Superposition. When two waves meet, the pressure at a point is the sum of the two. Here both waves are sound at nearly the same frequency. At some moments the two waves are in phase (their peaks line up): they add up and the sound is loud. A little later the waves have drifted out of step: the peak of one lines up with the trough of the other, they cancel, and the sound is quiet. Then they drift back into step again.

y = A sin(2π f₁ t) + A sin(2π f₂ t) = 2A cos(π (f₁ − f₂) t) × sin(2π f_avg t)

The sum is a wave of frequency f_avg = (f₁ + f₂)/2, which is the pitch you hear, whose amplitude (the envelope, the white curve) changes slowly as 2A|cos(π (f₁ − f₂) t)|. The loudness peaks every time the envelope reaches a maximum. This happens at the rate of f_beat = |f₁ − f₂| times a second (the cosine changes sign between peaks, but the loudness does not). So the time between two loud moments is 1/f_beat.

Unequal waves. If wave 2 is quieter, the waves can never cancel completely. The amplitude of the sum swings between A₁ + A₂ (loud) and |A₁ − A₂| (quiet), and the beats are less deep.

What you hear. A difference of about 1 to 15 Hz sounds like a single note whose loudness throbs. When the difference is bigger (about 15 to 40 Hz) the throbbing speeds up into a rough, buzzing sound, and when it is bigger still your ear can pick out the two notes. If the two frequencies are exactly the same there are no beats at all. Musicians use beats to tune instruments: they adjust a string until the beats slow down and stop.

The screen. It shows a short piece of the sound (the time at the bottom). The waves are drawn as pressure changes with time, and they slowly drift sideways. If too many cycles would be crowded on the screen, only the envelope is drawn.

The sound. Pure tones are made by your browser. Press “Both: beats” to hear the sum. Take care with the volume: it starts low. The sound stops when you scroll the page out of view.