Equivalence between general acoustic Willis media and conventional materials with embedded sources

Willis materials provide degrees of freedom to control mechanical waves unavailable in conventional materials, but our understanding of wave-matter interaction in these exotic media has been limited by their unconventional constitutive equations. This work derives an equivalence between the acoustic Willis wave equation inside a general inhomogeneous and anisotropic Willis medium and the well-known wave equation in conventional acoustic materials with embedded continuous distributions of monopole and dipole sources. It thus enables accurate and efficient computation of sound scattering from arbitrarily shaped general acoustic Willis materials in one, two, and three dimensions. The result is validated by showing in numerical simulations that realizable bulk Willis metamaterials, obtained by periodically replicating a labyrinthine cell, scatter sound identically to its equivalent material with embedded continuous source distributions. Furthermore, the equivalence provides insights into the physics of Willis materials. For example, it shows that multiple pairs of Willis coupling vectors produce exactly the same sound scattering regardless of excitation. It also directly shows whether the effective material parameters extracted from single Willis cell simulations maintain validity in bulk metamaterials based on that cell. This equivalence model will advance the design of Willis metamaterials and provide the tool to better understand the physics of Willis media.