  Linear and nonlinear mechanisms for conical wave propagation in
two-dimensional lattices are explored in the realm of phononic crystals. As a
prototypical example, a statically compressed granular lattice of spherical
particles arranged in a hexagonal packing configuration is analyzed. Upon
identifying the dispersion relation of the underlying linear problem, the
resulting diffraction properties are considered. Analysis both via a heuristic
argument for the linear propagation of a wavepacket, as well as via asymptotic
analysis leading to the derivation of a Dirac system suggests the occurrence of
conical diffraction. This analysis is valid for strong precompression i.e.,
near the linear regime. For weak precompression, conical wave propagation is
still possible, but the resulting expanding circular wave front is of a
non-oscillatory nature, resulting from the complex interplay between the
discreteness, nonlinearity and geometry of the packing. The transition between
these two types of propagation is explored.
