Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-12T19:54:48.619Z Has data issue: false hasContentIssue false
  • English
  • Français

A general wave equation for waves over rippled beds

Published online by Cambridge University Press:  21 April 2006

James T. Kirby
James T. Kirby
Affiliation:
Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FL 32611 USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A time-dependent extension of the reduced wave equation of Berkhoff 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 ± ∞ in the transverse direction, studied recently by Davies & Heathershaw and Mei. We then formulate and use coupled parabolic equations to study propagation over patches of arbitrary form in order to study wave reflection.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 162 , January 1986 , pp. 171 - 186
Copyright
© 1986 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berkhoff, J. C. W. 1972 Computation of combined refraction-diffraction. In Proc. 13th Intl Conf. Coastal Engng, vol. 2, pp. 471–490. ASCE.
Dalrymple, R. A., Kirby, J. T. & Hwang, P. A. 1984M Wave diffraction due to areas of energy dissipation. J. Waterway, Port, Coast, and Ocean Engng 110, 67–79.Google Scholar
Davies, A. G. & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidally varying topography. J. Fluid Mech. 144, 419–443.Google Scholar
Hasselmann, K. 1966 Feynmann diagrams and interaction rules of wave-wave scattering processes. Rev. Geophys. Space Phys. 4, 1–32.Google Scholar
Heathershaw, A. D. 1982 Seabed-wave resonance and sand bar growth. Nature 296, 343–345.Google Scholar
Kirby, J. T. 1984 A note on linear surface wave-current interaction over slowly varying topography. J. Geophys. Res. 89, 745–747.Google Scholar
Kirby, J. T. 1986 On the gradual reflection of weakly nonlinear Stokes waves in regions with varying topography. J. Fluid Mech. 162, 187–209.Google Scholar
Liu, P. L.-F. 1983 Wave—current interactions on a slowly varying topography. J. Geophys. Res. 88, 4421–4426.Google Scholar
Liu, P. L.-F. & Tsay, T.-K. 1983 On weak reflection of water waves. J. Fluid Mech. 131, 59–71.Google Scholar
Long, R. B. 1973 Scattering of surface waves by an irregular bottom. J. Geophys. Res. 78, 7861–7870.Google Scholar
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sand-bars. J. Fluid Mech. 152, 315–335.Google Scholar
Mitra, A. & Greenberg, M. D. 1984 Slow interactions of gravity waves and a corrugated sea bed. Trans. A8ME E: J. Appl. Mech. 51, 251–255.Google Scholar
Radder, A. C. 1979 On the parabolic equation method for water wave propagation. J. Fluid Mech. 95, 159–176.Google Scholar
Smith, R. & Sprinks, T. 1975 Scattering of surface waves by a conical island. J. Fluid Mech. 72,373–384.Google Scholar
Symonds, G. & Bowen, A. J. 1984 Interactions of nearshore bars with incoming wave groups. J. Geophys. Res. 89, 1953–1959.Google Scholar
Symonds, G., Huntley, D. A. & Bowen, A. J. 1982 Two-dimensional surf beat: long wave generation by a time-varying breakpoint. J. Geophys. Res. 87, 492–498.Google Scholar