Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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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  • [1] G. A. Athanassoulis, An expansion theorem for water-wave potentials, Journal of Engineering Mathematics 18, 181-194 (1984) MR 757802
  • [2] F. John, On the motion of floating bodies, II, Comm. Pure Appl. Math. 3, 45-101 (1950) MR 0032328
  • [3] J. T. Beale, Eigenfunction expansions for objects floating in an open sea, Comm. Pure Appl. Math. 30, 283-313 (1977) MR 0670432
  • [4] 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) MR 606488
  • [5] 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)
  • [6] 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)
  • [7] F. Ursell, Surface waves on deep water in the presence of a submerged circular cylinder, II, Proc. Camb. Philos. Soc. 46, 153-158 (1950) MR 0033403
  • [8] F. Ursell, The expansion of water-wave potentials at great distances, Proc. Camb. Philos. Soc. 64, 811-826 (1968)
  • [9] F. Ursell, Short surface waves due to an oscillating immersed body, Proc. Roy. Soc. London Ser. A 220, 90-103 (1953) MR 0059694
  • [10] F. Ursell, A problem in the theory of water waves, in Numerical solution of integral equations, Clarendon Press, Oxford (1974) MR 0479017
  • [11] 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)
  • [12] J. J. Stoker, Water waves. The mathematical theory with applications, Interscience, New York (1957) MR 0103672
  • [13] A. Zygmund, Trigonometric series, Vol. I, Cambridge Univ. Press, London (1959) MR 0107776
  • [14] I. Singer, Bases in Banach spaces, Springer-Verlag, Berlin (1970) MR 0298399
  • [15] J. R. Higgins, Completeness and basis properties of sets of special functions, Cambridge Univ. Press, London (1977) MR 0499341
  • [16] N. Bary, Sur les systèmes complets des fonctions orthogonales, Mat. Sb. 14 (56), 51-108 (1944) MR 0012161
  • [17] T. Kato, Perturbation theory for linear operators, Springer-Verlag, Berlin (1976) MR 0407617
  • [18] V. I. Smirnov, A course of higher mathematics, vol. V, Integration and functional analysis, Pergamon Press, Oxford (1964) MR 0168707
  • [19] R. R. Goldberg, Methods of real analysis, Blaisdell, New York (1964)
  • [20] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs, N. J. (1962) MR 0133008
  • [21] 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) MR 0218588
  • [22] 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) MR 0139950
  • [23] 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) MR 0246142
  • [24] A. E. Taylor, Introduction to functional analysis, Wiley, New York (1957) MR 0098966
  • [25] E. Hille and R. S. Phillips, Functional analysis and semi-groups, rev. ed., Colloq. Publ. vol. XXXI, Amer. Math. Soc., Providence, R. I. (1957) MR 0089373
  • [26] F. Ursell, On the rolling motion of cylinders in the surface of a fluid, Quart. J. Mech. Appl. Math. 2, 335-353 (1949) MR 0032330
  • [27] 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)
  • [28] M. J. Simon and F. Ursell, Uniqueness in linearized two-dimensional water-wave problems, J. Fluid Mech. 148, 137-154 (1984) MR 778139

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DOI: https://doi.org/10.1090/qam/872813
Article copyright: © Copyright 1987 American Mathematical Society

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