  In the framework of multiple-scattering theory, we show that the dispersion
relations of certain electromagnetic (EM) and elastic metamaterials can be
obtained analytically in the long-wavelength limit. Specific examples are given
to the two-dimensional metamaterials with cylindrical inclusions arranged in
square and triangular lattices. The role played by the lattice structure in
determining whether a dispersion relation is isotropic or not is shown
explicitly. Different lattice dependences between EM and elastic metamaterials
are also shown. In the case of isotropic dispersions, our results coincide with
those of isotropic effective medium theories obtained previously for EM and
elastic metamaterials, respectively, and, therefore, provide a more fundamental
support to those theories. In the case of elastic metamaterials with
anisotropic dispersions, our analytical results can provide an anisotropic
effective medium theory in the form of Christoffel's equation. In this case,
the isotropic effective medium theory can describe accurately the
angle-averaged dispersion relations. The properties of anisotropic dispersions
are discussed and verified by numerical calculations of a realistic elastic
metamaterial.
