= Solution
For $0<\omega<N$, put $\alpha=\omega/\sqrt{N^2-\omega^2}$. The <dispersion relation> permits $k=\pm\alpha m$. On the positive-frequency branch, for $k>0$,
$$
\omega=\frac{Nk}{\sqrt{k^2+m^2}},\qquad c_{gx}=\frac{Nm^2}{(k^2+m^2)^{3/2}}>0,\qquad c_{gz}=-\frac{Nkm}{(k^2+m^2)^{3/2}}.
$$
The <radiation condition> therefore selects $k=\alpha m$, since <energy> must travel into the fluid. Substitution into the <internal-wave polarization> relations gives the complex fields
$$
\boxed{u=\epsilon e^{i(\alpha mx+mz-\omega t)},\qquad w=-\alpha\epsilon e^{i(\alpha mx+mz-\omega t)},\qquad \sigma=\frac{iN^2\alpha\epsilon}{\omega}e^{i(\alpha mx+mz-\omega t)}.}
$$
The physical fields are their real parts; $v=0$. The vertical component of <group velocity> is negative, so this positive-$m$ <plane internal gravity wave> carries <energy> down and right, although its phase propagates in a different direction.
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