Sound Wave

Newton’s Formula And Laplace’s Correction

Newton gave a formula to calculate the speed of sound. But the value obtained using that formula did not match with the experimental value of the speed of sound. Later, Laplace corrected that formula and the value obtained using that corrected formula matched with the experimental value. This corrected formula is known as the Laplace’s correction for Newton’s formula.

Newton’s Formula

Newton assumed that when sound propagated through air, compressions and rarefactions occur very slowly. [Longitudinal Wave]

Due to the change in pressure, there is high temperature in compression region whereas low temperature in rarefaction region. Since the process is considered slow, the heat conducts from high temperature region to the low temperature region by which the temperature of the gas during the propagation of sound remains constant. This condition is an isothermal process and hence, it obeys Boyle’s law.

Therefore, for a given mass of gas at pressure $P$ and volume $V$, we have, \[PV=\text{Constant}\] Differentiating, \[PdV+VdP=0\] \[PdV=-VdP\] \[P=-\frac{dP}{\frac{dV}{V}} \text{ ___(1)}\]

Also, from the definition of bulk modulus of gas, \[B=\frac{\text{Volume Stress}}{\text{Volume strain}}\] \[=\frac{\text{Change in Pressure}}{\frac{\text{Change in volume}}{\text{Original Volume}}}\] \[=-\frac{dP}{\frac{dV}{V}} \text{ ___(2)}\]

The negative sign shows that the volume decreases as pressure increases.

From equation $(1)$ and $(2)$, \[B=P\]

The velocity of sound in a medium is given by, \[v=\sqrt{\frac{B}{ρ}} \text{ __(3)}\] Where, $ρ$ is the density of the gas.

From $(2)$ and $(3)$, \[v=\sqrt{\frac{P}{ρ}}\] This is the Newton’s formula for the speed of sound in gas.

At NTP, \[\text{Pressure (P)}=1.01×10^{5}\text{ Nm}^{-2}\] \[\text{Density of air at 0˚C (ρ)}=1.293 \text{ kgm}^{-3}\]

Thus, from Newton’s formula, \[v=\sqrt{\frac{P}{ρ}}=\sqrt{\frac{1.01×10^{5}}{1.293}}=280\text{ ms}^{-1}\]

Experimentally, the speed of sound in air at NTP was $332$ $\text{ms}^{-1}$. Thus, there was a great difference between the theoretical and experimental value.

Laplace’s Correction

After 120 years of Newton’s formula, Laplace corrected the formula for the velocity of sound in gaseous medium. Laplace suggested that the compression and rarefaction of the medium during propagation of sound occur very rapidly. Therefore, the heat developed at the compression region did not have enough time to be dissipated into the surrounding. There is no compensation of heat at these regions. The air in the path suffers temperature change but the heat is neither allowed to go out nor allowed to come in from outside. This situation is adiabatic rather than isothermal.

For adiabatic process, the relation between pressure and volume of the gas from which the sound is propagating is given by, \[PV^γ=\text{Constant} \text{ ____(4)}\] Where, $γ$ is the specific heat ratio of the gas.

Differentiating equation $(4)$, we get, \[PγV^{γ-1}dV+V^γdP=0\]

Dividing by $V^{γ-1}$, \[PγdV+VdP=0\] \[γP=-\frac{dP}{\frac{dV}{V}} \text{ ___(5)}\]

From equations $(2)$ and $(5)$, \[B=γP\] The velocity of sound in a medium is given by, \[v=\sqrt{\frac{B}{ρ}}\] \[\therefore v=\sqrt{\frac{γP}{ρ}}\]

This is the Laplace’s correction for Newton’s formula.

At NTP, the value $γ$ for air or diatomic gas is $1.40$. \[\therefore v=\sqrt{\frac{γP}{ρ}}\] \[=\sqrt{γ}×\sqrt{\frac{P}{ρ}}\] \[=\sqrt{1.40}×280\] \[=332 \text{ ms}^{-1}\]

This value matched with the experimental result. Hence, Laplace’s formula is the correct formula with correct assumptions.