State Newton’s formula for velocity of sound.

Newton’s formula for velocity of sound : 

i. Sound wave travels through a medium in the form of compression and rarefaction. At compression, the density of medium is greater while at rarefaction density is smaller. This is possible only in elastic medium.

ii. Thus, the velocity of sound depends upon density and elasticity of medium. It is given by \(v = \sqrt{\frac{E}{\rho}} ....(1)\)

Where, E is the modulus of elasticity of medium and ρ is density of medium.

Assumptions : 

1. Newton assumed that during propagation of sound wave in air, average temperature of the medium remains constant. Hence, propagation of sound wave in air is an isothermal process and isothermal elasticity should be considered.

2. The volume elasticity of air determined under isothermal change is called isothermal bulk modulus.

Calculations: 

1. For a gas or air, the isothermal elasticity E is equal to the atmospheric pressure P. Substituting this value in equation (1), the velocity of sound in air or a gas is given by

\(v = \sqrt{\frac{P}{\rho}}\).....(∵ E = P)

This is the Newton’s formula for velocity of sound in air.

2. But atmospheric pressure is given by, P = hdg

\(\therefore v = \sqrt{\frac{\text{hdg}}{\rho}} ....(2)\)

3. At N.T.P., h = 0.76 m of mercury, density of mercury d = 13600 kg/m³ and acceleration due to gravity, g = 9.8 m/s², density of air ρ = 1.293 kg/m³ 

4. From equation (2) we have velocity of sound,

\(v=\sqrt{\frac{0.76 \times 13600 \times 9.8}{1.293}}\) = 279.9 m/s at N.T.P.