Speed of Sound in a Gaseous Medium: Newton's Formula and Laplace's Correction

Sound travels through a gas such as air as a longitudinal wave, so the relevant elastic property is the gas's bulk modulus $B$ (which measures its resistance to compression) and the relevant inertial property is its density $\rho$, giving the general speed formula $v = \sqrt{\frac{B}{\rho}}$ The historical difficulty lay in deciding which bulk modulus to use, since a gas's bulk modulus depends on exactly how its compression takes place. Newton assumed that the compressions and rarefactions of a sound wave in air occur slowly enough for heat generated in a compression to flow away and be absorbed from a rarefaction, keeping the temperature of the gas constant throughout -- an isothermal process, obeying Boyle's law $PV=\text{constant}$. Differentiating this relation gives the isothermal bulk modulus as simply $B_{\text{isothermal}}=P$, the gas's own pressure, so Newton's formula for the speed of sound is $v_N = \sqrt{\frac{P}{\rho}}$ At standard temperature and pressure ($P\approx1.0\times10^{5}\ \text{Pa}$, $\rho\approx1.29\ \text{kg m}^{-3}$ for air) this predicts $v_N\approx280\ \text{m s}^{-1}$ -- roughly 15-16% lower than the experimentally measured speed of sound in air, about $332\ \text{m s}^{-1}$, a discrepancy Newton himself could not resolve. Laplace corrected the error over a century later, by pointing out that the compressions and rarefactions in a sound wave actually happen far too rapidly for any appreciable heat to be exchanged between neighbouring compressed and rarefied layers of air -- so the process is adiabatic, not isothermal, obeying $PV^\gamma=\text{constant}$, where $\gamma=C_P/C_V$ is the ratio of the gas's specific heat at constant pressure to that at constant volume. Differentiating this adiabatic relation gives the adiabatic bulk modulus as $B_{\text{adiabatic}}=\gamma P$, larger than the isothermal value by the factor $\gamma$, so Laplace's corrected formula is $v_L = \sqrt{\frac{\gamma P}{\rho}}$ For air, $\gamma\approx1.4$, and Laplace's formula raises the predicted speed by a factor $\sqrt{1.4}\approx1.18$ -- about 18% higher than Newton's value -- bringing the predicted speed to about $332\ \text{m s}^{-1}$, in close agreement with experiment. Using the ideal gas equation $PV=nRT$ (with …