In this dissertation we present a numerical study of 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 investigate impurities consisting of beads with different interaction exponents compared to the medium they are embedded in. We find that the impurities cause both reflection and transmission, including the formation of a multipulse structure, independently of whether the exponent in the impurities is smaller or larger than in the embedding medium. For wave propagation effects at interfaces and impurities we give an explanation in terms of quasi-particle collisions. Lastly, we study disorder and periodicity in the interaction potential exponents. Solitary waves in media with randomised interaction exponents experience exponential decay, where the dependence of the decay rate is similar to the case of randomised bead masses. For chains with interaction exponents alternating between two values we find qualitatively different propagation properties depending on the choice of the two exponents. We find regimes with exponential decay and stable solitary wave propagation with pairwise collective behaviour. For certain exponent values we observe a new type of stable confined wave, with a periodically changing wave form.