1800-150-456-789
FlagBACK
Flag Wave Motion> Two radio stations broadcast their progra...
question mark

Two radio stations broadcast their program’s at the same amplitude A and at slightly frequencies ω1 and ω2 respectively, where ω1 – ω2 = 103 Hz A detector receives the signals from the two stations simultaneously. It can only detect signals of intensity ≥ 2A2
(i) Find the time interval between successive maxima of the intensity of the signal received by the detector.
(ii) Find the time for which the detector remains idle in each cycle of the intensity of the signal.

Simran Bhatia , 11 Years ago
Grade 11
anser 1 Answers
Aditi Chauhan

Last Activity: 11 Years ago

Hello Student,
Please find the answer to your question
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.

Thanks
Aditi Chauhan
askIITians Faculty

star
LIVE ONLINE CLASSES

Prepraring for the competition made easy just by live online class.

tv

Full Live Access

material

Study Material

removal

Live Doubts Solving

assignment

Daily Class Assignments


Other Related Questions on wave motion

question mark
A particle on a speing executes shm when the particle is found at x=xmax/2 the speed of the partcle is 
A)vx=vmax 
B)vx=√3vmax/2
C)vx=√2vmax/2
D)vx=vmax/2
wave motion
0 Answer Available

Last Activity: 3 Years ago

question mark
displacement associated with three dimensional wave is given by sai x y is equal to 1 + 1 by 2 x + root 3 by 2 k x minus omega t show that the wave propagates along a direction making an angle 60 degree with the x-axis
wave motion
0 Answer Available

Last Activity: 3 Years ago

question mark
What do you mean by Energy of Simple Hormonic Motion?
 
wave motion
0 Answer Available

Last Activity: 3 Years ago

question mark
What is mean by motion and time define it's all types with there parts and their s.i. units.
wave motion
0 Answer Available

Last Activity: 3 Years ago

question mark
explain the effects of doppler effect in satellite communication
wave motion
0 Answer Available

Last Activity: 3 Years ago