are composed of a disordered or random collection of homogeneous layers. As an example, consider the model A variety of problems involving disordered systems can be formulated mathematically in terms of products of random transfer matri-shown in Fig. 1. Here we have chosen a pseudo-random ces, including Ising spin systems, optical and continuum mechanical sequence (to be described later) of layer thicknesses and wave propagation, and lattice dynamical systems. The growth or material property (elastic stiffnesses, dielectric constant, decay of solutions to these problems is governed by the Lyapunov etc). We can scale these to reflect typical values for differspectrum of the product of these matrices. For continuum mechanient applications. Figure 2 shows a 2D acoustic finite-differcal or optical wave propagation, the transfer matrices arise from the application of boundary conditions at the discontinuities of the ence wave propagation simulation for such a model; at a medium. Similar matrices arise in lattice-based systems when the given depth, the plot shows the energy recorded at that equations of motion are solved recursively. For the disordered latdepth as a function of time. The first energy seen at any tice mechanical system, on which we focus in this paper, the scatter-