Beats

If two sound waves have frequencies that are very close, then we can hear a phenomenon called beats. It occurs because the sum of two sine waves has a frequency that is the difference between the two frequencies.

The upper part of this figure shows two sine waves with almost identical frequencies. To add them, we use the trigonometric identity:

sin$\alpha$+sin$\beta$ = 2cos[($\alpha$-$\beta$)]sin[($\alpha$+$\beta$)]

The result of this addition is shown in the lower part of the figure. The mathematical form of this curve is obtained from the above equation with

$\alpha$ = 2$\pi$f1t, and $\beta$ = 2$\pi$f2t.

This leads to

sin(2$\pi$f1t)+sin(2$\pi$f2t) = 2 cos[$\pi$t(f1-f2)] sin[$\pi$t(f1+f2)]

The frequency in the sine term is the average of the two frequencies. The cosine term is very easy to hear because it is a modification of the amplitude at a very low frequency (like once per second).