Horizontal group-speed bound for a confined internal wave ($|c_g|\le \frac{\sqrt{\alpha g}}{\sqrt{(n\pi /h{)}^2+{\alpha }^2/4}}$)

Write $N=\sqrt{\alpha g}$ and $b_n^2=(n\pi /h{)}^2+{\alpha }^2/4$. For the signed rightgoing dispersion relation $\omega (k)=Nk/\sqrt{k^2+b_n^2}$, the horizontal phase velocity is $N/\sqrt{k^2+b_n^2}$ and the group velocity is $N{b}_n^2/(k^2+b_n^2{)}^{3/2}$. Its maximum is $N/b_n$ at zero wavenumber; the leftgoing branch has the opposite speed. The stationary phase method for an initially localized wave packet in this vertical mode gives appreciable large-time contributions on rays satisfying $|x|/t\le N/b_n$. This is an asymptotic propagation region, not exact compact support. Data with algebraic tails retain smaller contributions outside it, and the maximal-speed front needs a degenerate stationary-phase approximation.