Using that you can obtain this, Results. And it is interesting that it depends on, of course, the amplitude and cosine theta. Here, cosine theta is the angle that describes a scatterer. Okay? Let me go to the, Okay, sorry, this is a theta. Okay. The theta is the angle between wavelength, between the unit normal of the surface with respect to x-axis. When theta is pi over 2, we are over here. When theta equals 0, we are over here. Okay. Alright, over here, this is And that has to be evaluated on the scatterer surface, S0. And what he obtained, This is, we are study the scattering of the sphere. I mean otherwise, this is true for every case, but for simplicity, we apply this to a rigid sphere, okay? Then, evaluating this on surface S0, turns out to be like that. Okay, the velocity, this is the velocity, normal to this direction. As component of cosine theta, right, if theta is 0, that is large. But if theta is pi over 2, there is a general velocity. If the theta is pi, This part is moving in that direction, because the cosine pi is minus, and this part is moving that direction. Okay, that is interesting. Then what cosine theta says in 0. So it looks like this kind of shape. That look like somewhat similar way to breathing sphere. But it has a real part and an imaginary part, that is controlled by explanation jka cosine theta. Right? So if ka is small, that means ka is small, that means wavelength is large compared with the diameter, I mean radius of the sphere. In other words, the instant wavelength is large compared with the scaterrer's nominal size. Then mathematically we can expand, approximate, exponential jka cosine theta like that, 1 + jka cosine theta, wow. That is interesting, because first the part will move like that, and second part move like cosine squared theta, all right? Therefore first part contribution is due to the contribution as if the scatterer is moving like that. And second part on the other hand, because it is related with the cosine squared theta in this case, there's a sphere, this will move that way. And this point will move that way too. Therefore the scatterer is moving like that. That is similar with a trembling sphere. So as a conclusion, we can argue that the scattering of this sphere has two components. One is due to, as if the scatterer oscillates breathing sphere, and the second time is due to the scatterer as if it oscillates like a trembling sphere. So that is quite interesting. That, makes sense, also. It is only valid when ka is small. In other words, wavelength is large, compared with the scatterer size. Of course, you can see the other details in general ka. The near field meaning that if ka is small, that means wavelength is, Small compared with the scatterer size, then we can not approximate this, but we can write This one as, cos (ka cos theta) + jsin (ka cos theta). So the behavior will follow not this form, that form, if the wavelength is small compared with the scatterer. In this case, suppose that we have this scatterer and the wavelength is very small, then the wave we’ll see as if it bes big rigid wall, right? That's so called the ray theory, or to hold, in this case. Here's a scatterer and a small web is coming from here. Then this web will be reflected due to the presence of a rigid wall, and the reflection of light ray, okay? That is the foundation of ray theory. Okay? So this shows the directional component, of this. Depending on different wave number. Now, We have been successful to get the scatterer field. We didn't finish to calculate all the sound pressure due to this kind of velocity distribution. But using Kirchhoff Helmholtz integral equation, we can calculate the scattered sound field, because we do know the velocity on S0, okay? Okay then, what if we have this kind of a slit? If wave is coming over here, due to the presence of the impedance mismatch in space, there will be some interesting sound radiation, maybe some radiation, some radiation like that, some radiation like that. We would like to obtain the sound radiation pressure at any point of R. Okay, we can do a similar approach. How? There is a Pi, and there is a scatterer PSC, and a boundary condition over here, has to be, normal velocity should be 0. Over here, What is it? The boundary condition over here. That is, the velocity cannot be 0, or velocity can be any value. Pressure is also, Can have any value. The boundary condition we apply is the rigid wall or boundary equation written over there. And if Pi is coming this direction, we can say this is asymmetric, therefore we can solve this problem in this half space, then we can get a result. But, the pressure we obtained at r, Look very interesting, okay? This is the pressure. Okay, it is related with the velocity. But, it is related with this, that is nothing else, DR 1 over RDK, the monopole radiation. Instead of monopole radiation, it does have square root R, exponential jkr. This is different, but square root r, Decays slower than 1 over r, of course, yeah. So sound radiation from that kind of back flow in other words, some window radiation pattern would be quite different, but generally follow this function. That is somehow similar with what we obtained for the baffle of the piston case, simply shows the directivity pattern. Okay? The reason why we got a square root r behavior, is because what we try to get first, is when we have infinite two-dimensional slits. But if you have three-dimensional slits like this over here, we have this size and that size, then we have these two factors. So a is the size of x-axis, and b is the size of y-axis. And because we have scatterer in this axis as well as that axis, we have two different directivity patterns. And that is also very similar with what we obtained for the baffle of the piston case. So everything depends on this vector, ka and the kb, to simply measure of the size of window compared with the wavelengths. If the size of window is a small compared with the wavelength, that means we have a small window and we have a large wave number. What this will go? This will go to one and this will go to one. Therefore it behaves like 1 over r [INAUDIBLE] again. If the size of the window is getting larger and larger compared with the wavelengths, then the window is kind of the function that provide the some directivity. That makes sense, because if you have this kind of window, we are hearing sound coming from that window, then we could think of it, there is some speaker, speaker, speaker, speaker, speaker, speaker or speaker over there. Based on that observation, some people attempt to reduce sound coming from window, that we call, sometimes, active windows. Simply try to generate the sound using speaker on the rim of the window, in such a way to reduce the sound of radiation, I mean to reduce the sound of radiation, okay? Because the window only behaves as if this kind of shaping function mathematically. But as I told you already, generally behave like cold metal pole. So as a conclusion, as we saw in the baffle of the piston case, the fraction depends on, also scattering depends on, wavelengths, okay? Let's move to the sound barrier case. In other words, sound is coming this way, and we would like to estimate the sound is somewhere over here, and we can solve it. It looks like that for two-dimensional case. Two-dimensional case means that the sound barrier is infinitely long compared with the wavelength. Okay, which is the most case we have. Now it look like that, and the D theta looks like this. And interestingly over here again, it is proportional to 1 over square root of r, because we are handling two-dimensional case as we saw before, right? If you plot this result using, Using a well-known Fresnel number which is A+B-d, that means here's sound barrier, And here is the sound source. And this is A, here is a receiver, this is B, and this is d, okay? Then it look like that. So transmission loss of the sound barrier has a very simple expression, and this is the graph. Okay, now using this graph we can design the sound barrier. Or, in other words, we can determine the height of the sound barrier. Okay, say here is a house and here is a traffic road, the cars are moving with a high speed, and we would like to put some sound barrier over there. And then we would like to determine this height. To get transformation loss, say X transmission loss. X could be 20 dB, or 30 dB, and 40 dB, okay? And we can use this graph. Okay, if you want to reduce the transformation loss 30db then we follow this line. It says Fresnel number has to be, 10, 20, 30, 35. So this has to be 35. And we know the distance d, right? Right? And depending on the height from here to there, A and B can be determined, right? And the lambda we select, for example, traffic noise, look like [SOUND]. And this has to be frequency we want to control. And the corresponding wavelength would be 343, divide by frequency, and we can determine lambda. And using this lambda over there, we have a 35. We design A + B, okay? That will give us the height.
