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For oblique anti-plane shear waves in periodic layered elastic composites, itis shown that negative energy refraction is accompanied by positivephase-velocity refraction and positive energy refraction is accompanied bynegative phase-velocity refraction, and that, this happens over a broad rangeof frequencies. The composite's unit cell may consist of any number of layersof any variable mass-density and elastic shear modulus (with largediscontinuities).  Explicit series expressions for displacement, velocity, strain and stresscomponents, and energy-flux fields are given, and group-velocity vector iscalculated. The approach is based on a mixed variational principle where thedisplacement and stress components are viewed as independent fields subject toarbitrary variation. These fields are hence approximated independently, therebyensuring the necessary continuity conditions. The resulting computationalmethod yields the composite's frequency band structure and the associated modeshapes, in terms of the wave-vector components for any desired number offrequency bands.  The general results are illustrated using a two-phase and a three-phase unitcell with piecewise constant properties. It is shown that on their secondfrequency pass-bands, only the components of the phase and group velocitiesnormal to the layers are antiparallel, while the components along the layersare parallel. Therefore, both the two-phase and the three-phase compositesdisplay negative energy refraction with positive phase-velocity refraction andpositive phase-velocity refraction with negative energy refraction, dependingon how the composite is interfaced with a homogeneous solid.  The presented method is applicable and effective also when some or all of thelayers in a unit cell have spatially varying properties.

Anti-plane Shear Waves in Layered Composites: Band Structure and Anomalous Wave-refraction