Okay, let's start lecture number 13. What we learned in the last time was to study about the radiation from, for example, breathing sphere as well as the trembling sphere. We use the velocity potential that satisfy the governing equation. And we apply the boundary condition, in this case the velocity on the surface uniformly, oscillate and we do note the diameter or radius of sphere is A and we obtain pressure away from the source. This is R and T is time. And also we look at the property of each typical case in terms of the radiate of, the radiation impedence. And it was look like for the breathing sphere case. It has a two part. A real part and imaginary part and it look like kr squared. And the one plus kr squared and the imaginary part was minus jkr over one plus kr squared and it is proportional to of course c. That is the characteristic impedance. And this one looks like load 0C and U0. If you look at this, this is characteristic impedance and that is velocity. So this term is somehow related with the pressure. Minus jkr, over one minus jka and a over r squared exponential minus j zeta where zeta is equal to omega t minus k r minus a. For traveling speed, I will, if I write, then the pressure for the respective production and the time. So look like, interestingly, proportion of j over jc so thedifference between this pressure and that pressure already appear in this part because trembling sphere is some how related with imaginary parts. And you see that this the velocity in this direction. So actually the velocity that has to be effective to radiate sound would be uc and cosine zeta because that component is uc cosine zeta. Therefore, the radiation in this direction should be zero. So that means trembling sphere has some directivity. And then it is proportional a to the q and of course, zeta we have something that is pretty much related with ka and kr. k squared minus 2jka and then we have jk over r- r square and then we have exponential jkr- a. Okay, the radiation impedance that is the measure, physical measure and allow us to look at the physical meaning of the radiation characteristics of each case. So jr is row zero c and interestingly it is related with the kr to the force. Instead of kr to the scale. Four plus k r to the fours and the imaginary part is rather complicated and that is two kr plus kr two d [INAUDIBLE] and a four plus r to the 4th, okay, what I'm trying to say, by rewriting this complicated expression, which of course you can find it, your text, but the reason why I wrote this again is because let's look at this term. Okay, when A approach to 0, in other words the breathing sphere for example, tend to be very, tend to be very small and small then we will get theta is equal to omega t minus kr okay. And this term, ka, if ka is very small, that means a is small compared with the wavelength. This term tends to be very small and small then hold this expression of pole to monopole. And this one approach to dipole, okay. Another method from this rather simple representative solution is that the monopole and dipole is absolutely of course the one of typical solution that satisfy the equation. Okay, next question is, because we do know that the solution, a typical that satisfy the boundary condition, the equation, What or this solution is related with the general solution. Okay, for example, for example. If we want to investigate, this is very, very simple and representative solution. A second one that we can imagine, that could be very represent to the typical radiation problem would be baffled piston. [SOUND] Baffled piston case, okay. What if this has velocity applied like that? And the auction light with respect to the car minus the explanations j omega t. Okay, if this solution is related to that solution, that would be interesting and also that would be one of the beauty relating with the linear acoustics. Linear acoustics because it is linear. Super position of this solution could make the solution that corresponding to this case. If we do know the solution of this case then for example if we have a plate that is vibrating like that. All baffled, then you could think that this solution could be somehow similar with a solution that. As this oscillation plus that oscillation, it is different face arrow face, okay. So, if you know this solution then using that solution we can get a solution of more complicated case. So let me repeat again this is from the vendor solution if he finds the relation between this solution and that solution, that would be great. And further if you know the relation, I mean the solution that we can some what tackle to get the solution of this kind of general radiation problem, okay? So let us start to write the story. [LAUGH] Story how these can be Sad. What would be our story that connect all these 4, I mean 1, I mean the. What I mean is in this lecture we are trying to write the story that can connect the monopole dipole solution to piston driven baffled piston solution. And then how the solution can be related to rather general solution? So based on what we understood about this solution in terms of kr and ka. Far field and near field solution. And also we understand that the radiation impedance zr certainly represents the characteristics of this and we'd like to know how this radiation impedance is related, for example, this case, and then farther, to the more general case. So this is very interesting attempt. So let's begin with a one dimensional case to understand to write down the story that we desire to have.