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., E_S > E_L > E_G. So, the velocity of sound is maximum in solids and minimum in gases, hence v_steel > v_water > v_air.

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 (E_θ) = 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 (E_θ) = 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 ⇒ v_air = √ (1.41) x 280 ≈ 332 m/sec.