This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 134.148.29.34

This content was downloaded on 14/09/2014 at 12:17

Please note that terms and conditions apply.

Quantitative characterization of bandgap properties of sets of isolated acoustic scatterers

arranged using fractal geometries

View the table of contents for this issue, or go to the journal homepage for more

2014 Appl. Phys. Express 7 042201

(http://iopscience.iop.org/1882-0786/7/4/042201)

Home Search Collections Journals About Contact us My IOPscience

Quantitative characterization of bandgap properties of sets of isolated acoustic scatterers

arranged using fractal geometries

Sergio Castiñeira-Ibáñez1, Constanza Rubio2, Javier Redondo3, and Juan Vicente Sánchez-Pérez2*

1Dpto. Física Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain2Centro de Tecnologías Físicas: Acústica, Materiales y Astrofísica, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain3Paranimf 1, 46730 Gandia, Universitat Politècnica de València, Valencia, SpainE-mail: jusanc@fis.upv.es

Received January 9, 2014; accepted February 15, 2014; published online March 10, 2014

The improvement in the bandgap properties of a set of acoustic scatterers arranged according to a fractal geometry is theoretically quantified in thiswork using the multiple scattering theory. The analysis considers the growth process of two different arrangements of rigid cylinders in air createdfrom a starting cluster: a classical triangular crystalline array and an arrangement of cylinders based on a fractal geometry called a Sierpinskitriangle. The obtained results, which are experimentally validated, show a dramatic increase in the size of the bandgap when the fractal geometryis used. © 2014 The Japan Society of Applied Physics

The emergence of artificial periodic systems as devicescapable of controlling the propagation of wave fields,such as electromagnetic or elastic fields, has attracted

considerable interest during the last decade. The peculiarbehavior of waves when they are transmitted through theseperiodic materials, which are generally known as wavecrystals, is caused by the existence of bandgaps (BGs).1,2)

These BGs can be defined as ranges of frequencies, related tothe geometrical properties of the crystals, where the trans-mission of waves is forbidden.3) Destructive Bragg interfer-ence due to a multiple scattering process is the physicalmechanism of this attenuation phenomenon. In acoustics,where the periodic systems are generically called phononiccrystals, the BG properties have been used to design severalcontrol devices.4–7) However, the creation of large attenuationbands by means of Bragg interference, which is necessaryto design devices that are competitive with others based ondifferent physical principles, is not easy owing to the highnumber of parameters involved in the formation of the BGs.

Two ways of enhancing the BG properties have tradition-ally been considered. On the one hand, the possibility ofdesigning scatterers including new control mechanisms hasbeen developed. In this sense, some research in the fieldof environmental acoustics has proposed the constructionof mixed scatterers in such a way that Bragg interferenceis not the only mechanism involved in the creation ofwide attenuation bands. Thus, absorption or resonances areincluded in the design of these scatterers.8) On the other hand,the possibility of obtaining new arrangements of scatterersunlike the crystalline ones and with high performance in BGcreation has also been explored. Thus, the use of quasicrys-tals,9) quasiordered structures,10) or even amorphous struc-tures based on the concept of hyperuniformity11) in differenttypes of wave crystals has been analyzed. Joining both linesof research, acoustic barriers with high noise controlcapabilities due to the existence of wide attenuation bandshave been designed, constructed, and homologated.12)

Following this second line of research, some authorshave analyzed the possibility of using fractal geometries inthe design of wave crystals to obtain large BGs.13) Fractalgeometries have been used successfully by either adaptingthe shape of the scatterers14) or designing new arrays ofscatterers in which the increase in the size of the BG is dueto the coexistence of several identical arrays with a different

lattice constant, one within the other.15) In this last case, theresultant arrangements are usually called quasifractal struc-tures because only a few levels of the fractal geometry areachieved, although the fractal construction follows an infiniteiterative process. Nevertheless, even though it has beenqualitatively demonstrated that the size of the BG increaseswhen the scatterers are arranged following a quasifractalgeometry, it seems technologically interesting to quantify thisimprovement in order to examine the real possibilities ofusing these geometries to design technologically advanceddevices based on arrangements of isolated scatterers.

According to this last argument, in this work we present anexperimentally verified theoretical study devoted to quantify-ing the enhancement in the BG properties of a set of rigidisolated scatterers arranged according to a two-dimensionalfractal geometry. To properly organize the study, we followedtwo steps. First, we measured the variations in the BGproperties throughout the growth process of a quasifractalstructure created from the smallest possible starting cluster. Inthis growth process, the starting cluster is repeated as manytimes as necessary according to the fractal geometry selected.In this process, only translations of the starting cluster areneeded to create the quasifractal pattern. In the second step, werepeated the growth process but arranged the scatterersaccording to a classical crystalline pattern. In this case, thetypical translations and rotations of the starting clusternecessary in crystalline growth were used. The growth proc-esses were interrupted at the fourth level in such a way that thehighest samples can be analyzed according to our computa-tional and experimental capabilities. Finally, we quantitativelycompared the BG properties of the quasifractal and crystallinesamples. We took into account that the samples with the samelevel of growth in both cases (quasifractal and crystalline)must have the same external shape and size in order tocompare their BG properties, where the number of scatterersand their arrangement were the only differences allowed.

Although this study and the obtained results can be appliedto any type of wave field in ranges of frequencies where thephysics of the process is linear, we worked with a particulartype of phononic crystal called a sonic crystal,16,17) which isformed of rigid cylinders acting as scatterers in air. Thesetwo-dimensional arrays of scatterers are widely used inresearch for several reasons, such as their simplicity, theirsize, and the high symmetry of the cylinders, which provide a

Applied Physics Express 7, 042201 (2014)

http://dx.doi.org/10.7567/APEX.7.042201

042201-1 © 2014 The Japan Society of Applied Physics

low computational cost in theoretical simulations. Thus, wechose a starting cluster formed of three aluminum cylinderswith a radius of 0.02m and a length of 1m as the scatterers.These cylinders are arranged according to a triangular patternwith a lattice constant a = 0.0635m. We chose this patternbecause, among all the crystal lattices, it presents the largestBG owing to its degree of hyperuniformity.11,15) The inset ofFig. 1(a) shows a cross-sectional view of the cluster. Fromthis cluster, we grew a classical crystalline array following atriangular pattern. On the other hand, the quasifractal struc-ture was grown according to a fractal geometry called theSierpinski triangle, which has been successfully used in otherworks.15) The shape of this fractal is shown in Fig. 1(a); itsgeometry is obtained by removing triangles of different sizesfrom the starting triangle, bearing in mind that the fractallevel increases as the removed triangles become smaller. Herethe fifth level is shown. Note that the shape of each part of thefractal is always the same as that of the whole, which is animportant property of fractal symmetries. In addition to thisshape, in Fig. 1(a) we also show the quasifractal structureformed from the starting cluster and a picture of the experi-mental sample constructed to validate the theoretical results,which rises to the fourth level.

The calculations were implemented using the well-knownmultiple scattering theory.18–22) This analytical self-consistentmethod is used to obtain the scattered field of arrays ofisolated scatterers considering all the orders of scattering, andconsists of solving the following infinite system of linearequations:

Als �XN

j¼1

Xq¼1

q¼�1tjsAjq�ljsq ¼ tlsSls; ð1Þ

where Als represents the coefficients of the series of Hankelfunctions that characterize the scattered waves. With thismethodology, the acoustic pressure at the target point (x, y)for a distribution of cylinders at the frequency ¯ can beexpressed as

pð�; x; y; kÞ ¼ p inc þXN

l¼1

Xs¼1

s¼�1AlsHlsðkrlÞeð�s�lÞ; ð2Þ

where k is the wave number (k ¼ 2��=c), p inc ¼Ps �

sJsðkrÞe�s� is the incident plane wave, ðr; �Þ are the polarcoordinates of the point ðx; yÞ, and ðrl; �lÞ are the polarcoordinates of the point ðx; yÞ with respect to the cylinder l. Inthis work, the attenuation properties are represented using theinsertion loss (IL), which is defined as

IL ¼ 20 logjp incjjpj : ð3Þ

To quantify the attenuation properties of the consideredsamples, we used the attenuation area (AA) parameter in theanalyzed range of frequencies (0–6500Hz) and along the OXaxis (¥X direction of the sonic crystal). The AA parameterhas been used successfully in previous works related tosets of isolated cylinders in air10,15) and can be defined as thearea enclosed between the positive IL spectra and the 0 dBthreshold in the selected frequency range.

The experiments for ascertaining the validity of theanalytical results were conducted in an anechoic chamber8 © 6 © 3m3 in size. We obtained the pressure pattern at apoint located 1m behind the sample according to the schemeshown in the inset of Fig. 1(b), using a prepolarized free-field 1/2″ microphone (B&K 4189) and a sound source(GENELEC 8040A) emitting continuous white noise. The

cluster

r

Sierpinskitriangle

Quasifractalarrangement

(a)

(b)

a

a

S

XY

1m 1m R

Fig. 1. (a) Sierpinski triangle fractal geometry up to fifth level with cross-sectional view of largest quasifractal arrangement of cylinders considered (fourthfractal level). Picture shows experimental sample used in the measurements. Inset shows starting cluster. (b) Comparison of IL spectra obtained theoretically bymultiple scattering simulations (solid red line) and experimentally (dashed black line). Inset shows setup used.

Appl. Phys. Express 7, 042201 (2014) S. Castiñeira-Ibáñez et al.

042201-2 © 2014 The Japan Society of Applied Physics

frequency response of the sample was recorded from theFourier transform of the temporal signal obtained by themicrophone. The samples were constructed by hanging thealuminum cylindrical scatterers on a periodic frame withtriangular symmetry, as shown in the picture in Fig. 1(a). InFig. 1(b), we compare the experimental and theoretical ILspectra. The results exhibit fairly good agreement, so we canvalidate the theoretical model used.

To analyze the attenuation capabilities of the samples, inFig. 2 we show their AAs as a function of the filling fraction(ff ). To compare properly the BG results for both types ofsamples, the crystalline ones have been grown with anexternal triangular shape, as the quasi fractal ones have due tothe fractal geometry used. Note that, at each level of growth,the area occupied by both types of samples is the same. Thus,the couples CS1–FS1, CS2–FS2, CS3–FS3, and CS4–FS4have equal areas, which are greater when the level of growth(1, 2, 3, and 4 in ascending order) is higher. The numericalvalues of these areas are shown in Fig. 2 below the cross-sectional view of each crystalline sample. To represent thegrowth patterns of the samples, in all cases we used theff of the entire samples, defined as the sum of the areas ofall the cylinders forming each sample divided by the totalarea occupied by the sample. This definition differs from theclassical ff definition, but here it seems necessary becausethe rules of the crystalline state are not fulfilled for the fractalgeometries. Nevertheless, this definition allows us to deter-mine the area occupied by the cylinders that form each

sample divided by the total area occupied by the samplein the quasifractal cases, and coincides with the classicaldefinition of the ff used for crystalline arrangements.

Figure 2 shows both types of growth on the main graph.On the right, the growth pattern of the crystalline samples(squares connected by a solid black line) is shown. In thiscase, the ff remains constant throughout the growth process,ff = 36%, whereas the AA increases at the same time as thearea of the samples increases, which is typical behavior incrystalline growth. Thus, the largest crystalline sample (CS4)is formed by 528 cylinders occupying an area of A =

174.5 cm2, with an AA value of 39.580Hz0dB. On the otherhand, on the left, one can observe the growth pattern of thequasifractal samples (circles connected by a dotted blackline). As an aid to correct understanding of the fractal growth,in the transversal cut from the different samples, a blackrectangle indicates the previous sample. Note that the firstsamples in both growth processes are formed from thestarting cluster. In the quasifractal case, the trend observed inthe growth process indicates a decrease in the ff from 28 to14%, whereas the AA increases significantly in the process,from 3.158Hz0dB (FS1) to 64.954Hz0dB (FS4), with this lastvalue being that of the largest quasifractal sample analyzed.The AA, and hence the attenuation capabilities, of the FS4sample is clearly higher (164%) than the corresponding valueof the equivalent crystalline sample (CS4). On the other hand,FS4 is formed of 243 cylinders, 46% of those used in theconstruction of the CS4 sample. These results confirm that (i)

Cry

stal

line

gro

wth

FS4

243cyl.

FS2

FS1 CS1

TS1 TS2

TS3

Fractal samples (FS)

Transition samples (TS)

Crystalline samples (CS)

Startingcluster

CS2

CS3

CS4

528 cyl.

A=174,5cm2

(%)

FS3

Crystal-Fractaltransition

Fractal growth

A=86,5cm

A=42,5cm

A=20,5cm

2

2

2

Fig. 2. Attenuation capabilities vs filling fraction of crystalline (red squares connected by solid black line) and quasifractal (blue circles connected by dottedblack line) growth processes. Steps in creation of largest quasifractal structure (FS4) from a complete crystalline sample of equal size (CS4) are also shown(green pentagons connected by dashed black line). Cross-sectional views of each sample with its main characteristics, including the starting cluster, are alsoshown.

Appl. Phys. Express 7, 042201 (2014) S. Castiñeira-Ibáñez et al.

042201-3 © 2014 The Japan Society of Applied Physics

if a set of scatterers is arranged according to a fractal pattern,a rapid increase in the attenuation properties is achievedcompared with the crystalline case; (ii) if a device formedby a set of scatterers has to occupy a predetermined area, aquasifractal arrangement provides much more attenuationusing fewer scatterers than when the scatterers are arrangedaccording to a crystalline pattern. Additionally, in the maingraph of Fig. 2, one can observe the transition between thecrystalline and quasifractal growth processes (green penta-gons connected by a dashed black line). This transition canbe understood as the creation of the quasifractal FS4 froma complete crystalline sample (CS4), where the TS3, TS2,TS1, and FS4 samples are the four levels of the Sierpinskiquasifractal created. In this transition, one can observe adecrease in the ff when the AA increases in the samples,which maintains a constant external area throughout thetransition.

In summary, in this work we showed quantitatively thesuitability of arranging scatterers according to fractal geo-metries to obtain the largest BG instead of using the classicalcrystalline patterns. We compared the attenuation results,given by the AA parameter, obtained for a set of cylinders inair arranged using either fractal or triangular geometries. Theobtained results show that at only four levels of growth, thequasifractal samples exhibited significantly improved attenu-ation properties while using less than half of the scatterersrequired in the crystalline case. These results can be tech-nologically relevant in the design of wave control devicesusing arrangements of scatterers that are competitive withdevices based on different physical principles.

Acknowledgment This work was financially supported by the SpanishMinistry of Science and Innovation through project MTM2012-36740-C02-02.

1) E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).2) S. John, Phys. Rev. Lett. 58, 2486 (1987).3) J. V. Sánchez-Pérez, D. Caballero, R. Mártinez-Sala, C. Rubio, J. Sánchez-

Dehesa, F. Meseguer, J. Llinares, and F. Gálvez, Phys. Rev. Lett. 80, 5325(1998).

4) D. Torrent and J. Sánchez-Dehesa, New J. Phys. 9, 323 (2007).5) F. Wu, Z. Hou, Z. Liu, and Y. Liu, Phys. Lett. A 292, 198 (2001).6) L.-Y. Wu, L.-W. Chen, and C.-M. Liu, Physica B 404, 1766 (2009).7) J. O. Vasseur, P. A. Deymier, B. Djafari-Rouhani, Y. Pennec, and A.-C.

Hladky-Hennion, Phys. Rev. B 77, 085415 (2008).8) V. Romero-García, J. V. Sánchez-Pérez, and L. M. Garcia-Raffi, J. Appl.

Phys. 110, 014904 (2011).9) Y. Lai, X. Zhang, and Z.-Q. Zhang, J. Appl. Phys. 91, 6191 (2002).

10) V. Romero-García, E. Fuster, L. M. García-Raffi, E. A. Sánchez-Pérez, M.Sopena, J. Llinares, and J. V. Sánchez-Pérez, Appl. Phys. Lett. 88, 174104(2006).

11) M. Florescu, S. Torquato, and P. J. Steinhardt, Proc. Natl. Acad. Sci. U.S.A.106, 20658 (2009).

12) S. Castiñeira-Ibáñez, C. Rubio, V. Romero-García, J. V. Sánchez-Pérez, andL. M. García-Raffi, Arch. Acoust. 37, 455 (2012).

13) N.-K. Kuo and G. Piazza, Appl. Phys. Lett. 99, 163501 (2011).14) X.-J. Liu and Y.-H. Fan, Chin. Phys. B 22, 036101 (2013).15) S. Castiñeira-Ibáñez, V. Romero-García, J. V. Sánchez-Pérez, and L. M.

Garcia-Raffi, Europhys. Lett. 92, 24007 (2010).16) M. M. Sigalas, E. Economou, and M. Kafesaki, Phys. Rev. B 50, 3393

(1994).17) E. Economou and M. M. Sigalas, Phys. Rev. B 48, 13434 (1993).18) Y.-Y. Chen and Z. Ye, Phys. Rev. E 64, 036616 (2001).19) V. Twersky, J. Acoust. Soc. Am. 24, 42 (1952).20) D. Torrent and J. Sánchez-Dehesa, New J. Phys. 10, 023004 (2008).21) J. Mei, Y. Wu, and Z. Liu, Europhys. Lett. 98, 54001 (2012).22) J. Mei, C. Qiu, J. Shi, and Z. Liu, Phys. Lett. A 373, 2948 (2009).

Appl. Phys. Express 7, 042201 (2014) S. Castiñeira-Ibáñez et al.

042201-4 © 2014 The Japan Society of Applied Physics