  This papers investigates the scattering of oblique shear horizontal (SH)
waves off finite periodic media made of elastic and viscoelastic layers. It
further considers whether a Willis-type constitutive matrix (in temporal and
spatial Fourier domain) may reproduce the scattering matrix (SM) of such a
system. In answering this question the procedure to determine the relevant
overall constitutive parameters for such a medium is presented. To do this,
first the general form of the dispersion relation and impedances for oblique SH
propagation in such coupled Willis-type media are developed. The band structure
and scattering of layered media are calculated using the transfer matrix (TM)
method. The dispersion relation may be derived based on the eigensolutions of
an infinite periodic domain. The wave impedances associated with the exterior
surfaces of a finite thickness slab are extracted from the scattering of such a
system. Based on reciprocity and available symmetries of the structure and each
constituent layer, the general form of the dispersion and impedances may be
simplified. The overall quantities may be extracted by equating the scattering
data from TM with those expected from a Willis-type medium. It becomes evident
that a Willis-type coupled constitutive tensor with components that are assumed
independent of wave vector is unable to reproduce all oblique scattering data.
Therefore, non-unique wave vector dependent formulations are introduced, whose
SM matches that of the layered media exactly. It is further shown that the
dependence of the overall constitutive tensors of such systems on the wave
vector is not removable even at very small frequencies and incidence angles and
that analytical considerations significantly limit the potential forms of the
spatially dispersive constitutive tensors.
