# Velocity of Sound , Newton’s Formula and Laplace’s correction

### Velocity of sound waves :

Inertia and elasticity ; these are the two properties of matter that determine the velocity of sound. Analytically it has been shown that velocity of sound in a medium of elasticity E and density ρ is given by

$\displaystyle v = \sqrt{ \frac{E}{\rho}}$

From this expression it is clear that E is maximum in case of solid, then liquid and then gases.

Therefore, v is maximum in solids, then in liquids and then in gases.

For example:

vsteel ≈ 5000 m/s,

vwater ≈ 15000 m/s and

vair ≈ 330 m/s.

Note:

(a) The velocity of transverse mechanical wave is given by

$\displaystyle v = \sqrt{ \frac{\eta}{\rho}}\quad$ ; where η is modulus of rigidity and ρ is density of solid.

(b) As for sound vwater > vair , water is rarer than air for sound and denser for light. Rarer and denser medium for a wave is decided by its velocity of propagation not the density.

Lesser the velocity denser is the medium and vice-versa. This is why in travelling from air to water a beam of sound bends away from the normal while that of light towards the normal.

(c) In case of solids E involved in equation $\displaystyle v = \sqrt{ \frac{E}{\rho}}$ is Young’s modulus (Y) of elasticity while for liquids and gases bulk modulus (B) is used.

In Solid , $\displaystyle v = \sqrt{ \frac{Y}{\rho}}$

In liquids & gases , $\displaystyle v = \sqrt{ \frac{B}{\rho}}$

### Newton’s Formula & Laplace’s correction

Newton assumed that when sound propagates through air, temperature remains constant (i.e. the process is isothermal).

So, bulk modulus of elasticity B = BT = P

(isothermal bulk modulus BT of a gas is equal to its pressure).

$\displaystyle v_{air} = \sqrt{ \frac{P}{\rho}}$

Therefore at NTP ,

P = 1.01 × 105 N/m2 and ρ = 1.3 kg/m3

$\displaystyle v = \sqrt{ \frac{1.01\times {10}^5}{1.3}}$

= 279 m/s

The experimental value of v in air is 332 m/s at NTP. This discrepancy was removed by Laplace.

### Laplace’s Correction :

Laplace assumed that the propagation of sound in air is an adiabatic process not the isothermal.

B = Bs = γ P  ; [Adiabatic bulk modulus Bs of a gas = γ P]

Where $\displaystyle \gamma = \frac{C_p}{C_v}$ = 1.41 for air

$\displaystyle v_{air} = \sqrt{ \frac{\gamma P}{\rho}}$

$\displaystyle v_{air} = \sqrt{ \frac{1.41\times 1.01\times {10}^5}{1.3}} = 331.3 m/s$

Which is in agreement with the experimental value (332 m/s) thus,

We can conclude that sound waves propagate through gases adiabatically

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