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

Also Read :

→ Wave & Wave Function
→ progressive wave , Equation of a Plane Progressive Wave
→ Speed of Wave in a String
→ Sound Waves , Infrasonic & Ultrasonic Wave
→ Factor affecting Velocity of Sound
→ Displacement wave & Pressure wave
→ Pitch, Loudness & Quality of Sound Wave
→ Reflection, Refraction & Superposition of Waves
→ Standing Waves in a String
→ Vibration of strings
→ Vibration of air columns : Closed & Open organ Pipe
→ Beats
→ Doppler Effect

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