How does the

beat frequency (f

_{a}-f

_{b})/2 come into being? That most hated part of the

high school Mathematics curriculum,

trigonometry, holds the keys!

Suppose we have frequencies f+w and f-w, and we sound them together. If there's no phase difference between them (and if there is any, the same thing holds and we still get the same result), then we have the waveforms sin(2π(f+w)t) and sin(2π(f-w)t) sounding together. Retrieving some trigonometric identities from distant memory, we have that

sin(a+b)+sin(a-b) = sin(a)cos(b)+cos(a)sin(b) + sin(a)cos(b)-cos(a)sin(b) =

= 2sin(a)cos(b)

(that wasn't so bad, was it?).

So sin(2π(f+w)t)+sin(2π(f-w)t) = 2sin(2πft)cos(2πwt). In flamingweasel's example, f=251Hz and w=1Hz. We're going to be hearing a pure sine wave at frequency 251Hz (that's 2sin(2πft)), only it will be modulated (in exactly the same way that an AM signal is modulated) by a frequency of 1Hz. In other words, over the first half-second the 251Hz wave will have its amplitude squished down to 0, over the next half-second it will go back to full strength, and it will continue to cycle like that.