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Scattering of tidal waves by reefs and spits

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  26 February 2010

P. A. Krutitskii
P. A. Krutitskii
Affiliation:
Department of Mathematics, Faculty of Physics, Moscow State University, Moscow 119899, Russia.
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Abstract

The problem of scattering of tidal waves by reefs and spits of arbitrary shape is reduced to a skew derivative problem for the two-dimensional Helmholtz equation in the exterior of open arcs in a plane. The resulting boundary-value problem is studied by potential theory and a boundary integral equation method. After some transformations, the skew derivative problem is reduced to a Fredholm integral equation of the second kind, which is uniquely solvable. In this way the solvability theorem is proved and an integral representation of the solution is obtained. A uniqueness theorem is also proved.

MSC classification

Secondary: 35J25: Boundary value problems for second-order elliptic equations
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 355 - 370
Copyright
Copyright © University College London 2000

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References

1.Colton, D. and Kress, R.. Integral Equation Methods in Scattering Theory. (John Wiley and Sons, New York, 1983).Google Scholar
2.Crease, J.. Long waves on a rotating earth in the presence of a semi-infinite barrier. J. Fluid Mech., 1 (1956), 86–96.CrossRefGoogle Scholar
3.Crease, J.. The propagation of long waves into a channel in a rotating system. J. Fluid Mech., 4 (1958), 306–320.CrossRefGoogle Scholar
4.Davydov, A. G., Zakharov, Ye. V. and Pimenov, Yu. V.. A method of solving problems of the diffraction of electromagnetic waves by infinitely thin cylindrical screens (Russian). Dokl. Akad. Nauk SSSR, 261, (1981), 338–341.Google Scholar
5.Durand, M.. Layer potentials and boundary-value problems for the Helmholtz equation in the complement of a thin obstacle. Math. Meth. Appl. Sci., 5 (1983), 389–421.CrossRefGoogle Scholar
6.Gabov, S. A.. An angular potential and its applications. Math. U.S.S.R. Sbornbik, 32 (1977), 423–436.CrossRefGoogle Scholar
7.Gakhov, F. D.. Boundary Value Problems (Pergamon Press, Oxford); (Addison-Wesley Reading, Mass., 1966).CrossRefGoogle Scholar
8.Stoker, J. J.. Water Waves (Wiley-Interscience, New York, 1957).Google Scholar
9.Jeffreys, H.. The free oscillations of water in an elliptic lake. Proc. London Math. Soc. (2), 23 (1924), 455–476.Google Scholar
10.Krutitskii, P. A.. Dirichlet problem for the Helmholtz equation ouside cuts in a plane. Comp. Maths. Math. Phys., 34 (1994), 1073–1090.Google Scholar
11.Krutitskii, P. A.. Neumann problem for the Helmholtz equation outside cuts in a plane. Comp. Maths. Math. Phys., 34 (1994), 1421–1431.Google Scholar
12.Krutitskii, P. A.. Fast stratified flow over several obstacles, including wings. IMA J. Appl. Math., 57 (1996), 243–256.CrossRefGoogle Scholar
13.Krutitskii, P. A.. Wave propagation in a 2-D external domain with cuts. Applicable Analysis, 62 (1996), 297–309.Google Scholar
14.Krutitskii, P. A.. The Neumann problem for the 2-D Helmholtz equation in a multiply connected domain with cuts. Zeitschrift Analysis Anwendungen, 16 (1997), 349–361.CrossRefGoogle Scholar
15.Lamb, H.. Hydrodynamics. Cambridge Univ. Press (1932),.Google Scholar
16.Lifanov, I. K.. Singular Integral Equations and Discrete Vortices. VSP, Zeist (1996),.CrossRefGoogle Scholar
17.Muskhelishvili, N. I.. Singular Integral Equations. Noordhoff (Groningen, 1972). (3rd Russian edition: Nauka, Moscow, 1968).Google Scholar
18.Panasyuk, V. V., Savruk, M. P. and Nazarchuk, Z. T.. The Method of Singular Integral Equations in Two-Dimensional Diffraction Problems (Russian). Naukova Dumka (Kiev, 1984).Google Scholar
19.Proudman, J.. Diffraction of tidal waves on a flat rotating sheet of water. Proc. London Math. Soc. (2), 14 (1915), 89–102.CrossRefGoogle Scholar
20.Proudman, J.. Dynamical Oceanography. Methuen (London, 1953).Google Scholar
21.Sekerh-Zenkovich, S. Ya.. Diffraction of plane waves by a circular island. (Russian). Fizika Atmos. i Ocean., 4 (1968), 69–79.Google Scholar
22.Sretensky, L. N.. Theory of Wave Motions of a Fluid (Russian). ONTI (Moscow, 1936).Google Scholar
23.Srelensky, L. N.. Dynamical Theory of Tides (Russian). Nauka (Moscow, 1987).Google Scholar
24.Tuchkin, Yu. A. and Shestopalov, V. P.. Scattering of waves by a finite system of cylindrical screens of arbitrary profile with Dirichlet boundary condition (Russian). Dokl. Akad. Nauk SSSR, 285 (1985), 1107–1109.Google Scholar
25.Tuchkin, Yu. A.. Scattering of waves by an unclosed cylindrical screen of an arbitrary profile with Neumann boundary condition (Russian). Dokl. Akatl. Nauk SSSR, 293 (1987), 343–345.Google Scholar
26.Vladimirov, V. S.. Equations of Mathematical Physics. Marcel Dekker (New York, 1971).Google Scholar
27.Wilcox, C. H.. Scattering theory for the d'Alembert equation in the exterior domains. Lecture Notes in Mathematics 422, Springer (Berlin, 1975).Google Scholar
28.Wolfe, P.. An existence theorem for the reduced wave equation. Proc. Amer. Math. Soc., 21 (1969), 663–666.CrossRefGoogle Scholar
29.Zakharov, Ye. V. and Sobyanina, I. V.. One-dimensional integro-diffcrential equations of problems of diffraction by screens (Russian). Zh. Vychisl. Matem. Mat. Fiz., 26 (1986), 632–636.Google Scholar
30.Ikehata, M.. Enclosing a polygonal cavity in a two-dimensional object from boundary measurements. Inverse Problems, 15 (1999), 1231–1241.CrossRefGoogle Scholar
31.Ikehata, M. and Nakamura, G.. Slicing of a three-dimensional object from boundary measurements. Inverse Problems, 15 (1999), 1243–1253.CrossRefGoogle Scholar
32.Ikehata, M.. Reconstruction of obstacle from boundary measurements. Wave Motion, 30 (1999), 205–223.CrossRefGoogle Scholar
33.Ikehata, M.. How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms. J. Inverse and Ill-posed Problems, 7 (1999), 255–271.CrossRefGoogle Scholar
35.Ikehata, M.. Reconstruction of an obstacle from the scattering amplitude at a fixed frequency. Inverse Problems, 14 (1998), 949–954.CrossRefGoogle Scholar