Question

A point source emits sound equally in all directions in a non-absorbing medium. Two points $P$ and $Q$ are at distances of $9m$ and $25m$ respectively from the source. The ratio of the amplitudes of the waves at $P$ and $Q$ is

• A. $9:4$
• B. $1:22$
• C. $25:9$
• D. $625:81$

Answer 1

The correct option is C

$25:9$

Step 1: Given data

Distance from source, $r_1=9m$

Distance from source, $r_2=25m$

Step 2: Finding the ratio of the amplitudes of the waves

Intensity is the rate at which sound spreads over a surface of a given area, at some distance from the source. The intensity varies with the distance from the source and the power of the source.

$I=\frac{P}{4\pi r^2}$ ….(1)

Where,

$A=$Amplitude, $P=$Power, $r=$distance from the source.

$\Rightarrow$$I\propto P$ ….(2)

Also, $P=\frac{1}{2}\mu v{\omega }^2{A}^2$

Where, $\mu =$linear mass density, $v=$wave speed, $\omega =$angular frequency, $A=$amplitude of wave

$\Rightarrow$$P\propto A^2$ ….(3)

From (2) and (3) we get,

$I\propto A^2$ ….(4)

From (1) and (4) we get,

$A^2\propto \frac{1}{r^2}$

$$\begin{gathered} A\propto \frac{1}{r} \\ \frac{A_1}{A_2}=\frac{r_2}{r_1} \\ \frac{A_1}{A_2}=\frac{25}{9} \end{gathered}$$

Hence, option $\left(C\right)$ is correct.