Lesson Video: Beats

In this video we learn how beats are produced by sound waves of different frequencies, what beats are useful for, and how to calculate beat frequencies.


Video Transcript

In this video, we’re going to learn about beats. We’ll see what beats are, how they’re created, and how they can be useful.

To start out, imagine that you’ve planted some beats in your garden patch, and you’ve been carefully cultivating them for some time. Every day, you watch carefully as they grow. They seem to be taking a long time to mature. But one day, when you’re not even looking, they finally reach the point of being ready to harvest. When you eventually notice this, in your excitement, you run over to the garden patch and pull one up. These are gonna be some amazing beats.

Oh, but wait a second! This isn’t the type of beat we’re talking about. The beats we’re speaking of have a different meaning entirely. The beats we’re talking about have to do with sound waves that our ears detect. We might think of beats as something produced by a discrete periodic sound, like the beating of a drum. And it’s true that drums create beats. But when we use this word beats in a physics context, we’re speaking of something a bit different.

To consider just how beats are made in a physics context, we can start by recalling the relationship for a sinusoidal sound wave. This is a mathematical function that tells us the amplitude, 𝑦, of that sine wave as a function of position and time. In other contexts, we’ve frozen the time value of this function and just plotted it as it varies in space. To consider how beats are formed though, let’s do the opposite. Let’s freeze the position of this wave and instead let it vary in time, 𝑡.

So, we’re going to plot the wave height in meters as it relates to the time passed in seconds. And we can let the position 𝑥 be fixed at zero. Our plot of this sound wave as it varies in time might look like this. And depending on the particular frequency that this wave has, we might even be able to hear it, if its frequency falls within the range of about 20 to 20000 hertz. If we could hear this sound, it would sound like a solid constant tone, like the dial tone on a phone or some other steady sound.

If we listen to this sound, we wouldn’t hear any beats being produced. But all that changes if we added another wave that was at a different frequency. This second wave has the same amplitude as our original wave, but its frequency is much higher. It goes through more cycles per second. Physically, this might be something like having two speakers which are producing two tones of different frequencies and standing in such a way that we could hear both of those tones at the same time.

If both these frequencies were audible, the green would have a higher-pitched sound to our ear, and the blue would sound like a lower note. And because these two waves have the same amplitude, they would have the same loudness to our ear. The only difference would be the tone, or frequency, we would hear. It’s in this mixing of waves with different frequencies that beats are produced.

To see how, let’s recall the principle of wave superposition, which says that the combined amplitude of two waves, we can call them 𝑦 one and 𝑦 two, that mix is equal simply to the sum of those two amplitudes. This means that we can combine our blue and our green waves moving left to right, starting at 𝑡 equals zero. And at each point, the resulting amplitude of the wave, the combined wave, will equal the sum of the immediate amplitudes of the green and the blue wave.

For example, if we were to take this point where the blue and the green both have maxima, adding these waves together at this point in time gives a value twice as great as either of the waves individually. On the other hand, if we look at a point where the green and the blue waves destructively interfere, that is, where one has a maximum and the other has a minimum, at that point, the two waves would add up to zero. This would be the combined amplitude of the waves. If we add the blue and the green curves together at all points in time, we get a curve that looks about like this gold curve.

This function looks a little bit strange. It’s not a pure sine or cosine function. But notice that, just like the blue and the green curves, the gold curve has a periodic structure to it. When we look at this curve, we see there’s a peak and then two troughs and then a peak and two more troughs and another peak and two more troughs after that. So, even though this combined wave does look strange, there is an orderliness to it. We can say that this combined gold wave has its own frequency and, therefore, its own period.

If we call this frequency 𝑓 and then if we call the frequency of the green wave 𝑓 sub one and then the frequency of the blue wave 𝑓 sub two, then we can write that 𝑓, the frequency of the combined wave, is equal to 𝑓 sub one minus 𝑓 sub two. It’s this frequency 𝑓 that our ears would hear if we were standing in a place to listen to both 𝑓 one and 𝑓 two coming from speakers.

If we imagine listening to this combined wave, say we started out at the low point where its amplitude is zero, at that point, we wouldn’t hear any sound at all. It would be soft. But then the sound moves up an amplitude to a high point. Then, it goes back to a zero point, then back up to a high point, then back to zero. It’s over this period in between zero, or no points of sound, that we would hear a high peak of a sound. In other words, we would hear a beat.

That’s something that’s so interesting about this combined wave. It’s not that it’s just another tone with a constant amplitude like our blue and green waves that input to create it, but it’s its own new wave entirely that has null points and high points. Instead of hearing a steady tone, we hear beats. And the frequency of those beats is equal to the difference between the frequencies of the waves that combined to create the gold wave. This is how beats that our ears hear are created in physics, by the mixing of two sound waves that have different frequencies.

If the frequencies 𝑓 one and 𝑓 two are very close to one another, then the beat frequency will be very low. But if these frequencies are far apart, the beat frequency will be relatively high. If you’ve ever tuned a guitar or other instrument with a tuning fork, you’ve probably heard these beats being created.

If your instrument is in tune or very nearly so, then the beat frequency is extremely low. We effectively don’t hear any beats because 𝑓 one, let’s call that the frequency of the fork, and 𝑓 two, call that the frequency of our instrument, are effectively the same. But when 𝑓 two and 𝑓 one have a significant gap between them, we’re able to hear that with our ear and adjust our instrument. Let’s try out an example involving beat frequency.

A piano tuner hears a beat every 1.00 seconds when listening to a 164-hertz frequency tuning fork and a single piano string. What is the frequency of the string if it is of a lower frequency than the tuning fork?

In this scenario, the tuner is hearing a beat every 1.00 seconds. That means the beat has a frequency of 1.00 hertz. We can recall that beat frequency, 𝑓 sub 𝐵, is equal to the difference between the frequency of the one tone being mixed and the frequency of the other tone. And in case 𝑓 two is greater than 𝑓 one, we put absolute value bars around this difference. Since the tuner in our example hears a beat every 1.00 seconds, that means there’s one complete wave cycle each second, or that this beat frequency is 1.00 hertz.

By our beat frequency equation, we can say that this is equal to 164 hertz, the frequency of the tuning fork, minus the frequency of the piano string, which we’ve called 𝑓 sub 𝑠. Rearranging, we can say that 𝑓 sub 𝑠 is equal to 164 minus 1.00 hertz, or 163 hertz. That’s the frequency of this piano string before it’s tuned to the tuning fork.

Let’s summarize what we’ve learned so far about beats. We’ve seen that beats are created by sound waves with different frequencies overlapping. We also saw that the beat frequency created by two sound waves with different frequencies mixing is equal to the frequency of the first wave minus the frequency of the second wave. And then, in case that value is negative, we have absolute value bars around this difference.

And finally, we saw that beats are useful for helping us match frequencies between two different objects or systems. We might want to match the frequencies of sound produced by instruments in an orchestra or the frequencies of two different electrical signals. And beats can even help us with mechanical alignment of various parts, such as in automobiles or other vehicles. Overall, beats are a helpful and interesting concept and mathematically can be summarized in a straightforward way.

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