Last lecture. In the last lecture we reviewed the sensation on the hearing sensation system and we also talked about the sound tube testing. And some pressure the measure associated with the pressure is SPL in db scale and also we talked about the waiting and the why, we have to consider waiting, for example, dba scale and also we introduced the frequency scale which we use in acoustics. Or noise octal scale. And let us move our very classical problem that we handle for the case of a string. When we have a thin string and a thick string, if the characteristics in key terms of this one is z1, and z2. Applying the boundary condition over there, I used the coordinate x. And the boundary condition at x equals 0 would be velocity continuity and the force continuity. Then we found that the reflection coefficient which is simply the ratio between how much wave is reflected compared with the instant way. That was expressed by z1 + z2, z1- z2. And transmissions questions, we found that, Transmission coefficient is z1 + z2 and the 2z1. Okay, right. And also, we examine the, examine the driving point impedance or for infinite string then this was low lcs, so cs is this. The propagation speed of string and the rule area is density per unit string and interestingly, if we have a finite string then the driving point impedence for finding string is not same as this one but has imaginary part, only imaginary part and that is cs and the cotangent ko. And I think we have, you remember that I emphasized two things. So one is four infinite cases, deriving point in pwc is exactly the same as the characteristic in p terms of median. That means that when I drive this median. Because there is no face difference between force and velocity that means I can effectively drive this system. In this case, because impedance is force, generally force over the last day. In this case, because there is a 90-degree phase difference that expressed by j. That means there is always on reaction from the medium. Because again there is place difference and also it depends on kl and I emphasize the meaning of kl. If I rewrite in terms of wavelength that is two pi over lambda that is K therefore that is a major of land scale with the respect to wavelengths. So if wavelengths is very large, then l over lambda approach to small value then cotangent KL approach to 1/kL because cotangent kL is 1/tangent kL and kL is very small tangent kL approach to kL. So this approach to kL. So there are a lot of interesting phenomena associated with this expression as far as that expression. Also, the look at the impedance and impede, no, no, no, intensity, compare with the impedance intensity is pressure multiplied by velocity. And that exhibits high different characteristics associated with medium, right. Okay, let me then continue to expand what we have served in one dimensional string case to a general acoustic medium. So let's consider, for example, I have a medium and I have the characteristic impedance of this medium is z1 and suppose I have instant to it. And because of the discrepancy or discontinuity over here. There will be some reflection and I don’t know this is Pr and this is PI because this is instant way and there must be a sum transmitted way pt. And what I am going to show today is for example reflection quotient of this case and the transmission quotient of this case is exactly the same s this. What we observed for the one dimension string vibration case. Okay, and then I will move. The understanding based on this simple case to the case when we have, Mass. Okay. In this case if I have incident to right of course there would be some trans reflection and there will be some transmission. Obviously, this case and this case would be different because of the difference of this continuity. And we would like to know how much wave will be reflected and how much wave will be transmitted for this case. And also, we move on to the case when we have More general circumstances. For example, I have wire. But this wire has not only the mass, but also spring, and the damping over there. And now again we want to see how much wave will be refracted. Compared with the instance wave. And how much wave will be transmitted. Okay. And then we will, based on this understanding, we will move to the case when we have a wire that has spring and a damping, but what if I have a wave coming obliquely? Now I want to see what's going to happen. How much transmitted, how much reflected, and how much reflected, and how much transmitted. What is the angle difference between instant wave and the transmitted wave? Okay, of course we also talk about how much wave will be reflected and transmitted for this case and how much wave reflected and transmitted for this case too. Okay, so this is what we are going to handle in chapter three. The title of this chapter is Waves on Flat Surface of Discontinuity. And one would argue that this model is too simplified. For example when we have a plate Okay. When we have a plate, then there is an instant wave. And plate will vibrate or infinite plate will generate a wave. And then there will be some reflection And also there will be some transmission. The amount of reflection and the amount of transmission are definitely dominated by the characteristics of plates. Obviously plates means that it has a finite bending rigidity. In other words, there is some curvature produced by the bending rigidity. Therefore, how much reflected and how much transmitted is very much depends on the mass of plate as well as the bending rigidity of plate. Okay. Then let me ask you some simple questions. Suppose you have a same mass, And you're supposed to make a wire or partition that can have many more transmissions. What would be your choice? Do you like to have this kind of wire or that kind of wire? It's not easy to think right away. But, let me ask you again this question. I have a wire over here. And I'm saying, due to the discontinuity on this work, there's a some wave is reflected and some wave is transmitted. Why? Why some wave is reflected and why some wave is transmitted. To figure out why this kind of things happen, the easiest way or convenient way would be I mean, suppose you are one over the particle over here. Okay, everybody imagine that you are one of particle over there. Then what you will experience for this case, the wave this coming, this flew the particle will compressed and then Back to the volume width, back to the volume as it was. And then expand, and then compress again and again. This will move this wire because there is an either fluid particle, the moving of this wire would be opposed by this fluid of particle. So after what happen is there is a traveling wave, the fluid of particle is fluctuating over here and this fluid particle is also fluctuating and this wire will vibrate over here, right? So actually, physically what happened is when there was an incidence where the wire is vibrating. Because the wire is vibrating, there is a radiation due the vibration of a wire That radiation produced of transmission and the reflection. Therefore, if we think on how this play to reradiate compare with this wall. Then, you could have some good idea about the transmission and the reflection. So I'm trying to connect what I learn what we learned form this one dimension string case, to three dimensional radiation problem. So that is our objective up to a few weeks from now. So let us try to see what we can obtain. For the case of which we have only Planar, I mean this case. Let us start this case, okay? Let me write down this PI. That has a magnitude of capital PI And then for convenience lets you use explanation j and then I want to use minus j omega t minus k x Okay this form is due to the applied boundary condition. Because when x = 0 this will give me explanation minus j over the t. So we prefer to use this to apply boundary conditions. And then I can write this as also right going wave, so I use Pt, which is complex magnitude and then again use exponential And i used- j omega p- kx. And this I can write, this is Pr, let me use r, technical r explanation -j omega p. And [INAUDIBLE] x, over here that should be plus. Because it is propagating in this direction, relative to the x direction. Okay, I think everybody followed it. Now, our objective is to find out what is R and what is tau? In terms of the variable we just employed. Okay, including z1 and z2. Okay. Now, let me ask you one question. This medium is Z1 and the characteristic impedance of this medium is Z2. Do you think it's more general to assume that this is omega. If this is omega then this omega has to be different or same. Because we are handling linear systems, linear differential equation. If I decide with this system with omega one then this median has to be incited with the same omega. Right? With that argument, we can use the same omega. What about the k? What about the k? Notes that k is wave number. And that is the relationship, Omega over C, and Omega has to be saying, what about the speed of propagation at each region? The speed of propagation depends on The ratio between pressure and density. In other words, how much pressure is required a full unit volume change and depends on median. Therefore, we see Speed of propagation in this medium and speed of propagation in that medium has to be different. Therefore this k1 and this k has to be different so I have to use k1 over here and k1 over here and k2 over here. Okay, that's very important. So now, we set up all the assumed pressure field in terms of all necessary variables.