  We study solitary wave propagation in 1D granular crystals with Hertz-like
interaction potentials. We consider interfaces between media with different
exponents in the interaction potential. For an interface with increasing
interaction potential exponent along the propagation direction we obtain mainly
transmission with delayed secondary transmitted and reflected pulses. For
interfaces with decreasing interaction potential exponent we observe both
significant reflection and transmission of the solitary wave, where the
transmitted part of the wave forms a multipulse structure. We also investigate
impurities consisting of beads with different interaction exponents compared to
the media they are embedded in, and we find that the impurities cause both
reflection and transmission, including the formation of multipulse structures,
independent of whether the exponent in the impurities is smaller than in the
surrounding media. We explain wave propagation effects at interfaces and
impurities in terms of quasi-particle collisions. Next we consider wave
propagation along Hertz-like granular chains of beads in the presence of
disorder and periodicity in the interaction exponents present in the Hertz-like
potential, modelling, for instance, inhomogeneity in the contact geometry
between beads in the granular chain. We find that solitary waves in media with
randomised interaction exponents (which models disorder in the contact
geometry) experience exponential decay, where the dependence of the decay rate
is similar to the case of randomised bead masses. In the periodic case of
chains with interaction exponents alternating between two fixed values, we find
qualitatively different propagation properties depending on the choice of the
two exponents. In particular, we find regimes with either exponential decay or
stable solitary wave propagation with pairwise collective behaviour.
