 Solution

For $N^2>0$, write $K=\sqrt{k_x^2+k_z^2}$ and choose either signed branch $\omega =sN{k}_x/K$, with $s=\pm 1$. These two branches cover the roots of the squared dispersion relation. The group velocity of an internal gravity wave isIf instead the positive-frequency root is chosen for both signs of $k_x$, set $s=+1$ and multiply the displayed components by $\mathrm{sgn}(k_x)$, away from the cusp $k_x=0$.

Taking the dot product with the wave vector givesThus the group velocity, which describes the travel of a narrow wave packet and its energy, is perpendicular to the wave vector, and hence perpendicular to the phase velocity ${\mathbf{c}}_p=\omega \mathbf{k}/K^2$. This is the internal-wave phase and group velocity geometry. It also follows from phase-group orthogonality for degree-zero dispersion: the frequency depends only on the direction of $\mathbf{k}$, not its magnitude. For $k_z=0$, the group velocity is zero, consistent with the stationary extremum $|\omega |=N$.