Let's see the mechanical impedance or driving point impedance. Simply, I multiply 4 pi a-square- What is a 4 pi a square? That's the area of breathing sphere. Okay? Maybe, we call this mechanical impedance. Okay, then it look like that and then we have again. The famous ka square, divided by 1 plus ka square and the imaginary part is minus j ka over 1 plus ka square. Okay. When, first, when ka is very, very big. Then this mechanical impedance approach to- Sorry. It- We have to include the rho zer, rho zero C over there. Then this one will look rho zero c 4 pi a square. What it means? This physically means when breathing sphere oscillate with a small or large, large ka La, large ka. The, then the mechanical impedence, rho zero c 4pi a square, that means the whole surface of breathing sphere pushes the fluid as if it is plane wave because four pi a square is the total area Suppose you have breathing sphere or you somehow, you know, tear off the breathing sphere so that you have this kind of 4pi a square piston, and you are pushing the fluid. And then that is rho0 4pi a square. So that is interesting. And the second observation is ka is smaller than one. Then the mechanical impedance is equal to 4pi a square and a rho0 c and I have minus j. Because ka is small. 1 plus ka, ka over 1 plus ka square approach to ka. So, I have minus jka over here. Okay. What physically it means. So that is minus j, and I have a over here again, so I have, minus j, k. And the 4 pi a cube and a rho0 c, and what is related with a cube? a cube is volume. Okay? So, but, and a cubed multiplied by rho0 is, a volume multiplied by rho0 means something has to do mass. So, mass of total breathing sphere. That breathing sphere's mass should be. 4 over 3 pi a cube rho0 Therefore, this means that I have minus 3j omega mb or in text, I used md, I guess. Okay. What it means when ka is small the driving point impedance is imaginary part that you do know what it means physically. I mean the velocity and the pressure is out of phase and is proportional to mass of a whole breathing sphere. And it is proportional to -j omega. That is interesting. Because if there is a phase difference with the velocity with -j omega, that means I take the derivative of velocity with respect to time have for harmonic oscillation. Right. If I take a derivative with respect to time upon velocity, what do we get. That is acceleration. That means for the case when ka is small. The fluid is accelerating. Not in phase with the velocity. So, in a near field, actually the fluid is starting to accelerate, that makes sense in the far field. The acceleration is accumulated therefore the wave is propagating you know, following the real part of impedance sometime in a rho0 c. Okay. Now, let's look at another one. Another physics associated with, radiation of breathing sphere. So, let's see another measure, that, give us the physical understanding what would be. So we look at impedance. Okay? We also look at- Driving point appears to. What would be the physical measure that could be interesting to us? In chapter 2, we learned about the relation between pressure, density, and velocity. Okay. So we invited impedance to see the physics. And then what we did invite in chapter two, huh, Okay, intensity. Why? intensity is the power, what does it mean by power, that is force multiply by velocity. It means that how much energy per unit time. Through units surface we can generate. So that is interesting. And also where we look at, we look at energy. Okay. That is composed by two type, one is potential, and one is kinetic energy. And potential energy was p square over 2 rho0 c square. And kinetic energy is simply one-half rho0 v square. And intensity what we look at- intensity also has a real part and an imaginary part. When we look at the intensity, we observe that average intensity, in other words net power. This one have real part of P, that is the magnitude of pressure, complex magnitude. And U star, that is to conjugate the magnitude of- Of, of velocity. So lets look at them. What is average intensity for breathing sphere. What would it be? Okay, we know this and we know that. Okay, P I cannot remember but that is functional k over or minus jk, something like that. And u also has similar, one. So if I do that, what I got is one half rho0 c, u0 square, that is the magnitude of velocity. And I have ka square over 1 plus ka square, and I have a over r square. So, average intensity also depends on ka square one plus ka square. But it is related with a over r so that means that observe the net intensity. In, in the distance away from the center of the sphere has to be depends on one of r square. That makes sense because we assume that fluid medium is in the, the fluid medium does not have viscousity. Therefore, the only energy lost, would be caused by the spreading effect. Spreading effect means that the sound energy is, is propagating omnidirectionally all the way from the region, therefore to, to follow the conservation of energy as we away from the center of the breathing sphere which means, and r is getting larger and larger. The total energy on the surface that is 4pi r square has to be conserved therefore you have to get this kind of formulation as expression. So does that make sense. Okay. Okay, what would be a total power that I observe at r, at, at, at, at position r? that should be I average multiplied by 4pi r square. Okay? That is total energy I can observe at the position r. And actually I am integrating all the intensity and that has to be one half rho0 c, and u0 square and ka square, 1 plus ka square, and now I am multiplying 4 pi r square, that eliminate r square over there. And I have a square. Okay? So if I normalize this, average intensity, total average intensity, by- One half rho0 c and four pi a square that is the total surface of breathing sphere. Multiplies by u0 square. That is the energy, power that generated on the surface of breathing sphere, alright? And then if I do this, one half rho0 c, four pi a square, and u0 square ,this will go away. And 4 pi go away. a square go away. Everything go away. What I got is simply ka square over 1 plus ka square. That is beautiful. What does it mean? The total radiation power, normalized what supposed to be generated on the surface of sphere, is simply function of ka. And then if you draw this function. [BLANK_AUDIO] In say I_avg 4pi r square and normalized by this. They ka is very large and approach to 1. Okay, but what about when ka is small? That, obviously that is one, one plus ka square. When ka is very small, this one is much, much smaller than one. So, it should be one and this proportion to ka square, therefore total power increase like that but because this is square, and this ka, radiating power in the near-field follow 60 dB law. Okay, because that is square. If you take the dB scale this will provide 6dB per doubling ka. Well, that is interesting. It means as we get away from the center of a source, radiating power is increasing by 6dB. What is the practical usage of that? You can, you can, you can use the intensity probe or whatever and measure if it increases at 6dB per octave then you can so oh, this source behave like breathing sphere. Okay. Now that sometimes very useful, useful, observation. [BLANK_AUDIO]