On the growth and decay of acceleration waves in random media

We study the effects of material spatial randomness on the growth to shock or decay of acceleration waves. In the deterministic formulation, such wave s are governed by a Bernoulli equation d alpha/dx = -mu(x)alpha + beta(x)al pha(2), in which the material coefficients mu and beta represent the dissip ation and elastic nonlinearity, respectively. In the case of a random micro structure, the wavefront sees the local details: it is a mesoscale window t ravelling through a random continuum. Upon a stochastic generalization of t he Bernoulli equation, both coefficients become stationary random processes , and the critical amplitude oc, as well as the distance to form a shock x( infinity), become random variables. We study the character of these variabl es, especially as compared to the deterministic setting, for various cases of the random process: (i) one white noise; (ii) two independent white nois es; (iii) two correlated Gaussian noises; and (iv) an Ornstein-Uhlenbeck pr ocess. Situations of fully positively, negatively or zero correlated noises in mu and beta are investigated in detail. Particular attention is given t o the determination of the average critical amplitude (a,): equations for t he evolution of the moments of a, the probability of formation of a shock w ave within a given distance rc, and the average distance to form a shock wa ve. Specific comparisons of these quantities are made with reference to a h omogeneous medium defined by the mean values of the (mu, beta) process.