Four identical monochromatic sources A, B, C, D as shown in the (figure) produce waves of the same wavelength λ, and are coherent. Two receivers, R1 and R2, are at great but equal distances from B.

(i) Which of the two receivers picks up the larger signal?
(ii) Which of the two receivers picks up the larger signal when B is turned off?
(iii) Which of the two receivers picks up the larger signal when D is turned off?
(iv) Which of the two receivers can distinguish which of the sources B or D has been turned off?

           

R1B=d=R2B
AB=BC=BD=λ/2

Hint: The signal received by the two receivers depends on the phase of the signals.

Step 1: Find the signal received by R1.

(i) Consider the disturbances at the receiver R1 which is at distance d from B.
Let the wave at R1, because of A be yA = cos ωt. The path difference of the signal from A with that from B is λ/2 and hence, the phase difference is π.
Thus, the wave at R1, because of B is yB=cos (ωt-π)=-acos ωt
The path difference of the signal from C with that from A is λ and hence the phase difference
is 2π.
Thus, the wave at R1, because of C is yC= cosωt-2π=acosωt

The path difference between the signal from D with that of A is;

P=R1D-R1A=R1B2+BD2-R1B-AB
=d2+(λ2)2(dλ/2)=d1+λ4d21/2d+λ2
=d(1+λ28d2)d+λ2=λ28d2+λ2=λ2                  (d>>λ)

Therefore, the phase difference is π.

yD=a cos (ωt-π)=-a cos ωt

Thus, the signal picked up at R1 from all four sources is

yR1=yA+yB+yC+yD
=a cos ωt -a cos ωt+ a cos ωt-a cos ωt=0

Step 2: Find the signal received by R2.

Let the signal picked up at R2 from B be yB=a1 cos ωt

The path difference between signal at D and that at B is λ/2.

     yD=-a1cos ωt

The path difference between signal at A and that at B is 

(d)2+λ22-d=d1+λ24d21/2-d=λ28d

As d >> λ, therefore this path difference 0

and, phase difference = 2πλ18λ2d=0

Hence,    yA=a1 cos (ωt-ϕ)

similarly, YC=a1cos (ωt-ϕ)

The signal picked up by R2,

yA+yB+yC+yD=y=2a1cos(ωtϕ)|y|2=4a12cos2(ωtϕ)<I>=2a12

Thus R1 picks up the larger signal.

Step 3: Find the signal received by R1 and R2 when B is switched off.

(ii) lf B switched off, 

R1 picks up y=acosωt

IR1=12a2R2 picks up y=acosωtIR2=a2<cos2ωt>=a22

Thus R1 and R2 picks up the same signal.

Step 4: Find the signal received by R1 and R2 when D is switched off.

(iii) lf D is switched off,

R1 picks up y=acosωt

 IR1=12a2R2 picks up   y=3acosωtIR2=9a2<cos2ωt>=9a22

(iv) Thus R2 picks up a larger signal compared to R1 indicated B has been switched off and an enhanced signal at R2 indicated D has been switched off.