On the solvability of a two-dimensional water-wave radiation problem
Author:
G. A. Athanassoulis
Journal:
Quart. Appl. Math. 44 (1987), 601-620
MSC:
Primary 76B15; Secondary 35R35
DOI:
https://doi.org/10.1090/qam/872813
MathSciNet review:
872813
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Abstract: The existence of a unique weak solution for the two-dimensional water-wave radiation problem arising when a floating rigid body oscillates on the free surface is established for all but a discrete set of oscillation frequencies. The body boundary is taken to be of $C_ * ^{1,\alpha }$ class (see Sec. 2) and the body boundary condition is satisfied in the ${L^2}$-sense. The proof relies on an expansion theorem (Athanassoulis [1]) and on the property of the associated water-wave multipoles being a Riesz basis of ${L^2}\left ( { - \pi ,0} \right )$, a fact which is established in the present paper. Under stronger geometrical restrictions on the body boundary it is proved, using a method due to Ursell [10], that the weak solution is actually a classical one; that is, the velocity field is continuous up to ana including the body boundary.
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M. Lenoir and D. Martin, Étude théorique et numérique du problème linéarisé du mouvement sur la houle tridimensionnelle, École National Supérieure de Techniques Avancées, Paris, Report No. 124 (1980)
M. Lenoir, Méthod de couplage en hydrodynamique naval et application à la resistance de vagues bidimensionnelles, École National Supérieure de Techniques Avancées, Paris, Report No. 164, chapter III (1982)
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F. Ursell, The expansion of water-wave potentials at great distances, Proc. Camb. Philos. Soc. 64, 811–826 (1968)
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D. Euvrard, A. Jami, M. Lenoir, and D. Martin, Recent progress towards an optimal coupling between finite elements and singularity distribution procedures, Proc. 3rd Internat. Conf. Numerical Ship Hydrodynamics, Paris (1981)
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G. A. Athanassoulis, An expansion theorem for water-wave potentials, Journal of Engineering Mathematics 18, 181–194 (1984)
F. John, On the motion of floating bodies, II, Comm. Pure Appl. Math. 3, 45–101 (1950)
J. T. Beale, Eigenfunction expansions for objects floating in an open sea, Comm. Pure Appl. Math. 30, 283–313 (1977)
M. Lenoir and D. Martin, An application of the principle of limiting absorption to the motions of floating bodies, J. Math. Anal. Appl. 79, 370–383 (1981)
M. Lenoir and D. Martin, Étude théorique et numérique du problème linéarisé du mouvement sur la houle tridimensionnelle, École National Supérieure de Techniques Avancées, Paris, Report No. 124 (1980)
M. Lenoir, Méthod de couplage en hydrodynamique naval et application à la resistance de vagues bidimensionnelles, École National Supérieure de Techniques Avancées, Paris, Report No. 164, chapter III (1982)
F. Ursell, Surface waves on deep water in the presence of a submerged circular cylinder, II, Proc. Camb. Philos. Soc. 46, 153–158 (1950)
F. Ursell, The expansion of water-wave potentials at great distances, Proc. Camb. Philos. Soc. 64, 811–826 (1968)
F. Ursell, Short surface waves due to an oscillating immersed body, Proc. Roy. Soc. London Ser. A 220, 90–103 (1953)
F. Ursell, A problem in the theory of water waves, in Numerical solution of integral equations, Clarendon Press, Oxford (1974)
Y. S. Yu and F. Ursell, Surface waves generated by an oscillating circular cylinder on water of finite depth: Theory and experiment, J. Fluid Mech. 11(4), 529–551 (1961)
J. J. Stoker, Water waves. The mathematical theory with applications, Interscience, New York (1957)
A. Zygmund, Trigonometric series, Vol. I, Cambridge Univ. Press, London (1959)
I. Singer, Bases in Banach spaces, Springer-Verlag, Berlin (1970)
J. R. Higgins, Completeness and basis properties of sets of special functions, Cambridge Univ. Press, London (1977)
N. Bary, Sur les systèmes complets des fonctions orthogonales, Mat. Sb. 14 (56), 51–108 (1944)
T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin (1976)
V. I. Smirnov, A course of higher mathematics, vol. V, Integration and functional analysis, Pergamon Press, Oxford (1964)
R. R. Goldberg, Methods of real analysis, Blaisdell, New York (1964)
K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N. J. (1962)
J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 5th ed., Colloq. Publ. vol. XX, Amer. Math. Soc., Providence, R. I. (1969)
D. F. Harazov, On the spectrum of completely continuous operators depending analytically on a parameter, in topological linear spaces (in Russian), Acta Sci. Math. (Szeged) 23, 38–45 (1962)
I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Transl. Math. Mono. 18, Amer. Math. Soc., Providence, R. I. (1969)
A. E. Taylor, Introduction to functional analysis, Wiley, New York (1957)
E. Hille and R. S. Phillips, Functional analysis and semi-groups, rev. ed., Colloq. Publ. vol. XXXI, Amer. Math. Soc., Providence, R. I. (1957)
F. Ursell, On the rolling motion of cylinders in the surface of a fluid, Quart. J. Mech. Appl. Math. 2, 335–353 (1949)
D. Euvrard, A. Jami, M. Lenoir, and D. Martin, Recent progress towards an optimal coupling between finite elements and singularity distribution procedures, Proc. 3rd Internat. Conf. Numerical Ship Hydrodynamics, Paris (1981)
M. J. Simon and F. Ursell, Uniqueness in linearized two-dimensional water-wave problems, J. Fluid Mech. 148, 137–154 (1984)
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Article copyright:
© Copyright 1987
American Mathematical Society
