Name: Waves and Their Phases
Uploaded: 2019-07-19
Duration: 6 min 51 s
Description: In this video, we will learn how two or more waves can be studied in terms of their phases. Equations of two waves and graphs of two waves. Learn more!

Question

Four identical monochromatic sources $A, B, C, D$ as shown in the figure produce waves of the same wavelength $λ$ and are coherent. Two receiver $R_{1}$ and $R_{2}$ are at great but equal distances from $B$ . Which of the two receivers can distinguish which of the sources $B$ or $D$ has been turned off?

Accepted Answer

Consider all the disturbance at $R_{1}$
Let the wave from source A has zero path difference at $R_{1}$
$y A = a \cos ω t$
Path diff. between A and B is $λ / 2$ or $π$ so wave of B at $R_{1}$ is
$y B = - a \cos ω t$
path difference between wave A and C is $λ$ or $2 π$ .

So the wave from source C is

$y_{c} = a \cos (ω t - 2 π)$

or $y_{c} = a \cos ω t$

path difference between A and D = $R_{1} D - A R_{1}$

$= \sqrt{d^{2} + {(\frac{λ}{2})}^{2}} - (d - \frac{λ}{2}) = {(d^{2} + \frac{λ^{2}}{4})}^{\frac{1}{2}} - d + \frac{λ}{2}$

$= d {[1 + \frac{λ^{2}}{4 d^{2}}]}^{\frac{1}{2}} - d + \frac{λ}{d} = d [1 + \frac{λ^{2}}{8 d^{2}}] - d + \frac{λ}{2}$

Neglecting the term $\frac{λ^{2}}{8 d^{2}}, (∵ λ^{2} < < 8 d^{2})$

$∴$ path difference between A and D = $d + 0 - d + \frac{λ}{2} = \frac{λ}{2}$ or $π$

$∴ y_{D} = a \cos (ω t - π) = - a \cos ω t$

So all the signals picked up by $R_{1}$ from A, B, C and D

${y_{R}}_{1} = y_{A} + y_{B} + y_{C} + y_{D}$

$= a \cos ω t - a \cos ω t + a \cos ω t - a \cos ω t$

${y_{R}}_{1} = 0 s o <{I_{R}}_{1}> \approx 0$

So resultant signal at $R_{1}$ due to A, B, C and D sources is zero or no signal.

A signal at $R_{1}$ indicates that B has been switched off.
A signal at $R_{2}$ indicates that D has been switched off.

Important Questions on Wave Optics

Learn and Prepare for any exam you want !

Four identical monochromatic sources A,B,C,D as shown in the figure produce waves of the same wavelength λ and are coherent. Two receiver R1 and R2 are at great but equal distances from B. Which of the two receivers can distinguish which of the sources B or D has been turned off?

Important Questions on Wave Optics

Learn and Prepare for any exam you want !