= Solution
Linearize about rest and a flat surface. An appropriate small-amplitude scaling has $|\zeta|/h\ll1$ and $|u|/\sqrt{gh}\ll1$, with gradients on fixed common length/time scales. Products such as $(\zeta u)_x$ and $uu_x$ are then higher order. The <linearized shallow water equations> become
$$
\zeta_t+hu_x=0,\qquad u_t+g\zeta_x=0.
$$
Differentiate the first with respect to time and substitute the second:
$$
\boxed{\zeta_{tt}=gh\,\zeta_{xx},\qquad c=\sqrt{gh}.}
$$
The <shallow-water dispersion relation> therefore has a constant wave speed. In these equations $g$ is gravitational <acceleration>, despite the source's label “gravitational constant”; $gh$ has units of speed squared.
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