General acoustic Willis media modeled as conventional materials with embedded sources

Willis coupling vectors provide additional degrees of freedom to manipulate acoustic waves unavailable in conventional materials, but the additional material properties make it difficult to simulate the interaction between sound and Willis media especially when inhomogeneous media with arbitrary geometries are involved. To address this challenge, we establish the equivalence between the wave equation inside a three-dimensional, inhomogeneous, and anisotropic Willis medium to the well-known acoustic wave equation in conventional materials with embedded continuous distributions of monopole and dipole sources. We validate the equivalence in numerical simulations showing that conventional materials with embedded sources replace physical Willis metamaterials anisotropic in the Willis and mass density tensors without modifying the scattered sound distribution. Moreover, the equivalence offers insights into the physics of Willis materials. For example, it shows that various combinations of Willis coupling vectors can generate identical sound scattering for any excitation. It also shows whether effective material parameters extracted from single Willis cell simulations remain valid in bulk metamaterials obtained by replicating periodically that cell. These results suggest that the equivalence model can advance the design of Willis metamaterials and provide a tool to better understand the physics of Willis media.