Question

State Newton's formuls for velocity of sound in air. Point out the error and hence discuss Laplace's correction.

Answer 1

$\mathcal{SPEED\:\:OF\:\:SOUND}$

On the basis of observations, Newton obtained a formula for speed of sound in air as

$\sf{v=\sqrt{\dfrac{P}{\rho}}}$

where P is the isothermal elasticity of the air.

By Newton's formula the speed of sound at one atmosphere is

$\sf{V=\sqrt{\dfrac{1.013\times\:10^5}{1.29}}\cong\:280m/s}$

This value is less than experimental value of 332 m/s. Hence Newton's formula requires some correction, which was made by Laplace.

$\mathcal{LAPLACE'S\:\:CORRECTION}$

Laplace pointed out that when sound propagates in air the heat of the medium remains constant instead of its temperature. So he replaced isothermal elasticity by adiabatic elasticity $\sf{B_{ad}}$. The corrected formula is

$\sf{v=\sqrt{\dfrac{B_{ad}}{\rho}}}$

For adiabatic change, $\sf{PV^{\gamma}=constant}$

Differentiating both sides, we get

$\sf{P\left(\gamma\:V^{\gamma-1}\right)dV+V^{\gamma}dP=0}$

$\implies\sf{\gamma\:PdV+VdP=0}$

$\sf{\dfrac{dp}{\left(\dfrac{-dV}{V}\right)}=\gamma\:P}$

We have $\sf{\dfrac{dp}{\left(\dfrac{-dV}{V}\right)}=B_{ad}}$

$\sf{\therefore\:B_{ad}=\gamma\:P}$

where $\sf{\gamma=\dfrac{C_p}{C_v}}$ is the ratio of specific heats. Hence Laplace's formula for speed of sound in air (gas) is

$\sf{v=\sqrt{\dfrac{\gamma\:P}{\rho}}}$

For air $\gamma=\dfrac{7}{4}$, so the speed of sound in air at STP will be

$\sf{v=\sqrt{\gamma}{\dfrac{P}{\rho}}=\sqrt{\dfrac{7}{4}}\times\:280=332m/s}$

This value is in very close agreement with the experimental value.