Past exam of the fluids course of the University of Oxford / 2021 / 2 / b / Solution

For one-dimensional inviscid flow, conservation of mass and the Euler equation areSubstitute $u=U_0+v_1$ and the stated background-plus-perturbation expressions. Dropping products of perturbations gives the linearizationThe Taylor expansion of the equation of state givesSince the background coefficients are constant, $D_0$ commutes with ${\partial }_x$. Applying $D_0$ to the linearized continuity equation and using the linearized Euler equation therefore yieldsThis is the convected acoustic wave equation: in the coordinate $x-U_{0}t$, it becomes the ordinary one-dimensional wave equation. Positive $c_0^2$ is required for propagating sound waves.