  The refraction properties of phononic crystals are revealed by examining the
anti-plane shear waves in doubly periodic elastic composites with unit cells
containing rectangular and/or elliptical inclusions. The band-structure, group
velocity, and energy-flux vector are calculated using a powerful variational
method which accurately and efficiently yields all the field quantities over
multiple frequency pass-bands. Equifrequency contours and energy-flux vectors
are calculated as functions of the wave-vector. By superimposing the
energy-flux vectors on equifrequency contours in the plane of the wave-vector
components, and supplementing this with a three-dimensional graph of the
corresponding frequency surface,a wealth of information is extracted
essentially at a glance. This way it is shown that a composite with even a
simple square unit cell containing a central circular inclusion can display
negative or positive energy and phase-velocity refractions, or simply performs
a harmonic vibration (standing wave), depending on the frequency and the
wave-vector. Moreover that the same composite when interfaced with a suitable
homogeneous solid can display: 1. negative refraction with negative
phase-velocity refraction; 2. negative refraction with positive phase-velocity
refraction; 3. positive refraction with negative phase-velocity refraction; 4.
positive refraction with positive phase-velocity refraction; or even 5.
complete reflection with no energy transmission, depending on the frequency,
and direction and the wave length of the plane-wave which is incident from the
homogeneous solid to the interface. By comparing our results with those
obtained using the Rayleigh quotient and, for the layered case, with the exact
solutions, the remarkable accuracy and the convergence rate of the present
solution method are demonstrated. MatLab codes with comments will be provided.
