Question

The speed of sound in air at S.T.P is 300$$m{s}^{-1}$$. If the air pressure doubles, the temperature remaining the same, the speed of sound would become –
A) 1200 $$m{s}^{-1}$$
B) 600$$m{s}^{-1}$$
C) 300$$\sqrt{2}m{s}^{-1}$$
D) 300 $$m{s}^{-1}$$

Answer 1

Hint: We need to understand the relation between the speed of the sound waves in a medium and the temperature of the medium and its pressure conditions to find the solution for the change in the speed with the pressure change as per the problem.

Complete Solution:
We are given a situation in which the air pressure at a place is doubled without any change in the temperature. We know that the pressure and volume are related inversely to each other by Boyle's law in a constant temperature. According to Boyle's law, the pressure is inversely proportional to the volume and vice versa at a constant temperature.
i.e.,
$$P\propto {\displaystyle \frac{1}{V}}$$

Now, we know that the speed of a sound wave in a medium or a fluid medium is dependent of the pressure and the density of the medium at a constant temperature. The speed of the wave is directly proportional to the square root of the pressure of the fluid and inversely proportional to the square root of the density of the fluid which can be given as –
$$\begin{array}{rl}& v=\sqrt{{\displaystyle \frac{P}{\rho }}}\\ & \text{but,}\\ & \text{density, }\rho ={\displaystyle \frac{\text{Mass}}{\text{Volume}}}={\displaystyle \frac{m}{V}}\\ & \Rightarrow v=\sqrt{{\displaystyle \frac{P}{{\displaystyle \frac{m}{V}}}}}\\ & \Rightarrow v=\sqrt{{\displaystyle \frac{PV}{m}}}\\ & \text{Also,}\\ & V\propto {\displaystyle \frac{1}{P}}\end{array}$$
$$\begin{array}{rl}& \text{given,}\\ & P^{\prime }=2P\\ & \Rightarrow V^{\prime }={\displaystyle \frac{1}{2}}V\\ & \Rightarrow v^{\prime }=\sqrt{{\displaystyle \frac{P^{\prime }{V}^{\prime }}{m}}}\\ & \Rightarrow v^{\prime }=\sqrt{{\displaystyle \frac{2P{\displaystyle \frac{V}{2}}}{m}}}\\ & \therefore v^{\prime }=v=300m{s}^{-1}\end{array}$$

We understand that the speed of the sound is not affected by the change in pressure of the air. The speed of the sound remains to be 300 $$m{s}^{-1}$$. This is the required solution.

Hence, the correct answer is option D.

Note:
The sound waves are dependent on the pressure of the medium and the density when we consider fluids. The relation has extra relations when we consider the speed of the longitudinal waves in a solid which are dependent on the stress-strain moduli.