Transient acoustic wave propagation in bone-like porous materials using the theory of poroelasticity and fractional derivative: a sensitivity analysis

Abstract

In this paper, the transient acoustic wave propagation in a bone-like porous material saturated with a viscous fluid was investigated using Biot’s theory. Due to the interaction between the viscous fluid and solid skeleton, the damping behavior is proportional to a fractional power of frequency, i.e., the dynamic tortuosity was written in terms of the fractional power of frequency. Furthermore, to describe the viscous interaction of fluid and solid in the time domain, the fractional derivative was used. The fast and slow waves, which are the solutions to Biot’s equations, were described by fractional calculus in the time domain. The reflection and transmission operators were expressed in the Laplace domain and inverted into the time domain using Durbin’s numerical inversion. Once the numerical implementation was validated, the effects of porosity and viscosity on the stress, and reflected and transmitted waves were investigated. The results showed that by increasing the porosity the stress in a bone-like material filled with either air or bone marrow increases. The transmitted pressure decreases by increasing the porosity. The reflected pressure decreases for low viscous fluid when the porosity increases while it increases when the viscosity of the fluid is high. In addition, the results showed the importance of taking into account the fractional derivatives in the transient wave propagation in such porous materials.