Question

State Newton’s formula for the velocity of sound.

Explain Newtons formula for velocity of sound.

Answer 1

Newton’s formula for the velocity of sound:

1. Sound wave travels through a medium in the form of compression and rarefaction. At compression, the density of the medium is greater while at rarefaction density is smaller. This is possible only in an elastic medium.
2. Thus, the velocity of sound depends upon the density and elasticity of the medium. It is given by
   v = `sqrt("E"/rho)`    .....(1)
   Where, E is the modulus of elasticity of medium and ρ is the density of the medium.

Assumptions:

1. Newton assumed that during the propagation of the sound waves in the air, the average temperature of the medium remains constant. Hence, propagation of the sound waves 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 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 gas is given by
   v = `sqrt("P"/rho)     ...(because "E" = "P")` 
   This is Newton’s formula for the velocity of sound in air.
2. But atmospheric pressure is given by,
   P = hdg
   ∴ v = `sqrt("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 `rho` = 1.293 kg/m³
4. From equation (2) we have velocity of sound, 
   v = `sqrt((0.76 xx 13600 xx 9.8)/(1.293))` = 279.9 m/s at N.T.P