= Solution
Choose the rightgoing signed <dispersion relation> $\omega(k)=Nk/\sqrt{k^2+b_n^2}$. The horizontal <phase velocity> and <group velocity> are
$$
\boxed{c_p=\frac\omega k=\frac{N}{\sqrt{k^2+b_n^2}},\qquad c_g=\frac{d\omega}{dk}=\frac{Nb_n^2}{(k^2+b_n^2)^{3/2}}}.
$$
The leftgoing branch has their negatives. For a convention using positive frequency at both signs of $k$, multiply each displayed speed by $\operatorname{sgn}k$. For every nonzero $k$, $0<c_g<c_p$ on the rightgoing branch; the <horizontal group-speed bound for a confined internal wave> is $c_g\leq N/b_n$, approached as $k\to0$.
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