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Journal of Applied Mathematics
Volume 2015 (2015), Article ID 256084, 11 pages
http://dx.doi.org/10.1155/2015/256084
Research Article

Shear Current Effects on Monochromatic Water Waves Crossing Trenches

1Department of Civil and Environmental Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Republic of Korea
2Civil and Environment Division, Hyundai Engineering and Construction, Gye-dong, Jongno-gu, Seoul 110-920, Republic of Korea

Received 28 October 2014; Accepted 6 January 2015

Academic Editor: Michael Meylan

Copyright © 2015 Jun-Whan Lee 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.

Abstract

The reflection coefficients of monochromatic water waves over trenches with shear current are estimated analytically. The diffraction of waves by an abrupt depth change and shear current is formulated by the matched eigenfunction expansion method. The proper number of steps and evanescent modes are proposed by a series of convergence tests. The accuracy of the predicted reflection coefficients is checked by estimating the wave energy. Reflection and transmission characteristics are studied for various shear current conditions. The different combinations of strength, width of shear current, and incident wave angle with constant water depth topography are examined. The optimal figure of the trench with shear current is obtained by estimating the reflection coefficients for various sloped transitions. The resonant reflection of the water waves is found by multiarrayed optimal trenches and the interaction of sinusoidally varying topography with shear current.