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.
F. Ursell, Edge waves on a sloping beach, Proc. Roy. Soc. London Ser. A 214, 79–97 (1952)
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)
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
J. J. Stoker, Water Waves, Interscience, New York, 1957
J. J. Stoker, The formation of breakers and bores, Comm. Pure Appl. Math. 1, 1–87 (1948)
J. B. Keller, Surface waves on water of non-uniform depth, J. Fluid Mech. 4, 607–614 (1958)
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)
J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Ann. Phys. 9, 24–75 (1960)
Yu. A. Kravtsov, A modification of the geometrical optics method, Radiofizika 7, 664–673 (1964)
D. Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure. Appl. Math. 19, 215–250 (1966)
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)
P. Zhevandrov, Edge waves on a gently sloping beach: Uniform asymptotics, J. Fluid Mech. 233, 483–493 (1991)
J. W. Miles, Edge waves on a gently sloping beach, J. Fluid Mech. 199, 125–131 (1989)
V. P. Maslov and M. V. Fedoriuk, Semi-classical approximation in quantum mechanics, D. Reidel, Boston, MA, 1981
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)
P. Zhevandrov, Private communication, 1991
F. Ursell, Edge waves on a sloping beach, Proc. Roy. Soc. London Ser. A 214, 79–97 (1952)
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)
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
J. J. Stoker, Water Waves, Interscience, New York, 1957
J. J. Stoker, The formation of breakers and bores, Comm. Pure Appl. Math. 1, 1–87 (1948)
J. B. Keller, Surface waves on water of non-uniform depth, J. Fluid Mech. 4, 607–614 (1958)
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)
J. B. Keller and S. I. Rubinow, Asymptotic solution of eigenvalue problems, Ann. Phys. 9, 24–75 (1960)
Yu. A. Kravtsov, A modification of the geometrical optics method, Radiofizika 7, 664–673 (1964)
D. Ludwig, Uniform asymptotic expansions at a caustic, Comm. Pure. Appl. Math. 19, 215–250 (1966)
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)
P. Zhevandrov, Edge waves on a gently sloping beach: Uniform asymptotics, J. Fluid Mech. 233, 483–493 (1991)
J. W. Miles, Edge waves on a gently sloping beach, J. Fluid Mech. 199, 125–131 (1989)
V. P. Maslov and M. V. Fedoriuk, Semi-classical approximation in quantum mechanics, D. Reidel, Boston, MA, 1981
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)
P. Zhevandrov, Private communication, 1991
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Article copyright:
© Copyright 1994
American Mathematical Society
