= Solution
Apply <separation of variables> to <Laplace's equation>. The side-wall <Neumann boundary conditions> give horizontal factors $\cos(k_nx)$ with $k_n=n\pi/L$. The bottom condition fixes the vertical factor to $\cosh(k_n(z+h))$. Substitution into the surface condition in <1/a/i> gives the <normal modes of surface gravity waves in a rectangular tank>:
$$
\boxed{\phi_n=\cos(k_nx)\cosh(k_n(z+h))\,[A_n\cos(\omega_nt)+B_n\sin(\omega_nt)],\qquad\omega_n^2=gk_n\tanh(k_nh),\quad n\geq1}.
$$
The corresponding <surface elevation> follows from $\eta_n=-\partial_t\phi_n(x,0,t)/g$. The spatially uniform $n=0$ mode has zero frequency; its constant elevation changes total fluid volume, so it is absent when that volume is fixed. An additive constant in $\phi$ changes no fluid <velocity>.
Back to article page
