So this summarize what I wanted to argue today. For case when we have instant wave then because of the discontinuity we have reflected and PT. And we prove that, this is equivalent with having two PI which we called blocked pressure and the radiation due to the vibration of which will vibrate minus. Time so there will be some radiation. This is PT. So it's physically very straightforward. If you have efficient radiation, you have many, I mean big sound pressure outside. Therefore you have large PT. For the noise control purposes, You would like to have a small PT. In other words, no radiation. No radiation is the case when you have rigid wall and that is not practical or dismissable. You might want to have some water that has some filtered characteristics. In other words, what I meant by filtered is it has a vat and less radiation efficiency among certain frequency band. But has a good radiation efficiency in other band. Then you can use that band to control the noise that you want to control. So consider the frequency characteristics of this partition. You may design good partition by seeing the radiation of this partian. I used the same concept to argue that transmission loss of infinite partition or infinite plate is smaller than transmission coefficient of a finite plate. By introducing this radiation constant, and then we will use this radiation concept to explain later in the next chapter. Radiation scatter and defraction in unified order. So if you understand radiation you understand scattering, defraction, everything. So radiation is sort of key concept that explains almost everything, except the room acoustics, in acoustics. So, LM4 is simply is a two blocked pressure and radiation. Okay, if you have a more general partition that has spraying and damping. Using this concept, understand thing what's going to happen due to this kind of partition. It's rather simple because there is a same blocked pressure and then radiation of this partition that has not only mass but also spray and. Mass law is useless in most case because mass law if I use the frequency that is far beyond the resonant frequency of this partition, then it follows mass law. So we saw interesting picture like this. Follows mass law over here. Plot frequency follows 123 divide by mass per unit area or that sort of thing. I said last lecture this is the key idea, essential to understanding, this actually opens the understanding about the partition. Noise control problem. All right, back to this problem. Then again the transmission will be induced by the radiation of this kind of condition. And tau I will recall is proportionate to fluid loading. This is an interesting formula. Actually I got this formula from and for linear acoustics. This will holds for every case. And move on to the finite plate case but we just studied the transmission for infinite case. But I argued that transmission loss of infinite case certainly predicts upper bound Of all finite case, of plate case, therefore it is rather safe guideline to use transmission loss based on infinite plate case. Okay, we have a remaining sense. Sound does not impinge to the flat surface of this, in normal instance, or have some angle zeta i only. I mean, sound will Arrive at the flat surface of this continuity in many different directions. So how can you use this result to the case of which we have random instance case, in other words? This is normal instance case. Random instance means that sound comes from everywhere. How can we use this result to the random instance case? Conclusion? We just invited some traction factor, which is not big. Because think about it this way. If they have this instance case. Major radiation will be induced by the component abnormal to the surface. So that would be the major contribution. The correction factor due to the presence of this kind of wave in this case, very negligible. This case, there would be some contribution. So in the text you can find the several rarely practically used for correction factor. You can use it. Or if you are very brave and if you have enough time you may have assimilation. Or you may try to get the transmission coefficient for every instant case. So that's the end of our lecture. This summarized what I have taught. So let me see what we had before. Well, let me summarize. We begin with Snell's law. And because what the Snell's law physically says, it has to be matched on the boundary in such a way that the fluid of particle arranged over here has to move in the manner that the motion of the fluid of particle in this part and that part has to be same. In x direction, not y direction because we assume the. So we look at these Snell's Law in terms of wave number domain and advantage of looking at Snell's Law in wave number domain is certain exhibit. So-called exponentially decaying wave in x direction. And then we move to the case for the plate for having plate as at a discontinuity. And we look at the mathematical formulation that determines how much transmitted will be induced compared with the instant way. And we found that this kb has to be same as the wave number of freed particle in this direction. That has to be. As well as the wave number of transmitted wave in this direction too. And also we assume that this wave Zeta which is a function of y in this case. That has such an amplitude and the propagation look like exponential minus j omega t minus kb y. Using that assumed solution to the plate equation, we obtained interesting dispersion relation. Physically the wave on a plate according to that dispersion relation, says the long wave propagate slower than short wave. And high frequency wave propagate faster than Low frequency wave. And we. Argued that using. What I'll draw in this picture. Infinite plates case. And a finite plate case that transformation curve and a finite plate has to be larger than infinite plate. Therefore, per unit area, transmission loss, infinite plate case would be greater than equal to the finite plate case. Therefore our prediction based on infinite plate would be save those control guidelines. To understand why we got a more transmission curve for final plate case. Seeing what's going to happen. As if we are fluid particle then we can understand because if you oxalate this then the fluid over here and over there will more effectively influence the motion of the plate. And then I emphasize the gain. Every transmission curve shown to here is in terms. It can be expressed in terms of loading impedance and partition impedance. For the case of bending wave, partition impedance has two component. One is LM4 partition corruption that is minus j omega m. The other one, what is associated with bending rigidity? That's what I'm going to say today.
