  We develop an effective medium approach to characterize the propagation of
matter waves in periodic structures, such as graphene or semiconductor
superlattices. It is proven that the time evolution of the states that are not
more localized in space than the characteristic period of the structure can be
described exactly through an effective Hamiltonian, and that the electronic
band structure of the system can be exactly determined from the effective
Hamiltonian. As an illustration of the application of the method, we
characterize the mesoscopic response of graphene superlattices. It is shown
that these structures may be described using simply two effective parameters: a
dispersive potential, and an anisotropy tensor that characterizes the
pseudospin. Our model predicts that a graphene superlattice characterized by an
indefinite anisotropy tensor - such that the eigenvalues of the tensor have
opposite signs - may permit the perfect tunneling of all the stationary states
with a specific value of the energy when it is paired with a dual graphene
superlattice with positive definite anisotropy tensor.
