Propagation without wave equation toward an urban area model

Propagation in random media is a topic of great interest, whose application fields include, among others, the so-called last mile problem as well as the modeling of dense urban area radio communication channels. In this paper, a simple scenario for this issue is considered, with an optical-ray propagation across a medium of disordered lossless scatterers. The propagation medium behaves like a percolating lattice and the goal is to characterize statistically the propagation depth in the medium as a function of the density q of scatterers and of theta-the ray incidence angle. To the best of our knowledge, this approach is totally new. The problem is mathematically formulated as a random walk and the solutions are based on the theory of the martingale random processes. The obtained (approximate) analytical formulas have been validated by means of numerical simulations, demonstrating the applicability of the proposed model for a wide range of the global parameters q and theta. We believe that our results may constitute a promising first step toward the solution of more complicated propagation models and a wide class of communication problems.