Subwavelength Resonances in One-Dimensional High-Contrast Acoustic Media

Abstract

We propose a mathematical theory of acoustic wave scattering in one-dimensional finite high-contrast media. The system considered is constituted of a finite alternance of high-contrast segments of arbitrary lengths and interdistances, called the “resonators,” and a background medium. We prove the existence of subwavelength resonances, which are the counterparts of the well-known Minnaert resonances in three-dimensional systems. One of the main contribution of the paper is to show that the resonant frequencies as well as the transmission and reflection properties of the system can be accurately predicted by a “capacitance” eigenvalue problem, analogously to the three-dimensional setting. Moreover, we discover new properties which are peculiar to the one-dimensional setting, notably the tridiagonal structure of the capacitance matrix as well as the fact that the first resonant frequency is always exactly zero, implying a low-pass filtering property of a one-dimensional chain of resonators. Numerical results considering different situations with N = 1 to N = 6 resonators are provided to support our mathematical analysis and to illustrate the various possibilities offered by high-contrast resonators to manipulate waves at subwavelength scales.