 Solution

Linearize about rest and a flat surface. An appropriate small-amplitude scaling has $|\zeta |/h\ll 1$ and $|u|/\sqrt{gh}\ll 1$, with gradients on fixed common length/time scales. Products such as $(\zeta u{)}_x$ and $u{u}_x$ are then higher order. The linearized shallow water equations becomeDifferentiate the first with respect to time and substitute the second: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.