Internal waves in an exponentially stratified incompressible fluid (${\partial }_t^2({\nabla }^{2}w-\alpha w_z)+\alpha g{w}_{xx}=0$)

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.

Write $N=\sqrt{\alpha g}$ and $b_n^2=(n\pi /h{)}^2+{\alpha }^2/4$. For the signed rightgoing dispersion relation $\omega (k)=Nk/\sqrt{k^2+b_n^2}$, the horizontal phase velocity is $N/\sqrt{k^2+b_n^2}$ and the group velocity is $N{b}_n^2/(k^2+b_n^2{)}^{3/2}$. Its maximum is $N/b_n$ at zero wavenumber; the leftgoing branch has the opposite speed. The stationary phase method for an initially localized wave packet in this vertical mode gives appreciable large-time contributions on rays satisfying $|x|/t\le N/b_n$. This is an asymptotic propagation region, not exact compact support. Data with algebraic tails retain smaller contributions outside it, and the maximal-speed front needs a degenerate stationary-phase approximation.