Let the two radio waves be represented by the equation
y1 = A sin 2πv1 t
y2 = A sin 2πv2 t
The equation of resultant wave according to superposition principle
y = y1 + y2 = a sin 2πv1 t + A sin 2πv2 t
= A [sin 2 πv1 t + sin 2 πv2 t]
= A x 2 sin (2πv1 + 2 πv2) t/2 cos (2 πv1 + 2 πv2 )t/2
= 2A sin π(v1 + v2) t cos π(v1 – v2) t
Where the amplitude A’ = 2A cos π (v1 – v2) t
Now, intensity ∝ (Amplitude)2
⇒ I∝ A’2
⇒ I∝ 4A2 cos2 π (v1 + v2) t
The intensity will be maximum when
Cos2 π (v1 – v2) t = 1
Or, cos π (v1 + v2) t = 1
Or, π v1 – v2 t nπ
⇒ (ω1 – ω2)/2 t = nπ or, t = 2nπ/ ω1 – ω2
∴ Time interval between two maxima
Or, 2nπ/ ω1 – ω2 - 2(n – 1)π/ ω1 – ω2 or, 2π ω1 – ω2 = 2π/103 sec
Time interval between two successive maximas is
2π x 10-3 sec
(ii) For the detector to sense the radio waves, the resultant intensity ≥ 2 A2
∴ Resultant amplitude ≥ √2 A
Or, 2 A cos π (v1 – v2) t ≥ √2A
Or, cos π (v1 – v2) t ≥ 1/√2 or, cos [(ω1 – ω2) t/2] ≥ 1/√2
The detector lies idle when the values of cos[(ω1 – ω2) t/2] is between 0 and 1/√2
∴ (ω1 – ω2) t/2 is between π/2 and π/4
∴ t1 = π/ ω1 – ω2 and t2 = π/2 (ω1 – ω2)
∴ The time gap = t1 – t2
= π/ ω1 – ω2 – π/2 (ω1 – ω2) = π/2 (ω1 – ω2)
= π/2 x 10-3 sec.
