Question

$S_1$ and $S_2$ are two coherent sources of sound of frequency 110 Hz each. They 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 $330m/s$ then the resultant intensity at P is

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

Answer 1

Answer: (C)

$3I_0$

Wavelength $\lambda =\frac{v}{f}=\frac{330}{110}=3m$ as velocity and frequency are given.
The expression for intensity of two waves after interference is:
$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi$ .....(1) where $\varphi$ is the phase difference due to path difference ( $S_{2}P-S_{1}P$)
As seen from the figure,

$S_{2}P=\sqrt{3^2+4^2}=5$

$S_{1}P=4$

$\varphi =\frac{2\pi }{\lambda }(S_{2}P-S_{1}P)=\frac{2\pi }{\lambda }(5-4)=\frac{2\pi }{3}1=\frac{2\pi }{3}$

Substituting $\varphi$, $I_1=I_0$, $I_2=4I_0$ in (1)

$I=I_0+4I_0+2\sqrt{I_0\times 4I_0}\cos \frac{2\pi }{3}$

$=5I_0+4I_0(-\frac{1}{2})=5I_0-2I_0=3I_0$