CGWAVE: A Coastal Surface Water Wave Model of the Mild Slope Equation

Transcription

1 Tchncal Rport CHL-98-xx August 1998 CGWAVE: A Coastal Surfac Watr Wav Modl of th Mld Slop Equaton Zk Dmrblk Vjay Panchang U.S. Army Corps of Engnrs Unvrsty of Man Watrways Exprmnt Staton School of Marn Scncs 3909 Halls Frry Road Lbby Hall Vcksburg, MS Orono, ME by Approvd for Publc Rlas; Dstrbuton Is Unlmtd Prpard for Hadquartrs, U.S. Army Corps of Engnrs

2 CONTENTS PREFACE...v CONVERSION FACTORS, NON-SI TO SI UNITS OF MEASUREMENT...x 1. INTRODUCTION...1. BASIC EQUATIONS BOUNDARY CONDITIONS FINITE ELEMENT FORMULATION Opn Sa Problms Harbor Problms Altrnatv Opn Boundary Tratmnt Knmatc Paramtrs Floatng Docks ITERATIVE CG SOLUTION METHOD GENERATION OF FINITE-ELEMENT NETWORK PROCEDURE FOR STAND-ALONE USAGE OF CGWAVE Fl Dscrptons Raw Data Procssng Program rsol Vwng XYZ Data and Slctng Opn Boundary End Ponts XYZ Fl Format Program rsol Craton of Fnt Elmnt Grd Gomtry Boundary Condtons Th Gnral Informaton Fl (*.gn) Assmbly of Input Informaton - Program rform Th Wav Modl - Program cgwav Post-procssng th Output - Program trans Program trans Slctd Output at Ponts/Lns/Rctangular Sub-rgons Vwng Solutons n SMS Instructons for CGWAVE Intrfac n SMS (MODE- Usag) Cratng a Sz Functon Cratng th Background Scattr St Cratng th Coastln and Inlt Arcs Dfnng Domans Assgnng Rflcton Coffcnts Assgnng Opn Ocan Attrbuts Sttng th Actv Scattr Data Functon Cratng a Msh from Fatur Objcts Savng th CGWAVE.dat and.xyz (1-d) Fls Chckng Msh Qualty...7

3 Runnng CGWAVE EXAMPLES Shoal on a Slopng Bach Irrgular Wav Propagaton Ovr a Shoal Rctangular Harbor - Rsonanc Crcular Island Ponc d Lon Inlt, FL Barbrs Pont Harbor, HI Onslo Bay, NC Rfrncs SF98 v

4 LIST OF FIGURES Fgur 1. Dfnton Sktch of Modl Doman...11 Fgur. Dfnton Sktch...13 Fgur 3. A Typcal Elmnt...14 Fgur 4. Boundary Sgmnts...19 Fgur 5. Dfnton Sktch for Ln Intgrals Along th Opn Boundary Γ...3 Fgur 6. Dfnton Sktch for Floatng Structurs...41 Fgur 7. Modld Doman of Brkhoff t al. (198) Fgur 8. Wav Hght Comparsons for a Low Rsoluton Doman...77 Fgur 9. Wav Hght Comparsons for Incrasng Doman Rsolutons Fgur 10. Wav Hght Comparsons from Lnar and Non-lnar CGWAVE Runs on a Doman Rsoluton of 15 Ponts pr Wavlngth...79 Fgur 11. Modl Doman for Wav Propagaton ovr an Ellptc Shoal [aftr Vncnt and Brggs (1989)]...85 Fgur 1. Doman wthout Coastal Boundary Effcts...85 Fgur 13. Wav Hght Comparsons for M1 Input Condton...86 Fgur 14. Wav Hght Comparsons for M Input Condton...87 Fgur 15. Estmatd Wav Hghts for B1 Input Spctrum, 0 Drctonal Dscrtzaton...88 Fgur 16. Wav Hght Comparsons for B1 Input Condton, 4.14 Drctonal Dscrtzaton...89 Fgur 17. Wav Hght Comparsons for N1 Input Condton, 4.14 Drctonal Dscrtzaton...90 Fgur 18. Rctangular Rsonanc Harbor wth Sm-crcular Opn Boundary...93 Fgur 19. Thortcal and Numrcal Rsonanc Curvs for a Fully Rflctng Rctangular Harbor Fgur 0. Harbor Rspons Curvs for Varous Valus of Coastal Rflcton Coffcnt...95 Fgur 1. Harbor Rspons Curvs for Varous Valus of Frcton Factor Fgur. Cross Scton of Island and Shoal Bathymtry Fgur 3. Comparson of Analytc and Numrcal Wav Ampltuds...98 Fgur 4. Phas Dagram for Normal Incdnt Drcton, 15-scond Prod Wav Condton Fgur 5. Estmatd Wav Hght Dagram for Normal Incdnt Drcton, 15-scond Prod Wav Condton Fgur 6. Comparson Btwn Laboratory Data and CGWAVE Wav Hght Estmats along a Transct Fgur 7. Wav Phas Dagram for Normal Incdnt Drcton, 100-scond Prod Wav Condton Fgur 8. Estmatd Wav Hght Dagram for Normal Incdnt Drcton, 100-scond Prod Wav Condton v

5 Fgur 9. Wav Phas Dagram for Normal Incdnt Drcton, 35-scond Prod Wav Condton Fgur 30. Estmatd Wav Hght Dagram for Normal Incdnt Drcton, 35-scond Prod Wav Condton v

6 LIST OF TABLES Tabl 1. Summary of Input Informaton, Kyboard-slctd Output Opton...63 Tabl. Modl Input Data...81 Tabl 3. Drctonal Rsolutons Usd for Modl Input...8 Tabl 4. Modl Input Data...91 v

7 PREFACE Th rsarch dscrbd n ths rport was authorzd and fundd undr Work Unt 3969, Dvlopmnt of a Nw Gnraton Fnt Elmnt Harbor Wav Modl, of th Coastal Rsarch Program, sponsord by Hadquartrs, U.S. Army Corps of Engnrs. Admnstratv rsponsblty s assgnd to th U.S. Army Engnr Watrways Exprmnt Staton (WES) Coastal and Hydraulcs Laboratory (CHL), Dr. Jams R. Houston, Drctor, and Mr. Charls C. Calhoun, Jr., Assstant Drctor. Ms. Carolyn M. Holms, CHL, was th Program Managr, and Dr. Zk Dmrblk, Navgaton and Harbors Dvson (NHD), CHL, was th Prncpal Invstgator for th work unt. Dr. Dmrblk workd undr th admnstratv suprvson of Dr. Martn C. Mllr, Chf, Coastal Hydrodynamcs Branch, and Mr. Gn E. Chatham, Chf, NHD, CHL. Ths rport dscrbs th rsarch that has culmnatd n a stat-of-th-art, gnral-purpos wav prdctv modl calld CGWAVE. Issus that ar mphaszd n ths rport nclud thory and numrcal mplmntaton aspcts of ths modl and a st of xampls that llustrat th applcaton of CGWAVE to ral-world problms. A stp-by-stp usr s gud s also provdd n Scton 7 to facltat th modl s usag n projcts. Th study was prformd and th rport prpard ovr th prod 1 Jun 1995 through 15 August Dr. Dmrblk and a tam of rsarchrs ld by Dr. Vjay Panchang from th Unvrsty of Man, Orono, Man, dvlopd th numrcal modlng goals, concpts, and mthodology. Th Unvrsty of Man tam ncludd Drs. Bngy Xu and Davd Stwart, Mssrs. Luzh Zhao, Karl Schlnkr, Nshchy Chhabra, and W Chn. Th study tam compltd dvlopmnt, mplmntaton, and tstng of th modl. Th fld valdaton part of ths rsarch was collaboratd wth Dr. Mchl Okhro and Profssor Robrt Guza of th Scrpps Insttuton of Ocanography, La Jolla, Calforna. At th tm of publcaton of ths rport, Drctor of WES was Dr. Robrt W. Whaln. Commandr was COL Robn R. Cababa, EN. Th contnts of ths rport ar not to b usd for advrtsng, publcaton, or promotonal purposs. Ctaton of trad nams dos not consttut an offcal ndorsmnt or approval of th us of such commrcal products. Convrson Factors, v

8 Non-SI to SI Unts of Masurmnt Non-SI unts of masurmnt usd n ths rport can b convrtd to SI unts as follows: Multply By To Obtan acrs 4, squar mtrs dgrs (angl) radan ft mtrs knots (ntrnatonal) mtrs pr scond mls (US Statut) klomtrs nautcal mls 1.85 klomtrs x

9 1 INTRODUCTION Wav clmat plays a vry mportant rol n all coastal projcts. Howvr, n most cass, lttl (f any) wav data ar avalabl for ngnrng constructon and plannng. Fld obsrvaton and physcal modlng of wavs ar xtrmly dffcult, costly, and tmconsumng. Buoys ar far away from th projct st, and rmot-snsng nstrumnts do not systmatcally provd wav data at th dsrd rsoluton n th nar shor rgon. Snc no data-rcordng nstrumnt can antcpat futur sa stats, th dsrd sa-stat nformaton may b obtand and plans valuatd wth rlabl mathmatcal modlng tchnqus. It s ssntal to hav rlabl nformaton on wav condtons for many coastal and ocan ngnrng problms. Th most mportant wav condtons for dsgn and assssmnt n projct studs n th ara of ntrst nclud th wav hghts, wav prods and th domnant wav propagaton drctons. Typcally, ths wav paramtrs ar obtand from a wav transformaton modl that transfrs th wav data collctd at som rmot dp watr st to th locaton of th projct n th nar shor. As wavs mov from dpr watrs to approach th shor, ths fundamntal wav paramtrs wll chang as th wav spd changs and wav nrgy s rdstrbutd along wav crsts du to th dpth varaton btwn th transfr sts and th prsnc of slands, background currnts, coastal dfns structurs, and rrgularts of th nclosng shor boundars and othr gologcal faturs. Wavs undrgo th svrst chang nsd th surf zon whr wav brakng occurs and n th rgons whr rflctd wavs from coastln and structural boundars ntract wth th ncdnt wavs. Untl rcntly, th lnar wav ray thory was usd for wav transformaton by tracng rays from dp watr to th projct st nar shor. Th ffcts on wav propagaton of th wav hght and drcton along th wav crst ar gnord n th ray thory snc ths thory assums that wav nrgy propagats only along a ray and thus, 1

10 nrgy flux s consrvd btwn two adjacnt rays. As a consqunc of ths assumpton, ray thory braks down whn wav ray crossngs and caustcs occur bcaus th physcs of dffracton ar totally gnord n th numrcal ray modls. Startng n th arly 1980 s, coastal dsgnrs and rsarchrs hav rcognzd th mportanc of th combnd ffcts of rfracton and dffracton and bgun to dvlop mprovd thors and assocatd numrcal modls. Thr ar ndd svral wav thors avalabl that could adquatly dscrb th combnd rfracton and dffracton of wavs from dp watr to shallow watr (Dmrblk and Wbstr 199 and 1998). On of ths s th mld-slop quaton (MSE). Ths s a dpth-avragd, llptc typ partal dffrntal quaton whch gnors th vanscnt mods (locally manatd wavs) and assums that th rat of chang of dpth and currnt wthn a wavlngth s small, hnc th mld-slop acronym. Numrous MSE-basd numrcal modls hav bn dvlopd for prdctng th wav forcs on offshor structurs and studyng wav flds around th offshor slands. Numrcal, laboratory and fld tsts of th NSE modls hav shown that th MSE can provd accurat solutons to problms whr th bottom slop s up to 1:3. From a practcal standpont, th computatonal rqurmnts for solvng th MSE ar munch largr than thos for ray tracng. Th rasons for ths ar bcaus th MSE s a twodmnsonal quaton and has to b solvd as a boundary-valu problm wth approprat boundary condtons. Th ntr doman of ntrst must b dscrtzd and solvd smultanously and th lmnt sz has to b small nough that thr ar about 10 to 15 nods wthn ach wavlngth. Ths rqurmnts plac svr dmands on computr rsourcs whn applyng MSE modls to larg coastal domans. A dffcult problm n th prdcton of wavs nar shor s to dtrmn whr approxmatly th wav brakng (and brakr ln) occurs whn wavs ar nsd th sur zon. In numrcal modls prsntly usd, ths locaton s not known a pror, and s usually slctd wth an ad hoc crtra basd on th rato of wav hght to local watr

11 dpth. Bottom frcton and dsspaton from th surroundng land boundars (.. ntranc losss at th mouth of a harbor) may also b mprcally ncorporatd nto MSE modls. A smplfd vrson of th MSE s known as th parabolc approxmaton (PA), whch usually gratly rducs th xcssv computatonal dmands of MSE modl at th xpns of furthr assumptons and smplfcatons whch may rndr th numrcal prdctons naccurat and napproprat for many coastal and ocan ngnrng problms (Panchang t al. 1998). Th only purpos of adaptng th PA s to convrt th MSE to a st of smplr quatons that dscrb a wav propagatng n a prscrbd drcton whl stll takng both rfracton and dffracton n th latral drcton nto account. Th gratst advantag of PA s ts numrcal ffcncy, t can b solvd rathr asly by numrcal mans and thus could b usd for prdctng wav transformaton ovr a rlatvly larg coastal rgon. Whn rflcton s of major ntrst, as t s n harbors, th MSE should b usd snc th PA gnors rflcton. On must also b rmndd that th PA assums that th lngth scal of th wav ampltud varaton n th drcton of wav propagaton (x drcton) s much longr than that n th transvrs drcton (y drcton). Th PA s drvd on th assumpton that prcntag changs of dpth wthn a typcal wavlngth ar small compard to th wav slop. For dtals about PA modls, s Booj (1981), Lu (1983), Krby (1983), Lu and Tsay (1984), and Krby and Dalrympl (1984). Th PA has bn vrfd xtnsvly by laboratory studs and fld applcatons (Brkhoff t al. 198), Lu and Tsay (1984), Krby and Dalrympl (1984), Vncnt and Brggs (1989), (Dmrblk 1994, Dmrblk t al. 1996a and 1996b), and Panchang t al (1998). Th mld-slop wav quaton (also known as th "combnd rfractondffracton" quaton), frst suggstd by Eckart (195) and latr r-drvd by Brkhoff (197, 1976) and othrs, s now wll-accptd as th mthod for stmatng coastal wav condtons. It can b usd to modl a wd spctrum of wavs, snc t passs, n th lmt, to th dp and shallow watr quatons. Although th quaton was dvlopd n th md-svnts, computatonal dffcults prcludd th dvlopmnt of a modl for th 3

12 complt mld-slop quaton (xcpt for vry small domans). Typcally, coastal wav propagaton problms nvolv th modlng of vry larg domans. For xampl, consdr th cas of 1 scond wavs n watr of 15 m dpth. Th wavlngth L s about 136 m; an 8 km by 8 km doman s about 3600L n sz. Th dffcults assocatd wth solvng such larg problms spawnd th dvlopmnt of svral smplfd modls (.g. th "parabolc approxmaton" modls (Dalrympl t al. 1984; Krby, 1986), RCPWAVE modl (Ebrsol, 1985), EVP modl (Panchang t al 1988), tc.). Howvr, ths smplfd modls compromsd th physcs of th mld-slop quaton: thy modl only on- or two-way propagaton wth wak latral scattrng. Such modls ar hnc applcabl only to rctangular watr domans for a vry lmtd rang of wav drctons and frquncs. Most ralstc coastal domans wth arbtrary wav scattrng cannot b modld wth ths smplfd modls. Ths manual dscrbs a wav modl calld CGWAVE dvlopd at th Unvrsty of Man undr a contract for th U.S. Army Corps of Engnrs, Watrways Exprmnt Staton. CGWAVE s a gnral purpos, stat-of-th-art wav prdcton modl. It s applcabl to stmaton of wav flds n harbors, opn coastal rgons, coastal nlts, around slands, and around fxd or floatng structurs. Whl CGWAVE smulats th combnd ffcts of wav rfracton-dffracton ncludd n th basc mld-slop quaton, t also ncluds th ffcts of wav dsspaton by frcton, brakng, nonlnar ampltud dsprson, and harbor ntranc losss. CGWAVE s a fnt-lmnt modl that s ntrfacd to th SMS modl (Jons & Rchards, 199) for graphcs and ffcnt mplmntaton (pr-procssng and post-procssng). Th classcal supr-lmnt mthod as wll as a nw parabolc approxmaton mthod dvlopd rcntly (Xu, Panchang and Dmrblk 1996), ar usd to trat th opn boundary condton. An tratv procdur (conjugat gradnt mthod) ntroducd by Panchang t al (1991) and modfcatons suggstd by L (1994) ar usd to solv th dscrtzd quatons, thus nablng th modlr to dal wth larg doman problms. Ths manual provds a brf rvw of th basc thory n Sctons and 3, an ovrvw of how ths thory s mplmntd n 4

13 Sctons 4 through 6, a stp by stp gud for usng ths wav modl n Scton 7, and a st of xampls that wr run usng CGWAVE n Scton 8. 5

14 BASIC EQUATIONS Th soluton of th two-dmnsonal llptc mld-slop wav quaton s a wllaccptd mthod for modlng surfac gravty wavs n coastal aras (.g. Chn & Houston, 1987; Chn, 1990; Xu & Panchang, 1993; M, 1983; Brkhoff, 1976; Kostns t al., 1986; Tsay and Lu, 1983). Ths quaton may b wrttn as: whr Cg ( CCg η$ ) + σ η$ = 0 (1) C $η ( x, y ) = complx surfac lvaton functon, from whch th wav hght can b stmatd σ = wav frquncy undr consdraton (n radans/scond) C(x,y) = phas vlocty = σ/k C g (x,y) = group vlocty = σ / k =nc wth 1 kd n = + 1 snh kd () k(x,y) = wav numbr (= π/l), rlatd to th local dpth d(x,y) through th lnar dsprson rlaton: σ = gk tanh (kd) (3) Equaton 1 smulats wav rfracton, dffracton, and rflcton (.. th gnral wav scattrng problm) n coastal domans of arbtrary shap. Howvr, varous othr mchansms also nflunc th bhavor of wavs n a coastal ara. Th mld-slop quaton can b modfd as follows to nclud th ffcts of frctonal dsspaton (Dalrympl t al 1984; Chn 1986; Lu and Tsay 1985) and wav brakng (Dally t al 1985; D Grolamo t al 1988): 6

15 Cg ( CCg η$ ) + σ + σw + Cgσγ η$ = 0 (4) C whr w s a frcton factor and γs a wav brakng paramtr. Followng Dalrympl t al. (1984), w hav usd th followng form of th dampng factor n CGWAVE: n f r ak w = σ k 3π ( kd + snh kd) snh kd (5) whr a (= H/) s th wav ampltud and f r s a frcton coffcnt to b provdd by th usr. Th coffcnt f r dpnds on th Rynolds numbr and th bottom roughnss and may b obtand from Madsn (1976) and Dalrympl t al. (1984). Typcally, valus for f r ar n th sam rang as for Mannng s dsspaton coffcnt n. Spcfyng f r as a functon of (x,y) allows th modlr to assgn largr valus for lmnts nar harbor ntrancs to smulat ntranc loss. For th wav brakng paramtr γ, w us th followng formulaton (Dally t al 1985, Dmrblk 1994, Dmrblk t al. 1996b): γ = χ 1 d Γ d 4a (6) whr χ s a constant (a valu of 0.15 s usd n CGWAVE followng Dally t al (1985)) and Γ s an mprcal constant (a valu of 0.4 s usd n CGWAVE). In addton to th abov mchansms, nonlnar wavs may b smulatd n th MSE. Ths s accomplshd by ncorporatng ampltud-dpndnt wav dsprson, whch has bn shown to b mportant n crtan stuatons (Krby and Dalrympl 1986). Th nonlnar dsprson rlaton usd n plac of Equaton 3 s 7

16 5 [ 1 ] { } σ 1 = gk + ( ka) F tanh kd tanh kd + kaf (7) whr F F 1 cosh( 4kd) tanh ( kd) = 4 8snh ( kd) 4 kd = snh( kd) (8) 8

17 3 BOUNDARY CONDITIONS Along rgd, mprmabl vrtcal walls, no flow normal to th surfac gvs $ η / n = 0. Howvr, n gnral, th followng partal rflcton boundary condton appls along coastlns or prmabl structurs η$ = α η$ n (9) whr α = α + α s a complx coffcnt. For smplcty, α s gnrally rprsntd as 1 α = k 1 K 1+ K r r (10) whr K r s th rflcton coffcnt (Tsay and Lu, 1983; Chn and Houston 1987). Along th opn boundary whr outgong wavs must propagat to nfnty, th Sommrfld radaton condton appls lm kr kr r k η$ S 0 (11) whr $η S s th scattrng wav potntal. It s shown n M (1983) that th dsrd scattrd wav potntal $η S, whch s a soluton of th mld-slop quaton and satsfs th radaton condton Equaton 11, can b wrttn as: ( ) η$ = H (kr) α cos nθ+ β sn nθ S n= 0 n n n (1) 9

18 whr H n (kr) ar th Hankl functons of th frst knd. Th Hankl functons of th scond knd do not satsfy th Sommrfld radaton condton at nfnty and ar hnc xcludd from (1). Howvr, th η S gvn n (1) rqurs that th xtror doman b of constant dpth. Also for harbor problms (Fgur 1), th scattrd wav potntal as dscrbd by (1) dmands straght, collnar and fully rflctv coastlns n th xtror rgon. To ovrcom ths problms, Xu, Panchang and Dmrblk (1996) hav dvlopd an altrnatv schm n dalng wth th opn boundary condton. Ths conssts of usng th followng parabolc approxmaton along th opn boundary : η$ r S η$ η$ S + p S + q = 0 (13) θ whr p = k r + k r + k r + 0 k r and q = (14) k 0r In Equaton 14, k 0 can b takn as th wav numbr corrspondng to th avragd watr dpth along th opn boundary Γ. Wthn th modl doman Ω, th mld-slop quaton appls. Th parabolc approxmaton (13) wll b usd only along th sm-crcular arc Γ as th opn boundary condton. Th actual mplmntaton of ths boundary condtons s dscrbd latr. 10

19 y Opn Boundary Extror Γ Ω θ x Incdnt Wav Drcton A Modl Doman Land C A 1 Fgur 1. Dfnton sktch of modl doman 11

20 4 FINITE ELEMENT FORMULATION 4.1 Opn Sa Problms CGWAVE uss th fnt-lmnt mthod, whch s a powrful approach for modlng coastal phnomna n rgons of complx shap. In th cas of a group of scattrrs surroundd by an opn sa of constant dpth, th ncdnt wav may b wrttn as (Dmrblk and Gaston, 1985) kr cos( θ θi ) n η$ I = A = A εn J n (kr)cos n( θ θi ) (15) n= 0 whr A s th ampltud of th ncdnt wav, θ I s th ncdnt wav angl wth rspct to th x-axs, J n s th n-th ordr Bssl functons of th frst knd and 1 ε n = whn n = 0 whn n 0 (16) Th ncdnt drcton s dfnd such that th ncdnt wav travls n th postv x- drcton whn θ I s qual to zro; th x-drcton s obtand from th bathymtry data fl that s nput to th CGWAVE program. Ths bathymtry fl should b orntd such that th x-axs ponts to th ast. 1

21 As shown n Fgur, th ntr doman s sparatd nto two sub-domans. Doman Ω s th numrcal modl doman. Doman Ω 0 s th xtror doman xtndng to nfnty. W assum that complcatd topography, structurs, and slands, ar locatd nsd th crcular boundary Γ (n doman Ω). In Ω 0, th total wav potntal can b wrttn as th sum of ncdnt wav potntal and th scattrd wav potntal: Ω 0 Γ Ω B Fg. Dfnton sktch Γ η$ = η$ + η$ xt I S (17) For brvty, w wrt th govrnng Equaton 4 n th gnral form: ~ + ~ η $ bη $ = 0 (18) ( a ) whr ~ a CC g and ~ b C g σ + σw + C gσγ. C M (1983) has shown that th problm of solvng Equaton 18 wth boundary condtons dscrbd by (9) on coastlns/structurs and by Equaton 11 at nfnty s quvalnt to th statonary of th followng functonal J: ~ [ ( η$ ) Φ ] J = a b da a ds 1 ~ 1 α~ η $ + Ω Γ ~ 1 $ S ($ S + $ I ) a η$ S + η$ I η $ ds n η η η n B (19) Th soluton of th wav potntal can b found by mnmzng J ovr doman Ω. 13

22 In th fnt-lmnt mthod, w frst dscrtz th computatonal doman Ω nto a ntwork of smpl trangular lmnts. Th sz of ths lmnts should b much smallr than both th local wavlngth and th scal of local bathymtrc varaton. Fnr rsoluton s also dsrabl at placs whr th chang of ampltud n spac s rapd (.g. nar caustcs ). Ovr ach trangular lmnt, th wav potntal $η s approxmatd by th followng lnar two-dmnsonal functon $η, 3 η$ = N η$ = = 1 [ N 1 + N + N 3] η$ 1 η$ η $ 3 (0) whr $η rprsnt th wav potntals at th cornrs (nods) of th lmnt (Fgur 3) and N ar th lnar ntrpolaton functons: N a + b x + c y = (1) wth a = x y y x j k j k b = y y j k c = x x k j () j k Fg. 3 A typcal lmnt and = ara of lmnt = x x x y 1 1 y 3 y3 (3) 14

23 15 Not that for futur us, w gv N dx dy = 3 m N N N dx dy = 6 60, = j = k 60, = j or = k or j = k 1 60, j k j k Notc that n th abov formulatons, (, j, k) ar dnotd n a countr-clockws mannr. For lmnt, th followng rlatons can b stablshd for substtuton nto (19): = + = = $ $ $ η η η N x j N y r r (4) ( ) { } = = + = = = $ $ $ $ $ $ η η η η η η T 1 a 1 a a 1 a 3 a a 1 a a N x N y N x N x N x N x N x N x N x N x { } a 3 a 3 a 1 a 3 a a 3 a N x N x N x N x N x N x N x η $ (5) and

24 ( η ) = $ 3 = N η$ T { η } = 1 ( ) N1 ( N N ) ( ) 1 N 1 N 3 ( N N1 ) ( N ) ( N N3) { η } ( 3 1 ) ( N N N 3N ) ( N 3) $ $ (6) whr α = 1, s th dummy-ndx notaton and ( x x y), x 1,.. Equaton 5 rprsnts th sum for α = 1 and α =. Not that { $η } T = [ $η 1 $η $η 3 ] from Equaton 0. lmnt : W may also assum that th coffcnts ~ ~ ( a, b ) n Equaton 19 vary lnarly on 3 3 ~ ~ ~ ~ a = N a, b = N b (7) = 1 = 1 For th frst part of Equaton 19, w may wrt I 1 ~ ~ ( ( η$ ) η $ ) 1 = a b da = Ω 1 W T { η$ } [ K1]{ η$ } 1x3 3x3 3x1 (8) whr N N K ~ j ~ 1,, j = a N β β dxdy b N N N jdxdy x x β β α α (9) 16

25 whr β = 1,, 3 s anothr dummy-ndx notaton. Snc N x α N x j α 1 ( ) ( b b c c ) j j = + (30) and N dxdy β = 3 (31) w hav ~ a N N x β β α N x j α dxdy = ~ a + ~ a + ~ a 1 ( b b + c c ) 1 3 j j (3) Th scond trm of Equaton 9 s ~ b β N N N dxdy j ( ) j j ( j ) ~ ~ ~ = b N N dxdy + b N N dxdy + b N N N dxdy β ~ ~ ~ ( b b j b k ) = k j k (33) for j, and 17

26 ~ b β N N N dxdy j 3 ( ) k1 k1( ) k k( ) ~ ~ ~ = b N dxdy + b N N dxdy + b N N dxdy β ~ ~ ~ ( b bk1 bk) = (34) for = j, whr k1 & k ar th othr two nods of lmnt. Now, Equaton 9 can b wrttn as ~ a1 + ~ a + ~ a3 K1,, j = ( b b c c j + j) 1 ( b b j b k ) 60 ~ ~ ~ + + whn j ( b b k1 b k ) 30 3 ~ ~ ~ + + whn = j (35) Aftr computng th lmnt matrx [ K 1 ] for all th lmnts ( = 1,, 3,..., E), whr E s th total numbr of lmnts, w can assmbl thm nto a global matrx [K 1 ]. Equaton 8 bcoms { η T } [ ]{ η 1 1 } { η T } [ 1]{ η } 1 I1 = K = K $ $ $ $ Ω 1x3 3x3 3x1 (36) In (36) { η$ } T = { η$, η$, η$,, η$ } 1 3 N (37) whr N s th total numbr of nods n doman Ω 18

27 Th scond part of Equaton 19 s I 1 a = α ~ η$ ds (38) B whr B dnots all th coastln and ntrnal land boundars. Along a sgmnt p of th coastal boundars B (Fgur 4), th wav surfac lvaton functon $η and th varabl ~ a ar approxmatd by lnar functons as bfor: Modl Doman ModlM j p p p p p p η$ = N η$ + N η$ (39) j j ~ p p a N ~ p p a N ~ p = + a (40) j j Fgur 4. Boundary sgmnts Hr, w assum that j s th postv drcton of th boundary, countr-clockws for th coastln boundars and clockws for ntrnal land boundars. Frst, lt us fnd th xprsson for ( $η P ) on sgmnt P: P ( η$ ) p p p = ( N $ $ p η + N η ) j j { } ( ) ( ) P P P P P N N N j = η$ η$, j ( N P P j N ) ( N j ) P η$ (41) P η$ j P Equaton 38 can thn b wrttn as 19

28 I = = 1 B 1 a α~ η $ ds Nb 1 P= 1 P P P { η$, η$ j }[ K ] P η$ P η$ j (4) whr N b s th total numbr of nods along boundary B and P P (pont j) P P P K = α~ a N N N ds,, j b (pont ) b ~ P ~ P ( j ) 1 1 L P α a + a for j = 1 P P P αl ( 3a ~ + ~ a j ) for = j 1 j (43) Assmblng all sgmnts on coastal boundary B, w hav I N b 1 1 = $ $ η$ P= 1 η$ P P 1 {, P }[ K P B T B η ηj ] { } [ K P ]{ } = $ $ j η η (44) 1 N N b b N b N b 1 Th thrd part ntgral n Equaton 19 s ~ 1 ( ) a $ $ $ $ $ S S I ηs η η I η$ η + η + ds = I + I + I + I n n G 1 S I ~ $ a n ds 3 = η η $ S G S I - ~ $ a n ds I 4 = η η $ 4 G I I - ~ $ a n ds I 5 = η η $ 5 G S I ~ $ 6 = + aη η $ I ds I6 n G (45) 0

29 For smplcty, th opn boundary Γ s assumd to b a crcl of radus R. For computatonal purposs, th srs for th scattrd wavs (Equaton 1) and th srs for ncdnt wavs (Equaton 15) ar truncatd aftr a fnt numbr of trms. In prncpl, tral and rror should b prformd n modlng a crtan cas n ordr to choos th approprat numbr of trms. Hr, w assum that th srs wll b truncatd aftr m trms. By usng th orthogonalty of trgonomtrc functons: π sn nθsn mθdθ = 0 π 0 whn n m whn n = m (46) 0 whn n m π cosnθcosmθdθ = π whn n = m 0 0 π whn n = m = 0 (47) and substtutng $η S wth Equaton 1, th ln ntgral I 3 n Equaton 45 can b valuatd analytcally, as follows m π I 3 = kra 0H 0H 0 + ( + ) H H α α β n= 1 ~ ' ' n n n n whr k and ~ a can b takn as avrag valus along Γ and (48) d H H kr d kr H kr n n ( 1) ' n ( 1) ( ), H n ( ) ( ) r= R (49) For convnnc of mathmatcal manpulaton latr on, w dfn th followng vctor for th unknown coffcnts α and β : 1

30 T { µ } = { α 0, α 1, β1, α, β, α m, βm} 1 M..., (50) whr M = m +1. Th ntgral I 3 can now b rwrttn as I 1 T = { µ } [ K ]{ µ } (51) 3 3 whr [K 3 ] s a dagonal matrx of dmnson M by M: [ K 3] = π kra ~ ' ' ' ' ' dag { H 0H 0, H 1H1, H 1H1,..., H mh m, H mh m} (5) Th ntgral I 4 n Equaton 45 s I 4 = G ~ $ S aη η $ n ds NΓ m P P P P ' ' ka ~ ( N η$ + N j η$ j ) α 0H 0 + H n( α ncosnθp + βnsnnθp) ds P=1 sgmnt P n= 0 N m ka ~ Γ P P P ' ' = L ( η$ + η$ j ) 0H 0 + H n( ncosn P + nsnn P) α α θ β θ P=1 n= 0 (53) whr L P s th lngth of sgmnt P and Ν Γ s th total numbr of sgmnts (= total numbr of nods) along th crcular boundary Γ (Fgur 5). In Equaton 53, th valu of $η S / n s approxmatd by ts valu at th cntr of sgmnt P. Equaton 53 may b wrttn n matrx form : I Γ T { ˆ }[ ]{ µ } 4 = η K 4 (54)

31 whr { $η Γ } s th subst of { $η } for nods stuatd on boundary Γ : j P { η Γ T $ } { η Γ $, η Γ $, η Γ $,, η Γ = $ N } Γ 1xN Γ Ω θ P (55) and [K 4 ] s a fully populatd N Γ M matrx : Γ Fgur 5. Dfnton sktch for ln ntgrals along th opn boundary Γ ka ~ [ K 4] = ' 1 ' 1 ' 1 H 0L H n( cosnθn 1 + cosnθ1) L H n( sn nθn 1 + sn nθ1) L Γ Γ ' ' ' H 0L H n( cosnθ1 + cosnθ ) L H n( sn nθ1 + sn nθ ) L ' NΓ ' NΓ ' H 0L H n( n + N n N ) L H n( n + N n N ) L cos θ Γ 1 cos θ sn θ Γ Γ 1 sn θ Γ whr n = 1,,..., m. (56) Th nxt ntgral n Equaton 45 s I 5. Smlar to th tratmnt n I 4, w wll tak th cntr valu of $η I / n and assum lnar varaton of $η for a sgmnt on boundary Γ. Substtutng $η I n I 5 by Equaton 15, t s asy to fnd I 5 = ~ $ I aη η $ n ds N T Γ { Q5} { η$ } P P ( η + ηj ) ( θp θi ) [ ( θp θi )] = Γ kaa ~ P L $ $ cos xp krcos = Γ P=1 (57) 3

32 whr N { } T 1 { 5} = ~ aa ( q + 1), ( q 1 + ), N..., ( q N 1 + N ) Γ Γ Γ Γ Q k q L q L q L (58) and q [ ] ( ) kr ( ) = cos θ θ xp cos θ θ, P = 1,,..., N Γ (59) P P I P I Th last ntgral I 6 n Equaton 45 nvolvs both $η S and $η I. Hr, th Bssl- Fourr form of $η I (s Equaton 15) wll b usd and th ntgral can b found analytcally: I = 6 I G ~ $ S aη η $ n ds m = kaa ~ n ' εn J n (kr)cosn( θ θi ) H n( α ncosnθ+ βnsnnθ) ds n= 0 n= 0 π m m n ' = kaa ~ εn J n (kr)cosn( θ θi ) H n( α ncosnθ+ βnsnnθ) Rdθ n= 0 n= 0 = 0 T { Q } { µ } 6 Γ m (60) whr T { Q } 6 = πkraa ~ ' ' ' m ' m ' { J 0H 0, J1H1cos θi, J1H1sn θi,..., J mhmcosm θi, J mhmsnmθi} (61) Now, w hav valuatd all th ntgrals of th functonal J dfnd by Equaton 19 usng a lnar trangular lmnt ntwork. Collctng ths ntgrals togthr, w hav 4

33 J I I + I I I + I = T B T { η$ } [ K ]{ η$ } { η$ } [ K ]{ η $ } + { µ } [ K ]{ µ } 1 B T Γ T T Γ T { η $ } [ 4]{ µ } { 5} { η $ } + { 6} { µ } - K Q Q 1 3 (6) Snc J s statonary, th followng must b tru for th soluton of th problm : J η$ = 0 = 1,, 3,..., N (63) and J µ j = 0 j = 1,, 3,..., M (64) Ths rlatons gv B [ K1]{ η$ } [ K ]{ η $ } -[ K 4]{ µ } { Q5} = (65) and T [ K 3]{ } -[ K 4] { } { Q 6} µ η$ Γ = (66) From Equaton 66, w hav 1 1 T { m } = [ K 3] { Q 6} + [ K 3] [ K 4] { } $η Γ (67) 5

34 Substtutng {µ} n Equaton 65 by Equaton 67, Equaton 65 bcoms B 1 T Γ 1 [ K1]{ η$ } [ K ]{ η$ } -[ K 4][ K3] [ K 4] { η$ } = { Q5} [ K 4][ K 3] { Q 6} (68) Fnally, aftr propr assmblng, w hav [ A]{ $η } = { f} (69) Equaton 69 s th dsrd lnar systm of quatons, whch s th fnt-lmnt rprsntaton of th mld-slop quaton for opn sa problms. Notc that th boundary condtons, ncludng coastln boundars and th crcular opn boundary, ar all consoldatd n Equaton 69. Th soluton mthod usd n CGWAVE for Equaton 69 s dscrbd n th nxt scton. 4. Harbor Problms Th fnt-lmnt formulaton gvn abov s for opn-sa offshor problms. In cas of harbor problms, th formulaton s analogous. Th only dffrnc arss from th tratmnt of th opn boundary condton. Th classcal tratmnt of ths problms assums that th coastlns outsd th modl doman ar straght, collnar and fully rflctv. Th xtror wav fld s wrttn as $η xt = $η I + $η R + $η s, whr $η I, $η R, and $η s rprsnt th ncdnt, th rflctd, and th scattrd wav flds, rspctvly. Basd on th assumptons, w dfn (Dmrblk and Gaston 1985) η$ = η$ + η$ 0 I R = A + A krcos( θ- θi ) krcos( θ+ θi ) n = A ε J (kr)cosnθ cosnθ n=0 n n I (70) whr A s th ncdnt wav ampltud and θ I s th ncdnt wav angl wth rspct to 6

35 th xtror coastlns as shown n Fgur 1. Th scattrd wav potntal $η s n th xtror rgon must tak th followng form n ordr to comply wth th xtror coastln boundary condtons: $η = H (kr) α cosnθ S n n n= 0 (71) as shown n Xu, Panchang and Dmrblk (1995). Th corrspondng functonal for harbor problms has th sam form as Equaton 19 xcpt that $η I n Equaton 19 has to b rplacd by $η 0 (Equaton 70), $η S taks th nw form gvn by Equaton 71, and th opn boundary Γ rprsnts th smcrcl as shown n Fgur 1. Th fnt-lmnt formulaton of harbor problms can now radly b found n a mannr smlar to th opnsa problms dscrbd abov, by rplacng $η I and $η S wth Equaton 70 and Equaton 71 and prformng th boundary ntgraton for I 4 through I 6 from 0 to π. 4.3 Altrnatv Opn Boundary Tratmnt For most practcal cass, th fully rflctv straght coastln assumpton n th classcal tratmnt of th opn boundary condton s mpropr and th ffcts may substantal. Xu, Panchang and Dmrblk (1995) hav shown that t s prfrabl to us th parabolc approxmaton (Equaton 13) as th opn boundary condton. For harbor problms, along th opn boundary Γ (Fgur 1) w us (13) as th boundary condton for th scattrd wavs. Matchng th potntal and ts normal drvatv along Γ and usng th parabolc opn boundary condton (13), w hav th total potntal as and $η = $η 0 + $η S (7) η$ η$ η $ η$ 0 S 0 = + = pη$ S + n r r r η$ S q (73) θ 7

36 Usng Equaton 7 to lmnat $η s ylds η$ η$ = pη$ q + g (74) n θ whr η$ 0 g = + pη$ 0 + r η$ q θ 0 (75) Equaton 74 s th opn boundary condton n trms of total wav potntal $η. For ths parabolc boundary problm, th dsrd Jacoban functonal may b mor complcatd than that of Equaton 19. Thrfor th Galrkn fnt-lmnt formulaton s usd. Accordng to th Galrkn approach, th wav potntal $η s approxmatd by n = 1 η$ = η$ N (76) whr $η s th soluton $η at nod and N (x, y) s th lnar ntrpolaton functon for nod. Th unknown $η can b dtrmnd from th orthogonalty condtons btwn functon N and lft-hand sd of Equaton 18, that s ~ ( ( a η) + b η $ ) N d = 0 ( = 1,..., N ) ~ $ Ω Ω (77) Usng th dvrgnc thorm, Equaton 77 bcoms 8

37 + N ~ η$ a n ds + ~ a N d + ~ b N d = 0 η $ Ω η $ Ω (78) C Γ Ω Ω whr C and Γ dnots coastal boundars and opn boundars rspctvly, and $η / n s th normal drvatv of $η. Undr th lnar assumpton, th ntrpolaton functon N s to satsfy ( ) N x, y = 1 for (x, y) at nod 0 to 1 for (x, y) wthn lmnts surroundng nod 0 for (x, y) outsd lmnts surroundng nod (79) Thrfor, N can b rprsntd as N ( x, y ) = N ( x, y) (80) whr rfrs to thos lmnts around nod and N (x,y) s th lnar ntrpolaton functon corrspondng to an lmnt and on of ts nod. Whn (x, y) s at boundary, Equaton 80 bcoms N ( x, y ) = N P P1 P ( x, y ) = N ( x, y ) + N ( x, y) (81) P whr P 1 and P ar th boundary sgmnts to thr sd on nod. Th functon N P 0 for all othr sgmnts. Wthn an lmnt, th valu of $η s obtand by substtutng (80) nto (76), ( ) η$ x, y = η$ N + η$ N + η$ N (8) j j k k 9

38 whr (x,y ), j(x j,y j ), k(x k,y k ) ar th thr nods of lmnt (s Fgur 3) and functon N (x,y) s gvn by Equatons 1 through 4. Whn (x, y) s on a boundary sgmnt wth nods and j at th nds, Equaton 8 s smplfd to: ( ) P P η$ x, y = η$ N + η$ N (83) j j P P P P whr N = (s s) / L, N = (s s) / L, and s s th rlatv coordnat along P and j j has valus of s and s j at th two ndponts. Th lngth of th sgmnt s L P = s s j. Th followng rlatons rfrrng to lnar functon N (x,y) and N P (x,y) ar dvlopd for latr us: N N = + x N y j = 1 b + c j (84) N dxdy = 3 (85) N N j N kdxdy = 6 / 60 for = j = k / 60 for = j or j = k or k = / 60 for j k (86) P N s = 1, P L N P j s = 1 (87) P L 30

39 P P N ds = L P (88) N P3 ds = P 1 4 L P (89) ( P ) P P j ( j ) P (90) P P N N ds = N N ds = 1 P 1 L Now substtutng th Equatons 80 and 81 nto Equaton 78, w hav P P P N ~ η$ a ds + ~ ~ a η $ N d Ω - bη $ N d Ω = 0 ( = 1,..., N) n (I) (II) (III) (91) whr rfrs to thos lmnts around nod whr N (x,y)=0, and P rfrs to th two boundary sgmnts on ach sd of nod whn s a boundary nod. For lmnt wth thr nods, j, and k, whr $η, gvn by Equaton 8, s substtutd nto th th, j th and k th Equaton of 91, and hnc th scond trms com out ( Ω) + ( j Ω) j + ( k Ω) II = ~ a N d η$ ~ a N N d η$ ~ a N N d η$ = A η$ + A η$ + A η$ (9) j j k k k II = A η$ + A η$ + A η$ (93) j j jj j jk k II = A η$ + A η$ + A η$ (94) k k kj j kk k Lkws, th thrd trms n Equaton 91 bcom 31

40 ~ ~ ~ ( Ω) $ ( j Ω) $ j ( k Ω) III = bn d η bn N d η bn N d η$ = B η$ + B η$ + B η$ (95) j j k k k III = B η$ + B η$ + B η$ (96) j j jj j jk k III = B η$ + B η$ + B η$ (97) k k kj j kk k Assumng that th coffcnts a ~ and b ~ also vary lnarly on lmnt,.. ~ ~ a = a N ~ a N ~ + + a k N (99) j j k ~ ~ ~ ~ b = b N + b N + b k N (100) j j k thn usng th rlatons (84), (85) and (86), w hav A = ~ a N N dω = IJ 1 4 ~ ( ) I J b b + c c a N a j N j a k ~ + ~ + ~ N k dω I J I J = 1 1 ~ a ~ a ~ + j+ a k ( b Ib J + cicj ) (I, J =, j, k) (101) B IJ = b N b j N j b k ~ + ~ + ~ N k N I N Jd Ω = ~ ~ ~ ~ b + b j+ b k + b I for I = J 30 ~ ~ ~ ~ ~ b + b j+ b k + b I + b J for I J 60 (I, J =, j, k) (10) 3

41 By combnng th xprssons for $η at nods, j and k, th scond and thrd trms of Equaton 91 for lmnt can b rprsntd by mans of matrx, [ II + III ] = ([ A ] + [ B ]) { } $η ( = 1,,..., E ) (103) whr { η $ } { η $ η $ η T = $, j, k} (104) and th xprsson for [ A ] + [ B ] s th sam as that for [ K 1 ], s Equaton 35. An xprsson for th frst trm of Equaton 91 may b obtand by applyng crtan boundary condton and also usng Equatons 87 through 90. Ths gvs th followng rlatonshp whn nod s at coastal boundary, P P P P ( j j ) P P P P I C = α N a N + a N N + N ds ~ ~ j j η$ η$ P P P P P = C η$ + C η $ (105) j j Smlarly for nod j, P P P P P I = C η$ + C η $ (106) C j j jj j Thrfor, for sgmnt P wth nod and j, w hav th matrx formula P P P [ I C] = [ C ]{ } $η (107) 33

42 whr { η P T $ } { η$ η$ j} =, (108) P P P P P C IJ = α ~ a N + ~ a j N j N I N J ds P = ~ ~ ~ ( j I) P L α a + a + a for I = J 1 P L α( ~ a + ~ I a J ) for I J 1 (I, J =, j) (109) Whn nod s at opn boundary, boundary condton (Equaton 74) appls, and th frst trm n Equaton 91 bcoms P I = ~ η$ P Γ a p η$ + q g N ds θ P (110) (110) s Smlar to I C P (Equatons 107 and 108), th frst trm of th th and j th Equaton n [ I P ] = [ P P Γ Γ1 ]{ } 1 $η (111) whr P Γ 1 = IJ P L p ~ a + ~ a j+ a ~ I for I = J 1 L P I J 1 p ~ a + ~ a for I J (I, J =, j) (11) 34

43 Th scond trms n (110) ar and j P I = ~ η$ P Γ aqr N ds = ~ P η$ η$ N a qr N s s s s ds P a qr = ~ $ ~ η aqr ( η$ η $ j) (113) s L P P P I = ~ η$ ~ a qr Γ a qr ( η$ $ ) j j η (114) s L j P whr r s th radus of th smcrcl. Ths trms can also b wrttn n matrx form as [ I P ] [ P ]{ P } [ D P Γ = Γ + ] $η (115) whr P [ Γ ] = ~ a qr P L (116) and [ D ] = ~ η$ a qr s, η$ s j P T (117) Th thrd trms n Equaton 110 ar P I = ~ P 1 a gn ds = ~ P a g N ds = ~ a gl P P Γ 3 = IΓ 3 P P j 35

44 thrfor, [ I P ] [ P Γ Γ 3 3 ] = = 1 ~ 1 a gl P 1 (118) Th functon $η wll b obtand by frst computng th lmnt matrx [ A ] and [ B ] for lmnts = 1, E, and th boundary matrx [ ] C P P P P, [ Γ ], [ Γ ], [ Γ ] 1 3 sgmnts P = 1,, N P. Ths matrcs ar assmbld to obtan an N N systm of quatons, P P P P P P P ([ A ] [ B ]){ η$ } [ C ]{ η$ C} ([ Γ ] [ Γ ]){ η$ Γ} [ D ] [ Γ ] Ω = C Γ Γ Γ for Notc that T 0 P T [ D ] = ~ η$ η$ a qr a qr { A, A } 1 Γ s A s A =, ~ α 1 η$ η $ (119) whr A 1 and A ar two ponts that connct th opn boundary and coastal boundary and can b th wall boundary condton, (9), appls to $η / s. Thrfor, trm [ D P ] ncludd nto [ C P ]{ P C} $η and thn th assmbld quaton bcoms C Γ [ K1]{ η$ } + [ K ]{ η$ C} + [ K3]{ η$ } = { f} Γ (10) or [ A]{ $η } = { f} (11) Ths lnar systm of quatons may b solvd to obtan η. 36

45 4.4 Knmatc Paramtrs Onc a soluton for $η s obtand n CGWAVE, th maxmum wav vlocty, maxmum wav prssurs, wav phas angl and wav ampltud may b obtand from th valus of $η. Ths quantts ar obtand as follows. Th vlocty potntal for watr partcls of surfac watr wavs may b wrttn as ( ) = [ φ ω φ ω ] ( ) Φ x, y, z, t cos t + sn t Z z (1) 1 whr ( ) Z z = [ ( + h) ] ( ) cosh k z cosh kh (13) Th potntal may b xprssd n trms of $η by substtutng Φ = g ω η$ (14) nto Equaton 1; ths gvs whr ( ) η( x, y) Z( z) Φ x, y, z, t g t = R π + ω $ (15) ω η$ = η$ + η$ (16) 1 37

46 Ths xprsson may b wrttn as follows by sparatng th ral, $η 1, and magnary, $η parts on $η, and rplacng -π/-ωt wth α; ths gvs g Φ = [ + ] Z ω η α η α 1 cos sn (17) An xprsson for th vlocty of watr partcls s obtand by valuatng th gradnt of th xprsson for Φ n th last quaton; ths gvs v x g η ˆ ˆ 1 η = cosα + snα Z ω x x (18a) v y g η ˆ ˆ 1 η = cos sn Z y α + y α ω (18b) Ths xprssons contan th horzontal componnts of th vlocty. For smplcty, Z s takn as a local constant. Th magntud of th horzontal componnts of th vlocty s obtand by substtutng v x and v y from th last xprsson nto v = (v x ) + (v y ) (19) to obtan v g $ $ $ $ 1 1 = Z cos η + x η η + + y x y η α ω y sn α η$ η$ η$ η$ + sn x x y y 1 1 ( α) (130) Th maxmum horzontal vlocty occurs at locatons whr th drvatv of v wth rspct to α s qual to zro; ths occurs whr 38

47 + + + η$ η$ η$ η$ 1 1 sn x y x y ( α) η$ 1 η$ η$ + 1 η $ cos( α) = 0 x x y y (131) Rarrangng trms gvs α = 1 arctan η$ η$ η$ η$ x x y y η η η η $ $ $ $ 1 1 x y x y (13) Th magntud of th horzontal componnt of th vlocty wll hav ts mnmum and maxmum valus at ths valu of α and at α+π/. Th valu of v s calculatd at both of ths angls; th largr valu s th maxmum vlocty ovr all tms. Th prssur s obtand from th lnar form of th Brnoull quaton; Φ t P + + gz = constant (133) ρ Not that th prssurs assocatd wth vlocts (dynamc had, ½v ) ar gnord n ths lnar form. Th xprsson for Φ n Equaton 15 s substtutd nto ths xprsson and trms ar rarrangd to obtan t ( ) P = ρgz + ρgr η$ ω Z + constant (134) Th maxmum prssur ovr a wav cycl occurs whn th trm R($ η ωt ) s qual to H/. Th constant s chosn such that th hydrostatc prssur s qual to zro at z=0; thus 39

48 P = ρ gz + ρ g H max Z (135) Th wav phas angl β s obtand from η β = arctan $ η$ 1 (136) Th cosn of β vars from -1 to 1, and s wrttn as output. Th wav ampltud A s obtand from A = η$ + η$ 1 (137) A snap-shot of th sa surfac lvaton at tm = 0.0 s obtand from -ωt [ $ ] [ $ cos t + $ sn t] η = R η = η1 ω η ω (138) 4.5 Floatng Docks A floatng dock nsd th computatonal doman may b tratd followng th formulaton of Tsay & Lu (1983). If th dstanc from th bottom of th bd to th surfac of th floatng body s d( x, y) and th wav numbr corrspondng to ths dpth s k( x, y), th govrnng quaton for rgon undr th floatng body s ( $η ) p = 0 (139) whr 1 p = σ k 1 + kd snh kd (140) σ = gk tanh kd (141) Not that th scond trm of Equaton 1 has bn nglctd n Equaton 139. Ths 40

49 smplfcaton s known as th rgd ld approxmaton. Tsay and Lu (1983) hav dmonstratd that ths formulaton gvs vry good rsults whn compard wth som xact solutons. Equaton 139 s solvd n CGWAVE n plac of Equaton 1 for th rgon whr a floatng structur s prsnt n th doman Ω. d d Fgur 6. Dfnton sktch for floatng structurs 41

50 5 ITERATIVE CG SOLUTION METHOD CGWAVE provds th choc of choosng th wavlngth dpndnt rsoluton to solv th govrnng quaton. An n-dpth nvstgaton was prformd to dtrmn th snstvty of th soluton as a functon of wavlngth. Basd on our analyss, t s rcommndd that tn or mor ponts pr wavlngth to b usd. In rgons whr th chang n topography or wav-ampltud s rapd, a fnr grd s ncssary. To solv th llptc mld-slop quaton wthout approxmatng th physcs (as n parabolc modls), th problm has to b solvd smultanously ovr th ntr doman. Ths lads to a vry larg lnar systm of quatons (.g. Equaton 69). Drct mthods (.g. Gaussan lmnaton) ar oftn napplcabl du to xtrmly larg storag rqurmnt of th matrx [A]. Itratv mthods, on th othr hand, rqur mmory for only th non-zro lmnts n [A]. Snc [A] s hghly spars, tratv mthods can sgnfcantly nhanc th ablty of th llptc typ mld-slop quaton modls to handl larg doman problms. Howvr, most tratv procdurs rqur [A] to b dagonally-domnant or symmtrc and postv-dfnt. Unfortunatly, th coffcnt matrx [A] s not dagonally-domnant, nor symmtrc and postv dfnt. Panchang t al. (1991) rcommndd that th followng Gauss transformaton s appld to Equaton 69 : [ A * ][ A]{ } [ A * ]{ f} $η = (14) whr A * s th complx conjugat transpos of [A]. Xu (1995) has shown that th nw coffcnt matrx [A * ][A] s Hrmtan and postv-dfnt. Thrfor, th modfd conjugat-gradnt mthod, whch s oftn svral ordrs of magntud fastr than many othr schms, ncludng th tradtonal conjugat gradnt schm, s guarantd to convrg whn appld to Equaton 15. Th algorthm s mplmntd n CGWAVE as 4

51 follows: 1. Slct tral valus for { $η 0 } (.. 0 th traton) for all nods n modl doman whr th soluton s dsrd.. Comput for all ponts th rsdual {r 0 } = {f} - [A] { $η 0 }and th lft hand sd of Equaton 15 as {p 0 } = [A * ]{r 0 }. A r 3. Comput for th th traton th paramtrs α, dfnd as: α = [ * ]{ } [ A]{ p } 4. Updat { η$ } { $ + 1 = η} + α {p } for all ponts. 5. Chck for convrgnc of th soluton. Th crtron for convrgnc usd n CGWAVE s [A]{ η$ } {f} + 1 { η$ } + 1 < ε (143) whr ε s a prscrbd tolranc. If Equaton 16 s satsfd, stop. 6. If Equaton 16 s not satsfd, comput, for ach grd pont, {r +1} = {r } - α [ A]{ p }. * [ A ]{ r 7. Comput for th th + 1} traton : β = * [ A ]{ r } * 8. Comput { p + 1} = [ A ]{ r+ 1} + β { p }. 9. St = + 1, and go to stp 3.. In th procdur abov, th modul of an array { x } s dfnd as { x} N / = 1 x = 1 (144) 43

52 Ths procdur has bn dmonstratd to work vry wll n fnt-lmnt modls. In fact, ths tchnqus wr found to b far mor ffcnt n th prsnt fnt-lmnt runs than for th fnt-dffrnc studs of Xu & Panchang (1993) and Panchang t al. (1991). L (1994) has rcntly suggstd modfcatons to th tratv schms of Panchang t al. (1991) for nhancng th convrgnc. Comparson btwn ths two schms wr gvn n Xu and Panchang (1995). Although ths modfcatons rsultd n fastr convrgnc, t was notcd that, unlk th basc procdurs of Panchang t al. (1991), thy do not convrg to a soluton monotoncally. Instad, th rsdual rror dcrass n an oscllatory mannr as th tratons procd. Also, th gan n CPU tm vars from cas to cas and appars to b problm-spcfc, but t can sav ovr 50% of CPU tm for som applcatons. Ths schm has also bn ncludd n CGWAVE as an altrnatv choc, and consquntly, thr ar two typs of solvrs provdd n CGWAVE. Th solvng algorthm s slghtly dffrnt whn non-lnar mchansms ar ncorporatd nto th soluton. In ths cas, th soluton s frst solvd as f thr wr no non-lnar mchansms. Ths soluton s thn usd to prscrb ntal condtons for th non-lnar mchansms, and th systm of quatons ar modfd to ncorporat th rsultng non-lnar mchansms. CGWAVE thn solvs usng th modfd quatons. Ths tratv mthod of solvng, modfyng th systm of quatons wth non-lnar mchansms, and solvng agan contnus untl th rsultng soluton no longr changs btwn tratons. Th solvng algorthm s also dffrnt whn spctral wav condtons ar usd. In ths cas, a soluton s frst obtand for ach ndvdual spctral componnt. Th fnal soluton s obtand by lnar suprposton of th solutons for all spctral componnts. Ths suprposton s prformd aftr a spctral run s compltd, usng a post procssng program. 44

53 6 GENERATION OF FINITE-ELEMENT NETWORK CGWAVE rqurs a two-dmnsonal (D) trangular grd ntwork for ts fntlmnt calculatons. Although svral grd gnraton packags ar avalabl, thy ar not sutabl for llptc coastal wav modls for whch th sz of th lmnts must b rlatd to th wav lngth (whch vars wth local watr dpth) for propr rsoluton. A smcrcular opn boundary has to b cratd for spcal opn boundary tratmnt, and rflcton coffcnts, whch may vary from on part of th coastal boundary to anothr, ar also rqurd as nput data for th modl. To dal wth ths spcal problms, CGWAVE has bn ntrfacd wth th grd-gnrator assocatd wth th SMS (Surfac watr Modlng Systms) flow modlng packag. Th Engnrng Computr Graphcs Laboratory s dvlopng ths stat-of-th-art packag for th US Army Corps of Engnrs at Brgham Young Unvrsty. SMS contans a st of D hydrodynamc modls and a gnral purpos grd gnraton and vsualzaton packag. SMS ncluds an ffcnt fnt-lmnt grdgnrator. Howvr, ths grd-gnrator was orgnally dsgnd for othr typs of hydrodynamc modls. Thr utlty programs that hlp ntrfac CGWAVE wth th SMS grd-gnrator hav bn dvlopd for us outsd SMS, pror to th full ntgraton of CGWAVE nto SMS. Gvn a coars rctangular array of bathymtrc data, ths programs gnrat a wavlngth-dpndnt trangular nodal ntwork (basd on th usrspcfd rsoluton,.. th numbr of ponts pr wav lngth), automatcally construct th sm-crcular opn boundary, assgn rflcton coffcnts along th coastal boundars, lmnat unwantd land ponts, tc. Th rsultng grd and boundary data from SMS ar thn fltrd by anothr utlty program for us by th wav modl. Th output from th wav modl can b procssd and thn plottd by usng SMS. Ths maks modl mplmntaton vry ffcnt and allows th usr to vw a graphc rprsntaton of th soluton. 45

54 7 PROCEDURE FOR STAND-ALONE USAGE OF CGWAVE CGWAVE s an ffcnt and asy-to-us wav modl. Although mor advancd faturs ar antcpatd to volv n th futur, th currnt vrson s suffcntly usrfrndly and can provd th bas nformaton for shallow watr wav propagaton and transformaton for harbor/coastal problms or offshor opn sa problms. Th nformaton provdd by ths modl can b furthr usd to calculat wav forcs on structurs, stablty of brakwatrs, sdmnt/pollutant transport and can b usd to assst harbor/coastal protcton structur dsgn. In ths scton of th manual, stp-by-stp nstructons ar gvn for us of CGWAVE. Th procdur outlnd hr prtans to th usag of CGWAVE as standalon (.. outsd SMS); ths usag s rfrrd to as MODE-1. MODE-1 usag dos rqur th us of SMS for som procdurs. A smlar, automatd procdur s avalabl wthn SMS, and s rfrrd to as MODE-. Aftr succssful nstallaton of CGWAVE and SMS, th usr can procd wth applcatons of CGWAVE. In gnral, th MODE-1 applcaton of CGWAVE conssts of ght stps: Stp 1. Aftr a rgon for smulaton has bn chosn, a fl contanng (x,y,z whr z = dpth) data n rctangular fashon may b cratd from a map or othr data bas. Th nput fl format s dscrbd n dtal n scton 7.. of ths manual and n th Fl Formats scton of th SMS Rfrnc Manual. Stp. Th raw data gathrd n Stp 1 s procssd by th program rsol. Coastal and opn boundars ar dfnd and most xtranous data s lmnatd. Wavlngth dpndnt rsoluton s stablshd, and th rvsd (x,y,z) data st s prsntd for constructng th fnt lmnt grd. Stp 3. Th fnt lmnt grd s constructd usng SMS, and boundary condtons ar assgnd to th opn and coastal boundars. 46

55 Stp. 4. CGWAVE nput paramtrs ar spcfd. Ths paramtrs nclud: ncdnt wav condton, nonlnar mchansms (wav brakng, bottom frcton and nonlnar dsprson), choc of solvrs, choc of opn boundary condton, traton control paramtrs, tc. Stp 5. All rlatv nformaton ndd for nput to th modl s consoldatd nto a sngl nput fl. Ths procssng s don by program rform. Stp 6. Run th wav modl and obtan an output wav potntal fl. Stp 7. Procss th output fl (usng program trans ) to obtan ampltud, phas and knmatcs soluton fls. Stp 8. Us SMS to vw varous soluton mags usng a varty of contourng optons. 7.1 Fl Dscrptons Th followng fls contan sourc cod for th CGWAVE modlng packag: cgwav.f -- Fortran sourc cod of CGWAVE. cgwav.gn -- Tmplat of th gnral nformaton fl. rform.f -- Fortran sourc cod of rform. rsol.f -- Fortran sourc cod of rsol. trans.f -- Fortran sourc cod of trans. Gvn blow s a lst of xampl fls for applcaton of CGWAVE. Although on can us any nam for thm, th followng convnton s usd throughout ths manual. It s also rcommndd that th usr follow ths convnton for smplcty and consstncy. flnam.xyz -- Fl to b mportd to SMS and to b usd by program rsol. Contans (x,y,dpth) data dscrbng th bathymtry. flnam.bc -- Boundary condton fl cratd by SMS. flnam.go -- Gomtry fl cratd by SMS. 47

56 flnam.gn -- Gnral nformaton fl, txt format. Contans rqurd nput control paramtrs forcgwave. flnam.dat -- CGWAVE and trans nput data fl. Cratd by program rform. flnam.out -- CGWAVE wav potntal output fl. Input to program trans flnam.txt -- Txt output fl cratd by trans. Contans gnral nformaton about th projct modld, CGWAVE nput paramtrs, soluton status and complt output of x,y dpth, ampltud and phas at ach nod n txt format. flnam.sol -- Soluton output fl cratd by program trans. Contans nformaton about computd ampltud, phas and knmatcs. May b rad by SMS for graphc prsntaton of th soluton. ASCII txt format. flnam.pls -- Usr cratd nput fl for program trans whch provds nformaton about slctd nod ds, ponts, lns and rctangular sub-rgons whr output s dsrd. flnam.sl -- Slctd locatons output fl cratd by program trans. flnam.flw -- Floatng dock soluton output cratd by program trans. Of rlvanc only f thr s a floatng dock nsd th computatonal doman. Vlocty potntal output s avalabl n ths sub-doman, whch may b furthr usd (outsd CGWAVE) to calculat forcs and momnts. flnam.alp -- Anmatd flm loop for vw of varatons n ampltud and phas wth tm. For us n SMS. 7. Raw Data Procssng Program rsol Grd rsoluton s dpndnt on th local watr dpth. In gnral, 10 or mor nods pr (local) wavlngth ar dsrd. In most cass, a larg numbr of xtra nodal ponts must b cratd ovr th raw data ponts to satsfy th rsoluton rqurmnts. Hnc t s dsrabl to hav ths ponts gnratd bfor usng SMS to crat a fnt 48

57 lmnt ntwork. Nods wth ngatv or zro dpths (dry land) ar prohbtd wthn th computatonal doman. A mnmum watr dpth may b judcally chosn and rgons undr ths mnmum dpth (hghr n lvaton) should b lmnatd. For th opn boundary, t s ncssary to hav proprly dstrbutd ponts cratd on a smcrcl (or full crcl for offshor applcatons), and ponts outsd th opn boundary must b dscardd. Ths tasks ar thortcally trval, but vry tm consumng. Th utlty program rsol accomplshs ths tasks quckly and wth mnmal ffort on th part of th usr Vwng XYZ Data and Slctng Opn Boundary End Ponts Consdr as an xampl th raw data-st calld swansr.xyz. Ths raw data may b vwd n SMS. Ths can b don as follows. Run th SMS xcutabl fl. Whn SMS s opn, a tool paltt, graphcs wndow, and pull-down mnus at th top of th scrn ar vsbl. From th Fl mnu choos mport. Slct XYZ data from th Slct Import Format wndow. Slct th drv, drctory and fl swansr.xyz usng th Opn wndow. From th tool paltt slct th Crat Lnar Trangl tool Not that th nam of th tool that s currntly undr th mous pontr s dsplayd at th bottom of th SMS wndow. From th Elmnts mnu slct Trangulat. Ths rsults n a fnt lmnt trangular ntwork. Slct th Dsplay Optons tool from thr th tool paltt or from th Dsplay pull-down mnu. Th Dsplay Optons wndow allows th usr to vw or hd varous dsplay faturs. Toggl off Nods. Toggl th Contours on. Clck on th Optons button to th rght of th Contours toggl. Slct Color fll btwn contours, clck OK. Clck OK agan to xt th Dsplay Optons wndow. A color rprsntaton of th trangulatd grd s now vsbl. It s suggstd that th usr bcom famlar wth th functon of th tools and optons bfor attmptng any srous modlng ffort. Bfor xtng SMS, th usr should wrt down som nformaton that s usd to crat th smcrcular opn boundary. Pck a mnmum watr dpth and roughly stmat th startng and nd pont 49

58 coordnats (countr-clockws) of th smcrcular opn boundary. Snc th wav modl dos not allow zro or ngatv dpths n th computatonal doman, a mnmum dpth must b spcfd whn xcutng th program rsol. Mak sur that both smcrcl nd ponts ar nsd an ara whr watr dpth s gratr than th mnmum dpth on at last on nod of th background rctangl. In ths xampl,.0 (mtrs) s usd as th mnmum dpth and th ponts (3500.0,700.0),(100.0,450.0) as th startng and nd pont coordnats for th smcrcl. You may now qut th SMS (do not sav any fl nformaton at ths tm). 7.. XYZ Fl Format Th nput data fl for th program rsol must b a rctangular doman wth constant dx and dy for ach drcton. Th fl has th followng format: XYZ x1 y1 z1 x y z... xn yn zn For xampl, by vwng swansr.xyz wth a txt dtor, th followng wll b sn: XYZ

59 Not that ths s only th top porton of th swansr.xyz fl. Th frst ln must b XYZ followd by (x y z) data. Ths xyz fl should b orntd such that th x-axs ponts to th ast on th scrn whn th data st s dsplayd Program rsol Excuton of program rsol wll rsult n th craton of a nw XYZ data fl calld swanst.xyz, whch s to b usd as nput to SMS for furthr grd gnraton opratons. Ths fl has bttr rsoluton than th raw data fl (.g. swansr.xyz), and contans nformaton rgardng th smcrcular opn boundary. To bgn th xampl, run th rsol xcutabl. A program nformaton hadr wll b dsplayd on th scrn. Th followng nformaton wll appar aftr th hadr: Input flnam : (*.xyz) swansr.xyz (Ht <Entr> aftr ach rspons) Output Flnam: (*.xyz) swanst.xyz Input data spacng. Dx s: 00.0 Dy s: 51

60 00.0 Incdnt wav prod 30.0 Entr typ of opn boundary: Full Crcl (=0), Sm Crcl (=1) 1 Start from: End at (ccw): Ar thy Fxd (=0) or Approxmat (=1)? 1 Numbr of Nods pr Wavlngth : 10 Intrpolat btwn land and watr? (ys=0, no=1) 0 Mnmum Watr Dpth :.0 Mnmum Nod Spacng :.001 Startng pont of th smcrcl : End pont of th smcrcl : End of program, Prss Entr to contnu Som nots rgardng th abov qustons and answrs follow. Mnmum Watr Dpth. Whn spcfyng a mnmum watr dpth, th stmatd smcrcl nd ponts may or may not satsfy th mnmum dpth crtra. In most cass, 5

61 ths coordnats wll b thr nsd or outsd th doman (and rarly on th xact boundary ndcatd by th mnmum dpth.) By choosng Approxmat, th program rsol wll locat a boundary pont closst to th pont spcfd. Not also that program rsol rqurs that th locaton of th stmatd ponts should l nsd a rctangl (of th background or raw data ntwork) wth at last on cornr havng a dpth gratr than or qual to th mnmum dpth. Numbr of Nods pr Wavlngth. It s strongly rcommndd that at last 10 or mor ponts pr wavlngth b usd whnvr possbl. A rsoluton of 7 or 8 ponts pr wavlngth may yld a rasonabl soluton, but 6 should b tratd as an absolut mnmum. 15 ponts pr wavlngth ar suffcnt for most applcatons. Intrpolat Btwn Land and Watr. Othr than som dal cass wth constant dpth (.g. laboratory or analytcal tst cass), always answr ys to ths quston. Mnmum Nod Spacng. Ths fatur allows th usr to lmt th dnsty of nods (and, ndrctly, th total numbr of nods) placd wthn th doman. Ths may b ncssary for xtrmly shallow rgons or for vry larg domans. Ths fatur can contradct th Numbr of Nods pr Wavlngth paramtr, and should b usd wth cauton. It s rcommndd that th usr st ths paramtr to a vry small valu (.001 or smallr) for most applcatons. 7.3 Craton of Fnt Elmnt Grd A dscrpton of th stps nvolvd n cratng a fnt lmnt grd and constructng CGWAVE nput fls usng SMS follows. Ths procdur dscrbs MODE- 1 usag of CGWAVE. It s hghly rcommndd that th usr b fully famlar wth th MODE-1 usag, snc th automatd procdur for MODE- maks us of th stps dscrbd blow. In MODE-1, th usr nds to crat th followng fls: *.go (gomtry fl) 53

62 *.bc (boundary condton fl) *.gn (gnral nput fl) Ths fls ar usd by th program rform to construct th man CGWAVE nput data fl. Spcfc stps prtnnt to cratng ach of th abov fls ar provdd blow Gomtry Opn SMS. From th Fl mnu choos mport. Slct XYZ data from th Slct Import Format wndow. Slct th drv, drctory and fl swanst.xyz n th Opn wndow. From th tool paltt slct th Crat Lnar Trangl tool. From th Elmnts mnu slct Trangulat. Any nods and lmnts that l outsd th dsrd modl doman must b manually rmovd. Usng th Slct Elmnt and Slct Nods tools dos ths. Th <Shft> and <Ctrl> kys may b usd n combnaton wth ths tools to rmov multpl and groups of lmnts or nods. For xampl, a group of lmnts may b slctd for dlton as follows: Hold th <Ctrl> ky, clck (do not rlas) th mous button on th cntr of an lmnt. Drag th pontr across th group of lmnts to b dltd. An arrow bgnnng at th frst slcton pont should follow th pontr. Any lmnt touchd by ths arrow wll b slctd. Elmnts may b addd to (or rmovd from) a slcton group by holdng th <Shft> ky whl slctng an lmnt. Th slctd lmnts ar thn dltd wth th <Dl> ky. Othr slcton mthods ar avalabl n th Edt and Elmnts mnus. Rmmbr to clan up and/or rmov th lmnts n aras whr slands or structurs ar locatd. Aftr obtanng a satsfactory grd-ntwork, boundary condtons must b spcfd, as dtald n th followng scton. Slct Sav Gomtry n th RMA mnu. Nam th fl swanst.go. 54

63 7.3. Boundary Condtons Dfnng Opn and Coastal Boundars. Th boundary condtons for ths grd may now b spcfd. Pck Crat Nodstrng tool from th tool paltt. Ths tool s usd to dntfy th nods along th dg of th modl doman. Us th mous pontr to slct th start pont (nod) of th smcrcular opn boundary. (Th Zoom tool may b usd to hlp dntfy th start nod.) Prss and hold th <Ctrl> ky and slct th nd pont (nod) of th smcrcular opn boundary. Prss <Entr>. Th opn boundary has bn dfnd by a sngl nodstrng. For opn sa/offshor problms, whr opn boundary s a full crcl, smlar procdur appls. For ths xampl, w wll nxt dfn th coastln boundary usng a sngl strng. Procd wth th abov slcton procdur, ths tm usng th nd pont of th smcrcl as th nw start pont. Th nw nd pont s th start pont of th smcrcl. In som modlng stuatons, th usr may want to assgn dffrnt rflcton coffcnts to sctons of th coastln. If ths s th cas, a nodstrng must b cratd for ach ara of dffrng rflcton coffcnt. A nw nodstrng must always bgn wth th nd pont of th prvous strng. Th drcton of nodstrngs should b spcfd such that whn movng from th start pont to th nd pont of th nodstrng, th modld doman s on th lft sd of th nodstrng,.. countrclockws. Dfnng Islands Boundars. Island boundars ar dfnd n much th sam way as dscrbd abov, wth th followng xcptons. Island nodstrngs ar cratd such that th modld doman s on th rght sd of th nodstrng whn movng from th start pont to th nd pont,.. countr-clockws. A complt sland boundary must consst of two or mor nodstrngs, as SMS dos not allow a sngl nodstrng to bgn and nd on th sam nod. Islands should b dfnd aftr all opn and coastal boundars hav bn dfnd. Assgnng Rflcton Coffcnts. Pck Slct Nodstrng tool. A small slcton ara (box) should appar n th cntr of ach nodstrng. Plac th pontr on th slcton box for th opn boundary and slct t. Go to th RMA mnu and slct 55

64 Assgn BC.... In th RMA Assgn Boundary Condton wndow, pck th Had BC opton. St th lvaton (lowr part of th wndow) to 5.0. Ths valu s usd to tll CGWAVE that ths nodstrng s an opn boundary. Clck OK. Rpat ths procdur for all othr boundars. For all boundars othr than th opn boundary, th lvaton valu rprsnts th rflcton coffcnt and should b btwn 0.0 and 1.0, corrspondng to th rflcton coffcnt of that porton of coastal boundary (ncludng slands). Th valu of 1.0 for rflcton coffcnt should b usd for a fully rflctng boundary and 0.0 usd for fully absorbng boundary. Sav th boundary condton nformaton by slctng Sav BC... from th RMA mnu. For th xampl cas, nam th fl swanst.bc. Floatng Structurs. Floatng structurs ar dscussd n dtal n scton 4.5. Whn a floatng structur s ncludd nsd th doman, th lmnts contanng th floatng structur must b both markd and adjustd to sgnfy and account for th prsnc of th structur. Th lmnts contand by th ara of th floatng structur ar assgnd a Matral Typ ID of 3 as follows. Frst, slct all th lmnts wthn an ara boundd by th prmtr of th floatng structur. Thn slct Assgn Matral Typ from th Elmnts mnu. Clck on th Nw button n th Matrals Edtor untl th ID: changs to 3. Clck Slct. Nxt, modfy th dpth of th nods n ths ara as dscrbd n scton 4.5. Th nw dpths ar shown n Fgur 6 as d. Ths can b don usng th Transform Msh opton found n th Nods mnu. CGWAVE wll now rcognz ths ara as a floatng structur and trat t accordngly. Rnumbrng th Gomtry. Aftr compltng th grd gnraton and boundary dfnton, th grd must b rnumbrd. Usng th Slct nodstrng tool, slct any of th nodstrngs. From th Elmnts mnu choos Rnumbr. In th Rnumbrng Opts wndow, slct Band Wdth and thn OK. Aftr rnumbrng, th grd gnraton procss s complt. R-sav th gomtry and boundary condton fls usng th sam nams gvn bfor (swanst.go and swanst.bc). Som basc nformaton about 56

65 th grd may b vwd by clckng th Dsplay Modul Info tool, or by slctng Gt Info... from th Fl mnu. You may now qut SMS. 7.4 Th Gnral Informaton Fl (*.gn) A tmplat calld cgwav.gn s provdd for th bnft of th usr. A nw copy of ths fl should b mad for ach applcaton. Th contnts of th fl ar wrttn n ASCII txt and appar blow. Not that th ncdnt wav condtons contan data for a sngl (monochromatc) modl run. Spctral nput condtons may b nput by ncludng addtonal lns blow th frst (Angl/Prod/Ampltud) ln. Two mpty (blank) lns should follow th last (Angl/Prod/Ampltud) ln, sparatng t from th ) Opn Boundary... ln. Incdnt angls ar masurd n dgrs and th ncdnt drcton s dfnd such that th ncdnt wav travls n th postv x-drcton whn θ s qual to zro. PROJECT : Projct Nam (...blank ln) (...blank ln) 1) Incdnt Wav Condtons: (...blank ln) No. Incdnt Wav Angl Wav Prod (s) Incdnt Wav Ampltud (...blank ln) (...blank ln) ) Opn Boundary (Bssl Srs = 0, Parabolc = 1, Box = ) : 1 (...blank ln) If Bssl Srs, Numbr of Trms Usd : 50 If Parabolc, Extror Coastln Rflcton : 0.0 (...blank ln) (...blank ln) 3) Bottom Frcton (Ys=1,No=0)? 0 57

66 (...blank ln) If Ys, Frcton Coffcnt : 0.1 (...blank ln) (...blank ln) 4) Wav Brakng (Ys=1,No=0)? 0 (...blank ln) If Ys, Brakng Coffcnt : 0.15 (...blank ln) (...blank ln) 5) Nonlnar Dsprson Rlaton (Ys=1,No=0)? 0 (...blank ln) (...blank ln) 6) Solvr (cg standard == 0, modfcaton == 1) : 1 (...blank ln) Tolranc of Convrgnc for Lnar Equatons : 0.1E-8 Intrval for Chckng Convrgnc : 100 Maxmum Itratons for th Lnar Equatons : (...blank ln) (...blank ln) 7) Tolranc of Convrgnc for Nonlnar Mchansms : (...blank ln) Maxmum Itratons for th Nonlnar Mchansms : 8 (...blank ln) (...blank ln) nd of data Ths fl provds th basc ncdnt wav and boundary typ nformaton and varous control paramtrs for CGWAVE. Du to th programmng mthods usd to rad th data from ths fl, th txt must not b altrd. Th paramtr valus may b rplacd 58

67 wth valus that ar spcfc for th nw applcaton. All th numbrs must b prsnt and n th corrct locaton, whthr thy ar usd or not. 7.5 Assmbly of Input Informaton - Program rform Th program rform s usd for assmblng and convrtng all nput nformaton nto th format ndd by CGWAVE. It rads th gomtry, boundary condton and gnral nformaton fls and constructs a unfd nput data fl for CGWAVE. Th program also vrfs som componnts of th provdd data. Program rform maks th cntral part of CGWAVE portabl. In othr words, a workstaton may b usd to crat and manpulat th graphcs, and CGWAVE may b run on any othr platform (PC, manfram, tc.). Also, bcaus rform sorts and rwrts th data n clan, nstd groups, th sam data st can b usd to prform smulatons wth dffrnt ncdnt wav drctons, dffrnt combnatons of boundary rflcton coffcnts, dffrnt ncdnt wav ampltuds and prods (to som xtnt, snc rsoluton s wav prod dpndnt), tc. wthout gong through th troubl of dtng th gnral nformaton fl and rrunnng rform for ach cas. Ths s don by smply dtng a fw lns n th top sctons of th data (*.dat) fl. Anothr fatur of rform that s spcally usful s th ablty to chck th grd rsoluton for th ovrall doman. Hr, th rctangular harbor tst s usd to dmonstrat of th functon of rform. Run th program rform. Aftr th nformatonal hadr, th followng appars on th scrn: Gnral nformaton flnam: (*.gn) rct.gn Boundary condton flnam from SMS: (*.bc) rct.bc Gomtry flnam from SMS: (*.go) rct.go Output data flnam for CGWAVE: (*.dat) rct.dat 59

68 Do you want rsoluton analyss (ys/no)? ys Numbr of ponts pr wav lngth dstrbuton (%) From 0 to 5 :.0 From 5 to 6 :.0 From 6 to 7 :.0 From 7 to 8 :.0 From 8 to 9 :.0 From 9 to 10 :.0 From 10 to 11 :.0 From 11 to 1 :.0 From 1 to 13 :.0 From 13 to 14 :.0 From 14 to 15 :.0 From 15 up : End of program, Prss ENTER to contnu Th last st of nformaton followng Numbr of ponts pr wav lngth dstrbuton (%) s th rsoluton data for ths gomtry, calculatd from th grd-ntwork and wav prod spcfd by th usr n th gnral nformaton fl. As statd arlr, lss than 5 grd ponts pr wavlngth s not vald, ovr 15 ponts pr wavlngth s gnrally vry good. Th rsoluton data shown s for th ntr computatonal doman and t gvs th rsoluton dstrbuton for all th lmnts. Ths nformaton s output to th scrn only. Th usr may copy and rtan t manually for latr rfrnc. Ths fatur s also usful f th sam grd s to b usd agan wth a dffrnt wav prod; th rsoluton analyss wll hlp to dtrmn whthr or not grd modfcaton s ncssary. Ths may b th cas f th nw wav prod s smallr than th orgnal dsgn prod wav. 60

69 7.6 Th Wav Modl - Program cgwav Ths s th man part of th CGWAVE modl. Whn program cgwav s run for th rctangular harbor cas, th followng prompts appar: Input data flnam: (*.dat) rct.dat Output wav potntal flnam: (*.out) rct.out A block of nformaton rvwng nput paramtrs s prntd to th scrn. Ths s followd by mssags rgardng th procssng of nput nformaton. A hot start (or rstart) opton thn follows: A prvous soluton fl as ntal stat (y/n)? n Itratons Bgn, Plas Wat... Th modl wll run untl thr th convrgnc paramtr or th maxmum numbr of tratons paramtr s mt or xcdd. Th soluton wll thn b wrttn to th fl (*.out) spcfd arlr. Th hot start opton allows th usr to rstart th modl wth computatons bgnnng from an old CGWAVE output fl. Ths can b usful n th vnt that th modl s forcd to wrt a soluton (*.out) bfor th convrgnc crtra hav bn rachd. Ths partally convrgd soluton wll occur f th Maxmum Itratons for th Lnar Equatons paramtr s xcdd bfor th Tolranc for Convrgnc of Lnar Equatons paramtr s rachd. 7.7 Post-procssng th Output - Program trans Th output from program cgwav s a fl (.g. rct.out) whch contans grd nformaton and calculatd complx wav lvatons. Th program trans may b usd to convrt ths potntals nto a radly usabl format or to obtan output at dsrd locatons. 61

70 7.7.1 Program trans. Rformd data flnam (usd as nput to CGWAVE) : (*.dat) rct.dat Wav potntal fl (output from cgwav): (*.out) rct.out Plas slct on for output: g Gnral nfo only t Gnral nfo wth complt txt output n Non abov Th Gnral nfo only opton prnts txt nformaton about th nput paramtrs and numbr of tratons rqurd to obtan th soluton. Th Gnral nfo wth complt txt output opton adds th nod numbrs, locaton, computd wav ampltud and phas nformaton to th gnral nformaton output. Th namng convnton usd for th fl s flnam.txt. Crat output of ampltud and phas for SMS (ys/no)? Ths output provds nformaton n a format that s radabl by SMS and s usd for graphc vwng of ampltud, phas and sa surfac lvaton at tm = 0. Th namng convnton s flnam.sol. Ths convnton dvats from th SMS *.sol fl namng convnton n that th CGWAVE *.sol fls ar wrttn n ASCII txt format. Crat output of watr partcl knmatcs (ys/no)? Ths output provds maxmum horzontal vlocts and prssur gradnts. Th namng convnton s flnam-kn.sol. Crat tm srs output fl for vlocty and surfac lvaton anmaton (ys/no)? 6

71 An anmatd vw of th vlocty and surfac lvaton can b cratd for vwng n SMS. Th fl namng convnton s flnam.alp. Do you want output at slctd locatons (ys/no)? Ampltud and phas data can b xtractd at dscrt output ponts or along lns (transcts) or n rctangular sub-rgons. Th coordnats of th ponts, lns and subrgons may b spcfd drctly from th kyboard or rad from an nput fl. Th nput fl s most usful for collctng data at many ponts, as th procss of ntrng all th data pont nformaton by hand can bcom tdous. Th namng convnton for th output s flnam.sl. Th followng scton addrsss th us of th output at slctd locatons opton Slctd Output at Ponts/Lns/Rctangular Sub-rgons As statd abov, thr ar two avalabl mthods for spcfyng output locatons; by kyboard nput and by radng from a fl. Kyboard nput. Tabl 1 summarzs th nformaton ndd for th kyboard nput opton for xtractng CGWAVE prdctons at ponts, along lns and n rctangular subrgons. Rsults may b obtand for mor than on typ of locaton n a gvn trans run. Tabl 1. Summary of Input Informaton, Kyboard-Slctd Output Opton Mthod of Spcfyng Locaton Kyboard Input Informaton Ndd by Program trans Spcfc nods Total numbr of nods, nod numbrs (x,y) locatons (ponts) Total numbr of ponts, (x,y) coordnats Lns (data collctd along transcts) Total numbr of lns, (x,y) locaton of start pont, (x,y) locaton of nd pont, numbr of ntrvals* btwn ponts. 63

72 Rctangular sub-rgons (x,y) locaton of cornr ponts n clockws ordr bgnnng from uppr lft pont, numbr of ntrvals* btwn ponts 1 and and btwn ponts 3 and 4. *Numbr of ntrvals btwn ponts may b spcfd as dffrnt valus for ach ln. Program trans wll ask th usr a ys/no quston rgardng ach mthod of spcfyng th output locaton. A no answr skps to th nxt mthod. A ys answr wll rsult n addtonal prompts for nput nformaton. Aftr th nput s complt, th program progrsss to th nxt mthod. Input Fl Format. Th sam output locaton data gvn n Tabl 1 s also usd for th nput fl opton. Th nput fl format s gvn blow. Th namng convnton for ths fl s flnam.pls. Th % symbols ar commnts and ar not rad by program trans, howvr, lns bgnnng wth % should not splt th ndvdual data blocks. Th manng of ach numbr s dscrbd n th data block. Th sampl fl lstd blow ncluds all avalabl output optons, dnotd by thr flags &d for nod Ids, &pt for (x,y) ponts, &ln for transcts, &rc for rctangular sub-rgons. Th opton blocks do not nd to appar n th ordr shown, nor do all opton blocks nd to b prsnt. For xampl, f th usr wshs to spcfy data output from a sngl (x,y) locaton, thn th nput fl nd only contan th &pt flag on th frst ln, th quantty of ponts (1 n ths xampl) on th scond ln, th x,y locaton of th pont on th thrd ln and th nd of data statmnt on th last ln. Not that th nd of data statmnt must b th last ln of th fl. % Top of Fl % % Ths fl may b namd *.pls, for pont, ln and squar. % % Ths fl provds nformaton for slctd output locatons. % 64

73 % Crcular sland problm % % Slctd d flag % Total numbr of nod ds % d numbrs &d % % Slctd ponts flag % Total numbr of ponts % x y locaton of ach pont &pt % % Slctd Lns flag % Total numbr of lns % x y of start pont x y of nd pont % numbr of ntrvals btwn ponts &ln

74 % % Rctangular Sub-doman flag % uppr lft cornr % uppr rght cornr % lowr lft cornr % lowr rght cornr % numbr of ntrvals btwn frst two cornrs % numbr of ntrvals btwn last two cornrs &rc nd of data 7.8 Vwng Solutons n SMS Any of th *.sol fls cratd by program trans contan wav nformaton that can b rad, vwd and prntd by SMS. To do so, opn SMS and thn opn th corrspondng *.go fl. Th Data Browsr tool (also found undr th Data mnu) s usd to mport th soluton fl. Aftr opnng th Data St Browsr wndow, slct th Import. Whn th Import Data St wndow appars, slct Gnrc fl. Choos th approprat *.sol fl from th Opn wndow. Whn vwng Ampltud/Phas solutons, thr tm stp valus wll appar n th Tm wndow (to th rght of Scalar data st ) n th data st browsr. Th tm valu 0.0 corrsponds to wav ampltuds (whch hav valus gratr than or qual to 0.0). Tm 0.5 corrsponds to wav phass (valus from -1.0 to 1.0). Tm 1.0 corrsponds to sa surfac lvaton (both postv and ngatv valus). Aftr choosng th typ of soluton to vw, clck th Don button. Th soluton contours, or vctors f vctor data has bn slctd, wll appar 66

75 wthn th modld doman. Contours and vctors can b manpulatd usng th Dsplay Optons tool to provd a manngful mag. 67

76 7.9 Instructons for CGWAVE Intrfac n SMS (MODE- Usag) Cratng a Sz Functon 1. Import th.xyz data: Fl Mnu: choos Import Slct th XYZ data button. Clck OK. Fnd and slct an xyz data fl (bp.xyz, for xampl), clck OPEN.. In th Msh Modul follow ths stps n squnc: Data Mnu: choos Swtch Currnt Modl. Slct CGWAVE. Clck OK. CGWAVE Mnu: choos Crat Functons. Lav only th Wavlngth opton on (turn all othrs off). Entr dsrd valu for prod. Clck OK. Data Mnu: choos Data Calculator Slct a. Wavlngth n txt wndow (uppr lft). Clck Add to Exprsson. Slct / (dvd symbol) n th calculator. Entr dsrd numbr of lmnts n ach wavlngth (..,10) as dnomnator. Chang th ttl nw data st to sz (.. nam th nwly cratd functon). Clck on Comput. Clck Don. 68

77 7.9. Cratng th Background Scattr St 1. Crat Scattrd Data St Data Mnu: choos Msh->Scattrd Data Clck OK th nam scattr s assgnd to th st you just cratd.. Slct Intrpolaton Mthod. Swtch to Scattrpont Modul. Intrpolaton Mnu: choos Intrp Opts. Choos Invrs Dstanc Wghtd. Clck Optons. Choos Constant (Shphrd s Mthod). Clck OK (Optons Dalog). Clck OK (Intrpolaton Mthod Dalog) Cratng th Coastln and Inlt Arcs 1. Import th Coastln Arc. Go to th Map Modul and choos Fatur Objcts Covrags. Chang th Covrag typ on th bottom rght of th dalog to CGWAVE. Go to Fl Import and choos th Coastln button. Fnd th coastln fl and push OPEN. A brown Coastln typ arc wll b cratd.. Crat th Inlt Arc. Zoom n on th harbor ara of th arc usng th Zoom tool. Us th Crat Fatur Arc tool to crat an arc. Th arc should b constructd such that t spans th harbor ntranc and contans a sngl vrtx locatd nar th cntr of th harbor ntranc (th arc wll b mad up of two nd ponts and a vrtx). Not: zoom on th Inlt part of th harbor and mak sur that th nd nods of th nw arc ar placd xactly on on of th xstng coastln arc vrtcs on th South and North sds of th nlt. 69

78 Usng th Slct Fatur Arc tool, doubl clck on th nw arc. Choos Inlt typ whn th arc attrbuts dalog. Prss OK to xt th dalog. Dslct th arc by clckng on a blank ara of th scrn (t s ncssary to dslct any slctd arcs bfor buldng polygons) Dfnng Domans 1. Buld Polygons and Dfn th Harbor Doman In th Fatur Objcts mnu, slct Buld Polygons. Us th Slct Fatur Polygons tool to doubl clck on th harbor ara (whch s a nw polygon) and chang th CGWAVE typ to Harbor.. Dfnng th Doman from Coastln/Opn Boundary Intrscton Ponts. Us th Slct Fatur Vrtx tool to slct a vrtx on coastln corrspondng to th pont whr th coastln and opn boundary wll mt. If no vrtx xsts at ths locaton, an nw vrtx may b cratd usng th Crat Vrtx tool. Wth th vrtx slctd, go to Fatur Objcts mnu: choos Vrtcs <->Nods. Rpat stp for th rmanng coastln-opn boundary ntrscton pont. Swtch to Slct Fatur Pont /Nod tool. Clck th start pont of th opn boundary. Hold th SHFT ky and clck th nd pont of th opn boundary. Kp n mnd that th opn boundary wll b cratd n a countr-clockws drcton. In th Fatur Objcts mnu: choos Dfn Doman. Entr th dsrd Doman Typ and OK to xt. 3. Dfnng th Doman from th Harbor. Wth th Slct Fatur Vrtx tool, slct th vrtx n th mddl of th Inlt (rd) arc. Go to Fatur Objcts Dfn Doman and ntr th dsrd valus. Push OK. If a prompt appars sayng that t was unabl to crat th doman, t wll b ncssary to chang th sttngs n th Dfn Doman dalog. 70

79 7.9.5 Assgnng Rflcton Coffcnts Choos Slct Fatur Vrtx tool. Slct vrtcs on th coastln whr changs n th rflcton coffcnts ar dsrd. Go to Fatur Objcts Vrtcs<-> Nods. Ths splts th coastln arc. Each coastln arc can b assgnd ts own rflcton coffcnt. (Multpl vrtcs can b slctd and convrtd to nods at th sam tm by usng th Shft ky to multply slct th vrtcs). Doubl clck on ach coastln arc and spcfy th dsrd rflcton coffcnt for that sgmnt. (If mor than on arc wll hav th sam rflcton coffcnt, th arcs can b multply slctd usng th Shft ky and th attrbuts assgnd usng th Fatur Objcts Attrbuts command.) Assgnng Opn Ocan Attrbuts Choos Slct Fatur Arc tool. Slct on or mor opn boundary arcs. Slct Fatur Objcts Attrbuts command or doubl clck on a sngl arc and spcfy th arcs as Opn Ocan typ. (Not: f a rflcton coffcnt s dsrd along an ocan boundary to smulat a nar coastln typ stuaton, th ocan boundary can b dfnd as a coastln. Us th nstructons for assgnng coastln rflcton coffcnts) Sttng th Actv Scattr Data Functon Choos Fatur Objcts Buld Polygons. Doubl clck on th doman ara to slct th opn ocan ara of th doman and actvat th attrbuts dalog. Assgn t to b an Ocan typ. Slct th Dnsty rado button and clck on th Optons button. Slct th sz functon (cratd n scton 7.9.1) for Elmnt Sz and th lvaton functon for bathymtry. 71

80 Doubl clck on th Harbor polygon and rpat Dnsty opraton Cratng a Msh from Fatur Objcts D-slct all polygons by clckng somwhr ls on th scrn. Go to Fatur Objcts Map->D Msh and push OK at th prompt. Toggl th Dsplay Mshng Procss on f dsrd and push OK. Whn mshng s compltd, go to th Scattr Modul and push Dsplay Dsplay Optons and toggl th scattr pont symbols off for bttr vwng of th msh (f dsrd) Savng th CGWAVE.dat and.xyz (1-d) fls Go to th Msh Modul. Clck th Slct Nodstrngs button and slct a nodstrng (ths s ncssary for rnumbrng). Choos Elmnts Rnumbr and push OK. Push OK f promptd agan. Go to th CGWAVE mnu, choos Sav Smulaton. Slct fl nams for th.dat and.xyz (1-d) fls. If t s ndcatd that th 1-d fl cannot b savd, go to CGWAVE Modl Control and ncras th 1-d spacng or numbr of 1-d nods Chckng Msh Qualty Go to Scattr Modul. Choos Data Data Browsr and slct th sz functon usd to gnrat th ntwork. Choos Inrpolat..to Msh and ntr nam of functon for th nw ntwork to rprsnt th targt sz for ach lmnt. Go to Msh Modul. Choos CGWAVE Crat Functons and turn off all functons xcpt grd spacng. Clck OK. 7

81 Choos Data Data Calculator and buld sz grd functon, call t rror. Ths functon should b clos to zro. Aras wth larg valus ndcat nod spacng that dos not match th targt spacng. If th spacng s too far off, contact BYU or dt msh by hand. Anothr functon usful for assssng msh qualty s th rror/sz, call t pcnt_rr. Aras wth rd color pont to th lmnts, whch ar bggr than th targt sz, so rfnmnt may b warrantd Runnng CGWAVE CGWAVE can b run from CGWAVE Run CGWAVE. Clck on th fl button to gv th drctory that contans th CGWAVE xcutabl fl. 73

82 8 EXAMPLES Surfac watr wavs hav bn modld usng CGWAVE for a numbr of dalzd tstcass and ral practcal applcatons. A numbr of ths applcatons ar prsntd hr to both valdat CGWAVE and to dmonstrat ts ablty to modl wavs. 8.1 Shoal on a Slopng Bach Surfac watr wavs propagatng ovr th shoal prsntd by Brkhoff, t al. (198) wr modld. Ths tst cas dmonstrats th ablty of CGWAVE to smulat th ffcts of complx coastal bathymtrc faturs. Th shoal n Fgur 7 s orntd such that th major axs of th shoal s paralll to contours of th watr dpth. Data s collctd along th ght sctons (transcts) shown n Fgur 7. Normal ncdnt, plan wavs havng a prod of 1 scond ar modld for all runs. An ncdnt ampltud of 1.0 m s usd for modl runs usng th lnar dsprson rlaton. Runs that ar mad usng th non-lnar dsprson rlaton us an ncdnt ampltud of 0.03 m. Rsults wr compard for modl runs mad wth dffrnt grd rsoluton. Ths was accomplshd by varyng th numbr of nods n th computatonal doman and thus constructng dffrnt fnt lmnt grds. Nods wr locatd such that th numbr of nods pr wavlngth rmans constant throughout th doman. Grd dnsts from thr to fftn lmnts pr (local) wavlngth wr usd. Th rsultng fnt lmnt grds contan btwn 500 and nods and 5000 and lmnts, rspctvly. CPU tm s not takn nto account whn comparng th qualty of th modl rsults, as th longst soluton run s compltd n lss than four hours on a dsktop (00 MHz procssor) PC. 74

83 Fgurs 8, 9 and 10 show rsults for domans havng rsolutons of 3, 6, 8 and 15 nods pr wavlngth. Th numrcal soluton n Fgur 8 shows lttl corrlaton to th lab data. Th rsults n Fgurs 9 and 10 show a progrsson to th lab data as th rsoluton ncrass. Th lnar soluton n Fgur 10 matchs th data bttr than thr of th prvous rsolutons, howvr scton 7 dos not agr wth th data n th rgon 6 to 10 mtrs bhnd th shoal. A grd rsoluton of about 10 or gratr nods pr wavlngth s gnrally thought to b ncssary for an accurat soluton to most problms. Th non-lnar dsprson mchansm was appld only to grds contanng 15 nods pr wavlngth. Ths mchansm was not run for lowr rsoluton grds bcaus th lnar ampltud soluton s usd n th formulaton of th non-lnar ffct. Incluson of th non-lnar dsprson rlaton n th calculatons has a notcabl ffct on th soluton for all sctons (Fgur 10). Sctons 3 and 5 show a closr ft to th data n th aras to th lft and rght of cntr. Scton 7 shows th most markd mprovmnt n th rgon that bgns 6 to 7 m bhnd th shoal. Ths xampl problm showd that th rsults producd by th lnar run of CGWAVE ar good, provdd that a fn nough grd rsoluton s usd. Th bst rsults ar obtand wth th hghst grd rsoluton. Th addton of th non-lnar dsprson rlaton (wav-wav ntractons) mod shows a substantal mprovmnt n th modl stmats, yldng th bst comparson to th masurmnts

84 Fgur 7. Modld doman of Brkhoff t al. (198) 76

85 Fgur 8. Wav hght comparsons for a low rsoluton doman 77

86 Fgur 9. Wav hght comparsons for ncrasng doman rsolutons 78

87 Fgur 10. Wav hght comparsons from lnar and non-lnar CGWAVE runs on a doman rsoluton of 15 ponts pr wavlngth 79

88 8. Irrgular Wav Propagaton Ovr a Shoal Th hydraulc modl study of Vncnt and Brggs (1989) was usd to smulat spctral wav condtons n CGWAVE. Th drctonal spctral wav gnrator (DSWG) at th Coastal Engnrng Rsarch Cntr of th U.S. Army Engnrs was usd to smulat rfracton-dffracton of drctonally sprad rrgular wavs ovr an llptc shoal. Ths study showd that spctral condtons can produc sgnfcantly dffrnt rsults than monochromatc wavs. Th modl doman conssts of an llptc shoal surroundd by a rgon of constant dpth. Th orntaton of th shoal and placmnt of coastal boundars ar shown n Fgur 11. Th mnmum watr dpth ovr th shoal s approxmatly 0.15 m. Th dpth around th shoal s a constant m. Wav hght data was collctd along th nn sctons ndcatd n Fgur 11. Th spctral nput to CGWAVE was constructd usng a numrcally gnratd, two-dmnsonal spctra, E(f,θ) = E(f)D(θ). Exprssons usd for th nrgy spctrum, E(f), and th drctonal spctrum, D(θ), follow. Th nrgy spctrum s obtand usng th Txl Marsn Arslo (TMA) spctrum (Bouws t al. 1985, Panchang t al. 1990), 4 5 m ( f ) = α( π) f 15 + ( γ ) E g whr α = ( f f ) 4 f m xp. ln xp φ, f σ fm Phllps constant f m = pak frquncy γ = ( f h) pak nhancmnt factor (=1 for th Prson-Moskowtz spctrum) σ = shap paramtr (σ = σ a f f < f m and σ = σ b f f f m ) φ(f,h) s a functon that ncorporats dpth ffcts and s approxmatd by (Hughs 1984) as 0. 5( ω) for ω h < 1 φ= 1 0.5( ω h ) for 1 ω h 1.0 ω > for h whr ω h = πf(h/g) 1/. 80

89 Th drctonal spctrum s calculatd as follows (Vncnt and Brggs 1989), ( ) D θ j 1 1 = + xp π π j= 1 ( jσ ) m cos j ( θ θ ) whr θ m = man wav drcton = 0 j = numbr of trms n th srs (0 n numrcal calculatons) σ m = spradng paramtr m Th sam functons E(f) and D(θ) wr also usd n th laboratory study by Vncnt and Brggs (1989) to construct a numbr of frquncy and drctonal spctra (for dtals, s Vncnt and Brggs 1989 or Panchang t al. 1990). On spctral cas, B1, consstd of a broad frquncy spctrum; th othr cas, N1, had a narrow frquncy spctrum. Th nput data ar summarzd n Tabl. Two monochromatc and two spctral cass ar shown. Th ncdnt wav hght s [E(f)D(θ) f θ] 1/. Tabl. Modl Input Data Input Cas ID Prod (sc) Sgnfcant Wav Hght (m) α γ σ m Monochromatc M M Broad-drctonal B Narrow-drctonal N Th dscrtzaton of th drctonal spctrum s summarzd n Tabl 3. Othr rsarchrs hav usd th followng dscrtzatons. Panchang t al. (1990) usd a drctonal bn wdth ( θ) of 4.39 n a rang from 45 to +45, rsultng n 39 drctonal componnts for all spctra. Zhao and Anastasou (1993) found that as many as 63 componnts wr ndd for th B1 and N1 cass. L t al. (1993) producd rsults that suggst that an adquat soluton can b obtand wth as fw as fv drctonal componnts and fv frquncy componnts. In lght of th rsults of L t al. (1993), t 81

90 was dcdd that som xploraton of th ncssary numbr of drctonal componnts was n ordr. Tabl 3. Drctonal Rsolutons Usd for Modl Input Cas ID Numbr of Drctonal θ ( ) Componnts (bns) B N Tsts wr also prformd to quantfy th ffcts of th boundary condtons on th soluton. It was hypothszd that rflctons from th coastal boundars (smulatd walls of th wav tank) and mpropr tratmnt of ths fully absorbng boundars could caus sgnfcant varatons n th fnal soluton, spcally for componnts wth larg angls of ncdnc. Ths hypothss was tstd by comparng th rsults obtand from two domans. Doman A closly approxmatd th physcal modl and contand coastal boundars (Fgur 11). Intror and xtror boundars wr consdrd fully absorbng (K R = 0.0 and K EXT = 0.0). Doman B usd a full crcl as an opn boundary and contand all th dpth varatons (Fgur 1). It was found that th boundary condtons do not sgnfcantly ffct th rsults from CGWAVE n th ara of ntrst. Rsults ar only prsntd for Doman A. CGWAVE rsults for th monochromatc condtons M1 and M ar shown n Fgurs 13 and 14. Th wav ampltuds doubl bhnd th shoal and dcras on th sds of th shoal. Th lnar modl rsults show a smlar pattrn. Th non-lnar numrcal calculatons match th data bttr than th purly lnar rsults. Th data shows that th maxmum wav hght occurs furthr bhnd th shoal than prdctd by both lnar and non-lnar solutons (scton 7, Fgurs 13 and 14). Th CGWAVE rsults for th B1 spctral wav condton wth θ = 0 dscrtzaton s shown n Fgur 15. Th rsult xhbts fngr-lk aras of larg and 8

91 small wav hght radatng away from th shoal n th down-wav drcton that ar not prsnt n th lab data. Th modl rsults n Fgur 15 do not show th sam lvl of modulaton as th lab rsults. Th N1 solutons, although not shown, xhbt th sam down wav charactrstcs as th B1 solutons. Th numrcal soluton approachs th lab data whn th drctonal rsoluton s ncrasd. Fgurs 16 and 17 show th rsults for th B1 and N1 spctra, rspctvly, usng θ qual to Th wdths of th drctonal spctra wr lmtd to +/- 60 for B1 and +/-30 for N1 spctra, snc nglgbl amounts of nrgy ar outsd ths rang. Th fngr lk pattrns found n th cass wth larg θ wr not sn n ths rsults. Ths fndng agrs wth th obsrvatons mad by Zhao and Anastasou (1993) and Panchang t al. (1990) rgardng th wdth of drctonal bns ndd n th nput spctra. Th ffct of ampltud-dpndnt non-lnar dsprson was xamnd for th B1 and N1 cass. Zhao and Anastasou (1993)show an mprovmnt n th numrcal soluton. Th CGWAVE rsults ar ssntally unchangd (Fgurs 16 and 17). Ths occurs bcaus CGWAVE oprats on ach spctral componnt ndpndntly. Onc a lnar soluton for a gvn componnt s found, ths soluton ampltuds ar usd n th non-lnar dsprson rlaton and th nw soluton s found. Th nw ampltuds ar agan usd n th non-lnar mchansm. Ths tratons contnu untl a convrgnc crtron s rachd. Th fnal soluton to th componnt s savd and th modl s run for th nxt componnt. Th ncdnt wav hghts assocatd wth ach spctral componnt ar, at thr largst, an ordr of magntud smallr than th sgnfcant wav hght. Th wav hghts usd to comput non-lnar ffcts ar only a fracton of th sgnfcant wav hght, causng th non-lnar dsprson rlaton to hav a nglgbl ffct. For ths problm, CGWAVE producs lnar rsults that closly match th data for th monochromatc wavs propagatng ovr th shoal. Th ncluson of wav-wav ntracton has mprovd th qualty of ths soluton. CGWAVE also producs lnar rsults that closly match th data for th rrgular wav cass whn adquat drctonal 83

92 rsoluton s provdd n th nput spctra. Th soluton algorthm usd n th currnt vrson of th modl dos not compltly allow non-lnar ffcts to b accuratly rprsntd for th spctral wav modlng. Othr soluton algorthms should b valuatd n ordr to dtrmn whthr or not thy may yld rsults that mor closly match th lab data. 84

93 Fgur 11. Modl doman for wav propagaton ovr an llptc shoal [aftr Vncnt and Brggs (1989)] Fgur 1. Doman wthout coastal boundary ffcts 85

94 Fgur 13. Wav hght comparsons for M1 nput condton 86

95 Fgur 14. Wav hght comparsons for M nput condton 87

96 Fgur 15. Estmatd wav hghts for B1 nput spctrum, 0 drctonal dscrtzaton 88

97 Fgur 16. Wav hght comparsons for B1 nput condton, 4.14 drctonal dscrtzaton 89

98 Fgur 17. Wav hght comparsons for N1 nput condton, 4.14 drctonal dscrtzaton 90

99 8.3 Rctangular Harbor - Rsonanc In ths xampl, w xamn th nflunc of bottom frcton on dsprson of wav nrgy by frcton for rsonatng wavs n a rctangular harbor. Th opn boundary and harbor dmnsons ar shown n Fgur 18. Thortcal and lab data for ths cas wr prsntd n Ippn and Goda (1963), L (1971) and Chn (1986). Ippn and Goda (1963) showd that lnar thory prdcts ampltuds that ar too larg nar th rsonant frquncy of th harbor. Chn (1986) xamnd th ffcts of bottom frcton and coastal rflcton on th ampltuds. Laboratory data was collctd at th cntr of th back wall of th harbor (L, 1969). Modl nput condtons wr obtand from L (1969) and Chn (1986). Tabl 4 summarzs th nput wav condtons. Tabl 4. Modl Input Data Ampltud kl Ampltud Kl Th trm k s th wav numbr and l ( m) s th lngth of th harbor. Varatons n th frcton coffcnt and coastal rflcton wr consdrd. Wavs wr normally ncdnt to th xtror boundary for all modl runs. Rsults for a fully rflctng harbor ar shown n Fgurs 19, 0 and 1. Th rsults of lnar CGWAVE run wthout frcton show a good match to th analytcal soluton of Ippn and Goda (1963) (Fgur 19). Fgur 0 shows that th amplfcaton at 91

100 th rsonant paks dcrass as th coastal boundary rflcton coffcnt dcrass. Ths rflcton coffcnt dos not affct th frquncs at whch th rsonant pak occurs. Fgur 1 shows th ffcts of frcton on th CGWAVE prdctons. As th frcton paramtr s ncrasd, th soluton shows a progrsson toward th lab data at th frst rsonant pak. Th pak frquncy shows a small shft for th largr valus of th frcton paramtr. At th scond rsonant pak th solutons wth frcton match th soluton wthout frcton, ndcatng that nrgy dsspaton du to bottom frcton s lss mportant for short prod wavs than for long prod wavs. For ths problm, CGWAVE rproducs laboratory data by ncludng th ffcts of frctonal dsspaton and boundary absorpton. Th CGWAVE rsults show that harbor rsonanc s snstv to th nrgy absorbd at th coastal boundars and to dsspaton by frcton. Th choc of frcton paramtr and coastal rflcton coffcnt dos clarly affct th modl stmats n rsonanc cass. 9

101 Fgur 18. Rctangular rsonanc harbor wth sm-crcular opn boundary 93

102 Fgur 19. Thortcal and numrcal rsonanc curvs for a fully rflctng rctangular harbor 94

103 Fgur 0. Harbor rspons curvs for varous valus of coastal rflcton coffcnt 95

104 Fgur 1. Harbor rspons curvs for varous valus of th frcton factor 96

105 8.4 Crcular Island In ths xampl problm, w consdr th propagaton of long wavs around a crcular sland s smulatd. Homma (1950) and Jonsson, t al. (1976) showd that xtnsv scattrng occurs for long wavs. Ths cas tsts and dmonstrats th ablty of CGWAVE to handl strong scattrng ffcts wth a rlaxd (Xu t al. 1996) boundary condton. Ths boundary condton allows th modl to b appld to a gratr rang of modlng problms, as t dos not rqur full rflcton of th coastln outsd th modl doman. Th modld doman (Fgur ) conssts of a crcular sland on a shoal wth a parabolc varaton n dpth. Th sland has a radus of 10 km. Th opn boundary s placd at a radus of 35 km from th cntr of th sland. Th mnmum watr dpth s 444 m at th dg of th sland and ncrass to 4000 m at th outr dg of th shoal. Th doman has constant dpth byond th shoal. Incdnt wav prods of 40, 410 and 480 sconds ar consdrd. Fgur 3 shows th rsults for th 40 scond ncdnt wav condton. Ths fgur has bn dvdd n half du to symmtry of th soluton. Wavs ar ncdnt from th lft. Th uppr porton s th analytcal soluton, th lowr porton s th CGWAVE rsult. Th CGWAVE rsults match th analytcal soluton. Notably absnt ar sgns of artfcal boundary ffcts nar th opn boundary. Smlar rsults wr obtand for th longr wav prods, wth rsults matchng th analytcal data. 97

106 Fgur. Cross scton of sland and shoal bathymtry Fgur 3. Comparson of analytc and numrcal wav ampltuds 98

107 8.5 Ponc d Lon Inlt, FL Ths xampl shows th applcaton of CGWAVE to Ponc d Lon Inlt, Florda. Ths nlt s th subjct of a physcal modl study, allowng th comparson of laboratory data to CGWAVE modl runs. Wav-currnt ntracton may play a vry mportant rol on th wav flds to b prdctd n coastal nlts (Dmrblk t al. 1996b); ths capablty wll b avalabl n CGWAVE modl n th nar futur. Th coastln of th modld doman s orntd n roughly a north/south mannr. Th modld doman s boundd by a sm-crcl of.5 km radus. A jtty xtnds nto th doman from th north sd of th nlt. Th fnt lmnt msh for ths doman contans of about nods and lmnts. An xampl of a CGWAVE run of ths doman s gvn. Th wav phas dagram s shown n Fgur 4 and stmatd wav ampltuds ar shown n Fgur 5. Th ncdnt wav ampltud was 15 sconds. Th ncdnt wav drcton was normal to th coastln. Coastlns wr chosn to b fully nrgy absorbng (rflcton coffcnt of 0.0). Rsults from th numrcal modl ar compard to laboratory data n Fgur 6. Th data comparson s mad along a transct orntd paralll to th coastln and locatd svral hundrd mtrs to th south of th jtty. Th CGWAVE modl rsults ar for ths cas ar shown pror to fn tunng of th numrcal cod and adjustmnt of modl paramtrs. Th study s n progrss at th tm of ths wrtng. A dtald dscusson of th rsults s xpctd to appar n a rlvant scholarly journal at th concluson of th study. 99

108 Fgur 4. Phas dagram for normal ncdnt drcton, 15-scond prod wav condton 100

109 Fgur 5. Estmatd wav hght dagram for normal ncdnt drcton, 15-scond prod wav condton 101

110 Fgur 6. Comparson btwn laboratory data and CGWAVE wav hght stmats along a transct 10

111 8.6 Barbrs Pont Harbor, HI Ths xampl shows th applcaton of CGWAVE to th harbor and nar-by coastal rgons of Barbrs Pont Harbor, Hawa. A study s bng conductd to xplor harbor rspons to wav prods btwn 30 and 4000 sconds. Of partcular ntrst ar th contrbutons nfragravty wavs to th rsonanc of th harbor. Th modld doman conssts of th harbor ara and a rgon outsd th harbor of 3 km radus. Th radus of th ara outsd th harbor s masurd from th mouth of th harbor. Ths doman sz allows multpl wavlngths to b modld wthn th doman for all but th longst prod wavs. Wth approprat rsoluton (basd on th shortst prod wav) th grd contans nods and lmnts. Both monochromatc and spctral condtons ar usd as nput n ths modlng study. An xampl of th stmatd wav phass and ampltuds s gvn n Fgurs 7 and 8, rspctvly. Ths stmats ar basd on a normal ncdnt nput wav wth a prod of 100 s and ncdnt ampltud of 1 m. Th coastln has bn assumd to b fully absorbng for ths xampl cas. A dtald analyss of th rsults of ths study s n progrss at th tm of ths wrtng. Comparsons of CGWAVE wav hght stmats ar to b mad wth fld masurmnts n and around th harbor rgon. Th rsults and analyss ar to b documntd n rlvant journal artcls upon complton of th study. 103

112 Fgur 7. Wav phas dagram for normal ncdnt drcton, 100-scond prod wav condton 104

113 Fgur 8. Estmatd wav hght dagram for normal ncdnt drcton, 100-scond prod wav condton 105

114 8.7 Onslo Bay, NC Ths xampl shows th applcaton of CGWAVE to Nw Rvr Inlt n Onslo Bay, North Carolna. CGWAVE was appld to ths rgon to provd wav hght stmats for JLOTS (Jont Logstcs Ovr-Th-Shor) xrcss. A rctangular opn boundary s usd to sparat th modld doman from th opn ocan. Th modld ara outsd th nlt s 3 km along shor by km cross shor. Th nlt ara of th doman xtnds approxmatly 1.5 km nland. Th fnt lmnt msh for ths doman contans of about nods and lmnts. Th grd rsoluton s good (no lss than 10 nods pr wavlngth) for wavs of prod qual to or longr than 35 sconds ar. Th phas and ampltud rsults for a monochromatc wav smulaton ar shown n Fgurs 9 and 30, rspctvly. For ths modl run, ncdnt wavs wr normal to th coastln. Th ncdnt wav prod was 35 sconds. Th ncdnt wav hght was 1 m. Th coastln was assgnd a rflcton coffcnt of 0.4. Th altrnatng pattrn of wav phas s shown n Fgur 9. Wav hghts of m wr prdctd to th north sd of th nlt, rprsntd by th darkst color shown n Fgur

115 Fgur 9. Wav phas dagram for normal ncdnt drcton, 35-scond prod wav condton 107

116 Fgur 30. Estmatd wav hght dagram for normal ncdnt drcton, 35-scond prod wav condton 108