Newton's Formula: The velocity of sound in any medium is given by \(v=\sqrt{\frac E\rho}\), where E is the elasticity of the medium and ρ is the density of the medium. For air medium, the modulus of elasticity is Bulk's modulus of elasticity (B). The velocity of a sound wave in air is given by \(v=\sqrt{\frac B\rho}\).
Newton assumed that the formation of rarefaction and compression in the air is a slow process and the temperature is equal to the temperature of the surrounding i.e. the process is an isothermal process. The gas equation then is
PV = Constant, where P is the pressure of air and V is the volume.
On differentiating we get,
PdV + VdP = 0
Or, PdV = - VdP
Or, \(P=-\frac{dP}{\displaystyle\frac{dV}V}=\frac{Volumetric\;stress}{Volumetric\;Strain}=B\), Bulk modulus of elasticity of air.
So, Newton's formula for the velocity of sound in air is given by \(v=\sqrt{\frac B\rho}\) \(=\sqrt{\frac P\rho}\), where P is the pressure of air and ρ is the density of air.
For air, at NTP;
P = 1.013 x 105 Nm-2
ρ = 1.293 kg/m3
∴ \(v=\sqrt{\frac B\rho}\)
= \(\sqrt{\frac{1.013\times10^5}{1.293}}\)
= 280 m/s
But, the experimental value of the velocity of sound in air at NTP is about 332 m/s. So, the theoretical value does not agree with the experimental value.
Therefore, there must be something wrong with Newton's formula. A satisfactory solution to the discrepancy in Newton's formula was given by Laplace, which is known as Laplace correction.
Laplace correction: While deriving the above formula, Newton assumed that the propagation of sound takes place in isothermal conditions. But according to Laplace, the propagation of sound should take place in a rapid way so the process is an adiabatic process and the temperature at rarefaction and compression is different.
Since the adiabatic equation of state is
PVγ = Constant ------- (i)
Where, \(\gamma=\frac{C_p}{C_v}\) is a constant. On differentiating equation (i) we get
\(\gamma PV^{y-1}dV+V^\gamma dP=0\)
Or, \(\gamma PV^{y-1}dV=-V^\gamma dP\)
Or, \(\gamma=-\frac{V^\gamma dP}{V^{\gamma-1}dV}=-\frac{dP}{V^{-1}dV}=-\frac{dP}{dV/V}=-B_{adi}\)
Where Badi is the adiabatic bulk modulus of elasticity. A negative sign indicates that as the pressure increases, the volume decreases and vice versa. Hence the velocity of sound in air is:
\(v=\sqrt{\frac{B_{adi}}\rho}\)
∴ \(v=\sqrt{\frac{\gamma P}\rho}\)
This equation is called the Laplace equation for the velocity of sound in air.
For the air at NTP (Normal temperature and pressure)
γ = 1.4
P = 1.013 x 105 Nm-2
ρ = 1.293 kg/m3
∴ \(v=\sqrt{\frac{\gamma P}\rho}=\sqrt{\frac{1.4\times1.013\times10^5}{1.293}}=331.2\) m/s
This value is closely agrees with experimental value 331.2 m/s
