= Horizontal group-speed bound for a confined internal wave
{title2=$|c_g|\leq\frac{\sqrt{\alpha g}}{\sqrt{(n\pi/h)^2+\alpha^2/4}}$}
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 $Nb_n^2/(k^2+b_n^2)^{3/2}$. 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 $|x|/t\leq N/b_n$. 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.
Back to article page
