So lecture number 17, we will talk about Acoustically Large Space. Corresponding chapter is chapter five, the book which you are taking as a text book. Today, we will talk about very large space, meaning that the characteristic length of the space is very very large compared with the wavelength of interest. Okay, mainly we will introduce Sabine's Theory, which is very famous and well-known theory that describes the quality of room, or characteristics of room any, which can be regarded the Acoustic Large, can be measured by simple time constant, which is reverberation time or reverberation period. I think I already mentioned very briefly what it means, but today I'd like to derive this, and I will conclude. This will be point one six one and sizable and open area window. AS can be regarded as summing up or the absorption coefficient and area of the surface. And let's see how we can get this resolved. Under what kind of condition we can get this result. Okay? Suppose we have a room that look like this, and maybe in the lecture hall you have stairs and you have some seats, whatever. And those can be regarded as for example, some surfaces impedance, say Z_1 and you have another Sophy's Z_2 and probably some other sophistries Z_3 and so on so on. And suppose you have orchestra playing over here that radiates sound, and this sound will meet the surface distribution of impedance mismatch that I cannot hear Z_1, Z_2, Z_3, and it of course will be reflected back and forth like that, so eventually all the sound will be mixed up in this space. So, if you look at the energy that is of course in general, function of space and time, it is consisted by two components as we know. One is the kinetic energy, which is one half [inaudible] and the other one would be what we call potential energy. This is just the energy we derive by squeezing the unit volume of compressed air. And then, we can find the kinetic energy by squeezing the unit volume of air by the pressure, we'll get this result. Okay. And if you see the certain volume over here, right away, within a certain period of time of this energy that I can see that sort of mean energy inside of this volume. And during that period of time could be regarded as, one over delta v, integral dv, and then I also take the average with respect to time from t to t plus T, capital T is the period, I mean, time duration that I'm going to take on average. And E(r,t)dt. If you look at this energy which delta v is very large compared with wavelength cube of interest, and Lambda is wavelength, and also the T is large compare with the frequency that I have interest. This energy would look like indeed in space if this is the space from the sound source, then will look like very much constant if I'm very very far away from the sound source. For example, I'm saying over here. What kind of song do you want to hear from me? I didn't prepare this demonstration this time. Maybe, for example, if I sing like, right? Then what you hear over there and over there is sort of summing up all the kinetic and potential energy. And what you hear for example, over there and over there would be, in some sense very much an average sense, average with respect to that volume, and average with respect to the period would be quite constant, so look like that. And then, at some stage look like that because if you are close to me, then what you hear would be different with what he can hear. So there could be some fluctuation. And then, at some point would look like that, from here to there, there, what you hear is most likely direct wave. So, I call this is direct field, and I call this field reverberant. From here to there, I call reverberant field and I call this field diffuse field. Okay, that's very important concept. Or you could say SPL, very much similar. So, in direct field, sound pressure level would decay, six dB per octave. Octave means by increasing twice of the distance the sound SPL will decrease by six dB. So, if this energy is diffused or satisfy diffuse sound field, then we can say the following things. Also note that, the rate of energy change with respect to time, has to be balanced by intensity. So if you apply those equation to one dimensional case, if there is some war impedance mismatch over here, say Z_1 and Z_2, and if you supply the sound by using speaker over there, then some energy will go out. So rate of increase of energy has to be balanced to what coming in minus what coming out. What coming out is a smaller than what coming in, then the E dt will be positive, but if what coming in is smaller than what coming out, then the E dt would be negative. Okay? So expanding this right here, and I suppose that we have diffuse field as well as direct field inside, then we can write the following things.
