On the Singular Integral Equation Connected with the Stokes Gravity Waves
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1 Poceedns o e Wold Coness on Enneen 7 Vol I WCE 7 Jly London U On e Snl Inel Eqon Conneced w e Sokes Gy Wes Nno sl sc Te none snl nel eqon ssoced w e Sokes y wes n e ncompessle Ele ld s sded Te exsence o e solon s poed nd e ppoxme solon s consced y mens o ple Index Tems Snl Inel Eqon Sokes Gy Wes S I INTRODUCTION okes y we s peodc sce we o pemnen om on n nsced ld lye o consn men dep In s cse sy nd compesson plys n nsncn ole nd e ld s ssmed o e ncompessle Ele ld 3 4 Te welen s smll s comped w e men dep I s ssmed e o-om o e eseo s l nd e moon s wo-dmensonl In e ely wok o e o e snl nel eqon o e Sokes ee ondy ws oned 5 In e pesen wok s ssmed e peod o e we s e smll Te nel eqon s smpled nd e exs-ence o e solon s poed y mens o skelsl eoy nd Sde s xed pon pncple Te ppoxme pole o e we s consced y sn ple II STTEENT OF THE PROBLE Te coodne sysem xoy mon w e we s en Te xs Ox psses lon e oom nd e xs Oy psses o e mxmm pon o e we emclly e polem s sed s ollows 3 STOES PROBLE Fnd e peodc ce : y y x sc s conoml mppn o e e D y x on e sp q q cons en e ollown condon olds nscp eceed c 7; esed c 7 7 Nno sl s w I Jksl Tls Se Unesy Unesy S 86TBILISI GEORGI pone: ; e-ml: nnk@ mlcom y cons wee x y s complex poenl s speed poenl s sem ncon s complex speed nd q e e dene pose consns s y cceleon Hee we consde e Sokes Polem o e symmec peodc peked wes w e peod nd w e condon q Te cse ws consdeed y deen os III SOLUTION OF THE PROBLE In e wok o e o 5 e Sokes polem ws edced o e ollown snl nel eqon 3 sn 3 d d wee s n nknown ncon o e skelsl- esel H clss s e nc o wc s e Weesss e-ncon o e ndmenl peods nd q Weesss e-ncon s epesenle y e sees c nd s e ollown popees: sn ; exp ; 3 ISBN: ISSN: Pn; ISSN: One WCE 7
2 I s meomopc ncon w e smple poles ; n m n m T mn 3 s dole qs-peodc ncon e wee nd e e ddends o 4 wee s e Weesss sm-ncon Hn ond one peod o e pole o e Sokes we wll e en y e oml 3 / 4 s e ncon s symmec y sn e popees o e-ncon we cn ewe e eqon n e om 6 sn 3 d 5 wee ; d d 6 / / / / 7 ssme e solon o 5 s epesenle n e om 8 9 sn sn wee s n nknown pose ncon o Holde clss By e epesenon 7 s oos We now sppose s e smll nd kn no ccon e popees o e-ncon e omls nd we cn ewe eqon 5 n e ollown om ; d d wee / 4 s e Weesss sm-ncon ccodn o e popees o sm-ncon e kenel s wekly snl s e nel eqon w espec o Le s ewe s ; d d Usn e epesenon 3 nd popees o encon we on wee sn sn sn 3 sn sn sn / Poceedns o e Wold Coness on Enneen 7 Vol I WCE 7 Jly London U ISBN: ISSN: Pn; ISSN: One WCE 7
3 sn sn 8 sn sn sn 4 Te kenel s e lomc snly e kenel s connos on e se nd lm Conseqenly lm d 5 Now le s consde e s em o e oml d d 6 Tkn no ccon oml 3 nd nsen sn no s em o 6 we on sn d d 7 By sn e epesenons nd popees o e Ccy ype nels s esy o conclde e ncon s Hölde connos n elons o e clss H e pon nd lm 8 Now le s sppose 9 Insen 9 no e -nd sde o we on ; B d Hence nsed o eqon we consde e ppoxme eqon Le s nlye s eqon Le e ; C S s onded se o ncons om ccodn o e popees o e kenel nd oml 8 we on d wee mx mx d lso e ollown oml s ld ; d B B Tkn no ccon e popees o e ncons n nd nd d nd oml 8 we conclde e se o ncons S s nomly onded nd nomly connos Conseqenly cell condons olds nd e opeo B s compleely connos Poceedns o e Wold Coness on Enneen 7 Vol I WCE 7 Jly London U ISBN: ISSN: Pn; ISSN: One WCE 7
4 Poceedns o e Wold Coness on Enneen 7 Vol I WCE 7 Jly London U lso ccodn o e oml 8 nd skelsl nd k 3 e ncon on e - nd sde o s Hölde connos n nd elons o e clss H e pon Hence we e poed e ollown THEORE Te opeo B on e - nd sde o s compleely connos n e spce C e nel eqon s e connos solon elons o e Hölde clss nd lm Now le s poe e solon o eqon exss Pn e noon n e nel o e nd sde o we cn epesen s nel eqon s d Te ollown eoem s ld B 3 THEORE Tee exss e solon o eqon 3 o e clss C n e ll ; ; 4 Tkn no e ccon e epesenon 3 we on B d 4 5 Te oml 5 mples sses e condon mx mx d 4 6 Hence ccodn o Sde s xed pon pncple nd ssy e condon 6 ee exss e connos solon o eqon 3 nd conseqenly o eqon ccodn o nd Teoem we conclde e ncon en y e omls 8 9 s e solon o eqon n e ll 4 IV CONCLUSION Tee exs e Sokes y peodc wes o e om sn sn 4 / 3 / 3 / 3 wee s Hölde connes pose ncon s e smll pose nme s om e condon 6 s e smll one o e ppoxme solons o e eqon s nd conseqenly e ppoxme solon o e eqon o e ype 8 9 s sn sn ccodn o 4 e coespondn Sokes we wll e en y sn sn 4 / 3 / 3 7 / 3 Below e p o 7 s consced y mens o ple- o e deen pmees nd s en n Fe Fe nd Fe 3 ISBN: ISSN: Pn; ISSN: One WCE 7
5 Poceedns o e Wold Coness on Enneen 7 Vol I WCE 7 Jly London U 3 4 d; / 3 C F Te p o 7 n cse o ; ; 5 wee C s e dene consn nd e e y smll pose consns In 7 e ppoxme solon o s eqon s en CNOWLEDGENT Te o s el o Poesso Vkn Psl o e sel emks F Te p o 7 n cse o ; ; 5 F3 Te p o 7 n cse o ; ; 5 REFERENCES G H Sokes On e Teoy o Ocslloy Wes Tnscons o e Cmde Plosopcl Socey olviii no 847 pp 97-9 GB Wmn Lne nd None Wes Wley-Inescence NY-London-Toono 974 c 33 3 Len e nd BV S Te Polems o Hydodynmcs nd e emcl odels Nk oscow 98 c5rssn 4 S Ck Hndook o Osoe Enneen Elsee 5 5 Nsl On e none plne ondy le polem Repos o VI olx noi 995 pp Nsl On Sokes none nel we eqon Inel eods n Scence nd Enneen edos B Bem C Consnd Ses Resec Noes n emcs Sees CRC pp -4 7 Nsl On e Sokes None Wes n D Recen dnces n emcs nd Componl Scence ed Ime Rds ol58 6 pp TB Benjmn PJOle New Hmlonn sce symmees nd conseon lows o we wes Jonl o Fld ecncs ol5 98 pp EV Bldko PHTylo New sympoc descpon o none we wes n Lnn coodnes Jonl o Fld ecncs ol56 6 pp H nd R Con Volesnen Ue llemene Fnconeneoe nd Ellpsce Fnconen Spne Be 99 c NI kelsl Snl Inel Eqons P Noodo Gonnen 953 c L Lsenk nd V Soole Elemens o Fnconl nlyss oskow Nk 965 c5 3 S k Elemens o emcl Pyscs oskow Nk 968 c3 Rssn PPENDIX In e wok o e o 6 y mens o e conoml mppn meod e Sokes polem ws edced o e none nel eqon o e deen om n new le ISBN: ISSN: Pn; ISSN: One WCE 7
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