Simulating Wave Reflection Using Radiation Boundary

Abstract

A finite element wave propagation model for linear periodic waves in a coastal sea region is developed. The model includes refraction, diffraction, and reflection of gravity waves on water over arbitrary bathymetry. The authors discuss the use of the computer model in simulating waves using a number of classical examples involving a circular pile, submerged shoal, and breakwater. The method of solution involves complex potential theory. The equation used by Berkhoff (1976; Mathematical Models for Simple Harmonic Linear Water Waves—Wave Diffraction and Refraction. Delft: Delft Hydraulics Laboratory, Delft University of Technology, p. 111) and the authors within the domain is elliptic in type allowing wave trains to cross, thereby producing amphidromic points. An amphidromic point is the two-dimensional version of a node in a one-dimensional standing wave caused by imperfect reflection or wave train interference. Radiating and partially reflecting boundaries are modelled by the authors, using a parabolic equation developed in different ways by Radder (1979; On the parabolic equation method for water-wave propagation, Journal of Fluid Mechanics, 95, 159–176) and Booij (1981; Gravity waves on water with non-uniform depth and current. Delft: Delft Hydraulics Laboratory, Delft University of Technology, p. 127), allowing the passage of energy through a boundary over arbitrary bathymetry. Radder (1979) and Booij (1981) develop this equation in the domain as an alternative to the elliptic equation. Berkhoff (1976) uses a downstream radiation boundary condition based on Hankel functions for the shoal problem, valid only in constant depth. The upstream boundary condition of Berkhoff (1976) for the same shoal problem is derived using the wave ray method. The limitation of the wave ray method is that for general purposes the rays frequently cross, resulting in no solution. The method used by the authors has the advantage of simplicity in that the boundary conditions are very simple to implement but none of the physical features are lost.