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We use an asymptotic expansion introduced by Benilov and Pelinovsky to study the propagation of a weakly nonlinear wave pulse through a stationary random medium in one space dimension, where the wave motion is governed by a hyperbolic system of conservation laws. With respect to a realization-dependent reference frame, the leading order solution is non-random, as in the linear theory of O'Doherty and Anstey. This self-averaging of the leading order solution means that closure problems do not arise in the nonlinear theory. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower order dissipative term that describes the effects of double scattering of waves off fluctuations in the medium. We apply the asymptotic expansion to the propagation of sound waves in gas dynamics and nonlinear elasticity, and discuss the influence of random fluctuations on shock formation.