Slow sound propagation in lossy locally resonant periodic structures

We investigate the sound propagation in an air-filled tube periodically loaded with Helmholtz resonators. By tuning the Helmholtz with the Bragg resonance, we study the efficiency of slow sound propagation in the presence of the intrinsic viscothermal losses of the system. While in the lossless case the overlapping of the resonances results in slow sound induced transparency of a narrow frequency band surrounded by a strong and broadband gap, the inclusion of the unavoidable losses imposes limits to the slowdown factor and the maximum transmission. Experiments, theory and finite element simulations have been used for the characterization of acoustic wave propagation. Experiments, in good agreement with the lossy theory, reveal the possibility of slowing sound at low frequencies by 20 times. A trade-off among the relevant parameters (delay time, maximum transmission, bandwidth) as a function of the tuning between Bragg and Helmholtz resonance frequency is also presented.


I. INTRODUCTION
Locally resonant acoustic metamaterials 1 derive their unique properties e.g. negative effective mass density 2 and negative bulk modulus 3 , from local resonators contained within each unit cell of engineered structures. Due to these effective parameters, a plethora of fascinating phenomena have been proposed over the last years, including negative refraction, super-absorbing sound materials, acoustic focusing, and cloaking (see Ref. [4] and references therein).
Although the inclusion of losses in locally resonant structures is very important, their role has been underestimated while in some studies totally ignored. Loss is not only an unavoidable feature, but also it may have deleterious consequences on some of the novel features of metamaterials 5 including double negativity and cloaking. Recent works on both photonic [6][7][8] and phononic [9][10][11][12] periodic structures show that the dispersion relation can be dramatically altered. In particular, flat propagating bands corresponding to slow-wave propagation, acquire an enhanced damping as compared to bands with larger group velocities 7,9 .
The aim of this work is to study the influence of losses on slow sound propagation in periodic locally resonant structures. For this reason, we theoretically and experimentally analyze the sound propagation in a tube periodically loaded with Helmholtz resonators (HRs) taking into account the viscothermal losses 13 . In particular, we investigate configurations where the Bragg resonance frequency due to periodicity and the frequency of the Helmholtz resonators either coincide or are very close to each others. In the first case, a super-wide and strongly attenuating band gap is created. This property has recovered interest during last years in different branches of science including elastic waves 14 , split-ring microwave propagation 15 , sonic crystals 16 , and duct acoustics 17 , among others. In acoustics, this tuning, first studied by Sugimoto 18 , is of great importance for sound and vibration isolation 17,19 . In the case of slightly detuned resonances, i.e. once the Bragg and the local resonance are slightly different, an almost flat band appears, a feature which is particular useful for slow waves applications 20 .
Here, we make use of this detuning to theoretically and experimentally examine the effect of losses in the slow sound band. We focus on both periodic systems and finite periodic arrays with N side HRs. Using the transmission matrix method, we characterize the group index, the bandwidth, and the slow-wave limits of these structures, showing good agreement with experiments. The limit of the slow sound due to losses is of relevant importance for the design of narrow-band transmission filters and switches. Moreover, it could also open perspectives in the way to control the nonlinear effects at the local resonances 21 , which could increase the functionality of the acoustic metamaterials leading to novel acoustic devices for the sound control at low frequencies.

II. THEORY
The propagation of linear, time-harmonic acoustic waves in a waveguide periodically loaded by side branches has been first studied in Ref. [22]. Using Bloch theory and the transfer matrix method, one can derive the following dispersion relation (see also Refs. [18], [23]): where q is the Bloch wave number, k is the wave number in air, L the lattice constant, Z b the input impedance of the branch (see Ref. [23] for the case of HR branch), and Z 0 = ρ 0 c 0 /S the acoustic impedance of the waveguide where S is its cross-sectional area; ρ 0 , c 0 the density and the speed of sound in the air respectively, and j = √ −1.
The transmission coefficient through a finite lattice can be derived using the transmission matrix method. For the case of N side branches, the total transmission matrix can be expressed as follows 19,24   P 1 where represent the transmission matrices for the propagation through a length L in the waveguide and through a resonant branch respectively. P 1 (U 1 ) and P 2 (U 2 ) are the pressure (and respectively volume velocity) at the entrance and at the end of the system. Considering the previous equations, the pressure complex transmission coefficient can then be calculated 19 as t = 2 The sound waves are always subjected to viscothermal losses on the wall and to radiation losses. Viscothermal losses are taken into account by considering a complex expression for the wave number. In our case, we used the model of losses from Ref. [13], namely we replace the wave number and the impedances by the following expressions by setting s = r/δ where δ = 2µ ρ 0 ω is the viscous boundary layer thickness, µ being the viscosity of air, χ = √ P r with P r the Prandtl number, β = (1 − j)/ √ 2, γ the heat capacity ratio of air and r the radius of the considered tube. Radiation losses, which appear at each connection between the waveguide and the HRs, are accounted for through a length correction of the HRs neck defined in the description of the experimental set-up.

III. EXPERIMENTAL APPARATUS
The experimental apparatus that we used in this work to calculate the dispersion relation of periodic systems and the transmission coefficient of a finite periodic locally resonant system is shown in Fig. 1. Each HR is made of a neck (cylindrical tube with an inner radius R n = 1 cm and a length l n = 2 cm), and a cavity (cylindrical tube with an inner radius R c = 2.15 cm and a variable length, l c ). We use different configurations through this work from 3 to 6 HRs, loaded periodically along a cylindrical waveguide with an inner radius R = 2.5 cm, 0.5 cm wall thickness, and total length of 3 m. The last HR is always connected at a distance of L/2 cm from the end of the set-up, x end , where L = 30 cm is the constant distance between adjacent resonators. The sound source is a piezo-electric buzzer embedded The input impedance measurement setup (see Fig. 1), together with the transmission matrix method allow us to experimentally evaluate the dispersion relation of periodic systems. To do that we use the measured input impedance Z = P 1 U 1 and the transfer impedance Z T = P 2 U 1 . From Z, one can calculate the acoustic impedance at the position x 0 , Z = P 1 U 1 , which is located L/2 = 15 cm from the first HR as well as the Z T = P 2 U 1 (see Fig. 1). Then, the impedance matrix of the symmetric structure from x 0 until the rigid end is given by: This setup can be also used for the experimental calculation of the transmission coefficient.
To do that we replace the rigid end termination by an anechoic termination made of a 10 m long waveguide partially filled with porous plastic foam to suppress as much as possible the back propagative waves. In this case, the microphone is placed at a distance of L=30 cm after the last HR, namely the same distance as between the source and the first HR.
Using the transmission matrix method, considering anechoic termination, and assuming the symmetry of the structure, the complex transmission coefficient t reads as follows We start by studying the coupling between the Bragg and the resonance band gap taking into account the presence of the viscothermal losses. In Fig. 2 Fig. 2(e)). We define a detuning length parameter, ∆l = l 0 − l c , where l 0 corresponds to the cavity length at which f 0 = f B . Thus, ∆l measures how far we are from the complete overlap between the Bragg and the HR resonance. As shown in Fig. 2(e), if ∆l < 0 (∆l > 0) the HR resonance approaches the Bragg's one from lower (higher) frequencies. According to Ref. [18], for the case of ∆l = 0 a wide band-gap appears in the region f B (1 − (κ/2) 1/2 ) < f < f B (1 + (κ/2) 1/2 ), where κ = Sclc SL measures the smallness of the cavity's volume relative to the unit-cell's volume. For our case, white dashed line in Fig. 2(e), the above expression predicts a band gap for 416.5 < f < 727.7 Hz, which is in very good agreement with the experiments (Fig. 2(a)-(b)) for the case of ∆l 0 cm. However, it is very difficult to find in practice the case of ∆l = 0 because one needs to control either the length of the cavity or the lattice constant with a high precision. When ∆l ≈ 0 one can observe that the lossless theory (green dotted line in Fig. 2(a)) predicts a flat branch inside the band gap. This branch is drastically reduced once losses are introduced.
We continue by studying the detuned case, i.e., the case ∆l = 0. In this situation, as shown in Fig. 2(e) between the Bragg and the resonance bands, there is a range of frequencies with small attenuation (small Im(qL)). For example the real part of the complex dispersion relation for the case ∆l = −0.4 cm (see Fig. 2(c)), shows an almost flat real band, which means slow sound propagation. We introduce the group index as a slowdown factor from the speed of sound c, defined as n g ≡ c/υ g where υ g = ∂ω ∂Re(q) is the group velocity. In lossy periodic structures, the real and imaginary parts of the group velocity correspond to propagation velocity and pulse reshaping respectively (see 8 and references within). Negative values of n g correspond to negative group velocity induced by the losses, as it has been also reported in Ref. [3]. Figure 2(f) shows the theoretical and experimental group index obtained from the data shown in Fig. 2(c). It is worth noting that through the bandwidth of the transmitted frequencies (marked in Fig. 2(f) with the double arrow) the group index is n g > 20. This slowdown factor is comparable with the results of previously reported experiments 28 . Figure 2(e) shows also that the bandwidth of the transmission band depends on ∆l. Now, we investigate in more detail this slow sound propagation in lossy finite lattice of HRs. In particular, the slow sound propagation is characterized using a delay time and we study the trade-offs among this delay and the transmission losses in a finite lattice. Fabry-Pérot resonances inside the finite structure are clearly observed. For these resonant frequencies there is no reflection at the entrance of the tube. However for the lossy case, Fig. 4(b), the behavior is dramatically modified: the resonances are destroyed and reflections appear at the entrance and thus standing waves are generated. Therefore the system is not any more completely transparent. presented showing that the near-zero group velocity theoretically predicted disappears due to losses. Finally, using simulated acoustic wave fields into the structure, we have pointed out the presence of reflected waves in the lossy case oppositely to the lossless case.
We believe that this experimental and theoretical study shows the great importance of losses in acoustic wave propagation through periodic locally resonant structure and con-tributes to very promising research in the field of acoustic metamaterials, acoustic transmission filters and slow wave applications. ACKNOWLEDGMENTS We acknowledge V. Pagneux and A. Maurel for useful discussions. GT acknowledges financial support from FP7-People-2013-CIG grant, Project 618322 ComGranSol. VRG acknowledges financial support from the "Pays de la Loire" through the post-doctoral programme.