Okay, now let's move to another case. The question is what is going to happen if you have wave instant of [INAUDIBLE]? To surface of this continuity. What if there is wave of likely instant to for example, partition, general partition. And what if the wave instant to, for example, plate. That is our next topic of discussion. Okay, to see the fundamental physics associated with oblique instant, let's see the following graph. Okay, the first question is is instant to and the r is reflected angle and the t is transmitted angle. What is the relation between i and R intersected t. That is governed by very well known law and I'm sure some of my students know this. Surprisingly when you are in elementary school, that is called Snell's Law. Okay? Maybe some of the students feel very guilty that you do not remember Snell's Law, but it doesn't really matter, okay? Let's see some detail physics associated with the Snell's flow. Okay instant wave this flat surface of discontinuity with this angle, okay. Then it propagated this way, therefore this is the wavelength of instant wave I mean suppose it propagate this way like that, and this is the point B and this is point A. Okay this is propagate, [SOUND] at certain time, the incident wave approach, I mean reach over there, A. Right? And then it travels. What the distance? This distance. Okay. And the reflected wave reflected and propagated that direction, therefore this is the wavelengths due to the reflection. And those are transmitted waves therefore this is the wavelengths of transmitted wave. And projected distance upon this discontinuity has to be same, right? Projected distance of zeta t over there, and projected distance of this wavelength over there, and projected distance of this length over there has to be same, right? Geometric curve identity. So let's express that one in terms of, in terms of the geometry. Okay, delta y, delta y, delta y has to be same. Lambda i/sine theta or back to here, I can say delta y, and this is the angle, so what is the relation between lambda 2 and lambda t and delta y? Lambda t, this is lambda t. And this is a theta T. So therefore this is a theta T, so therefore delta Y, psi theta is lambda T, right? So, this is true. So, I can say delta y is the lambda i over sine theta i, and this is true and this is true. And this is the Snell's Law. And of course this is true too. Move shows this and this is not interesting, so let's se what it means, by that physically okay. So let's look at this one and that one. This is interesting, because depending on c0 and c1, you can have interesting phenomena. Let's see over there. If I convert this in wave number form, and wave number is omega/C0. So therefore this term is k0 sine omega 0, and this term is k1 sine omega. Okay, I sine omega I, sine theta I. So this is simply the projected wave number on the surface. Right? So if K0 is very, is larger than KI, okay. This is the k0 and I draw the picture over there. At a certain angle this is k0 sine 0. So that is the k0. And sign set to zero is this lens, as that lens has to be equal to K1 sign set at zero. So in this case, the K1 sign set to zero over there has to be the same as K set to 1 has to 90 degrees. So if the angle of incidence is larger then this critical angle then this has to go over there. And it is not possible for k1 sin theta 1 can be this so geometrically there is a very interesting phenomena happen. So in this case, how mathematically can we achieve this case? This the case. Going over there, this is the case we should have imaginary angle that has to be greater than pi over 2, okay. This is the case, because I mean K K zero, sine zeta zero is larger than this. Therefore, the angle has to be greater than pi over two. And that angle has to be something that represent the physics that we do not know. And this angle, we call critical angle. Okay. And this happened physically for a case when we have a medium. K is omega over c. So when c is small that means I am in error. When c is large, that means there is water. Now if some instant angle is greater than critical angle, then there is no transmission. To the water. Actually what happened is the surface where there's a surface wave is generated. So if there is a sound wave coming from water, in this case, K zero. Is larger than k1 in this case, in any case her is a transmission. Let me continue to this rather strange concept in the next electro, and let me summarize what we've learnt today. We've reviewed the Mass Law and we studied this Mass Law is quite relevant with the Mass Law we observe in single system in high frequency. So we were sufficiently persuade that the mass law is valued only for the high frequency reason, and the essential elements I attempted, I have attempted to. And to you into this lecture is transmission is composed by two part one is fluid and the other one is. The fundamental physical basis to get that formulation is observing that PI plus PR on the left hand side of partition is equal to PT minus J omega Y multiply by Z zero, that means that the transmission is have transmission can be considered as two times the instant pressure to that is blocked pressure plus the radiation of partition. And the radiation of partition introduced fluid loading impedance. That is written over there. That's the central part that I attempted to deliver to you guys. And then we show that everything we learned today can be seen as this picture. This picture shows an absolute region, and stiffness controlled region, and damping controlled region. And then we introduce the Interesting physical behavior happen we have oblique instant aware. And next lecture where we talk a little bit more about the physics associated with oblique instant and I would like to persuade you the instant angle has to be same as, I mean reflected angle has to be same as the eastern angle. Of course transmitted angle be different by seeing those snares low in a wave number low Which is not, I mean, very well known. So, I personally think that looking in wave number provide more physical insight. Okay.
