 Solution

Irrotationality permits a velocity potential with $\mathbf{u}=({\varphi }_x,{\varphi }_z)$. Incompressibility gives Laplace's equation in the moving domain $z<\eta (x,t)$. Neglecting surface tension and taking the free-surface pressure as constant atmospheric pressure, the full surface conditions areThe kinematic condition makes the free surface material; the dynamic condition is Bernoulli's equation on that surface. A time-dependent additive gauge in $\varphi$ has absorbed the spatially uniform Bernoulli constant. Deep below the surface, impose $\nabla \varphi \to 0$ as $z\to -\infty$, together with suitable horizontal decay. Initially $\eta (x,0)={\eta }_0(x)$ and the fluid is at rest, so $\nabla \varphi (x,z,0)=0$; its initial constant may be chosen as zero.