Near-Field Acoustic Holography: The Frame Drum

Transcription

1 Nea-Field Acusic Hlgaphy: The Fae Du Gégie Tel Uivesiy f Illiis Ubaa-Chapaig Sue REU 00 Advis: D Seve Eede Absac A phase-sesiive seup f ea-field acusic hlgaphy was esablished i de deeie he chaaceisic esace des f a vibaig ebae. The expeie was caied u usig a fae du as he vibaig syse f iees. By exciig he ebae a ceai fudaeal fequecies ad ecdig he cplex sigals f he pessue ad paicle velciy fields acss he duhead we deled 3-D epeseais f he vibaig suface. Celais bewee a eige-fequecy ad is cespdig eige-de f esace ae descibed hughu his ep. Iduci ad Theeical Descipi A pecussi isue played by sikig a ighly seched ebae aached a cicula fae is called a ebaphe. The asie aue f is sud eveals es wih -peidic wavefs which will heefe have -haic paials ad piches e less ideeiae. T udesad ad epese such uique acusical behavi we cduced u expeieal wk usig a fae du as he vibaig syse f iees. F a cicula ebae wih displacee a he buday he ideal vibaial del is well kw ad ca be deived aheaically. v ( ). ( ) exp( iω ) δ ( ) This is he expessi cespdig he ebae i whee is he displacee f he ebae f is es psii v is he velciy ω is he agula fequecy is he psii alg he adius ad is he agle he x-axis. The lef had 0

2 side f he equai descibes a pi like peubai ha scillaes wih fequecy psiied a 0 ( () δ is a w-diesial dela-fuci wih diesi / ). Hece lkig a diffee ha 0 he equai f he displacee beces: ( ) ( ) 0. v [i] Whee he Laplacia pea f a w diesial pble is give by. Iseig he Laplacia i [i] ( ) ( ) 0. v Assuig ( ) ( ) ) ( T whee T() exp(iω ) we bai a ie idepede equai: ( ) ( ) 0. ω v Makig he fllwig sepaai f vaiables ( ) ( ) ( ) θ P R he equai beces: () () ( ) ( ) 0 v P P R R ω. The depedecy is equal a csa; hus we ca sepaae he ad es i w disic diffeeial equais: ( ) ( ) ( ) ( ) R k R P P Wih v f λ (ω/π)(π/k) ω/k. Slvig he equai f P( ): ( ) ( ) ( ) 0 si cs ± ± P β α.

3 3 α ad β ae abiay csas which saisfy he fllwig cdii: β α. Nw lkig a () R : () () 0 R k R This equai is called he Bessel equai aed afe he Gea aheaicia Fiedich Wilhel Bessel he slui is f he f: ( ) ( ) ( ) k Y B k J A R Whee J (k) ad Y (k) ae Bessel fucis f he s ad d kid especively. Alhugh f u pupse Y ca be a slui due he fac ha i diveges whe appaches 0. Theefe B 0 (we ige he Bessel fucis f he d kid). J has vaish he edge (buday) f he ebae eaig ha J (xkr) 0 whee R is he adius f he du. As figue shws us hee ae uliple pis whee J equals ze. The adius f he ebae beig csa hee exis a ifiie ube f diffee k-vecs ha saisfy he buday cdii. We ae hse k whee cespds he h ze f J. Gaphically efes he de ube f a Bessel fuci ad epeses he ube f des. Ne ha he abve deivai was paly ake f D Eede s lecue es {}. We have w sufficie ifai bai he eige-de sluis f ( ): The eige-de slui f he epal wave equai is f he fllwig f(s): Figue : Bessel fucis f he s ad d kid. T he lef he fis five Bessel fucis f he s kid (J) ae shw. The igh figue shws he fis five Bessel fucis f he d kid (Y). Ne ha Y diveges as appaches ze. () () () )] ( exp[ ) cs( ) si( ; ; ) cs( ) si( i T T c b c b c b T ω ω δ ω ω ω

4 The cplee eige-de slui f he w-diesial sadig wave a -D cicula ebae is hus give by:... ( ) ( ) ( ) R A A ( ) P J J ( k ( k ( ) T ( ) R)exp[ i( δ )]exp[ i( ω R)[ α cs( ) β si( )][ b )] cs( ω Wih eige-fequecies eige-waveleghs ad eige-eegies f: ) c si( ω )] f ω /π vk /π v / λ λ π / k πr / x E 4 Mω A ad I he cplee slui A cespds he apliude abiaily deeied by he fuci geea. Ne ha if we excie he duhead a 0 we ca ly geeae he dal vibais which pssess a ai-de a his pi. I de geeae he eige-des e has ve he psii whee he dive fce acs d a disace alg he adius. Ne ha all J 0 des ae degeeae wheeas all he he des ( > 0) pssess -flds degeeacies because f he w spaial degees f feed (x-y - ) ad he aial syey f he cicula ebae. The fllwig expessi helps fidig he (ideal) eige-fequecy ha e us apply he duhead i de eveal e specific al de (alhugh diffee f u expeieal ehd f fidig eige-fequecies): ω vk ν π π Oce a eige-fequecy is kw we ca exaie he cplex sud field by calculais f he pessue ad paicle velciy fields assciaed i. The cplex paicle velciy u ( ) (a 3D cplex vec quaiy) is deived f he cplex pessue p ( ) (a cplex scala quaiy) via Eule s equai f iviscid fluid flw: u ( ) p ( ) [ii] ρ 4

5 I cylidical/pla cdiaes: ˆ ρ ˆ ϕ ẑ whee ρ x y ρ ρ ϕ z ϕ a ( y x) ad ˆ ρ csϕxˆ siϕyˆ ˆ ϕ siϕxˆ csϕyˆ. F a cplex haic sud field he cplex pessue eige-fucis (a z 0) assciaed wih he dal vibais f a ideal (i.e. pefecly cplia) du ebae f adius R ae sluis he 3D wave equai f he cplex pessue p ( ) ( v ) p( ) 0 (phase velciy v ω k) ad ae f he geeal f: i ( ρϕ 0 ) ( ρ) cs( ϕ ϕ) ( ξ ρ ) cs( ϕ ϕ) ( ω k z) z iϕ p p z pj k e e ( ω k z) i z iϕ p p J R e e h k is he adial wave-ube f he ( ) de deig he ( ) Whee ze ( ξ k R) f he Bessel fuci J( ξ ) 0 a he adial buday ρ R ad ϕ is he ai agle abu he z-axis f he eie disibui i he x-y plae defied by he x-y lcai f he divig fce pi. The fid he paicle velciy f [ii] i( ω k ) ( ) ˆ ˆ z z iϕ p p 0 ˆ ρϕz ρ ϕ z pj( k ρ) cs( ϕ ϕ) e e ρ ρ ϕ z i( ω k z) i p ˆ ˆ z ϕ ρ ϕ zˆ pj( ξ ρ R) cs( ϕ ϕ) e e ρ ρ ϕ z h Nw ( ρ ) { ( ρ) ( ρ) } J k k J k J k ρ ρ ad f 0: J0( k0 ρ ) k0 J( k0 ρ ) Hece f a haic sud field: e z ikz z ik e ϕ cs( ϕ ϕ ) si ( ϕ ϕ ) ikz z z iω iω u ( ) u ( ) e e iω u ( ) iωu ( ) e iωu ( ). Thus he adial agula ad heigh cpes f he cplex paicle velciy ae fud be: 5

6 k p ( ρϕ 0 ) { ( ρ) ( ρ) } cs( ϕ ϕ) ( ω k z) ρ i z iϕ p u z J k J k e e iωρ p ρ ( ρϕ 0 ) ( ρ) si( ϕ ϕ) ( ω k z) ϕ i z iϕ p u z J k e e iωρ kp ( ρϕ 0 ) ( ρ) cs( ϕ ϕ) ( ω k z) z z i z i p u z J k e e ϕ ωρ Theefe he apliudes cpes f u ae: k p p ρ ad u ρ u ϕ ωρ ωρ u kp z z ωρ We ca w cay u w idepede spaial sca easuees f he cplex sud fields i pxiiy he du suface a a heigh z d (<<R) abve he duhead. Fuhe aheaical descipi f he cplex sud field ea he suface is give i D Eede s lecue es {}. Ahe way ap he behavi f he pessue ad paicle velciy sud fields is use Fas Fuie Tasf echiques wih he use f a ppaga G. This is explaied i Nea field acusic hlgaphy wih paicle velciy asduces by Fi Jacbse ad Yag Liu. Recdig he cplex pessue ad cplex paicle velciy (ai lecules velciy) ve a suiable spaial age uliaely allws iagig he vibai des f a duhead. We shwed ha ly w idices ae ecessay uiquely deeie dal vibai haics f a cicula ebae due is -diesial aue. Fig. : The fis () eige-des f a ideal du ebae whee adjace egis scillae 80 u f phase each he. The ube belw each de cespds he ideal fequecy ai f a esace de alized J0. Muliply hese ubes by (.405 / π R) T/ σ cve acual fequecy (R beig he ebae adius T he esi acss he duhead σ he ass pe ui aea). 6

7 We clealy see ha he haic specu f a du (ad pecussis i geeal) diffes f als evey he isue whse haics ae iege uliples f he lwes esace de (fudaeal fequecy). These fequecy-elaed pheea ae well descibed by T.D. Rssig i The Physics f Musical Isues (973) {3}. May he physical quaiies ay be deived f he spaial easuees f he cplex pessue ad cplex paicle velciy fields ea he vibaig ebae. Ag hse ae he cplex acusic ipedace ( Z P/U ) he cplex sud ~ ~ ~ iesiy ( I P.U * ) as well as sucual wave ube idicais ad phase ifai. A isighful descipi f hese ay physical quaiies is give i D Eede s lecue es {4}. ~ ~ ~ Appaaus ad Expeieal Pcedue The exaci f fudaeal acusic quaiies such as he cplex pessue ad cplex paicle velciy is cucial f a pfud udesadig f a cplex acusic sud field ad leads a accuae 3-D epeseai f a -ideal vibaig syse. I de ack such acusic pheea a specific daa-acquisii (DAQ) syse was esablished. I descibes a phase sesiive seup f ea-field acusic hlgaphy (NAH). Cpue f daa pcessig Lck-i Aplifies Thps ds - Mized aslaial sages Pessue ad paicle velciy icphes (abve ad belw he du) Fae du Fig.3: Picue ake iside he lab shwig he DAQ syse f acusic hlgaphy. 7

8 Tw sall supe-ages (eah ages) ae psiied abve ad belw he du ebae a a desied psii. A fuci geea is used dive a siusidal cue a cil siuaed beeah he ages. A exisig egaive ipedace cicui (NIC) akes he fla (fequecy-elaed) cue upu a sable ipu pwe level. I esues a csa ageic field dive he ages. Ne ha he csa cue NIC eliiaes phase shifs (ie delayed espse f he duhead f is dive fce) due iducace f he cil. The cue uig hugh he cil iduces a scillay ageic field which causes he ages ve up ad dw pducig vibais acss he duhead f ay ipu fequecy. Tw pais f pessue ad paicle velciy icphes siuaed abve ad belw he du ebae ae used exac he acusical quaiies eeded udesad ad chaaceize he sud field ad vibaial behavi f he duhead. T exac such physical quaiies f he icphes each icphe was absluely calibaed i a Lp 94.0 db sud field a f KHz usig a NIST-ceified Exech caliba. Up calibai hei upu vlage culd he be elaed eihe pessue paicle velciy (expessed i RMS Pa /s ahe ha abiay RMS vls). [I a Lp 94.0 db sud NTP: p.0 Pa (RMS) ad u.4 /s (RMS)]. The pessue ad paicle velciy icphes ha ae placed abve he duhead ae called he sca ics because hey ae aached he ized ad cpueized aslaial sages i de spaially sca he eie duhead suface by akig 3x3 easuees wih c seps. These sages called Thps ds ae accuae a icee. The icphes placed udeeah he duhead ( i ics ) hld a diffee pupse. Because vibaig syses such as a cicula ebae have a able depedece weahe-elaed quaiies such as he epeaue huidiy ad aspheic pessue a de-lckig syse had be esablished i de keep ack f he esa fequecy (fc(tph)) usig digial-echiques i he PC s DAQ cde. Theefe a pai f pessue ad paicle velciy icphes was psiied belw he ebae de-lck a desied esa fequecy hldig he phase f he pessue i icphe csa a 90.0_0.5 degees. Whe he du vibaes a a eige-de he pessue beeah he ages is a a axiu ad eiely iagiay. T access a desied eige-de f vibai he fequecy is swep uil he pessue i he cplex plae is aliged wih he psiive egaive iagiay axis depedig he pe-defied paiy f he ages. The cplex sigal f each icphe is se a Lck-i Aplifie (hus fu LIA s). Each LIA easues he apliudes f he eal/i-phase ad iagiay/90 -u-fphase cpes f a cplex haic (i.e. peidic) sigal bh elaive sable sie wave f efeece. Me heeical descipi f LIA s is give i D Eede s lecue es {5}. Uliaely he ecded cplex sigals f each easuee ad all he phase ifai baied hugh daa acquisii is pcessed by a cpue. Ne ha he eal pa f he sud field elaes he sud ha ppagaes e s ea decayig hugh ie wheeas he iagiay pa cespds scillay vibais ea he du suface. The kw phase-shifs effecs (which we ll discuss lae) ae ceced i he fflie daa aalysis. As saed i he hey ly ceai des ay be geeaed depedig whee he duhead is beig excied ad a which fequecy ae he ages dive (i.e. fequecy 8

9 a which he duhead is vibaig). Hweve chaaceisic des f vibai ca ly be bsevable if he ebae vibaes a a eige-fequecy. Hece he fis sep f he expeieal pcedue is called a fequecy sca. The pupse is aiai he sca icphes a csa psii abve he ages ad he sweep he fequecy ve a pe-defied age. F u if we chse he fequecy sca g f 0.5 Hz aud khz wih Hz seps. Alhugh s f wha appeas ieesig ad eleva ccus bewee ~00 Hz ad ~800 Hz. We cduced hee fequecy scas a he cee half adius ad edge f he du (each sca lass abu fu hus). By he aalyzig he ecded pessue specu ve he give age f fequecies ad disiguishig he lcal axia f pessue e ca quickly ideify he esace peaks f he duhead each peak cespdig a eige-fequecy f his divig fce psii. The secd pa csiss f a spaial sca. I is wha peis us iage he vibai des (eige-des) f he du ebae. Oce he eige-fequecies ae kw e ca sabilize he age vibais a desied fequecy exciig he duhead a is cespdig de f esace. The he aslaial sages suppig he P/U icphes will sca he duhead (as peviusly descibed) allwig 3D epeseai f he vibaig syse ad evealig he elai bewee a J eigede ad is eige-fequecy. The fllwig figues display 3-D pls f he fis eige-des f a pefecly cplia ebae. These J Bessel fucis f he haic specu f all vibaig cicula ebaphes. A ideal delig f hese al des is shw belw whee he fis clu epeses (0) des ad he fis w () des ec...: 9

10 Ivesigai f a Fae Du - Resuls ad Aalysis The bjec f iees is a iches fae du ade by Re wih syheic calfski called fibesky 3. Is ebae is ade u f w layes (PET fil ad Tyvek) jied he wde fae by a yl web sip. The fae du suface was psiied a abu 4 abve ad belw he i ad sca icphes especively. Ne ha due he eplacee f he cil igh udeeah he ages he i icphes wee placed each side f he cil addig disace f he icphes he divig fce psii. As saed i he expeieal pcedue a fequecy sca has fis bee caied u. The pupse is fid he csucive ad desucive iefeece paes bewee he fuci geea ipu fequecies ad he duhead espse. The hey ifs ha ly ceai des ae accessible depedig whee is siuaed he divig fce pi. Theefe we chse cduc a fequecy sca a hee diffee psiis alg he adius f he du ebae. Fig.4: Fequecy sca pefed a he cee f he duhead. I shws he RMS values f he pessue (lef) ad paicle velciy (igh) apliudes. Apliudes f he pessue ad paicle velciy fields idicae whee he iefeeces ccu. The psiive peaks deed by cicles desigae esace des. Resace ca be chaaceized by he edecy f a acusic syse absb e eegy whe i is dive a a fequecy ha aches e f is w eige-fequecy f vibai (i.e. esa fequecy). Each psiive peak ca be elaed a eige-de f vibai. Sice he ebae was fis dive a he cee e ay quickly cclude ha he hee fis peaks displayed he gaph pbably efe he hee fis (0) des especively J0 (76.7 Hz) J0 (46. Hz) ad J03 (66.6 Hz). Ne ha w fequecies f esace ha ae vey clse each he ay geeae a cbiai f w disic eige-des if he ebae is beig dive a e f hese fequecies. Havig w 0

11 clse-by eige-fequecies is pbably due he fac ha he duhead pssesses a fiie siffess (i.e. pefecly cplia). As eied i he pevius seci he ex sep is sabilize he divig fce a a fequecy f iees. The fis fequecy peak was fud be 76.7 Hz. Befe saig a spaial sca f he fis eige-de a his fequecy cdiis wee als ecded: T he vlage apliude f he fuci geea was 4.0 V he LIA s sesiiviy was 000 V he abie epeaue was 0.7 C he elaive huidiy 63% ad he aspheic pessue 740 Hg. Vaiais i hese weahe depeda quaiies sgly ifluece he esa fequecy. The fllwig gaphs pvide ifai abu he difig eige-fequecy ad he efeece phase f he ages psiied a he cee f he du. Fig.5: The lef picue epeses he difig esa fequecy ad he igh picue shws he efeece phase (wih a egaive paiy) bh i fuci f he easuee ube (3x3). The fial epeaue was ecded be 3.3 C (.6 C wae ha whe he sca saed). Ne ha f a lw de such as J0 i akes abu 4 hus cplee a spaial sca. The lef gaph eveals ha he eige-fequecy is beig slighly shifed duig he sca peid. The dif i abie epeaue is picipally espsible f shifig he eige-fequecy; hweve he use f he de-lckig i icphes allws keepig ack f he esa de f vibai. The efeece phase was ughly lcked bewee -88 ad -90. F a ideal ebae he fequecy ad phase wuld be csa hughu he eie sca peid. The ex sep is he dal aalysis which I ecall is he pcess f descibig he dyaic ppeies f a elasic sucue i es f is eige-des f vibai. Oce a spaial sca is de e ca bai 3-D epeseais f ay acusical quaiies eleva u sudy. Disibui f he cplex pessue ad he cplex paicle velciy acss he du suface pvide isighful ifai abu he excied al de.

12 Fig.6: 3-D epeseai f he iagiay pas f he pessue (lef) ad paicle velciy (igh) baied hugh scaig he eie du suface wih a dive fequecy f ughly 76 Hz. The exciai pi (i.e. ages psii) is a he cee f he du. Oe ca iediaely cclude ha he fis baied eige-fequecy cespds he J0 eige-de als called beahig de which is he lwes esace de f a cicula vibaig syse. Ne ha all he ecded apliudes hughu his ep ae -ea-squaed (RMS) values. May he epeseais f cplex acusical quaiies (paicle displacee ad acceleai acusic ipedece sud iesiy ad pwe eegy desiy liea ad agula eu desiies phase ifai ) ay be baied hugh fflie daa aalysis f he easued pessue ad paicle velciy fields. A iage epeseai was csiued hugh scaig he vibaig suface a a desied de-lcked eige-fequecy. Figue 4 shwed he fis psiive fequecy peak be elaed he fis eige-de f vibai. The ex w axia igh als eveal ew eige-des f highe de. Duig his secd spaial sca we had lwe he LIA s sesiiviies ad he vlage apliude f he fuci geea.5 V peve he sca f veladig ad sppig. Phase ffses wee fud be 58 f he i icphes -3 f he pessue ad -6 f he paicle velciy sca icphes. These phase cecis ae added he aw daa i he fflie aalysis. Alhugh caefully ecded cdiis ad LIA s sesiiviies f he eaiig des will be lised hughu he es f he aalysis.

13 Fig.7: Sae pls as Fig.5 f a de-lcked eige-fequecy f ughly 46. Hz (/- 0.5 Hz). The efeece phase is lcked aud he desied -90 (egaive paiy). Accdig he heeical explaai e wuld expec his de f esace be he secd (0) de J0. The fequecy ai f his de alized J0 is abu.4 which sees clse he expeced heeical ai f.96.takig a clse lk a he pessue ad paicle velciy epaiis: 3

14 Fig.8: Lef ae he eal pas f he pessue (p) ad paicle velciy (b). Thei iagiay pas ae displayed he igh had side. The spaial sca f he secd eige-fequecy f 46. Hz eveals disubig esuls. Eve hugh we exclusively expeced he secd eige-de be a (0) de due he exciai beig a he cee f he du he p ad b lef pls clealy shws ha he de displayed is J0 bu J. Hweve he iagiay cpes (igh) f he pessue ad paicle velciy fields shw csideable eseblace wih he J0 eigede. Thus i sees bh des ae beig geeaed a a fequecy f ughly 46 Hz. The eas why we wee als able excie he du a is J eige-de is due he fac ha w eaby fequecies f esace ae clse eugh each he s ha bh f he eige-des ae beig excied a he sae ie. A pbable cause f such aifac is ha he du ebae is pefecly cplia. Me ifai abu he picipal esace-shif effecs due a -ideal syse is give i The Physics f Musical Isues by Fleche ad Rssig {3}. The ex fequecy peak shw i figue 4 is a 66.6 Hz. The cespdig expeced eige-de shuld be J03. alhugh he fequecy ai is 3.75 which is clse he J5 de if we cside he heeical ais. Fig.9: Iagiay pessue (lef) ad paicle velciy (igh) f a fequecy lcked aud 66 Hz. 4

15 As heeically expeced he displayed esace de which cespds he eigefequecy f 66 Hz is he J03 Bessel fuci. Alhugh e ay ice ha he apliude f he ue ig is hgeus alg is peiee. I is divided i fu lups f siila apliudes which is a aifac f he J Bessel fuci. The J de shuld be diecly excied whe he divig fce is placed a he cee because i pssesses a de a his pi. Is heeical fequecy ai is 3.50 clse he ai f f he J03 de. These w eige-des ae pesuably vey clse each he ad ae beig geeaed siulaeusly whe dive aud 66 Hz. Oce agai a pbable cause f such aifac is due he fiie siffess f he ebae (as peviusly eied i he aalysis f he pevius de a 46 Hz). F a cleae iepeai f he esa fequecies we pled he esuls f hee fequecy scas a diffee ages psiis alg he du adius: Fig.0: Fequecy sca f hee diffee psiis f he dive fce: A he cee (blue - sae as figue 4) half-adius (gee) ad a he edge (ed). As peviusly saed e shuld be able excie he duhead a ly ceai des f vibai depedig whee he ebae is beig excied. A pl f he pessue apliudes f diffee ages psiis ay help us uecessaily wase ie. If e lks a he gee cuve he pie psiive peaks (belw ~700Hz) sesibly velap wih peaks f he ed cuve. Hweve he ed cuve pssesses axia which d velap wih hse f he he cuves. Theefe ly five e spaial scas ae ecessay cve he fae du haic specu f he fequecy age f iees (< 700 Hz). Ne ha afe 700 Hz pubea aifacs f he syse such as shell esace ay ale he accuacy ad claiy i chaaceizig a eige-de. Wheeas belw 00 Hz ise due he achiey ad he veilai syse see diae he icphes sigal. 5

16 Five spaial scas wee he pefed a Hz 50 Hz 575 Hz 65 Hz ad 685 Hz (psiive peaks f he ed cuves cespdig ye ukw des). Ne ha hese fequecies ae jus saig values ad ae e less appxiae due he ifluece f cliae cdiis as descibed ealie. Eige-des geeaed by hese five fequecies ca all be excied by psiiig he divig fce a he edge f he du ebae (.5 c f he cee). Fig.: 3-D pl f he iagiay pessue (p-lef) iagiay paicle velciy (b-lef) eal pessue (p-igh) ad eal paicle velciy (b-igh). The ages wee psiied a he edge f he duhead ad iiially dive a abu Hz. The eige-de ha cespds a fequecy f appxiaely Hz is udeiably he J eige-de. The fequecy ai is.75 elaively clse he heeical ai f.59. A lup he y ai-de shws a defaul i he cuve shess. This lup cicides wih he pi whee he ebae is beig dive. The ages psii als cicides wih he quae f he du ha des pssess a suppig clapig a (which daps acusical eegy u f he syse). Takig a clse lk a he eal cpes f he pessue ad paicle velciy fields we bseve a hgal ai f he sucue i he x-y plae. The iagiay disibuis pssess a dal lie i he x-dieci wheeas i has bee shifed i he y-dieci f he eal pas. 6

17 The ex fequecy f iees is abu 50 Hz. The iagiay disibui bai f he spaial sca is shw belw i figue. The eseblace wih he J3 eige-de is sigifica eugh cclude ha his esa de is well defied a his ipu fequecy ad cuplig f w eaby des k place. Fig.: Iagiay pa f he pessue (lef) ad paicle velciy (igh). The ages ae a he edge ad he divig fequecy is ughly lcked aud 50 Hz. The hee eaiig fequecies (575 Hz 65 Hz ad 685 Hz) uexpecedly celae uclea cbiais f w eaby des f esace. The fllwig gaphs display he iagiay pa f he pessue ad paicle velciy fields. Fig.3: Vibai de f he duhead whe dive a 575 Hz. 7

18 Fig.4: Vibai de f he duhead whe dive a 65 Hz. Fig.5: Vibai de f he duhead whe dive a 685 Hz. All hese eaiig esa des ae ill defied ad eai sewha ukw. A aheaical del ucuple ixed clse-by des f esace (i de disce e al de f he he) was discussed by D Eede ad ay seday be added his lecue es. N-uif esi acss he du ebae he siffesselaed cdiis such as ai-ladig bedig siffess ad siffess shea ae piaily espsible f cbiais f eaby eige-des. The fllwig figue is a able f all he des sudied hughu he expeie. Tw ws f he sae cl idicae eaby esa fequecies (hece he s plausible cuplig f w eige-des). A sigle w cespds a pefecly defied al de. 8

19 Eigede ideal ai easued pediced (deg) J()/J0 fequecy Fequecy Phase ffse J(0) Hz J() Hz J() Hz J(0) Hz J(3) Hz J() /65 Hz J(4) /575 Hz J() Hz J(03) Hz J(5) J(3) Hz J(6) Hz Fig.6: Fequecy elaed ifai f each fud eige-de. Discussi ad Cclusi The aalysis f a 3-D cplex sud field assciaed wih u phase-sesiive eafield acusic hlgaphy ehd is esablished fflie ce he scas ae fiished. This is whee he exaci f acusical quaiies eleva u sudy akes place. Hweve a lage au f pls wee shw i his ep. Paicle displacee ad acceleai acusic ipedece sud iesiy ad pwe eegy desiy liea ad agula eu desiies all f hese physical quaiies wee skipped i he esul aalysis eve hugh hey wee all pcessed. Thei spaial ifai was sill ecded ad pled f he sudied psiis f he divig fce. A esseial subjec has e less bee kep uuched hughu his ep he ccep f phase ceci. A few aifacs f he syse/seup ay geeae phaseshifs effecs ad hece us be ceced. The picipal causes f phase-shifs culd be lised as such: The phase-shif effecs due eaby/velappig esaces ad he pssible -liea ixig f eaby eige-des f a eal du. The delays due elecics saic iefeece ec... Alhugh elecic ppagai ie effecs he phase ae usually egligibly sall (a few asecds cpaed echaical peids f a few illisecds a esa eige-fequecy belw KHz). Als us be csideed he ppagai ie effec f he icphes placed a a heigh z abve/belw he duhead which gives ise fequecy-depeda phase shifs ϕ -kz. The paicle velciy icphes als has fequecy depeda phase shifs alhugh quie sall ve u age f iees (<0 ). Theefe i is ecessay cec all kw phaseshif effecs i he fflie daa aalysis. As descibed i he expeieal pcedue shifs f he esace fequecy due epeaue aspheic pessue ad huidiy ae beig ceced by de-lckig he esace. Phase-shif effecs due he iedelayed espse f he duhead f is dive fce ae als eliiaed: The -D echaical equai f i: a x bv x kx Fdive (whee Fdive is f he cilages) he cil beig dive by a csa NIC which eves phase-shifs due 9

20 he iducace f he cil. Phase ceci is a cucial sep i he pcedue f u expeie. This pape pays a DAQ syse which descibes a phase-sesiive seup f eafield acusic hlgaphy. May specificiies f he seup wee hugh u ad desiged by D Seve Eede hiself. Eve hugh he appaaus was iiially develped iage he sud field f ay vibaig syses we fcused u expeie a specific pecussi isue called fae du. The baied esuls evealed easable siilaiies wih he heeical (ideal) del f vibai. N-uif esi acss he duhead ebae--shell cuplig asyeic clapig f he du shell spaial sabiliy f he excied syse sadig waves ad iefeeces ea he seup csa difs f he cdiis all hese diffee cicusaces ay cause he bseved del divege f he heeical del f a pefecly cplia vibaig duhead. Ackwledges Abve all I wuld like expess y gaiude y REU advis D Seve Eede whse assisace paiece pfud kwledge ad isighful explaais pvided e wih a deepe udesadig f physical acusics ad added csideably y eseach expeiece. I als wish hak Ada Was f claifyig a few bscue cceps hughu he eseach Ty Pis as he REU cdia ad Kaie Bule f shaig he wkspace. This REU pga was ade pssible hugh supp f he Naial Sciece Fudai (PHY ). Refeeces {} Eede Seve. Maheaical Musical Physics f he Wave Equai. Lecue es. {} Eede Seve. The 3-D cplex sud field Assciaed wih Nea-field Acusic Hlgaphy f a Vibaig Du Head. Upublished. {3} Neville H. Fleche ad Thas D. Rssig. The Physics f Musical Isues ( d edii -973). {4} Eede Seve. Physical Quaiies Assciaed wih a Cplex Sud Field. Upublished. {5} Eede Seve. Measuee f Cplex Sud Fields Pa : The Use f Lck-I Aplifies f Phase-Sesiive Measuees f Cplex Haic Sud Fields. Lecue es. 0