Nonlocal description of sound propagation through an array of Helmholtz resonators

A generalized macroscopic nonlocal theory of sound propagation in rigid-framed porous media saturated with a viscothermal fluid has been recently proposed, which takes into account both temporal and spatial dispersion. Here, we consider applying this theory capable to describe resonance effects, to the case of sound propagation through an array of Helmholtz resonators whose unusual metamaterial properties such as negative bulk moduli, have been experimentally demonstrated. Three different calculations are performed, validating the results of the nonlocal theory, relating to the frequency-dependent Bloch wavenumber and bulk modulus of the first normal mode, for 1D propagation in 2D or 3D periodic structures.Comment: 19 page

Dynamic compressibility of air in porous structures at audible frequencies

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Nonlocal dynamics of dissipative phononic fluids

We describe the nonlocal effective properties of a two-dimensional dissipative phononic crystal made by periodic arrays of rigid and motionless cylinders embedded in a viscothermal fluid such as air. The description is based on a nonlocal theory of sound propagation in stationary random fluid/rigid media that was proposed by Lafarge and Nemati [Wave Motion 50, 1016 (2013)WAMOD90165-212510.1016/j.wavemoti.2013.04.007]. This scheme arises from a deep analogy with electromagnetism and a set of physics-based postulates including, particularly, the action-response procedures, whereby the effective density and bulk modulus are determined. Here, we revisit this approach, and clarify further its founding physical principles through presenting it in a unified formulation together with the two-scale asymptotic homogenization theory that is interpreted as the local limit. Strong evidence is provided to show that the validity of the principles and postulates within the nonlocal theory extends to high-frequency bands, well beyond the long-wavelength regime. In particular, we demonstrate that up to the third Brillouin zone including the Bragg scattering, the complex and dispersive phase velocity of the least-attenuated wave in the phononic crystal which is generated by our nonlocal scheme agrees exactly with that reproduced by a direct approach based on the Bloch theorem and multiple scattering method. In high frequencies, the effective wave and its associated parameters are analyzed by treating the phononic crystal as a random medium.United States. Office of Naval Research (N00014-13-1-0631

Transients in porous media: asymptotic time-domain Green functions and limits of current frequency-domain models

Time domain responses of porous media have been studied by some authors, but generally the possible descriptions have been given in the frequency domain. The aim of this paper, limited to materials with rigid skeleton considered as equivalent fluids, is to compare the descriptions by Johnson-Allard ($JA$%) as well as by Pride-Lafarge ($PL$) with i) some analytical, approximate formulas, based upon asymptotic high frequency expansion ; ii) the exact formula by Zwikker and Kosten for the case of cylindrical pores. The paper starts with a short summary of the statement of the different general full frequency models ($JA$ and $PL).$ The Green function in the time domain is shown to exhibit interesting properties of materials. In particular the maximum response depends on one dimensionless parameter only, which is denoted $\xi$ and is the ratio of the travelled distance to the product of the \textquotedblleft frozen\textquotedblright\ sound speed and a characteristic viscous relaxation time. The distance $\xi$ is related to a time domain Stokes number. The numerical computation of the Green function is done by FFT, with some precautions, because of the importance of the higher frequencies on the response shape. The $PL$ description is shown to be the best full frequency general model, but some discrepancies with the exact model appear at short times or short distances. When the distance $\xi$ increases from zero, the asymptotic expansion shows that the maximum of the Green function decreases first as $1/\xi ^{2}$, then exponentially

Multiple Scattering and Visco-Thermal Effects on 2D Phononic Crystal

In this paper, we are interested in the transition between regimes here either visco-thermal or multiple scattering effects dominate for the propagation of acoustic waves through a 2D regular square array of rigid cylinders embedded in air. An extension of the numerical method using Schl\"omilch series is performed in order to account for visco-thermal losses. Comparison withexperimental data and results from classical homogenization theory allows to study the transition between a low frequency limit (where viscous and thermal effects dominate) and a high frequency regime (where multiple scattering effects become predominant). For this particular geometry, a large frequency domain where visco-thermal and multiple scattering effects coexist is found

Low frequency in situ metrology of absorption and dispersion of sound absorbing porous materials based on high power ultrasonic non-linearly demodulated waves

Abstract The present work is related to acoustic in situ free-field measurements of sound absorption in porous materials, such as cellular plastic foams, glass-wool or recycled felt materials. The emphasis is given towards fine metrology of absorption in view of potential industrial applications. A powerful ultrasonic array working at 40 kHz is used. It enables to measure absorption acoustical data down to 100 Hz due to the exploitation of the non-linear ultrasonic demodulation phenomenon in air. Fine measurements of acoustic absorption are compared to numerical predictions based on the ''equivalent-fluid'' model (when the squeleton frame is motionless), and to some measurements performed on a Brü el and Kjaer impedance tube. Dispersion curves are also measured and compared to the numerical predictions for some automotive felt materials which are compressed at various ratios. Data obtained with a dedicated portable instrument are also discussed for the same type of materials and configurations

PROPAGATION DU SON DANS LES MATERIAUX POREUX A STRUCTURE RIGIDE SATURES PAR UN FLUIDE VISCOTHERMIQUE : définition de paramètres géométriques, analogie électromagnétique, temps de relaxation

In the context of airborne acoustics, the role of characteristic lengths ^, ^', is highlighted during impedance measurements, for various porous absorbent materials such as reticulated polyurethane foams or glass wool. In addition, we look at the notion of geometric parameter, the symmetry between viscous and thermal effects, and the role of viscous and thermal relaxation times. An electromagnetic analogy is used to clarify the position of the problem. The relationship between the various orders of approximation and the successive moments of the relaxation time distribution is shown. This reveals the “universal theory” as order 0 of approximation. Finally, the effects of deviation from this theory are discussed.Dans un contexte d'acoustique aérien, le rôle des longueurs caractéristiques ^, ^', est mis en évidence lors de mesures d'impédance, pour divers matériaux poreux absorbants tels que mousses réticulées de polyurethane ou laines de verre. Au delà, une réflexion est menée concernant la notion de paramètre géométrique, la symétrie entre effets visqueux et thermiques, le rôle des temps de relaxation visqueux et thermiques. On utilise une analogie électromagnétique pour préciser la position du problème. On montre la relation entre les divers ordres d'approximation et les moments successifs de la distribution des temps de relaxation. Ceci fait apparaître la "théorie universelle" comme l'ordre 0 d'approximation. Finalement, les effets de déviation par rapport à celle-ci sont abordés

Comments on ‘‘Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media’’

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Acoustic wave propagation in viscothermal fluids, an electromagnetic analogy

International audienceAcoustic wave propagation in viscothermal fluids, an electromagnetic analogy, Chapter, (68 pages), in a book, Fundamentals of acoustic waves propagation in periodic structures, metamaterials and porous media, to appear in editions Springer, 2020. Abstract: First, we recall the Navier-Stokes-Fourier model linearized equations, which govern the propagation of small amplitude, long wavelength waves in viscothermal uids; we specify how these equations are derived from several thermodynamic simplifications, and examine some of their solutions. Then, we analyze the general pattern of macroscopic nonlocal equations of propagation of small amplitude electromagnetic waves in effective homogeneous media, taking into account both the temporal and spatial dispersion. We argue that we lack a whole thermodynamics to fully precise all intervening quantities; proceeding by analogy, we then suggest that for the general acoustics of a homogeneous fluid, an analogous general pattern of nonlocal equations of propagation would arise, if we had sufficient thermodynamics. These ideas are finally implemented to obtain, within the available Navier-Stokes-Fourier's model, a nonlocal description of compressional waves