Transparency robust to disorder in a periodic array of Helmholtz resonators
TTransparency robust to disorder in a periodic array of Helmholtz resonators
O. Richoux and V. Pagneux
LUNAM Universit´e, Universit´e du Maine, CNRS, LAUM,UMR 6613, Av. O. Messiaen, 72085 Le Mans, France
A. Maurel
Institut Langevin, LOA, CNRS 7587, ESPCI, 1 rue Jussieu, 75005 Paris, France (Dated: October 13, 2018)In this paper, the influence of disorder on 1D periodic lattice of resonant scatterers is inspected.These later have multiple resonance frequencies which produce bandgap in the transmission spec-trum. One peculiarity of the presented system is that it is chosen with a nearly perfect overlapbetween the Bragg and the second hybridization bandgaps. In the case of a perfect ordered lattice,and around this overlap, this produces a narrow transparency band within a large second bandgap.As expected, the effect of the disorder is generally to increase the width of the bandgaps. Never-theless, the transparency band appears to be robust with respect to an increase of the disorder. Inthis paper, we study this effect by means of experimental investigations and numerical simulations. a r X i v : . [ phy s i c s . op ti c s ] O c t I. INTRODUCTION
Phononic crystals have experienced an increasing interest in recent years because of their potential applicationsto acoustic filters [1], the control of vibration isolation [2], noise suppression, and the possibility of building newtransducers [3]; for a review see [4]. It is thus of interest to understand which properties of such structures aresensitive to inherent imperfections in their design and which are not. Besides, one can also address the question ofwhether or not the disorder can make new interesting properties to appear.It is usual to characterize a random medium in terms of an effective -homogeneous- medium. For random pertur-bation of homogeneous free space one finds that the dispersion relation K ( ω ) departs from the dispersion relation k ( ω ) in free space without disorder, and the imaginary part of the effective wavenumber K indicates how much theopacity due to disorder is important [5]. In the case of photonic or phononic crystals, the band structure of the unper-turbed medium is more complicated, with a wavenumber Q of the Bloch Floquet mode being either purely real (passband) or complex (stop band). The addition of disorder modifies the band structures of these periodic-on-averagesystems [6–12], and generally, produces an increase in the band gap width [13]. Among periodic media, the caseof periodic arrays of resonant scatterers is very attractive since the resonances inherent to the individual scatterersproduce strong modifications of the wave propagation; owing to these modifications in the wave properties may helpto design materials with unusual properties. Such arrays present band gaps around the resonance frequencies of anindividual scatterer. Because periodically located, Bragg resonances are also produced, resulting in a complex bandgap structure. Overlapping two types of gaps, a resonant scatterer gap and a Bragg gap, have been shown to produceinteresting phenomena, as the creation of a super wide and strongly attenuating band gap used for structure isolation[14–17] and slow wave application [18].In this paper, we consider the propagation of an acoustic wave in a periodic array of Helmholtz resonators connectedto a duct in the plane wave regime (low frequency regime with one propagating mode in the duct). The correspondingmodel describes the 1D propagation of the pressure field p ( x ) through resonant point scatterers (Kronig- Penneysystem) [19, 20] p (cid:48)(cid:48) + k p = (cid:88) n V n ( k ) δ ( x − nd ) p ( x ) , (1)where d is the periodicity of the array and where V n ( k ) encapsulates the effect of the n th resonator of the array. Thedisorder is introduced by varying the volume of the Helmholtz resonators. When an overlap between a Bragg gap and aresonant bandgap is produced, a narrow transparency band appears within the resulting large bandgap. Unexpectedly,we found that this transparency band is robust with respect to the disorder. Indeed, first, for small disorder, thetransmission decreases; but increasing further the disorder induces an increase in the transmission. We have carriedout experiments whose results show qualitatively this behavior. To get further informations, with a broader range ofthe disorder parameter, numerical calculations are shown, that confirm the transparency induced by disorder. Thepaper is organized as follow: in Part II, the 1D model and the CPA result for the randomly perturbed system arediscussed. The experimental results are presented in Part III, and this is completed by numerical calculations, in PartIV. Finally, a discussion is proposed in Part V. II. PROPAGATION IN 1D PERIODIC AND PERTURBED HR ARRAY
At low frequencies, when only one mode can propagates in the duct, the propagation of acoustic waves in an arrayof Helmholtz resonators periodically located with spacing d (Fig. 1) can be described by p (cid:48)(cid:48) + k p = (cid:88) j V j ( k ) δ ( x − jd ) p ( x ) , (2)where p is the pressure field and k = ω/c (the time dependance e − iωt is omitted, ω is the angular frequency and c the sound velocity in free space). The potential is V j ( k ) = − sS w k n sin k n (cid:96) cos k c L + α cos k n (cid:96) sin k c L cos k n (cid:96) cos k c L − α sin k n (cid:96) sin k c L , (3)with α = Sk c / ( sk n ), where S w , S, s are the area of the main waveguide, of the cavity and of the neck, respectively. (cid:96) and L denote the length of the neck and of the cavity respectively (see Fig. 1(b)). The wavenumbers are k m = k [1 + βδ/R m ], with m = w, c, n (waveguide, cavity and neck respectivelly) and R m the corresponding radius, with β = (cid:2) γ − P r − / (cid:3) (1 + i ) / √ δ = (cid:112) ν/ω the viscous boundary layer depth ( ν the cinematic viscosity).The term proportional to β in the wavenumber k is a good model of the viscous and thermal attenuation of soundin the duct. We can notice that, with s (cid:28) S w , the strength of the Helmholtz scatterer is small except at resonances.Approximating k n and k c by k , and thus omitting the attenuation, these cavity resonances correspond to a vanishingterm D ( k ) ≡ cos k(cid:96) cos kL − α sin k(cid:96) sin kL , and they are of two types: i) the typical Helmholtz resonance occurringat low frequency, say for k(cid:96) → k H = 1 / √ α(cid:96)L and ii) the resonances in the cavity (hereafter referred asvolume resonances), near kL = nπ . For instance, for n = 1, k V L = π + 1 α tan( π(cid:96)/L ) . (4)For a single resonator, these resonances produce a vanishing transmission. When the resonators are organized in a per-fect periodic array, band gaps are created around the resonance frequencies, according to Bloch Floquet wavenumber Q becoming purely imaginary, Q being given bycos Qd = cos kd + V k sin kd. (5)When disorder is introduced in the volume cavity by changing the length L n of the n th cavity, L n = L (1 + (cid:15) n ) and (cid:15) n ∈ [ − (cid:15)/ (cid:15)/ K using CPA approach [10]cos Kd = cos kd + (cid:104) V (cid:105) k sin kd, (6)where (cid:104) . (cid:105) denotes the ensemble average for all realizations of the { (cid:15) n } n -values. The resulting transmission coefficientis T N = e ikd − B e − ikd e ikd − iKNd − B e − ikd + iKNd , (7)where we have written p ( x ≥ N d ) = T N e ik ( x − Nd ) (the incident wave is e ikx ) and with B ≡ − e i ( k − K ) d − e i ( k + K ) d . (8)Obviously, the above results obtained from CPA recover the perfect periodic case when (cid:15) = 0.In the following, we present the experimental set-up to realize the lattice of Helmholtz resonators. Comparisonsbetween the measured transmission and the above CPA- result, Eq. (7) is presented. III. EXPERIMENTAL RESULTSA. Experimental set-up
The experimental set-up (fig. 1) consists in a 8 m long cylindrical waveguide with an inner area S w = 2 × − m and a 0 . N = 60 Helmholtz resonators periodicallydistributed, with inter-distance d = 0 . s = 7 . × − m and a length (cid:96) = 2 cm) and by a cavity with variable length. The cavity is a cylindrical tube withan inner area S = 1 . × − m and a maximum length L max = 16 . Z , defined as the ratio of the acoustic pressure p and theacoustic flow u (the product of the velocity by the area cross section) at the entrance of the lattice, as described in[24]. This allows to get the reflection coefficient R defined as p = (1 + R ) p + owing to u = u + + u − with u + = p + /Z w , u − = − p − /Z w , where the index + and − denote the parts of the quantity associated to right- and left-going waves: R = Z − Z w Z + Z w . (9)At the output, an anechoic termination made of a 10 m long waveguide partially filled with porous plastic foamsuppresses the back propagative waves. This ensures the output impedance to be close to the characteristic impedance Z w = ρc/S w . Finally, a microphone is used to measure the pressure p e at the end of the lattice. d Ss LS w FIG. 1. Picture of the experimental set up (left panel). Shematic of the experimental setup (right panel).
Using line matrix theory, ( p, u ) and ( p e , u e ) are linked by the transfer matrix through (cid:18) pu (cid:19) = (cid:18) A BC D (cid:19) (cid:18) p e u e (cid:19) (10)with p = Zu ( Z being measured) and u e = p e /Z w (the acoustic flow is deduced from p e because of the anechoictermination). Then, the transmission coefficient T defined as p e = T p + , is calculated using that u = (1 + R ) p + /Z bydefinition of R and from above, u = [ C + D/Z w ] p e = [ C + D/Z w ] T p + , from which T = 2 Z t Z + Z w , (11)where Z t ≡ [ C + D/Z w ] − is deduced from the measured ( p e , u )-values.When considered, the disorder in the lattice is introduced through the variable lengths L n , n = 0 , . . . , N of thecavities, and L n = L (1 + (cid:15) n ) is used with a normal distribution of (cid:15) being chosen for each realization and for eachresonator cavity, with (cid:15) n ∈ [ − (cid:15)/ (cid:15)/ V n in Eq. (3).The transmission coefficients are measured for ten different distributions with same standard deviation, and themean value (cid:104) T (cid:105) is taken. B. Experimental observations
The transmission coefficient T in the perfect periodic case is presented in the Fig. 2 for a cavity length L = 0 . k H )for kd/π ∈ [0 .
15; 0 . kL close to π and 2 π ); these are for kd/π in [0 .
64; 0 . .
22; 1 . kd/π ∈ [1; 1 . L = 0 . L = d , the volume resonance k V , with k V ∼ π/L , and the Bragg frequency k B = π/d are very close, resulting in an almost perfect overlap of thetwo corresponding band gaps, previously labeled b and c, visible here in the range kd/π ∈ [0 .
98; 1 .
12] (frequencyrange [1600; 1800] Hz). The first band gap, associated to the Helmholtz resonance k H is almost non affected by thechange in L while the volume resonance with k V L (cid:39) π (previously labeled d) is sent to higher frequency (not visiblein our plot). A noticeable feature is the existence of a small transparency band inside the large stop band near kd = π , FIG. 2. Transmission coefficient for an ordered lattice with a cavity length L = 0 .
165 m and lattice spacing d = 0 . a feature already observed in other system where such overlapping is realized [16–18]. This feature, in addition to themain behavior of T , is accurately captured by our analytical expression, Eq. (7), in the perfect periodic case, thuswith constant unpertubed potential V (and K = Q ). FIG. 3. (a) Transmission coefficient of an ordered lattice for a cavity length L = 0 . d = 0 . (cid:15) = 0 .
08. (c) Mean value of the transmission coefficientfor a disordered lattice with (cid:15) = 0 .
1. (c) Mean value of the transmission coefficient for a disordered lattice with (cid:15) = 0 .
18. Theblue line corresponds to the experimental case obtained with 10 averages and the red line to our analytical prediction with 100averages (except for (a) without disorder).
We now consider several amplitudes (cid:15) of disorder in the scattering strength of the resonators, as previously described.The measured transmission coefficients |(cid:104) T (cid:105)| are reported in Fig. 3(b,c,d) for respectively (cid:15) = 0 . , . , . K of the effective Bloch mode acquires an imaginary part due to the disorder (inaddition to the attenuation) in the ex-pass bands of the perfect ordered case. In counterpart, in the ex-stop bands ofthe perfect ordered case, the imaginary part of the wavenumber decreases, resulting in an increase of the transmission[6].In the second stop band, an interesting behavior can be noticed, although very qualitative at this stage: inside thesecond band gap, around kd = π , the small transparency band remains visible, since we observe a peak of transmissionrobust to disorder. This trend is confirmed by the analytical model (red curves on Fig 3).In the following section, we use numerical calculations to get further insights on this induced transparency near kd = π . IV. NUMERICAL INSPECTION OF THE INDUCED TRANSPARENCY
We now present results from numerical experiments of the propagation in the array of Helmholtz resonators. Thisis done by solving Eq. (2), with variable V n values. The disorder is introduced by using L n = L (1 + (cid:15) n ) in Eq. (3).To calculate p ( x ), we implement a method based on the impedance, as describe in [9]. For each frequency, N = 10 realizations of the disorder with same amplitude (cid:15) are performed. The effective transmission (cid:104) T (cid:105) is calculated byaveraging the transmission coefficients (cid:104) T (cid:105) = 1 /N (cid:80) T n , where the T n are the transmission coefficients for eachrealization.The main result is presented in Figs. 4. In Fig. 4(a), |(cid:104) T (cid:105)| is shown in a 2D plot as a function of kd and (cid:15) andFig. 4(b) shows several transmission curves for given (cid:15) -values. Clearly, with 10 realizations of the disordered, theaveraged systems have converged. The transparency robust to disorder is quantitatively confirmed: For the largestvalues of disorder, the transmission near kd = π increases with the disorder. FIG. 4. (a) Mean value of the transmission coefficient in function of the disorder. (b) Mean value of the transmission coefficientfor (cid:15) = 0 .
08 (blue), (cid:15) = 0 . (cid:15) = 0 .
18 (black) and (cid:15) = 0 . V. DISCUSSION
Robustness of transparency to disorder could appear counterintuitive with regards to the usual influence of disorderin wave propagation. Indeed, the presence of disorder is known to break the wave propagation and to avoid anytransmission. In this study, robustness of transparency is the result of the mixing of two different physical phenomena: (1) the non-exact overlap of the Bragg and hybridization band gaps which generates, in the periodic case, a narrowpassband located inside a band gap and (2) the presence of disorder on potential which prevents the wave propagationinside the media. In the periodic case, one of the edges of transparency band due to overlap is located at kd/π = 1which corresponds to the Bragg frequency [16, 18]. Because disorder is injected in the potential (resonance frequency ofthe Helmholtz cavity), the edge of the transparency band is not affected [12]. As a consequence, the narrow passbandis very sensitive to disorder and disappears from its upper edge remaining the lower edge unchanged and creating apeak of transparence for kd/π = 1. On the contrary, with no overlapping, the Bragg bandgap increases with disorderonly from its upper edge. The lower edge belongs to a passband. In this case, there is no peak of transparency. Thisconfiguration can be used to filtering applications by tuning very narrow filter by injection of disorder in the system.
VI. CONCLUSION
In this paper, we reported an experimental and numerical characterization of a periodic-on-average disorderedsystem. The usual widening of the band gaps of disordered arrays is observed. On the other hand, when nearlyperfect overlap between the Bragg and the scatterer resonance frequencies is realized, evidence of robust transparencyhas been shown.
Acknowledgments . This study has been supported by the Agence Nationale de la Recherche through the grant ANRProCoMedia, project ANR-10-INTB-0914. V.P. thanks the support of Agence Nationale de la Recherche through thegrant ANR Platon, project ANR-12-BS09-0003-03. [1] V. Romero-Garcia, C. Lagarrigue, J.-P. Groby, O. Richoux and V. Tournat, J. Phys. D , 305108 (2013).[2] M.I. Hussein, K. Hamza, G.M. Hulbert and K. Saitou, Waves in Random and Complex Media , 491 (2007).[3] T.Z. Wu and S.T. Chen, IEEE Trans. Microwave Th. and Tech. , 3398 (2006).[4] J.H. Page, A. Sukhovich, S. Yang, M.L. Cowan, F. Van Der Biest, A. Tourin, M. Fink, Z. Liu, C.T. Chan and P. Sheng,Phys. Stat. Sol. B , 3454 (2004).[5] C. M. Linton and P.A. Martin, J. Acoust. Soc. Am. (6), 3413-3423 (2005).[6] V. D. Freilikher, B. A. Liansky, I. V. Yurkevich, A. A. Maradudin and A. R. McGurn, Physical Review E (6), 6301(1995).[7] L. I. Deych, D. Zaslavsky and A. A. Lisyansky, Phys. Rev. Lett. (24), 5390 (1998).[8] P. Han and C. Zheng, Phys. Rev. E (4), 041111 (2008).[9] A. Maurel and V. Pagneux, Phys. Rev. B (5), 052301 (2008).[10] A. Maurel, P. A. Martin and V. Pagneux, Waves in Random and Complex Media (4), 634-655 (2010).[11] F. M. Izrailev, A. A. Krokhin and N. M. Makarov, Physics Reports (3), 125-254 (2012).[12] A. Maurel and P. A. Martin, The European Physical Journal B (11), 1-10 (2013).[13] S. H. Chang, H. Cao and S. T. Ho, Quantum Electronics, IEEE Journal of Quantum Electronics (2), 364-374 (2003).[14] C. Croenne, E. J. S. Lee, H. Hu and J. H. Page, AIP Advances , 041401 (2011).[15] Y. Xiao, B. R. Mace, J. Wen and X. Wen, Phys. Lett. A , 1485 (2011).[16] N. Sugimoto and T. Horioka, J. Acoust. Soc. Am. (3), 1446 (1995).[17] C. E. Bradley, J. Acoust. Soc. Am. (3), 1844 (1994).[18] G. Theocharis, O. Richoux, V. Romero-Garcia, A. Merkel and V. Tournat, Limits of slow sound propagation and trans-parency in lossy locally resonant periodic structures, accepted by New Journal of Physics (2014).[19] O. Richoux and V. Pagneux, EPL (1), 34 (2002).[20] O. Richoux, E. Morand and L. Simon, Annals of Physics, (9), 1983-1995 (2009).[21] O. Richoux, V. Tournat and T. LeVanSuu, Phys. Rev. E, (2), 026615 (2007).[22] The sensor is developed jointly by CTTM (Centre de Transfert de Technologie du Mans, 20 rue Thal`es de Milet, 72000Le Mans, France) and LAUM (Laboratoire d’Acoustique de l’Universit´e du Maine, UMR CNRS 6613, Avenue OlivierMessiaen, 72085 Le Mans Cedex 9, France)[23] A.D. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications, Acoustical Society of America (1989).[24] C. A. Macaluso and J. P. Dalmont, J. Acoust. Soc. Am.129