Resonance Analysis of One‐Dimensional Acoustic Media: A Propagation Matrix Approach

ABSTRACT The propagation matrix is a natural and powerful tool for analyzing one‐dimensional wave propagation. While it has been widely applied to study wave phenomena in specific one‐dimensional media, a systematic framework based on this method for characterizing scattering resonances and deriving their asymptotics in general settings has been lacking. This work fills this gap by developing such a framework for general one‐dimensional acoustic media. Specifically, by combining the propagation matrix with Möbius transformations, we characterize 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. In this way, we provide a new and self‐contained framework, distinct from variational methods (Feppon et al., SIAM Journal on Applied Mathematics , 83, no. 2 (2023): 625–665), for analyzing the asymptotics of one‐dimensional scattering resonances and facilitating the analysis of high‐frequency resonances.