Effective parameters of clusters of cylinders embedded in a nonviscous fluid or gas

This work presents a comprehensive analysis of the homogenization method recently used by Torrent et al. Phys. Rev. Lett. 96, 204302 2006 to get the effective parameters of two-dimensional clusters of rigid cylinders in air. Here, the method is developed by studying the scattering of sound by clusters of fluid cylinders embedded in a nonviscous fluid or gas. This general problem is studied in the long wavelength limit homogenization by multiple scattering theory. Asymptotic relations are derived and employed to formulate a method of homogenization based on the scattering properties of the cluster. Exact formulas for the effective parameters i.e., effective sound speed and effective density are obtained as a function of the location of each cylinder and the physical parameters density and speed of sound of cylinders and the embedded medium. Results for several fluid-fluid composite systems are reported. The case of rigid infinite density cylinders in air is deeply analyzed, showing results for ordered and disordered lattices. It is concluded that the method provides a tool of designing acoustic metamaterials with prefixed refractive properties.