So let's consider what's going to happen in rev, reverberant. Recall again the, when I generate a sound over here. It's two type of energy. We can think, one is direct. And, and the other one is one is a reverberant. And the rate of change of this energy, total energy inside the room, would be integral of this with respect to dv and dv, I integrate them again. So that is. Physical meaning of that is I am trying to see the rate of change of the whole energy inside of, of, of the field. Or I want to see the energy inside of sound field. In terms of two component. One is direct, energy due to direct sound. And, another one is energy due to reflect, reverberate sound. That's, that's, that's, that's pretty reasonable. And then I can say, and this energy will be, Will be balanced by following two powers because this, this induces the energy loss. Okay, for example, let's see there is a one dimensional case. I have a speaker over here, and I am putting sound over here. That make direct. And then if there is a some impedance over here, then it will reflect it. But some of this one would go out. And that calls a Pi out direct. Some power that go out without having any reflection or reverberation. And the some power will go out after having reverberation. So I, there will be some Pi out reverb. Okay? So I can write here is a, and also this is Pi in, direct. This is what, what power coming in from, from the from the source inside of the room. Okay. Pi in direct, in short, I will use d and rate of, rate of increase of energy would be, this one minus what is coming out. If coming out is larger than this, then this would be, of course, negative, if Pi coming out is smaller than this. Then, the inside of energy is increasing, okay? So, then I can write over here, this has to be balanced by Pi out direct. Plus Pi out reverb. [BLANK_AUDIO]. And in, if I, if I, if I, if I, Making a sound inside over here continuously, in other words in steady state mathematically it means d dt is 0 because in steady state, everything is constant during certain interval of time, therefore, I can write Pi in direct minus Pi out direct plus, I put parentheses over there, Pi out reverve has to be 0. Or Pi in direct has to be, this direct has to be balanced by this one and that one. That is pretty reasonable, that means I, I spent $10 per day and I got $10 per day. Then there's no increase of the money in my account, okay, in steady state. Right, so, now I have Pi in direct and I have Pi out direct. I have Pi out reverb and I also have energy direct and I have energy reverb. And we would like to know the relation between these things. And again, between that. That's our objective. All right [COUGH]. First. [BLANK_AUDIO]. For Sabine's theory for diffuse field or roughly for reverberant field, what we found is that from Sabine [SOUND], we found that there is a relation between Pi in direct and Pi out direct. What is coming in and what is coming out? [BLANK_AUDIO]. Okay, remember, when you have a sound of an energy generated, the sound is going to be decayed by absorption, and that is related with open area window, absorption is related with open area window. So we can we can write the following things. Pi in is proportional to Pi out in the beginning because that is obvious when we have large power in then large power out would be anticipated. And. This proportionality can make equality by introducing absorption coefficient. [BLANK_AUDIO]. Yeah, that makes sense, there is power coming in and there's power coming out. Of course power coming out in steady state cannot be larger than [LAUGH] power coming in and therefore our power has to be less than one. ' Kay. And that is related with absorption coefficient. Therefore, [SOUND] for a steady state. For a steady state, Pi out reverb has to be equal to 1 minus alpha bar Pi in direct, I was confused. So I have to write Pi out direct and Pi in direct. That's good, okay. And this true, right? So we have a relation between, [COUGH] we have a relat, relation between Pi out, Pi out reverb and Pi in direct. We have this relation. We have this relation. And eventually, we want to have the relation between reverberant energy and direct energy. And how much is going to, how much direct energy is related with the the reverbaration? But, by using the Sabine's theory, we got this relation. Okay, where. Absorption coefficient, is equal to total area, and the open area, open window area, again. Okay now, and they are attempting to now, power in direct. We would like to relate this, this is power in direct with the power out direct somehow. Okay? So power in direct, is, of course, 4 pi r square and intensity at position r. That is obvious. [BLANK_AUDIO] And energy direct is obviously one half rho0 velocity average means mean square velocity this is related with acoustic, kinetic energy in the particle, and plus P mean square average divided by 2 rho0 c square, that is the acoustic potential energy as you know before. And for far field approximation, assumption gives us, this is, just a P square average divide by 2 rho0 c square because in the far field, in the far field the characteristic impendence of medium is equal to P over U for far field because there is only plane wave. So that is true, and this intensity is also P square average over rho0 c. Why? The intensity is pressure multiplied by velocity, okay? And the velocity is 1 over rho0 c right, so again, far field approximation says, this is true. Therefore I can write e direct is, as you can see over here we have this and that is Pi in direct, therefore we can write this is. 1 over 4 pi r square, Pi in direct divide by c. Okay. That is good. So we have this relation. That is, I found e direct is related with, what, this one. So we have a relation between this and that. We have a relation of this. Okay. And also, we would like to recall that, we would like to recall that the reverb, we have to work for the this one, okay? This one is the what is remain. And we know that the energy change of reverberant sound field with respect to time multiplied by v is the total energy change inside of the room. Okay? That has to be, total energy change of reverberant sound, sound energy, it has to be balanced by what? I'm talking about reverberant sound. And that has to be balanced by Pi out reverberant. If it increase, if it decrease, then Pi out has to be. Increased, if it increase Pi out has to be decreased, so there is minus. And, from Sabine's theory we found that this is, is equal to minus e reverb divide by tau. That's familiar equation, and the tau is related with, of course, the 4V c As. That's what we found in the last lecture. Therefore, we are getting close to the to the conclusion. Therefore now we have this rela, this relation, too. So we have this, the relation between this, and tau is also obviously related with that. So we will be able to get the relation between e reverb is equal to Pi out reverb I'm putting this one. This one is equal to that one so that one is V. Pi out, divide 1 over v and then multiply tau. And that is correct, right? And just using this one over there and the e reverb has to be Pi out reverb divide by v and then I have a tau over there. All right? That's not having any significant, physical meaning. And we know that there is a relation between Pi out reverb and Pi in. Where is it. Pie out reverb and Pi in. That is the Pi out reverb and Pi in that is Pi out reverb has to be equal to 1 minus alpha bar and Pi in direct. Therefore what I can get is. From this relation is tau divide by v, 1 minus alpha bar and Pi in direct. So I got and. And using the relation of tau over here I get, this has to be equal to 4 over C As 1 minus alpha bar and the Pi in direct. Okay. So, this is very nice because, this gives the relation between reverberant energy and the Pi in direct, okay. So physically means that, the what is power in due to the source is directly related, related with what I hear as a reveberant sound field, okay? And that makes sense. And as you can see over here, if open area window is large, that means there is a lot of damping or assumptions, strictly speaking. Then the power in, power in, does not provide. As much energy to the e reverberant and that makes sense, and if alpha is 1, what it means? What it means? Everything is go away, therefore you e reverb has to be 0. Okay? That's good. So it certainly relates what we actually want to have in the beginning. We want to have is some measure that relates the space. Or characteristics from the source to the listener, where we can sense the direct energy as well as a reverberant energy. And the reverberant energy is related with Pi in d, so there is some possibility that we can have such relation that we, we, we want to have in the first place. Okay, and recall that the total energy. As to, we have a two part, one is direct and one is reverb. In the beginning we know that e direct is also related with. Pi in direct, of course, divide by 4 pi r square times C. Okay. Then for using that relation we get, this is, one plus r square and 16 pi over As. 1 minus alpha bar. And then we can say, because this is the same as e direct, and here is r square. And let me write down there r0 square where r0 square is inverse of that. That is what? 16 pi 1 minus alpha and As. And of course r0 is square root of this. Okay, let's see what it is over here, the total sound field has two components. one is, of course, direct, and the other one is r direct multiply r over r0 square. Okay? So this part indicate how much direct field that we can hear. And this part. Indicate how much reverberant we hear depending on the position of r. And if r0 is very large, that means, there is very small reverberant effect. If r0 is very small, there is very large reverberant effect. And that r0 is related with this value and that makes sense. It related open area window, so if an open area window is large that means. We have more direct field because everything is going out very quickly, and if alpha is which is the the the measure of absorption coefficient in the mean scale in a total room. So if alpha is what one, that what it means. Open area window and a total area win, total area surface of the room is the same. That means there is no possibility to have reverberation, so in this case r0 is 0. Okay, what it means? That this goes to infinity. That means we have. Zero possibility to have a reverberant, reverberant sound effect. This is somewhat confusing, but we call this as radius of reverberation. If radius of reverbaration is very large. That means, you have quite large direct field. If radius of reverberation is small, then inversely you have a small region or distance that you can, you can hear the direct sound. Okay. [COUGH] Let's see what we can, what we can have in the picture, in the graph. Okay. And as you can see here, the total energy is composed by direct, direct as well as reverberant. This is reverberant one. So this is total energy. This point, is of radius of gyration, radius of reverberation, is where we got equally likely energy of sound direct, I mean equal, equal participation of direct and reverberant sound. Over here, what you can see is the we hear more direct sound compared with reverberant sound. If you go over here. You will hear more reverberant sound than direct sound, and this is what we call radius of reverberation. The radius of reverberation is good measure to express, the dependency of the location of people. Okay? Because depends on where, where, where you are in radius, the radius of, radius of reverberation could be different. Therefore this is good measure describing the, the sound inside of the, acoustically large room.
