Inverse problems in air-saturated porous media via reflected waves
Résumé
Direct and inverse scattering problem in air filled porous materials is solved. A simple ultrasonic reflectivity method is proposed for measuring the porosity of porous materials having a rigid frame. The proposed method is based on 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 time domain model of wave propagation was initially introduced by the authors [Z. E. A. Fellah and C. Depollier, J. Acoust. Soc. Am. 107, 683 (2000)]. The viscous and thermal losses of the medium are described by the model devised by Johnson et al. [D. L. Johnson, J. Koplik, and R. Dashen, J. Fluid. Mech. 176, 379 (1987)] and J. F. Allard (Chapman and Hall, London, 1993)] modified by a fractional calculus-based method applied in the time domain. The reflected acoustic wave by air filled porous media can be approximated by the reflected wave at the first interface. The expression of the reflection coefficient is reduced to a simple expression depending on porosity, tortuosity, and incidence angle. An expression of the porosity function of the tortuosity and incidence angle is obtained by solving the inverse problem. Experimental and numerical validation results of this method are presented. This method has the advantage of being rapid and efficient.
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