Okay, let's move to the trembling sphere case. Okay, please, please follow what I'm, the way I'm approaching to introduce trembling sphere, which is not easy, okay? Okay, I have a sphere, okay? To describe sphere, we need coordinate. So, I invited coordinate as before, xyz. And here is the center of a sphere. And then, we assume that this sphere is trembling in this direction. Okay? Now, then look at the fluid particle over here. Even though I oscillate this way, the only way these fluid particles oscillate is this direction. [SOUND] Because the fluid does not have damping, does not have viscosity. This direction has nothing to do with the oscillation of this fluid. So, let's suppose this one at certain position A has a velocity of UC. [BLANK_AUDIO] And, use the cordinate. Okay. That is, like that and let's say this is, this is phi and this is theta and this is r. Okay, that looks beautiful. And then, of course, the boundary condition that will determine the unknown amplitude. We didn't, we didn't assume the field yet. But mostly, r equal a should be equal to UC cosine theta. This is UC and this is theta, therefore UC cosine theta is this velocity. Okay? Now, what velocity potential phi look like? Okay. Then we can envisage, or we can, we can imagine that because this is oscillating in, in, in, in this direction. [SOUND] So that is somewhat similar that suppose we have two spheres. And this one has different phase. Then when it push that way, this one is breathing and this one shrink. When this one is coming over this way, then this one is breathing and this one is shrinking. There could be resemble with what we are seeing over there. So we can say that the breathing sphere is proportionate to 1 over a to the jkr. Why? Because we are using minus j omega t, minus kr. So, this term is plus kr. Right? And the distance between these two is getting smaller and smaller and smaller. Mathematically we can write that is d/dr, e to the jkr over r. So I can, I can argue that the solution that mimic physically and mimic the, Physical behavior of a trembling sphere, or have this form. Or you can, you can solve using separation of variables for the polar coordinate expression of, of the Laplace equation. But, I can argue physically that the solution that associate with the cordinate r would have this behavior. And what about the dependency with respect to phi. It all dependence if respect to phi because breathing sphere is up and down. So there is no phi dependency. The only dependency with respect to angle would have something to do theta. Okay lets invite some unknown coefficient to A. And this one depends on theta. Okay, let's see it. If theta is zero, okay? That has a maximum. If theta is 95 degree, Pi over two oh 90 degree, that should be. Zero. So maximum zero or I could say every, you know, magnitude dependency I put into A, then I can say, oh, that should be following cosine theta. Because cosine theta has theta equal 0. It should be 1. When theta equal pi over 2 or 90 degree, it should be 0. So, I can bravely say, this is cosine theta. That's nice. So, using that physical argument, I can say, I will write over again the velocity potential has to be some magnitude. Sorry, some magnitude and cosine theta, and then d/dr e to the jkr over r, that is the candidate of a solution. Observing, what's going on physically of the surface of a trembling sphere. And last, I have assumed solution, I have boundary condition. Then I can solve it. I can get pressure, how? The pressure which will depends on r and theta, I can get that is minus rho0 dphi/dt. And I can get you know, velocity in r and theta direction by taking derivative of those potential functions with respect to r or with respect to theta. Okay. And that is trivial. But as we did before for breathing sphere, let's look at the impedance of trembling sphere, it looks like the following. That is proportional to, again, rho0 c which is the characteristic impedance of medium and scale factor- That is interesting. That is proportional kr to the fourth and four plus kr to the fourth that's the real part and I have a imaginary part that is. Two kr and then I have kr to the cube, and then I have four plus kr to the fourth. Okay, that's interesting [BLANK_AUDIO] The real part remember the real part of the breathing sphere was a function of, okay. kr or ka to the scale, this case the real part of trembling sphere is like that and this is the radiation impedance. [BLANK_AUDIO] On, when theta equal Okay, let me ask you. What is the radiation impedance when theta equal Pi over two? [BLANK_AUDIO] Mm? Zero. Nothing is radiating. So, this formula is valid only when the when question. So, this radiation impedance is, does not depend on theta, right, um? Hm, Okay. Looks like, this, this is not correct. But if you drive the, driving point I'm in, the radiation impedance the following what I just showed it is true. Show the simple model tells us unknown physics. This case that says the radiation impedance independent of theta. Wow, wow, wow because radiation impedance is actually the ratio between pressure and velocity and the dependency of velocity. On theta, a dependency of pressure on theta in this case, same. Therefore, the dependency is cancelled out. Therefore what we get is radiation impedance that is independent of theta. Wow. That's good. That's interesting. What does it mean? Meaning, okay, based on this, we can also see the radiation impedance on the surface of a radiator. And r equal a, we got, simply saying but different, only that is dependent on ka, function of ka. And if you draw this, we will see the, some of the interesting graphs look like. [SOUND] Look like this. Okay. I have kr Zr divided by rho0 c again in the far field approach to one, but in this case it look like that. So, that is kr to the forth one plus kr to the forth, that's the real part. And the imaginary part look like 2 kr plus kr to the cube divided by 1 plus kr to the forth. And it has maximum, somewhere over here, and then goes like that, and this is when we have a square root 2, and that is square root 2, too. And the other one is, for trembling sphere. PI average as we did before that is, ka to the fourth, one plus ka to the fourth. Therefore, the average power increases as we go- As we get a larger, larger, breathing, size of the trembling sphere look like that. And actually this is because this is power of fourth, the increase of average power is 12dB per octave. Okay, so that is interesting. That is interesting. And this is only an average power. But, but it has cosine theta dependency, right? So, it has a directivity involved, which I will talk more detail. Okay? This is all about I want to say today and [BLANK_AUDIO]
