Dispersion relations of generalized one-dimensional phononic crystals

TL;DR

This paper introduces a versatile and comprehensive method for calculating exact and approximate dispersion relations in one-dimensional resonant phononic crystals, accommodating various structures and scatterer types using a unified mathematical framework.

Contribution

It develops a general matrix-based algorithm employing plane wave expansion and iterative analytical expressions, applicable to diverse 1D phononic crystal configurations and scattering scenarios.

Findings

Method accurately computes dispersion relations for complex 1D phononic structures.

Algorithm demonstrates high efficiency and versatility through numerical examples.

Convergence conditions are rigorously established for the proposed method.

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.