The analysis of resonant frequencies and blow-up estimates of close-to-touching subwavelength resonators in the two-dimensional Helmholtz system

TL;DR

This paper studies wave scattering in a 2D Helmholtz system with close-to-touching resonators, revealing unique resonant behaviors and gradient blow-up rates, which differ from 3D cases.

Contribution

It provides the first detailed analysis of sub-wavelength resonant modes and blow-up estimates for closely spaced resonators in 2D Helmholtz systems.

Findings

Identification of two distinct sub-wavelength resonant modes

Asymptotic characterization of resonant frequencies

Quantitative estimates of wave field blow-up rates

Abstract

In this paper, we investigate wave scattering by a pair of closely spaced inclusions embedded in a homogeneous medium, characterized by a high contrast physical parameters. The system is modeled by the two-dimensional Helmholtz equation. We show that this configuration exhibits two sub-wavelength resonant modes, whose frequencies display distinct leading-order asymptotic behaviors. These findings differ significantly from those in the three-dimensional Helmholtz setting. Furthermore, we provide a quantitative analysis of the gradient blow-up rates for the wave field localized between the two resonators.