Question

Calculate the number of beats heard per second if there are 3 sources of frequencies (n -1), n and (n +1) Hz of equal intensity sounded together.

Answer 1

Let us assume each disturbance has an amplitude ‘A’ then the resultant displacement is given by,

y = A sin 2π(n – 1)t + A sin2πnt+ A sin2π(n + 1)t

i.e. y= 2A sin 2πnt cos2πt + A sin2πnt

y= A(1 + 2cos2πt) sin2πnt

∴ Resultant amplitude: A(1 + 2 cos2πt)

Amplitude is maximum when cos 2πt = 1

i.e., when 2πt = 2πk k = 0,1,2……….

i.e., when t = 0, 1, 2, 3……….

∴ Time difference between successive maxima = 1 s 

Similarly, amplitude is ‘0’ when cos2πt = -1/2

i.e., when cos 2πt = 2πk + 2π/3

k = 0, 1, 2………….

i.e. when t = k + 1/3

i.e., when t = 1/3, 4/3, 7/3 ……

Again the time difference between successive minima = 1 s 

∴ The frequency of beats is also 1 Hz. 

Thus, one beat is heard per second.