# Amar Bose: 6.312 Lecture 22

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AMAR BOSE: OK. Last time we-- I hope you can hear me today-- last time we developed the expression for the number of normal modes in a rectangular room, and we got a vector interpretation of the modes that could exist, in which the vector length was proportional to frequency, and the direction was equal to the direction that you would have to launch a plane wave in the room to cause a standing wave, which represents one of the normal modes.

So we had an expression for the number of modes below a given frequency. It was 4/3 pi, volume of the room over c cubed, f cubed. The number of normal modes in the room below frequency f. If I look at the density of these normal modes, just consider the derivative of nf with respect to f, drops down a 3, cancels this 1. And we get 4 pi, volume of the room, over c cubed times f squared.

Now this is what you saw in the demonstration that we had. Well, you really didn't get a chance to see it well, but as the oscillator was turned up and you were watching the scope, if you could turn it at a uniform speed, you'd find out that the trace would jump around much, much faster. In other words, the density of the normal modes is proportional to the square of the frequency.

Which means that most of the normal modes are up at the high end, if you wish. When we computed what your room would have, an 18-- I think we took, or something like-- 18' by 25' by 8 foot high room, we got 12 million modes below 10 kilohertz. Well, if you went to just 15 kilohertz, the ratio of the square is 15 to 10, 2 and 1/2 times or 2 and 1/4 times.

So the normal modes you have are many, many, many in a room. And that raises some very interesting questions. In our vector picture of our frequency space, we had last time developed the following. There were points along the x-axis separated by, I think it was, c over 2lx, and the y-axis-- whoops, let's make this x, make this y, and make this z. c over 2lz, and finally, c over 2lx. That's the spacing on these lines. This is your fx, your fy, your fz.

Each one of these points and each point out in the space-- in other words, if I were to go up here like this and project out this particular point here, the 1, 1, 1 point, would come out to here somewhere. Something like that. This vector, this cube which represented one unit this way-- not cube, but-- represented one unit this way, one unit this way, one unit this way, the end vector for all of that would come up to here. The length of that vector would be the frequency of that mode. The angle of that vector would be the angle that you launch it on.

Now from this pattern, we were able to go to a giant 1/8 of a sphere and say that the number of normal modes in a room was essentially, approximately, the number of these little boxes or cells in that 1/8 of a sphere. Now it's a heck of a lot, and it's a heck of a lot more than what you would think of first. If you thought of a room in which you had a mode just going back and forth like this, one going up and down if this were a rectangular room, and one going this way. Those modes are the sum of this-- all the ones, let's say, below some big frequency f. All the points along here, some of these points, and some of those points, which is a heck of a lot less than the sum of the points that are out here in this 1/8 of a sphere coming out from this, this, and this plane, less than radius f.

Now a few things you can see about this from this diagram right away. The worst dimensions to build a room for acoustic purposes, i.e. for communication or for music, would be a cube. Because in a cube, lx, ly, lz, are all the same. And you then have modes that can exist this direction, this direction, and this direction, all of the same frequency. So at that particular frequency, you get enormous peaks and dips.

So the worst dimension-- the standing waves are the things that cause you the problems, so I'll tell you where they're good and where they're bad in a minute, hopefully. But generally, they cause you troubles. And the worst of all worlds is to have your troubles in triplicate, one, two, three all the same length. So all of your modes then that represented modes going along axes, would in fact have common frequencies.

The next worst would be if the cells along one axis, let's say, were twice what they are in the other. Because then this frequency here would be equal to this frequency here. It would be the first mode in this direction, but that frequency, the length of the vector, is the same as the frequency over here.

So rooms with integral multiple dimensions are going to give more trouble acoustically than rooms with the square root of 5 as your dimension. So when you build your home one day, tell the carpenter you want the length of the room the square root of 5 times the width. See what he says. So you can see all of this just from the frequency pattern that we have.

Now somebody asked the question after the lecture last time, and I thought I answered it, and one of the teaching assistants was eavesdropping and he said, I think you missed the question entirely. I don't think what you answered was at all what was asked. And he told me what he thought was asked, and I believe he was right. Namely, if-- someone asked this question-- if I'm not right on the question, correct me, all right? If you could only have these different modes that are intersection of these projected out axes, how could somebody speak in a room, because his voice has all the frequencies? How could you play a violin in the room, you'll be missing most of the frequencies of the violin?

SPEAKER 1: The other frequencies decay faster and don't set up resonances. But there's still-- if you're standing there, there's no walls in between you and me, than of course I'll receive all the frequencies before they hit the walls.

AMAR BOSE: You'll receive all the frequencies before they hit the walls, that sounds sort of reasonable. But-- yeah, we'll say that that's true, but we haven't yet heard about direct field and reverberant field. However, you actually receive all the frequencies after they hit the walls, too. Now you're thinking about this problem, it feels to you very complex. Oh my god, how can you answer this? Those are the frequencies, and now we're speaking with all frequencies. Yeah?

SPEAKER 2: Well, a voice or a violin produces a huge number of frequencies, no matter what particular tone it is trying to produce, if you only get, you know, half of those frequencies, that's still enough to reconstruct the sound in your mind.

AMAR BOSE: That's also true, but it does turn out to get them all. Well-- almost. Yeah?

SPEAKER 3: How about if they were only being sent in one particular direction, you might miss some, but since they are being spread in all directions, you've got millions of modes.

AMAR BOSE: You have millions of modes, yeah, but you know, there is no mode in this room that falls at, let's say, this point in this box. Now what I want you to do for a second is go back to what we talked about on the first day, namely, when the problem looks complex, cut it down to a simpler one. And if you cut this problem down to the problem of a tube, one-dimension-- we'll assume waves going only in this direction.

Remember that if we got any waveform at all started in this-- obviously, when I close the tube it has to be 0, but I could be any time waveform of any sort-- close the ends of the tube. We could have gotten that going by having a loudspeaker out here and then shutting the door suddenly. There it is, all the frequencies that the loudspeaker were broadcasting are in there.

Because you can drive a tube at one end with any frequency you want. And you can calculate what the envelope for that frequency, the standing wave envelope is, and whatnot. Close the door, remember, what happens? Only the normal modes can exist. But the normal modes make up this waveform by Fourier. And then the waveform moves around. So after the source stops, then whatever the waveform was in the room is made up by all the normal modes, and only those frequencies exist after the source stops.

Now you're going to get an example of this on the last day that I guarantee you, you will not forget. Said positively, you will always remember once you hear it. The effective normal modes, we're going to let you hear a sound that was sent into a room, recorded on good equipment at the time, played back, recorded again, played back. I'm not going to tell you what's going to happen, because I'd have no credibility if I told you right now what's going to happen. But it'll be interesting.

Now I'll just tell you that the normal modes in the room can degenerate sound at one heck of a rate, and yet, they are extremely vital to the sound that composers have produced or written for production. Why? Because they have written the sound, the music, to be played in a closed environment. If you take an orchestra and just park it outside, don't even bother with the shell, the violins sound like they were Woolworths instead of Stradivariuses. The whole thing is flat. It's uninteresting.

So the composer of music has composed for real rooms. And his music sounds good in it, most of times. Now if you went through those twice, like recording out there and coming back in, it would sound anything but good. I mean the entire piece, ruined. So a little lesson that what is good in some quantity might be poison in a little bit larger quantity. And you'll get to experience that.

So we have learned-- and by the way, what's good, what musicians have composed for in the past, isn't good for speech, the environment in general. It has much more, as we're going to calculate today, reverberation time than you would like to have for speech. Speech is very interesting. For example, I may have told you this, if you put a headset on somebody--did I tell you this? You put a headset on, and you take a microphone, you record them, and you put a delay in. And you crank up the delay and you ask them just to read a newspaper. And you get the delay beyond 50 milliseconds, and they can't read anymore. They'll just stutter.

So when you feedback sound with a delay comparable to a syllable length and greater, it's totally confusing. And so it is if you had the sound in here with, let's say, heavy reflections in the neighborhood of 7,500, 200 milliseconds, it would sound like I was talking through a barrow, which it almost does now as a result. But you wouldn't be intelligible at all.

So what environments we've learned to exist in are the environments that we've learned to communicate in. A very interesting example, the Bundestag building in Bonn, the German government building, two and a half years ago it was built and the acoustics were done by one of the world-renowned companies, German government moved in, 24 hours later, out. They could not communicate. They simply were not intelligible from one across tables.

And so there's still a lot-- and you're going to get some demonstrations of these kinds of thing on the last day-- but there's still an enormous amount that needs to be done to understand, or really, to achieve the kind of communication, kind of environment that you would like to have. If you played the music of Bach in some of the very modern concert halls, like even Kresge, it's not like it should be. If you play Stravinsky in a cathedral, doesn't work either. So with time, basically, the places that music was composed for changed, and so did the compositions. Yeah?

SPEAKER 4: I'm not quite following you. Are you saying that when you have a source of sound with many frequencies, first of all, do you hear all those frequencies, and if you do or your don't, does your mind reconstruct the rest, or is it that you hear them in one ear and not the other or something?

AMAR BOSE: OK, if you're in here, and there's a loudspeaker here, you hear everything except where there is a null at that particular frequency, in the standing wave, let's say. But all frequencies that this loudspeaker radiates exist in this, too. You here them all. When the source stops, you hear the normal modes that they can produce.

Take this again, make it simpler yet. Go back to this. Current source. You can have any arbitrary source here having all frequencies, you can put white noise into this thing, shut the source off, and only the natural frequency of this thing can exist there after. It's the ringing frequency effect. Now this is the same as this, only thing is there are a lot more ringing frequencies, we calculated. Any frequency which has an integral number of half wavelengths in here, and there are a heck of a lot more over here. Yeah?

SPEAKER 4: [INAUDIBLE PHRASE] in a room something to be worried about? So you can't hear-- if [UNINTELLIGIBLE PHRASE] you can't hear a certain frequency, sort of in our demo from last time.

AMAR BOSE: OK, yeah. You got a demo in which there was a loudspeaker up here, and then you could move your head closing one ear, and you found out there was a null at some point, maybe 20 dB difference between moving it a little bit. It turns out that is not a problem when you're listening to music. If it were a problem, of course, orchestra seats would all be dentist chairs, and maybe there would only be one seat in the audience.

Basically what happens is that if you reproduce the spectrum with an average-- let's say that I have two spectra that are coming to the ears, A and B that come. A comes to both ears, B comes to both ears. And they agree with an average spectrum-- an averaging criterion of about 10%-- over the center frequency at any point. In other words, when you're here, average over 10% of it. When you're here, average over-- this is frequency, and this is the spectrum vertically.

So if they average out over about 10%, you won't hear the difference. Maybe we'll get on the last day or so you can talk about some of this. This is in the realm of the psychoacoustics. And thank god, because if that weren't true, if you could distinguish it, then it would be a sad situation to hear anything, and music wouldn't have been composed anywhere else but outdoors. Any other questions? We'll go on to what we-- OK.

Now today we're going to derive one of the most important-- important in a sense, most commonly used-- parameters in the acoustics of architecture, namely, reverberation time. Now we're going to do this using statistics. Now why-- you've all studied statistics of some sort, why do people use statistics? What's going on? You don't ask why you study things?

SPEAKER 4: We don't have a model that we can be sure of. We only have a model that we know is correct for a certain probability.

AMAR BOSE: OK. You don't have a model. Meaning that it's very complex, perhaps. A situation is so complex that it's not easy to model. In other words, if you have a bunch of molecules in a box here, and you're going to find out when they collide with the walls, if you had all the initial-- you had each one, and you had them knowing which way they were going with what velocity, you could get a start to maybe the first collision, or something, but it's so complex that you wouldn't have a model that was useful. So that's one reason.

Another reason might be that even if you could know all those initial velocities of the different molecules and the velocity direction and speed, it'd be a heck of a job to trace them all around. Just like it would be if I wanted to look at the sound radiating from me right now and trace every single collision that is here. So it's too involved to make computations.

And a third reason, maybe you only want averages in the end. Just like we talked about a minute ago, certain averages are important. So these are basically the reasons that you would use statistics for any kind of a model. Now the important thing is that statistics involve only averages at the input. You avoided all this complicated model, the complicated computations. So they only give you averages at the output. That's basic. So looking at a room and the reflections is almost, or quite similar to the molecules running around in a box. Here goes a wave over there, it hits, comes off at the angle of incidence, goes over here, hits, goes back, et cetera.

So now what we'll do is talk about how we set up this model. In physics, you may have remembered the mean distance between collisions, and if you know the velocity, you know the mean time between collisions. Same kind of thing over here. And we assume that when the soundwave collides, in this case, that there's some loss to it.

We assume that. We didn't have to assume that, but we're assuming that because, A, in the real world, there is some loss, and after you make a sound, as we did one day clapping up there-- I can't do it here because of the absorption-- it decays. And that's because some of the energy is absorbed hitting each wall. So what people have traditionally-- how the field was developed was to define alpha as an energy absorption coefficient. Energy absorbed per collision, collision with a wall, over energy incident.

Now this definition was made for a given material averaged over all angles of incidence. In other words, if I do it for the blackboard, I go this angle, this angle, this angle, measure the reflection, and then average, with an equal waiting for all the angles. Because in the statistical model, things are so complex when you get out there, you assume that the waves that are bouncing around this room are, on the average, coming from every direction. I may be speaking, some of the sound goes right there, but eventually some of it will come back from here. So this energy absorption is defined, the coefficient for a given material, as this ratio averaged with equal weighting over all angles of incidence.

Now that's for a given material, for this blackboard, but it's different for that wall, it's different for the ceiling. So you define an average coefficient for the materials of the room because I'm not going to chase these waves all around here. And I'm going to say, well, OK, this is very absorbing. This is about 1% of the energy hits a blackboard comes back-- I mean, is absorbed. So now I'm going to say, I'm going to define an average absorption per collision. I don't care now whether it hits that or it hits that. That's the only way I can do it if I don't want to have to trace all the rays.

So this is defined as follows. By the way, what you do in science again and again is you make a hypothesis. You might look at this thing that's written down, say, gee, I wonder why they did that. You make certain that seems reasonable, you make a model based on that, you make the calculations and predictions, like reverberation time, and if it works, you keep the model. When you first find out that it doesn't work, out goes the model, and you replace it by a better way, hopefully.

So here's how they decided to do it. Alpha 1 for the given material, times the area of that material, alpha 2 times the area-- I'll put this down first, then we'll-- s total, or s of the room or whatever you want to call it. It's reasonable to say that if that area up there is 3 times bigger than this area, that the probability that sound is going to hit there is 3 times higher than it is if it's going to hit here. And that's no doubt what motivated this kind of a definition for the average absorption per collision in the room. Average absorption, energy absorption, per collision.

Now that doesn't include one thing. This is a property of the room, but you came into the room, and you altered the acoustics. And by tradition, by a model that was first thought up, what they do for that is they add, for things that are brought into the room-- and this is approximate, but for the kinds of things that it's used for, it, again, turns out to be so.

An alpha s to the numerator, to the numerator only for people. For example, there's an alpha s, an equivalent alpha s for you, and every time you enter the room, somebody puts that in the numerator to find out what the average alpha bar is, which is this expression per collision. So objects that are brought into the room, an alpha s is added to the numerator, just by tradition, but it works.

And this wasn't made, for example, this wasn't an expression that was derived for your little office or something, it was more or less all done for public buildings. Yeah?

SPEAKER 1: Do you think there's some condition where the room would have to be sort of square, because how would I-- say this wall right here, I wouldn't count the absorption coefficient of the wood, and then count how it goes in and then out and then in and then out as being more and more area for that.

AMAR BOSE: Yeah. I mean, for example, what this would be considered-- there's about a foot and a half of fiberglass behind this kind of thing-- and you would have for this surface, you would just use what you call an average absorption coefficient for this. It will be a function of frequency. But to find that out, you would hit this surface at a given frequency with the wave coming this way, then with a wave coming this way, a wave coming-- and you'd classify this as a total surface. You'd never get down to the kind of definition that is represented here. You'd take an average over all that.

SPEAKER 1: [INAUDIBLE PHRASE] like a square room, aren't there some places where more sound hits against? Like maybe in the center of the walls, because [UNINTELLIGIBLE PHRASE].

AMAR BOSE: No. Well, first of all, by the way, all of this-- I hadn't said it yet-- is going to be for large, irregular rooms. We'll get to that in a minute. But they're not for, for example, a focused room. How many of you have been in the Capitol building in Washington D.C.? OK, maybe 10 or 15%. Well, the thing is a dome, as you know, and a dome does what? Remember hard surfaces are to sound exactly like headlight reflector is too light, like a mirror is too light.

And so if you have a dome like that, there are two places, famous places, on the floor where the sound goes like this, and with a huge crowd in the room, if one person stands over here-- these things are about 40 feet, 35 feet apart or so-- and another person stands here, you can whisper and hear it. And for a long time, so the story goes-- at least the story they tell when you go through the building-- was that one of the two political parties learned about this and a desk of a very key person was here, and these people didn't talk, they just listened. These were the opposite party, over here. So a lot of leaks went on.

And it's very effective. I mean you could really, literally, just whisper. Well you get that kind of a focusing. Now that is not considered in what we're doing. That's an aberration. So this is a large, sort of a random-shaped room, if you want to say it. And the assumption is equal probability. When you get to rooms that are like that, that assumption is invalid, and you've got to treat it separately.

OK, here we go. Let's see. We'll make a model like this. These are walls that it hits. Let's say I have a wave of energy, density D0 that comes in, plane wave that hits this wall. It comes off at an angle that is equal to the angle of incidence. It comes down here, then its energy density is D0 times 1, minus alpha bar.

I've now assumed that alpha bar represents each collision, because I've taken the average over this. So it hits this, alpha bar is absorbed. Comes over here, hits, alpha bar is absorbed, 1 minus alpha bar comes off. Hits this and you get D0 times 1 minus alpha bar squared. And then hits this one, you get D0 1 minus alpha bar cubed, et cetera, on, and on, and on, and on.

So now what we want to do is get an expression for this D0, the energy density. As you can see, it's going down. So we want to find out how fast it goes down. Energy density in this plane wave is proportional to the square of pressure, so pressure is decreasing with time. And now all of this, so that we don't get confused, there's not a continuous source. There was a plane wave which came in here and that's it. It's not replenished anymore. Somebody shut it off. It hit here, it went over here, it went over here, just like the tube now. We launched a plane wave in here and we shut the door. So that's not being replenished. We'll talk about what happens when it's replenished soon enough, not today. OK.

We know that-- well, we don't know, I'll have to put it down. Another hypothesis which is rather amazing when you think about it, and it's only been verified in a couple of rooms because the calculations are so difficult. And it wasn't verified for a long time after it was a hypothesis. And you don't know that it's really so for any of the rooms, but that the mean distance between collisions with a surface is 4 volume of the room, 4 volume, over the area-- I'll just call it volume. OK, you know it's v equals volume of the room. s equals, in this case, area of room. 4v over s.

And as I say, for some cases where you can calculate it, it really turned out to be that. But before that was done, this was used. And again, you make a whole model, and at the end you see if the model gives the results that you can measure. Well, if we know this, and we know the velocity of the sound wave, we know the mean time between collisions. So the mean time between collisions, t prime, is equal to d over the velocity. So it's equal to 4v over the velocity of sound times s.

OK, now that's enough that we should be able to find out how fast this thing decays. Yes?

SPEAKER 5: [INAUDIBLE PHRASE]

AMAR BOSE: [UNINTELLIGIBLE PHRASE]? Yeah, what did I call it?

SPEAKER 5: [INAUDIBLE PHRASE]

AMAR BOSE: Mean distance between collisions, yeah, same thing. The equivalent in physics [UNINTELLIGIBLE PHRASE]. OK, so we have-- I can get an expression for D of, let's see-- D of n, huh? D of n is D0 1 minus alpha bar to the n. After the third collision, it's cubed. After the n-th collision, that's what's left.

And mind you, alpha bar is a positive number. It's not greater than one, in spite of the fact that you can add things like this for the people that are in the room. If you add a reasonable number, the numerator can never become greater than the denominator. See, this is not-- this alpha s for people, for objects-- I'll just say, it's mainly used with people-- is added only to the numerator, and not to the denominator, but don't get scared, alpha can't get greater than one. A window, by the way, makes 1, because the energy goes out the window and doesn't come back. And so-- we don't care for our model inside whether it got absorbed and transformed to heat or whether it disappeared.

OK, so remember when we did a tube, we had a little bit of absorption at each end, and the pulse went down? That was just preparation for this. We're going to do the same kind of thing. We have this for all the n. We know time. So we can relate it to a decaying exponential. So the way to do it is to have a function which only has meaning at these discrete ends, but then you make an envelope function of that, which gives you the shape of it, and is valid at each of the instants where you had reflections.

So let t equal nt prime. t prime, remember, is the mean time between the collisions at the walls. So in this case, I could write this as a function of time, saying that t of n-- instead of n reflections, I'll just say t of nt prime. At that instant of time, this is the energy that came down from that, or that existed on this third reflection, which would now occur at a time nt prime. Any questions? OK.

So nt prime, let's let that equal to t from this. D0 1 minus alpha bar. And n then becomes t over t prime. Now anything to a power-- what happened? Yeah. Anything to a power can be expressed as an exponential. We did that one way when we had the absorption at two ends of the tube. We'll do it another way here.

To get it as an exponential, of course, you know that the logarithm of any quantity x is the exponent to which you have to raise e, if it's a natural log, to get x. So you can express anything you want. In this case, 1 minus alpha is equal to ln e to the 1 minus alpha bar. So the logarithm of this is the quantity that you must raise e too to get this.

Well, that's clearly 1 minus alpha. So D of t is equal D0 1 minus alpha bar-- whoops, D0 ln e to the 1 minus alpha bar t over t prime equals D0 times ln e to the 1 minus alpha bar. And I'll put in for t prime this thing up here, 1 minus alpha bar, 4vcst. Oops, I have a logarithm I forgot. Wait, no. What did I do here? Hold it. ln, all of this stuff, is equal to ln of this. That's in there. OK. It seems all right.

OK. So now what we're getting, we can see right away. What have I done? I've--

SPEAKER 6: 1 minus alpha bar is equal to e to the log 1 minus alpha bar.

AMAR BOSE: One minus alpha bar is equal to--

SPEAKER 6: e.

AMAR BOSE: Yeah, yeah, yeah, yeah, yeah, yeah. Sorry. We'll start from here. Let's see. e to the ln 1 minus alpha bar. Yes, of course. OK. Yeah, sorry about that. OK. So e to the ln 1 minus alpha bar times t over t prime. And that's equal to e to the ln-- putting in now again just for the t prime-- 1 minus alpha bar, t prime is 4v over cs times t.

Now it's characteristic in acoustics to define one more parameter that's used very often, and it only simplifies this expression. And that definition is as follows, a prime definition is minus s times ln 1 minus alpha bar. It's related to absorption, and it's called, by the way, called metric-- assuming the metric measure-- metric absorption coefficient.

Now it has in it the average absorption of the room and the area of the room. And it's defined as a negative simply-- if you look at this, alpha is between 0 and 1, so the logarithm is negative, with the negative sign out here, a prime is a positive quantity. OK? So that is normally something that is just used in acoustics. We don't have to use it, but it simplifies these expressions. And so this becomes then D0 e to the-- let's see-- e to the minus a prime c over 4v times t. Yes. OK.

So now we have this continuous function D, energy, which is that. Now I'm going to say that D is proportional to p squared-- energy density we calculated before for the plane wave that was going around the room. So I can write the expression.

Now, be careful. Up until now, we have never had complex amplitudes as a function of time. And we said if you ever calculated one, then go back to go and start again. Exception, we're going to use it here. Reason is, there you have this sine wave which is the plane wave that's coming around in the room, and it's decaying at such a slow rate compared to it. So it's convenient to extend that notion of a complex amplitude, which would've characterized the amplitude, i.e. the magnitude [? and the ?] phase of a sine wave, and say, OK, it's just getting a little bit smaller in time. So that's the only time we do this.

p of t is just the square root of this business, of this. So that's p0 e to the minus a prime. Just put a factor of a half in here. Over 8vct. So now we have an envelope which is going down like this-- much slower, actually, but big time scale. This then, remember a is a positive quantity, so you have a decaying exponential of this. Now this, in essence, because, watch out now, there's going to be a little bit of manipulation in the getting from here to the answer in logarithms, which is the way it's normally displayed. But the meat of it is all contained right here.

You're seeing that pressure is decaying. And what you want to know is how long it takes for this pressure to go down to a certain level, and that how long-- that quantity of how long is going to be defined as the reverberation time. It's going to actually be defined as the time that it takes this sound pressure to drop to 1,000th of its value. But for the ear, 1,000th of its value isn't the end of the world. It's only 60 dB, right?

OK, so let's define the sound pressure level-- SPL, we'll call it. When you measure these sound pressure levels-- well, first of all, sound pressure level is defined as the logarithm of the sound pressure relative to some reference, your choice. Normal referenda, there are two that are sometimes used, but the most common one I believe, the reference sound pressure you use is the one that corresponds to the minimum, if you were in a totally quiet room, the minimum sound pressure that you could hear at the most sensitive frequency of the ear, which is about 3 kilohertz or so, 3 and 1/2.

So if you were in a totally quiet room and they raised the level until you could just hear it, that'd be about 2 times 10 to the minus 1/5 newtons per meter squared. That's common reference number, but whatever the reference number here isn't going to be important anyway for the calculation. So SPL is defined as 20 log to the base 10 of p of t over a p reference. And just let me mark so that's not room or something, p reference equals p.

OK, so now let's just put that expression in here, and we're on our way. SPL is equal to 20 log to the base 10 of p0 over pr-- p0 e to the minus a prime c over 8v times t. OK, logarithm of a product is the sum of the logarithms, so I can write this fellow as 20 log to the base 10 of p0 over p reference, plus 20 log to the base 10 of the rest of it, e to the minus a prime c, 8v times t.

Now this is SPL at time 0. This is SPL of t, it's a function of t as the right-hand side. And this is SPL of 0, at time 0. Because at time 0 you had-- where the heck did it go-- oh, at time 0 you had D0 going into the room, or p0, the same. So now we have SPL of t is SPL at time 0 plus this thing over here. And now we'll try to sort that out.

OK, let's write this SPL of t is SPL of 0 plus 20 log to the base 10 with an e inside. What a mess. If it was ln, we would be all set, but you know the logarithm to the base b of m is log to the base, some other base that you might like, c of m times the logarithm to the base b of c. So we can get out of this mess here.

We have log to the base 10. Let's make believe it was log to the base e, and just put the correction factor on the end of it. OK so plus 20 times ln-- maybe I'll do it in two steps. ln e to the minus a prime c over 8v times t, times the log to the base 10 of e. And this thing, I think, is something like 0.434.

So SPL time t is equal to SPL of 0 plus-- logarithm of e to the something is the something. 20, and that's the something, is a minus sign up there. So 20 a prime c over 8v times t times 0.434. Now SPL of times t, you can see-- well, let me go another step further. SPL at time zero is a bigger number than SPL at times t, because the whole thing is decaying, OK? So let's write SPL of 0 minus SPL of t equals SPL of 0-- what the heck is all this stuff here.

SPL of t equals this, yeah. SPL of 0 minus SPL-- wait a minute, wait a minute, wait a minute. Can't have two equal signs in the same expression here. SPL-- what the heck did I do? Yeah, this was a minus sign and that was an equals sign. OK. So now SPL of 0 minus SPL of t would be equal to 20 times a prime c over 8v times t times 0.434. OK. See what I would rather get rid of up here. Yeah.

Now we put in the key number, which we defined as a very particular t, equals the time for the SPL to drop by 60 dB, or the sound pressure to drop to 1,000th of its value. Reverberation time, by definition, time for SPL to drop by 60 dB. OK. So that says SPL of 0 minus SPL cap t must be 60 dB. And I plug-in the cap t here. So this side must be 60 dB, with cap t plugged in here. This is 60 and cap t is in here, I can solve for cap t.

So 60 equals 20 a prime c over 8v times cap t times 0.434. Or t equals-- and let's see, I know what this answer is, it would just be nice to know I got something similar to it. Yeah, OK. Let me write the thing out first. Is 60 divided by 20 a prime c times 8v divided by 0.434. OK. If I have not made any mistakes, this should come out to about-- somebody'll have to check it out. 55v, roughly, 55v over a prime c, that's the reverberation time. See if that-- is that OK?

OK, now this number is probably the most important number as used by acoustical architects. It varies in public halls from short times like Kresge, maybe 1.8 seconds, to longer ones in other concert halls-- 2.2 is a long one. If you went over to the Briggs-- I'm not sure the current one, but the men's shower room over there, it's about 10 seconds. And you can't hear anything.

If you've ever been in there and you're trying to talk to somebody else, you're screaming, which is making it worse anyway. And the whole thing is so reverberant, everybody shouts at everybody and there's just one huge amount of noise. That's what happens when the reverberation gets to long. If it gets too short, this is what they call dry sound. Dry meaning that it's like outside. I don't know why the word dry came, but when they say a room is too dry, it's usually this thing is too short, and it's usually that it will sound more or less like outside.

So now you look at this, and you see, first of all, what happens when you increase the absorption? Has that been here all this time? What happens when you increase the absorption that's actually in the room? Let's see if I have something-- yes, here it is. If I increase the absorption, this quantity gets magnitude wise bigger. This gets bigger, this gets bigger, and therefore t goes down. So you increase the absorption, it goes down.

You take the same building with the same materials and enlarge it, just decide you're going to enlarge it, guess what happens? Reverberation time goes up. Not unexpected, because it takes a longer time to reach each wall. And so there's a longer time. If you doubled the time it took to reach each wall, you'd expect to go down at half the slope.

So all that I told you, would tend to say, if you went in with a sound pressure level to a room-- this room or any other room-- and you put a microphone there, put a source here, maybe a loudspeaker, and shut it off all of a sudden and watched the sound level meter that the microphone was hooked to, that you would see it going down like this. And the time that it took to go down by 60 dB would be the reverberation time. And this height here would be 60 dB.

Well, turns out if you go into the room and you put a sine wave into the room, some frequency, that's the kind of thing you see. If you put random noise into the room, just hook the thing through a noise generator, you don't see that. You see something like this. Any idea why? There are discrete-- in other words, you see a line whose slope changes in time to a different slope, and it goes straight again.

SPEAKER 7: [INAUDIBLE PHRASE]

AMAR BOSE: Reflection?

SPEAKER 7: Reflection from the wall?

AMAR BOSE: Yeah, well reflection from the wall is all in the average absorption coefficient, which caused the thing to go down. Let me give you-- I will repeat a clue. I just told you, I hooked it up to a noise generator. I didn't hook it up to a sine wave. Yeah?

SPEAKER 8: The [UNINTELLIGIBLE PHRASE]

AMAR BOSE: Right. In other words, one of the frequencies-- the sine wave of one frequency would decay like this. The sine wave of another frequency, if I raised it to the same level before I shut the source off, would go down like this. Another frequency would go down like that. So what happens is, when you put them all together, the one that falls fastest gives you this kind of a thing, until it falls down below the other modes, assuming it had enough amplitude to show up. It falls down, and then you get to the frequency which was decaying slower and slower.

And so if you see something with two abrupt slope changes when you're looking at the SPL, watch out. Because it says you have a reverberation time that is pretty drastically varying with different frequencies in the room. And we're going to see, as we go on, that can be related to how the room absorbs some frequencies much more than it absorbs others, and can be a very unbalanced kind of situation. Yeah?

SPEAKER 1: Why would it be discrete? Because it seems like if you're putting in white noise, and you have absorption coefficients that are functions of frequency-- presumably some continuous function of frequency-- then I don't see how it could ever be discrete. Like, it should still be--

AMAR BOSE: Well, these all belong to normal modes of the room, which aren't the modes-- which are the modes that exist after you shut the source off. Remember that? You put all frequencies in, but after you shut it off, only certain ones exist. OK. Yes?

SPEAKER 9: [UNINTELLIGIBLE] that question with the normal mode, aren't there so many normal modes, though, that it would be continuous?

AMAR BOSE: No, it actually turns out, wherever your microphone is, it's going to pick out some of the normal modes, it's going to be at the null of other ones, and you get enough separation. And it turns out, also, that these things are not-- normally, what you'll do in a concert hall-- by the way, there's a book if you're interested in what happens in concert halls in reverberation time, there's a a book by Beranek called Musical Acoustics, I believe, in which he's gone around, or he's gathered the data from major concert halls all over the world, and they measure reverberation time at like 100 hertz, 200, 500, 1,000, et cetera.

There's another thing, by the way, that I really want you to do. In chapter 10, look at one page in there in which they have these absorption coefficients alpha for the different materials, and you'll see how they vary with frequency. In general, they go up with frequency. And they vary from things like this surface here to, say, 1% reflection at, let's say, 100 hertz. Even at 10 kilohertz, this thing is probably only a couple of percent per reflection. To things like 40, 50% when you get velour curtains, if you have them high density enough. But looking over that gives you a good feel for what kind of absorption you're going to get from different materials in a room. So that if you go to fix up your living room, for example, or a bigger place than that, you'll know at least what to select. Yes?

SPEAKER 9: Does the absorption coefficient include other factors like how far away you hung the curtain from the wall?

AMAR BOSE: No. The absorption is of the curtain itself, and it doesn't include that. And so that's a factor you have to store as knowledge. In other words, some of the things will be-- somewhere they're listed, like if you plaster it on here as opposed to if you have it here, but not in general, I think. Yeah.

OK, so let me just see if there's anything-- is there anything I should cover that I didn't on this? My voice is about to go down 60 dB, so-- OK. I think that's all we need to know at this point about reverberation time. Good.