Stochastic differential equation for wave diffusion in random media

In this work, we present a statistical analysis of the wave motion through random media with perfect spatial disorder of inclusions. It is assumed that such a disorder can be tackled with the random potential function theory, whence the propagation of waves naturally turns to a diffusion process. The associated Itoô drift-diffusion process, and its Fokker-Planck equation are derived. It is found that the “ensemble” wave, i.e., the collective wave motion, fluctuates in space as a geometric Brownian motion. Finally, the effect of a double-well potential with random (vibrating) valleys is studied qualitatively by the Monte Carlo method. In practice, this situation occurs for high concentration and perfect dispersion of conductive/dielectric fillers, i.e., whose location and orientation are completely randomized.