Anomalous Diffusion in Acoustic Phononic Crystals

In recent years numerous experimental studies have indicated that field transport processes in complex media can be governed by physical mechanisms characterized by hybrid behavior. This behavior is not typically accounted for in classical field transport theories based on integer order differential or integral models. Examples of these phenomena are given by anomalous transport or hybrid transport. The occurrence of anomalous transport phenomena has also been observed in wave fields interacting with random scattering media and metamaterials. The present study investigates the propagation of sound in a typical acoustic metamaterial configuration made of a square lattice of hard cylinders in air. By using a combination of deterministic, stochastic, and fractional continuum models, we theoretically and numerically explore the contributions of different transport modalities to the overall propagation of sound. Results suggest that anomalous transport can arise even in periodic media when in presence of anisotropic bandgaps. The analysis of the acoustic intensity reveals the characteristic formation of heavy-tailed distributions that are an important indicator of hybrid transport and of fractional dynamic behavior. The intensity distribution can be used to extract the equivalent fractional order of the metamaterial and to represent the response of the medium by a space-fractional diffusion equation.