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Reflection of nonlinear deep-water waves incident onto a wedge of arbitrary angle

Published online by Cambridge University Press:  17 February 2009

T. R. Marchant  and
A. J. Roberts
T. R. Marchant
Affiliation:
Department of Mathematics, University of Wollongong, N.S.W. 2500, Australia.
A. J. Roberts
Affiliation:
Department of Applied Mathematics, University of Adelaide, S.A. 5000, Australia.
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Abstract

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Wave reflection by a wedge in deep water is examined, where the wedge can represent a breakwater of finite length or the bow of a ship heading directly into the waves. In addition, the form of the solution allows the results to apply to ships heading at an angle into the waves. We consider a deep-water wavetrain approaching the wedge head on from infinity and being reflected. Far from the wedge there is a field of progressive waves (the incident wavetrain) while close to the wedge there is a short-crested wavefield (the incident and reflected wavetrains). A weakly-nonlinear slowly-varying averaged Lagrangian theory is used to describe the problem (see Whitham [16]) as the theory includes the nonlinear interaction between the incident and reflected wavetrains. This modelling of a short-crested wavefield allows the nonlinear wavefield to be found for broad wedges, as opposed to previous theories which are applicable to thin wedges only.

It is shown that the governing partial differential equations are hyperbolic and that the solution comprises two regions, within which the wave properties are constant separated by a wave jump. Given the wedge angle and the incident wavefield, the jump angle and the wave steepness and wavenumber of the short-crested wave-field behind the wave jump can be determined. Two solution branches are found to exist: one corresponds to regular reflection, while for small amplitudes the other is similar to Mach-reflection and so it is called near Mach-reflection. Results are presented describing both solution branches and the transition between them.

Type
Research Article
Information
The ANZIAM Journal , Volume 32 , Issue 1 , July 1990 , pp. 61 - 96
Copyright
Copyright © Australian Mathematical Society 1990

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