Question

Two radio stations broadcast their programmes at the same amplitude A, and a slightly different frequencies ν₁ and ν₂ respectively, where ν₂ – ν₁ = 10³ Hz. A detector receives the signals from the two stations simultaneously. It can only detect signals of intensity > 2A². 

(a) Find the time interval between successive maxima of the intensity of the signal received by the detector. 

(b) Find the time for which the detector remains idle in each cycle of the intensity of the signal.

Answer 1

(a) The expression for the superposition of two waves having the same amplitude is

The resultant wave has the slowly varying amplitude

A′ = 2A cos (ω₂– ω₁) t/2.

It is maximum when (ω₂– ω₁) t = 0, 2π, 4π,...

Hence the time interval between successive maxima is

Thus, the time interval during which the amplitude becomes less than √2A is

During this time interval the detector remains idle for each cycle of the intensity of the signal.