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When sound propagates in a porous medium, it is attenuated via several energy loss mechanisms which are switched on or o as the excitation frequency varies. The classical way of measuring acoustic energy loss in porous materials uses the Kundt impedance tube. However, due to its short length, measurements are made in the steady state harmonic regimes. Its lower cuto frequency is often limited to a few hundreds of Hertz.Two long acoustic waveguides were assembled from water pipes and mounted to create test-rigs for the low-frequency acoustic characterization of monolayer and stratied airsaturated poroelastic materials. The rst waveguide was straight and had a length of 120 m, while the second was coiled to gain space and was 135 m long. The long waveguides appeal to very low frequency measurements using impulsive acoustic waves (with rich spectral content) because the incident waves can be separated in time from echoes o the extremities of the guides. The transmission coecient of porous materials recovered using the two waveguides compared well with those from the transfer matrix method (TMM) used here in combination with Biot's 1962 theory to describe propagation in porous dissipative media. This wave-material interaction model permitted the recovery of the properties of poroelastic materials from transmitted acoustic waves propagating in air. The parameters involved are the Young's moduli, Poisson ratio and microstructural properties such as tortuosity and permeability. Being able to descend to lower frequencies guarantees the correct verication of the magnitude of the measured transmission coecient which approaches unity towards the static frequency. The coiled and straight 1 waveguides were found to be equivalent and provided data down to frequencies of the order of ≈ 12 Hz.

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.

In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shear-thickening fluid becomes more shear thickening. This approach has many useful applications in industry, for the investigation of blood flow and fractal fluid hydrology.

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