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1 W r InchOpn, h world s lding publishr of Opn Accss books Buil b sciniss, for sciniss 3,500 08,000.7 M Opn ccss books vilbl Inrnionl uhors nd diors Downlods Our uhors r mong h 5 Counris dlivrd o TOP % mos cid sciniss.% Conribuors from op 500 univrsiis Slcion of our books indd in h Book Ciion Ind in Wb of Scinc Cor Collcion (BKCI) Inrsd in publishing wih us? Conc book.dprmn@inchopn.com Numbrs displd bov r bsd on ls d collcd. For mor informion visi

2 Chpr Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw José Simão Anuns do Crmo Addiionl informion is vilbl h nd of h chpr hp://d.doi.org/0.577/6866 Absrc Numricl modls r usful insrumn for suding compl suprposiion of wv wv nd wv currn inrcions in cosl nd surin rgions nd o invsig h inrcion of wvs wih compl bhmris or srucurs buil in nrshor rs. Morovr, sinc hir pplicions r significnl lss pnsiv nd mor flibl hn h consrucion of phsicl modls, h r convnin ool o suppor dsign. Th bili of h sndrd oussinsq nd Srr or Grn nd Nghdi quions o rproduc hs nonlinr procsss is wll known. Howvr, hs modls r rsricd o shllow wr condiions, nd ddiion of ohr rms of disprsiv origin hs bn considrd sinc h 990s, priculrl for pproimions of h oussinsq-p. To llow pplicions in grr rng of ν 0 / λ, ohr hn shllow wrs, whr ν 0 is h wr dph rs nd λ is h wvlngh, nw s of Srr-p quions, wih ddiionl rms of disprsiv origin, is dvlopd nd sd wih h vilbl d nd wih numricl soluion of oussinsq-p modl, lso improvd wih disprsiv chrcrisics. Δplici nd implici mhods of fini diffrnc r implmnd o solv boh pproimions of oussinsq nd Srr ps wih improvd disprsiv chrcrisics. fini lmn mhod is lso implmnd o solv n nsion of oussinsq-p modl h ks ino ccoun wv currn inrcions. pplicion mpls o solv rl-world problms r shown nd discussd. Th prformncs of boh HΓ modls r comprd wih primnl d of vr dmnding pplicions, nml: (ξ) highl nonlinr solir wv propging up slop nd rflcing from vricl wll nd (ξξ) priodic wv propging in inrmdidph wrs upsrm rpzoidl br, followd b vr shllow wrs ovr h br, nd gin in inrmdi-dph wr condiions downsrm. Kwords: Δndd oussinsq quions, Δndd Srr quions, Γisprsiv chrcrisics, Inrmdi wrs, Numricl mhods 05 Th Auhor(s). Licns InTch. This chpr is disribud undr h rms of h Criv Commons Aribuion Licns (hp://crivcommons.org/licnss/b/3.0), which prmis unrsricd us, disribuion, nd rproducion in n mdium, providd h originl work is proprl cid.

3 4 Nw Prspcivs in Fluid Dnmics. Inroducion In rcn dcds, significn dvncs hv bn md in dvloping mhmicl nd numricl modls o dscrib h nir phnomn obsrvd in shllow wr condiions. Indd, no onl our undrsnding of h phnomn hs significnl improvd bu lso h compuionl cpbiliis h r vilbl hv lso incrsd considrbl. In his con, w cn now us mor powrful nd mor rlibl ools in h dsign of srucurs commonl usd in cosl nvironmns. h nd of h 970s, du o h lck of sufficinl dp knowldg, bu bov ll for lck of compuing powr, h us of h linr wv hor for h simulion of phnomn, such s rfrcion nd diffrcion of wvs, ws common prcic. In h 980s, ohr modls h k ino ccoun no onl h rfrcion bu lso h diffrcion procss hv bn proposd nd commonl usd b [] rkhoff ζρ., 98, [] Kirb nd Γlrmpl, 983, [3] ooij, 983, [0] Kirb, 984, nd [5] Γlrmpl, 988, mong mn ohrs. Howvr, s h r bsd on h linr hor, hos modls should no b uilizd in shllow wr condiions. s nod in [6], vn h im modls bsd on h Sin-Vnn quions (s lso [3] Sin-Vnn, 87) wr λrquρ usd ξ prζcξcζρ ζppρξcζξτs. Hτwvr, ζs νζs η wξdρ dmτsrζd, ξ sνζρρτw wζr cτdξξτs ζd λτr sτm ps τλ wζvs, mτdρs ηζsd τ ζ τ-dξsprsξv ντr, τλ wνξcν ν Sζξ-Vζ mτdρ ξs ζ ζmpρ, ζr ρξmξd ζd ζr τ usuζρρ ζηρ τ cτmpu sζξsλζcτr rsuρs τvr ρτμ prξτds τλ ζζρsξs (s lso [33]). Nowds, i is gnrll ccpd h for prcicl pplicions h combind grvi wv ffcs in shllow wr condiions mus b kn ino ccoun. In ddiion, h rfrcion nd diffrcion procsss, h swlling, rflcion nd brking wvs, ll hv o b considrd. lso ccording o [6] ζ umηr τλ λζcτrs νζv mζd ξ pτssξηρ τ mpρτ ξcrζsξμρ cτmpρ mζνmζξcζρ mτdρs. Indd, no onl hs hr bn gr improvmn in our horicl knowldg of h phnomn involvd bu lso h numricl mhods hv bn usd mor fficinl. Th gr dvncs md in compur chnolog, spcill sinc h 980s, improving informion procssing nd nbling lrg mouns of d o b sord, hv md possibl h us of mor mhmicl modls, of grr compli nd wih fwr rsricions. Indd, onl modls of ordr ( =ν 0 / λ, whr ν 0 nd λ rprsn, rspcivl, dph nd wvlngh chrcrisics) or grr, of h oussinsq or Srr ps [4, 35], r bl o dscrib h nir disprsiv nd nonlinr inrcion procss of gnrion, propgion nd run-up of wvs rsuling from wv wv nd wv currn inrcions. I is lso worh o poin ou h in mor compl problms, such s wv gnrion b sfloor movmn, propgion ovr unvn booms, nd ddd brking ffcs, high-frqunc wvs cn ris s consqunc of nonlinr inrcion. In h ps fw rs, h possibili of using mor powrful compuionl fciliis, nd h chnologicl voluion nd sophisicion of conrol ssms hv rquird horough horicl nd primnl rsrch dsignd o improv h knowldg of cosl hdrodnmics. Numricl mhods imd for h pplicions in nginring filds h r mor sophisicd nd wih highr dgr of compli hv lso bn dvlopd.

4 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ In Scion, h gnrl shllow wr wv hor is usd o dvlop diffrn mhmicl pprochs, which r nowds h bsis of h mos sophisicd modls in hdrodnmics nd sdimnr dnmics. Δnsions of h quions for inrmdi wr of h mor gnrl pprochs (ordr ) r prsnd in Scion 3. Numricl formulions of h modls nd ppropri pplicion mpls r prsnd in Scion 4. Firsl, submrgd dissipion plform o proc h ugio lighhous, siud h Tgus sur mouh, Lisbon, is dsignd nd sd numricll. Th scond pplicion is rl-world problm concrning cosl procion, using submrgd srucur o forc h brking wvs offshor. Th hird mpl shows h giions sblishd in h por of Figuir d Foz, Porugl, for diffrn s ss, wih h objciv of dsigning nw prociv jis, or nd h ising ons. Finll, h fourh mpl shows h prformnc of h ndd Srr quions for h propgion of wv in vr dmnding condiions. Conclusions nd fuur dvlopmns r givn in Scion 5.. Mhmicl formulions W sr from h fundmnl quions of h Fluid Mchnics, wrin in Δulr s vribls, rling o hr-dimnsionl nd quζsξ-irroionl flow of prfc fluid [Δulr quions, or Nvir-Soks quions wih h ssumpions of non-comprssibili (dρ / d =dξvv =0), irroionli (rτv =0) nd prfc fluid (dnmic viscosi, µ =0)]: u + v + w = 0 z u + uu + vu + wu = - p z v + uv + vv + wv = - p z w + uw + vw + ww = - p r - μ z z u = w ; v = w ; v = u z z r r () wih p =0 z =η(,, ), w =η + uη + vη z =η(,, ), nd w =ξ + uξ + vξ z = ν 0 + ξ(,, ). In hs quions ρ is dnsi, is im, μ is grviionl cclrion, p is prssur, η is fr surfc lvion, ξ is boom, nd u, v, w r vloci componns. Γfining h dimnsionlss quniis ε =ζ / ν 0 nd =ν 0 / λ, in which ζ is chrcrisic wv mpliud, ν 0 rprsns wr dph, nd λ is chrcrisic wvlngh, w procd wih suibl nondimnsionl vribls: ( ) ' = l, ' = l, z' = z ν, h' = h ζ, ' = ν, ' = μν l = c l, p' = p rμν, ( ) ( ) ( ) ( ) l ( ) l ( 0 ) u' = u ζ μ ν = uν ζc, v' = v ζ μ ν = vν ζc, w' = w ζν μ ν = w ζc, whr c 0 rprsns criicl clri, givn b c 0 =(μν 0 ) /, nd, s bov,η is fr surfc lvion, ξ rprsns bhmr, u, v nd w r vloci componns, nd p is prssur. In

5 6 Nw Prspcivs in Fluid Dnmics dimnsionlss vribls, wihou h lin ovr h vribls, h fundmnl quions nd h boundr condiions r wrin [5]:. Fundmnl quions u + v + w = 0 z εu + ε uu + ε vu + ε wu = -p z εv + ε uv + ε vv + ε wv = -p z ε w + ε uw + ε vw + ε ww = -p - d z z u = s w ; v = s w ; v = u z z b c () b. oundr condiions ( ) w = + u + v, z = - + w = h + uh + vh, z = h b p = 0, z = h c (3) Ingring h firs quion () bwn h bd + ξ nd h fr surfc εη, king ino ccoun 3() nd 3(b), ilds h coninui quion (4): ( ) ( ) ( ) éη ε ξ é ξ εη u é ξ εη v = 0 (4) whr h br ovr h vribls rprsns h vrg vlu long h vricl. Thn, ccping h fundmnl hpohsis of h shllow wr hor, =ν 0 / λ < <, nd dvloping h dpndn vribls in powr sris of h smll prmr, h is ξ= 0 ξ ( s ), for ξ ( h ) λ = å λ λ = u,v,w,p,,,a (5) whr A=u + v, from coninui (), nd wih 3() nd 3(b), h following quions r obind: ( ) w = - z + - A + w (6) * ( h ) w = A + w (7) ** *

6 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ whr h simpl nd doubl srisk rprsn h vribls vlus h boom nd h surfc, rspcivl. Of () w obin, succssivl [34]: ( ) ( ) u = u,, v = v,, ( )( ) ( )( ) ( )( ) ( )( ) u = - z + - A + z + - A + w + u * * v = - z + - A + z + - A + w + v * * (8) (9) so h h vrg vlus of h horizonl componns of h vloci, on h vricl, r givn b: ( 6)( ) * s s h u = u + u A * 4 ( s )( h 0 0 )( A w ) O( s ) ( 6)( ) * s s h v = v + v A * 4 ( s )( h 0 0 )( A w ) O( s ) (0) On h ohr hnd, king ino ccoun h, from (5) nd (9) w obin: * ( s ) ( h ) λ = λ + O for λ = u, v,,, w 0 () ( s 3)( h ) ** u = u A * 4 ( s )( h )( A w ) O( s ) ( s 3)( h ) ** v = v A * 4 ( s )( h )( A w ) O( s ) () Rprsning b Γ=w + εuw + εvw + εww z h vricl cclrion of h pricls, w g Γ=w 0 + εu 0 w 0 + εv 0 w 0 + εw 0 w 0z + O( ), nd from (6), (7) nd () h following pproch is obind: ( ) G = - z + - éa ua va A * * * + éw uw vw + + O + ( s ) (3)

7 8 Nw Prspcivs in Fluid Dnmics in which h rms wihin h firs prnhsis (srigh prnhsis) rprsn h vricl cclrion whn h boom is horizonl, nd h rms wihin h scond prnhsis (srigh prnhsis) rprsn h vricl cclrion long h rl boom. I should b nod h quion (d) cn b wrin s: s G = -p z - (4) whr, b vricl ingrion bwn h boom nd h surfc, h prssur p on h surfc is obind s: p p ( s ) ( s ) = h G + ** ** = h G + ** ** (5) which, long wih (b) nd (c), llow us o obin [34]: ** ( u uu vu wu ) h z ( s ** ) ** ( v uv vv wv z ) h ( s ** ) G = G = 0 (6) or vn, givn h ( λ s ) ** = λ s ** ε( λ z ) ** η s, whr λ =(u, v) nd s =(,, ): ( ) ( ) u + u u + v u + h + s G = 0 ** ** ** ** ** ** v + u v + v v + h + s G = 0 ** ** ** ** ** ** (7) dvloping prssions (7) in scond pproch (ordr in ), h following quions of moion (8) r obind (for dils s [34]): u + uu + vu + h { é( 3)( ) ( ) ( 3)( ) P P } é Q ( )( ) Q s h h 4 + s h + + h - s + = v + uv + vv + h s é { ( 3)( h ) ( ) ê P ( 3)( h ) P ú } 4 s éh Q ( )( h ) Q s 0 ( h ) A = ( ) P = + - A - ua - va - Q = w + uw + vw ( ) w = + u + v A = u + v (8)

8 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ whr, likwis, h br ovr h vribls rprsns h vrg vlu long h vricl. In dimnsionl vribls nd wih solid/fid boom (ξ =0), h compl s of quions is wrin, in scond pproch: ( ) ( ) ν + νu + νv = 0 u + uu + vu + μh ( ) ( ) ( ) ( ) + é 3 ν + P 3 νp ν Q νq = v + uv + vv + μh ( ) ( ) ( ) ( ) + é 3 ν + P 3 νp ν Q νq = ( ) P = ν A - ua - va - A Q = w + uw + vw w = u + v A = u + v (9) whr ν =ν 0 ξ + η is ol wr dph. Th on-dimnsionl form (HΓ) of h quion ssm (9) is wrin, lso wih fid boom: ( uν) ν + = 0 ( ) ( ) ( ) ( ) νu + νuu + μνh + éν P 3 + Q + ν P + Q = 0 P = - ν u + uu - u Q = u + uu + u (0) ssuming ddiionll rliv lvion of h surfc du o h wvs (ε =ζ / ν 0 ) hving vlu clos o h squr of h rliv dph ( =ν 0 / λ), i.. O(ε) = O( ), from h ssm of quions (8), nd h sm ordr of pproimion, h following pproch is obind, in dimnsionl vribls: ( ) ( ) ν + νu + νv = 0 ( ) ( ) ( ) ( ) ( ) ( ) u + uu + vu + μh - é P + ν P + ν Q = r r v + uv + vv + μh - é 6 P 3 ν P ν Q = r r () whr ν r =ν 0 ξ is h wr column high rs, P nd Q r givn b P = (ν 0 ξ)(ū + v ) nd Q =(ūξ + v ξ ). Th momnum quions r wrin s:

9 30 Nw Prspcivs in Fluid Dnmics h ( 6) ( ) ( 3) ( ) ( 3) ν ( u v ) ( ) ν ( u v ) 0 r r u + uu + vu + μ + ν u + v - ν u + v r r = h ( 6) ( ) ( 3) ( ) ( 3) ν ( u v ) ( ) ν ( u v ) 0 r r v + uv + vv + μ + ν u + v - ν u + v r r = () (3) wih ξ =0, h compl ssm of quions (4) is obind: ( ) ( ) ν + νu + νv = 0 h ( 3) ( ) ( ) ν ( u v v v r ) h ( 3) ( ) ( ) νr ( v u u u ) u + uu + vu + μ - ν u + v + ν u r r = 0 v + uv + vv + μ - ν u + v + ν v r r = 0 (4) Furhr simplifing h quions of moion (8), rining onl rms up o ordr in, i.., nglcing ll rms of disprsiv origin, his ssm of quions is wrin in dimnsionl vribls: ( ) ( ) 0 ν + νu + νv = u + uu + vu + μh = 0 v + uv + vv + μh = 0 (5) pprochs (9), (4) nd (5) r known s Srr quions, or Grn & Nghdi, oussinsq nd Sin-Vnn, rspcivl, in wo horizonl dimnsions (HΓ modls). Th clssicl Srr quions (9) [7] r full nonlinr nd wkl disprsiv. oussinsq quions (4) onl incorpor wk disprsion nd wk nonlinri nd r vlid onl for long wvs in shllow wrs. s for h oussinsq-p modls, lso Srr s quions r vlid onl for shllow wr condiions. 3. Drivion of highr-ordr quions 3.. Wkl nonlinr pprochs wih improvd disprsiv prformnc 3... Nwτμu s ζpprτζcν To llow pplicions in grr rng of ν 0 / λ, ohr hn shllow wrs, [7] inroducd highr-ordr disprsiv rms ino h govrning quions o improv linr disprsion

10 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ propris. rdfining h dpndn vribl, [30] chivd h sm improvmn wihou h nd o dd such rms o h quions. Following [36], h ndd ussinsq quions obind b [30] r drivd in his scion, using h nondimnsionlisd scld quion ssm () (3) s h sring poin, rhr hn h procdur prsnd b Nwogu. For simplici rsons, nd wihou loss of gnrli, h on-dimnsionl ndd oussinsq quion ssm is drivd. For consisnc wih h prvious work (Scion ), h coninui quion () in wo vricl dimnsions (, z) is ingrd hrough h dph: z z ò w dz = - z ò u dz (6) - + ξ - + ξ æ ö w - w = - udz + u - u z z z - + ç ò è - + ξ ø - + z (7) Γnoing w z b w, wih ξ =0 nd using Libniz rul, h boundr condiion 3() z = + ξ givs: z æ ö w = -ç udz ç ò (8) è - + ξ ø Subsiuing (8) in h irroionli quion (), u z = w, ilds: æ z ö u = s w = -s udz z ç ò (9) è - + ø Considring Tlor sris pnsion (30) of u(, z, ), bou z = z α, whr h horizonl vloci u α (, ) dnos h vloci dph z α, u (, z α, ), his Tlor pnsion is ingrd hrough h dph from + ξ o z, ilding (3) (for dils s [36]): ( ) ( z - z ) α u = u + z - z u + u +L α α z z= z zz (30) α z= zα z ò - + ξ ( z - z ) ( - + z ) é udz = ( z + - ) u + ê - úu ê ú ( z - z ) ( - + z ) 3 3 é + ê + úu + L zz ê 6 6 ú z= z z z= z (3)

11 3 Nw Prspcivs in Fluid Dnmics Subsiuing (3) in quion (9) givs: ì é( z z ) ( z ) ü u = - s ( z z í + - ) u + ê - úu + L z ý (3) ê ú î z= z þ Γiffrniing quion (3) wih rspc o z, noing from (9) h u z =O( ) : ( s ) 4 ( z - z ) ì ü u = - s zz íu + ( z - z ) u + u + L z z= z zz ý z= z î þ = - s u + O (33) Γiffrniing quion (33) wih rspc o z nd noing h boh u z nd u zz r O( ) : { ( ) } ( s ) z= z z= z u = - s u + z - z u + L = O (34) 4 zzz z zz Rpd diffrniion of his prssion will produc prssions for h highr drivivs of u wih rspc o z nd show hm o b of O( 4) ordr or grr. Subsiuing quions (33) nd (34) bck in quion (3): ì ( - + z ) u = -s ( - + z z ) u - u + O s î z= z ( z ) u O ( ) ( ) í z 4 = -s é s ü ý þ (35) Subsiuing quions (33), (34) nd (35) in h Tlor sris pnsion (30) producs n prssion for h horizonl vloci componn u: ( z - z ) ü ( ) ý ( ) ì u = u -s ( z - z ) é( - + z ) u + u + O s î þ 4 α í α α α (36) Subsiuing h horizonl vloci (36) in quion (8) lds o n prssion for h vricl vloci componn w:

12 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ {( ) w = z u 3 ( z ) ( z ) éæ ö ç z u ( u ) é ü s ê ú ê 6 ú þ è ø - ý + O 4 ( s ) (37) Using h vlociis u (36) nd w (37) in h vricl momnum, quion (d) ilds: { é( ) α ( )} z ( ) ε - z + - ξ u + O + p + + O ε = 0 (38) This cn b rrrngd o giv n prssion for h prssur p: ( ) ( ) 4 - p = - s é z + - u + O s, s z (39) Ingring hrough h dph from z o h fr surfc εη: ìé z ü - p + p = h - z + s í + - z u + O s, s h z ê î þ 4 ( ) ú ý ( ) (40) Using h fr surfc boundr condiion 3(c) for h prssur, nd dnoing p z simpl s p givs: ìé z ü p = h - z + s íê + - z u + O s, s î þ 4 ( ) ú ý ( ) (4) Subsiuing h quions for h horizonl vloci componn (36), h vricl vloci componn (37) nd h prssur (4) in h horizonl momnum quion (b) givs h oussinsq momnum quion: ( z - z ) ( ) ì ü -s ( - ) é( - + ) + ý î þ ìé z ü + u u + h + s + - z u = O s, s î þ u z z z u u í 4 íê ( ) ú ý ( ) (4)

13 34 Nw Prspcivs in Fluid Dnmics or ì ü u + u u + h + s z é - u + u = O s, s z ( ) ( ) ý ( ) 4 í î þ (43) Th scond quion of h oussinsq ssm is dvlopd b firs ingring h coninui quion () hrough h dph + ξ: h æ ö udz - u h u w w 0 h h = ç ò è - + ø (44) Using Libniz Rul nd h kinmic boundr condiions h bd z = + ξ 3() nd h fr surfc z = εη 3(b) givs: h æ ö h + udz 0 = ç ò (45) è - + ø From h prssion (36) for h horizonl vloci u: h ò - + h ì é ( z z ) - udz = ( - + h ) u -s ( z ) u í ê ú é - + ê ú î - + εη 3 é ü ( z - z ) 4 + ê ú u O ý + ( s ) ê 6 ú - + þ (46) or h ò - + ( - + z ) ìé z udz = ( - + h ) u -s ( z ) u í ê - ú é - + ê ú î 3 é 3 z ( - + z ) ü 4 + ê- + úu O ( s, s ý + ) ê 6 6 ú þ (47) ilding:

14 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ h ò - + ( - ) ìé udz = ( - + h ) u -s ( ) z ( z ) u í ê ú é - + ê ú î 3 é ( - ) ( - ) z ( - ) z ü 4 + ê + + ú( u ) O ( s, s ý + ) ê 6 ú þ (48) nd hus h ò - + ìé - udz = ( - + h ) u + s z ( ) ( ) u íê + ú - é - î é z ( - ) ü 4 + ê - ú( - )( u ) O ( s, s ý + ) ê 6 ú þ (49) Subsiuing quion (49) in h fr surfc quion (45) givs h oussinsq coninui quion: ìæ - ö h + é( h ) u s z ( ) é( ) u íç è î ø æ z ( - ) ö ü 4 + ç - ( - )( u ) O( s, s ý = ) ç 6 è ø þ (50) Rurning o dimnsionl vribls, unscld form, quions (43) nd (50) r wrin: z u + u u + μh + z ( ν u ) + ( u ) = 0 (5) r h éæ ν ö æ z ν r r + é( ν h 0 ) u ( ) ( ) 0 + z ν ν u ν u + êç + + r r - = r ú ê ç 6 è ø è ø ú ö (5) Sing h rbirr dph z α =αν 0, whr α 0, h ssm (5) (5) is rwrin: u u u μ ν ν ( u + + h + é ) ( ) 0 r r ν u + = r (53)

15 36 Nw Prspcivs in Fluid Dnmics é æ ö æ ö 3 h + é( ν h ) u ν ( ν u ) ν ( u ) 0 + r + êç = r r ç 6 r ú ê è ø è ø ú (54) In wo dimnsions, h quion ssm (53) (54) is wrin:. ( ν h ) h + Ñ é + u r ì æ ö æ ö 3 ü + Ñ íç + ν Ñ éñ.( ν ) + ν (. ) 0 r r - Ñ Ñ = r ý u ç 6 u î è ø è ø þ (55)... r r r ( ) μ h ν é ( ν ) ν ( ) u + u Ñ u + Ñ + Ñ Ñ u + Ñ Ñ = 0 u (56) whr rprsns h wo-dimnsionl grdin opror wih rspc o h horizonl coordins (, ) ( = /, / ), nd h vloci vcor u(,, )=(u, v) rprsns h wodimnsionl vloci fild dph z =αν Bjξ ζd Nζdζτπζ s, ζd Lξu ζd Su s ζpprτζcνs Sring from h sndrd oussinsq quions nd doping h mhodolog inroducd b [7] [] prsnd nw pproch h llows for pplicions unil vlus of ν 0 / λ o h ordr of 0.5, nd sill wih ccpbl rrors in mpliud nd phs vloci up o vlus of ν 0 / λ nr n ddiion nd subrcion procss, using h pproimion u = μ η nd considring disprsion prmr β in h momnum quion of h HΓ ssm (4), ji nd Ndok obind n improvd s of oussinsq quions for vribl dph, wih β vlu obind b comprison of h disprsion rlion of h linrizd form of h rsuling quion wih scond-ordr Pdé pnsion of h linr disprsion rlion ω / μπ =nh(πν r ). In ordr o improv disprsion nd linr sholing chrcrisics in h ji nd Ndok s quions, [3] inroducd wo uning prmrs, α nd γ, so h β =.5α 0.5γ. Th nonlinri in h prvious oussinsq-p modls ws improvd b Liu nd Sun dding highr-ordr rms ccur o h ordr of O( ε ). Th HΓ sndrd oussinsq quions nd h pprochs of ji nd Ndok nd Liu nd Sun r idnifid wihin h following ssm of quions (57) for wr of vribl dph: ν ( νu) 0 h {( ) ( g ) 6} ( 6) ( ) + = u + uu + μ ν u r - - g μν h + + ν u + μν h r r r ( ) h r r ( s ) ν u μν = O (57)

16 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ fr linrizion of h quions ssm (57), [3] obind h following disprsion rlion (58): πν é ( 6 r + - g ) ( ) ( g ) w π ν r = (58) μπ + + π ν / - + π ν / 6 r r Compring quion (58), wrin in rms of h phs spd (59) C + ( - g 6 )( πνr ) ( )( ) ( g )( ) w é = = μν ê ú r π ê πν / πν / 6 ú r r (59) wih h linr disprsion rlion ω / μπ =nh(πν r ), using h pproch (60) ( πνr ) ( πν ) w é C = = ( μν ) nh Aξr r ( πνr ) = μν ê ú + O é r ( πν) π ê ê ú 5ú + r (60) llows o obin h bs vlus for h prmrs α nd γ: α =0.308 nd γ = Considring ppropri vlus for h uning prmrs α nd γ, w cn idnif wihin h quion ssm (57): Th sndrd oussinsq quions b sing α =γ =0 Th ji nd Ndok quions considring α =γ =0.0 Th Liu nd Sun quions wih α =0.308 nd γ = visul comprison of numricl rsuls of h ndd oussinsq pproimion (57), wih α =0.308 nd γ = ), shown in [, ] nd [3], wih similr sud prformd b [37], using h ndd oussinsq (53) (54) modl (Nwogu s pproch, wih α = 0.53) shows no rlvn diffrncs in h grphs. In his rgrd, i is worh rmmbring [4] ζρντuμν ν mντds τλ drξvζξτ ζr dξλλr, ν rsuρξμ dξsprsξτ rρζξτs τλ νs dd Bτussξsq quζξτs ζr sξmξρζr, ζd mζ η ντuμν τλ ζs ζ sρξμν mτdξλξcζξτ τλ ν scτd-τrdr Pζdé ζpprτξmζ τλ ν λuρρ dξsprsξτ rρζξτ. 3.. Full nonlinr pprochs wih improvd disprsiv prformnc Insd of using h horizonl vloci crin dph, ohr nsions of h oussinsq quions hv bn md b using h vloci ponil on n rbirr dph, lso wih uning prmr. Wi ζρ. (995) [39] usd h Nwogu s pproch o driv oussinsqp modl which rins full nonlinri b including O( ε ) rms no considrd in h

17 38 Nw Prspcivs in Fluid Dnmics Nwogu s (53) (54) ssm, nd hus improving h nonlinri o O(ε)=. Wi ζρ., nd lr [6] Gobbi ζρ., drivd fourh-ordr oussinsq modl in which h vloci ponil is pproimd b fourh-ordr polnomil in z. In rms of non-dimnsionl vribls, boh considr h boundr vlu problm for ponil flow, givn b: 0, j + m Ñ j = - + h zz j + m Ñν. Ñ j = 0, z = - + z - é h + j + ( ) ( ) 0, ê Ñ j + j z z ú = = h m h + Ñj. Ñh - j = 0, z = h z m z (6) whr, s bov, z is h vricl coordin sring h sill wr lvl ν 0 (, ) nd poining upwrds, scld b picl dph ν 0, nd η is h wr surfc displcmn scld b rprsniv mpliud ζ. Th wo dimnsionlss prmrs ε nd µ r dfind s ε =ζ / ν 0 nd µ =(π 0 ν 0 ), wih rprsniv wv numbr π 0 =π / λ, so µ =(π). Tim is scld b π 0 (μν 0 ) /, nd φ, h vloci ponil, is scld b εν 0 (μν 0 ) /. W us h nondimnsionl wr lvl ν ξ insd of ξ. Ingring h firs quion of (6) ovr h wr column, nd using h ppropri boundr condiions, h coninui quion is obind: h + Ñ. M = 0 (6) εη whr M = +ξ φdz. Rining rms o O ( µ ), nd dnoing φα s h vlu of φ z = z α (, ), n pproim vloci ponil is givn b:. ( ) m, ( z z) ( ν ) ( z z ) O( 4 - ) j = j + m - Ñ Ñ j + - Ñ j + m (63) Subsiuing quion (63) ino (6), mss flu consrvion quion is obind [39]:.. ì æ ö ì ì é z h + Ñ íç ν + h z ( ν j m j ) j è - ø íñ + íñ ê Ñ Ñ + Ñ - ú î î î ê ú é ( ν h ) ν hν ( h ) üüü ê - - ú + Ñ éñ. ( ν Ñj ) - Ñ Ñ j = 0 - ýýý 6 þþþ (64)

18 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ Similrl, subsiuing (63) ino h hird quion of (6), momnum quion is obind in rms of h vloci ponil. Thn, givn h z = z α, u α = φ α, full nonlinr vrsion of oussinsq p modl in rms of η nd u α is wrin:.. ( ( ) ) ú ( u ) ì ì ìé z h + Ñ í + h í + m í - - h + h Ñ Ñ 6 î î î ê ú é üüü + êz + ν - Ñ Ñ. ν = O ú þ þþ æ ö ç ν ν ν u - ê è ø ( h ) é ( u ) ( m - - ýýý ) (65). ( ) O( 4 ) u + u Ñ u + Ñ h + m R + m S = m (66) whr.... é R = z Ñ( Ñ u ) + z Ñ éñ ( ν u ) - Ñ ( h ) Ñ + hñ ( ν - - ) ê u u ú (67) ì S = Ñ í - u. Ñ Ñ. u + - u. Ñ Ñ. u î ( z h )( ) é ( ν ) éz ( h ) - ( )( ) ê ú + Ñ é { Ñ.( ν u ) + hñ. u - } ü ý þ (68) I should b nod h Nwogu s pproimion is rcovrd b nglcing highr-ordr rms. Numricl compuions show h his modl grs wll wih soluions of h full ponil problm ovr h rng of rlvn wr dphs. high-ordr, prdicor corrcor, fini diffrnc numricl lgorihm o solv his modl ws dvlopd nd is prsnd in [5] for h on- nd wo-lr modls (s Scions 3.. nd 3..). Considring on lr onl, h corrsponding numricl modl h is h bsis of h COULW VΔ modl is prsnd lr, in Scions 4..3 nd To improv h ccurc of numricl modls, i hs bn common prcic h us of highordr polnomils o pproim h vricl vloci dpndnc. Howvr, his rquirs vr lbor nd pnsiv numricl clculion procdurs. diffrn pproch is suggsd b [6], which consiss of using qudric polnomils, mchd inrfcs h divid h wr column ino lrs. In his rgrd, i is worh mnioning [6] "νξs ζpprτζcν ρζds τ ζ s τλ mτdρ quζξτs wξντu ν νξμν-τrdr spζξζρ drξvζξvs ζssτcξζd wξν νξμν-τrdr pτρτmξζρ ζpprτξmζξτs".

19 40 Nw Prspcivs in Fluid Dnmics 3... Mζνmζξcζρ mτdρ λτr τ-ρζr Γfining h prmrs S =.u nd T =. ( ξ) u + ( / ε)( ν ξ / ), h modl uss h following pproch for h coninui quion (o compu η vlus), in nondimnsionl vribls:. ν- h + + Ñ é( h + ν ) u - ìé ( h + ν ) π -. í ê 6 ú î é h - ν ü + ê - + úñ ý = ê ú þ h + ν- - m Ñ ê - ú ÑS - 4 ( h ν ) π T O ( m - ) (69) whr u = horizonl vloci vcor, π =α ν + β η, α nd β r cofficins o b dfind b h usr. Th ind mns on-lr modl. To compu h vloci componns (u, v), h following pproch of h momnum quion is solvd, in nondimnsionl vribls:. u ì π ü + u Ñ u + Ñ h + m í Ñ S + π ÑT ý î þ é ( u. π ) T π ( u. T ) π ( u. π ) S π é æ T ö + Ñ( u. Ñ S ) ú + m êt ÑT - Ñ h ç ú ú è ø æ h S ö + m Ñ ç hs T - -h u. ÑT è ø é h 4 + m ( ) ( ) S. S Ñ ê - u Ñ ú = O m + m Ñ Ñ + Ñ Ñ + Ñ Ñ (70) Th horizonl vloci vcor is givn s: U ìz - π 4 = u - m S z π T O í Ñ + - Ñ + î ü ( ) ý ( m ) þ (7) This on-lr modl, ofn rfrrd o s h full nonlinr, ndd oussinsq quions in h lirur (.g. [38]), hs bn mind nd pplid o significn n. Th wkl nonlinr vrsion of (69) (7) (i.., ssuming O(ε)=O( µ ), hrb nglcing ll nonlinr disprsiv rms) ws firs drivd b [30]. Through linr nd firs-ordr nonlinr nlsis of h modl quions, Nwogu rcommndd h z = 0.53ν, nd h vlu hs bn

20 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ rcommndd nd dopd b mos rsrchrs who us hs quions. Rfs. [] nd lr [38] usd h Nwogu s pproch o driv high-ordr modl which rins "full nonlinri" b including rms no considrd in h Nwogu s pproch. Compring Liu's nd Wi nd Kirb's quions, hr r som diffrncs h cn b ribud o h bsnc of som nonlinr disprsiv rms in Wi nd Kirb [9]. Th bov on-lr quions (69) nd (70) r idnicl o hos drivd b [] Mζνmζξcζρ mτdρ λτr wτ-ρζrs Γils bou h wo-lr mhmicl modl r prsnd in [5] nd [4]. Th modl consiss of coninui quion, momnum quion for h uppr lr nd mching quion for h vloci in h lowr lr. Ths quions r solvd using n ppropri numricl mhod o compu vlus for h fr surfc lvion η nd h vloci componns (u, v). In dimnsionl vribls, his s of quions is wrin ([4]):. η + Ñ é( h z ) ( z ν ) r - u + + u ì 3 3 éz + ν ( z + νr ) π é r z - ν ü r - Ñ. íê - úñ S + ê - ( z + νr ) π T úñ ý 6 îê ú ê ú þ 3 3 ìéh - z ( h - z ) π éh - z ü - Ñ. íê - úñ S + ê - ( h - z ) π Ñ T = 0 ú ý ê 6 ú î þ. u + Ñ ( u u ) + μñh é π æ h ö + ê Ñ S + π ÑT - Ñ S - Ñ( ht ) ú ç ê è ø ú é h + Ñ ê ( T + hs ) + ( π -h )( u. Ñ ) T + ( π -h )( u. Ñ) S + ( T + hs ) R R n S η λ T ú é Ñ - Ñ u æ π ö h -Ñ Ñ S + π Ñ T + Ñ Ñ S + hñ ç æ T ö 0 ç ú = è ø è øú (7) (73) u π - z π - z + Ñ S + ( π - z ) Ñ T = u + Ñ S + ( π - z ) ÑT (74) whr S =.u, T =ζ(s S ) + T, S =.u, nd T =.(ν r u ). R η = brking-rld dissipion rm, R λ = λ / (ν r + η) u η u η ccouns for boom fricion, whr u η = vloci vlud h

21 4 Nw Prspcivs in Fluid Dnmics sfloor, nd λ = boom fricion cofficin, picll in h rng of 0 3 0, ν T = consn dd viscosi, =( /, / ), π = 0.7ν r = vluion lvl for h vloci u, ζ = 0.66ν r = lr inrfc lvion, s = vluion lvl for h vloci u, nd η = fr surfc lvion HD Srr s quions wih improvd disprsiv prformnc To llow pplicions in grr rng of ν 0 / λ, ohr hn shllow wrs, nw s of ndd Srr quions, wih ddiionl rms of disprsiv origin, is dvlopd nd sd in [] nd [] b comprisons wih h vilbl s d. Th quions r solvd using n fficin fini diffrnc mhod, whos consisnc nd sbili r sd in h work b comprison wih closd-form solir wv soluion of h Srr quions. From h quion ssm (0), b dding nd subrcing rms of disprsiv origin, using h pproimion u = μ η nd considring h prmrs α, β nd γ, wih β =.5α 0.5γ, llows o obin nw ssm of quions wih improvd linr disprsion chrcrisics: ( uν) ν + = 0 ν u + uu + μ( ν + ) + ( + )( W u - νν u ) - ( + b ) u 3 ν + μw ( ν + ) - μνν ( ν + ) - b μ ( ν + ) - νν uu 3 ν u u uu ν u ν u ν ( ) ( ) ( ) ( ) ν + ( W + ν ) uu + u = 0 (75) fr linrizion of h quion ssm (75), h disprsion rlion (58) is obind. s for h oussinsq pproch obind b Liu nd Sun, quing quion (58) wih h linr disprsion rlion ω / μπ =nh(πν r ), using h pproch (60), vlus of α =0.308 nd γ = r obind, so h β =0.0. I should b nod h h Srr s quion ssm (0) is rcovrd b sing α =β =0. 4. Numricl formulions nd pplicions 4.. HD Boussinsq-p pprochs 4... WACUP umrξcζρ mτdρ n nsion of h oussinsq modl (4) o k ino ccoun wv currn inrcions hs bn drivd nd prsnd in [5]. This modl is nmd W CUP (ζ HD WAv Pρus CUrr

22 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ Bτussξsq-p mτdρ). Wih dimnsionl vribls, king mn quniis of h horizonl vloci componns U =(u + u c ) nd V =(v + v c ), whr ind c rprsns currn nd u =U - u c nd v =V - v c, h finl s of hs quions r wrin s follows: ( ) ( ) 0 ν + νu + νv = (76) νr ( ) ( ) U + UU + VU + μ ν + - U + V 3 νr æ ö - éuc ( U + V ) + vc ( U + V ) + ν + u + v r c c 3 ç è ø τ τ s η - ν ( U + U ) - + = 0 ρ ρν ρν (77) νr ( ) ( ) V + UV + VV + μ ν + - U + V 3 νr æ ö - éuc ( U + V ) + vc ( U + V ) + ν + u + v r c c 3 ç è ø τ τ s η - ν ( V + V ) - + = 0 ρ ρν ρν (78) whr τ s nd τ η rprsn srsss on surfc nd boom, rspcivl. Th sndrd oussinsq modl (4) nd h ndd ssm of quions (76) (78) r solvd in [8] nd [5], rspcivl, using n fficin fini lmn mhod for spil discrizion of h pril diffrnil quions. Firsl, h (U, V) drivivs in im nd hird spil drivivs r groupd in wo quions. This mns h n quivln ssm of fiv quions is solvd insd of h originl quion ssm (76) (78). Th finl quion ssm ks h following form: ν + νu + Uν + νv + Vν = 0 (79) ( ) r + u r + v r = -uu - vu - μ ν + c c νr æ ö + é u U + U + v V + V - ν + u + v c c r c c 3 ç è ø é s η + ên + ν ( + u + v ) ( U U ) 0 ρ r c c 3 ú = rν rν ( ) ( ) ( ) ( ) (80)

23 44 Nw Prspcivs in Fluid Dnmics ( ) s + u s + v s = -uv - vv - μ ν + c c νr æ ö + é u U U v V V ν u v c c r c c 3 ê ç + + ú è ø ( ) ( ) ( ) ( ) é s η + ên + ν ( + u + v ) ( V V ) 0 ρ r c c 3 ú = rν rν (8) ( ) νr U - U + U = r (8) 3 ( ) νr V - V + V = s (83) 3 I should b nod h wkl vricl roionl flows wr ssumd, which sricl corrspond o limiion of h numricl mhod. s h vlus of vribls ν, U, V, r nd s r known im, w cn us numricl procdur bsd on h following sps o compu h corrsponding vlus im + (for dils s [5]):. Th quion (79) llows us o prdic h vlus of vribl ν(ν p + ), considring h known vlus of h, U nd V im in h whol domin.. Δquions (80) nd (8) mk i possibl o prdic h vlus of vribls r (r p + ) nd s (s p + ), king ino ccoun h vlus of U, V, r, s nd ν + =0.5ν + 0.5ν p + known for h whol domin. 3. Soluions of quions (8) nd (83) giv us h vlus of h mn-vrgd vloci componns U nd V (U + nd V + ), king ino ccoun h prdicd vlus of r nd s (r p + nd s p +, rspcivl). 4. Δquion (79) llows us o compu h dph ν im + (vlus of ν + ) considring h vlus of vribls ν, U +05 =0.5U + 0.5U + nd V +05 =0.5V + 0.5V + known for h whol domin. 5. Δquions (80) nd (8) llow us o compu h vlus of vribls r nd s im + (vlus of r + nd s + ), king ino ccoun h vlus of r, s ν +05 =0.5ν + 0.5ν +, U +05 =0.5U + 0.5U + nd V +05 =0.5V + 0.5V + known for h whol domin. Th Prov-Glrkin procdur is uilizd o chiv soluions for h unknowns ν, r nd s. ccording o h wighd rsidul chniqu, minimizion rquirs h orhogonli of h rsidul R J o s of wighing funcions W ξ,j, i..,

24 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ ò W R dd = 0, ξ =, L, ξ, J J (84) D whr h gnrl form of h wighing funcions pplid o hs quions is dfind s: ( ) d ( ) W = N + d N + N, ξ =, L, ξ, J ξ uξ vξ (85) ξ ξ nd whr h cofficins δ uξ, nd δ vξ r funcions of: (ξ) h locl vlociis U nd V; (ξξ) h rio of h wv mpliud o h wr dph, nd (ξξξ) h lmn lngh. To illusr his procdur, compl soluion of quions (79) (80) nd (8) is prsnd hr in dil. Inroducing in quion (79), h pproimd vlus r givn b: p» pˆ = å N p (86) = ξ ξ h following rsidul R 79 is obind: R = νˆ + νu ˆ ˆ + Uν ˆ ˆ + νv ˆ ˆ + Vν ˆ ˆ (87) 79 Th R 79 rror minimizion lds o h following quion: ò D ò ˆ ˆ ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) W R dd = W ν + νu + Uν + νv + Vν dd ξ,79 79 D ξ,79 é = ò W êå N ( ν ) + å( N ) U å N ν + å N U å( N ) ν D ê dd = ξ,79 j j π π j j π π j j j= π= j= π= j= ( ) ( ) + N V N ν + N V N ν π π j j π π j j ú π= j= π= j= ú å å å å 0 (88) In h mri form, his quion m b wrin s: Aν + Bν = 0 (89) whr

25 46 Nw Prspcivs in Fluid Dnmics ò ζ = W N dd ; ξ, j =, L, ξ, j D ξ,79 j ò ( ) ( ) j ò å π η = W N U N dd + W N U N dd ξ, j D ξ,79 π π D ξ,79 π j π= π= ò ( ) å ( ) + W N V N dd + W N V N dd ; ξ, j =, L, D å å ξ,79 π j ξ,79 π π π D j π= π= ò (90) Th rsidul R 80 is wrin s: R = rˆ + uˆ rˆ + vˆ rˆ + uu ˆ ˆ + vu ˆ ˆ 80 c c ˆ ˆ ( ˆ νr + μ ν + ) - é( uˆ ) ( Uˆ + Uˆ ) + ( vˆ ) ( Vˆ + Vˆ c c ) 3 ˆ é + ν ˆ ˆ ˆ ˆ ˆ r ê + u + u + v + v c c c c ( ˆ ˆ s η - n U + U ρ ) - + rν rν ( ) ( ˆ ) ( ) ˆ ( ) ( ˆ ) ( ) ˆ ( ) ú (9) I should b nod h h rm 3 ν r(ξ + u c ξ + v c ξ )(V + V ) of quion (80) is of ordr 4 or grr. For his rson, i is no considrd in h numricl dvlopmns. Similrl, nd for h sm rson, considring λ =ν r / 3, ll rms involving λ in h numricl dvlopmns r omid. Th Grn s horm is usd o solv h scond drivivs prsn in quion (80) (rsidul w obin: R 80 ), nd in quions (8) o (83), i.., considring p^ =(U^, V^ ), from p^ + p^ n ( ˆ ˆ é ) ( ) ( ) å ò W p + p d D = W N N p d D ξ, ρ ò + D D ξ, ρ ê j j j ú (9) j= nd so n W ξ,ρ j= (N ) j + (N ) j p j d = (W ) ξ,ρ (N ) j p j d j= (W ) ξ,ρ j= (N ) j p j d + W Γ ξ,ρ (N ) j p j dγ ; p =(U, V ) j= (93) Rining rms up o ordr 3, h R 80 rror minimizion lds o:

26 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ ò D ì é W R d W N r N u N r î ê D = ξ,80 80 ò í ξ,80 êå ( ) + j π ( c ) ( ) j å D πå j j j= π= j= ( ) ( ) ( ) ( ) å å å å å å + N v N r + N u N U + N v N U π c π j π π j π π j j j j π= j= π= j= π= j= ( ν ) r ρ + μå( N ) ( ν + ) ú - ρ ( ) ( c ) ( c ) j j å N å N é u U + v V π π π j= ú ρ= 3 π= é + W ê N N + N ν N u N ê ( ν ) r π å å( ) ( ) å ( ) å( ) ( ) å( ) ( ) ξ,80 π j j ρ r ρ π c π j j π= j= ρ= π= j= + å N ( ν ρ r ) ( ) ( c ) ( ) ( ) ρå N v π π å N ú j j ρ= π= j= ú é + N ( ν ) N ( u ) ( % ρ r π c ) + Nπ ( vc ) ( % å ) ρ êå π å n π ú - U ρ ρ= π= π= ü -W N é,80 ( ) ( ) 0 ξ å π ê - N ν dd = s η j j π π ú å ý π= j= ν þ (94) whr ( % ) = -( W ) ( N ) ( ), ( % ) = -( W ) ( N ) ( ) å ξ,80 j j ξ,80 j j j= j= å (95) nd é p = - ê W N + W N p p = U V ξ, J j ξ, J j j ê j= j= ú ( ) ( ) ( ) ( ) ú, (, ) å å (96) In h mri form, his quion m b wrin s: Ar + Br = C (97) whr ζ =,,,, ξ, j ò W N dd ξ j = L ξ,80 j (98) D é η =,,,, ξ, j ò W N u N N v N d ξ j D ξ,80 êå + π c π j π c π ú D = L (99) j π= π= ( ) ( ) å ( ) ( )

27 48 Nw Prspcivs in Fluid Dnmics ì é c = - ò íw êå N u å( N ) U + å N v å( N ) U î ê ξ D ξ,80 π π j j π π j j π= j= π= j= ( ν ) r ρ + μå( N ) ( ν + ) ú - å Nρ å( N ) ú 3 é( uc ) U + ( vc ) V é N u N + W ê ê N N + N ν + å N ( ν ρ r ) ( ) ( ) ( ) ( ) ρå N v π c π å N ú j j ρ= π= j= ú é + N ( ν ) N ( u ρ r π c ) ( % ) + Nπ ( vc ) ( % å ) ρ ρ êå π å π ú -n U ρ= π= π= ü -W N é,80 ( ) ( ) N ν d 0, ξ,. ξ å π ê - D = = s η j j ý π π ú å K π= j= ν þ j j π π π j= ρ= π= ( ν ) r π å å( ) ( ) å ( ) å( ) ( ) å ( ) ( ) ξ,80 π j j ρ r ρ π c π j π= j= ρ= π= j= j (00) Th rsidul R 8 is wrin s: ˆ ˆ ˆ ˆ ν æ r U U 8 ö R = U rˆ 3 ç è ø (0) ccording o Glrkin s procdur, fr using ingrion b prs (or Grn's horm) o rduc h scond drivivs, h R 8 rror minimizion lds o h following quion: W ξ,8 R 8 = d {W ξ,8 N j U j j= (ν + r ) π N π π= 3 (W ) ξ,8 j= (N ) j + (W ) ξ,8 (N ) j U j j= W ξ,8 N j r j }d N (ν r ) π j= Γ π W 3 ξ,8 (N ) j U j dγ (0) Th ls rm of (0) cn b wrin s: Γ N π (ν r ) π 3 (ν W ξ,8 (N ) j U j dγ = r ) p Γξ N p W 3 q,8 (N ) r U r dγ ξ (ν + r ) p Γ N p W 3 q,8 U dγ (03)

28 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ whr p, q =,,, bing h numbr of nods in h corrsponding lmn sid coincidn wih h boundr domin, Γ ξ rprsns h lmn sids wihin h domin, wih h corrsponding ingrl null bcus h rsuling lmn conribuions r qul, bu wih opposi signls, nd Γ rprsns h lmn sids coincidn wih h boundr domin. ccordingl, n quivln form of quion (0) m b wrin s follows: W ξ,8 R 8 = d {W ξ,8 N j j= ν ( + r ) π N π π= 3 (W ) ξ,8 j= (N ) j + (W ) ξ,8 (N ) j }U j d j= = W ξ,8 N j r j d + N (ν r ) p j= Γ p W 3 q,8 U dγ p, q =,, (04) In h mri form, his quion m b wrin s: AU = B (05) whr ( νr ) ì é,,8 ( ) ( ),8 ( ) ( ) ü π ζ = ξ j ò íwξ å N + j å Nπ ê W ξ å N + W N d D j å,8 úý D 3 ξ j î j= π= ê j= j= ú þ (06) η ξ = W ξ,8 N j r j d + N (ν r ) p j= Γ p W 3 q,8 U dγ p, q =,,. (07) s rcommndd in [5] ζ suξζηρ μrξd ξs τrmζρρ crucξζρ τ ν succss τλ ζ λξξ ρm mτdρ. I τur cζs, ν λτρρτwξμ ruρs mus η λuρλξρρd λτr ξs μrζξτ.. Eρm sξd ρτwr νζ ν ρτcζρ dpν. b. Mξξmum τλ 0 τ 5 ρms pr wζv ρμν. c. Cτurζ umηr ζρwζs ρτwr νζ τ ξ ν wντρ dτmζξ Rζρ cζs sud usξμ ν WACUP mτdρ Th forificion of S. Lournço d Cbç Sc (lighhous of ugio) Tgus sur (Porugl) hs ndurd ovr h cours of four cnuris h coninuous cion of wvs nd currns,

29 50 Nw Prspcivs in Fluid Dnmics s wll s bhmric modificions rsuling from h movmn of significn quniis of snd in h r whr i is locd. Wih h innion of prvning h dsrucion of his forificion, svrl sudis wr conducd o vlu h bs procion srucur. Th sudis hv ld o prociv srucur which consiss of circulr dissipion plform, wih lvl of m (HZ) nd bou 80 m rdius (Figur ). Figur. Submrgd dissipion plform o proc h ugio lighhous, siud h mouh of h Tgus sur. Th wv currn oussinsq-p modl W CUP ws usd o obin h im-dpndn hdrodnmic chrcrisics of h join cion of rlivl common wv ovr h high flow id currn. Th wv chrcrisics r: wv high, H = 3.5 m, priod T = s nd dircion = 80. Figur shows h fr surfc lvl whn h wv pprochs h plform nd is cion on h ugio lighhous. Figur 3 shows fr surfc wr lvl, in qusisionr s. s cn b sn, h wv cion on h forificion ws drsicll rducd s consqunc of h wv brking, rfrcion nd diffrcion on h dissipion plform consrucd round h forificion. Figur. Prspciv of h fr surfc obind b simulion round h ugio lighhous, siud in h Tgus sur, Porugl, in rnsin condiion.

30 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ Figur 3. Procion of h ugio lighhous siud h mouh of h Tgus sur. Prspciv of h fr surfc obind wih numricl simulion, in qusi-sd s COULWAVE τ-ρζr mτdρ numricl schm similr o h of [38] nd [39] is uilizd b [5], wih h inclusion of r nonlinr rms. sicll using h sm high-ordr prdicor corrcor schm, [5] dvlopd numricl cod (COULW VΔ) bsd on Nwogu s quions for on nd wo lrs. Prmrizions of boom fricion nd wv brking hv bn includd in h cod, s wll s moving boundr schm o simul wv runup nd rundown. fini diffrnc lgorihm is usd for h gnrl on- nd wo-lr modl quions. ccording o [9], ν quζξτs ζr sτρvd usξμ ζ νξμν-τrdr prdξcτr-cτrrcτr scνm, mpρτξμ ζ νξrd τrdr ξ ξm pρξcξ Adζms-Bζsνλτrν prdξcτr sp, ζd ζ λτurν τrdr ξ ξm Adζms-Mτuρτ ξmpρξcξ cτrrcτr sp [3]. lso in ccordnc wih [9], h implici corrcor sp mus b ird unil convrgnc cririon is sisfid. In ordr o solv numricll h nondimnsionl quions (69) (7), hs r prviousl rwrin in dimnsionl vribls. Thn, o simplif h prdicor corrcor quions, h vloci im drivivs in h momnum quions r groupd ino h dimnsionl form (for dils s [5]): π -h U = u + u + π - ν u - u + ν u r r ( h )( ) h éh ( ) π -h V = v + v + ( π -h )( ν v r ) - h éh v + ( ν v r ) ê ú (08) (09) whr subscrips dno pril drivivs. For rsons of sbili nd lss irions rquird in h procss of convrgnc, h nonlinr im drivivs, risn from h nonlinr disprsion rms η(.(ν r u α ) + ν r / ε ) nd ( η / ).uα, cn b rformuld using h rlions:

31 5 Nw Prspcivs in Fluid Dnmics... é æ ν ö é æ ν ö é æ ν ö Ñ h ç Ñ ( ν u ) + = Ñ h ç Ñ ( ν u ) + - Ñ h ç Ñ ( ν u ) + ê è øú ê è øú ê è øú r r r r r r... æ ö æ ö h h Ñ ç Ñ u u u = Ñ Ñ - Ñ Ñ ç è ø è ø ( hh ) (0) () Th plici prdicor quions r: ( ) + Δ - - η = η + E - E + E ξ,j ξ,j ξ,j ξ,j ξ,j () ( ) ( ) 3( ) ( ) + Δ U = U + F - F + F + F - F + F ξ,j ξ,j ξ,j ξ,j ξ,j (3) ξ,j ξ,j ξ,j ( ) ( ) 3 ( ) ( ) + Δ V = V + G - G + G + G - G + G ξ,j ξ,j ξ,j ξ,j ξ,j (4) ξ,j ξ,j ξ,j whr ( h ) ( h ) E = -ν - é + ν u - é + ν v r r r ì éæ ö æ ö ü + í( h + νr ) êç ( h - hν + ν r r ) - π S + ç ( h - νr ) - π T úý î è 6 ø è ø þ ì éæ ö æ ö ü + í( h + νr ) êç ( h - hν + ν r r ) - π S + ç ( h - νr ) - π T úý î è 6 ø è ø þ (5) ( ) ( ) h ( h ) F = - é u + v - μ - πν - π ν + Eν + ν é - ée( hs T ) + - ( π h )( us vs ) ê - + ú - é( π - h )( ut + vt ) - é( T + hs) ê ú r r r r (6) h - π F = v - ( π - h )( ν v r ) + h éh v + ( ν v r ) ê ú (7)

32 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ ( ) ( ) h ( h ) G = - é u + v - μ - πν - π ν + Eν + ν ê ú é - ée( hs T ) + - ( π - h )( us + vs ) ê ú r r r r - é( π - h )( ut + vt ) - é( T + hs) ê ú (8) h - π G = u - π - ν u + u + ν u r r ( h )( ) h éh ( ) (9) nd ( ) ( ) S = u + v ; T = ν + ν u + ν v (0) r r r ll firs-ordr spil drivivs r diffrncd wih fourh-ordr ( 4 = 4 ) ccur quions, which r fiv-poin diffrncs. Scond-ordr spil drivivs r pproimd wih hr-poin cnrd fini-diffrnc quions, which r scond-ordr ccur. Th fourh-ordr implici corrcor prssions for h fr surfc lvion, η, nd h horizonl vlociis, u nd v, r: ( ) + Δ η = η + E + E - E + E ξ,j ξ,j ξ,j ξ,j ξ,j ξ,j () 4 + ( ) ( ) ( ) + Δ U = U + F + F - F + F + F - F ξ,j ξ,j ξ,j ξ,j ξ,j ξ,j () ξ,j ξ,j 4 + ( ) ( ) ( ) + Δ V = V + G + G - G + G + G - G ξ,j ξ,j ξ,j ξ,j ξ,j ξ,j (3) ξ,j ξ,j 4 s nod in [5], ν ssm ξs sτρvd η λξrs vζρuζξμ ν prdξcτr quζξτs, ν u ζd v ζr sτρvd vξζ (08) ζd (09), rspcξvρ. Bτν (08) ζd (09) ξρd ζ dξζμτζρ mζrξ ζλr λξξ dξλλrcξμ. Tν mζrξcs ζr dξζμτζρ, wξν ζ ηζdwξdν τλ νr (du τ νr-pτξ λξξ dξλλrcξμ), ζd ν λλξcξ Tντmζs ζρμτrξνm cζ η uξρξzd. A νξs pτξ ξ ν umrξcζρ ssm, w νζv prdξcτrs λτr η, u ζd v. N, h corrcor prssions r vlud, nd gin u nd v r drmind from (08) nd (09). lso in ccordnc wih [5], ν rrτr ξs cζρcuρζd, ξ τrdr τ drmξ ξλ ν ξmpρξcξ cτrrcτrs d τ η rξrζd. Tν rrτr crξrξζ mpρτd ξs ζ duζρ cζρcuρζξτ, ζd rquξrs νζ ξνr

33 54 Nw Prspcivs in Fluid Dnmics å å w - w* ε w - w* m < or < ε + + w 00 w (4) In hs prssions, w rprsns n of h vribls η, u nd v, nd w is h prvious irions vlu. Th vlu of h rror is s o 0 6. Linr sbili nlsis prformd b [38, 8] nd [40] show h < / ( c) o nsur sbili, whr c is h clri Rsζrcν sud usξμ COULWAVE mτdρ For h nlsis concrning cosl procion, h mn currns round submrgd srucur (rificil rf) r nlsd. Th oupu of h COULW VΔ modl corrsponds o h vloci vlus dph 0.53 ν r blow h wr surfc. Th vloci his dph is usd o drmin h vloci clls nr h shorlin h could giv n indicion of h sdimn rnspor. Γivrgn clls indic rosion nr b h shorlin nd convrgn clls indic sdimnion. Th numricl simulions o sud h HΓ bhvior of h hdrodnmics round h rf hv bn don for four css: wo rf gomris, vring h rf ngl (45 o nd 66 o ), nd wo wv condiions. Th chrcrisics of h simulions for h diffrn css r dscribd in Tbl. Rf ngl ( ) H (m) T (s) Numbr of grid poins pr wvlngh Grid siz (m) Tim sp (s) C C C C Tbl. Min chrcrisics of h simulions prformd. s rcordd in [8], ν cτmpuζξτζρ dτmζξ ξs ζrτud 870 m ξ ν ρτμ-sντr ζd 670 m ξ ν crτss-sντr dξrcξτ wξν ζ cτsζ τd spζcξμ τλ ζrτud = =.0 m ζd ζ Cτurζ umηr τλ 0.5. Tν τζρ sξmuρζξτ ξm wζs 800 s. A λρζ ηττm ξs pρζcd ξ λrτ τλ ν sρτp wνr wζvs ζr μrζd usξμ ν sτurc λucξτ mντd ([39] Wξ ζρ., 995). Tν sτurc λucξτ ξs ρτcζd ζ =80 m ζρτμ ν dξrcξτ (Figur 4). Two spong lrs r usd, on in fron of h offshor boundr o bsorb h ougoing wv nrg, nd h ohr on h bch, boh wih widh of 0.5 ims h wvlngh of h incidn wv. Th numricl rsuls obind b h modl r h im sris of h fr surfc lvion, h wo vloci componns, u nd v, nd h wv brking rs (Figur 5).

34 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ Figur 4. Schmic rprsnion of h simuld gomr (no scl) [8]. C: H = 4.0 m, T = 5 s, 45º C: H = 4.0 m, T = 5 s, 66º C3: H =.5 m, T = 9 s, 45º C4: H =.5 m, T = 9 s, 66º Figur 5. Vloci prns of css C o C4 [8].

35 56 Nw Prspcivs in Fluid Dnmics Rsuls of his simulion r dscribd in [9], including h obsrvd phnomn, such s ζρτμ ν rλ, ζ ξcrζs τλ ν wζv νξμν ξs τηsrvd, ξ τppτsξξτ τ ν sξuζξτ wξντu ν rλ, τwξμ τ ν dcrζs τλ ν dpν ξ ν rλ zτ. Mτrτvr, τwξμ τ ν ξcrζs ξ wζv νξμνs, ν wζv ηrζπξμ τccurs ζrρξr (ζd ξ μrζρ τvr ν rλ) ξ cτmpζrξsτ wξν ν sξuζξτ wξντu ζ rλ. Frτm Fξμur 5, ξ ξs cρζr νζ ν prsc τλ ν rλ sξμξλξcζρ ζρrs ν wζv νξμνs, wξν ν wζv νξμν ξcrζsξμ ζρτμ ν rλ ζs ζ cτsquc τλ dcrζsξμ wζr dpν. Figur 5 lso shows for ll css h convrgn clls ppr, indicing possibl sdimnion nr h shorlin, suggsing h h chosn gomris (Figur 4) r dvngous for boh cosl procion nd improving surf condiions. nw, s rfrrd in [8], morphologicl sud should b don in ordr o confirm hs rsuls. 4.. HD Srr s sndrd modl 4... Numrξcζρ λτrmuρζξτ Th quion ssm (9) is solvd in [7] using n plici fini diffrnc mhod bsd on h McCormck im-spliing schm. For his purpos, h quions r wrin in h following form: ν + P + Q = 0 (5) { } ( ) ( ) é( b ) 3 = -( μ + b + ) ν - + Rdξv( ν μrζdu) P + up + vp + μ + + ν ( ) ( ) é( b ) 3 = -( μ + b + ) ν - + Rdξv( ν μrζdv) η { } Q + uq + vq + μ + + ν η (6) (7) whr P =νu, Q =νv, α =d ν / d ; β=d ξ / d nd h boom fricion rms, τ η nd τ η, r obind hrough (8): η P P Q Q P Q = μ + nd = μ + (8) 7 3 η 7 3 K ν K ν In ordr o ppl h McCormck s mhod, quions (5) (7) r spli ino wo ssms of hr quions hroughou h O nd O dircions. Th corrsponding oprors L nd L k h following form [7]:

36 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ Opror L Opror L ( ) é( ) ν + P = 0 (9) { 3 } ( ) P + up + μ + b + ν = - μ + b + ν - + R P (30) η ( ) é( ) ( ) Q + uq = RQ (3) ν + Q = 0 (3) ( ) P + vp = RP (33) { 3 } ( ) η Q + vq + μ + b + ν = - μ + b + ν - + RQ (34) Considring h gnric vribl F, h soluion im ( + ), for h compuionl poin (ξ, j), is obind from h known soluion F ξ, j hrough h following smmric pplicion: + æ D ö æ D ö æ D ö æ D ö æ D ö æ D ö æ D ö æ D ö F = Lç Lç Lç Lç Lç Lç Lç Lç F è 4 ø è 4 ø è 4 ø è 4 ø è 4 ø è 4 ø è 4 ø è 4 ø ξ, j ξ, j (35) whr ch opror, L nd L, is composd of prdicor corrcor squnc nd rprsns gnric im. In h pplicion (35) of igh prdicor corrcor squncs, lrnl bckwrd nd forwrd spc diffrncs r usd. fr ch prdicor nd ch corrcor of h pplicion F, h vlus of h vlociis (u, v) r updd nd h vlus of h vricl cclrions, α nd β, r rclculd (for dils s [7]) Rζρ cζs sud This modl ws pplid o compu h giion sblishd in h por of Figuir d Foz, Porugl, for diffrn s ss. This por is,50 m long nd 400 m wid, pproiml. Is vrg dph is of h ordr of 7 m, wih pproiml m hroughou h our por bsin. Two simulions wr prformd, considring h s of rs s iniil condiion in boh css, lhough for diffrn idl highs. Th firs cs corrsponds o h nrnc of sinusoidl wv in h opn s (inpu boundr), wih mn wv high H = 3.90 m, priod T = 5 s, wvlngh λ = 47 m, nd dircion Φ = 60 W. Th scond cs corrsponds o mn wv high H = 4.80 m, priod T = 7.5 s, wvlngh λ = 73 m, nd dircion Φ = 58. W. Figurs 6 nd 7 show prspciv viws of h fr surfc compud in h bsin 69. s nd 360 s fr ciion. s cn b sn, zon wih srongr giion is obsrvd in h our hrbor in h scond cs.

37 58 Nw Prspcivs in Fluid Dnmics Figur 6. Por of Figuir d Foz. Prspciv viw of h fr surfc compud 69. s fr ciion, for mn wv high, H = 3.90 m, priod T = 5 s, wvlngh λ = 47 m nd dircion Φ = 83 W. Tid high, 3.35 m (HZ). Figur 7. Por of Figuir d Foz. Prspciv viw of h fr surfc compud 360 s fr ciion, for mn wv high H = 4.80 m, priod T = 7.5 s, wvlngh λ = 73 m nd dircion Φ = 58. W. Tid high,.65 m (HZ) HD Srr s nsion modl Numrξcζρ sτρuξτ Th quion ssm (75) is solvd using n fficin fini diffrnc mhod, whos consisnc nd sbili wr sd in [] nd [] b comprison wih closd-form solir wv soluion of h Srr quions. For his purpos, h rms conining drivivs in im of u r groupd. Th finl ssm of hr quions is rwrin ccording o h following quivln form (SΔRIMP modl) [, 3]: ( uν) ν + = 0 ì q + uq - éu + + ν u + + u - ν u î ê ν uu ( ) ( ) ( )( ) ( ) í ν + éμ( ) νuu + W + b h - μνν h - b μ h 3 æ b ö - é( ν u ) + ν u - ν uu + - ν u u ç è 3 ø ( ν ) + b 3 + η r = 0 ν é + ( ) + W u - ( + ) ν ν u - ( + b ) u = q 3 c W = h + ν + ( ) d ü úý þ (36)

38 Modling of Wv Propgion from Arbirr Dphs o Shllow Wrs A Rviw hp://d.doi.org/0.577/ To compu h soluion of quion ssm (75) (vlus of h vribls ν nd u im + ), w us numricl procdur bsd on h following schm, islf bsd on h ls quion ssm (36), for vribls ν, q nd u. Knowing ll vlus of ν ξ nd u ξ, ξ =, N, in h whol domin im, h quions (36c) nd (36d) r usd o obin h firs vlus of q ξ nd Ω ξ in h whol domin. Thn, w coninu wih h following sps, in which h ind p mns prdicd vlus (s lso [, ] nd [3]):. Th firs quion (36) is usd o prdic h vribl vlus ν pξ im + (ν + pξ ), in h whol domin.. Th scond quion (36b) mks i possibl o prdic h vribl vlus q pξ im + (q + pξ ), king ino ccoun h vlus ν ξ+ =0.5(ν + ξ + ν pξ ), nml for Ω ξ in h whol domin. 3. Th hird quion (36c) mks i possibl o compu h mn-vrgd vlociis u ξ + + im +, king ino ccoun h prdicd vlus ν pξ nd q + pξ, nml for Ω ξ in h whol domin. 4. Th firs oprion (sp ) is rpd in ordr o improv h ccurc of h vribl vlus ν ξ im + (ν ξ + ), using h vlus ū ξ + =0.5 (u ξ + u ξ + ) in h whol domin. 5. Finll, h scond oprion (sp ) is rpd in ordr o improv h ccurc of h vribl vlus q ξ im + (q + ξ ), king ino ccoun h vlus ν + ξ =0.5(ν + ξ + ν ξ ) nd ū ξ + =0.5 (u ξ + u ξ + ) in h whol domin. ch inrior poin i, h firs, scond nd hird-ordr spil drivivs r pproimd hrough cnrd diffrncs nd h im drivivs r pproimd using forwrd diffrncs. Th convciv rms (uν ) nd (uq) in quions (36) nd 36b) r pproimd hrough cnrd schms in spc nd im for vribls h nd q. ch im, hs rms r wrin in h following form: æ ν - ν + ν - ν ö æ u - u ξ+ ξ- ξ+ ξ- ξ+ ξ- ö ç 4D ç D è ø è ø +D +D +D ( uν) = u + ξ ( ν + ν ξ ξ ) æ q - q + q - q ö æ u - u ξ+ ξ- ξ+ ξ- ξ+ ξ- ö ç 4D ç D è ø è ø +D +D +D ( uq) = u + ξ ( q + q ξ ξ ) (37) (38) ll fini diffrnc quions r implici. Thrfor, h soluion of ssm (36) rquirs, in ch im sp, h compuion of fiv hr-digonl ssms of N- quions (sps o 5), which r sil compud using h hr-digonl mri lgorihm (TΓM ), lso known s h Thoms lgorihm. Th sbili condiion o b obsrvd cn b prssd in rms of h Courn/CFL numbr, nd is givn b:

39 60 Nw Prspcivs in Fluid Dnmics D C = μν <.0 R D (39) Bτudζr cτdξξτs W ofn prscrib n influ on h lf boundr, usull mono- or bi-chromic wv flow. Th iniil condiion for his (mono-chromic) influ is: ζ ì æ p ö cos( w) é æ p öü h ( 0,) = ísin( w) nhç - + sin( w) - ê - nh ç - úý î è w ø w è w øþ ζ μ é æ p ö u ( 0,) = cos( w) ênhç - + ú w è w ø (40) (4) whr ζ is h wv mpliud nd h ζν is usd s rmp funcion. I incrss smoohl from 0 o, o cr smooh sr. For lrg, h boundr condiion rducs o: ζ η ( 0,) = sin( ω) (4) μ u ( 0,) = cos( w) (43) w If w wn o void rflcions h righ boundr (oupu), h domin is ndd wih dmping rgion of lngh L dζmp. In his cs, rms lik m() η nd m() u m b ddd o h coninui (36) nd momnum (36b) quions, rspcivl. Th lngh of h dmpd rgion is chosn such h w do no s n significn rflcions Sτρξζr wζv rζvρρξμ up ζ sρτp ζd rλρcξτ τ ζ vrξcζρ wζρρ Δprimnl d nd numricl rsuls r vilbl for solir wv propging on h bhmr shown in Figur 8 [, ]. I shows consn dph bfor = 55 m nd slop :50 bwn = 55 m nd = 75 m. n imprmbl vricl wll is plcd = 75 m, corrsponding o full rflcing boundr condiions. solir wv of mpliud 0. m is iniill cnrd = 5 m. Th compuionl domin ws uniforml discrizd wih spil sp = 0.05 m. zro fricion fcor hs bn considrd. Compuions wr crrid ou wih im sp =0.00 s. Figur 9 comprs numricl im sris of surfc lvion nd s d = 7.75 m. Figur 9 shows wo pks; h firs on corrsponding o h incidn wv, nd h scond o h rflcd wv. Th ndd Srr modl prdicions for boh pks gr wll wih h msurmns. RMSE vlus qul o m nd 0.07 m wr found in firs nd scond pks, rspcivl, for h wv high. Rgrding h phs, hr is loss of pproiml 0.05 s nd of 0.0 s in hos pks (for dils s [] nd []).