= Solution
For <irrotational flow>, $\mathbf u=\nabla\phi$. Incompressibility gives the <Laplace equation>
$$
\boxed{\nabla^2\phi=0},
\qquad 0<x<L,quad0<y<L,quad z<0.
$$
The rigid side walls impose no penetration:
$$
\phi_x=0\quad(x=0,L),
\qquad
\phi_y=0\quad(y=0,L),
$$
and decay in the infinitely deep fluid requires $\nabla\phi\to0$ as $z\to-\infty$.
Linearizing the kinematic condition from part (a) about $z=0$ gives
$$
\boxed{\eta_t=\phi_z\quad(z=0)}.
$$
The <Unsteady Bernoulli equation>, evaluated at the surface and linearized about hydrostatic equilibrium, gives the dynamic condition
$$
\boxed{\phi_t+g\eta
=-\frac{p_0}{\rho}cos\frac{\pi x}{L}
\cos\frac{2\pi y}{L}\cos(\omega t)quad(z=0)}.
$$
These are the <linearized free-surface boundary conditions>. Initial values for $\eta$ and $\phi$ specify the transient; below we also record the initially quiescent solution.
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