- [Voiceover] What's up everybody?
I wanna talk to you about beat frequency,
and to do so let me talk to you
about this air displacement versus time graph.
So this is gonna give you the displacement
of the air molecules for any time at a particular location.
So say you had some speaker and it was playing
a nice simple harmonic tone and so it would sound
something like this.
(tone playing)
That's 440 hertz, turns out that's an A note.
People use that a lot when they're tuning instruments
and whatnot so that's this sound would sound like,
and let's say it's sending this sound out
and at a particular point, one point in space,
we measure what the displacement of the air is
as a function of time.
Let's just say we're three meters
to the right of this speaker.
Just so we have a number to refer to,
so there's air over here, the air's chillin,
just relaxin and then the sound wave comes by
and that causes this air to get displaced.
It moves back and forth.
A minuscule amount but some amount,
and if we graphed that displacement
as a function of time we would get this graph.
So in other words this entire graph
is just personalized for that point in space,
three meters away from this speaker.
So why am I telling you this?
Well because we know if you overlap two waves,
if I take another wave and let's just say this wave
has the exact same period as the first wave,
right so I'll put these peak to peak
so you can see, compare the peaks, yep.
Takes the same amount of time
for both of these to go through a cycle,
that means they have the same period,
so if I overlap these, in other words if I took
another speaker and I played the same note next to it,
if I played it like this I'd hear constructive interference
cause these are overlapping peak to peak,
valley to valley perfectly.
This note would get louder if I was standing here
and listening to it and it would stay loud the whole time.
It would just sound louder the entire time,
constructive interference,
and if I moved that speaker forward a little bit
or I switched the leads, if I found some way
to get it out of phase so that
it was destructive interference,
I'd hear a softer note, maybe it would be silent
if I did this perfectly and it would stay silent or soft
the whole time, it would stay destructive in other words.
So if you overlap two waves that have the same frequency,
ie the same period, then it's gonna be constructive
and stay constructive, or be destructive
and stay destructive, but here's the crazy thing.
Let me get rid of this.
What if we overlapped two waves that had different periods?
What would happen then?
Let's just try it out.
So let me take this wave, this wave has a different period.
Look it, if I compare these two peaks,
these two peeks don't line up,
if I'm looking over here the distance
between these two peaks is not the same
as the distance between these two peaks.
It's hard to see, it's almost the same,
but this red wave has a slightly longer period
if you can see the time between peaks
is a little longer than the time between peaks
for the blue wave and you might think,
"Ah there's only a little difference here.
"Can't be that big of a deal right?"
It kind of is.
It causes a new phenomenon called beat frequency,
and I'll show you why it happens here.
So if I overlap these two.
So now you take two speakers,
but the second speaker you play it
at a slightly different frequency from the first.
What would you get?
Let's just look at what happens over here.
They start out in phase perfectly overlapping, right?
Peak to peak, so this is constructive,
this wave starts off constructively
interfering with the other wave.
So you hear constructive interference,
that means if you were standing at this point
at that moment in time, notice this axis is time not space,
so at this moment in time right here,
you would hear constructive interference
which means that those waves would sound loud.
Sound really loud at that moment,
but then you wait, this red waves got a longer period.
So it's taking longer for this red wave
to go through a cycle, that means they're
gonna start becoming out of phase, right?
The peaks aren't gonna line up anymore.
When this blue wave has displaced
the air maximally to the right,
this red wave is gonna not have done that yet,
it's gonna take a little longer for it to try to do that.
So these become out of phase, now it's less constructive,
less constructive, less constructive,
over here look it, now the peaks match the valleys.
This is straight up destructive, it's gonna be soft,
and if you did this perfectly
it might be silent at that point.
You wait a little longer and this blue wave
has essentially lapped the red wave, right?
You waited so long the blue wave has gone through
an extra whole period compared to the red wave,
an so now the peaks line up again,
and now it's constructive again
because the peaks match the peaks
and the valleys match the valleys.
So at that point it's constructive
and it's gonna be loud again so what you would hear
if you were standing at this point three meters away,
you'd first at this moment in time hear the note be loud,
then you'd hear it become soft
and then you'd hear it become loud again.
You'd hear this note wobble,
and the name we have for this phenomenon
is the beat frequency or sometimes it's just called beats,
and I don't mean you're gonna hear Doctor Dre
out of this thing that's not the kind of beats
I'm talking about, I'm just talking about
that wobble from louder to softer to louder.
Actually let me just play it.
Let me show you what this sounds like.
So if we play the A note again.
(tone playing)
That's the A note.
Let me play, that's 440 hertz, right?
That's a particular frequency.
Let me play just a slightly different frequency.
I'll play 443 hertz.
(tone playing)
And you're probably like that just sounds
like the exact same thing, I can't tell the difference
between the two, but if I play them both
you'll definitely be able to tell the difference.
So I'm gonna play them both now.
Here's the 443 hertz, and here's the 440.
(two tones playing)
And you hear a wobble.
This thing starts to wobble.
So let me stop this.
So that's what physicists are talking about
when they say beat frequency or beats,
they're referring to that wobble
and sound loudness that you hear
when you overlap two waves that different frequencies.
This is important, it only works
when you have waves of different frequency.
So what if you wanted to know the actual beat frequency?
What if you wanted to know how many
wobbles you get per second?
So how often is it going from constructive
to destructive back to constructive?
If that takes a long time the frequency is gonna be small,
cause there aren't gonna be many wobbles per second,
but if this takes a short amount of time,
if there's not much time between constructive
back to constructive then the beat frequency's
gonna be large, there will be many wobbles per second.
How would you figure out this beat frequency,
I'll call it FB, this would be how many times
this goes from constructive back to constructive per second.
So if it does that 20 times per second,
this thing would be wobbling 20 times per second
and the frequency would be 20 hertz.
So how do you find this if you know
the frequency of each wave,
and it turns out it's very very easy.
I'm just gonna show you the formula in this video,
in the next video we'll derive it
for those that are interested,
but in this one I'll just show you what it is,
show you how to use it.
So the beat frequency if you wanna find it,
if I know the frequency of the first wave,
so if wave one has a frequency, f1.
So say that blue wave has a frequency f1,
and wave two has a frequency f2,
then I can find the beat frequency
by just taking the difference.
I can just take f1 and then subtract f2,
and it's as simple as that.
That gives you the beat frequency.
Now you might wonder like wait a minute,
what if f1 has a smaller frequency than f2?
That would give me a negative beat frequency?
That doesn't make sense we can't have a negative frequency
so we typically put an absolute value sign around this.
You should take the higher frequency minus the lower,
but just in case you don't just stick
an absolute value and that gives you
the size of this beat frequency,
which is basically the number of wobbles per second,
ie the number of times it goes from constructive
all the way back to constructive per second.
That's what this beat frequency means
and this formula is how you can find it.
Now I should say to be clear,
we're playing two different sound waves,
our ears really just sort of gonna hear one total wave.
So these waves overlap.
You can do this whole analysis using wave interference.
You write down the equation of one wave,
you write down the equation of the other wave,
you add up the two, right?
We know that the total wave is gonna equal
the summation of each wave at a particular point in time.
So at one point in time if we take
the value of each wave and add them up,
we'd get the total wave, what would that look like?
What would the total wave look like?
It would look like this.
If we just add it up you'd get a total wave
that looks like this green dashed wave here.
Right over here, they add up to twice the wave,
and then in the middle they cancel to almost nothing,
and then back over here they add up again,
and so if you just looked at the total wave,
it would look something like this.
So the total wave would start with a large amplitude,
and then it would die out because they'd become destructive,
and then it would become a large amplitude again.
So you see this picture a lot
when you're talking about beat frequency
because it's showing what the total wave
looks like as a function of time
when you add up those two individual waves
since this is going from constructive
to destructive to constructive again,
and this is why it sounds loud
and then soft and then loud again to our ear.
So what would an example problem look like for beats?
Let's say you were told that there's a flute,
and let's say this flute is playing a frequency
of 440 hertz like that note we heard earlier,
and let's say there's also a clarinet.
They play it, they wanna make sure they're in tune,
they wanna make sure they're jam sounds good
for everyone in the audience,
but when they both try to play the A note,
this flute plays 440, this clarinet plays a note,
and let's say we hear a beat frequency,
I'll write it in this color, we hear a beat frequency
of five hertz so we hear five wobbles per second.
In fact if you've ever tried to tune an instrument
you know that one way to tune it is to try
to check two notes that are supposed to be the same.
You can tell immediately if they're not the same
cause you'll hear these wobbles,
and so you keep tuning it until
you don't hear the wobble anymore.
As those notes get closer and closer,
there'll be less wobbles per second,
and once you hear no wobble at all,
you know you're at the exact same frequency,
but these aren't, these are off,
and so the question might ask,
what are the two possible frequencies of the clarinet?
Well we know that the beat frequency
is equal to the absolute value
of the difference in the two frequencies.
So if there's a beat frequency of five hertz
and the flutes playing 440, that means the clarinet
is five hertz off from the flute.
So the clarinet might be a little too high,
it might be 445 hertz, playing a little sharp,
or it might be 435 hertz, might be playing a little flat.
So we'd have to tune to figure out
how it can get to the point where
there'd be zero beat frequency,
cause when there's zero beat frequencies you know
both of these frequencies are the same,
but what do you do?
How does it know?
How does the clarinet player know which one to do?
You kind of don't sometimes.
Sometimes you just have to test it out.
Let's say the clarinet player assumed,
all right maybe they were a little too sharp 445,
so they're gonna lower their note.
So they start to tune down, what will they listen for?
They'll listen for less wobbles per second.
So if you become more in tune in stead of,
(imitates wobbling tone)
you would hear,
(imitates slowing wobble)
right, and then once you're perfectly in tune,
(hums tone)
and it would be perfect, there'd be no wobbles.
If this person tried it and there were
more wobbles per second then this person would know,
"Oh, I was probably at this lower note.
"cause if I'm at 435, and I go to say 430 hertz,
"that's gonna be more out of tune."
Now the beat frequency would be 10 hertz,
you'd hear 10 wobbles per second,
and the person would know immediately,
"Whoa, that was a bad idea.
"I must not have been too sharp.
"I must've been too flat."
So now that you know you're a little too flat
you start tuning the other way,
so you can raise this up to 440 hertz
and then you would hear zero beat frequency,
zero wobbles per second, a nice tune,
and you would be playing in harmony.
So recapping beats or beat frequency occurs
when you overlap two waves that have different frequencies.
This causes the waves to go from being
constructive to destructive to constructive over and over,
which we perceive as a wobble in the loudness of the sound,
and the way you can find the beat frequency
is by taking the difference of the two frequencies
of the waves that are overlapping.