On the growth and decay of acceleration waves in random media

Martin Ostoja-Starzewski, Jerzy Trebicki

Research output: Contribution to journalArticlepeer-review


We study the effects of material spatial randomness on the growth to shock or decay of acceleration waves. In the deterministic formulation, such waves are governed by a Bernoulli equation da/dx = -β(x)a + β(x)α2, in which the material coefficients H and βrepresent the dissipation and elastic nonlinearity, respectively. In the case of a random microstructure, the wavefront sees the local details: it is a mesoscale window travelling through a random continuum. Upon a stochastic generalization of the Bernoulli equation, both coefficients become stationary random processes, and the critical amplitude αc as well as the distance to form a shock x become random variables. We study the character of these variables, especially as compared to the deterministic setting, for various cases of the random process: (i) one white noise; (ii) two independent white noises; (iii) two correlated Gaussian noises; and (iv) an Ornstein-Uhlenbeck process. Situations of fully positively, negatively or zero correlated noises in fi and βare investigated in detail. Particular attention is given to the determination of the average critical amplitude (ac), equations for the evolution of the moments of a, the probability of formation of a shock wave within a given distance x, and the average distance to form a shock wave. Specific comparisons of these quantities are made with reference to a homogeneous medium defined by the mean values of the {μ,β}x process.

Original languageEnglish (US)
Pages (from-to)2577-2614
Number of pages38
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number1987
StatePublished - 1999
Externally publishedYes


  • Acceleration wave
  • Bernoulli equation
  • Mesoscale
  • Random media
  • Stochastic mechanics
  • Wave propagation

ASJC Scopus subject areas

  • General Mathematics
  • General Engineering
  • General Physics and Astronomy


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