Homogenizing the time-harmonic acoustics of bone: The monophasic case

For the interrogation of cancellous bone using ultrasound, we undertake a derivation of the time-harmonic, acoustic equations, idealizing the bone as a periodic arrangement of a Kelvin-Voigt viscoelastic porous matrix containing a viscous fluid. Since we are interested in acoustics, rather than filtration, we assume that the fluid is slightly compressible. Moreover, we study the monophasic case. A priori estimates are obtained for the time-harmonic equations. By letting the characteristic size of the inhomogeneities tend to zero and passing to the limit in the sense of the two-scale convergence, the effective equations for the monophasic vibrations are obtained, for which we prove existence and uniqueness.