Numerical extraction of three-dimensional acoustic polarizabilities for acoustically small scatterers

Abstract

Polarizability is a convenient descriptor for scattering from low-ka inhomogeneities in both electromagnetism, where it originated, and in acoustics, where it is increasingly used in the design of metamaterial elements. In two or three dimensions, the polarizability is represented by a block matrix that couples the local pressure and particle velocity fields to the lowest-order components of the multipole expansion, usually truncated to dipole order. A fully dense acoustic polarizability matrix has components that couple spatially uniform pressure and velocity oscillations to dipole and monopole scattering, respectively. This unique coupling leads to acoustic bianisotropy, also known as Willis coupling, for a collection of scatterers. The design of Willis materials necessitates computationally efficient methods to extract all components of the polarizability matrix and, at present, only two-dimensional analytical extraction methods are found in the literature. In this work, we present an algorithm to extract all components of the three-dimensional polarizability matrix from a scatterer with arbitrary geometry and composition. The method presented here is numerically efficient and yields results that are in agreement with an analytically obtained benchmark polarizability, providing a useful tool for acoustic metamaterial design.