Dispersion relations of generalized one-dimensional phononic crystals

Abstract

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 (1D) phononic crystal}, in which {different} types of {waves} and scatterers{/resonators} {can be} {considered} by adjusting certain parameters. The generalized {1D 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 utilizing 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.