= Solution
For positive wavenumber $k$, the <phase velocity> is $c_p=\omega/k$, the speed of a fixed phase. The <group velocity> is $c_g=d\omega/dk$, the leading speed of a narrow wave packet. Differentiating the supplied <dispersion relation> gives
$$
\boxed{c_p=\sqrt{\frac gk\tanh(kH)}},
$$
$$
\boxed{c_g=\frac{g[\tanh(kH)+kH\operatorname{sech}^2(kH)]}{2\omega}
=\frac{c_p}2\left(1+\frac{2kH}{\sinh(2kH)}\right)}.
$$
For fixed $k>0$, the shallow-depth limit uses $\tanh s/s\to1$:
$$
\boxed{\lim_{H\to0}\frac{c_p^2}H=g}.
$$
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