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

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 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 two media comprising the composite have the same modulus but different densities. The homogeneous half-space can have any chosen modulus and density. An approximate formulation based on a stochastic variational principle is formulated and solved explicitly in the particular case of an exponentially decaying correlation function, generalizing a previous solution in which the homogeneous medium had the same modulus as the composite. The main novelty of the present work is that the mean energy fluxes in the composite and reflected from the boundary are calculated, demonstrating explicitly the large backscatter associated with the mean-zero component of the reflected signal and the systematic transfer of energy from the decaying mean transmitted waves into the mean-zero disturbance as distance from the boundary increases.