Numerical Model for Shoaling and Refraction of Third-Order Stokes Waves over an Irregular Bottom

摘要

This numerical model solves for water wave height, angle, and number directly on a rectangular grid. Required input are the deepwater wave height, period, and direction and the bathymetry in the region of interest. The model employs a finite difference scheme. The irrotationality equation of the wave number vector is solved for the wave angle, and the conservation of energy flux equation is solved for the wave height. Iteration is required. A closed form expresssion, to third-order, for the time-averaged, vertically integrated energy flux is derived. Stokes' second definition of wave celerity is used in the derivation to reduce the number of intermediate calculations. Expressions for the wave energy and the group velocity are also derived. The model is written such that both first-order (linear) and third-order stokes wave theory model computations may be conducted. The modeling process begins at higher intermediate depth, or deep water, and waves are propagated shoreward until an Urseell number of 25 or another, user-specified, value is reached. The model is applied for the following cases: (a) comparison of small amplitude and finite amplitude wave refraction and shoaling on a plane beach, (b) refraction and shoaling over an irregular bottom configuration, and (c) comparison of the model shoaling predictions to laboratory data of Iverson (1951).