Amar Bose: 6.312 Lecture 21
AMAR BOSE: OK, as far as room modes are concerned, we started with a one-dimensional tube. We went to two dimensions last time. Just to quickly summarize some of these results, we found that if we launched a wave like this and we drew just like we did in the case of the one-dimensional, we-- well, first of all we found out that no matter if these are right angles in the room, no matter which way we launch a wave, a wave eventually through two reflections, will come back on it, which is the criterion for a standing wave. Then the only thing you have to do is make sure that the standing wave fits, namely, that fits within the boundaries such that there's 0 normal velocity at all surfaces.
And so we looked at this criterion, then we drew the nulls of the standing wave, and we defined-- this distance here is lambda over 2 between two nulls in the standing wave, just like in a single tube. And this distance, if the angle here is theta, this distance is lambda over 2 cosine theta. Similarly, this distance is lambda over 2 cosine phi, where phi is the complement here, the angle from here to here.
This is what we did last time. And we pointed out that-- or we developed last time that the reflected wave angle of incidence is equal to angle of reflection, and therefore, if I draw the lines for the nulls of the standing wave in this direction, they will fall on top of these. They will be equally spaced. And if there is to be 0 velocity at all the corners, there will be an equal number of those in the same length lx. The room length here was lx. And up here, lw.
Then we defined something that was very artificial. And we're going to make use of this artificial quantity, not to be confused with any kind of a real wavelength at all. But this distance here, which is lambda over 2 cosine theta, we defined lambda x equals lambda over cosine theta. So this is lambda x over 2. This is the real lambda. Similarly, we do it for the y-axis, lambda y-- it's just a definition, that's all-- is lambda over cosine phi. Only definitions at this stage.
Now we will also introduce, purely definition, we'll only see the definition. We'll see the usefulness of this in a few minutes. The following relationship, lambda x fx equals c. That's the definition of fx in terms of lambda x. lambda y fy equals c. We do that, and we substitute for lambda x here. We would have here lambda over cosine theta. So we would get-- let's see, lambda x I substitute for, I get lambda over cosine theta.
Well, let's do it. lambda over cosine theta times fx is equal to c, or-- pardon me-- lambda over c is f. Because, of course, for real frequencies and real wavelengths, lambda f is equal to c, right? So this would tell us that fx equals f cosine theta. Why? Because I took cosine theta over the other side, and I had left on this side lambda over c, but lambda over c is f.
Or is it? Did I get it upside down? Let's see. No, that's right. OK, and similarly, we have fy-- just by the exactly the same kind of a step-- equals f cosine phi. All sorts of interesting relationships just coming out of this definition and this for each axis. Now the criterion last time that we mentioned was that an integral number of these distances, lambda over 2 cosine theta, or lambda x, an integral number of the must lie in this dimension of the room, an integral number of these must lie in the top dimension, so that we get 0 velocity everywhere. Yes?
SPEAKER 1: Lambda over 2 cosine theta or lambda over cosine theta?
AMAR BOSE: Lambda x. Where is this. Lambda x is this, lambda over cosine theta.
SPEAKER 1: Down there, you said--
AMAR BOSE: Oh, oh, oh, thank you very much. I did that yesterday, too-- Tuesday, or Thursday. OK. So we had the criterion-- oh, I can put it here-- n times lambda x over 2 is lx. An integral number of lambda x's over 2, which is this distance here, fits in lx. Now another integral number of lambda y over 2 must be ly by the same argument. Now since I have an n here, and that could be independent of the n here, I'm just going to put in an nx here and an ny here just to relate that this integer n belongs to the x-dimension and this one belongs to the y-dimension.
OK. Now we have enough just from that definition to get a very interesting interpretation of what goes on in the room. Namely, let's square this equation, both sides, let's square this equation and add, see what happens. f squared times cosine squared theta plus cosine-- I've just gathered the terms-- cosine squared phi-- but that's this squared plus this, so that must be equal to fx squared plus fy squared.
Now this, of course, is equal to unity. So we have f squared is equal to fx squared plus fy squared. Or f is equal to square root of fx squared plus fy squared. And we have fx f cosine theta, fy is f times the cosine theta of that direction. That's enough to say we can think of f as a vector. We can think of f as a vector with components along the x-axis, f cosine phi, the direction cosine, f time this direction cosine.
In other words, we could think of f-- this is an interesting space here-- if you think of f this way and fx out here-- well, just put it this way-- fx, fy, and this is theta, and this is phi, of course. So what enables us to do this is this equation and the realization that these are just components along those two axes.
Now the very interesting thing about this is that if I pick an n-- and I'll just write one more equation to make this clear, but I'm going to state at the beginning, then after I write the equation, I'll say it again. Turns out, if I pick an nx, any integer, from the room, then I determine lambda x. And of course, from that relation on the top, I could determine fx. So I pick any integer for this, and I know fx, once I know the room. Pick any integer for this, I know fy. From here, I know f, which is exactly the length of this vector. And interestingly enough, I know the angle, because I know fx and I know f, and so I know the angle.
So what it says is picking any-- I'll make it a little clearer by just getting rid of the lambda x in here in a minute-- but what it's really saying is if you pick any integers for nx and ny, that will tell you what the allowable normal mode in the room is. Both its magnitude and the direction you would have to launch it into the room such that it would satisfy all the boundary conditions. So it really is a normal mode, something that will run forever with no losses in the room.
Let me just put the extra expression here. Let's substitute for lambda x here c over fx first. Take this equation, nx c over fx over 2 is equal to lx. Or fx-- this is a times 2-- fx. So solving for fx, I get nx c over 2lx. Let's see if that's right, now. fx, yeah. And then similarly, fy is equal to ny just by getting rid of the lambda up here, ny c over 2ly.
So pick an nx, pick an ny, the two integers as you choose, that gives you fx, that gives you fy, and now, you go over to this vector picture, and that tells you completely-- or you can go back to these equations-- but that tells you completely what frequency is allowable. This is the only frequency in the room. These are our components that we think of, but they have nothing to do with the frequency that's propagating in this room. There is only one frequency, and that is f. But pick an n1 and an n2, and you have the length of f, which is the actual frequency. And then you have the angle at which you have to launch it. Any questions?
At first when you look at this, even though the equations that get you to this point are extremely simple, but you have to play-- if you're like most people, you have to play with them a little bit to really realize that you haven't lifted yourself up by your bootstraps or something. You just have to go over it a few times, but it starts very simply by defining a lambda x over 2 as the projection of these nulls in the standing wave, saying you have to have an integral number of them in here.
And then calling that thing a lambda x, and then maybe one more definition that lambda x fx, which is another fictitious thing, the f is equal to c. All of that, that's all, brings you right down to here, which says if you know fx and fy, which you do if you picked the integers, you know this, and then you know because of the direction cosines, you know the rest of it.
OK. Now let's look at this space. If we had a room-- well, I'm looking at the vector f now, and the space with the fx projections on this axis, and fy projections on this axis. Every time I choose a different integer, I get a different fx, which in turn, if I had a different fx, if I came out to here, let's say-- suppose this particular one was for fx equal 1, then if I chose fx equal 2, I'd be out here on this axis, and fy equal 1, I'd be having a frequency of a larger magnitude than this one, and a smaller angle.
So I can look at this in the following way, I can break this axis up into equal increments here, and call the space between these increments just this quantity here, because I up that by 1 each time I raise the index or the integer. So c over 2lx is the spacing between here, between here, between here. Similarly, I could-- it won't be the same, because the room ly isn't the same as lx, but the spacing between these will be c over 2ly.
And now if I do this across here, every intersection here, every intersection is an allowable frequency. In other words, this intersection is this frequency. And this frequency we would normally call-- let's see, this is the x-axis and that's the y-axis, so this one along here had two units. So nx was 2-- oh, it happened to be 2-- and ny was 1. So this would be nx is 2, so that's f1-- I'm sorry, 2, 1. In other words, it would normally be fnx, fny, like that.
Just to give a label to this particular normal mode, so each one has a label by its point of location, how many units along the x-axis, how many units along the y-axis. And these are called the normal modes of the room in the very same sense that for parallel lc, the normal mode is the frequency at which things can oscillate back and forth between the l and c. Or in a tube, you have a bunch of normal modes, an infinite number, all of those that fit into the tube, closed tube at both ends, with 0 velocity at both ends.
Now it's interesting, if there is only one dimension to the tube, then you have these as possible frequencies, this one, this one, this one, this one. That's exactly what we saw, because in the one-dimensional case, we found that the allowable frequencies where in fact c divided by 2l, the length of the tube. So similarly in this two-dimensional room, c over 2ly would be the frequencies that you could have for every c over 2ly going this way. That would be just like what you saw in a single tube with nulls.
Now it's immediately clear if you look at this that there's a heck of a lot more modes. Forget about the third dimension in this room. If this didn't have any risers for seats, I could have waves going back and forth here, I could have waves going this way. If I looked at all those different allowable frequencies below some maximum frequency, say 1 kilohertz or 10 kilohertz, and then I looked at all the ones along here below 10 kilohertz, that would be all the ones such that the distance from here to here is 10 kilohertz long, the vector. The sum of those is a lot less than all the points out here in space.
So there are a lot more modes that you get, that you launch in that room up here that are going in different directions than there are ones that go back and forth and back and forth. Now when you clapped before up there, when you heard that "brrr" [rolling r's], that represented just one of the modes that was slapping back and forth. Not one of the modes, but all of them together, it was an impulse. But that represented, if you wish, the sum of the modes along here. And you can't do it along here, this direction, because there's fiberglass up there.
Now if you wanted to find, for example, approximately how many modes would be-- well, let's go back to, just to tie it down again, let's go back to here. If you wanted to find out how many allowable modes there were below 10 kilohertz, you could count them all up. They would be exactly-- this is one, because this vector's much less than 10 kilohertz, and this is the second one, this is the third one, this is a fourth one, et cetera. You go out here to a vector length, a frequency length of 10 kilohertz, and you count all the points on the axis here.
Similarly, if you wanted to find the mode that could propagate strictly up and down, you'd go up here to 10 kilohertz, and all the points that you had here, which would be 10 kilohertz divided by that distance, the number of them, that would be the number of modes that you could have in that direction. Now if you wanted to find out the number of modes that you could have in this room below 10 kilohertz, what you would do is you would come along way out here-- I mean, this is too big of a diagram now. You'd come along and you'd make a radius that went along here at the desired frequency.
Remember, the magnitude of the vector is the frequency. And when you did that, of course, some of them would appear-- you'd have some sticking out of the thing. You wouldn't get an exact answer, because some of the modes would be contained inside, in other words, this wouldn't be an allowable mode, this one would. So you'd have some error going along the border when you just draw a nice circle there and the frequencies are defined in terms of rectangular matrix.
But when you're out there with the 1,000 of these things in here, there's a real easy way to do it. And you just think of what would happen if you took this grid that I've drawn here and shifted it over here 1/2 a cell, a cell being this length, shifted the whole grid that way, and shifted it, let's say, this way 1/2 a cell. Then all of these frequency points would lie in the middle of the cells.
And so an easy way to count the approximate number in a large circle is how many cells are in the large circle. And the way you'd do that is you take the area of the large circle and you divide by the area of a cell. Each cell has the same area. This one, this one, this one, this one. So the area of a cell is simply this quantity times this. So you take this times this, divided into the area of this circle here. The area of the circle is what? pi r squared over 4. pi f squared over 4.
Divided by the area of one of these would tell you approximately how many allowable frequencies, i.e., normal modes could exist in the whole room, considering all angles, below a frequency equal to this circle, plus and minus those ones that fall in and out of the circle, but that would be negligible as the circle gets large. Well, we're going to do that for the three dimension because that's where the result is clear.
But if there are any questions, we can figure out where we are, because I'm just going to take a jump and tell you what happens. It's pretty simple what happens when we we go to three dimensions. But this is pretty important up to here. From one dimension to two, if we understand that, three is easy. Again, the equations are fittingly simple, but you have to think about introduction of these [? quadrants. ?] OK?
All right, now what happens, you can already, I think, imagine, when you go to three dimensions? Then this way in the room, depending on how you launched the plane wave, you have the same kind of thing like this in the vertical direction. You have to have an integral number of the half wavelength projections along the vertical axis. Just one extension to what you already had in the two dimensions. So that because you have to have those integral number of those projections so that you'll get no normal velocity up there in the corner, also-- in all corners of a rectangular room. All of this stuff is only valid for a totally rectangular room.
So when you do that, you just come down with exactly the same kind of thing with the direction cosines. You just have one more of them. And this becomes an fx squared, fy squared, fz squared. So for three dimensions, f squared is summation of-- square root of fx squared plus fy squared plus fz squared. And exactly what you had before, you can write this as-- [UNINTELLIGIBLE].
Let's see, I hope I wrote it somewhere before. I didn't. I think I'll write it this way. If I put in for fx and fy here, f, and here I put in the expressions that I have derived right down here. I can take this c/2 out of the square root. And I get nx over lx, quantity squared plus ny over ly, quantity squared. Reason I didn't use this-- we did it all in pictures, and so here are the equations, of course. Just substituting fx in terms of nx and lx. This becomes this.
Again, this tells you, pick any integer, bang, you have f right away. You have f from that. If you pick any integer, you also had fx. So you have f and fx, you have cosine theta, which is all you need. So this, then, could similarly be written c over 2 nx over lx squared plus ny over ly squared plus nz over lz squared.
Pick your integers, you already know your room that you're dealing with, so you know lx, ly, lz, you get the allowable modes. And you would normally call the modes an f, nx ny nz. In other words, if this was 1, 2, and 3, it would be the f123 mode. That's just a way of designating it. How many of you have had microwaves? Just a couple. Remember the modes, they can exist in a container? A copper container, for example. It's the same thing. It's exactly the same thing. You get all the different modes-- not quite, but you get these kind of modes just by doing the same kind of analysis.
OK. now in three dimensions, it's a little harder to draw this thing, but let's try. Now I'll take, this'll be the x-axis, this'll be the y-axis-- this is the vector f we're now drawing-- xyz. So this distance here is-- how did we do it-- c over 2lz. c, lz. And along here we'll-- this distance here is c over 2ly. And finally, distance here is c over 2lx-- l sub x.
And now when you have picked an integer n, that means nx, you go so many units along here, ny you go so many units along here, nz you go so many units along here. And you think of this thing as three planes, one here, one here, and one coming out there. So you're somewhere out here for an allowable fx. The frequency that's allowable is the length of the vector to this point, and the angle at which the wave has to be launched to satisfy all the boundary conditions is exactly the angle of this vector, just as it was over there, the direction cosines to each axis having established that.
So let's see. If you had, let's say-- oh, let's just take one mode here. I'll take one that's here. I'll go up a distance this much here. Let's see-- hang on a second. Whoops. So I went 1 unit up on the z-axis, 1, 2, 3 units over on the y-axis, 2 units here, and that established this point. And so the magnitude of the frequency f would be given by this length, and the angle would be exactly given by the angle of that vector. And f would be-- let's see, x is 2, y is 3, and z is 1. f231 mode.
Now let's look at the volume of a cell here. The volume of a cell, if I took 1 unit up-- let's see, this is one cell. I could take one anywhere. And the volume of the cell is c over 2lx times c over 2ly times c over 2lz. Now that is what? c cubed over 8 times the volume of the room. So the volume of a cell is constant over the volume of the room.
Now if I want to find out how many normal modes are allowable in a given room below a certain frequency-- there's obviously, if you don't put a limit on frequency, there's an infinite number of normal modes just in the single tube of this length. Because all you have to have is n lambda over 2 is equal to the length of the tube, so you can go as high as you want for n, and the lambda just keeps getting smaller. So there's an infinite number in any direction.
But if you now want to say how many normal modes are there in this room below a frequency of, let's say, 10 kilohertz or something, then you go with a radius out here-- this is a corner, OK? So this is 1/8 of a sphere that you would run this radius over. You'd take a string here and attach it, which was 10 kilohertz long by our measure, and move that over this space, and that would define an 1/8 of a sphere actually over here.
And all you have to do then is find out how many of these cells belong-- I can fit, never mind what happens to the very edge, it's way out there. And it's very, very-- it's a huge distance compared to, there's so many cells you don't care if you screw up one, but you won't, even on the average on the border. Some will be partly in, some will be partly out. So how many of these cells are in 1/8 of a sphere of radius f, where f is the frequency that you want to find out how many normal modes there are below that frequency in the room? So let's see.
1/8 of a sphere. 1/8 of a sphere of radius f has volume what? 1/8 4/3 pi times the radius. But the radius now is the radius of the thing-- the radius is the maximum frequency-- so f cubed. 1/8 of a sphere-- that's the 1/8-- 4/3 pi r cubed is the volume of a sphere. So that is the volume of the sphere. All I have to do is divide that by the volume of the cell, find out how many cells are there, and we're in. We have the number of normal modes, approximately, that can exist.
This breaks down when you get down to very small number. If you were going to take a small radius, then the cell is big compared to that. But then, if you're going to do that, you can count. You don't need any-- so volume of a sphere divided by volume of a cell is equal to 1/8 pi-- 1/8 times 4/3 times pi f cubed-- we got the volume of a cell, did we, yeah-- c cubed times 8, that's upstairs, times volume of the room, and that's all.
So let's take a look at that. 8 go away, so that looks like n, which is the normal modes below f. By this symbol, n sub f means the allowable normal modes or frequencies that can exist in this room, satisfying all the boundary conditions. We can set up standing waves in this room below a frequency f-- that's what I mean by n sub f. So n sub f then equals 4/3 pi f cubed over c cubed times volume of the room.
Now let's just take a look at, roughly, a number, so you can get calibrated to a living room. Let's say you have a room lx, the length, let's say, is 25 feet. Living room, just say ly is oh, let's say 15 feet, and lz is ceiling height or something-- 8 feet. OK, so volume of the room is equal to just the product of this-- 25 times 15 times 8. So let's see what happens here. And let's ask for f is equal to just 10 kilohertz, OK? Not even the full audio band. Just 10 kilohertz. 10 to the fourth hertz.
So we want nf, which is the number of modes below frequency f, which is 10 kilohertz, in this living. So nf then is equal to 4/3 pi f cubed, which is 10 to the fourth cubed, over c cubed-- let's take c, we only need approximate answers here, c's about 1,100 and something feet per second, so we'll take it as 1,000. This is 1,000, so cubed. Let's take care of c, times the volume of the room, times 25 times 15 times 8.
OK, let's see what we have. pi and 3-- near the same. 4 times-- only I can do that-- 4 times that is 100. 4 10 to the fourth times 3 10 to the cubed, 10 to the 12th, 10 to the ninth, so that's 10 to the third up here, times 100, which is from these two, is 10 to the fifth times 15 times 8, 40, 8 [UNINTELLIGIBLE PHRASE] 120. So that is 1.2 times 10 to the seventh, or 12 times 10 to the sixth modes.
So in your living room, below 10 kilohertz only, there are about 12 million normal modes. And remember now, these are the points, these are the allowable frequencies, and-- well, never mind that-- all these, every time you go through something like this it's a null and there's a peak in between. So in terms of peaks and dips, you've got at least-- peaks or dips-- at least 12 million of those in your room.
Now for those of you are audiophiles and have just gone out and bought your amplifier flat from DC-to-Light and paid a lot for it, you've got a problem. These normal modes-- even with losses, I mean, the way we've drawn it where those 0's are up there at the intersection of the troughs, they are 0's, so there's nothing, no response. But in reality, what happens, just like the standing wave in its single tube with a little bit of loss on each end, instead of coming down the envelope, instead of coming down to 0, just came down like that.
Well, in a normal average living room, the difference between the peak and the dip is at least 20 dB. So you've now got 12 million modes just for up to 10 kilohertz. And you've bought your pre-amp flat, or your amplifier, you've paid a lot of money, what to do? Now the audiophile tells you, well, it's easy, you go buy an equalizer, a graphic equalizer, they push out the different parts of the spectrum and push them down.
Now you could decide to buy one. Now they're very expensive if they have 12 million knobs on them, and it takes an enormous amount of time to adjust them all, even if you know what you're doing. And then when you've adjusted them all, you have to buy a dentist's chair, because it only works when you're right here. If you move over here, all those modes have changed as you will shortly see and experience.
So what actually happened is Bell Telephone-- I think I told you this, but it was earlier and you didn't see the picture like you can see now-- Bell Telephone developed that graphic equalizer, which is something that just gives you the opportunity to take a frequency response, and at each frequency where you have a knob associated with it, you can alter the response up or down at that frequency, make a bunch of peaks and dips in there. If you have 10 knobs, you can do it with 10 frequencies.
They did that for a very valid reason. And the reason was, when you're speaking and you have loudspeakers that are behind you, perhaps, the loudspeaker comes back, and through the resonances of the room, there's certain frequencies where that loudspeaker will peak right at your microphone and give you feedback, and cause the big squeal you've heard when people get up to talk sometime. And the only thing that they can do to cut the squeal is cut the volume. Well, if you cut the volume and it's a huge ballroom or something, you don't get the coverage. So the name of the game, and this is very important in public address, is intelligibility, not quality.
So Bell got the idea that, all right, fine, we'll turn the gain up until it squeals. This is a set up that's still used. People come setting up audio systems for PA like this, they turn it up until it squeals, find what frequency it's squealing, and then cut that frequency down a little bit. And then they can turn the gain up a little bit. And then it squeals at some other frequency. Then they put a dip there and turn it up a little bit more. Well, all of this together can get you maybe 6 or so dB, maybe a little bit more, of additional gain. And that's not trivial, so it gives you more coverage.
Now of course, the frequency response that you have in your system has all these holes in it, in your nice flat system. But you can do a lot to speech and still make it intelligible. In fact, if you take a speech waveform, something like this, and you decide to put it through an infinite clipper and you get out this. You get out just to a two level waveform and listen to the two level waveform, it's intelligible-- not pleasant, but it's intelligible. And that's the name of the game in public address.
Now, this thing doesn't do that, and it's much better, and so-- but not understanding the reasons for this, the whole hi-fi industry jumped on equalizers and said, oh boy, we can use those things now to compensate for the normal modes of your room, all the resonances in the room. And they have not the slightest idea how that works, that whether you perceive a normal mode depends on where you are in the room, and that there are a heck of a lot of them. So you can buy these things for cars, you can buy them for everything, and the only excuse for them is people like to play with buttons.
Now why don't we set something up here so we can actually hear what goes on in the room. Meantime, if there are any questions? Yes?
SPEAKER 2: In an equalizing system with speech where you're cutting out [? the resonant frequencies ?], if the microphone were moved or something was changed, would all of that get out of whack?
AMAR BOSE: Yeah. All gone. Because if you take-- I'm going to show you this in a minute-- but if you take a microphone here, and you move it here, the frequency response from any given source is totally different in the room. And this might-- by the way, this gets us a little bit into the psychoacoustics-- but this is pretty devastating when you think, as I said, that you bought all your equipment flat, supposedly, that what happens now all these peaks and dips exist in the room that you're going to hear in a minute, and how come you can tolerate different seats in the symphony hall?
How come you can move to the next seat adjacent where there's a totally different frequency response and still enjoy it? In fact, not even realize it's changed? If you went towards the corners or towards the walls, you'll know it changes, but you might think about this a little bit. Especially when you hear the experiment, and then we'll talk about why it really is. Oh I have to remember something else to tell you. That'll remind me, yeah. Does this work?
SPEAKER 3: [INAUDIBLE PHRASE]
AMAR BOSE: What's this? OK, can you get a trace on that thing? Hello, hello, hello, hello. You'd better give me this, pull it out here. OK. Yeah, maybe the overall intensity if you cut it down and up the contrast they might be able to see it.
SPEAKER 3: [UNINTELLIGIBLE PHRASE]
AMAR BOSE: OK.
SPEAKER 3: [UNINTELLIGIBLE PHRASE]
AMAR BOSE: Well, let's hope it [? grew. ?] Is it working? Yeah, OK. OK, now set the sweet time so you can see the sine wave that you're talking about.
SPEAKER 3: [UNINTELLIGIBLE PHRASE].
AMAR BOSE: Yeah, that's all right. Let me-- if you have a good-- do you have a trigger, a useful trigger on this thing? Just trigger it with the-- oh, there is a very low frequency. I think it's coming out of the air conditioning, that's the problem. It's superimposed on an extremely low frequency. Yeah, oh boy. Yeah, I don't know if that can be turned off or not. Let me just check. Can that air conditioner be-- no? OK.
Too bad. All right, but I think you can still see what goes on. Now what you want to do is just get the other trace out of the way, and we'll put this in the center so we can see it juggling. But what we want to look at is the amplitude of the sine wave. And now bring this trace down to the center, so all the jumping around will-- yeah, I can hear this thing. It's terrible.
OK, now just sweep through the frequencies, and what you're going to notice is that the height of the sine wave goes all over the map. Not the position, that's the air conditioner, but the amplitude of the sine wave. Now it's almost 0. Goes through 0 and comes big. Stop it when it goes to a 0 next, pretty near a 0. It's hard. Try.
Now you should be out there hearing things going up and down. This isn't exactly a high hi-fi speaker here, but I guarantee you that what you're hearing, the up and down part, is not due to the loudspeaker.
Now actually, if you were able to go slow enough, Carlos, what you hear, what most of you are hearing, when you see it large, it is not necessarily loud for you. Your peaks and dips are at a different place. Because just as if you were sitting in a tube here, the microphone-- let's suppose you're at a normal mode-- you're sitting here and the microphone's sitting here. It'll be high.
At some other frequency, when the standing wave pattern is squeezed up here, this might be high and this might be low. So the correlation-- turn the volume up a little bit if you can stand it. Gee, there's still-- vacuum tube--those things, by the way, with glass on them, we call vacuum tubes. Prehistoric-- yeah, if it doesn't produce smoke, it's OK.
SPEAKER 4: [UNINTELLIGIBLE PHRASE]
AMAR BOSE: OK. Yeah. That'll overcome the-- yeah, you'll have to hang on to your ears. There's a 0 in between that. But you have to be-- that's pretty good. What's another problem is there's a foot and a half of fiber glass behind these two walls, and so the damping is not very good. And so that's why the standing wave ratio here is not so high. If you did this in a room without the damping, woo.
OK, so now go up in frequency and you'll watch that the modes become much more dense in frequency. This is going to hurt your ears a little bit, watch out. Now it's going up and down like anything. Yeah, keep going. Where are you in frequency? 8 kilohertz. OK.
Now let me try-- turn it down here. I'm going to try something, but I don't expect this to work because there's not a corner in the room. There's all fiberglass. But pick a low frequency, couple hundred hertz or something, maybe. Now first of all, I'm going to walk along. I'm going to keep away from that air conditioner. You should see this thing change as I go along. Does it?
Now if this were a real corner, what would happen is remember at the corner, all the velocities are 0, therefore all the pressures are high? First of all, I can't get within a foot and a half of the corner, and half of it's absorbed. But did it go up? It did? That's luck. But if you go into the corner, you will hear all the normal modes, because every one of them peaks there.
This is why if you take a loud speaker that is a velocity source, for example, which most of them are as far as-- I mean, electromagnetic ones are as far as the pressure in rooms is concerned. And you put it in a corner where the velocity is 0, then it's like a single tube. If you put it where the velocity envelope is very small, and you put this much of a signal, then you get a huge signal out when you're further away from that.
Because your source chooses the velocity, the shape of the standing wave is determined by the rest of the tube, or in this case, the room. And so if you take a normal loudspeaker and you place it in the corner, it will sound very, very, bassy and whatnot, because all of those modes peak there. That's bigger. Yeah, this is 200 hertz, so I have to move a bit to get it.
Now try a high frequency. Try like a 1 kilohertz. Now 1 kilohertz wavelength is a foot approximately. Put it-- yeah, roughly. It's hard on your ears, but. Just a small motion should cause a huge motion. Here it's almost 0. By the way, you don't have a high patch filter, do you?
SPEAKER 3: I think there is.
AMAR BOSE: There is? Because you may be able to filter out the air conditioner. Here, look. Node here. Oh, god, it's hard as heck to get it. Peak here. Now I want to ask you to do just one thing. That's about a kilohertz?
SPEAKER 3: Yeah.
AMAR BOSE: A kilohertz, so quarter wavelength from peak to dip. Close one ear. Now, unfortunately, all of you are going to foul it up a little bit for your neighbors, because you're moving objects in the room, but I think it will still work. Close one ear, so that you now are measuring with basically one ear, and move slowly. And you should go between a peak and a dip. I hope nobody's looking in now. They'll think it's a prayer meeting.
Could you hear it? Let's see. I think I don't need it. Yeah? Oh, you have that little patch filter? Yeah, oh, OK. What we did is just filter out the air conditioner, a lot of the air conditioner frequencies. So now-- OK, thanks.
Now what you think about is within the distances that you moved or I moved, you can move in any concert hall, and you won't notice anything. You won't notice a difference in sound. Now if you, by the way, if you happen to open two ears instead of just one, then of course for the higher frequencies you have a probability that it might be a peak here and a null over here, which would be helpful to you, but try-- I've done this many times, try in a concert hall to do the same thing. Just plug one ear and then move your head around if you don't mind what people think of you, and music is still sounding like music.
Now think about the next thing. This is only magnitude of transfer function that you were seeing up here. Every time you go through a resonance like this, it's 180 degrees phase shift. And now the audiophiles think that phase is very important. I wonder how they listen in a room, because the phase of all these modes coming to you, reflections coming to you, from all over the place, all with different phase. You move that cart away, the phase has just changed for most of the frequencies for me to you.
And now, another one of the myths is-- how many of you have ever heard of, they're still on the market, time-aligned loudspeakers. Anybody heard that? Fortunately, only a few. What it means is the people market these things-- it was very big, I don't know, comes and goes, but within the last decade or so it was very big-- have a big box here with tweeters and woofers, and the woofer being a bigger speaker, a lot of the sound generating area is back a little bit. So a time-aligned speaker, if they have three of them, are very carefully designed to have the tweeter back further than the mid-range.
So the idea is that all three of these things will come to you time aligned, so the phase will be right. Now even if cones didn't break up into all these modes and everything else, just think about this. By the way, the most amazing thing is engineers are the biggest suckers for this kind of an argument. They really go for it, and it sounds good to them, and that's it. Because you've learned the condition for fidelity transmission is flat magnitude and uniform phase.
Then the output of a box which has flat magnitude and uniform phase is of course just input delayed by the slope of the phase. So that sounds like motherhood and apple pie. Everybody who's an engineer believes it. Well, if this were important to you, if you didn't now know anything about all these normal modes which are fouling up everything anyway, the engineer doesn't think of the following thing.
OK, suppose the average distance from all of these things is the same, and they all come to you time aligned. There's the loudspeaker down there. Now you better sit on axis with the loudspeaker, because when you stand up, of course, all these distances are going to be different. And so next time somebody's selling you one of these things, just go down like this and go up, and if the world changes maybe he has a point about time alignment. If it doesn't change, this is all nonsense, and it won't.
So this is the kind of thing that fills magazines of hi-fi. In fact, I'm sometimes tempted to give just one lecture on hi-fi folklore. It is amazing. The English and the Japanese journals are the worst, Americans are third. Basically everything from monster cable that you see, which if you had approximately a mile between the receiver and the speakers, which is pretty rare, it would make a difference, transmission line considerations. Because think of what the audio wavelength is. Just think of what the electrical wavelength is for electrical transmission line at 10 kilohertz, and find out how long you have to be to make a difference.
So there are papers that say that digital sound-- English journals are full of papers saying the digital sound is bad for the brain. And of course, if you drive this analog device with a digital signal, no digital stuff is coming out anyway, all right? So there's an amazing amount of this all over the place.
OK, let's see. Well, if you can ask any questions on the material, or if you want to ask any questions about folklore, I'd be happy to supply answers to either.
By the way, how many of you have heard of the Hans Christian Andersen story, the emperor's new suit? OK, a number of you. I should relate it, perhaps, just for one reason. And it'll make all the rest of the things more clear. This is a story of Hans Andersen, for those of you who don't know it, basically two quote unquote tailors came along, and they had this great idea that they would weave this marvelous suit for the emperor. And they very carefully did all of this, such that they brought the staff in early to see it, and they were all lectured that only the pure in heart can see this thing.
And they were busy pulling thread, and turning the wheel, and everything. But everybody who came in was told, only if you're pure in heart will you see this. And of course, nobody wanted to admit that they weren't pure, so everybody saw it. And so they worked up the staff, and finally brought the emperor in. By this time, he'd heard the same thing from so many people who had seen it, of course, who had seen the cloth, and it was so beautiful, that he saw it too, of course. He didn't want to admit he didn't see it.
So he got all dressed up, and he went down in a parade, I don't remember if he was on his horse, or what. And everybody else-- it was all announced to the public ahead of time that this fantastic new clothes you could only see if you're pure in heart. So everybody's there admiring the thing, and a little kid says, daddy, he's naked. And of course, the child was assumed to be pure, and so then everybody saw that there wasn't any suit.
Well, you can take that story-- by the way, a roommate of mine here at MIT when we were going through, when he handed in his master's thesis, the professor, [UNINTELLIGIBLE], scanned it over quickly, looked right at him, and said, I want you go to library tonight, and I want you to dig out and Hans Christian Andersen's Emperor's New Clothes. Well he hadn't read the story, and he came home that night, and boy, he was white. God, he says, my professor told me to read this when I handed in my thesis, and I read it, and I know what he thinks of my thesis now. Fortunately, it was the sense of humor of the professor. But it caused a couple of days that were pretty bad, I'll tell you.
But in the hi-fi world, there is a heck of a lot of this kind of thing also, that can't you hear that, you hear it? And people, oh yeah, I don't want to be dumb, I hear it, sure, I hear it. But there's a standard psycho-- there are a number of ways you can get around these things. There are standard psychoacoustic tests. A very simple one is what we call an Ax type test. In other words, you'll give a person samples of two things, and x will be either A or B. B being a different thing than A, a different sound than A.
And if you want to hear-- let's suppose A and B were two audio signals that you were listening to somehow, anyhow, in a room, and they differed only by some phase shift that went into the loudspeaker, let's say an all pass network which had some phase shift in it. And then you play A and then you play B right after that, that's one pair. And people are told, press the button if the second one is different than the first. Then you can put a given phase shift in B that isn't in A, and you just go down and you give them a whole bunch of samples.
And you know what you did here, you know that this was A and this was A and this B, et cetera. And you correlate their button pressing, which is their perception of whether the thing is B or not, and you can right away tell whether they perceive the difference or whether they don't. So you can tell whether that phase shift is important or whether it's totally irrelevant.
As it turns out, by the way, Phillips in Holland did some experiments along this direction, the first ones I'd heard of any way. I was visiting around, they did them early, about 1960. And they had to take an all pass network-- all pass networks are networks which have poles and 0's, like so, so the magnitude is always 1. In other words, at any frequency-- remember, the magnitude of the transfer function is the product of all the 0 vectors to their frequency divided by the product of all the pole vectors.
Well, if the darn thing's symmetrical, the magnitude is always 1. But as you go up here in the axis, these pole vectors wind this way, which is negative phase, and the 0 vectors wind this way, which is more negative phase. So what happens is the magnitude is 1, and the phase goes like so. Every time you pass one of these things, it goes-- well, if you come and pass it here, it's a whole 180, so 360 jumps. Wild.
Well, they made these networks as best as they could make them, with the pole and 0 real close to the axis so you had a really sharp jump. And with a great deal of effort, the only thing they could find out that was audible was the ringing of this pole when it got so close was such that if you pronounced the word with a stop constant, you know, stop, it would go stop-bing, but on no music could you ever hear the thing at all.
Networks like this will actually cause a square wave, for example, going in to come out some time like a triangle, sometime like just a hodgepodge of stuff that goes around like that. At each different frequency, it'll be a totally different picture, no resemblance to the input at all. And hi-fi equipment used to be all judged on-- one of the important judges was the transit response. If you put in a square wave and you got out a square wave, that was a good hi-f.
Well it turns out that by just changing the phase alone, you can totally foul up the output. In other words, you just put it through a network which has uniform magnitude and phase, you can make this output not identifiable on the transient response to an input. But you can't hear the difference. Now if you could, there are a lot of-- once you understand what you just experienced in normal modes, there are a lot of reasons that you would expect that's true.
Because when you moved your head, you just made a good 10, 12 million phase shifts that are very intense all change. In other words, you made things maybe 10 million of them change 360 degrees when you moved from here to here. So if this business of time alignment or-- because, by the way, in a loudspeaker, these conventional things, high frequencies come at you this way, low frequencies go to the wall, so they all travel, by the time it gets to your ear-- and that's what we're going to see in a while-- by the time it gets to your ear, they've all traveled different distances. What the woofer is sending you has traveled a totally different distance than what the tweeter is traveling to you.
So all of those reasons would lead you to suspect that the monaural phase-- I'm not talking about phase difference between ears, but monaural phase-- is something that's not going to foul up any music performance, otherwise all musicians would have composed for just one person. OK. Yeah.