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For a tank $0<x<L$, $-h<z<0$, let the left wall move by $\mathrm{Re}(c{e}^{-i\omega t})$ with small amplitude, while the right wall and bottom remain fixed. A periodic velocity potential $\mathrm{Re}(f{e}^{-i\omega t})$ has $f_x(0,z)=-i\omega c$. Add the harmonic functionto obtain $G=f+Q$ with homogeneous wall and bottom Neumann boundary conditions. Thus $G=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (k_{n}x)\cosh (k_n(z+h))$, where $k_n=n\pi /L$. The linearized free-surface boundary conditions give $G_z-\lambda G=H$ at $z=0$, with $\lambda ={\omega }^2/g$ andIts Fourier cosine coefficients $H_n=(2/L){\int }_0^{L}H(x)\cos (k_{n}x)dx$ give the title expression for $n\ge 1$ and $a_0=-H_0/\lambda$. In particular $H_n=2i\omega c\lambda /(L{k}_n^2)$ for $n\ge 1$. The denominator vanishes at the corresponding normal modes of surface gravity waves in a rectangular tank, giving resonance. Exact undamped resonance produces secular growth instead of a bounded periodic response. Free transients do not generally decay in this inviscid model.

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 $\mathbf{u}=\nabla \varphi$ with a velocity potential satisfying ${\nabla }^2\varphi =0$. Impermeability gives ${\varphi }_z=0$ at the bottom, ${\varphi }_x=0$ at the $x$ walls, and ${\varphi }_y=0$ at the $y$ walls.

For the actual surface $z=\eta (x,y,t)$, the exact kinematic boundary condition for a free-surface graph isThe Unsteady Bernoulli equation, with atmospheric pressure at the surface and its time-dependent constant absorbed into $\varphi$, givesFor small disturbances about fluid at rest, $\varphi$ 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 thereforeor, after eliminating the elevation,

The separated normal modes of surface gravity waves in a rectangular tank havewhere $m,n$ 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, ${\varphi }_z=k_{mn}\tanh (k_{mn}h)\varphi$; substitution into the surface condition givesThe elevation isReal 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.

Apply separation of variables to Laplace's equation. The side-wall Neumann boundary conditions give horizontal factors $\cos (k_{n}x)$ 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 i gives the normal modes of surface gravity waves in a rectangular tank:The corresponding surface elevation follows from ${\eta }_n=-{\partial }_t{\varphi }_n(x,0,t)/g$. 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 $\varphi$ changes no fluid velocity.