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

This article is cited in 18 scientific papers (total in 18 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  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  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
    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
  • Математический сборник Sbornik: Mathematics (from 1967)
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