Homogenization of acoustic metamaterials of Helmholtz resonators in fluid

Abstract

By using a two-step homogenization approach, we derive analytical formulas of effective mass density e and effective bulk modulus B e for two-and three-dimensional acoustic metamaterials of Helmholtz resonators ͑HRs͒ in fluid. A negative B e is found at certain frequencies due to the monopolar resonance, leading to a low-frequency acoustic band gap. A unified picture is presented for metamaterials of HRs and three-component metamaterials of negative e . Our work supports recent observations in a one-dimensional array of HRs ͓N. Fang et al., Nat. Mater. 5, 452 ͑2006͔͒ and presents important high-dimensional extensions for exploring more fascinating phenomena. DOI: 10.1103/PhysRevB.77.172301 PACS number͑s͒: 42.25.Bs, 41.20.Jb, 46.40.Ϫf, 62.30.ϩd Recently, acoustic and/or elastic metamaterials ͑artificial structured materials 1,2 ͒ have received considerable interest due to their exotic acoustic and/or elastic properties. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] In particular, their effective mass density and modulus can be singly or simultaneously negative, 3-13 allowing intriguing phenomena for sound waves such as low-frequency band gaps, 3-10 negative refraction, and superlensing. 1,[12][13][14] Negative effective mass density or modulus can occur at certain frequencies if an appropriate resonance is included into the structures. 3-13 A famous example is the threecomponent phononic crystal with locally resonant structures, 3 which exhibits a negative effective mass density e due to a dipolar resonance. 4,5 Very recently, a lowfrequency band gap was demonstrated in a one-dimensional ͑1D͒ array of Helmholtz resonators ͑HRs͒. 6 By using further retrieval analysis of wave-scattering coefficients, 7-9 a negative effective bulk modulus B e was confirmed in the 1D metamaterials of HRs. 9 Unlike other designs for negative B e using air bubbles or soft-rubber spheres, 12,13 this metamaterial consists of a rigid body but still possesses a sizable working frequency range. However, high-dimensional metamaterials of HRs, which are essential for more fascinating phenomena, 6,14,15 remain unexplored.In this Brief Report, we study two-dimensional ͑2D͒ and three-dimensional ͑3D͒ acoustic metamaterials consisting of cylindrical and spherical HRs in fluid ͑such as water or air͒, respectively. By using a two-step homogenization approach, we can derive analytical formulas for the e and B e within coherent-potential approximation ͑CPA͒. 18 The accuracy of these formulas is confirmed by accurate retrieval results using multiple-scattering techniques. We show that our metamaterials of HRs can exhibit a negative B e at certain frequencies due to a monopolar resonance. Systematic analyses are done for the frequency range of negative B e and the low-frequency-limit behavior. Based on our derivations, a unified picture is presented for metamaterials of HRs and three-component metamaterials of negative e .Our HRs are cylindrical and/or spherical rigid shells with uniformly distributed slits and/or holes, as shown in Fig. 1...

Scattering by coupled resonating elements in air

“…Similar approximations have been used for a periodic array of circular scatterers [27]. The fluid layer is defined through the thickness h l = hF and density ρ l = ρ/F 2 with filling fraction F = Nd/(2πr o ) satisfying the following conditions:…”

Elastic metamaterials and dynamic homogenization: a review

“…An excellent review [64] points out that the majority of metamaterial homogenization studies published in the last decade failed to respect basic causality and passivity properties and in some cases also violated the second law of thermodynamics. Another broad technique for determining effective properties is through the Coherent Potential Approximation method [65] and its enhancements [66,67]. The metamaterial under study is embedded in a matrix, which has the properties of the effective media.…”

Composite Acoustic Medium with Simultaneously Negative Density and Modulus

“…In comparison with the numerous theoretical and experimental research results on electromagnetic metamaterials, research efforts on the acoustic metamaterials are relatively few and mainly concentrated on theoretical aspects [10,15,16,17,18,19,20]. Only, recently, Fang et al reported realization of negative modulus in a metamaterial consisting of Helmholtz resonators [11].…”