Question

$S_1$ and $S_2$ are two coherent sources of sound of frequency $110\text{Hz}$ each and have no initial phase difference. The intensity at a point P due to $S_1$ is $I_0$ and due to $S_2$ is $4I_0$. If the velocity of sound is $330\text{m/s}$, then the resultant intensity at P is

• A. $I_0$
• B. $9I_0$
• C. $3I_0$
• D. $8I_0$

Answer 1

The correct option is C $3I_0$

The distance between point 'P' & source $S_2$ is $S_{2}P=\sqrt{3^2+4^2}=5\text{m}$
Path difference between waves arriving at 'P' from $S_1\text{and}{S}_2$ is,
$\Delta x=S_{2}P-S_{1}P=5-4=1\text{m}$
Phase difference between interferring waves at 'P' will be,
$\Delta \varphi =K\Delta x=\frac{2\pi }{\lambda }\times \Delta x$
$\Rightarrow \Delta \varphi =\frac{2\pi }{3}\times 1=\frac{2\pi }{3}(\because \lambda =\frac{V}{f}=\frac{330}{110}=3\text{m})$
Resultant intensity at 'P'
$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \Delta \varphi$
$\Rightarrow I=I_0+4I_0+2\sqrt{4I_0^2}\cos \left(\frac{2\pi }{3}\right)$
$\Rightarrow I=5I_0-2\times 2I_0\times \frac{1}{2}(\because \cos \frac{2\pi }{3}=\frac{-1}{2})$
$\Rightarrow I=3I_0$