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This paper focuses on analysing acoustic structures containing locally reacting resonators, aiming to establish a framework for discussing sound properties related to scattering and transmission in a periodic resonator array. The paper introduces an analytical model for the input surface impedance of non-uniform resonator arrays, which is a challenging task when addressing sound propagation to the opposite side of the array. To address this challenge, a mode-matching method is employed within a one-dimensional periodic structure combined with the decomposition of the initial mirror-symmetric system into symmetric and antisymmetric sub-systems. This procedure leads to simplified analytical solutions of transmission problems about non-uniform resonator arrays, and its accuracy is verified through numerical simulations. The proposed method provides insights into mode shapes and amplitudes, thereby complementing numerical simulations and enabling a comprehensive understanding of sound fields beyond conventional results such as pressure fields and sound absorption coefficients alone. The model of a periodic varying-height resonator array presented herein involves developing mathematical equations or simulations to capture the physical characteristics and interactions of the resonators. This approach empowers researchers to predict and comprehend the non-uniform array's transmission behaviour and performance characteristics, with diverse applications spanning various systems, including electromagnetic devices.
Shen et al. (Sun,) studied this question.