beats

Strike two piano strings tuned almost — but not quite — to the same note, and you hear a single tone that slowly swells loud, fades to silence, and swells again, throbbing once or twice a second. That pulsing waxing-and-waning is beats. It is what two oscillations of nearly equal frequency do when they add together: they drift in and out of step, reinforcing then cancelling, over and over.

Add two cosines of close frequencies, cos(omega_1 t) + cos(omega_2 t). A trig identity rewrites the sum as 2 cos((omega_1 - omega_2)/2 · t) cos((omega_1 + omega_2)/2 · t). Read this as a fast oscillation at the average frequency (omega_1 + omega_2)/2, wrapped inside a slow amplitude envelope that wobbles at the much lower frequency (omega_1 - omega_2)/2. The slow envelope is what your ear hears as the throbbing; the loudness peaks (beats) come twice per envelope cycle, so the beat frequency is the full difference |omega_1 - omega_2|.

In a driven system, beats appear when you push a lightly damped oscillator at a frequency close to (but not equal to) its natural frequency before the transient has died out: the driven response and the system's own ringing have nearly the same frequency and beat against each other. Beats are also the everyday tool of musicians, who tune by listening for the beats between two notes to slow to nothing — when the throbbing stops, the frequencies match exactly.