Spark image
When two waves of slightly different frequency overlap a phenomenon known as beats results. In sound the resulting effect is a kind of throbbing sensation.

The final displacement (y):
y = y1 + y2 = a (2pf1t + sin 2pf2t) = 2a cos 2p([f1 f2]t/2 x sin 2p([f1 + f2]t/2)

(a) amplitude varies with time with a frequency [f1 f2]/2
(b) The ear is sensitive to the intensity and not the amplitude of a vibration:

Beat frequency (f) = 2 x amplitude frequency = [f1 f2]
© Keith Gibbs 2011