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Laplace Correction

Laplace Correction :- Laplace modified Newton’s formula assuming that propagation of sound in air is an adiabatic process and not an isothermal process. This is because the propagation of sound waves occurs over such short time scales, there isn’t sufficient time for heat exchange to take place between compressed layer of air and surroundings and also between rarefied layer of air and surroundings.

As a result, the sound propagates under adiabatic conditions and not under isothermal conditions.

Now in an adiabatic process,

$\displaystyle P{{V}^{\gamma }}=\text{constant}$

$\displaystyle \Rightarrow \text{d(}P{{V}^{\gamma }}\text{)=0}$

$\displaystyle \Rightarrow P\gamma {{V}^{\gamma -1}}dV+{{V}^{\gamma }}dP=0$

$\displaystyle \Rightarrow \gamma P=-\frac{VdP}{dV}=-\frac{dP}{\left( \frac{dV}{V} \right)}={{B}_{adiabatic}}$

Here $\displaystyle \gamma =\frac{{{C}_{P}}}{{{C}_{V}}}$

$\displaystyle \therefore {{B}_{adiabatic}}=\gamma {{P}_{atm}}$

$\displaystyle \Rightarrow v=\sqrt{\frac{B}{\rho }}=\sqrt{\frac{{{B}_{adiabatic}}}{{{\rho }_{air}}}}=\sqrt{\frac{\gamma {{P}_{atm}}}{{{\rho }_{air}}}}$

For air γ = 1.4, therefore

$\displaystyle \Rightarrow v=\sqrt{\frac{1.4\times 1.013\times {{10}^{5}}}{1.29}}=279\sqrt{1.4}=330m{{s}^{-1}}$

Which is in good agreement with the experimental value (332 m/s). This in turn establishes that sound propagates adiabatically through gases.

The velocity of sound in air at NTP is 332 m/s which is much lesser than that of light and radio–waves (= 3 x 10⁸ m/s). This implies that :–

(a) If we set our watch by the sound of a distant siren it will be slow.

(b)If we record the time in a race by hearing sound from starting point it will be lesser than actual.

(c)In a cloud–lightening, though light and sound are produced simultaneously but as c > v, light proceeds thunder.

In case of gases :–

$\displaystyle {{v}_{sound}}=\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma PV}{M}}=\sqrt{\frac{\gamma nRT}{M}}=\sqrt{\frac{\gamma RT}{{M}/{n}\;}}$

$\displaystyle \Rightarrow {{v}_{sound}}=\sqrt{\frac{\gamma RT}{{{M}_{w}}}}$

And from kinetic-theory of gases :–

$\displaystyle {{v}_{rms}}=\sqrt{\frac{3RT}{{{M}_{w}}}}$

$\displaystyle So\text{ }\frac{{{v}_{sound}}}{{{v}_{rms}}}=\sqrt{\frac{\gamma }{3}}$

Laplace Correction

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