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- [Instructor] If you've got a medium

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and you disturb it, you can create a wave.

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And if you create a wave in a medium

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that has no boundaries,

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in other words, a medium that's so big,

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this wave basically
never meets the boundary,

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then there's nothing really stopping you

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from making a wave of any wave length

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or frequency whatsoever.

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In order words, there's not really

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any naturally preferred wave lengths,

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they're all pretty much as
good as any other wave length.

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However, if you confine
this wave into a medium

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that has boundaries, this
wave is gonna reflect

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when it meets the boundary

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and that means it's gonna
overlap with itself.

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And when this happens,
you can create something

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that are called standing waves.

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And we'll talk about what
these mean in a minute

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but the reason we care about them,

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is because when standing waves happen,

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they select preferred wave
lengths and frequencies.

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Only particular wave
lengths and frequencies

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are gonna set up these standing waves

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and what ends up happening

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is that these often become dominant

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and that's why these standing waves

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are important to study.

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So, let's study some standing waves.

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Let's take a particular example.

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Let's say you've got a string,

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whoa, not that many strings.

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One string here and you nail

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this string down at both ends.

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So, you're gonna prevent
any motion from happening

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at the end of this string.

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This string can wiggle in the middle

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but it can't wiggle at the end points.

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And this isn't that crazy.

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A guitar string is basically
a string fixed at both ends.

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Piano strings are strings
fixed at both ends.

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So, the physics behind standing waves

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determines the types of
notes you're gonna get

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on all of these instruments.

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And by the way, this point over here,

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we're basically making
sure that it has no motion.

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So, by nailing it down,

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what I really mean is that
there's gong to be no motion

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at this end point and no
motion at this end point.

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And instead of calling
those no motion points,

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physicists came up with a name for that.

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They call these nodes.

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So node is really just a fancy word

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for not moving at that point.

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So, for this string, there's
gonna be nodes at each end.

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And we'll see, when you
set up a standing wave,

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it's possible that there's
nodes in the middle as well

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but there don't have to be.

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For this string though,

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we're making sure that there
have to be nodes at each end.

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So, why does a standing wave
happen and how does it happen?

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Well, let's say, you give
the end of the string here

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a little pluck and you
cause a disturbance,

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that disturbance is
gonna move down the line

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because that's what waves do.

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It's gonna come over here.

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Once it meets a boundary,

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it's gonna reflect back to the left.

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Now, it turns out, when
a string hits a boundary

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where it's fixed, when it
hits a node, in other words,

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it gets flipped over.

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So, you might have tried
this before with the hose.

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If you send a pulse down the line

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and you try to see how it reflects,

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it gets reflected upside down.

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It doesn't matter too
much for our purposes,

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but every time it's gonna reflect,

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it flips its direction
and it keeps bouncing.

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Now, let's say instead of
sending in a single pulse,

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we send in a whole bunch of pulses, right.

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We send in like a simple harmonic wave.

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Now when this thing reflects,

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it's gonna reflect back on top of itself

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because this leading edge
will get reflected upside down

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this way and it's gonna meet all the rest

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of the wave behind it and overlap with it,

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creating some total wave

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that would be composed of the
wave traveling to the right,

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plus the wave traveling to the left.

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And you can use super
position and interference

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to figure out what that is,

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which, for most wave lengths,
is just gonna be a mess.

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So, if you just send in
whatever wave length you want

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and let it reflect back in on itself,

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the total wave you get might
not really be anything special.

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It might just be sort of a mess in here.

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Nothing really all that interesting.

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However, there'll be
particular wave lengths

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that set up a standing wave,

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and I'll show you what that
looks like in a minute.

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How do you find these
special wave lengths?

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You simply ask yourself,

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what wave lengths could
I draw on this string

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so that there was a node at each end?

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What wave length would
fit inside of this region

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and have a node at both ends.

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So, instead of trying to add up

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a complicated super position of waves,

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we can figure out the special wave lengths

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simply by drawing them
and seeing which ones fit.

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So, let's try it out.

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Let's get rid of all this.

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Alright, what wave
lengths would fit in here?

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Well, we know what a simple
harmonic wave looks like.

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It looks something like this.

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So, the question we need to ask

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is if we start at the zero point,

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because I wanna make sure I
have a node at the left end,

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what might the shape
of this graph look like

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so that I reach a node
at the other end as well?

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Well, the first possibility, look at it,

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I start at a node, when
do I get to a node again?

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I get to a node when it
takes this shape right here.

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There's another node.

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So, the first possibility,

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which is gonna be the
longest, largest possibility,

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would be a wave that just
kind of looked like this.

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Looks kinda like a jump rope.

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This would be the first
possible standing wave

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you can set up on this string.

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And that means it's special,

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it's called the fundamental wave length.

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This is the big daddy.

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This guy dominates all
the other wave lengths

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we're gonna meet.

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Yeah, there's other standing
waves you can set up on here

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but this one's the big alpha dog

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and if you let this string
vibrate however it wants,

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it's gonna pick the
fundamental wave length.

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And so, what would we see happen?

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The string's not just gonna
be suspended in air like this,

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it's gonna be moving around,

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but these are called standing waves

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because this peak no longer looks

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like it's moving right or left.

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This peak is just gonna move up and down,

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so a lot of times when we
draw these standing waves,

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we draw a dash line underneath here

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that mirrors the bold line

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because all this peak's
gonna do is go from the top

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to the bottom, then back to the top,

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it's just gonna oscillate.

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It's gonna look like a jump rope

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but it's not revolving,

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it's just moving up, then
down, then up, then down,

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it takes this shape,
then it would be flat,

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then it might look something like this

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and then it comes down to here

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and then it goes back up,

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and it keeps going up and down.

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We call it standing,
it's more like dancing.

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It's kinda like a dancing wave

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but we call them standing waves

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because these peaks
don't move right or left.

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So, that's the fundamental wave length.

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What would the next possibility look like?

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Let's see, we gotta go from a node.

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We know we have to go from node

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all the way to another node.

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That was the first one.

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So, let's just keep going
and go to the next one.

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And that would be the next
possible standing wave

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because it'd have to fit within here.

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What would that look like?

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It would come up, it would go down

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and then it would come back up,

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so that it meets this
node on the other end.

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That would be the next wave length.

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Sometimes this is called
the second harmonic.

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Second 'cause it's the second possibility,

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harmonic because these are resonances

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and this term is used a lot

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when you talk about resonances
with musical instruments.

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What would the third harmonic look like?

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Well, we gotta start at a node,

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we go to this one, that
was the fundamental,

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this is the second harmonic,

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so that's gonna be the third harmonic.

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So, this one's gonna come up,

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it's gonna go down, it's gonna go back up

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and then it's gonna come back down

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and that would be the third harmonic.

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And you can see you could keep going here.

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You can create infinitely many of these.

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But let's analyze what's going on up here.

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What's actually happening
in these standing waves?

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Note that there's gonna be points,

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like right here on this third harmonic,

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and if I draw its mirror
so that I can get this.

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So, this is what it would look like

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maybe a little after, actually
exactly one half of a period

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after this bold line.

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So, you wait, this peak
moves down to here,

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this valley moves up to this peak,

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this peak moves down to here,

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they're oscillating back and forth.

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But note, this point
right here just stays put.

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That's not even gonna move.

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That's a node and so is
this point right here.

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These points are happening

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'cause when those waves line back up,

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remember, the wave travels to the right,

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bounces back to the left,

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and at this point right here
and this point right here,

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you're getting destructive interference

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between those two waves.

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Similarly, at these points,

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where you're getting the
maximum displacement,

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the two waves are lining up in such a way

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that they're interfering constructively.

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So, the nodes are the destructive points

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where the wave cancels.

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That makes sense 'cause there's
nothing happening there,

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there's no motion.

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And these maximum displacement points

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are the constructive points.

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We should give those a name.

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What do you think we call those?

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If you guessed anti-node,
then you're right.

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These are called anti-nodes

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because that's where
there's the most motion.

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Now, you often have to
figure out mathematically,

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in terms of the length of the string,

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what are the actual wave
lengths you can get.

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So, drawing the picture
allows you to find those.

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But how do you actually
get them mathematically?

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Well, I've kind of created
a horrible mess here.

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So, let me clean this up.

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Sorry about that.

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Get rid of all that.

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Let me just add some strings in here.

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And so this doesn't get too abstract,

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let's just say the length of this string,

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it's pretty big, let's
just say it's 10 meters.

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A really long string, you
secure it at both ends.

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So, the first standing
wave looked like this.

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Jump rope mode, looked like that.

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Now, if the string has
a length of 10 meters,

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what would be the wave
length of this wave?

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You might say 10 meters but no.

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This is not one whole
wave length, remember,

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If we looked at this wave
pattern we had over here

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that we were using,

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this is one entire wave length.

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You had to go all the way to here

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to get through a whole wave length,

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this was only half of a wave length.

258
00:08:37,842 --> 00:08:40,832
So, this jump rope is only
half of a wave length.

259
00:08:40,832 --> 00:08:42,254
What would a whole wave length look like?

260
00:08:42,254 --> 00:08:44,941
This would extend all the way out here,

261
00:08:44,941 --> 00:08:46,487
all the way back up, I
can't really get there

262
00:08:46,487 --> 00:08:47,791
so I don't go off screen.

263
00:08:47,791 --> 00:08:49,447
That would be a whole wave length.

264
00:08:49,447 --> 00:08:52,447
So, this would be 10 meters
and then another 10 meters.

265
00:08:52,447 --> 00:08:54,184
That means that the wave
length of this wave,

266
00:08:54,184 --> 00:08:56,747
even though a whole wave
length isn't fitting in here,

267
00:08:56,747 --> 00:08:59,910
if there was a whole wave
length on this string extended,

268
00:08:59,910 --> 00:09:02,670
this wave length would be 20 meters.

269
00:09:02,670 --> 00:09:03,766
What was the next wave?

270
00:09:03,766 --> 00:09:05,262
Remember it looked like this.

271
00:09:05,262 --> 00:09:07,185
It had one node in the middle,

272
00:09:07,185 --> 00:09:09,146
whereas this first fundamental wave length

273
00:09:09,146 --> 00:09:10,385
had no nodes in the middle.

274
00:09:10,385 --> 00:09:12,201
And again, if this string is 10 meters,

275
00:09:12,201 --> 00:09:13,986
what's this wave length equal?

276
00:09:13,986 --> 00:09:14,819
Well, that's easy.

277
00:09:14,819 --> 00:09:15,967
This is one whole wave length.

278
00:09:15,967 --> 00:09:17,636
So, that would just be 10 meters.

279
00:09:17,636 --> 00:09:19,538
So, the wave length
here would be 10 meters,

280
00:09:19,538 --> 00:09:21,556
'cause one whole wave length fit exactly

281
00:09:21,556 --> 00:09:23,895
within the string's length of 10 meters.

282
00:09:23,895 --> 00:09:26,135
And the third harmonic
looks something like this.

283
00:09:26,135 --> 00:09:27,831
It has two nodes in the middle.

284
00:09:27,831 --> 00:09:30,466
Note, you keep picking up
another node in the middle.

285
00:09:30,466 --> 00:09:32,873
Fundamental has no nodes in the middle.

286
00:09:32,873 --> 00:09:35,199
Second harmonic has
one node in the middle.

287
00:09:35,199 --> 00:09:36,983
Third harmonic'll have two.

288
00:09:36,983 --> 00:09:38,879
The fourth will have three and so on.

289
00:09:38,879 --> 00:09:40,519
So, what is this wave length?

290
00:09:40,519 --> 00:09:42,447
This one's a lot harder
for people to figure out.

291
00:09:42,447 --> 00:09:43,280
So, let's look at this.

292
00:09:43,280 --> 00:09:45,583
One wave length is all the way to here.

293
00:09:45,583 --> 00:09:48,255
So, this is a wave length.

294
00:09:48,255 --> 00:09:50,401
But our string is this long.

295
00:09:50,401 --> 00:09:53,590
So, what fraction of this
length is this wave length?

296
00:09:53,590 --> 00:09:56,543
Well, look at, this wave
length is two thirds

297
00:09:56,543 --> 00:09:58,592
of the entire length of the string.

298
00:09:58,592 --> 00:10:00,244
So that means we could just
say that this wave length

299
00:10:00,244 --> 00:10:03,244
is two thirds of 10 meters which is,

300
00:10:04,409 --> 00:10:07,426
we could write it as 20 meters over three,

301
00:10:07,426 --> 00:10:08,259
and we'll keep going here.

302
00:10:08,259 --> 00:10:09,452
I'll draw the rest.

303
00:10:09,452 --> 00:10:10,768
This is the fourth harmonic.

304
00:10:10,768 --> 00:10:11,922
How big is this wave length?

305
00:10:11,922 --> 00:10:14,354
Well, this wave length
covers half of the string.

306
00:10:14,354 --> 00:10:16,090
So, this wave length is gonna be

307
00:10:16,090 --> 00:10:17,570
half the length of the string

308
00:10:17,570 --> 00:10:21,088
and that's gonna be half
of 10 which is five meters.

309
00:10:21,088 --> 00:10:21,921
And we can keep going.

310
00:10:21,921 --> 00:10:24,197
I can draw the fifth harmonic down here.

311
00:10:24,197 --> 00:10:26,330
And it would look like this
and you could ask yourself,

312
00:10:26,330 --> 00:10:27,804
how big is this wave length?

313
00:10:27,804 --> 00:10:29,010
Well, this wave length, let's see.

314
00:10:29,010 --> 00:10:30,701
One, two, three, four, five,

315
00:10:30,701 --> 00:10:32,644
we got five of these humps in here.

316
00:10:32,644 --> 00:10:35,556
So, this wave length's gonna be two fifths

317
00:10:35,556 --> 00:10:37,018
of this entire length.

318
00:10:37,018 --> 00:10:41,068
So, I'm just gonna write
lambda is two times 10

319
00:10:41,068 --> 00:10:44,503
would be 20 meters, so two fifths of 10

320
00:10:44,503 --> 00:10:46,589
would be 20 meters over five,

321
00:10:46,589 --> 00:10:49,082
oh, which we could
simplify as four meters.

322
00:10:49,082 --> 00:10:52,165
But what if they asked you
for like the 43rd harmonic?

323
00:10:52,165 --> 00:10:53,685
If they're like, hey,
what's the wave length

324
00:10:53,685 --> 00:10:55,261
of the 43rd harmonic?

325
00:10:55,261 --> 00:10:58,042
I don't wanna sit down and
draw like 43 of these things

326
00:10:58,042 --> 00:10:59,840
and try to figure out what fraction it is.

327
00:10:59,840 --> 00:11:00,693
And you don't have to.

328
00:11:00,693 --> 00:11:01,694
There's a pattern here.

329
00:11:01,694 --> 00:11:03,010
So, let me show you the pattern.

330
00:11:03,010 --> 00:11:04,149
So, this is gonna look kinda weird

331
00:11:04,149 --> 00:11:06,317
but I'm gonna write this
first fundamental wave length

332
00:11:06,317 --> 00:11:08,984
as two times 10 meters over one.

333
00:11:10,533 --> 00:11:12,501
And then I'm gonna write
the second harmonic

334
00:11:12,501 --> 00:11:15,749
as two times 10 meters over two.

335
00:11:15,749 --> 00:11:17,181
And I'll write this third harmonic

336
00:11:17,181 --> 00:11:20,014
as two times 10 meters over three.

337
00:11:21,069 --> 00:11:22,854
This fourth harmonic is gonna be

338
00:11:22,854 --> 00:11:25,993
two times 10 meters over four.

339
00:11:25,993 --> 00:11:28,329
This fifth harmonic could
be written equivalently

340
00:11:28,329 --> 00:11:32,496
as two times 10 meters over
five, since that's 25ths.

341
00:11:33,474 --> 00:11:35,026
And now, hopefully, you see the pattern.

342
00:11:35,026 --> 00:11:37,924
You realize, okay, I see
what's going on here.

343
00:11:37,924 --> 00:11:41,634
If I want the wave length
of the nth harmonic,

344
00:11:41,634 --> 00:11:44,554
n could be like the first,
the second, the third,

345
00:11:44,554 --> 00:11:49,001
so n is really just an integer
one, two, three and so on.

346
00:11:49,001 --> 00:11:50,833
I could figure out what
that wave length would be

347
00:11:50,833 --> 00:11:54,409
simply by taking two times
the length of the string,

348
00:11:54,409 --> 00:11:55,465
I'm gonna write it as L,

349
00:11:55,465 --> 00:11:58,261
so this applies to any
string of any length

350
00:11:58,261 --> 00:12:01,209
as long as it's got
nodes at the end points.

351
00:12:01,209 --> 00:12:03,865
So, take two times L and
then just divide by n.

352
00:12:03,865 --> 00:12:05,679
So, in other words, if
I want the wave length

353
00:12:05,679 --> 00:12:08,236
of the 84th harmonic, I'll just take

354
00:12:08,236 --> 00:12:11,769
two times the length of my
string and divide by 84.

355
00:12:11,769 --> 00:12:13,665
If I wanted the 33rd harmonic,

356
00:12:13,665 --> 00:12:16,009
I'd take two times the
length of the string over 33

357
00:12:16,009 --> 00:12:19,126
and that would give me the
wave length of that harmonic.

358
00:12:19,126 --> 00:12:22,068
Now you should remember when
we derived this equation,

359
00:12:22,068 --> 00:12:24,518
we drew these pictures and these pictures

360
00:12:24,518 --> 00:12:27,634
all assume that the end points are nodes.

361
00:12:27,634 --> 00:12:30,139
So, this equation assumes you have a node.

362
00:12:30,139 --> 00:12:32,169
Node, standing wave on a string,

363
00:12:32,169 --> 00:12:34,593
which honestly, is almost always the case,

364
00:12:34,593 --> 00:12:38,289
since on all instruments with
a string both ends are fixed.

365
00:12:38,289 --> 00:12:41,570
So recapping, when you confine
a wave into a given region,

366
00:12:41,570 --> 00:12:43,787
the wave will reflect off the boundaries

367
00:12:43,787 --> 00:12:45,249
and overlap with itself

368
00:12:45,249 --> 00:12:47,945
causing constructive and
destructive interference.

369
00:12:47,945 --> 00:12:50,913
For particular wave lengths,
you can set up a standing wave,

370
00:12:50,913 --> 00:12:53,857
which means the wave just
oscillates up and down

371
00:12:53,857 --> 00:12:55,521
instead of left to right.

372
00:12:55,521 --> 00:12:56,633
In these standing waves,

373
00:12:56,633 --> 00:12:59,730
the points where there's
no motion are called nodes.

374
00:12:59,730 --> 00:13:01,401
And the points of maximum displacement

375
00:13:01,401 --> 00:13:02,900
are called anti-nodes.

376
00:13:02,900 --> 00:13:04,414
You can find the possible wave lengths

377
00:13:04,414 --> 00:13:07,201
of a standing wave on a
string fixed at both ends

378
00:13:07,201 --> 00:13:08,864
by ensuring that the standing wave

379
00:13:08,864 --> 00:13:10,870
takes the shape of a simple harmonic wave

380
00:13:10,870 --> 00:13:12,894
and has nodes at both ends,

381
00:13:12,894 --> 00:13:14,574
which if you do, gives you a formula

382
00:13:14,574 --> 00:13:16,121
for the possible wave lengths

383
00:13:16,121 --> 00:13:18,345
for a node node standing wave

384
00:13:18,345 --> 00:13:20,897
as being two times the
length of the string

385
00:13:20,897 --> 00:13:22,914
divided by the number for the harmonic

386
00:13:22,914 --> 00:00:00,000
you're concerned with.