It seems somewhat counter-intuitive that when you multiply two sine waves together (fbig x fsmall) that you get the same result as when you add a sine wave of frequency fbig+fsmall to a sine wave of frequency fbig-fsmall.

This Excel spreadsheet performs these two operations and plots the result, showing the eqivalency.

For example mixing two signals with a frequency of 20 Hz and 1 Hz, respectively yields

while adding two signals, one with a frequency of 19 Hz and the other with a frequency of 21 Hz yields

The signal with a frequency of 19 Hz has an initial phase offset of 90 degrees, while the signal with a frequency of 21 Hz has an initial phase offset of -90.

Mathematically, where f1 is the frequency in hertz and is greater than f2,

sin( (pi/180) * (0 + (360 * f2 * t) ) ) * sin( (pi/180) * ( 0 + (360 * f1 * t) ))

is equal to

(0.5 * sin( (pi/180) * (90 + (360 * (f1-f2) * t) ) ) ) + (0.5 * sin( (pi/180) * (-90 + (360 * (f1+f2) * t) ) ) )

Without the phase shifts, the result is subtley different

Other phase shifts produce the same basic shape with other suble differences.