Homogenizing the ultrasonic response of wet cortical bone

We outline the mathematical model of the time-harmonic ultrasonic response of wet cortical bone. Using two-scale asymptotics, we derive an effective model of acoustic wave propagation in a two-phase medium modeling a fine mixture of linear piezo-elastic solid and a viscous, Newtonian, ionic bearing fluid. Following the works of Moyne and Murad, and Lemaire et. al. for the quasi-static case, we develop a two-scale homogenization method for the dynamical system, albeit the time harmonic case. The idea is to connect the bulk pressure to the small displacement by an assumption used in acoustics, i.e. the pressure p ≈ - ρfa2f divu; where ρf is the fluid density, af the speed of sound in the fluid, and u is the displacement. We investigate several time scales; one is associated with high frequency domination which leads to different hierarchies. The ratio ε of a typical size of the microstructural inhomogeneity and the macroscopic length scale is a small parameter of the problem. Another possibly small parameter is the Peclet number which influences the type of effective equations which are obtained. A brief asymptotic analysis is presented.