Explicit Dirichlet–Neumann operator for water waves

Abstract

An explicit expression for the Dirichlet–Neumann operator for surface water waves is presented. For non-overturning waves, but without assuming small amplitudes, the formula is first derived in two dimensions, and subsequently extrapolated to higher dimensions and with a moving bottom. Although described here for water waves, this elementary approach could be adapted to many other problems having similar mathematical formulations.