- [Teacher] What's up everybody?
In this video, I wanna derive
this formula right here.
This is the formula for beat frequency,
so remember the way you get beats
or beat frequency is by overlapping
two waves that have different frequencies.
So when you do this, you get beats,
and I'm not talking about Dr. Dre beats.
These are a little less exciting.
This is a sound wave that goes from being loud,
i.e. constructive to destructive, soft,
and then all the way back to loud, constructive again
and the number of times it does that per second
is the beat frequency.
The number of those wobbles per second
is the beat frequency, and the way
you find that beat frequency is you
just take the difference in the two
individual frequencies of the two sound waves
that are overlapping, but why is this
the formula for beat frequency?
Why isn't it something more complicated?
Why did it come out to be this?
That's what I wanna show you how
to derive in this video,
and the way we'll approach it
is we're just gonna say all right,
instead of finding that, let's find this first.
Let's find the beat period.
So if beat frequency is number
of wobbles per second, then the beat period
would just be the number of seconds per wobble.
In other words, how long does it take
for this process to go from constructive
all the way back to another constructive?
'Cause if we can find this, if we can
find this beat period, then we know
that the frequency is always just
one over the period.
So if I can find the beat period,
all I have to do is take one over that,
and I'd get the beat frequency.
So, how do I get this?
How do we find this beat period?
We find it by recognizing that
this beat period is the time it takes
for these waves to shift in such a way
that these peaks originally overlap
and then they overlap again later on in the process.
So let me clean this up.
Let's get rid of this.
Let's just look at the two waves individually.
So assuming they start off in phase, perfectly constructive.
Right, so I'm gonna assume over here
they started off constructive.
We really just wanna know how long
does it take for them to become constructive again
all the way over to here?
We're looking for this time.
And the way we'll find this is by
recognizing if these waves started off in phase,
every time the red wave goes through
one whole cycle, so in the time it takes
the red wave to complete one period,
it is now this far in front of the blue wave,
and by in front of, I just mean,
happening that much later than the
blue wave's crest is happening.
So these crests are now out of phase
a little bit by this amount.
What would that amount be?
That's just the difference in periods
between the two waves.
So that's just gonna be the period of wave one.
We'll make the red wave, wave one,
and we'll make the blue wave, wave two.
That way we're not gonna get a negative
number here or anything weird
'cause the period of the red wave
is a little longer than the period of the blue wave.
So if we take this difference,
this is how out of phase in time
these two peaks are getting over time.
So after, if I waited one cycle,
they would be exactly that far apart in time.
If I waited two cycles, they'd be
two times that far apart.
If I waited three cycles,
they'd be three times that far apart.
How does this help us?
Well look, now we can just say,
all right if we wanna know how far
apart these two crests are in time
at any given moment, I'll just take
this difference in period, and I'll multiply
by the number of cycles that the red wave
has gone through.
So if the red wave has gone through three cycles,
I'd plug in three here,
and then three times this difference
would tell me those peaks are gonna be
spaced apart in time by that much time,
this whole time right here.
This is how far apart in time the two crests are.
So they start off right on top of each other.
After one cycle they're that far.
After two cycles that far,
three cycles that far, four cycles that far,
five cycles that far.
Now they start to get so far apart
in phase that this red wave is overlapping
with the next blue peak, i.e. the blue wave
has basically lapped the red wave at this point,
and the blue wave has advanced in time
through so many cycles that it's next peak
is overlapping with this red peak right here,
and that's when it becomes constructive again.
So this is the condition for this to become
constructive again is that this red wave
is now so out of phase, it's out of phase
one whole blue period because now it's
gonna overlap with that blue peak
of the next peak.
So the next peak sneaks in here,
and now the two overlap again.
So, really if I wanna know the time
between constructive to constructive again,
I just wanna know when this distance,
the two peaks have been spaced apart.
When that just equals one entire period of the second wave.
So when this is true, I know that I've
waited long enough for these to become constructive again
because the blue wave will have exactly lapped
the red wave, and then they overlap
peak to peak again, but you might be concerned.
You might be like, wait a minute,
there's no time in here.
How am I gonna solve for the time it took
when there's no actual little t.
These are just constants, right?
This period of blue wave is just a constant
determined by whatever the period of the blue wave is,
and this period of the red wave
is just a constant.
Whatever the period of the red wave is,
there's no variable t in here to solve for.
Here's how we sneak t in here.
This number of cycles the red wave has gone through
isn't necessarily an integer.
Maybe it's an integer.
Maybe the red wave goes through exactly,
in fact, on this picture, we can count
the red wave went through one, two,
three, four, five, six cycles exactly
before it became in phase again
with the blue wave, but that doesn't always happen.
That was just because I wanted to draw this nicely
and make it so that you could see this line up well.
Maybe this isn't an integer.
Maybe this happens after 4.2 cycles or 3.1 cycles.
It'd be nice if we had a more general way
to write the number of cycles the red
wave has gone through and we do.
Think about it.
The number of cycles the red wave has gone through
is just gonna equal the time that you've waited
divided by the period of the red wave.
I'll call that period one 'cause that's
what we called it up here.
This makes sense 'cause if I wait
one whole period one, then I'd have period
one over period one gives me one whole cycle
that the red wave has gone through.
If I wait two periods, I'd have two times T1 up here.
The T1s would cancel each other and I'd get two.
Two cycles is what it's gone through.
So this gives me an opportunity
to write the number of cycles
the red wave has gone through as a function
of whatever time it happens to be.
Now I can plug in times where I
don't get an integer number of cycles.
I can plug in at 4.3 seconds,
how many cycles has the red wave gone through?
I take 4.3 seconds, I divide by the period
which is the time for one cycle,
and I get the number of cycles
the red wave has gone through.
So I'm gonna replace this nR over here
with this expression.
I'll just copy this, and I'm gonna
rewrite nR as the time that you've waited
divided by the period of that red wave,
and now all I have to do is solve
for the time little t because that will
be telling me the time that it took
for these two waves to get out of phase in time
by a whole period of the second wave
which is the requirement for it to go
from constructive all the way back to constructive
which is what we were looking for.
That was the beat period up here.
So if I solve this for little t,
when I solve this for little t,
I'll get little t, the time that it takes
in order for these two peaks to get
spaced out a whole second period
is gonna be, I'll multiply both sides by T1,
I get T1 times T2, and then divided by,
I'll have to divide by this parenthesis term.
So I'll have to divide by T1 - T2.
And I'm claiming that this actually is the beat period.
It's the beat period because we stuck
in this condition that they're gonna
be spaced apart one whole period
of the second wave, and if they started in phase,
they're gonna end in phase,
and that's gonna be the time between constructive points.
So this right here is the beat period.
This time we solve for is the beat period.
So I'm gonna take this.
I'm just gonna say this beat period is equal to this.
So this is the beat period for these two waves.
So here's a cool formula for beat period,
but we didn't want the beat period.
Remember we wanted to derive this formula
for beat frequency.
So the way we find frequency is we say
that the frequency is one over the period.
And let me clean this up a little bit.
Let me get rid of this.
So if we solve now, if we take one
over the beat period, that's just gonna equal
one over this right hand side
which means I just get the difference in the periods
divided by the product of the two periods
because I just flip that right hand side,
and I can expand this now.
I've T1 minus T2 over T1 times T2.
So if I expand the top, this is
gonna be equal to T1 over T1 times T2,
and then minus T2 over T1 times T2.
And now something magical happens.
The T1s here cancel, and the T2s over here cancel,
and I get that one over the beat period
is gonna be equal to one over the period
of the second wave minus one over
the period of the first wave.
But look at this.
One over the beat period, what's one over the beat period?
That's just the beat frequency.
So one over beat period is just beat frequency.
And what's one over the period of the second wave?
That's just the frequency of the second wave.
And what's one over the period of the first wave?
That's just the frequency of the first wave.
And we get our beat frequency formula.
This is the formula we were trying to find.
Remember over here we wanted to show
that the beat frequency formula looked like this,
and it does.
It looked just like this,
and the reason is you had to wait
exactly that long, the beat period.
In order for these peaks to get
back into phase again, you had to wait
for the peaks to get out of phase
by one whole period of the second wave.
Now you might be like, wait,
we need the absolute values in here, right?
I mean we could put these in here.
We were careful at the beginning.
I made sure that I took a difference
between a larger period and a smaller period.
So I took the larger minus the smaller
which made sure we had a positive number,
but you could throw this in here
just in case you don't know which period
or which frequency is larger.
But that's how you get it,
that how you determine the beat frequency,
and that's why the formula looks the way it does.
So recapping, we were able to determine
and derive a formula for the beat frequency
by figuring out the time we'd have to wait
in order for these waves to become
one whole period out of phase
which is another way of saying how long
we have to wait for them to become back in phase.