Question

Two identical speakers, emit sound waves of frequency ${10}^3\text{Hz}$ uniformly in all directions.
The audio output of each speaker is $\frac{9\pi }{10}\text{mW}$. A point $P$ is at a distance of $3\text{m}$ from $S_1$ and $5\text{m}$ from $S_2$. Resultant intensity of sound at $P$ is

• A. $34\mu W/m^2,$ if the sources are incoherent.
• B. $4\mu W/m^2,$ if the sources are coherent but out of phase.
• C. $25\mu W/m^2,$ if $S_1$ is switched off.
• D. $64\mu W/m^2,$ if the sources are coherent and in phase.

Answer 1

Answer: (A), (B), (D)

The correct options are
A $34\mu W/m^2,$ if the sources are incoherent.
B $4\mu W/m^2,$ if the sources are coherent but out of phase.
D $64\mu W/m^2,$ if the sources are coherent and in phase.

Intensity at a distance $r$ from a sound of power output $P$ is given by $I=\frac{P}{4\pi r^2}$
$\therefore I_1=\frac{0.9\pi }{4\pi (3{)}^2}=\frac{1}{40}{\text{mW/m}}^2=25{\mu \text{W/m}}^2$
$I_2=\frac{0.9\pi }{4\pi (5{)}^2}=\frac{9}{1000}{\text{mW/m}}^2=9{\mu \text{W/m}}^2$
For incoherent interference,
$I=I_1+I_2=34{\mu \text{W/m}}^2$

For coherent interference,
when $\varphi =\pi ,$
$I=(\sqrt{I_1}-\sqrt{I_2}{)}^2=4{\mu \text{W/m}}^2$
When $\varphi =0$
$I={(\sqrt{I_1}+\sqrt{I_2})}^2=64{\mu \text{W/m}}^2$