= Internal waves in an exponentially stratified incompressible fluid
{title2=$\partial_t^2(\nabla^2w-\alpha w_z)+\alpha g w_{xx}=0$}
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 equations
$$
u_x+w_z=0,\quad \rho_0u_t=-p_x,\quad \rho_0w_t=-p_z-g\rho',\quad \rho'_t=\alpha\rho_0w.
$$
Eliminating <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 $\partial_t^2(\nabla^2W-\alpha^2W/4)+\alpha gW_{xx}=0$. Between rigid boundaries at $z=0,h$, the <normal modes> have
$$
W\propto\sin(n\pi z/h)e^{i(kx-\omega t)},\qquad \omega^2=\frac{\alpha gk^2}{k^2+(n\pi/h)^2+\alpha^2/4},\quad n\geq1.
$$
The vertical <velocity> is $e^{\alpha z/2}W$; the exponential factor compensates for the decreasing background <mass density> in the mode structure.
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