High frequency homogenization for a one-dimensional acoustic black hole lattice

A hybrid theoretical framework was developed to understand the wave propagation properties of acoustic black hole (ABH) lattices, based on a combination of WKB theory and the high-frequency homogenization method. Explicit approximate expressions were obtained for the associated dispersion curves. Firstly, a matrix formulation to model wave propagation in a one-dimensional lattice was developed, where the wave components of the flexural displacement were described as WKB solutions of the flexural wave equation. The standing wave frequencies and the corresponding wave shape functions required by the high-frequency homogenization analysis were obtained accordingly. Secondly, the effective dispersion relation of the ABH lattice was obtained analytically by utilizing high-frequency homogenization. Based on the dispersion results obtained from the hybrid framework, the group velocity and hence the modal density of the ABH lattice were obtained analytically. Finally, the proposed technique was validated through several numerical examples.