Internal wave

Remarks

Eliminating $u$, $v$, $p$ and $b$ gives the single equation $\partial_t^2 \Delta w + N^2 (\partial_x^2 + \partial_y^2) w = 0$, and inserting a plane wave gives the dispersion relation. If $\Theta$ is the angle between the wavevector and the horizontal, the relation is $\omega = N \cos \Theta$, so the frequency depends only on the direction of the wavevector and $0 < \omega \le N$.

Example: a wave with $m = k$ and $l = 0$ has wavevector at $45$ degrees and frequency $\omega = N / \sqrt{2}$.

Non-example: a disturbance forced at a frequency $\omega > N$ cannot propagate as an internal wave; the relation has no real wavevector, and the response decays away from the source.

The group velocity is perpendicular to the wavevector, so energy moves along the crests. This is visible in the St. Andrew's cross pattern made by an oscillating cylinder in a stratified tank. In a rotating fluid with Coriolis parameter $f$ the relation becomes $\omega^2 = \big( N^2 (k^2 + l^2) + f^2 m^2 \big) / (k^2 + l^2 + m^2)$.