×

zbMATH — the first resource for mathematics

Propagation of transient acoustic waves in layered porous media: Fractional equations for the scattering operators. (English) Zbl 1099.74035
Summary: Acoustic waves scattering from a rigid air-saturated porous medium is studied in the time domain. The medium is one-dimensional and its physical parameters are depth-dependent, i.e., the medium is layered. The loss and dispersion properties of the medium are due to the fluid-structure interaction induced by wave propagation. They are modeled by generalized susceptibility functions which express the memory effects in the propagation process. The wave equation is then a fractional telegraph equation. Two relevant scattering operators – transmission and reflection operators – give the scattered fields from the incident wave. They are obtained from Volterra equations which are fractional equations for the scattering operators.

MSC:
74J20 Wave scattering in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
26A33 Fractional derivatives and integrals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fellah, Z. E. A. and Depollier, C., ?Transient acousticwave propagation in rigid porous media: A time-domain approach?, Journal of the Acoustical Society of America 107, 2000, 683-688. · doi:10.1121/1.428250
[2] Johnson, D. L., ?Recent developpments in the acoustic properties of porous media?, in Proceedings of the International School of Physics ?Enrico Fermi? Course XCIII Frontiers in Physical Acoustics, D. Sette(ed.), North Holland, 1986, pp. 255-290.
[3] Allard, J. F., Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials, Chapman & Hall, London, 1993, pp. 255-290.
[4] Lafarge, D., Lemarnier, P., Allard, J. F., and Tarnow, V., ?Dynamic compressibility of air in porous structures at audible frequencies?, Journal of the Acoustical Society of America102, 1997, 1995-2006. · doi:10.1121/1.419690
[5] Johnson, D. L., Koplik, J., and Dashen, R., ?Theory of dynamic permeability and tortuosity in fluid-saturated porous media?, Journal of Fluid Mechanics176, 1987, 379-402. · Zbl 0612.76101 · doi:10.1017/S0022112087000727
[6] Champoux, Y. and Allard, J. F., ?Dynamic tortuosity and bulk modulus in air-saturated porous media?, Journal of Applied Physics70, 1991, 1975-1979. · doi:10.1063/1.349482
[7] Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives, Gordon andBreach, Yverdon, Swittzerland, 1993. · Zbl 0818.26003
[8] Hanyga, A. and Rok, V., ?Wave propagation in micro-heterogeneous porous media: A model based on an integro-differential waveequation?, Journal of the Acoustical Societyof America107, 2000, 2965-2972. · doi:10.1121/1.429326
[9] Cascaval, R. C., Eckstein, E. C., Frota, C. L., Goldstein, J. A., ?Fractional telegraph equations?, Journal of Mathematic Analysis and Application276, 2002, 145-159. · Zbl 1038.35142 · doi:10.1016/S0022-247X(02)00394-3
[10] Kristensson, G. and Krueger, R. J., ?Direct and inverse scattering in the time domain for a dissipative wave equation: I. Scatteringoperators?, Journal of MathematicalPhysics27, 1986, 1667-1682; Kristensson, G. and Krueger, R. J., ?Direct and inverse scattering in the time domain for a dissipative wave equation: II. Simultaneous reconstruction of dissipation and phase velocity profiles?, Journal of Mathematical Physics27, 1986, 1683-1693. · Zbl 0595.45018
[11] Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 2, Interscience, New York, 1962. · Zbl 0099.29504
[12] Stokes, G. C., ?On the intensity of the light reflected and transmitted through a pile of plates?, in Mathematical and Physical Papers, Vol. 2, Cambridge University Press, Cambridge, 1883.
[13] Fellah, Z. E. A., Depollier, C., and Fellah, M., in preparation.
[14] Kirsh, A. and Kress, R., ?Two methods for solvingthe inverse acoustic scattering problem?, Inverse Problems4, 1988, 749-770. · Zbl 0698.65076 · doi:10.1088/0266-5611/4/3/013
[15] Weng, C. C., Waves and Fields in Inhomogeneous Media, IEEE Press, New York, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.