The last lecture, we actually reviewed a very well known law, which is Snell's Law. That essentially determines How the, >> [SOUND] >> If you have two different regions, characterized by specific impedance of media, let's call this is instant web. It has certain pressure amplitude complex pi, and then propagating in the direction that has instant angle Zed to I, so this has to be wave number beta multiplied by R, and this wave number beta K Of course, has three components, kx, ky, and kz in the direction of in its vector IGK if I use the rectangular coordinate. But essentially, it says that it is propagating into space with wave number K, okay. This is simply the projection of wave number upon x, y, and z direction. Of course in two dimensional case for 2D case, k has only kx and ky, and in this case, kx has to be k. That's the wave number, number of wave per unit length propagating in this direction. K, and if I use the coordinate, That is x and y, then it means that is kx cosine k cosine theta i. And the ky has to be k sine theta. >> Okay. And also, the reflected wave, Pr, has to be complex magnitude Pr, and then I have exponential j omega t. Maybe I use plus over here, k dot r, and then recognize that that has to be kr, that has to be ki. And this has to be ki, okay? The transmitted wave can be written similarly, that it has complex magnitude, and propagating in Certain wave number that has to be k t. Okay. [COUGH] So, this is what we assumed that if you have a two different media, we assume that there will be some wave reflected and some wave is transmitted. Of course, that has to be reflected angle, and that has to be transmitted angle. Okay, The issue, the question is, what is the relation between theta i, theta r and theta t? And how the translated y with respect to pr and reflected y with respect to pi, has to be. The first one we call transmission corruption, the second one we call reflective corruption. [SOUND] Okay, of course there can be found from where? For the physical condition at interference, what is the physical condition of interference? Pressure has to be z, the velocity has to be z. But because this is the oblique instance case. We have a two component. One is pressure in X direction, the other one is pressure in Y direction, the velocity in X direction and velocity in Y direction. And because notes that the fluid is inviscid. [SOUND] >> [COUGH] >> What do I mean by inviscid? Simply, there is no viscosity, because there is no viscosity, the pressure in Y direction. Okay, that's not necessarily to be same. In other words, because there is no viscosity, if I excite a fluid over there, then this fluid will not move. Okay? The only pressure that has to be matched on this surface on this continuity has to be pressure in X direction, and the velocity in X direction, and the velocity in Y direction. Okay, that gives us the solution about this relation, and this relation, as well as this relation. Okay? Having an understanding the [INAUDIBLE] associate with [INAUDIBLE] will go to the case. When we have different material, different type of flat surface of this continuity. For example, what if we have infinite plate? >> Okay, we have an infinite plate and suppose the wave is instant of from this direction, and the wave Then the y will be reflected. And of course the y will be transmitted. We would like to especially know the this. Look like. And also we want to know Pt in terms of the characteristics of this boundary. Because this is plate, plate means the physical sort of component that can transmit the random rigidity. Compare with a membrane, that can transmit the attention. Okay and we would like to relate this one with what we learned before. Tau is related with what? Fluid loading impedance, in the previous chapter. [COUGH] Partition impedance plus impedance. That's what we learned and that can be applied to the general partition case when we have normal instance, and because that is related with this one is related with a radiation. Into fluid medium. Okay? So we would like to see how this kind of flat surface of this. In other words, if the plate, infinite plate, carriers the wave, which you will call bending wave. In this direction, bending wave, will properate in this direction. And the wave number of bending wave, kb, has to be related with wave number of this and that and that one, and that is our interest. Provide kb, how this one is related with, related with ki kr and a kt. That would be our interest. Okay? And then our final objective of this lecture would be. We started the transmission loss and transmission corruptions, how the pt is related to pr. Or the infinite case, how is the infinite case related with the finite case? Okay, the transmission question or transmission loss, that is determined or predicted by assuming the plane is infinite. Predict upper bound or lower bound of transmission loss. Okay, so for example if we want to. Predict transmission loss of this kind of case [COUGH] What is related with tau, infinite, tau, finite, how those are related? Okay? This is very interesting approach. So, I want to compare this with this. Per unit area. Okay, I'm saying transmission question the per unit area of this case. Infinite case and the finite case. Okay, one simple way to look at this, is let's look at the transmission loss of this region. Note that all case [COUGH] transmission lost, transmission collection is obtained as if this is radiating sound through the medium. Okay, if radiation of this kind of plate is larger than the radiation of this kind in an infant case then we have bigger transmission loss or transmission corruption before the final case right? But if you look at this case, compare with that case, and suppose we have a fluid particle over here, and fluid particle over here. Okay, because of the etch effect. Okay, suppose we have very large structure for having the concept Rather than the exaggerated way. Obviously I can effectively radiate more radiation compared with that case. So I can argue that the transmission coefficient, which is the ratio between Pt and Pi. Well the finite k has to be larger than transmission coefficient of infinite case. So therefore I can say the transmission coefficient in DV scale, that is one over t infinity or t finite not only to the scale. That is transformation law. For infinite case, always bigger than transmission laws for the finite. Therefore, what we can estimate RTL transmission laws By assuming that the plate is infinite always predict upper bound of transmission loss of finite k, so therefore what we, Estimate transmission loss based on infinite is very useful. It's very useful because as, because if we have upper bounds transmission losss, maybe I use equality over there. I over estimate the transmission loss, therefore I can argue that what we estimated is very safe estimation in terms of transmission loss. So this is useful, this is therefore very useful. So that will be our conclusion of this lecture. Let me begin with the issue that I just got this, and the review what we have, in this relation, since using PowerPoint. Okay as I said before the instant wavelengths could be like this because the instant wave is propagating in this direction. And the reflected wavelengths has to be this amount. And transmitted wave has to be this amount. As you can see here geometrically it shares the same identity over there. So lambda t, lambda t has to be same as the lambda I sine of the instance angle and the lambda r sine theta r. Sorry, the other way around. So this has to be saying, that gives me the famous Snell's Law. And it is interesting if you look at this Snell's Law in terms of wave number, that says ki sine theta-i equals to kr sine theta-r, and that has to be same as kt sine theta-t. Personally I think that is interesting. Because that means in terms of wave number, the projected wave number upon the surface of this continuity, the surface of this continuity has to be same. That makes sense, number of waves propagating on the surface of this continuity has to be same. If I say again number, wave numbers of an incident wave and number of waves of reflected wave, and number of waves of transmitted wave has to be same. So if you use that concept, okay, so this is incident wave and this is, for example, reflected wave. The interesting thing physics can be observed very easily over here. So sine theta zero is for example this is K zero and the K zero sine theta zero has to be this length. Has to be same as the k1. K1 sine theta i, that is this amount. So this one has to be same as this one. No problem until k, I mean if it reaches to the some critical angle, over there, then this has to be reached this amount therefore in this case, theta i, theat 1 has to be 90 degree. So if it exceed the critical frequency dramatically it has to be this amount. Therefore ki there is no way to make ki say sine zeta, i has to be this amount. So one simple way is to make zeta i is greater than pi over 2. I'm introducing some imaginary part over there, okay, mathematically. And you will see we need to put the minus sign over there, okay? So that produce interesting phenomena because we have the wave propagating more then angle pi over 2. And because of this contribution we have evanescent wave. In other words, wave is decreasing with respect to, for example if you have this interface in respect to that distance, now for the wave is very confined in the vicinity of the interface. So that is very interesting phenomena which can be seen by using the approach.