We present a comprehensive method for determining both exact and approximate dispersion relations for one-dimensional resonant phononic crystals, applicable to a wide range of structures, regardless of their specific characteristics. This general framework employs a unified mathematical model, referred to as generalized one-dimensional phononic crystal, in which different types of waves and scatterers/resonators can be considered by adjusting certain parameters. The generalized one-dimensional phononic crystal consists of both a host one-dimensional homogeneous elastic material with physical properties represented in matrix form and an arbitrary set of scatterers within the unit cell, including resonators (discrete and continuous), small material inclusions or variations in cross-sectional area. Based on general assumptions, and imposing the periodicity and Bloch solutions, we develop a matrix-based algorithm using the plane wave expansion method to derive the solution. Additionally, we propose an iterative procedure that provides analytical expressions for the first- and second-order terms, particularly useful in the context of weak scattering. The convergence conditions of the method are rigorously defined. The efficiency of the approach is demonstrated through several numerical examples, highlighting its versatility in different waveguide configurations and scattering scenarios.
Lázaro et al. (Fri,) studied this question.
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