Two coherent sound sources emit waves of the same angular frequency that reach a…

PHYSICS · Multi-part · CBSE · Physics · Waves · Waves

Problem

Two coherent sound sources emit waves of the same angular frequency

\omega

that reach a listener along the same straight line. Each source can be adjusted to provide the same displacement amplitude

A

at the listener’s position. (i) Starting from the individual wave displacements, derive an expression for the resultant intensity

I

at the listener in terms of

A

\theta

, and a constant of proportionality

k

(where

I = k A_{\text{res}}^{\,2}

and

A_{\text{res}}

is the resultant amplitude). (ii) Sketch qualitatively the variation of

I

with

\theta

from

0

2\pi

, indicating the positions of maxima and minima. (iii) If

A = 2.0\,\text{cm}

and

k = 1.0

in consistent units, calculate

I

for

\theta = 0^{\circ}

60^{\circ}

, and

180^{\circ}

. (iv) Using your result, prove that the contrast between successive maxima and minima is greatest when the individual amplitudes are equal.