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Mathematical Problems in Engineering
Volume 2011, Article ID 607196, 20 pages
Research Article

On Step Approximation for Roseau's Analytical Solution of Water Waves

1Department of Marine Environmental Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan
2Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701, Taiwan

Received 24 November 2010; Accepted 3 March 2011

Academic Editor: Mohammad Younis

Copyright © 2011 Chia-Cheng Tsai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An indirect eigenfunction marching method (IEMM) is developed to provide step approximations for water wave problems. The bottom profile is in terms of successive flat shelves separated by abrupt steps. The marching conditions are represented by the horizontal velocities at the steps in the solution procedure. The approximated wave field can be obtained by solving a system of linear equations with unknown coefficients which represents the horizontal velocities under a proper basis. It is also demonstrated that this solution method can be exactly reduced to the transfer-matrix method (TM method) for a specific setting. The combined scattering effects of a series of steps can be described by a single two-by-two transfer matrix for connecting the far-field behaviors of both sides for this method. The solutions obtained by the IEMM are basically exact for water wave problems considering step-like bottoms. Numerical simulations were performed to validate the present and commonly used methods. Furthermore, it also shows that the solutions obtained by the IEMM converge very well to Roseau's analytical solutions for both mild and steep curved bottom profiles. The present method improves the converges of the TM method for solving water wave scattering over steep bathymetry.