# zbMATH — the first resource for mathematics

Modelling resonant arrays of the Helmholtz type in the time domain. (English) Zbl 1402.74087
Summary: We present a model based on a two-scale asymptotic analysis for resonant arrays of the Helmholtz type, with resonators open at a single extremity (standard resonators) or open at both extremities (double-sided resonators). The effective behaviour of such arrays is that of a homogeneous anisotropic slab replacing the cavity region, associated with transmission, or jump, conditions for the acoustic pressure and for the normal velocity across the region of the necks. The coefficients entering in the effective wave equation are simply related to the fraction of air in the periodic cell of the array. Those entering in the jump conditions are related to near field effects in the vicinity of the necks and they encapsulate the effects of their geometry. The effective problem, which accounts for the coupling of the resonators with the surrounding air, is written in the time domain which allows us to question the equation of energy conservation. This is of practical importance if the numerical implementations of the effective problem in the time domain is sought.

##### MSC:
 74Q05 Homogenization in equilibrium problems of solid mechanics 74J15 Surface waves in solid mechanics 35Q74 PDEs in connection with mechanics of deformable solids 76Q05 Hydro- and aero-acoustics
Full Text:
##### References:
 [1] Helmholtz, HL, $$On the sensations of tone as a physiological basis for the theory of music$$ (ed. and transl. AJ Ellis). New York, NY: Dover. (Original German publication, 1863), (1954) [2] Valière, JC; Palazzo-Bertholon, B.; Polack, JD; Carvalho, P., Acoustic pots in ancient and medieval buildings: literary analysis of ancient texts and comparison with recent observations in French churches, Acta. Acust. United. Acust., 99, 70-81, (2013) [3] Xiong, L.; Bi, W.; Aurégan, Y., Fano resonance scatterings in waveguides with impedance boundary conditions, J. Acoust. Soc. Am., 139, 764-772, (2016) [4] Romero-García, V.; Theocharis, G.; Richoux, O.; Merkel, A.; Tournat, V.; Pagneux, V., Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators, Sci. Rep., 6, 19519, (2016) [5] Jiménez, N.; Huang, W.; Romero-García, V.; Pagneux, V.; Groby, JP, Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption, Appl. Phys. Lett., 109, 121902, (2016) [6] Yang, X.; Yin, J.; Yu, G.; Peng, L.; Wang, N., Acoustic superlens using Helmholtz-resonator-based metamaterials, App. Phys. Lett., 107, 193505, (2015) [7] Lemoult, F.; Fink, M.; Lerosey, G., Acoustic resonators for far-field control of sound on a subwavelength scale, Phys. Rev. Lett., 107, 064301, (2011) [8] Lerosey, G.; De Rosny, J.; Tourin, A.; Fink, A., Focusing beyond the diffraction limit with far-field time reversal, Science, 315, 1120-1122, (2007) [9] Maznev, AA; Gu, G.; Sun, SY; Xu, J.; Shen, Y.; Fang, N.; Zhang, SY, Extraordinary focusing of sound above a soda can array without time reversal, New J. Phys., 17, 042001, (2015) [10] Ingard, U., On the theory and design of acoustic resonators, J. Acoust. Soc. Am., 25, 1037-1061, (1953) [11] Kergomard, J.; Garcia, A., Simple discontinuities in acoustic waveguides at low frequencies: critical analysis and formulae, J. Sound Vib., 114, 465-479, (1987) [12] Schwan, L.; Umnova, O.; Boutin, C., Sound absorption and reflection from a resonant metasurface: Homogenisation model with experimental validation, Wave Motion, 72, 154-172, (2017) [13] Schweizer, B., The low-frequency spectrum of small Helmholtz resonators, Proc. R. Soc. A, 471, 20140339, (2014) · Zbl 1371.78024 [14] Schweizer, B., Resonance meets homogenization, Jahresbericht der Deutschen Mathematiker-Vereinigung, 119, 31-51, (2017) · Zbl 1359.78029 [15] Lamacz, A.; Schweizer, B., Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. (http://arxiv.org/abs/1603.05395), (2016) [16] Pham, K.; Maurel, A.; Marigo, JJ, Two scale homogenization of a row of locally resonant inclusions—the case of anti-plane shear waves, J. Mech. Phys. Solids, 106, 80-94, (2017) [17] Felbacq, D.; Bouchitté, G., Theory of mesoscopic magnetism in photonic crystals, Phys. Rev. Lett., 94, 183902, (2005) [18] Lombard, B.; Maurel, A.; Marigo, JJ, Numerical modeling of the acoustic wave propagation across an homogenized rigid microstructure in the time domain, J. Comp. Phys., 335, 558-577, (2017) · Zbl 1375.76169 [19] Marigo, JJ; Maurel, A., Second order homogenization of subwavelength stratified media including finite size effect, SIAM J. Appl. Math., 77, 721-743, (2017) · Zbl 1375.35028 [20] Delourme, B., Modèles et asymptotiques des interfaces fines et périodiques en électromagnétisme. Doctoral dissertation, Université Pierre et Marie Curie-Paris VI, Paris, France, (2010) [21] Delourme, B.; Haddar, H.; Joly, P., Approximate models for wave propagation across thin periodic interfaces, J. Math. pures Appl., 98, 28-71, (2012) · Zbl 1422.76160 [22] Marigo, JJ; Maurel, A., Homogenization models for thin rigid structured surfaces and films, J. Acoust. Soc. Am., 140, 260-273, (2016) [23] Marigo, JJ; Maurel, A.; Pham, K.; Sbitti, S., Effective dynamic properties of a row of elastic inclusions: the case of scalar shear waves, J. Elast., 128, 265-289, (2017) · Zbl 1374.74106 [24] Jiménez, N.; Romero-García, V.; Pagneux, V.; Groby, JP, Quasi-perfect absorption by sub-wavelength acoustic panels in transmission using accumulation of resonances due to slow sound, Phys. Rev. B, 95, 014205, (2016) [25] Bakhvalov, NS; Panasenko, G., $$Homogenisation: averaging processes in periodic media: mathematical problems in the mechanics of composite materials, Mathematics and its Applications$$, 36, (2012), Springer Science & Business Media [26] Marigo, JJ; Pideri, C., The effective behavior of elastic bodies containing microcracks or microholes localized on a surface, Int. J. Damage Mech., 20, 1151-1177, (2011) [27] Maurel, A.; Marigo, JJ; Ourir, A., Homogenization of ultrathin metallo-dielectric structures leading to transmission conditions at an equivalent interface, J. Opt. Soc. Am. B, 33, 947-956, (2016) [28] Hewett, DP; Hewitt, IJ, Homogenized boundary conditions and resonance effects in Faraday cages, Proc. R. Soc. A, 472, 20160062, (2016) · Zbl 1402.35269 [29] Marigo, JJ; Maurel, A., An interface model for homogenization of acoustic metafilms. In $$Handbook of metamaterials properties$$ (eds RV Craster, S Guenneau), vol. 2, ch. 12, pp. 599-644. Singapore: World Scientific Publishing Company, (2017) [30] Romero-García, V.; Theocharis, G.; Richoux, O.; Pagneux, V., Use of complex frequency plane to design broadband and sub-wavelength absorbers, J. Acoust. Soc. Am., 139, 3395-3403, (2016) [31] Quan, L.; Zhong, X.; Liu, X.; Gong, X.; Johnson, PA, Effective impedance boundary optimization and its contribution to dipole radiation and radiation pattern control, Nat. Commun., 5, 3188, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.