00:04:01.820 --> 00:04:04.820 plugging in V1 and V2. 00:04:04.820 --> 00:04:07.820 And there are two more examples we can actually 00:04:07.820 --> 00:04:12.110 make use of this equation we obtained last time to discuss 00:04:12.110 --> 00:04:14.540 what would be the physics outcome 00:04:14.540 --> 00:04:16.290 of this kind of situation. 00:04:16.290 --> 00:04:20.750 So the first example which we can actually discuss 00:04:20.750 --> 00:04:24.460 is that, OK, now I have this string. 00:04:24.460 --> 00:04:31.190 Assume, though, they're connected to a wall. 00:04:31.190 --> 00:04:33.530 So this time we can say, oh, wait, wait, wait, wait 00:04:33.530 --> 00:04:34.030 a second. 00:04:34.030 --> 00:04:35.960 There's only one string now, right? 00:04:35.960 --> 00:04:38.570 But last time we were solving two strings, right? 00:04:38.570 --> 00:04:42.310 But what I'm doing now is to treat the wall 00:04:42.310 --> 00:04:44.300 as if it's a string. 00:04:44.300 --> 00:04:47.720 But this wall is really massive. 00:04:47.720 --> 00:04:54.080 Therefore, the rho l, or the mass per unit length, 00:04:54.080 --> 00:04:55.410 is really large. 00:04:55.410 --> 00:04:55.910 OK. 00:04:55.910 --> 00:04:59.540 It goes to infinity. 00:04:59.540 --> 00:05:02.570 If that's the case, if you're set this idea-- 00:05:02.570 --> 00:05:06.990 this is still a two string system-- 00:05:06.990 --> 00:05:09.200 then I can now go ahead and calculate 00:05:09.200 --> 00:05:11.510 what will be the velocity. 00:05:11.510 --> 00:05:17.240 The velocity V2 goes to 0. 00:05:17.240 --> 00:05:21.340 Then I can go ahead and plug into my equations. 00:05:21.340 --> 00:05:25.610 So we spend a lot of time in the last lecture 00:05:25.610 --> 00:05:27.650 to obtain those equations. 00:05:27.650 --> 00:05:32.900 And we will find that if I have V2 goes to 0, 00:05:32.900 --> 00:05:36.560 I have r equal to minus 1. 00:05:36.560 --> 00:05:40.400 And the tau, which is actually related 00:05:40.400 --> 00:05:43.010 to the amplitude of the transmitted wave, 00:05:43.010 --> 00:05:47.070 is equal to 0, which you can see from this equation. 00:05:47.070 --> 00:05:48.590 What does that mean? 00:05:48.590 --> 00:05:51.890 That means once we solve that question, 00:05:51.890 --> 00:05:57.800 we also know what would be the outcome of this experiment, 00:05:57.800 --> 00:05:59.090 this physical situation. 00:05:59.090 --> 00:06:02.390 When we have a string attached to a wall 00:06:02.390 --> 00:06:06.920 and we have an incident wave, as a function of time, what 00:06:06.920 --> 00:06:10.370 is going to happen afterward is that this wave is 00:06:10.370 --> 00:06:13.930 going to propagate and hit the wall and get 00:06:13.930 --> 00:06:17.210 refracted completely. 00:06:17.210 --> 00:06:21.230 The amplitude ratio is minus 1. 00:06:21.230 --> 00:06:25.280 Therefore, all the energy is refracted 00:06:25.280 --> 00:06:30.800 by the wall in this highly idealized situation. 00:06:30.800 --> 00:06:33.150 So that's kind of interesting. 00:06:33.150 --> 00:06:37.440 The second example is also very interesting. 00:06:37.440 --> 00:06:42.740 So if you have a string attached to a massive ring, 00:06:42.740 --> 00:06:51.850 and this ring can go up and down freely without any friction, 00:06:51.850 --> 00:06:55.750 you can again say, no, no, no, this is again a single string 00:06:55.750 --> 00:06:57.040 system, right? 00:06:57.040 --> 00:07:00.490 But what I'm going to argue now is that, OK, 00:07:00.490 --> 00:07:05.680 there's another string which is so light mass per unit length 00:07:05.680 --> 00:07:09.340 is close to 0. 00:07:09.340 --> 00:07:13.000 So it is actually in the air. 00:07:13.000 --> 00:07:16.150 If I do that, what is going to happen? 00:07:16.150 --> 00:07:21.790 The rho l is going to go to the limit of 0, 00:07:21.790 --> 00:07:27.070 because the right-hand side you almost cannot see it. 00:07:27.070 --> 00:07:31.300 And the V2, which is the velocity 00:07:31.300 --> 00:07:36.130 of the transmitted wave, goes to infinity. 00:07:36.130 --> 00:07:38.300 If that's the case, you can then again, 00:07:38.300 --> 00:07:40.930 plugging into this equation we obtained, and then 00:07:40.930 --> 00:07:46.030 you conclude that r would be equal to 1 00:07:46.030 --> 00:07:50.270 and tau will be equal to 2. 00:07:50.270 --> 00:07:51.590 So what does that mean? 00:07:51.590 --> 00:07:55.600 This means that if you have this end, which 00:07:55.600 --> 00:07:59.230 is the open end, attached to a ring which 00:07:59.230 --> 00:08:04.740 can move up and down, then you get that refraction. 00:08:04.740 --> 00:08:07.280 The amplitude of the refractive wave 00:08:07.280 --> 00:08:11.950 doesn't change, because i is equal to 1. 00:08:11.950 --> 00:08:16.980 So it's still in the positive direction we defined. 00:08:16.980 --> 00:08:19.060 But now, it goes backward. 00:08:19.060 --> 00:08:22.700 Again, all the energy is actually refracted, 00:08:22.700 --> 00:08:25.000 as you can see from this equation. 00:08:25.000 --> 00:08:27.370 But the curious case is that-- 00:08:27.370 --> 00:08:33.350 the strangest thing is that the tau is equal to 2. 00:08:33.350 --> 00:08:35.159 That's kind of strange, right? 00:08:35.159 --> 00:08:37.347 Tau is equal to 2. 00:08:37.347 --> 00:08:38.180 What does that mean? 00:08:38.180 --> 00:08:43.840 That means you are going to predict a transmitted wave 00:08:43.840 --> 00:08:49.250 with amplitude exactly the two times the incident wave, 00:08:49.250 --> 00:08:52.390 and it's going to be propagating in the right-hand side 00:08:52.390 --> 00:08:55.180 and the speed goes to infinity. 00:08:57.970 --> 00:08:59.070 What does that mean? 00:08:59.070 --> 00:09:02.650 Does that mean the energy is not conserved? 00:09:02.650 --> 00:09:07.660 Have we found the cure of the energy crisis? 00:09:07.660 --> 00:09:12.530 Because now-- I can actually take all those energy. 00:09:12.530 --> 00:09:14.480 I can design this thing, and then this thing 00:09:14.480 --> 00:09:17.290 will bounce around all over the place. 00:09:17.290 --> 00:09:20.180 And that is going to emit energy. 00:09:20.180 --> 00:09:22.550 Oh my god, we solve all the problem. 00:09:22.550 --> 00:09:25.650 You should be really excited about it, right? 00:09:25.650 --> 00:09:28.430 No? 00:09:28.430 --> 00:09:33.430 But unfortunately, rho l goes to 0. 00:09:33.430 --> 00:09:38.540 So there's actually nothing oscillating out of this system. 00:09:38.540 --> 00:09:43.100 So therefore, there is no additional energy radiated out 00:09:43.100 --> 00:09:44.360 of this system. 00:09:44.360 --> 00:09:45.290 Too bad. 00:09:45.290 --> 00:09:48.530 Go back to work. 00:09:48.530 --> 00:09:49.520 All right. 00:09:49.520 --> 00:09:52.220 So that's actually what we discussed last time, 00:09:52.220 --> 00:09:54.920 and I hope that complete the loop. 00:09:54.920 --> 00:10:00.800 And today, before we actually move to sound wave, 00:10:00.800 --> 00:10:04.000 I would like to talk about, very briefly, 00:10:04.000 --> 00:10:08.060 harmonic progressing waves. 00:10:08.060 --> 00:10:13.580 So now, we can see that harmonic progressing wave looks really 00:10:13.580 --> 00:10:15.740 beautiful, as you can see here. 00:10:15.740 --> 00:10:23.820 And it can be described by a cosine kx minus omega t plus 5. 00:10:23.820 --> 00:10:25.600 5 is actually the face. 00:10:25.600 --> 00:10:30.620 And you can always write it in different forms. 00:10:30.620 --> 00:10:33.770 And since we have learned how to describe 00:10:33.770 --> 00:10:37.670 in general the progressing wave, this is just to remind you 00:10:37.670 --> 00:10:39.880 that, OK, there's no proper notion 00:10:39.880 --> 00:10:43.800 to describe a harmonic progressing wave. 00:10:47.450 --> 00:10:52.520 So since we have learned about waves, 00:10:52.520 --> 00:10:58.160 which involve oscillation in the transverse direction. 00:10:58.160 --> 00:11:00.540 So basically, we always say, OK, things 00:11:00.540 --> 00:11:05.000 are oscillating up and down in the case of string. 00:11:05.000 --> 00:11:07.990 Before I start, though, the sound wave, there's 00:11:07.990 --> 00:11:12.830 a different kind of wave which we can also see 00:11:12.830 --> 00:11:15.480 very often in the daily life. 00:11:15.480 --> 00:11:19.220 This is called longitudinal waves. 00:11:19.220 --> 00:11:25.100 For example, I can have a spring wave, and I can actually-- 00:11:25.100 --> 00:11:26.890 imagine I have a spring wave. 00:11:26.890 --> 00:11:28.220 And I can do this. 00:11:28.220 --> 00:11:31.250 I oscillate in the horizontal direction. 00:11:31.250 --> 00:11:37.100 Then that can produce displacement with respect 00:11:37.100 --> 00:11:39.200 to the equilibrium position. 00:11:39.200 --> 00:11:43.970 And this kind of behavior is like a density wave. 00:11:43.970 --> 00:11:46.253 We call it longitudinal waves. 00:11:49.040 --> 00:11:53.640 This is exactly what is happening with sound wave. 00:11:53.640 --> 00:11:55.180 So what is actually sound wave? 00:11:55.180 --> 00:12:00.830 Essentially, a collection or motion of air molecules. 00:12:00.830 --> 00:12:04.350 And they are actually oscillating back and forth. 00:12:04.350 --> 00:12:11.840 And we may use that to extend energy all over the place. 00:12:11.840 --> 00:12:15.560 And today we are going to discuss the sound wave. 00:12:15.560 --> 00:12:18.950 And by the way, just for simplicity, 00:12:18.950 --> 00:12:24.500 because drawing all those dots really take a lot of time. 00:12:24.500 --> 00:12:27.470 So what we sometimes do is that, OK, 00:12:27.470 --> 00:12:32.150 we can now draw the pressure, the amplitude of the pressure, 00:12:32.150 --> 00:12:35.260 or the amplitude of the displacement 00:12:35.260 --> 00:12:39.770 of individual molecules in the discussion 00:12:39.770 --> 00:12:40.800 as a function of time. 00:12:40.800 --> 00:12:45.190 So if we draw the amplitude as a function of time 00:12:45.190 --> 00:12:49.310 or as a function of location, then it 00:12:49.310 --> 00:12:52.830 looks exactly the same as what we discussed before 00:12:52.830 --> 00:12:56.600 for the transverse waves. 00:12:56.600 --> 00:12:58.550 So just some clarification. 00:12:58.550 --> 00:13:02.370 It's not like the molecules are going up and down. 00:13:02.370 --> 00:13:05.810 They are going back and forth, and it's just a matter 00:13:05.810 --> 00:13:08.270 presenting that these are. 00:13:08.270 --> 00:13:12.010 So this is actually an example of a travelling wave 00:13:12.010 --> 00:13:14.250 in the longitudinal direction. 00:13:14.250 --> 00:13:16.470 And you can see that it is actually 00:13:16.470 --> 00:13:20.970 the density which is actually changing as a function of time. 00:13:20.970 --> 00:13:23.450 And as you can see, it's actually 00:13:23.450 --> 00:13:29.410 traveling at a fixed speed and going in the right direction 00:13:29.410 --> 00:13:33.460 of the blackboard. 00:13:33.460 --> 00:13:38.110 So those being said, we can actually 00:13:38.110 --> 00:13:41.780 get started with a concrete example. 00:13:41.780 --> 00:13:46.400 So I would like to discuss with you now a system, 00:13:46.400 --> 00:13:53.676 which is like you have a tube, with cross section area, A. 00:13:53.676 --> 00:13:58.850 So A is actually the area of the cross section. 00:13:58.850 --> 00:14:01.850 And I can now wonder-- 00:14:01.850 --> 00:14:07.550 now the physics question I'm asking is, what would be the-- 00:14:07.550 --> 00:14:14.700 what would be the behavior of the air inside this tube? 00:14:14.700 --> 00:14:18.730 So before I go ahead and solve this problem, 00:14:18.730 --> 00:14:21.450 I need to define and give you some more information 00:14:21.450 --> 00:14:26.580 about this tube and also the condition or the environment 00:14:26.580 --> 00:14:29.050 this tube is living in. 00:14:29.050 --> 00:14:32.220 So the first information I would like to give you 00:14:32.220 --> 00:14:38.080 is that the pressure, the room pressure, 00:14:38.080 --> 00:14:42.100 is actually P0 in this example. 00:14:42.100 --> 00:14:44.970 So the P0 is actually the room pressure. 00:14:44.970 --> 00:14:51.600 And I can now define coordinates is the x direction is actually 00:14:51.600 --> 00:14:53.910 in the horizontal direction pointing 00:14:53.910 --> 00:14:56.280 to the right-hand side. 00:14:56.280 --> 00:15:03.630 And now, I can actually try to describe a small unit volume 00:15:03.630 --> 00:15:09.150 inside the tube by location x. 00:15:09.150 --> 00:15:15.180 And the width of this volume, I call it delta x. 00:15:15.180 --> 00:15:23.092 And if I go ahead and prepare this system and at time, t, 00:15:23.092 --> 00:15:25.890 something is happening to this system-- 00:15:25.890 --> 00:15:29.960 so now, you need length. 00:15:29.960 --> 00:15:32.210 You need volume, I was discussing. 00:15:32.210 --> 00:15:38.150 This get displaced with respect to the equilibrium position. 00:15:38.150 --> 00:15:40.730 So that means, assuming that something happened 00:15:40.730 --> 00:15:46.210 at the t equal to t, the left inside edge of the volume 00:15:46.210 --> 00:15:50.760 is shifted toward a positive direction, which is described 00:15:50.760 --> 00:15:54.540 by wave function psi x. 00:15:54.540 --> 00:15:57.510 And the position of the right-hand side 00:15:57.510 --> 00:16:06.240 edge of this volume is shifted to side x plus delta x. 00:16:06.240 --> 00:16:09.750 So something happened to this system. 00:16:09.750 --> 00:16:12.570 We can also say that-- we can also 00:16:12.570 --> 00:16:16.230 describe this system, the pressure of this system, 00:16:16.230 --> 00:16:18.030 by P function. 00:16:18.030 --> 00:16:22.230 P of x is actually equal to P0, which 00:16:22.230 --> 00:16:25.860 is actually room pressure-- 00:16:25.860 --> 00:16:29.190 P0 is the baseline room pressure-- 00:16:29.190 --> 00:16:39.500 plus some kind of displacement in pressure, psi P. 00:16:39.500 --> 00:16:45.010 So now we describe the pressure acting on the left inside edge 00:16:45.010 --> 00:16:49.470 of the small unit volume, and the right-hand side, 00:16:49.470 --> 00:16:52.000 you can also do the same thing. 00:16:52.000 --> 00:16:59.530 P of x plus delta x will be equal to P0 plus psi P, 00:16:59.530 --> 00:17:02.450 describing the displacement or how 00:17:02.450 --> 00:17:07.300 offset the pressure is as a function of x but now evaluated 00:17:07.300 --> 00:17:10.780 at x plus delta x. 00:17:10.780 --> 00:17:17.329 So once we have all of those elements defined-- 00:17:17.329 --> 00:17:19.000 these are essentially just a copy 00:17:19.000 --> 00:17:24.280 of what I have in the slide and those are a reminder here-- 00:17:24.280 --> 00:17:30.310 now, we can actually calculate the motion of all the molecules 00:17:30.310 --> 00:17:38.670 in this volume, because I have pressure, I have displacement. 00:17:38.670 --> 00:17:41.770 The displacement is described by psi, 00:17:41.770 --> 00:17:45.880 end position is described by psi, the change in pressure 00:17:45.880 --> 00:17:50.260 is described by psi P. And now I can go ahead and apply, 00:17:50.260 --> 00:17:52.270 for example, Newton's law. 00:17:52.270 --> 00:17:54.070 Then I can calculate what would be 00:17:54.070 --> 00:18:02.830 the acceleration for all the molecules inside this volume. 00:18:02.830 --> 00:18:05.320 But wait a second. 00:18:05.320 --> 00:18:09.260 That sounds all great, but I don't know yet 00:18:09.260 --> 00:18:13.040 how to relate pressure and the volume, 00:18:13.040 --> 00:18:16.790 because pressure is actually expressed by psi P 00:18:16.790 --> 00:18:20.510 and the volume is related to psi. 00:18:20.510 --> 00:18:25.070 I need to know is actually the relation between pressure 00:18:25.070 --> 00:18:28.190 and the displacement or pressure between psi 00:18:28.190 --> 00:18:30.650 so that I can make progress. 00:18:30.650 --> 00:18:33.830 So that this actually the main discussion which I would 00:18:33.830 --> 00:18:37.610 like to do in this lecture. 00:18:37.610 --> 00:18:41.970 So given those information, I can now 00:18:41.970 --> 00:18:47.540 calculate what is actually the change in this little volume. 00:18:47.540 --> 00:18:55.190 So I can calculate the change in volume, 00:18:55.190 --> 00:19:00.380 which is described by delta V. But delta V 00:19:00.380 --> 00:19:05.180 can be actually calculated by a, which is actually 00:19:05.180 --> 00:19:08.700 the area of the cross section, times 00:19:08.700 --> 00:19:19.970 psi x plus delta xt minus psi xt. 00:19:19.970 --> 00:19:27.620 So basically, just calculate how much the boundary is actually 00:19:27.620 --> 00:19:28.850 displaced. 00:19:28.850 --> 00:19:35.630 And if we always take very small amplitude approximation, 00:19:35.630 --> 00:19:39.500 then basically this expression is 00:19:39.500 --> 00:19:47.870 roughly equal to A partial psi partial x times delta x, 00:19:47.870 --> 00:19:50.990 where the delta x is really very small. 00:19:50.990 --> 00:19:55.280 So a very small volume I was talking about. 00:19:55.280 --> 00:19:59.165 And I can also calculate the pressure. 00:20:05.400 --> 00:20:06.990 What is the pressure difference? 00:20:06.990 --> 00:20:10.620 The pressure difference is between the pressure 00:20:10.620 --> 00:20:14.280 acting in the left-hand side edge and the pressure which 00:20:14.280 --> 00:20:17.380 is acting on the right-hand side edge. 00:20:17.380 --> 00:20:20.090 So I can now calculate pressure difference, 00:20:20.090 --> 00:20:32.500 delta P. Delta P would be minus psi P x plus delta x t plus psi 00:20:32.500 --> 00:20:36.230 P x t. 00:20:36.230 --> 00:20:40.740 So basically, one is essentially the pressure 00:20:40.740 --> 00:20:44.030 pushing the body in the right-hand side. 00:20:44.030 --> 00:20:45.530 The other one essentially pushing it 00:20:45.530 --> 00:20:47.950 in the left-hand side direction. 00:20:47.950 --> 00:20:52.504 Again, I can take very small delta x approximation. 00:20:52.504 --> 00:20:54.170 And basically, what you are going to get 00:20:54.170 --> 00:21:00.360 is minus partial psi P partial x delta x. 00:21:03.790 --> 00:21:06.910 So we have prepared all of those information 00:21:06.910 --> 00:21:11.020 about volume and the pressure. 00:21:11.020 --> 00:21:13.540 As I mentioned before, the big question 00:21:13.540 --> 00:21:17.380 which we would like to ask is, how do I 00:21:17.380 --> 00:21:23.110 relate pressure and the volume so that I can make progress? 00:21:23.110 --> 00:21:26.530 If I can relate pressure and volume, 00:21:26.530 --> 00:21:28.990 then I can know what is the relation 00:21:28.990 --> 00:21:32.290 between psi P and the psi. 00:21:32.290 --> 00:21:35.260 Then I can ask you to make use of Newton's Law. 00:21:35.260 --> 00:21:40.870 Then I can calculate the resulting equation of motion. 00:21:40.870 --> 00:21:46.900 So there's two possible interesting scenarios 00:21:46.900 --> 00:21:50.530 which we can relate temperature-- so sorry, 00:21:50.530 --> 00:21:53.380 relate pressure and the volume. 00:21:53.380 --> 00:21:58.870 The first one was proposed by Newton. 00:21:58.870 --> 00:22:05.060 Newton said that, OK, this is an interesting phenomena. 00:22:05.060 --> 00:22:09.820 In my opinion, although you actually displaced 00:22:09.820 --> 00:22:12.820 this volume-- 00:22:12.820 --> 00:22:19.780 make the displacement for those molecules in the tube-- 00:22:19.780 --> 00:22:23.620 but because the heat was conducted from one region 00:22:23.620 --> 00:22:25.510 to the other region, all those regions 00:22:25.510 --> 00:22:27.840 are connected to each other. 00:22:27.840 --> 00:22:34.460 And the speed of this heat transfer is so fast. 00:22:34.460 --> 00:22:37.400 It's really fast, like instant. 00:22:37.400 --> 00:22:42.670 This heat is actually transferred from one direction 00:22:42.670 --> 00:22:43.360 to the other-- 00:22:43.360 --> 00:22:46.530 one position to the other position. 00:22:46.530 --> 00:22:55.030 Therefore, over the course of this evolution, 00:22:55.030 --> 00:22:59.040 the temperature should be unchanged. 00:22:59.040 --> 00:23:03.610 No matter what you do to the air inside the tube, 00:23:03.610 --> 00:23:05.500 the temperature should be unchanged, 00:23:05.500 --> 00:23:11.860 because Newton thinks that heat should be-- 00:23:11.860 --> 00:23:13.725 the speed of the heat distribution 00:23:13.725 --> 00:23:15.880 is really, really fast. 00:23:15.880 --> 00:23:18.550 Much faster than all those vibration 00:23:18.550 --> 00:23:21.760 happening in the tube. 00:23:21.760 --> 00:23:25.270 If that is the case-- 00:23:25.270 --> 00:23:27.050 that is the case-- 00:23:27.050 --> 00:23:36.300 then that means we can use ideal gas law. 00:23:41.670 --> 00:23:44.825 P times V is equal to nRT. 00:23:50.480 --> 00:23:56.570 I hope that you have learned this before in 8.01 and 8.02. 00:23:56.570 --> 00:23:58.750 If that's the case, that means-- 00:23:58.750 --> 00:24:00.890 so all those things are constant, 00:24:00.890 --> 00:24:05.770 because we assume that temperature is unchanged. 00:24:12.900 --> 00:24:17.080 Therefore, the right-hand side is essentially a constant. 00:24:17.080 --> 00:24:23.290 Therefore, P times V would be some kind of constant. 00:24:23.290 --> 00:24:32.710 The V would be proportional to 1/P. 00:24:32.710 --> 00:24:35.320 So that essentially is the first idea, 00:24:35.320 --> 00:24:38.390 which is coming from Newton, in order 00:24:38.390 --> 00:24:43.450 to relate pressure and the volume. 00:24:47.240 --> 00:24:52.120 The second idea is coming from Laplace. 00:24:52.120 --> 00:24:56.350 Laplace says, OK, he has a different opinion 00:24:56.350 --> 00:24:58.210 on this matter. 00:24:58.210 --> 00:25:04.570 He think that this essentially is an adiabatic process. 00:25:04.570 --> 00:25:06.210 What does that mean? 00:25:06.210 --> 00:25:12.220 That means the heat flow from the compressed region 00:25:12.220 --> 00:25:17.430 to the other region is really negligible, 00:25:17.430 --> 00:25:20.190 because the oscillation is really fast 00:25:20.190 --> 00:25:24.980 and the speed of the transfer of the heat 00:25:24.980 --> 00:25:30.330 is really slow compared to the time scale of the oscillation. 00:25:30.330 --> 00:25:35.970 Therefore, in Laplace's opinion, he 00:25:35.970 --> 00:25:40.485 thinks that the whole process is adiabatic process. 00:25:50.790 --> 00:25:55.150 If that's the case, which I will show you later, 00:25:55.150 --> 00:26:01.490 that means you have this relation between pressure 00:26:01.490 --> 00:26:02.580 and the volume. 00:26:02.580 --> 00:26:05.890 P times V to the gamma. 00:26:05.890 --> 00:26:09.140 Gamma essentially related to the decrease 00:26:09.140 --> 00:26:13.650 of freedom of the molecule, which we will discuss later 00:26:13.650 --> 00:26:15.910 in the class. 00:26:15.910 --> 00:26:19.615 This would be equal to constant. 00:26:24.070 --> 00:26:27.490 So the very interesting thing of this lecture 00:26:27.490 --> 00:26:33.460 is that we are going to be able to test which one is correct. 00:26:33.460 --> 00:26:37.930 You will be able to see if Newton win or Laplace win. 00:26:40.690 --> 00:26:44.080 So as I mentioned before, one is assuming 00:26:44.080 --> 00:26:49.540 the heat transfer, the speed of the heat propagation, 00:26:49.540 --> 00:26:54.490 is really, really much larger than the speed of oscillation. 00:26:54.490 --> 00:26:57.010 The other viewpoint from Laplace is 00:26:57.010 --> 00:27:01.420 that the heat flow is actually really negligible 00:27:01.420 --> 00:27:05.350 compared to the oscillation we are talking about here. 00:27:05.350 --> 00:27:09.340 And now, as usual, I would like to have a vote now. 00:27:09.340 --> 00:27:15.910 How many of you support Newton's idea? 00:27:15.910 --> 00:27:22.716 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11-- 00:27:30.020 --> 00:27:34.604 so 15 of you vote for Newton. 00:27:38.510 --> 00:27:42.060 How many of you saying Laplace is correct? 00:28:01.990 --> 00:28:04.950 How about the others? 00:28:04.950 --> 00:28:06.390 OK, very good. 00:28:06.390 --> 00:28:12.510 So we have a majority of you support the idea of Laplace, 00:28:12.510 --> 00:28:15.550 and some of you actually support Newton. 00:28:15.550 --> 00:28:19.590 And we are going to see what is going to happen in the lecture 00:28:19.590 --> 00:28:21.510 today. 00:28:21.510 --> 00:28:26.160 So let's go ahead and apply these two ideas. 00:28:26.160 --> 00:28:29.730 So PV gamma equal to constant. 00:28:29.730 --> 00:28:34.760 So in the case if ideal gas law, gamma is equal to 1. 00:28:34.760 --> 00:28:37.927 Therefore, I just have to work on these function of form. 00:28:37.927 --> 00:28:39.510 And then also later we will figure out 00:28:39.510 --> 00:28:43.500 what is gamma all together. 00:28:43.500 --> 00:28:48.360 So that's consider only small vibration. 00:28:48.360 --> 00:28:56.940 Small vibration means-- small vibration means 00:28:56.940 --> 00:29:01.170 that I have psi P, which essentially, 00:29:01.170 --> 00:29:07.470 from this definition, psi P is the change 00:29:07.470 --> 00:29:12.840 in pressure with respect to the room pressure, P0. 00:29:12.840 --> 00:29:18.060 So psi P, assuming that's much, much smaller than P0. 00:29:21.210 --> 00:29:27.810 And also, I assume that the changing volume, delta V, which 00:29:27.810 --> 00:29:30.870 I calculated there, is much, much smaller 00:29:30.870 --> 00:29:36.600 than P0, which is essentially the original volume 00:29:36.600 --> 00:29:42.050 of this little area-- 00:29:42.050 --> 00:29:48.420 original volume of this area I was working here. 00:29:48.420 --> 00:29:49.260 All right. 00:29:49.260 --> 00:29:51.055 So that's the two assumption. 00:29:54.660 --> 00:30:00.920 Before I change the position of the boundary, which essentially 00:30:00.920 --> 00:30:09.340 is the upper graph, if I change anything, I have P0 V0 gamma. 00:30:09.340 --> 00:30:17.310 This is equal to some constant, C. And gamma can be equal to 1 00:30:17.310 --> 00:30:19.900 so that you have ideal gas law. 00:30:23.180 --> 00:30:28.530 After the vibration initially happened, 00:30:28.530 --> 00:30:34.470 after the wall's initially displaced from the equilibrium 00:30:34.470 --> 00:30:38.110 position, what I'm going to get is-- 00:30:38.110 --> 00:30:47.910 I will have P plus delta P times V0 plus delta V to the gamma. 00:30:47.910 --> 00:30:56.080 This is equal to C. 00:30:56.080 --> 00:31:01.120 So based on those, actually I have already 00:31:01.120 --> 00:31:14.650 calculated delta P and delta V. So in this case, this delta P 00:31:14.650 --> 00:31:17.190 should be-- 00:31:17.190 --> 00:31:19.040 OK, this is actually not the delta P 00:31:19.040 --> 00:31:23.020 I was talking about there, so I should change it to delta P. 00:31:23.020 --> 00:31:27.120 Because that's, essentially, the difference 00:31:27.120 --> 00:31:31.440 between the resulting pressure and the original pressure. 00:31:31.440 --> 00:31:34.416 It's not the difference between the left-hand side pressure 00:31:34.416 --> 00:31:35.790 and the right-hand side pressure, 00:31:35.790 --> 00:31:39.180 which essentially is showing there. 00:31:39.180 --> 00:31:43.960 So this delta P, which I have already defined, 00:31:43.960 --> 00:31:49.190 essentially called psi P. Therefore, I 00:31:49.190 --> 00:31:59.120 will derive this to be P plus psi P. 00:31:59.120 --> 00:32:04.130 And I can now also copy these. 00:32:04.130 --> 00:32:08.150 You are going to get V0 plus delta V to the gamma. 00:32:08.150 --> 00:32:12.160 And this is equal to C. 00:32:12.160 --> 00:32:17.470 So as I mentioned before, I'm considering small vibration. 00:32:17.470 --> 00:32:21.880 Delta V is much, much smaller than V0. 00:32:21.880 --> 00:32:27.875 Therefore, this expression can be written as P-- 00:32:31.540 --> 00:32:33.040 sorry, this should be P0. 00:32:33.040 --> 00:32:36.100 I'm making some mistake here. 00:32:36.100 --> 00:32:44.160 This expression should be written as P0 plus psi P. 00:32:44.160 --> 00:32:46.810 V0 to the gamma. 00:32:46.810 --> 00:32:58.620 1 plus gamma delta V over V0, because delta V is much, 00:32:58.620 --> 00:33:02.120 much smarter than V0. 00:33:02.120 --> 00:33:05.360 Can everybody follow? 00:33:05.360 --> 00:33:11.510 So here I will already take small vibration approximation. 00:33:11.510 --> 00:33:15.440 And basically, I can now rewrite this thing. 00:33:15.440 --> 00:33:17.570 I just expend all those terms. 00:33:17.570 --> 00:33:22.400 Basically, I call this equation number two. 00:33:22.400 --> 00:33:36.406 Equation number two will become P0 V0 gamma plus gamma delta V 00:33:36.406 --> 00:33:48.850 V0 gamma to the minus 1 P0 plus psi P V0 00:33:48.850 --> 00:34:04.130 to the gamma plus gamma delta V psi P V0 to the gamma minus 1. 00:34:04.130 --> 00:34:05.580 So there's no magic. 00:34:05.580 --> 00:34:08.870 Essentially, it's just expanding these terms. 00:34:08.870 --> 00:34:11.750 Then, basically, you are going to get four terms. 00:34:11.750 --> 00:34:18.020 And basically, if you do write down the equation number 00:34:18.020 --> 00:34:20.190 two especially, that is essentially 00:34:20.190 --> 00:34:21.320 what you are going to get. 00:34:24.199 --> 00:34:27.620 So we are making progress, and we 00:34:27.620 --> 00:34:30.139 would like to simplify things. 00:34:30.139 --> 00:34:34.090 And you can quickly identify the hardest term, 00:34:34.090 --> 00:34:36.230 P0 V0 to the gamma. 00:34:36.230 --> 00:34:38.389 I know what is the value of that, right? 00:34:38.389 --> 00:34:42.350 That essentially is the original situation, 00:34:42.350 --> 00:34:52.000 and that is essentially equal to C. 00:34:52.000 --> 00:34:56.409 Let's take a look at also this term. 00:34:56.409 --> 00:35:00.010 This term is proportional to what? 00:35:00.010 --> 00:35:05.590 Proportional to delta V, which is a very small quantity, 00:35:05.590 --> 00:35:12.400 and proportional to psi P, which is another very small quantity. 00:35:12.400 --> 00:35:17.540 Therefore, taking a small vibration approximation, 00:35:17.540 --> 00:35:19.960 I would just simply ignore this term. 00:35:22.590 --> 00:35:24.920 Is everybody following? 00:35:24.920 --> 00:35:26.270 All right. 00:35:26.270 --> 00:35:29.080 So this term, this original term, 00:35:29.080 --> 00:35:32.580 is equal to C based on this expression. 00:35:32.580 --> 00:35:38.540 So we start from the system before the vibration happened. 00:35:38.540 --> 00:35:42.380 After the vibration happened there's a change in pressure, 00:35:42.380 --> 00:35:49.880 there's a change in delta V. But if you multiply PV 00:35:49.880 --> 00:35:53.680 to the gamma, this expression is still 00:35:53.680 --> 00:35:56.240 equal to C, some kind of constant. 00:35:56.240 --> 00:36:02.060 And then, now I do small vibration approximation, 00:36:02.060 --> 00:36:05.990 and I drop the term which is actually proportional 00:36:05.990 --> 00:36:10.730 to delta V times psi P. And basically what I get 00:36:10.730 --> 00:36:17.760 is that C is equal to C plus gamma delta P V0 00:36:17.760 --> 00:36:23.510 gamma minus 1 P0 plus psi P V0 to the gamma. 00:36:28.480 --> 00:36:32.890 This term, these two constants cancel. 00:36:32.890 --> 00:36:36.940 And now, I can actually move one of the terms 00:36:36.940 --> 00:36:38.280 to the left-hand side. 00:36:38.280 --> 00:36:40.620 Then basically, what I am going to get 00:36:40.620 --> 00:36:48.700 is psi P V0 to the gamma would be equal to minus gamma delta V 00:36:48.700 --> 00:36:51.670 gamma 0 gamma to the-- 00:36:51.670 --> 00:36:59.260 V0 to the gamma minus 1 times P0. 00:36:59.260 --> 00:37:02.935 We can't immediately cancel V0 to the gamma. 00:37:06.660 --> 00:37:09.000 Therefore, what are we going to get? 00:37:09.000 --> 00:37:18.195 I'm getting psi P would be equal to minus gamma P0 over V0 00:37:18.195 --> 00:37:26.002 delta V. Everybody following? 00:37:33.410 --> 00:37:35.300 So this is essentially the expression. 00:37:35.300 --> 00:37:38.220 And we also know why essentially is 00:37:38.220 --> 00:37:42.470 delta V. Based on this expression, 00:37:42.470 --> 00:37:48.920 delta V is essentially A partial psi partial x delta x, 00:37:48.920 --> 00:37:52.310 if we look at the upper board, which we actually just 00:37:52.310 --> 00:37:54.620 derived a moment ago. 00:37:54.620 --> 00:38:00.580 Therefore, I can write, replace delta V by that expression. 00:38:00.580 --> 00:38:05.090 Then basically what I get is psi P would be equal 00:38:05.090 --> 00:38:15.396 to minus gamma P0 A delta x divided by V0 partial psi 00:38:15.396 --> 00:38:17.180 partial x. 00:38:17.180 --> 00:38:21.360 I'm just plugging in the expression for delta V 00:38:21.360 --> 00:38:22.930 to that equation. 00:38:22.930 --> 00:38:25.460 A lot of mathematics, but all of them 00:38:25.460 --> 00:38:27.740 should be pretty straightforward. 00:38:27.740 --> 00:38:29.890 You don't actually have to copy because all of them 00:38:29.890 --> 00:38:31.440 are in the lecture note. 00:38:31.440 --> 00:38:32.810 OK? 00:38:32.810 --> 00:38:34.010 All right. 00:38:34.010 --> 00:38:36.800 So here, you can see I can, again, 00:38:36.800 --> 00:38:39.620 simplify this expression. 00:38:39.620 --> 00:38:44.210 A is actually the cross section of the tube, 00:38:44.210 --> 00:38:48.470 and delta x is the width in the x direction. 00:38:48.470 --> 00:38:53.400 So A times delta x is just V0. 00:38:53.400 --> 00:38:55.070 Oh, very good! 00:38:55.070 --> 00:39:00.040 I get this very simple expression, minus gamma P0 00:39:00.040 --> 00:39:01.776 partial psi partial x. 00:39:04.760 --> 00:39:09.920 So we have achieved our goal to simplify the expression 00:39:09.920 --> 00:39:15.050 and to find the relation between psi P and the psi. 00:39:15.050 --> 00:39:19.660 That's actually what originally we were hoping to do, 00:39:19.660 --> 00:39:22.340 and we have achieved that. 00:39:22.340 --> 00:39:26.080 And I call it equation number three here. 00:39:26.080 --> 00:39:29.590 Don't forget what is psi and psi P. Psi 00:39:29.590 --> 00:39:34.270 P is the amount of change in pressure, 00:39:34.270 --> 00:39:39.400 and the psi is the amount of displacement of the wall-- 00:39:39.400 --> 00:39:42.120 of the molecule in the volume. 00:39:45.600 --> 00:39:50.460 So now, I'm really close to my solution, 00:39:50.460 --> 00:39:55.170 because now I can now calculate the force acting 00:39:55.170 --> 00:39:58.260 on this little volume, because now I 00:39:58.260 --> 00:40:01.860 know what is the pressure. 00:40:01.860 --> 00:40:05.250 So what is the F total? 00:40:05.250 --> 00:40:08.700 The F total is essentially delta P, 00:40:08.700 --> 00:40:11.760 which is the difference in pressure 00:40:11.760 --> 00:40:15.011 from the left-hand side end compared to the right-hand side 00:40:15.011 --> 00:40:15.510 end. 00:40:15.510 --> 00:40:17.705 So that's what we calculated before. 00:40:20.640 --> 00:40:30.240 Now this, A partial psi P partial x delta x. 00:40:30.240 --> 00:40:32.625 We also know what would be the mass. 00:40:35.820 --> 00:40:41.300 We know that the little mass in this volume, delta m, 00:40:41.300 --> 00:40:49.100 will be equal to rho, which is the density of the air, 00:40:49.100 --> 00:40:53.970 times A, which is the cross section, times delta x. 00:40:53.970 --> 00:40:59.130 That will give you the little area, V0. 00:40:59.130 --> 00:41:05.640 So rho times A times delta x will be your delta m. 00:41:05.640 --> 00:41:08.370 We are almost there. 00:41:08.370 --> 00:41:12.750 I have the m, I have the force, what kind of law 00:41:12.750 --> 00:41:15.782 do I need to use to get my equation of motion? 00:41:15.782 --> 00:41:16.740 AUDIENCE: Newton's law. 00:41:16.740 --> 00:41:18.360 YEN-JIE LEE: Newton's law, right? 00:41:18.360 --> 00:41:20.100 Newton's law. 00:41:20.100 --> 00:41:23.670 So F is equal to m times a, right? 00:41:23.670 --> 00:41:26.950 So therefore, I can now calculate 00:41:26.950 --> 00:41:33.400 and essentially I can now plug in rho times A times delta x. 00:41:33.400 --> 00:41:34.780 What is A? 00:41:34.780 --> 00:41:38.630 A is essentially psi double-dot. 00:41:38.630 --> 00:41:42.630 That's essentially describing the displacement with respect 00:41:42.630 --> 00:41:45.420 to the equilibrium position. 00:41:45.420 --> 00:41:53.400 And now I know this is equal to force, which is A times delta x 00:41:53.400 --> 00:41:57.056 partial psi P partial x. 00:42:02.620 --> 00:42:05.680 And you can see that both ends you have a delta x. 00:42:05.680 --> 00:42:07.960 So I can cancel that. 00:42:07.960 --> 00:42:11.140 Both ends you have an A, so now I can cancel that. 00:42:11.140 --> 00:42:15.130 Then basically you get rho psi double-prime. 00:42:15.130 --> 00:42:20.240 This is equal to partial psi P partial x. 00:42:23.580 --> 00:42:26.755 From the beginning we're talking about the relation between psi 00:42:26.755 --> 00:42:32.430 P, the displacement in pressure, and psi, how much 00:42:32.430 --> 00:42:35.430 the molecules are displaced. 00:42:35.430 --> 00:42:38.460 And then we have the solution here. 00:42:38.460 --> 00:42:41.550 If you assume the relation which was 00:42:41.550 --> 00:42:46.230 given by Newton or by Laplace, basically, you 00:42:46.230 --> 00:42:53.490 can conclude that this would be equal to gamma 00:42:53.490 --> 00:43:00.370 P0 partial square psi P partial x square. 00:43:10.560 --> 00:43:11.490 All right. 00:43:11.490 --> 00:43:14.220 And I can now put all the constants 00:43:14.220 --> 00:43:15.300 to the right-hand side. 00:43:18.880 --> 00:43:24.880 Basically, what you get is psi double-prime. 00:43:24.880 --> 00:43:27.640 So here it should be partial square psi partial x square, 00:43:27.640 --> 00:43:33.500 because I replaced psi P by psi. 00:43:33.500 --> 00:43:36.230 And I must miss one-- 00:43:36.230 --> 00:43:39.320 I must miss one negative sign somewhere. 00:43:39.320 --> 00:43:40.220 AUDIENCE: Over there. 00:43:40.220 --> 00:43:40.670 YEN-JIE LEE: Where? 00:43:40.670 --> 00:43:41.570 AUDIENCE: [INAUDIBLE] 00:43:41.570 --> 00:43:43.320 YEN-JIE LEE: Oh, this essentially-- 00:43:43.320 --> 00:43:44.736 there's a minus sign there, right? 00:43:44.736 --> 00:43:49.692 AUDIENCE: [INAUDIBLE] On the left side. 00:43:49.692 --> 00:43:51.150 YEN-JIE LEE: On the left-hand side. 00:43:51.150 --> 00:43:51.450 Yeah. 00:43:51.450 --> 00:43:51.790 That's right. 00:43:51.790 --> 00:43:52.498 AUDIENCE: Oh, no. 00:43:52.498 --> 00:43:55.020 There should be A. That's wrong. 00:43:55.020 --> 00:43:57.630 YEN-JIE LEE: Yes, you are right. 00:43:57.630 --> 00:44:00.010 And the minus sign should belong there. 00:44:00.010 --> 00:44:02.870 So that actually-- sorry for that. 00:44:02.870 --> 00:44:06.700 So there should be a minus sign here. 00:44:06.700 --> 00:44:08.840 And there should be a minus sign here. 00:44:08.840 --> 00:44:12.790 And after I plug in equation number three, 00:44:12.790 --> 00:44:16.900 then I get psi double-prime equal to gamma P0 00:44:16.900 --> 00:44:21.730 over rho partial square psi partial x square. 00:44:21.730 --> 00:44:24.620 Any other problems you find? 00:44:24.620 --> 00:44:26.080 Not yet? 00:44:26.080 --> 00:44:28.270 OK. 00:44:28.270 --> 00:44:30.700 So look at this equation. 00:44:30.700 --> 00:44:31.390 Oh my god! 00:44:31.390 --> 00:44:32.710 What is this equation? 00:44:32.710 --> 00:44:34.300 AUDIENCE: Wave. 00:44:34.300 --> 00:44:37.000 YEN-JIE LEE: Wave equation. 00:44:37.000 --> 00:44:38.500 Again. 00:44:38.500 --> 00:44:39.040 Again. 00:44:39.040 --> 00:44:42.010 Wave equation. 00:44:42.010 --> 00:44:43.810 You can say that, huh, I'm not surprised, 00:44:43.810 --> 00:44:48.930 because this system is used so many times. 00:44:48.930 --> 00:44:50.530 I have learned this so many times, 00:44:50.530 --> 00:44:53.200 but I am still surprised that this 00:44:53.200 --> 00:44:57.970 is so identical to the physics which 00:44:57.970 --> 00:45:02.160 we have been studying for the strings 00:45:02.160 --> 00:45:07.030 for over the few lectures. 00:45:07.030 --> 00:45:08.710 So that's very nice. 00:45:08.710 --> 00:45:12.550 And now, the question we have an answer 00:45:12.550 --> 00:45:17.230 is that, OK, what essentially is gamma? 00:45:17.230 --> 00:45:20.090 What is essentially gamma? 00:45:20.090 --> 00:45:26.960 So gamma, in the case of the adiabatic process, 00:45:26.960 --> 00:45:33.260 gamma is actually equals to alpha plus 1 minus alpha. 00:45:33.260 --> 00:45:39.620 And alpha is related to the number of degrees of freedom. 00:45:39.620 --> 00:45:43.190 So if you haven't done this before, 00:45:43.190 --> 00:45:47.870 I have a concrete proof of the adiabatic process. 00:45:47.870 --> 00:45:51.890 And this is actually coming from the first law 00:45:51.890 --> 00:45:54.660 of thermodynamics. 00:45:54.660 --> 00:45:57.650 And basically, you will be able to conclude 00:45:57.650 --> 00:46:02.645 that gamma will be equal to alpha plus 1 divided by alpha. 00:46:05.180 --> 00:46:07.220 The value of alpha-- 00:46:07.220 --> 00:46:11.090 the value you get for alpha is related 00:46:11.090 --> 00:46:15.680 to how many degrees of freedom you can actually 00:46:15.680 --> 00:46:17.930 have in this system. 00:46:17.930 --> 00:46:28.700 For example, if I have a system which is made of atomic gas-- 00:46:28.700 --> 00:46:33.140 so there's only one atom in a molecule-- 00:46:33.140 --> 00:46:36.530 and basically you have 3 degrees of freedom. 00:46:36.530 --> 00:46:40.160 So you can move this thing in the horizontal direction. 00:46:40.160 --> 00:46:45.130 You can move this atom upside down or back and forth. 00:46:45.130 --> 00:46:48.920 So there are three degrees of freedom. 00:46:48.920 --> 00:46:54.860 And if you calculate alpha, that will give you 3/2. 00:46:54.860 --> 00:46:59.240 And basically, if you calculate gamma according to the equation 00:46:59.240 --> 00:47:03.620 you are going to get 1.67. 00:47:03.620 --> 00:47:08.470 On the other hand, if you have atomic gas, 00:47:08.470 --> 00:47:12.740 that means you have more degrees of freedom. 00:47:12.740 --> 00:47:15.320 So basically, you have not only the translation 00:47:15.320 --> 00:47:16.940 of degrees of freedom-- 00:47:16.940 --> 00:47:23.270 the three ones which are identical to the atomic gas-- 00:47:23.270 --> 00:47:27.000 you can also have two rotational degrees of freedom. 00:47:27.000 --> 00:47:32.540 So you can have these two atoms rotating like this, 00:47:32.540 --> 00:47:37.030 you can have that rotating like that. 00:47:37.030 --> 00:47:41.330 The trickiest thing is this one vibration degree of freedom, 00:47:41.330 --> 00:47:46.460 as you learned from the couple equations before. 00:47:46.460 --> 00:47:52.070 But this one, very trickily, is not excited at all 00:47:52.070 --> 00:47:52.880 at low temperature. 00:47:52.880 --> 00:47:55.640 You have to go to really, really super high temperature 00:47:55.640 --> 00:48:03.470 so that this actually contribute to the overall degrees 00:48:03.470 --> 00:48:04.380 of freedom. 00:48:04.380 --> 00:48:08.410 So therefore, you have a total of available degrees 00:48:08.410 --> 00:48:10.010 of freedom of 5. 00:48:10.010 --> 00:48:17.140 And if you calculate the gamma, you basically will get 1.4. 00:48:17.140 --> 00:48:22.610 So let's now calculate what will be 00:48:22.610 --> 00:48:28.145 the resulting speed of light. 00:48:28.145 --> 00:48:29.010 Sorry, no. 00:48:29.010 --> 00:48:30.140 Not speed of light. 00:48:30.140 --> 00:48:38.890 The speed of sound So if the temperature remain unchanged, 00:48:38.890 --> 00:48:44.380 if you take this equation here, this is a wave equation. 00:48:44.380 --> 00:48:51.050 Therefore, I know how to calculate the speed of sound. 00:48:51.050 --> 00:48:54.340 The speed of sound will be equal to-- 00:48:54.340 --> 00:48:56.790 P will be equal to the square root 00:48:56.790 --> 00:49:02.200 of gamma P0 divided by rho. 00:49:02.200 --> 00:49:09.160 So I have figured out the rho and the room pressure for you. 00:49:09.160 --> 00:49:17.290 So the P0 will be 10 to 5 kilogram ms squared, 00:49:17.290 --> 00:49:20.950 and the rho will be equal to 1.2. 00:49:20.950 --> 00:49:24.480 Rho is actually the density of the air. 00:49:24.480 --> 00:49:29.890 It's essentially 1.2 kilogram per meter cubed. 00:49:29.890 --> 00:49:38.230 If I have a gamma equal to 1, which is the case for ideal gas 00:49:38.230 --> 00:49:43.300 law temperature unchanged, if I calculate the resulting 00:49:43.300 --> 00:49:46.090 speed of sound you are going to get something 00:49:46.090 --> 00:49:51.590 like 389 meters per second. 00:49:51.590 --> 00:49:55.060 So that's the prediction for Newton. 00:49:55.060 --> 00:50:02.960 And the second case, if we have-- 00:50:02.960 --> 00:50:06.100 if we are believing what Laplace actually said, 00:50:06.100 --> 00:50:08.980 the heat flow is really, really negligible 00:50:08.980 --> 00:50:12.340 compared with the speed of oscillation, 00:50:12.340 --> 00:50:17.410 then we have, as we discussed last slide, 00:50:17.410 --> 00:50:20.140 gamma would be equal to 1.4. 00:50:20.140 --> 00:50:23.710 Therefore, you would be able to calculate the resulting 00:50:23.710 --> 00:50:29.650 speed of sound, and that is actually 342. 00:50:29.650 --> 00:50:31.650 So those are the predictions. 00:50:31.650 --> 00:50:35.770 And what I'm going to do now is to really demonstrate 00:50:35.770 --> 00:50:43.420 that we can actually measure the speed of sound in front of you. 00:50:47.460 --> 00:50:49.210 So the first thing which I will need to do 00:50:49.210 --> 00:50:54.510 is to switch so that you can see the camera. 00:50:54.510 --> 00:50:59.860 And now, I have a set up here. 00:50:59.860 --> 00:51:03.980 Basically, this set up is like the following. 00:51:03.980 --> 00:51:08.000 So basically, very similar to the setup we have here. 00:51:08.000 --> 00:51:12.580 But at one end, we actually have a speaker 00:51:12.580 --> 00:51:15.440 which produced sound wave. 00:51:15.440 --> 00:51:17.200 So this is essentially what we have. 00:51:21.950 --> 00:51:29.280 This is the tube, and we have, one end, 00:51:29.280 --> 00:51:37.150 there's a speaker attached to here, produce a sound wave. 00:51:37.150 --> 00:51:41.190 And basically, the amplitude will look like this. 00:51:41.190 --> 00:51:47.350 So basically, this will create some kind of standing wave 00:51:47.350 --> 00:51:49.660 inside the tube. 00:51:49.660 --> 00:51:55.320 And I have another device, which is actually a microphone. 00:51:55.320 --> 00:52:00.280 A microphone is connected to this scope, which shows 00:52:00.280 --> 00:52:04.450 you the amplitude of the-- 00:52:04.450 --> 00:52:09.100 basically, the amplitude measured by this microphone. 00:52:09.100 --> 00:52:14.890 And you see that if I move this, as a function of position 00:52:14.890 --> 00:52:17.770 you see that the am is changing. 00:52:17.770 --> 00:52:25.360 It's getting smaller when it is actually hitting the note here, 00:52:25.360 --> 00:52:31.920 because here there's almost no oscillation in the air. 00:52:31.920 --> 00:52:37.250 Therefore, you will measure a very small signal 00:52:37.250 --> 00:52:39.590 at that position. 00:52:39.590 --> 00:52:42.100 And if you continue to move, then you 00:52:42.100 --> 00:52:45.740 can see that, aha, I move away from the note, 00:52:45.740 --> 00:52:48.890 therefore I see some kind of maxima. 00:52:48.890 --> 00:52:54.870 Then, I see that this amplitude is dropping again. 00:52:54.870 --> 00:52:59.520 If I continue-- if I continue, then say that, aha, again, 00:52:59.520 --> 00:53:06.700 this amplitude is increasing to a very large value. 00:53:06.700 --> 00:53:12.710 Then it decreases to a minima around the note. 00:53:12.710 --> 00:53:16.340 So what I going to do now is to measure 00:53:16.340 --> 00:53:20.040 the distance between notes. 00:53:20.040 --> 00:53:26.930 And since the sound waves, which I actually put into the system, 00:53:26.930 --> 00:53:31.580 have a frequency of-- 00:53:31.580 --> 00:53:32.720 let me see. 00:53:32.720 --> 00:53:35.010 The frequency I put in-- 00:53:35.010 --> 00:53:39.120 the frequency I put in is actually 1 kilohertz. 00:53:42.800 --> 00:53:48.070 With the location of the note, I can know what will be, what? 00:53:48.070 --> 00:53:52.820 What will be the wavelengths of the sound waves. 00:53:52.820 --> 00:53:56.120 So therefore, I can now measure the distance 00:53:56.120 --> 00:53:59.750 between those three notes. 00:53:59.750 --> 00:54:03.830 Then I would be able to measure the wavelengths. 00:54:03.830 --> 00:54:08.450 Then I would be able to know who is correct-- 00:54:08.450 --> 00:54:14.570 if Newton is correct or Laplace is correct. 00:54:14.570 --> 00:54:16.580 So let's do that. 00:54:16.580 --> 00:54:20.360 So let me find the first minima. 00:54:20.360 --> 00:54:25.920 So the first minima is around 64. 00:54:25.920 --> 00:54:29.690 64 centimeter. 00:54:29.690 --> 00:54:31.640 And you can see that. 00:54:31.640 --> 00:54:37.130 Say stop when you see that it's reaching the minima again. 00:54:46.040 --> 00:54:46.610 Stop. 00:54:46.610 --> 00:54:48.320 OK, very good. 00:54:48.320 --> 00:54:50.630 So this is actually the first note, the location 00:54:50.630 --> 00:54:53.210 of the first note, and I was trying 00:54:53.210 --> 00:54:55.760 to find the next note so that I can actually 00:54:55.760 --> 00:54:57.560 readout the wavelengths. 00:54:57.560 --> 00:55:00.770 Now, this will increase again. 00:55:06.910 --> 00:55:10.600 And reach-- stop? 00:55:10.600 --> 00:55:12.180 Is that a stop sign? 00:55:12.180 --> 00:55:14.090 OK, very good. 00:55:14.090 --> 00:55:15.290 All right. 00:55:15.290 --> 00:55:24.860 So I get the value, which is actually 30 centimeter. 00:55:24.860 --> 00:55:27.200 So now I can calculate what would be the lambda. 00:55:27.200 --> 00:55:31.880 The lambda is actually 64 minus 30. 00:55:31.880 --> 00:55:35.040 Then, what I get is 34 centimeters. 00:55:38.570 --> 00:55:46.070 And if I calculate the velocity of the sound wave, then 00:55:46.070 --> 00:55:51.530 basically I have F times lambda, and that will give you-- 00:55:51.530 --> 00:55:55.820 this is actually equal to 0.34 centimeter. 00:55:55.820 --> 00:56:00.920 So that would give you 340 meters per second. 00:56:00.920 --> 00:56:01.500 Oh my god. 00:56:01.500 --> 00:56:07.056 This is so close to the prediction of Laplace. 00:56:07.056 --> 00:56:08.960 First of all, this is amazing. 00:56:08.960 --> 00:56:09.560 Why? 00:56:09.560 --> 00:56:13.490 Because with this looks really crappy thing, 00:56:13.490 --> 00:56:19.360 I can measure speed of sound. 00:56:19.360 --> 00:56:23.980 Secondly, ooh, the measurement is really great. 00:56:23.980 --> 00:56:26.870 It match within 1%. 00:56:26.870 --> 00:56:30.290 You guys did a good job of stopping me. 00:56:30.290 --> 00:56:32.360 Very nice. 00:56:32.360 --> 00:56:38.860 And finally, very unfortunately, people who voted for Newton 00:56:38.860 --> 00:56:40.520 is wrong. 00:56:40.520 --> 00:56:43.220 So what happened is the following. 00:56:45.884 --> 00:56:49.850 What happened is that Newton actually 00:56:49.850 --> 00:56:54.680 assumed that the speed of propagation of the heat 00:56:54.680 --> 00:56:59.210 is really fast, but actually that's not true. 00:56:59.210 --> 00:57:03.130 Because, for example, I'm standing here and heating up 00:57:03.130 --> 00:57:05.600 the air. 00:57:05.600 --> 00:57:09.590 In the last few lecture, I even heat up the air 00:57:09.590 --> 00:57:13.412 by some kind of fire in front of you. 00:57:13.412 --> 00:57:14.870 But you don't feel the heat, right? 00:57:14.870 --> 00:57:18.320 So the heat propagation is really not really, really fast 00:57:18.320 --> 00:57:21.560 compared to the speed of vibration. 00:57:21.560 --> 00:57:24.440 The vibration is really quick, because it's really 00:57:24.440 --> 00:57:31.100 vibrating up and down 1,000 times per second. 00:57:31.100 --> 00:57:35.950 So that means what is actually much more reasonable 00:57:35.950 --> 00:57:43.160 is to describe this process is adiabatic process. 00:57:43.160 --> 00:57:45.080 So we will take a five minute break 00:57:45.080 --> 00:57:47.390 here so that we can take questions, 00:57:47.390 --> 00:57:50.730 and then we'll come back at the 39. 00:58:03.590 --> 00:58:06.800 So welcome back, everybody. 00:58:06.800 --> 00:58:12.170 So we can see that from the last-- 00:58:12.170 --> 00:58:15.980 so from the discussion we had before the break, 00:58:15.980 --> 00:58:22.070 we see that the sound wave can be described by something which 00:58:22.070 --> 00:58:24.230 we are now very familiar with-- 00:58:24.230 --> 00:58:26.430 the wave equation. 00:58:26.430 --> 00:58:32.600 And also, we know what is the speed of the sound, which, 00:58:32.600 --> 00:58:36.290 based on this wave equation, the speed of sound 00:58:36.290 --> 00:58:44.990 is actually equal to square root of gamma P0 over rho. 00:58:44.990 --> 00:58:51.840 So gamma is actually obtained from this discussion 00:58:51.840 --> 00:58:56.750 of how many degrees of freedom we have. 00:58:56.750 --> 00:58:59.150 So the first case we discussed is, 00:58:59.150 --> 00:59:04.060 if you have a single atom of which you make your air, 00:59:04.060 --> 00:59:10.790 then basically the gamma is actually higher. 00:59:10.790 --> 00:59:15.800 On the other hand, if you have diatomic gas, 00:59:15.800 --> 00:59:18.590 then the gamma is actually slightly smaller. 00:59:18.590 --> 00:59:22.520 It's 1.4. 00:59:22.520 --> 00:59:29.990 So what would happen if I change the air in my lung 00:59:29.990 --> 00:59:35.990 to monatomic atom? 00:59:35.990 --> 00:59:39.050 So what is going to happen is that the speed of sound 00:59:39.050 --> 00:59:41.870 is going to be increased. 00:59:41.870 --> 00:59:43.910 The speed of sound will increase. 00:59:43.910 --> 00:59:46.730 And I have a fixed sized of lung. 00:59:46.730 --> 00:59:49.250 I didn't increase the size of my lung. 00:59:49.250 --> 00:59:53.525 Therefore, the frequency of my sound will what? 00:59:53.525 --> 00:59:56.320 Will increase. 00:59:56.320 --> 00:59:59.880 So how about we do that experiment and see if it works? 00:59:59.880 --> 01:00:07.520 So here, I have a balloon here, which is full of helium. 01:00:07.520 --> 01:00:09.080 Let me see if it works. 01:00:09.080 --> 01:00:12.740 I'm not sure if it will work, but let's see. 01:00:12.740 --> 01:00:14.040 Fingers crossed. 01:00:14.040 --> 01:00:16.580 We'll see what happens. 01:00:16.580 --> 01:00:21.050 Now I'm going to do a measure of operation 01:00:21.050 --> 01:00:25.190 to replace all the air in my lung by this. 01:00:48.960 --> 01:00:51.126 Does my sound change? 01:00:51.126 --> 01:00:51.625 [LAUGHTER] 01:00:51.625 --> 01:00:53.450 No? 01:00:53.450 --> 01:00:55.171 Didn't work. 01:00:55.171 --> 01:00:56.270 Let me do that again. 01:01:06.750 --> 01:01:08.140 I speak more aggressive. 01:01:08.140 --> 01:01:10.230 [LAUGHTER] 01:01:11.275 --> 01:01:13.501 Did you hear any difference? 01:01:13.501 --> 01:01:14.000 No. 01:01:20.318 --> 01:01:21.786 (HIGH PITCHED) Any difference? 01:01:25.520 --> 01:01:26.120 Works? 01:01:26.120 --> 01:01:26.660 Works now? 01:01:29.270 --> 01:01:30.797 Very good. 01:01:30.797 --> 01:01:31.880 Maybe we should use some-- 01:01:36.260 --> 01:01:38.402 maybe we should use that sound to go over 01:01:38.402 --> 01:01:39.360 all the lecture, right? 01:01:42.800 --> 01:01:44.450 It's a very dangerous experiment, 01:01:44.450 --> 01:01:47.600 because you are replacing all the air in your lung. 01:01:47.600 --> 01:01:48.505 So you may choke. 01:01:51.290 --> 01:01:55.894 Fortunately, I survived this experiment and hope you enjoy. 01:01:55.894 --> 01:01:58.738 [APPLAUSE] 01:02:03.960 --> 01:02:06.240 What happened is the following. 01:02:06.240 --> 01:02:13.460 So basically-- basically, the gamma becomes large. 01:02:13.460 --> 01:02:17.400 Therefore, the speed of sound in my lung becomes large. 01:02:17.400 --> 01:02:21.110 Therefore, the frequency of my sound 01:02:21.110 --> 01:02:24.590 increased and you hear some really strange sound. 01:02:24.590 --> 01:02:25.640 OK, very good. 01:02:25.640 --> 01:02:29.870 So before the end, I would like-- 01:02:29.870 --> 01:02:31.750 poor Newton. 01:02:31.750 --> 01:02:35.870 Before the end, I would like to discuss with you something 01:02:35.870 --> 01:02:40.620 which I hope I would not see again in the exam 01:02:40.620 --> 01:02:42.650 but I saw before. 01:02:42.650 --> 01:02:49.640 So if I create a progressing wave in a single-- 01:02:49.640 --> 01:02:56.540 in a closed end, open end tube, this progressing wave 01:02:56.540 --> 01:03:00.068 is going to be propagating at the speed of what? 01:03:00.068 --> 01:03:01.490 The speed of what? 01:03:01.490 --> 01:03:02.170 AUDIENCE: Sound. 01:03:02.170 --> 01:03:03.170 YEN-JIE LEE: Sound, yes. 01:03:03.170 --> 01:03:06.380 It's going to be propagating at the speed of sound, 01:03:06.380 --> 01:03:08.525 and you would reach the boundary. 01:03:11.880 --> 01:03:16.356 The question is, will we see this? 01:03:16.356 --> 01:03:23.880 Would this progressing wave just simply leak out of the tube. 01:03:26.910 --> 01:03:28.965 How many of you think that's going to happen? 01:03:31.700 --> 01:03:35.330 I hope I will never see that again in the exam. 01:03:35.330 --> 01:03:36.415 This would never happen. 01:03:39.050 --> 01:03:40.620 Why? 01:03:40.620 --> 01:03:49.370 That means you will have a super narrow collimated progressing 01:03:49.370 --> 01:03:52.370 wave going straight out of a tube, 01:03:52.370 --> 01:03:58.110 and that will not actually match the boundary conditions 01:03:58.110 --> 01:04:00.500 at the end of the tube. 01:04:00.500 --> 01:04:03.740 So that means, basically, first of all, 01:04:03.740 --> 01:04:08.190 there was no refraction. 01:04:08.190 --> 01:04:12.470 That means that all the energy is transferred 01:04:12.470 --> 01:04:14.370 outside of the world. 01:04:14.370 --> 01:04:18.190 And according to what we discussed before, 01:04:18.190 --> 01:04:21.420 what you would expect is that, OK, now you suddenly 01:04:21.420 --> 01:04:29.040 change to environment which you have really very large volume. 01:04:29.040 --> 01:04:33.240 Therefore, you it will be very difficult to change 01:04:33.240 --> 01:04:36.410 the pressure outside of the tube. 01:04:36.410 --> 01:04:38.160 Because what you are actually connected to 01:04:38.160 --> 01:04:43.470 is a reserve of infinite number of molecules outside. 01:04:43.470 --> 01:04:48.320 It's going to be really hard to change the pressure. 01:04:48.320 --> 01:04:51.300 Therefore, apparently this behavior 01:04:51.300 --> 01:04:53.850 doesn't match the boundary condition. 01:04:53.850 --> 01:04:57.000 And therefore, what you should expect 01:04:57.000 --> 01:05:02.530 is something like this, which I can show you here. 01:05:02.530 --> 01:05:06.220 So in this case, you have both side opened. 01:05:06.220 --> 01:05:10.540 What is going to happen is that at the boundary-- actually, 01:05:10.540 --> 01:05:14.530 it's like the case of hitting a wall, 01:05:14.530 --> 01:05:19.000 because outside of the tube you have really, 01:05:19.000 --> 01:05:23.600 really large volume, huge amount of air out of it. 01:05:23.600 --> 01:05:26.680 Therefore, it's like hitting a wall. 01:05:26.680 --> 01:05:29.590 The amplitude of the progressing wave 01:05:29.590 --> 01:05:34.030 changes sine and goes back through the tube. 01:05:34.030 --> 01:05:37.070 And of course, this system is actually not perfect. 01:05:37.070 --> 01:05:41.590 Therefore, there can be some leaking out-- 01:05:41.590 --> 01:05:46.390 some energy leak out of the tube, which essentially 01:05:46.390 --> 01:05:50.320 must be happening, because we can actually roll the tube 01:05:50.320 --> 01:05:52.000 and we can hear the sound. 01:05:52.000 --> 01:05:54.210 That is because some of the sound wave 01:05:54.210 --> 01:05:56.110 actually leaks out of the tube. 01:05:56.110 --> 01:05:59.530 And this process will go over and over again. 01:05:59.530 --> 01:06:02.290 And this progressing wave is going 01:06:02.290 --> 01:06:06.850 to be going back and forth, like that. 01:06:06.850 --> 01:06:11.730 So I hope that after this demonstration everybody 01:06:11.730 --> 01:06:15.136 will expect that, OK, this will be not-- 01:06:15.136 --> 01:06:21.370 the result will be like this progressing wave 01:06:21.370 --> 01:06:25.900 is going to be reflected because of the boundary condition 01:06:25.900 --> 01:06:29.768 and also change sine in terms of amplitude. 01:06:35.450 --> 01:06:37.280 So what we have learned today-- 01:06:37.280 --> 01:06:39.930 so it's already close to the end. 01:06:39.930 --> 01:06:46.390 We have learned example of a longitudinal wave 01:06:46.390 --> 01:06:49.790 And basically, longitudinal wave is actually 01:06:49.790 --> 01:06:54.140 in the form of density wave in the example 01:06:54.140 --> 01:06:55.940 which we covered today. 01:06:55.940 --> 01:07:00.080 And the mathematical description of the sound wave 01:07:00.080 --> 01:07:03.350 is going to be almost identical to what 01:07:03.350 --> 01:07:09.170 we have learned from the string case, which we actually 01:07:09.170 --> 01:07:13.370 discussed last time. 01:07:13.370 --> 01:07:15.770 There are two boundary conditions 01:07:15.770 --> 01:07:20.330 which I would like to briefly discuss before we 01:07:20.330 --> 01:07:23.310 end the lecture today. 01:07:23.310 --> 01:07:32.380 So in the case of open end, as we discussed before, 01:07:32.380 --> 01:07:34.750 we can have a system which contains 01:07:34.750 --> 01:07:40.400 a closed end and an open end. 01:07:40.400 --> 01:07:43.900 What will be the boundary condition for a closed end? 01:07:43.900 --> 01:07:47.080 So the closed end have a wall here. 01:07:47.080 --> 01:07:52.438 Therefore, when you have your molecule, 01:07:52.438 --> 01:07:55.540 the air molecule oscillating back and forth, 01:07:55.540 --> 01:08:02.590 when they are actually close to the wall, they cannot vibrate. 01:08:02.590 --> 01:08:03.090 Why? 01:08:03.090 --> 01:08:04.790 Because it's hitting the wall. 01:08:04.790 --> 01:08:07.420 It cannot vibrate so that actually, 01:08:07.420 --> 01:08:11.010 the boundary condition at the closed end, 01:08:11.010 --> 01:08:17.279 where you have a wall closing the tube, is psi equal to 0. 01:08:20.790 --> 01:08:25.370 And on the other hand, if you have an open end-- 01:08:25.370 --> 01:08:32.520 if you have an open end, that means outside of the tube 01:08:32.520 --> 01:08:34.770 the pressure is equal to what? 01:08:34.770 --> 01:08:36.460 It's equal to P0. 01:08:36.460 --> 01:08:39.300 The room pressure. 01:08:39.300 --> 01:08:42.250 And you have so many stuff there. 01:08:42.250 --> 01:08:46.260 Therefore, it's not possible to actually change 01:08:46.260 --> 01:08:52.020 the pressure dramatically at the edge of the open end. 01:08:52.020 --> 01:08:54.050 Therefore, what will be the condition? 01:08:54.050 --> 01:08:59.910 Psi P. Psi P is again the displacement with respect 01:08:59.910 --> 01:09:00.840 to the room. 01:09:00.840 --> 01:09:05.189 Pressure will be equal to 0. 01:09:05.189 --> 01:09:09.810 Based on what we actually have learned 01:09:09.810 --> 01:09:16.060 from this expression-- sorry for that, my finger slipped. 01:09:16.060 --> 01:09:24.410 From this expression, psi P is equal to minus gamma P0 d psi 01:09:24.410 --> 01:09:32.800 dx So psi P is proportional to partial psi partial x. 01:09:32.800 --> 01:09:38.240 Therefore, this boundary condition 01:09:38.240 --> 01:09:45.000 actually translates to partial psi partial x equal to 0. 01:09:47.609 --> 01:09:52.260 So this issue looks really familiar to you, 01:09:52.260 --> 01:09:56.480 because in terms of psi, if you forget about this system, 01:09:56.480 --> 01:09:59.940 what those boundary conditions mean to you 01:09:59.940 --> 01:10:04.140 is exactly the same as you have some kind 01:10:04.140 --> 01:10:09.450 of a wall in the left-hand side and it's connected to a string, 01:10:09.450 --> 01:10:11.090 and the right-hand side of the string 01:10:11.090 --> 01:10:15.030 is connected to a massless ring which can actually 01:10:15.030 --> 01:10:17.140 move up and down. 01:10:17.140 --> 01:10:20.780 These two systems, if you actually 01:10:20.780 --> 01:10:23.430 don't look at the detail-- 01:10:23.430 --> 01:10:28.620 only look at the wave functions and the boundary conditions-- 01:10:28.620 --> 01:10:31.950 they are identical. 01:10:31.950 --> 01:10:35.250 So that's actually the first lesson we learn from here. 01:10:35.250 --> 01:10:39.330 So when I talk about sound wave or when 01:10:39.330 --> 01:10:41.910 you think about sound wave problem, 01:10:41.910 --> 01:10:44.760 there's nothing to be afraid of anymore, 01:10:44.760 --> 01:10:48.760 because that's actually the same as what we have learned 01:10:48.760 --> 01:10:52.050 with wall and a string system. 01:10:52.050 --> 01:10:54.090 That's the first thing we learned. 01:10:54.090 --> 01:11:00.660 Secondly, that only works when I write my wave function 01:11:00.660 --> 01:11:03.390 in the form of psi. 01:11:03.390 --> 01:11:08.180 So now I can actually get the first normal mode 01:11:08.180 --> 01:11:11.965 would be like this if I plot psi as a function of x. 01:11:11.965 --> 01:11:13.570 The second normal mode-- 01:11:13.570 --> 01:11:16.600 doesn't surprise you-- will look like this, 01:11:16.600 --> 01:11:18.830 et cetera, et cetera. 01:11:18.830 --> 01:11:22.330 If I plot psi-- 01:11:22.330 --> 01:11:25.810 if I plot psi as a function of x. 01:11:31.120 --> 01:11:35.110 On the other hand, we also know that psi P 01:11:35.110 --> 01:11:41.950 is proportional to d psi partial psi partial x. 01:11:41.950 --> 01:11:50.370 Therefore, you can also plot psi P as a function of x. 01:11:50.370 --> 01:11:54.720 Then what you are going to get is something like this. 01:11:54.720 --> 01:12:01.870 In the closed end, the psi P is actually reaching the maxima, 01:12:01.870 --> 01:12:03.640 because it's got the wall. 01:12:03.640 --> 01:12:06.980 Therefore, it can actually produce pressure 01:12:06.980 --> 01:12:08.080 on top of the wall. 01:12:08.080 --> 01:12:13.090 But you cannot move the position of all of those molecules 01:12:13.090 --> 01:12:14.650 in front of the wall. 01:12:14.650 --> 01:12:16.230 Therefore, that makes sense. 01:12:16.230 --> 01:12:20.110 You will see exactly in the opposite direction, 01:12:20.110 --> 01:12:25.360 if you plot the amplitude as a function of x, 01:12:25.360 --> 01:12:28.870 you see a picture which is almost like flipped. 01:12:28.870 --> 01:12:32.800 Of course, you can also do the same thing for the second one. 01:12:32.800 --> 01:12:35.920 And basically, what you are going to get is something like. 01:12:35.920 --> 01:12:39.880 The second normal mode, et cetera, et cetera. 01:12:39.880 --> 01:12:46.930 So be careful about the matching between the boundary 01:12:46.930 --> 01:12:53.020 condition obtained from the tube and string-wall system. 01:12:53.020 --> 01:12:54.430 They are identical. 01:12:54.430 --> 01:12:56.830 Open corresponds responds to open, 01:12:56.830 --> 01:13:00.160 closed corresponds to closed when you express 01:13:00.160 --> 01:13:04.060 your equation of motion in terms of psi. 01:13:04.060 --> 01:13:07.390 On the other hand, if you change that to psi P, 01:13:07.390 --> 01:13:10.200 then the relation is actually flipped. 01:13:10.200 --> 01:13:13.450 Thank you very much, and I hope you enjoyed the lecture today. 01:13:13.450 --> 01:13:16.920 And I will see you next week.