So far, we analyzed... We calculated the periods
of lots of oscillators: pendulums, springs,
rulers, hula hoops. We gave them a kick,
moved them off equilibrium, and then they were oscillating at their own
preferred frequency. Today, I want
to discuss with you what happens if I force upon
a system a frequency of my own. So we call that
forced oscillations. I can take a spring system,
as we have before. This is x equals zero,
this is x, and we have the spring force,
very familiar, minus kx. But now this object here,
which is mass m, I'm going to add
a force to it, F zero, which is
the amplitude of the force, times the cosine of omega t. So I'm going to force it
in a sinusoidal fashion with a frequency that I choose. This frequency is not the frequency with which
the system wants to oscillate. It is the one that I choose,
and I can vary that. And the question, now, is
what will the object do? Well, we have
Newton's Second Law-- ma equals minus kx plus that
force, F zero cosine omega t. a is x double dot,
so I get x double dot, plus-- I bring this in-- k over m times x equals
F zero divided by m times the cosine omega t. Now, the question is what is the solution
to this differential equation? It's very different
from what we saw before, because before,
we had a zero here. Now we have here
a driving force. It's clear that if you wait
long enough that sooner or later that system will have to start
oscillating at that frequency. In the beginning, it may be
a little different. In the beginning, it may want
to do its own thing, but ultimately,
if I take you by your arms and I shake you back and forth,
in the beginning you may object, but sooner or later,
you will have to go with the frequency
that I force myself upon you. And when we reach that stage,
we call that the steady state as opposed to the beginning, when things are
a little bit confused, which we call
the transient phase. So in the steady state,
the object somehow must have a frequency which is
the same as the driver, and it has some amplitude A. And I want to evaluate with you
that amplitude A. So this is my trial function that I'm going to put
into this differential equation. x dot equals minus A
omega sine omega t. x double dot equals minus A
omega squared cosine omega t. And so now I'm going
to substitute that in here, so I'm going to get minus A
omega squared cosine omega t plus k over m times A
cosine omega t, and that equals F zero divided by m
times the cosine of omega t. And that must always hold. So therefore I can divide out
my cosine omega t. I can bring the A's together, so I get A times k over m
minus omega squared equals F zero divided by m. Now, this k over m is something
that we are familiar with. If we let the system do
its own thing-- we bring it away
from equilibrium and we don't drive it-- then we know that omega squared,
which I will give the zero, equals k over m. This is the frequency
that we have dealt with before. This is the driving frequency--
it's very different. And so I'm going to substitute
in here for k over m omega zero squared, and so I find, then,
that the amplitude of this object here at the end
of the spring will be F zero divided by m divided by omega zero squared
minus omega squared. And this amplitude has
very remarkable characteristics. First of all, if I drive the
system at a very low frequency so that omega is much,
much smaller than omega zero, we call omega zero often
the natural frequency. It is the one that it likes. If you have omega much,
much less than omega zero, this goes--
omega zero squared is k over m-- so you get an amplitude A
which is F zero divided by k. If you go omega way
above the natural frequency or, let's say,
omega goes to infinity-- it becomes very, very large-- then downstairs becomes very,
very large, so A goes to zero. But now, what happens when
omega is exactly omega zero? Then the system goes wacky. Look what happens. The downstairs becomes zero and
the amplitude goes to infinity. And that's
what we call resonance. So if we drive it
at that frequency, the system goes
completely berserk. I can make a plot of A
as a function of frequency. When I say "omega," you can obviously always change
to hertz, if you prefer that, because omega is two pi times F,
so you can do it either in hertz or you can do it in radians
per second, of course. So if I make a plot of the amplitude versus
frequency omega, then at low values-- I have here
F zero divided by k-- is the amplitude. When I hit
the resonant frequency, the natural frequency
of the system, it goes out of hand,
it goes to infinity. The moment that omega is
larger than omega zero, notice that
the amplitude becomes negative. A negative amplitude
simply means that you get all of a sudden
a phase change of 180 degrees, so the object is 180 degrees
out of phase with the driver. I will not expand on that too
much today, but it is negative, and so it comes up here,
and then it goes here to zero for very high frequencies
of omega. So something very spectacular
is going to happen at the resonant frequency
of the system. In practice, of course, the amplitude will not go
to infinity, and the reason for that is that there is always friction
of some kind. There is always damping, but
you get a very high amplitude but not infinitely high. So if I make you a more
realistic plot of the amplitude, and I will take now the
absolute value, the magnitude, so we don't have to worry
about it getting negative-- we don't have to worry about
the 180-degree phase shift-- then you would get a curve
that looks like this. And here, then,
if this is frequency F, then here you would get the natural frequency
when things go out of hand. And depending upon
how much damping there is, this curve would look either
very narrow and very spiky-- it goes very, very high, then there is
very little damping-- or if there is a lot
of damping in the system, it's more like this. So the narrower this... We call this
the resonance curve. The narrower that is,
the less damping there is. I have here a system
on the air track which is an object
that I can drive with a frequency
that I can choose. Here is a spring...
object mass m, and here is another spring. It's fixed on this side,
right there, and here I'm going to drive it, so I have here this variable
force that you see there. And what I want to show you now
is that first I will drive it at a frequency which is way
below the resonant frequency. Then you will see an amplitude,
not very large. I will then drive it way
above the resonant frequency. Again, you will see an amplitude
which is very low. If I can go very high frequency, you will see
that it almost stands still, and then I will try to hit
the resonant frequency, and that will be... It's about one hertz,
the resonant frequency, if I just let this object
do its own thing, just this. Now you see
the natural frequency-- it's about one hertz-- but now I'm going to drive it
here with a system, and we are going to pull
on this spring with a frequency
that we control. So let me start. You see here the frequency. You see an indicator, very low,
way lower than one hertz. And when you look at the way
that the system responds, if you wait long enough when
the transients are died out, you will see
that they go hand in hand, that the amplitude is
in phase with the driver. That's why we have
plus amplitude here. But the phase is
not so important today. See, they go hand in hand. Very small amplitude,
roughly at zero divided by k, which is the spring constant
of the spring. Now I'll go way above resonance,
and this system will slip here, so don't pay attention
to the arrow anymore. Way above resonance. You see, it starts to slip. Look at the amplitude--
very modest, very small. But we went over this curve, so first we probed it here
and now we're probing it here. Look, it's almost not moving
at all, almost standing still, and I'm driving it
at a high frequency now. And now I'm going to find you
this resonance, which is near one hertz... which is somewhere here. And look-- very high amplitude. If we're not careful, then we
can actually break the system. Very high amplitude--
I'm trying to scan over it now, go a little bit off resonance. Now I'm back on resonance. See what a huge amplitude! Better turn it off. So you see here the response
when I drive a system. When my system is
a little bit more complicated-- for instance,
if I had two masses here so I would add one here, spring constant k,
spring constant k-- I could repeat this experiment, and if I did that, I would find
two resonant frequencies. And if I do it
with three objects, I would find
three resonant frequencies. If I did it with five, I would
find five resonant frequencies. And when I make, then, this curve of A amplitude
as a function of frequency-- either hertz or in radians per
second, whichever you prefer-- then if I had three objects
there in a row, you would see
something like this. And depending upon how many
of these objects you have, you get more
and more resonances. And these resonances
can all be found by driving the system
and searching for them. If I go to a system
whereby I have an infinite number
of these masses... We call them
coupled oscillators; these oscillators are coupled
through the springs. An infinite number
of coupled oscillators would be a violin string. Here's a violin string. And the reason why I call it "infinitely" number
of oscillators is that I can think
of each atom or each molecule as being driven, as being connected by springs
to the neighbor. And so it's an infinite number
of coupled oscillators. And so when I start to shake
this system, I would expect
a lot of resonances, and that's what I want
to explore with you now. In the case here, that the objects move in the
same direction of the spring... I call this the y direction and
I call this the x direction, so the spring is
in the x direction, the objects, the beats
are in the x direction, and the oscillations are
in the x direction. We call those
longitudinal oscillations. There is also a way
that you can have transverse oscillations,
transverse... whereby the motion is
in the y direction, whereas the beats are
in the x direction. I could even do that
with this system-- I could make them
oscillate like this, because the springs will
obviously also work if I do this with the system. And that's the way I want
this violin string or piano string
to oscillate now, because that's
the only meaningful way that I can make it oscillate. And so I wonder-- if I'm going to drive it here
by shaking it up and down, searching for
the resonant frequencies-- what I will be seeing. If I go with
very low frequencies, the string laughs at me,
the string does nothing. It's just bored,
doesn't respond to me. I am somewhere here
in the resonance curve. But then when I increase
the frequency slowly, I hit the very first frequency
at which it likes to oscillate. I call that f1. And when I look at the string,
and you will see that shortly, the string will oscillate
like so. It will go up here,
and it will go down here. And all it will do is this. (makes swishing sounds) That's its first resonance. And I call this n equals one,
often called the first harmonic. And then I go
to higher frequencies, and it will not do very much. It's very unhappy. And then all of a sudden,
I hit a second resonance. I call that f2. And the second resonance
will show up like this. This point of the string
will not move at all. When this part is up,
this part will be down. I call that s... n equals two. And it oscillates like so. (makes swishing sounds) And this point
which is not moving at all, we call that a node. I go a little bit
beyond that second resonance, and nothing happens-- the
string will be quite unhappy, sort of flutters a little bit until I hit
another resonance, f3, and then is the next resonance. I will see two nodes appear,
one here and one here, and the string
will oscillate like so. This goes up and down,
180 degrees out of phase, and these two ends are in phase. This is n equals three. And I can go on like that
and add nodes, and I will show you
some of that very shortly. The frequency that I generate-- I call that f of n,
n is an integer; it could be one, two, three
or four-- is linear in f1. In other words,
if f1 were 100 hertz, then f2 would be 200 hertz
and f3 would be 300 hertz. We call n equals one-- we call
that the first harmonic. Some books also call it
the fundamental. I will call it
the first harmonic. And we call n equals two
the second harmonic. And so on-- n equals three
is the third harmonic. So we're going to get a series
of discrete frequencies which were equally spaced. f1 depends on the length
of the string, on the tension-- I will write "tension"; don't
confuse this with period-- and it depends
on the mass of the string. Without going into the detail
how the dependence is, these are the parameters that
determine the first harmonic. I have here a very special
violin string or piano string, whatever you want to call this. And we can generate
these resonant frequencies by searching for them, and I would need assistance
from a student. Would you be willing to help me? So, you hold one end of
the piano string in your hand. Don't let it go, please,
don't let it go. I promise you
I won't let it go either. Okay, so I put
a certain tension on it, and I start to shake
at a very low frequency. Look how low, and look how happy
this string is-- nothing. It just laughs at me,
ignores me, it doesn't like me. Now I'm going to increase
the frequency. And now I'm hitting the first
resonance-- it's coming up. There it is, and that's exactly the shape that you see
on the blackboard-- very clear. I shake it here. I have exactly
the resonance at one. Let me now try to hit
the second harmonic. If I go up a little over the f1,
then nothing much happens. It's hard for me to see. Is this... is this
the second harmonic, or is this already the third? (chuckling ):
It's already the third. Let me see
whether I can get the second. I think I got it now; do I? Okay, so here you see
the second harmonic. You see, indeed, that node,
that point standing still, and the amplitude is enormous. That's a characteristic
for a resonant frequency. We call it also
a normal mode frequency or a natural frequency. It's all the same thing. Let me now try to generate
one that is very high, as high as I possibly can, and you tell me
which harmonic it is. All you have to do is count
how many nodes there are, not counting my hand
and his hand, and you add one, and that is
the harmonic that I generate. The system is not too happy. You see, you feel it sort
of when it comes... Yeah, yeah, yeah, yeah... No, no, no, no, no... See, now I'm off resonance. It's very hard for me to hit
resonance, but I will get it. There it comes, there it comes,
there's no question now! I got it, I'm on resonance now!
Look at it! Clearly I'm on resonance! So, how many did you count? STUDENT:
Six. LEWIN:
Six harmonic? Looked 12 to me, but okay,
you count better than I can. Okay, thank you very much. So, you see here how
the system responds to a driver, and a complicated system
like a string has many, many
of these resonant frequencies, which we call the normal modes. Now, when you have a violin,
you have four strings. They all have the same length. There are musical instruments
like a piano, whereby the length is different. Violin-- four strings,
all the same length. You can set the tension. The tension is changed,
normally, when you tune the instrument
before you start playing, but these four strings
all have a different mass, and that gives them
a very different frequency. Now, if you play the violin, you cannot change the tension,
of course, during the playing. That would be
a little difficult, although there are
instruments that exist where actually
the playing depends on the tension of the string. With the violin, that's not
the case, so with the violin, the only option you have
while you are playing it is to make
the string length shorter, and you do that by moving
your finger along the string. By making it shorter,
the frequency goes up, and by making it longer,
the frequency goes down. Now, how do you excite
a musical instrument like a piano string
or a violin string? There is nothing
that is driving it exactly at
that resonant frequency, so if you want to get a 440
hertz out of a violin string, then you are not driving it
exactly at the 440 hertz, like we are doing. Well, that's true,
but if you take a bow and you rub the string
with a bow, then in a way what you're doing
is you're exposing that string to a lot of possible
frequencies, not just one. But it is a rubbing action. I could also rub it
with my finger, or I could pluck it, or I could
kick it, and what it does is it ignores all the frequencies
which are not at resonance, but it picks out the ones
which are at resonance. So striking it with a bow
is effectively exposing it to a whole large spectrum
of frequencies, and it picks out
the ones that it likes. In fact, if you strike
a violin string with a bow, you may excite it simultaneously in the first harmonic
and in the second, and even in the third harmonic,
or even higher harmonics. And that makes the difference
between the various instruments. That gives it
the special tone quality. Depends upon the cocktail, the combination of the various
harmonics that you excite. Now, if you go
to woodwind instruments, then the situation is
quite different. I have here a sound cavity. It's a box that has length l,
and there is air inside here. And I want to see whether this system has
resonant frequencies. So I put in here
a little loudspeaker to generate sound
with different frequencies and search for resonances,
and there are. The resonances
of this system, however, are different from the string
in that sense. It's not the box
that will resonate here, but it is the air itself
that starts to resonate. Air acts like a spring. In fact, in that sense, it is very parallel
to our spring system. It is also
a longitudinal oscillation, whereas that is
a transverse oscillation. So you make
pressure waves inside, and if you do that
at the right frequency, the air acts like a spring and then you get
strong reactions, which are the resonances. In this case, the nth mode,
the nth harmonic, is given by n times the velocity
of sound divided by 2L. And v is about 340 meters per
second at room temperature, so it is the sound speed. Notice it's again linear in n. In other words, if an instrument
produces a certain frequency-- let us say that
L equals 25 centimeters-- then you can calculate
the frequency f1. You know what the speed
of sound is, 340, so we take n equals one, so the frequency
that we would get equals 340 divided by 2L,
which is 0.5 in meters, and so that gives me 680 hertz. So that would be this big,
it would give you 680 hertz, but the second harmonic, f2,
will be double that. So it will be 1,360 hertz,
and so on. Now, this system which
is closed on both sides wouldn't be
a very good musical instrument because your sound
would not come out. So what people do,
they make them open, a sound cavity which is
just open on both sides. Like this, for instance--
this is open on both sides. And even though
it may surprise you, if we put a little
loudspeaker here, we can excite the air column,
even in this open system, in a complete similar way than we do it here
in this closed system, and we get exactly
the same series of frequencies that I put here
on the blackboard. There are also
musical instruments which are open on one side
and closed on the other. So this is called
an open-open system, and this would be called
a closed-open. A clarinet is closed-open. Here I can also get a series
of resonant frequencies, even though here, the series
wouldn't be exactly like this; it's a little different. It doesn't matter now
how different, but it is a little different. But you get, again, a whole
series of resonant frequencies. So, again, you see that
if I make the system longer, then I get lower frequencies. So if I make the length
one meter, yay big, then I would get a frequency f1, which is about four times lower
than the one that I have there, which is 170 hertz. And if the second harmonic were,
then, 340 hertz, and so on... So when you see an organ
in a church, you see all these organ pipes,
different lengths. The long ones have
the very low tones, and the short ones have
the very high tones. And that is the way
that these instruments work. I have here a wind organ which I
have demonstrated to you before. It is open on both sides, and because it is corrugated
in a very special way, when I blow wind past this,
it will go into resonance. That's to say, wind here is like a spectrum
of all possible frequencies. It's like the bow on the violin. And then it picks out
the frequencies that it likes. However, if I increase
the speed of the wind, I can try to force it
into higher frequencies. So at low wind speeds, I am more likely to hit
the lower harmonics. At high wind speeds, I am more likely to hit
the higher harmonics. This one is 75 centimeters long,
it's open-open; and if I were able to hit
the first harmonic, that would be a frequency
of about 240 hertz. I may not be able to excite the lowest harmonic, the
first harmonic, but I'll try. But certainly the higher harmonics
are very easy to excite, and I can make you
even listen simultaneously to more than one frequency, just like the violin string,
when you strike it, can simultaneously oscillate
in a combination of these modes. So let me swirl this around. I'll first start low. (tube hums medium tone) This may be the second harmonic. 480 Hz (tube hums higher tone) This is definitely way higher. (tube hums even higher) (tube hums higher,
then starts going back down) (tube returns to first tone) This is again the second harmonic. 480 Hz Were there times that
you heard two simultaneously? (tube hums two tones) I hear two. If you play a flute,
that... you do the following: You make holes in here, and when you take
your fingers off both holes, then the effective length
of the flute is this long and you get a high tone. If you put your finger
on this one, then the effective length
of the flute is this long and you get a lower tone. When you put your finger
on both, the effective length
of the flute is this long and you get an even lower tone. I have here a very special
flute, open on both sides, and here you see the two holes. We will first close the two. That gives us
the lowest frequency... (plays low tone,
then up and down triad) (plays short tune
on low tones) ...that you play the flute. So it's simply a matter
of making the instrument longer or shorter. I have another example
here of an instrument which is open here
and here is closed. This is my version
of a trombone. There's a piston here. So this is a system
that is like this, but I can move this in and out, and so when I have it far in,
the frequency would be high... (plays high tone) and when I have it here... (plays low tone) (tones gradually get higher) So you see,
it's directly related to the length of this trombone. And if you learn how to play,
you can try to play a tune. I'll try. (plays "Jingle Bells") (class laughs quietly) (tune continues) (tune ends) (class applauds) LEWIN:
Thank you. Resonances are all around us. When you drive your car,
you may have noticed that all of a sudden, you hear
some crazy rattle somewhere. It could be your mirror,
or it could be something else, it could be an ashtray, because you are driving it,
and you are exciting it with the frequency
with which the wheels go around and you may hit
a resonant frequency. You go a little slower
and the rattle stops, but something else
may start to rattle. The reason
why the first rattle stopped, because you go off resonance, but you may go on resonance
of another object. All objects that you have
in your room have preferred
resonant frequencies. Whether they are
the pots or the pans or whether it is
your refrigerator, or anything you can think of, everything has
resonant frequencies. Your body has
resonant frequencies. If I took you in my hands
and I would start to shake you, then if I do it
at low frequency, not much would happen,
but there would be one frequency that your arms begin
to move like this, like they're
physical pendulums, right? And if I hit that frequency, then indeed there would be
a strong response, and so your body has
many resonant frequencies: your arms, your legs,
your head, everything. We also experience, all of us,
emotional resonances-- a small input, a huge output. Falling in love is
an emotional resonance. If someone touches a sensitive
nerve, that is a resonance. Someone could say
something to you and it could be
a very sensitive issue for you, and you go nonlinear! Your response is
unbelievably strong! That, in my view, is also a form
of an emotional resonance. I have here two tuning forks. And these tuning forks
are designed in such a way that all I would have to do
is just bang them and they will pick out
their own resonant frequencies. The tuning fork is
very simple, like this. I give it a kick, and the kick is like dumping a whole spectrum
of frequencies on it, and it picks out the ones that
it likes, which is this one, and that's the one
at which it will resonate. It's possible that I can excite
it at higher frequencies, at higher harmonics, but that's a little hard,
even, with a tuning fork. So this one is 256 hertz. (plays low tone) So the prongs move
256 times per second, and this one is 440 hertz. (plays higher tone) And you hear no overtones--
you don't hear higher harmonics. If I take something as simple
as a wineglass like this, it has many, many
normal-mode frequencies, many resonant frequencies. The lowest one is
very easy to excite, and I will do that. I will rub it with my finger. Rubbing is like striking
a string with a bow. It's all the same-- I expose it
to lots of frequencies, it ignores them all, it picks
out the one that it likes, the resonant frequency, the normal mode,
the natural frequency. And what the glass will do, in its lowest harmonic--
in its first harmonic-- it will sort of oscillate,
like this, and I will show that to you
later in this lecture in slow motion. But let me first make you
listen to the frequency. It's around 430... 470 hertz. I have to wash my hands because
I have chalk on my fingers and because of the chalk,
I will not be able to excite it. I have to really rub it
with some liquid, and therefore
my hands have to be chalk-free. Chalk is just too greasy--
let me try. (plays high tone) Very clear. I remember when I was a student,
we had after-dinner speakers. If we got bored,
we would all do this. Let me tell you,
that makes a lot of noise. (tone continues) So, again, I'm not exciting it
with the 470 hertz. I'm dumping
the whole frequency... the whole spectrum
of frequencies on it, and it picks out
the one that it likes, which, in this case,
is the 470 hertz. Resonances can be destructive, and rumor has it
that there are singers, women who take a wineglass
and they do exactly what I did. They do this. (plays high tone) They listen carefully, they generate that frequency
with their voice, they increase the
volume of the voice, and then the rumor has it
that bang, the glass goes. In other words, the amplitude
of the glass becomes so large that you get so high
so close to resonance and so much power goes into it because of the volume
of the voice that the glass breaks. I'm going to try to break
a glass with you, and you'll see
that it is not easy. I have here a wineglass. It's almost the same
as the one I have there. And we can illuminate
that glass with a strobe light, which you see here. And I'm going to show you there the display
of that strobe light. And the reason why we strobe it
is that we want you to see, as we excite the exact
frequency of the glass, we want you to see
the motion of the glass. And the way we can make you see
the motion is by strobing it not exactly at the frequency
of the sound but a little bit
different frequency. So you will see
the stroboscopic motion, then, of the glass. So I can... already
I will make it darker shortly, but I want you to see
at least most of this. I can generate, then,
the 470 hertz, which is very close
to the resonant frequency. (plays high tone ) This is the tone
that we will use. We will increase the volume, and then we will try to hit
that resonance just right. We may be off by a few hertz. We have to be right on
within a hertz, and then we'll see whether
we can make the glass break. Now, I want to warn you, the sound is going
to be very strong, so you may... as the time goes
on, as we go to higher volume, you may want to turn, you
may want to close your ears. In fact, I will use this
to protect my ears, and I will even use this
to protect my eyes in case the glass might break, which I doubt whether it will,
but who knows? All right,
so let's make it very dark. (tone continues) So, you see here the glass. It's not doing very much. And I'm now going to increase
the volume of the sound. (tone gets louder) I'm going to cover my ears now. We already begin
to see some motion. I'm not sure
that I am on resonance. (tone continues,
becomes louder) We increase the volume. (tone continues, louder) (tone continues) (tone continues) I'm now changing the frequency. (similar tone playing) Getting very close. (tone continues,
getting slightly louder) Getting very close. (shatters loudly) (tone fades away) (lower tone plays
as slow-motion replay begins) (shatters slowly) I mentioned that resonances
can be very destructive, and there are some striking
examples in history. In fact, you may have... When there's a storm, you may
have seen the traffic signs, which... just a sign on a pole, that the traffic sign
starts doing this. This is strange,
because there's a wind. Wind is like a spectrum
of all kinds of frequencies. It's like blowing air
into a musical instrument. And then this traffic sign
just picks out the frequency that it likes,
and then, if the wind is strong enough,
it could be very destructive. And the most striking example
of destruction is that of a bridge
which was built. And that is the Tacoma Bridge,
on the West Coast, which is very dramatic, and, of course, I will have
to show you that movie. You will see what
the wind can do to a bridge. So we'll start this movie,
and then we'll make it dark. FILM ANNOUNCER:
On the first of July, 1940, a delegation of citizens met
in Washington State. The weather was beautiful,
the occasion historic, and the speechmaking and fanfare
altogether appropriate. This was the grand opening
of the Tacoma Narrows Bridge. From the beginning, the bridge, which spanned Puget Sound
between Seattle and Tacoma, was traveled in style,
as well it should have been. The Tacoma Narrows Bridge was one of the longer
suspension bridges on earth. And if somebody hadn't
overlooked something, it probably would have remained one of the longer
suspension bridges on earth. The problem wasn't that
right from the beginning, a lot of people didn't pay
a lot of attention to details-- they did. But somewhere along the line,
and this was obvious in the end, it looks as if someone forgot
the significance of resonance. (low rumbling as wind blows) Among other things,
the Tacoma Narrows Bridge was the most spectacular
Aeolian harp in history. Unfortunately,
its first performance was destined to run
only about four months. (sirens wailing) In the meantime, she was
a beautiful bridge... beautiful, but a little strange. Even before construction
was completed, people observed
its peculiar behavior. That was because
even in a light breeze, ripples ran along the bridge. After a while, one of the local humorists
called her "Galloping Gertie." And for fairly obvious reasons,
the name stuck, at least until
the seventh of November, 1940. Then as now, Seattle and Tacoma
were sports-minded cities. For four months,
a regional sport was to drive across
the bridge on a windy day. While some claimed it was
like riding a roller coaster, others found it
a little disconcerting to see the car
in front disappear. How popular
this bridge sport was, or to what extent it might have
spread across the country, is anybody's guess. On November 7, 1940, the winds
were relatively moderate, about 40 miles per hour. A new mode appeared. Rather than ripple,
the bridge began to twist. (wind blowing) A wind of 40 miles per hour
is not too strong, but it was strong enough to start the bridge
twisting violently. And at 11:00 a.m., it fell. (low roaring) (loud crashing) (rumbling as wind blows) (rumbling continues) LEWIN:
Amazing-- amazing
what resonance can do. With musical instruments,
we have... we're stuck to the speed
of sound in air, which is 340 meters per second. And when I speak to you,
I have here a sound cavity which has a certain size,
a certain shape. And while I speak, this size and
this shape changes all the time. And that gives my voice
a very characteristic sound that makes it possible for me
to make a low sound. It makes it possible for me
to make a high sound. And no matter how I speak, you will say,
"Yeah, there's no question. That's Walter Lewin--
that's clear." It's very recognizable. Each one of you and I have
a very well recognizable voice. The situation would change if I could change the speed
of sound in my system. And helium, which has a very different
molecular weight from air, has a way higher speed
than the speed of sound in air. It's about 2.7 times higher,
2.7 times 340 meters per second. So if I fill my system
with helium-- apart from the fact that
there's no oxygen in helium, so I wouldn't survive very long;
that's a detail... (class laughs) LEWIN:
If I fill my system with helium, then, of course, my vocal cords
and my sound cavity here, they don't know that. So they do the same thing that they normally do
when they speak to you. However, since the speed
of sound is much higher here in my sound cavity, the frequency that it
will produce is way higher, and I sound very differently. The problem, however,
is, as I mentioned, there is no oxygen in helium, and yet if I only take
a little bit of helium, it doesn't work very well and
you don't hear much difference, so I have to really take a lot
of helium, and I will try that. But for that I pay
a small price, and the small price
is that you can faint. I'll try just to stay away
from the fainting level. So, here is helium, and I have
to let my air out first and then get helium in and maybe
let the helium out again and let the helium in again. And during that time
I can't speak, but then, when I think
I'm full enough with helium, I will say a few words to you. So, let's get the pressure up
to the level that I like. (exhaling) (helium hissing from hose) (inhales) (helium hissing again) (inhales again) (in very high voice):
I told you it would sound
very differently. (class laughs and applauds) Well, I hope you enjoyed this,
and I'll see you Wednesday. Thank you. (class laughs) (exhales)
