Question

What is the effect of temperature on velocity of sound?

Answer 1

Sound waves travel through any material medium with a speed that depends on the properties of the medium. As sound waves travel through air, the elements of air vibrate to produce changes in density and pressure along the direction of motion of the wave. If the source of the sound waves vibrates sinusoidally , the pressure variations are also sinusoidal. The mathematical description of sinusoidal sound waves is very similar to that of sinusoidal waves on strings.

Complete step by step answer:
Factors affecting speed of sound in gas:
Effect of pressure: The speed of sound in a gas is given by
$v=\sqrt{\dfrac{\gamma P}{\rho }}$
We know that $PV=nRT=\dfrac{m}{M}RT$
At constant temperature,
$P\Delta V=\dfrac{\Delta m}{M}RT$
$\Rightarrow P=\dfrac{\Delta mRT}{\Delta VM}$
$\Rightarrow P=\rho \dfrac{RT}{M}$
$\Rightarrow \dfrac{P}{\rho }$ = constant
Therefore, with the change in pressure , the density also changes in such proportion, so that $\dfrac{p}{\rho }$ remains constant. Hence, pressure has no effect on the speed of sound in a gas.

Effect of density: For two gases of densities ${{\rho }_{1}}$ and ${{\rho }_{2}}$ at the same pressure with ratios of specific heats ${{\gamma }_{1}}$ and ${{\gamma }_{2}}$,
$\dfrac{{{v}_{1}}}{{{v}_{2}}}=\sqrt{\dfrac{{{\gamma }_{1}}}{{{\gamma }_{2}}}\times \dfrac{{{\rho }_{2}}}{{{\rho }_{1}}}}$
Effect of temperature
We have, $\dfrac{P}{\rho }=\dfrac{RT}{M}$
$\therefore v=\sqrt{\dfrac{\gamma RT}{M}}$..........clearly, $v\propto \sqrt{T}$.

Hence the speed of sound in a gas is proportional to the square root of its absolute temperature.

Note: The speed of sound waves in a medium depends on the compressibility and density of the medium. If the medium is a liquid or a gas and has bulk modulus B and density $\rho$, the speed of sound wave in that medium is,
$v=\sqrt{\dfrac{B}{\rho }}$
It is interesting to compare this expression with the equation for the speed of transverse waves on a string, $v=\sqrt{\dfrac{T}{\mu }}$ . In both cases, the wave speed depends on an elastic property of the medium and on an inertial property of the medium. In fact, the speed of all mechanical waves follows an expression of the general form,
$v=\sqrt{\dfrac{\text{elastic property}}{\text{inertial property}}}$