To what extent randomly layered material can be used for modelling ultrasonic wave propagation in cancellous bone

It has been suggested that a periodical two-layered medium might by considered as the simplified model of wave propagation in cancellous bone. In the present paper an approach is proposed to describe a one-dimensional problem of wave propagation in the cancellous bone modeled as a randomly layered material composed of two kinds of alternating elastic layers having randomly distributed thicknesses and different acoustic properties. A compact form of the dispersion equation and the relationships for reflection coefficients and acoustic impedance of the randomly layered medium is obtained. The wave parameters are functions of acoustical properties of the layers and the distributions of their thicknesses. The obtained equations are used to derive the analytical expressions for three special cases: for long waves; for a periodic medium composed of layers of constant thicknesses; and for random medium characterized by uniform distribution of layer thicknesses. Special attention is paid to the frequency analysis of the wave parameters characterized by strong and weak acoustic inhomogeneity. It is shown that although the developed model is a rough simplification of the cancellous bone architecture, it has potential for future research, particularly for higher frequency range, when the scattering effects play the predominant role in wave attenuation.