 Solution

For $\omega >N$, the dispersion relation gives $k^2=-m^2{\omega }^2/({\omega }^2-N^2)$. Put $\kappa =m\omega /\sqrt{{\omega }^2-N^2}>0$. Boundedness as $x\to +\infty$ selects $k=i\kappa$, rather than the exponentially growing root. The internal-wave polarization relations now yieldAgain take real parts and set $v=0$. This is an evanescent wave, decaying over distance ${\kappa }^{-1}$; there is no propagating far-field internal gravity wave at this frequency. The pressure amplitude $P=-i\omega u/\kappa$ is in quadrature with the horizontal velocity, so the time-averaged horizontal pressure-work flux vanishes. The equality $\omega =N$ is a singular limiting case and is outside the two frequency ranges considered here.