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.