Unsteady edge waves generated by time-harmonic pressure: exact solutions

An analytical description of unsteady edge waves over a uniform slope is proposed. It is assumed that the waves are excited by time-harmonic external pressure with inhomogeneous spatial distribution. The problem is considered in Lagrangian variables. An exact solution of the hydrodynamic equations is obtained. It generalizes the stationary Gerstner–Constantin solution. The proposed model describes the dynamics of coastal splashes of arbitrary initial shape. An important feature of the new solution is that it describes waves both propagating and standing in the longshore direction.