  Solutions of first-order nonlinear hyperbolic conservation laws typically
develop shocks in finite time even with smooth initial conditions. However, in
heterogeneous media with rapid spatial variation, shock formation may be
delayed or avoided. When shocks do form in such media, their speed of
propagation depends on the material structure. We investigate conditions for
shock formation and propagation in heterogeneous media. We focus on the
propagation of plane waves in two-dimensional periodic media with material
variation in only one direction.
  We propose an estimate for the speed of the shocks that is based on the
Rankine-Hugoniot conditions applied to a leading-order homogenized (constant
coefficient) system. We verify this estimate via numerical simulations using
different nonlinear constitutive relations and layered and smoothly varying
periodic media. In addition, we discuss conditions and regimes under which
shocks form in this type of media.
