= Solution
Use a <plane wave> with phase $kx+mz-\omega t$, and take $N>0$, $k,m\ne0$. The positive-<frequency> branch of the given hydrostatic <dispersion relation> is $\omega=N|k|/|m|$. The <wavevector> is $\mathbf K=(k,m)$. Differentiating with respect to its components gives the <group velocity>, which carries the <wave packet> and its <energy>:
$$
\boxed{\mathbf c_g=\nabla_{\mathbf K}\omega
=\left(\frac\omega k,-\frac\omega m\right).}
$$
The normal <phase velocity>, which follows a crest in its normal direction, is
$$
\boxed{\mathbf c_p=\frac\omega{k^2+m^2}(k,m).}
$$
Consequently,
$$
\mathbf K\cdot\mathbf c_g=0,\qquad
\mathbf c_p\cdot\mathbf c_g=0,\qquad
\frac{c_{g,z}}{c_{g,x}}=-\frac{k}{m}.
$$
Thus the <group velocity> lies along the constant-phase lines $z=-(k/m)x+\mathrm{constant}$, perpendicular to the <phase velocity>. For $k,m>0$, crests move upward and rightward, whereas <energy> moves downward and rightward. Choosing $m<0$ gives upward and rightward <energy> propagation and downward and rightward normal phase propagation. Conjugate or oppositely directed components supply the other outgoing directions from a localized source.
\Image[/past-exam-of-the-geophysical-fluid-dynamics-course-of-the-university-of-oxford/2024-internal-wave-propagation.png]
{title=Internal-wave crests, normal phase propagation, and perpendicular energy propagation for positive horizontal and vertical wavenumbers}
The speeds are
$$
\boxed{|\mathbf c_g|=\frac N{|m|}\sqrt{1+\frac{k^2}{m^2}},\qquad
|\mathbf c_p|=\frac{N|k|}{|m|\sqrt{k^2+m^2}}.}
$$
The hydrostatic limit is $|k/m|\ll1$. In that limit <energy> propagates almost horizontally, with $|c_{g,x}|=N/|m|$ and $|c_{g,z}|=N|k|/m^2$, while normal phase propagation is almost vertical and has speed approximately $N|k|/m^2$. The horizontal speed at which a crest intersects a fixed height is $\omega/k=c_{g,x}$; this is not the horizontal component of the normal <phase velocity> vector. For example, $N=10^{-2}\ \mathrm{s^{-1}}$, vertical <wavelength> $2\pi/|m|=10\ \mathrm{km}$ and horizontal <wavelength> $2\pi/|k|=1000\ \mathrm{km}$ give $|c_{g,x}|\simeq15.9\ \mathrm{m\,s^{-1}}$ and $|c_{g,z}|\simeq0.159\ \mathrm{m\,s^{-1}}$.
These are local speeds in a uniform medium, without a background current. The non-rotating approximation requires $|f|\ll\omega$, while the <hydrostatic approximation> requires $\omega\ll N$. At lower <frequencies>, rotation changes the <dispersion relation> to that of <inertia-gravity waves>.
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