Acoustic Wave. Hongbin Ju Acoustic Wave Equations. u 0. (4)

Transcription

1 Acouic Wv Hogbi Ju Th dicuio of diff mod d hi icio ovid imo ifomio o oud ouc. Oc h oud ouc dmid h k i o obi h oud fild b olvig h couic wv quio. I hi ch w ud how o olv h oud fild i o c. Soud i duc i o b dicud i ch. Th oud fild fom ulig h i h-dimiol 3- c i fi ivigd. Th coc of moool diol d quduol ioducd. Bd o h moool oud oluio o of h mo imo chiqu o olv couic quio i ioducd: h 3- fucio d h foml igl oluio. Uig h igl oluio h oud fild fom movig ouc cill oig ouc i udid. Th wo-dimiol - fucio i lo bifl dicud. Acouic Wv Equio Fo h couic mod i uifom idl flow: u u S 3 u. Th h ioiol ioic li Eul quio. ibl wih ubci m flow quii d u. All oh vibl ubio vibl. loci Poil d Acouic Wv Equio Th mo imo coquc of Eq. u i h ioducio of h vloci oil. Show i Fig. i cool volum boudd b i ufc S i d ou ufc S o. C i clod coou i. C i ducibl if i c hik o oi wihou hvig o co boudi. I hikig h fom ufc S c. If i coou i ducibl h gio i igl-cocd. Ohwi i i mulil-cocd. If h volum i Fig. i h-dimiol d h dimio of h i bod fii h if igl-cocd. O h oh hd if h volum i wo-dimiol h i

2 bod d o ifii i h hid dicio. A coou cloig h bod i o ducibl; hfo i mulil-cocd. Thi gum how h diffc of h mhmicl m wh dlig wih wo-dimiol d h-dimiol oblm. Fig. Cool volum boudd b i ufc S i d ou ufc S o. Suo i igl-cocd gio h wo oi O d P d wo h C d C oiig h wo oi. C d C fom clod coou C C C d h i ufc S c wih coou C h boud. B lig h Sok hom w kow h ciculio oud h coou i zo ic u. Thfo u dl u dl. 5 C C Th m h li igio i ol fucio of h oiio of O d P. I do o dd o h h. Thfo w c dfi fucio of oiio : P O u dl. 6 Wh P och O dl O w hv: dl dl O O u dl.! C A dl dl O u dl h u. 7

3 i h vloci oil fucio. Acouic vloci i h gdi of h vloci oil. I mulil-cocd gio coou C C C m o b ducibl. Ud hi cicumc h Sok Thom do l d ciculio oud C m o b zo v fo u. Th u dl u dl. C C Th li igio dd o h h goig fom O o P. loci oil dfid b Eq.6 h mulil vlu oi ddig o how m oud h h go. i o log fucio of oiio ic i m hv mulil vlu h m oiio. I hi c ificil bi uch S b i Fig. mu b id o mk h gio igl-cocd. Fig. Aificil bi i mulil-cocd cool volum. Th bfi of ioducig h vloci oil i wofold: h umb of vibl i ducd d h ioiol quim i uomicll ifid ic. O didvg i h mhmicll h vloci oil qui high od mooh h h vloci ilf. I i h gdi of vloci oil o h oil ilf h h hicl mig. Sic i dfid b h il igl [Eq.6] fucio of im * ddd o do o ffc h quio d li. Wih h wl ioducd vloci oil Eq. d c b wi : 3

4 8 9 Aum h couic mdium i uifom. Th h diviv o h lf id of Momum quio Eq.9 c b ichgd. Eq.9 c b ducd o: i.. f f d wh f i fucio of im. Accodig o Eq.7 fucio of im ddd o do ffc h couic vloci. Thfo Eq. c b wi iml :. Subiuig i io coiui quio 9 w hv: wh. 3 u i u i u. i i Th h couic quio i uifom m flow. I i lo ifid b oh couic vibl uch. Th oivil oluio of o 3 i h couic wv. Th ivil oluio o cood o voici wv o o wv. Th o ouc i li Eul quio 3. Th coodig couic quio 3 o i homogou il diffil quio. I dcib h ogio of oud wv i uifom flow. A oi i h mdium mu b zo. Ohwi h oud ouc hi oi. Th

5 ouc fom l cio uch flow icio o fom cig of oh mod du o ouifomi i h flow o fom oli icio bw diff mod. Soluio Uiqu d Boud Codiio I igl cocd gio Eq.3 h uiqu oluio if h oml vloci o ll h boud ufc S i d S o i Fig. cloig h of i i cibd ch7.doc. Th uiqu of h oluio c b ovd b h g igl mhod Pic989.7 Bchlo Ch.7&.8. If h ouid ufc S o i ifii h oud fild vih ccodig o h culi quim. Culi m h i o oud bfo i ch h obv. If h gio i mulil cocd w fi mk i igl cocd b iig bi i Fig.. Th h oluio i uiqu wh h oml vloci diibuio i cibd o ll ufc: S i S o d bi S b. Pcibig oml vloci o S b i quivl o h flu co h ificil bi o ciculio oud h i bod. Wh i h coc ciculio fo icul oblm? Th modl w bg wih [Eq.] um ivicid mdium. If h i bod ufc i mooh zo ciculio i of umd Mo&Igd968.. If h i h dg o h i ufc vloci h h dg go o ifii if h mdium i ivicid. I hi c vicoi c o b glcd h h dg. o volv d hd h h dg du o h ffc of vicoi; h ciculio i iducd oud h i boud. Thfo focig h Ku codiio h h dg i h w o h c ciculio du o h vicou ffc. Bu hi codiio c ol b lid oc o h dg uch h ilig dg of ifoil. icoiuiifii i llowd oh h dg uch h ldig dg of h ifoil. Pcibig oml vloci i ol uffici o c boud codiio. A uiqu oluio c lo b dd if vloci oil o u i o h ufc. Acull li combiio of h oml vloci d u c b giv h boud ufc. Thi i h imdc boud codiio which will b dicud i oh ch. I o c h ou ufc S o i ifii. I umicl l i i imoibl o h comuio domi o ifii. Uull h boud i lcd f w fom h ouc gio d oim quio ud o u culi uch h Sommfld diio Boud quio: lim fo - io mdium lim fo 3- io mdium 5 o h diio boud codiio coidig m flow b Tm: 5

6 M co M i. 6 I i od h Eq.6 i diff fom wh M=. Th boud codiio fomll mk o gio fii. Soud d b ibig Sh Soud d b dill ibig Sh i Sio Mdium M Flucuio H w will dicu h oud fild i io mdium gd b l ouc: vibig h. Coid h wih diu. Th h ufc vib dill wih uifom mliud d h. Wh h vibio mliud i mll h boud codiio c b d b uifom vloci h omil ufc: u v. 7 Accodig o h uiqu of oluio dicud i h viou cio h oud fild i uiqul dmid wih h boud codiio i 7. I h hicl coodi m h gdi d h Llci civl: i i i i i i i. Th boud vloci i 7 d h oud fild hicll mmic. Thfo wv quio d c b imlifid: Th gl oluio h d Almb oluio of 8 i: 8. 9 F E. 6

7 F d E wo bi fucio ig civl h ou-goig d i-comig hicl wv. Th quim of culi clud fucio E. Thfo h oluio i: wh F F F u u F 3 u df F. d Th oud ouc i h mchicl vibio of h h. Th volum of h h chg i ufc vib. I i quivl o h icio of fluid io h mdium. Th of h icd volum i Subiuig io boud codiio 7 w hv v. F F. 5 Th fi od li odi diffil quio OE 5 c b olvd uig h mhod of io of vibl d h mhod of viio of m c.f. Adi. Suo h i o vibio h h oluio i: Thfo h i vloci oil i: F ' ' d'. 6 d ; 7 d h oud u: d. 8 i h dd im h ufc oi clo o h obv. 7 d 8 giv h oud fild gd b vibig h. Th l wh log. Th i o hicl d mhmicl mig fo. Th foc of h h cig o h fluid i zo ic h u i hicll mmic. 7

8 Moool Th ouc i comc wh i dimio i mll comd wih oud wvlgh i... I i uful o ivig h oud fild fo comc ouc. W um h mliud of volum icio i co wh h h hik io oi. Pfomig igio b w hv. d d No h d. 9 Thi oimio i v imo d will b ud dl hf. Al 9 i 7 d 8 h 3 u u 3 3 u wh. Eq.33 dcib h oud fild fom comc ouc i.. ouc ulig low fquci. Acull h l fo fquc ic h ouc gio i ducd o cocul oi. Thi ouc i clld moool. Th dicivi of h oud fild fom moool i omidiciol. Accodig o 3 h couic vloci h wo diff chciic i wo gio. I h f fild h domi ffc i fom h cod m o h igh hd id of Eq.3: h volum icio o h cclio of h vibig ufc. Th cclio i blcd b couic u [Eq.3]. Th f fild i h oud fild. I h fild h domi ffc i fom h volum icio i 8

9 h fi m. loci i h fild i much lg h h couic u. Acouic u i high od qui. Th o fo h og vloci oud h ouc i h w k co wh hikig h ouc o m h boud codiio Eq.. W cll hi kid of ouc h vloci div. loci co b blcd b oud u i h ouc gio wh Eq&3 duc o u d. Th flow i l icomibl i hi gio. No voici wv i uifom flow uo o flucuig u. I i oibl hi m voici wv? I c b how h h cul of vloci i lo zo. Thfo h m o h igh hd id of Eq.3 oloidl ioiol vloci fild. Thi gio i of fd h oil fild of h bod. Th oil fild dc f w fom h bod. I h ffc o oh bod ol wh h i oimi. Th Nohomogou Acouic Equio Acouic quio uch 3 d oluio 33 vlid vwh c h ouc oi. Now w o wi h wv quio h lo fomll li hi ouc oi. Th wv quio hould hv hi fom: S 33 wih oluio:. 3 i h ouc oiio. Th ouc fucio S i o gul fucio. I c ol b dfid i h igl. Th igio of i h h do iclud h ouc oi i zo. If h h wih diu i cd h ouc oi fom h u ivgc Thom w hv: S 3 d d d iddd d d d i idd 3 d d. 35 Alhough h igd i h volum igio i igul h ouc oi i i igbl. A 9

10 S. 36 Th ihomogou wv quio i: 37 Th oud ouc i h im diviv of m icio. I i od h i h ihomogou quio h il coodi i o h fild o obv coodi. i h ouc coodi. I i m o h il vibl. o wih c o o. Suoiio li o li quio 37. Aum h diibuio of volum icio h h ihomogou quio d i oluio : 38 d. 39 No i i h il im divio o h whol im diviv of h ouc i 38 d 39. Boh h quio d h oluio l i h whol fild. Soud d b Tvl Ocillig Sh i Sio Mdium Momum Flucuio d iol Coid igid h vl ocillig log h z-i. Th ocillio d of h h c i v c i z dicio. Th volum of h h do chg. Th ol volum of h mdium co im. So i i o moool. Th oud i gd b h lio of h h. I i h hick oi i ubomchi. Th oblm c b d movig ouc oblm h dil i h followig cio of hi ch. Howv if ocillio d v c i mll o fi od i i quivl o vibig h wih fid c d h d h ufc: u vc co. No h co ddc of h ufc d comd o h uifom ufc d i h moool 7. Boud codiio d h oud fild o log hicll mmic. Id h immic bou h z-i. Th dd mhod o olv h oud fild

11 fom bi vibio of h i h io of vibl. Mo&Igd Fo li oblm lik hi w c iml u h mhod of uoiio. Suo w hv wo dill vibig h o z h wih of volum icio h oh z h wih of volum icio. Th ol of h volum icio i zo. Thi diffc i. Accodig o 8 h oud u fom h wo ouc i: d. d d civl h dic fom h obvio oi o h u d low h c: hco h hco h. A h w k mliud of h co h fom Eq. d 9 co h co h d h h h d h. 3 Th dil vloci c b obid b: u d'. Fom 3 d w c h co ddc of h oud fild. Th oud fild h wo lob wih og oud i h z dicio. Th wo lob hv h m mgiud bu 8 ou of h. Th foc of h h cig o h mdium i i z dicio:

12 F z co i d d h. 5 3 A Eq.3 bcom: h co. 6 Comd wih h moool oluio 3 6 how h uod oud fild of wo moool wih ooi gh. Thi i clld diol. I gh i h. To mi h hicl mig of h diol oud fild 6 w d h volum ouc z h fo mll h: h co O h. 7 d d O h No: d d. Th ouc z h c b umd o b z lu cocio. Th cocio i du o h chg of dic hco d h chg of h dd im h co wh h ouc i oiiod. Ali h imil io o h ouc z h. If w igo h diffc of i mliud h um of h wo ouc giv h fi m i 6. Thi m h oud ouc du o h diffc of h dd im of h wo ouc. O h oh hd if w igo h diffc of h dd im d coc o h diffc i h w obi h cod m of oluio 6. Thi m i mll i h ffild. I h ffild h oud i mil gd b h dd im ffc of h wo moool. W c wi oluio 6 i h fom wih h il diviv: d hco d. 8 Th il diviv i o h fild coodi o h ouc coodi. Bcu of h dio of h oud ogio fom h ouc o h obv h oud fild obvio im h il diibuio. Th gdi of hi il diibuio h obvio oi i h dicio of h diol dmi h oud u hi oi. Th gdi iclud wo ffc: h dd im diffc d h ogio dic i mliud. I h f fild h dd im ffc i domi; hfo o m k ou of h il diviv i 8.

13 I m of F z h wv oluio fo h diol i: 3co d Fz d. 9 Soluio 6 c lo b dicl divd b fomig h Tlo io o moool oluio 3. To gliz h oluio uo h diol i iud wih h io vco h h 5 h h. To fi od Tlo io w hv: h. 5 I Ci coodi h oluio h hi fom: h. 5 5 d 5 u glizd 8. Th oud h obvio oi i dmid b h gdi of h moool oud fild i h dicio of h diol. To obi h ihomogou wv quio fo h diol o m m o ig ov h cd h ouc i h moool c. Bu h igio i zo ic h i o flow fom h wo moool. Th coc w i o bgi wih h ihomogou wv quio fo moool 37: h h h. 53 uduol Momum Flu Flucuio 3

14 Coid wo diol wih h m gh h bu i ooi dicio. Th dic bw h wo diol i q cd i Fig.3. Accodig o 5 h uod oud u i: h 5 q q. q Fig.3 A quduol. To fi od h Tlo io of 5 i: h q I Ci coodi w c wi h oluio :. 55 h q i. 56 i h d q c b i dicio. Th wo cil c. If h d q lll h quduol i logiudil. A logiudil quduol h wo lob i oud fild. Comd wih diol h wo lob fom logiudil quduol i h id of ou of h. If h d q dicul h quduol i ll. Th fou lob i h oud fild of quduol. 55 c b wi o: h q h q. 57 A quduol c b dcomod io wo logiudil quduol. Logiudil quduol i h bic of ll quduol.

15 5 Th ihomogou quio fo quduol i: q h q q h. 58 I umm oud ouc c b d b uoiio of moool. Wh moool clo o ch oh modl uch diol o quduol mo oi. A quduol c lw b d io wo logiudil quduol. Logiudil quduol i h bic of quduol. Muliol Eio iol i comod of wo moool wih qul gh bu ooi ig whil quduol i comod of fou moool wih qul gh bu ooi ig. Moool diol quduol c. h bic of oud ouc. If ouc gio i comc comd o h oud wvlgh ouc c b d b h um of h bic ouc. L bgi wih h oud fild fom wo moool: o wih gh h z d h oh wih z h. Accodig o 3 h ol oud u i: co h O d d h. 59 To od of h O h couic fild i gd b moool wih gh ] [ hi gomic c d diol wih gh ] [ h. Accodig o 37 h ihomogou quio i: ] [ ] [ O h h h h. 6

16 6 Suo h N moool wih gh N. Th ol oud u i: N. 6 Aum h ouc gio i mll i ll dimio comd wih h oud wv lgh d i h gomic c of h ouc gio. Th oud u fom h - h moool c b dd bou :!. 6. Thfo h ol oud u fom ll h moool i: N N N!. 63 To h zo-h od h N moool c b d b o moool wih h ol gh of ll h moool. Th diol m o hvig quivl diol i 6 wh c b cho i h middl of h wo moool. If ll h moool ocill h m fquc ˆ l i h

17 7 ˆ! ˆ ˆ i N i N i N. 6 Th h oud ouc c b d b o moool o diol d o quduol c. Th quivl moool h h gh of h um of ll h moool. Th diol i i dicio of h vco um of ll h moool. Fucio i Th imiol O Sc wih No Flow W hv dicud h moool d i high od diviv. Th h of oud ouc i flow: volum flucuio foc d vicou ocillio. Th c b civl modld moool diol d quduol. Sic high od ol c b coucd b moool i i il o olv h couic fild of moool. Th oluio o h couic wv quio of moool wih ui gh i clld h fucio. Sig ouc gh ui volum ul wih ui gh miig oud i h ihomogou couic quio 37 d i oluio 3 w hv: i h fucio i io mdium i h o c. I h oud h obvio oi d im gd b ul ouc oi ld im. Th fucio i Eq.66 i i Ci coodi. fucio i clidicl coodi d i hicl coodi lo uful i licio d will b dicud l. Oo h ubci o licil idic h h lf id of h quio i od o h obvio coodi. Th fucio i of wi o licil idic h fild obv coodi d im d h ouc coodi d miig im. Thi i v imo i h followig co.

18 A o obi h h-dimiol Llci quio d i fucio: If h mdium mov co vloci u ihomogou wv quio: ifi h mo gl. 697 Obvioul h fucio ifig 7 i diff fom 66. Th fucio h h imo oi i6 fo oof: Culi: h i o oud bfo h ouc g i ld: wh. 7 cioci:. 7 I hi icul c uifom mdium h fucio d i doi h m fucio. I i lf-doi o mmic. I gl h doi fucio i diff fom i oigil: uch i h h flow i Tm& Auiul lo ifi: i igul wih od O h ouc oi. Igl io of Acouic Equio: Kichhoff Fomul 8

19 I h viou cio w hv dicud h iml ouc d h fucio. Th oud fild fom coiuoul diibud moool w bifl miod i Eq.38 d 39. H w will ud diibud ouc i mo dil. Suo w hv coiuoul diibud ouc fild q i h movig mdium. Th ihomogou couic wv quio i [f.3&38]: q o 75 q. 76 Wih h fucio h il diffil couic wv quio c b fomd io igl couic wv quio. Th igio will b od o ouc coodi d im. Th fucio ifig 7 id of Eq.7 will b ud. A i Fig. cool volum i boudd b ufc S icludig i ufc S i d ou ufc S o. Th ufc c b bi wh o boud codiio focd. Subc Eq.7 mulilid b fom Eq.76 mulilid b d ig hi quio ov volum fom im o + + m icluiv of : q d d d d. 77 Accodig o Scod Idi: dsd. 78 dd S Soud u w cho h couic vibl i Eq.75&76. i iglvlud fucio of oiio o m if h gio i igl-cocd o mulilcocd. If vloci oil i cho h couic vibl o coc i h i igl-vlud ol i igl-cocd gio. If h gio i mulil-cocd ificil bi S b hould b id o mk h gio igl-cocd i Fig. d h ufc igio i 78 hould iclud S b. Howv h oml dicio o h wo id of h ufc hv ooi dicio; h igio o h bi i zo. Thfo quio 78 li fo wll. 9

20 To ig iml c. d d i 77 w bgi wih h Sio Cool olum i Sio Mdium Coid io cool volum liv o h obv wihou m flow i... Sic do v wih im h igio od c b chgd: dd d d. d Th igio i zo quid b h culi of h fucio 7. Th w obi h fmou Kichhoff fomul i io volum wihou m flow: S q dd dsd d if i i volum d o hufc; if i ouid of volum. 79 Th Kichhoff fomul lo clld h Kichhoff-Hlmholz Igl Thom w fi ublihd i 88. Alhough h fomul i d o oud u i hold fo oh couic vibl. Th fi m o h lf id i h oud fild fom h diibud ouc i h volum if h w o ufc S i d S o. I hi c fucio 66 li d h Kichhoff fomul duc o 39 h w bd o h icil of uoiio. Th cod m o h lf id of 79 h ffc of h hicl boud ufc o h oud fild fcio flcio c. o h coibuio of h ouc fom ouid of h ufc. Th l m o h lf id i h ffc of h iiil codiio which i zo if o oud i iiill. If i o h ufc h volum igl bou h l fucio i 77 i udfid. A limi li o 79 will b fomd o div fomul licbl fo o h ufc.

21 Th il diffil quio 76 dcib h liohi of ouc d h oud i. O h oh hd Kichhoff fomul 79 coc h ouc gio o i f fild oud fild. I icoo i igl quio h ffc of h ouc h hicl boud o bi mbl ufc d h iiil codiio. ll h Kichhoff fomul i o oluio. Th ukow d i h volum igl d i h ufc igl i 77. Accodig o h oluio uiqu hom d co b cibd iddl o h ufc. O mu b olvd wh h oh i cibd. Sio Cool olum wih Uifom M Flow Th wo w o iclud uifom m flow u io h Kichhoff fomul. Th fi i o follow h m ocdu fo 79 oldi976. No u. W hv h Kichhoff' fomul fo io volum i uifom m flow: q dd u u dsd Si So. 8 d if i i h volum d o h ufc; if i o i h volum. No h m i h ufc igl comd o h o-flow Kichhoff fomul Eq.79. Th cod mhod i o mk u of h li fomio of h coodi. Wihou lo of gli w blih coodi wih h -i lll o h m flow h wv quio Eq.75 i: u u Th iml li fomio i h lil fomio: q. 8

22 u Wih 8 8 c b fomd o quio i io flow. All h l fo h io mdium l. Howv h o of h oucobv chg f h fomio. A io ouc i h coodi m mov i h oh coodi m. Th Loz- fomio c b ud o void i: 3 3 d u 83 wh M M u. I hi m h dimio i dicio i dild i uboic flow whil h im i comd d hifd. Th mou of h im hif vi wih. Wih hi fomio couic quio 75 d h fucio quio bcom: q 3 u [No:.] Acouic quio 8 h h m fom i io mdium d h o of h oucobv do chg. Eq.8 i ow widl ud i licio. Moio Tfomio 83 i lo clld h Pdl-lu Tfomio ic h couic quio i ducd o h Pdl-lu quio fo comibl d flow: M. 3 Somim i i fd o h Km-Ti Tfomio. Movig Cool olum wih Uifom M Flow Suo cool ufc S mov wih im. Accodig o h Libiz ul d d d d wh i h movig vloci of h ufc. Th w hv S ds

23 3 S S S dsd u d dsd u dsd d d d d dd u dd dd Th Kichhoff' fomul fo movig ufc i m flow i. i o i h volum if o h ufc; d ii h volum if d dsd u u dd q S 86 Thi fomul duc o 79 o 8 wihou m flowmovig ufc. I i fo gl ouc. To l i fo ouc fom flow i ocouic modificiodivio dd uch oldi vio d h FW-H quio. Alicio of Kichhoff Fomul A w hv miod h Kichhoff' fomul h igl io of h diffil couic quio. ll h o h oluio. Th i ukow vibl i h ufc igl of 79 8 d 86: h ufc igio i could wih h oud fild. Th volum igio m lo iclud h ukow vibl if h i og icio bw h ouc d h oud. If h i o obc i io mdium d h oud h o bck ffc o i ouc q 79 duc o: dd q. 87

24 I hi c h Kichhoff' fomul i h oluio. Th oud fild c b clculd b hi fomul uig h o-c fucio 66 i h Ci coodi i h o c. A ml i h oi dicio. Noi fom ubulc i modld b h Lighhill Acouic Alog. Th Lighhill i umd kow d h oud i g um o bck ffc. I om iuio h coulig of h ouc d i gd oud o h ufc c b limid. If h ufc i igid h o-io boud codiio mu b ifid. Th S dsd. If h fucio lo ifi h o-io boud codiio h ufc h S S i dsd o. Eq. 79 duc o Eq.87: h ufc coulig i limid d Eq.87 i h oluio. Bu h fucio i Eq.87 mu if h boud codiio h ufc. Chck ou Mo&Igd fo how o dvlo h fucio i duc uig h oml mod. Chck ou How998:.6.65 fo h comc fucio which giv h ldig od m fo h oud oducd b ouc olid bod. I gl i i difficul o fid fucio ifig h boud codiio h ufc. If h i o volum ouc i io mdium Eq.79 duc o: dsd. 88 S Th ufc h wo oibl ffc o h oud fild. O i h diffcio: h oud og owd h ufc d i diffcd. Th oh i h obc ufc c oud ouc. No m which of ffc giv d wh h i m flow o h bod ufc h oud fild c b clculd fom Eq.88. Howv ccodig o h oluio uiqu hom o d uiqu oluio ol o of d o li combiio of h wo c b cibd o of h ufc. Th co b cibd imuloul. Wh o i cibd h oh h o b olvd. Thfo ih o o h ufc i ukow d d o b olvd.

25 5 To olv h igl quio 88 l' u h obvio oi o h ufc i Fig.. Wh h ouc oi och h obvio oi h fucio i igul. W will how h igl mi fii d i h um of h icil vlu of h igio d. b Fig. Obv o h ufc. Fi w dfom h ufc h obv i Fig.b. Th ufc h obv i uhd io h obc o fom wo w ufc: S d g. Ag7 S i h oigil ufc S cludig mll oud ufc wih diu. g i hlf h ufc wih diu. Th obv i i h w volum o Eq.88 li: g S d ds dsd '. 89 Alig h f-c fucio 66 h cod igl i 89 i: g g g g g g g d d d d d d dsd dsd ds dsd dsd dsd i i i 3. Suo h obv li o mooh of S wh i mooh. A

26 Pluggig i io 89 w hv dsd g. dsd S '. 9 Th igl i ov h oigil ufc S cludig h igul oi h obv. I i clld h icil igl. Thi how h igl ov S i h um of i icil vlu d. Eq.9 i h iml: S ' dsd o S. 9 Thi i h boud ufc igl quio o oud u. Cigho.l Pic989.8 Af i i olvd h oud fild c b comud fom 88. Th ufc c b dicizd o Eq.9 i olvd umicll. Thi i h boud lm mhod BEM. Log-BEM.df BEM5.df If h i ufc i mooh modl wihou ciculio li. I hi modl h lif o h bod i zo. If h ufc h h dg uch i ifoil h modl wihou ciculio will giv iguli h dg. I hi c h Ku codiio mu b focd o mov h iguli which i quivl o dd ciculio d lif o h mdium d h bod. Oh mhod iclud CF o CAA mhod. O ml i h oud fom. A ificil ufc c b u oud h f ough o h h li couic quio hold ouid h ufc. Th flow fild wihi h ufc i kow fom oh mhod uch im o umicl oluio. Soud Fild fom Movig Souc I hi cio w will u Kichhoff fomul 87 o ivig h ffc of movig ouc i o c. Th fucio fo io ouc i io mdium i h Ci m 66 i ud i h li. A io ouc ocill bu i im vgd oiio do mov. Soud Fild fom Movig Souc wih Co loci 6

27 Th iml c i ll h ouc mov wih co vloci U. Eblih fm movig wih h ouc. Th movig fm i cho o coicid wih h fid fm :. A im h coodi of h ouc oiio i h fid fm i U. 9 Suo h gh of h ouc obvd i h movig fm i A. To fcili h lio w um ll h ouc ocill h m gul fquc co-fquc umio whil hi gh v wih oiio i.. i A l{ Aˆ }. 93 To l fucio 66 h ouc mu b dfid i h fid fm: q ˆ i A U A U. 9 Fig.5 Effc of movig ouc. Emiig h ouc i Eq.9 i h fid fm o m fid i o ol i ocill i bu i mliud lo vi i Aˆ U. Eio 9 fomll h wo ffc of h ouc: h ocillio d h covcio. Th hicl mig c b lid b Fig.5. A im h ouc i i h lliic gio how i h figu. L obv h oi mkd b. I coodi i h fid d movig fm d civl. Sho im d l h ouc gio d h mkd oi mov b dic U d o w oiio. Bu h fid obvio oi i h movig fm mov b dic U d. A h mliud of h ouc i Aˆ U. I vi ol wh A ˆ i o uifom. Th oud ouc i h fid fm i im dd v i i d i h movig fm wh i i ouifom. To g h oud h ouc gh i h fid fm mu v wih im. I i uful o ivig h im viio of h ouc fid oi. Th ouc 7

28 gh obvd o ol vi wih h ouc ocill bu lo vi du o h ouc moio how b h fi gum of A i Eq.9 hfo U q A A A U A. 95 Th im chgig of qui i h wo fm diff: Covioll w do h im im obvd i h movig fm obvd i h fid fm. Th. d h U. 96 h ouc ocillio. U i h ouc moio ffc. Th diffc bw d i h ouc moio ffc. I c b glcd ol wh ll of h h codiio ifid: ouc ocillio i o mll; h ouc mov lowl U ; d 3h ouc gh i fil uifom A. If ouc i d i h movig fm uch h d foc o oig bld h fi codiio i o m. Sic h ouc i o h bld ufc i il diibuio i highl ouifom hfo ih h 3 d codiio i ifid. A d foc g o oud if i do mov. Th movig ffc i h ol oud gio mchim i hi c. I hould b igod ud cicumc i ubomchi oi l. Th ffc of h movig ouc i ow mid. Subiuig ouc gh 9 d fucio 66 io h Kichhoff fomul 87 w hv: A U dd. 97 olum iclud ll h ouc i h fid fm. I occui h whol c d i o fucio of im. Th fi gum i h ouc gh flc h U 8

29 9 ffc of ouc moio. To hi ffc licil w m cov h il igl fom h fid fm o h movig fm : d d U A U d d U A U 98 Th igio od c b chgd i h l ic h whol c i o fucio of im. Fo h wo fucio f U A d g U w hv f gd f g' gdg f g'. 99 i h h oo of g i.. U. i h ouc miig im fo h oud chig h obv im. If h ouc movig d i uboic h i ol o oo =. If h d i uoic h wo oo =. I i o ighfowd o olv fom v fo h iml c of uifoml movig ouc. Wh h ouc do mov i h m dd im bfo. No M U U U U U U g ' M M co.

30 i h gl bw h ouc movig dicio d h ouc-obv vco h miio im. M U i h couic Mch umb o ouc Mch umb. I i o h cul Mch umb ic h ouc movig vloci U i o h flow vloci. M i h Mch umb owd h obv. Subiuig Eq.99 d Eq. io Eq.98 h fil io fo oud u fom movig ouc i im domi i: Fo hmoic ouc U A d. *5 ' M i Aˆ d. * ' M i h dic bw h obv d h ouc miio im. Fomll h oud fild fom movig ouc i h oud fild if h ouc i io dividd b M. Th moio h wo mo ffc. Th fi i h mliud fco M. I i oll du o h ouc moio d gd wh h il igio ov i h fid fm i covd o h ov i h movig fm. Th comli of h il igio ov i Eq.97 du o A U i livd i ic h fi gum i A i o log fucio of im. Howv h comli do di. I i u movd o h mliud fco M. Th oh mo ffc of moio i o h miio im which ffc mliud fco d M d mol imol ffc h h d h fquc. Fo d ouc A A M. Th i o oud ic do o v wih im. O h oh hd if h ouc mov h d M im dd d h oud i gd. Fo h hmoic ouc i h movig fm i 93 if h ouc do mov h oud mliud h obvio oi i Aˆ d. Th fquc i h oud fild i h m i h ouc: d d. Th ffc of h ouc moio o h oud mliud d h h ml d i. Th oud mliud i chgd hough M co d. Th oud civd b 3

31 obv i fo of h ouc co i og h h civd b obv bhid h ouc co. O h oh hd h ouc moio ffc o fquc i hough A. To how i l um h ouc moio d i uboic d i h f fild: U U O. 3 Th dd im i: M co. *5- i M co Accodig o Eq. h oud u h im fco. Th oud fquc mud i h fid fm i M co. Tim i h fid fm i comd hfo i fquc ic wh h ouc mov owd h obv. Thi i h ol ffc hough h d M i h ol fco. Wh M h ol fco d h oud fild bcom igul d cil m i dd. Th obvio im i comd i which c b lid fuh fom. Aum U h U M. Th M. 5 M Bcu of h ouc moio d o log qul which h lo b how b 96. Chgig of qui i cio im d miio im diff. Ecl kig fquc i h cuc of ocillio u o h m. Th qui h ocillio h co mliud. Mhmicll h Foui fom of igl giv Foui como wih co mliud. Thfo h viio of mliud du o d M co i lo chg h 3

32 fquc d giv h ol ffc. W m coid hi h ol ffc hough mliud. Th viio of mliud cood o Foui como diff fquci. Thfo h ffc of h ouc moio o oud mliud d h o cl d i. Th oud fild i i h um ov h oibl miio im fom. Fom ow o w will omi hi ummio fo covic. Soud Fild fom Abiil Movig Souc All l fo h oud fild fom uifoml movig ouc vlid fo biil movig ouc c h Aoimio U ' d'. 6 Soluio i coci d h cl hicl mig. I i h bi fo dvloig umicl mhod i im domi. f. o ch.doc fo dil. I c hdl b dicl ud i fquc domi wihou oimio. Two of oimio uull md. Th fi i o mliud fco M. M wh U i mll uch i Eq.-3 i Blk. Howv hi oimio c ol b md wh h oh wo of h h codiio dicud vioul m: h ouc ocillio i o mll d h ouc gh i fil uifom. Obvioul hi i o h c i ubomchi ic ouc o bld ufc ogl dicoiuou i c. M wh h igio i h fid fm i covd o h igio i h movig fm. I im-domi mhod igio imlmd i h hicl domi. I i covi o u h movig fm. Thfo M lo i h mhod. I fquc domi mhod oluio d i mod. Souc could wih mod h which c il cov bw h fid d h movig fm. Thfo fid fm i uull dod i fquc domi mhod d hdlig of M i voidd. Th cod of oimio i o h miio im d h dic. Mo of h comli of movig ouc oi dicio com fom fidig h miio im i fom h cio im. Thi i o v fo ouc movig co vloci. O w o g oud i i o comu h oud fild i ouc im id of cio im. c b dicl comud fom b. Thi g i licbl i im-domi umicl imulio. f. ch.doc fo im domi 3

33 mhod I l h cio im i fd. I i o voidbl o fid h miio im. Fig.6 Aoimio of b ' fo fidig h dd im. dig o h codiio h wo oimio fo d. I h f fild c b oimd b Eq.3 d h miio im b. I i of ud fo high-d oo. Fo lowl movig ouc. h dd im i i h h i igo d i Eq. i lcd b ' how i Fig.6: ' fom which h dd im c b obid dil: No ' '. 7 " "co " U M ' M If M h o of h oimio i: M co. ' M co. I i uifid fo lowl movig ouc. Thi oimio i widl ud i low d ubomchi oi l. Aoh oimio i of md o mliud fco. I h f fild h movig ffc o i of igod. A co bw h obv d h gomic c of h ouc gio c b ud. 33

34 Soud Fild fom oig Souc I hi cio w will l i h ivigio of oud fild gd b oig ouc. I i covi o u h clidicl coodi how b Fig.7 i h fm movig wih h ouc: z 8 d i h fid fm: z. 9 Fig.7 Clidicl coodi. Th ouc o gul d lio bw h wo fm i: o h z l. Th clidicl coodi. 9 A d h m. Igoig h movig ffc i mliud Sic h movig ffc i igod i h mliud hi li ol li fo ouc ocillig high fquc i h movig fm. Th oud u obvio oi wih clidicl coodi o o zo c b vlud b Eq.. Th fi oimio w mk i i h f fild o oim i h mliud b. Wh h oig d i mll M i m ul o glc h movig ffc i h mliud i.. M i Blk c... Th i h f fild 3

35 35 ' d A. Wih M umd i h ouc h m b oimd b ' how i Fig.6 o h h dd im c b obid fom Eq.7 i which co i co ] i i [ ] co co [ ' o o o o o o o o z. i h gl bw d h z-i o i. Fo hmoic ouc i h movig fm: i A ˆ chg o 93? bcom: ' co i ˆ ik i ik d A o wh k. Th Jcobi Ag io: i iz z J i co i iz z J i i of ud o d l wv um of clidicl wv. cf. Mo&Igd968 Eq Wih hi io vibl bou d i h oud u io c b d d h oud u i clidicl coodi i: ' i ˆ i i ik d k J i A o. Fo h igl fquc obvd i h movig fm h i mulil fquci i h fid fm du o h ol ffc hough h: ig

36 . A ch fquc h i iig mod i h cicumfil dicio wih iig mod umb. Oc h fquc d h mliud of h ouc i h movig fm kow oud u c b comud fom. If h ouc i d i h movig fm uch d foc o oud i comud fom hi fomul. Thi i du o h umio of M. Soluio ol li fo ouc wih high fquc ocillio. Fo d ouc i h movig fm ffc of M mu b id. Fig.8 Aifoil ubcd o icid flow. Suo oo wih B bld o i h wk of o wih v. To ivig h ouc chciic l coc o o oig bld i Fig.8. Th odmic of h flow ov h bld dd imil o h icid vloci W : W U. 3 U i h um vloci i h fid fm i h bld oig vloci. Th ouc gh dd o W d i dicio gl of ck hfo i fucio of U h wk flow of h um o. I h fid fm h wk flow i iodic i wih iod. I c b d b hi Foui i: I h oig fm h wk bcom: U z U m m z im. im im z Um z. 5 m W Obvd oiio z i h movig fm h icid vloci ocill wih im mliud U z fquc m m 5-36

37 wh h bld mov hough h wk. Th oio of h oo bld fom h flow il ouifomi i h o wk io im ouifomi. Fquc of h ouc obvd i h movig fm i h o bld ig fquc BPF d i hmoic ddig o h oio d of h o liv o h oo d h umb of o v id of h umb of h oo bld. Th bld od o h icid wk wih h ouc gh i fom imil o 5: m im im Am z. 6 Th ic dl fucio i o idic h h oud ouc ol i wh h bld occui h c. Th B oo bld d b B. Th oud ouc h h bld i: m im im Am z [ B]. 7 If w coc o h mh Foui como of h wk i 5 h ouc i: Aˆ m i Th coodig oud u i Sic m B A z i J m m cf. Fomul.doc h m A z k im i B o o zo m im Am z... B. i im i m B i m B B m kb kbm J im kbm m k [ B] 8 i io i m [ B] ikb i kbm o ikb d dz d dz

38 Thi ul i imil o Eq.- of Blk. Th oud fquci obvd i h fid fm kb h oo bld-ig fquc BPF d i hmoic. Th dd o h oio d of h oo liv o h o d h umb of oo bld. Th oll diff fom h ouc fquc m which i h o bld-ig fquc d i hmoic. Coodig o ch fquc kb h i iig mod i h cicumfil dicio: kb m. Siig mod i dfid oiiv if i og i oiiv o dicio. I c b howd h h fquc obv kb i h diffc bw h ouc fquc m d h iig mod m kb oig fquc m kb. Thi i coi wih h fc h h im i h fid fm i h diffc of h im i h movig fm d h chgig du o h ouc gdi: U. I i clld h fquc cig. Th diio coffici J m i i dmid b h iig mod d h ouc fquc m. Fo d ouc i h oig fm m m kb m i co ccodig o 9 i.. o oud. Th ol ffc hough mliud o M w igod wh divig 9. A w hv miod h ouc diibuio o h oig bld ufc i highl ouifom i c. Effc of M co b glcd v M i mll. Kig h movig ffc i mliud To k h ol ffc hough mliud i i b o modl h ouc dicl i h fid fm Mo&Igd Th h ouc i io i h Kichhoff igio. I h fid coodi h wk i Eq..Suo im oo bld occui h c fid oi z. Th ouc h bld i ooiol o U z. Sic h oo i oig d h occuc fquc of bld hi locio ib i B oo BPF. Th im fco i h. i h hmoic of h BPF. Th m gum li oh locio c h h i im lg. Thfo h coc im fco i q z A m ib m z d h modl of h ouc gh i: im ib A m m z i m B ib. bgi wih ic = cood o d ouc which g o oud. Thi io flc h ouc moio ffc w dicud vioul. I i oducd b h ouc moio d h ouc gh gdi?. Th diffc of h ouc modl i h fid fm d i h movig fm 7 i h i h fid fm h ouc i coiuoul diibud wih whil i h movig fm i i dic wih 38

39 i i ol wh h bld occu h oiio. I h fid fm h ouc i oi i h ouc gio h oo o. Subiuig ouc modl io h Kichhoff fomul w obi h oud u fom h mh como of h wk: m A m z i m B ib d d dz Wih h oimio fo lowl movig ouc d h Jcobi-Ag io h oud u i m i.. A k A m m z z ik o i m B ib ib ib ib k ib i co k i J B i o. d d dz i m Bk d d dz m o o zo A m z i Bm o ib ib Bm i J B i Bm d dz. Thi i h diff ul fom 9. I i h oo bld ig fquci i J B m B i id of h o BPF i 9. Th oud fquci h oo bld ig fquc d i hmoic: B. Th iig cicumfil mod of h oud fild i m B h diffc of h o wk hmoic d h oo hmoic lo clld h Sofi-Tl mod. Fo ch fquc i.. fid h mulil iig mod fo diff m. Fo ch iig mod h oud u i: m A z J m i Bm ib ib Bm o Bm B i ddz i. Th cicumfil iod of h iig mod i m B. Th oio d h d of h iig mod i B m B. Alhough h ouc w modld i h fid fm o dvlo h oluio omim i i mo covi o dmi h ouc i h oig fm ih 39

40 umicll o hoicll. Subiuig 9 io h ouc i h oig fm i: i m B im q z A z. 3 m 3 v.7? If w olv h gubld icio oblm i fquc domi i h oig fm im w obi h ouc gh qˆ m z wih im fco. I c b dcomod io A z ccodig o hi liohi: m m qˆ m i m B z Am z. Oc A m z i dmid h oud u of hi iig mod c b dmid b h igio i h ouc gio of hi ouc gh mulilid b wighig fucio i B m ib B m o i J B m B i. d J B m B i i h wighig fucio flc h imoc of h ouc locio o h oud fild. Th ouc lg diu g og oud. Th i o oud log h i c wh B m. Thi cull i h ul of ouc moio ffc. Wh B m h oud ivig h i ccl ch oh wh h ouc o. O h oh hd wh B m oud wv foc ch oh which hould b voidd o cool h oo oi. A om dil locio J B m B i =. Thi i lo h ffc of ouc moio. Wh h ouc o of h locio o h oud i cclld log h obvio oi h gl. Soud gd b wo oo Th imil li lo li fo wk gd b fo oo id of o. I h fm movig wih h fo oo h wk i: U w w z U m m z im w. 5 w i h cicumfil gl i h fm oig wih h fo oo. Suo h fo oo o d w h i h fid fm h wk i: U z m U m z im w. 6

41 I h fid fm uo im f oo bld occui h c z. Sic bld hi oi h f oo Bld Pig Fquc B h ouc h im fco: ib i h im lg diff cicumfil locio. i h hmoic of h f oo BPF. Th ouc gh i modld : q z m m A A m m z z. im w i m B ib im 7 Fquc i h fid fm i modl 3 h bgi wih. m B m w. iff fom h ooo Subiuig ouc modl 7 io h Kichhoff fomul mkig u of h oimio d h Jcobi-Ag io h oud u i h f-fild i: m z o i.. o o A m A m k z m i m B z m o o zo A z ik o im i Bm w im i Bm w k i im i co o J k i i Bm o im i Bm w Bm i J Bm m i m d d dz i m Bk d dz d d dz. 8 Thi oluio duc o w. I duc o h oluio of fo oo icig wih f o. I h ooo c h m b mulil iig mod o fquc. H fo h oooo c o iig mod m cood o ol o fquc vic v. Th oud fild hv mod oh h m o cific fquc ol wh i i cd b ouifomi uch lic i duc duig ogig oc. Th fquc obv B m i h diffc bw h ouc m w

42 fquc m w d h iig mod m B oig fquc m B. Thi i coi wih h fc h h im i h fid fm i h diffc of h im i h movig fm d h chgig du o h ouc gdi: U. I i clld h fquc cig. Bd o oluio 8 oud ii of mod m dd o wo fco: ouc gh A z d ouc locio i diio coffici m J B m m i. Uull h low h iig mod umb B m h high h diio coffici J B m m i. If B m i lg J B m v m i i lg. Thi mod i clld 'cu-off'. J B m bcom imo ol wh h fquc ic o om lvl. Th h mod i 'cu-o'. Blk986 ol..88 Thi i how h 'cu-o' d 'cu-off' homo i lid wh h ' fucio i h Ci coodi 66 i ud. Th ' fucio d i clidicl coodi will b giv l. Th 'cu-o' hom will b lid i lighl diff w. Th o of h f oo i oughl iodic i h cicumfil dicio i.. A m z i og wh B m i clo o B d i hmoic. Thfo bd o B d w c im h h mod who B m i mll d clo o B d i hmoic h og oud ii. To cool oi B d mu b lcd cfull. A ml l um d B. Fo m h iig mod umb -8-8 wh =3 Th iig mod umb bolu vlu clo o B i 8 hfo h og oud c b cd mod m=. Fo m h iig mod umb -6 =3 Th iig mod umb bolu vlu clo o B= d 6 hfo h og oud c b cd mod m= d 3. I hi ml o of combiio of m d oduc iig umb qul o. d B digd o void hi hig. Emiig w fid o cuoff mchim of h oud ogio i h f c. Thi i diff fom oo gig oud i duc. I duc ol limid umb of mod c og. Oh mod vc. Fo h oo-lo d u gd oud i og ol wh h i d ch o cd uoic. Hubbd Th ouc i h fm oig wih h f oo i: q z A z i m B im w m. 9 m

43 If w olv h gubld icio oblm i fquc domi i h oig fm im w w obi h ouc gh qˆ m z wih im fco. I c b dcomod io A z ccodig o hi liohi: m qˆ m i m B z Am z. 3 Oc A m z i dmid h oud u of hi iig mod c b dmid b Eq.8. Fucio i Clidicl Coodi i O Sc Wh w h ' fucio i clidicl coodi w m f g. ' fucio i clidicl coodi: Mo&Igd Eq I i divd fom h Ci fucio uig h Jcobi-Ag io. l oluio i clidicl coodi: Mo&Igd.36 I licio h fucio i Ci coodi i of ud d h Jcobi Ag io i ud o f i o h clidicl coodi wh dd. I hould giv h m ul uig h fucio i clidicl coodi dicl. J Y: good fo dig wv hfo i duc H: good fo ogig wv i o c Shicl coodi m: oi wh h ouc gio i cocd i mll gio of c i ll h dimio Clidicl coodi m: oi wh h ouc gio i dd i o dicio d cocd i h oh wo. Ci coodi m: oi wh h gio dd i dimio. Aoi coodi m lo cho bd o h ouc igio wh h ouc gio i o comc. I hi c h ouc igio i dd. Th oi coodi m i cho fo h covic of umicl igio ' fucio i hicl coodi: Mo&Igd Eq.7..3 Fucio i Two imio Th fucio i h-dimiol io mdium i o c giv b Eq.66 i Ci coodi d b ** i clidicl coodi. A oi ouc i wo dimio c b modld li ouc i h dimio. Thfo h 3

44 fucio c b obid b igig log h li ouc. Th fucio c lo b obid b dicl olvig h couic quio. Th cod och i ud i hi cio. Th o-homogou wo-dimiol quio fo h fucio i: i 3 Th oi ouc ocill cicul fquc. Th oluio h h m fquc o w c dfi: ˆ i. 3 Fucio i Sio Mdium W fi dicu h c u. Th oluio i ill mmic. Pol coodi i dod: g [ ]. Th immic quio i h ol coodi m i: d ˆ dˆ ˆ d d. 33 Th coodig homogou quio of 33 lcig h igh hd id b i h Bl quio. I gl oluio c b coucd b h wo lil idd Bl fucio: ˆ AJ BY. 3 J d Y h zoh od fi d cod kid of Bl fucio civl. A d B dmid b boud codiio wo dil locio. Soluio 3 dig wv uch i ul duc ch6.doc. Fo o c w dicuig 3 i o oi choic. Id w u h hid kid of Bl fucio: ˆ AH BH. 35 H d H Hkl fucio. I h f fild

45 i H i H. Coidig h im fco i 3 o c h H wv ogig owd. d H wv ogig owd B i dmid b h boud. I h cu c h i ol ouwd ogig wv hfo B=. A i dmid b h boud. Th igio of boh id of Eq.33 o cicul h do iclud h ouc oi i zo. If h cicul wih diu i cd h ouc oi h S ˆ Fom h u ivgc Thom w hv: A Thfo A ˆ ds. dh d A H dd d H i[l.59] d i H H. d ˆ i A i H. 36 Thi i h wo-dimiol fucio i io mdium i Ci coodi i fquc domi. I h h logihmic iguli h oigi. Th fucio i clidicl coodi i fd o Eq i Mo&Igd968. 5

46 6 I h limi of icomibl mdium. Th o-homogou quio i: ˆ. O m fid i fucio b fomig h limiig li of 36. Bu h coc w i o u h imil ocdu fom 33 o 36 which ld o hi fucio i wo-dimiol io icomibl mdium: l ˆ. Fucio i Suboic M Flow Cf.How Wh u d u w c u h Pdl-lu fomio 83 o fom Eq.3 io: ] [ u i u i. 37 Accodig o 36 h oluio o 37 i wo dimio i: i u i H i. I h oigil coodi i i: ˆ H i M i 38 Eq.38 i h fucio i uboic m flow fo Eq.3. Fucio i Th imiol O Sc wih M Flow Eq.69 i h o-homogou quio fo h ' fucio i h-dimiol flow:

47 7 Alig h Pdl-lu fomio 83 i Eq.69 o obi. 39 Th ' fucio i io mdium i ovidd b Eq.66. Thfo h oluio o Eq.39 i h dimio i. vig i o h oigil coodi o obi h ' fucio fo uifom m flow: ] [ ] [ M M M M M M No : mlificio of m flow. C.f. Michlk&Michl979 Eq.3. Ikd l 6 Eq.8 Nfi-Yzdi l Eq.. How998 Eq..7.7 fo fquc domi.