Dispersive effects for nonlinear waves in periodic media

First-order hyperbolic systems do not describe dispersive waves, but dispersive effects can arise in such systems as a result of periodically- or randomly-varying coefficients or ambient state. I will discuss examples of this in applications including elasticity and shallow water waves. Effective dispersion can arise due to both reflection and diffraction of waves, and it can drastically alter the long-term evolution of solutions, for instance by leading to solitary waves or by destabilizing shock waves. I will review what is known about these effects and some open questions.