Shape of a wave front in a heterogenous medium

Wave propagation in a heterogeneous medium, characterized by a distribution of local elastic moduli, is studied. Both acoustic and elastic waves are considered, as are spatially random and power-law correlated distributions of the elastic moduli with nondecaying correlations. Three models-a continuum scalar model, and two discrete models-are utilized. Numerical simulations indicate the existence, at all times, of the relation, alpha=H, where alpha is the roughness exponent of the wave front in the medium, and H is the Hurst exponent that characterizes the spatial correlations in the distribution of the local elastic moduli. Hence, a direct relation between the static morphology of an inhomogeneous correlated medium and its dynamical properties is established. In contrast, for a wave front in random media, alpha=0 (logarithmic growth) at short times, followed by a crossover to the classical value, alpha=1/2, at long times.