 Solution

Let the velocity perturbation be $(u,w)$, pressure perturbation be $p$, and density perturbation be ${\rho }^{\prime }$ about the hydrostatic background $\hat{\rho }(z)$. The linearized inviscid Boussinesq approximation givesStable stratification means ${\hat{\rho }}_z<0$, and the buoyancy frequency isDifferentiate the momentum equations to eliminate $p$, use incompressibility, and then use the density equation to eliminate ${\rho }^{\prime }$. This yieldsFor $w\propto e^{i(kx+mz-\omega t)}$, the internal gravity wave dispersion relation is ${\omega }^2=N^2{k}^2/(k^2+m^2)$.