Okay this will be Lecture Number 12. We will continue to talk about Two spheres, one is breathing and the other one is Trembling sphere. So in the last lecture. We found that for breathing sphere. [BLANK_AUDIO] Breathing means that the fluid particles. Are surrounding this field will oscillate with the same phase. [SOUND] And if I invite to coordinate x, y, z, whose. origin is locate at the theta of sphere, okay, normally. It has dependency with respect to this angle and that angle as well as the distance from the center, of a sphere. And, the wave will propagate through this phase. And since, the breathing sphere is symmetric, means that. Does not have a dependency of a theta and phi the governing equation will simply follow [SOUND] this homogeneous equation, therefore we get phi would be proportional to 1 over r and then unknown amplitude we use A and then it oscillates with respect to time. Harmonically, we assume, therefore, we can write minus j omega t and then it is moving away from the center. Therefore, we can say the spatial dependence would be kr. [COUGH] and this A unknown, complex amplitude [SOUND] can be obtained by inviting. Another equation, logically, and that another equation should come from boundary condition. [SOUND]. Said B, C, Boundary Conditions. What kind of boundary condition it is. Okay. When we talk about, for example, limp wall, incident wave and reflective wave and there is a transmitted wave. And now we apply the two boundary condition, one is pressure continuity boundary condition or the pressure difference between two surfaces must follow the Newton's Second Law and another boundary condition is fluid particle, the velocity must be continuous on both sides. [BLANK_AUDIO] So in this case the boundary co, condition, what we can obviously write is the boundary condition associated with the kinematic boundary condition, natural boundary condition or forced boundary condition. Because what we obviously know that the fluid and particle, for example over here. [BLANK_AUDIO] Should be the the velocity of what will follow. By this assumed solution. So that will determine the assumed amplitude A. So let's write the boundary condition at any, at the surface in r equal A. This is r equal A. So say the velocity in r direction, that depends on r and. Has to be some magnitude U zero and exponential minus a omega t. Sorry, this r has to be a. Okay? So, equating this, with dphi/dr, that is the velocity, in our direction, gives us. Therefor we got Phi. The velocity potential. And the using the relation between velocity potential and velocity. Which is, I said over there and the pressure with respect to r and t can be obtained by rho zero dphi/dt. So we obtained pressure. And those velocity and the pressure, was a function of kr or kr square. That's what we learned in the last lecture [COUGH]. And these. Very useful to see the physics through radiation impedance, Zr. So let's look at radiation impedance of breathing sphere. [SOUND]. Which is simply the pressure, pressure at r, divided by velocity at any point along the space r. And we found that, this is very interesting, because it is related with the rho0 c, that is specific impedance of medium. And we remember that if it is rho0 c. Then impedance has only real path, and also, it is equal to rho0 c means that it radiates some as if it is plane wave. But, because the breathing sphere is not. Always same as the plane wave, it is modified and the modification factor that follows to this, to this one, is somehow related with. 1 plus ka square and ka square. That was real part and have imaginary part that, I remember that was, minus j and numerator is ka instead of ka square for the real part. And at 1 plus ka, square same denominator. So, this is a scale factor, different with rho0 c. [BLANK_AUDIO] Okay, that is quite interesting [COUGH]. And then we look at. This, scale factor graphically. So seeing this in terms of kr and the significance of kr, that we have to realize. Is that k r is two phi over lambda r. This is k r. Physically means that we are seeing the distance, how far away from the center of the sphere with respect to wavelength of interest. And the real path, behaves like ka scale over one plus ka scale. [SOUND] So if I normalize the radiation impedance with rho0 c, which is the Characteristic impedance of medium that sees, means I want to see the radiation impedance compared with the plane wave. That's very interesting observation. That's very interesting way to see, to see the physics associated with the breathing sphere. Okay, now we observed that, as k goes high and bigger and bigger, this value will approach to 1. This is 1. So in this region, the approach to 1 that means. Breathing sphere case, in a far field. Meaning that when we are seeing the wave, wave very far away, far away in terms of wavelength, it,it behaves as plane wave. So, this is plane wave region or we use the, important terminology that we call this field is FAR Field. Far means I am far away from the breathing center of the breathing field. With respect to wavelength. And very close to the this point, when kr is very small, which means that wave, the distance is very close to the breathing sphere with respect to wavelength. The graph has to follow this and it look like, this behavior. So, obviously we can see that the real part in the near field is much smaller than, the real part of breathing sphere in the FAR field. That means, when I, when I push, I mean when I breathe, when I breathe so, so, so to generate a wave. The, the fluid particle that is in phase with pressure is getting smaller and smaller as we approach to the breathing sphere. And, and this kind of behavior is always happen in, most cases. [COUGH] Okay. And let's look at the the imaginary part, which is ka over 1 plus ka square. And We found that this one and imaginary part of shares the common point that is, if you look at the book, [NOISE] that is. One, yeah, if I put 1 and that is one half and this graph look like that. So imaginary part increase up to an kr is one. What, what, what's the meaning of kr equal one? kr equal one means 2 phi times r divided by lambda is 1 so that means r over lambda, is 1 over 2 phi. That means I am away from the center, okay? 1 over 2, 2 phi times wavelength. Okay? So you if, if, if we D, I mean that is approximately we are in the distance which is like a 1-20th of wavelength. That case, the participation of real part and im, imaginary part is equally likely. And then, is suddenly decrease, I mean the imaginary contribution of imaginary, imaginary part is, is getting smaller and smaller. So, for the noise control purposes, we would like to use this kind of behavior. For radiation purposes, we would like to use this FAR field behavior and this region we call NEAR field. So what it means, when you hear this sound very new to the speaker for example what you would hear is this kind of behavior, but if you're far away from the speaker, speaker, you would hear this kind of behavior. That's what you learned in the last lecture. Okay, emphasize again. Mathematically, this one has two parts and the scale is kr. Okay, what if I, I, What if I substitute r with a? What does it mean? Then, when. [BLANK_AUDIO] When, r equal a, simply Z r equal a. That would be equal to rho0 c and then I have, I have ka square. Sorry, this should be kr, mistake this is kr and kr. So this is [LAUGH] kr and kr. I confused. Nobody pick up my mistake this time. Okay. That has to be ka square divided by 1 plus ka square minus j ka divided by 1 plus ka square. Therefore. It depends on ka. What's the meaning of ka? Ka is 2 phi a over lambda. What does it mean by this, compare with that, all over lambda? That this. The size of radiator, size of radiator has to be, has to be seen in terms of wavelength. Okay, that's interesting. Okay, I have a breathing sphere and I want to generate like 100 hertz. Then the wavelength would be more than 3 meters, 3.4, usual cases. And if size is like this size then, a over lambda would be very small. That means real part is negligible and imaginary part is dominate. What does it mean? That means, when we have radiator who's size is much, much smaller than radiator it does not, it is not governed by real part, it is governed by imaginary part. That means. It's not good radiator. We will see more details about that. And basically graph will follow. The graph which we saw over there. But in this case the, wave has to be seen in terms of ka. And this is. Radiation impedance when r equal a divided by rho0 c. See, the physical meaning of this, is exactly same as the driving point impedance that we studied in, in, in, before. So again. The real part approach to one as k, ka is getting larger and large, and the imaginary part approach to. [SOUND] Zero as k becomes larger and larger. And again this is FAR field. FAR field, FAR field, do you think that is correct with this correction. No, no longer field concept. This area, we say, efficient radiation [SOUND]. Because we have only. We are part of the approach to 1, this is very important concept and this area. It's not efficient [SOUND] and those regions are emphasized again, controlled by k. Okay, then practically it implies many things. For a sub-woofer, of course woofer does not, does not normally look like a breathing sphere. But for low frequency sound generator, normally. For example some HI-FI audio system that want to generate like a 20 Hertz up to few 100, like 10 par. The wavelength is so large. So to, to, to, to make efficient radiator, we have to go this region, that means the radiator has to be very large, therefore sub-woofer normally is very large. Okay? To get a high, you know, radiation. For example, for Twitter. 1 Kilohertz or 5 Kilohertz, the wavelength of 5 Kilohertz is what? 6 Centimeter or 7 Centimeter, very small. Okay? Therefore. We don't have to design a radiator, big radiator. Small radiator would be sufficient enough, okay? Let's sort of review of the last lecture. Also, we put some. I mean more physical insight about impedance. Okay, okay, let's look at this, with respect to some other physical measure. [BLANK_AUDIO]
