Write the equation for the speed of a sound wave according to Newton and explain the correction made by Laplace.

(N/A) According to Newton,the speed of sound in an ideal gas is given by:
$v = \sqrt{\frac{P}{\rho}}$ ... $(1)$
Laplace pointed out that the pressure variations during the propagation of sound waves occur so rapidly that there is insufficient time for heat exchange to maintain a constant temperature. Therefore,these processes are adiabatic rather than isothermal.
For an adiabatic process,an ideal gas satisfies the relation:
$P V^{\gamma} = \text{constant}$
Differentiating both sides:
$\Delta(P V^{\gamma}) = 0$
$P(\gamma V^{\gamma-1} \Delta V) + V^{\gamma} \Delta P = 0$
$\gamma P \Delta V + V \Delta P = 0$
$\gamma P = -\frac{\Delta P}{\Delta V / V} = B$
where $B$ is the adiabatic bulk modulus.
Substituting $B = \gamma P$ into the general formula for the speed of sound $v = \sqrt{\frac{B}{\rho}}$,we get the Laplace correction:
$v = \sqrt{\frac{\gamma P}{\rho}}$ ... $(2)$
Here,$\gamma = \frac{C_P}{C_V}$ is the ratio of specific heats. For air,$\gamma = 1.4$. Using this formula at $STP$,the calculated speed of sound is approximately $331.3 \ m/s$,which matches experimental results.