 Solution

Linearize about water at rest with uniform surface elevation $h_0$. Write $h=h_0+\eta$, $u=v$, $S_0=S(x,h_0)$ and $T_0=S_h(x,h_0)$. To first order the shallow water equations becomeDifferentiate continuity in time and eliminate $v_t$:For a time-harmonic standing disturbance, use $\eta (x,t)=\mathrm{Re}[\eta (x)e^{-i\omega t}]$. ThenDividing by $S_0$ provesThis identifies the coefficient as the local long-wave wavenumber. It is constant under the fixed-shape geometry used in part (a), when $c_0$ is independent of $x$; that assumption is needed for the constant-coefficient mode calculation in part (c).