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On symmetric wedge mode of an elastic solid

Part of: Waves Equations of mathematical physics and other areas of application Elastic materials

Published online by Cambridge University Press:  11 January 2021

ALEXANDER NAZAROV Open the ORCID record for ALEXANDER NAZAROV [Opens in a new window] ,
SERGEY NAZAROV  and
GERMAN ZAVOROKHIN
ALEXANDER NAZAROV
Affiliation:
Department of Mathematics and Mechanics, St.Petersburg State University, Universitetskii Prospect, 28, 198504, St.Petersburg, Russia, emails: al.il.nazarov@gmail.com; srgnazarov@yahoo.co.uk St.Petersburg Department of the Steklov Mathematical Institute, Fontanka, 27, 191023, St.Petersburg, Russia, email: zavorokhin@pdmi.ras.ru
SERGEY NAZAROV
Affiliation:
Department of Mathematics and Mechanics, St.Petersburg State University, Universitetskii Prospect, 28, 198504, St.Petersburg, Russia, emails: al.il.nazarov@gmail.com; srgnazarov@yahoo.co.uk Institute for Problems in Mechanical Engineering of RAS, Bol’shoy Prospekt V.O., 61, St. Petersburg, 199178, Russia
GERMAN ZAVOROKHIN
Affiliation:
St.Petersburg Department of the Steklov Mathematical Institute, Fontanka, 27, 191023, St.Petersburg, Russia, email: zavorokhin@pdmi.ras.ru
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Abstract

The existence of a symmetric mode in an elastic solid wedge for all admissible values of the Poisson ratio and arbitrary interior angles close to π has been proven.

Keywords

Wedge wavesguided acoustic waveslocalised solutionsdiscrete spectrum

MSC classification

Primary: 35Q74: PDEs in connection with mechanics of deformable solids
Secondary: 74B05: Classical linear elasticity 74J99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 2 , April 2022 , pp. 201 - 223
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Dedicated to Professor V.M. Babich on the occasion of his 90th anniversary, with admiration

References

Agmon, S. & Nirenberg, L. (1963) Properties of solutions of ordinary differential equations in Banach space. Comm. Pure. Appl. Math. 16, 121–239.CrossRefGoogle Scholar
Amrouche, C. & Girault, V. (1994) Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czechoslovak Math. J. 44, N1, 109–140.CrossRefGoogle Scholar
Babich, V. M. (2010) A class of topographical waveguides. Algebra Anal. 22, N1, 98–107 [in Russian]; English transl.: St. Petersburg Math. J. 22 (2011), N1, 73–79.Google Scholar
Birman, M. S. & Solomjak, M. Z. (2010) Spectral Theory of Self-Adjoint Operators in Hilbert Space, 2nd ed., revised and extended. Lan’, St. Petersburg, 2010 [in Russian]; English transl. of the 1st ed.: Mathematics and Its Applications. Soviet Series, vol. 5, Kluwer, Dordrecht etc. 1987.Google Scholar
Biryukov, S. V., Gulyaev, Yu. V., Krylov, V. V. & Plessky, V. P. (1995) Surface Acoustic Waves in Inhomogeneous Media, Springer-Verlag, Berlin – Heidelberg – New York.CrossRefGoogle Scholar
Bonnet-Ben Dhia, A. S., Duterte, J. & Joly, P. (1999) Mathematical analysis of elastic surface waves in topographic waveguides. MMMAS 9, 755–798.Google Scholar
Duvaut, G. & Lions, J.-L. (1972) Les Inéquations en Mécanique et en Physique, Dunod, Paris.Google Scholar
Glazman, I. M. (1963) Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, Fizmatlit, Moscow [in Russian]; English transl.: Israel Program for Scientific Translations, Jerusalem, 1965.Google Scholar
Gohberg, I. C. & Krein, M. G. (1965) Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Fizmatlit, Moscow [in Russian]; English transl.: Translations of mathematical monographs, AMS, 1969.Google Scholar
Il’in, A. M. (1989) Matching of Asymptotic Expansions of Solutions to Boundary Value Problems, Nauka, Moscow [in Russian].Google Scholar
Kamotski, I. V. (2008) Surface wave running along the edge of an elastic wedge. Algebra Anal. 20, N1, 86–92 [in Russian]; English transl.: St. Petersburg Math. J. 20 (2009), N1, 59–63.Google Scholar
Kiselev, A. P. (2004) Rayleigh wave with a transverse structure. Proc. R. Soc. Lond. A 460, 3059–3064.CrossRefGoogle Scholar
Kondrat’ev, V. A. (1967) Boundary problems for elliptic equations in domains with conical or angular points. Trans. Mosc. Math. Soc. 16, 227–313.Google Scholar
Kozlov, V. A. & Maz’ya, V. G. (1988) Spectral properties of the operator bundles generated by elliptic boundary-value problems in a cone. Funct. Anal. Appl. 22, N2, 114–121.CrossRefGoogle Scholar
Krylov, V. V. (1989) Conditions for validity of the geometrical-acoustic approximation in application to waves in an acute-angle solid wedge. Sov. Phys. Acoust. 35, 176–180.Google Scholar
Krylov, V. V. (1990) Geometrical-acoustics approach to the description of localized vibrational modes of an elastic wedge. Sov. Phys. Tech. Phys. 35, 137–140.Google Scholar
Krylov, V. V. (1991) On the existence of a symmetric acoustic mode in a quadratic solid wedge. Moscow Univ. Phys. Bull. 46, N1, 45–49.Google Scholar
Lagasse, P. E. (1972) Analysis of a dispersion-free guide for elastic waves. Electron. Lett. 8, N15, 372–373.CrossRefGoogle Scholar
Lagasse, P. E. (1973) Higher-order finite element analysis of topographic guides supporting elastic surface waves. J. Acoust. Soc. Am. 53, 1116–1122.CrossRefGoogle Scholar
Lagasse, P. E., Mason, I. M. & Ash, E. A. (1973) Acoustic surface waveguides analysis and assessment. IEEE Trans. Microwave Theory Tech. 21, 225–236.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. (2012) Course of Theoretical Physics, 3rd ed., Vol. 7. Theory of Elasticity, Elsevier.Google Scholar
Lord, Rayleigh (J.W. Strutt) (1885) On waves propagating along the plane surface of an elastic solid. Proc. London Math. Soc. 17, 4–11.Google Scholar
Maz’ya, V. G., Nazarov, S. A. & Plamenevskii, B. A. (2000) Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vols. 1–2, Birkhäuser, Basel.Google Scholar
Maz’ya, V. G. & Plamenevskii, B. A. (1977) On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr. 76, N1, 29–60 [in Russian]; English transl. in: Elliptic Boundary Value Problems. AMS Transl. Ser. 2 123 (1984), AMS, Providence, RI, 57–88.Google Scholar
McKenna, J., Boyd, G. D. & Thurston, R. N. (1974) Plate theory solution for guided flexural acoustic waves along the tip of a wedge. IEEE Trans. Sonics Ultrasonics 21, N3, 178–186.CrossRefGoogle Scholar
Moss, S. L., Maradudin, A. A. & Cunningham, S. L. (1973) Vibrational edge modes for wedges with arbitrary interior angles. Phys. Rev. B 8, N6, 2999–3008.CrossRefGoogle Scholar
Mozhaev, V. G. (1989) Ray theory of wedge acoustic waves. Moscow Univ. Phys. Bull. 44, N5, 38–42.Google Scholar
Mozhaev, V. G. Unexpectable relations in the theory of surface wave propagation in topographical slightly perturbated media. In: Conference “Acoustoelectronic Surface Wave Devices for Signal Processing”, Cherkassy, USSR, Proceedings, Moscow, VINITI, pp. 12–13 [in Russian].Google Scholar
Mozhaev, V. G. (1991) Simple analytic relations describing symmetric acoustic wedge waves in obtuse wedges. In: 15th All-Union Conference “Acousto-Electronics and Physical Acoustics of Solids”, Part 2, Leningrad, LIAP, pp. 8–10 [in Russian].Google Scholar
Nazarov, S. A. (1999) The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes. Russ. Math. Surv. 54, N5, 947–1014.CrossRefGoogle Scholar
Nazarov, S. A. (2010) Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold. Siberian Math. J. 51, N5, 866–878.CrossRefGoogle Scholar
Nazarov, S. A. (2011) Asymptotic formula for an eigenvalue of the Dirichlet problem in a cranked waveguide. Vestnik St. Petersburg Univ. Math. 44, N3, 190–196.CrossRefGoogle Scholar
Nazarov, S. A. (2014) Asymptotics of eigenvalues of the Dirichlet problem in a skewed Ƭ-shaped waveguide. Comput. Math. Math. Phys. 54, N5, 811–830.CrossRefGoogle Scholar
Nazarov, S. A. & Plamenevsky, B. A. (1994) Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin, New York.CrossRefGoogle Scholar
Parker, D. F. (1992) Elastic wedge waves. J. Mech. Phys. Solids 40, N7, 1583–1593.CrossRefGoogle Scholar
Pupyrev, P. D. (2017) Linear and Nonlinear Wedge Waves in Solids. PhD Thesis, Moscow [in Russian].Google Scholar
Pupyrev, P. D., Lomonosov, A. M., Nikodijevic, A. & Mayer, A. P. (2016) On the existence of guided acoustic waves at rectangular anisotropic edges. Ultrasonics 71, 278–287.CrossRefGoogle ScholarPubMed
Shanin, A. V. (1997) Excitation and scattering of a wedge wave in an elastic wedge with angle close to 180°. Acoust. Phys. 43, N3, 344–349.Google Scholar
Šnol’, È. È. (1957) On the behavior of the eigenfunctions of Schrödinger’s equation. Mat. Sb. (N.S.) 42(84), N3, 273–286 [in Russian].Google Scholar
Tiersten, H. F. & Rubin, D. (1974) On the fundamental antisymmetric mode of the wedge guide. In: Proceedings of IEEE Ultrasonics Symposium, pp. 117–120.CrossRefGoogle Scholar
Vainberg, M. M. & Trenogin, V. A. (1969) Theory of Branching of Solutions of Nonlinear Equations, Nauka, Moscow [in Russian].Google Scholar
Van Dyke, M. (1964) Perturbation Methods in Fluid Mechanics, Academic Press, New York – London.Google Scholar
Williams, M. J. (1952) Stress singularities resulting from various boundary conditions in angular corners of plate in extension. J. Appl. Mech. 19, N2, 526–528.CrossRefGoogle Scholar
Zavorokhin, G. L. & Nazarov, A. I. (2010) On elastic waves in a wedge. Zapiski Nauchnykh Seminarov POMI 380, 45–52 [in Russian]; English transl.: J. Math. Sci. 175 (2011), N6, 646–650.Google Scholar
Zavorokhin, G. L., Nazarov, A. I. & Nazarov, S. A. (2018) The symmetric mode of an elastic solid wedge with the opening close to a flat angle. Doklady Physics 63, N12, 526–529.CrossRefGoogle Scholar