Past exam of the waves and compressible flow course of the University of Oxford / 2023 / 2 / a

Write the density and pressure as ${\rho }_0(z)+{\rho }^{\prime }(x,z,t)$ and $p_0(z)+p^{\prime }(x,z,t)$, respectively. The background satisfies hydrostatic balance, $p_0^{\prime }=-g{\rho }_0$, and ${\rho }_0^{\prime }/{\rho }_0=-\alpha$. Linearizing the Euler equations for an inviscid fluid, incompressibility and advection of density givesDifferentiate the horizontal equation in $x$ and use incompressibility to obtain $p_{xx}^{\prime }={\rho }_0{w}_{zt}$. Differentiate the vertical equation in time and twice in $x$:Dividing by ${\rho }_0>0$ gives the equation for internal waves in an exponentially stratified incompressible fluid,The first-derivative term retains the variation of inertial density with height; no Boussinesq approximation has been made.

For $w=e^{\alpha z/2}W$, direct differentiation givesThus the transformed partial differential equation and its impermeable-wall boundary conditions areThe positive exponential factor does not change where the vertical velocity vanishes.