Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Linear water waves over a gently sloping beach

Authors: S. M. Sun and M. C. Shen
Journal: Quart. Appl. Math. 52 (1994), 243-259
MSC: Primary 76B15
DOI: https://doi.org/10.1090/qam/1276236
MathSciNet review: MR1276236
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Abstract: The objective of this paper is to justify rigorously the ray method originally developed by Keller [6] for linear water waves over a two-dimensional gently sloping beach. The approximate formula for eigenvalues of the linear water wave problem and the uniform ray method expansion at and near a shoreline are all consequences of the justification.

References [Enhancements On Off] (What's this?)

  • [1] F. Ursell, Edge waves on a sloping beach, Proc. Roy. Soc. London Ser. A 214, 79-97 (1952)
  • [2] A. S. Peters, Water waves over sloping beaches and the solution of a mixed boundary value problem for $ \Delta \varphi - {k^2}\varphi = 0$ in a sector, Comm. Pure Appl. Math. 5, 81-108 (1952)
  • [3] M. Roseau, Contribution à la théorie des ondes liquids de gravité en profondeur variable, Publications Scientifiques et Techniques du Ministère de l'Air, no. 275, Paris, 1952
  • [4] J. J. Stoker, Water Waves, Interscience, New York, 1957
  • [5] J. J. Stoker, The formation of breakers and bores, Comm. Pure Appl. Math. 1, 1-87 (1948)
  • [6] J. B. Keller, Surface waves on water of non-uniform depth, J. Fluid Mech. 4, 607-614 (1958)
  • [7] M. C. Shen, R. E. Meyer, and J. B. Keller, Spectra of water waves in channels and around islands, Phys. Fluids 11, 2289-2304 (1968)
  • [8] J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Ann. Phys. 9, 24-75 (1960)
  • [9] Yu. A. Kravtsov, A modification of the geometrical optics method, Radiofizika 7, 664-673 (1964)
  • [10] D. Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure. Appl. Math. 19, 215-250 (1966)
  • [11] M. C. Shen and J. B. Keller, Uniform ray theory of surface, internal and acoustic wave propagation in a rotating ocean or atmosphere, SIAM J. Appl. Math. 28, 857-875 (1975)
  • [12] P. Zhevandrov, Edge waves on a gently sloping beach: Uniform asymptotics, J. Fluid Mech. 233, 483-493 (1991)
  • [13] J. W. Miles, Edge waves on a gently sloping beach, J. Fluid Mech. 199, 125-131 (1989)
  • [14] V. P. Maslov and M. V. Fedoriuk, Semi-classical approximation in quantum mechanics, D. Reidel, Boston, MA, 1981
  • [15] N. G. Askerov, S. G. Krein, and G. L. Laptev, A class of non-self-adjoint boundary value problems, Soviet Math. Dokl. 5, 424-427 (1964)
  • [16] P. Zhevandrov, Private communication, 1991

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DOI: https://doi.org/10.1090/qam/1276236
Article copyright: © Copyright 1994 American Mathematical Society

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