Abstract:
It is established that the problem of elastic oscillations of a homogeneous anisotropic semicylinder (console) with traction-free lateral surface (Neumann boundary condition) has no eigenvalues when the console is clamped at one end (Dirichlet boundary condition). If the end is free, under additional requirements of elastic
and geometric symmetry, simple sufficient conditions are found for the existence of an eigenvalue embedded in the continuous spectrum and generating a trapped elastic wave, that is, one which decays at infinity at an exponential rate. The results are obtained by generalizing the methods developed for scalar problems, which however require substantial modification for the vector problem in elasticity theory. Examples are given and open questions are stated.
Bibliography: 53 titles.
Keywords:
homogeneous anisotropic semicylinder, trapped waves, point spectrum on the continuous spectrum, artificial boundary conditions.
This publication is cited in the following 13 articles:
S. A. Nazarov, “Raspredelenie mod sobstvennykh kolebanii v plastine, zaglublennoi v absolyutno zhëstkoe poluprostranstvo”, Matematicheskie voprosy teorii rasprostraneniya voln. 53, Zap. nauchn. sem. POMI, 521, POMI, SPb., 2023, 154–199
S. A. Nazarov, “Elastic Waves Trapped by a Semi-infinite Strip with Clamped Lateral Sides and a Curved or Broken End”, Mech. Solids, 58:7 (2023), 2619
S. A. Nazarov, “Elastic Waves Trapped by Semi-Infinite Strip with Clamped Lateral Sides and a Curved or Broken End”, Prikladnaya matematika i mekhanika, 87:2 (2023), 265
S. A. Nazarov, “Two-Dimensional Asymptotic Models of Thin Cylindrical Elastic Gaskets”, Diff Equat, 58:12 (2022), 1651
S. A. Nazarov, “Threshold resonances and virtual levels in the spectrum of cylindrical and periodic waveguides”, Izv. Math., 84:6 (2020), 1105–1160
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S. A. Nazarov, “Finite-dimensional approximations of the Steklov-Poincaré operator in periodic elastic waveguides”, Dokl. Phys., 63:7 (2018), 307–311
S. A. Nazarov, “Discrete spectrum of cranked quantum and elastic waveguides”, Comput. Math. Math. Phys., 56:5 (2016), 864–880
S. A. Nazarov, “Eigenmodes of a thin elastic layer between periodic rigid profiles”, Comput. Math. Math. Phys., 55:10 (2015), 1684–1697
S. A. Nazarov, “Localization of longitudinal and transverse oscillations in a thin curved elastic gasket”, Dokl. Phys., 60:10 (2015), 446–450
Nazarov S.A., “Asymptotics of the natural oscillations of a thin elastic gasket between absolutely rigid profiles”, Pmm-J. Appl. Math. Mech., 79:6 (2015), 577–586
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