Approximate determination of the transmission and reflection coefficients for water-wave flow over a topography

The objective of this paper is to present a semi-analytical method for the calculation of the transmitted and the reflected parts of an incident harmonic progressive wave in a homogeneous fluid layer over a general topography. The domain of definition of the problem is properly divided into suitable subdomains. Elementary solutions of the problem, satisfying the field equation and a subset of the boundary conditions as well, are obtained in each subdomain by the combined use of the techniques of separation of variables and finite integral transforms. These solutions are given in the form of infinite series involving an infinite number of unknown coefficients. The series are truncated and only a finite number of the coefficients is determined so as to satisfy approximately the remaining boundary conditions. Numerical illustrations indicate that the method yields highly accurate results and that the accuracy of the procedure is improved by increasing the number of the unknown coefficients taken to account with the presence of certain optimum degrees of freedom.