Shape of a Wave Front in a Heterogenous Medium S. Mehdi Vaez Allaei 1 and Muhammad Sahimi 2, * 1 Institute for Advanced Studies in Basic Sciences, Gava Zang, Zanjan 45195-1159, Iran 2 Mork Family Department of Chemical Engineering and Materials Science, University of Southern California, Los Angeles, California 90089-1211, USA (Received 24 July 2005; published 23 February 2006) 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,   H, where  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,   0 (logarithmic growth) at short times, followed by a crossover to the classical value,   1=2, at long times. DOI: 10.1103/PhysRevLett.96.075507 PACS numbers: 62.30.+d, 42.15.Dp, 47.56.+r, 91.60.Lj The relation between the static and dynamical properties of heterogeneous materials and media has been a problem of fundamental interest for decades. In an inhomogene- ous multiphase material one is interested [1], for example, in the relation between the material’s morphology—the shape, size, spatial distribution, and the local or micro- scopic properties of the material’s constituent phases— and any dynamical process that takes place in the material, such as transport of electrical current, fluid, or stress. Thus, characterization of the morphology of an inhomogeneous medium and measuring and modeling its macroscopic properties have been problems of great interest for a long time. Wave propagation in inhomogeneous media has been an important means of gaining information on the me- dia’s morphology, contents, and properties. Understand- ing how waves propagate in heterogeneous media is fun- damental to such important problems as earthquakes; underground nuclear explosions; the morphology of oil, gas, and geothermal reservoirs; oceanography; and mate- rials and medical sciences [2– 4]. For example, seismic wave propagation and reflection are utilized [2] to not only estimate the hydrocarbon content of a potential oil field, but also the spatial distribution of its porosity and fractures and faults. The same essential concepts and techniques are used in such diverse fields as materials science and medicine. The focus of this Letter is on how acoustic and elastic waves propagate in an inhomogeneous medium. In par- ticular, the relation between the shape of a wave front (WF) and the morphology of an inhomogeneous medium in which the wave is propagating is studied. Consider, as an example, propagation of seismic waves in rock which represents a highly heterogeneous medium. In this prob- lem, the interference of the waves that have undergone multiple scattering, caused by the rock’s heterogeneities, strongly influences how the waves propagate in the rock. The shape of the WF is influenced by two properties: (1) the change in the wave transmission direction caused by the heterogeneity, and (2) slow down or speed up of the wave speed. Whereas the shape of a propagating WF in a homogeneous medium is smooth and well defined, the same is not true about propagation of the same waves in rock. We study propagation of acoustic as well as elastic waves in strongly heterogeneous media using three distinct models, namely, a continuum scalar model, and two dis- crete models, one scalar and one vector model. Both spa- tially random and power-law correlated spatial distribu- tions of the local elastic moduli are considered. One main goal of the Letter is to explore the possibility of the ex- istence of a relation between the shape of a propagating WF and a medium’s spatial distribution of its local elastic moduli. Propagation of acoustic waves was previously studied by several groups [5–7]. However, previous studies [5–7] considered neither the type of systems that we model in the present Letter, nor studied the issues that we investigate. In addition to its general importance, our study is motivated in part by recent works that indicate that, (1) the distribution of the elastic moduli of heterogeneous rock follows a fractional Brownian motion (FBM) [8], a stochastic distri- bution with long-range, nondecaying correlations charac- terized by a power-law correlation function (see below), and (2) depending on the spatial distribution of the elastic moduli, one may have localized acoustic waves in any dimensions [9], leading to pinning of the WF and rough- ening of its shape. The continuum model. —Acoustic wave propagation in a medium with a distribution of elastic constants is described by the scalar wave equation: @ 2 @t 2 x;t r xrx;t  0; (1) which represents the continuum limit of a system with off- PRL 96, 075507 (2006) PHYSICAL REVIEW LETTERS week ending 24 FEBRUARY 2006 0031-9007= 06=96(7)=075507(4)$23.00 075507-1  2006 The American Physical Society