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Approximations to wave trapping by topography

Published online by Cambridge University Press:  26 April 2006

P. G. Chamberlain  and
D. Porter
P. G. Chamberlain
Affiliation:
Department of Mathematics, The University of Reading, PO Box 220, Whiteknights, Reading RG6 6AX, UK
D. Porter
Affiliation:
Department of Mathematics, The University of Reading, PO Box 220, Whiteknights, Reading RG6 6AX, UK
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Abstract

The trapping of linear water waves over two-dimensional topography is investigated by using the mild-slope approximation. Two types of bed profile are considered: a local irregularity in a horizontal bed and a shelf joining two horizontal bed sections at different depths. A number of results are derived concerning the existence of trapped modes and their multiplicity. It is found, for example, that the maximum number of modes which can exist depends only on the gross properties of the topography and not on its precise shape. A range of problems is solved numerically, to inform and illustrate the analysis, using both the mild-slope equation and the recently derived modified mild-slope equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 325 , 25 October 1996 , pp. 357 - 376
Copyright
© 1996 Cambridge University Press

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References

Berkhoff, J. C. W. 1973 Computation of combined refraction-diffraction. Proc. 13th Intl Conf. on Coastal Engng, July 1972, Vancouver, Canada, pp. 471–490. ASCE.
Bonnet-Ben Dhia, A.-S. & Joly, P. 1993 Mathematical analysis of guided water waves. SIAM J. Appl. Maths 53, 1507–1550.Google Scholar
Booij, N. 1983 A note on the accuracy of the Mild-Slope equation. Coastal Engng 7, 191–203.Google Scholar
Chamberlain, P. G. & Porter, D. 1995a The modified mild-slope equation. J. Fluid Mech. 291, 393–407.Google Scholar
Chamberlain, P. G. & Porter, D. 1995b Decomposition methods for wave scattering by topography with application to ripple beds. Wave Motion 22, 201–214.Google Scholar
Evans, D. V. & Kuznetsov, N. 1996 Trapped modes. In Gravity Waves on Water of Finite Depth (ed. J. N. Hunt). Computational Mechanics (to appear).
Ince, E. L. 1944 Ordinary Differential Equations. Dover.
Jones, D. S. 1953 The eigenvalues of ∇2u — λu = 0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684.Google Scholar
Lambert, J. D. 1992 Numerical Methods for Ordinary Differential Systems. Wiley.
Lavrentiev, M. A. & Chabat, B. V. 1973 Effets Hydrodynamiques et Modeles Mathematique. Mir, Moscow.
LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley.
Phillips, G. M. & Taylor, P. J. 1973 Theory and Applications of Numerical Analysis Academic Press.
Porter, D. & Chamberlain, P. G. 1996 Linear wave scattering by two-dimensional topography. In Gravity Waves on Water of Finite Depth (ed. J. N. Hunt). Computational Mechanics (to appear).
Porter, D. & Staziker, D. J. 1995 Extensions of the mild-slope approximation. J. Fluid Mech. 300, 367–382.Google Scholar
Smith, R. & Sprinks T. 1975 Scattering of surface waves by a conical island. J. Fluid Mech. 72, 373–384.Google Scholar
Stokes, G. G. 1846 Report on recent researches in hydrodynamics. Report on 16th Mtg Brit. Assoc. Adv. Science, Southampton, pp. 1–20. Murrey, London. Also in Mathematical and Physical Papers, vol. I, p. 167, Cambridge University Press, 1880.