Propagation of Longitudinal Waves in Circularly Cylindrical Bone Elements

Abstract

Two-phase poroelastic material is taken as a model of the living bone in the sense that the osseous tissue is treated as a linear isotropic perfectly elastic solid, and the fluid substances filling the pores as a perfect fluid. Using Biot’s equations, derived in his consolidation theory, four coupled governing differential equations for the propagation of harmonic longitudinal waves in circularly cylindrical bars of poroelastic material are derived. A longer manipulation reduces the task of solution to a single ordinary differential equation with variable coefficients and a regular singular point. The equation is solved by Frobenius’ method. Three boundary conditions on the curved surface of the bar, expressing the absence of external loading and the permeability of the surface, supply a system of three linear equations in three unknown coefficients. A nontrivial solution of the system gives two phase velocities of propagation of longitudinal waves in agreement with the finding of Biot for an infinite medium. A simplification to the purely elastic case yields the elementary classical result for the longitudinal waves.