Acoustic bianisotropy in a subdivided meta-layer

Willis materials are a class of metamaterials exhibiting momentum-strain coupling. Asymmetry at the microscopic level and nonlocal effects at the mesoscale level give rise to Willis coupling, which is a form of acoustic bianisotropy. Resonant Willis meta-layers incorporate resonators into an isolated element to effectively obtain Willis coupling over a narrow frequency band. By subdividing a meta-layer into multiple resonator shapes, Willis coupling can be obtained over multiple frequency bands. A finite subdivision rule yields self-similar resonator shapes, which translates into resonance frequencies exhibiting a power-law dependence. Furthermore, manipulation of the internal asymmetry allows for tuning of the Willis coupling. This article reports on the measurement and modeling of resonant Willis meta-layers having parallel resonators. Measurements are obtained with a transmission impedance tube, and predictions are based on a lumped-element analysis. With resonator shapes based on subdivisions of the equilateral triangle, Willis coupling is obtained over several frequency bands in the range of 500-1500 Hz. For a subset of the resonators, Willis coupling is tuned by mirroring the asymmetric location of some resonator necks across the element's midplane. Directionally dependent acoustic absorption is observed, along with tuning of the absorption curves by introducing non-uniformity to the internal sample asymmetry.