When two sources of sound of nearly equal frequencies are sounded together, the amplitude of resultant wave due to superposition of waves at a point is in space. Sometimes the amplitude is maximum and sometimes, it is minimum. As intensity id directly proportional to the square of amplitude, the intensity also varies periodically with time from maximum to minimum and to maximum again. Let two waves be,
y1 = a sin 2πv1t,
and y2 = a sin 2πv2t
are superimposed. On superposition we get a resultant waves as,
y = y1 + y2
= a sin πv1t + a sin πv2t
= a(sin πv1t + sin 2πv2t)
The resultant wave is simple harmonic wave with amplitude
A = 2a cos 2π({v1 - v2}/{2})t
and frequency = ({v1 + v2}/{2})
Now, the intensity is maximum, when
The intensity between successive, maxima is {1}/{v1 - v2} and frequency of beats = (v1 - v2).
The amplitude is minimum when
The intensity between successive minima is {1}/{v1 - v2} and frequency = {v1 - v2}. It implies that intensity is maximum (v1 - v2) times and minimum (v1 - v2) times in second.
Thus, the frequency of beats of motion
= (v1 - v2).
