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.
This publication is cited in the following 35 articles:
Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Mahran Rihani, “Maxwell's equations with hypersingularities at a
negative index material conical tip”, Pure Appl. Analysis, 7:1 (2025), 127
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Ruming Zhang, “High Order Complex Contour Discretization Methods to Simulate Scattering Problems in Locally Perturbed Periodic Waveguides”, SIAM J. Sci. Comput., 44:5 (2022), B1257
S. A. Nazarov, “Propagating and standing Rayleigh waves near rivet chains connecting Kirchhoff plates”, Siberian Math. J., 62:6 (2021), 1084–1099
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Sonia Fliss, Patrick Joly, Vincent Lescarret, “A Dirichlet-to-Neumann approach to the mathematical and numerical analysis in waveguides with periodic outlets at infinity”, Pure Appl. Analysis, 3:3 (2021), 487
S. A. Nazarov, “Scattering of Low-Frequency Elastic Waves in An Infinite Kirchhoff Plate”, J Math Sci, 252:5 (2021), 664
V. A. Kozlov, S. A. Nazarov, A. Orlof, “Trapped Modes in Armchair Graphene Nanoribbons”, J Math Sci, 252:5 (2021), 624
S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160
S. A. Nazarov, “Anomalies of acoustic wave scattering near the cut-off points of continuous spectrum (a review)”, Acoust. Phys., 66:5 (2020), 477–494
S. A. Nazarov, “Almost Complete Transmission of Low Frequency Waves in a Locally Damaged Elastic Waveguide”, J Math Sci, 244:3 (2020), 451
G. Leugering, S. A. Nazarov, J. Taskinen, “Umov-poynting-mandelstam radiation conditions in periodic composite piezoelectric waveguides”, Asymptotic Anal., 111:2 (2019), 69–111
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
S. A. Nazarov, “‘Blinking’ and ‘gliding’ eigenfrequencies of oscillations of elastic bodies with blunted cuspidal sharpenings”, Sb. Math., 210:11 (2019), 1633–1662
S. A. Nazarov, “Strange behavior of natural oscillations of an elastic body with a blunted peak”, Mech. Sol., 54:5 (2019), 694–708
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
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
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
V. A. Kozlov, S. A. Nazarov, “Waves and radiation conditions in a cuspidal sharpening of elastic bodies”, J. Elast., 132:1 (2018), 103–140
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
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