 Hydrostatic internal-wave propagation geometry

For a positive-frequency non-rotating internal gravity wave with $\omega =N|k|/|m|$, differentiation gives group velocity $(\omega /k,-\omega /m)$. Its wavevector $(k,m)$ is orthogonal to this vector, so energy travels along constant-phase lines and perpendicular to normal phase velocity. The horizontal crest-intersection speed $\omega /k$ is distinct from the horizontal component $\omega k/(k^2+m^2)$ of that normal phase velocity. The hydrostatic approximation requires $|k/m|\ll 1$.