Speed of sound

Remarks

Linearise the compressible Euler equations about a fluid at rest with uniform density $\rho_0$. Then small perturbations $p'$ of the pressure solve the wave equation $\partial_t^2 p' = c^2 \Delta p'$, so they travel at speed $c$. Example: for an ideal gas with constant ratio of specific heats $\gamma$, isentropic changes satisfy $p = K \rho^\gamma$, so $c^2 = \gamma p / \rho = \gamma R T$, with $R$ the specific gas constant and $T$ the temperature. For air, $\gamma = 1.4$ and $R \approx 287\ \mathrm{J/(kg\,K)}$, and at $T = 293\ \mathrm{K}$ this gives $c \approx 343\ \mathrm{m/s}$. Non-example: Newton's value $\sqrt{(\partial p / \partial \rho)_T} = \sqrt{p / \rho} \approx 290\ \mathrm{m/s}$ uses the derivative at constant temperature instead of constant entropy. It is too low, because sound waves are too fast for heat to flow between compressed and expanded regions.