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Derivation of beat frequency formula

In this video David derives the formula for beat frequency. Created by David SantoPietro.

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Video transcript

- [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.