How does air pressure affect the speed of sound in relation to its kinetic theory etc?
I have tried searching this but i have not found a suitable answer as other websites simply related to air pressure waves.
[Physics] How does air pressure affect the speed of sound
acousticsairpressurespeed
Best Answer
To first order, the speed of sound is not affected by pressure. Pressure waves can be shown to fulfill the D'Alembert wave equation $(c_S^2\,\nabla^2 - \partial_t^2)\psi=0$ where the wavespeed $c_S$ is given by:
$$c_S = \sqrt{\frac{K}{\rho}}$$
where $K$ is the bulk modulus of the medium in question and $\rho$ its density. Now, for an ideal gas, the bulk modulus $K$ is in most conditions proportional to the pressure; if the compression is adiabatic (good approximation for high frequency sound, as there is little time for heat to shuttle back and forth in the gas), then $K=\gamma\,P$, where $\gamma$ is the Heat Capacity Ratio or Adiabatic Index. However, from the ideal gas law $P\,V=n\,R\,T$ we have:
$$\rho = \frac{n\,M}{V} = \frac{P\,M}{R\,T}$$
where $M$ is the mean molar mass of the gas in question in kilograms. Thus the pressures cancel out in the speed of sound:
$$c_S = \sqrt{\frac{\gamma\,R\,T}{M}}$$
Thus we see that the speed is also weakly affected by the humidity - more water in the air lowers the mean molecular mass. If we put $M=0.029$, $T=300K$ and $\gamma = 1.4$ for air, we get $c_S=347{\rm m\,s^{-1}}$.