Beat frequency | Physics | Khan Academy

What's up, everybody? I want to 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 going to 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: 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 what the sound would sound like, and let's say it's sending the 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 3 m to the right of the speaker, just so we have a number to refer to. So there's air over here. The air is chilling, just relaxing, 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. If we graft 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, 3 m 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, i.e., the same period, then it's going to 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 at it; if I compare these two peaks, these two peaks 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 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 wave's got a longer period, so it's taking longer for this red wave to go through a cycle.

That means they're going to start becoming out of phase, right? The peaks aren't going to line up anymore. When this blue wave has displaced the air maximally to the right, this red wave is going to not have done that yet; it's going to 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, now the peaks match the valleys; this is straight up destructive. It's going to 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, and 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 going to be loud again. So what you would hear if you were standing at this point, 3 m 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 going to hear like Dr. 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—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, 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 going to play them both now.

Here's the 443 Hertz, and here's the 440, and you hear a wobble. It starts, 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 in sound loudness that you hear when you overlap two waves that have 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 going to be small because there aren't going to 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 is going to 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, it turns out it's very, very easy.

I'm just going to 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 want to find it, if I know the frequency of the first wave—so if wave 1 has a frequency F1, so say that blue wave has a frequency F1, and wave 2 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, i.e., 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 going to 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 going to 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 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 become destructive, and then 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 then let's say there's also a clarinet; they play it. They want to make sure they're in tune; they want to make sure their 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.

More as those notes get closer and closer, there will 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 5 hertz and the flute's playing 440, that means the clarinet is 5 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, it might be playing a little flat.

So, it would 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 frequency, you know both of these frequencies are the same. But what do you do? 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 going to 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 instead of whoa, whoa, whoa, whoa, you would hear whoo, whoo, whoo, right? And then once you're perfectly in tune, whoa, 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 going to 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 have 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.