Question

A point source emits sound of the same power in all directions in a non-absorbing medium. Two points $P$ and $Q$ are at a distance of $9\text{m}$ and $25\text{m}$ respectively from the source. The ratio of the amplitude of the waves at $P$ and $Q$ is

• A. $\frac{3}{5}$
• B. $\frac{5}{3}$
• C. $\frac{9}{25}$
• D. $\frac{25}{9}$

Answer 1

The correct option is D $\frac{25}{9}$

We know that the intensity of sound is given by,
$I=\frac{1}{2}\rho {\omega }^2{A}_o^{2}v$,
where, $\rho$ is the density of medium, $\omega$ is the angular frequency of sound, $A_o$ is the displacement amplitude of particles of medium and $v$ is the speed of sound.
For the same medium and same source of sound,
$I\propto A_o^2$
So, $\frac{I_1}{I_2}=\frac{A_{o1}^2}{A_{o2}^2}$ ..........(1)
Also, we know,
$I=\frac{P}{A}=\frac{P}{4\pi r^2}$,
where, $P$ is power, $A$ is surface area and $r$ is distance from the source.
For same power,
$I\propto \frac{1}{r^2}$
So, $\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}$ ..........(2)
From (1) and (2),
$\frac{A_{o1}}{A_{o2}}=\frac{r_2}{r_1}=\frac{25}{9}$