Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section D / D12 / c / Solution

Small sound disturbances in a gas couple density, pressure and longitudinal velocity. Linearizing mass conservation and the pressure-gradient force about a uniform state givesEliminating velocity yields ${\partial }_t^2{\rho }^{\prime }=c_s^2{\nabla }^2{\rho }^{\prime }$. When local heat exchange is negligible over an oscillation, $c_s^2=(\partial p/\partial \rho {)}_s$. For an ideal gas,The adiabatic index is $\gamma =C_p/C_V$; an isothermal disturbance instead has $c_T^2=p_0/{\rho }_0$. Pressure and velocity in a progressive plane wave are in phase and obey $p^{\prime }={\rho }_0{c}_{s}u$, with average energy flux $⟨p^{\prime }u⟩=p_{\mathrm{peak}}^{\prime 2}/(2{\rho }_0{c}_s)$. Particle oscillation speed is not the sound propagation speed. Viscosity, heat conduction and molecular relaxation dissipate sound; changes in active internal degrees of freedom can cause dispersion. The continuum acoustic model requires wavelength large compared with the molecular mean free path and small fractional perturbations.