An ultrasonic reflectivity method is proposed for measuring porosity and tortuosity of porous materials having a rigid frame. Porosity is the relative fraction by volume of the air contained within a material. Tortuosity is a geometrical parameter which intervenes in the description of the inertial effects between the fluid filled the porous material and its structure at high frequency range. It is generally easy to evaluate the tortuosity from transmitted waves, this is not the case for porosity because of its weak sensitivity in transmitted mode. The proposed method is based on measurement of reflected wave by the first interface of a slab of rigid porous material. This method is obtained from a temporal model of the direct and inverse scattering problems for the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material having a rigid frame [Z. E. A. Fellah, M. Fellah, W. Lauriks, and C. Depollier, J. Acoust. Soc. Am. 113, 61 (2003)]. Reflection and transmission scattering operators for a slab of porous material are derived from the responses of the medium to an incident acoustic pulse at oblique incidence. The porosity and tortuosity are determined simultaneously from the measurements of reflected waves at two oblique incidence angles. Experimental and numerical validation results of this method are presented.

This paper provides a temporal model of the direct and inverse scattering problem for the propagation of transient ultrasonic waves in a homogeneous isotropic slab of porous material having a rigid frame. This new time domain model of wave propagation takes into account the viscous and thermal losses of the medium as described by the model of Johnson et al. [D. L. Johnson, J. Koplik, and R. Dashen, J. Fluid. Mech. 176, 379 (1987)] and Allard [J. F. Allard (Chapman and Hall, London, 1993)] modified by a fractional calculus based method applied in the time domain. This paper is devoted to the analytical calculus of acoustic field in a slab of porous material. The main result is the derivation of the expression of the scattering operators (reflection and transmission) which are the responses of the medium to an incident acoustic pulse. In this model the reflection operator is the sum of two contributions: the first interface and the bulk of the medium. Experimental and numerical results are given as a validation of our model.

Ultrasonic wave propagation in human cancellous bone is considered. Reflection and transmission coefficients are derived for a slab of cancellous bone having an elastic frame using Biot’s theory modified by the Johnson et al. model for viscous exchange between fluid and structure. Numerical simulations of transmitted waves in the time domain are worked out by varying the modified Biot parameters. The sensitivity of each physical parameter used in the theory has been studied in transmission. Some parameters play an important role in slow wave waveform, such as the viscous characteristic length and pore fluid bulk modulus. However, other parameters play an important role in the fast wave waveform, such as solid density and shear modulus. We also note from these simulations that some parameters, such as porosity, tortuosity, thickness, solid bulk modulus and skeletal compressibility frame, play an important role simultaneously in both fast and slow waveforms compared to other parameters which act on the waveform of just one of the two waves. Experimental results for slow and fast waves transmitted through human cancellous bone samples are given and compared with theoretical predictions.

Wave propagation of acoustic waves in porous media is considered. The medium is assumed to have a rigid frame, so that the propagation takes place in the air which fills the material. The Euler equation and the constitutive relation are generalized to take into account the dispersive nature of these media. It is shown that the connection between the fractional calculus and the behavior of materials with memory allows time-domain wave equations, the coefficients of which are no longer frequency dependent, to be worked out. These equations are suited for direct and inverse scattering problems, and lead to the complete determination of the porous medium parameters.