Abstract

Description: A half-space x_2 > 0 is occupied by a two-component statistically-uniform random composite with specified volume fractions and two-point correlation. It is bonded to a uniform half-space x_2 < 0 from which a plane wave is incident. The transmitted and reflected mean waves are calculated via a variational formulation that makes optimal use of the given statistical information. The problem requires the specification of the properties of three media: those of the two constituents of the composite and those of the homogeneous half-space. The complexity of the problem is minimized by considering a model acoustic-wave problem in which the three media have the same modulus but different densities. It is formulated as a problem of Wiener–Hopf type which is solved explicitly in the particular case of an exponentially decaying correlation. A striking feature in this case is that the composite supports exactly two mean acoustic plane waves in any given direction. Each decays exponentially. At low frequencies the rate of decay of one wave is much slower than that of the other; at higher frequencies the decay rates of the two waves are comparable. Thus, in general, there are two transmission coefficients and one reflection coefficient, and the conditions of continuity of traction and displacement of the mean waves do not suffice to determine them: the solution absolutely requires a more complete calculation, such as the one presented.