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  • It's amusing, after your morning coffee or tea,

  • to start tapping on the cup with your spoon.

  • It's pretty much the same pitch everywhere, tun-tun-tun, whatever it is, and then maybe

  • it's the characteristic pitch of this cup, and if you take another cup and

  • another spoon, for that matter, maybe you get a different sound.

  • Who knows? So maybe it's always the same pitch from this particular cup

  • if tapped by this spoon. But then, I started tapping somewhere else.

  • Those four points emit a common pitch,

  • And those four points also emit a common pitch,

  • but a higher pitch than those four points. We'll explore what's causing this,

  • but you might first think that it has to do with the handle. That seems to be a very

  • suspicious culprit, and thank you very much for pointing that out too.

  • Of course it has to do with the handle, but-- if it has to do with the handle, wouldn't you have

  • thought that this half closer to the handle and this half farther from the handle

  • should behave differently?

  • But that's not how the symmetry is broken.

  • Indeed, the point next to the handle and point farthest from the handle

  • behave exactly the same way. Whereas 45 degrees off, you get the higher pitch.

  • It has to do with the handle, perhaps, but the symmetry-breaking pattern is not so naive

  • as we at first thought. Let's understand what's happening, and in order to do this we'll

  • break it up in two stages. First, we'll understand why any quadruplet of four

  • points that form the vertices of a square always give you the same pitch,

  • a common pitch, and then we'll understand why this quadruplet emits a

  • higher pitch than those four points.

  • Okay, so let's forget about the handle. So the handle-- pretend the handle's not there.

  • When we tap this point, we are making this point vibrate. We're setting it in motion

  • like this.

  • Well, roughly speaking,

  • the point diametrically opposite can react in one of the two ways. When this

  • point is doing that, the other point can go like this,

  • Or go like this. We say in phase

  • or out of phase. Now, this reaction-- this response-- is essentially moving the

  • cup back and forth, back and forth, sliding it as a total rigid body, as a

  • whole body, and it does not really (to a good approximation) deform the cup.

  • But, of course, the sound has to do with the deformation of the cup, how it's vibrating altogether.

  • So, so this is not producing sound-- that's not what you're hearing--

  • what you're hearing is this response when this point moves back and forth

  • this goes in and out and at the same time like this

  • - Professor, why does that make a sound, by the way? - Because sound means that somebody

  • (for example, it can be your throat, it can be for example birds chirping,

  • it can be a cup chiming) is vibrating very, very fast and shaking air around it, and when you

  • shake the air like this

  • the air next to it gets compressed and bounces off because the air has some

  • elasticity, and then that bounce compresses the next part of the air and bounces off

  • and there's a wave that propagates like this towards the camera and eventually

  • reaches the ear and inside the eardrum this wave is shaking the eardrum and

  • that's what you hear as a sound. Okay. So the vibration and deformation ultimately, of

  • the very rapid deformation of the cup, is producing the sound. And this response is not

  • really deforming the cup, whereas this one is, so. And this is what's primarily

  • responsible for the sound. On the other hand, the cup as a whole doesn't want to

  • change its volume if it can help it, it wants to stay as incompressible as

  • possible. In other words, when those two go in, well, by reaction these two are pushed out

  • in order to keep the same volume, and if these go out these two are pulled in so that

  • you get this kind of rhombus-type oscillation, pum-pum-pum-pum-pum, and that's why any

  • quadruplet of four points that form the vertices of a square always sing in unison.

  • It doesn't matter which of the four points you start exciting, all the other

  • three follow, and four together go into this losange

  • or rhombus vibration. Well, so we now understand why this quadruplet of points and

  • this quadruplet of points each

  • emits a common pitch. Now, why this quadruplet of points are higher-pitched than this

  • quadruplet of points are lower-pitched-- in order to understand that we must

  • resurrect the handle that we have been neglecting so far. So the handle comes back.

  • You see, when-- let's say that we excite one of the four points, those,

  • those four points. As we saw, it doesn't matter which of the four points you

  • excite because all of them work in unison. When those four points are

  • vibrating together, they have to take the handle, drag the handle with them, because

  • you see, one of the points, vibrating points, is attached to the handle, so the

  • handle must move back and forth while those four points are vibrating. In contrast

  • when those four points are made to vibrate, you remember what happens-- when this goes in,

  • this goes out and this goes in, this goes out. The point just in between is

  • what we call a node. In other words, to a good approximation, it's stationary,

  • it doesn't move in between, so as far as these quadruplet of points are concerned,

  • the handle is invisible-- it's as if the hand were not there because it's attached to

  • a point that's not doing any vibration, vibrational motion. So we now have two

  • elastic systems-- you can think of them as springs if you like-- this one and this one.

  • One of them is attached to some heavy mass that he has to drag along; the other

  • one is not attached to anything because this is invisible to those four points.

  • And they are made of the same stiff material that is the cup. So imagine two

  • springs, one of them is attached to a heavy mass, the other one is not attached

  • to a heavy mass, and when you let them go what you hear-- the heavy one goes *wmmm*

  • [imitates low-frequency oscillations]

  • in a very sluggish way, whereas the back goes hee-ho, hee-ho, and that's the difference of the

  • pitch that we hear. And that is why these four points emit a low pitch whereas those four

  • points emit the higher pitch. So far, we saw where the handle was and we try to figure out

  • what this sound pattern was like. The inverse problem is also interesting

  • by which I mean, since we're talking about the pitch of the sound that we hear,

  • let's imagine ourselves in a pitch dark room. Say that we walk into this room.

  • We don't know where the handle is, but imagine that we are allowed to go around

  • the cup, tapping it everywhere, and record the sound pattern that we get from each point.

  • From that recorded data of the sound pattern can we reconstruct where the handle was?

  • That is the inverse problem. A bit more abstractly speaking, instead of saying

  • that we know the cause and then by solving whatever mathematical model,

  • partial differential equations, whatnot, you try to predict what the effect

  • is going to be, we are going the other direction. We have this observed data,

  • effects if you like, which we know to be true, I mean, we just see this, and we

  • then try to figure out what on earth is causing, is responsible for the

  • production of this effect. So this is the inverse problem rather than our forward

  • problem. And if you think about it much of the scientific endeavor is really

  • about solving an inverse problem. We wanted to get to the bottom of things,

  • want to recover the cause, and this very, very simple everyday morning breakfast

  • example shows that the inverse problem is not always naively solvable. Indeed

  • you probably agree with me that if I took a cup and rotated 90 degrees, the

  • basic pattern that we hear

  • in a north-south-east-west and so on and in between will be the same.

  • We don't know where the handle is up to 90 degree rotation. You can be here, here,

  • here, here. So we can narrow it down to four possible positions

  • Yes. Well, okay, so there is some sort of ambiguity, you know, as regards the

  • position because the handle can be here or here or here or here and you'll get the same

  • pattern of the noise. But there's something even worse.

  • You can't even tell the number of handles because instead of having a large handle

  • here you could have two maybe medium handles, and you agree that the pattern

  • of the vibration will be the same, or indeed four small handles here and here

  • and here and you'd probably get still the same, what we call the ground state or the

  • same kind of vibration pattern. So you can't even tell the number of handles

  • and you can't tell the position of handles but up to symmetry, you can actually

  • start saying something and this is a very typical solution in the

  • typical situation. You wanted to "mod out"-- you have to sort of neglect and

  • somehow take out this mentally and then start solving the problem. Anyway, so this

  • very simple example shows that the universal problem is on the whole not always

  • solvable. But you can actually say something when you sort of start

  • handling the symmetry in a clever way.

  • - If we turned the lights out, and I was trying to figure out the cup, can I do it?

  • - We in general can't, but we can say something up to symmetry. You know,

  • we know that for example there aren't three handles because that would break the symmetry in

  • a completely different manner. In fact, ever since I noticed this phenomenon

  • I've been looking for a cup with 3 handles because that would be very interesting

  • and recently a friend of mine, Brian White of Stanford, got me 2 cups

  • and one of them has three handles. Now, I'm-- unfortunately it's not a very nice cup

  • in-- I mean, it is a very beautiful cup-- but for our purposes, for our

  • purposes it's not a very well-performing cup because it doesn't really chime nicely.

  • And in between.

  • So it's somewhat interesting, I mean, those three points

  • and those three points in between emit a common pitch, but in between...

  • I'm not sure that I can do this. You can't really hear the difference.

  • It's a very very small difference.

  • Yeah. So in between there's a slight rising of the pitch that I'd like to

  • hear more clearly, so we need a larger cup right away. And this is a very choppy

  • chunky mug from Stanford's math department, but if you take a really well-made thin

  • and delicate china cup, this chime is so much more beautiful, so I recommend it.

  • Arrows 3. Well, it's 3 to the 3 to the 3. You ain't seen nothing yet. OK, so

  • here we go, 3, 4 arrows, one more arrow.

  • Well, what does this mean? 3, 3 arrows of 3, 3 arrows...

It's amusing, after your morning coffee or tea,

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コーヒーカップのバイブレーション - Numberphile (Coffee Cup Vibrations - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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