The Physics Philes, lesson 114: We Got da Beats
Remember last week we learned all about wave interference. As I mentioned at the end of that post, we stuck to waves that have the same frequency. But that’s not always how things work out in the real world. Something slightly different happens when we have two waves overlap that have slightly different frequencies, and all we have to do is apply some concepts we already know and love.
Let’s think for a moment about these two waves, specifically a point at which those two waves overlap. We can use the principle of superposition – basically just adding the waves together – to find the displacement at a particular time. When we do this to two waves with slightly different frequencies, we find that at certain times the waves are in phase, and certain times the waves are out of phase. When the waves are in phase, their amplitudes add up. When they are out of phase, they cancel each other out.
The resulting wave looks like a sinusoidal wave with varying amplitude. The amplitude is at its maximum when the two waves are in phase and the amplitude is zero when the waves are out of phase. If we’re talking about sound waves, we can hear these differences. The amplitude variations cause the loudness to vary. These variations are called beats. The loudness will also vary with frequency, and we call that frequency the beat frequency.
We can determine the beat frequency from just the frequencies of the original two waves. Let’s say that we have two waves of slightly different frequencies, fa and fb. The frequency fa is larger than fb. The period of fa is Ta, the period of fb is Tb, and Ta is smaller than Tb. These two waves start out in phase and are in phase again when fa has gone through one more cycle than fb and the time at which this happens we will call the period of the beat. Let the letter n represent the number of cycles fa has gone through in the time that corresponds to the period of the beat. In that case, the number of cycles fb will have gone through in the same time is n-1. That means that the period of the beat is equal to the number of cycles multiplied by the frequency’s period:
Doing a little algebra and eliminating the n, we get:
We know from previous lessons that the period is the reciprocal of the frequency. That means that the period of the beat must be the reciprocal of the beat frequency:
That means that:
Oh cool! The beat frequency is just the difference between the frequency of the two original waves.
We can hear the beats if the beat frequency is less than 6 or 7 Hz. For example, if an instrument had that kind of beat frequency we would hear it as out of tune. However, once we start getting above 7 Hz, we can’t distinguish individual beats anymore. Instead we hear it as consonance and dissonance, depending on the ratio of the two tones we start out with. Sometimes, we can hear something called the difference tone which has a pitch equal to the beat frequency. You could hear this in addition to the two original tones.
So that’s pretty neat, right? It’s cool that we know how things we can’t see interact with each other in ways we can measure. Next week we’ll shift gears a little as we start a discussion on the Doppler effect.
Featured image credit: Alice Henneman via Flickr