# Factors affecting Velocity of Sound

Factor affecting Velocity of Sound

$\displaystyle v_{air} = \sqrt{\frac{\gamma P}{\rho}}$

From Kinetic theory of gases ; $\displaystyle \frac{P}{\rho} = \frac{RT}{M}$

$\displaystyle v_{air} =\sqrt{ \frac{\gamma RT}{M} }$

Therefore velocity of sound in a gas is of the order of rms speed of gas moecules

However , $\displaystyle v_{rms} = \sqrt{\frac{3 RT}{M}}$

Effect of temperature:   In a gas

$\displaystyle v = \sqrt{\frac{\gamma RT}{M}}$

$\displaystyle v \propto \sqrt{T}$

i.e. with increase in temperature velocity of sound in a gas increases .

Let us find velocity of sound in air at t°C.

At NTP , vair = v0°C = 332 m/s

$\displaystyle \frac{v_t}{v_0} = \sqrt{\frac{273+t}{273 + 0}} = (1+\frac{t}{273})^{1/2}$

When t is small-

$\displaystyle \frac{v_t}{v_0} = (1+\frac{t}{546})$

$\displaystyle v_t = v_0(1+\frac{t}{546})$

Putting v0 = 332 m/s ; we have, vt = (332 + 0.61 t) m/s

i.e. for small temperature variations at 0°C, velocity of sound changes by 0.61 m/s when temperature changes by 1°C.

Effect of pressure:   In a gas ;

$\displaystyle v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma RT}{M}} = \sqrt{\frac{B}{\rho}}$

Change in pressure has no effect on velocity of sound in a gas, so long as temperature remains constant; because ;

$\displaystyle \frac{P}{\rho} = constant \quad$;As long as temperature is constant.

As long as temperature is constant. i.e. when P increases , ρ  decreases vice-versa and net result is that v remains constant.

Effect of Relative humidity:  When humidity increases, there is an increase in the relative number of water molecules and hence a decrease in the molar mass (avg. molecular wt.), and the speed of sound increases.

Note: The speed of sound in air is not affected by Amplitudes, frequency, phase, loudness, pitch, or quality.