Stokes Approximation Solutions for Steep Standing Waves

Abstract

A hybrid analytical-numerical method for standing waves in water of any depth exactly satisfies the field equation, the bottom boundary condition, the periodic lateral boundary conditions and the mean water level constraint. The wave height and the kinematic and dynamic free surface boundary conditions are imposed numerically, as a problem in nonlinear optimisation. The algorithm is confirmed against an existing fifth-order analytical theory. The method extends the available predictive range for standing waves to near-limit waves in deep, transitional and shallow water. The limitations of the numerical method are clearly identified. The limit wave can not be predicted but near-limit extreme wave indicators for wave height, wave number and crest elevation are defined over the complete range of water depths.