483 Shar Wav Propagaion in PizolriPizolri Composi layrd sruur Absra Th propagaion bhavior of ar wav in pizolri omposi sruur is invsigad by wo layr modl prsnd in his approah. Th omposi sruur ompriss of pizolri layrs of wo diffrn marials bondd alrnaivly. Disprsion quaions ar drivd for propagaion along h dirion normal o h layring and in dirion of layring. I has bn rvald ha hiknss and lasi onsans hav signifian influn on propagaion bhavior of ar wav. Th phas vloiy and wav numbr is numrially alulad for alrnaiv layr of Polyvinylidn Difluorid (PVDF and Lad Zirona Tiana (PZT H in omposi layrd sruur. Th analysis arrid ou in his papr valuas h ff of volum fraion on h phas vloiy of ar wav. Kywords Pizolri, Shar wav, disprsion quaion, volum fraion, PVDF. Anu Mli Gaur * a Din Singh Rana b a,b Dparmn of Insrumnaion (I.I.E, Kurukra Univrsiy Kurukra, Haryana, India369 *Email:gauranu@gmail.om Phon: +9744389 Rivd 3.3.4 In rvisd form 8.4.4 Apd.6.4 Availabl onlin 7.8.4 INTRODUCTION Th phnomnon of pizolriiy disovrd by Prr and Jaqus Quri has provn o b limligh in dvlopmn of ponial marial for nw lass of snsors and auaors. In rn yars pizolri marials has drawn muh anion owards appliaion in surfa aousi wav (SAW miro snsors, nrgy harvsing sruur, halh monioring sysms, ransdurs and auaors (Du al. 7. SAW dvis basd on pizolri omposis hav nhand lromhanial rspons and high snsiiviy in omparison o singl marial sruur. Pizolri omposis hav found maor appliaion in snsing and masurmn indusris also invariably. Th dynami rspons of Saw snsor is valuad by analyzing h wav propagaion and vibraion parn in hs pizolri basd omposi sruurs. Numrous rsarhrs hav invsigad h propagaion bhavior of ar wav in pizolri omposi du o is vas appliabiliy in saw dvis. Qin al. (4 invsigad h propagaion bhavior of horizonally ar wav in pizolri polymr omposi sruur. Th dis
484 Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur prsiv and anuad hararisi of wav propagaion in hin pizolri layr boundd o subsra is sudid by Dua al. 9. Thy onlud ha hs hararisis ar inflund by visous dissipaion onsidrably. Wang and Zhao (3 obaind h disprsion rlaion for pizolrilasi omposi plas. Shar wav propagaion in pizolri omposi sruur was disussd xnsivly by Qian al. (4 and Singh al. (3. Som rsarhrs hav disussd h ff of volum fraion on h sop band ff (Piliposian al. and Pang al. 8. Shar horizonal aousi Wavs a h Boundary of Pizolri Crysals wr invsigad by Pyaakov (. Propagaion propris of SHwavs in a pizo rami layrd sruur and ff of volum fraion on phas vloiy was disussd by Vaish al. (3 and 9. Du al. (9, Zakharnko (3 and Ni al. ( sudid h propagaion propris of wav propagaion in layrd pizomagni pizolri sruur. Th ff of imprf bonding on inrfaial wavs in dissimilar pizolri omposi has bn sudid by Huang, and Li xnsivly (. Rnly som rsarhrs hav rvald h imprf bonding is maor aus in disprsion of ar wav propagaion in PEPM inrfa (Mlkumyan al. 8, Huang al. 9, and Rahman al. 4. In pas dad, rsarh work fousd muh on ar wav propagaion in pizolri pizomagni inrfa, bu no work was rpord so far on ar wav propagaion in pizolripizolri omposis. Th obiv of his papr is o invsiga h propagaion bhavior of ar wavs in pizolri omposi sruur. This sudy is fousd o obain disprsion quaion for propagaion of wav in dirion normal o layring and in dirion of layring. Th influn of layr hiknss and lasi onsan on ar wav propagaion has also bn numrially valuad by onsidring h inrfa of wo marial PVDF and PZTH. Th ff of wav numbr and dimnsional lss frquny has bn plod o ow h variaion bwn h quaniis. This work provids a horial framwork for dsigning and dvlopmn of PEPE omposi sruur for snsor and ransdur appliaions. PROBLEM FORMULATION AND CONSTITUTIVE EQUATIONS Th pizolri layrd (PE sruur is own in Figur. Th omposi sruur ompriss of pizolri layrs bondd prfly alrnaivly of wo diffrn marials. Ths bondd layrd hav hiknss of h and h rspivly. Shar wav propagaion is onsidrd o b propagaing ihr in dirion normal o layring i.. in posiiv dirion of x axis or in dirion of layring i.. in posiiv dirion of y axis wih poling dirion akn along h z axis. h Dirion of layring Y X PE layr I PE layr II Dirion Normal o layring h h Figur : Shmai of priodi Pizolri Pizolri (PE layr I, PE layr II Layrd Sruur. Lain Amrian Journal of Solids and Sruurs (4 483496
Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur 48 For ar wav propagaing in xy plan, h onsiuiv sysm of quaions rfrd o pizolripizolri inrfa an b wrin as (Qian al. 4: ì si iklkl kie ü = k í ý D = klkl + kek î þ ì si iklkl kie ü = k í ý D = kl kl + kek î þ ( ( Whr σ i, ε i ar srss and srain snsor, D, and E k ar displamn and lri fild innsiy. ikl, ' ikl, ki, ' ki, ϵ k, ϵ ' k ar lasi, pizolri and dilri onsans for pizolri mdiums rspivly. Th moion quaion for PEI and PEII an b rprsnd in quaion (3 as (L 4 and Sun al. 968 ì s í D î i, = ru ü i ý = þ i, (3 Th srain nsor ε i and lri fild innsiy E i an b rprsnd as ì í E î ü = ( u + u ý = x i þ i, i i, i (4 Whr u i and r rprsns h mhanial displamn in i h dirion and mass dnsiy. Elrial ponial funion an b xprssd as. Th onsiuiv sysm of quaions for PEI mdia ar s = + + E x x y 3 z 3 z s = + + E y x y z 3 y s = + + E x x y z 33 x = E zy 44 zy y = E zx 44 zy y xy = 44 xy D = + E x x x D = + E y y y D = + + + E z 3 x 3 y 33 z 33 z ( Lain Amrian Journal of Solids and Sruurs (4 483496
486 Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur Th onsiuiv sysm of quaions for PEII mdia ar s = + + E s = + + E s = + + E zy = 44zy Ey zx = 44zy Ey xy = 44xy Dx = x+ Ex Dy = y+ Ey D = + + + E x x y 3 z 3 z y x y z 3 y x x y z 33 x z 3 x 3 y 33 z 33 z (6 ( ( Whr 44 =, 44 = For ar wav propagaion, h mhanial displamn and lrial ponial funion omponns in x, y and z dirion an b xprssd as u, u, w = w( x, y,, = ( x, y,. Eliminaing u and E from h quaion ( and (6, w g h sysm of quaions (7 and (8 ì í î æ ö æ ö w w w 44 ç + + ç + = r çè x y ø çè x y ø æ ö æ ö + + èç x y ø èç x y ø w w ç ç ü ý þ (7 ì í î æ ö æ ö w w w 44 ç + + ç + = r çè x y ø çè x y ø æ ö æ ö + + èç x y ø çè x y ø w w ç ç ü ý þ (8 For inrfa a x bwn PEPE layrs, h following boundary ondiions mus saisfy as w (, y = w (, y, (, y = (, y (, y = (, y, D (, y = D (, y zx zx x x (9 For all inrfas bwn PEPE layrs, h following boundary ondiions mus b saisfid as follow w ( h, y = w ( h, y, ( h, y = (h, y ( h, y = ( h, y, D ( h, y = D (h, y zx zx x x ( Lain Amrian Journal of Solids and Sruurs (4 483496
Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur 487 3 SOLUTION W will disuss h wo ass for propagaion of ar wavs. Th sysm of quaions (7 and (8 is solvd for hs wo ass of propagaion bhavior. 3. Propagaion along h dirion normal o h layring For ar wav propagaing along h posiiv dirion of x axis, soluion of sysm of quaions (7 and (8 an b xprssd in following form ik( x w x, W x ik, ( x ( = ( ( x = F ( x ik( x w x, W x ik, ( x ( = ( ( x = F ( x ( ( Whr k is wav numbr, is propagaion vloiy of ar wav, i =. W x, W x, F x, F x ar som undrmind funions. ( ( ( ( Subsiuing quaion ( and ( in (7 and (8 w g ( ( ( ( + + F + F F = r 44 W ikw k W ik k k W W + ikw k W F + ikf k F ( ( ( ( W ikw k W ik k k W 44 + + F + F F = r W + ikw k W F + ikf k F (3 (4 Th soluion of quaions (3 and (4 an b drmind as ( + / ikx (/ ikx W = G + H ikx ( + / ikx (/ ikx F = ( G + H x + ( G + H ( + / ikx (/ ikx W = G + H ikx ( + / ( / F = ( G + H x + G + H ikx ikx ( ( (6 = ( + /, = ( + / 44 r 44 r Whr and ' rprsns h bulk ar wav vloiy in PEI and PEII mdia rspivly. Lain Amrian Journal of Solids and Sruurs (4 483496
488 Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur Th ompl soluion of mhanial displamn and lrial funion in PEPE inrfa an b obaind by subsiuing quaion ( and (6 in ( and ( whih an b xprssd in following form as ( + / ikx (/ ikx ik( x w (, x = G H + úû ikx ( + / ( / (, x = ( G + Hx + G + H ik ( x ikx ikx ( ( + / ikx (/ ikx ik( x w ( x, = G H + úû ikx ( + / ( / ( x, = ( G + Hx + G + H ik ( x ikx ikx ( ú û ú û (7 (8 3. Propagaion along h dirion of h layring For h wav propagaion in posiiv dirion of y axis, h mhanial displamn and lrial ponial funion rprsnd as ik( y w x, y, W x ik,, ( y w ( = ( ( xy = F ( x,, W ik( y x,, ik x ( y ( x y = ( ( xy = F ( (9 ( Th ompl soluion of mhanial displamn and lrial funion is obaind and rprsnd as sysm of quaions in ( and ( ibx ibx ik( y w (,, x y = G H + úû kx kx (,, xy = G + H + G + H ú û w ( x y G H bx bx = +,, ibx ib ik x ( y ( ú ( ik y kx kx bx bx ik ( y ( xy,, = ( G + H + ( G + H ú úû ú û ( ( Whr b = k, b = k Lain Amrian Journal of Solids and Sruurs (4 483496
Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur 489 4 SHEAR WAVE PROPAGATION AND DISPERSION RELATION 4. Propagaion along h dirion normal o h layring Subsiuing h quaions (7 and (8 in quaions ( and (6 givs ikx ikp ( + / ( / ( ( ikx ikx = H + G H ik x ú û ikx ik( x = H zx D x ikx ikp ( + / ( / ( ( ikx ikx = H + G H ik x ú û ikx ik( x = H zx D x (3 (4 Whr P 44 + 44, + P = = Using h boundary ondiions (9 in sysm of quaions (7, (8, (3, and (4, provids h following linar algbrai quaions wih undrmind offiin as G, G, Gʹ, Gʹ, H, H, Hʹ, and Hʹ G + H G H G G + G + H G H G H + H H QG + QH ikp ikp H + H ia ia ikh ( b ikh G ( + b + H G H ia ia ikh G + hh + G + H G ikh ikh + h ( b ikh ( + b H G H i i a a ikh ikh ( b ikh H ( + b + G H H Q G + Q H ikp ikp ikh H + H ( For solving h linar algbrai quaions, w hav inrodud h following faors kh kh P a =, b =, Q =, w = k P For obaining h soluion of quaions (, h drminan of offiins of 8x8 marixs mus b qual o zro. Th drminan of sysm of quaions ( provids as (Chrisnsn 979 and Sun al. 968 sin( asin( b + Qos(hk Qos( aos( b + Q sin( asin( b (6 Lain Amrian Journal of Solids and Sruurs (4 483496
49 Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur Th abov quaion an b rprsnd in muh simplr form ( +Q æ ö æ ö æ ö æ ö kh kh kh kh os(hk os os + sin sin ç ç Q ç ç è ø è ø è ø è ø æ ö ç ø æwh ö ækh ö ækh ö ( + Q ækh ö kh èç ø çè ø çè ø Q çè ø çè os os os + sin sin = (7 Whr h=h +h is oal hiknss of pizolri bondd layrs. Th quaion (7 rprsns h disprsion rlaion for ar wav propagaing in dirion normal o layring. 4. Propagaion along h dirion of h layring For Shar wav propagaion along h dirion of layring, subsiuion of quaion ( and ( in quaion ( and (6 provids h srss and lrial displamn omponn as ibx ibx ( ( kx kx = ib P G + H + k G + H ik( y zx kx kx ik( y ( D = k G + H x bx bx ( ( kx kx = bp G + H + k G + H ik( y zx kx kx ik( y ( D = k G + H x úû úû (8 (9 Using h boundary ondiions ( in sysm of quaion (, (, (8 and (9, provids h following linar algbrai quaion wih undrmind offiin as G, G, Gʹ, Gʹ, H, H, Hʹ, and Hʹ G + H G H G + H G H G H k k k k G + H G + H + QG QH + G H ibp ibp ibp ibp G H G + H ibh ibh bh bh G + H G H ibh ibh kh kh kh kh b h bh G + H + G + H G H G H k k k k G + H G + H + QG QH + G H ibh ibh kh kh b h bh kh kh ibp ibp ibp ibp kh kh kh kh G H G + H (3 For obaining h soluion of quaion (3 h drminan of offiin marix mus b quad o zro, whih provids h following disprsion rlaion (Qin al. 4 Lain Amrian Journal of Solids and Sruurs (4 483496
Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur 49 {{[ sinh( kh o( kh + sinh( kh o( kh ][ r os( b h sinh( b h + Prsin( b h o( b h ]} 44 {[ sinh ( kh + sinh ( kh ][ r sinh(b h + P r sin(b h ]}} ( / 44 4 ( ( ( ( ( úû ( r r sin ( b h sinh ( b h + P [ ( ( ] 44 44 r r os b h o b h {( ( ( [ ( ( ]}.. / sinh kh sinh kh sinh b h sinh b h { } P + sinh kh sinh kh + o kh o kh (3 Pb r =, r =, Q = Pb Equaion (3 known as h disprsion quaion for ar wav propagaing along h dirion of layring. Th influn of layr hiknss, volum fraion, xisn of sop band and numrially analysis will b onsidrd in prding sion. NUMERICAL ANALYSIS AND DISCUSSION Th propagaion hararisi of ar wav basd on disprsion rlaion drivd in quaion (7 is invsigad by numrially analysis arrid ou in his sion for wo diffrn pizolri marials. Th marials usd in numrial alulaion ar PVDF and PZTH. Tabl liss h propris of pizolri marials. Propris Pizolri onsan Dilri onsan Elasi Consan Mass Dnsiy Marials (C/m (E F/m 44 (E N/m ρ(e3 kg/m 3 PVDF.6.6.9.78 PZTH 7 77.3 7. Tabl : Marial Propris usd in Numrial Calulaion Fig 7 ar h disprsion urvs owing h ff of variaion in layr hiknss on h irular frquny ω and wav numbr k, for propagaion along h dirion normal o layring. Thr xiss a rlaionip bwn wav numbr and irular frquny whih an b xprssd as k=ω/, whr is propagaion vloiy. Th valu of in numrial ompuaion is akn as m/s. I an b obsrvd from h figur and 4 ha h irular frquny drass wih inras in hiknss of pizolri layrs. Th wav numbr found o b drasing wih inras in hiknss of inrfa layr h. Th variaion of wav numbr wih hiknss is own in figur, 6 and 7. Th numbr of ar wav mods found o b inrasd, as h hiknss of boh pizolri layrs bom qual as vidn from h figur 4 and 7. Lain Amrian Journal of Solids and Sruurs (4 483496
49 Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur 4 (rad/s 3 3 4 h (mm Figur : Cirular frquny ω vs. h for h =.mm. 4 w (rad/s 3 3 4 h (mm Figur 3: Cirular frquny ω vs. h for h =.mm. 4 (rad/s 3 3 4 h (mm Figur 4: Cirular frquny ω vs. oal hiknss h for h =h. Lain Amrian Journal of Solids and Sruurs (4 483496
Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur 493 k Non dimnsional Wav numbr 4 3 3 4 h (mm Figur : Wav numbr k vs. h for h =.mm. k Non dimnsional wav numbr 4 3 3 4 h (mm Figur 6: Wav numbr k vs. h for h =.mm. k Non dimsional Wav numbr 4 3 3 4 h (mm Figur 7: Wav numbr k vs. oal hiknss h for h =h. Lain Amrian Journal of Solids and Sruurs (4 483496
494 Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur To sudy h variaion parn of wav numbr kh on ωh / ', w assumd a nw variabl volum fraion, whih is dfind as γ=h /(h +h, whr h and h ar hiknss of PEPE layrs in omposi sruur. Th urvs ar plod for diffrn valu of γ ranging from..8 as dpid in figur 8 and 9. For propagaion along h dirion normal o layring, i is obsrvd from h urvs ha numbr of sop bands inrass wih inras in valu of volum fraion. Bu as h volum fraion inrass, h dras in widh of sop band is obsrvd from h urvs own blow. wh /' Non dimnsional frquny wh /' Non dimsional frquny kh Non dimnsional Wav numbr kh Non dimnsional Wav numbr Figur 8: Sop band ff of propagaion normal o layring for γ (a. (b.4. wh /' Non dimnsional frquny wh /' Non dimnsional frquny kh Non dimnsional Wav numbr kh Non dimnsional Wav numbr Figur 9: Sop band ff of propagaion normal o layring for γ (a.6 (b.8. For invsigaing h influn of volum fraion on phas vloiy, figur (a and (b plod for wo valus of irular frquny Hz and Hz wih h fixd a mm. I is larly obsrvd from h urvs, h phas vloiy dras gradually wih inras in valu of volum fraion γ. Th urvs ar own for wav propagaion along h dirion normal o layring. Lain Amrian Journal of Solids and Sruurs (4 483496
Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur 49 Phas vloiy (m/s 3 3..4.6.8 Non dimnsional Volum fraion Phas vloiy (m/s 4 3..4.6.8 Non dimnsional Volum fraion Figur : Phas vloiy ( vs. volum fraion (γ for irular frquny (a Hz (b Hz. 6 CONCLUSIONS In his papr, w hav invsigad Shar wav propagaion in pizolri omposi sruur having wo pizolri layrs bondd oghr alrnaivly. Th disprsion quaions wr obaind hrough analyial mhod for propagaion along and in h dirion of layring. Th limiaion of his invsigaion, w hav assumd hr was no iniial srss prsn in ihr of h layrs in pizolri omposi sruur. Fuur work an b don o sudy h propagaion bhavior in PE sruurs in h prsn of iniial srss. Th numrial rsuls obaind from his sudy, w draw following onlusions (a Whn Shar wav propagas in dirion normal o layring, h sop band ff xiss and numbr of sop band found o b inras wih inras in volum fraion. (b For h as of wav propagaion in dirion of layring, no sop band xiss. ( Thr is signifian influn of volum fraion on h phas vloiy. W found ha, phas vloiy drass subsqunly wih inras in volum fraion. So w onlud ha hr xiss a linar rlaionip bwn volum fraion and phas vloiy. Th rsuls found o b usful in dvloping a nw lass of SAW snsor basd on PVDF PZT omposi wih improvd rspons and highr snsiiviy. This an b ahivd by obaining h dsird propagaion of ar wav by sling h appropria marial, hiknss, lasi onsans and ohr boundary ondiions. Rfrns Chrisnsn, R.M.: Mhanis of Composi Marials, WilyInrsin, Nw York, (979. Du, J., Xian, K., Wang, J.: SH surfa aousi wav propagaion in a ylindrially layrd pizomagni /pizolri sruur, Ulrasonis 49, 3 38 (9. Lain Amrian Journal of Solids and Sruurs (4 483496
496 Anu Mli Gaur and Din Singh Rana / Shar Wav Propagaion in PizolriPizolri Composi layrd sruur Du, J., Jin, X., Wang, J., al.: SH wav propagaion in a ylindrially layrd pizolri sruur wih iniial srss, Aa Mhania 9, 9 74 (7. Dua, J., Xian, K., Wanga, J.,. al.: Lov wav propagaion in pizolri layrd sruur wih dissipaion, Ulrasonis 49, 886 (9. Huang, Y., Li, X.F., L, K.Y.,.al.: Inrfaial ar horizonal (SH wavs propagaing in a wophas pizolri/pizomagni sruur wih an imprf inrfa, Philosophial Magazin Lrs 89, 9 3 (9. Huang, Y., Li, X.F,.al.: Inrfaial wavs in dissimilar pizolri ubi rysals wih an imprf bonding. IEEE Transaions on Ulrasonis, Frrolris, and Frquny Conrol 8, 6 6 (. L, U.: Spral Elmn Mhod in Sruural Dynamis, Inha Univrsiy Prss, Kora, (4 Mlkumyan, A., Mai, Y.W.,.al.: Influn of imprf bonding on inrfa wavs guidd by pizolri/pizomagni omposis, Philosophial Magazin 88, 96 977 (8. Ni, G., Liu, J., Fang, Q.,.al.: An Shar horizonal (SH wavs propagaing in pizolri pizomagni bilayr sysm wih an imprf inrfa, Aa Mhania 3, 999 9 (. Pang, Y., Liu, J., Wang, Y.,.al.: Wav propagaion in pizolri/pizomagni layrd priodi omposis, Aa Mhania Solida Sinia, 48349 (8. Piliposian, G.T., Avisyan, A.S., Ghazaryanb, K.B.,.al.: Shar wav propagaion in priodi phononi/phooni pizolri Mdium, Wav Moion 49, 34 (. Pyaakov, P.A.: Shar Horizonal Aousi Wavs a h Boundary of Two Pizolri Crysals Sparad by a Liquid Layr, Aousial Physis 47, 739 74 (. Qian, Z., Jin, F., Wang, Z.,. al.: Disprsion rlaions for SHwav propagaion in priodi pizolri omposi layrd sruurs, Inrnaional Journal of Enginring Sin 4, 673 689 (4. Qian, Z., Jin, F., Wang, Z.,. al.: Lov wavs propagaion in a pizolri layrd sruur wih iniial srsss, Aa Mhania 7, 4 7 (4. Rahman, N.U., Alam, M.N.: Fini lmn modling for bukling analysis of hybrid pizolri bam undr lromhanial loads, Lain Amrian Journal of Solids and Sruurs, 77789 (4. Singh, B.M., Rokn, J.: Propagaion of SH wavs in layrd funionally gradin pizolri pizomagni sruurs, Philosophial Magazin 93, 697 (3. Sun, C.T., Ahnbah, J.D., Hrrmann, G.,.al.: Coninum hory for laminad mdium, Journal of applid mhanis 3, 467473 (968. Vaih, A.K., Dahiya, A.: Shar wavs in a pizorami layrd sruur, Aa Mhania 4, 77 744 (3. Vaih, A.K., Gupa, V.: Vibraions of porous pizolri rami plas, Journal of Sound and Vibraion 3, 78 797 (9. Wang, H, M., Zhao, Z, C.: Lov wavs in a wolayrd pizolri/lasi omposi pla wih an imprf inrfa, Arhiv of Applid Mhanis 83, 43 (3. Zakharnko, A.A.: Fundamnal mods of nw disprsiv SHwavs in pizolromagni pla, Pramana Journal of Physis 8, 8987 (3. Lain Amrian Journal of Solids and Sruurs (4 483496