The Acoustical Physics of a Standing Wave Tube

Transcription

1 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets The Acustical Physics f a Stadig Wave Tube A typical cylidrical-shaped stadig wave tube (SWT) {aa impedace tube} f legth L ad diameter D Lwith ifiitely rigid walls ad clsed eds is shw he figure belw: Sie-Wave Fucti Geeratr Acustical Trasducer ~ Stadig Wave Tube (SWT): i A e e i p u B e e p, u -mics lcated at Sud eergy is iput t the SWT at the psiti = 0, e.g. usig a sie-wave fucti geeratr cected t sme id f acustical trasducer, such as a wafer-thi pie-electric trasducer (r a ludspeaer). Ideally-speaig, the trasducer shuld have frequecy-depedet phase-shift(s) relative t the drivig sie-wave fucti geeratr. Hwever, he real wrld, such devices d t,0 exist. At frequecies belw the lwest cutff frequecy f the SWT ( f.84 v D~ 3300 H fr v 345m sad D 6 cm) ly -D type plae waves ca prpagate he SWT. Pressure (p) ad differetial/particle velcity (u) micrphes are c-lcated at the geeric psiti alg the symmetry axis f the SWT. They are used t recrd the cmplex istataeus ttal pressure ad the istataeus cmplex -D lgitudial/-cmpet f the ttal particle velcity at that lcati assciated with the presece f right- ad left-mvig acustic travelig plae waves prpagatig he SWT. The resultat istataeus cmplex pressure stadig wave at the pit is thus a liear superpsiti f these tw travelig plae waves: p t Ae Be Ae e Be e Ae Be e, = 0 i it i it i i it where the cmplex, frequecy-depedet waveumber i vi the detes cmplex cjugati, i.e. i ad i ad thus i A {extremely} imprtat micr-detail here is that i rder t be able t crrectly cmpare theretical predicti(s) t experimetal data, the chice f usig e vs. e he thery is i fact t arbitrary. I the UIUC Physics 93POM/406POM SWT experimet, i rder t btahe ecessary phase-sesitive ifrmati the cmplex ature f pressure (p) ad -D particle velcity (u) as a fucti f frequecy, the electrical sigals utput frm the pressure ad particle velcity micrphe preamplifiers are each iput t separate lc-i amplifiers (SRS mdel # DSP- 830) which als use the sigal utput frm the sie-wave fucti geeratr as the referece sigal fr each f the lc-i amplifiers. I ur SWT experimet we have explicitly selected 0 referecig -- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved. c ;. The use f esures that we are always apprpriately mathematically describig decayig expetial atteuati phemea, i.e. fr > 0, usig e fr bth right- ad left-travelig waves (as ppsed t uphysical, expetially grwig phemea, i.e. with distace). L e = L

2 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets f the tw lc-i amplifiers t the fucti geeratr s sie-wave sigal, ad thus because f the way a lc-i amplifier wrs, als implicitly meas that we have selected the e sig cveti. Had we istead selected e.g. 80 referecig f the lc-i amplifier t the sie-wave referece sigal, the we wuld have implicitly istead selected the e sig cveti. Because f ur e chice i referecig f the lc-i amplifiers t the fucti geeratr s sie-wave sigal, bth the istataeus cmplex pressure ad istataeus cmplex particle velcity precess (i.e. rtate) it cuter-clcwise (CCW) he cmplex plae as time icreases, sice e cst isit it whereas e cst isit. Geerically, the istataeus cmplex pressure (SI uits: Pascals) ad istataeus cmplex particle velcity (SI uits: m/s) ca be writte as: i p it pt, pt, ipt, p e e ad r i The real parts f the cmplex istataeus pressure p, t -D particle velcityu, t u u, t u, t iu, t u e i e r i ad/r cmplex istataeus are i-phase (if +ve) r 80 ut-f phase (if ve) relative t the referece sigal utput frm the sie-wave fucti geeratr; the imagiary parts f the cmplex p t, u, t are +90 istataeus pressure ad/r cmplex istataeus -D particle velcity ut-f-phase (if +ve) r 90 ut-f phase (if ve) relative t the referece sigal utput frm the sie-wave fucti geeratr, as shw belw {fr a geeral case/geeric situati} he s-called phase diagram i.e. the cmplex plae, at time t = 0: it Istataeus Cmplex Particle Velcity Amplitude, r, i, t, ta ut, ut, u t u t iu t u i r Imagiary u p Amplitude f Sie-Wave Fucti Geeratr Sigal Real Istataeus Cmplex Pressure Amplitude, r, i, p t p t ip t p t, ta pi t, pr t, Real Fr small amplitudes, the istataeus cmplex pressure ad istataeus 3-D cmplex vectr particle velcity are related t each ther via Euler s equati fr cmpressible, iviscid fluid flw {iviscid fluid flw meas that ay/all viscus/dissipative frces iertial frces}: ur, t t p r, t where = mass vlume desity f the fluid ad, u r t ur, t 0 3 is assumed. Fr {be-dry} air at NTP,.04g m. Fr the SWT with -D lgitudial particle velcity measuremet, the crrespdig -D Euler s equati fr plae waves prpagatig u, t t p, t. he SWT reduces t -- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved.

3 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets The ttal istataeus cmplex pressure at the psiti assciated with the presece f a stadig plae wave he SWT is the istataeus liear superpsiti f a (verall) right-prpagatig cmplex travelig plae wave ad a (verall) left-prpagatig travelig plae wave:, i i p t A e B e e The verall istataeus cmplex pressure amplitude is i fact a liear superpsiti f a ifiite umber f idividual right- ad left-mvig cmplex travelig pressure waves with cmplex amplitudes a ad b, 0,,,3... respectively, each f which are assciated with the sie-wave sigal utput frm the acustical trasducer (lcated at = 0) at times earlier tha t = 0. Thus, mathematically the cmplex amplitudes assciated with right- ad left-mvig cmplex amplitudes ca each be represeted by the ifiite series A a ad B b. 0 0 Bth f these series represetatis ca be represeted graphically via phasr diagrams he cmplex plae, e.g. fr A a as shw he figure belw fr t = 0: 0 Imagiary a A a A 0 a a a a 3 Real ad Precisely a resace f the SWT, the idividual cmplex amplitudes a b assciated with the idividual right- ad left-mvig travelig waves respectively, are perfectly i phase with each ther, i.e. all f the idividual relative phases 0 0 ad thus the verall phases 0 ad A 0 a B b 0 phasr diagrams fr A a ad B b 0 a, 0, graphically crrespdig t straight-lie. -3- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved. 0 b

4 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets Explicitly writig ut the cmplex pressure amplitudes A a ad B b 0 assciated with the verall right- ad left-mvig cmplex plae waves: i 0 i i i 3 i 0 i i i 3 i A Ae Ae Ae Ae A e e e e A e Ad: i0 i i i3 i0 i 3 i... i i... 0 B Ae Ae Ae Ae A e e e e A e A The multiplicative phase factr e assciated with the th term i each f the tw ifiite series arises frm the fact that each such ctributig wave had t rigiate at a earlier time, t 0 i i rder fr all such waves t arrive simultaeusly at the = psiti at the time t = t. Nte that sice e rtates cmplex quatities CCW he cmplex plae as the time t icreases, the sig he i argumet f the e phase factr assciated with waves arrivig at the = psiti at the time t = t frm the earlier time t 0 must be egative. Sice the elapsed time fr rud trips f right- r left-mvig waves prpagatig he SWT is t tt L v L v, the the cmplex phase shift assciated with rud trips f right- r left-mvig waves prpagatig he SWT is t L v i i L i i L L il L ad thus e e e e e. Thus: L il 4 L 4iL 6 L 6iL A A... L il Ae e Ae e Ae e A e e B Nte that sice the ed walls f the SWT (lcated at = 0 ad = L respectively) are assumed t be ifiitely rigid, we have tacitly/implicitly assumed that additial phase shift(s) f the right-/ left-mvig travelig waves ccurs up reflecti at the ed walls. If such reflecti-iduced phase i shifts were t ccur, the additial phase factrs e i L ad e wuld eed t be icluded he abve expressis i rder t explicitly tae it accut/prperly mathematically describe geeral/geeric phase shifts assciated with reflecti f the idividual right- ad left-mvig plae waves at {-perfectly rigid} ed-walls f the SWT. Is it pssible t btai a aalytic, clsed-frm expressi fr the ifiite series assciated with the cmplex amplitudes B? The aswer is a mst defiite yes! A ad Defiig t L 0 ad x L, ad tig that t ix t t e e e cs xi e si x the aalytic/clsed-frm expressis fr the tw series the RHS f this relati, fr t > 0 are [] : t six e si x 0 csh t cs x ad: t siht e cs x 0 csh t cs x. -4- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved.

5 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets Thus: t t t ix sih t si x e cs xi e si x e e i csh tcs x csh t cs x Hece the aalytic/clsed-frm expressi fr cmplex A B is: L L sih si L il A = B A e e A i 0 csh L cs L csh L cs L is: Thus, the verall istataeus cmplex pressure amplitude p t,, cs p t A e B e e A e e e A e e sih L si L Ae cs i e csh L cs L csh L cs L i i it i i it it Nte that the physics assciated with stadig plae acustic waves iside a SWT is similar t that assciated with stadig plae electrmagetic/visible light waves iside a Fabry-Pert etal with semi-trasparet/partially-silvered ad/r dielectric-cated plae-parallel mirrrs! it Defiig: F L L sih si i csh L cs L csh L cs L ca be writte as:, F cs The p t, p t A e e The -D cmplex particle velcity is f the geeral frm, cmplex pressure p t, via the -D Euler equati:,, u t p t t u t i A e e Thus:, F cs si Sice v this relati ca als be writte as: u, t i A F e cssi e v Fr iviscid fluid flw, te that r equivaletly, that. it u t u e ad is related t the it it -5- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved.

6 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets The cmplex lgitudial particle displacemet {frm its mial equilibrium psiti} t, related t the cmplex lgitudial particle velcity u, tvia u, t, t t. t, A e cs si e v Thus: F it is The cmplex specific acustic lgitudial impedace f the SWT tube at the psiti is defied as the rati f cmplex pressure t cmplex lgitudial particle velcity: it p, t p e p cs i v u, t u e u cs si Nte that the cmplex specific acustic impedace f the SWT is purely imagiary, ad is als a time-idepedet quatity, sice p t, ad u, t have the same time-depedece factr e. The SI uits f cmplex specific acustic lgitudial impedace are Pa-s m g s -m, als w simply as acustic hms, als w as Rayls (i hr f Lrd Rayleigh). The -D cmplex lgitudial acustic itesity I at the psiti is defied as: A I p, t u, t p u i F e cs cs si v The cmplex lgitudial acustic itesity I he SWT is purely imagiary ad is als a time-idepedet quatity. The SI uits f cmplex acustic itesity I are Watts m. The time-averaged ttal acustic eergy desity tt a the idividual time-averaged acustic ptetial eergy desity acustic ietic eergy desity i a w : w at the psiti is the additive sum f ptl a w ad the time-averaged, tt ptl i p t p a a a, 4 v 4 4 v 4 w w w u t u The time-averaged eergy desities are purely real, time-idepedet quatities. The SI uits f 3 eergy desity are Jules m. -6- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved.

7 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets Precisely at e f the resat frequecies f the SWT f v v L, with bth f the p ad u mics lcated e.g. at = L, the L L, ad hece L L, si Lsi 0 ad si Lsi 0 ad thus the cs cs, cs cs istataeus verall cmplex pressure amplitude at = L the th resace f the SWT becmes: L L sih p L, t Ae e csh L it cshx Nw sice cth x tah x sih x ad usig the fact that [] csh x tah x, the we see that sih x sihx cth x ad thus if tah x csh x x L we see that the abve expressi fr p L, t the SWT resaces ca equivaletly be writte as: L p L, t Ae cth L e Thus, we see that there are pressure ati-des at bth = 0 ad = L the resaces f the SWT fr ifiitely rigid/clsed ed walls. The istataeus -D lgitudial particle velcity at = L the resaces f the SWT is: Agai, usig the relati cth x L sih L v L u L, ti Ae e csh sihx tah x csh x = L e f the resaces f the SWT ca be rewritte as: it the lgitudial particle velcity at L it u L, ti Ae cth L e v is 90 Nte that a resace f the SWT, u L, t. ut-f-phase relative t p L, t The istataeus -D lgitudial particle displacemet at = L a resace f the SWT is: L sih L v L L, t Ae e csh it -7- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved.

8 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets Agai, usig the relati cth x sihx tah x csh x = L e f the resaces f the SWT ca be rewritte as: L it L, t Ae cthle Thus we see that the resaces f the SWT, L, t u L, t ad is 80ut f phase relative t p L, t the lgitudial particle displacemet at is 90ut f phase relative t. I terms f a phasr diagram, the cmplex pressure p L, t velcityu L, t ad cmplex lgitudial displacemet,, cmplex lgitudial particle L t the resaces f the SWT as bserved at = L ad at time t = 0 are rieted as shw he figure belw: Imagiary p u dd dd eve p u.b. All e +it phasrs rtate CCW i lc-step with icreasig time, t eve eve dd Real The cmplex specific lgitudial acustic impedace at = L the resaces f the SWT is purely imagiary ad time-idepedet: p L, t L i v u L, t The cmplex lgitudial itesity at = L the resaces f the SWT is als purely imagiary ad time-idepedet: A L I L p cth L u L i e L v -8- Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved.

9 UIUC Physics 93POM/Physics 406POM The Physics f Music/Physics f Musical Istrumets Nte that sice there are acustic stadig waves preset he SWT, the cmplex lgitudial sud itesity I must be purely reactive (i.e. purely imagiary). A -er value assciated with the real cmpet f I is due t a actual {time-averaged} flux, r flw f eergy dw the SWT this cat be due t a stadig wave ly a travelig sud wave ca have this! The time-averaged ttal eergy desity at the psiti = L a resace f the SWT is: p L, t w L w L w L u L, t tt ptl i a a a 4 v 4 A L A L e cth L e cth L v v 4 4 A cth 4 v tt ptl i L Or: w L w L w L e L a a a Nte that sice ptl a i w L w L. a fr iviscid fluid flw, the resaces f the SWT we see that Refereces: [] Table f Itegrals, Series ad Prducts, I.S. Gradshtey ad I.M. Ryhi, page 4, Academic Press, Ic I this referece, please te that the expressi t si x e si x 0 csh t cs x is factually i errr he d ad 3 rd quadrats (where cs x 0 ). The crrect expressi, valid i all fur quadrats is: t si x e si x 0 csh t cs x [] ibid, page Prfessr Steve Errede, Departmet f Physics, Uiversity f Illiis at Urbaa-Champaig, Illiis All rights reserved.