Slow Interactions of Gravity Waves and a Corrugated Sea Bed

[+] Author and Article Information
A. Mitra

Department of Environmental Resources Engineering, Humboldt State University, Arcata, Calif. 95521

M. D. Greenberg

Department of Mechanical and Aerospace Engineering, University of Delaware, Newark, Del. 19711

J. Appl. Mech 51(2), 251-255 (Jun 01, 1984) (5 pages) doi:10.1115/1.3167608 History: Received September 01, 1982; Revised June 01, 1983; Online July 21, 2009


The effect of a corrugated sea bed on the linear theory of gravity water waves is considered. By straining the time variable, a perturbation solution is found in ε (the ratio of corrugation amplitude to mean water depth), through first order, for a wave system that is arbitrarily oriented with respect to the corrugations. That solution breaks down when the wave number k normal to the corrugation is a half-integer multiple of the wave number 2ω of the corrugations, i.e., when k = (ω, 2ω, .... Of these singularities, the first (k = ω) appears at the first order. To obtain a uniformly valid zeroth-order solution we include a zeroth-order reflected wave system, and obtain an alternation between incident and reflected waves on a time scale of order 0(ε−1 ). As representative of the other singular wave numbers, we consider k = 3ω, which singularity appears at the third order, and obtain a uniformly valid solution through second order (for the shallow water limit). Nonlinear effects are considered to the extent of noting that the zeroth-order linear and nonlinear results are identical, even for the first singular wave number k = ω.

Copyright © 1984 by ASME
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