A Single-Integral Representation for the Green Function of Steady Ship Flow in Water of Finite Depth

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1 A Sigl-Itgrl Rprsttio for th Gr uctio of Stdy Ship low i Wtr of iit Dpth Thi Nguy & Xio-Bo Ch Costl Systs Sttio, P City, L 7 (USA x: Eil: NguyTC@csc.vy.il Rsrch Dprtt, BV, 7bis, Plc ds Rflts, 9 Courbvoi (rc Coprd to th dp wtr cs, thr r vry fw works dvotd to th Gr fuctio of th stdy ship flow i wtr of fiit dpth. O clssicl forultio of this Gr fuctio is giv i Whus (97. Its o-uifority wh th wtr dpth tds to ifiity is highlightd i Ch & Zho ( who obti w uifor forultio by rovig costt trs, o of which is ifiit i gitud. Both of ths forultios coti doubl itgrl which is difficult to vlut uriclly. I th dp wtr cs, this probl is llvitd by th dvlopt of sigl-itgrl rprsttios for th Gr fuctio. So of ths rprsttios, such s th thr giv i Noblss (98, r uch or bl to uricl coputtio. It is dsirbl to driv siilr sigl-itgrl xprssios for th fiit-dpth cs. Ufortutly, th pprochs i Noblss (98 cot b xtdd to fiit dpth. I this ppr, w ploy th tchiqu usd i Sorodi (97 to rduc th doubl itgrl of th fiit-dpth Gr fuctio to sigl itgrls by trtig th -itgrtio s th ir itgrtio d vlutig it usig Cuchy s rsidu thor. Th rsultig sigl-itgrl rprsttio cotis ifiit sris which covrg slowly for kr >>. I this cs, thr ltrtiv xprssios for th sris r dvlopd, d thy r wll suitd to uricl coputtio.. Stdy, fiit-dpth Gr fuctio Usig o of th fors of th ship-otio Gr fuctio dvlopd i Ch & Nguy (, w c rdily obti xprssio for th stdy Gr fuctio by tkig th liit of th forr s th frqucy pprochs zro. Itroducig coordit syst ovig with th sourc t vlocity U log th positiv x- xis th (x, y-pl coicids with th fr surfc d th positiv z-xis poits upwrd, w c writ th stdy Gr fuctio for sourc t (x s, y s, z s d fild poit t (ξ, η, ς s th su S G G G ( S G ccouts for th fr-surfc ffcts, d G is dfid i trs of sipl sigulritis s follows S G ( / r ( ζ zs / r ( ζ zs ( Hr, w hv orlizd th vribls by th wtr dpth H d dfid r x y ( s x, y ( ξ x s, η y. Th fr-surfc copot G is giv by th followig doubl itgrl G A cosh k( ξ cosh k( z / cosh k s li ε with k k i( k x k y A D iε sig( k k, d D ( k k th k with th dpth roud ubr. Usig polr ourir vribls (k,, w c writ Eq. ( s ik ysi A G li ε D / k iε sig(cos D ( k cos k th k. Itrchg th ordr of itgrtio d tkig th liit ε, w hv ik ysi G A (5 L k cos th k pth L, s show i igur, is dpdt o th xistc d loctio of th zros of th disprsio fuctio D. Lt s itroduc th criticl wvubr k which corrspods to wvs with phs vlocity U. k is giv iplicitly by vlocity U. Wh k rl xis fro - to. Wh k th k / k. or >, k is dfid to b zro sic thr r o wvs with phs < k, thr r o pols i th itgrd of (5, d pth L is siply stright li o th > k, thr r four pols giv by, ± cos ( th k /( k d, ± cos ( th k /( k (6 ( (

2 Pth L ow hs idttios roud ths pols, d th loctio of th idttios i th uppr or lowr hlf of th coplx pl is dtrid by th liit ε d is show i igur. k<k -pl k>k -pl - pth L - pth L igur. Dfiitio of pth L for k < k d k > k I th xt sctio, w will obti sigl-itgrl rprsttio for G by vlutig th itgrl Θ blow usig Cuchy rsidu thor. ik ysi (7 ik L k cos th k. Evlutio of itgrl Θ i Th itgrl Θ c b vlutd usig Cuchy s rsidu thor. Usig th trsfortio, w c rwrit Θ i trs of th w vribl s follows: ik [ x( iy( ] f ( d C C ( th k /( k is th rtio of th phs vlocity of wvs with wvubr k to th sourc s vlocity. Th pth of itgrtio C is ow uit circl ctrd t th origi of th -pl d hs idttios corrspodig to thos of pth L s show i igur. Th itgrd f hs fiv sigulritis. our r sipl pols ssocitd with th zros of th doitor. Th riig sigulrity is t d is sstil sigulrity. Th four sipl pols of f r giv by:, ± d, ± (9 Ths pols r ithr ll rl or ll coplx dpdig o whthr is grtr th or lss th uity. or >, th sourc s vlocity is grtr th th phs vlocity of ll wvs, d is lss th uity for ll k. All four sipl pols r coplx i this cs. or <, k is th wvubr for wvs with phs vlocity qul to th sourc s vlocity. Thrfor, wh k k. Also, > for k < k d < for k > k. Thus, th pols r rl wh k < k d coplx wh k > k s illustrtd i igur. Not tht sic w dfi k for >, th coditio k > k is lwys tru, d igur is lso vlid for th suprcriticl cs. d (8 k<k -pl k>k -pl pth C pth C igur. Dfiitio of pth C i th -pl To pply Cuchy s rsidu thor to Eq. (8, w d to obti th rsidus of f for ll sigulritis isid pth C. W will first cosidr th rsidu t th sstil sigulrity. Th rsidus for th sipl pols will b trtd xt for th two css wh k < k d k > k. Lt s itroduc th polr coordits (R,

3 ψ R x y, x R cosψ, d y R siψ. With ths vribls, w c rwrit th trs ivolvig x d y i Eq. (8 s: x( iy( R [cosψ ( i siψ ( ] R( γ γ ( γ (8 bcos iψ. Usig Eq. ( d Eq. 9.. of Abrowitz & Stgu (97, th xpotil tr of Eq. ik [ x( iy( ] ( f ( ( ( iγ J ( kr Th rsidu of f t is siply th cofficit of th - tr of th Lurt sris d is giv by rs ( f, ( ( J Sic k for >, th coditio k < k c oly occurs for subcriticl roud ubrs. Th vlocity rtio i this cs is grtr th uity, d th sipl pols i Eq. (9 r rl with d loctd isid pth C s show i igur. Ths pols do ot corrspod to y zros of th disprsio fuctio D, d, thrfor, thir rsidus s to cotribut oly to th locl ffcts. Th rsidus of f t d r sily obtid, d thir su is giv by cos(xk cosh( yk i si( xksih( yk rs ( f, rs ( f, (5 Applyig Cuchy s rsidu thor to Eq. (8 d usig th rsults i Eqs. ( d (5, w obti th followig xprssio for th itgrl Θ vlid for k < k i xk cosh( yk i si(xksih( cos( yk ( γ ( (6 J ( kr, w hv cos, d th rsidu t, s giv i Eq. (, c b siplifid to kr [ i γ ( i γ ] ( si( J si Usig Eqs. (7 d (8, w obti th followig xprssio for th itgrl Θ vlid for k > k si(kxcos i iky si ( γ si( J ( kr si (9 With Θ dfid for ll vlus of k, w c ow rwrit th fr-surfc copot G i Eq. (5. Mkig us of th sytry proprty of G bout th xis y, it s sy to show tht th coplx prt of G vishs. Th fil for of G vlid for ll roud ubrs is giv by ( i γ J ( kr urthror, w c xpd th doitor of f i trs of Tylor sris bout [ ( ( ( ] ( Substitutig Eqs. ( d ( ito (8, w obti th followig Lurt sris xpsio of f bout Rsidus t sipl pols for k < k Rsidus t sipl pols for k > k [ ] ( γ ( kr ( Th vlocity rtio is lss th uity i this cs, d ll sipl pols i Eq. (9 r coplx. Ths pols r th trsfors of th four zros,,, d of th disprsio fuctio D. Oly two of th pols, d, li isid pth C, d thir rsidus cotribut to th fr-fild wvs. Th su of th rsidus t d is giv by ikysi rs ( f, si( kx cos rs ( f, (7 si Sic i rs ( f, γ ( kr (8

4 G k k k A cos( xkcosh( yk A si( cos cos( si kx ky k si ( ( ( cos(ψ J si( cos(ψ J ( kr ( kr (. Altrtiv xprssios for th ifiit sris Both ifiit sris i Eq. ( covrg slowly wh kr >>, but th vlu of k i th first sris is liitd by k, d i ost pplictios th coputtio of this sris dos ot prst jor difficulty. Th scod sris, howvr, is hrdr to vlut i its currt for wh kr is lrg sic xcssiv ubr of trs is dd. Thrfor, for kr >>, w dvlop thr ltrtiv rprsttios tht r or bl to uricl coputtio. Th scod ifiit sris i ( c b xprssd i trs of sris of th for h kr α ( si(α J ( h ( d α ± ψ. Th first rprsttio of α is obtid by cosidrig scod-ordr ordiry diffrtil qutio stisfid by α d kig us of th ourir si trsfor. It s writt s I ht si(α (cos h isi h α si( hcosα dt ( (si α t it t it I {} dots th igiry prt of th xprssio isid th brckts. Usig th bov itgrl rprsttio, w c dvlop th followig syptotic xpsio h cosα, α si( hcosα [si ( cos ]( si / h h h α ( th first 5 cofficits of th sris r giv by, si α /8 9si α /8, si α /8, 6 5 5si α /8 5si α /8 75si α /, si α /8 5si α /8 5si α / 675si α /768 Th scod rprsttio i ( is uch bttr suitd for uricl coputtio th th sris i ( wh hsi α >>. Howvr, it is ot usful wh hsi α <. To coplt (, w dvlop th followig scdig sris usig th Tylor sris of si( α d th rcurrc rltio for th Bssl fuctios th first 5 cofficits b r giv by b J ( b J ( h /95 J b J ( h / J ( h /(6 ( h /(6h 5J ( h /(68h b J ( h /5 9J ( h /(h 9J ( h /(68h 5J ( h /(6h, ( α hsiα b ( hsi α (5 5J b J ( h /5 J ( h /(h ( h /(h 9J ( h /(h 5J ( h /(5h Eq. (5 c b usd wh h si α <, but wh hsi α O(, Eq. ( is or pproprit d c b vlutd usig th lgorith (5..5 i [] for ifiit itgrls with xpotilly dcrsig itgrds. Rfrcs [] Abrowitz, M. & Stgu, I. A. (967 Hdbook of Mthticl uctios. Dovr Publictios. [] C X. B. & Nguy, T. ( Ship-otio Gr fuctio i fiit-dpth wtr. ourth Itrtiol Cofrc o Hydrodyics, Yokoh (Jp. [] C X. B. & Zho, R. ( Stdy fr-surfc flow i wtr of fiit dpth. Proc. 6 th Itrtiol Workshop o Wtr Wvs d lotig Bodis, Hiroshi (Jp. [] Noblss,. (98 Altrtiv itgrl rprsttio for th Gr fuctio of th thory of ship wv rsistc. Jourl of Egirig Mth 5 (. [5] Sorodi, A. I. (97 Wvs t th fluid surfc durig th otio of subrgd llipsoid of rvolutio. Applid Mthtics & Mchics 6, 8-5. [6] Whus, J. V. (97 Th wv rsistc of Ships. Advcs i Applid Mchics, 9-5., (6

5 8th IWWWB L Croisic (rc 6-9 April Qustio by : J.N. Nw Wht is th liit of your w sigl itgrl wh th dpth tds to ifiity? Author s rply: or th purpos of kig dpth tds to ifiity, w y writ th w sigl itgrl i othr for by usig ship's lgth s rfrc lgth istd of wtrdpth. Th liit of th odifid sigl itgrl is th th Gr fuctio i dp wtr wh th dpth tds to ifiity d th wvubr i th itgrl is kpt costt. S s th clssicl itgrl rprsttio, thr is o-uifority bout th wvubr tds to zro. This pculir proprty hs b trtd i Ch & Zho ( by rovig costts bddd i th forultios.

Linear Algebra Existence of the determinant. Expansion according to a row.

Linear Algebra Existence of the determinant. Expansion according to a row. Lir Algbr 2270 1 Existc of th dtrmit. Expsio ccordig to row. W dfi th dtrmit for 1 1 mtrics s dt([]) = (1) It is sy chck tht it stisfis D1)-D3). For y othr w dfi th dtrmit s follows. Assumig th dtrmit