This will be Lecture number 15. Today I will review So called Kirchhoff- Helmholtz Equation And then I will briefly show how this equation can be used for, as we already studied, but what we learned is not easy to understand. So we will quickly review how we use this for Understanding the sound radiation from baffled piston. And then based on this understanding we will quickly review how this was applied for the radiation From a plate vibration Here we will talk about the relation between the free space wave number and the wavelength number in x, y and z direction. How the distribution of wave number in x and y really affect the sound population in g direction. Okay And then we will again talk about how this Kirchhoff- Helmholtz integral equation is used to get the sound pressure At any point of interest in space. Essentially that is the result of application of Helmholtz-Kirchhoff integral equation. On this problem, knowing that we do have information about the velocity on the plate. Okay, then we will go to Scattering Problem. Okay, a scattering problem Can be addressed as following. For instance if you have this kind of scatter, you have an instant way of coming. And then due to the presence of the impedance mismatch in space, due to The presence Of impedance Mismatch in space. For example if this is rigid wall defining normal vector like that, surface normal from the surface or surface normal and surface normal. Then we scatter, there will be some scattered wave. Total wave of course, because we are handling linear acoustics, total field would be Pi + Psc. Okay, knowing that we are handling this in frequency domain. Frequency domain approach. Okay then, we will, Introduce how we can solve this problem upon knowing the Pi. We would like to get the Psc, scattered sound pressure due to the presence of impedance mismatch in space. This is the sort of whole story that we are going to talk about today. So let me begin with reviewing Kirchhoff- Helmholtz integral equation very quickly, then. Okay, this is well-known Kirchhoff- Helmholtz integral equation. It looks like a complicated but for convenience I just separated the tree expression. One is due to the contribution of pressure, Pressure fluctuation on the boundary over here. And this is the contribution due to the velocity fluctuation on the boundary. We can write this equation, not separating Sp and Su boundary condition of course. Okay, then, Okay, essentially, it says there's two contribution. One is due to the pressure on the boundary, r0 is the boundary surface. So the pressure on the boundary will be propagated using this propagator, okay, Green's function. And this is related with the velocity because the Euler equation says, gradient pressure is simply rho 0 dv / dt. Velocity is the, V is velocity. So this part says the velocity is propagated using this grid's function. And we attempted to see the sound pressure at any point to r, when we have the velocity generated by the baffled piston, which has the velocity distribution look like that. Okay, to solve this problem we've modified this problem as if we have two symmetric velocity per fluctuations like a braiding sphere. Then due to the symmetry we can automatically satisfy this rigid condition. And by selecting the Green's function that Satisfies this condition, in other words we can select the Green's function that That would be this up here on this surface, but has some value on another surface. So essentially, it loos like that. So simply, that means the velocity, velocity on the surface would propagate by the function. 1 over R creates function, grace function. And summing up all the propagation will result the pressure at R. And this can be used to any type of baffled piston case. For example, if they have a baffle and if they have a plate we can use that function approach. Because UN in this case would be something related with the sine or cosine. Okay, we will see that later on. And then what we do in the beginning to look at first the some pressure at this point along this line, okay? Then of course R can be written as like this. Z is the distance from here to there, and zeta is this distance and of course R is the distance from this infinitesimal element that is 2 pi zeta d zeta because the lens over here is 2 pi zeta, and this lens is d zeta, therefore this is the area over here. And integration of those can be quickly done by noting that if you change variable z square zeta square to maybe alpha square, then this will be z k and alpha, and then this is alpha. If you take a derivative with respect to the variable. In this case, zeta is a variable. Therefore, in this case, 2 zeta d zeta is equal to 2 alpha d alpha. Therefore, I have to write over here pi then 2 zeta d zeta is 2 alpha d alpha. Therefore that would be alpha d alpha. Alpha goes away. Therefore we have this very simple intuition and then we can do it. So, it turns out the simple mathematics gives us this expression. Physical meaning of this expression simply says that sound observed at z, in other words there is a baffle. This is the z axis so sound observed that any point of z is, composed by two component. One is this one that is e to the k z that means that sound coming from at the center of the piston. And this is R0 is the distance from here to there, so another sound is coming from at the rim of baffled piston. So that is quite interesting. Of course, in this area, the E to the KZ and E to the KR0 Both result constructively or destructively so there will some oscillation but later on there is an exponentially, I mean linearly decaying sound waves over here. So in the far field it looks like e to the k z time, radiation. In other words, far field means that I am very far away from the source. In other words, kz is very large, okay? If kz is large, then you've got this result. And this shows simply that the sound is radiating like monopole, okay? As we expected, all right. And then expanding what we learned to the case when we would like to observe the sound at the point which is not on z but anywhere other than z axis. And result shows this interesting result and again shows that it is radiating sound like one over r e to the j k r over r that means monopole but somehow shaped by this factor. This is a necessary function for a force decline. The argument is ka sine theta. This provides directivity. Okay? Okay, this is the case which we have the baffled piston but it is oscillating with UN but if you look at the radiation impedance that looks like this. Remember, radiation impedance of monopole? The real part of monopole radiation look like k. If you look in ka schedule that look like ka scale 1 plus ka scale, right? And imaginary part will look like ka over one plus ka squared. Was it two? Maybe. So, I mean this one, this one may look like this. And this one may look like that. Where this case because we have a baffled piston, we have some oscillation, but in a far field over here, the radiation pattern or radiation impedance is very much similar with the gradient sphere, I mean in some part between breathing and trembling sphere. But generally the radiation pattern is quite similar but it can be also embesieged in certain extent if you oxalate some fluid with a baffle, and baffle the piston, baffled piston oxalates the fluid. Yeah, in a far field and if ka is very large. In other words wave lengths is small. Compare with the radiators size, then the fluid particle on the surface of radiator or it could be oxalate very effectively to propagate in a far field. So that is this way. The only difference is somewhere over here. But in average size it's pretty much close this radiation pattern. So that is quite interesting result. And we expand that observation to see the radiation from this kind of plate that has a distributed velocity over the surface of a radiator. But again, because we are handling linear acoustics, That means we can use the principle of superposition. Therefore, we can regard this complicated velocity distribution as the sum of these normal mode, for example. And we do know the baffled piston result. Therefore, we can regard this kind of the vibration would be sum of this plus and that, then this can be regarded as these two type of baffled piston. Of course, we can say that this plate is vibrating with this amplitude, but in a far-field. The radiation pattern should be very similar with the radiation of this baffled piston, so we can use this result. And again, using similar concepts, you can see that due to the interference between the nodes of the radiated sound field we will get what will look like that. If we have more modal contribution, then the radiated sound at certain points is getting smaller and smaller and smaller due to the interference in fact. That's what we learned in the last lecture. Again, we know that the k squared, that is, a free space k wave number, should be kx square plus ky square plus kz square. That is generally called dispersion relation. The way to get dispersion relation is simply plot. Our general solution p(x,y,z,t) should be capital P(x,y,z) and then exponential -j omega t to wave equation. Okay, then we will get this dispersion relation. Assume that P(x, y, z), As the some coefficient times exponential jk z and z, and then exponential j, sorry, k x and x, y and y, exponential jk z and z. Then we will get this. And this is a very interesting wave number plot that expresses how the sound is radiating away from the plate, okay? If you looked at the wave number along this line, that means we have certain kx but 0 ky. 0 ky means we have, What? In y direction, say if we say this is x direction, this is y direction. If ky is 0, what it means? ky is 0, what it means? ky is simply the number of waves per unit length in y direction. Therefore, ky is 0 means that there is no wave. In other words, in y direction there is a DC. In other words, in y direction everything comes back and forth. So in this case we have a plate, but there is only wave in x direction. Plus, minus, plus, minus, things like that. As we away from the center we will have more waves along x side. And in this region, we have similar behavior while we have many, many waves in y direction, but rather a DC component in x direction. We call this as edge mode region. This is edge mode region and this is corner mode region. If you look at the radiation efficiency, corner mode region is not efficient at all, as we can see over there. But we didn't look at how the sound pressure at any point can be obtained. The vibration of plate, Un(x 0, y 0)=Umn, sine k x, x 0, and sine k y, y 0. Then the sound pressure using Kirchhoff Elovich recursion sound pressure, p(x,y,z) would be -jk rho 0 c Umn over 2 pi. And I have integration part where I'm integrating 0 to ly and 0 to lx. And I have sine kx x 0 and sine ky y 0. And then this would propagate like a monopole R to jkR, and I'm integrating dx 0 dy 0. Meaning that I have a plate. This is x and y, and this is z. And we use the notation, this surface is S 0. Therefore, this is region of y 0 and x 0 as well as x and y, okay? So the only difference is this part compared with the other case. And then we can calculate it using MATLAB software.