 Solution

For irrotational flow, $\mathbf{u}=\nabla \varphi$. Incompressibility gives the Laplace equationThe rigid side walls impose no penetration:and decay in the infinitely deep fluid requires $\nabla \varphi \to 0$ as $z\to -\infty$.

Linearizing the kinematic condition from part (a) about $z=0$ givesThe Unsteady Bernoulli equation, evaluated at the surface and linearized about hydrostatic equilibrium, gives the dynamic conditionThese are the linearized free-surface boundary conditions. Initial values for $\eta$ and $\varphi$ specify the transient; below we also record the initially quiescent solution.