Journal of Fluid Mechanics

Nonlinear water waves at a submerged obstacle or bottom topography

John  Grue a1
a1 Department of Mathematics, Mechanics Division, University of Oslo, P.O. Box 1053, Blindern, Norway

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Nonlinear diffraction of low-amplitude gravity waves in deep water due to a slightly submerged obstacle is studied experimentally in a wave channel and theoretically. The obstacle is either a circular cylinder or a rectangular shelf. The incoming waves (with wavelength λ) undergo strong nonlinear deformations at the obstacle when the wave amplitude is finite. An infinite number of superharmonic waves are then introduced to the flow. Their wavelengths far away from the obstacle are λ/4, λ/9, λ/16,…, due to the dispersion relation being quadratic in the wave frequency. The superharmonic wave amplitudes grow with increasing incoming wave amplitude up to saturation values. They are found to be prominent at the obstacle's lee side and vanishingly small at the weather side. The second- and third-harmonic wave amplitudes are, surprisingly, in several examples found to be comparable to the incoming wave amplitude. Up to 25% of the incoming energy flux may be transferred to the shorter waves. The theoretical model accounts for nonlinearity by the Boussinesq equations in the shallow region above the obstacle, with patching to linearized potential theory in the deep water. The theory explains both qualitatively and quantitatively the trends observed in the experiments up to breaking.

(Published Online April 26 2006)
(Received April 4 1991)
(Revised February 27 1992)