  This work presents a multiple scattering formulation of two dimensional
acoustic metamaterials. It is shown that in the low frequency limit multiple
scattering allows us to define frequency-dependent effective acoustic
parameters for arrays of both ordered and disordered cylinders. This
formulation can lead to both positive and negative acoustic parameters, where
the acoustic parameters are the scalar bulk modulus and the tensorial mass
density and, therefore, anisotropic wave propagation is allowed with both
positive or negative refraction index. It is also shown that the surface fields
on the scatterer are the main responsible of the anomalous behavior of the
effective medium, therefore complex scatterers can be used to engineer the
frequency response of the effective medium, and some examples of application to
different scatterers are given. Finally, the theory is extended to
electromagnetic wave propagation, where Mie resonances are found to be the
responsible of the metamaterial behavior. As an application, it is shown that
it is possible to obtain metamaterials with negative permeability and
permittivity tensors by arrays of all-dielectric cylinders and that anisotropic
cylinders can tune the frequency response of these resonances.
