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Surface-wave propagation over sinusoidally varying topography

Published online by Cambridge University Press:  20 April 2006

A. G. Davies  and
A. D. Heathershaw
A. G. Davies
Affiliation:
Institute of Oceanographic Sciences, Crossway, Taunton, Somerset, TA1 2DW
A. D. Heathershaw
Affiliation:
Institute of Oceanographic Sciences, Crossway, Taunton, Somerset, TA1 2DW
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Abstract

Surface waves travelling in water of finite depth may be scattered by a region of undulating bottom topography. The present study is concerned with the idealized two-dimensional situation in which long-crested surface waves are incident upon a patch of long-crested regular bottom ripples. The principal question examined concerns the amount of incident wave energy that is reflected by the ripple patch. Linear perturbation theory is used to show that the reflection coefficient is both oscillatory in the quotient of the length of the patch and the surface wavelength, and also strongly dependent upon the quotient of the surface and bed wavelengths. In particular, there is a Bragg resonance between the surface waves and the ripples, which is associated with the reflection of incident wave energy. A secondary question concerns the nature of the wave field in the immediate vicinity of the ripple patch. In resonant cases, it is shown how the partially standing wave on the upwave side of the ripple patch gives way, in an almost linear manner over the patch itself, to a progressive transmitted wave on the downwave side. The theoretical predictions are compared with an extensive set of laboratory observations made in a wave tank. Comparisons relating both to the reflection coefficient, and also to the wave field over the ripple patch, are shown to give consistently good agreement. Finally, the implications of the results for sediment transport on an erodible bed are examined.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 144 , July 1984 , pp. 419 - 443
Copyright
© 1984 Cambridge University Press

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References

Beckmann, P. & Spizzichino, A. 1963 The Scattering of Electromagnetic Waves from Rough Surfaces. Pergamon.
Dalrymple, R. A. & Fowler, J. F. 1982 Bragg scattering by pile-supported structures. J. Waterway, Port, Coastal and Ocean Div. ASCE 108 (WW3), 426–429.Google Scholar
Davies, A. G. 1980 Some interactions between surface water waves and ripples and dunes on the seabed. Inst. Oceanogr. Sci. Rep. 108.Google Scholar
Davies, A. G. 1982a The reflection of wave energy by undulations on the seabed. Dyn. Atmos. Oceans 6, 207–232.Google Scholar
Davies, A. G. 1982b On the interaction between surface waves and undulations on the seabed. J. Mar. Res. 40, 331–368.Google Scholar
Davies, A. G. & Heathershaw, A. D. 1983 Surface wave propagation over sinusoidally varying topography: theory and observation. Inst. Oceanogr. Sci. Rep. 159.Google Scholar
Fitz-Gerald, G. F. 1976 The reflexion of plane gravity waves travelling in water of variable depth. Phil. Trans. R. Soc. Lond. A 284, 49–89.Google Scholar
Fortuin, L. 1970 Survey of literature on reflection and scattering of sound waves at the sea surface. J. Acoust. Soc. Am. 47, 1209–1228.Google Scholar
Goda, Y. & Suzuki, Y. 1977 Estimation of incident and reflected waves in random wave experiments. In Proc. 15th Intl Conf. on Coastal Engng, 1976, pp. 828–845. ASCE.
Heathershaw, A. D. 1982 Seabed-wave resonance and sand bar growth. Nature 296, 343–345.Google Scholar
Jeffreys, H. 1944 Motion of waves in shallow water. Note on the offshore bar problem and reflexion from a bar. Ministry of Supply Wave Rep. 3 (London).Google Scholar
Kellum, F. W. 1956 An electronic gage for measurement of small waves and ripples. Bull. Beach Erosion Board, US Army Corps of Engrs 10, 32–40.Google Scholar
Kennedy, J. F. & Falcón, M. 1965 Wave-generated sediment ripples. MIT Dept Civ. Engng, Hydrodyn. Lab. Rep. 86.Google Scholar
Kreisel, G. 1949 Surface waves. Q. Appl. Maths 7, 21–44.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Langhorne, D. N. 1982 A study of the dynamics of a marine sandwave. Sedimentology 29, 571–594.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. & Black, J. L. 1969 Scattering of surface waves by rectangular obstacles in waters of finite depth. J. Fluid Mech. 38, 499–511.Google Scholar
Mitra, A. & Greenberg, M. D. 1984 Slow interactions of gravity waves and a corrugated seabed. Trans. ASME E: J. Appl. Mech. (in press).Google Scholar
Newman, J. N. 1965 Propagation of water waves past long two-dimensional obstacles. J. Fluid Mech. 23, 23–29.Google Scholar
Nielsen, P. 1979 Some basic concepts of wave sediment transport. Tech. Univ. Denmark, Inst. of Hydrodyn. and Hydraul. Engng. Series Paper 20.Google Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximations behind them. In Waves on Beaches (ed. R. E. Meyer), pp. 95–121. Academic.
Scott, T. 1954 Sand movement by waves. US Army Corps of Engrs, Beach Erosion Board, Tech. Memo. 48.Google Scholar
Short, A. D. 1975 Multiple offshore bars and standing waves. J. Geophys. Res. 80, 3838–3840.Google Scholar
Sleath, J. F. A. 1975 Transition in oscillatory flow over rippled beds. Proc. Inst. Civ. Engrs 59, 309–322.Google Scholar
Stafford, R. B. 1972 Investigation and procedure for calibration of CERC laboratory wave gage FWK Model-1. Washington, DC, Coastal Engng Res. Center (unpublished document).
Ursell, F. 1947 The reflection of waves from a submerged low reef. Admiralty Res. Lab., Teddington, Middx, Rep. ARL/R5/103.41/W.Google Scholar