Okay, this is the equation based on the Kirchhoff Helmholtz equation. Of course, we have a different scale factor. And we would like to know the pressure along this line. Okay, then the exponential jkR tom over R changing like this. Of course, I have un that is constant and this is simply the disk slip. Okay, I use the zeta over here and the zeta has to vary from zero to A over here. And then the area would be because this length is D zeta. And this is 2 pi zeta. Therefore, that is this one. And that corresponds to the s0, okay? And I have un and this is the R and R has to be z scale. This is a z, this is a z at the scale. That's good, that's good. It is interesting, if I substitute this certain so it's a variable like z squared plus 8 squared is a alpha squared. Then as you can see over here we have this vector, 2 pi delta. Okay and this delta would be here, to delta d delta would be two alpha, e alpha and that allow us to calculate this very easily and that gives us this result, very simple result this okay? Pressure on the line of z is and that is interesting. If you see the pressure divided by u and this is somehow related with impedance that is simply proportional to rho 0 c, but having this factor. Okay? If you don't have this factor, then piston, baffle the piston behave as if Planar. Okay, the first term is exponential jkz. What is this? That is wave coming from the center of the piston, because we use the coordinate, this is z. And this term is wave coming from the center of the piston. And forget about minus, looking at this. And that is the wave coming from, this is, R zero, coming from the rim of the piston. So as I said before, two waves is coming, one is from here and the other one coming from there, all right? If I getting away from the center that means z approach to sorry. This is r0. This is the distance of r0. r0 is this distance. I'm sorry, so if away from the center if I away from the baffle the piston. That means z approach to r0. Our general approach to then what we will have. Zero? Because both terms are equal, hm? If we are far away from the center, from the piston then z approach to r0, is it correct? Or is it r0 or is it this? This is this. So if z is very, very large compared with A then the pressure we will get is very small. Is it true? Yes it is true because the radiation pattern is like a motor port and we are far away from the center. The pressure has to be diminished. So that's okay, okay? That's okay. Right, this is another approximation. [NOISE] When the z is significantly large compared with the a. This can be approximated like that. Okay, what is that? Okay, looking at this term 1 over z to the ktejke. That is monopole, okay. That is monopole. So again when z is large compared with the size of a baffle, sound baffle that behaves like a monopole as we anticipated before okay, that's fine. Now we would like to see the pressure at any point other than z axis. So sound, I have a baffled piston over here. I already looked at the sound at the center, over here. But I want to see how the sound propagates in space, other than this axis. Okay? Other than this axis. Over here, over there, over there. One thing we can immediately know it should be symmetric with respect to z axis, yeah? Should be symmetric in other words, if the pressure over here is one, then the pressure over here is one, and around this circle, the pressure has to be same, right? And what other things we can find? Okay, let's see time by time. Okay this is velocity and this is characteristic impedance, okay? If you somehow got to a driving point impedance of this piston. I suggest to one of you guys try to do that. What is the driving point impedance of a baffled piston? Last night I found that I didn't do it, strangely, in the past 20 years. But I should do it. Because looking at the driving point impedance of piston or the interesting in connection with. We have looked at all the radiation pattern. By looking at driving point impedance but not for the piston. So why don't we do that? Okay, anyway in this related roger C that is the characteristic impedance of medium and it looked like one of our integer kr that is monopole. But it is scaled by this interesting function. J1, that is the Bessel function, first kind. Divided by the elemental Bessel function. What it look like? If ka is very large, in other words, Or sorry, if ka is very small, in others wavelength is large compare with the baffle size of a. This term look like uniform in terms of theta. But when k getting large and large, what we can see this function look the following. Okay, this is when k equal zero very small ka when it is getting larger and large, you will see some directivity pattern and when ka equal 20 you will see very spiky directivity and theta equal 0 of some side lobes over there. So in this case, is a high frequency component, high frequency case. That's why you will see, you were here or you were experienced some good directivity for. That's why you need two to experience directivity. Okay, here you can experience also the directivity, good directivity. But over here, when ka is small. Your speaker is now provided a directivity. That's why for a sub woofer sometimes you don't need many when you are poor student especially. You don't need to buy two sub woofers because woofer does not provide the directivity uniform. All right? That is interesting.