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00:00:01,196 --> 00:00:05,166
- If you blow air over
the top of a soda bottle,

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you get a tone.

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I've got a soda bottle right here.

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I'm going to show you.

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Listen to this (inhales):

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(whistling tone)

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So the question is, why
does it make that noise?

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How come you get that loud sound?

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And it has to do with something
called standing waves,

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or a very closely related
idea is resonance.

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And so we're going to talk about this.

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How do these work?

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So let's go into here.

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What I've really got ...

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I'm going to model this.

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I'm going to say that I've
just got a soda bottle.

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I'm going to model it
like it's just a tube,

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a simple tube, and one end is closed.

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This is important.

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This end over here is closed.

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I tried to shade it in,

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to show you that this end is blocked off.

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That's the bottom of the soda bottle.

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This is on its side.

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And this end over here is open.

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And so what happens,
you've got this closed end,

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you've got this open end,
you got air in between.

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What's the air do?

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Well, when I blow over the top,

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the air starts to move around.

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But this air on the closed
end, it's pretty much stuck.

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If it tried to ...

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It wants to oscillate back and forth,

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that's what these air
molecules want to do,

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but every time it tries to oscillate,

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it just bumps into this
closed end, loses its energy.

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And when I try again,
bumps in, loses its energy,

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so it doesn't really go anywhere.

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Whereas, on this side, this
side's open and, shoot,

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this air can just dance like crazy,

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oscillate back and forth.

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It wouldn't go that far.

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I'm exaggerating here.

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So you can see it.

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But this end will oscillate
much more than this other end,

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this closed end.

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The air just stays there.

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In the middle, it'll oscillate somewhat,

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somewhere in the middle.

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And so, if you wanted to see this,

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I made a little animation
so you can see this happen.

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Here's what it would look like:

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you see that the closed end,
the air's not doing anything.

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At the open end, the air
can oscillate wildly.

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And in the middle, it's a varying amount

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that gets smaller and smaller

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as you get toward that closed end.

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Okay, so that's this bottle.

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That's how we're modeling
this bottle here.

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So, you could do this for ...

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If you cut the bottom out,

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if you cut the bottom out of the bottle,

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you'd also be able to
set up a standing wave.

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It would look like this.

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So let's say we've got
an open end on both ends.

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So now we got an open end on both ends.

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This side is open,

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this side here is open.

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This means the air now, on this side,

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isn't stuck anymore.

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This air can oscillate like crazy.

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This air over here can
oscillate like crazy,

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and it turns out, if
you blew over a bottle

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that was cut open on both ends,

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or if you just had a PVC pipe
and you blow over the top,

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you'd get another resonance.

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You'd get another standing wave.

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And in the middle, this air
molecule would just stay still.

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These would oscillate
like crazy on the ends

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and this is what that looks like.

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It looks a little bit like this.

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So both ends oscillating like crazy

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and then right in the middle,
air not really moving at all.

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And so, this is a standing wave.

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It's a standing wave.

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I don't want ...

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I actually don't like that name.

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I like the name dancing wave.

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I mean, the air is still moving.

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This air is moving back and forth.

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This air is not.

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But lots of the air's
moving back and forth.

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And they call it a standing
wave because, no longer ...

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remember, with a wave ...

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with a wave, you had
this compressed region.

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And what it looked like,

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it looked like the compressed region

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was moving down the
line with some velocity.

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So this is a moving wave.

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But when you set up a standing wave ...

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I'll show you again.

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This standing wave doesn't really ...

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this one here, say ...

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it's not really ...

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the compressed region doesn't look

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like it's moving down the line.

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Everything just kind of
bounces back and forth.

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So, how do we describe
this mathematically?

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That's the hard part.

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This is the part that drives people crazy.

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What we could do, we
could try to draw a line.

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Let's draw some lines that represent where

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all the air particles are in
their equilibrium position.

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Equilibrium position's a fancy name for,

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this is if the air was ...

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this is our open, open tube

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and this is what the air is at

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when you're just not messing with it.

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The air is in this
position, just hanging out.

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I'm going to draw these lines here,

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so that we know, this is
where they want to be.

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And when they get displaced
from that position,

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we'll be able to tell how
far they've been displaced.

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So if you did this, if
you took a PVC pipe,

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you blew over the top,

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this is what the air looks like before.

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Sometime afterward, it
might look like this.

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Now the air's displaced.

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So check it out.

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This one got displaced
all the way over to there.

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This one went to there.

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This one went to there.

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This one went just a smidgeon over.

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This one didn't go anywhere.

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This one went to the right.

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And this one went to the right,

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and this one went to the right.

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And this one went way over to the right,

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because it's at the open end.

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So you've got varying
amounts of displacement

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at different points.

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So what we're going to do,
we're going to graph this.

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I'm going to make a graph
of what this thing is doing.

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So let's make a X axis, a horizontal axis,

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This will represent where
I am along the tube.

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And then we'll make a vertical axis.

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This will represent how much
displacement there actually is.

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So this top end, so this
will be displacement,

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the amount of displacement
of the air molecule.

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And then this is just
position along the tube,

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where exactly am I along the tube?

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I'm just going to call it X.

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And so, if we graph this,
what are we going to get?

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Well, what we're going to get is,

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right here, this air
molecule at this X position

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has displaced a lot to the left.

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And usually leftward's negative.

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So on this graph,

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I'm going to represent
it down here somewhere.

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I'm going to just pick a point down here.

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And I'm going to graph ...

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I'll pick a different color
so we can see it better.

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I'm going to graph this.

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That's a lot of displacement.

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This one didn't displace at all.

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That one's just right in the middle,

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so that's got to be right on the axis,

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because that represents zero displacement.

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This is zero displacement over here.

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And then over here,
displaced a lot to the right,

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so that would be a lot of
displacement to the right.

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And in between, it's varying amounts

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and it would look like this.

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You'd get a graph that
looks something like that.

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And what is this?

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This is a standing wave.

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This is what we'd see.

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But it wouldn't stay like this.

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These particles would ...

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This one in the middle
keeps on not doing anything,

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but this one over here would then move

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all the way this way and
oscillate back and forth.

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And so what you would see this shape do,

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if you played this in time,

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this would start to move
back to equilibrium,

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so this spot would start
to move up to here.

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You'd get another point in
time where it looked like this,

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everything not nearly displaced
as much as it was before.

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And then you wait a little
longer, it goes flatline.

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Everything's back to its
equilibrium position.

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Then this one over here
starts to move to the right,

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so now it's a little
bit further to the right

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than its equilibrium position.

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And eventually it flip-flops like this,

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and so you'd get a graph like that.

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And so this is what happens.

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If you watched this graph,

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this graph would dance up and down.

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This part would move
all the way to the top

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and then all the way to the bottom.

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And it's good to know

202
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that does not represent an air
molecule moving up and down.

203
00:07:04,533 --> 00:07:07,300
These air molecules do
not move up and down.

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They move left and right.

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And this graph that we're drawing

206
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represents the amount they
have displaced left or right.

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And so this graph, this peak
called a standing wave '

208
00:07:18,966 --> 00:07:21,069
because this peak does not ...

209
00:07:21,069 --> 00:07:23,266
this looks like a peak right on a wave.

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On this graph right here,

211
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this peak does not move to the right.

212
00:07:27,634 --> 00:07:30,034
It dances up and down now.

213
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That's what this thing does.

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I wish we could have
called them dancing waves,

215
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but they're called standing waves.

216
00:07:35,233 --> 00:07:37,000
These peaks move up and down.

217
00:07:37,000 --> 00:07:39,500
And the node, this guy
just stays right here.

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00:07:39,500 --> 00:07:40,966
If this was a regular traveling wave,

219
00:07:40,966 --> 00:07:42,945
you'd see this node move to the right,

220
00:07:42,945 --> 00:07:45,000
you'd see this peak move to the right.

221
00:07:45,000 --> 00:07:46,266
It doesn't do that anymore.

222
00:07:46,266 --> 00:07:48,305
And so we call this a standing wave.

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00:07:48,305 --> 00:07:49,579
And I already said it,

224
00:07:49,579 --> 00:07:52,966
but this point in the middle
is given a special name.

225
00:07:52,966 --> 00:07:56,866
This point right here is called the node.

226
00:07:56,866 --> 00:07:59,567
This is a node, and
these points at the end,

227
00:08:00,100 --> 00:08:02,334
this location here and this location here

228
00:08:02,334 --> 00:08:06,100
that oscillate wildly,
are called antinodes.

229
00:08:06,533 --> 00:08:09,266
So the antinodes are points
where it oscillates wildly

230
00:08:09,266 --> 00:08:12,633
and the nodes are points where
it doesn't oscillate at all.

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00:08:12,633 --> 00:08:13,933
This particle does not move

232
00:08:13,933 --> 00:08:16,900
and this point on the
graph just stays at zero.

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00:08:16,900 --> 00:08:20,433
So the tricky part is,

234
00:08:20,433 --> 00:08:24,600
how do we represent this mathematically?

235
00:08:24,600 --> 00:08:26,300
This is how we represent it graphically.

236
00:08:26,300 --> 00:08:28,133
How do we represent this mathematically?

237
00:08:28,133 --> 00:08:30,567
Let me clean this up a little bit.

238
00:08:30,567 --> 00:08:33,799
The question is, how much
of a wavelength is that?

239
00:08:33,799 --> 00:08:35,399
How much of a wavelength is this?

240
00:08:35,400 --> 00:08:38,989
Well, if you remember,
one whole wavelength ...

241
00:08:38,989 --> 00:08:41,933
I'm going to draw a whole
wavelength over here.

242
00:08:41,933 --> 00:08:45,766
One entire wavelength
looks something like this.

243
00:08:45,766 --> 00:08:47,400
So here's a graph just to represent

244
00:08:47,400 --> 00:08:49,766
a wave versus X.

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00:08:49,766 --> 00:08:50,640
An entire wavelength is

246
00:08:50,640 --> 00:08:53,700
when it gets all the way
back to where it started.

247
00:08:53,700 --> 00:08:55,533
So from some point in the cycle

248
00:08:55,533 --> 00:08:57,083
all the way back to
that point in the cycle

249
00:08:57,083 --> 00:08:58,300
would be one wavelength.

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00:08:58,300 --> 00:08:59,666
How much of a wavelength is this?

251
00:08:59,666 --> 00:09:01,100
Well, look, this is only ...

252
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It starts at the bottom

253
00:09:02,767 --> 00:09:05,538
and then it makes it to the top.

254
00:09:05,538 --> 00:09:06,266
But that's it.

255
00:09:06,266 --> 00:09:07,501
It stops there.

256
00:09:08,066 --> 00:09:12,512
So the question is, well,
is that a whole wavelength?

257
00:09:12,512 --> 00:09:14,133
No, that's only half of a wavelength.

258
00:09:14,133 --> 00:09:16,811
So if we wanted to know,
how much of a wavelength

259
00:09:16,811 --> 00:09:20,401
is this in terms of the
length of this tube?

260
00:09:20,401 --> 00:09:22,400
Say this tube has a length L.

261
00:09:23,066 --> 00:09:25,775
For this first one, we'd realize that,

262
00:09:25,775 --> 00:09:27,566
okay, that's half a wavelength.

263
00:09:27,566 --> 00:09:32,566
So L, 1/2 of a wavelength
is fitting into a length L.

264
00:09:32,701 --> 00:09:36,267
So this 1/2 of a wavelength
equals a certain distance.

265
00:09:36,267 --> 00:09:38,934
The distance that 1/2
of a wavelength equals

266
00:09:38,934 --> 00:09:41,333
for this first standing wave we've set up

267
00:09:41,333 --> 00:09:44,866
is just 1/2 lambda.

268
00:09:45,900 --> 00:09:50,833
What that means is, well,
then lambda equals two L.

269
00:09:50,833 --> 00:09:51,866
So this is it.

270
00:09:51,866 --> 00:09:56,235
The lambda of this wave is two L.

271
00:09:56,235 --> 00:09:58,807
And we call that the
fundamental frequency,

272
00:09:58,807 --> 00:10:00,800
or the fundamental wavelength.

273
00:10:00,800 --> 00:10:03,000
And it's a special name because
this is the one you'll hear.

274
00:10:03,000 --> 00:10:05,533
If you blow over a tube, this
is the one that you'll hear.

275
00:10:05,533 --> 00:10:06,400
It's going to sound loud.

276
00:10:06,400 --> 00:10:08,167
This is the wavelength you'll hear.

277
00:10:08,167 --> 00:10:10,133
But that's not the only
one you can set up.

278
00:10:10,133 --> 00:10:12,333
The only requirement
here is that these ends

279
00:10:12,333 --> 00:10:14,266
are going to oscillate like crazy.

280
00:10:15,293 --> 00:10:17,434
Ends, we know, have to be antinodes.

281
00:10:17,434 --> 00:10:20,066
So, in this case, we had
a node in the middle,

282
00:10:20,066 --> 00:10:21,933
two antinodes at the end.

283
00:10:21,933 --> 00:10:24,633
The question is, what other
standing wave could you set up?

284
00:10:24,633 --> 00:10:25,477
Another one would be,

285
00:10:25,477 --> 00:10:27,276
okay, got to be antinode here,

286
00:10:27,600 --> 00:10:29,466
got to be antinode on the other end,

287
00:10:29,466 --> 00:10:31,333
but you might have multiple
nodes in the middle

288
00:10:31,333 --> 00:10:33,233
instead of just one node.

289
00:10:34,294 --> 00:10:36,433
Say we did something like this.

290
00:10:37,809 --> 00:10:40,233
Say we had a wave like that.

291
00:10:40,233 --> 00:10:43,900
Now, antinode on this
end, antinode on this end,

292
00:10:43,900 --> 00:10:45,868
it's got to be because, a open, open tube,

293
00:10:45,868 --> 00:10:47,639
the open ends have to be the antinodes

294
00:10:47,639 --> 00:10:49,600
for the displacement of the particle.

295
00:10:49,600 --> 00:10:52,266
And now we've got two nodes
in the middle, though.

296
00:10:52,266 --> 00:10:54,433
So we've got two nodes in the middle,

297
00:10:54,433 --> 00:10:55,300
two antinodes.

298
00:10:55,300 --> 00:10:57,033
How much of a wavelength is this?

299
00:10:57,033 --> 00:10:57,833
Let's check it out.

300
00:10:57,833 --> 00:10:59,767
So this was a whole
wavelength, the whole blue.

301
00:10:59,767 --> 00:11:02,733
So this green is all the way up to the top

302
00:11:02,733 --> 00:11:04,500
and then all the way to the bottom.

303
00:11:04,500 --> 00:11:06,033
Look -- that's a whole wavelength.

304
00:11:06,033 --> 00:11:09,100
So in this case, L, the
length of this tube,

305
00:11:09,100 --> 00:11:12,400
is equaling one whole
wavelength for the second ...

306
00:11:12,400 --> 00:11:14,300
this is called the second harmonic.

307
00:11:14,766 --> 00:11:15,802
This is also set up.

308
00:11:15,802 --> 00:11:17,471
You don't hear it as much.

309
00:11:17,471 --> 00:11:21,000
But if you were to
analyze the frequencies,

310
00:11:21,000 --> 00:11:22,901
you'd see that there's a
little bit of that frequency

311
00:11:22,901 --> 00:11:24,833
in there, too, a little
bit of that wavelength.

312
00:11:24,833 --> 00:11:27,466
So in this case, lambda equals L.

313
00:11:27,466 --> 00:11:29,763
So this is called the second harmonic.

314
00:11:29,763 --> 00:11:32,366
So I'm going to call this lambda two.

315
00:11:32,366 --> 00:11:34,400
Lambda two just equals L.

316
00:11:34,400 --> 00:11:38,400
This is the second harmonic.

317
00:11:38,400 --> 00:11:40,400
And you can find the third harmonic.

318
00:11:40,400 --> 00:11:42,066
Let's see; what else would be possible?

319
00:11:42,066 --> 00:11:43,300
Let's try another one.

320
00:11:43,800 --> 00:11:45,402
You know it's got to
be antinode on this end

321
00:11:45,402 --> 00:11:48,539
because it's open, antinode
on this end because it's open,

322
00:11:48,539 --> 00:11:50,715
but instead of having just one
or two nodes in the middle,

323
00:11:50,715 --> 00:11:51,505
I could have three.

324
00:11:51,505 --> 00:11:53,200
So I'm going to come all
the way up to the top,

325
00:11:54,500 --> 00:11:56,600
and I'm going to come all the
way back down to the bottom,

326
00:11:57,300 --> 00:11:57,900
and then I'm going to go

327
00:11:57,900 --> 00:11:59,533
all the way back up to the top again.

328
00:11:59,533 --> 00:12:03,300
This is antinode on this
end, antinode on this end,

329
00:12:03,300 --> 00:12:07,600
now you got one, two,
three nodes in the middle.

330
00:12:07,600 --> 00:12:09,366
And so, how much of a wavelength is this?

331
00:12:09,366 --> 00:12:10,233
Let's try it out.

332
00:12:10,233 --> 00:12:12,000
Let's reference our one wavelength.

333
00:12:12,000 --> 00:12:13,567
So it starts at the bottom,

334
00:12:13,567 --> 00:12:16,067
and then it goes all
the way up to the top,

335
00:12:16,067 --> 00:12:18,033
and then it goes all the
way down to the bottom,

336
00:12:18,033 --> 00:12:19,166
but this one keeps going.

337
00:12:19,166 --> 00:12:21,000
This is more than a whole wavelength.

338
00:12:21,000 --> 00:12:23,100
Because that's just this part.

339
00:12:23,100 --> 00:12:24,567
That's one whole wavelength.

340
00:12:24,567 --> 00:12:26,233
Now I got to go all the
way back up to the top.

341
00:12:26,233 --> 00:12:30,233
So this wave is actually
one wavelength and a half.

342
00:12:30,233 --> 00:12:33,001
This amount is one extra
half of a wavelength,

343
00:12:33,001 --> 00:12:35,566
so this was one wavelength and a half.

344
00:12:35,566 --> 00:12:38,966
So in this case, L, the
total distance of the tube,

345
00:12:38,966 --> 00:12:40,233
that's not changing here.

346
00:12:40,233 --> 00:12:42,467
The total distance of the tube is L.

347
00:12:42,467 --> 00:12:44,566
This time, the wavelength
in there is fitting,

348
00:12:44,566 --> 00:12:47,100
and one and 1/2 wavelengths fit in there.

349
00:12:47,100 --> 00:12:49,401
That's 3/2 of a wavelength.

350
00:12:50,456 --> 00:12:55,456
That means the lambda
equals two L over three.

351
00:12:55,700 --> 00:12:58,333
So in this case, for lambda three,

352
00:12:58,333 --> 00:13:02,059
this is going to be
called the third harmonic.

353
00:13:02,059 --> 00:13:06,236
This is the third possible
wavelength that can fit in there.

354
00:13:06,236 --> 00:13:08,167
This should be two L over three.

355
00:13:08,167 --> 00:13:10,533
And so, it keeps going.

356
00:13:10,533 --> 00:13:12,406
You can have the fourth
harmonic, fifth harmonic,

357
00:13:12,406 --> 00:13:14,594
every time you add one more node in here,

358
00:13:14,594 --> 00:13:16,200
it's always got to be antinode on one end,

359
00:13:16,200 --> 00:13:18,321
antinode at the other.

360
00:13:18,321 --> 00:13:19,401
These are the possible wavelength,

361
00:13:19,401 --> 00:13:22,534
and if you wanted the possible,
all of the possible ones,

362
00:13:22,534 --> 00:13:23,734
you can probably see the pattern here.

363
00:13:23,734 --> 00:13:28,012
Look: two L, and then just
L, then two L over three,

364
00:13:28,012 --> 00:13:30,143
the next one turns out
to be two L over four,

365
00:13:30,143 --> 00:13:32,300
and then two L over five, two L over six,

366
00:13:32,300 --> 00:13:36,000
and so, if you wanted to just
write them all down, shoot ...

367
00:13:36,000 --> 00:13:37,718
lambda n equals --

368
00:13:37,718 --> 00:13:42,300
this is all the possible
wavelengths -- two L over n,

369
00:13:42,300 --> 00:13:47,300
where n equals one, two,
three, four, and so on.

370
00:13:48,670 --> 00:13:51,567
And so, look at, if I had
'n equals one' in here,

371
00:13:51,567 --> 00:13:52,445
I'd have two L.

372
00:13:52,445 --> 00:13:53,239
That's the fundamental.

373
00:13:53,239 --> 00:13:55,202
You plug in n equals one,
you get the fundamental.

374
00:13:55,202 --> 00:13:58,572
If I plug in n equals
two, I get two L over two.

375
00:13:58,572 --> 00:13:59,366
That's just L.

376
00:13:59,366 --> 00:14:00,652
That's my second harmonic,

377
00:14:00,652 --> 00:14:02,700
because I'm plugging in n equals two.

378
00:14:02,700 --> 00:14:05,267
If I plug in n equals three,
I get two L over three,

379
00:14:05,267 --> 00:14:06,766
that's my third harmonic.

380
00:14:06,766 --> 00:14:09,400
This is telling me all
the possible wavelengths

381
00:14:09,892 --> 00:14:12,300
that I'm getting for this standing wave.

382
00:14:12,300 --> 00:14:13,733
So that's open, open.

383
00:14:13,733 --> 00:14:15,104
In the next video,

384
00:14:15,104 --> 00:00:00,000
I'm going to show you how to
handle open, closed tubes.