Acoustic Band-Gap Formulation in Infinite Periodic Porous Media With a Multi-Layered Unit Cell: Multi-Scale Asymptotic Method

This article introduces a numerical formulation for studying frequency band structure in multi-periodic acoustic composite structures. Herein, multi-periodic acoustic composite structures are defined as periodically-layered acoustic media wherein each layer is composed of periodically-repeated unit fluid cells, especially those arising from the study of rigid-frame porous materials. Hence, at least two periodic scales (microscopic and mesoscopic, respectively) comprise the macroscopic acoustic media. Under the Floquet-Bloch’s condition, exploitation of the multi-periodicity allows for efficient generation of dispersion curves via a multi-scale asymptotic method (for homogenizing the mesoscale) coupled to original acoustic transfer matrix methods (for the macroscale). The combined numerical formulation results in a general analysis procedure for evaluating complex dispersion relationships. The dispersion curves can be used to reveal frequency stop bands wherein the wave vector is highly imaginary—i.e., plane waves experience rapid attenuation. The formulation is applied to four infinite, multi-periodic acoustic composite structures in order to demonstrate the formulation’s utility and to reveal novel properties, particularly those which can be influenced by design of the mesoscopic unit cell.