On the Frequencies of Scattering on Traps with Micro-Inhomogeneous Structures—Helmholtz Resonators

This paper studies the vibration of a body with a hole (trap), which is caused by a plane acoustic wave. We assume that the body contains a periodic microstructure of small inclusions, which can partially reflect acoustic waves, and consider the scattering problem for the acoustic waves, which is modeled by an appropriate boundary value problem for the Helmholtz equation. Our goal is to describe the asymptotic behavior of the spectrum of this boundary value problem as the period of the microstructure and the diameter of the hole tend to zero. The scattering frequencies are characterized by the poles of the analytic continuation of the resolvent of the corresponding boundary value problem.