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

Download details:

IP Address: 131.215.70.231

This content was downloaded on 16/03/2016 at 20:45

Please note that terms and conditions apply.

Visco-thermal effects in acoustic metamaterials: from total transmission to total reflection and

high absorption

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

2016 New J. Phys. 18 033003

(http://iopscience.iop.org/1367-2630/18/3/033003)

Home Search Collections Journals About Contact us My IOPscience

New J. Phys. 18 (2016) 033003 doi:10.1088/1367-2630/18/3/033003

PAPER

Visco-thermal effects in acoustic metamaterials: from totaltransmission to total reflection and high absorption

MiguelMolerón,Marc Serra-Garcia andChiaraDaraioDepartment ofMechanical and Process Engineering, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland

Keywords: acousticmetamaterials, viscous and thermal losses, extraordinary transmission, Fabry–Perot resonance

AbstractWe theoretically and experimentally investigate visco–thermal effects on the acoustic propagationthroughmetamaterials consisting of rigid slabswith subwavelength slits embedded in air.Wedemonstrate that this unavoidable lossmechanism is notmerely a refinement, but that it plays adominant role in the actual acoustic response of the structure. Specifically, in the case of very narrowslits, the visco–thermal losses avoid completely the excitation of Fabry–Perot resonances, leading to100% reflection. This is exactly opposite to the perfect transmission predicted in the idealised losslesscase.Moreover, for awide range of geometrical parameters, there exists an optimum slit width atwhich the energy dissipated in the structure can be as high as 50%. This work provides a clear evidencethat visco–thermal effects are necessary to describe realistically the acoustic response of locallyresonantmetamaterials.

1. Introduction

Metamaterials are artificial structuredmaterials inwhich the presence of resonances in themicro/meso-scaleleads to unprecedented properties [1, 2]. In recent years,metamaterials consisting of rigid slabs withsubwavelength perforations have attracted considerable attention due to their ability to achieve normalised-to-area transmission (i.e., transmission normalised to the fraction of area occupied by the holes) significantly biggerthan unity, a phenomenon known as extraordinary acoustic transmission (EAT) [3]. This phenomenon,analogue to extraordinary optical transmission [4], can be achieved bymeans of different physicalmechanisms,such as the excitation of Fabry–Perot (FP) resonances in the holes [3, 5–7], the acoustic Brewster angle [8, 9], orthe acoustic analog to the supercoupling effect in density–near–zerometamaterials [10]. Promising applicationsto this fascinating phenomenon have been suggested, including acoustic collimators [11], superlenses [12],highly efficient Fresnel lenses [13], beam shifters [14], passive phased arrays [15] and invisibility cloaks [16].

Amain limitation in the practical realization of EAT and other unconventional phenomena in locallyresonantmetamaterials arises from the unavoidable presence of viscous and thermal boundary layers at thesolid-fluid interface [17, 18], which can induce important losses. However, only a few papers have investigatedboundary layer effects inmetamaterials. In [19], it was demonstrated that visco–thermal dissipation has a stronginfluence in the slow sound propagation inwaveguides with side resonators, hindering the formation of near–zero group velocity dispersion bands. This feature was exploited later to design low frequency acoustic absorbers[20].More recently, visco–thermal dissipation inmicroslits has been used to enhance the attenuation ofmetamaterials [21], and important boundary layer effects have also been reported in phononic crystals [22, 23].

The goal of the present work is to investigate visco–thermal losses in acousticmetamaterials consisting ofrigid slabswith subwavelength slits. Previous studies have already proven that this dissipationmechanismmaysignificantly attenuate the otherwise perfect transmission peaks associated to FP resonances [8, 24], while thenonresonant EATmechanismbased on the Brewster angle remainsmuch less affected [8]. However, theseworkslack a clear theoretical analysis of the behaviour of the system in the presence of losses and do not describecompletely the physicalmechanisms governing the reduction of transmission, in relation to reflection and/ordissipation. This paper complements these earlier studies, providing an experimental and theoretical analysis ofthe acoustic transmission, reflection, and absorption in the presence of visco–thermal dissipation. Our results

OPEN ACCESS

RECEIVED

6November 2015

REVISED

4 January 2016

ACCEPTED FOR PUBLICATION

1 February 2016

PUBLISHED

1March 2016

Original content from thisworkmay be used underthe terms of the CreativeCommonsAttribution 3.0licence.

Any further distribution ofthis workmustmaintainattribution to theauthor(s) and the title ofthework, journal citationandDOI.

© 2016 IOPPublishing Ltd andDeutsche PhysikalischeGesellschaft

demonstrate that this lossmechanism avoids completely the excitation of Fabry–Perot resonances in gratingswith very narrow slits, which leads to 100% reflection. In addition, we prove that the optimumabsorption isabout 50%,which is attributed to the critical coupling of the slit cavity resonators with the hostmedium [25–28].

2. Experimental setup

Figure 1(a) shows the samples under experimental consideration, whichwere fabricated using 3Dprinting(StratasysObjet500). Thematerial usedwas a rigid thermoplastic (Veromaterials (c))withmanufacturerspecifiedmass density 1.17–1.18 g cm−3 andmodulus of elasticity 2–3 GPa. The samples are rectangular blockswith an air channel connecting the input and output sides. Sample A has a straight channel, while in samples B toE the channels describe a zigzag path. Forwavelengthsmuch bigger than the height of the corrugations, thezigzag channel behaves similarly to a straight slit with effective length Leff, which is approximately equal toshortest path taken by thewave to pass through the structure [29, 30]. The samples were placed between twoaluminium tubes with square cross-section and 34mm inner side, as shown infigure 1(b). The square cross-section artificially imposes periodic boundary conditions in the transverse directions. Since, from the point ofview of the planewaves traveling inside the tubes, the samples’ geometry is constant along the z-direction, nomomentum can be excited in this direction. Therefore, the pressure field is always constant in z, meaning thatthe structure is equivalent to a 2D slabwith a periodic array of slits along the y-direction, see figure 1(c). Therelevant geometrical parameters of this equivalent system are the slit width w 2.7mm= , the grating periodd=34mmand the effective grating thickness Leff, L L 52 mmeff = = for Sample A, L L2.08eff = for SampleB, L L3.16eff = for SampleC, L L4.24eff = for SampleD and L L5.32eff = for Sample E. The transmissionT p pt i

2∣ ∣= , reflection R p pr i2∣ ∣= , and absorption coefficient A R T1= - - weremeasuredwith 4

microphones (G.R.A.S. 40BD)using the two-port technique [31], where p p,i r and pt are respectively thecomplex amplitude of the incident, reflected and transmitted planemode [see figure 1(b)].Wemeasured thesequantities using phase sensitive detectionwith a sinusoidal wave as reference signal, injected to a loudspeaker(Clarion SRE 212 H) on the left extremity. A 150mmthick absorbing foamwas placed on the right side tominimise backward reflections. The testing frequency rangewas limited to 1 5[ ]- kHz. The lower limit isimposed by the loudspeaker, which is not able to radiate sound below approximately 1kHz. The upper limit is

Figure 1. (a) Samples under test, characterised by the slit width w 2.7 mm= , the grating period d=34mmand the effective gratingthickness L L 52 mmeff = = for Sample A, L L2.08eff = for Sample B, L L3.16eff = for Sample C, L L4.24eff = for SampleD andL L5.32eff = . The cover of the samples has been removed to reveal the internal structure. (b) Schematic of the experimental setup. (c)2Dperforated slab equivalent to the one studied experimentally.

2

New J. Phys. 18 (2016) 033003 MMolerón et al

imposed by the cutoff frequency of the first high-ordermode in the ducts, approximately 5kHz, so that only theplanemode excites the samples.

3.Model

The acoustic propagation through the grating depicted infigure 1(c) ismodelled using amultimodal approachdeveloped in previous works by the authors [13, 32].We express the acoustic pressure field, p x y,( ), as amodaldecomposition,

p x y A B y, e e , 1n

nx

nx

nn n( ) ( ) ( ) ( )ȷ ȷå f= +b b-

whereAn andBn are respectively themodal amplitude of the n-th forward and backwardmode, nb are thepropagation constants, and yn ( )f are the eigenfunctions. In the surrounding space with periodic boundaryconditions, the eigenfunctions are

yd

n1

e , , 2nn d k y2 sin( ) ( )ȷ [ ( )] f = Îp q+

where k cw= is thewavenumber in free space,ω is the angular frequency, c is the sound speed, and θ is theincidence angle with respect to x of the impinging planewave (here 0q = ). Assuming rigid boundaries, theeigenmodes of the slit are given by,

yw

n

wy

wn

2cos

2, 3n

n,0( ) ( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥ f

d p=

-- Î

with n,0d theKronecker delta ( 1n,0d = for n=0 and 0n,0d = otherwise).In the absence of losses, the propagation constants of the slitmodes are given by the dispersion relation

k n wn2 2 2( )b p= - . The effect of the viscous and thermal losses can be taken into account by introducing an

additional term into these propagation constants (see [33]),

kn

w

k

w

22 Im Re 4n n n n

2 22

,0 ȷ( )( { } { }) ( )⎜ ⎟⎛⎝

⎞⎠b

pd e e= - + - -

where

n

wk1 , 5n v t

2

( )⎜ ⎟⎡⎣⎢

⎛⎝

⎞⎠

⎤⎦⎥e

pe e= - +

kl1

2, 6v

vȷ( ) ( )e = +

and

kl1

1 2, 7t

tȷ( )( )

( )eg

=+-

In equations (5)–(7), 1.4g = the adiabatic specific heat ratio of air, lv is the viscous characteristic length and lt isthe thermal characteristic length. At standard conditions, c 344 m» /s, lv and lt are respectivellyl 4.5 10v

8= ´ - mand l 6.2 10t8= ´ - m (see [33]).We note that, according to the time convention chosen in

this paper (e tȷw- ), we only keep solutions to equation (4) fulfilling Re , Im 0n n{ } { } b b .Writing the continuity equations of pressure and normal velocity at the interface between the grating and the

surrounding space leads to the reflection and transmissionmatrices, R and T (see [13, 32] for details), defined asA ARR I=

and A ATR I=

, where AI

, AR

and AT

are row vectors containing the incident, reflected and

transmittedmodal amplitudes. Finally, the energy reflection, transmission and absorption coefficients are givenrespectively by

RA A

A A

R Y

R Y

e

e, 8R R

I I

t

t

{ ( ) }{ ( ) }

( )*

*=

TA A

A A

R Y

R Y

e

e, 9T T

I I

t

t

{ ( ) }{ ( ) }

( )*

*=

A R T1 , 10( )= - -

where ckY diag n{ }b r= and superscripts ‘t’ and ‘*’ indicate respectively the transpose and the complexconjugate. The series of equation (1)was truncated to 40modes in the free, periodic space and 5modes in the slit,fromwhich only the fundamental one [n= 0 in equations (2)–(4)] is propagative.

3

New J. Phys. 18 (2016) 033003 MMolerón et al

4. Results

We start our analysis by studying the influence of Leff in the acoustic response of the samples. Figures 2(a)–(e)show the experimental (solid lines) and numerical (dash-doted lines) transmission coefficients. The losslesstransmission coefficients (doted lines), obtained by replacing nb with k n wn

2 2 2( )b p= - in equation (4), arealso shown. The lossless transmission coefficients exhibits the perfect transmission peaks typical of FPresonances at f sc L2 eff» , with s a positive integer. However, when visco–thermal effects are included in themodel, we observe a strong attenuation of the resonance peaks as Leff increases, which is in good agreementwiththe experimental results.We also observe a downshift of the resonance frequencies compared to the lossless case,due to the slowing down of thewave because of the dissipation [24]. Thesefigures represent afirst and clearevidence that neglecting visco–thermal effects leads to a poor description of the actualmetamaterial response.

Figures 2(f)-(j) show the absorption coefficients as a function of frequency for the different samples.Experimental and numerical results are in good qualitative agreement and demonstrate a strong dissipation atthe FP resonance frequencies. The attenuation peaks reach between 33%and 55% in experiments, and between30%and 50% in the numerical results.We also notice an increase of absorptionwith frequency, which is due tothe fact that Im 0{ }b ( 0b is the propagation constant of the fundamental slitmode) also increases with frequency,approximately as f , see equation (4).

Although the agreement between experimental and numerical results is globally good, particularly in termsof the amplitude of the absorption peaks, we also observe some discrepancies. The experimental absorptioncoefficient is in general higher than the numerical one, which can be atributed to additional losses in theexperimental setup, as visco–thermal losses in the aluminium tubes (these effects are onlymodelledwithin theslits), material losses, or losses due to energy leakage between the different pieces forming the experimentalapparatus.We also observe features in the experimental curves that are not observed in the numerical results.

Figure 2. (a) to (e) show the transmission coefficients for samples A to E, respectively. (f) to (j) show the absorption coefficients forsamples A to E, respectively. The shaded regions in (h), (i) and (j) represent the frequency range inwhich vibrationalmodesmay exist.An example of thesemodes at f=2750Hz is shown in (i).

4

New J. Phys. 18 (2016) 033003 MMolerón et al

The experimental curves around the fourth peak infigures 2(h)-(j) exhibit an additional peak, not observed inthe numerical curve. The origin of these peaks is the excitation of vibrationalmodes of the samples, which isconsistent with the downshift of the peak frequencies as the height of the internal corrugations increases.Weverified this through the computation of the samples’ vibrationalmodes using ComsolMultiphysics. Theshaded regions infigures 2(h)–(j) represent the frequency range inwhich vibrationalmodes were obtained,accordingly to the range ofmechanical parameters provided by themanufacturer. An example of thesemodes atf=2750Hz is displayed infigure 2(i).

It is remarkable to achieve such high absorption using visco–thermal effects, considering that the slit width isabout 2 orders ofmagnitude bigger than the viscous and thermal boundary layers ( 10 m5~ - [34]). Thisbehaviour, which is consistent with recent observations byWard et al. [24], is rather surprising as intuitionsuggest that visco–thermal effects would only become relevant when the slit width is of the same order as theboundary layers’ thickness.

To quantify themaximumamount of energy that can be absorbed in these structures by the combination ofFP resonances and visco–thermal losses, we have computed the amplitude of the first absorption peak,Ares, as afunction of the geometrical parameters. Figure 3(a) showsAres versus Leff andw for d=34mmfixed, andfigure 3(d) shows the same quantity versus d andw for L 52 mmeff = fixed.We observe that the parameterhaving a stronger influence on the response is the slit widthw. For any Leff and d considered,Aeff exhibits amaximumbetween w 0.5 mm= andw=2mm, atwhich the structure dissipates about 50%of the incidentenergy.

The vanishing ofAres after itsmaximum is due to the fact that the acoustic impedance of the surroundingmedia, Z c d1 r= , approaches that of the slit cavity, Z ck w2 0r b= , as w d . This inhibits the formation of ahigh amplitude standingwave in the slit cavity and reduces the ability to dissipate energy. However, thevanishing ofAres as w 0 is less straightforward. In principle, visco–thermal losses (given by Im 0{ }b ) and thestrength of the FP resonance (provided by the impedance ratio Z Z1 2) increase as w 0 when consideredseparately. Hence, one should expect the dissipation to increase also in this region.However, inspecting thetransmission and reflection coefficients at resonance, respectivellyTres [figures 3(b) and 3(e)] andRres

[figures 3(c) and 3(f)], we observe that the structure behaves as a perfect reflector as w 0 , meaning that verylittle energy is stored (and therefore dissipated) in the resonator. Remarkably, although the dramatic drop in thetransmission is a direct consequence of the presence of losses in the system, this drop is not reflected in anincrease of dissipation.

To explain this counterintuitive behaviour, we derive analytical expressions for the reflection andtransmission coefficients. To accomplish this, we reduce ourmodel to only the fundamentalmode in both theslit cavity and the free, periodic space [n= 0 in equation (1)]. The amplitude reflection and transmissioncoefficients of thismode, respectively r0 and t0, take the following form

r1 e 1

1 e 111

Z

ZL

Z

ZL Z

Z

0

2

22

2

eff

eff

12

22

0

1

2

0 1

2

( )( ) ( )

( )( )

ȷ

ȷ=

- -

+ - -

b

b

-

-

and

t4 e

1 1 e, 12

Z

ZL

Z

Z

Z

ZL

0 2 22

eff

eff

1

2

0

1

2

1

2

0( ) ( )( )

ȷ

ȷ=

+ - -

b

b

fromwhichR andT are obtained as R r02∣ ∣= andT t0

2∣ ∣= . In the absence of losses ( k0b = ) the FP resonancesappearwhen e 1L2 eff0ȷ =b , or equivalently when k s Leffp= , which leads toR=0 andT=1, regardless of Leff,w, and d [3, 5–7, 30]. However, when visco–thermal losses are accounted for, 0b is Re Im0 0 0ȷ{ } { }b b b= + ,and FP resonances appearwhen e 1L2 Re eff0ȷ { } =b . In such case, the reflection and transmission coefficientsbecome

r1 e 1

1 e 113res

Z

ZL

Z

ZL Z

Z

0,

2 Im

22 Im

2

eff

eff

12

22

0

1

2

0 1

2

( )( ) ( )

( )( )

{ }

{ }=

- -

+ - -

b

b

and

t4 e

1 1 e. 14res

Z

ZL

Z

Z

Z

ZL

0,

Im

2 22 Im

eff

eff

1

2

0

1

2

1

2

0( ) ( )( )

{ }

{ }=

+ - -

b

b

-

-

5

New J. Phys. 18 (2016) 033003 MMolerón et al

Contrary to the conservative case (equations (11) and (12)), we see that the acoustic response at resonancedepends on the geometrical parameters, both on Leff andw/d (through the impedance ratio Z Z w d1 2 µ ). Forw d , that is Z Z 01 2 , one has R 1 andT 0 , which is consistent with the numerical results infigure 3.Remarkably, this result holds for any L 0eff > as long as dissipation is present in the slit (Im 00{ }b > ), which isalways true in realistic situations. In otherwords, the reflection always tends to 1 in slabs such that w d ,regardless of the thickness Leff. This is visible infigure 3(c). Another implication of equations (13) and (14) is that,whenw is very small (say w 1< mm), high transmission can be achieved only if the period d is comparable tow,that is when Z Z 11 2 . Thismeans that EAT (i.e. transmission considerably bigger than unitywhennormalised to the ratiow/d) cannot be achieved in slabs with very narrow slits.

In order to obtain an experimental evidence for this behaviourwe have fabricated and tested two additionalsamples. The new samples are identical to sample A, but the slit width is equal to 0.7mmand 1.7mm.Figure 4(a) to 4(c) show, respectively, the experimental absorption, transmission and reflection coefficients.Thesefigures confirm the behaviour described previously infigure 3: the reflection (transmission) increases(decreases)monotonically as w 0 , and there is an optimumw thatmaximises the absorption. For aquantitative comparison of experiments with theory, figures 4(d)–4(f) show, respectively,Ares,Rres andTres as afunction ofw. Solid lines represent the analytical results obtainedwith equations (11) and (12). Dashed lines

Figure 3. (a)numerical absorption, (b) transmission and (c) reflection coefficients at resonance as a function of Leff andw, for aconstant d=34mm. (d)Numerical absorption, (e) transmission and (f) reflection coefficients at resonance as a function of d andw,for a constant L 52 mmeff = .

6

New J. Phys. 18 (2016) 033003 MMolerón et al

represent the numerical result obtainedwith themultimodalmethod, equations (8) and (9). The experimentaldata, obtained from themaxima ofA andT, orminima ofR infigures 4(a)-(c) is representedwith dots.Horizontal error bars in experimental results represent the standard deviation of the actual slit width from thedesired values,measured at four different points along the slit. This deviationwas less than 0.1mmfor allsamples. The trend exhibited by experimental results agrees verywell with both numerical and analytical results,either in absorption, transmission and reflection, which corroborates the theoretical predictions.

The optimum0.5 absorption found infigures 3(a), 3(d) and 4(d) is a characteristic of systemswith openresonators fulfilling the critical coupling condition [25], a phenomenon that has recently attracted considerableattention in acoustics [26–28]. This condition refers to the situation inwhich the energy dissipated in theresonator is equal to the energy radiated to the hostmedium. In the case of a symmetric resonator and one–sidedexcitation (as considered in this paper) critical coupling leads to 0.5 absorption [26–28]. To corroborate thisphenomenon in our system, we use the quality factor of the Fabry–Perot transmission peaks,

Q E E EVT rad( )= D + D , where E is the energy stored in the slit cavity resonator, EVTD is the energy dissipatedper cycle due to visco–thermal effects and EradD is the energy radiated per cycle. This quantity is computed asQ f fres= D , where fres is the resonance frequency and fD is the half–power bandwidth. The quality factor can

also be expressed as Q Q Q1VT

1rad

1= +- - - , with Q E EVT1

VT= D- and Q E Erad1

rad= D- . In the absence of visco–

thermal losses (Q 0VT1 =- ) computing the quality factor of the system gives directly the term Qrad

1- . Once Qrad1- is

known, the term QVT1- is given by Q Q QVT

1 1rad

1= -- - - . Notice that critical coupling occurs when Q QVT1

rad1=- - .

The red and blue curves infigure 4(d) represent, respectively, Qrad1- and QVT

1- . As expected, it is observed that the

optimum0.5 absorption (w= 1.20mm) is very close to the crossing between Qrad1- and QVT

1- (w= 1.24mm). It isinteresting to note that critical couplingmay lead to 100%absorption using a reflecting surface close to the slab[26, 27], or using asymmetric resonators [28]. These configurations provide an interesting approach for thedevelopment of high performance sound absorbingmaterials.

Figure 4. (a)Experimental absorption, (b) transmission and (c) reflection coefficient for sample Awith slit width 0.7mm (solid line),1.7mm (dashed line) and 2.7mm (dotted line). (d)Absorption, (e) transmission and (f) reflection coefficient at resonance as afunction ofw. Dots represent the experimental data, obtained from themaxima infigure (a) and (b) and theminima infigure (c). Solidlines represent analytical results (equations (13) and (14)) and dashed lines represent numerical results (equations (8) and (9)). The redand blue curves in (d) represent Qrad

1- and QVT1- , respectively.

7

New J. Phys. 18 (2016) 033003 MMolerón et al

5. Conclusion

In summary, visco–thermal effects are essential to describe realistically the acoustic response ofmetamaterialscomposed of rigid slabswith subwavelength slits. Due to the presence of this lossmechanism, the behaviour ofthe structure at the FP resonances depends completely on the geometrical parameters, which can be adjusted toachieve high transmission, high absorption or high reflection.Ourworkmay have important implications in thedesign of acousticmetamaterials. For instance, the inability to obtain sharp FP resonances in slabs with verynarrow slits compromises the practical realization of resonant EAT at ultrasonic frequencies. On the other hand,understanding and exploiting this property gives the possibility to design subwavelengh sized, tailorable devicesproviding high transmission, high reflection or high absorption. From amore general perspective, we expectthat our workwill result inwidespread consideration of this unavoidable lossmechanism in acousticmetamaterials research.

References

[1] Craster RV andGuenneau S 2012AcousticMetamaterials: Negative Refraction, Imaging, Lensing andCloaking (Heidelberg: Springer)[2] Deymier PA2013Acoustic metamaterials and phononic crystals (Heidelberg: Springer)[3] LuM-H, LiuX-K, Feng L, Li J, HuangC-P, ChenY-F, ZhuY-Y, Zhu S-N andMingN-B 2007Phys. Rev. Lett. 99 174301[4] EbbesenTW, LezecH J, GhaemiHF, ThioT andWolff PA 1998Nature 391 667[5] Christensen J,Martin-Moreno L andGarcia-Vidal F J 2008Phys. Rev. Lett. 101 014301[6] WangX2010Appl. Phys. Lett. 96 134104[7] ZhouY, LuM-H, Feng L,Ni X, ChenY-F, ZhuY-Y, Zhu S-N andMingN-B 2010Phys. Rev. Lett. 104 164301[8] D’AguannoG, Le KQ, TrimmR, AlùA,Mattiucci N,Mathias AD andAközbekN2012 Sci. Rep. 2 340[9] Maurel A, Félix S andMercier J-F 2013Phys. Rev.B 88 115416[10] Fleury R andAlùA2013Phys. Rev. Lett. 111 055501[11] Christensen J, Fernandez-Dominguez A I, Leon-Perez F,Martin-Moreno L andGarcia-Vidal F J 2007Nat. Phys. 3 851[12] Zhu J, Christensen J, Jung J,Martin-Moreno L, YinX, Fok L, ZhangX andGarcia-Vidal F J 2011Nat. Phys. 7 52[13] MolerónM, Serra-GarciaM andDaraio C 2014Appl. Phys. Lett. 105 114109[14] Wei P, Liu F, Liang Z, XuY, Chu ST and Li J 2015Europhys. Lett. 109 14004[15] Li Y, JiangX, Liang B, Cheng J-c andZhang L 2015Phys. Rev. Appl. 4 024003[16] Zhao J, ChenZN, Li B andQiuC-W2015 J. Appl. Phys. 117 214507[17] Kirchhoff G 1868Ann. Phys. 210 177[18] Rayleigh L 1901Philos.Mag. 1 301[19] Theocharis G, RichouxO, Romero-García V,Merkel A andTournat V 2014New J. Phys. 16 093017[20] Groby J-P,HuangW, LardeauA andAuréganY 2015 J. Appl. Phys. 117 124903[21] RuizH, Claeys CC,Deckers E andDesmetW2016Mechanical Systems and Signal Processing 70–71 904–18[22] Duclos A, LafargeD and PagneuxV 2009Eur. Phys. J.: Appl. Phys. 45 11302[23] GuildMD,García-ChocanoVM,KanWand Sanchez-Dehesa J 2014 J. Acoust. Soc. Am. 136 2076[24] WardGP, Lovelock RK,MurrayAR J,Hibbins AP, Sambles J R and Smith JD 2015Phys. Rev. Lett. 115 044302[25] CaiM, PainterO andVahala K J 2000Phys. Rev. Lett. 85 74[26] LeroyV, Strybulevych A, LanoyM, Lemoult F, Tourin A and Page JH 2015Phys. Rev.B 91 020301[27] Romero-García V, Theocharis G, RichouxO,Merkel A, Tournat V and PagneuxV 2016 Sci. Rep. 6 19519[28] Merkel A, Theocharis G, RichouxO, Romero-García V andPagneuxV 2015Appl. Phys. Lett. 107 244102[29] Liang Z and Li J 2012Phys. Rev. Lett. 108 114301[30] Li Y, Liang B, ZouX-Y andCheng J-C 2013Appl. Phys. Lett. 103 063509[31] AbomM1991Mech. Syst. Signal Proc. 5 89[32] MolerónMandDaraioC 2015Nat. Commun. 6 8037[33] BruneauAM, BruneauM,Herzog P andKergomard J 1987 J. Sound. Vib. 119 15[34] Morse PMand IngardKU1968Theoretical Acoustics (NewYork:McGraw-Hill)

8

New J. Phys. 18 (2016) 033003 MMolerón et al