Beats and superposition

Superposition is the statement that when two waves overlap in the same medium, their displacements simply add at every point, moment by moment, and each wave continues on afterward exactly as if the other had never been there. Waves do not collide, merge or bounce off one another the way particles do; they pass straight through each other, and the pattern seen while they overlap is nothing more than the running sum of the two.

Beats are what that addition produces when the two frequencies are close but not identical. At some moments the two waves are in phase and reinforce, producing a loud peak; a short time later they have drifted out of phase and nearly cancel, producing a quiet dip. Because the two frequencies are slightly different, this in-phase, out-of-phase cycle repeats steadily, and the ear hears it as a slow throbbing in loudness layered on top of the underlying pitch — a beat. If the two frequencies were exactly equal there would be no drift and therefore no beating at all, so beats require a difference, not a match.

The beat frequency is simply |f₁ − f₂|, the plain difference between the two component frequencies — not their average, which is instead roughly the pitch of the combined tone that the throbbing is layered on. Musicians use this directly: two strings tuned close but not quite together produce an audible beat, and tuning consists of adjusting one string until that beat frequency drops toward zero, at which point the two are in tune.