Beats

Two guitar strings that are almost, but not quite, in tune make a sound that throbs: wah… wah… wah. As the tuner tightens one, the throbbing slows and finally stops. That throbbing is called beats.

Two close frequencies#

Take two sound waves of equal amplitude and slightly different angular frequencies ω₁ and ω₂, travelling the same way. At a fixed point, say x = 0, the total excess pressure is

where $\bar\omega = (\omega_1+\omega_2)/2$ is the average and $\Delta\omega = \omega_1 - \omega_2$. Read it as a wave at the average frequency whose amplitude slowly changes:

The beat frequency

The amplitude, and so the loudness, peaks whenever |cos(Δω t/2)| = 1. That happens every time Δω t/2 grows by π, so the time between peaks is 2π/Δω = 1/|ν₁ − ν₂|.

Using beats#

Tuning instruments

Play the instrument with a reference note and adjust it until the beats slow down and disappear: then the two match.

Finding an unknown frequency

Beats tell you the difference in frequency but not which is higher. To decide, change one source a little and see whether the beats speed up or slow down. Sticking a little wax on a fork’s prong makes it heavier, which lowers its frequency.