Resonance analysis of one-dimensional acoustic media: a propagation matrix approach

TL;DR

The paper develops a comprehensive one-dimensional theory for acoustic resonances in high-contrast media via a propagation-matrix approach. It recasts resonances as zeros of an explicit trigonometric polynomial and establishes their global distribution, uniform resonance-free region, and duality under density-contrast transformation. Through a Newton-polygon analytic framework, it recovers the discrete capacitance-matrix description of subwavelength Minnaert resonances and extends the theory to non-reciprocal systems, connecting subwavelength behavior to capacitance matrices in both Hermitian and non-Hermitian settings. The work highlights fundamental dimensional differences with higher-dimensional cases and lays groundwork for future extensions to non-subwavelength resonances and non-Hermitian media.

Abstract

This work analyzes the scattering resonances of general acoustic media in a one-dimensional setting using the propagation matrix approach. Specifically, we characterize the resonant frequencies as the zeros of an explicit trigonometric polynomial. Leveraging Nevanlinna's value distribution theory, we establish the distribution properties of the resonances and demonstrate that their imaginary parts are uniformly bounded, which contrasts with the three-dimensional case. In two classes of high-contrast regimes, we derive the asymptotics of both subwavelength and non-subwavelength resonances with respect to the contrast parameter. Furthermore, by applying the Newton polygon method, we recover the discrete capacitance matrix approximation for subwavelength Minnaert resonances in both Hermitian and non-Hermitian cases, thereby establishing its connection to the propagation matrix framework.

Figures (6)

• Figure 1: A chain of $N$ resonators, with lengths $(\ell_j)_{1\leq j\leq N}$ and spacings $(s_{j})_{1\leq j\leq N-1}$.
• Figure 1: Zeros of $f(k; \sigma)$ for the configuration $\bm{t} = (0.8, 0.9, 1, 1.1, 1.2, 1.3, 1.4)^\top$ and $\sigma = 0.8$. All zeros are confined to the strip $C_1(\sigma) < \mathfrak{Im} k < C_2(\sigma)$. In the region $|\mathfrak{Re} k| \leq 5$ and $|\mathfrak{Re} k| \leq 50$, the argument principle yields 25 zeros and 245 zeros, giving a density of 2.5 and 2.45 zeros per unit length, respectively. These empirical densities align closely with the theoretical one $\|\bm{t}\|_1 / \pi \approx 2.451$ established in Theorem \ref{['thm: f(z;mu) zeros']}.
• Figure 1: The lower boundary of the convex hull
• Figure 1: Structure of $Q$ with $a_L=1,a_1=0,a_2=2,\cdots, a_{2N-l-1}=1,a_R=2$.
• Figure 2: Zeros of $f(k; \sigma)$ with sufficiently small $\delta$. Panels (b)-(g): Zoom-in views near $k_1$-$k_6$ with different orders $n(k_i)$. We see that exactly $n(k)$ zeros are located near each $k \in E$, where the set $E$ is given in Lemma \ref{['thm: f(k;0) form']}.

Theorems & Definitions (27)

• Lemma 2.1
• Theorem 2.2
• Remark 2.3
• Proof 1: Proof of Theorem \ref{['thm: resonant frequency property']}
• Theorem 3.1
• Example
• Lemma 3.2
• Theorem 3.3
• Proof 2
• Remark 3.4