 Internal waves in an exponentially stratified incompressible fluid

For a quiescent two-dimensional incompressible inviscid fluid with mass density ${\rho }_0(z)=r{e}^{-\alpha z}$ and $\alpha >0$, hydrostatic balance and density advection give the linear perturbation equationsEliminating pressure and density perturbations gives the title partial differential equation. It retains the variation of inertial density rather than invoking the Boussinesq approximation. The transformation $w=e^{\alpha z/2}W$ gives ${\partial }_t^2({\nabla }^{2}W-{\alpha }^{2}W/4)+\alpha g{W}_{xx}=0$. Between rigid boundaries at $z=0,h$, the normal modes haveThe vertical velocity is $e^{\alpha z/2}W$; the exponential factor compensates for the decreasing background mass density in the mode structure.