TY - JOUR

T1 - Waves Trapped by Semi-Infinite Kirchhoff Plate at Ultra-Low Frequencies

AU - Nazarov, S. A.

N1 - Publisher Copyright: © 2020, Allerton Press, Inc.

PY - 2020/12

Y1 - 2020/12

N2 - Abstract: In this paper, a semi-infinite Kirchhoff plate with a traction-free edge, which rests partially on a heterogeneous Winkler foundation (the Neumann problem for the biharmonic operator perturbed by a small free term with a compact support), is considered. It is shown that, for arbitrary small ε > 0, a variable foundation compliance coefficient (defined nonuniquely) of order ε can be constructed, such that the plate obtains the eigenvalue ε4 that is embedded into a continuous spectrum, and the corresponding eigenfunction decays exponentially at infinity. It is verified that no more than one small eigenvalue can exist. It is noteworthy that a small perturbation cannot prompt an emergence of an eigenvalue near the cutoff point of the continuous spectrum in an acoustic waveguide (the Neumann problem for the Laplace operator).

AB - Abstract: In this paper, a semi-infinite Kirchhoff plate with a traction-free edge, which rests partially on a heterogeneous Winkler foundation (the Neumann problem for the biharmonic operator perturbed by a small free term with a compact support), is considered. It is shown that, for arbitrary small ε > 0, a variable foundation compliance coefficient (defined nonuniquely) of order ε can be constructed, such that the plate obtains the eigenvalue ε4 that is embedded into a continuous spectrum, and the corresponding eigenfunction decays exponentially at infinity. It is verified that no more than one small eigenvalue can exist. It is noteworthy that a small perturbation cannot prompt an emergence of an eigenvalue near the cutoff point of the continuous spectrum in an acoustic waveguide (the Neumann problem for the Laplace operator).

KW - near-threshold eigenvalue embedded into continuous spectrum

KW - semi-infinite Kirchhoff plate

KW - small perturbation

KW - threshold resonance

KW - Winkler foundation

UR - http://www.scopus.com/inward/record.url?scp=85101746780&partnerID=8YFLogxK

U2 - 10.3103/S002565442008018X

DO - 10.3103/S002565442008018X

M3 - Article

AN - SCOPUS:85101746780

VL - 55

SP - 1328

EP - 1339

JO - Mechanics of Solids

JF - Mechanics of Solids

SN - 0025-6544

IS - 8

ER -