Kelistrikan dan Kemagnetan

**Topics covered:**

Traveling Waves

Standing Waves

Musical Instruments

**Instructor/speaker:** Prof. Walter Lewin

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So today, I will start with a general discussion on waves, as an introduction to electromagnetic waves, which we will discuss next week.

We'll start with a very down-to-earth equation, Y equals one-third X.

And I'm going to plot that for you, so here is Y and here is X, and that's a straight line through the origin, Y equals one-third X.

Suppose, now, I want this line to move.

I want this line to move with a speed of 6 meters per second in the plus X direction.

All I will have to do now is to replace X in that equation by X - 6T.

Notice the minus sign.

I will go, then, in the plus X direction.

The equation then becomes Y equals one-third times X - 6T.

So look at it at T equals 1.

At T equals 0, you already have the line.

At T equals 1, you now have Y equals 1/3 X - 2.

That means, here it will intersect at - 2, and there it will intersect at + 6, and the line parallel to the first one, this line is now T = 1, and this is T = 0.

And it has moved in this direction, with a speed of 6 meters per second.

And so what this is telling us, that if we ever want something to move with a speed V in the plus X direction, then all we have to do in our equations to replace X by X - VT, and if we want it to move in the minus X direction, then we replace X by X + VT.

That's all we have to do.

So now, I'm going to change to something that is a real wave.

I now have Y = 2, times the sin 3X.

That's a wave.

It's not moving, not yet.

So I can make a plot of Y as a function of X, and that plot will be like this.

This is zero, so when the sine is zero, this is pi divided by 3, and this is 180 degrees, and it's again zero, this is 2 pi divided by 3, it's again zero.

And lambda, which we call the wavelength, lambda, in this case, is from here to here, that is 2 pi divided by 3, this goes also from here to there.

I will introduce a symbol K that you will often see, we call that the wave number, and K is simply defined as 2 pi divided by lambda.

So in our specific case, K is 3.

This here is K.

If you know this number, you can immediately tell what the wavelength is.

Now, I want to have this wave move.

I want to have a traveling wave.

And I want to have it move with 6 meters per second in the plus X direction.

So the recipe is now very simple, all I have to do replace this X by X - 6T.

So now I get Y equals 2 sin [3(X-6T)].

And if you now look at this curve, this equation, and you plot it a little bit later in time than T0 -- this is already T0 -- a little later in time, you will see that, indeed, it has moved in the plus X direction.

And it's moving with a speed of 6 meters per second.

So this equation, when you look at it, holds all the characteristics of the oscillation.

It holds the amplitude.

This 2 is the amplitude.

This is - 2, that's the amplitude.

This information, K, holds the information on the wavelength, and this information tells you what the speed is.

And the minus sign, which is important, tells you that it's going in the plus X direction, and not in the minus X direction.

Can we make such a traveling wave?

Yes, we can do that, actually, quite easily.

Suppose I have here a rotating wheel -- rotate with angular frequency omega, and let this has a radius R, and I give it 2 units, so that I get the same amplitude that I have here.

And I attach to this a string, and I put some tension on the string, so that I create a wave as I rotate it, and the string is attached here, and as it rotates, the wave is going to propagate into the string with a velocity, let's say, V.

So I can generate a traveling wave.

The period of one oscillation -- if you were here on the string, you're going up, you're going down, you're going up, you're going down, that's all you're doing, when the wave passes by -- the period of one whole oscillation is obviously 2 pi divided by this omega.

The wavelength lambda that you are creating -- from here to here is lambda -- well, if you know the speed with which it is traveling, and you know it has been traveling capital T seconds, one oscillation, that's a distance lambda.

So this is V times T.

But this is also V divided by F, if F is the frequency in Hertz.

And so the frequency F is then also given by the speed divided by lambda.

And so I can write down this equation now in a somewhat different form, Y equals 2 times the sine, and now I bring the 3 inside, so I get 3X-18T.

This 18 is now that omega.

This is omega T.

In here is all the timing information.

Omega, the period T, everything is in here.

Here is all the spatial information.

This is K.

In here is the information about lambda.

And so if I know omega, and I know K, then I can also find the velocity, which is omega divided by K.

So everything is in here, omega divided by 3 gives me back my 6 meters per second.

So once you have the equation, I can ask you any question about that wave, and you should be able, then, to answer.

Wavelength, frequency, in hertz, in radians per second, speed, everything.

You may ask me now, "Why do you discuss this with us?" Well, we are coming up to electromagnetic waves next week, and electromagnetic waves, you're going to see lambdas, you're going to see omegas, you're going to see capital Ts, you're going to see frequency, you're going to see Ks, everything you see there you're going to see next week.

One exception, that Y, the displacement Y, will not be in centimeters or meters, but it will be an electric field, a traveling electric field, volts per meter.

Or a traveling magnetic field, tesla.

But other than that, all these quantities will return in exactly the same way.

Now I want to discuss with you a standing wave first, because standing waves are going to be important.

This is a traveling wave.

And now comes something even more intriguing, which is a standing wave.

Suppose I have a wave traveling in this direction, and I call that Y1, and Y0 is the amplitude, sine (K X - omega T).

And notice now, I have all the symbols that we are familiar with.

We have the K here, we have the omega here, and we have the amplitude here.

And the minus sign tells me, [wssshhht], it's going in the plus direction.

But I have another wave.

And the wave is exactly identical, in terms of amplitude, in terms of wavelength, in terms of frequency, identical, but it's traveling in this direction.

And so this is Y2, which is Y0 sine (KX + omega T).

This plus sign tells me it's going in this direction.

And so if this is a string, the net result is the sum of the two.

So I have to add them up.

So Y = Y1 + Y2.

So I have to do some trigono- trigonometric manipulation, and this is what I leave -- I'll leave you with that, that's high school stuff -- you add the two up and you'll find 2 Y0-- notice that the amplitude has doubled -- times the sine (K X) times cosine (omega T).

That's the sum of those two.

And this is very, very different from a traveling wave.

Nowhere will you see K X - omega T any more.

K X is here, separate under the sine, and omega T is separate under the cosine.

All the timing information is now separate from the spatial information.

And so what does a standing wave like this look like?

Well, let's -- a bracket here.

Let's make a drawing of such a standing wave.

So here we have Y, and here we have X.

Let's only look at the sine K X for now.

If X is 0, the sine is always 0, so this point will never move.

But if K X is 180 degrees, it's also 0, always.

So lambda over two will never move.

X is lambda, when this is 360 degrees, it will never move.

- lambda / 2, will never move.

So what will it look like?

Well, you're going to see something like this, let's take the moment when T equals 0, so when cosine omega T is plus 1.

So we're going to have a curve like this, so this goes up to 2Y0 like this -- and this here is then my 2Y0.

These points will never move, they will always stand still.

There's nothing like a traveling wave.

If it's a traveling wave, these points will see the wave go by, they will go up and down, they never do that.

They sit still.

They have a name.

We call them nodes.

Let's now look at little later.

Let's look at T equals one quarter of a period.

Now, the cosine is 0.

So there's not a single point on the string that is not 0.

So the string looks like this.

If you took a picture of the string, you wouldn't even know it's oscillating.

It would be just a straight line.

And now, if we do -- look a little later, and we look at T equals one-half the period, then the cosine is -1.

So now the curve will look like this.

And so what does it mean?

If we just look what's here happening, this is what's going to happen.

The string is just doing this, and there are points that stand still.

Nothing is going like this, nothing is going like this.

You see this point going up and down, up and down, up and down, and this will do the same, and these nodes will do nothing.

So that is what a standing wave will look like, and I think the name standing wave is a very appropriate name, very descriptive, because it's really standing, it's not -- it's not moving.

At least, not traveling along the X direction.

Can we make a standing wave?

Yes, we can, and I will do that today.

A standing wave can be made by shaking -- or rotating, in that fashion -- a string.

So here I have a string, I -- say I attach the string to the wall there, and I move it up and down here.

So a wave goes in -- I do just this, like the rotating disc -- the wave travels, but the wave is reflected, and so I have a wave going in and I have a wave coming back, so I have now two waves going through each other.

And if the conditions are just right, then these reflective waves -- this one will reflect, when it arrives here, it will reflect again, it goes back again, and it will continue to reflect -- so if the conditions are just right, then these reflective waves will support each other, and they will generate a large amplitude -- as I will demonstrate to you -- but that's only the case for very specific frequencies, and we call those resonance frequencies.

The lowest possible frequency for which this happens -- which we call the fundamental -- will make the string vibrate like this.

So the whole thing goes.

[wssshhht], [wssshhht], [wssshhht], and we call that the fundamental.

We call that also the first harmonic.

If now I increase the frequencies, then I get a second resonant frequency, and a node jumps in the middle -- there is already a node here, and there is a node here, because this motion of my hand here is very small, as I will demonstrate to you, for all practical purpose you can think of this being a node -- and so now the string in the second harmonic will oscillate like this.

[Wssshhht], [wssshhht], [wssshhht], [wssshhht], so this is the second harmonic.

And if we go up in frequencies, then -- this should be right in the middle, by the way -- and if I go up in frequency one step more, then I get another resonance whereby we get an extra node, and so we get the third harmonic, and you just can go on like that.

You get a whole series of resonance frequencies.

And so, for the fundamental, lambda 1 -- the 1 refers to the first harmonic -- is 2L, if L is the length of my string.

This is L.

You only have half a wavelength here, so L is 2L.

But we know that the frequency is the velocity divided by the wavelength -- we see that there, frequency is velocity divided by the wavelength -- so the frequency F1 is the velocity divided by lambda 1, so that's divided by 2L.

So that's the frequency in the fundamental for which this resonance phenomenon occurs.

For the second harmonic, lambda 2 equals L.

You can tell, you see a complete wavelength here.

And F2, that frequency, is going to be twice F1.

And F3 is going to be 3 F1.

And if you want to know, for the Nth harmonic, N being Nancy, then lambda of N equals N -- 2L/N.

Substitute in N1, and you find the wavelength for the first harmonic.

Substitute for N2, and you find the wavelength for the second harmonic.

And so on.

And the frequency for the Nth harmonic -- N stands for Nancy -- is N times V divided by 2L.

So here you see the entire series of frequencies and wavelengths for which we have resonance.

Unlike in our LRC system that we discussed last time, where you had one resonance frequency, now you have an infinite number of resonance frequencies, and they are at very discrete values, equally spaced.

I want to demonstrate this to you with a violin string, it's a very special violin string, it's here on the floor, it's a biggie, and I need some help from someone.

You helped me before, would you mind helping me again?

So here is, uh, one end of the string, which you're going to hold, you're going to be a node, believe it or not.

Hold it better, two hands -- no, much better.

You will see shortly, why -- no, no, no, much better.

That's it.

And walk back a little, walk further.

Yes, that's good, hold it.

I will put on a white glove, and there is a reason for that, because I want you to be able to see my hand when we're going to make it dark, so that you will convince yourself that my hand, which is generating the wave, is hardly moving at all.

For practical purposes, it's a node, and yet we get these wonderful resonance phenomenon.

So I'm going to make it very dark so that the UV will do its job, and you can see the string better, that's the only way we can make you see the string well.

All right.

Don't let go, er- under any circumstances, you will hurt me if you do that.

Of course, if I let go first, then [pfft], I will hurt you, but that's not my plan.

OK, so let's try to go a little bit further back.

Let's try to, uh, find, first the -- the fundamental.

And I'll try to find it by exciting just the right frequency with my hand.

There it is.

I think I got it.

That's the fundamental.

And look how little my hand is moving here.

And you will see a very large amplitude in the middle.

And so these reflected waves, one runs to him, it runs back at me, it runs back at him, keeps reflecting many times, they support each other in a constructive way, that's what resonance is all about.

And now I'll try to find the second harmonic -- so you'll see another node coming in at the middle.

It's easier for you to see than for me, actually.

And it's not always easy to find the -- no, no, no, I'm too low frequency, I have to go up.

I think I got it now.

Is this it?

Yes, one extra node in the middle?

Speak out up, please.

[chorus of agreement] Ah, that's better.

Now I can hear you, thank you.

Um, there are three nodes now.

My friend there is a node, I'm a node, and then there is one in the middle.

If you subtract 1, the 3-1 is 2, then it's the second harmonic.

And so now I will try to generate a very high frequency, in resonance, and then you count the number of nodes, subtract one, and then you know which harmonic I was able to generate.

But I will try to -- not so easy to get a resonance in there.

No, I'm off resonance.

No.

Pengembangan Perkuliahan

1. Buatlah sebuah Esai mengenai materi perkuliahan ini

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5. Kembangkan produk tersebut dengan senantiasa meningkatkan kualitasnya#### Staff

Visualizations:

Prof. John Belcher

Instructors:

Dr. Peter Dourmashkin

Prof. Bruce Knuteson

Prof. Gunther Roland

Prof. Bolek Wyslouch

Dr. Brian Wecht

Prof. Eric Katsavounidis

Prof. Robert Simcoe

Prof. Joseph Formaggio

Course Co-Administrators:

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Technical Instructors:

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Course Material:

Dr. Peter Dourmashkin

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Dr. Sen-Ben Liao

#### Acknowledgements

The TEAL project is supported by The Alex and Brit d'Arbeloff Fund for Excellence in MIT Education, MIT iCampus, the Davis Educational Foundation, the National Science Foundation, the Class of 1960 Endowment for Innovation in Education, the Class of 1951 Fund for Excellence in Education, the Class of 1955 Fund for Excellence in Teaching, and the Helena Foundation. Many people have contributed to the development of the course materials. (PDF)

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