# Velocity of Longitudinal Wave

## Velocity of Longitudinal Wave

1) Velocity of Sound in any Elastic Medium: It is given by $$v=\sqrt{\frac{E}{\rho }}=\sqrt{\frac{Elasticity\,of\,the\,medium}{Density\,of\,the\,medium}}$$.

a) In solids, $$v=\sqrt{\frac{\gamma }{\rho }}$$; Where Y = Young’s modulus of elasticity.

b) In a liquid and gaseous medium, $$v=\sqrt{\frac{B}{\rho }}$$; where B = Bulk modulus of elasticity of liquid or gaseous medium.

c) As solids are most elastic while gases least i.e., ES > EL > EG. So, the velocity of sound is maximum in solids and minimum in gases, hence vsteel > vwater > vair.

d) The velocity of sound in case of extended solid, $$v=\sqrt{\frac{B+\frac{4}{3}\eta }{\rho }}$$; B = Bulk modulus, η = Modulus of rigidity, ρ = Density.

2) Newton’s Formula:  He assumed that when sound propagates through air temperature remains constant i.e. the process is isothermal. For isothermal process:

B = Isothermal elasticity (Eθ) = Pressure (P) ⇒ $$v=\sqrt{\frac{B}{\rho }}=\sqrt{\frac{P}{\rho }}$$.

For air at NTP: P = 1.01 x 10⁵ N/m² and ρ = 1.29 kg/ m³ ⇒ $${{v}_{air}}=\sqrt{\frac{1.01\times {{10}^{5}}}{1.29}}$$ ≈ 280 m/sec.

However, the experiment value of sound in air is 332 m/ sec which is greater than that given by Newton’s Formula.

3) Laplace Correction: He modified Newton’s formula assuming that propagation of sound in gaseous medium is adiabatic process. Fro adiabatic process:

B = Adiabatic Elasticity (Eθ) = γP ⇒ $$v=\sqrt{\frac{B}{\rho }}=\sqrt{\frac{{{E}_{\phi }}}{\rho }}=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}$$.

For air, γ = 1.41 ⇒ vair = √ (1.41) x 280 ≈ 332 m/sec.