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Mat. Sb., 2014, Volume 205, Number 7, Pages 43–72 (Mi msb8286)  

This article is cited in 23 scientific papers (total in 23 papers)

Umov-Mandelshtam radiation conditions in elastic periodic waveguides

S. A. Nazarovab

a Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg
b St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We study settings of the problem of elasticity theory on wave propagation in an elastic periodic waveguide with radiation conditions at infinity. We present a mathematical theory for energy radiation conditions based on Mandelshtam's energy principle and the Umov-Poynting vector, as well as using the technique of weighted spaces with detached asymptotics and the energy transfer symplectic form. We establish that in a threshold situation, that is, when standing and polynomial elastic Floquet waves appear, the well-known limiting absorption principle, in contrast to the energy principle that is being applied, cannot identify the direction of the wave's motion.
Bibliography: 37 titles.

Keywords: elastic periodic waveguide, Mandelshtam's energy radiation condition, Umov-Poynting vector, energy transfer symplectic form.

Funding Agency Grant Number
Russian Foundation for Basic Research 12-01-00348


DOI: https://doi.org/10.4213/sm8286

Full text: PDF file (818 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2014, 205:7, 953–982

Bibliographic databases:

UDC: 517.956.8+517.956.227+539.3(3)
MSC: Primary 35Q74; Secondary 74B05
Received: 26.09.2013 and 13.02.2014

Citation: S. A. Nazarov, “Umov-Mandelshtam radiation conditions in elastic periodic waveguides”, Mat. Sb., 205:7 (2014), 43–72; Sb. Math., 205:7 (2014), 953–982

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. Fliss, P. Joly, “Solutions of the time-harmonic wave equation in periodic waveguides: asymptotic behaviour and radiation condition”, Arch. Ration. Mech. Anal., 219:1 (2016), 349–386  crossref  mathscinet  zmath  isi  elib  scopus
    2. S. A. Nazarov, “Almost standing waves in a periodic waveguide with a resonator and near-threshold eigenvalues”, St. Petersburg Math. J., 28:3 (2017), 377–410  mathnet  crossref  mathscinet  isi  elib
    3. S. A. Nazarov, J. Taskinen, “Elastic and piezoelectric waveguides may have infinite number of gaps in their spectra”, C. R. Mec., 344:3 (2016), 190–194  crossref  isi  elib  scopus
    4. S. A. Nazarov, K. Ruotsalainen, P. Uusitalo, “Localized waves in carbon nano-structures with connected and disconnected open waveguides”, Mater. Phys. Mech., 29:2 (2016), 116–124  isi
    5. S. A. Nazarov, “The asymptotic behaviour of the scattering matrix in a neighbourhood of the endpoints of a spectral gap”, Sb. Math., 208:1 (2017), 103–156  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. S. A. Nazarov, “Open waveguides in a thin Dirichlet lattice: II. Localized waves and radiation conditions”, Comput. Math. Math. Phys., 57:2 (2017), 236–252  mathnet  crossref  crossref  isi  elib
    7. G. Cardone, T. Durante, S. A. Nazarov, “The spectrum, radiation conditions and the Fredholm property for the Dirichlet Laplacian in a perforated plane with semi-infinite inclusions”, J. Differential Equations, 263:2 (2017), 1387–1418  crossref  mathscinet  zmath  isi  elib  scopus
    8. S. A. Nazarov, J. Taskinen, “Radiation conditions for the linear water-wave problem in periodic channels”, Math. Nachr., 290:11-12 (2017), 1753–1778  crossref  mathscinet  zmath  isi  scopus
    9. S. A. Nazarov, ““Wandering” eigenfrequencies of a two-dimensional elastic body with a truncated cusp”, Dokl. Phys., 62:11 (2017), 512–516  crossref  crossref  mathscinet  isi  elib  scopus
    10. S. A. Nazarov, “Wave scattering in the joint of a straight and a periodic waveguide”, J. Appl. Math. Mech., 81:2 (2017), 129–147  crossref  mathscinet  zmath  isi  scopus
    11. G. Cardone, T. Durante, S. A. Nazarov, “Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation”, J. Math. Pures Appl., 112 (2018), 1–40  crossref  mathscinet  zmath  isi  scopus
    12. V. A. Kozlov, S. A. Nazarov, “Waves and radiation conditions in a cuspidal sharpening of elastic bodies”, J. Elast., 132:1 (2018), 103–140  crossref  mathscinet  zmath  isi  scopus
    13. A. Kirsch, A. Lechleiter, “A radiation condition arising from the limiting absorption principle for a closed full- or half-waveguide problem”, Math. Meth. Appl. Sci., 41:10 (2018), 3955–3975  crossref  mathscinet  zmath  isi  scopus
    14. T. Dohnal, B. Schweizer, “A Bloch wave numerical scheme for scattering problems in periodic wave-guides”, SIAM J. Numer. Anal., 56:3 (2018), 1848–1870  crossref  mathscinet  zmath  isi  scopus
    15. S. A. Nazarov, “Finite-dimensional approximations of the Steklov-Poincaré operator in periodic elastic waveguides”, Dokl. Phys., 63:7 (2018), 307–311  mathnet  crossref  crossref  isi  scopus
    16. S. A. Nazarov, ““Wandering” natural frequencies of an elastic cuspidal plate with the clamped peak”, Mater. Phys. Mech., 40:1 (2018), 47–55  crossref  isi
    17. S. A. Nazarov, “Various manifestations of Wood anomalies in locally distorted quantum waveguides”, Comput. Math. Math. Phys., 58:11 (2018), 1838–1855  mathnet  crossref  crossref  isi  elib
    18. Leugering G., Nazarov S.A., Taskinen J., “Umov-Poynting-Mandelstam Radiation Conditions in Periodic Composite Piezoelectric Waveguides”, Asymptotic Anal., 111:2 (2019), 69–111  crossref  mathscinet  zmath  isi  scopus
    19. S. A. Nazarov, “Finite-dimensional approximations to the Poincaré–Steklov operator for general elliptic boundary value problems in domains with cylindrical and periodic exits to infinity”, Trans. Moscow Math. Soc., 80 (2019), 1–51  mathnet  crossref
    20. S. A. Nazarov, “‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings”, Sb. Math., 210:11 (2019), 1633–1662  mathnet  crossref  crossref  adsnasa  isi
    21. Nazarov S.A., “Strange Behavior of Natural Oscillations of An Elastic Body With a Blunted Peak”, Mech. Sol., 54:5 (2019), 694–708  crossref  isi
    22. V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped modes in armchair graphene nanoribbons”, Matematicheskie voprosy teorii rasprostraneniya voln. 49, Zap. nauchn. sem. POMI, 483, POMI, SPb., 2019, 85–115  mathnet
    23. S. A. Nazarov, “Rasseyanie uprugikh voln na malykh chastotakh v beskonechnoi plastine Kirkhgofa”, Matematicheskie voprosy teorii rasprostraneniya voln. 49, Zap. nauchn. sem. POMI, 483, POMI, SPb., 2019, 142–177  mathnet
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