Dynamic homogenization of an acoustic metamaterial: Effective parameters of a rectangular lattice of cylinders

We present a dynamical homogenization model for the propagation of acoustic waves in periodic arrangements of scatterers. The model is based on the plane wave expansion method and leads to the effective parameters associated with the propagation. Effective mass density, effective bulk modulus and effective Willis term are frequency and wave vector dependant. The case of a rectangular lattice of cylinders is investigated. The homogenization model in the local approximation fits with the plane wave expansion method for small wave numbers. The branches of the dispersion curves are correctly found because of the anisotropy of the mass density.