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Three‐Dimensional Surface Water Waves Governed by the Forced Benney–Luke Equation
Author(s) -
Curtis Christopher W.,
Shen Samuel S. P.
Publication year - 2015
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12091
Subject(s) - bathymetry , euler equations , waves and shallow water , mechanics , mathematics , classical mechanics , physics , mathematical analysis , geology , thermodynamics , oceanography
This paper studies the propagation of three‐dimensional surface waves in water with an ambient current over a varying bathymetry. When the ambient flow is near the critical speed, under the shallow water assumptions, a forced Benney–Luke (fBL) equation is derived from the Euler equations. An asymptotic approximation of the water's reaction force over the varying bathymetry is derived in terms of topographic stress. Numerical simulations of the fBL equation over a trough are compared to those using a forced Kadomtsev–Petviashvilli equation. For larger variations in the bathymetry that upstream‐radiating three‐dimensional solitons are observed, which are different from the upstream‐radiating solitons simulated by the forced Kadomtsev–Petviashvilli equation. In this case, we show the fBL equation is a singular perturbation of the forced Kadomtsev–Petviashvilli equation which explains the significant differences between the two flows.