= Solution
{c}
For $k>0$, <phase velocity> is the speed of a fixed phase, $c_p=\omega/k$. The <group velocity> $c_g=d\omega/dk$ transports a narrow-band wave envelope. With $q=kH$,
$$
\boxed{c_p=\sqrt{\frac{g\tanh q}{k}},\qquad
c_g=\frac g{2\omega}(\tanh q+q\,\operatorname{sech}^2q)
=\frac{c_p}{2}\left(1+\frac{2q}{\sinh2q}\right).}
$$
As $H\to0$ with $k$ fixed, $\tanh(kH)/(kH)\to1$. Therefore
$$
\boxed{\lim_{H\to0}\frac{c_p^2}{H}=g.}
$$
Both velocities tend to the shallow-water speed $\sqrt{gH}$ at leading order.
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