On the critical amplitudes of acceleration wave to shock wave transition in white noise random media

The problem of transition of acceleration waves into shock waves is addressed in the context of weakly random media. A class of random media is modeled by a vector white noise random process representing two material coefficients appearing in the Bernoulli equation governing the evolution of acceleration waves. The problem of shock formation, which involves a stochastic competition of dissipation and elastic nonlinearity, is treated using a diffusion formulation for the Markov process of the inverse amplitude. The first four moments of the critical inverse amplitude are derived explicitly as functions of the means and crosscorrelations of the underlying vector random process. It is found that the Stratonovich as well as the Itô interpretation of the stochastic Bernoulli equation lead to an increase of the average critical amplitude of the random medium problem over the critical amplitude of the deterministic homogeneous medium problem. Probability distribution of the critical inverse amplitude is found to be, in general, of Pearson's Type IV.