= 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 yield
$$
\boxed{u=\epsilon e^{-\kappa x+i(mz-\omega t)},\qquad w=-\frac{i\kappa}{m}\epsilon e^{-\kappa x+i(mz-\omega t)},\qquad \sigma=-\frac{N^2\kappa}{m\omega}\epsilon e^{-\kappa x+i(mz-\omega t)}.}
$$
Again 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.
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