A general wave equation for waves over rippled beds. (English) Zbl 0596.76017
Summary: A time-dependent extension of the reduced wave equation of J. C. W. Berkhoff [Proc. 13th Int. Conf. Coastl. Eng., Vol. 2, 471-490 (1972)] is developed for the case of waves propagating over a bed consisting of ripples superimposed on an otherwise slowly varying mean depth which satisfies the mild-slope assumption. The ripples are assumed to have wavelengths on the order of the surface wavelength but amplitudes which scale as a small parameter along with the bottom slope. The theory is verified by showing that it reduces to the case of plane waves propagating over a patch of sinusoidal ripples, which vary in one direction and extend to \(\pm \infty\) in the transverse direction, studied recently. We then formulate and use coupled parabolic equations to study propagation over patches of arbitrary form in order to study wave reflection.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76M99 | Basic methods in fluid mechanics |
Keywords:
time-dependent extension; reduced wave equation; mild-slope assumption; plane waves; patch of sinusoidal ripples; coupled parabolic equations; wave reflectionReferences:
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