Abstract
A fractional-order wave equation is established and solved for a space of three dimensions using spherical coordinates. An equivalent fluid model is used in which the acoustic wave propagates only in the fluid saturating the porous medium; this model is a special case of Biot’s theory obtained by the symmetry of the Lagrangian (invariance by translation and rotation). The basic solution of the wave equation is obtained in the time domain by analytically calculating Green’s function of the porous medium and using the properties of the Laplace transforms. Fractional derivatives are used to describe, in the time domain, the fluid–structure interactions, which are of the inertial, viscous, and thermal kind. The solution to the fractional-order wave equation represents the radiation field in the porous medium emitted by a point source. An important result obtained in this study is that the solution of the fractional equation is expressed by recurrence relations that are the consequence of the modified Bessel function of the third kind, which represents a physical solution of the wave equation. This theoretical work with analytical results opens up prospects for the resolution of forward and inverse problems allowing the characterization of a porous medium using spherical waves.
Highlights
Acoustic absorbers, such as plastic foams or fibrous materials [1,2,3,4], are widely used for sound insulation, passive sound control [1], and noise reduction in the aeronautical and building industries [5]
We show in this paper, by using analytical computational developments, that Green’s function of the porous medium and the solution of its fractional wave equation can be expressed by recurrence relations and depend on
The analytical calculation of the inverse Laplace transforms of expressions (43) and (44). Makes it possible to obtain an analytical form in the time domain of the solution to the fractional spherical-wave equation. This solution allows us to predict the ultrasonic wave in spherical coordinates propagating in a porous medium by knowing the wave p(0, t) incident in the medium given by the initial conditions and the properties of the material represented by the wave number k(s), which essentially depends on the porosity, tortuosity, viscous and thermal characteristic lengths, and viscous and thermal surfaces
Summary
Acoustic absorbers, such as plastic foams or fibrous materials [1,2,3,4], are widely used for sound insulation, passive sound control [1], and noise reduction in the aeronautical and building industries [5]. Knowing the experimental incident signal and the solution of the propagation equation (Green function of the medium) makes it possible to simulate the acoustic field propagating in the porous material, as well as to predict the waves reflected and transmitted by a slab of material [19]. We show in this paper, by using analytical computational developments, that Green’s function of the porous medium (in spherical coordinates) and the solution of its fractional wave equation can be expressed by recurrence relations and depend on. Green’s function of a unidirectional propagation problem and its fractional wave equation solution These results open up new perspectives for the prediction of sound absorption in porous media and their ultrasonic characterization, since the resolution of the propagation equation is the basis for solving the direct and inverse problems
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