Skip to main content
SpringerLink
Log in

Rayleigh Waves for Elliptic Systems in Domains with Periodic Boundaries

  • PARTIAL DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We consider formally self-adjoint elliptic systems of partial differential equations generating formally positive operators and having the polynomial property. Sufficient conditions that ensure the existence of Rayleigh surface waves in the Neumann problem on a half-space with periodic boundary are found. We give examples of specific problems of mathematical physics in which our sufficient conditions are simplified or turn into a criterion and study problems in the theory of plates and piezoelectrics that are not covered by general results. The latter problem requires a major modification of the approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The ellipticity of the operator \(\mathcal {L}(\nabla ) \) follows from the polynomial property (20); see [23, Ch. 5, Sec. 1].

  2. In this case, the boundary \(\partial \Pi \) is not Lipschitz; however, the prism \(\Pi \) itself is representable as a union of Lipschitz domains, and this property is sufficient for all reasoning below.

  3. Our constructions are also suitable for the general elliptic systems considered. Complex conjugation is not needed in problems of elasticity theory.

REFERENCES

  1. Nečas, J., Les méthodes en théorie des équations elliptiques, Paris: Masson, Prague: Academia, 1967.

  2. Lions, J.-L. and Magenes, E., Problèmes aux limites inhomogènes et applications, Paris: Dunod, 1968.

    MATH  Google Scholar 

  3. Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of Mathematical Physics), Moscow: Nauka, 1973.

  4. Mittra, R. and Lee, S.W., Analytical Techniques in the Theory of Guided Waves, New York: Macmillan, 1971.

    MATH  Google Scholar 

  5. Wilcox, C.H., Scattering theory for diffraction gratings, in Appl. Math. Sci. Ser., Singapore: Springer Sci. Bus. Media, vol. 46, 1997.

  6. Rayleigh, J.W.S., On waves propagated along the plane surface of an elastic solid, Proc. London Math. Soc., 1885, vol. 17, no. 253, pp. 4–11.

    Article  MathSciNet  Google Scholar 

  7. Lamb, H., On waves in an elastic plate, Proc. R. Soc., 1917, vol. A93, pp. 114–128.

    MATH  Google Scholar 

  8. Stoneley, R., Elastic waves at the surface of separation of two solids, Proc. R. Soc. London A, 1924, vol. 106, pp. 416–428.

    Article  Google Scholar 

  9. Viktorov, I.A., Zvukovye poverkhnostnye volny v tverdykh telakh (Sonic Surface Waves in Solids), Moscow: Nauka, 1981.

  10. Kaplunov, J.D., Kossovich, L.Y., and Nolde, E.V., Dynamics of Thin Walled Elastic Bodies, San Diego, CA: Academic Press, 1997.

    MATH  Google Scholar 

  11. Mikhasev, G.I. and Tovstik, P.E., Lokalizovannye kolebaniya i volny v tonkikh obolochkakh. Asimptoticheskie metody (Localized Vibrations and Waves in Thin Shells. Asymptotic Methods), Moscow: Nauka, 2009.

  12. Konenkov, Yu.K., On the “Rayleigh”-type flexural wave, Akust. Zh., 1960, vol. 6, pp. 124–126.

    MathSciNet  Google Scholar 

  13. Grinchenko, V.T. and Meleshko, V.V., Specific features of energy distribution in a thin rectangular plate under edge resonance, Dokl. Akad. Nauk USSR. Ser. A, 1976, no. 7, pp. 612–616.

  14. Kim, J.-Y. and Rokhlin, S.I., Surface acoustic wave measurements of small fatigue cracks initiated from a surface cavity, Int. J. Solids Struct., 2002, vol. 39, pp. 1487–1504.

    Article  Google Scholar 

  15. Zakharov, D.D. and Becker, W., Rayleigh type bending waves in anisotropic media, J. Sound Vib., 2003, vol. 261, pp. 805–818.

    Article  MathSciNet  Google Scholar 

  16. Kamotskii, I.V., Surface wave running along the edge of an elastic wedge, St. Petersburg Math. J., 2009, vol. 20, no. 1, pp. 59–63.

    Article  MathSciNet  Google Scholar 

  17. Kamotskii, I.V. and Kiselev, A.P., An energy approach to the proof of the existence of Rayleigh waves in an anisotropic elastic half-space, Prikl. Mat. Mekh., 2009, vol. 73, no. 4, pp. 645–654.

    MathSciNet  Google Scholar 

  18. Zavorokhin, G.L. and Nazarov, A.I., On elastic waves in a wedge, J. Math. Sci., 2011, vol. 175, no. 6, pp. 646–650.

    Article  MathSciNet  Google Scholar 

  19. Krushynska, A.A., Flexural edge waves in semi-infinite elastic plates, J. Sound Vib., 2011, vol. 330, pp. 1964–1976.

    Article  Google Scholar 

  20. Nazarov, A., Nazarov, S., and Zavorokhin, G., On symmetric wedge mode of an elastic solid, Eur. J. Appl. Math., 2021, vol. 33, no. 2, pp. 201–223.

    Article  MathSciNet  Google Scholar 

  21. Lawrie, J. and Kaplunov, J., Edge waves and resonance on elastic structures: an overview, Math. Mech. Solids, 2012, vol. 17, no. 1, pp. 4–16.

    Article  MathSciNet  Google Scholar 

  22. Nazarov, S.A. and Plamenevsky, B.A., Elliptic Problems in Domains with Piecewise Smooth Boundaries, Berlin–New York: De Gruyter, 1994.

    Book  Google Scholar 

  23. Nazarov, S.A., The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes, Russ. Math. Surv., 1999, vol. 54, no. 5, pp. 947–1014.

    Article  Google Scholar 

  24. Nazarov, S.A., Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators, J. Math. Sci., 1998, vol. 92, no. 6, pp. 4338-4353.

  25. Nazarov, S.A., Non-self-adjoint elliptic problems with a polynomial property in domains possessing cylindrical outlets to infinity, J. Math. Sci., 2000, vol. 101, no. 5, pp. 3512–3522.

    Article  MathSciNet  Google Scholar 

  26. Agranovich, M.S. and Vishik, M.I., Elliptic problems with a parameter and parabolic problems of general type, Russ. Math. Surv., 1964, vol. 19, no. 3, pp. 53–157.

    Article  MathSciNet  Google Scholar 

  27. Kondrat’ev, V.A., Boundary problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 1967, vol. 16, pp. 227–313.

    MathSciNet  MATH  Google Scholar 

  28. Agmon, S., Douglis, A., and Nirenberg, L., Estimates near the boundary for solutions of elliptic differential equations satisfying general boundary conditions. 2, Commun. Pure Appl. Math., 1964, vol. 17, pp. 35–92.

    Article  MathSciNet  Google Scholar 

  29. Solonnikov, V.A., General boundary value problems for Douglis–Nirenberg elliptic systems. II, Proc. Steklov Inst. Mat., 1968, vol. 92, pp. 269–339.

    Google Scholar 

  30. Gohberg, I.C. and Krein, M.G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., Providence, R.I.: Am. Math. Soc., 1969, vol. 18.

  31. Birman, M.Sh. and Solomyak, M.Z., Spectral Theory of Self-Adjoint Operators in Hilbert Space, Math. Appl. (Sov. Ser.), Dordrecht: Reidel, 1987, vol. 5.

  32. Nazarov, S.A., Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides, Izv. Math., 2020, vol. 84, no. 6, pp. 1105–1160.

    Article  MathSciNet  Google Scholar 

  33. Kamotskii, I.V. and Nazarov, S.A., Exponentially decreasing solutions of diffraction problems on a rigid periodic boundary, Math. Notes, 2003, vol. 73, no. 1, pp. 129–131.

    Article  MathSciNet  Google Scholar 

  34. Lekhnitskii, S.G., Teoriya uprugosti anizotropnogo tela (Theory of Elasticity of an Anisotropic Body), Moscow: Nauka, 1977.

  35. Bertram, A., Elasticity and Plasticity of Large Deformations, Berlin–Heidelberg: Springer, 2005.

    MATH  Google Scholar 

  36. Nazarov, S.A., Asimptoticheskaya teoriya tonkikh plastin i sterzhnei. Ponizhenie razmernosti i integral’nye otsenki (Asymptotic Theory of Thin Plates and Rods. Dimension Reduction and Integral Estimates), Novosibirsk: Nauchn. Kniga, 2002.

  37. Langer, S., Nazarov, S.A., and Specovius-Neugebauer, M., Affine transformations of three-dimensional anisotropic media and explicit formulas for fundamental matrices, J. Appl. Mech. Techn. Phys., 2006, vol. 47, no. 2, pp. 229–235.

    Article  MathSciNet  Google Scholar 

  38. Kamotskii, I.V. and Nazarov, S.A., Elastic waves localized near periodic families of defects, Dokl. Ross. Akad. Nauk, 1999, vol. 368, no. 6, pp. 771–773.

    MathSciNet  MATH  Google Scholar 

  39. Shoikhet, B.A., On asymptotically exact equations for thin plates of complex structure, Prikl. Mat. Mekh., 1973, vol. 37, no. 5, pp. 914–924.

    MathSciNet  Google Scholar 

  40. Ciarlet, P.G., Mathematical Elasticity, II: Theory of Plates, Studies in Mathematics and Its Applications, Amsterdam, vol. 27, 1997.

  41. Dauge, M., Djurdjevic, I., Faou, E., and Rössle, A., Eigenmode asymptotics in thin elastic plates, J. Math. Pures Appl., 1999, vol. 78, no. 9, pp. 925–964.

  42. Nazarov, S.A., On the asymptotics of the spectrum of a thin plate problem of elasticity, Sib. Math. J., 2000, vol. 41, no. 4, pp. 744–759.

    Article  Google Scholar 

  43. Dauge, M. and Yosibash, Z., Eigen-frequencies in thin elastic 3-D domains and Reissner–Mindlin plate models, Math. Meth. Appl. Sci., 2002, vol. 25, no. 1, pp. 21–48.

    Article  MathSciNet  Google Scholar 

  44. Mikhlin, S.G., Variatsionnye metody v matematicheskoi fizike (Variational Methods in Mathematical Physics), Moscow: Nauka, 1970.

  45. Parton, V.Z. and Kudryavtsev, B.A., Elektromagnitouprugost’ p’ezoelektricheskikh i elektroprovodyashchikh sred (Electromagnetoelasticity of Piezoelectric and Electrically Conductive Media), Moscow: Nauka, 1988.

  46. Tiersten, H.F., Linear Piezoelectric Plate Vibrations, New York: Springer, 1964.

    Google Scholar 

  47. Suo, Z., Kuo, C.-M., Barnett, D.M., and Willis, J.R., Fracture mechanics for piezoelectric ceramics, J. Mech. Phys. Solids, 1992, vol. 40, no. 4, pp. 739–765.

    Article  MathSciNet  Google Scholar 

  48. Nazarov, S.A., Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate, J. Math. Sci., 2003, vol. 114, no. 5, pp. 1657–1726.

    Article  MathSciNet  Google Scholar 

  49. Nazarov, S.A., Ruotsalainen, K.R., and Silvola, M., Trapped modes in piezoelectric and elastic waveguides, J. Elasticity, 2016, vol. 124, no. 2, pp. 193–223.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, 199178, Russia

    S. A. Nazarov

Authors
  1. S. A. Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Nazarov.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nazarov, S.A. Rayleigh Waves for Elliptic Systems in Domains with Periodic Boundaries. Diff Equat 58, 631–648 (2022). https://doi.org/10.1134/S0012266122050044

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266122050044

Search

Navigation