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

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

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

Bibliographic databases:

Document Type: Article
UDC: 517.956.8+517.956.227+539.3(3)
MSC: Primary 35Q74; Secondary 74B05

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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• http://mi.mathnet.ru/eng/msb8286
• https://doi.org/10.4213/sm8286
• http://mi.mathnet.ru/eng/msb/v205/i7/p43

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
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
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
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
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
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
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
9. S. A. Nazarov, ““Wandering” eigenfrequencies of a two-dimensional elastic body with a truncated cusp”, Dokl. Phys., 62:11 (2017), 512–516
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
11. Cardone G. Durante T. Nazarov S.A., “Embedded Eigenvalues of the Neumann Problem in a Strip With a Box-Shaped Perturbation”, J. Math. Pures Appl., 112 (2018), 1–40
12. Kozlov V.A., Nazarov S.A., “Waves and Radiation Conditions in a Cuspidal Sharpening of Elastic Bodies”, J. Elast., 132:1 (2018), 103–140
13. Kirsch A., Lechleiter A., “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
14. Dohnal T., Schweizer B., “A Bloch Wave Numerical Scheme For Scattering Problems in Periodic Wave-Guides”, SIAM J. Numer. Anal., 56:3 (2018), 1848–1870
15. Nazarov S.A., “Finite-Dimensional Approximations of the Steklov-Poincaré, Operator in Periodic Elastic Waveguides”, Dokl. Phys., 63:7 (2018), 307–311
16. Nazarov S.A., ““Wandering” Natural Frequencies of An Elastic Cuspidal Plate With the Clamped Peak”, Mater. Phys. Mech., 40:1 (2018), 47–55
17. S. A. Nazarov, “Various manifestations of Wood anomalies in locally distorted quantum waveguides”, Comput. Math. Math. Phys., 58:11 (2018), 1838–1855
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