Convected acoustic wave equation ($({\partial }_t+U_0{\partial }_x{)}^2{\rho }_1=c_0^2{\partial }_x^2{\rho }_1$)

For a uniform background velocity $U_0$ and speed of sound $c_0$, linearized conservation of mass, the Euler equation and the local equation of state $p_1=c_0^2{\rho }_1$ give this wave equation. In the coordinate $x-U_{0}t$ it is the ordinary one-dimensional wave equation. Its two characteristic curves travel at laboratory velocities $U_0+c_0$ and $U_0-c_0$; both point downstream in supersonic flow.