Acoustic Scattering by a Cross-Shaped Hole on a Smooth Boundary Surface Near a Resonance Point of the Interior Domain

Abstract

INTRODUCTORY REMARKSThe Dirichlet problem corresponding to a resonator with walls of finite thickness is a wellknown singularly perturbed boundary value problem for the Helmholtz equation and has been studied quite comprehensively. The resonator has the form Ω ε = Ω i ∪ Ω e ∪ κ ε , where Ω i and Ω e are the disjoint components of the complement of a domain diffeomorphic to a spherical layer and κ ε is a small hole that joins these components and has the size of the order of ε 1. It was shown in [1] that the analytic continuation of the solution of this problem into the lower complex half-plane of the spectral parameter has poles with small imaginary part close to the eigenvalues of the Dirichlet Laplacian −∆ Ωi . The poles of a resonator with finite walls were analyzed in [2-6] by various methods as ε → 0. There is a closely related problem of resonance excitation of a domain through a hole on an acoustically smooth boundary surface and the problem of scattering by a hole on an acoustically smooth boundary surface near the resonance of the interior domain, both problems corresponding to a resonator with infinitely thin walls. For a resonator with infinitely thin walls, the singularly perturbed Dirichlet boundary value problem for the Helmholtz equation admits numerical analysis on the basis of a boundary integral equation of the first kind. (See [7,8] for the case of resonance excitation and [7,9] for the case of resonance scattering by a hole.) In what follows, in the framework of the formalism developed in [7,9], we perform a numerical analysis of scattering by a cross-shaped hole on an acoustically smooth boundary surface near a resonance of the interior domain; moreover, we give a more detailed description than in [9] of the structure relations underlying the numerical analysis of the singularly perturbed Dirichlet problem for the Helmholtz equation corresponding to a resonator with infinitely thin walls.