We consider the linear problem of the steady oscillations of an ideal unbounded fluid with a free surface in the presence of a system of submerged horizontal cylindrical bodies of arbitrary cross section. We propose a criterion of the unique solvability of the problem and, based on this criterion, develop an algorithm for finding trapped unforced modes of fluid oscillations (solutions to the homogeneous boundary value problem) for bodies of a given arbitrary shape. The algorithm is applicable to the oscillations frequencies located on the continuous spectrum, as well as outside it. We also discuss the numerical realization and reliability of the obtained numerical results. As an illustration, we give examples of numerical experiments which are compared with known results. Bibliography: 28 titles. Illustrations: 5 figures.
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References
N. G. Kuznetsov, V. G. Maz’ya, and B. R. Vainberg, Linear Water Waves: a Mathematical Approach, Cambridge Univ. Press, Cambridge (2002).
O. V. Motygin, “On unique solvability in the problem of water waves above submerged bodies” [in Russian], Zap. Nauchn. Semin. POMI 369, 143–163 (2009); English transl.: J. Math. Sci., New York 167, No. 5, 680–691 (2010).
O. V. Motygin and P. McIver, “On uniqueness in the problem of gravity-capillary water waves above submerged bodies” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465, No. 2106, 1743–1761 (2009).
F. Ursell, “On head seas travelling along a horizontal cylinder,” J. Inst. Math. Appl. 4, 414–427 (1968).
V. G. Maz’ya, “Boundary integral equations,” In: Encyclopaedia of Math. Sciences, Analysis IV 27, pp. 127–222, Springer, Berlin (1991).
F. Ursell, “The expansion of water-wave potentials at great distances,” Proc. Camb. Philos. Soc. 64, No. 3, 811–826 (1968).
O. V. Motygin, “On trapping of surface water waves by cylindrical bodies in a channel,” Wave Motion 45, 940–951 (2008).
D. A. Indeitsev, N. G. Kuznetsov, O. V. Motygin, and Yu. A. Mochalova, Localization of Linear Waves, St. Petersb. Univ. State, St. Petersb. (2007).
C. M. Linton and P. McIver, “Embedded trapped modes in water waves and acoustics,” Wave Motion, 45, No. 1–2, 16–29 (2007).
P. J. Cobelli, V. Pagneux, A. Maurel, and P. Petitjeans, “Experimental study on water-wave trapped modes,” J. Fluid Mech. 666, 445–476 (2011).
F. Ursell, “Trapping modes in the theory of surface waves,” Proc. Camb. Philos. Soc. 47, No. 2, 347–358 (1951).
D. S. Jones, “The eigenvalues of ∇2 u + λu = 0 when the boundary conditions are given on semi-infinite domains,” Proc. Camb. Philos. Soc. 49, No. 4, 668–684 (1953).
F. Ursell, “Mathematical aspects of trapping modes in the theory of surface waves,” J. Fluid Mech. 183, 421–437 (1987).
S. A. Nazarov, “Simple method for finding trapped modes in problems of the linear theory of surface waves,” Dokl. Akad. Nauk, Ross. Akad. Nauk 429, No. 6, 746–749 (2009); English transl.: Dokl. Math. 80, No. 3, 914–917 (2009).
P. McIver and D. V. Evans, “The trapping of surface waves above a submerged, horizontal cylinder,” J. Fluid Mech. 151, 243–255 (1985).
R. Porter and D. V. Evans, “The trapping of surface waves by multiple submerged horizontal cylinders,” J. Eng. Math. 34, 417–433 (1998).
D. V. Evans and R. Porter, “Trapped modes about multiple cylinders in a channel,” J. Fluid Mech. 339, 331–356 (1997).
M. J. Simon, “On a bound for the frequency of surface waves trapped near a cylinder spanning a channel,” Theor. Comput. Fluid Dyn. 4, No. 2, 71–78 (1992).
O. V. Motygin, “On frequency bounds for modes trapped near a channel-spanning cylinder,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 456, No. 2004, 2911–2930 (2000).
O. V. Motygin, “Estimates of the trapped-mode frequencies of oscillation of a liquid in the presence of submerged bodies” [in Russian], Prikl. Mat. Mekh. 69, No. 5, 818–828 (2005); English transl.: J. Appl. Math. Mech. 69, 733–742 (2005).
M. McIver, “Trapped modes supported by submerged obstacles,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 456, No. 2000, 1851–1860 (2000).
R. Porter, “Trapping of water waves by pairs of submerged cylinders,” Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 458, No. 2019, 607–624 (2002).
M. Abramowitz and I. Stegun (Eds.) Handbook of Mathematical Functions, Dover, New York (1965).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
V. P. Trofimov, “The root subspaces of operators depending analytically on a parameter” [in Russian], Mat. Issled. 3, No. 9, 117–125 (1968).
M. S. Birman and M. Z. Solomyak, Spectral Theory of Selfadjoint Operators in Hilbert Space [in Russian], Leningr. Univ. Press, Leningrad (1980); English transl.: Reidel, Dordrecht (1987).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1976).
G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer, Berlin (1993).
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Translated from Problems in Mathematical Analysis 55, March 2011, pp. 65–80.
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Motygin, O.V. Trapped modes in a linear problem of the theory of surface water waves. J Math Sci 173, 717–736 (2011). https://doi.org/10.1007/s10958-011-0269-y
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DOI: https://doi.org/10.1007/s10958-011-0269-y