Question

Two identical speakers emit sound waves of frequency of ${10}^3$ Hz uniformly in all directions. The audio output of each speaker is $9\pi /10$ mW. A point '$P$' is at a distance 3 m from the speaker $S_1$ and 5m from speaker $S_2$. Resultant intensity at $P$ is $I_g$. Match the items in Column I with the items in Column II:

Answer 1

Intensity at a distance $r$ from a source of power output $P$ is given by
$I=\frac{P}{4\pi r^2}$
$I_1=\frac{0.9\pi mW}{4\pi (3{)}^2}=\frac{1}{40}mW=25\mu W/m^2$
$I_1=\frac{9\pi mW}{40\pi (5{)}^2}=\frac{9}{1000}mW=9\mu W/m^2$
For incoherent source,
$I_R=I_1+I_2+2\sqrt{I_1{I}_2}=(25+9)=34\mu W/m^2$
For coherent source, $\Delta =0$
$I_R=I_1+I_2+2\sqrt{I_1{I}_2}$
$(\sqrt{I_1}+\sqrt{I_2}{)}^2=(5+3{)}^2=64\mu W/m^2$
For $\delta =\pm \pi ,I_R=(\sqrt{I_1}-\sqrt{I_2}{)}^2$
$=(5-3{)}^2=4\mu W/m^2$
If the speaker $S_2$ is switched off, $I_R=I_1=25\mu W/m^2$