beats

Beats are the slow, periodic rise and fall of amplitude that occurs when two oscillations of nearly equal frequency are superimposed. If two signals have frequencies f₁ and f₂, their sum oscillates at the average frequency (f₁ + f₂)/2 with an envelope that pulses at the beat frequency |f₁ − f₂|.

Musicians use beats constantly. When tuning a guitar string against a reference tone, you hear a pulsing wah-wah-wah that slows down as the two frequencies approach each other and vanishes when they match. Piano tuners listen for beats between strings that are supposed to sound the same note.

In coupled pendulums, beats appear as energy transfer. Start one pendulum swinging and the other still; the first gradually stops while the second picks up the motion, then the process reverses. The beat frequency is the difference between the two normal-mode frequencies of the coupled system. The phenomenon is the time-domain signature of two close spectral lines — the same mathematics appears in quantum mechanics, radio engineering, and acoustics.