Okay, let's see for the flexure wave case. Again, the instant wave is propagating that way. Therefore the incident wave length is like that, and reflected wave has to be like that, and transmitted wave has to be like that. And interestingly, in this case this has to be same as the lambda b, okay? Then lambda b is the bending wavelengths, that is related with a kb in terms of this dispersion relation. And then this dispersion relation is not same as the dispersion relation of that we saw that is k equals omega over c. That is the dispersion relation of served in fluid compressed over fluid like air or water. Let's see how we get the dispersion relation of bending wave. Dispersion relation of bending wave. Remember how we get this dispersion relation? Can anybody remember? How we get this dispersion relation? Okay, we have a governing equation. That relates how the spatial distribution, spatial gradient of spatial gradient that is related with temporal change of some pressure, okay? And we know that the one solution that satisfy this equation, has to be plane wave. And we argue that we have complex pressure, and then exponential and minus j, omega t, minus k vector r. For simple case, for plane wave, one dimensional case, that has to be exponential minus j omega t, minus k, x. Or if I omit the notation of x, that is kx. And then plug this one over there, give us dispersion relation. Dispersion relation is nothing but determining the relation between temporal oscillation and spatial oscillation. In this case, spatial oscillation and temporal oscillation is pretty much linear. But that would not be true for the medium like plate. We will see. So let me go back what we did, we assume Pi, and Pr, and Pt. And then we know that there's a bending wave propagating into y direction if you use this x and y coordinate, okay? And then what we'll do is just apply the boundary condition as we did before. What kind of boundary condition will we do? I will say the pressure over here and the pressure over here, that would be the pressure over here would be Pi + Pt Pr and the pressure over here would be Pt. So I will say that Pi + Pr- Pt, which is the net force acting on this plate per unit area. And that will introduce the vibration or wave propagation of a plate. And also we can say that the velocity induced by in Pt and, Pr has to be the same as the velocity and induced by the plate. Also we can say the velocity induced by Pt in extraction has to be the same as the velocity of plating extraction. So this is assumed some pressure field subscript i r and t is simply indicating instant, and reflecting, and transmitted wave. And I assume that the bending wave is propagating in y direction. This is y direction. This is y direction. With wave number kb. And I just use the Newton's Second Law on the plate. This is the net pressure, in other words force acting per unit volume at x = 0. That has to be balanced by the bending wave, that is, m is mass per unit in an area of the plate. And b is bending rigidity. Interestingly, for the plate, this term has to be replaced by the tension per unit length. This term has very interesting form because it has d to the fourth to the dy to the fourth due to the bending rigidity. This is typical, bending rigidity. We assume, of course, the simple plate, or Euler bending, our Euler beam, or Euler plate, which means that plane section remain plane. Physically that means when I bend the plate, the plate when it was like that, should remain as plane that means the shear effect upon the cross section is negligible. In other words, this means that the thickness of plane is very small compared with what? Thickness of plate is small compared with wavelengths, in this case. So if you have a very high frequency impinging to the plate, then this assumption would not be hold. So, that's another very important note I would like to give you. So but for example in all the wave frequency, like 1 kilohertz, the wavelength is order of more than 30 centimeter. So like this, right? Therefore, if we are considering the plate that has this thickness, that's the usual case, then this assumption pretty practical. So let's go on to 10 kilohertz. Then we have 3.4 centimeter plate. In this case, this assumption might not be practically acceptable, but it could produce very reasonable result. So again, emphasizing that the bending, everything has to be scaled by, I mean everything in association with the length scale has to be scaled by wavelengths. Absolute scale like 1 centimeter and 10 centimeter and 1 meter thickness of plate does not have any meaning in acoustics or wave propagation in general. Okay, and using the assumed solution, this, we obtain this kind of dispersion relation. That is very interesting to demonstrate to this dispersion relation saying that k b to the 4th is proportional to omega square. In other words, what? What does it mean? Okay, let's draw some picture. Okay, this is the dispersion relation. In the beginning, we had k b to the 4th, that is m over b, omega square. [COUGH] That is same as this. Okay, that look like this. And this line is k equals omega over c. This expresses the dispersion relation we saw for the medium like water and air. Okay, because this k equal omega over c, therefore what we can obtain over here, that is omega 1. And this is k1, and slope of this is 1 over c. Or you may see this one over that one would be indicating speed of propagation. And that has to be same in this case. But if you look at this curve, this one, this one over this one, and this one over that one is different. That means speed of propagation at each frequency is different. In other words, the medium is dispersive. Dispersive medium. In fact, in fact, in this case, this is higher frequency compared with that frequency, or higher frequency, higher frequency, this one. Okay, this one over that one is higher than this one over that one. Therefore, higher frequency wave propagates faster than lower frequency wave. So when I make this kind of two waves, then this wave will propagate faster than this wave because of this dispersion relation. So when you generate a bending wave in a plate, when you actually measure it, you will see the higher frequency will arrive at the position of your measurement, then the low frequency. And this is the case of bending wave. But for water wave, it's different. The water waves is exactly different way. For water wave, longer wave propagate faster than short wave. In other words, low frequency wave propagate faster than high frequency wave. So when it is right. I mean short frequency wave, that is long wavelength, propagate faster than high frequency wave, that is short wave, okay. So what you see in a sea shore. Then you will see very long wave, that could have more than one hour or for example. More than 20 minutes and 30 minutes. The c line will move up very slowly and the. That means there's some wave is traveling from far away from us. So the people who live very long time on a seashore, they can predict storm by just observing that kind of a long wave in advance. Okay, that is very interesting related with dispersive medium and dispersion relation. And actually this is not very much related with what we are going to study in linear acoustics. But have, anyway it is interesting so therefore I introduce to you. Okay, mathematically, we can say the velocity due to incident y, this is velocity due to incident. And this is the velocity due to reflected, In x direction. Okay, this is the velocity due to the incident wave in x direction, that's why I put cosine theta i, that means I multiply cosine theta i over here, divide by specific impedance of medium, that is velocity, because the impedance is related p over velocity. So this is the velocity and this is the velocity of reflected wave. The reason why I put the minus sign is because as we talked before, reflected wave in x direction has to be the negative sign compared with the Instant wave of instant wave. And that has to be the same as the velocity of plates. That is simply I am taking rate of change of displacement of plate which is a multiply exponential minus plus j omega t minus kb. Why, because the bending wave is propagating in a y direction. That's why I got minus j omega because I take a derivative with respect to time. And I'm evaluating those on x equals zero, that's why I'm not only jkby, that has to be the same as the fluid particle on the transmission side. That is again, pt over z0 and multiply cosine theta t, and that has to be same as this. And interesting that this theta i and the sine theta i, that is the exponential term. And this exponential term has to be same. In other words, KI sine theta I, sine theta, and KI sine theta R, and the KT sine theta T has to be the same. Therefore, to get the solution, this has to be the same. Right you have explanation of something, explanation of something, and you have some RPA1, RPA 2, RPA3, this exponent has to be the same, otherwise we cannot get the solution, all right? So this is the same and that has to be a whole, and that has to be whole. The last one is important because this is that term, okay. So, resolve the solution and regard tau, okay. And here is an interesting attempt because of you move this d0 to there. Then which is a symbol of what we have for the link case. And for the case, we have 2Z0 and 2Z0 m + j omega m minus that minus j omega comes from the [INAUDIBLE] equation. And as I said before, generally we can say, z, tau composed by two components. One is partition impedance, and the other one is fluid loading impedance. In this case, these are fluid loading impedance, and these are partition impedance. Therefore, we can say this is partition impedance. And interestingly, this partition impedance has two components, one is [INAUDIBLE] contribution. Simply saying that the transmission corruption is related to one over minus j omega one, again, it follows mass law. What is mass law? Mass law simply say, later on, transmission lost will be increased 60 b per of frequency or mass doubling the mass. So generally, it will follow this mass law, but it is related with this partition impedance that is interesting because it is proportionate to one over omega. So, let's look at some more detail about this. So, we found that total transmitted wave looked like that, okay. You see interesting, this term it is propagating in y direction with wave number of kb that makes sense. Because in y direction, the previous k when we do not have any structure on the interference because of impedence assumption, there is no population in y direction. But this time we do have some interface that forces our fluid to move in y direction. So when you employ the, when you employ the plate, you are actually providing some mechanism that allow fluid to populate in y direction. Not only in x direction, but you also provide the mechanism that can move fluid particle in y direction. And number of waves per unit lens is lamda B. And associate wave number is KB. And this is what you can see mathematically. What I explained physically is expressed mathematically over here disappear, okay. And then what does this mean? This x, this describes the propagation in x direction. Okay, that's very interesting. It propagate with, let me write it again in more exaggerated way, bigger way. k0 square root, 1 minus kb, over k0 square x. Okay, it is propagating in x direction. Depending on the magnitude of the wave number in the bending wave number in kb and k0. k0 is simply another over c0 and c0 is the speed of propagation in l or water simply surrounding fluid. So that is interesting, because if kb over k0 is, smaller than 1, then this one is greater than, I mean is positive, therefore there's a wave propagation in x direction, k0 multiplied by this amount. However, if kb over k0 is larger than one, we don't know what it means physically. But mathematically, we can observe that if kb over k0 is larger than 1, than this whole term is negative. That means this term as imaginary part. Imaginary part much apply by imaginary part or produce by minus sign. We disregard plus sign. That means the wave in x direction exponentially decay in this case. That is interesting, that means that if you use a plate in certain frequency band, some wave is exponentially decaying in x direction. In other words, transmission question in x direction will be exponentially decay, and that would be very interesting. We might attempt to find some application area based on this interesting phenomenon. Okay? So let's look at some more details about this. And interestingly, kb over k0 also involves in magnitude of scale. Okay? That means the magnitude scale, that has to be scaled all over this square root value. If kb over k0 is larger than 1, this one has to be imaginary part. So, the pressure, magnitude of pressure would be rather different with the magnitude of pressure on kb over k0 is smaller than 1. So let's look at some detail about this. Of course, this is not major part of the lecture I would like to deliver today, but it's certainly interest and some students might have an interest to use this kind of peculiar, rather interesting behavior in certain application. Okay, if I convert this wave number into frequency domain or in terms of speed of propagation, then it'll look like that. So if cb is greater than c0, That means bending wave speed is greater than c0. Recall that is the case of supersonic. Then, this one is real value and there is wave propagating as we saw often in x direction. But if c0 over cv is greater than 1, in other words, if cv is smaller than c0, that we call subsonic case. Then the wave in x direction decays exponentially, okay, things like that. So that we call supersonic and subsonic case, which is often very important in certain structure. Okay. This explains more. This is the plot that come from the dispersion relation. So speed of propagation of bending wave is proportional to square root of omega. Therefore, as we increase the frequency, the speed of propagation increase the square root of the frequency, so therefore, the curve will look like that. So this is the frequency when the speed of propagation is the same as the speed of propagation of air or water. It's the case when we have kb equal, has the same direction in the ky. And this is the case when speed of bending wave is larger than speed of medium, then we have a supersonic case. And this is the case when we have subsonic case.