A closed basin rings in seiche modes

A long rectangular basin has uniform undisturbed depth \(h\), length \(L\), and rigid vertical end walls at \(x=0,L\). In the inviscid linear shallow-water approximation, the free-surface displacement \(\eta(x,t)\) and depth-averaged horizontal velocity \(u(x,t)\) obey \[ \eta_t+h u_x=0,\qquad u_t+g\eta_x=0. \] The walls impose \(u(0,t)=u(L,t)=0\). Eliminate either field to obtain the wave equation and derive the admissible free-surface eigenfunctions, wavenumbers, frequencies, and periods. Explain why the wall condition becomes a Neumann condition on every nonzero-frequency \(\eta\) mode, and identify the special zero mode. Reconstruct both \(\eta\) and \(u\) for \[ \eta(x,0)=\eta_0\cos(\pi x/L) +\frac{\eta_0}{2}\cos(2\pi x/L),\qquad u(x,0)=0. \] Verify both first-order equations rather than checking only the derived wave equation, and state a nondispersive phase relation between the surface and velocity amplitudes. For \(L=1.00\,\mathrm{km}\), \(h=25.0\,\mathrm m\), \(g=9.81\,\mathrm{m/s^2}\), and \(\eta_0=0.200\,\mathrm m\), compute the wave speed and the first two modal periods. At one quarter of the fundamental period, compute \(\eta\) and \(u\) at \(x=L/3\). Check the shallow-water parameters \(k_1h\) and \(k_2h\).

Differentiate continuity in time and momentum in space before eliminating \(u_{xt}\).