A unified theory for modeling water waves

Abstract

This chapter presents a unified theory for modeling water waves. The different types of waves in water, varying from ripples on a placid pond, breaking of shoaling waves on a beach, billows on a stormy sea and in the ocean interior, to geophysical waves and devastating tsunamis, are truly extensive. Euler's equations are adopted to describe three-dimensional, incompressible, inviscid long waves on a layer of water of variable depth, which may vary with the horizontal position vector. Four sets of theoretical models for describing fully nonlinear fully dispersive (FNFD) unsteady gravity-capillary waves on water of variable depth in terms of four sets of basic variables are obtained. It is found that for determining solutions to the model equations for initial-boundary value problems with external forcing by surface pressure and seabed motion, effective numerical schemes are useful. It is observed that for two-dimensional irrotational water waves in particular, an alternative closure relation is derived by applying Cauchy's contour integral formula. The modeling of FNFD waves in water of uniform depth is also elaborated.

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