Factors affecting speed of sound

As sound travels through the open atmosphere, three environmental factors affect its speed: pressure, temperature, and humidity.

1. Effect of pressure. By Laplace's formula, $v=\sqrt{\gamma P/\rho}$. Writing $\rho=M/V$ for a fixed mass M of gas occupying volume V, this becomes $v=\sqrt{\gamma PV/M}$. At CONSTANT temperature, Boyle's Law says $PV=\text{constant}$; since M and γ are themselves constants for a given gas sample, the whole quantity under the square root stays fixed, so v itself stays constant -- a change of pressure ALONE, with temperature held fixed, has NO effect on the speed of sound. This can also be seen using the ideal gas equation $PV=nRT$ (n = number of moles), which gives $v=\sqrt{nRT\gamma/M}$: with temperature T fixed, v again depends only on T, not on P by itself. (Both derivations agree: for a gas obeying the ideal-gas law, a pressure change has no effect on the speed of sound UNLESS it is accompanied by a temperature change.)
2. Effect of temperature. Writing the speed at temperature $T_0$ as $v_0$ and at T as v, and using $v\propto\sqrt{T}$ (from the ideal-gas form above, since $\gamma$, $R$ and $M$ are fixed), $\dfrac{v}{v_0}=\sqrt{\dfrac{T}{T_0}}$ -- the speed of sound in air is directly proportional to the square root of the absolute temperature, so it increases as air gets hotter. Taking $T_0=273$ K and $T=(273+t)$ K for a Celsius temperature t, and expanding the square root for SMALL t using the binomial approximation, this simplifies to $\dfrac{v}{v_0}\approx1+\alpha t$ with $\alpha=\dfrac{1}{546}$; using $v_0=332$ m/s at 0°C gives $v=v_0+\dfrac{v_0}{546}t\approx v_0+0.61t$ -- for every 1°C rise in temperature, the speed of sound in air increases by about 0.61 m/s. This linear approximation holds well for modest temperature changes (up to about 50°C); at 22°C, for instance, it predicts a speed close to the often-quoted 344 m/s. …