= Solution
Put $\lambda=\omega^2/g$ and
$$
Q=\frac{i\omega c}{2L}\big[(z+h)^2-(L-x)^2\big],\qquad G=f+Q.
$$
This polynomial is a <harmonic function>: $Q_{xx}+Q_{zz}=0$. Also $Q_x(0,z)=i\omega c$, $Q_x(L,z)=0$ and $Q_z(x,-h)=0$. Thus $G$ satisfies <Laplace's equation> with homogeneous <Neumann boundary conditions> on both side walls and the bottom. Its <Fourier cosine series> consequently has the form
$$
\boxed{G(x,z)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(k_nx)\cosh(k_n(z+h)),\qquad k_n=\frac{n\pi}{L}}.
$$
The surface condition becomes $G_z-\lambda G=H(x)$, where
$$
H(x)=\frac{i\omega c}{L}\left[h-\frac\lambda2\big(h^2-(L-x)^2\big)\right].
$$
For a nonzero forcing frequency away from the natural frequencies, the coefficients of the <oscillating-wall forcing of a rectangular tank> are
$$
H_n=\frac2L\int_0^L H(x)\cos(k_nx)\,dx,\qquad a_0=-\frac{H_0}{\lambda},\qquad a_n=\frac{H_n}{k_n\sinh(k_nh)-\lambda\cosh(k_nh)}\quad(n\geq1).
$$
Here $k_0=0$ in the definition of $H_0$. In fact $H_n=2i\omega c\lambda/(Lk_n^2)$ for $n\geq1$, since $\int_0^L(L-x)^2\cos(k_nx)dx=2L/k_n^2$. The zero-mode coefficient ensures that the mean <surface elevation> is $hc/L$ in complex amplitude, exactly the first-order <volume conservation> adjustment to the moving wall.
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