Question

Sound waves are travelling in a medium whose adiabatic elasticity is E and isothermal elasticity E' . The velocity of sound waves is proportional to
A. \[E\]
B. \[\sqrt E \\ \]
C. \[\sqrt {E'} \\ \]
D. \[\dfrac{E}{{E'}}\]

Answer 1

Hint:The sound wave is a mechanical wave which needs a medium to travel through. It travels by causing the regions of variable pressure in the medium. As pressure changes, the temperature of the medium will change.

Formula used:
\[v = \sqrt {\dfrac{E}{\rho }} \]
where v is the speed of the sound wave in a medium with density\[\rho \]and elasticity E.

Complete step by step solution:
When a sound wave is travelling in a medium then it creates a consecutive region of the maximum and minimum pressure. The region where the particles of the medium are close to each other, the pressure is high and the region where the particles of the medium are far away from each other that region is of low pressure. The region of high pressure is called the compression and the region of low pressure is called the rarefaction.

The energy of the sound is travelled along the direction of the motion of the sound wave by the vibration of the medium particle about the mean position. The motion of the particles of the medium depends on the sensitivity of the medium against the pressure variation. The pressure per unit fractional change in volume of the medium in which sound is travelling is characterised by the term elasticity.

The adiabatic elasticity of the medium is given as E and the thermal elasticity is given as E’. The heat transfer by the sound wave due to conduction is proportional to the rate of change of temperature per unit horizontal displacement. But the temperature gradient due to motion of the sound wave is insignificant so the sound is approximately an adiabatic process.

So, the velocity of the sound in the medium depends on the adiabatic elasticity of the medium rather than the thermal elasticity. The velocity of the sound in the medium is given as \[v = \sqrt {\dfrac{E}{\rho }} \], where \[\rho \] is the density of the medium. Hence, the velocity of the sound is proportional to \[\sqrt E \].

Therefore, the correct option is B.

Note: Adiabatic is the process for which heat transfer is assumed to be zero and in an isothermal process the temperature is constant. But due to pressure variation the temperature of the medium changes.