## Shock waves in random media

Shock waves appear generically in the solution of nonlinear hyperbolic conservation laws. Their presence is important in many nonlinear wave models, including those of fluid dynamics, water waves, traffic flow, and elasticity. While simulations are often based on the assumption of a homogeneous background state, many nonlinear wave processes occur in the context of random spatial or temporal fluctuations, at least at the microscale. It has been shown recently that periodic modulation of a medium can have dramatic effects on the propagation of shock waves (see Ketcheson & LeVeque 2012). Indeed, strong variations in the underlying medium tend to destabilize shock waves and lead instead to the formation of solitary waves through the mechanism of effective dispersion. Preliminary experiments indicate that the formation of shock waves is also delayed in the presence of random material fluctuations (see Fouque & Nachbin 2003).

The purpose of this project is to investigate the behavior of shock waves in random materials, using asymptotic analysis and numerical simulations. Starting with one-dimensional models of elasticity and water waves, effective equations will be developed based on homogenization. Pseudospectral simulations of these equations will be compared with fihite volume simulations of the full spatially-varying material problem in order to understand the influence of random materials on shock wave propagation.