Nonlocal description of sound propagation through an array of Helmholtz resonators

Navid Nemati; Anshuman Kumar; Denis Lafarge; Nicholas X. Fang

doi:10.1016/j.crme.2015.05.001

Nonlocal description of sound propagation through an array of Helmholtz resonators

Navid Nemati ¹ ; Anshuman Kumar ¹; Denis Lafarge ²; Nicholas X. Fang ¹

¹ Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
² Laboratoire d'acoustique de l'université du Maine, UMR 6613, avenue Olivier-Messiaen, 72085 Le Mans cedex 9, France

Comptes Rendus. Mécanique, Acoustic metamaterials and phononic crystals, Volume 343 (2015) no. 12, pp. 656-669.

Abstract
Résumé

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, which enables the description of 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, related to the frequency-dependent Bloch wavenumber and bulk modulus of the first normal mode, for 1D propagation in 2D or 3D periodic structures.

Une théorie macroscopique nonlocale générale de la propagation du son dans les milieux poreux à structure rigide saturés par un fluide viscothermique a récemment vu le jour. Tenant un compte complet des dispersions, tant temporelles que spatiales, elle décrit entièrement les résonances. Nous l'appliquons ici au cas de la propagation du son dans un réseau de résonateurs de Helmholtz, dont les propriétés non usuelles (modules de compressibilité négatifs) ont été établies expérimentalement. Trois calculs différents sont présentés, qui valident les résultats de la théorie non locale, relatifs au nombre d'onde et module de compressibilité, qui sont fonctions de la fréquence, du mode de Bloch principal (le moins atténué), pour une propagation 1D en géométries périodiques 2D ou 3D.

Received: 2015-01-22
Accepted: 2015-05-22
Published online: 2015-06-18

DOI: 10.1016/j.crme.2015.05.001

Keywords: Helmholtz resonators, Acoustic metamaterials, Nonlocal description, Spatial dispersion, Viscothermal fluid, Negative modulus
Mots-clés : Résonateur d'Helmholtz, Métamatériaux acoustiques, Description non locale, Dispersion spatiale, Fluide viscothermique, Module de compressibilité négatif

Author's affiliations:

Navid Nemati ¹; Anshuman Kumar ¹; Denis Lafarge ²; Nicholas X. Fang ¹

¹ Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
² Laboratoire d'acoustique de l'université du Maine, UMR 6613, avenue Olivier-Messiaen, 72085 Le Mans cedex 9, France

@article{CRMECA_2015__343_12_656_0,
     author = {Navid Nemati and Anshuman Kumar and Denis Lafarge and Nicholas X. Fang},
     title = {Nonlocal description of sound propagation through an array of {Helmholtz} resonators},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {656--669},
     publisher = {Elsevier},
     volume = {343},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crme.2015.05.001},
     language = {en},
}

TY  - JOUR
AU  - Navid Nemati
AU  - Anshuman Kumar
AU  - Denis Lafarge
AU  - Nicholas X. Fang
TI  - Nonlocal description of sound propagation through an array of Helmholtz resonators
JO  - Comptes Rendus. Mécanique
PY  - 2015
SP  - 656
EP  - 669
VL  - 343
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crme.2015.05.001
LA  - en
ID  - CRMECA_2015__343_12_656_0
ER  -

%0 Journal Article
%A Navid Nemati
%A Anshuman Kumar
%A Denis Lafarge
%A Nicholas X. Fang
%T Nonlocal description of sound propagation through an array of Helmholtz resonators
%J Comptes Rendus. Mécanique
%D 2015
%P 656-669
%V 343
%N 12
%I Elsevier
%R 10.1016/j.crme.2015.05.001
%G en
%F CRMECA_2015__343_12_656_0

Navid Nemati; Anshuman Kumar; Denis Lafarge; Nicholas X. Fang. Nonlocal description of sound propagation through an array of Helmholtz resonators. Comptes Rendus. Mécanique, Acoustic metamaterials and phononic crystals, Volume 343 (2015) no. 12, pp. 656-669. doi : 10.1016/j.crme.2015.05.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2015.05.001/

References
Cited by

[1] D. Lafarge; N. Nemati Nonlocal Maxwellian theory of sound propagation in fluid-saturated rigid-framed porous media, Wave Motion, Volume 50 (2013), pp. 1016-1035

[2] C. Zwikker; C.W. Kosten Sound Absorbing Materials, Elsevier Publishing Company, Inc., New York, 1949 reprinted 2012 by the NAG (Nederlands Akoestisch Genootschap)

[3] D.L. Johnson; J. Koplik; R. Dashen Theory of dynamic permeability and tortuosity in fluid-saturated porous media, J. Fluid Mech., Volume 176 (1987), pp. 379-402

[4] Y. Champoux; J.F. Allard Dynamic tortuosity and bulk modulus in air-saturated porous media, J. Appl. Phys., Volume 70 (1991), pp. 1975-1979

[5] D. Lafarge; P. Lemarinier; J.F. Allard; V. Tarnow Dynamic compressibility of air in porous structures at audible frequencies, J. Acoust. Soc. Am., Volume 102 (1997), pp. 1995-2006

[6] R. Burridge; J.B. Keller Poroelasticity equations derived from microstructure, J. Acoust. Soc. Am., Volume 70 (1981), pp. 1140-1146

[7] A.N. Norris On the viscodynamic operator in Biot's equations of poroelasticity, J. Wave-Mater. Interact., Volume 1 (1986), pp. 365-380

[8] M.Y. Zhou; P. Sheng First principles calculations of dynamic permeability in porous media, Phys. Rev. B, Volume 39 (1989), pp. 12027-12039

[9] D.M.J. Smeulders; R.L.G.M. Eggels; M.E.H. van Dongen Dynamic permeability: reformulation of theory and new experimental and numerical data, J. Fluid Mech., Volume 245 (1992), pp. 211-227

[10] J.L. Auriault Dynamic behavior of a porous medium saturated by a Newtonian fluid, Int. J. Eng. Sci., Volume 18 (1980), pp. 775-785

[11] J.L. Auriault; C. Boutin; C. Geindreau Homogenization of Coupled Phenomena in Heterogenous Media, ISTE and Wiley, 2009

[12] E. Sanchez Palencia Nonhomogeneous Media and Vibration Theory, Lectures Notes in Physics, vol. 127, Springer, Berlin, 1980

[13] A. Bensoussan; J.L. Lions; G.C. Papanicolaou Asymptotic Analysis for Periodic Structure, North-Holland, Amsterdam, 1978

[14] R.V. Craster; J. Kaplunov; A.V. Pichugin High-frequency homogenization for periodic media, Proc. R. Soc. Lond. A, Volume 466 (2010), pp. 2341-2362

[15] T. Antonakakis; R.V. Craster; S. Guenneau; E.A. Skelton An asymptotic theory for waves guided by diffraction gratings or along microstructured surfaces, Proc. R. Soc. Lond. A, Volume 470 (2013), p. 20130467

[16] C. Boutin; A. Rallu; S. Hans Large scale modulation of high frequency acoustic waves in periodic porous media, J. Acoust. Soc. Am., Volume 132 (2012), pp. 3622-3636

[17] C. Boutin Acoustics of porous media with inner resonators, J. Acoust. Soc. Am., Volume 134 (2013), pp. 4717-4729

[18] M. Yang; G. Ma; Y. Wu; Z. Yang; P. Sheng Homogenization scheme for acoustic metamaterials, Phys. Rev. B, Volume 89 (2014), p. 064309

[19] Y. Wu; Y. Lai; Z.Q. Zhang Effective medium theory for elastic metamaterials in two dimensions, Phys. Rev. B, Volume 76 (2007), p. 205313

[20] C. Boutin Sound propagation in rigid porous media: non-local macroscopic effects versus pores scale regime, Transp. Porous Media, Volume 93 (2012), pp. 309-329

[21] J.R. Willis Exact effective relations for dynamics of a laminated body, Mech. Mater., Volume 41 (2009), pp. 385-393

[22] L.D. Landau; E.M. Lifshitz Electrodynamics of Continuous Media, Course of Theoretical Physics, vol. 8, Elsevier, Butterworth–Heinemann, Oxford, 2004

[23] N. Fang; D. Xi; J. Xu; M. Ambati; W. Srituravanich; C. Sun; X. Zhang Ultrasonic metamaterials with negative modulus, Nat. Mater., Volume 5 (2006), pp. 452-456

[24] S. Zhang; L. Yin; N. Fang Focusing ultrasound with an acoustic metamaterial network, Phys. Rev. Lett., Volume 102 (2009), p. 194301

[25] S. Zhang; C. Xia; N. Fang Broadband acoustic cloak for ultrasound waves, Phys. Rev. Lett., Volume 106 (2011), p. 24301

[26] N. Nemati; D. Lafarge Check on a nonlocal Maxwellian theory of sound propagation in fluid-saturated rigid-framed porous media, Wave Motion, Volume 51 (2014), pp. 716-728

[27] J.F. Allard; N. Atalla Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, John Wiley & Sons, 2009

[28] F. Hecht New development in FreeFem++, J. Numer. Math., Volume 20 (2012), pp. 251-265

[29] M.R. Stinson The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape, J. Acoust. Soc. Am., Volume 89 (1991), pp. 550-558

Cited by Sources:

Comments - Policy