Euler s Equation for Inviscid Fluid Flow

Transcription

1 UIUC Phsics 406 Acusicl Phsics f Music Eulr s Equin fr Inviscid Fluid Flw Eulr s quin fr inviscid (i.. dissipinlss) fluid flw is spcil/limiing cs f h mr gnrl {nn-linr} Nvir-Ss quin which prsss Nwn s nd lw f min fr {cmprssibl} fluid flw. Th N-S qn, in h bsnc f rnl driving frcs is: Du 4 r, pr, 3B ur, ur, D Th w dissipiv rms n h righ-hnd sid f h Nvir-Ss quin nn-r u nd h curl f h vrici f u r sscid wih h cfficin f shr viscsi f h grdin f h divrgnc f h pricl vlci h pricl vlci fluid, nd h cfficin f bul viscsi f h fluid B, bh f which hv SI unis f Pscl-scnds (P-s). Du Th im driviv rm n h lf-hnd sid f h Nvir-Ss quin, is h D cmpl pricl cclrin sscid wih n infinisiml vlum lmn V f fluid {.g. ir} cnrd n h spc-im pin r,. Frm dimnsinl nlsis, n h Du g- m/ s N D r, 3 3 is frc dnsi. Th rm u is nwn s D m m D h cnvciv (r subsniv, mril) driviv, cmpud frm sinr bsrvr s rfrnc frm,.g. fid in h lbrr: D D ur, ur, ur, ur, Eulr s quin fr inviscid fluid flw is firs-rd lin hmgnus diffrnil quin, rising frm cnsidrin f mmnum cnsrvin in lsslss/dissipinlss cmprssibl fluid (liquid r gs), h in h bsnc f rnl driving frcs dscribs h rlinship bwn cmpl prssur p r, nd cmpl pricl vlci u in h 3 cmprssibl fluid, f vlum mss dnsi r, gm. Eulr s quin fr inviscid fluid flw is hus vlid fr fluids whr h viscsi f h fluid nd/r h cnducin f h in h fluid r bh r {r cn bh b pprimd s bing ngligibl}: Dur, ur, r, r, u r, u r, pr, D -- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

2 UIUC Phsics 406 Acusicl Phsics f Music Inviscid fluid flw in cmprssibl liquid r gs ccurs whnvr h mgniud f inril Finril cing n n infinisiml vlum lmn V f h fluid cnrd n h pin Fviscus cing n h fluid, R F F. Fr i wll frcs r in h fluid r lrg in cmprisn h dissipiv frcs.g. fluid wih high Rnlds numbr: inril viscus w frm n bunding/cnfining surfcs is n such mpl f n inviscid fluid. In nlg wih lcric chrg cnsrvin, h mss cninui quin fr fluid flw dscribs cnsrvin f mss vr spc-im pin r, wihin h vlum V f h fluid: r, ru, r, 0 r: whr: J u g m -s r, J 0 is h 3-D vcr cusic mss currn dnsi. Fr vrd cmpl sund filds S in ir ( NTP) h w r cnsidring in his curs (in h udi frqunc rng: 0 H f 0 KH ), picl sund prssur lvls r: SPL r, L r, 0lg p r, p 34 db. p 5 Th rfrnc sund vr-prssur mpliud is p 0 RMS Pscls RMS N m in {bn-dr} ir NTP, nd w hv shwn in prvius P406POM lcur n h sund vr-prssur mpliud f p.0 RMS Pscls crrspnds sund prssur lvl f SPL Lp 0lg0 p p 94 db 34 db in {bn-dr} ir NTP. N h sund vr-prssur mpliud f p.0 RMS Pscls is hn h mbin msphric prssur 5 5 Pm.030 Pscls NTP, r: p P m 0. A sund vr-prssur mpliud h is s p r, P.030 RMSPsclscrrspnds lrg s h msphric prssur islf, 5 n lms unimginbl sund prssur lvl f p 0 m h n vr-prssur mpliud f p r, 0 RMSPscls pin lvl f p 0 pin 0 m SPL L 0lg p p 94 db! {N crrspnds sund prssur SPL L 0lg p p 0 db, which is h hrshld fr pin... } Nn-linr ffcs in ir bcm incrsingl nicbl vr-prssur mpliuds grr p 00 RMS Pscls P.030 Pscls, which crrspnds sund hn 5 nl prssur lvl f p 0 nl m SPL L 0lg p p 34 db (S grph blw). -- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

3 UIUC Phsics 406 Acusicl Phsics f Music Th nn-linr rspns in ir fr lrg prssur vriins SPL ' s 34 db riss frm h nn-linr rlin bwn h prssur nd h dnsi f ir. Fr dibic chngs in ir prssur (rlvn fr sund prpgin in ir fr udi frqunc sunds {i.. f < 0 KH}): P Pm p cnsn {whr fr i CP CV 75.4}. Th rlin P r, f ir is shwn in h bwn {bslu} prssur nd vlum mss dnsi figur blw, whr quilibrium (i.. n sund is prsn) Pm P nd m : W cn prss h insnnus bslu prssur P s Tlr sris pnsin bu h quilibrium prssur Pm P nd mss dnsi m cnfigurin: P P Pr, P r, r,... r, r, P P P... r Fr smll prssur vriins pr, r, vr-prssur p r, nd h vlum mss dnsi, Pm firs rd linr rlinship iss bwn r, fr ir: P pr, Pr, P Pr, r, r, -3- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

4 UIUC Phsics 406 Acusicl Phsics f Music A mhmicl smn sscid wih h cnsrvin f mss wihin n infinisiml vlum lmn V f ir f quilibrium vlum V is givn b: V V cnsn. Thus, h vlumric srin (rlvn fr sund prpgin in ir) is: VV r: VV, hnc firs rdr h vr-prssur: P P V V pp PP B V V whr B P is h dibic bul mdulus f ir NTP}. Hwv fr dibic chngs, h bslu ir prssur P cnsn nd hus: B P P, hnc: P P V V p P B B P P s V V Th frcinl chng in vlum mss dnsi is nwn s h cndnsin: s Thus, fr vrd udi sund vr-prssur mpliuds p r, 00 RMSPscls { SPL 34 db }, h rspns f ir s mdium fr sund prpgin is vr nrl linr. This in urn implis h fr vrd sund vr-prssur mpliuds, h vlum mss r,.04gm s 0}. 3 dnsi f ir NTP is nrl cnsn, i.. {i.. Hwv fr vrd udi sund vr-prssur mpliuds, wih smll prssur vriins p r, P, sinc: r, r,, hus: r, r, r, r, is h {incrmnl} vlum mss dnsi mpliud sscid wih h prsnc f h cusic sund fild, h im drivivs r, r, 0 nd s 0. Hwv fr r,, h nn-linr ru, r, rm in h mss cninui quin cn b linrid: ru, r, r, ur, u u u nglc -4- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

5 UIUC Phsics 406 Acusicl Phsics f Music Hnc, fr vrd udi sund filds, h linrid mss cninui quin is: r, u 0 N ls h fr vrd udi sund filds, h linrid cmpl cusic mss currn J r, u r, g m -s. dnsi is: Liwis, fr vrd udi sund filds, h nn-linr Eulr quin cn liwis b r, r, r, w firs m h pprimin: linrid. Fr, wih Du r, D Du u u u D A scnd pprimin h w nw m fr vrd udi sund filds is h i cn b shwn h h mgniud f h nn-linr rm u r, u is vr smll in cmprisn u rm, nd hnc cn b nglcd. Thus, h linrid h mgniud f h vrsin f Eulr s quin, vlid fr SPL 34 db (vr-prssur mpliuds p r, 00 RMSPscls) bcms: u p r, r: u p r, Th Hlmhl Thrm lls us h h vcril nur f n rbirr vcr fild F r is full-spcifid/uniqu if.) lim F r 0 nd b.) h divrgnc.nd. h curl f F r r r bh nwn, i.. F rc r nd F r D wih h rsricin h F r D r, sinc h divrgnc f h curl f n vcr fild is lws r. 0 Fr h 3-D pricl vlci u fluid such s i fr vrd vr-prssur mpliuds f p r sscid wih sund wvs prpging in n inviscid, 00 RMSPscls, w shwd bv h h linrid mss cninui quin (prssing cnsrvin f mss), lls us h h spil divrgnc f h 3-D pricl vlci fild is qul h ngiv f h nrmlid ( frcinl) im r f chng f h vlum mss dnsi: r, u -5- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

6 UIUC Phsics 406 Acusicl Phsics f Music Wh is h curl f h 3-D pricl vlci fild, ur,??? Phsicll, h curl f vlci fild is fn sscid.g. wih rin nd/r vlci shr such s h vlci fild v sscid wih whirlpl, r vr in wr. Fr his rsn, h curl v r, is smims nwn s/clld h vrici. f vlci fild Hwv in n inviscid fluid (i.. n which is dissipinlss/hs r viscsi) such s i v 0, bcus n inviscid fluid cnn suppr vlci shrs nd/r u 0 fr vrd p r, 00 RMSPscls. Firs, w u wih rspc im: h vrici vrics in h inviscid fluid. W cn plicil shw/prv h udi sund vr-prssur mpliuds in ir NTP f h pril driviv f u u u Hwv h Eulr quin fr inviscid fluid flw is: p r, u u p Hwv h curl f h grdin f n rbirr sclr fild f r, i.. f 0, hus: u ur, p 0 This lls us h: u cnsn fcn n vrici in h inviscid fluid ( u 0, hus: is ls lws. Thus, if fr n im, hr is ), hn i mus rmin = 0 fr ll im. Q.E.D. If w h im driviv f bh sids f h {linrid} mss cninui quin, nd h divrgnc f bh sids f h {linrid} Eulr quin: u p u nd: p r, pr, c nd hn using h {linrid} dibic rlinship bwn cmpl vrprssur, p nd r, pr, r, pr,. mss dnsi,, w ls hv h rlin: c Hnc, w bin h {linrid} wv quin fr cmpl vrprssur: p, 0 c p r c -6- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

7 UIUC Phsics 406 Acusicl Phsics f Music If w nw h spil grdin f bh sids f h linrid mss cninui quin, nd h im driviv f bh sids f h linrid Eulr quin, nd gin us h {linrid} dibic rlinship bwn cmpl vrprssur, p nd mss dnsi, r, pr, r, pr,, hn:, w ls hv h rlin: c r, pr, u nd: c c u r Cmbining hs w quins, w bin: p r u c c, ur,, p If h cmpl vcr cusic pricl vlci fild is irrinl (i.. u r u u u using h vcr rlin u wv quin fr cmpl vcr pricl vlci: u u 0 c Th Cmpl Pricl Vlci Pnil, r,, 0), hn, w ls bin h {linrid} Sinc n inviscid (i.. dissipinlss) fluid ds n suppr vrici, i.. u r hn sinc h curl f h grdin f n rbirr sclr fild f r, f 0, w cn wri u u, whr u r, vlci pnil sscid wih u. Thn r, 0. u u, 0 is ls lws i.. is h cmpl pricl N h sinc u nd h grdin prr ˆ ˆ ˆ {in Crsin crdins} hv SI unis f ms nd m rspcivl, h cmpl vlci pnil u r, hs SI unis f m s. Phsicll, n ls h lins/cnurs {nd/r 3-D surfcs} f cnsn u K i cnsn r hus {cmpl!} quipnils, which r {vrwhr} u. prpndiculr h cmpl pricl vlci is h cusic nlg f N ddiinll h u r, wih u u 0 h lcrsic pnil r sscid wih h lcrsic fild Err sinc in lcrsics Er r0 {whrs in lcrdnmics, E B 0}., -7- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

8 UIUC Phsics 406 Acusicl Phsics f Music Epliing h nlg f h cncp f lcricl vlg i.. diffrnc in lcricl b b b - b r d E r d w cn ls dfin cmpl pnil pricl vlci pnil diffrnc ( pricl vlci vlg ) s: r d u r d b b,, b - b u u u u Frm h mss cninui quin: u nd: u, hn fr vrd udi sund vr-prssur mpliuds in {bn-dr} ir NTP f p r, 00 RMSPscls{ SPL 34 db }, hn: u r, r,, which cn b wrin s u r, r, ; his is Pissn s quin fr h cmpl pricl vlci pnil! Thus, w cn hus slv {crin clsss f} cusicl phsics prblms simpl b slving Pissn s quin u r, r, fr h cmpl pricl vlci pnil r,, subjc h bundr cndiin(s) {nd/r iniil cndiins = 0} sscid wih u h spcific prblm using chniqus/mhdlg similr h usd fr slving Pissn s quin r 0 in E&M prblms! N h {gin} using h {linrid} dibic rlinship bwn cmpl vrprssur r, pr, r, pr,. Hnc fr nd mss dnsi, w ls hv: c vrd udi sund filds, h bv diffrnil quin fr h cmpl vlci pnil r, pr,. cn quivlnl b wrin s: u c If u u, h {linrid} Eulr quin cn b wrin s: ur, ur, u r, p r,, which implis h: p r,, nd u r, pr, p r, r, hnc h:. Frm bv, w ls hv: c, hus:, u r c r, r,, bu frm h bv Pissn quin: u r,, hus, w bin h wv quin fr h cmpl vlci pnil:, 0 c u r, u r c u -8- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

9 UIUC Phsics 406 Acusicl Phsics f Music Drivin f Eulr s Equin fr Inviscid Fluid Flw frm Nwn s Scnd Lw f Min: W cn driv Eulr s quin fr inviscid fluid flw using Nwn s nd lw f min Fn m nd h sm im gin sm usful insigh in h phsicl mning f pricl u. vlci, Cnsidr n infinisiml vlum lmn 3 V m bundd b h surfc S cnrd n h spc-im pin r, {= cnr f mss f h infinisiml vlum lmn V} cnining {bn-dr} ir NTP, in hrml quilibrium wih h ir surrunding i, nd wih quilibrium 3 vlum mss dnsi.04 g m, s shwn in h figur blw: ŷ r ẑ V r, Bunding Surfc, S ˆ Rhr hn wr in h fid lbrr rfrnc frm, w dlibrl chs wr in rfrnc frm h is c-mving wih h infinisiml vlum lmn V f ir. N h h prssur p sscid wih h infinisiml vlum lmn V s msurd in h cmving rfrnc frm f h infinisiml vlum lmn V is h sm prssur s msurd p is inrinsicll sclr quni. in h fid lbrr frm, his is bcus prssur Th ir { NPT} cnind wihin h infinisiml vlum lmn V is sic / 5 quilibrium bslu prssur f n msphr, i.. pm.030 Pscls nd fini mprur T = 0 C (= 93.5 K). A h micrscpic lvl, h ir mlculs wihin h h 3 infinisiml vlum lmn V ch hv mn hrml nrg E Twhr B Juls Klvin nd cllid rndml wih ch h undrging Brwnin rndm-wl p mins. Supps h sund wv wih vr-prssur mpliud p r, 00 RMSPscls { SPL 34 db } is incidn n h {iniill sic} ir cnind wihin h infinisiml vlum lmn V. Whn h vr-prssur mpliud p is insnnusl grr (lss) hn h mbin prssur p m, h ir cnind wihin V mmnril cmprsss (pnds), rspcivl. N h cncpull, h surfc S h bunds h infinisiml vlum lmn V is ndwd wih mgicl prpris, in h i is ficiius, Gussin-p surfc (.g. s cmmnl usd in E&M prblms), h nur f h bunding surfc S ls is n which pnds nd/r cnrcs s h ir cnind wihin h infinisiml vlum lmn V pnds r cnrcs. Oprinll his mns w nd nl p rc f linr/lding-rdr rms in vrius pnsins -9- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd. ml B

10 UIUC Phsics 406 Acusicl Phsics f Music Furhrmr, if h nur f incidn sund wv is such s cus h ir mlculs wihin h infinisiml vlum lmn V cllcivl mv in givn dircin, i.. b displcd b cllciv 3-D disnc r, frm is quilibrium psiin, wih cllciv vlci u, h mgicl Gussin surfc S c-mvs wih h ir nd cllciv cclrin cnind wihin V. An infinisiml vlum lmn f si.g. cubic micrn 3 V m is sisicll lrg nugh fr ur purpss. Th ir cnind wihin his infinisiml vlum lmn V is in hrml quilibrium wih islf nd wih h ir surrunding i. Avgdr s numbr 3 N A mlculs ml nd rcll h n ml f {bn-dr} NTP hs ir mn/vrg mlr mss f m 8.97gms ml. Thus, fr vlum mss dnsi f ir ml g m NTP hr r 4.06 cm ml, r ~ 5 billin mlculs f ir pr cubic micrn NTP. Th vrg/mn vlci vcr sscid wih h mn/vrg hrml nrg Uh f his numbr f ir mlculs cnind wihin h infinisiml vlum h 3 lmn V is uml 0, hwvr h hrml nrgis Eml BT m uml sscid wih individul ir mlculs cnind wihin V m b such h individul mlculs wihin V lv hrugh h bund surfc S vi iing hrugh n f h p, bm r sid surfcs sscid wih S. Hwv n f h hr mgicl prpris ndwd wih h c-mving surfc S sscid wih h ir cnind wihin h infinisiml vlum lmn V is h if n ir mlcul lvs (nrs) h bunding surfc S givn pin r ml n n sid f h vlum lmn wih vlci vcr uml rml,, i insnnusl nrs (lvs) h surfc S gin wih vlci vcr u cnj ml rml,, bu n h hr sid f h cnj vlum lmn, is cnjug pin r ml rliv h cnr pin r, f h infinisiml vlum lmn, V. Thus h l ir mss m ir, h vrg/mn linr mmnum P U r ll cnsrvd b his mgicl ir nd h vrg/mn hrml nrg prpr f h ficiius Gussin surfc S bunding h infinisiml vlum lmn V. Frm Nwn s nd lw f min, Fn m, w cn clcul h frc(s) cing n h ir p. Th mss wihin h infinisiml vlum lmn V du n vr-prssur mpliud f ir cnind wihin h infinisiml vlum lmn V is m V g lls us h F m r h: r, F r, m F r, V n h n n {n} frc pr uni vlum cing n h infinisiml vlum lmn s: f r, F r, V f. n n. Thus h cclrin n. Nwn s nd lw. W dfin h N, l us (iniill) cnsidr nl h -cmpnn f h n frc du n vr-prssur p cing n h infinisiml vlum lmn V f i s shwn in sid viw in h figur blw: -0- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

11 UIUC Phsics 406 Acusicl Phsics f Music F F ˆ F F ˆ nˆ ˆ r, nˆ ˆ ˆ N h hr w mus b mindful f h nur f h cmprssiv frc(s) du h {smll} vr-prssur p cing n h infinisiml vlum lmn V nml, h hrml quilibrium f h ir cnind wihin h vlum V, s wll s ll hr djcn / nighbring infinisiml vlum lmns f ir mus b minind ll ims during his prcss. Th rsricin h p r, 00 RMSPscls{ SPL 34 db } fr hrmnic/pridic vr-prssur mpliuds wih frquncis in h udi rng f humn hring (0 H < f < 0 KH) gurns h hrml quilibrium hlds during his prcss. Frm hrmdnmic prspciv, his is clrl rvrsibl, dibic, nd hnc isnrpic prcss. Th infinisiml vcr r lmns sscid wih h (LHS) nd (RHS) f h infinisiml vlum lmn V r: A Anˆ A ˆ m nd A ˆ ˆ An A m. N h h uni nrml vcrs nˆ ˆ nd n ˆ ˆ sscid wih hs w surfcs, b cnvnin, pin uwrd frm/prpndiculr h surfc S. Th -frc cing n h LHS surfc lcd is: F F ˆ p A p A ˆ. Th -frc cing n h RHS surfc lcd is: F F ˆ p A p A ˆ. F F F p p A. Th n -frc cing n h infinisiml vlum lmn V is: ˆ -- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd. n Th n -frc pr uni vlum cing n h infinisiml vlum lmn V A is: p Fn p p A ˆ p fn ˆ V A In h limi h h vlum V f h infinisiml vlum lmn 0: p r, f, ˆ n r W cn rp his nlsis fr h - nd -cmpnns f h n frc pr uni vlum du h vrprssur mpliud cing n h infinisiml vlum lmn V f i h rsuls r similr: p r, p r, f, ˆ n r nd: f, ˆ n r

12 UIUC Phsics 406 Acusicl Phsics f Music Th l n 3-D vcr frc pr uni vlum is hrfr: f,, ˆ, ˆ, ˆ n r fn r f n r f n r p p p ˆ ˆ ˆ ˆ ˆ ˆ p p Thus w hv: f nd: fn r, fr, pr,, hnc: p. Rcll h (fr p r, 00 RMSPscls{ SPL 34 db }, h pricl cclrin is h im r f chng f h pricl vlci u i.. u wih p r, 00 RMSPscls{ SPL 34 db }: u p QE.. D., hnc w bin Eulr s quin fr inviscid fluid flw, vlid fr ir Cmplifing his quin, w hv: u p Alhugh his rlinship bwn h cmpl pricl cclrin u nd cmpl prssur p r,, pricl vlci ws drivd in h c-mving/cnr-f-mss rfrnc frm sscid wih h infinisiml vlum lmn V cnrd n h spc-im pin 5 r,, suprimpsd n p f sic prssur fild pm.03 0 Pscls, i cn b sn i h fr smll, hrmnic/pridic vr-prssur mpliud vriins,.g. pr, p r p r p h ch f hs quniis r h sm in h lbrr rfrnc frm. wih, m {frm quilibrium nd cmpl pricl r sscid wih h cllciv, rndm-hrml W cn nw ls s h h cmpl pricl displcmn r, m psiin}, cmpl pricl vlci u m s cclrin u m s nrg-vrgd-u min f h ir mlculs cnind wihin h infinisiml vlum lmn V bundd b h {c-mving} surfc S cnrd n h spc-im pinr,., -- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

13 Cmpl Sund Filds S UIUC Phsics 406 Acusicl Phsics f Music : Th cusicl phsics prpris sscid wih n rbirr vrd udi cmpl S r, b sund fild cn b cmpll/uniqul drmind h spc-im pin msuring w phsicl quniis sscid wih h cmpl sund fild: (.) h cmpl vr-prssur p r, h spc-im pin r,.nd. (b.) h cmpl pricl vlci u h spc-im pin r, wih: limu r0, u nd: u r r - sclr quni, - 3-D vcr quni, 0{r = cnsn}. Cmpl Sund Fild Quniis: Wring in h Tim-Dmin vs. h Frqunc-Dmin I is rml imprn whnvr wring wih n/ll cmpl sund fild quniis undrsnd/disinguish s whhr n is wring wih such quniis in h im-dmin vs. wring wih such quniis in h frqunc-dmin h r n h sm/indnicl Cmpl quniis in h im-dmin vs. hir frqunc-dmin cunrprs r rld b Furir rnsfrms f ch hr bcus im (unis = scnds) nd frqunc f (unis = /sc = H) r s-clld Furir cnjug vribls f ch hr. W hus us h nin S vs. S indic im-dmin cmpl sund fild vs. frquncdmin cmpl sund fild h spc-pin r, rspcivl. Hw d w nw whhr w r wring in h im-dmin vs. h frqunc dmin? A im-dpndn insnnus vlg signl V V cs p,.g. upu frm prssur snsiiv micrphn plcd h pin r ˆ, ˆ, ˆ in h sund fild f ludspr {lcd h rigin 0,0,0 } nd drivn b sin-wv funcin gnrr (f ngulr frqunc f) + pwr mplifir is picl mpl f im-dmin signl i is bsrvbl.g. n n scillscp, which displs h insnnus vlg signl V r, V r, cs pr, upu frm h micrphn s funcin f im,. mning h (vrll) phs p W spcif, fr clri/dfininss s h h scillscp rc f h displ f h p-mic signl V r, V r, cs pr, is riggrd rnll b h snc signl upu frm h sin-wv gnrr which srvs s h rfrnc signl nd hus givs phsicl f h p-mic signl, which is dfind rliv h im- FG dmin sin-wv vlg signl VFG V cs upu frm h sin-wv gnr sinc (b indusr sndrd, h psiiv-ging dg f ) h TTL-lvl snc signl upu frm h sinwv gnrr is usd in-phs riggr h sr f h scillscp rc displing h micrphn signl V - r, V r, cs r,. pmic p -3- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

14 UIUC Phsics 406 Acusicl Phsics f Music FG N h h insnnus im-dmin vlg signls VFG V cs nd V r, V r, cs pr, r purl rl quniis. W cn cmplif hs insnnus im-dmin quniis b dding qudrur/imginr rms hm: FG FG FG FG i V V cs iv sin V cs isin V nd: FG V r V r r iv r r pmic pmic pmic - -,, cs p,, sin p, - i p V cs isin V p p A {dul-chnnl} lc-in mplifir is mnifsl frqunc-dmin dvic h is ruinl usd in mn ps f phsics primns simulnusl msur h rl (i.. in-phs) nd imginr/qudrur (i.. 90 u-f-phs) cmpnns f cmpl hrmnic (i.. singlfrqunc) signl, rliv rfrnc sin-wv signl f h sm ngulr frqunc f. In h bv mpl, w culd.g. ddiinll simulnusl cnnc h micrphn s im-dmin upu signl V r, V r, cs pr, h inpu f h lc-in mplifir nd hn ls cnnc h TTL-lvl snc upu f h sin-wv gnrr h rfrnc inpu f h lc-in mplifi which is phs-lcd h cul insnnus {imdmin} sin-wv vlg signl V V cs upu frm h sin-wv FG gnrr. FG Th lc-in mplifir hn upus frqunc-dmin rl ( X ) nd imginr ( Y ) cmpnns f h cmpl p-mic signl h r rspcivl in-phs (90 u-fphs) rliv h lc-in mplifir s rfrnc inpu signl in his cs, h TTL-lvl snc signl upu frm h sin-wv gnrr: - ip ip V cs p ip ip sin,, sin i, p r V r p r pmic X R V p-mic R V V R V R cs p isin p Y Im V p-mic Im V V Im V Im cs p Thus, w s h h lc-in mplifir upus h rl (i.. in-phs) nd imginr/qudrur {i.. 90 u-f-phs) cmpnns f h frqunc-dmin cmpl vlg mpliud sscid wih h prssur micrphn s upu signl, bind h pin r in h (cmpl) sund fild f h ludspr: V R V iim V V cs p iv sin p i p V cs isin V p p -4- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

15 UIUC Phsics 406 Acusicl Phsics f Music In h -D R-Im cmpl pln, h cmpl frqunc-dmin phsr digrm fr V r is sic (i.. ds n r) nd pprs s shwn blw: cmpl, Im Y V V sin p p Im V V R V cs V p R X In h cmpl im-dmin, h nir phsr digrm fr cmpl V r h cmpl pln ngulr frqunc. rs CCW in, Pls s/rd Phsics 406 Lc. Ns 3 Pr fr ddiinl dils n hw lc-in mplifirs wr, nd hir us(s) in h lbrr Grphicll, h rl nd imginr frqunc-dmin cmpnns f h cmpl vlg mpliud signl upu frm h p-mic migh l smhing li h shwn in h figurs blw, fr pur (i.. singl-frqunc) sin-wv signl upu frm h sin-wv gnrr + pwr mplifir driving ludspr: V r V r V r r i r ip,,, cs p, sin p, X V r R, p-mic p, cs, V r r Frqunc-Dmin Y V r V r r Im, p-mic, sin, p -5- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

16 UIUC Phsics 406 Acusicl Phsics f Music N h h ngulr frqunc spis in h bv figur sscid wih h rl nd imginr cmpnns f h cmpl frqunc-dmin mpliud V r in fc -D dl-funcins {in ngulr-frqunc spc}, which cn b mhmicll rprsnd s V cs p nd V sin p, rspcivl. N n f h mn inrsing/inriguing prpris f h -D dl funcin: Sinc f, hnc d df, nd hus: d f f df f f df f f df f f df f f df N furhr h h -D dl funcin (i.. sc/rdin) nd h h -D dl funcin f f frqunc, unis f invrs frqunc, nd f hs phsicl unis f invrs ngulr hs phsicl f f (i.. scnds), sinc h -D ingrls d f df r bh dimnsinlss Th bv cmpl frqunc-dmin rsul(s) shuld b cmprd wih hir cmpl im-dmin cunrprs: -, -, -,, i p r, i p r pmic pmic i V pmic r V r V r V cs isin p p X R V R V V R -, Rcs, sin,, cs, i p r i p r p mic V r r i r,, p p V r r p - ip ip pmic Y Im V p-mic Im V V Im V r r isin, Im cs,, sin, p V r r p p As mnind bv, h frqunc-dmin cunrprs f cmpl im-dmin quniis FG i such s V FG V p, nd V -,, i r pmic r V r r bind b ing h Furir rnsfrm f h im-dmin quniis, nd vic-vrs. -6- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

17 Wh is Furir rnsfrm? UIUC Phsics 406 Acusicl Phsics f Music Fr cninuus cmpl im-dmin funcins f, h Furir rnsfrm f h cmpl im-dmin funcin f i h cmpl frqunc-dmin is: f f d whr is rd s {dumm} vribl in h ingrin vr {ll} im, frm. Th invrs Furir rnsfrm f cninuus cmpl frqunc-dmin funcin f i f f d whr f is rd s {dumm} vribl in h ingrin vr {ll ngiv.nd. psiiv} ngulr frquncis: h im-dmin is:. N ls h h fcr f pprs hr pr-mulipling h lr ingrl vr h ngulr frqunc vribl bcus w r using h ngulr frqunc f in h ingrl rhr hn h frqunc f islf s {dumm} vribl f ingrin chnicll sping, frqunc f (sc ) nd im (scnds) r ru Furir cnjug vribls f ch h nd n ngulr frqunc f (rdins/sc ) nd im (scnds). Fr mnchrmic/singl-frqunc ( hrmnic) sund filds h rlinship bwn gnric cmpl im-dmin vs. cmpl frqunc-dmin quniis is simpl givn b f f. Thus,.g. h rlins bwn cmpl im-dmin vs. cmpl i frqunc-dmin sclr vr-prssur nd/r 3-D cmpl vcr pricl vlci r: i p p p ip i nd: u u u ˆu ˆu ˆ i u iu u i i i u ˆ u ˆ u ˆ i Thr r svrl usful rlins sscid wih Furir rnsfrms which w lis hr: Tim-Dmin: h f bg Frqunc Dmin: h f bg Linri: Trnslin: h f i h f Mdulin: i h f h f Scling: h f h f * h f * Cnjugin: h f -7- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

18 UIUC Phsics 406 Acusicl Phsics f Music Cmpl Spcific Acusic Immincs - Adminc nd Impdnc f Mdium: Th mdium (slid, liquid r gs) in which sund wvs prpg hs sscid wih i h prpr f hw s (r hw difficul) i is fr sund wvs prpg hrugh h mdium h s-clld cmpl spcific cusic immincs cmpl spcific cusic dminc nd/r cmpl spcific cusic impdnc (h rciprcl f cmpl spcific cusic dminc) giv us such infrmin. Fr prpgin f -D sund wvs in mdium, h cmpl spcific cusic immincs i.. cllcivl h cmpl spcific cusic dminc nd/r cmpl spcific cusic impdnc r bh wll-dfind quniis. Th r dfind in nlg h cmpl frm f Ohm s Lw (V IZ, I VY ) s usd.g. in lcricl circui hr, sinc cmpl vrprssur p is h nlg f cmpl AC vlg V, nd pricl vlci u is ~ h nlg f cmpl AC lcric currn I {N h J u g s-m is h cmpl J I A nqv v Amp m Cul sm - is h cusic mss currn dnsi}, whrs cmpl lcricl currn dnsi}. N ls h bh J Cmpl Sclr Elcricl Immincs: Cmpl Elcricl Impdnc: ; Cmpl Elcricl Adminc: ; nd J V ; Z Ohms Vls Amps I ; I ; V ; r 3-D vcr quniis. Y Simns Ohms Amps Vls V ; i i, If w wri u hs rlins using cmpl plr nin: I ; i i I I, hn, ning h cncllin f V V i im dpndnc fcrs: Z Y ; ; V ; I ; I ; V ; i i V V i i I I i i I I i i V V I I I i V V V iv I iz Z Z i i I I I ii V iy Y i Y V V V Nw: Z Y r: Y Z, nd w s h: Z V I Y hnc: Y i Y i Z i Z Y Z Z Z. Thus: V ; V Z ; = Z I ; I Y nd:, I ; I Y ; = Y V ; V Z -8- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

19 UIUC Phsics 406 Acusicl Phsics f Music Cmpl 3-D Vcr Spcific Acusic Immincs: p Acusic P-s m Cmpl Spc. Acus. Impdnc: Rl 3 u, Ohms N-s m r u Acusic m P-s Cmpl Spc. Acus. Adminc: Rl 3 p Simns m N-s N h h cmpl spcific cusic immincs nd r, r, vcr quniis. Th cmpl 3-D vcr spcific cusic dminc r, ur, pr, r 3-D is clrl mhmicll wll-dfind vcr quni: u u, u r u,, ˆ, ˆ, ˆ ˆ ˆ ˆ r r r r p p p p u, ˆ, ˆ, ˆ r u r u r u p p whr: u,, r u r u,,,, r r p r, pr, pr, Th cmpl 3-D vcr spcific cusic impdnc p u m iniill sm li mhmicll lss wll-dfind vcr quni. Hwv n phsicl/cmmn sns grunds, w nw h.g. h mgniuds f h cmpl 3-D vcr spcific cusic r, mus bh b invrin (i.. unchngd) undr simpl immincs, nd crdin rnsfrmins.g. rins nd/r rnslins f h lcl crdin ssm, s wll s invrin undr.g. simpl rins f h sund surc undr invsigin. Cnsidr simpl, -D mnchrmic/singl-frqunc sund fild such s n cusic i rvling pln wv prpging.g. in h lcl ˆ dircin. Thn u u 0, wih i p p 0, whrs u u 0. Th cmpnns f h cmpl 3-D vcr i spcific cusic dminc r u,, r p r u i p u p 0, 0. whrs -9- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

20 UIUC Phsics 406 Acusicl Phsics f Music Obviusl, if w crr u.g. simpl rin f ur lcl 3-D crdin ssm, h r, will chng ccrdingl, hwvr h mgniud * r, r, r, r, r, r, will n chng. individul,, cmpnns f Liwis, h individul,, cmpnns f will chng ccrdingl undr simpl rin f ur lcl 3-D crdin ssm, hwvr h mgniud * r, r, r, r, r, r, will n chng. W hus wri h cmpl 3-D vcr spcific cusic impdnc crdins s fllws: p ˆ ˆ r, r,,, ˆ r r u * * * p u p u p u * * u u u u u * * * p u, ˆ, ˆ r u r u ˆ u * * * pr, u, ˆ, ˆ, ˆ r u r u r u u u whr: *,,.g. in Crsin * * p ru, r, p r u pru, r,,,,, r r u u u Hnc, h chnicl/mhmicl issu hr is h rinliin f n rbirr, gnric cmpl rciprcl 3-D vcr quni: * * u u u * u u u * u prllling h which is dn fr n rbirr, gnric cmpl rciprcl sclr quni: I cn b sn h indd: p p p p p p p * * * * u u r p p,, nd ls h: -0- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

21 UIUC Phsics 406 Acusicl Phsics f Music r r r r r r *,,,,,,,,,, p u,,, r p r u r p u u u u p r u r u r u r u p p u u r r u,, p r, ur, Hwv w ls s fr h individul,, cmpnns f h cmpl 3-D vcr spcific cusic immincs h:, u u r * pr, r, pru,, r u, u r * pr, r, pru,, r, u u r * pr, r, pru,, r Addiinll, h prssins fr h cmpl 3-D vcr spcific cusic immincs: u u, u r u,, ˆ, ˆ, ˆ ˆ ˆ ˆ r r r r pr, pr, pr, pr, nd: * * * p pr, u, ˆ, ˆ r u r ur, ˆ,, ˆ, ˆ, ˆ r r r r u u cn b sn mhmicll bhv prprl.g. undr rbirr rins f h lcl 3-D crdin ssm, s wll s fr rins f 3-D sund surcs, nd ls fr cmpl 3-D sund filds cmpsd f.g. n rbirr suprpsiin/linr cmbinin f hr muullrhgnl prpging mnchrmic pln rvling wvs prpging in h ˆ, ŷ nd ẑ p r, p r, p r, p r,. dircins, wih cmmn sclr cmpl prssur, 3 -- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

22 UIUC Phsics 406 Acusicl Phsics f Music N ls h bh h im-dmin cmpl prssur p r, 3-D pricl vlci u nd h im-dmin cmpl sscid.g. wih singl frqunc ( hrmnic) sund fild i will in gnrl hv im dpndnc f h frm. Thus, sinc h 3-D spcific cusic i immincs r dfind s ris f hs w quniis, h fcr in h bh h numrr r, u r, pr, p u nd h dnminr f h ris nd cncls fr hrmnic/singl-frqunc cmpl sund filds, hus w s h h cmpl 3-D vcr spcific cusic immincs r in fc im-indpndn quniis In fc, h r mnifsl frqunc dmin quniis! i u u u Tim Dmin: i p, Frqunc Dmin p r p p r p r Tim Dmin:,, u u i i p u Frqunc Dmin Cmpl 3-D Spcific Acusic Immincs (fr Hrmnic Sund Filds): p Cmpl Spcific Acusic Impdnc: Rl u Tim-indpndn quni! Frqunc-dmin quni! Cmpl Spcific Acusic Adminc:, Tim-indpndn quni! Frqunc-dmin quni! u r Rl p r r,, Th im-indpndn cmpl spcific cusic immincs r 3-D vcr frquncdmin quniis. Thir 3-D -- Crsin frqunc-dmin cmpnns cn b plicil wrin u s: ˆ ˆ ˆ,, r r u, u r u u ˆ ˆ ˆ p p p p *,, p r,, ˆ, ˆ, ˆ r r r r u r r * * * p u p r u r p u p u ˆ ˆ ˆ u u u u,,, -- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

23 UIUC Phsics 406 Acusicl Phsics f Music r clld cmpl spcific N, w plin wh nd cusic impdnc nd dminc, rspcivl. As mnind bv, r immincs spcificll sscid wih h prpgin mdium. And, in rdr vid cnfusin, hr {lrd} iss w hr cusic imminc quniis, nwn s h cmpl 3-D vcr cusic impdnc nd h cmpl 3-D Y, which r sscid wih h cusics f vcr cusic dminc sund wvs prpging insid ducs (i.. pips) wih crss-scinl r S s dfind blw: Cmpl 3-D Acusic Immincs (fr Hrmnic Sund Filds): Cmpl 3-D Acusic Impdnc: 3, P-s m Rl m 5 u S Nsm - p r Y Cmpl 3-D Acusic Adminc: u S mps - Y Rl -m 3 p m N-s N h h quni U u S m s m m 3 s 3 m s. bcus f is dimnsins is nwn s h vlum vlci, Insid duc f crss scinl r S, h cmpl 3-D vcr spcific cusic immincs nd r hus rld h cmpl 3-D vcr immincs Y b h rlins: nd r: S nd, Y, r r S S nd Y,, r r S Frm h bv rlins, sinc h cmpl 3-D vcr spcific cusic immincs nd r mnifsl frqunc dmin quniis, w s h h Y r ls cmpl 3-D vcr cusic immincs nd mnifsl frqunc dmin quniis. nd -3- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

24 UIUC Phsics 406 Acusicl Phsics f Music Phsicll, jus s h cmpl sclr lcricl impdnc Z is msur f n lcricl I J S C s whn dvic impd h flw f cmpl sclr AC lcricl currn cmpl sclr AC vlg V is pplid crss h rminls f h lcricl dvic, h cmpl 3-D vcr cusic impdnc is msur f h cusicl mdium s bili impd h flw f cmpl cusic mss currn J S u S g s p pin r. I fr cmpl vr-prssur Similrl, jus s cmpl sclr lcricl dminc Y Z is msur f h s wih which n lcricl dvic dmis h flw f cmpl sclr AC lcricl currn I whn cmpl sclr AC vlg V is pplid crss h rminls f h lcricl dvic, h cmpl Y is msur f h s wih which n 3-D vcr cusic dminc cusicl mdium s dmis h flw f cmpl sclr cusic mss currn I J S u S g s in h prsnc f cmpl vr- p h pin r. prssur Anhr w gin sm phsicl insigh in h nur f cmpl 3-D vcr spcific cusic impdnc p u nd cmpl 3-D vcr spcific cusic dminc u p f mdium sscid wih hrmnic sund fild is imgin phsicl siuin whr w s h {mgniud} f h cmpl p p.0pscl. sclr vr-prssur b cnsn/fid vlu,.g. Thn, fr hrmnic sund fild, if h cmpl 3-D vcr spcific cusic impdnc p u h pin r hppns b vr high, fr fid cmpl sclr vr-prssur p, his lls us h h cmpl 3-D vcr pricl vlci u h pin mus hrfr b vr smll, nd hnc h crrspnding cmpl 3-D vcr cusic J u h pin mus ls b vr smll. mss currn dnsi Cnvrsl, if fr hrmnic sund fild h cmpl 3-D vcr spcific cusic impdnc p u h pin r hppns b vr lw, fr fid cmpl sclr vr-prssur p, his lls us h h cmpl 3-D vcr pricl vlci u h pin mus hrfr b vr lrg, nd hnc h crrspnding cmpl 3-D vcr cusic J u h pin mus ls b vr lrg. mss currn dnsi -4- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

25 UIUC Phsics 406 Acusicl Phsics f Music Fr hrmnic sund fild, if h cmpl 3-D vcr spcific dminc u p h pin r hppns b vr high, fr fid cmpl sclr vr-prssur p, his lls us h h cmpl 3-D vcr pricl vlci u h pin mus hrfr b vr lrg, nd hnc h crrspnding cmpl 3-D J u h pin mus ls b vr lrg. vcr cusic mss currn dnsi Cnvrsl, if fr hrmnic sund fild h cmpl 3-D vcr spcific cusic dminc u p h pin r hppns b vr lw, fr fid cmpl sclr vr-prssur p, his lls us h h cmpl 3-D vcr pricl vlci u h pin mus hrfr b vr smll, nd hnc h crrspnding cmpl 3-D vcr cusic mss currn dnsi J u h pin mus ls b vr smll. Th Rl nd Imginr Cmpnns f Cmpl 3-D Vcr Spcific Acusic Immincs: As in h cs fr AC lcricl circuis, h cmpl sclr lcricl impdnc Z nd cmpl sclr lcricl dminc Y Z cn b wrin u plicil in rms f hir rl nd imginr cmpnns: Z R ix whr R R Z is h rsisnc nd X ImZ Y G ib whr G R Y is h cnducnc nd B ImY Similrl, fr h cs cmpl hrmnic sund fild cusic impdnc r is h rcnc. is h suscpnc. S r nd cmpl 3-D spcific cusic dminc r r cn b wrin u plicil in rms f hir rl nd imginr cmpnns: r i whr:, R, Im, h cmpl 3-D vcr spcific r r r is h 3-D spcific cusic rsisnc h pin r nd: is h 3-D spcific cusic rcnc h pin r. r g r ib r whr:,,, g R is h 3-D spcific cusic cnducnc h pin r nd: b Im is h 3-D spcific cusic suscpnc h pin r. -5- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

26 UIUC Phsics 406 Acusicl Phsics f Music Fr hrmnic/singl-frqunc sund filds, w cn bin prssins fr h rl nd imginr nd dminc in rms f h rl nd imginr prs f cmpl sclr vr-prssur p nd u frm hir rspciv dfiniins p u u p. prs f frqunc-dmin cmpl 3-D vcr spcific cusic impdnc cmpl 3-D vcr pricl vlci nd Supprssing h frqunc-dmin rgumn fr ninl clri s s, nd wring wih nl n f h hr vcril cmpnns,, r, fr cmpl 3-D vcr spcific cusic dminc: r i u ur iui ur iui pr ip pu i r r pu i i pu r i pu i r i i p pr ipi pr ipi pr ip i p p Thus w s h fr,, r : r p u pu r r i i R p i nd: p u pu pu pu p p r i i r i r r i Im Liwis, fr cmpl 3-D vcr spcific cusic impdnc: * pr ipi ur iui pr ipi ur iu r i i pru r pu i i pu i r pu pu r i i i u u u u u Thus, w s h fr *,, r : r p u pu r r i i i nd: R u p u pu u i r r i Im Ning h: * * u u u * p p p nd h: * * p u p u * pu u u u W s h: u pu pu p nd h: i u pu pu p r r r r i i i i r r i r quivlnl h: r r r: r r nd h: i i r: i i -6- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

27 Thus, w s h fr givn r nd w s h fr givn UIUC Phsics 406 Acusicl Phsics f Music cmpnn f,, r p u pu : r r i i i nd: R u,, r cmpnn f p u pu u i r r i Im : r p u pu r r i i R p i nd: p u pu pu pu p p r i i r i r r i Im s wll s: * nd: * r quivlnl: * nd: *. I cn b sn frm hs dfiniins h in gnrl h individul vcril cmpnns,, r d n pin in h sm dircin in spc. h: nd Sinc p u, nhr usful rlin is: u p : u u u p u p p u. * p p u p u u u u Similrl, sinc, hn: N h h bv prssins fr h rl nd imginr cmpnns f cmpl cusic spcific impdnc nd/r dminc givn in rms f linr cmbinins f h rl nd imginr cmpnns f cmpl sclr cusic vr-prssur nd cmpl vcr pricl vlci. As w hv discussd prviusl, h phsicl mning f h rl nd imginr cmpnns f cmpl sclr cusic vr-prssur nd cmpl vcr pricl vlci r rspcivl h in-phs nd 90 (qudrur) cmpnns rliv h driving sund surc. Hwv his is n h phsicl mning f h rl nd imginr cmpnns f cmpl cusic spcific immincs, bcus f h bv-dfind linr cmbinins f cmpl sclr cusic vr-prssur nd cmpl vcr pricl vlci. W shll s/lrn h h phsicl mning f h rl nd imginr cmpnns f cmpl cusic immincs prpris f h phsicl mdium in which cusic disurbncs prpg r rspcivl sscid wih h prpging nd nn-prpging cmpnns f cusic nrg dnsi. Th rl nd imginr cmpnns f h cusic spcific immincs r fn clld h civ nd rciv cmpnns f h cmpl sund fild, rspcivl, sinc (s bv): r i g ib nd: -7- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

28 UIUC Phsics 406 Acusicl Phsics f Music W cn gin furhr/ddiinl insigh in h nur f cmpl nd b wriing ur primr cusic frqunc-dmin vribls in cmpl plr nin frm: Cmpl sclr prssur: i p p p ip p r r i Cmpl 3-D vcr pricl vlci: u ur iui ur iui ˆ ur iui ˆ ur iui ˆ i, i u u r iu, u ˆ u ˆ u r ˆ Cmpl 3-D vcr spcific cusic dminc: r ii r ii ˆ r ii ˆ r ii ˆ i, i r i, ˆ ˆ r ˆ Cmpl 3-D vcr spcific cusic impdnc: r ii r ii ˆ r ii ˆ r ii ˆ i, i r i, ˆ ˆ r ˆ Thus, fr hrmnic/singl-frqunc sund filds w s h fr givn, h: cmpnn f u p Similrl, fr givn p r u r,, r *,, u,, r i u i u u i p iu i u p i pu i p p p : cmpnn f i p iu i p u p u ip u i pu u r, u -8- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

29 UIUC Phsics 406 Acusicl Phsics f Music W ls s h fr hrmnic/singl-frqunc sund filds h -phs: whrs h -phs: pu p u u p u p p u, in nlg similr rlins bind.g. fr cmpl AC lcricl circuis! i p Th phsr rlin(s) in h cmpl pln fr p pr ipi p i u, u ur iu i u, r i i i r i i nd i r shwn in h figur blw, fr h spcil/limiing cs f pu 90 pu, whr h impdnc phsr cmpnn is bc--bc wih h dminc phsr cmpnn {n.b. fr h mr gnrl cs whr pu 90 pu, hn nd r n bc--bc}: u Im p p R u 90 pu p u If w nw h csin f h w phss nd : cs cs pu csp u nd: cs cs cs cs cs u p u p (cs() vn fcn()) W s h whn: cs cs h: 0 pu up, i.. h: p u. Whn: cs cs 0 h: 90 pu up, i.. h: p u 90. Whn: cs cs h: 80, i.. h: 80. pu up p u -9- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

30 UIUC Phsics 406 Acusicl Phsics f Music Summr f Vrius Frqunc-Dmin Sund Fild Phsicl Quniis: Cmpl sclr prssur: i p p pr ipi pr Cmpl 3-D vcr pricl displcmn: r ii r ii ˆ r ii ˆ r i i ˆ i i i ˆ ˆ ˆ Cmpl 3-D vcr pricl vlci: u ur iui ur iui ˆ ur iui ˆ ur iui ˆ i, i u u r iu, u ˆ u ˆ u r ˆ Cmpl 3-D vcr pricl cclrin: r ii r ii ˆ r ii ˆ r ii ˆ i, i r i, ˆ ˆ r ˆ Cmpl 3-D vcr spcific cusic dminc: r ii r ii ˆ r ii ˆ r ii ˆ i, i r i, ˆ ˆ r ˆ Cmpl 3-D vcr spcific cusic impdnc: r ii r ii ˆ r ii ˆ r ii ˆ i, i r i, ˆ ˆ r ˆ -30- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

31 UIUC Phsics 406 Acusicl Phsics f Music Fr vrd hrmnic/singl-frqunc sund filds, if h 3-D vcr cmpl u is nwn/msurd, hn sinc h 3-D i u u, nd h 3-D vcr cmpl i r,, whr: is h 3-D vcr u, frqunc-dmin pricl vlci mpliud vcr cmpl im-dmin pricl vlci im-dmin pricl displcmn cmpl frqunc-dmin pricl displcmn mpliud, nd sinc hn: u d u d u d u i i i i Bu sinc: i r,, w s h: u i u Liwis, sinc: i i i u u i u iu i iu Bu sinc:, w ls s h: -3- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.

32 UIUC Phsics 406 Acusicl Phsics f Music Lgl Disclimr nd Cprigh Nic: Lgl Disclimr: Th uhr spcificll disclims lgl rspnsibili fr n lss f prfi, r n cnsqunil, incidnl, nd/r hr dmgs rsuling frm h mis-us f infrmin cnind in his dcumn. Th uhr hs md vr ffr pssibl nsur h h infrmin cnind in his dcumn is fcull nd chnicll ccur nd crrc. Cprigh Nic: Th cnns f his dcumn r prcd undr bh Unid Ss f Amric nd Inrninl Cprigh Lws. N prin f his dcumn m b rprducd in n mnnr fr cmmrcil us wihu prir wrin prmissin frm h uhr f his dcumn. Th uhr grns prmissin fr h us f infrmin cnind in his dcumn fr priv, nn-cmmrcil purpss nl. -3- Prfssr Svn Errd, Dprmn f Phsics, Univrsi f Illinis Urbn-Chmpign, Illinis All righs rsrvd.