JEE Main & Advanced Physics Wave Mechanics Velocity of Longitudinal Wave (Sound Wave)

(1) Velocity of sound in any elastic medium : It is given by  \[v=\sqrt{\frac{E}{\rho }}=\sqrt{\frac{\text{Elasticity of the medium}}{\text{Density of the medium}}}\]

(i) In solids \[v\,=\,\sqrt{\frac{Y}{\rho }}\]; where Y = Young's modulus of elasticity

(ii) In a liquid and gaseous medium \[v\,=\,\sqrt{\frac{B}{\rho }}\]; where B = Bulk modulus of elasticity of liquid or gaseous medium.

(iii) As solids are most elastic while gases least i.e. \[{{E}_{S}}>{{E}_{L}}>{{E}_{G}}\]. So the velocity of sound is maximum in solids and minimum in gases, hence   

\[{{\upsilon }_{steel}}>{{\upsilon }_{water}}>{{\upsilon }_{air}}\]  5000 m/s > 1500 m/s > 330 m/s

(iv) The velocity of sound in case of extended solid (crust of the earth) \[v=\sqrt{\frac{B+\frac{4}{3}\eta }{\rho }};\] B = Bulk modulus;  \[\eta =\] Modulus of rigidity;  \[\rho =\]Density

(2) Newton's formula : He assumed that when sound propagates through air temperature remains constant. i.e. the process is isothermal. For isothermal process

B = Isothermal elasticity \[({{E}_{\theta }})=\] Pressure\[(P)\Rightarrow v=\sqrt{\frac{B}{\rho }}=\sqrt{\frac{P}{\rho }}\]

For air at NTP : \[P=1.01\times {{10}^{5}}N/{{m}^{2}}\] and \[\rho =\text{ }1.29kg/{{m}^{3}}\].

\[\Rightarrow \] \[{{v}_{air}}=\sqrt{\frac{1.01\times {{10}^{5}}}{1.29}}\approx 280\,m/s\]

However the experimental value of sound in air is 332 m/sec which is greater than that given by Newton's formula.

(3) Laplace correction : He modified Newton's formula assuming that propagation of sound in gaseous medium is adiabatic process. For adiabatic process

B = Adiabatic elasticity \[({{E}_{\phi }})=\gamma P\]

\[\Rightarrow \] \[v=\sqrt{\frac{B}{\rho }}=\sqrt{\frac{{{E}_{\phi }}}{\rho }}==\sqrt{\frac{\gamma P}{\rho }}=\sqrt{\frac{\gamma RT}{M}}\]

For air : \[\gamma =1.41\Rightarrow {{v}_{air}}=\sqrt{1.41}\times 2.80\approx 332\,\,m/\sec \]

(4) Relation between velocity of sound and root mean square velocity : If sound travel in a gaseous medium then \[{{\upsilon }_{sound}}=\sqrt{\frac{\gamma RT}{M}}\] and r.m.s. velocity of gas \[{{\upsilon }_{rms}}=\sqrt{\frac{3RT}{M}}\]

So  \[\frac{{{\upsilon }_{rms}}}{{{\upsilon }_{sound}}}=\sqrt{\frac{3}{\gamma }}\] or \[{{\upsilon }_{sound}}={{[\gamma /3]}^{1/2}}{{\upsilon }_{rms}}\]