Incorporation of macroscopic heterogeneity within a porous layer to enhance its acoustic absorptance

Abstract

We seek the response, in particular the spectral absorptance, of a rigidly-backed periodically-(in one horizontal direction) inhomogeneous layer composed of alternating rigid and macroscopically-homogeneous porous portions, submitted to an airborne acoustic plane body wave. The rigorous theory of this problem is given and the means by which the latter can be numerically solved are outlined. At low frequencies, a suitable approximation derives from one linear equation in one unknown. This approximate solution is shown to be equivalent to that of the problem of the same wave incident on a homogeneous, isotropic layer. The thickness h of this layer is identical to that of the inhomogeneous layer, the effective complex body wave velocity therein is identical to that of the porous portion of the inhomogeneous layer, but the complex effective mass density, whose expression is given in explicit algebraic form, is that of the reference homogeneous macroscopically-porous layer divided by the filling factor (fraction of porous material to the total material in one grating period). This difference of density is the reason why it is possible for the lowest-frequency absorptance peak to be higher than that of a reference layer. Also, it is shown how to augment the height of this peak so that it attains unity (i.e., total absorption) and how to shift it to lower frequencies, as is required in certain applications.

On the constant constitutive parameter (e.g., mass density) assumption in integral equation approaches to (acoustic) wave scattering

“…Thus, to compare the answers furnished by the four forms of the IE technique, we were obliged, at the outset, to make the assumption of a constant-density obstacle. But this was not sufficient, because none of the four IE techniques enable the obtention of a mathematically-tractable solution for a macroscopicallyhomogeneous (this term is used either for a medium containing no macroscopic objects or for a porous medium such as a solid network of connected microscopically-thin rods within a fluid for which a homogenization scheme enables its constitutive properties to be approximated [64,100] by those of an 'effective' homogeneous fluid or solid) obstacle of general shape. Consequently, we imposed the additional assumption that the shape of the obstacle be canonical, this meaning that the governing PDE is separable in a given coordinate system and that the boundary of the obstacle can be represented by an equation k =const., where k is one of the coordinates in the chosen coordinate system.…”

“…A second remark has to do with the 'closeness' of the solution of the scattering problem of the 'new' obstacle (i.e., the homogeneous one with canonical-shaped virtual boundary) to that of the problem dealing with the 'old' obstacle (i.e., the one with several objects or one object with arbitrary shape). Homogenization, and, in particular, its mixture formula version, has been shown to give rise to very useful (in both the inverse and forward problem contexts [68,99,100]), and often precise, approximations of the solution, especially at low frequencies. This fact constitutes another, important, justification of the procedure adopted in our study.…”

“…In the geophysics community, this assumption appears a priori to be less justified, but is nonetheless frequently made for reasons of 'simplicity' or 'mathematical convenience' [75]. In the porous media acoustics context (notably for applications dealing with the absorption of airborne sound) it has been found [100] that the effective (bulk) density of typical air-saturated porous materials can be complex, dispersive, and quite different (real part) from the density of air. Sometimes, theoretical studies of scattering are undertaken in a quite general setting (i.e., without assuming constant mass density), but the mass density is taken to be either continuously-varying [90] or constant in the final, numerical stage [32].…”

Resonant amplified seismic response within a hill or mountain

“…In recent years, research in these fields has largely switched to the near-field, stimulated by the discovery of such spectacular effects as SERS (Surface-Enhanced Raman Scattering) [211,240,315,345,365], enhanced frequency-selective (total) absorption (as applied notably to energy harvesting [172,198] or noise reduction [121,122,357,198]), and negative refraction [180,207] in media (recently dubbed metamaterials [368,381]) bounded by, or including, periodic structures [90,211,210,220,77,222,365,108,316,295,6,366,367,203,345,294,112,134,204,76,81,198,199,54,80,97,134,207,261].…”

Forward and inverse acoustic scattering problems involving the mass density

“…Such simple models are often the outcome of certain assumptions and approximations in what is called the domain integral formulation (treated in depth in [94]) of the forward-scattering problem. These assumptions usually have to do with: 1) treating an elastic wave problem (in a solid or porous medium) as an acoustic wave problem [13,27,41,34,42,8] (in a so-called equivalent fluid), 2) treating a microscopically-inhomogeneous (e.g., porous) medium as a macroscopically-homogeneous (effective) medium [7,18,17,19,5,58,93], 3) treating the bioacoustic, marine acoustic, electromagnetic and geophysical problem as one in which the mass density, or another constitutive property, is constant everywhere (i.e., is the same and spatially-constant within the obstacle as well as in the host) [65,30,64,73,22,23,45,74,66,27,84,55,52,15,34,96,7,29,57,5,78,36,37,86,96], 4) treating a 3D problem as a 2.5D, 2D or even 1D problem [72,24,94] (and many other references) 5) treating the potato as a sphere...…”