So lets start to track our breathing sphere. I have a sphere which is perfectly round, and coordinate and measuring from the center of a sphere. That I call r is the measure of the distance from the center of sphere. Assuming this one is vibrating with magnitude U0, and frequency of omega. [COUGH] And this will vibrate and radiate sound. And suppose the radius of this sphere is a. Okay. before we start to see this radiation problem theoretically, let's close our eyes and then suppose we are the particle somewhere over here. And suppose we are the particle somewhere over there, and suppose we are seeing the fluid of particle nearby. Your cell. And the wave if each fluid particle experience Depends on wavelengths okay, as well as frequency. And we remember that the radiation depends on frequency as well as wavelengths. Because the averting propagating is related to whether this relationship, that simply say k=omega/c. Okay? And what about the radiator size? If radiator size is a small, compare with wavelength that means our order C the size over radiator with respect to wavelength, that is pretty much related with KA. If KA is a small, that means radiator size is small compared with the wavelength. Okay radiation size is small compared with the wavelength. In this case radiator size is a ,for example, the wavelength is this amount. The characteristics of radiating sound are quite different compared with case when ka is larger than the radiator size. Is a large compared with wave lengths that means, this breathing sphere radiate to wave lengths like this. Okay, in both, I mean, in two cases would be quite different. And obvious since that those characteristics has to be measured with respect to k a, as we saw before. Okay, so let us see how this kind of things can come out, by studying this radiation In analytic way. Okay, let's invite the velocity potential phi, that would be depending on r as well as time. And say this will have some arbitrary amplitude. And obviously it will decay 1/r, and then exponential minus j(omega t-kr). How do I know it depends on 1/r? Okay, there are several ways to understand that all depends on 1 over r. First theoretically, this velocity potential function. Satisfy this governing equation. You can derive this from the general governing equation And this [INAUDIBLE] has very complicated in form in terms of theta phi and R. And we do know that there no dependence with respect to Zeta and phi. Than we will come up with the this equation. And this is interesting, because that means R phi will be. Plain wave. So I can write that this is e time exponential minus j minus kr. Therefore, I can write this. So the result come from the solution of equation. And we know that this governing equation is held for every case. And i just converted this governing equation to this rather simple case which is in this case which is the theta and phi dependants. In this case there's no theta and phi dependants. Okay? You see, it is amazing that in all the days, the acoustician always starts with this rather unusually looking simple case. And then they found the physics and then they expand to understand rather complicated their case. So we have this and taking a derivative with the respect to space I can get a velocity r. So let's take the derivative of this solution with respect to r. This will give me a. One over r square and minus I'm taking derivative. With respect to r of 1/r, as you know this will give me minus 1 over r square. And then I have exponential minus j. I will use theta, and theta is simply the aggregation of j omega t. Only that t-kr. And then second term. I have A and 1/r and then I have to take a derivative of this argument with respect to r. This will give us. What? -j-k that is +jk right? And then I have exponent j. Okay that is the velocity in r direction at any point. So let's summarize. Rearrange this expression in rather compact form. And I can say this is A, and I have -1/r squared over here and I have plus r and jk. Okay, is it correct? Okay. And then I have explanation minus J, omega T, minus K R. Okay, that is velocity. Okay, then, what would be the velocity at At r = a. Then this formula gives me the velocity at r = a has to be a minus one over a squared. Plus j k. Over a. And I have exponential minus j (omega t- k a), that is the velocity. And that velocity has to be same with this. Because I assume that breeding sphere will vibrate with a magnitude U zero, with frequency of omega. So what I can do here. U zero, exponential minus g omega t, has to be equal to A minus one over a square, plus jk over a, exponential plus jka and exponential minus j omega t. So this relation determine the assumed amplitude A. Therefore We can write the amplitude A has to be, this is common, has to be U0 divided by Minus 1/a squared +jk/a and I have explanation -jka. that has to be the amplitude of the velocity potential. Okay, so that is enter. We are getting. Getting formula that certainly starts to exhibit the physics behind it. Because it has a and k and we got k a over there. But it's not very interesting. Because it just say the exponential jk is the real part, what is a real part? A real part is the imaginary part is, it is still, it has some meaning but it is not really exciting at the moment. So, let's change this form so that we can see, or we can at least we can at least feel the physics. Then I can change this just like that. If I multiply A square, right, both sides, on both side, then I will get minus one, plus jka, and now I have a to the 2 and then e to the -jka. That is the amplitude of A. Okay that means The velocity potential would be the phi (r,t) has to be equal to u0/- 1 + jKa. a square, and that is a. And 1 over r exponential minus j theta. Okay, that is velocity potential. Because we have velocity potential we can obtain pressure and a velocity, okay. What would be, then, Velocity in our direction, okay? Let's try to do it, I hope everybody is following me. Everybody is following me, okay. I mean this is the least derivation you have to do to do the acoustics. So what is that will give me a half Taking a derivative with respect to r will give me -1/r squared. And I have explanation of over here. And I will take a derivative with respect to theta. Omega t, minus kr. So this will give me what? I take derivative with respect to r, then I have j minus, j minus, will give me plus jk and this is what I got, right? Additional time, additional time minus JK. Okay now [COUGH] this is velocity in our direction. So if I write velocity >> [INAUDIBLE] >> Here? Okay, right. Plus J K R. I think that is correct. Okay. So therefore velocity in r direction. Can be written as. U0 a square -1 + j k a and I have minus -1 /r to the 2nd + j k a / r and then I have e to the -j k a, okay. And then I have exponential -j omega t- Kr. Okay? Is it correct? Now, what is the impression? For this pressure. Okay pressure at point r with respect to time. Is = -rho0 infinity. And if we know the function phi that is As we saw over there So I will write again, U zero, a squared, minus 1 j, k, a. And then, taking a derivative with respect to time, I have Roe zero over here. Take a derivative with respect to time will give me -j omega. Okay? And then I have exponential -j, Omega t minus k. And I have exponential minus jk here, all right? Okay, okay now. So, I have pressure and I have velocity. And it looks very interesting. Let's force to look at the pressure term. Okay, pressure term.