Acoustic and flexural wave energy conservation for a thin plate in a fluid
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1 cousc nd fleurl wve energy conservon for hn ple n flud rryl MCMHON 1 Mrme vson efence Scence nd Technology Orgnson HMS Srlng W usrl STRCT lhough he equons of fleurl wve moon for hn ple n vcuum nd flud re well nown s no esy o fnd dscusson of energy conservon for ple fleurl wves prculrly ley wves where ple nd flud cn echnge energy. Nor re formule esly found for cousc nd fleurl wve nec energy densy poenl energy densy nd energy densy flu ncludng he effec of ley wves. Ths pper derves formule for cousc nd fleurl energy denses nd energy densy flues nd fnds he energy conservon equon for he coupled hn ple flud sysem eywords: cousc le Flud I-INCE Clssfcon of Subjecs Number(s): INTROUCTION Conservon of energy s somemes useful consrn n undersndng srucurl vbron problems. For nsnce s useful o now f rdon from srucure no flud cn be gnored for ner feld couscs becuse he energy densy flues whn he srucure re much lrger hn he cousc energy densy flu. Even for he smples sysem of fleurl wves for n nfne hn ple n flud he bes nown e boos do no derve from he wve equon formule for ple wve energy denses nd energy densy flues (1 ). Ths pper flls hs bsc nformon gp by dervng formule for cousc nd fleurl wve energy densy nd energy densy flu nd n energy connuy equon for fleurl wves coupled o cousc wves. Regrdng he cse of ley wve whereby fleurl wve coess wh n cousc wve close o ple hs pper dscusses energy conservon where for emple n cousc ley wve rvels wy from ple bu s mplude decreses eponenlly wh dsnce from he ple nd ncreses eponenlly wh dsnce long he ple.. WVE ENERGY ENSITY N ENERGY ENSITY FLUX RELTIONS Consder n soled lossless sysem (.e. no eernl ppled forces nd no energy dsspon) ssfyng wve equon of he form u v (1) where u ( nd v( re comple funcons reled o he wve mplude. We see o denfy from eqns. (1) he energy densy U( nd energy densy flu ( h ssfes he conservon of energy equon U () Equon () smply ses h he re of chnge n energy densy U s due o grden n he energy densy flu. Well nown physcl conceps llow U ( o be denfed s he sum of nec energy U ( nd poenl energy U ( so h U( ) U ( ) U ( ) () I s empng o ssume h ( s sum of nec energy densy flu ( nd poenl energy densy flu (. However denfyng formule for ( nd ( unmbguously s 1 drryl.mcmhon@defence.gov.u Iner-nose 1 ge 1 of 9
2 no s srghforwrd s for U ( nd U (. Insed he dervons below dvdes ( no prs denoed ( nd ( h re no unmbguously reled o ( nd ( bu ( ) ( ) ( ) () Energy denses nd energy densy flues re rel qunes h re qudrc n u ( nd v (. Equon () cn be consruced from eqn. (1) usng only he rel prs u u u / nd v v v /. Ths defnes U( nd ( pplcble ny poson nd me bu ofen he rndom phse verged energy densy nd energy densy flu re lso useful. n energy conservon equon s derved from he rel pr of eqn. (1) u v (5) Mulply he LHS of eqn. (5) by v he resul ercs he spl grden nd me dervve erms from he denes u u v u v v (6) v 1 v v (6b) We hen ge n energy conservon equon of he form U U (7) where 1 U v (8) u v (8b) U v u (8c) The generlsons of eqns. (1) o (8) o hree dmensons re srghforwrd by replcng by spl vecor r replcng he one dmenson spl dervve by he vecor grden operor nd replcng he second order spl dervve by he Lplcn operor. lso he energy densy flu Γ becomes vecor. The generlsed wve equon s v u (9) The energy conservon eqn. () generlses o U. Γ (1) Equons () nd () re smply eended o funcons of vecor r. Eqn. (8) for he nec energy densy s unchnged. The formul (8b) for he energy densy flu s replced by Γ v u (11) Smlrly he hree dmensonl eenson of eqn. (8c) s U. Γ v. u (1). COUSTIC WVES IN FLUI.1 cousc energy densy nd energy densy flu relons The cousc wve equon s obned by usng n eqn. (9) he relonshps ny comple quny q s q q q where q denoes he rel pr nd q denoes he mgnry pr. Iner-nose 1 ge of 9
3 u( r p( r (1) 1 v( r p( r (1b) c where p( r s he comple cousc wve pressure nd c s he speed of sound n he bul flud (.e. lrge dsnce from ny elsc or oher surfce). The fcor ensures h U nd Γ hve uns of energy densy nd energy densy flu respecvely. For plne wve of frequency f /( ) nd flud densy he verge energy densy flu (Fhy nd Gordno Chper eqn. (.16) ()) cn be used o fnd c (1) From eqns. (8) (11) nd (1 b) we fnd spl nd me dependen qunes U c 1 p p p p p (15) c 8c 1 p p p p p p p p p p p p 1 Γ c c c (15b) Equons (1) nd (1 b) led o p p U p p p p p 1 1. Γ c c 8c (16) Comprng he LHS nd RHS of eqn. (16) mples hu nd Γ re gven by 1 U p p p (16b) c Γ (16c) Equon (16c) shows h he ol cousc energy densy flu s jus Γ Γ gven by eqn. (15b).. lne wve cousc energy densy nd energy densy flu relons Consder sngle comple pressure wve p( r p ep. r (17) where f for frequency f s he comple cousc wvenumber vecor s n rbrry phse nd he pressure mplude p s rel. wve wh comple wvenumber does no usully occur n solon bu rvellng sndng nd evnescen wves re needed ogeher o blnce energy densy nd energy densy flows. I s possble for o be comple ner surfce such s ple. Subsung eqns. (1 b) nd (17) no he wve eqn. (9) for he cse where p s poson nd me ndependen we fnd he consrn on. (18) where / c. Sepre consrns on he rel nd mgnry prs of eqn. (18) re (19). (19b) Eqn. (19b) shows h n cousc wve cn hve nonero mgnry pr for he wvenumber (e.g. n evnescen wve) provded he rel pr nd mgnry pr of he comple wvenumber vecors re orhogonl. Eqn. (19) consrns he relve mgnudes of he rel nd mgnry prs of he wvenumber vecor ner surfce o conform wh he wvenumber n he bul flud fr from he surfce. The energy densy nd energy densy flu re me dependen wh flucuon frequency f. The energy densy nd energy densy flu muully chnge wh me nd blnce ou o ssfy eqn. (1). The dels of he dervons re srghforwrd usng eqns. (17) (15 b) nd (16b). For brevy we use For surfce wve nercng wh he pressure wve n flud s he sum of he phse of he surfce wve nd he phse of he pressure wve componen rvellng perpendculr o he surfce. Iner-nose 1 ge of 9
4 phse r for cousc energy densy nd flu vrons defned by r r r r. r r r (). (b) The me dependen relonshps for he energy denses nd energy densy flues for sngle comple wvenumber plne cousc wve re U Γ U r p ep. r1 cos r r p ep. r 1 cos( r c U r p ep. r c r p ep. r (1 cos( r ) sn( r c (1) c (1b) S (1c) (1d) mples surfce cousc source (or sn) o be presen mng he poenl energy densy lrger hn he nec energy densy by he moun U S (.e. U U US ). From (1d) hs surfce source/sn lso gves Γr wvenumber perpendculr o bu osclles n sgn verges o ero nd hence mes no ne conrbuon o he verge energy densy flu. ll energy denses vry eponenlly wh dsnce boh prllel o nd perpendculr o surfce so f we requre so h he cousc energy densy decreses wh dsnce from he surfce hen for of wve rvellng wy from he surfce he energy densy ncreses long he surfce n he drecon h wvefron propges nd for of wve rvellng owrds he surfce he energy densy decreses long he surfce n he drecon h wvefron propges.. FLEXURL WVES FOR N INFINITE THIN FLT LTE.1 Fleurl wve energy densy nd energy densy flu relons The wve equon for hn ple fleurl wves (Fhy nd Grdono Chper 1 pp. 6 7 ()) s obned wh eqn. (1) by he relons ( u( () v( M ( (b) where ( s he comple perpendculr devon of he ple from he verge plne = poson nd me s he ple bendng sffness per un re M s he mss per un re of he ple nd s he fcor 1/ M h gves correc energy densy uns. The epresson (8) for U n erms of ( usng eqn. (b) leds o M M U ( ( ( ( () 8 From eqn. (8b) for n erms of ( nd ( usng eqns. ( b) leds o ( ( ( ( ( ( ( ( (b) To fcle denfyng he poenl energy densy nd energy densy flu from eqn. (8c) we use he deny v u v v u u () Then eqn. (8c) s ssfed by ssumng r Snce Γ Γ he subscrp s dropped. Iner-nose 1 ge of 9
5 Iner-nose 1 ge 5 of 9 v u U (5) v u (5b) From eqns. ( b) nd (5) we fnd 8 U (6) Hence we hve 8 U (6b) Eqn. (6b) s conssen wh he sc verson of he poenl energy densy sored n ben hn ple derved n e boos such s h n Lndu nd Lfsh eqn. (11.6) p. 6 (5). From eqns. ( b) nd (5b) we fnd (6c). lne fleurl wve energy densy nd energy densy flu relons Consder for n nfne fl ple sngle comple fleurl wve propgng long he -s wh dsplcemen n he + drecons ep ep ep ) ( (7) where f for frequency f s he comple fleurl wvenumber s n rbrry phse nd he wve mplude s rel nd consn. Noe h sngle wve wh comple wvenumber does no usully occur n solon owng o he need o conserve energy 5 bu he eenson of hs heory o coheren superposons of mulple fleurl wves s srghforwrd. Subsung eqn. ( b) nd (7) no he wve eqn. (1) we fnd f where M f / whch when sepred no rel nd mgnry prs gves f (8) (8b) The wo wve ypes ssfyng (8 b) re f (9) f (9b) Wve ype eqn. (9) s he unenued fleurl wve for ple n vcuum nd eqn. (9b) s he correspondng evnescen fleurl wve. Smlr o eqns. ( b) we use fleurl wve energy densy nd flu phses ) ( nd. Subsung eqn. (7) no he energy densy nd energy densy flu formule ( b) nd (6b c) gves M U cos 1 ep ) ( () cos 1 sn ep ) ( (b) 5 For nsnce hrmonc pon or lne force ppled o n nfne ple crees boh rvellng wve wh rel wvenumber nd n evnescen wve wh n mgnry wvenumber. These wo wves re correled by her common drvng force.
6 U ( (c) cos ep sn cos sn ( ep (d) The ol energy densy U U U nd ol energy densy flu from eqns. ( b c d) nong h M f re gven by f cos sn ep f U( ep (1) ( (1b) Tng he me dervve of U nd spl dervve of hen subsung hem no he LHS of eqn. () we fnd U ( ( () ep f sn 1 cos For free unforced ple eqns. (8 b) pply nd hen clerly he RHS of () s ero. lso from eqns. (8 b) he ol energy densy eqn. (1) nd energy densy flu eqn. (1b) re me ndependen for he unforced ple. Ineresngly eqn. (1b) shows he energy densy flu s lwys me ndependen for sngle plne fleurl wve ssfyng eqn. (7). For he evnescen wve by eqns. (8b) nd (1b) we fnd ( lhough ( nd ( re nonero bu cncel ech oher ou. I s lso neresng from eqn. (1b) h ( cn be negve even f s posve (.e. he ne energy densy flu cn be n he oppose drecon o he wve velocy). 5. FLEXURL WVES FOR N INFINITE THIN FLT LTE IN FLUI 5.1 Chrcersc equon for ple flud dsperson relons The equon of moon for hn ple wh one sde n nonvscous flud nd he oher n vcuum s derved by ddng me nd spce dependen cousc pressure genered by he ple o he LHS of eqn. (1) fer subsung eqns. ( b). Ths cousc pressure p( he ple surfce s ssumed o be proporonl o ple cceleron ( n he + drecon fer defnng comple flud mss densy M M M. Hence we hve p( M ( () For he cousc wve genered by fleurl wves eqn. (17) s now replced by p( p f ep f () The ple s cceleron crees pressure grden n he flud so for n cousclly hrd ple surfce 6 p( ξ ( p( (5) nd hence from eqns. () nd (5) M (6) M (6b) M (6c) 6 Ths oms compresson nd sher wves n he ple merl h couple o he flud cousc nd ple fleurl wves. Iner-nose 1 ge 6 of 9
7 The ple dsplcemen ( s sll gven by eqn. (7). Combnng eqns. (7) () nd () he mplude of he pressure wve s gven by p f M (7) nd he phse dfference beween cousc nd fleurl wves s f (7b) The chrcersc equons for hn ple n flud re derved by ddng he flud conrbuon (ofen clled flud lodng) o he ple mss densy nd so eendng eqns. (8 b) o M f 1 (8) M M f (8b) M The nd componens of he comple cousc wvenumber re consrned by eqns. (19 b) nd led o (9) (9b) Eqns. (9 b) llow nd o be reled o nd hence be elmned from eqns. (8 b) h cn hen be solved. Wheres for hn ple n vcuum here re only he wo wve mode dsperson relons (9 b) eqns. (8 b) cn be rerrnged o ffh order polynoml equon n suggesng here re fve dfferen ple flud modes wh dfferen dsperson relons (Crghon (7)). One well nown soluon he Soneley-Schole wve s n unenued subsonc wve wh pressure decresng eponenlly wh dsnce from he ple. Oher ple-flud modes re ley wves wh comple wvenumbers mplyng h n nfne ple-flud sysem s closed nd does no hve nurl modes correspondng o cousc fr feld plne wves movng wy from or owrds he ple. Some ple-flud modes re ley below he concdence frequency 7. ll fve ple flud modes conrbue o vbron eced by hrmonc pon or lne force (Fe nd Lu (8) nd Chpmn nd Soron (9)) whch hen leds o rdon from he regon ner he econ re. 5. le flud energy densy nd energy densy flu conservon equon Conservon of energy equons for closed ple flud sysem re derved by eensons o equons derved n Secon. Flud lodng eends eqn. (7) o U U U F WF () whereu F s n cousc energy densy n he flud reled below o M nd WF s he ple flud cousc energy densy rnsfer re reled below o M. Eqn. () s derved srng wh he flud loded eensons o eqns. (1) nd (5) u 1 v v (1) where ( b) sll denfy u( nd v( n erm of ( nd M M () M M Mulplyng he LHS of eqn. (1) by v nd followng he sme procedure ledng o eqn. (7) we fnd eqn. () wh he er erms denfed s M ( ) M U ( ) F () W M M F ( ) ( ) ( ) (b) Then subsung he plne fleurl wve eqn. (7) no eqn. ( b) we obn 7 The frequency where he fleurl wve phse speed equls he speed of sound n he flud. Iner-nose 1 ge 7 of 9
8 M U F ( ep 1 cos () M WF ( ep 1 cos (b) Usng eqn. ( b) n () we obn he ple flud eenson o eqn. () U ( ( U F WF WF ( (5) WF ( ep 1 f sn f 1 cos (5b) When he chrcersc eqns. (8 b) re ssfed we hve W F ( nd recover eqn. () for closed ple-flud sysem conservng energy. IfW F ( re nconssen wh closed pleflud sysem nd W F ( needs o be cncelled by n eernl energy source o mnn overll energy conservon. ddng n eernl energy conrbuon ( o he energy densy rnsfer re on fleurl wves on boh sdes of (5) o cncel ( gves W F U ( ( U F ( WF ( WEF ( WF ( WEF ( (6) n emple of forced ple flud vbron s cousc bsorpon nd scerng by ple n flud. The pplcon of eqns. (5b) nd (6) o hs problem s n noher pper by he uhor (6). 6. SUMMRY Srng wh lner wve equon eqn. (1) ble o represen boh flud nd hn ple wve med Secons deduced generl formule for energy densy nd energy densy flu from her conformy wh he energy conservon eqn. (). Wheres fundmenl mechncs prncples redly llow he nec nd poenl energy densesu ndu respecvely o be unmbguously denfed wo energy densy flues nd re needed o descrbe he propgon of nec nd poenl energy. Secons ppled he energy densy nd energy densy flu formule o cousc felds bu ng no ccoun he ner feld effec of surfce resulng n comple wvenumber vecor. comple wvenumber nroduces eponenl chnges n wve mplude wh dsnce prllel nd perpendculr o he plne of he surfce. The cousc wve equon requres he rel nd mgnry prs of he wvenumber vecor o be orhogonl o ech oher. In he cse of sngle plne wve wh comple wvenumber epressons for he nec poenl nd surfce energy denses re derved. The energy densy flu vecor hs componens prllel o nd perpendculr o he wvefron correspondng o he mgnry nd rel prs of he wvenumber vecor respecvely. Secon derves formule for he energy densy nd energy densy flu for fleurl wves of n nfne hn fl ple n vcuum. The ne energy densy flu for sngle fleurl plne wve s me ndependen n conrs o he flu of n cousc wve whch hs me dependen pr wh frequency f. lso unle n cousc wve comple wvenumber fleurl wve cn hve n energy densy flu n he oppose drecon o he wve propgng drecon defned by he rel pr of he wvenumber. Secon 5 combnes he heory of Secons nd o derve wve dsperson nd energy conservon relons for n nfne hn fl ple flud sysem couplng cousc nd fleurl wves. Wheres ple n vcuum hs only wo nurl wve modes (propgng nd evnescen wves) ple n flud hs fve nurl wve modes. The energy conservon equon for he ple-flud sysem s eended n Secon 5. o nclude eernl forces such s cousc econ by plne wve reed n noher pper by he uhor (6). REFERENCES 1. Junger M. C Fe. Sound Srucures nd Ther Inercon. nd edon. MIT ress; Fhy F. J. Sound nd Srucurl Vbron: Rdon Trnsmsson nd Response. cdemc ress; Cremer L Hecl M. nd Ungr E. E. Srucure orne Sound. nd edon. Sprnger-Verlg; W EF Iner-nose 1 ge 8 of 9
9 . Fhy F Grdono. Sound nd Srucurl Vbron: Rdon Trnsmsson nd Response. nd edon. Elsever ( novel eboo); Lndu L. Lfsh E.M. Theory of Elscy. rd edon. ergmon ress; McMhon. cousc forcng of fleurl wves nd cousc felds for hn ple n flud roceedngs of Iner-nose 1 Melbourne usrl November Crghon. G. The free nd forced wves on n nfne hn flud-loded elsc ple. J. Sound Vb. 1979; 6: p Fe Lu Y. N. The nerfeld response of lne drven flud loded ple. J. cous. Soc. m. 1985; 78: p Chpmn C. J Soron S. V. The forced vbron of n elsc ple under sgnfcn flud lodng. J. Sound Vb. 5; 81: p Iner-nose 1 ge 9 of 9
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