As I mentioned in the last lecture, we would like to review what we learned in the last lecture, okay? I don't know whether this review really make you understand. The diffraction and scattering and radiation problem, at the same time. But let us try at least, the message or objective we have, [SOUND] in this chapter. As the name implies, we would like to understand radiation, Scattering, and diffraction, phenomena at the same time. The reason why we attempted to understand these three look very different acoustic behavior, along with the sort of unified concept is because for example is a radiation is, as the name implies, the sound radiating from the body because of its vibration. That is we normally call radiation. Okay? And the scattering, is what we normally say when you have incident wave, and it meet some object in space, and then it is scattered. And then it radiate. So in this case, we said total sound field has two components, one is incident sound, and the other one is scattered sound. And we found that in the last lecture, Pt has to satisfy rigid wall bound condition, in other words, dP dot n, this is the normal direction vector surface normal has to be zero, in other words, the velocity of the surface has to be zero. And this has to be Pt. So if you use this, and if you know Pi, then we can certainly calculate Psc. That's the basic concept. And then in the last lecture for simplicity. As a special case of this scattering, we consider the scattering due to the presence of sphere and space whose radius is a. And the scattered field does depend not only a, but also angle theta, strictly speaking, the scattering field does depend on ka, then measures the size of the scatterer with respect to wavelength lambda. Okay. For the radiation case, we learned the breathing sphere case that has a radius of a, in this case two as well as trembling sphere case, or we, we, we consider the case, when it trembles this way. And we found that radiation, for the trembling as well as the breathing case does depend on ka as well as depends on the observation distance from the center of the either breathing or the trembling sphere. In other words if you're in far field, in other words if kr is very large. That means we are observing sound and, and in the far field with respect to wavelengths of interest. And also the radiation does depend on the scale of ka also. The scales radiator size with respect to wavelength. And you remember that, the real part of this radiation is proportional to, kr, ka square, 1 plus ka square, and the imaginary part is simply ka, 1 plus ka square thing like that. Okay? All right. And then we move to the case in which we have slit. We start with a two dimensional slit, in other words, we have a infinite, you know,length scale In this, in this direction and we found that, that, if he regards the size of a slit, Okay, in this case a, this is, this is, this is- Think that is a. Then the sound of radiation from this slits. This also depends on kr and kr as well as the angle. Okay, and all this case we can solve using famous Kirchhoff-Helmholtz integral equation. Okay, for example, we can write G dp dn minus dG dn P, on the surface, that is radiated. So if you select G. G as the monopole and the make somehow dGdn disappeared and we can use the simpler form that predicts allow to predict some pressure at any point and that integral called Rayleigh integral. Rayleigh integral first kind or second kind, is nothing but when we have this one is zero or this one is zero, okay? That means if you do know the velocity of the surface of a radiator, we can always predict a sound. Okay? You have a powerful computer and you can do it. An, any shape of the radiation or scattering you can calculate. Okay? But what I'm, what I, what I want to emphasize using this lecture is, even if you got the result I mean you have to be able to understand the physics that results try to give you. Okay, to understand the physics it is necessary to understand the radiation or scattering, diffraction of the simpler case. So, we can expand this case to the case for which we have this kind of slit and this kind of opening, okay. The reason why we can say all these cases, can be considered along with the unified concept, because this is nothing but the case, the wave that is changed due to the presence of impedance discontinuity in space, okay? This is the case when we have, you know. When, radiator is looking at the space, with the respect to radiation impedance, and it, and this is the case, when we have impedance mismatch along this surface. And this is the case, when we have impedance, special impedance mismatch over here. Impedance of the wall in this case, is infinity, impedance over here is a pressure release impedance. And this is the case when impedance is infinity because, this is a rigid wall. When we have this kind of impedance mismatch, we can also use this, Kirchhoff-Helmholtz integral equation. That describes how the wave is propagating in space. And then we argue that because we do know the sound radiation from this kind of slit. Or this kind of opening, as well as this. We can also get a sort of analytic result, for the case when we only have this, okay, that's the case a very very long then you can get a solution by looking at the half space of this case that provide us the diffraction problems, analytic solutions of the diffraction problem. And also we can invite some, another boundary condition over here, and boundary condition over there, and split it out. Half space solution would give us the analytic solution of the diffraction problem. I mean you guys have a very strong computer, you don't have to, I mean you can calculate the sound field that this kind of, due to this kind of impedance, impedance distribution in space by using this. Either you can use a Rayleigh integral of first kind or second kind. We can get it, but as I mentioned before, it does not provide you any physical insight at all. I do know you have such powerful calculation of, powerful calculation ability but, I'm trying to provide to you some. sound powerful understanding, basis of acoustics. Okay, those, sound understanding depends on ka, kr, sometimes the J1,. sine something divided by something. Sometimes it depends on sinc function, blah blah blah, as you saw before. And I will repeat in somebody to show you how these things are expressed in analytic form today. And this part as I mentioned you before is the most difficult part. [BLANK_AUDIO]