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- [Voiceover] What's up everybody?

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I wanna talk to you about beat frequency,

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and to do so let me talk to you

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about this air displacement
versus time graph.

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So this is gonna give you the displacement

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of the air molecules for any
time at a particular location.

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So say you had some
speaker and it was playing

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a nice simple harmonic
tone and so it would sound

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something like this.

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(tone playing)

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That's 440 hertz, turns
out that's an A note.

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People use that a lot when
they're tuning instruments

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and whatnot so that's this
sound would sound like,

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and let's say it's sending this sound out

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and at a particular
point, one point in space,

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we measure what the
displacement of the air is

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as a function of time.

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Let's just say we're three meters

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to the right of this speaker.

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Just so we have a number to refer to,

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so there's air over
here, the air's chillin,

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just relaxin and then
the sound wave comes by

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and that causes this air to get displaced.

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It moves back and forth.

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A minuscule amount but some amount,

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and if we graphed that displacement

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as a function of time
we would get this graph.

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So in other words this entire graph

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is just personalized
for that point in space,

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three meters away from this speaker.

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So why am I telling you this?

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Well because we know if
you overlap two waves,

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if I take another wave and
let's just say this wave

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has the exact same
period as the first wave,

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right so I'll put these peak to peak

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so you can see, compare the peaks, yep.

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Takes the same amount of time

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for both of these to go through a cycle,

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that means they have the same period,

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so if I overlap these,
in other words if I took

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another speaker and I played
the same note next to it,

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if I played it like this I'd
hear constructive interference

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cause these are overlapping peak to peak,

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valley to valley perfectly.

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This note would get louder
if I was standing here

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and listening to it and it
would stay loud the whole time.

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It would just sound
louder the entire time,

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constructive interference,

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and if I moved that speaker
forward a little bit

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or I switched the leads,
if I found some way

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to get it out of phase so that

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it was destructive interference,

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I'd hear a softer note,
maybe it would be silent

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if I did this perfectly and
it would stay silent or soft

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the whole time, it would stay
destructive in other words.

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So if you overlap two waves
that have the same frequency,

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ie the same period, then
it's gonna be constructive

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and stay constructive, or be destructive

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and stay destructive, but
here's the crazy thing.

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Let me get rid of this.

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What if we overlapped two waves
that had different periods?

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What would happen then?

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Let's just try it out.

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So let me take this wave, this
wave has a different period.

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Look it, if I compare these two peaks,

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these two peeks don't line up,

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if I'm looking over here the distance

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between these two peaks is not the same

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as the distance between these two peaks.

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It's hard to see, it's almost the same,

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but this red wave has a
slightly longer period

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if you can see the time between peaks

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is a little longer than
the time between peaks

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for the blue wave and you might think,

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"Ah there's only a little difference here.

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"Can't be that big of a deal right?"

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It kind of is.

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It causes a new phenomenon
called beat frequency,

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and I'll show you why it happens here.

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So if I overlap these two.

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So now you take two speakers,

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but the second speaker you play it

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at a slightly different
frequency from the first.

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What would you get?

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Let's just look at what happens over here.

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They start out in phase
perfectly overlapping, right?

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Peak to peak, so this is constructive,

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this wave starts off constructively

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interfering with the other wave.

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So you hear constructive interference,

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that means if you were
standing at this point

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at that moment in time, notice
this axis is time not space,

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so at this moment in time right here,

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you would hear constructive interference

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which means that those
waves would sound loud.

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Sound really loud at that moment,

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but then you wait, this red
waves got a longer period.

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So it's taking longer for this red wave

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to go through a cycle, that means they're

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gonna start becoming out of phase, right?

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The peaks aren't gonna line up anymore.

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When this blue wave has displaced

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the air maximally to the right,

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this red wave is gonna
not have done that yet,

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it's gonna take a little longer
for it to try to do that.

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So these become out of phase,
now it's less constructive,

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less constructive, less constructive,

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over here look it, now the
peaks match the valleys.

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This is straight up
destructive, it's gonna be soft,

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and if you did this perfectly

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it might be silent at that point.

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You wait a little longer
and this blue wave

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has essentially lapped
the red wave, right?

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You waited so long the
blue wave has gone through

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an extra whole period
compared to the red wave,

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an so now the peaks line up again,

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and now it's constructive again

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because the peaks match the peaks

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and the valleys match the valleys.

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So at that point it's constructive

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and it's gonna be loud
again so what you would hear

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if you were standing at this
point three meters away,

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you'd first at this moment in
time hear the note be loud,

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then you'd hear it become soft

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and then you'd hear it become loud again.

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You'd hear this note wobble,

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and the name we have for this phenomenon

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is the beat frequency or
sometimes it's just called beats,

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and I don't mean you're
gonna hear Doctor Dre

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out of this thing that's
not the kind of beats

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I'm talking about, I'm just talking about

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that wobble from louder
to softer to louder.

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Actually let me just play it.

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Let me show you what this sounds like.

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So if we play the A note again.

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(tone playing)

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That's the A note.

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Let me play, that's 440 hertz, right?

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That's a particular frequency.

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Let me play just a slightly
different frequency.

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I'll play 443 hertz.

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(tone playing)

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And you're probably like that just sounds

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like the exact same thing,
I can't tell the difference

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between the two, but if I play them both

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you'll definitely be able
to tell the difference.

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So I'm gonna play them both now.

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Here's the 443 hertz, and here's the 440.

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(two tones playing)

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And you hear a wobble.

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This thing starts to wobble.

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So let me stop this.

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So that's what physicists
are talking about

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when they say beat frequency or beats,

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they're referring to that wobble

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and sound loudness that you hear

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when you overlap two waves
that different frequencies.

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This is important, it only works

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when you have waves of
different frequency.

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So what if you wanted to know
the actual beat frequency?

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What if you wanted to know how many

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wobbles you get per second?

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So how often is it going from constructive

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to destructive back to constructive?

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If that takes a long time the
frequency is gonna be small,

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cause there aren't gonna
be many wobbles per second,

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but if this takes a short amount of time,

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if there's not much time
between constructive

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back to constructive
then the beat frequency's

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gonna be large, there will
be many wobbles per second.

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How would you figure
out this beat frequency,

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I'll call it FB, this
would be how many times

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this goes from constructive
back to constructive per second.

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So if it does that 20 times per second,

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this thing would be
wobbling 20 times per second

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and the frequency would be 20 hertz.

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So how do you find this if you know

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the frequency of each wave,

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and it turns out it's very very easy.

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I'm just gonna show you
the formula in this video,

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in the next video we'll derive it

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for those that are interested,

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but in this one I'll
just show you what it is,

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show you how to use it.

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So the beat frequency
if you wanna find it,

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if I know the frequency of the first wave,

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so if wave one has a frequency, f1.

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So say that blue wave has a frequency f1,

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and wave two has a frequency f2,

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then I can find the beat frequency

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by just taking the difference.

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I can just take f1 and then subtract f2,

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and it's as simple as that.

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That gives you the beat frequency.

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Now you might wonder like wait a minute,

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what if f1 has a smaller
frequency than f2?

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That would give me a
negative beat frequency?

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That doesn't make sense we
can't have a negative frequency

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so we typically put an absolute
value sign around this.

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You should take the higher
frequency minus the lower,

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but just in case you don't just stick

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an absolute value and that gives you

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the size of this beat frequency,

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which is basically the
number of wobbles per second,

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ie the number of times
it goes from constructive

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all the way back to
constructive per second.

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That's what this beat frequency means

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and this formula is how you can find it.

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Now I should say to be clear,

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we're playing two different sound waves,

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our ears really just sort of
gonna hear one total wave.

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So these waves overlap.

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You can do this whole analysis
using wave interference.

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You write down the equation of one wave,

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you write down the
equation of the other wave,

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you add up the two, right?

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We know that the total wave is gonna equal

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the summation of each wave at
a particular point in time.

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So at one point in time if we take

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the value of each wave and add them up,

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we'd get the total wave,
what would that look like?

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What would the total wave look like?

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It would look like this.

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If we just add it up
you'd get a total wave

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that looks like this
green dashed wave here.

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Right over here, they
add up to twice the wave,

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and then in the middle they
cancel to almost nothing,

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and then back over here they add up again,

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and so if you just
looked at the total wave,

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it would look something like this.

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So the total wave would
start with a large amplitude,

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and then it would die out because
they'd become destructive,

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and then it would become
a large amplitude again.

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So you see this picture a lot

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when you're talking about beat frequency

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because it's showing what the total wave

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looks like as a function of time

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when you add up those two individual waves

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since this is going from constructive

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to destructive to constructive again,

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and this is why it sounds loud

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and then soft and then
loud again to our ear.

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So what would an example
problem look like for beats?

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Let's say you were told
that there's a flute,

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and let's say this flute
is playing a frequency

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of 440 hertz like that
note we heard earlier,

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and let's say there's also a clarinet.

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They play it, they wanna
make sure they're in tune,

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they wanna make sure
they're jam sounds good

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for everyone in the audience,

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but when they both try to play the A note,

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this flute plays 440, this
clarinet plays a note,

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and let's say we hear a beat frequency,

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I'll write it in this color,
we hear a beat frequency

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of five hertz so we hear
five wobbles per second.

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In fact if you've ever
tried to tune an instrument

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you know that one way to tune it is to try

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to check two notes that are
supposed to be the same.

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You can tell immediately
if they're not the same

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cause you'll hear these wobbles,

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and so you keep tuning it until

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you don't hear the wobble anymore.

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As those notes get closer and closer,

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there'll be less wobbles per second,

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and once you hear no wobble at all,

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you know you're at the
exact same frequency,

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but these aren't, these are off,

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and so the question might ask,

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what are the two possible
frequencies of the clarinet?

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Well we know that the beat frequency

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is equal to the absolute value

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of the difference in the two frequencies.

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So if there's a beat
frequency of five hertz

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and the flutes playing 440,
that means the clarinet

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is five hertz off from the flute.

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So the clarinet might
be a little too high,

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it might be 445 hertz,
playing a little sharp,

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or it might be 435 hertz,
might be playing a little flat.

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So we'd have to tune to figure out

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how it can get to the point where

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there'd be zero beat frequency,

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cause when there's zero
beat frequencies you know

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both of these frequencies are the same,

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but what do you do?

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How does it know?

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How does the clarinet
player know which one to do?

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You kind of don't sometimes.

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Sometimes you just have to test it out.

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Let's say the clarinet player assumed,

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all right maybe they were
a little too sharp 445,

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so they're gonna lower their note.

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So they start to tune down,
what will they listen for?

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They'll listen for less
wobbles per second.

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So if you become more in tune in stead of,

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(imitates wobbling tone)

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you would hear,

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(imitates slowing wobble)

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right, and then once
you're perfectly in tune,

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00:10:46,437 --> 00:10:47,561
(hums tone)

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and it would be perfect,
there'd be no wobbles.

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If this person tried it and there were

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more wobbles per second
then this person would know,

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00:10:54,182 --> 00:10:55,990
"Oh, I was probably at this lower note.

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"cause if I'm at 435, and
I go to say 430 hertz,

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"that's gonna be more out of tune."

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Now the beat frequency would be 10 hertz,

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you'd hear 10 wobbles per second,

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and the person would know immediately,

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"Whoa, that was a bad idea.

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"I must not have been too sharp.

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00:11:11,506 --> 00:11:12,834
"I must've been too flat."

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00:11:12,834 --> 00:11:14,357
So now that you know
you're a little too flat

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you start tuning the other way,

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so you can raise this up to 440 hertz

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00:11:19,478 --> 00:11:21,743
and then you would hear
zero beat frequency,

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zero wobbles per second, a nice tune,

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and you would be playing in harmony.

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So recapping beats or
beat frequency occurs

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when you overlap two waves that
have different frequencies.

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This causes the waves to go from being

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constructive to destructive
to constructive over and over,

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which we perceive as a wobble
in the loudness of the sound,

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and the way you can
find the beat frequency

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is by taking the difference
of the two frequencies

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of the waves that are overlapping.