TY - JOUR

T1 - Localized waves propagating along an angular junction of two thin semi-infinite elastic membranes terminating an acoustic medium

AU - Лялинов, Михаил Анатольевич

PY - 2023/9/1

Y1 - 2023/9/1

N2 - We study existence of localized waves that can propagate in an acoustic medium bounded by two thin semi-infiniteelastic membranes along their common edge.The membranes terminate an infinite wedge that is filled by the medium, and are rigidly connected at the points of their common edge. The acoustic pressure of the medium in the wedge satisfiesthe Helmholtz equation and the third-order boundary conditions on the bounding membranes as well as the other appropriateconditions like contact conditions at the edge. The existence of such localized waves is equivalent to existence ofthe discrete spectrum of a semi-bounded self-adjoint operator attributed to this problem.In order to compute the eigenvalues and eigenfunctions, we make use of an integral representation (of the Sommerfeld type) for the solutions and reduce the problem to functional equations. Their non-trivial solutions from a relevant class of functions exist only for somevalues of the spectral parameter. The asymptotics of the solutions(eigenfunctions) is also addressed.The far-zone asymptotics contains exponentially vanishing terms. The corresponding solutionsexist only for some specific range of physical and geometrical parameters of the problem at hand.

AB - We study existence of localized waves that can propagate in an acoustic medium bounded by two thin semi-infiniteelastic membranes along their common edge.The membranes terminate an infinite wedge that is filled by the medium, and are rigidly connected at the points of their common edge. The acoustic pressure of the medium in the wedge satisfiesthe Helmholtz equation and the third-order boundary conditions on the bounding membranes as well as the other appropriateconditions like contact conditions at the edge. The existence of such localized waves is equivalent to existence ofthe discrete spectrum of a semi-bounded self-adjoint operator attributed to this problem.In order to compute the eigenvalues and eigenfunctions, we make use of an integral representation (of the Sommerfeld type) for the solutions and reduce the problem to functional equations. Their non-trivial solutions from a relevant class of functions exist only for somevalues of the spectral parameter. The asymptotics of the solutions(eigenfunctions) is also addressed.The far-zone asymptotics contains exponentially vanishing terms. The corresponding solutionsexist only for some specific range of physical and geometrical parameters of the problem at hand.

UR - https://www.mendeley.com/catalogue/6b25b52d-c840-379b-a1a5-a6c9990c5c2e/

U2 - 10.1134/s1061920823030068

DO - 10.1134/s1061920823030068

M3 - Article

VL - 30

SP - 345

EP - 359

JO - Russian Journal of Mathematical Physics

JF - Russian Journal of Mathematical Physics

SN - 1061-9208

IS - 3

ER -