Laplace's correction

Laplace's key insight was that the compressions and rarefactions of a sound wave are NOT a slow process, as Newton had implicitly assumed, but a genuinely RAPID one -- for a sound of frequency 256 Hz, for instance, the air at any point is compressed and rarefied 256 times every single second. Heat is generated in a compression (where the gas is momentarily squeezed) and absorbed back out in the neighbouring rarefaction, but this cycle happens far too fast for that heat to actually diffuse away to (or in from) neighbouring regions -- so the total heat content of any small parcel of air effectively stays constant through the cycle. A process with no heat transfer in or out is, by definition, an ADIABATIC process, not an isothermal one -- so the correct elastic modulus to use in Newton's $v=\sqrt{E/\rho}$ is the ADIABATIC modulus of elasticity, not the isothermal one Newton had assumed.

Always remember: in an isothermal process temperature stays fixed while heat can still flow in or out; in an adiabatic process there is neither heat transfer NOR mass transfer into or out of the system.

The adiabatic elastic modulus of air works out to $E=\gamma P$, where P is the medium's pressure and γ (the adiabatic ratio) is the ratio of air's specific heat at constant pressure to its specific heat at constant volume, $\gamma=\dfrac{C_p}{C_v}$; for air, $\gamma\approx1.41$. Substituting this corrected modulus into Newton's formula gives the Newton-Laplace formula for the speed of sound in air,

$v=\sqrt{\dfrac{\gamma P}{\rho}}$ …