Willis coupling in one-dimensional layered bulk media

Willis coupling, which couples the constitutive equations of an acoustical material, has been applied to acoustic metasurfaces with promising results. However, less is understood about Willis coupling in bulk media. In this paper a multiple-scales homogenization method is used to analyze the source and interpretation of Willis coupling in one-dimensional bulk media without any hidden degrees of freedom, or one-dimensional layered media. As expected from previous work, Willis coupling is shown to arise from geometric asymmetries, but is further shown to depend greatly on the measurement position. In addition, a discussion of the predicted material properties, including Willis coupling, of macroscopically inhomogeneous media is presented.