Laplace's Correction

In 1816, Laplace resolved the roughly 16% gap between Newton's isothermal prediction and the experimentally observed speed of sound by pointing out a flaw in Newton's underlying assumption. Sound's compressions and rarefactions actually occur far too rapidly for heat to have time to exchange between neighbouring regions of the gas -- and air is, in any case, a poor conductor of heat -- so temperature does NOT remain constant during sound propagation; the process is instead adiabatic. Under adiabatic conditions the gas obeys Poisson's law rather than Boyle's law, $PV^{\gamma}=\text{constant}$, where $\gamma=C_P/C_V$ is the ratio of the specific heat at constant pressure to the specific heat at constant volume. Differentiating this relation, $\gamma PV^{\gamma-1}dV+V^{\gamma}dP=0$, gives $-V\,dP/dV=\gamma P=K_A$, defining $K_A$ as the adiabatic bulk modulus of air; substituting into the general elastic-wave formula gives Laplace's corrected result, $v=\sqrt{K_A/\rho}=\sqrt{\gamma P/\rho}$. Since air is mainly composed of the diatomic gases nitrogen and oxygen, taking $\gamma=1.4$ and applying it to Newton's theoretical 280 m/ …