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Existence of edge waves along three-dimensional periodic structures

Published online by Cambridge University Press:  06 July 2010

SERGEY A. NAZAROV  and
JUHA H. VIDEMAN
SERGEY A. NAZAROV
Affiliation:
Institute of Mechanical Engineering Problems, Russian Academy of Sciences, VO, Bol'shoi pr., 61, 199178 St. Petersburg, Russia
JUHA H. VIDEMAN*
Affiliation:
CEMAT/Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal
*
†Email address for correspondence: [email protected]
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Abstract

Existence of edge waves travelling along three-dimensional periodic structures is considered within the linear water-wave theory. A condition ensuring the existence is derived by analysing the spectrum of a suitably defined trace operator. The sufficient condition is a simple inequality comparing a weighted volume integral, taken over the submerged part of an element in the infinite array of identical obstacles, to the area of the free surface pierced by the obstacle. Various examples are given, and the results are extended to edge waves along periodic coastlines and over a periodically varying ocean floor.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 659 , 25 September 2010 , pp. 225 - 246
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Birman, M. S. & Solomjak, M. Z. 1987 Spectral Theory of Self-Adjoint Operators in Hilbert Space. Reidel.CrossRefGoogle Scholar
Bonnet-Ben Dhia, A.-S. & Joly, P. 1993 Mathematical analysis of guided water waves. SIAM J. Appl. Math. 53, 1507–1550.CrossRefGoogle Scholar
Evans, D. V. & Fernyhough, M. 1995 Edge waves along periodic coastline. Part 2. J. Fluid Mech. 297, 307–325.CrossRefGoogle Scholar
Evans, D. V., Levitin, M. & Vassilev, D. 1994 Existence theorems for trapped modes. J. Fluid Mech. 261, 21–31.CrossRefGoogle Scholar
Evans, D. V. & Linton, C. M. 1993 Edge waves along periodic coastline. Q. J. Mech. Appl. Maths. 46, 644–656.CrossRefGoogle Scholar
Garipov, R. M. 1967 On the linear theory of gravity waves: the theorem of existence and uniqueness. Arch. Ration. Mech. Anal. 24, 352–362.CrossRefGoogle Scholar
John, F. 1950 On the motion of floating bodies. Part II. Commun. Pure Appl. Math. Anal. 3, 45–101.CrossRefGoogle Scholar
Jones, D. S. 1953 The eigenvalues of ∇2u + λu = 0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684.CrossRefGoogle Scholar
Kamotskii, I. V. & Nazarov, S. A. 1999 Elastic waves localized near periodic sets of flaws. Dokl. Ross. Akad. Nauk. 368, 771–773 (translation in Dokl. Phys. 44, 715–717).Google Scholar
Kamotskii, I. V. & Nazarov, S. A. 2003 Exponentially decreasing solutions of the problem of diffraction by a rigid periodic boundary. Mat. Zametki 73, 138–140 (translation in Math. Notes 73, 129–131).Google Scholar
Kondratiev, V. 1967 Boundary value problems for elliptic equations in domains with conical or angular points. Trans. Moscow Math. Soc. 16, 227–313.Google Scholar
Kundu, P. K. & Cohen, I. M. 2004 Fluid Mechanics, 3rd edn. Elsevier Academic.Google Scholar
Kuznetsov, N., Maz'ya, V. & Vainberg, B. 2002 Linear Water Waves: A Mathematical Approach. Cambridge University Press.CrossRefGoogle Scholar
Ladyzhenskaya, O. 1985 The Boundary Value Problems of Mathematical Physics. Springer.CrossRefGoogle Scholar
Linton, C. M. & McIver, M. 2002 The existence of Rayleigh–Bloch surface waves. J. Fluid Mech. 470, 85–90.CrossRefGoogle Scholar
Linton, C. M. & McIver, P. 2007 Embedded trapped modes in water waves and acoustics. Wave Motion 45, 16–29.CrossRefGoogle Scholar
McIver, P., Linton, C. M & McIver, M. 1998 Construction of trapped modes for wave guides and diffraction gratings. Proc. R. Soc. Lond. A 454, 2593–2616.CrossRefGoogle Scholar
Nazarov, S. A. 2008 Concentration of trapped modes in problems of the linearized theory of water waves. Mat. Sbornik 199, 53–78 (translation in Sbornik: Math. 199, 1783–1807).Google Scholar
Nazarov, S. A. 2009 a A novel approach for detecting trapped surface waves in a canal with periodic underwater topography. C. R. Mec. 337, 610–615.Google Scholar
Nazarov, S. A. 2009 b A simple method for finding trapped modes in problems of the linear theory of surface waves. Dokl. Ross. Akad. Nauk. 429, 1–4 (translation in Dokl. Math. 80, 1–4).Google Scholar
Nazarov, S. A. 2009 c Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains. In Sobolev Spaces in Mathematics II: Applications in Analysis and Partial Differential Equations (ed. Maz'ya, V.), International Mathematical Series, vol. 9, pp. 261–309. Springer.CrossRefGoogle Scholar
Nazarov, S. A. 2009 d Sufficient conditions for the existence of trapped modes in problems of the linear theory of surface waves. Zap. Nauchn. Semin. St.-Petersburg Otdel. Mat. Inst. Steklov 369, 202–223 (translation in J. Math. Sci., in press).Google Scholar
Nazarov, S. A. & Plamenevsky, B. A. 1994 Elliptic Problems in Domains With Piecewise Smooth Boundaries. Walter de Gruyter.CrossRefGoogle Scholar
Nazarov, S. A. & Taskinen, J. 2008 On the spectrum of the Steklov problem in a domain with a peak. Vestnik St.-Petersburg Univ. 1, 56–65 (translation in Vestnik St.-Petersburg Univ. Math. 41, 45–52).Google Scholar
Nazarov, S. A. & Taskinen, J. 2010 On essential and continuous spectra of the linearized water-wave problem in a finite pond. Math. Scand. 106, 141–160.CrossRefGoogle Scholar
Nazarov, S. A. & Videman, J. H. 2009 A sufficient condition for the existence of trapped modes for oblique waves in a two-layer fluid. Proc. R. Soc. Lond. A 465, 3799–3816.Google Scholar
Porter, R. & Evans, D. V. 1999 Rayleigh–Bloch surface waves along periodic gratings and their connection with trapped modes in waveguides. J. Fluid Mech. 386, 233–258.CrossRefGoogle Scholar
Porter, R. & Evans, D. V. 2005 Embedded Rayleigh–Bloch surface waves along periodic rectangular arrays. Wave Motion 43, 29–50.CrossRefGoogle Scholar
Porter, R. & Porter, D. 2001 Interaction of water waves with three-dimensional topography. J. Fluid Mech. 434, 301–335.CrossRefGoogle Scholar
Roseau, M. 1958 Short waves parallel to the shore over a sloping beach. Commun. Pure Appl. Math. 11, 433–493.CrossRefGoogle Scholar
Stokes, G. G. 1846 Report on recent researches in hydrodynamics. In 16th Meeting of the British Association for the Advancement of Science, pp. 1–20. Murray. Reprinted in Mathematical and Physical Papers, vol. 1, pp. 157–187 (1880). Cambridge University Press.Google Scholar
Sukhinin, S. V. 1998 Waveguide, anomalous, and whispering properties of a periodic chain of obstacles. Sib. Zh. Ind. Mat. 1, 175–198 (in Russian).Google Scholar
Ursell, F. 1951 Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47, 347–358.CrossRefGoogle Scholar
Ursell, F. 1952 Edge waves over a sloping beach. Proc. R. Soc. Lond. A 214, 79–97.Google Scholar
Ursell, F. 1987 Mathematical aspects of trapping modes in the theory of surface waves. J. Fluid Mech. 183, 421–437.CrossRefGoogle Scholar
Wilcox, C. H. 1984 Scattering Theory for Diffraction Gratings. Springer.CrossRefGoogle Scholar