The Experimental Realization of an Acoustic Cloak in Air with a Meta-Composite Shell

31 Pages·2017·10.88 MB·English

applied
sciences
Article
The Experimental Realization of an Acoustic Cloak in
Air with a Meta-Composite Shell
Shiuh-KuangYang1,Jau-ChoLin1,2,*andJyin-WenCheng3
1 DepartmentofMechanicalandElectro-MechanicalEngineering,NationalSunYat-senUniversity,
Kaohsiung80424,Taiwan;[email protected]
2 ArchitectureandBuildingResearchInstitute,MinistryofInterior,NewTaipeiCity23143,Taiwan
3 CepstrumTechnologyCorporation,Kaohsiung80245,Taiwan;[email protected]
* Correspondence:[email protected];Tel.:[phone]
AcademicEditor:VitalyiGusev
Received:10March2017;Accepted:25April2017;Published:29April2017
Abstract: Anisotropic cloak shells can be used for the spatial transformation of a space to alter
the propagation of acoustic waves by redirecting them along a pre-determined path. This paper
outlinesthedesign,fabrication,andexperimentalanalysisofacircularacousticcloakshellmade
of meta-composite material for in-air applications. Based on the three-dimensional coordinate
transformation,wefirstdesignedananisotropiccirclemeta-compositecloakshellaccordingtoits
impedance values. The cloak shell comprises various layered structures with cavities and tubes,
respectively, providing acoustic mass and compliance for the provision of anisotropic material
properties. Secondly,weconductednumericalandexperimentalanalysesunderpracticeworking
conditionstodemonstratetheefficacyoftheacousticcloak.Thestructureofthecloakshell,fabricated
bythree-dimensionalprinting(3Dprinting),isexperimentallyevaluatedinasemi-anechoicroom
withafree-fieldenvironment. Thesimulationandexperimentalresultsdemonstratetheacoustic
cloakingeffectsinthescatteringfarfield. Besidesthescatteringfield,thesoundfieldmeasurement
resultsobtainedwiththeregionenclosedbytheshellalsoshowstheabilitiesofthecloakshellin
alteringthedirectionofwavepropagationalongapre-determinedpathinair.
Keywords: metamaterials;acousticcloaks;meta-compositecloakshell;transformationacoustic
1. Introduction
In1968,Veselago[1]proposedtheconceptofmetamaterialspossessingpropertiesnotfoundin
typicalmaterialsinEMwaves. Sincethen,numerousmetamaterialshavebeendeveloped,including
negativerefractionmaterials,perfectlenses,andinvisibilitycloaks(Shelbyetal.[2]andPendryet
al. [3,4]). In 2000, Liu et al. [5] used a lead sphere and silicon composite material to construct a
resonantsonicmaterialwithuniquematerialproperties. Leeetal.[6]andHuangetal.[7]reported
onamembranousmetamaterialstructure. CummerandSchurig[8]developedacousticcloaksusing
transformationacoustics. Throughthetransformationofspatialcoordinates, incidentsoundfrom
all directions can be guided to pass around an acoustic cloak shell. The object within the cloaked
regionisunaffectedbyincidentacousticwavesandisthereforesaidtobeacousticallycloaked. In2007,
CummerandSchurig[8]usedtheexactequivalenceofacousticsandEMwavesintheconstruction
of two-dimensional (2D) acoustic cloaks. They paved the way for the application of coordinate
transformationtechniquestoprovidecloaksforacousticwaves. Three-dimensionalacousticcloaking
referstoacousticscattering,whichwasdevelopedin2008[9],therebydemonstratingtheexistence
ofthree-dimensional(3D)transformationcloaks. In2013,Chenetal.[10]expandedthestrategyof
transformation acoustics from simple cylindrical shapes to general coordinate systems. This was
achievedbyovercomingthedifficultiesinvolvedinimpedancematchingalongtheboundaryofthe
Appl.Sci.2017,7,456;doi:10.3390/app7050456 www.mdpi.com/journal/applsci
Appl.Sci.2017,7,456 2of31
cloakshellinthebackgroundmedia. Theirresultshaveprovidedareferencebywhichthematerial
parametersofvariousacousticcloakscanbederived. Inthefieldofoptics,AndkjærandSigmund[11]
employedtopologyoptimizationfordesigningall-dielectriccloaksin2011. In2013,Fujiietal.[12]
used a level set-based topology method that provides discrete cloaking configurations with good
performance. Otomori et al. [13] proposed a method for the design of an electromagnetic cloak
madeofferritematerialin2013. FujiiandUeta[14]describedthetopology-optimizedcarpetcloaks
made of dielectric material in 2016. Lan et al. [15] experimentally demonstrated a unidirectional
electromagneticcloakdesignbytopologyoptimization. TheyfabricatedaTefloncloakingsampleby
apreciseengravingmachine. Urzhumovetal.[16]reportedamicrowaveX-bandcloakingdevice
in2013.
Previousresearchonacousticcloaksusedtheconceptofcoordinatetransformationtodevelop
designparametersforcloakshells. Coordinatetransformationaltersthematerialpropertiesofacloak
shellbymakingthemanisotropicwithspatiallyvaryingmassdensityandbulkmodulus;however,
the complexity of such materials has hampered the realization of acoustic cloaks. In recent years,
Zhangetal.[17]usedanetworkofacousticcircuitelementstocreateacylindricalcloakshellforuse
inwateroverafrequencyrangeof52–64kHz. Theyusedthesamemetamaterialsforthefocusing
of ultrasound images with a flat lens [18]. Carpet cloaks in air have also been researched. Popa
and Cummer [19] and Zigoneanu et al. [20] used metamaterial structures comprising a periodic
arrangement of perforated plastic plates, and used the impedance tube method to measure the
transmission properties. Zhu et al. [21] and Popa et al. [22] performed numerical investigations
andexperimentson2Dtriangularacousticgroundcarpetcloaksbasedonperforatedplasticplates.
Sanchisetal.[23]recentlyusedageneticalgorithmtooptimizetheringstructureof3Dacousticcloaks
toreducethescatteringofthefieldinordertomaketheacousticcloakworkable. In2013,Andkjær
andSigmund[24]investigatedacloakthatconcealsanaluminumcylinderacousticallyforairborne
sound based on a gradient-based topology optimization algorithm. In 2014, Zigoneanu et al. [25]
usedtheconceptofcarpetcloakswithametamaterialcomprisingastructureofperforatedplatesfor
thefabricationofapyramidaldeviceasanacousticcarpetcloakoperatingat3kHz. Thepyramidal
cloakingdevicecomprisedfourperforatedplatecomponentstoenable3Dcloaking.
Theoreticalandpracticalguidelineshavebeendevelopedforthedesignofacousticcloaksin
water; however,littleworkhasbeencompletedonthedevelopmentofcloakshellsworkinginair.
Enabling cloaking by the transformation of spatial coordinates (transformation acoustic) requires
spatially anisotropic material properties along three coordinate axes. Furthermore, a cloak shell
workinginairrequiresadensityorbulkmodulusoflessthanunityinsomeareasoftheshell;this,
however,isnoteasytorealizephysically. Themetamaterialsdesignedforuseinwaterareinapplicable
inair. Previousattemptstodevelopmetamaterialsaimedathidingobjectssubmergedinwaterwere
unabletoprovidetheanisotropicmaterialdensitythatwouldberequiredinair. Eventhescattering
cancellationtechniquesproposedbySanchisetal.[23]aredifficulttoimplement,andtheresulting
cloakfailswhenthehiddenobjectisreplacedwithanotherobjectofadifferentshape. Thisfailurecan
beattributedtothescatteringinteractioncausedbytheincongruitybetweenthenewobjectandthe
designoftheringstructurewithinthecloakshell.
The present study focused on the design, fabrication, and evaluation of acoustic cloak shells
in air. This research involved the physical realization of the acoustic cloak in air by means of the
proposedcloakshelldesignedtospatiallyvarytheimpedancevaluesofthecloakshell. Inaddition
tothecloakdesign, wehavealsopresentedanewmeta-compositestructurematerialforpractical
applications. Finally,arealcloakshellwasproducedby3Dprintingandsubjectedtobothnumerical
andexperimentalanalyses.
Appl. Sci. 2017, 7, 456   3 of 31 
Appl.Sci.2017,7,456 3of31
2. The Design of the Cloak Shell with a Meta‐Composite Shell 
2.12.. TThhe eDDeseigsing onf oa fCtihrceuClalro CakloaSkh Sehllelwl bityh TahrMeee‐Dtai-mCeonmsiponoasli tCeoSohrdeilnlate Transformation 
Acoustic cloaking is an important application in acoustic engineering. Developing the means to 
2.1. TheDesignofaCircularCloakShellbyThree-DimensionalCoordinateTransformation
control the propagation of sound waves has long been a goal of acoustic engineers. Transformation 
acoustiAcsc o[8u]s htiacvcel obaekeinn gusisedan inim thpeo rfotarnmt aopf pal icclaotaiokn shinelal ctoou csotinctreonlg tihnee eprrinopg.agDaetvioenlo opfi nsgouthned mweaavness to
byc oimntpraorlttihneg parnoispoatgroatpioicn moaftseoruianld pwroapveerstiehsa.s Tlhonisg kbineedn oaf gdoeavlicoef caacno ubset iacpepnlgieinde aesr sa. Tnroainses fiosromlaatotiro n
fora croesuisdteicnsti[a8l] bhuaivldeinbegesn anudse sdchinoothlse info trhme ovficainciltoya okf shhieglhlwtoacyosn otrr oalirtphoerptsr;o hpoawgaetvioern, iotf rseoquunirdesw tahve es
abbilyityim top aarffteincgt saonuinsodtsr oinp itchem aautedriiballep frroeqpueretniecsy. rTahnigsek. indofdevicecanbeappliedasanoiseisolator
forTrheissi dpenaptiearl bpurielsdeinntgss aan ndosvcehlo omlsetian‐cthoemvpiocisnitiety sotrfuhcitguhrwe abyassoedr aoirnp otrrtasn;shfoowrmevateiro,nit raecqouuisrteiscst,h  e
in awbihliitcyht othaef ftercatnssofournmdsatiinonth oefa supdaibtilael fcroeqourdeinncaytersa nsgqeu.eezes the transformed space into the cloak 
shell reTghioisnp, aapse srhporwesnen itns aFingouvreel m1ae. tTa-oc opmropdousictee satnru acctuoruestbiacs celdoaokn tfroarn asf othrmreaet‐idoinmaecnosuisotnicasl, oinbwjechti,c h 
weth aeptpralinesdf ocromoardtioinnaotef strpaantisaflocromoartdiionna tiens tshqeu eraezdeiaslt hdeirtercatniosfno,r meed, assp awceeliln atos tthhee cvloeratkicsahl edllirreecgtiioonn,, as
u
showninFigure1a. Toproduceanacousticcloakforathree-dimensionalobject,weappliedcoordinate
e , as shown in Figure 1b. The coordinate transformation causes the real domain along the  e  
wtransformationintheradialdirection,e ,aswellastheverticaldirection,e ,asshowninFigureu1b.
u w
diTrehcetiocono trod ibnea tceomtrapnrsefsosremd aintitoon thcaeu csleosakth sehreelal lrdegoimona,i nreasluolntigngth iene thde irfoecrmtioantioton boef cao tmrapnrsefsosremdeidn to
u
regthioenc lionaskidseh. eTlhlree rgeiaoln d,oremsualinti nisg siinmtuheltafonremouastliyo ncoomfaptrreasnsesdfo irnm tehde rveegritoicnailn dsiirdeec.tiTohne, reeal, dthoemreabiny is
w
exsteimnduilntagn tehoeu esflyfeccot mofp trhees scelodaikn ftrhoemv e2rDti ctoal 3dDir. eTchtiuosn,, tehwe, cthloearkebedy erexgteionnd iinng Ftihgeureeff 1ecbt ios ffrtheee fcrloomak tfhreo m
eff2eDctsto o3f Din.cTidheuns,t twheavcelosa, kmeadkrienggi oitn ainn iFdiegaulr peo1sbitiisonfr efoerf rtohme ptlhaeceemffeecntst ooff oinbcjeidctesn rtewquavireisn,gm calokainkginigt.a n
In idtehael phoosriitzioonntfaolr tphleanpela, ceams esnutgogfeosbtejedc tsbrye qCuhireinng  ectl oaakl.i n[g1.0I]n, twhee huosriezdo ntthael pslapnaeti,aals  scuogogrdesinteadteb y
traCnhsefonremtaatli.o[n10 a]n,wd egouvseerdntihneg sepqautaiatliocno oerqduinivaateletnrcaen stofo drmetaertimoninaen tdheg omvaetrenriinagl peaqruaamtieotneresq ρu iavnadle λn cien to
thed etrtaernmsfionremt hdeommaaitne roiaf lap 2aDra cmooetredrisnaρtea ncdircλlei ncltohaekt aras nfosflolorwmsd: omainofa2Dcoordinatecirclecloak
asfollows:

(a) 

(b) 
FigFuigreu r1e. 1D.eDsiegsnig onf ocfirccilrec lcelocaloka uksuinsgin tgratnrasnfosrfmoramtiaotnio ancoaucosutisctsi:c (sa:)( aa)ftaefrt ecroocordoirndaintea tteratnrasfnosrfmoramtiaotni,o n,
ReRgieognio Bn nBon loonlognegr eerxeisxtiss tins itnheth teratnrasnfosrfmoremde ddodmoamina,i nth,ethreebreyb myamkainkgin tgheth oebojebcjte cwtiwthiitnh iinnainccaecscseisbslieb le
to tioncinidceidnet nwtawvaevs.e Ts.hTish misamkeask ethset hmeamteartiearl ipalropproerpteiersti eosf othfet hcelocaloka skheshlle (lRle(Rgeiognio An)A a)naisnoitsrootproicp iacnadn d
inhinohmoomgoegneenoeuosu. s(.b()b G)Geoemometertirci ctrtraannssffoorrmmaattiioonn iinn tthhee vveerrtitcicaalld diriercetcitoino,ne, we,wwh, ewrehinertehine stphaec espinacRee ginio n
ReCginono lCo nngoe rloenxgisetrs einxitshtes tirna nthsefo trrmanesdfodrommeadi nd.oTmhiasino.p Tehraisti oonpeersasteinotnia ellsysecnotniavlelryt scothnevseprtasc ethwe isthpianceth e
wittrhainns ftohrem terdandsofomrmainedin dtoomthaeineq iunitvoa ltehnet oefqauitvhainlepnlta toef ina othrdine rptolaatec hiine voeradceoru tsoti cacinhvieisvieb ilaictyoufostricth e
three-dimensionalobject.
invisibility for the three‐dimensional object.
Appl.Sci.2017,7,456 4of31
 eu df 0 0 
f deu
 (cid:16) (cid:17)−1 
ρ=  0 f df 0  (1a)
 eu deu 
 (cid:16) (cid:17)−1 
0 0 eu df
f deu
e (cid:18)df (cid:19)−1
λ = u (1b)
f de
u
where ρ and λ are the mass density and bulk modulus, respectively, relative to the background
media; e , e , and e are the position parameters of the coordinate system, as shown in Figure 1b.
u v w
e isanonnegativerealnumberdefiningtheradialposition,e liesbetween0to2πande definesthe
u v w
verticalposition. Function f isthecoordinatetransformfunctionintheradialdirectiontobedefined.
Further,theverticalspaceistransformedandcompressedalongtheverticaldirection,asshownin
Figure1b. Forcoordinatetransformationintheverticale direction,thetransformationfunctioninthe
w
verticaldirectionisselectedaccordingtothefollowing: e/ = g(e ) = δ·e . Hereδisthescalefactor
w w w
(realconstant)intheverticale direction. Inaccordancewiththemethodsoutlinedin[10],andusing
w
thewaveequationequivalencebetweenthecoordinatetransformeddomainandtheoutsidefield,the
cloakshellmaterialparametersofbothradiale andverticale directionsinthetransformeddomain
u w
canbewrittenas(seetheSupplementaryMaterials):
 eu df 0 0 
f deu
 (cid:16) (cid:17)−1 
ρ=  0 f df 0  (2a)
 eu deu 
 (cid:16) (cid:17)−1 
0 0 (δ)2eu df
f deu
e (cid:18)df (cid:19)−1
λ = u . (2b)
f de
u
Followingtheabovemethod,wedesignedthe3Dacousticcirclecloakshellworkinginair. Firstly,
in the horizontal plane, we selected the transform function f of the cloak shell in the horizontal
e −e planease/ = R +e (R −R )/R ,wheretheradialpositionsare R = 3and R = 6,and
u v u 1 u 2 1 2 1 2
the cloak shell region along the radial direction is 3 < e < 6. The transform function f leads to
u
e/ = f(e ) = 3+0.5e , 4.5 < e/ < 6, e/ = e , and e/ = e . Theabovecoordinatetransformation
u u u u v v w w
procedure in the radial direction compresses the region in the real domain within e < 3 in the
u
horizontale −e planeintoregione/ > 3inthetransformeddomain. Thisprocessmakesregion
u v u
e <3intherealdomainnon-existentinthetransformeddomainandtransformsregione <3intoa
u u
cloakedregion.Objectsthatonewishestocloakcanbeplacedinsidethecloakedregiontoprotectthem
frombeinghitbyincomingsoundwaves. Secondly,intheverticale direction,werandomlyselected
w
δ =0.4toobtainthecoordinatetransformationfunctionintheverticaldirection,e/ = g(e ) =0.4e .
w w w
SubstitutingthetransformationfunctionintoEquations(2a)and(2b)providesthematerialparameters
ofthe3Dtransformedcloakshellasfollows:
 
0.33 ∼0.5 0 0
ρ=  0 2 ∼3 0 ρ (3a)
  0
0 0 0.21 ∼0.32
λ =1.33λ ∼2λ . (3b)
0 0
According to the above results, the material properties of the cloak shell along e , e , and
u v
e are spatially varying in three directions. The circular acoustic cloak shell is composed of an
w
anisotropic inhomogeneous material. Since the acoustic impedance of the material is the primary
factordeterminingthepropagationbehaviorofwavesthroughthemedium; therefore,weusethe
Appl. Sci. 2017, 7, 456   5 of 31 
Secondly, in the vertical  ew direction, we randomly selected 0.4 to obtain the coordinate 
e/  g(e ) 0.4e
transformation function in the vertical direction, w w w. Substituting the transformation 
function into Equations (2a) and (2b) provides the material parameters of the 3D transformed cloak 
shell as follows: 
0.33~0.5 0 0 
 
ρ 0 2~3 0 0  (3a)
 0 0 0.21~0.32
 
λ1.330 ~20.  (3b)
According to the above results, the material properties of the cloak shell along  e ,  e , and  e  
u v w
are spatially varying in three directions. The circular acoustic cloak shell is composed of an anisotropic 
inhomogeneous  material.  Since  the  acoustic  impedance  of  the  material  is  the  primary  factor 
Appl.Sci.2017,7,456 5of31
determining the propagation behavior of waves through the medium; therefore, we use the guiding 
principle in the design of the cloak shell material. The structural design of the cloak shell is divided 
ignutoi dthinrgeep lraiynecrisp laeloinngt htheed reasdigianl odfirtehcetioclno aaks pshloetltlemd iante Friigaul.rTe h2e. Tshtreu tchturerea lladyeesrisg n(eof =th 3e, c4l.o5,a aknsdh e6l;l 
u
eis d=i v0i~d2edi)n ctoontshtrreuectl atyheer csloaalokn sghethlle rreagdioianl; dthireercetfioorne,a tshpeylo wtteildl bine pFhigyusricea2ll.yT rheealtihzreede blayy perrosp(oeusi=ng3 ,
4v.5,and6;e =0~2π)constructthecloakshellregion;therefore,theywillbephysicallyrealizedby
v
meta‐composite structures. 
proposingmeta-compositestructures.

Figure2.Thecloakshellregionisdividedintothreestructurallayers.Thethreelayers(eu=3,4.5,and
F6i;geuvr=e 20.~ T2πhe) rcelopareks sehnetltlh reegcliooank iss hdeilvlirdeegdio inntwo itthhraene isstorturcotpuircaml laatyeerriasl. pTrhoep tehrrteiees .layers (eu = 3, 4.5, 
and 6;  e  = 0~2) represent the cloak shell region with anisotropic material properties. 
v
SubstitutingthespatiallyvariedcloakshellmaterialparametersρandλinEquations(3a)and
(cid:112) (cid:112)
(3b)Sinutbostrietluattiinogn sthheip szpa=tiaρll·yc v=arρiedλ c/loρa=k sheλllρ m(thateerdieafil npiatrioamnoetfesrps ecρifi canadco ust iicni mEqpueadtaionncse )(,3wa)e 
canobtaintherequiredratioofspecificacousticimpedanceofthecloakshell,z ,ine ,e ande ,as
and  (3b)  into  relationship  z c  /   (the  definition shoefl l specuificv  acouwstic 
showninFigure3a–c. ThediagraminFigure3isplottedaccordingtotherelationshipsbetweenthe
iCmapretedsaianncec)o, owrde icnaant eosbatnaidn tthheec ryelqinudirreicda lractoioor odfi nsapteecisfyics taecmouosntitch iemhpoerdizaonnctea lopf ltahnee calosaSkx s=heell, cozs(e,) 
u shellv
iann deS,y  e= eaunsdin (eev),, ares sspheocwtivne ilny. FAigsusrheo w3an–cin. TFhigeu driea3g,rathme rina tiFoigoufrteh e3 sips epcliofitcteadc oaucsctoircdiimngp etdoa tnhcee 
u v w
valuesofthecloakshellvariesspatiallyinthreedifferentdirections. Acomparisonofthevaluesin
relationships between the Cartesian coordinates and the cylindrical coordinate system on the 
Figure3atothoseinFigure3cshowsthat,althoughtheyexhibitthesamedistribution,thevariation
horizontal plane as  Sx e cos(e ) and  Sy e sin(e ), respectively. As shown in Figure 3, the 
ratiosinthe eu, ev, and ewudirectiovnsdiffer. Theiumpedavncevaluesofthecloakshellvaryspatially
riantitoh eofe thaen dspeecidfiicr eacctoiounssti,ci nimcrepaesdianngcfer ovmaluthees ionfn tehrem colostatko sthheello uvaterriems ossptaltaiyaellry. Iinn tthhereee ddiifrfeecrteionnt ,
u w v
dtihreecitmiopnesd. aAn cceomvaplaureisiosna ocfo tnhseta vnatl.uFeus ritnh eFrimguorree ,3ba ytos uthbostsiet uinti nFgigtuhree d3ecs sighnoewds mthaatte,r aialtlhpoaurgahm tehteeyrs 
(cid:112)
eρxhainbdit λthoef sthame cel odaikstsrihbeulltioobnt, atihnee dvianriEatqiouna tiroantio(3s )iinn ttoheth eep,h aese, vaenlodc iety r edliarteicotniosnhsip diffλe/r.ρ ,Tthhee 
u v w
designedphasevelocitiesinthee ,e ,ande directionswithinthecloakshellareobtained,asplotted
impedance values of the cloak sheull vvary spawtially in the  e  and  e  directions, increasing from the 
inFigure4. AsshowninFigure4,varyingthevelocityaloungdirectiwone whilekeepingitconstantin
v
e
idnnireercmtioonste utoa llothwes  foourttehremmosatn ilpauyleart.i oInno fththee  prvo pdaigraetcitoionnd, irtehceti oinmopfetdhaenacceo uvsatilcuwe aivse sai nctohnesteaun–te.v 
Fpulartnheerwmitohrien, tbhye cslouabkstsithuetliln.gT htehes pdateisailgvnaerdia tmioantseroifalt hpeaprhamaseetevresl ocρityainndsi deth oefc ltohaek  cslhoealkl resghieolnl 

arerelatedtothecoordinatetransformation. Thecoordinatetransformationmakesthecloakshell
obtained in Equation (3) into the phase velocity relationship /, the designed phase velocities 
becomeanisotropicandresultsinthespatialvariationsofthe phasevelocity. Throughthismechanism,
wavesinsidethecloakshellcanbeguidedalongapre-determinedpath(asshowninFigure5,PathA);
i.e.,thepropagationpathofthewavescanbemadetoavoidtheoriginaltransmissionpaththrough
thetransformedregionoccupiedbythecloakedobject. Intheverticaldirectione ,theeffectofthe
w
coordinatetransformationincreasesthephasevelocityoftheacousticwaves,asshowninFigure4c.
Variationsinthephasevelocitycausetheacousticwavestopropagatesmoothlyupward(asshown
inFigure5,Pre-DeterminedPathB),therebyproducingacloakedspaceintheverticale direction.
w
Inthisway,thecloakshellcausestheacousticwavestopropagatealongthepre-determinedpathsand
avoidhittingthecloakedobject.
Appl. Sci. 2017, 7, 456   6 of 31 
e e e
in the  ,  , and   directions within the cloak shell are obtained, as plotted in Figure 4.  
u v w
As shown in Figure 4, varying the velocity along direction  e  while keeping it constant in direction
v  
e  allows for the manipulation of the propagation direction of the acoustic waves in the  e –e  
u u v
planewithin the cloak shell. The spatial variations of the phase velocity inside the cloak shell region 

are related to the coordinate transformation. The coordinate transformation makes the cloak shell 
become  anisotropic  and  results  in  the  spatial  variations  of  the  phase  velocity.  Through  this 
mechanism, waves inside the cloak shell can be guided along a pre‐determined path (as shown in 
Figure 5, Path A); i.e., the propagation path of the waves can be made to avoid the original 
transmission path through the transformed region occupied by the cloaked object. In the vertical 
direction  e , the effect of the coordinate transformation increases the phase velocity of the acoustic 
w
waves, as shown in Figure 4c. Variations in the phase velocity cause the acoustic waves to propagate 
smoothly upward (as shown in Figure 5, Pre‐Determined Path B), thereby producing a cloaked 
space in the vertical  e  direction. In this way, the cloak shell causes the acoustic waves to 
w
propagate along the pre‐determined paths and avoid hitting the cloaked object. 
Appl.Sci.2017,7,456 6of31

(a)
Appl. Sci. 2017, 7, 456   7 of 31 
Appl. Sci. 2017, 7, 456   7 of 31 

(b)

(c)  
(c)
Figure 3. The required ratio of specific acoustic impedance value  z  of the cloak shell (relative to 
FFthiiggeuu brraeec 3k3..g TTrohhueen rrdeeqq suupiierrceeiddfi crra aattciiooo uoosff tsiscpp eiemcciifpfiiecc daaaccnoocuuess tvtiicac liuimmep)p:e e(dada)a ncnclcoeea vkva aslulhueee llz zrsssehhhgeeellillllo onof fit nthh eeec clloo adakkir seshcheteilollln ((r;r ee(lblaa)tt iiivvnee t thtooe  
tthhee bbaacckkggrroouunndd ssppeecciifificc aaccoouussttiicc iimmppeeddaannccee vvaalluuee)):: ((aa)) ccllooaakk sshheellll rreeggiioonn iinn euedui rdecirteiocnti;o(nb;) (ibn) tihne theve 
u
deeirve  cddtiiirroeencc;ttii(oocn)n;i; n ((cct))h  iienn e ttwhheed  ireeewct  ioddniirr.eeccttiioonn..  
v w

(a)  
(a)
Figure4.Cont.

(b)  
(b)

(c)  
(c)
Figure 4. Designed variation in wave phase velocity (relative to background) inside the cloak shell: (a) 
Figure 4. Designed variation in wave phase velocity (relative to background) inside the cloak shell: (a) 
in the  e  direction of the cloak shell region; (b) in the  e  direction; (c) in the  e  direction.  
in the  eu direction of the cloak shell region; (b) in the  ev direction; (c) in the  ew direction.  
u v w
The direction of incoming waves can be guided by altering the wave phase speed using the 
The direction of incoming waves can be guided by altering the wave phase speed using the 
anisotropic material properties of the cloak shell. 
anisotropic material properties of the cloak shell.
Appl. Sci. 2017, 7, 456   7 of 31 

(c)
Figure 3. The required ratio of specific acoustic impedance value  z  of the cloak shell (relative to 
shell
the background specific acoustic impedance value): (a) cloak shell region in  e  direction; (b) in the 
u
e  direction; (c) in the  e  direction. 
v w
Appl.Sci.2017,7,456 7of31

(a)

(b)

(c)
Figure 4. Designed variation in wave phase velocity (relative to background) inside the cloak shell: (a) 
Figure4. Designedvariationinwavephasevelocity(relativetobackground)insidethecloakshell:
in the  e  direction of the cloak shell region; (b) in the  e  direction; (c) in the  e  direction.  
(a)intheeuudirectionofthecloakshellregion;(b)intheevdirectiovn;(c)intheewdirection.Twhedirection
oTfhien cdoimreicntgiown aovfe sinccaonmbiengg uwidaevdesb ycaanlt ebrein gguthideewd abvye pahltaesriensgp ethede uwsainvge tphheaasnei ssoptreoepdi cumsinatge rtihael 
Appl. Spacrnio.i p2s0oe1trr7toi, e7ps,i 4co5 fm6t ha teecrlioala kprsohpeellr.ties of the cloak shell.  8 of 31 

FFiigguurree 55.. PPrree-‐ddeetteerrmmiinneedd ppaatthh ffoorr wwaavvee pprrooppaaggaattiioonn aarroouunndd tthhee ccllooaakk sshheellll.. TThhee aanniissoottrrooppiicc mmaatteerriiaall 
pprrooppeerrttiieess ooff tthhee ccllooaakk sshheellll ccaauussee tthhee ddiirreeccttiioonn ooff wwaavvee pprrooppaaggaattiioonn ttoo cchhaannggee ffrroomm iittss oorriiggiinnaall 
ddiirreeccttiioonn ttoo aa pprree-‐ddeetteerrmmiinneedd ppaatthh.. 
2.2. Meta‐Composite Structural Design for a Three‐Dimensional Circle‐Type Cloak Shell 
2.2. Meta-CompositeStructuralDesignforaThree-DimensionalCircle-TypeCloakShell
AAss sshhoowwnni ninF Figiguurere3 ,3t,h tehsep sepceificcifaicc oaucostuicstiimc pimedpaendcaenvcael uveaslu,zessh,e llz,svheallr,y vianrsyo mine saormeaes aorfetahse ocfl otahke 
schloeallk, wshheilclh, wnehcicehss niteacteessstihtaetuess ethoef uascelo oafk as chleolalkc asphaeblll ecaopfapbrloev oidf ipnrgovthidesinegs ptheceisfiec sipmecpiefidca inmcepevdaalunecse. 
Hvaelruee,sw. eHperroep, owsee apmroeptao-sceo ma pmoestitae‐csotrmupctousrieted setsriugcnteudret odpersoigvnideed dtioff eprreonvtiidmep dedifafenrceenvt ailmuepsedinanthcee 
evua,leuve,sa nind etwhed ireect,i oens,t harnodu gehth ediuresectoiofnssp etchirfiocusgthru cthtuer aulsceo mofp osnpeencitfsi.c Asstrpurcetvuiroaul sclyomdepsocnreibnetds.  
u v w
inFigure2,theanisotropiccloakshellisactivebyathree-layermeta-compositestructure. Figure6
As  previously  described  in  Figure  2,  the  anisotropic  cloak  shell  is  active  by  a  three‐layer 
explainsthecompositedblockelementsthatformthecloakshellbythelayermeta-compositestructure.
meta‐composite structure. Figure 6 explains the composited block elements that form the cloak shell 
Theblockelementsinthelayeraredesignedaccordingtothespatiallyanisotropicpropertiesofthe
by the layer meta‐composite structure. The block elements in the layer are designed according to the 
cloakshellthroughimpedancedesign.
spatially anisotropic properties of the cloak shell through impedance design. 

Figure 6. The three layers of meta‐composite structures forming the cloak shell region. Each block 
element in the layer is designed according to the spatially anisotropic properties of the cloak shell 
through impedance design. 
Acoustic structural engineering [26–28] can be used to develop structural devices, such as tubes 
and cavities, in order to provide acoustic mass and acoustic compliance, respectively. Thus, the 
impedance value can be adjusted by altering the configuration of this structural device. As shown in 
Figure 7a, we assembled block elements with cavities and tubes in three directions, the tubes acting 
as acoustic mass and the cavity in the center ensuring acoustic compliance. Each block element
Appl. Sci. 2017, 7, 456   8 of 31 

Figure 5. Pre‐determined path for wave propagation around the cloak shell. The anisotropic material 
properties of the cloak shell cause the direction of wave propagation to change from its original 
direction to a pre‐determined path. 
2.2. Meta‐Composite Structural Design for a Three‐Dimensional Circle‐Type Cloak Shell 
As shown in Figure 3, the specific acoustic impedance values,  z , vary in some areas of the 
shell  
cloak shell, which necessitates the use of a cloak shell capable of providing these specific impedance 
values. Here, we propose a meta‐composite structure designed to provide different impedance 
e e e
values in the  ,  , and   directions through the use of specific structural components.  
u v w
As  previously  described  in  Figure  2,  the  anisotropic  cloak  shell  is  active  by  a  three‐layer 
meta‐composite structure. Figure 6 explains the composited block elements that form the cloak shell 
by the layer meta‐composite structure. The block elements in the layer are designed according to the 
Aspppal.tSiacil.l2y0 1a7n,i7s,o4t5r6opic properties of the cloak shell through impedance design.  8of31

Figure6. Thethreelayersofmeta-compositestructuresformingthecloakshellregion. Eachblock
Figure 6. The three layers of meta‐composite structures forming the cloak shell region. Each block 
elementinthelayerisdesignedaccordingtothespatiallyanisotropicpropertiesofthecloakshell
tehlreomuegnht iminp tehdea lnacyeerd eiss idgnes.igned according to the spatially anisotropic properties of the cloak shell 
through impedance design. 
Acousticstructuralengineering[26–28]canbeusedtodevelopstructuraldevices,suchastubes
Acoustic structural engineering [26–28] can be used to develop structural devices, such as tubes 
and cavities, in order to provide acoustic mass and acoustic compliance, respectively. Thus, the
and cavities, in order to provide acoustic mass and acoustic compliance, respectively. Thus, the 
impedancevaluecanbeadjustedbyalteringtheconfigurationofthisstructuraldevice. Asshown
impedance value can be adjusted by altering the configuration of this structural device. As shown in 
in Figure 7a, we assembled block elements with cavities and tubes in three directions, the tubes
Figure 7a, we assembled block elements with cavities and tubes in three directions, the tubes acting 
actingasacousticmassandthecavityinthecenterensuringacousticcompliance. Eachblockelement
as acoustic mass and the cavity in the center ensuring acoustic compliance. Each block element 
comprisesasinglecavityinthecenterwithfourtubesindifferentdirections. Theseblockelements
provide the acoustic impedance value z . In the structure of tube, the physical law governing the
b
motion of air inside the tube [26,28] is f(t) = M(du(t)/dt), where M is the mass of volume fluid
inside the tube. After dividing the cross-sectional area A of the tubes, the motion of air leads to
(cid:0) (cid:1)
f(t)/A = (M/A)·{d[u(t)A]/dtA},whichcanbefurtherexpressedas p(t) = M/A2 ·[dU(t)/dt].
(cid:0) (cid:1)
Usingtheabove,therelationshipcanbefurtherexplainedasfollows: p(t) = M/A2 ·[dU(t)/dt]
(cid:2) (cid:3)
= (ρ l A)/A2 ·[dU(t)/dt]= M [dU(t)/dt] = jwM U. Inthestructureofthecavity,thephysical
0 e A A
lawsgoverningthecompressionofairbeingactedonbyaforceinacavity[26,28]leadtothefollowing
(cid:82)
equation: p(t) = (1/C ) U(t)dt = U/jwC = −jU/wC . The value M is the acoustic mass
A A A A
or acoustic inertance, C is the acoustic compliance. Using the above properties, a block element
A
comprisingtubesandcavitystructuresprovidestheacousticimpedancevalueZ intherespectivee ,
b u
e ,ande directionsasfollows:
v w
1 (cid:18)ρ l (cid:19) 1(cid:18)ρ c2(cid:19)
Z =jwM − j=jw 0 e − 0 j. (4)
b A wC A w V
A
Appl. Sci. 2017, 7, 456   9 of 31 
comprises a single cavity in the center with four tubes in different directions. These block elements 
z
provide the acoustic impedance value  . In the structure of tube, the physical law governing the 
b
motion of air inside the tube [26,28] is  f(t) M(du(t)/dt), where M is the mass of volume fluid 
inside the tube. After dividing the cross‐sectional area A of the tubes, the motion of air leads to 
   
f(t)/A(M/A) du(t)A /dtA
,  which  can  be  further  expressed  as 
pt M / A2dUt/dt. Using the above, the relationship can be further explained as follows:
p(t) M /A2dUt/dt =  l A/ A2dUt/dt =  M dUt/dt jwM U .  In  the 
0 e A A
structure of the cavity, the physical laws governing the compression of air being acted on by a force 
in a cavity [26,28] lead to the following equation:  p(t)(1/C )U(t)dtU/jwC jU/wC . 
A A A
M C
The value   is the acoustic mass or acoustic inertance,   is the acoustic compliance. Using the 
A A
above properties, a block element comprising tubes and cavity structures provides the acoustic 
impedance value  Z  in the respective  e ,  e , and  e  directions as follows: 
b u v w
1 l  1c2 
Zb  jwMAwC j jw A0 e w V0 j   (4)
A .
The  first  part,  jwM  is  the  acoustic  impedance  value  supplied  by  the  tube  structure;  
A
j(1/wC )
the second value,  , is the acoustic impedance value supplied by the cavity structure.  
A
In Equation (4),  j 1,  w is the circular frequency,    is the density of the background 
0
medium (air), and c is the wave speed in air. Values A represent the cross section of the tubes and 
value V is the volume of the cavity. Because tubes connected the cavity, forming as the flanged end 
in the cavity side, the effective length  l  is the original tube length plus the correction value of the 
Appl.Sci.2017,7,456 e 9of31
flanged end [27] as shown in Figure 7b. 

Appl. Sci. 2017, 7, 456   10 of 31 
(a)
(b)
Figure 7. Structure of the meta‐composite material used to make up the cloak shell: (a) the block 
Figure7. Structureofthemeta-compositematerialusedtomakeupthecloakshell: (a)theblock
elements include cavities and tubes arranged as circular elements in three layers. The inset on the 
elementsincludecavitiesandtubesarrangedascircularelementsinthreelayers. Theinsetonthe
upper  right  shows  the  details  of  single  block  elements  with  rectangular  cavities  and  tubes.  
upperrightshowsthedetailsofsingleblockelementswithrectangularcavitiesandtubes.Thebottom
The bottom right inset exhibits both the cloak shell layered structures and cylinder object inside the 
rightinsetexhibitsboththecloakshelllayeredstructuresandcylinderobjectinsidethecloakedregion;
cloaked region; (b) the part slices of the meta‐composite material layer exhibit the interior cavity and 
(b)thepartslicesofthemeta-compositemateriallayerexhibittheinteriorcavityandtubeindirections.
tube in directions. The coefficient a explains the cross section dimension of and individual element, 
Thecoefficientaexplainsthecrosssectiondimensionofandindividualelement,andlerepresentsthe
aenffde ctlieve rleepnrgetshesnotsf tthhee etuffbeectciovne nleenctgetdhsw oift hthteh etucbaev ictoy.nnected with the cavity. 
e e e
ETqhueafitirosnt p(a4r)t ,pjwroMvideiss tihmepaecdouanstciec ivmaplueedsa nocf etvhael ucelosaukp pshlieeldl biny the,t ube, satrnudc ture; dthireesceticoonnsd.  
A u v w
Tvhaleu √efr,ejq(1u/ewncCyA )o,f isththe emaecotau‐cstoimcipmopsieted asntrcuecvtualrue eiss uspept laiesd 1b0y00t hHezc,a vainty usntrduiscttourrtee.dI nfrEeqquueanticoyn f(o4r), 
j = −1,wisthecircularfrequency,ρ isthedensityofthebackgroundmedium(air),andcisthe
human hearing. Using the relationshi0p  Z A  Z  in conjunction with the specific cavity 
b i shell
wavespeedinair. ValuesArepresentthecrosssectionofthetubesandvalueVisthevolumeofthe
dimensions, the cloak shell can be represented using a three‐layer metamaterial comprising a 
cavity. Becausetubesconnectedthecavity,formingastheflangedendinthecavityside,theeffective
sletrnugctthurlaeli bsltohcek oerleigmineanlt,t uasb eshloenwgnt hinp Fluigsutrhee 7c. oNrroetcet itohnatv  aAlui eoaf2t hies flthaen cgreodsse‐nsedct[i2o7n]aal sarsehao owfn thien 
iFnidgiuvried7uba.l element forming the cloak shell layers, where the element size a of a specific layer in the 
studyE qisu aptrieosnen(t4e)d pinro vFiidguesre im7bp. eTdhaen cdeesvigalnu epsaroafmtehteersc lwoaekre sdheesllcriinbedeu ,ine vT,abalned 1.e wBedcairuesceti othnes .
mTheetafmreaqtueerinacly  sotfruthcetumree tap-rcoovmidpeoss itaecostursutcictu rmeaissss eatnasd 10a0c0ouHszti,ca ncoumndpilsiatonrctee drfartehqeure nthcyanf orachouumstainc 
rheesaisrtinangc.eU, stihneg sthheellr eisla atibolne sthoi psto|Zre |a·cAou=sticZ enerignyc roantjhuenrc tthioann wcoitnhstuhmees peenceifirgcyc aavnidty itd himase na slioowns‐,lothsse 
b i shell
nclaotaukres.h Feilglucaren 8b eexrepplareinses nttheed iunsteinrigoar dtherteaeil-sla oyfe rthme emtaemtaa‐tceormialpcoosmitep rmisaintegriaals tsrtuructcuturarleb, lsohcokweilnegm tehnet ,
interior details of block elements with rectangular cavities and tubes sliced by a horizontal surface in 
the  e –e  plane and by a vertical surface in the  e –e  plane, respectively. Based on the principle 
u v u w
of Equation (4), the combination of cavity and tubes of the block element provides the acoustic 
impedance value  Z  in the  e ,  e , and  e directions, respectively. The tubes connected the 
b u v w 
cavity without extending inside the interior of the cavity. In the design, the cloak shell structure 
element unit is composed of cavity and a pair of tubes series connected in the respective direction. 
They give the impedance performance as the Helmholtz resonator [29]. Table 1 lists the design 
parameters of the structure. A total of 144 block elements were used in the cloak shell. The volume of 
the cavities was set as 27,000  mm3 for each layer. The largest size of the cavities was 30 mm,  
and the wavelength of the design frequency was 343 mm (1000 Hz, c = 343 m/s); therefore, the size of 
the elements did not exceed 1/10 of the wavelength. The cloak shell structure elements were 
designed based on the principles of the long wavelength limit [30] and the structure of the cloak 
shell could then be treated as an effective medium for incident waves. As per the anisotropic nature 
of the acoustic impedance values in three different directions, the designed block element tube 
lengths in three directions belonging to the cloak shell are plotted in Figure 9. The spatial variations 
of  the  tube length around  the  circumference  of the  circle‐type  shell  agree  with  the required
Appl.Sci.2017,7,456 10of31
asshowninFigure7. Notethat A = a2isthecross-sectionalareaoftheindividualelementforming
i
thecloakshelllayers,wheretheelementsizeaofaspecificlayerinthestudyispresentedinFigure7b.
ThedesignparametersweredescribedinTable1. Becausethemetamaterialstructureprovidesacoustic
massandacousticcomplianceratherthanacousticresistance,theshellisabletostoreacousticenergy
ratherthanconsumeenergyandithasalow-lossnature. Figure8explainstheinteriordetailsofthe
meta-compositematerialstructure,showingtheinteriordetailsofblockelementswithrectangular
cavitiesandtubesslicedbyahorizontalsurfaceinthee –e planeandbyaverticalsurfaceinthee –e
u v u w
plane,respectively. BasedontheprincipleofEquation(4),thecombinationofcavityandtubesofthe
blockelementprovidestheacousticimpedancevalueZ inthee ,e ,ande directions,respectively.
b u v w
Thetubesconnectedthecavitywithoutextendinginsidetheinteriorofthecavity. Inthedesign,the
cloakshellstructureelementunitiscomposedofcavityandapairoftubesseriesconnectedinthe
respectivedirection. TheygivetheimpedanceperformanceastheHelmholtzresonator[29]. Table1
liststhedesignparametersofthestructure. Atotalof144blockelementswereusedinthecloakshell.
Thevolumeofthecavitieswassetas27,000mm3 foreachlayer. Thelargestsizeofthecavitieswas
30mm,andthewavelengthofthedesignfrequencywas343mm(1000Hz,c=343m/s);therefore,
thesizeoftheelementsdidnotexceed1/10ofthewavelength. Thecloakshellstructureelements
weredesignedbasedontheprinciplesofthelongwavelengthlimit[30]andthestructureofthecloak
shellcouldthenbetreatedasaneffectivemediumforincidentwaves. Aspertheanisotropicnatureof
theacousticimpedancevaluesinthreedifferentdirections,thedesignedblockelementtubelengths
inthreedirectionsbelongingtothecloakshellareplottedinFigure9. Thespatialvariationsofthe
Appl. Sci. 2017, 7, 456   11 of 31 
tubelengtharoundthecircumferenceofthecircle-typeshellagreewiththerequiredimpedanceofthe
Appl. Sci. 2017, 7, 456   11 of 31 
cloakshellinFigure3. Throughthecombinationofthecentercavityanddifferenttubelengthsinthe
impedance of the cloak shell in Figure 3. Through the combination of the center cavity and different 
eu,evi,tmuapbneed dlaeenwncgdet hiorsfe  ticnht eiot hcnleos a,ekth s, eheepllr,  oinapn Fodisg euedrem  3d.ei rTteahc-rtcoioounmgsh,p  ttohhseei  tcpeorsmotprbouinsceatdtui ormen epotfar e‐tchcoeism ceeplnyotsesiritme c asuvtrliautytce tausnrtdeh  depirafefnecirisseeonltty r opic
u v w
matetrsuiibamelu pllearntoegpst ehrtsht iieen s a,thnweis hoetiurco,h peipcvr , omavnaiddte ereiatwhl  edpisrrohepcetelilrotsnietsrs,u , tchwteuh pricerhof poporrsoethdvei dm3ee Dttah‐ceco oomsrhdpeolilns iatsetter usttcrrtuaucnrtesu frofeor rmp rtaehcteiiso en3lyDc  loak.
Italssociomdouersldacitnreiasbt eet shtretah neasnfaoinsroimstroaottpirooincp  cimcloamatkea.r tiIeatrl  aialpslropo dproeesrptciereirsbt,ie esws thhoiefc chal noipasrokotvrsoihdpeeilc l tmchaean tesbrhieaelal lpc hrsotirepuvecrettduiersien  odffo icrrle octahtkley  sth3heDrll o ugh
theimccoaponer ddbiean naacctehe itedrvaeensdisg fionnrd.miraetciotlny  cthloraoku.g Iht  tahlseo i mdepsecdriabnecse  tdhees aignnis. otropic material properties of cloak shell 
can be achieved indirectly through the impedance design. 
TabTleab1l.eD 1e. sDiegsnigpna praarmameteetresrso offb blloocckk eelleemmeennttss uusesded ini nthteh celocalok askheslhl. ell.
Table 1. Design parameters of block elements used in the cloak shell. 
Layer  Element Size a  Cavity Volume  Tube Inner Diameter c Element Number/per 
ElementSizea CavityVolume Element
LayerNNLuuamymbebre err  Eleme(mntm S)i ze a  Cba v× ibty × V bo (lmumme3)   TubTeu IbnenI(enmrn Dmeri)a Dmieatmere tcerc(mElmem) ent LNauymerb er/per 
(mm) b×b×b(mm3) Number/perLayer
Num1b er  (m5m5 )  b × b 2×7 b,0 (0m0 m3)  (m11m.2)8   La2y8e r 
1 1  5555  2277,,000000  e ‐direc1t1io.2n8  11.2181.28  28  28
u
2 22   414.4211.2.2   222777,,00,0000000   eeuve‐‐udd-idrireiercectticoitoinon n  11.92 8  11.28 5522   52
3  43.9  27,000  eve‐vd-idreircetci1ot1ino.2 n8  9  9 64 
3 43.9 27,000 11.28 64
3  43.9  27,000  11.28  64 
Block 
eBlelomcekn t 
Blockelement
deimlemenesniot ns 
dimensions
dimensions 

Figure 8. The interior details of the meta‐composite material structure. The block elements include 
Fciagvuirtiee s8 .a nTdh et uinbteesr aiorrr adnegteadil sa so cf itrhceu lmare etale‐mcoemnptso isnit teh mreea tlearyiaelr ss.t Truhcet uinrsee. tT ohne  tbhleo lcekf te slhemowens ttsh ein dcelutadiels  
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u v
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u w
TThhee yr iegxhhti binits tehte  pdreetsaeilnst os f dtheeta cillso aokf  sthheell  lbalyoecrke de lsetmruecntutsr essl iicnesdid eb yth ea  cslouarkfaecde  reing iothne.  Tehue –coemwb ipnlaatnioen.   
Tohfe cya vexithyi baint dth teu dbeetsa oilfs  tohfe t hbelo cclkoa ekl esmheelnl tla pyreorvedid setsr uthcteu arecos uinsstiicd eim thpee cdlaonakcee dv arelugeio nZ. Th ien c sopmebciifnica tieon , 
b u
oef ca, vaintdy  aend tduibreesc toiof nths,e r belsopcekc teilveemlye.n t provides the acoustic impedance value  Zb in specific eu, 
v w 
e , and  e directions, respectively. 
v w