Okay. Suppose we have a monopole over here, okay. Which is the case when breathing sphere's size is very small. So the result in in in, shows the for the monopole case is quite similar with the breathing sphere case. And the pressure, as you see over there, is 1 over r e to the jkr. Okay, that's the simplest source case, that has a very interesting mathematically es, at least an interesting characteristics. One is, e to the jkr, meaning that it is oscillating in space, and the other one is 1 over r. 1 over r is interesting because when r goes to infinity, the magnitude of sound decay, I mean goes to zero. When r goes to zero, the magnitude of sound go to infinity. So that is, has a singular, I mean, sort of full behavior at the center. And that is very good in, in some case because it is very mathematically easy to handle. And we look at the velocity over there, and then we found that it is related with e to the jkr over r, that is the the spatial sound distribution in space, and it is related rho0 c, we do know that pressure divided by velocity is, is, is impetus, okay? If we do not have this part, then imped-, radiation impedance is the same as rho0 c that is the characteristic impedance of medium. So you remember that if the radiation impedance is the same as rho0 C, that physically means you are propagating sound in space as if you are exciting the fluid in one dimension, one dimensional fluid. In other words, the P and the velocity does not have any phase difference physically meaning that as you push the fluid, the fluid is just responding you without having any phase difference. Okay, you see it is easy to drive the fluid, I've drive the fluid this way, then the fluid particle would have the velocity in this direction at the same time. But if it does have a phase difference, that means when I drive the fluid, then there is some reaction from the fluid, because of the presence of j over there. And the impedance, we'll look at it, look like that. There is some real part and imaginary part. If we, if we, if we multiply one minus j, 1 over kr, then we will get the familiar expression that we had before. Okay, if we go to the dipole case, this is the case when we have a trembling sphere in this, in this direction. The expression rather be different with what we saw for the, the breathing sphere or, or monopole case. Okay, looking at in terms of radiation impedance, is quite complicated compared with the radiation impedance of monopole or breathing sphere case. Okay, look at the pressure, I mean velocity. The distinct difference between breathing sphere and trembling sphere, or monopole and a dipole is the following. First, it has this term, this is quite different with what we saw for the monopole case, it does depends on kr also, but in different manner. And that the other one is a cosine theta. [BLANK_AUDIO] cosine theta. Okay, that means if theta is pi over 2, then there is no radiation in this direction, and a lot of the, the radiation in this direction, so very good radiation in this direction, getting smaller and small, like that. So that means the trembling sphere does radiate very effectively in, in, in this direction, but less likely in this direction because we assume that there is no viscosity in the fluid. And viscosity of the you know, air, and viscosity of water is very, very small, kinematic viscosity in fact, of air, is larger than kinematic viscosity of water. Right? Because kinematic viscosity is mu divided by the density and density of water is very, very high, compared with the air, there, thus for kinematic viscosity of water is, is smaller than kinematic viscosity of air. So, here we have a cosine theta dependency. Now let's start to see what we can feel, you just close your eyes and then start to, I mean, I mean, try to be a fluid particle on trembling sphere, okay? Let's imagine. Okay, to be a expert in acoustics, the best way is to always try to be a fluid particle in space and time. Imagine, imagine. And try to be you itself as a fluid particle, okay? Now let's do it. For trembling sphere, you're sitting over here, and you're sitting over there. Let's say this is the case A, this is the case B. What do you feel if you, if you are on case A? You do not feel much excitation. Something is, is teasing you maybe somewhere over there, but does not really provide a, a lot of pressure enough to be excited. But if you're here, you fill a large excitation with respect to omega, but if you have, if you, if you are over hear, you will feel some excitation larger than over here A, but smaller then over B. That provide you the directivity. [BLANK_AUDIO] Okay? Let's keep that feeling in, in, in, in, in mind, where quadrapole case, in this case as you hear using the our famous demonstration, you will you, you'll have a very small radiation efficiency as you can expect because those four sources are fighting each other, right? With different phase, 90 degree different phase. And here, as you can see, you have two different angular dependencies, theta and phi. Okay? Because you have quadrapole radiation, okay? You have angle dependency in this as well as that direction. So, the directivity look like you have four you know, this kind of shape in space. Okay, that's all right. Let's move on. Our radiation case that we studied. Suppose we have a baffled piston, okay? What we can anticipate, before we get this, we, we look at the solution, is, if the size of baffle, which is a in this case, is small compared with the wavelength lambda, that is ka, then the solution has to approach which solution? trembling sphere's a solution. [BLANK_AUDIO] Or, or, in, in, or in this case, you have to see the monopole solution as well. So let's see. The pressure, this is obtained by using Rayleigh integral equation, that is, which is the special form of Kirchhoff-Helmholtz integral equation, it look like that. Okay, it is interesting because it is propagating with e to the jjkz and z is the distance from the center of the radiator, okay? So, it is propagating with e to the jkz and, it is inversely proportional to z, so at the center of the radiator, it look like propagating sound as if I am seeing the monopole. That makes sense. And there is another scale factor that is jka, okay? Meaning that, if the size of the radiator, that is a, is small compared with wavelength, this contribution is getting smaller and small, approaching to real monopole. Okay that is interesting. So this is what I observed along this line, and another interesting thing is that, as we talked you before, This expression, in general, when z is, when z, when I obs, when I, when z is very large compared with the wavelengths, we will observe, we will get this solution, and this solution certainly also provide interesting physical insight. Meaning that, what is this? This is exponential jkr0, r0 is the distance from the very rim of the radiator to the observation point. So this term essentially says, essentially describe the wave coming from the rim of the radiator. And that the other term, this, this is the way you come in from the center of the radiator. So that means physically, what I observe from the line z, I feel two waves. One is coming from the center of the rim, center of the radiator, the other one is coming from the rim of the radiator. Okay. And that the other things in between just cancel out. Noise control purposes if you, if you only, if you only are, are, are, have an interest in the controlling noise at the center, you only have to put the sound absorbing or put the some control on the rim as well as the center. That will do. Now, if you, if you do anything in between, it's useless. Okay, that's good. And then we move to the more general case in other words what if I'm observing sound away from the center, em, eh, eh, z. symmetric axis. And we would like to have a more general solution. And we found that, oh, it depends on this function. So let's look at this. It still depends on generally, 1 over r behavior. 1 over r, e to the jkr behavior. Wow. Even if it radiates the sound, and a circular radiator, even if it is circular radiator, generally it follows to monopole radiation, but it's scaled by this. J1 ka sine theta and ka sine theta. What is this? J1 is- [BLANK_AUDIO] Bessel's function of the first kind, so we need to know the behavior of this function, and I will show you later on. Before I go into that, what is ka? ka is seeing the size of radiator compared with the wavelength. And the theta, what is a theta? Theta is the angle between this, all right? So when theta is 0, I'm on z axis. When theta goes to pi over 2, I'm on x axis. When theta go to pi I'm on negative z axis. Okay? And also it is a scaled by ka. Let's look at how it what it means. So, look at radiation impedance. It has two parts. One is this one, and the other one is this one. And this one behave like that over there. [BLANK_AUDIO] Wow it is interesting. So when ka is getting large and large, the real part of the radiation impedance approach to one as we expected, like monopole. [BLANK_AUDIO] Right? What it,it, what it means physically by ka is large. The size of the radiator is small compared with the wavelength. So, this is small. Therefore, we can see the sound as if it is coming from a monopole that makes sense. These are some oscillation in, in, in, in around ka is a small, but not much. So in average sense, in follows, it follows the sort of breathing sphere case, okay? I mean, we, we, we engineering purposes, we do not really care about this kind of detailed information, right? [BLANK_AUDIO] Huh? So, solution certainly try to persuade us you don't have to worry about the detail, the general behavior of circular piston radiation is in the order of sense, it follows the monopole or breathing sphere solution. As you can see here, this one too. [BLANK_AUDIO] Okay? Let's move on to the other more complicated case. For the for the vibrating plates, famous kx ky, this version relation you have many you know, discrete point of the dispersion relation, and this is k. So depending on where you're located, over here or over there, this is the if, if you are here, you, you have good radiation in z direction, if you're over there, you have bad radiation in this case, okay? And we studied the how it is radiated. And back to diffraction and scattering problem. Okay we have a slit, and we have this slit that has a size of b. Two dimensional case, for simplicity. And then what we found is that this kind of sinc function and this one, e to the jkr, so in space it is propagating as if plain wave, but it is a scaled but not 1 over r, on square root of 1 over r. [BLANK_AUDIO] Why? Because it has two dimensions. If you, if you go back to the solution, a membrane solution for example, then you will see when you hit the membrane in the middle, the, the, the wave is propagating like a square root 1 over r. Reason? Because it is two dimensional case. Now, how this function look like, this function we said the sinc function. The sinc function has a maximum at zero, and then oscillate like that. Then we will see the details of sinc function later on, okay. That is a little bit different with the, the function we saw before, the, that says function of force kind divide by the argument. The only difference is a sinc function oscillate with same period, but the, Bessel's function of first kind, divided by the argument is oscillate, but different certainly elongated the the, the period. But that's not really matter. [BLANK_AUDIO] Now, it is interesting the, it is oscillating, but with respect to this very interesting scale. kb over 2. So what is kb? So that kb simply we are trying to look at the size of a slit in terms of wave lengths again. If kb is a small. If kb is a small, in other words, if the size of b is small compared with the wavelength, then this term go to approximately one. And it is al, also behave as if e to the jkr divide by square root, square root of r. Okay. Similar but only decaying not with respect to 1 over r, but with respect to 1 over root, square root of r. If you have slit that has length of b, as well as a over here, does, there's a contribution of two sinc function. One describes the directivity in, in, in a, in, in, in x direction, and the other one describes the directivity in y direction. It makes sense. So, we can have a certain intermediate conclusion. Diffraction and scattering does depends on the size of the slit, like a sinc function. all right? So, suppose we have a window over there, and then you have a car coming over there. Then you will, you will get a noise through the window. Okay, at the center of the window, you have a very large radiation, but it depends on where you locate in this angle as well as that angle, right? Maybe if you're wise, you pick up the, a certain angle that you do not hear the traffic noise, but it does depend on frequency because it does depend on k multiplied b, and k multiplied a. Right? So I do recognize there is some people who is starting to have active window. I don't know whether it's going to be working in practice. What you do is, you change the impedance of the window by putting some sound speaker, speaker at the edge of the window, in such a way to reduce the sound radiation efficiency. Okay? But generally it does depend on the sinc function, as you can see over here there is e to the jkr over r as I mentioned. So everything we observed is, there is e to the jkr over r behavior but is scaled or shaped by this, this sinc function, or Bessel's function of first kind, divided by the argument of the Bessel's function. So, in, I mean, we can see that oh, every scattering problem diffraction problem can be seen in one family, what, what is? It's the jkr of r, but there is a shaping function that depends on, you know, radiators, geometrical shapes, that is scaled by, for example, ka and kb. Okay? This is the case that the wave we can obtain when we have a circular opening in space. Okay. In this case, it depends on sinc function. In this case, it certainly depends on J1 some argument divided by argument and this argument certainly describes the way in which we see the size of radiation in terms of wavelength. You see, many of you are sick about talking wavelengths, wavelengths, wavelengths, but, that's the physics. And the angle of theta, okay? When theta equals zero, you are seeing the wave on this line and ka is, is, is large. Okay, a is a small, and you are seeing the radiator or the opening is small and small like this size, then you will see only you know, 1 over r, into jkr, monopole type radiation. Now let's look at the sinc function as well as the Bessel's function of first kind over here. It look very similar. It look very similar as you can see over there. As ka getting large and large, meaning that the wavelength is, is, is getting smaller and small compared with the size of, of the radiator, right? Then we have very rapid fluctuating behavior as we, we can see. Okay, sinc function in this case, is red line over here, and the Bessel's function of first kind divided by the argument, I mean, uses in this case the the blue line. As you can see, the Bessel's function case, is this is the case of that, it, it does not decay as rapidly as the sinc function case, but order of magnitude is very similar, okay? Now, let's move to the scattering. [BLANK_AUDIO]