Sound

Beats

two close notes throb

Two notes close in frequency drift in and out of step, and their sum throbs in loudness at the difference of the two frequencies.

Example

Two notes close in frequency drift in and out of step, and their sum throbs in loudness at the difference of the two frequencies. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Two notes close in pitch

Play two notes whose frequencies are close — here 5 and 3 hertz. They drift in and out of step: sometimes the crests line up and add, sometimes they oppose and cancel, over and over.

fa=5 Hz,fb=3 Hzf_a = 5\ \text{Hz}, \quad f_b = 3\ \text{Hz}
Two close waves and their sum: a beat patternTwo waves of slightly different wavelength and, in purple, their sum, which swells and shrinks in a repeating envelope.sum

The loudness throbs at the difference

Where the two add you hear a loud swell; where they cancel, a moment of quiet. These throbs come at the beat frequency, the difference of the two: 5 minus 3 is 2 hertz — two loud swells every second. (The picture shows the pattern across space; the throb you HEAR is this pattern in time.)

fbeat=∣fa−fb∣=5−3=2 Hzf_{\text{beat}} = |f_a - f_b| = 5 - 3 = \hl{2}\ \text{Hz}

Wider frequency gaps throb faster

The beat rate is the frequency gap. Notes that are very close throb slowly; notes farther apart throb faster, until the separate pitches stop sounding like a slow beat.

fafbfbeat5 Hz4 Hz1 Hz5 Hz3 Hz2 Hz7 Hz3 Hz4 Hz\begin{array}{c|c|c}f_a & f_b & f_{\text{beat}} \\ \hline 5\ \text{Hz} & 4\ \text{Hz} & 1\ \text{Hz} \\ 5\ \text{Hz} & 3\ \text{Hz} & 2\ \text{Hz} \\ 7\ \text{Hz} & 3\ \text{Hz} & 4\ \text{Hz}\end{array}
waves 5 Hz against 3 Hz gives a clean beat frequency of 2 Hz, with the sum drawn as the honest point-by-point combination.