= Solution
Take the undisturbed <free surface> as $z=0$, the bottom as $z=-h$, and the walls as $x,y=0,a$. For an <incompressible flow> which is an <irrotational flow>, write $\mathbf u=\nabla\phi$ with a <velocity potential> satisfying $\nabla^2\phi=0$. Impermeability gives $\phi_z=0$ at the bottom, $\phi_x=0$ at the $x$ walls, and $\phi_y=0$ at the $y$ walls.
For the actual surface $z=\eta(x,y,t)$, the exact <kinematic boundary condition for a free-surface graph> is
$$
\eta_t+\phi_x\eta_x+\phi_y\eta_y=\phi_z\quad\text{at }z=\eta.
$$
The <Unsteady Bernoulli equation>, with atmospheric pressure at the surface and its time-dependent constant absorbed into $\phi$, gives
$$
\phi_t+\frac12|\nabla\phi|^2+g\eta=0\quad\text{at }z=\eta.
$$
For small disturbances about fluid at rest, $\phi$ and $\eta$ are first order. Products of first-order quantities and Taylor corrections from evaluating at $z=\eta$ are second order. The linear surface conditions are therefore
$$
\boxed{\eta_t=\phi_z,\qquad \phi_t+g\eta=0\quad\text{at }z=0,}
$$
or, after eliminating the elevation,
$$
\boxed{\phi_{tt}+g\phi_z=0\quad\text{at }z=0.}
$$
The separated <normal modes of surface gravity waves in a rectangular tank> have
$$
\phi=\Phi_{mn}\cos\frac{m\pi x}{a}\cos\frac{n\pi y}{a}
\frac{\cosh(k_{mn}(z+h))}{\cosh(k_{mn}h)}e^{i\omega t},\qquad
k_{mn}=\frac\pi a\sqrt{m^2+n^2},
$$
where $m,n$ are nonnegative integers not both zero. The horizontal factors satisfy the wall conditions, and the vertical factor satisfies both the <Laplace equation> and the bottom condition. At the top, $\phi_z=k_{mn}\tanh(k_{mn}h)\phi$; substitution into the surface condition gives
$$
\boxed{\omega_{mn}^2=gk_{mn}\tanh(k_{mn}h),\qquad m,n\geq0,\quad(m,n)\ne(0,0).}
$$
The elevation is
$$
\eta=-\frac{i\omega\Phi_{mn}}{g}\cos\frac{m\pi x}{a}\cos\frac{n\pi y}{a}e^{i\omega t}.
$$
Real physical <standing waves> are the real parts or linear combinations of these complex modes. The uniform $m=n=0$ mode has zero frequency and would change the mean depth; conservation of the fixed water volume excludes it from the disturbances. Modes with equal $m^2+n^2$ are degenerate and can be superposed.
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