A time-domain numerical modeling of two-dimensional wave propagation in porous media with frequency-dependent dynamic permeability

Wave equations for porous media described by the Biot model

Single-mode equivalent space-time representations of the acoustic wave propagating in a Biot poroelastic medium have previously been found only for asymptotic cases: In the low frequency regime, where the viscous skin depth is greater than the characteristic pore size, the time domain equivalent is represented with integer order temporal and spatial loss terms, whereas in the high frequency regime, it is represented with fractional order temporal and spatial loss terms. In the current work, a time domain representation in terms of a partial differential equation is proposed for all three wave solutions of the Biot model across all frequencies, and it is derived from the material response function of the Biot poroelastic medium with suitable approximations for the compressional modes and the dynamic permeability. The dynamic permeability in the time domain is represented by a fractional pseudo-differential operator. Optimal correction factors are introduced into the wave equation to compensate for the discrepancies in the compressional wave dispersion and attenuation. Additionally, the method for incorporating the squirt flow mechanism into the wave equation via the Extended Biot poroviscoelastic model is described. The proposed wave equation has a physical basis and satisfies the passivity criterion.

Sound absorption by acoustic microlattice with optimized pore configuration

The great progress in material science and nano-micro fabrication enables the applications of metamaterials with well-defined and well-organized microstructures for noise reduction. However, what intrinsic morphology of the metamaterial would result in optimum sound absorbing efficiency remains uncertain. This work presents a microlattice metamaterial, comprising well-defined and organized material morphology in terms of pore size and porosity, for generating optimum sound dissipation. A compact governing equation is established and verified experimentally to show that the optimum sound absorption can only be reached when the pore size equals twice the thickness of a viscous boundary layer.

Design of broadband time-domain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models

A methodology to design broadband time-domain impedance boundary conditions (TDIBCs) from the analysis of acoustical models is presented. The derived TDIBCs are recast exclusively as first-order differential equations, well-suited for high-order numerical simulations. Broadband approximations are yielded from an elementary linear least squares optimization that is, for most models, independent of the absorbing material geometry. This methodology relies on a mathematical technique referred to as the oscillatory-diffusive (or poles and cuts) representation, and is applied to a wide range of acoustical models, drawn from duct acoustics and outdoor sound propagation, which covers perforates, semi-infinite ground layers, as well as cavities filled with a porous medium. It is shown that each of these impedance models leads to a different TDIBC. Comparison with existing numerical models, such as multi-pole or extended Helmholtz resonator, provides insights into their suitability. Additionally, the broadly-applicable fractional polynomial impedance models are analyzed using fractional calculus.

A generalized recursive convolution method for time-domain propagation in porous media

An efficient numerical method, referred to as the auxiliary differential equation (ADE) method, is proposed to compute convolutions between relaxation functions and acoustic variables arising in sound propagation equations in porous media. For this purpose, the relaxation functions are approximated in the frequency domain by rational functions. The time variation of the convolution is thus governed by first-order differential equations which can be straightforwardly solved. The accuracy of the method is first investigated and compared to that of recursive convolution methods. It is shown that, while recursive convolution methods are first or second-order accurate in time, the ADE method does not introduce any additional error. The ADE method is then applied for outdoor sound propagation using the equations proposed by Wilson et al. in the ground [(2007). Appl. Acoust. 68, 173-200]. A first one-dimensional case is performed showing that only five poles are necessary to accurately approximate the relaxation functions for typical applications. Finally, the ADE method is used to compute sound propagation in a three-dimensional geometry over an absorbing ground. Results obtained with Wilson's equations are compared to those obtained with Zwikker and Kosten's equations and with an impedance surface for different flow resistivities.

Optimization on microlattice materials for sound absorption by an integrated transfer matrix method

Materials with well-defined microlattice structures are superlight, stable, and thus bear great potential in sound absorption. An integrated transfer matrix method (TMM) is proposed to evaluate the sound absorbing efficiency of these lattice materials, in which a massive number of micropores are densely placed. A comparison between integrated TMM and conventional TMM reveals that the proposed approach offers better predictions on sound absorption of microlattice. This approach is then employed to optimize the microlattice material to determine the best pore and porosity that lead to maximum absorbing efficiency capability and minimum required thickness to attain a target sound absorption.