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- [Teacher] What's up everybody?

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In this video, I wanna derive

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this formula right here.

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This is the formula for beat frequency,

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so remember the way you get beats

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or beat frequency is by overlapping

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two waves that have different frequencies.

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So when you do this, you get beats,

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and I'm not talking about Dr. Dre beats.

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These are a little less exciting.

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This is a sound wave that
goes from being loud,

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i.e. constructive to destructive, soft,

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and then all the way back
to loud, constructive again

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and the number of times
it does that per second

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is the beat frequency.

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The number of those wobbles per second

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is the beat frequency, and the way

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you find that beat frequency is you

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just take the difference in the two

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individual frequencies
of the two sound waves

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that are overlapping, but why is this

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the formula for beat frequency?

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Why isn't it something more complicated?

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Why did it come out to be this?

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That's what I wanna show you how

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to derive in this video,

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and the way we'll approach it

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is we're just gonna say all right,

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instead of finding that,
let's find this first.

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Let's find the beat period.

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So if beat frequency is number

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of wobbles per second,
then the beat period

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would just be the number
of seconds per wobble.

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In other words, how long does it take

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for this process to go from constructive

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all the way back to another constructive?

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'Cause if we can find this, if we can

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find this beat period, then we know

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that the frequency is always just

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one over the period.

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So if I can find the beat period,

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all I have to do is take one over that,

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and I'd get the beat frequency.

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So, how do I get this?

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How do we find this beat period?

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We find it by recognizing that

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this beat period is the time it takes

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for these waves to shift in such a way

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that these peaks originally overlap

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and then they overlap again
later on in the process.

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So let me clean this up.

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Let's get rid of this.

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Let's just look at the
two waves individually.

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So assuming they start off in
phase, perfectly constructive.

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Right, so I'm gonna assume over here

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they started off constructive.

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We really just wanna know how long

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does it take for them to
become constructive again

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all the way over to here?

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We're looking for this time.

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And the way we'll find this is by

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recognizing if these waves
started off in phase,

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every time the red wave goes through

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one whole cycle, so in the time it takes

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the red wave to complete one period,

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it is now this far in
front of the blue wave,

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and by in front of, I just mean,

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happening that much later than the

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blue wave's crest is happening.

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So these crests are now out of phase

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a little bit by this amount.

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What would that amount be?

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That's just the difference in periods

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between the two waves.

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So that's just gonna be
the period of wave one.

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We'll make the red wave, wave one,

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and we'll make the blue wave, wave two.

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That way we're not gonna get a negative

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number here or anything weird

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'cause the period of the red wave

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is a little longer than the
period of the blue wave.

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So if we take this difference,

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this is how out of phase in time

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these two peaks are getting over time.

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So after, if I waited one cycle,

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they would be exactly
that far apart in time.

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If I waited two cycles, they'd be

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two times that far apart.

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If I waited three cycles,

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they'd be three times that far apart.

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How does this help us?

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Well look, now we can just say,

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all right if we wanna know how far

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apart these two crests are in time

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at any given moment, I'll just take

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this difference in
period, and I'll multiply

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by the number of cycles that the red wave

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has gone through.

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So if the red wave has
gone through three cycles,

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I'd plug in three here,

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and then three times this difference

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would tell me those peaks are gonna be

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spaced apart in time by that much time,

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this whole time right here.

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This is how far apart in
time the two crests are.

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So they start off right
on top of each other.

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After one cycle they're that far.

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After two cycles that far,

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three cycles that far,
four cycles that far,

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five cycles that far.

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Now they start to get so far apart

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in phase that this red wave is overlapping

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with the next blue
peak, i.e. the blue wave

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has basically lapped the
red wave at this point,

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and the blue wave has advanced in time

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through so many cycles that it's next peak

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is overlapping with this
red peak right here,

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and that's when it becomes
constructive again.

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So this is the condition
for this to become

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constructive again is that this red wave

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is now so out of phase, it's out of phase

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one whole blue period because now it's

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gonna overlap with that blue peak

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of the next peak.

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So the next peak sneaks in here,

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and now the two overlap again.

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So, really if I wanna know the time

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between constructive
to constructive again,

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I just wanna know when this distance,

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the two peaks have been spaced apart.

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When that just equals one entire
period of the second wave.

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So when this is true, I know that I've

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waited long enough for these
to become constructive again

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because the blue wave
will have exactly lapped

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the red wave, and then they overlap

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peak to peak again, but
you might be concerned.

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You might be like, wait a minute,

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there's no time in here.

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How am I gonna solve for the time it took

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when there's no actual little t.

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These are just constants, right?

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This period of blue
wave is just a constant

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determined by whatever the
period of the blue wave is,

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and this period of the red wave

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is just a constant.

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Whatever the period of the red wave is,

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there's no variable t
in here to solve for.

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Here's how we sneak t in here.

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This number of cycles the
red wave has gone through

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isn't necessarily an integer.

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Maybe it's an integer.

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Maybe the red wave goes through exactly,

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in fact, on this picture, we can count

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the red wave went through one, two,

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three, four, five, six cycles exactly

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before it became in phase again

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with the blue wave, but
that doesn't always happen.

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That was just because I
wanted to draw this nicely

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and make it so that you
could see this line up well.

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Maybe this isn't an integer.

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Maybe this happens after
4.2 cycles or 3.1 cycles.

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It'd be nice if we had a more general way

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to write the number of cycles the red

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wave has gone through and we do.

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Think about it.

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The number of cycles the
red wave has gone through

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is just gonna equal the
time that you've waited

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divided by the period of the red wave.

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I'll call that period one 'cause that's

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what we called it up here.

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This makes sense 'cause if I wait

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one whole period one, then I'd have period

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one over period one
gives me one whole cycle

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that the red wave has gone through.

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If I wait two periods, I'd
have two times T1 up here.

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The T1s would cancel each
other and I'd get two.

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Two cycles is what it's gone through.

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So this gives me an opportunity

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to write the number of cycles

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the red wave has gone
through as a function

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of whatever time it happens to be.

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Now I can plug in times where I

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don't get an integer number of cycles.

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I can plug in at 4.3 seconds,

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how many cycles has the
red wave gone through?

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I take 4.3 seconds, I divide by the period

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which is the time for one cycle,

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and I get the number of cycles

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the red wave has gone through.

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So I'm gonna replace this nR over here

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with this expression.

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I'll just copy this, and I'm gonna

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rewrite nR as the time that you've waited

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divided by the period of that red wave,

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and now all I have to do is solve

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for the time little t because that will

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be telling me the time that it took

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for these two waves to
get out of phase in time

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by a whole period of the second wave

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which is the requirement for it to go

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from constructive all the
way back to constructive

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which is what we were looking for.

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That was the beat period up here.

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So if I solve this for little t,

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when I solve this for little t,

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I'll get little t, the time that it takes

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in order for these two peaks to get

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spaced out a whole second period

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is gonna be, I'll
multiply both sides by T1,

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I get T1 times T2, and then divided by,

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I'll have to divide by
this parenthesis term.

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So I'll have to divide by T1 - T2.

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And I'm claiming that this
actually is the beat period.

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It's the beat period because we stuck

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in this condition that they're gonna

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be spaced apart one whole period

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of the second wave, and
if they started in phase,

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they're gonna end in phase,

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and that's gonna be the time
between constructive points.

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So this right here is the beat period.

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This time we solve for is the beat period.

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So I'm gonna take this.

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I'm just gonna say this beat
period is equal to this.

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So this is the beat period
for these two waves.

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So here's a cool formula for beat period,

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but we didn't want the beat period.

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Remember we wanted to derive this formula

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for beat frequency.

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So the way we find frequency is we say

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that the frequency is one over the period.

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And let me clean this up a little bit.

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Let me get rid of this.

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So if we solve now, if we take one

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over the beat period,
that's just gonna equal

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one over this right hand side

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which means I just get the
difference in the periods

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divided by the product of the two periods

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because I just flip that right hand side,

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and I can expand this now.

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I've T1 minus T2 over T1 times T2.

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So if I expand the top, this is

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gonna be equal to T1 over T1 times T2,

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and then minus T2 over T1 times T2.

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And now something magical happens.

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The T1s here cancel, and
the T2s over here cancel,

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and I get that one over the beat period

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is gonna be equal to one over the period

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of the second wave minus one over

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the period of the first wave.

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But look at this.

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One over the beat period,
what's one over the beat period?

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That's just the beat frequency.

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So one over beat period
is just beat frequency.

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And what's one over the
period of the second wave?

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That's just the frequency
of the second wave.

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And what's one over the
period of the first wave?

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That's just the frequency
of the first wave.

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And we get our beat frequency formula.

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This is the formula we
were trying to find.

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00:08:54,808 --> 00:08:56,615
Remember over here we wanted to show

261
00:08:56,615 --> 00:08:59,280
that the beat frequency
formula looked like this,

262
00:08:59,280 --> 00:09:00,113
and it does.

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00:09:00,113 --> 00:09:00,946
It looked just like this,

264
00:09:00,946 --> 00:09:02,043
and the reason is you had to wait

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00:09:02,043 --> 00:09:04,371
exactly that long, the beat period.

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00:09:04,371 --> 00:09:06,003
In order for these peaks to get

267
00:09:06,003 --> 00:09:07,992
back into phase again, you had to wait

268
00:09:07,992 --> 00:09:09,482
for the peaks to get out of phase

269
00:09:09,482 --> 00:09:12,184
by one whole period of the second wave.

270
00:09:12,184 --> 00:09:13,261
Now you might be like, wait,

271
00:09:13,261 --> 00:09:15,568
we need the absolute
values in here, right?

272
00:09:15,568 --> 00:09:17,058
I mean we could put these in here.

273
00:09:17,058 --> 00:09:18,894
We were careful at the beginning.

274
00:09:18,894 --> 00:09:20,616
I made sure that I took a difference

275
00:09:20,616 --> 00:09:23,237
between a larger period
and a smaller period.

276
00:09:23,237 --> 00:09:25,341
So I took the larger minus the smaller

277
00:09:25,341 --> 00:09:27,118
which made sure we had a positive number,

278
00:09:27,118 --> 00:09:28,448
but you could throw this in here

279
00:09:28,448 --> 00:09:30,370
just in case you don't know which period

280
00:09:30,370 --> 00:09:31,833
or which frequency is larger.

281
00:09:31,833 --> 00:09:32,666
But that's how you get it,

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00:09:32,666 --> 00:09:34,554
that how you determine the beat frequency,

283
00:09:34,554 --> 00:09:37,531
and that's why the formula
looks the way it does.

284
00:09:37,531 --> 00:09:39,636
So recapping, we were able to determine

285
00:09:39,636 --> 00:09:42,216
and derive a formula
for the beat frequency

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00:09:42,216 --> 00:09:44,482
by figuring out the time we'd have to wait

287
00:09:44,482 --> 00:09:46,056
in order for these waves to become

288
00:09:46,056 --> 00:09:48,831
one whole period out of phase

289
00:09:48,831 --> 00:09:50,424
which is another way of saying how long

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00:09:50,424 --> 00:00:00,000
we have to wait for them
to become back in phase.