Complex Variable Method and Applications in Potential Flows

Transcription

1 HAPTER 6 hpt 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ompl l Mthod nd Applctons n Potntl lows o ctn tp o low t s possl to ntodc compl ls to d n th solton o th low polm. ompl l cn onl sd th ottonl low tht s two-dmnsonl nd pssl n tsn coodnt sstm o pol coodnt sstm.. th low mst sts oth φ nd ψ. o no oth coodnt sstm cn compl l sd. 6. Nomnclt nd Alg o ompl ls Etnd th l nm sstm ncldng thn w consd nms o th om psnt ponts on th - pln. cos sn Ag tn / R Rl pt Im Imgn pt Imgn s Th compl nm cn lso psntd s cto n pln n odd p o l nms. Rl s

2 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ompl conjgt: onjgt s dnd s Ag Ag s th lcton o th pont n th l s. R Im - - Th compl l cn lso pssd n pol coodnts n. Snc cos sn thn cos sn onsd: d cos sn sn cos d Ddd cos sn t two sd Thn: dcos sn cos sn dcos sn cos sn dcos sn cos sn sn cos d cos sn sn cos d cos sn sn cos d d cos sn sn cos d cos sn Intgtng oth sds lds to: dcos sn cos sn d ln cos sn cos sn Tho cos sn - 7 -

3 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Usng th ponntl om w s tht th podct o two compl nms nd s gn Ag I thn I... cos sn cos sn. In gnl om nown s D Mo s thom: cos sn n cos n sn n Sml ltons cn wttn o n compl pln. Empl: o th ς - pln ζ Rcos φ snφ R ζ R ; φ tn / φ - 8 -

4 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6. Anltc ompl nctons I s compl ncton n th -pln w dn dt o compl ncton lm z o ll possl. Tht s snc o th dt shold n nd th lmt mst st ndpndnt o how. A compl nctons tht dntl clld nltc nctons. o th ponts n th compl domn o -pln wh th s nltcl clld gl ponts o th ncton. o th ponts n th compl domn wh th ncton s not nltcl clld sngl ponts o th ncton. Th ls o l ncton o on l clcls cn gnll ppld to nltc nctons. d d g d dg d d g d d g g g d d g g g g o g W dg d dw d d d.. n nltcl ncton o n nltc ncton s nltc

5 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ch- Rmnn ondtons 6.3. ch- Rmnn ondtons n tsn oodnt Sstm onsd: I sts thn th som condtons tht nd mst sts: z lm onsd th pth long nd long gn. o th pth z z lm lm lm Tho lm lm lm o th pth z z s lm lm lm

6 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS lm lm lm Snc shold th sm n oth pths tho Sch condtons o n nltc ncton clld ch- Rmnn ondtons.

7 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ch- Rmnn ondtons n Pol oodnt W consd compl nltc ncton n pol coodnt sstm s : I sts thn th som condtons tht nd mst sts: z lm onsd th pth long nd long gn. o th pth z z lm lm lm Tho lm lm lm o th pth z z lm lm lm Tho d

8 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS lm lm lm shold th sm n oth pths tho Sch condtons o n nltc ncton clld ch- Rmnn ondtons n pol coodnt sstm.

9 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Scnt ondtons o ompl ncton to Anltc onsd: nd d d d d d d d d d d om nd w cn gt nd clld conjgtd ncton. ch Rmnn condtons pod th ncss condton o nltc. W to ontnt o th st od dt tms sch s condtons o to nltc. pods th scnt Ths stss ch Rmnn condtons nd h contnos st od dts thn s n nltc ncton nd cn pssd s

10 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Som Popts o Anltc nctons o compl ncton wh Accodng to ch Rmnn condtons s nltc thn nd onsl ncton stss ch Rmnn condtons thn cn wttn s ncton o onl. Som popts o nltc nctons: Lt compl poston l ncton ς s nltc ncton nd ς compl Thn:. ς s ncton o.. ς ς. ς hs dnt l o l o... ς s n. dς 3. dz dς 4. t pont n th pscd domn s ndpndnt o th pth sd to ch th dz pont wh th dt s ng ltd. 5. ς ς z t stss ch Rmnn condtons: ; Thn

11 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Tho Smll.. oth th l nd mgn pts o n nltc ncton sts Lplc ton. Th oth conjgt hmonc ncton. 6. onsd two mls o cs gn const nd const wh nd th l nd mgn pts o n nltc ncton. I w consd two dcton on tngntl to c long nd th oth noml to t n som nnown dcton thn n n t n n t ˆ ˆ ˆ Gdnt o const c s noml to th c! Snc: j ˆ ˆ j ˆ ˆ ˆ ˆ ˆ ˆ j j Tho const nd const othogonl

12 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.4 ompl Potntl nd ompl loct ompl potntl Accodng to th contnt ton o dl low o ottonl lows o -D ncompssl low th stm ncton ψ dtmnd ψ ψ ; nd I th -D ncompssl low s ottonl potntl ncton φ wll t: φ Tho: φ φ φ ψ φ ψ Whn ψ nd φ ltd th o ltons t s possl to om compl ncton s ln comnton o ψ nd φ nd cll t W th compl potntl: φ ψ o φ ψ How snc ψ nd φ sts th ch Rmnn condtons tho w cn psnt s wh Th o concpt s ld o n two l ls nd sml to ψ nd φ. Assm nd ς And nd sts ch Rmnn condtons thn ς cn psntd s ς wh

13 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ompl loct I φ ψ d. o th pth nd long d tho d d d d Along pth d d d d φ ψ d. Pth nd long d tho d d d d Along pth d d d φ ψ d d d / d.. W s compl potntl s clld compl loct. d d cos cos sn cos sn cos sn cos sn cos cos sn sn sn cos sn d sn sn sn sn cos sn cos cos cos cos sn cos sn cos.. compl loct d d n pol coodnt sstm

14 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.5 Unom lows 6.5. Unom low to th Rght o -D ncompssl low wth th loct dstton s gn : ; Is th low phscll possl? Th contnt o ncompssl low s Tho th low s phscll possl. Th stm ncton ψ cn dtmnd ψ ψ ; ψ ψ ψ ψ Tho ψ ψ d d onst Is th low s ottonl? ˆ ˆj ˆ z Tho th low s ottonl. Snc th low s ottonl potntl ncton φ wll t: φ φ φ φ φ d osnt d Tho φ W φ ψ In tsn sstm W In pol sstm W

15 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.5. Unom low t n Angl o Attc o α Snc W dn Y Y Th compl ncton wll Snc cosα snα snα cosα cosα snα snα cosα α X X Thn th potntl compl wll cosα snα snα cosα cosα snα snα cosα cosα snα Tho cosα snα α snα cosα cosα snα snα cosα In tsn coodnt sstm In pol coodnt sstm α α - 4 -

16 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.6 -D Soc nd Sn D Soc Dnton: A -D soc s dnd s nnt ln om whch low sss long dl lns. To clclt th olm low t om th soc Thn th loct ld wll : Is th low phscll possl? Th contnt o dl low s z X Y Tho th low s phscll possl. To dtmn stm nctonψ. ψ ψ ψ d d Ths ψ const const Lt s ssm ψ whn const Tho ψ o sn low ψ - 4 -

17 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Is th low ottonl? ˆ ˆ ˆ z z Tho th low s ottonl. Snc th low s ottonl potntl ncton φ wll t: To dtmn potntl nctonφ Ths φ φ φ ln d d φ ln const osnt Lt s ssm φ whn const ln Tho φ ln o sn low φ ln Tho o soc low th compl potntl wll : φ ψ ln ln ln ln.. ln o sn low: ln To dtmn th loct ld om th potntl compl o soc low - 4 -

18 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS d d dln d To dtmn th potntl compl ncton ntgtng th loct ld In pol coodnt sstm: d d. Intgtng th o ton w cn h ln const W ssm R ln const ln const const ln whn tho Ths ln

19 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.6. A Soc t - Pls Sn t Wthot constnt tm th psson o soc low wll ln o soc sttd t poston n th compl pln th psson wll ln P o soc t - nd sn t soc ln sn ln comnd soc sn ln ln ln ln ln X To dtmn th gomt o th stmlns o soc sn comnd low th ψ nd φ gn th psson o ψ nd ψ ψ comnd ψ ψ tn o tn

20 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS tn o tn Thn tn tn tn tn tn Tho tn Thn tn comnd ψ I tn tn tn c c c c c c c c ψ ψ Th ccls wth cnt t c nd ds o c R.. th ccls on Y s. To dtmn th gomt o th so-potntl lns Epotnt lns: Snc ln ln ln φ φ φ Snc nd Tho / ln 4 ln ln φ

21 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS o const φ lns 4 4 φ 4 Tho th const φ lns ccls wth cnt t nd ds o 4 R.. th ccls on X s cpt.

22 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS D Dolt Dnton: A dol s otnd whn soc nd sn o l stngth ppoch ch oth so tht th podc o th stngth nd th dstnc pt mns constnt... const soc comnd ln 4 ln 4 ln ln sn Usng L Hoptl s ols Whn d d d d 4 lm 4 lm ln 4 lm ln 4 lm Tho o -D dolt sn cos comnd.. ψ φ sn cos

23 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.8 -D ot Dnton: A -D ot s mthmtcl concpt tht ndcd loct ld gn : ; ; Is th low ld phscl possl? Th contnt o dl low s z Tho th low s phscll possl. To dtmn th stm nctonψ. ψ ψ ln ψ d onst d Ths ψ ln const Whn ssm ψ whn thn ln const Tho th stm ncton wll ψ ln ln o ψ ln Is th low ottonl? ˆ ˆ ˆ z z Tho th low s ottonl. Snc th low s ottonl potntl ncton φ wll t:

24 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS φ φ φ d osnt d Y Ths φ const Lt s ssm φ whn const Thn φ X I s dnd s th cclton o ccl cont-clocws sondng th -D ot Thn c dl c ˆ ˆ d Is th Stos thom stll pplcl? Thn ot t! Tho: ψ ln ; φ o -D ot low: φ ψ ln ln ln { ln ln ln ln ln.. ln Th snglt o ths psson s loctd t. Th potntl compl o post ot loctd t wll ln I th ot s ott n clocws thn th potntl compl wll : ln ln ln }

25 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS D low ond cl lnd low ond ccl clnd cn dcomposd s nom low pls dolt low. o nom low to th ght o dol low st t Th potntl compl o th comnd low wll sn cos sn cos sn cos sn cos sn cos Tho: sn cos ψ φ Thn loct ld wll sn cos cos cos 3 φ φ To dtmn th stgnton ponts: At stgnton pont nd o sn sn Whn cos Whn

26 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS cos I w dn Th two stgnton ponts wth coodnts o ng nd To dtmn th gomt o th stmln pssng th stgnton ponts: Th stmln pssng th stgnton pont wll sn stgn ψ Thn th ton o th sold od wll A cl wth ds R sn sn s X Tho th low ld s to th low ond clnd. To smmz th low ond ccl clnd: W sn cos ψ φ sn cos To lton o th low loct ld om th potntl compl Snc d d sn cos sn cos sn cos

27 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Tho sn cos Pss cocnt dstton on th sc o th ccl clnd Snc P P p At th sc o th clnd sn Accodng to Bnoll s ton P P P P Pss cocnt dstton on th sc o th ccl clnd wll : 4sn sn P P p Angl Dg. p

28 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS low ond Spnnng lnd low ond spnnng clnd cn dcomposd s th comnton o nom low -D dolt nd ot low. Th potntl compl o nom low A dol low st t A -D ot n clocws ln 3 I w m Thn th potntl compl o th comnd low wll : ln ln 3 Whn w st thn th potntl compl wll o non-spnnng clnd. d d } } sn cos { } sn cos sn cos { ln Thn: cos sn

29 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS To dtmn th stgnton ponts: Snc nd t th stgnton pont cos o o ASE. I sn sn 4 I w gt o. Ths s th cs o non-spnnng clnd w stdd o. Y Snc > 4 thn s hs to n th 3 d nd 4 th dnt. Snc sn Thn th spnnng loct o th clnd cn not too hgh n od to nd stgnton pont n th sc o th clnd. Whn th stgnton ponts on th sc o th clnd X Y s X sn 4 s ± Y s Th solton s ld onl whn o 4. Whn > 4 th 4 stgnton pont wll not on th sc o th spnng clnd n mo

30 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ASE I < Tho s mpossl ASE 3 I 4 4 ± In od to m th solton phscll possl t hs to > 4.. > 4. X Y

31 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ch Intgl Thom onsd th compl ncton ς not ncssl nltc dnd n ctn gon n th -pln. Th ln ntgl o ς om to long ln s d ς. Th l o ths ntgl n gnl dpndnt on th pth nd th nd ponts nd. Snc d d d Thn } { } { } { d d d d d d d ς W ntstd onl n compl ntgls whos ls dpnd onl on th nd ponts nd not on th pth tht jonts thm. o ths to t th ntgnd o th ntgl on th ght hnd sd o th o ton mst n ct dntl. } { } { d d d d Sppos th o psson cn wttn s d d λ β B compson w cn wt ths s: d d d d d β β β d d d d d λ λ λ X Y

32 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS.. β λ λ β B dnttng nd tng w gt: nd whch th ch- Rmnn condtons. In oth wods th compl ncton ς shold nltcl o th compl ntgl to ndpndnt o th pth. ch ntgl thom I ς s nltc n smpl conncts gon thn ς wll th ncton o onl.. ς ς nd th ntgl ς d s ncton o th nd ponts onl... ς d d Thn o closd c ς d. Th sttmnt s t whn th no ponts o nclosd c wh dς d To consd th cs whn sngl pont s nclosd n th nclosd c Th post sns o ntgton s tht whch th gon nclosd c s to th lt. dς onsd gon wh ς s nltc t pont cpt wh d th sngl pont. o n closd c not nclosng ς d. z.. s I th closd c ncloss th sngl pont w nclosd ths pont n closd c nd ocs o ttnton on th nltc gon twn nd

33 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS W nd ths gon nto closd gon ondd contnos c ntodcng th ct thn w ppl th ch ntgl thom pomng th ln ntgl long th ond n postl sns lws png th nclos gon to th lt. ς d M ς d ς d ς d ς d ς d M In lmtng sns: ς d ς d nd thn Sngl pont t Also M d ς d ς ς d nd M ς d ς d M Tho ς d ς d.. ll th ln ntgls o ll closd ccts nclosng th sngl pont l. I c ncloss n sngl ponts Sngl pont t 3 3 Thn ς d ς d ς d K ς d n M

34 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Intgl o compl ncton o ς n onsd compl ncton n ς n ± ± L Dς d n o n< n Dς d n n n n Tho s sngl pont. Whn n> Dς d n n s gl pont. I w consd closd c n th om o ccl.. R n d cl R R cl R n n R n n n n d n R d n cl R R o n R cl R n d n R d n n n n n n n R n R o n d cos n sn n n n n Whn n d d R R d As dscd o o n closd c nclosng th sm sngl pont th ln ntgl wll th sm. Tho o n closd c d In smm: Gn n ς n ± ± n d whn whn L n n o s not nclosng nd s nclosng

35 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6. Blss Intgl Lw W cn clclt oc ctng on sold od lcd n ld low n ltng loct componnt on th sc o th od nd thn ntgtng pss sng th Bnoll ton. Tho th dclt to clclt odnmc ocs s to condct th od shp ntgton. Blss ntgl lw stts tht th compl potntl s nown o low ond od thn t s possl to lt th ocs nd th tnng momnt ctng on th od mns o smpl conto ntgls. oc on od: Lt th compl potntl dscng th low ot -D od whos ond n th -pln s closd c. Thn th componnt o th nt oc on th od otnd om X Y d d d Wh dnots p nt noml to -D pln. d P ds d P ds sn β d P ds cos β d P sn β ds P cos β ds Psn β ds P cos β ds Pds sn β cos β Pds β Y ds β X Snc d d d ds cos β ds sn β ds β Tho d P d om Bnoll s ton P const P const Thn d const d - 6 -

36 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Intgtng d ond th od th closd c d const d const d d d β cos sn d ds β β ds Tho cos sn β β ds Thn th conjgt o s gn cos β sn β ds cos β sn β ds β ds On th od- stmln th loct s gn d W wh cos β ; sn β d Thn d d Tho cos β sn β β β β ds β cos β β ds β sn β ds β d d ds β ds d d d Momnt: locws momnts post stllng dw M R d d Assm post ocs thn dm d d d P ds sn β d P ds cos β - 6 -

37 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS cos sn d d const d d P P ds P ds d d dm β β R R R R sn cos R cos sn d dz d ds ds ds ds ds ds d d d const d d d d const d d const dm M β β β β β β β β β oc nd Momnt on spnnng ccl clnd o spnnng clnd th potntl compl ncton s ln Tho d d Snc d d d d d d d d d d d d d d d * Tho:

38 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS } 4 R{ } 4 R{ } R{ } 4 R{ 4 R{ } R{ } R{ } 4 R{ } R{ } R{ } R{ } R{ R R 3 3 d d d d d d d d d d d d d d dz d M Tho: M

39 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.3 onoml Tnsomton o th cs nd n th -pln tnsom to c ς nd ς n th ς -pln conoml tnsomton s dnd s sch tht: Two ln lmnts nd cd n th -pln ntsctng t ngl β pods ln sgmnts AB nd D n th ς -pln ntsctng t th sm ngl β wth th sm sns nd hng th sm to o lngth s th ognl lmnts:. ngl β n -pln β n ς -pln β d B β. cd AB D D o ctn pont o th c tnsomton: A -pln ζ-pln Gn th nltc dς dς mppng ς ponts n th ognl compl -pln wh o gl d d ponts on th -pln. dς Ponts n th ognl compl -pln wh ctcl ponts on th -pln. d dς Th ponts n th ognl compl -pln wh sngl ponts on th -pln. d ς Anltc ncton dς Nt o ponts n - d pln nd Rgl ponts tnsomton onoml tnsomton Anltc ncton tcl ponts Not conoml tnsomton Anltc ncton Sngl ponts Not conoml tnsomton At ctcl pont th tnsomton hs popt o mltplng ngl n wh n s th n d ς oth o th dt n whch st com non-zo t th ctcl pont. d

40 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS I tnsomton s conoml tnsomton t hs ollowng popts:.lplc ton s nnt... Lplc ton n th -pln wll tnsomd nto Lplc s ton n ς -pln podd ths two plns ltd conoml tnsomton. o mpl: Lt ς ς conoml tnsomton. Both φ nd ψ solton o Lplc ton n -pln. ψ ψ φ φ Tho: nd Snc s nnt nd conoml tnsomton ψ ψ φ φ Tho: nd In oth wods compl potntl n th -pln s lso ld compl n th ς -pln nd c s.. loct ld onsd th compl potntl ς φ ψ d d dς dς W Wˆ ς d dς d d 3. Th stngths o th cclton nd soc nnt o conoml tnsomton s th nt stngth o ll otcs nsd th contos. s th nt stngth o ll soc nd sns nsd th contos. ˆ nds ˆ j ˆ ˆ nds ˆ j ˆ dˆ dj ˆ d d ˆ ds d d n Y d ds -pln ê n ds d d d X

41 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS ˆ tds d d W d d d d d d d Now consd tnsomng nto ς nd dw dw dς dς W Wˆ ς d dς d d Thn W d W ς d W ς dς ˆ dς d ˆ How to s conoml tnsomton to sol polms o odnmcs. Pos polm n -pln wth compl od gomt sch s ol.. Mp od to smpl od n ς -pln sch s ccl clnd 3. nd solton to th Lplc ton n ς -pln... nd ς n th ς -pln sch tht ς sts oth ond condtons t nnt tnsomd om -pln. Lt Im ς s constnt long th sc 4. Tnsom th solton c to -pln... Knowng th tnsomton ς ς nd ς φ ψ. Tnsom ς sts ς ς n ς. Empl: ς -pln s th low ond ccl clnd o ds spnnng wth cclton cold com low ond n ol thogh conoml tnsomton. ln

42 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS 6.4 Joows Tnsomton nd Joows Aol Th tnsomton ncton o Joows tnsomton s ς ς wh s l. o lg ls o w wll h ς whn. In oth wods om th ogn th mppng s dntcl nd th compl loct s th sm n oth th plns om th ogn. Tho nom low o ctn mgntd s ppochng od n th -pln t sm ngl o ttc nom low o th sm mgntd nd ngl o ttc wll ppochng th cospondng od n ς - pln. nd th sngl nd ctc ponts o th Joows tnsomton: dς d Sngl pont t s sll wthn od dς d tcl pont t ± Snc ± nd o I w s tsn n ς -pln nd pol n -pln ς Thn: cos sn cos sn cos sn cos sn

43 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS s. o ccl loctd n th ogn wth ds n -pln Wht t wll n ς -pln thogh th Joows tnsomton? Whn cos Whn } Whn }

44 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS s o ccl loctd n th ogn wth ds > n -pln Wht t wll n ς -pln thogh th Joows tnsomton? wh >. Snc ς ς cos sn cos cos sn sn Snc cos sn It s th ton o n llps wth mjo s nd sm-mno s o. - -pln ζ-pln

45 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS s 3. Joows Aols om nd odnmcs pont o w th most ntstng pplcton o th Joows tnsom s to n ost ccl. I w consd ccl slghtl ost om th ogn long th ngt l s on otns smmtc Joows ol. z pln ς z z ζ-pln. - Th ton o th ost ccl s z - wh th constnt s smll nm. I th clnd s dsplcd slghtl long th compl s s wll on otns cmd ol shp. A β z pln ζ-pln ς z z B A B - H th ponts A nd B th ntcpts o th dsplcd ccl on th l s nd th cospondng ponts n th tnsomd pln. Th ngl β s th ngl omd th ln jonng th pont A o B nd th ogn wth th l s. I ltng low ot th ognl ccl hd n mposd th Joows tnsomton wold h gntd ltng low ot th Joows ol; z pln ς z z ς- pln - 7 -

46 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Althogh sch low s mthmtcll possl n lt t m not lstc. Th stgnton ponts on th clnd mp to stgnton ponts tht not lws lstc. o nstnc th stgnton pont on th top sc o th ol cnnot st s std lght snc th loct wold tnd to nnt s on mos clos to th tlng dg. Th onl mns o mng lstc low s to mpos th Ktt condton wh th stgnton pont s ocd to st t th tlng dg ths mng th stmlns low smoothl om ths pont. Ths s don djstng th l o otct stngth sch tht th stgnton ponts on th clnd sd t th clnd s ntcpts o th l s. In ths cs whn th clnd s tnsomd on stgnton pont wll ocd to th tlng dg. z pln ς z z ζ - pln Th lt oc gntd th ltng low o th clnd s popotonl to th cclton ot th clnd mposd th ddd ot low ccodng to th Ktt-Joows lton L. Th ltng oc on th sltng Joows ol s not cl. To lt th lt th cclton s ndd nd tho th loct ld. Th loct lds n ch pln cn ltd to ch oth thogh th chn l o dntton. I th ltng low ot th clnd s dnd s ncton wh z n th z- pln nd ζ n th ζ -pln th locts n ch pln ; ς ˆ W W ς z ς B chn l: ς ς z ς W W ς Usng th Joows tnsomton; ς ll th loct ld clos to th clnd nd ts tnsomd contpt dssml s on wold pct. How th w om ths ojcts th loct lds com dntcl s th mgntd o z coms lg thn th constnt l o. Snc th - 7 -

47 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS cclton cn clcltd ot n closd pth ncldng pths om th ojct sc th ccltons mst th sm n oth plns. clnd Joows ot stngth Th ppopt ot stngth to mpos th Ktt condton mst dtmnd. onsd th ltng low ot clnd. Th loct n th dcton s sn R H R s th ds o th clnd sc. Ths loct s zo on th sc o th clnd t th stgnton ponts. At th ponts o -β. z pln sn β R 4 R sn β I th ld s ottd α to smlt n ngl o ttc 4 sn β α R -β Snc th cod lngth o th Joows ol s 4 th lt cocnt cn wttn L 4 Rsn α β c 4 L Mng th ssmpton tht R L sn α β α β - 7 -

48 HAPTER 6 OMPLEX ARIABLE METHOD AND APPLIATIONS IN POTENTIAL LOWS Empl A Joows ol s omd dsplcng ccl o ds -.8 l s nd.5 mgn s. nd ot stngth α o nd m/s L t α o nd α o clnd β sn.5 O.87.5 β O tn stgnton pont.987 4sn Rsnαβ L sn L sn