Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

On the solvability of a two-dimensional wave-body interaction problem


Authors: G. A. Athanassoulis and C. G. Politis
Journal: Quart. Appl. Math. 48 (1990), 1-30
MSC: Primary 35Q35; Secondary 31A35, 76B15
DOI: https://doi.org/10.1090/qam/1040231
MathSciNet review: MR1040231
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Abstract: The two-dimensional, deep-water, wave-body interaction problem for a single-hulled body, floating on the free surface of an ideal liquid, is considered. The body boundary may be nonsmooth and may intersect the free surface at arbitrary angles. The existence of a unique solution representable by a multipole-series expansion is proved for all but a discrete set of oscillation frequencies. The proof is based on the property of the associated multipoles to be a basis of the space $ {L^P}\left( { - \pi , 0} \right)$, $ 1 < p \le 2$. Strict estimates of the form $ {D_n} = O\left( {n^{ - \alpha }} \right)$ are also obtained for the coefficients of the multipole-series expansion for piecewise smooth $ \left( {0 < \alpha < 2} \right)$ and smooth $ \left( {\alpha = 2} \right)$ body boundaries.


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  • [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972.
  • [2] G. A. Athanassoulis, An expansion theorem for water-wave potentials, J. Engrg. Math. 18, 181-194 (1984) MR 757802
  • [3] G. A. Athanassoulis, On the solvability of a two-dimensional water-wave radiation problem, Quart. Appl. Math. 44, 601-620 (1987) MR 872813
  • [4] G. A. Athanassuolis, P. D. Kaklis, and C. G. Politis, The limiting values of added masses of a partially submerged cylinder of arbitrary shape, J. Ship Research 32 1, 1-18 (1987)
  • [5] G. A. Athanassoulis, P. D. Kaklis, and C. G. Politis, Low-frequency oscillations of a partially submerged cylinder of arbitrary shape, National Technical University of Athens, Dept. of Naval Arch. and Marine Engrg., Report No. D1-88 (1988)
  • [6] G. A. Athanssoulis and C. G. Politis, On a radiation problem for two-dimensional floating bodies of arbitrary shape with corners, Scientific and Methodological Seminar on Ship Hydrodynamics, Varna, Bulgaria (1985)
  • [7] G. A. Athanassoulis and C. G. Politis, On the solvability of a two-dimensional wave-body interaction problem, National Technical University of Athens, Dept. of Naval Arch. and Marine Engrg., Report No. 2 (1987)
  • [8] N. K. Barry, A Treatise on Trigonometric Series, vol. I, Pergamon Press, 1964
  • [9] J. T. Beale, Eigenfunctions expansions for objects floating in an open sea, Comm. Pure Appl. Math. 30, 283-313 (1977) MR 0670432
  • [10] M. B. Count, On the hydrodynamic behaviour of ocean wave energy absorbing structures--A theoretical treatment, Central Electricity Generating Board, England, Report R/M/N960 (1977)
  • [11] K. Doppel and G. C. Hsiao, On weak solutions of the floating body problem, University of Delaware, Dept. of Math. Sciences, Center for the Math. of Waves, Report No. 87-17 (1987)
  • [12] P. L. Duren, Theory of $ {H^p}$-spaces, Academic Press, 1970 MR 0268655
  • [13] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Math. Monographs, vol. 18, Amer. Math. Soc., 1969 MR 0246142
  • [14] G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Transl. Math. Monographs, vol. 26, Amer. Math. Soc., 1969 MR 0247039
  • [15] P. Guevel and J. M. Kobus, Flotteurs cylindriques horizontaux soumis á des oscillations forcées de très faibles amplitudes, Bulletin d' Association Technique Maritime et Aeronautique 75, 183-204 (1975)
  • [16] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1965
  • [17] D. F. Harazov, On the spectrum of completely continuous operators depending analytically on a parameter, in topological linear spaces, (Russian), Acta Sci. Math. (Szeged) 23, 38-45 (1962) MR 0139950
  • [18] F. John, On the motion of floating bodies. II: Simple harmonic motions, Comm. Pure Appl. Math. 3, 45-100 (1950) MR 0037118
  • [19] P. D. Kaklis and G. A. Athanassoulis, The low-frequency heaving motion of a cylinder on the free surface of a liquid, SIAM J. Appl. Math. 49 2, 1-17 (1987) MR 988610
  • [20] Y. Katznelson, An Introduction to Harmonic Analysis, Dover, 1968 MR 0248482
  • [21] R. E. Kleinman, On the mathematical theory of the motion of floating bodies--an update, David Taylor Naval Ship Research and Development Center, Report 82/074 (1982)
  • [22] N. G. Kuznetsov and V. G. Maz'ya, Problem concerning steady state oscillations of a layer of fluid in the presence of an obstacle, Soviet Phys. Dokl. 19, 341-343 (1974)
  • [23] M. Lavrentiev and B. Chabat, Effet hydrodynamique et modèles mathémtiques, Editions MIR, Moscou, 1980
  • [24] M. Lenoir, Une application du principe d'absorption limite au problem bidimensionnel du mouvement sur la houle, in Mèthodes de couplage en hydrodynamique navale et application à la résistance de vagues bidimensionnele, École Nationale Supérieure de Techniques Avancées, Rapport de Recherche 164 (1982), pp. III.1-III.29
  • [25] M. Lenoir and D. Martin, An application of the principle of limiting absorption to the motion of floating bodies, J. Math. Anal. Appl. 79, 370-383 (1981) MR 606488
  • [26] F. D. Lesley, Personal communication, 1985
  • [27] A. I. Markushevich, Theory of Functions of a Complex Variable, Chelsea Publ. Co., 1977 MR 0444912
  • [28] C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975 MR 0507768
  • [29] C. Pommerenke, On univalent functions, Bloch functions and VMOA, Math. Ann. 236, 199-208 (1978) MR 0492206
  • [30] C. Pommerenke, Personal communication, 1985
  • [31] F. Reisz and B. Sz. -Nagy, Functional Analysis, Frederick Ungar Publ. Co., 1955 MR 0071727
  • [32] G. E. Schober, Personal communication, 1985
  • [33] M. J. Simon and F. Ursell, Uniqueness in linearized two-dimensional water-wave problems, J. Fluid Mech. 148, 137-154 (1984) MR 778139
  • [34] I. Singer, Bases in Banach spaces I, Springer-Verlag, 1970 MR 0298399
  • [35] S. Steinberg, Meromorphic families of compact operators, Arch. Rational Mech. Anal. 31, 372-380 (1968) MR 0233240
  • [36] A. E. Taylor, Introduction to Functional Analysis. John Wiley, 1958 MR 0098966
  • [37] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen Publ. Co., Tokyo, 1959 MR 0114894
  • [38] F. Ursell, On the heaving motion of a circular cylinder on the surface of a fluid, Quart. J. Mech. Appl. Math. II 2, 218-231 (1949) MR 0030866
  • [39] F. Ursell, On the rolling motion of cylinders in the surface of a fluid, Quart. J. Mech. Appl. Math. II 3, 335-353 (1949) MR 0032330
  • [40] F. Ursell, Short surface waves due to an oscillating immersed body, Proc. Roy. Soc. London Ser. A 220, 90-103 (1953) MR 0059694
  • [41] F. Ursell, The decay of the free motion of a floating body, J. Fluid Mech. 19, 305-319 (1964) MR 0163538
  • [42] F. Ursell, A problem in the theory of water waves, Numerical Solution of Integral Equations, edited by L. M. Delves and J. Walsh, Clarendon Press, 1974 MR 0479017
  • [43] M. Vullierme-Ledard, Vibrations engendrées par les oscillations forcées d'un corps rigide immergé dans un fluide incompressible présentant une surface libre, C. R. Acad. Sci. Paris A 296, 611-614 (1983) MR 705174
  • [44] S. E. Warschawski, On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc. 12, 614-620 (1961) MR 0131524
  • [45] S. E. Warschawski and G. E. Schober, On conformal mapping of certain classes of Jordan domains, Arch. Rational Mech. Anal. 22, 201-209 (1966) MR 0192039
  • [46] J. V. Wehausen and E. V. Laitone, Surface waves, Encyclopedia of Physics, vol. IX, Springer-Verlag, 1960, pp. 446-778 MR 0119656
  • [47] J. V. Wehausen, Methods for boundary-value problems in free surface flows, David Taylor Naval Ship Research and Development Center, Report 4622 (1974)

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DOI: https://doi.org/10.1090/qam/1040231
Article copyright: © Copyright 1990 American Mathematical Society

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