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Speed of Sound in Air

Speed of Sound in Air :- The formula for velocity of sound in air was first obtained by Newton. He assumed that when sound propagates through a gas the temperature remains constant (i.e. the process is isothermal). So a compressed layer of air gradually loses heat to the surroundings and a rarefied layer gains heat from the surroundings, so that temperature of air remains constant.

We know in an isothermal process,

$\displaystyle PV=\text{constant}$

$\displaystyle \Rightarrow \text{d(PV)=0}$

$\displaystyle \Rightarrow PdV+VdP=0$

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

$\displaystyle \therefore {{B}_{isothermal}}={{P}_{atm}}$

Now we know that speed of longitudinal waves is given by $\displaystyle v=\sqrt{\frac{B}{\rho }}$ , so

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

At S.T.P. P_atm = 1.013 × 10⁵ N/m² and ρ_air = 1.29 kg/m³, putting these values, we get,

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

However this value is about 16 % less than the experimental value of sound in air which is about 332 m/s. In order to remove this large discrepancy between theoretical and experimental values of velocity of sound, Laplace modified Newton’s formula and this modification is called Laplace’s Correction.

Speed of Sound in Air

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