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- [Instructor] So imagine
you've got a wave source.

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This could be a little
oscillator that's creating a wave

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on a string, or a little
paddle that goes up and down

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that creates waves on water,

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or a speaker that creates sound waves.

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This could be any wave source whatsoever

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creates this wave, a nice
simple harmonic wave.

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Now let's say you've got
a second wave source.

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If we take this wave
source, the second one,

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and we put it basically right
on top of the first one,

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we're gonna get wave interference

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because wave interference
happens when two waves overlap.

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And if we want to know what the
total wave's gonna look like

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we add up the contributions
from each wave.

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So if I put a little backdrop in here

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and I add the contributions,

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if the equilibrium point is right here,

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so that's where the wave would be zero,

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the total wave can be found by adding up

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the contributions from each wave.

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So if we add up the contributions
from wave one and wave two

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wave one here has a value of one unit,

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wave two has a value of one unit.

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One unit plus one unit is two units.

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And then zero units and
zero units is still zero.

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Negative one and negative
one is negative two,

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and you keep doing this
and you realize wait,

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you're just gonna get a really
big cosine looking wave.

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I'm just gonna drop down to here.

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We say that these waves are
constructively interfering.

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We call this constructive interference

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because the two waves
combined to construct a wave

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that was twice as big
as the original wave.

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So when two waves combine and form a wave

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bigger than they were before,

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we call it constructive interference.

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And because these two
waves combined perfectly,

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sometimes you'll hear this
as perfectly constructive

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or totally constructive interference.

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You could imagine cases
where they don't line up

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exactly correct, but you
still might get a bigger wave.

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In that case, it's still constructive.

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It might not be totally constructive.

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So that was constructive interference.

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And these waves were constructive?

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Think about it because
this wave source two

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looked exactly like wave source one did,

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and we just overlapped them
and we got double the wave,

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which is kinda like alright, duh.

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That's not that impressive.

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But check this out.

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Let's say you had another wave source.

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A different wave source two.

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This one is what we call Pi
shifted 'cause look at it.

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Instead of starting at a maximum,

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this one starts at a minimum

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compared to what wave source one is at.

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So it's 1/2 of a cycle ahead of or behind

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of wave source one.

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1/2 of a cycle is Pi because
a whole cycle is two Pi.

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That's why people often
call this Pi shifted,

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or 180 degrees shifted.

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Either way, it's out of
phase from wave source one

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by 1/2 of a cycle.

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So what happens if we overlap these two?

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Now I'm gonna take these two.

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Let's get rid of that there,
let's just overlap these two

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and see what happens.

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I'm gonna overlap these two waves.

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We'll perform the same analysis.

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I don't even really need the
backdrop now because look at.

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I've got one and negative one.

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One and negative one, zero.

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Zero and zero, zero.

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Negative one and one, zero.

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Zero and zero, zero and
no matter where I'm at,

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1/2, a negative 1/2, zero.

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These two waves are gonna add up to zero.

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They add up to nothing,

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so we call this destructive interference

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because these two waves
essentially destroyed each other.

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This seems crazy.

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Two waves add up to nothing?

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How can that be the case?

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Are there any applications of this?

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Well yeah.

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So imagine you're sitting on an airplane

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and you're listening to the annoying roar

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of the airplane engine in your ear.

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It's very loud and it might be annoying.

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So what do you do?

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You put on your noise
canceling headphones,

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and what those noise
canceling headphones do?

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They sit on your ear, they
listen to the wave coming in.

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This is what they listen to.

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This sound wave coming in,
and they cancel off that sound

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by sending in their own sound,

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but those headphones Pi shift the sound

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that's going into your ear.

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So they match that roar
of the engine's frequency,

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but they send in a sound that's Pi shifted

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so that they cancel and your
ear doesn't hear anything.

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Now it's often now completely silent.

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They're not perfect, but
they work surprisingly well.

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They're essentially
fighting fire with fire.

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They're fighting sound with more sound,

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and they rely on this idea
of destructive interference.

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They're not perfectly,
totally destructive,

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but the waves I've drawn
here are totally destructive.

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If they were to perfectly cancel,

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we'd call that total
destructive interference,

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or perfectly destructive interference.

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And it happens because
this wave we sent in

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was Pi shifted compared to
what the first wave was.

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So let me show you something interesting

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if I get rid of all this.

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Let me clean up this mess.

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If I've got wave source one,

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let me get wave source two back.

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So this was the wave that was
identical to wave source one.

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We overlap 'em, we get
constructive interference

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because the peaks are lining
up perfectly with the peaks,

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and these valleys or troughs
are matching up perfectly

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with the other valleys or troughs.

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But as I move this wave
source too forward,

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look at what happens.

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They start getting out of phase.

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When they're perfectly lined
up we say they're in phase.

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They're starting to get out of phase,

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and look at when I move it forward enough

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what was a constructive
situation, becomes destructive.

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Now all the peaks are
lining up with the valleys,

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they would cancel each other out.

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And if I move it forward a little more,

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it lines up perfectly again
and you get constructive,

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move it more I'm gonna get destructive.

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Keep doing this, I go from constructive

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to destructive over and over.

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So in other words, one way to
get constructive interference

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is to take two wave sources
that start in phase,

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and just put them right
next to each other.

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And a way to get destructive
is to take two wave sources

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that are Pi shifted out of phase,

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and put them right next to each other,

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and that'll give you destructive

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'cause all the peaks match the valleys.

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But another way to get
constructive or destructive

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is to start with two
waves that are in phase,

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and make sure one wave gets moved forward

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compared to the other, but how far forward

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should we move these in order to get

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constructive and destructive?

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Well let's just test it out.

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We start here.

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When they're right next to
each other we get constructive.

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If I move this second wave source

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that was initially in
phase all the way to here,

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I get constructive again.

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How far did I move it?

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I moved it this far.

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The front of that speaker moved this far.

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So how far was that?

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Let me get rid of this.

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That was one wavelength.

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So look at this picture.

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From peak to peak is
exactly one wavelength.

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We're assuming these waves
have the same wavelength.

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So notice that essentially what we did,

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we made it so that the
wave from wave source two

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doesn't have to travel as far

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to whatever's detecting the sound.

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Maybe there's an ear here,

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or some sort of scientific
detector detecting the sound.

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Wave source two is now
only traveling this far

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to get to the detector,
whereas wave source one

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is traveling this far.

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In other words, we made
it so that wave source one

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has to travel one wavelength further

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than wave source two does,

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and that makes it so that they're in phase

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and you get constructive
interference again.

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But that's not the only option,

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we can keep moving wave
source two forward.

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We move it all the way to here,

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we moved it another wavelength forward.

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We again get constructive interference,

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and at this point, wave source one

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is having to make its wave
travel two wavelengths further

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than wave source two does.

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And you could probably see the pattern.

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No matter how many wavelengths
we move it forward,

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as long as it's an integer
number of wavelengths

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we again get constructive interference.

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So something that turns out to be useful

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is a formula that tells us alright,

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how much path length
difference should there be?

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So if I'm gonna call this X two,

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the distance that the
wave from wave source two

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has to travel to get to
whatever's detecting that wave.

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And the distance X one, that
wave source one has to travel

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to get to that detector.

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So we could write down
a formula that relates

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the difference in path length,
I'll call that delta X,

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which is gonna be the distance
that wave one has to travel

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minus the distance that
wave two has to travel.

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And given what we saw up here,

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if this path length
difference is ever equal

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to an integer number of wavelengths,

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so if it was zero that was when they

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were right next to each
other, you got constructive.

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When this difference is
equal to one wavelength,

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we also got constructive.

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When it was two wavelengths,
we got constructive.

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It turns out any integer
wavelength gives us constructive.

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So how would we get
destructive interference then?

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Well let's continue with this wave source

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that originally started in phase, right?

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So these two wave sources
are starting in phase.

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How far do I have to move
it to get destructive?

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Well let's just see.

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I have to move it 'til
it's right about here.

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So how far did the front
of that speaker move?

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It moved about this far, which
if I get rid of that speaker

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you could see is about
1/2 of a wavelength.

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From peak to valley,
is 1/2 of a wavelength,

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but that's not the only option.

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I can keep moving it forward.

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Let's just see, that's constructive.

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My next destructive happens here

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which was an extra this far.

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How far was that?

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Let's just see.

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That's one wavelength,
so notice at this point,

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wave source one is having to
go one and 1/2 wavelengths

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further than wave source two does.

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So let's just keep going.

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Move wave source two, that's constructive.

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We get another destructive here

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which is an extra this far forward,

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and that's equal to one more wavelength.

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So if we get rid of this you
could see valley to valley

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is a whole nother wavelength.

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So in this case, wave
source two has to travel

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two and 1/2 wavelengths
farther than wave source two.

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Any time wave source one has to travel

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1/2 integer more wavelengths
than wave source two,

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you get destructive interference.

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In other words, if this
path length difference here

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is equal to lambda over
two, three lambda over two,

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which is one and 1/2 wavelengths.

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Five lambda over two, which
is two and 1/2 wavelengths,

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and so on, that leads to
destructive interference.

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So this is how the path length differences

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between two wave sources can determine

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whether you're gonna get constructive

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or destructive interference.

257
00:09:22,967 --> 00:09:25,184
But notice we started
with two wave sources

258
00:09:25,184 --> 00:09:26,446
that were in phase.

259
00:09:26,446 --> 00:09:28,330
These started in phase.

260
00:09:28,330 --> 00:09:31,166
So this whole analysis down here assumes

261
00:09:31,166 --> 00:09:35,588
that the two sources started
in phase with each other,

262
00:09:35,588 --> 00:09:37,946
i.e. neither of them are Pi shifted.

263
00:09:37,946 --> 00:09:39,827
What would this analysis give you

264
00:09:39,827 --> 00:09:41,661
if we started with one
that was Pi shifted?

265
00:09:41,661 --> 00:09:43,437
So let's get rid of this wave two.

266
00:09:43,437 --> 00:09:45,241
Let's put this wave two back in here.

267
00:09:45,241 --> 00:09:46,090
Remember this one?

268
00:09:46,090 --> 00:09:49,231
This one was Pi shifted relative to

269
00:09:49,231 --> 00:09:50,920
relative to wave source one.

270
00:09:50,920 --> 00:09:53,394
So if we put this one in here,
and we'll get rid of this,

271
00:09:53,394 --> 00:09:56,435
now when these two wave sources
are right next to each other

272
00:09:56,435 --> 00:09:58,118
you're getting destructive interference.

273
00:09:58,118 --> 00:10:00,840
So this time for a path length
difference of zero, right?

274
00:10:00,840 --> 00:10:02,837
These are both traveling the same distance

275
00:10:02,837 --> 00:10:04,155
to get to the detector.

276
00:10:04,155 --> 00:10:06,349
So X one and X two are gonna be equal.

277
00:10:06,349 --> 00:10:07,812
You subtract them, you'd get zero.

278
00:10:07,812 --> 00:10:10,140
This time the zero's giving us destructive

279
00:10:10,140 --> 00:10:11,255
instead of constructive.

280
00:10:11,255 --> 00:10:13,099
So let's see what happens
if we move this forward,

281
00:10:13,099 --> 00:10:15,591
let's see how far we've
gotta move this forward

282
00:10:15,591 --> 00:10:17,243
to again get destructive.

283
00:10:17,243 --> 00:10:18,788
We'd have to move it over to here.

284
00:10:18,788 --> 00:10:20,185
How far did we move it?

285
00:10:20,185 --> 00:10:21,058
Let's just check.

286
00:10:21,058 --> 00:10:23,757
We moved the front of
this speaker that far,

287
00:10:23,757 --> 00:10:25,880
which is one whole wavelength.

288
00:10:25,880 --> 00:10:27,426
So if we get rid of this,
we had to move the front

289
00:10:27,426 --> 00:10:29,394
of the speaker one whole wavelength,

290
00:10:29,394 --> 00:10:31,309
and look at again it's destructive.

291
00:10:31,309 --> 00:10:34,525
So again, zero gave us
destructive this time,

292
00:10:34,525 --> 00:10:36,603
and the lambda's giving us destructive,

293
00:10:36,603 --> 00:10:39,685
and you realize oh wait, all
of these integer wavelengths.

294
00:10:39,685 --> 00:10:42,437
If I move it another
integer wavelength forward,

295
00:10:42,437 --> 00:10:44,955
I'm again gonna get
destructive interference

296
00:10:44,955 --> 00:10:47,499
because all these peaks
are lining up with valleys.

297
00:10:47,499 --> 00:10:50,994
So interestingly, if two
sourcese started Pi out of phase,

298
00:10:50,994 --> 00:10:52,149
so I'm gonna change this.

299
00:10:52,149 --> 00:10:56,333
Started Pi out of phase,
then path length differences

300
00:10:56,333 --> 00:10:58,336
of zero, lambda, and two lambda

301
00:10:58,336 --> 00:10:59,921
aren't gonna give us constructive,

302
00:10:59,921 --> 00:11:01,656
they're gonna give us destructive.

303
00:11:01,656 --> 00:11:02,891
And so you could probably guess now,

304
00:11:02,891 --> 00:11:04,646
what are these path length differences

305
00:11:04,646 --> 00:11:06,782
of 1/2 integer wavelengths gonna give us?

306
00:11:06,782 --> 00:11:07,617
Well let's just find out.

307
00:11:07,617 --> 00:11:09,778
Let's start here, and
we'll get rid of these.

308
00:11:09,778 --> 00:11:10,611
Let's just check.

309
00:11:10,611 --> 00:11:12,847
We'll move this forward
1/2 of a wavelength

310
00:11:12,847 --> 00:11:13,680
and what do I get?

311
00:11:13,680 --> 00:11:14,886
Yup, I get constructive.

312
00:11:14,886 --> 00:11:19,327
So if I move this Pi shifted
source 1/2 a wavelength forward

313
00:11:19,327 --> 00:11:21,275
instead of giving me destructive,

314
00:11:21,275 --> 00:11:23,046
it's giving me constructive now.

315
00:11:23,046 --> 00:11:26,125
And if I move it so it goes
another wavelength forward

316
00:11:26,125 --> 00:11:29,114
over to here, notice
this time wave source one

317
00:11:29,114 --> 00:11:31,638
has to move one and
1/2 wavelengths further

318
00:11:31,638 --> 00:11:32,647
than wave source two.

319
00:11:32,647 --> 00:11:34,413
That's 3/2 wavelengths.

320
00:11:34,413 --> 00:11:35,981
But instead of giving
us destructive, look.

321
00:11:35,981 --> 00:11:37,529
These are lining up perfectly.

322
00:11:37,529 --> 00:11:40,039
It's giving us constructive,
and you realize oh,

323
00:11:40,039 --> 00:11:43,841
all these 1/2 integer wavelength
path length differences,

324
00:11:43,841 --> 00:11:45,214
instead of giving me destructive

325
00:11:45,214 --> 00:11:46,605
are giving me constructive now

326
00:11:46,605 --> 00:11:49,908
because one of these wave
sources was Pi shifted

327
00:11:49,908 --> 00:11:50,877
compared to the other.

328
00:11:50,877 --> 00:11:52,978
So I can take this here,
and I could say that

329
00:11:52,978 --> 00:11:55,829
when the two sources
start Pi out of phase,

330
00:11:55,829 --> 00:11:57,959
instead of leading to destructive

331
00:11:57,959 --> 00:11:59,915
this is gonna lead to
constructive interference.

332
00:11:59,915 --> 00:12:02,364
And these two ideas are the foundation

333
00:12:02,364 --> 00:12:05,115
of almost all interference
patterns you find

334
00:12:05,115 --> 00:12:06,980
in the universe, which is kind of cool.

335
00:12:06,980 --> 00:12:09,319
If there's an interference
pattern you see out there,

336
00:12:09,319 --> 00:12:10,733
it's probably due to this.

337
00:12:10,733 --> 00:12:12,948
And if there's an
equation you end up using,

338
00:12:12,948 --> 00:12:15,923
it's probably fundamentally
based on this idea

339
00:12:15,923 --> 00:12:17,775
if it's got wave interference in it.

340
00:12:17,775 --> 00:12:18,990
So I should say one more thing,

341
00:12:18,990 --> 00:12:21,905
that sources don't actually
have to start out of phase.

342
00:12:21,905 --> 00:12:23,553
Sometimes they travel around.

343
00:12:23,553 --> 00:12:25,944
Things happen, it's a crazy universe.

344
00:12:25,944 --> 00:12:28,926
Maybe one of the waves get
shifted along its travel.

345
00:12:28,926 --> 00:12:31,382
Regardless, if any of them get a Pi shift

346
00:12:31,382 --> 00:12:33,439
either at the beginning or later on,

347
00:12:33,439 --> 00:12:35,854
you would use this second
condition over here

348
00:12:35,854 --> 00:12:38,256
to figure out whether you get
constructive or destructive.

349
00:12:38,256 --> 00:12:40,120
If neither of them get a phase shift,

350
00:12:40,120 --> 00:12:43,782
or interestingly, if both
of them get a phase shift,

351
00:12:43,782 --> 00:12:45,351
you could use this one
'cause you could imagine

352
00:12:45,351 --> 00:12:46,756
flipping both of them over,

353
00:12:46,756 --> 00:12:49,507
and it's the same as not
flipping any of them over.

354
00:12:49,507 --> 00:12:51,892
So recapping, constructive
interference happens

355
00:12:51,892 --> 00:12:53,971
when two waves are lined up perfectly.

356
00:12:53,971 --> 00:12:55,375
Destructive interference happens

357
00:12:55,375 --> 00:12:58,209
when the peaks match the valleys
and they cancel perfectly.

358
00:12:58,209 --> 00:13:00,508
And you could use the
path length difference

359
00:13:00,508 --> 00:13:03,399
for two wave sources to
determine whether those waves

360
00:13:03,399 --> 00:13:06,162
are gonna interfere
constructively or destructively.

361
00:13:06,162 --> 00:13:07,920
The path length difference
is the difference

362
00:13:07,920 --> 00:13:11,548
between how far one wave has
to travel to get to a detector

363
00:13:11,548 --> 00:13:13,857
compared to how far
another wave has to travel

364
00:13:13,857 --> 00:13:15,204
to get to that same detector,

365
00:13:15,204 --> 00:13:17,179
assuming those two
sources started in phase

366
00:13:17,179 --> 00:13:20,046
and neither of them got a Pi
shift along their travels.

367
00:13:20,046 --> 00:13:22,053
Path length differences
of integer wavelengths

368
00:13:22,053 --> 00:13:23,770
are gonna give you
constructive interference,

369
00:13:23,770 --> 00:13:26,389
and path length differences
of 1/2 integer wavelengths

370
00:13:26,389 --> 00:13:28,127
are gonna give you
destructive interference.

371
00:13:28,127 --> 00:13:30,751
Whereas if the two sources
started Pi out of phase,

372
00:13:30,751 --> 00:13:33,624
or one of the got a Pi phase
shift along its travel,

373
00:13:33,624 --> 00:13:35,830
integer wavelengths for
the path length difference

374
00:13:35,830 --> 00:13:37,709
are gonna give you
destructive interference.

375
00:13:37,709 --> 00:13:40,491
And 1/2 integer wavelengths
for the path length difference

376
00:13:40,491 --> 00:00:00,000
are gonna give you
constructive interference.