Question

$S_1$ and $S_2$ are two coherent sources of sound of frequency $110Hz$ 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. $8I_0$
• B. $3I_0$
• C. $9I_0$
• D. $I_0$

Answer 1

The correct option is B $3I_0$

We know that:$\lambda =\frac{\upsilon }{f}$

Given values,

coherent sources of sound of frequency

$f=110Hz$

velocity of sound is $\upsilon =330m/s$

Wavelength $\lambda =\frac{\upsilon }{f}=\frac{330}{110}=3m$

Now, $I=I_1+I_2+2\sqrt{I_1{I}_{\left(2\right)}}\cos \varphi$

Here, $I$= resultant intensity

$\varphi$= phase difference by path

difference $S_{2}P-S_{1}P$

$S_{1}P=\sqrt{4^2}=4$

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

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

Substitute the values,

$I_1=I_0,I_2=4I_0$ and $\varphi =\frac{2\pi }{3}$

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

$I=5I_0+2\sqrt{4I_0^2}(-\frac{1}{2})$

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

$I=5I_0-2I_0$

$I=3I_0$

Final Answer: (c)