Dispersion Relation and Wave Loads on a Vertical Cylinder in Water due to First Order Diffraction
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1 Dspeson Relon nd Wve Lods on Vecl Clnde n We due o s Ode Dffcon Dbu B Absc Hee we consde e fs ode wve dffcon b clndcl sucue. Te clndcl sucue s ccul vecl sufce pecn n we of fne dep. s we se e bound vlue poble n es of veloc poenl funcon. Te foulon of e wve-sucue necon s bsed on e ssupon of ooeneous del ncopessble nd nvscd flud. Incluson of oonl llows us o noduce e veloc poenl funcon wc ssfes e Lplce equon n e flud don. Te nonlne fo e wve-sucue necon poble ses fo e fee sufce bound condons. We use e sepon of vbles eod o solve e fs ode poble b wn e ol veloc poenl s e su of e ncden veloc poenl nd e sceed veloc poenl. o e cobned fee sufce condon we deve e dspeson elon. Afe devn e pessue usn Benoull s equon we obn e nlcl epesson fo e fs ode foce on e clnde b nen e pessue ove e weed sufce. Nuecl esuls fo e dspeson elon nd wve lods fo vous deps o dus os e pesened. Inoducon Te copuon of e we wve foces on offsoe sucues s one of e n neess n desnn sfe offsoe sucues. Te sucue be fed o flon s se-eed sucue n se. Tee s le nube of sucues wc e coposed of ubul ebes lke ccul clndes. Wen e sucue spns snfcn oun of wvelen e ncden wves undeo scen o dffcon. Dffcon of wves needs o be consdeed wle evlun e wve foces. Den nd Dlple [] pesened evew of poenl flow dodncs. Soluons fo sndn nd poessve sll plude we wves povde e bss fo pplcon o nueous pobles of enneen nees. Te dscussed e foulon of e lne we we eo nd developen of e sples wo-densonl soluon fo sndn nd poessve wves. Debn [] dscussed eoecl sudes of nonlne we wves ove e ls few decdes. Hs wok s pl devoed o e ecl eo of nonlne we wves w pplcons. He suded e eo of nonlne sllow we wves nd solons w epss on eods nd soluons of sevel evoluon equons e oned n e ounl of Mecl Scences & Mecs Educon 7
2 eo of we wves. onson [4] descbes e ecl des nd ecnques e decl elevn o we wve eo. Bennn w e noducon of e ppope equons of flud ecncs oee w e elevn bound condons e des of nondensonlson scln nd spoc epnsons e befl eploed. Rn [6] pesened n noducon o e ecl nd pscl specs of e eo of we wves. He dscussed e wve eo of A nonlne wve eo of Sokes dl dncs n sllow we. He enoned bou e dncs of flon offsoe sucues. In e pesen wok we pesen e bound vlue poble fo fed clnde n we of unfo dep. Te veloc poenl funcon ssfes e Lplce equon. We use e sepon of vbles eod o solve e fs ode poble b wn e ol veloc poenl s e su of e ncden veloc poenl nd e sceed veloc poenl. Te soluon s obned n es of Bessel s funcons. o e cobned fee sufce condon we deve e dspeson elon. Afe devn e pessue usn Benoull s equon we obn e nlcl epesson fo e fs ode foce on e clnde b nen e pessue ove e weed sufce. Nuecl esuls fo e dspeson elon nd wve lods fo vous dep o dus os e pesened. Veloc Poenl uncon Hee we consde fed vecl clnde n we of fne unfo dep. Te clnde eends fo se bed - o e fee sufce. We dep s. s e fee sufce elevon funcon nd s e dus of e clnde. Incden wve s popn lon posve -decon. Te ncon wve ncden upon e sufce of e clnde undeoes dffcon o scen. To evlue e wve lods n e clnde we need o consde e effec of e ncden wve nd e dffced wve. Te clndcl coodne sse w vecll upwds fo e sll we level SWL esued dll fo e fo -s nd fo e posve -s s used. o Cesn coodnes -plne epesens e sll we level SWL nd -s posve upwd fo e SWL. Cesn nd Clndcl coodnes e eled b sn. Te foulon s bsed on e ssupons of del ncopessble nd nvscd flud. We ssue se floo s fl nd oonl nd sued -. Te equon of connu fo flud w veloc v nd dens s ven b v ounl of Mecl Scences & Mecs Educon 8
![Rn [6] pesened n noducon o e ecl nd pscl specs of e eo of we wves. He dscussed e wve eo of A nonlne wve eo of Sokes dl dncs n sllow we. He enoned bou e dncs of flon offsoe sucues.](/docs-images/47/20796029/images/page_2.jpg "In e pesen wok we pesen e bound vlue poble fo fed clnde n we of unfo dep. Te veloc poenl funcon ssfes e Lplce equon.")
3 o ncopessble flud e connu equon s v o n ncopessble nd nvscd flud w oonl oon we cn noduce veloc v suc Equons nd eld e veloc poenl ssfes e Lplce equon n e flud don.e. 4 We wll ssue s e onc. Te oonl foce coponens lon decons e ven b P d d P sn d d 6 especvel. Hee s e P s e pessue on e cuved sufce of e clnde wc cn be copued fo e veloc poenl usn Benoulls equon. s e fee sufce elevon funcon. Bound Vlue Poble n es of Veloc Poenl nd ee Sufce Elevon uncons 5 Te bound vlue poble n es of s cn be epessed s nd Govenn Equon: 7 Dnc fee sufce bound condon e fee sufce 8 : ounl of Mecl Scences & Mecs Educon 9
4 ounl of Mecl Scences & Mecs Educon Knec fee sufce bound condon e fee sufce : 9 Snce e clnde s fed e nol veloc s eo so e bod sufce bound condon on e cuved sufce: n wee n s e ouwd nol. Assun se floo s fl nd oonl e boo bound condon - cn be epessed Te veloc poenl nd e sufce elevon cn be wen n es of Sokes epnson s ϕ O nd O Hee ka s e densonless sll pee wee k e wvenube A wve plude. Sub-nde s used o epesen e fs epnson e coespondn o lne ppoon nd sub-nde s used fo e second ode ppoon. A e fee sufce we ve so. Epndn b Tlos eoe bou we ve e odfed veloc poenl e fee sufce s ϕ O Dnc fee sufce bound condon now s ven b O Knec fee sufce bound condon now cn be wen s
5 ounl of Mecl Scences & Mecs Educon O Now copn e coeffcens of we cn sepe e fee sufce bound condons fo vous ode. o e fs ode e dnc fee sufce bound condon s nd e knec condon becoes Elnn e fs ode fee sufce elevon funcon fo e equons nd e cobned fee sufce condon cn be epessed 4 Te pessue P s deened fo Benoulls equon P wee. Subsun sees epnson fo e veloc poenl we ve O P 5 Snce e ncden wve s popn n -decon -coponen of e oonl foce vnses nd e -coponen s ven b d d P 6 Wn e -nel s e su of we obn
6 ounl of Mecl Scences & Mecs Educon d d d 7 Now we we... wee epesens e - ode conbuon. Tus e fs ode foce coponen s ven b d d 8 s Ode Dffcon Poble Te bound vlue poble fo fs ode poenl $\p_{}$ becoes Govenn Equon: 9 Cobned fee sufce bound condon: Bod sufce bound condon: Boo bound condon: We ssue e s e onc.e. [ ] e Φ Re wee. Φ s e cople veloc poenl nd s e fequenc of e wve. Now we solve e lne bound vlue poble b decoposn
7 e cople veloc poenl no ncden poenl sceed poenl Φ S. s follows Φ I. nd I S Φ Φ Φ Te cople ncden veloc poenl cn be obned s 4 I A k Φ β k k nd e cople sceed veloc poenl cn be deved s S A k Φ β H k k H Hee β β ; k nd H k k k e Bessel nd Hnkel funcons of fs knd of ode especvel [ 5]. We use e sepon of vbles eod o obn ese soluons b wn Φ R Θ Z. Ts elds Bessel equon fo e ncden poenl we obn e soluon n es of Bessel funcon of fs knd nd fo e sceed poenl we obn e soluon n es of Hnkel funcon of fs knd. Sceed poenl ssfes e don condon. Tus e fs ode veloc poenl cn be wen s A k Re β { k k H k e H 5 Dspeson Relon fo e s Ode Teo Cobned fee sufce condon fo e fs ode cople veloc poenl s ven b Φ Φ on ounl of Mecl Scences & Mecs Educon
![of fs knd of ode especvel [ 5]. We use e sepon of vbles eod o obn ese soluons b wn Φ R Θ Z.](/docs-images/47/20796029/images/page_7.jpg "Ts elds Bessel equon fo e ncden poenl we obn e soluon n es of Bessel funcon of fs knd nd fo e sceed poenl we obn e soluon n es of Hnkel funcon of fs knd.")
8 Now on we ve sufce Φ k n k Φ. o e cobned fee condon we cn we k n k Φ. Φ. Tus e dspeson elon cn be wen s k n k 6 wc descbes elon beween e wvenube k nd e fequenc. To obn copuonl esuls we consde non-densonl pees nd k./ so we ve n 7 Now we pesen e nuecl esuls of e dspeson elon fo vous dep o dus os. We consde e follown os: / / / 5. Te follown ble descbes e vlues of fo wc n. s Ode oce Coponen s ode cople veloc poenl s Φ A k β { k k H k H Snce n d n 8 n nd ppes n e foce copuon we need o consde onl es nvolvn n e epesson of Φ.. So fo nd we obn ounl of Mecl Scences & Mecs Educon 4
9 ounl of Mecl Scences & Mecs Educon 5 { [ Re 4 Re Re H e k k A H e k k A H H e k k A Tus e fs ode foce s { } { } k k A d d k k k A d d n sn 4 sn sn Re 4 Defnn α nd sn α we cn we n k - k A 4 α 9
10 wee α n Non-densonl coponen s 4 -α n k A Tus e u non-densonl fs ode foce s ven b 4 n k A Nuecl Resuls Now we pesen e nuecl esuls of e dspeson elon fo vous deps o dus os. We consde e follown os: / / / 5. Tble I descbes e vlues of fo wc n. ue I pesens e dspeson elon s funcon fo / fo vous. Zeos cn be obseved fo s p wc ces w e vlues enoned n ble I. Now we pesen e copuonl esuls fo e fs ode non-densonl oonl foce coponen ven b equon. Hoonl coponen s non-densonled b dvdn b A. Vous dep o dus os we consde e / / / 5. ue II nd fue III depc e nondensonl oonl foce coponen fo / 6 nd / 4. We dspl e pcl esuls fo e non-densonl oonl foce coponen fo vous fo pcul dep o dus o /. I s been dspled n fue IV. Now we pesen e non-densonl u oonl foce coponen ven b e equon. o / / / 5. ue V depcs e u non-densonl oonl foce coponen. ounl of Mecl Scences & Mecs Educon 6
11 o ll e bove esuls s obvous e foce coponen s oe effecve wen e pee nes beween.5 nd.5. Le e dep s le e foce. Te effec s nelble fo le. Tble I Zeos of Dspeson elon 7 fo vous nd / ue. Zeos of Dspeson elon 7 fo vous w / ounl of Mecl Scences & Mecs Educon 7
12 ue II Non-densonl oonl foce coponen fo vous / w / 6 ue III Non-densonl oonl foce coponen fo vous / w / 4 ounl of Mecl Scences & Mecs Educon 8
13 ue IV Non-densonl oonl foce coponen fo vous w /. ue V Non-densonl u oonl foce coponen fo vous /. Dbu B P.D. Te Unves of Tes-Pn Aecn USA ounl of Mecl Scences & Mecs Educon 9
14 Refeences AbowM. nd Seun I. A. Hndbook of Mecl uncons Dove Publcons New ok 97. Den R. G. nd Dlple R. A. Wve Mecncs fo Ennees nd Scenss Pence-Hll Inc New ese 984. Debn L. Nonlne We Wves Acdec Pess London 994. onson R.S. A Moden Inoducon o e Mecl Teo of We Wves Cbde Unves Pess Cbde 997. McLcln N. W. Bessel uncons fo Ennees Clendon Pess 955. Rn M. WATER WAVES: Reln Moden Teo o Advnced Enneen Pcce Ofod Unves Pess Ofod 994. ounl of Mecl Scences & Mecs Educon
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