Transmission and reflection at the boundary of a random two-component composite

A half-spacex 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-spacex 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 function. Possible generalizations are discussed in a brief concluding section.