Past exam of the theoretical mechanics course of the University of Oxford / 2009 / B / 4 / a / Solution

The inviscid equations are ${\rho }_t+\nabla \cdot (\rho \mathbf{u})=0$ and $\rho [{\mathbf{u}}_t+(\mathbf{u}\cdot \nabla )\mathbf{u}]=-\nabla p$. Write $\rho ={\rho }_0+{\rho }^{\prime }$, $p=P({\rho }_0)+p^{\prime }$ and retain only linear terms. With $c_0^2=P^{\prime }({\rho }_0)>0$, barotropy gives $p^{\prime }=c_0^2{\rho }^{\prime }$. Irrotational perturbations allow ${\mathbf{u}}^{\prime }=\nabla \varphi$. The momentum equation integrates to $p^{\prime }=-{\rho }_0{\varphi }_t$, after using the time-dependent gauge of $\varphi$ to remove the spatially constant integration term. ThusSubstitution into ${\rho }_t^{\prime }+{\rho }_0\Delta \varphi =0$ gives the acoustic wave equation $\boxed{{\varphi }_{tt}=c_0^2\Delta \varphi }$.