Effective acoustic equations for a two-phase medium with microstructure

We study acoustic wave propagation in a two-phase medium in which the solid phase is a linear elastic material, and the fluid phase is assumed to be a compressible Newtonian barotropic fluid. Assuming that properties of the medium change rapidly on the small scale ε, we analyze the microscopic nonlinear Navier-Stokes equations and show that they can be linearized when ε tends to zero. Using a variant of Tartarʹs method of oscillating test functions, we derive effective acoustic equations which turn out to be viscoelastic. In order to treat disordered materials occurring in nature, we develop a new approach to describing geometry of a nonperiodic medium with length scale separation. Our approach is not based on probabilistic considerations. Instead, we postulate that certain inequalities hold uniformly on the microscale.