Okay, let's move this plot. Okay, you have a general vibrating surface. Okay, then you can say using super position. You can say the vibration of this would be a super position of these modes, okay. And each mode can be regarded as this piston driven radiation, and this piston driven radiation, so on, so on, right? So the vibration of this kind of plate could be obtained by using into the equation that we used for piston driven baffler piston case. So let's do it. How can you do it? It's simple. Because just we can predict a sound by using exponential jkR or a un dS0, this un is simply this. You can do it by using your laptop computer. You can get the result, but to understand what is inside of the result, You have to understand what it means. So let's have some demonstration. Okay this is the plate. Okay, and then, No, not significant? [LAUGH] I made this speaker and we have four speakers. Okay in the beginning four speaker has same phase, so it radiates like a baffled piston like that. And then by using the switch over there the baffled piston will auxiliate, and then radiate sound like that, and the sound plays a radius sound like that. And then last case each piston each speaker has different face, so the radiation would be small. Okay, let's see. So everything in equal face. This radiation. This radiation and this radiation, it's strange, and it sounds like it's the same. You can hear, okay. Here, because I am over here. [LAUGH] You already have the distraction. Okay, okay everybody got what I intended to show? You feel there's some reduced radiation, okay. The radiation when we have all different phases, you got less radiation. You experience that? So that means a lot, really. That means a lot. Right, if you have a plate. If you want to have less radiation, in other words if you want to control the noise coming from certain structure, what do you have to do? Simply if you are able to auxiliate the structure with different phases then the radiation wold be reduced very much. Okay, another one. You saw the radiation from the baffled piston, okay. And you know that the sound coming effectively from the center and the rim. And if you want, if you have some sound reducing material in your hand. Where do you have to put it? Center or rim? If you put sound reduction in a rubber or whatever, somewhere in between the center and the rim it's useless. If you're a clever noise control engineer, if you do know the physics that has to do baffled piston. You have to put, or reduce, vibration at the rim, or center. That is a fact, okay. Now if we have a plate and that is auxiliating all different phase, expanding what you learn from the baffled piston where the sound is radiating? At the rim. Okay, if we have a structure and that is auxiliating with many modes, most effective sound power would come from the rim of the structure. Therefore where you have to put sound vibration resilient to material on the structure. You have to put at the rim of the structure not anywhere. Of course if you are rich then you can put many sound resilient material all over the plate. But if you are clever, Those control engineer or if you, at least if you listen my lecture you don't have to put all the sound in resilient to material any other place except the rim or center, okay? So that's what I'm going to say using this theory, and the result which we got. This picture vividly shows what I demonstrated now, okay. This sound is radiating like a motorboat only diminishing it's magnitude as you away from the source like one over decay over, you can see. Right here sound is very strong over up to here, but getting smaller and small because of the interference due to this phase difference and more diminishing, rapidly diminishing of sound because of the phase difference between this four elements. So another thing you could think that to control the sound you may generate many modes. Shift the modes to the higher mode. If it is possible, normally it is not possible, okay? So it really says many things, okay. The other way to see what I just explained is to see those things theoretically or mathematically. So say we have. This kind of pressure field. Auxiliating time harmonically but it is propagating in space, x, y, and z, right? This is just expressing propagation of any sound in space. First seen in terms of XYZ coordinate, okay? And then of course it has to satisfy wave equation. Right? So taking a derivative with respect to space twice will give this term, and taking a derivative with respect to time would give me omega scale, that is k scale. Wow, that is interesting. kx squared + ky squared + kz squared has to be same as k squared k is omega over c. So free space wave number. So if I select kx and ky, kz has to be determined automatically. Due to this relation, dispersion relation. And four, the plate that is vibrating with this k wave number. Okay, wave number is number of waves per unit lengths multiplied by 2 pi. Okay, say if you have a one wave that is m = 1 then kx will be pi over Lx. Because kx is 2 pi over lambda and the lambda in this case, what is it? Because in this case Lx is this and when m equals 1, wavelengths is half wavelengths of Lx, well that makes sense. If m=2, then the wave is like this. And m=3, the wave is like that. And similarly m = 1,2,3 represents that physical configuration. So it means that if I have a plate that is auxiliating with this wave number, in other words I have a plate and it is auxiliating with this wave number. The wave in z direction, wave in z direction must follow this dispersion relation kz square is = K square- kx squared- ky squared. What does it mean? What does it mean? The graphical explosion, explanation is following. Okay now. This is ky and that is kx. As we saw before, kx is m pi over Lx and M can be 1, 2, 3, 4, 5. Therefore, the kx, ky is not continuous value it is this great value over here and over there and over there, okay? Let me write down k squared = kx squared, sorry,- kx +. Okay. This circle is a k. When we are here that means kx squared plus ky squared is larger than k squared, therefore we got kz. Imaginary. That means wave propagation in z direction there has to be zkzz. That's the wave propagating in z direction because kz imaginary. Therefore this, simply the wave in z direction simply exponentially decay. So in this region, Wave does not effectively radiate. And that's why we call this a radiation circle. Whether or not kx and ky scale is larger than radiation, I mean the k square. And in this region, we want k squared is larger than kx + ky squared therefore kz squared is always real. Therefore, there is radiation exponential jkzz. So this is the region where we can have good radiation. Another interesting point, look at this, kx, it's very large, again, a ky is small, right? And that means, in this case, I have a plate. Because kx is large, that means m is large. That means in x direction there are many modes. In y direction, y is a small, there is small, less small. Therefore in this case over here it look like that, but over there, mode would look like, This. In two case, as you can see here, there's some cancellation due to the face difference. And only the radiation come from this edge, we call edge mode region. And again, this is edge mode region again. And over here because kx and ky is equally likely, the vibration pattern would look like that. And we all have corner modulation. In other words, only corner mode effectively radiate the sound. So as you can see, using simply this dispersion relation. Okay we just saw for example, in other words we just see the wave number of dominant expression. We can see all the physics that is associated with this interesting one. Okay, that is interesting. But also, it is the foundation of acoustic holography. [LAUGH] Yeah. Why? I measure pressure on xy plane, I transform the information of xy plane to kx and ky plane. Okay, as we saw before, and wave on this region will propagate effectively but a wave in this region would not be effectively propagate. In other words, this region where we'll see the evidence into other words explanation. Became where this region we have propagating wave. So what we only can do is transform the disk to kx ky plane and then propagate using this function and then back propagating we can estimate. We can see the sound pressure at any z Right? So that is the basis of acoustic work.