I will summarize what we have learned about the normal modes
of the string fixed at both ends, which is very relevant
today for musical instruments. So, suppose I have a string
with length L, mass per unit length mu,
and tension T, then I know that the speed of
propagation is the square root of T divided by mu.
And, the wavelength is always this P to propagation divided by
the frequency, speed of propagation times the
period of one oscillation that I avoid capital T's for period.
So I will just write it down as frequency.
And so, in its lowest mode, recall that the fundamental,
first harmonic, you will get then this
situation, N equals one, and lambda one is then clearly
2L if this length is L. And, the frequency is then V
divided by 2L. So, that's the frequency for
the lowest possible mode, the first harmonic.
If we go to the second harmonic, and the picture
changes. We get a node in the middle.
And we have here N equals two. So, lambda two is now L.
And so, F2 is now, again, V divided by the
wavelength. So that becomes,
now, V divided by L. And so, we can write down,
now, the general equation for the N's mode,
N being Nancy. Lambda of L is then 2L divided
by N. In the frequency in the Nth
mode, that is the Nth harmonic, is then NV divided by 2L.
And, this V is given there. So, what you see here,
that if the fundamental here were, for instance,
100 Hz, then the second harmonic would be 200 Hz,
the third harmonic 300 Hz. They come in series one,
two, three, four, five, and so on.
Now, clearly, there is the possibility that I
would have one end of that string open.
We call this then closed-closed,
or fixed-fixed. But suppose I have the same
length, L, but here I have this infamous rod which is
frictionless with the infamous ring, which has no mass.
And it is fixed here, and now I want to know what the
normal modes are. And in the lowest mode,
you will get this. The angle is 90°.
DY DX must be zero, there.
And then, it oscillates like this back and forth.
And so, now you have that N equals one, which is now the
fundamental first harmonic. Lambda one is now 4L lambda
one. And therefore,
the frequency that you now generate is twice as low is now
V divided by 4L. So, you can go one step
further. You can ask,
now, for the second harmonic. So here, again,
is that rod. And now I introduce another
node here. And so, now the string would
look like this. This is, again,
90°. And as it oscillates,
it would move like this. And so now, this is N equals
two. This is the second harmonic.
You can just look at it. Lambda two is four thirds times
L, right? You need to make it longer.
And therefore, F2 is now three divided by 3V
divided by 4L. And so, now we can write down
the general recipe for the Nth mode, N being Nancy.
So, we will find, then, that lambda N is now 4L
divided by 2N minus one. And that, now,
changes the picture quite dramatically because if you
follow here that you go from lambda one to lambda two,
the change is not by a factor of two.
But it changes by a factor of three.
And so the frequency in the Nth mode, which is the velocity
divided by lambda N is 2N minus one times the velocity divided
by 4L. And so, what you see now:
if we take both systems, and suppose L were the same,
mu were the same, and T were the same,
then F1 in this mode, this instrument is half the
frequency of F1 there because there you have downstairs a 2L,
and here you have a 4L. So, for instance,
if that were 100 Hz, with the same length here you
would get 50 Hz if all other things are the same.
And then the second harmonic would be 150 Hz because if N
becomes one, you get a three upstairs.
If N becomes two, you get a five upstairs.
So now the ratio is one, three, five,
seven, whereas there the ratio is one, two, three,
four, etc. Now, if I take a string in
isolation and I oscillated, I almost hear no sound.
There is not enough to air that is displaced by the string.
And so, what we do is we have to mount it on a surface that
starts to vibrate with it. And you see that on all musical
instruments. I will demonstrate that to you
in a simple way with a tuning fork.
I have here a tuning fork. And if I just excite this
tuning fork, it is 440 Hz. You'll you're practically
nothing. I hit it now,
I can hear it. I'm very close.
But most of you cannot hear it. However, the moment I put it on
the surface, the surface starts to vibrate with it.
So, I drain energy faster out of the tuning fork,
but you get more volume, more sound.
A larger surface oscillates, so the pressure wave is
stronger. So I'll demonstrate that.
Do you hear it now? Do it here, big difference,
right? Huge difference when I put it
on the surface. I have here a music box which I
bought many years ago in Austria.
And, it has these prongs. And, when I rotate it,
I may be able to hear it. But you won't hear it.
Maybe some of you are very close.
Can you hear it? Good for you.
Now listen. [MUSIC PLAYS] Big difference.
Now the whole surface oscillates.
Again, you drain energy faster, of course.
But you get more sound. I can put it on here.
[MUSIC PLAYS] Big difference. And of course you will see that
in the design of all musical instruments.
Needless to say, the design of the sounding
boards which are connected to the strings are,
of course, secrets that the company are very fond of and not
telling you. I am not aware of musical
instruments, of stringed instruments, whereby one end of
the string is attached to a frictionless rod,
and whereby that end is attached to a mass-less ring.
So, I will restrict myself in discussing musical instruments,
string instruments, to the one's whereby both ends
are fixed. And so, therefore,
the frequency, F1, that is the one that I am
interested in, is the speed of propagation,
which is T divided by mu, and then one divided by 2L.
So, these are the key players in the design of instruments.
If you make L longer, you get a lower pitch.
If you make L shorter, you get a higher pitch.
And, we all know that. If you make L shorter,
you get a higher pitch. If you make the tension higher,
you get a higher pitch. If you make the mass per unit
length lower, you get a higher pitch.
And so, that's key in designing musical instruments.
If you take a piano, it is 88 keys.
The lowest frequency is 28 Hz. The highest is 4,000 Hz.
It covers seven octaves. And when you hit a key,
a hammer actually comes down onto the string.
And it excites the string in the accommodation of normal
modes. In fact, in many cases,
when you hit one key without you realizing it,
you hit more than one string simultaneously.
But that's a detail now which I will not further expand on.
If you take a Steinway grand piano, even though it has only
88 keys, it has 216 strings. And so, the idea of the piano,
then, is you change the length of the strings.
So that's a player. The shorter the lengths,
the higher the frequency. And you can change mu.
And, when you open a can oh, and you look at the various
strings, you immediately see that some strings are as thick
as my pinky. You have a huge mu,
and others are very thin. They have a very low value.
And that will give you, then, higher pitch.
The tension of the strings in the piano are approximately all
the same. They are quite high.
They are near 200 N per string. So, a grand piano,
then the total force, all the strings together is
something like 45,000 N, which is an immense force,
when you think about it. Now, if you go to violins and
cellos and a bass, they have four strings.
Then the length is a given. But if you look at the various
strings, and we will see a violent later,
you will see that one string is much thicker than the other.
So, you play with mu. You change the mu.
So, you only have four fundamentals then.
If you go to a Museum of Fine Arts in Boston,
they have a wonderful collection of musical
instruments. You'll see musical instruments
with just two strings, and you will also see musical
instruments with one string. But I have never seen one,
as I said earlier, whereby one end of the string
[can/can't?] freely move. But maybe they do exist.
So, now comes the issue, if you have built an
instrument, how do you tune it? Well, with the piano you ask a
piano tuner maybe once a year or once every year to come into the
piano for you. And what the candidates in or
does it simply changes the tension in the string,
which is a major job, of course, if you have 216
strings. But the violin,
and guitar, and cello, and bass, it is the player
herself or himself who is doing the tuning.
We will see that later today. They will actually change the
tension in the string. And then, they listen carefully
to get just the right tone. And, they do that before they
start playing. If you play the piano,
with all due respect for piano players, all they have to do is
hit the right keys in the right sequence.
That's all there is to it when you are piano player.
Now, think about the violinist, or a guitar,
or a cello. They cannot just hit one
string. They have to change the length
of the string all the time to change the fundamental.
And that's the whole idea when you have the violin and you
strike it with a bow. You rub the string,
and the string will start to oscillate in the fundamental in
the higher harmonic. By making the string no shorter
with your hands, you increase the pitch.
And so, playing there is that you as a player continuously
have to move the length of the string and just hit the right
length. It goes beyond my imagination
that anyone can do that. I cannot.
I tried it when I was young. I took violin lessons.
I was a total disaster. If you take a harp,
it is in a way like a piano except there's no hammer.
But you pluck the string. And now there is something very
special about the harp. You can decide where you pluck
the string. And that makes a difference in
the percentage of higher harmonics, in which higher
harmonics you excite. And we will deal with that in
8.03 when we do Fourier analysis.
You will see, then, that if you pluck a
string in the middle, do you get a different series
of higher harmonics than when you pluck at,
say, a certain distance of 20 cm from the end.
It's a very interesting part. In fact, the piano is designed
in such a way that the hammer hits the string about 1/7 of its
length from one end. And that is done to suppress
the seventh harmonic. Now, why the seventh harmonic
has to be depressed? That is up to you to decide.
Maybe we don't like seventh harmonics.
I don't know. It beats me.
But in any case, it's purposely done to suppress
the seventh harmonic. There is a musical instrument
that has only one string. One was designed by my daughter
when she was a nursery school. That's a long time ago.
And I have it here. I'm very fond of it.
I demonstrated it many times. It's called a washtub bass,
I think. So, you see here the one
string. And you see here,
then, the surface that is necessary to make you hear the
string. In this case,
it is, I think, a box from Kentucky Fried
Chicken or something like that. So, I need three hands to play
this. So the only thing that I can do
now, I can't change mu. It's a given.
I can't change the length. That's a given.
So, what is the only thing I can do?
I can change the tension, and that's the way you play it.
And for that, I need three hands.
But I will try to do it with my mouth.
And then, you listen. When I increase the tension,
you will hear the frequency go up.
When I decrease the tension, it goes down.
You could hear it, right?
Clearly different in frequencies.
Pythagoras, who lives in the sixth century B.C.
discovered that musical notes are very pleasing when the
length of the strings come in simple ratios.
This is quite remarkable when you think of that.
Two-two-one give you an octave. Three-two-two give you a fifth,
and four-two-three gives you a fourth.
And, the evolution of Western music is based on that.
When you take a piano, the piano has the octave:
one-two-two. It has the fourth,
four-two-three, and it has the fifth,
three-two-two. And somehow these are very
pleasant intervals when you play them in combination for our
Western ears. It doesn't mean that it is also
true for other cultures. The ancient Greek astronomers,
even all the way up to Kepler in the 17th century belief that
the musical instruments could also explain orbits of planets.
And this was known as the music of the spheres.
It was believed that the movement of the planet produces
music, but our ears were just not sensitive enough to be able
to hear that music. And of course it shouldn't
surprise you that all this was mixed up with religion.
The fact that it was pleasing to hear music that comes in
simple ratios must show the hand of God, if you want to believe
that. I now want to turn towards wind
instruments. And we're going to put a wind
instrument here, which is nothing but a tube,
length, L. Let's start with one that is
closed on both sides, length L.
And, if it is closed on both sides, then the pressure here on
this side and this side can build up, as we discussed last
time. So, there's going to be
pressure anti-nodes. And, there are going to be
nodes for the motion of the molecules.
If I have here a sound cavity which is open on both sides and
I have here a pressure node, and I have here a pressure
node, the pressure can never be higher than the ambient pressure
because it's connected with the universe.
When I say pressure, I always mean overpressure over
and above 1 atmosphere. And so, it's immediately clear
that the solutions that would work out here for
closed-closed-system must be exactly identical to this.
So, you do get precisely here that lambda N is,
again, 2L divided by N. There is no difference.
And, F of N is NV divided by 2L.
But, of course, the enormous differences that
the V is God-given, that is, the speed of sound in
air, the air column begins to oscillate.
And so V, now, is approximately 340 m per
second, which is nonnegotiable. So, there is no way that you
can manipulate mu NT to change the speed of propagation.
The speed of sound in a gas, something that I really
wouldn't want you to remember, I will return to that at the
end of my lecture, is given by this equation.
But you may forget it as far as I'm concerned.
R is the gas constant. T is the temperature of the gas
in degrees Kelvin. And gamma, if any of you
remember that from 8.01, is the ratio of specific heat
at constant pressure divided by specific heat at constant
volume. But, if you've never had that,
that's fine, too.
Just take my word for it. I will give you some numbers
for gamma later. And then, M is the molecular
weight. So, this is the speed of sound
in a gas. So, it's a given for air.
There's nothing you can do about it.
Room temperature, it's given.
It changes with temperature, which is interesting.
So, the way you get the system going, you must somehow blow air
past maybe in this direction, or so, and then the column
starts to excite. It's very hard to see why that
happens, but then the column gets into normal modes.
And you may get a whole collection of various normal
modes, a superposition of many normal modes.
Now, there is, in the case of sound cavities,
what is very easy to make a system which is closed at one
end and which is open at the other.
That is very easy, which you cannot do here.
You can do that here. And so, therefore,
in this case, that one side of the cavity is
closed and the other one is open, there are musical
instruments. The clarinet comes very close
to be almost closed at one side and open at the other.
But there are many musical instruments which are open on
both sides. So, then you would have here
that F of M equals 2M minus one times V divided by 2L.
So then you have, indeed, this case which we
dismissed for stringed instruments.
[That's?] divided by 4L. And, the easiest way to
demonstrate that to you, to take my pen,
this is the cover of my pen. It is closed on this side,
believe me. And it's open here.
And it is about 3 cm long. You may not like the frequency.
It's a very high-pitched frequency.
And it may be a cocktail of more than one mode.
That is a musical instrument. Not the best,
but it is a musical instrument: closed and open on one end and
the other. So, now, it is important that
we get some feeling for the frequencies that you can produce
with the various wind instruments.
And so, here, I have listed for you the
length, L. And this is,
then, an open-open system. Of course, it's the same for a
closed-closed system. But a closed-closed system
doesn't make a very nice musical instrument because the sound is
not coming out. So that's not the idea.
So, therefore, it's open-open.
Or, it is closed-open. And you see here the length.
And, of course, the speed of sound is a given.
So, I can't change that. And this is,
then, the fundamental, the first harmonic,
covering the entire range of your hearing all the way from 17
Hz. So, you would take a 10 m long
tube open on both sides to get in the fundamental the 17 Hz.
And for a closed-open system, that would then be easier.
So you would get away with half the length to get the same.
You would get half the frequency.
That's basically what a closed-open system does.
And so, we'll keep that on so that we can understand when we
hear the various frequencies, why the frequencies are as
higher as low as they are. Now here, we have a tuning fork
which is 256 Hz, which is mounted on a sound
cavity which is closed on one side and open on the other.
And that is done in order to get perfect matching between the
resonance frequency, the fundamental of the box,
and the 256 Hz. You can do that,
of course, if you're only interested in one particular
frequency. So, for this case,
then, you would have that the frequency, F,
is the velocity. So we go to this case now,
to the case of this equation. So, the frequency,
this is the first harmonic, now, is V divided by 4L.
So, that is 256. So, that is 340,
approximately, divided by 4L.
And then, L comes out to be about 33 cm.
And, indeed, if you measure the lengths of
this, you will find that it is roughly 33 cm.
And so, now you have a case where you have perfect matching.
So, when I hear this tuning fork, then the box starts to
oscillate exactly at that frequency.
So you get a very large sound. So you drain the energy very
quickly. You do see often tuning forks
which are mounted, instead, this way.
You can also have a box like this, open and open on both
sides. Then it has to be twice as
long. So, for very high frequency
tuning forks, you often see it open-open.
If we have to do it for this one, it would have to be twice
as long, and that is just unpractical, and not necessary,
of course. Now, with musical instruments,
you produce an infinite number of tones.
So there is no way that you can, of course,
design your sounding board that it resonates with every single
frequency. And so, this is where the
secrecy of the manufacturer comes in.
And they're not going to tell you how they do that.
And therefore, some instruments are better
than others. And you pay for that.
If you have a barrel filled with liquid, I remember that in
France they did this technique that I'm telling you about,
and the wine was up to here, say, and they wanted to know,
you couldn't look into the barrel.
You wanted to know where the level of wine was.
They would knock on this. And the sound here would be
very different from the sound here because here's air and
here's liquid. So it's obvious that where you
have liquid the residences are very different from the
residences here. So, that's one way that you can
tell what the level is of the wine.
When you have a cold, which I happen to have today,
I remember as a kid I would go to the doctor and he would ask
me to inhale. And then he would knock on my
chest. And I would exhale and inhale
and knock on my chest. And then he would be able to
tell whether there is fluid in my lungs are not.
I don't know how sensitive that is.
It's not my field, but that's the way it was done.
And maybe you have experienced the same thing,
that sometimes also, I don't know why,
they knock on your back even. Now, I have developed,
in fact I can even say invented a method to test for the
presence of brains. And, the way I do that is that
I strike a tuning fork and put the tuning fork on someone's
head. And you can imagine if it's
empty here, then you get the same effect as you have what the
wine barrel. And then you hear very clear
sound. But if there are brains here,
like the liquid, you get nothing.
And I have patented that. So you can't try that at home.
But I can demonstrate it to you.
You hear much? Oh.
You hear much? Maybe I shouldn't be lecturing
8.03, then. Now, of course,
in my case, I think I passed the test within reason.
It is not so clear, of course, that all the
students would pass that test. And I was wondering whether
anyone has the courage that I may try that.
Anyone of you? You are afraid,
right? You're worried that it will all
come out. Do you mind?
You don't mind. You're a strong man.
You have to be quiet, otherwise I can't tell the
difference. [LAUGHTER] I say no more.
I think it's better that I teach at 8.03 than you do.
So, when you design a wood instrument, the only thing you
have to play with is L because T is no longer negotiable.
And so, with organ pipes, when you go to churches and you
see organ pipes, you see a whole zoo of these
organ pipes: open-open and closed-open.
And every fundamental that you want to excite,
you need one pipe. So, it's a huge number of
types. It's, in a way,
like the piano. If you take a flute,
then you make it longer and shorter simply by drilling holes
in it. And if you hold your hands on
both holes, then this is the length of the flute which gives
you, then, a lower frequency than when you take one hand off
because now it's shorter. This is connected with the
universe, so here, no pressure can build up.
So this becomes a pressure node.
So it's shorter. And if you take this one off,
it's even shorter, and so the pitch will go even
up. That's the basic idea behind a
flute. And, I can demonstrate that to
you. [MUSIC PLAYS] Do you hear the
pitch go down? [MUSIC PLAYS]
I have here a flute-like instrument which is open on this
side, and it's also open on here where we passed the air by.
So, you can consider this to a very good approximation is open,
open. It is 16.6 cm,
and so if you want to know what the fundamental is,
well, then you apply this equation, N equals one.
And then, you get something like 1,024 Hz.
But, if you may get closed at one end, and you apply this
equation, you would only get half that.
So you only get 512 Hz. It's a big difference,
factors of two. So this is, then,
the fundamental for open-open. [MUSIC PLAYS] And now comes
open-closed. [MUSIC PLAYS]
Factor of two difference. I have, here,
the very special tube. It is open and open on both
sides, and if you don't believe me, Annie, I can see you.
Can you see me? Can you?
Can you see me open and open on both sides?
And it is corrugated, which of course is very
important why it works so well. And this has a length of 77 cm.
And so, you can calculate, now, using this equation,
N equals one, what the fundamental is.
And, the fundamental is about 220 Hz, the first harmonic.
And, since it's open-open on both sides, the second harmonic
will be 440 Hz. In the third harmonic is 660.
And, by twirling this around, with a little luck,
you can get a wind flow past here which only excites the
lowest mode. Sometimes you hear the lowest
one and the second. But, when you twirl it faster,
you get higher harmonics. And I want to demonstrate this
to you. I will first try to excite the
220, which is the lowest mode possible, which is the first
harmonic, the fundamental. And, I'll try to make it clean,
just only the fundamental. That's it.
It's about 220 Hz. [MUSIC PLAYS] 440,
660, 880, 660, 440, I can't get above 880.
[APPLAUSE] There are ways that you can
change the length of a wind instrument.
One is by making holes in it. Another one is by really
physically changing the length. And there was one instrument
which is well-known for that, which is a trombone.
So, you actually changed, this is a system which is open
and it is closed at the end in this case.
You can see that. It's like a piston,
so it's closed. And so, now by changing the
length of the cavity, you can change the fundamental.
So, we get it this way. [MUSIC PLAYS] [APPLAUSE]
We recognize instruments by the cocktail of the harmonics that
they generate, and depending upon how you
excite them. I mentioned already the
plucking of the harp, but also the way,
for instance, you blow on musical
instruments, I don't know whether any of you have ever
tried to play on the trumpet, but if I gave you a trumpet,
chances are that you would get no sound out of it at all.
You have to hold your lips in a special way.
You have to know how to spit in there the right way.
It goes like [MAKES NOISE], something like that.
So, that is also a way that you can excite certain harmonics in
relation to the fundamental. And that, then,
determines the sound quality of your instrument.
And, I'd like to demonstrate this to you now in various ways
that the different instruments have different sound quality.
What it comes down to is that if you ask each instrument to
play, for instance, 440 Hz tone,
you will see that the violin cannot just simply produce 440,
but it will automatically also, at the same time,
generate 880, and made that as the second
harmonic, and maybe a higher harmonic.
And that's different for different instruments.
And that's, of course, the idea behind the tone
quality. And the way that we're going to
demonstrate that you is as follows.
We have here a microphone. And I will first make you
listen to a tuning fork, for instance,
440 Hz, which will give you, then, this signal [times?] the
amplitude of the membrane of the microphone.
If, however, an instrument were to generate
440 in addition higher harmonic, you would get the sum of this
signal and, of course, the higher harmonics.
And so, you will see no longer a nice sinusoid.
But, you see on top of the sinusoid the higher harmonics.
So, let me now, first, show you the 440 Hz.
Here's the microphone. [SOUND PLAYS] Boring:
just one frequency. Nothing rich about it.
256 [SOUND PLAYS], there's no indication for any
higher harmonics. That's the way that tuning
forks are designed. If I take this flute,
[SOUND PLAYS] as far as I can tell only the fundamental,
no higher harmonics. Now, maybe if you blow in a
very special way, maybe I can excite higher
harmonics. But this is now,
oh, I want to show you 4,000 Hz so that you can easily hear.
This is 4,000 Hz. [SOUND PLAYS] Right,
so you get the idea, so you know now what you're
going to see. But you don't know yet what
you're going to hear. Neither do I.
We have six students who are very brave.
I would like all of them to come down now.
And, they're going to demonstrate their musical
instruments. So, all of you come down,
please. Yes, you got your instrument
there. Just come here,
that's fine. Don't worry.
Come here. All right, where is the
violinist? There's the violinist.
Yeah, oh, boy. Oh, boy.
[LAUGHTER] The first person who is going to demonstrate the
violin is going to be Mark [Avarro?].
And, I'm going to ask Mark first to only produce what he
thinks, then, is 440 Hz.
We have, for you, a special chair.
We will bring the chair very shortly.
So let me remind you, then, of the 440.
This is 440. Come a little closer and
produce 440. Now, look.
Did you see that you could really see the 440 in there?
It was the same spacing as my tuning fork, but there were many
higher harmonics there. So it [wants?] more.
And they, then, give you all these wiggles.
Now, the audience is yours. And now, I'll give you 20 or 30
seconds, anything you want. Play anything you love.
Go ahead. [MUSIC PLAYS]
You are fantastic. [APPLAUSE] Is Sharon here?
Sharon, so, Sharon is going to demonstrate to us the flute.
And, maybe you can show it to the class.
See, her flute has more holes than mine.
I only have two, and she has quite a few more.
And she opens and closes them with valves.
Sharon, you may not be able to exactly get the 440 with this
instrument, but that's OK. Come close and show us your
440. Excellent.
Did you see? You could really see the 440,
but you could see more than that.
It separates it from the violin, of course.
The audience is yours. Make us happy.
[MUSIC PLAYS] Terrific.
[APPLAUSE] Now, we have a very special guest,
which is Shawn [Augin?], who was so kind to go through
the trouble of bringing the cello.
Now, you think, Shawn, that you can come close
to 440? Try that.
And if you can do it just in a fundamental.
You may have to tell us whether you had to shorten the string
with your finger. You do have to do that?
Go ahead. [MUSIC PLAYS] Try it once more.
I can see the underlying 440. And, you shortened the string?
Oh, there was no finger on the string?
So, it is the fundamental of that string, the middle A of the
piano. Show us yours.
You practiced yours? Oh, you didn't,
good for you. So, you prefer not to play?
That's fine. Thank you very much.
Now we have a Chinese clarinet from [Nyugen Choo?],
and she told me that the nice thing about it is that it's not
really a clarinet. So, can you make a 440?
Well, since it's on a clarinet, I think you have good reasons
to be off. OK, try it.
[Use your shoulder clamp?]. See all these holes in here?
One, two, three, four, it even says F there.
[MUSIC PLAYS] Very nice. I do see overtones.
The audience is yours.
[MUSIC PLAYS] It's all right.
I also did very poorly on the trombone.
That's the way it goes. [MUSIC PLAYS]
You're doing great. [MUSIC PLAYS]
Wonderful. [APPLAUSE] Where is the saxophone?
There is the saxophone. Colin Johnson.
Colin, 440. [MUSIC PLAYS] Man.
You got all the way down there? 440.
[MUSIC PLAYS] You can't see it, but we can.
That's very nice. If you stay there,
the audience is yours. [MUSIC PLAYS]
Terrific. [APPLAUSE] Now,
you get the hero, Aston.
Aston is going to play the 440 Hz tone on his percussion.
Show us yours. But, we had email exchange
about it, and you said you could produce 440 Hz.
Can you show the percussion for one thing?
Oh, no, we're going to do percussions, right?
Go ahead. Small percussion.
[SOUND PLAYS] I think it's nowhere near 440.
What do you think? Oh, it has 440 overtones.
It can never have a 440 overtone.
The overtones are always higher.
That maybe an undertone, 440.
Go ahead. [SOUND PLAYS] Let's call it
440. Well, that's why you're here,
right? Man, we can't wait.
20 seconds, it's your show. [MUSIC PLAYS]
That's closer to 440 than what I've seen.
You could see it. We need one more bang.
[SOUND PLAYS] Thank you all.
Great. [APPLAUSE] If there's any
singer in the audience who wants to try, be my guest.
I think this is a great moment for the break.
While we have a break, feel free to play around with
the wind organs, and also with the tuning forks,
be a little careful. You may also try the trombone
and the flutes. And then, five minutes from
now, we will reconvene.
So, what we have been looking at our one-dimensional standing
waves, normal mode solutions to the famous wave equation.
In the case of transverse motion, we have D2Y,
DX squared is one over V squared times D2Y [G?] squared.
So, this is the wave equation, Y now in this direction.
So, for instance, if we have a string,
and we have a string with length L and it is fixed on both
ends than the boundary conditions determine the precise
allowed solutions. We call them [igan?] states in
quantum mechanics. And I can write down,
now, the displacement, Y, in the N'th mode as a
function of X and T. And, we give the amplitude of
the N'th mode, say, a value that can differ
from mode to mode. So, we get A of N,
and then we get the sine of N pi X divided by L.
And then, we get the cosine of omega N times T.
So, this is the time domain, which makes it shake with that
frequency. And this is now the part that
makes sure that the N's are always fixed.
If you make X equals zero, Y is zero.
And if you make X equals L, Y is zero.
And, that gives you, then, a whole zoo of possible
frequencies because omega N is always V times K of N.
And so, that then becomes V times N pi divided by L.
And so, you see the ratios, one, two, three,
four, answer one NVL. That's right.
Now if we go the [SOUND OFF/THEN ON] of longitudinal --
-- then if we have a system which is open-open like we've
discussed before, and it has length L,
then you get exactly the same solutions, provided that you
replace what we have Y there, you have to replace that by P,
P being the overpressure because at the ends of an
open-open system, the pressure can never become
higher or lower than ambient. And so, you have pressure
nodes. And so, you get exactly the
same solution that you have there, provided that you replace
this by P, and this you replace maybe by some capital P,
which is then the amplitude. But, this and this is the same
except, of course, that omega, this V is
nonnegotiable in the case of sound.
If you have a closed system, so now you close one end.
Then you want the boundary conditions at P is zero at the
open end, and P has to be an anti-node at the closed end
where the pressure can build up. That gives you,
then, different values for this K of N.
The sine may even have to be changed to a cosine it depends
on where you define X equals zero.
That's not so important now. But what is important now that
K of N in that equation now gets the form 2N minus one times pi
divided by 2L. So, everything comes from
boundary conditions. And now I want to make the step
from one-dimensional to two-dimensional.
If we go two-dimensional, which is easy to do,
and we can even demonstrate that.
So, this has length LX, and this has length LY.
It could be a [frame?] with a [soap film?],
a membrane, which can oscillate.
And, we'll assume that the membrane is attached everywhere
to the frame. So, that is the boundary
conditions, everywhere attached. And so, we want to know now
what the normal modes are for this system.
So now, our one-dimensional wave equation becomes a
two-dimensional wave equation. And so, if this is my
coordinate system, XYZ, then now we get that D2Z
DX squared plus D2Z DY squared is now one over V squared times
D2Z DT squared. That, now, is a two dimensional
wave equation. And when you put in your
boundary conditions, which in this case,
suppose we call this zero, so, for X equals zero,
N for X equals LX, you want Z to be zero because
the membrane cannot move. And, for Y equals zero and for
Y equals LY, you also want Z to be zero.
Then, you can now, in analogy with that,
you can write down now the solution for Z as a function of
X, Y, and T. Now you had to introduce to
quantum numbers if I call N Nancy and M a quantum number.
So, now, you can get Z, and now I have two integers
that can change. So, M now goes one,
two, three, etc. Nancy is one,
two, three, etc. And this is now a function of
X, Y, and T. It's two-dimensional.
So, I have to introduce some kind of an amplitude,
which can be different for the different modes,
Mary Nancy. And now I get something
extremely similar to this, except I get two sines.
So, I get here sine of M pi X divided by LX times the sine of
Nancy pi Y divided by L of Y times the cosine of omega MN
times T. And you see that it meets the
boundary conditions. If you put in X equals zero or
X equals L of X, you get a zero.
If you put Y equals zero or Y equals L of Y,
you get your zero. And, if you substitute this
solution, which we clearly, intuitively know,
it has to be correct, if you substitute that in that
wave equation, then you get the connection
between the K values, N omega, and then you will find
that omega MN, which is always the velocity
times K value of M of N becomes, now, V times the square root of
Mary pi divided by LX squared plus Nancy I divided by LY
squared. This, now, sets the entire
range of all possible normal mode frequencies.
So, now you can play with Mary M and you can play with Nancy N,
and you can cocktail that in all combination that you want
to. And, for instance,
if we ask this membrane to oscillate in M equals one N
equals one mode, then it means that the entire
membrane comes to you, and goes back,
and comes to you, and goes back.
That's the lowest possible mode.
All points are in phase, and they all have the same
frequency. That's the characteristic for a
normal mode. If, for instance,
we had Mary equals two, and Nancy equals one,
then we would have in the direction of Mary,
we would have plus and minus. This part would come to you,
but this part goes away from you.
So, a plus and a minus, and this now is a nodal line.
So, we get a nodal line. So, it's not anymore a nodal
point, which you get in one-dimensional solutions,
but you get a nodal line. This stands still.
And then, the membrane oscillates like this and is
attached to the edge. And if, for instance,
you wanted to know the Mary equals two Nancy equals two
mode, so M equals two, N equals two,
then you would get two nodal lines plus-minus minus-plus.
If this comes out of the blackboard, this comes out of
the blackboard, this goes into the blackboard,
this goes into the blackboard. And, in French,
you will see some remarkable, wonderful pictures of a guy
with beautiful hair from the 50s whereby you see the oscillations
of the soap film in these various modes.
It's really very, very exciting,
very interesting. And I want to demonstrate this
to you in a way that I cannot possibly calculate.
I doubt whether there are many people who can,
although maybe some civil engineers can do that,
because this system is way more complicated than the membrane
that we have here that is attached to the frame.
What we have is a square plate, and it is only fixed in the
middle. That's where it's held.
That's the only point that is not moving.
That is here. But, what we do,
we drive the thing. Actually, there is a piston
below here which drives that middle point.
And, it oscillates it. And then, you get an infinite
number of frequencies whereby the system goes into normal
mode. We can change the frequency.
And the way we can make you see these frequencies is by putting
a powder on it, which we will do.
And then, the places where there are nodal lines,
the powder stays. And, the places where there are
anti-nodes, the power is literally thrown off.
And, it accumulates at the nodes.
And, that's the idea of the next demonstration.
We call these the Chladni plates.
And after the lecture, I would invite you to actually
play here with this violin bow. So, here we have similar
Chladni plates which have different boundary conditions.
This is a square, triangle, and this is a circle.
And, you can put stuff on there.
And then, you can try to hit these resonant frequencies with
the bow. And you get remarkable
patterns. So, let's first go to our
system, which is driven. So, that is very controlled.
We can adjust the frequencies almost any way we want to.
And, I will start with the low frequency.
Sometimes the sound may be very high.
I may have to adjust the sound. You can hear it.
We'll start somewhere I think in the range of about,
we are close to 300 Hz. And then, we'll increase the
frequency. And I will add powder as
necessary. And so, what you're looking at
now a two-dimensional normal modes, which give you nodal
lines. [SOUND PLAYS] Wow,
I'm right on resonance. That's an accident.
We didn't try that. That's a beautiful resonance.
Who on earth could have predicted these nodal lines?
I will go to a higher frequency.
I will lower the volume a little.
Wow.
Look at that. You see where the anti-nodes
are. It's thrown off.
And it accumulates at the nodes.
Hey, that's a weird one coming up.
My goodness, I've done it many times.
Every time I do it, I see different ones.
No, the normal mode frequencies are often so close together that
you can very easily skip one and go over one.
Isn't that amazing? You look at the lines here.
It's completely ridiculous. [LAUGHTER]
This is a different one, right?
Different from the previous one.
I'll increase the volume. When you are coming up to a
resonance, you can even hear it because the sound volume
increases.
Holy smokes! Isn't that amazing?
You try to calculate that. I would like to think,
though, that aero and astro people who design airplanes
[would have rings?], and people who build bridges,
that they should be able to calculate these resonant
frequencies because you may get destruction.
Look at this one. I've never seen this one in my
course. [LAUGHTER] This is completely
bizarre. All right, so you get the idea.
I really would like you to play with that, with the ones there
at the end of the lecture to see whether you can excite them.
[APPLAUSE] So, now, I want to make the
step to three dimensional wave equations.
This was a problem that Professor Mavalvala,
who is sitting in the back, put on the final exam of 8.03
this spring this year. If you don't like it,
there she is. So, this problem,
who has suggested that problems you, Nergis?
Don't tell them. Here we have a sound cavity in
three dimensions: A, B, C are the length.
And, this is our coordinate system, X, Y,
Z. And, it is open in this
direction. And, it is open there.
So, it is a box which is open on both sides in the Z
direction. And, the rest is all closed.
So, now the three dimensional wave equation,
D2P DX squared, P is the overpressure,
plus D2P DY squared is D2P DZ squared equals one over V
squared times D2P DT squared. That is the three dimensional
wave equation. And so, it should really be no
problem for you or me to write down immediately the solution of
the normal modes. You must make sure that you
have pressure nodes in the Z direction at Z equals zero,
and at Z equals C. And, you must make sure that
you have anti-nodes in the X and the Y direction where the
pressure can build up. That's the boundary condition
for a closed system. And so, the solution,
then, for the pressure in terms now of three quantum numbers,
L as in Lion, M as in Mary,
N as in Nancy is now some kind of an amplitude which can be
chosen differently for the different modes.
So, I give them, then, the index LMN.
And then, you get the cosine of pi times L times X divided by A,
which is my L of X. And then, I get the cosine of
pi times Mary times Y divided by B.
And then, I get the sine of pi times Nancy times Z divided by
C. And then, I get the cosine of
omega LNN times T. And so, you see when you look
there, convince yourself that you meet all the boundary
conditions. This is the one that is the
open end. That means if you substitute Z
equals zero, this one will make it zero.
That's where you have the pressure node where it's
connected with universe. No pressure can build up.
But if you substitute in here Z equals C, you also get zero.
And, you get the anti-nodes here.
So, this is a solution that immediately presents itself.
If you have followed what we did 1D and what we did 2D,
this is the obvious solution. And so, the system can
oscillate in a combination of all these normal modes.
But what is interesting now is that L can be zero,
one, two, three, because there's nothing wrong
with making L zero. The cosine remains one then.
Nothing goes to zero. And Mary can go zero,
one, two, three. It's only Nancy that cannot be
zero because if Nancy is zero, then there is no pressure
everywhere. There is no wave.
So, you see here that you now allow for zero,
zero, one modes, which will give you certain
frequencies. And, the frequencies that you
will get, you will find by substituting this solution back
into the differential equation into the wave equation,
which gives you, then, always the connection
between omega and K. And so, that connection is then
that omega LMN becomes V, which is the speed of sound
times the square root. If you are ready now,
we get pi L divided by A squared plus pi N divided by B
squared plus pi Nancy divided by C squared.
And so, the lowest possible frequency for this system is the
zero-zero-one mode. If I assume that C is the
largest dimension in the system, so that's the length,
L, I didn't put the length in there, but C is sort of the
length of my musical instrument, so you will get,
then, that zero-zero-one mode would give me,
then, omega zero-zero-one. So, this is not there.
This is not there. I call this now L to make the
connection with my musical instruments.
And so, I get V times N times pi divided by L.
And, that is precisely what we had before for a musical
instrument. For a sound cavity,
which is open and open at both sides, or which is closed and
closed on both sides, and so what you see now is that
if you have a musical instrument, even though it has
dimensions, A and B, that the lowest frequency that
you get is always dictated, then, by L.
But not only that, if A and B are much,
much smaller than C, then the second harmonic is
almost certainly going to be 002.
So, by making this one go to two, you are still at a lower
frequency than by making M go to one.
And, the third harmonic is very likely to be here.
And then there comes a time, of course, that the next
frequency comes in when you change the L and the M's.
And, you can get a whole zoo of frequencies.
And that is what that problem was all about.
In spring 2004, it's on the Web,
and the solutions are also on the Web.
The final exam was one of several problems.
If I return, now, to the speed of sound in
gas, R is the universal gas constant.
You may have had that in 8.01. T is the temperature in degrees
Kelvin. What I want to mention,
though, what is interesting: a friend of mine was a flute
player in an orchestra. And I asked him whether they
have to take into account the change in temperature when they
start playing. And he said yes,
they actually do. They tune their instrument to
adjust it to the temperature by making their estimate a longer
or little shorter. So, the temperature which
changes the velocity of sound must, of course,
be taken into account if you are in an orchestra,
and you have the flutes, and you have the wind
instruments together with the violin.
So, the violence can change the tension and the strings,
but the wind instruments also adjust to the temperature.
Now, if we have air and gamma, air is a diatomic molecule.
Nitrogen and oxygen, for those of you who have taken
8.01, and when this was covered for a diatomic gas,
gamma is about 7/5, which would make it roughly a
1.4. And, the molecular weight,
[while?] oxygen two is 32, nitrogen two is 28.
You have 80% nitrogen and you have 20% oxygen.
So, anything you take, you take 2930 is fine with me.
So, that is, then, the molecular weight for
air. So, now, suppose we do the same
for helium. Now, helium is a mono-atomic
gas. And, a mono-atomic gas,
gamma is about five thirds, which is about 1.66.
But the molecular weight, which in this case is atomic
weight, and helium is just a single atom, is four.
And, consequently, the speed of sound and helium
is roughly three times the speed of sound in air.
You can just substitute those numbers in there.
And you'll find that it's about three times higher.
Now, my voice is some kind of a musical instrument,
not unlike the violin. It has strings,
which are my vocal chords, which oscillate,
and these strings are connected to a sounding board which is my
larynx, which is my throat. And that creates a voice which
you recognize. You say, yeah,
that is Walter Lewin. My sounding board,
just like a sounding board of the violin, resonates at certain
frequencies better than at others.
And that gives you, then, the characteristic tone
of that particular violin, that particular person.
However, when I inhale helium, my poor throat has no idea
about that. And so it starts to resonate at
different frequencies. And therefore,
my voice will be very different because of the difference in
speed of sound because remember, the resonance frequency of
sound cavities, their resonance,
their frequency, depends directly on the
velocity, V. I have hear omega,
but the same is true for F, of course.
Now, there's only one problem, and the problem for me,
not for you, is that there is no oxygen in
helium. And if I take in too much
helium, I'll be on the floor. If I take in too little,
it won't work. And so, there is always a very
fine line. But I'm not joking about it
that this is really an experiment that can make me
extremely dizzy because will be without oxygen for some time.
And, I'm actually unaware of it because you just inhale that
helium. It feels great.
[LAUGHTER]
The person who's going to talk to you next you'll think is not
Walter Lewin.
I wasn't joking, really.
I hope you're going to have a good weekend. 1061 1:24:37,688 See you.