Acoustic skin effect with non-reciprocal Willis materials

The Willis material model, coupling kinetic energy with a potential one, is shown to equip an extraordinary capacity in characterizing complex acoustic and elastic wave phenomena of metamaterials. This model has been further extended to active systems via breaking the symmetry between two coupling coefficients, leading to odd or non-reciprocal Willis material models [Quan et al., Nat. Commun. 12(1), 2615 (2021)]. In this work, through a 2D homogenous non-reciprocal acoustic Willis material (NRAWM), we demonstrate that the bulk local mode, referred to as skin effect in non-Hermitian systems, can survive on boundary of NRAWMs under proper conditions. The direction of the localization is closely related to the intrinsic direction embedded in the NRAWMs, and the localization is robust and topologically protected. To validate the prediction, a 2D discrete lattice made of non-local active acoustic scatterers is proposed and then homogenized as a NRAWM based on the retrieval method. The far-field radiation patterns of the local modes for both the 2D discrete lattice and the homogenized 2D NRAWM are evaluated, and they are in good agreement with each other. This work paves the way to design and explore the rich wave phenomena in non-Hermitian acoustic systems.