Latticed pentamode acoustic cloak (supplementary Info)

Transcription

1 Lattied petamode aousti loak (supplemetay Ifo) Yi Che, Xiaoig Liu ad Gegkai Hu Key Laboatoy of yamis ad Cotol of Flight Vehile, Miisty of Eduatio, Shool of Aeospae Egieeig, Beiig Istitute of Tehology, Beiig 8, Chia We peset i this supplemetay mateial aalytial solutios of wave atteig of layeed ylidial loak with pefet ad impefet PM mateials. Coside elastodyami equatios fo solid mateial σ C: u, u σ. (S) The elasti teso of pefet PM mateial fo ylidial loak is haateized by C S S, S K e e K e e, (S) K whih has oly oe o-zeo eigevalue, ad the mateial a sustai oly oe stess states popotioal to S, i.e. σ ps, whee the ala p is amed as pseudo pessue. Fom Eq. (S) ad Eq. (S), the wave equatio of the pefet PM mateial a be expessed i the pseudo pessue as p S: [ ( ps )]. Fo poblem i pola system, it eads. p p p p p, (S3) whee K ad ad ig dietios, espetively, K ae the phase veloities alog the adial K / the wave speed of bakgoud fluid. Fo impefet PM mateial, the vetoial elasti wave whih is goveed by K K K K K G u u u u u u,, (S4) u, u must be osideed. S. Satteig of ylidial layeed loak with pefet PM mateials

2 Suppose the loak is assembled fom pefet PM layes whih ae umbeed fom the ie side to the oute by ~. The th ( ) PM laye oupies ig egio ad is haateized by ( K, K, ). A iidet plae wave p exp ik x with exp it i time depedee is exited i the bakgoud fluid, whee k / is the wave umbe. The iidet ad atteed pessue i bakgoud egio ( the fist kid, espetively, ) ae expessed i Bessel futio ad Hakel futio of os, os (S5) p a J k p b H k i with a ( ) i ( ) beig the kow iidet oeffiiet. The pseudo pessue i eah PM laye satisfies equatio Eq. (S3) i pola system. The solutio a be expaded as summatio of Bessel equatio of o-itege ode v os g, ad g v, g g k g v satisfies whee K, K field i the th PM laye is the obtaied as, ad k. The pseudo pessue whee p, a J k b H k os (S6) v v J v ad H v desigate Bessel futio ad Hakel futio of o-itege ode v fo the th laye. Employig the otiuity oditio of omal veloity ad tatio aoss the itefaes betwee PM layes (fo bakgoud fluids, [ p], [ p ], ( ~ ) e S e S. S I), the tasmittae elatio fo the oeffiiets of adaet layes a be obtaied as, a a( ) Y, Y A, (,... ) (S7) b b ( ) s J ( k ) h ( ) s H ( k ) f ( ) ( ) ( ) v ( ), v s J ( k ) h ( ) s H ( k ) f ( ) v ( ) ( ) v ( ) v i s H ( k ) h ( ) s H ( k ) h ( ) ( ) v ( ) v s J v k f s( ) J( ) v k( ) f ( ) ( ) ( ) ( ) ( )

3 whee A s, ( J v( k ) H v ( k ) H v ( k ) Jv( k )) h k k ( ) ( s s ) H v( ) s H v ( ) f ( ) ( s s ) J ( k ) s J ( k ) v v s K, s K ad the pime deotes deivative with espet to. The solutio of atteig oeffiiets b i the bakgoud fluids a be fialized by omplemetig the soud-had oditio (adially fix the displaemet) at the loak s ie bouday, T b a b a T a T b ( p) T e S,, T Y (S8) ( s s ) J ( k ) s J ( k ) a v v ( s s ) H ( k ) s H ( k ) b v v The pseudo pessue i eah laye a the be detemied by Eqs. (S6-S8). S. Satteig of ylidial layeed loak with impefet PM mateials Fo the ylidial loak with impefet PM mateial, the pessue i bakgoud fluid still takes Eq. (S5), while the wave field iside the impefet PM mateial obeys elasti wave Eq. (S4). The poblem is atually to solve aousti atteig of a ylidial shell made of gaded othotopi solids, ad is hadly to get a losed fom aalytial solutio. To this ed, a semi-aalytial poedue, state spae appoah [, ], is employed. Coespodig to ylidial expasio of the iidet wave, we adopt followig mode expasios of displaemet ad stess iside the impefet PM loak shell u u os, u u si os, si, os (S9) Fo the th ode, T { u, u,, } is defied as the state vaiable sie they ae oeted laye by laye by the otiuous oditio aoss itefaes. By usig Eq. (S9) ad Eq. (S4), ad afte a tedious maipulatio, goveed by a set of odiay diffeetial equatio, is deived to be d d P (S)

4 whee P KK KG ( ) K K K K K ad K / K ( ) /. Eq. (S) a be umeially solved by dietizig the loak (eithe otiuously vayig o layeed) shell futhe ito suffiietly thi layes, hee P a be egaded as a ostat matix i eah thi laye. The state vaiable at the fot ( ) ad bak ( ) of the th thi laye is oeted by Aumulatively, ( ) exp ( ) P ( ), (S) at the oute ad ie side of the loak is oeted as ( ) T ( ), T exp(( ) P (( ) / )). (S) At the oute bouday of the loak, the otiuous oditio betwee the solid ad bakgoud fluid is fulfilled by u ( ) ( aj ( k ) b H ( k )) / ( ) ( a J ( k ) b H ( k )), ( ). At the ie bouday of the loak, adially suppoted bouday oditio is efoed i osistet with that fo the loak with pefet PM mateial, u ( ), ( ). The atteig oeffiiet i the bakgoud fluid a the be detemied as, b a ( T T T T ) J( k ) ( T T T T ) K k J ( k ) ( T T ( ) k H ) T 3 T 43) H k ( T T 43 T 3 T 4) K ( k I the umeial alulatio, the loak shell is dietized ito about thi layes. S.3 etemiatio of TSCS (S3) Havig the atteig oeffiiet b, the total atteed eegy a be alulated usig followig itegal alog a ile with adius elosig the loak shell,

5 p E p ds p p d Im Im C. (S4) Substitutig Eq. (S5) ito above equatio, ad osideig the idetity Im H ( k ) H ( k ) / ( ), Eq. (S4) a be poved to be idepedet o. The total atteed eegy ad TSCS ae evaluated as E b b ( ), TSCS ( ) k (S5) Fo the full wave simulatio of the lattied loak, Eq. (S4) is dietly used i the TSCS alulatio by umeial itegatio. S.4 Effetive desity of the PM mateial Sie the PM lattie is solid ad thee is o esoae happes i the osideed fequey egime, the effetive desity a be simply alulated by its volume aveage. As a ustifiatio, we plot i the figues below the dyami effetive desities of the PM mateial alog the - ad theta- dietio withi ka=[, Pi]. We adopt the same method etievig the dyami effetive desity as that i [3]. The left pael ad ight pael ae the desities etieved fom the most ie laye ad most oute laye of the loak, espetively. It is see fom the figue that the desity aisotopy of the PM mateial is egligible ad vey lose to the volume aveage. Refeees: [] W. Che, Z. Bia, ad H. ig, It. J. Meh. Si. 46, 59 (4). [] S. Hashemiead ad M. Raabi, J. Soud Vib. 3, 8 (7). [3] R. Zhu, X. Liu, G. Huag, H. Huag ad C. Su, Phys. Rev. B, 86, 4437().