Abstract
In this paper, we present a fractal (self-similar) model of acoustic propagation in a porous material with a rigid structure. The fractal medium is modeled as a continuous medium of non-integer spatial dimension. The basic equations of acoustics in a fractal porous material are written. In this model, the fluid space is considered as fractal while the solid matrix is non-fractal. The fluid–structure interactions are described by fractional operators in the time domain. The resulting propagation equation contains fractional derivative terms and space-dependent coefficients. The fractional wave equation is solved analytically in the time domain, and the reflection and transmission operators are calculated for a slab of fractal porous material. Expressions for the responses of the fractal porous medium (reflection and transmission) to an acoustic excitation show that it is possible to deduce these responses from those obtained for a non-fractal porous medium, only by replacing the thickness of the non-fractal material by an effective thickness depending on the fractal dimension of the material. This result shows us that, thanks to the fractal dimension, we can increase (sometimes by a ratio of 50) and decrease the equivalent thickness of the fractal material. The wavefront speed of the fractal porous material depends on the fractal dimension and admits several supersonic values. These results open a scientific challenge for the creation of new acoustic fractal materials, such as metamaterials with very specific acoustic properties.
Highlights
The acoustic equations were written for a fractal porous material with a rigid structure, using the concept of the nabla operator in a non-integer dimensional medium
The problem of a layer of porous material was treated by taking into account the two interfaces of the material, making it possible to derive the reflection and transmission operators in the time domain
The results obtained allowed us to conclude that a fractal porous material can be considered as a non-fractal porous material, but with a effective thickness depending on the fractal dimension
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Carpinteri et al [15,16,17] suggested interesting models with fractional derivatives (socalled local derivatives) of non-integer orders with respect to spatial coordinates for the description of fractal materials. The concern with these models noted by Tarasov [7,8] is that there are not enough differential equations with these fractional derivatives that are solved for various fractal material problems. The generalizations of the gradient, divergence, curl, and Laplace vector operators for fractional and non-integer spaces to describe anisotropic fractal media are not taken into account by the product measure approach. A discussion on the impact of the fractal dimension on the propagation is given
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