Second-order theory and calculations of motions and loads of arbitrarily shaped 3D bodies in waves

:;l,i..t*'i'e; I '1J.a{1\

i '+ ' . t? i

' { , - IL

Sr.conrl-+rdcr Theor') ' aud C.alculat ions of ' I \ {ot i<rns and[,oads of Arbitrarily Shaped 3D Bodies in \\ 'aves

Georgios N. Zaraphonitis & Apostolos D. Papanikolaou*

Nalional Tcchnical Llniversiry' of Athcns. Departmcnt of Naval Architecture andMarine Engineering. Patission 42. Athens 10682. Greece

ABSTRACT

Thts poper deals v,ith the development of a contplete second-ovder theo;'v forthe evaluation of motions and loads of arbitrailt'shaped 3D bodies in wsvesatf nite water depth. The developed second-order potential theon, leads to thesolutiott of integral equations for the evaluation of second-order potentialsand conesponding second-order pressures,.forces. motions, etc. The developedalgorithm for the treatment of the second-order inhomogeneity of the free-surfoce boundary condition allows the application of the related computerprogram to bodies of arbitrarv shape, in contrast to previous theoriesapplicable only to cLrisvmmetric bodies. The paper includes typical numeicalresults for the second-order motions and loads .for vaious bodies of bothsr i sv nt m etric a n d n on ar i s1, nt m et ri c sh a p e.

Kel t v 'orc ls : nonl inear u 'ave loads. nonl inear mot ions and responses.second-order potent ia l . general l l , shaped mar ine st ructures.

I INTRODUCTION

Nonlinear effects due to the interaction of floating bodies with incidentwaves are often very irnportant for the design of offshore structures. In

*Prolessor of NTUA and visit ing Professor. University of Hawaii at Manoa. Dept ofOcean Engineering (April l99l).

r 65Marine Structures 0951-8339/93/$06.00 @ 1993 Elsevier Science Publishers Ltd. Eneland.Printed in Great Britain.

l ( '6 6r , , , r : , '1() .s l t ' . Zr i r ' , tVl tot t i t i . t , Apo.sto l r t .s D. Pol t t t r : ik t , l tou

t l i c las t c lecac ie . g rca t c l l i ) r t has bcen r l cvo tcd to t l t c cva lua t ion o [ ' t i i esecond-orc ic r v t ' loc i tv Io ten t i l i l and o1 ' sccout l -o rdcr fo rccs 3q1 i1111 t . r l )vert ical cy' l inclers l rnc' l l r r isvnrnrctr ic bodies. The rr-cl t r i rcd solut ion ol ' t i r rsccond-orderboundan'r ' l t lue prol) lcnr is, conrparcci to the solut ion of thef i rst-order problc-rn. much more compl icated due to an inhomogeneoustcrm appearing in its free-surface boundary condition. Because of thbcomplication, many early attempts failed to satisfy completely ttrcsecond-order free-surface boundary condition. In a well-known pape4Molinr describes a method for the calculation of the second-order forceson axisymmetric bodies subject to monochromatic incident waves.u&ich avoids the calculation of the second-order potential expiicitly.However, the evaluation of the second-order potential leading to secondorder pressures, wave profiles and runup is by itself of great interescompared with the evaluation of integral values only, like forces andmotions. In the last decade, many authors dealt with the second-orderproblem, some of them solving directly for the potential and others onlyfor the second-order forces. Among them, Eatock-Taylor and Hungicalculate directly the second-order wave force acting on a verticdbottom-mounted cylinder by monochromatic *a.red.'{im and Yue3'1,calculate the second-order potential around axisymmetric bodies fsmonochromatic and bichromatic waves. Lokens calculates the seconGorder velocity potential for three-dimensional bodies of general shapetaking into account the influence of the free-surface inhomogeneity in anapproximative way. In the following, an exact theoretical-numericalmethod for the calculation of the second-order velocity potential and allrelated physical quantities of interest for three-dimensional bodies ofgeneral shape subject to monochromatic or bichromatic incident waveswil l be presented.

2 PROBLEM FORMULATION

We consider a rigid body of general shape, floating on the free surface ofa fluid a nd being subject to an incident regular wave (see Fig. I ). Let ft be

Fig. l. Geometry of boundary-value-problem.

Sccct : t t l -Ltn l r ' r t l tL 'on ' , [or - i l ) bodic. ; i r ; r r ' r r r ' , r 161

lhc c lon l r in ( ) l - t l re f lu id . Sp the l ice sur iacc 01 ' t i rc f lu i r l . S1y the bot tornsurf l ice r l t colrsiaut clcpth,H. unrl Sr' thc u'c-t lr-t l surJhcc of t lre l .ot lr ' . \ \ 'cirrtr<rclrrce an irrcrt ial coorcl inate s),stern (,OA'\-Z) u,i t l i O ir point on t l icfree surlace at rest antl OZ vert ical ancl posit ive upu'ards. a secondcoordinate system (CX')"2') paral lel to the incrt ial one, where C is apoint fixed on the body and a body-fixed coordiqate system (Cryz),which coincides with (CX')/'Z') at rest (Fig. 1). Let N be the unit vectornormal to the instantaneous wetted surface positive outwards, expressedin the inertial coordinate system, whereas d is the same vector expressedin the body fixed coordinate system.

We assume the f luid to be incompressible and inviscid and the f luidmotion to be irrotational. so that it can be described by a velocitypotential. In order to calculate this velocity potential, we have to solve aboundary-value problem consisting of the Laplace equation in the fluiddomain and appropriate conditions on the boundaries of the domainfrom the body up to infinity. The resulting boundary-value problem is aswell known nonlinear; thus we introduce a regular perturbationexpansions method to decompose it into a series of liqgar boundary-value problems. The velocify potential and all the unknown quantities ofthe problem are expanded in a power series of a small parameters whichis proportional to the incident wave steepness:

@ = 5 .@t t ) + r 2O(2 ) + s3O(3 ) + . . . ( l )

The boundary equations on the instantaneous position of the freesurface of the fluid Se and the wetted surface of the body Sq, are expandedin Taylor series about the position of these surfaces at the state of rest (SF.and S1yn. respectively). Collecting terms with the same order of e, \4,eformulate a series of boundary-value problems of different order. Herewe present the boundarv-value problems of f irst and second orcler:

First-order boundary value problem:

a2o)( l ) a2o{ l } a2o(r}- I - - - . | - - = f l

dX2 Ay2 aZ2

ao( r ) a2Q(r )R - f - - . ; = g- oz o I '

V@(r ) . r to l = 6 t t l . f f ro r

ao(r)-----; = uO L

Radiation condition

( in A)

(at Se6)

(at Svys)

(at Ss)

(at infinity)

l 6 s ( le, r r r r i r , . r N. 7,unt ; ' I i t t t t i I is . Apo,t t ' I t t . t D. I tapunikoI t , , t t

S t'<'o r t d -ord (' r | ) ( ) u n d d n' ra I t1 ( p ro b lat t t :

drq)( r ) a :d)( l ) a :o( r )ej '--

*-,F * - '

: t)

do{r ) aro( : )8 a z

* d r t

= A \ - l

VO(2).fr(o) : B(2)ao(2)dZ :U

Radiation condit ion

( i n O )

(at Sep)

(at Swo)

(at Ss)

(at infinity)

where

and

3 EVALUATION OF POTENTIALS _ MONOCHROMATICWAVES

3.1 First-order potential

Let(!r)1r; andxllt,i : l, . . . ,6 be the real and complex first-order motionrepresentations of the body (ff r)(r) = Rel.fl')e -j,' l, i = 1, . . ., 6) with j thetime-complex imaginary unit. For the solution of the first-order problemwe introduce complex velocity potentials and we decompose the first-order potential into several components, namely the incident wave, thewave diffraction and the six radiation potentials

+: *lr+;.#] -*41oo"';' (2al

j, Z x'i"o,)"-:'{ rll

I

A Q \ - Io5

I o*g . l t rr IBe\ - 6 t : r . f r roy + i ( r ) . f r r r r - V@(r) . f r r r r - | vO\ l , . i r r r | . f r (o) (2b)

L o*i' ' i"' JIn the above equat ions.g is the gravi tat ional accelerat ion and i{r) and i( : )are the frrst and second order velocities of a point at the wetted surface ofthe body respectively.

o(,) - n. {(oi', + o$) -

St, t ru i i i -or , l t , r t / t t t ' r r ' l i t r - l I ) l to i i t . t i r t

l - l t c i r i e i t i . : r r 1 \ \ ' i r vc p ( ) t L ' l l i i i t l l t l t l ) o i l t t f : l l ' .

I (r9

\ ' . 7 ,1 ' i n { ) i s cqu r l t o

c : , l r ( l ' . \ ' . 2 1 - - 1 q r r g c o s h ( A ( Z l / 1 ) ) " t A t \ c . s / i +

) \ i r ' / i ) ( l )' ' a ) c o s h ( k / / )

wircre i , -r ;s the wave frequency, k t I t e wave number. o*, the u ave a mpl i tudeandB is the angle ofwave inciderrce (0 = 180" for u 'aves rtrnning in thenegative X direction).

Let G' (P, q be Green's function pulsating with frequency @. where Pis a field point in O with coordinates IXr,Y".Zplr andQ is a source point inO with coordinates [Xq, ya, Zalr.6we also define for later use G2(P, Q ) asGreen's funct ion pulsat ing with frequency 2a. For the evaluat ion of thediffraction and radiation potentials. we express them through adistribution of pulsating sources over Syy6 with source density o(Ql

4 t t t ( p ) =

The source density o( ' ) is calculated from a Fredholm integral equat ionof second type. so that @(r)(P) satisfies the wetted surface boundarycondit ion.

I r r , r , , n . A G , ( P . Q ) , ^ ^ d @ " ' ( P )- o ( l ) ( p ) * - l - | [ r t t ' ( e ) " : J i : : : d S o : 2 - - 6 r r r . f r r o r ( 6 ]

- , , JSu,, , , - '

aMo)(P) AN(o)

ln practical situations the solution of eqn (6) is obtained throughdiscretization of the rvetted surface by N triangular or quadrilateralelements of constant source density olr) over their area 65,, andtransformation of eqn (6) to a system of 1/ l inear equat ions with Nunknou'ns. Thus. eqn (5) turns to

, r r r (p ) ol, 'tG,(P. 0,,)ds,,

3.2 Second-order potential

The second-order potential can be decomposed in the following way:

@(2)(x, y, Z, t) = A|t(x,y, z) + Re lO!')(x, y. 21e-zi',7 + 6(r)r (8)

where

A(2) - - a ! "sk-

2sinh(2kH\

In the fol lou' ins we wil l restr ict our attention to the calculation of

ot" I-. Io"'(Q)c'(P' o)dsa

. , A '

=aYA - /a / l 4

n = |

(s)

(71

l l ( ) G, ' , , ; - ' ios l t ' . Z,u, tJ ' l tor t i t i t , .41t t t . t io lo, l ) . I toJturr i l : t , l t tot r

( ) . r ) ( . . \ . ) ' . Z ) . s incc t l r i s i s I l r c on l r '1c l 'n r o i - t i r c sccont l -o rdcr I )o te n t ia l \ ' , ' i th; r co l r t i ' i l ) i l t i r rn lo lhc s t -c t t i t t l -o r t l c r l i l r c r ' s . L t t g= i r r ( t ) a r t i l 7 j r ) be t l t c re - l t lr i i l t l complex scco l t ( i -o f ( l c r t l ro t i i l l . t s o1 ' l l t c ' l )o ( l ) ' ( r= j t ' ( r ) : g ' i ; ) +P..-{gf]rg-r i" ' ' } ) . \ l 'c c iccortr i ' ,115s 0l t) into l l tcrm for thc sccolrd-or( lcrrn : ident wa!e po ter t t ia l . s ix rad ia t ion po ten t ia ls and a second-ord t : r'scatter ing' potenl ial :

6

e9 = 09,, +O!? - zjroZx\1)e,,

The seconcl-order'scatterin-t'potential @!2] rnust not be confused with thediffraction of the second-order incident wave potential. Instead, as canbe seen from the boundary conditions on the water free surface and thebody wetted surface. it contains the diffraction of the second-orderincident wave potential, but also terms resulting from the interaction ofall first-order potentials given in eqn (3) among themselves.

The second-order incident wave Dotential is eiven bv

Q*)6,Y,2) : -Y# cosh(2k(Z + H)),21k(x rosq + rs inB) (10)

sinho(kI l )

The six second-order radiation potentials satisfy boundary valueproblems similar to the hrst-order ones, and they can be evaluated in thesame way. Thus. it only remains to calculate O!?.

The second-order'scattering' potential satisfies the following boundaryvalue problem:

(in o)

(at S"4)

(at Sp,,)

(at Ss)

(at infinity)

vtoltJ : ov4!i).fr(o) - b,

\ t l l

- r , ' r d @ \ i '- 4 a l - O : s ' * 8 - d Z - = o ,

oPrS' : od Z

Radiation condition

where

i t ' . ' , , . d / A 6 ( t t , , , , \o2 : - "zZQ" 'AZ\s aZ

- , 'Q" ' )

and

bz = -vO!1) .frtur + d{]r . frtor * I1

(e)

) a ( 2 )+ iar(v@( r))2 - g "4! + 4a,e\' l" o L

( l l )

6r t l . f r t t r - V41t t ) . f r1 t1

lot l , ' ' ' r - t r r l-llt'o',' ' T"'|^

l rol ' '7"r I

l 7 r

( l l )

It can be proved that the second-order'scattcring' potential satisfies thefol lowing integral cquat ion:

^ofltpt + [ | ollrota*o#'9) otoJ s , ' , J ' - ' - d M o t ( q )

= I*. I ",rr.

e)b,(e)dso - ,! I,,,lo,rr,s)az(s)dSs

( l 3 )

where m = 2n for P€S1y6 and m = 4n for Pe O.The main diff iculty in eqn (13) is the calculation of the last integral on

the iree surface. extending from the body's waterline to infiniry. After thisintegral has been calculated, eqn (13) can be solved in the same way as weused to solve the integral equation of the first-order problem.

3.3 Calculation of the free surface integral

It is already shown that. in order to evaluate the second-order'scattering'potential we have to calculate first a series of integrals along the freesurface Seu. of the following form:

I(P,) = (14 )

for ai i P,. rvhere P, i = l . 2. . . . . N is the centroid of the i th element used forthe discretization of the wetted surface of the body. For the evaluation ofintegral (14) we introduce a circle with center O and radiusRo. so that thewhole body lies within that circle (Fig. 2). In this way, we divide the free

I,,,, I o,rr,.s)a2 (s) dsg

Fig. 2. Definit ion of frec-surface domains.

l7 l 6t ' r . , r i1 ios N. Zar i tp l toni t i .s , .4 loqolor [ ) . I ) t tpot t i l ; t , !ut t t t

suf l i rcc in to t r io p l r r l s : onc l )c t \ \ ' cen t i re boc lv 's \ \ ' a tc r l i r re and t i te c i rc le( S ' , , , , ) a l r ( l l r t ( ) i l t c r o l l e f r o n ) t h l t l c i r c l c t o i r r l i t r i t r ' ( S r , . , , ) . T h c i n t c g r a l u i l lbc t ' l r l cu l r r tc t l nurncr icu l l f in thc l l r s t par t a r rd ana l l ' t i cu l l f in t l te Sc 'co l t ( lpa f t .

- fhc intcr:rat ion outside the cirelc wi l l be perlonl led in a polar

coordinate svstem with centcr O (sec Zaraphonit is & Papanikolaou).r6For the anali,tical calculation of the integrals, a2(S) must be written in amore sui table form than that given in eqn (11).

Let 4$) be the sum of the first-order diffraction and radiationpotentials. The first-order potential then takes the following form:

O(r ) _ @! , , +O$) ( 15 )

Subst i tut ing eqn (15) in eqn ( l l ) we decompose rz2(S) as fol lows:

a : (S) = ass(S) * a1s(S) (16)

The term dss coffieS from the interaction of O$) with itself, whereas theterm ays stems from the interaction of @$) with p!r). The correspondingallterm disappears trivially.

For the representation of Green's functions appearilrg in eqns (7) and(13) we use John's series expansion.8'e For a large horizontal distance rbefween source and field point we keep only the first term of that series.since the modified Bessel functions appearing in the higher-order termstend rapidly to zero.

G(P,q = z i r { ro - u=g * ,coshk(Zr+ l l ) coshk (Zq + H)Hs(kr )

( l 7 )

where u : co)lg is the frequency number.Wc express the Hankel funct ion appearing in eqn (17) in polar

coordinates using the so-cal led Grafs theorem.r0 Subst i tut ing the resultthereof in eqn (7). we can f inal ly express @!j ' in the formrr

og)(s) H,(kR)(Ac,cos(n0) * ls,sin(n 9)) ( l 8 )

where (R, 0, 0) are the coordinates of point S C SFo in a polar coordinatesystem with center at the point O and z axis vertical and positive upwardsand Ag, andAs,, ft = 0, 1,2. . .. are constant coefficients, with subscripts sand s related to cos(n0) and sin(nd), respectively.

Subst i tut ing eqn ( l8) in the equat ions for a1s and using Grafs theoremto express GzQ, Q) in polar coordinates, we can prove the followingequat ion :

@

=YL

n = 0

Si'cortJ-rtrdcr tlrtor.t' -fttr -: [) ltrnlic.s lr u rrlrr 173

( r . (1 ' ] .S) i r1s( .S)r i0 = t i r l , , , ( l ) )H, , (k l t )1 / , , , (A ' .1 i ) .1 , , + , , , (kR), / /i l = l j D t = \ )

q i

+ ) ) A; , , (P)H,(kR)H, , , (k)R)Jv,__| (kR)/- /-

n - 0 r r r = 0

( le )where"I, * ̂ andJ1, - nrlsre the Bessel functions of the first kind and ordern * m a n d l n - m l . r e s p e c t i v e l y . a n d A : . , , ( P ) a n d A , . , , , ( P ) n . m = 1 . 2 . . . . .co are appropriate functions of point P. The expressions for thesefunctions are omitted here because of their complexity. but they are givenin Zaraphonitis.rl

Substituting eqn (18) in the expression for rzss w€ can prove thefollowing equation:

I crtp.s)a$J(s)d9 = nh11(p)Cil0r10(kR)r10(kR)Ho(krR)J - -

+ | n h,,(P)C;.,,H,(kR)H,,(kR)Ho(k:R)

@ @

+ t l rrL,(P)H,(kR)H,,(kR)H,,*, ,(krR)/ - Ln = 0 m = 0

+ t I nE,.,,(P)H,,(.kR)H,,,(kR)Hp,-,,,1(k:R)L. .L

n = 0 n t = 0

where ft3 is the wave number corresponding to wave frequency 2a.Cf ,, ,are constant coeff icients. /zs(P) and E:,,(P), n. m = l. 2.. . . , co areappropriate functions of point P. The equations defining Cf . ,,. holPy and8i.,,(P) are very lengthy and they are omitted here, but they are given inZaraphonit is.rr l t remains now to calculate a series of integrals in theaxial direction. of the form

I H^(kR)H.(kR)HlkrR)Rd^R and.rRn

(2t\where n,m,l = 0, l , 2, . . . , o. For the expression of the Bessel and Hankelfunctions in (21) we use recurrence relations and Hankel asvmntotic

I,

(20)

I H,(kR)H*(k2R)^kR)RdRJR,,

I7J 6, , rXir ; . r l t ' .T,r t ra l t l tot t i t i . t ,Apostol t t .sD. Pnputr ikolaou

crpAn: ions . ' ( 'unc l f in r r l l r ' \ \ ' c reduce the in tegra ls (21) i r r lo a se l - i cs o [in teqra ls o f thc lo rnr

(22)

For the calculat ion of Ln.n = 0, l , 2 we further employ Fresnel integralsand recur rence re la t ions . t r ina l l y , fo r n :3 ,4 ,5 . . . . we express L , byGamma functions, which we calculate using appropriate asymptoticexpansions and recurrence relations.

.1 EVALUATION OF POTENTIALS _ BICHROMATIC WAVES

In the following, in order to cover the irregular wave case, we firstconsider the fundamental problem of bichromatic incident waves.

Let the body be subject to an incident wave train, consisting of twoharmonic waves with frequencies ao and @r,(ao ) rait). The time-dependent part of the second-order velocity potential in-cludes harmonicterms with frequencies 2a.ro, 2@6,ao * ar6 and @o - o)h.In this paragraphwe will describe briefly a method for the evaluation of the second-orderpotential oscillating with frequencies @, = @o * a;6 and (D6 = o)o - o)6(sum- and difference-frequency problems).

We decompose these second-order potentials in a way analogous toeqn (9)

6

a':,\ = ol1h+ Q1l,o- j(a"+ rr)I x',=rIn4,,roi = I

(23)

u'here subscripts , altd ,1 stand for the sum- and difference-frequencyp roblems. respectivel.v. The second-order su m- a n d difference-frequencyincident wave potentials are already known (see, e.g. ReL l2), while theradiation potentials can be evaluated in a way completely analogous tothat of the first-order problem. [t remains then to evaluate the second-order sum- and difference-frequency'scattering' potential p(f]70.

The procedure for the evaluation of the second-order sum- anddifference-frequency'scattering' potential is completely analogous tothat of the case for monochromatic incident waves. An integral equationsimilar to eqn (13) is formulated. The integral over the free surface iscalculated numerically berween the water line and a circle surroundingthe body, and analyically outside this circle. To perform the analyticalintegration, Green's functions, hrst-order potential and the free surfaceinhomogeneity are expressed in polar coordinates as in eqn (18). The

Stt t t t rd-order t l tLun l i t r 3[ ) futd i t : / l xr / r ' r , . r l i5

i n tegra t ion in the c i rcumleren t ia l anc l a r ia l d i rec t ion is con tp le tc lvan i t logous to the case o l a monochromat ic i l r c ic len t u 'a l ' c . Thr . de t i r i l s < l f 'th is p rocedurL- Are g i ren bv Zaraphon i l i s . r t

5 CALCULATION OF PRESSTJRES. FORCES,AND N, lO' f lOn-S

5.1 Calculat ion of pressures

Once the ve loc i tv po ten t ia l has been ca lcu la tec l . the f lu id l r r r ' ssurL- i sg iven bv Bernou l l i ' s equat ion

l , = - p g : - p Q , - r : R l v O l '

where subscript , stands fbr the part ial der ir , 'at ive withAccording to the emploved perturbat ion proceclure ( lcan he expanded in to pou 'e r sc r iL 's in e :

p : p , , , , * e p , , ) + t : 7 r ( t ) + . . .

5.2 Calculation of forces

The hydrodvnamic force F and the nloment, i l .^.r t . . l on the bocl1,by thel luid can be calculated bv integrat ion of t l - re l )uid pressu re over the wetteclsurface of the bodr ' :

F = - I, , Jpi idS

and p( ii f

t lJ.lo J

t . we

M--

( 1 4 )

respect to t ime.) . the pressure ; r

r r i r

x N)ds (26)

t l Th r

Expanding eqns (26) into pou'er series inexpress ions: l l

i r t t : pgA , ( - { ! "+d ' . " . r - { \ "1 ' o )T + p

i t : r -

1 ; - i r r . V O l r ) ) D d S

der ive the lo l low ing

I f o l r , ids (27a)

fi" '))f + lrR"'t11l,, ',.* I dG\"' +

I I oo'r)F i i d.s

- ' f 6 i ' ,1' [ . , . t /- J r r t e o s u

oir'i ltt.s + l.j 'Il

iII

p 8 A _ (

*01,,

*0L,,

l / h ( i,t trgi t t.t .\ l u ru lt l t t, n i t i :. .. l l ttt.tto l r tr I) l\t pLt rt t k t t i , t r t tt

w h e r e r 1 , 1 i s t h c ' n r a s s o l t h c - h o d t . . { , , t h e w t t e r p l a n c l t r c i l . . \ - 1 : I , r 1 . . r ' 1 . : 1 , 1 1thc- coord ina tcs o l ' the c r ' r r te r o l ' l ' l oa ta t ion in the bodr - l l r c 'd coord in r tcsvs tern . f1 ; the l lo t io l l ( ) l ' the c r 'n te r o l - In i l ss o l ' the b<tdr in the inc ' r l ia lcoorc l ina le svs te ln andf l ; , , i s i t s sc 'cond der iva t ive u ' i th respec t to t in re : r /the c l i s tuncc- o f the l ' rec sur l i t c . ' t ' ron t ( 'a t s ta tL - o l ' r cs t . ( * the re la t i v r 'ntot iol t i r r the' \ 'cr t ical r l i rect i t ' l t o l- thc l ' rec sur lhcc ui th resl ' rcct to thcl - roc ly 's water l ine . d the t rng le o l ' inc l ina t ion o l - the *c ' t ted sur f 'ac r ' to thevcrt ical c l i rect iorr at the u'ater l inc. R is the transformation matr ix f iornthe (C. r r ' : ) to (CX 'Y 'Z ' ) coord ina te s rs tenr (see c ' .g . Papan iko laou &Zaraphon i t i s ) . r r and L = [0 .0 . I ] i s the un i t vec tor in thc \e r t i ca l d i rec t ionar rd subscr ip t $ '1 in the i r r tegra l s ig r t n rear ts in tepra t i ( rn o le r rv i i le r l ine .

Le I Mry be the to ta l moment exer tec l on the boc iy bv thc l lu id andsrav i tv ac t ion . expressec l in the (CX 'Y 'Z ' ) coord ina tc s \s tem. I t can beproved lha t :

l-ltr\l

* pst\"\ r '

(

[ - r , - l ( f ( ] I I L , . l l= o*'."''" 1 ;; J. PRq':' l" L ''" -;;

]. I i ]{I1.,I l 'o:' rr.xti).rs] )

J S , , , J

-L . .Itl.jIL

f - ( :n - : . ; )

[-"'.r [ ".lr t f \ = pg.4*6\ ' ' I . r , \* t ;os.t-a({!" '+s!")I-r , , I

L" l L"]I r , . l ]

t.L;;ll* orrt'' I ,

l{

L , .

0

( l t la )

lur td

+ p.{s'!t'

0

0

- ( ; - l l

L

t - -| - ( ,I

t

- L ,

L t .

0

o

l ''].I

Set t t t td -on lc r t l t t r t r t l i t r . l [ ) h t t l i t . : 1 /? 1111r ' ( , . \

*0J,, , . i*1' ' , . 'x t)d's. l , L . / ' ivo'r) l : ( \ : x D)..1,s

l r ;

. - r r r r ( . i x i ) , ,S R ^ ( l /

c()st /

H,,{'," + F$l + tril+ rf} i - l. 2. . .6 (l9b)

* u L., [ ' i " ' ' volr )( i t )ds -

+ R ( r ) ( - t ( i X ^ F , r ) ) * p r t r l l d l r ' ) l l r ih )

In (28a. b ) / i s the c l i sp laced vo lume o f the boc ly ' a t res t . I i s thc ine r t iamat r ix in the bod1, f i xec l coorc l ina te svs tem. d the vec tor o f 'ansu la rve loc i tv w i th respec t to the iner t ia l coord ina te svs ten t . . \ -n = [_ r r . . l ,H . :n ] laud - r -6 : [ -xc ; . _1 ,< ; . i c , ] f the co t t rd ina te i o f the center o f 'buovancr , andmass in the body-f ixed coordinate system and L,, the waterplane arean-loments of inert ia with respect to the i and j axes in lhe horlv- l lxc-clcoorcl inate system too.

5.3 Equat ions of motion

The tb l low ing equat ions o l mor ion can be proved to be ra l i c l to thesecond-order for a bodv free to move in waves:

M,,i',1,' t r - l l \ ^ l l lH, ,€ ) " + f .U i i - l . l . 1 l 9 a )

' r r tQ. ' \ \ I

o

tL, - |

: \ -/-

6

\- ,r,r tt:l -) l v r , , \ , t t

-

L

where subscripts a. H. u denote terms related to potent ial . hvdrostat ic andinert ial forces (and moments) and lM,, l is the general ized body inert iamat r ix :

lM,il =

I , l 000M2 , . , -M . r ' r ,

0M0-Mzc ,0Mxr ;

0 0 lll M.r'c, -Mrr, 0

0 -M:, . , M.t ,r , 1, ,

M:rt () -Mxc, 1.,

- Myr, ll,l.r,;; 0 1.r 1r,. 1,,

and [Ht , l i s thc hvdros ta t i c coe t - f l c ien ts mat r ix . I t can be provc-d rhu t17,, = (') cxccpt li)r

(r

t/-

I , . 1 ' ,

1.. 1,,

I

I \ (;(,()t-{t().\ N'. Zurttpltortiti.s. .,1Jto.ttttlo.r D. paputtikolttou

11, , = - pg.4, , . H4 = - pg(L: : - f L ' ( :o - : r ; ) ) .

H . . - p g ( L , , I V ( : o - ; t ; ) )

l l , t = I t r . : -pg.{* . I , r . H: , : H. . = pg.4, , .yr . H. l : = H. . , = Lr :

I j ' i , ' : p I ioir) tds' ancl r t i ' [ : p l ' / '01r,1.. x t i )( i .s' ' t * - ' J1 . . , J

i ' * '= p l f o ; ' , l l , ts * \pI f lvo, , , l , t , t -s- ' r * . J ' '

J r * . , l

+ p I l ' r .T ' ' , .vol , ' ) t , . l .s _ ' , r r6 i , , ] ' i ^u,J r * , . J i . ' J r r . , - " c o s 9

r t f i : pI i * l ' ' , . . 'x i )d.s * topI f loo, , , l , ( . . -x i i ) . ' rsJ s * , J ' ' J . r u , J

* o 1,, , , [ ,^-", .voi ' , )( ,r ,x i )ds - ]o*f , , , ,X t t f f ) . l

{Fir i . r - ' ,1! . . . . F ' ; i l , = l ; peA*, l tg=!, , ' + 6l ' , ' ) [0. 0. l . . r ,e. - . rp. 0 l r

[f '* ' ," r-{;!. r l;\ l t = Pttrf t ' - M{!!i:nld r l

l f i i ! . f i , ! . f ( ; ' o l . = - ; i r r ra ld r r r - td l , ; ) + . f c X (R , r )F r r r ) -

. \ ( i x Md- lR l l t t ' t( l r -

dl,t ' : l-di"{!". +tL"t\". -4(r)f(r) lr

6 DISCUSSION OF RESULTS

The rr iethod described in the previous paragraphs.has been used lbr theeva lua t ion o f the second-order po ten t ia ls . p ressures . fo rces and mot ionsfor a number of bodies f loat ing l reely on the free surface or being f ixed inspace and exc i ted bv monochromat ic o r b ichromat ic waves . For theca lcu la t ion o f Green 's func t ions we used a method presented ear l ie r bvthe second o l - the au thorsT anc l the we i l -known program FINGREEN(sc 'e Rc ' f . l4 ) . A compar ison u , i th o ther theore t ica lo rexper imenta l resu l tshas bee-n c lonc . in a l l cascs u l rc re such rcsu l ts u ,e rc ava i lub lc - . l -hcasrecmcn l u ' i th thc 'se resu l ts u 'as sa t is fac to rv to exce l len t . Some- t rp ica lre su l ts thc rco f u , i l l he sh<tu ,n ncr t .

\ , t ' , t t < l - , ' r . / , ' 1 1 1 , , ' 4 1 ) t r . l l \ 1 t , 4 1 1 , , / , r l r , / r , . l ; , )

l - h e ' s e c o n d - o r c l c r l b r c e s a n d n t ( ) m c n l s ( c q n s ( 2 7 ) i r n r l ( 1 , \ ) ) c . l n s i s t o lseccln t l -orcL-r 1c r ln s duc- to dou ble p roclucts <l l ' f l rst-o rclcr r lLr i l n t i t ie s ( Plr r tI ) . f i rout l t ---Kn' lov l i l rces due to t l rc 'sc'concl-orclr ' r inci i icr t t uavc yrotcrr t i r r l( l ' }a r t I l ) . and secc lnd-orc lL ' r tL 'n ls c lue to thc 'sc i l l t r l r i r ru '1 " ro1e111 la lo1 . lnor ( l l ' r to conr l la rL - spcc i l i c tc rn ts thereo l . thcsc l r rs t t t rn l rs c luc to 4 \ \ ' l r r - t 't l econt l - t t t sed l i r r t l re r in lo te rms duc to thc u ' t ' t t cc l s r r r l i r cc l i r rc ins (P . r r ll l l ) . and tL ' rms duc to the l - ree sur l ) tce lo rc ing (Par t lV) .

In F igs . i (a ) anc l 3 (h) u ,e p resr -n t theorc t i ca l resu l ts lb r thc rea l i rn r li rnag inarv par t o l - the seconc l - r t rc le r h r t r i zon ta l lb rcc 'c r i t ' t c ( l i r )mot tochro lna t ic w 'aves on a bo t tom-nrountc r - l vc r t i c i r l cv l indcr . l i r r a l lu i t lc l e p t h e q u a l t o I ' l ( r t i m e s t h e c v l i n t l e r ' s r a c l i u s 1 H : l . l 6 R 0 ) . T h e c l a s h c t ia n d s o l i d l i n e s r e p r e s e n t t h e ( n u n t e r i c a l ) r e s u l t s o l M o l i n a n c l M a r i o n . r 'rv l i i l e the svmbols a re the rcsu l ts o f the present rne th( )c1 . The c iashec i l ine

^ _ D ^ / t ^ A

+ - - - T o t c f o r c .

r," {rY]}'''P e R o o l r o

a --- Pq( (lV)

+ - - - l o t o i F o r c c

I

I

2 . 4

( t l )

F i g . 3 . S c c o n t l - o r r l c r h o r i z o n l l l r r i r r c l i r r c c i r r n P l i 1 1 1 , 1 1 ( ) l h ( ) l t { ) l r - n ) ( ) u n t c \ l f ' , I i r ) ( i ( ' r

r c l I pu n . ( b ) i r n l r g i r t i r r r 1 l t r t .

( a )

I S { )

and thc s \n rb( ) l * g i rc thc to ta l s r 'eon( l -o r ( l c r lb rcc . u ,h i le thc so l i r l l i ncatrd svnrbr>l A st i t r td t i r r thc s1- lc 'c i l ic pur l o1' thr- sccond-<)rclc-r l i t rcr 'c lLrc t t rthc f ' ree sur f t rcc inhonrogc 'ne in (Par l IV) . As can bc sccn l l - i t rn l l t csel igurc 's . the i r g rc - t ' rnen l hc tu 'e r 'n lhc luo nrc l l rods is vc r r gor l r l anr l thei rnpc l r tance o l 'Pur l IV verv s icn i t l c ln t .

F igures -1 (a)anc i .1 ( b )shou thc rc ' i r l lp r l i r las i la ry p i r r t o l the sL-cor r l -o r t le r hor izon ta l fb rce uc t ing on a rec tangu lar barse (so-ca l lec l 'Fa l t inscn 's

barge') f ixed on the t l 'c-c sur lace being cxci tecl br a rnor. tochronlr t ic \ \ ,ave.T l r c c l i m e n s i c l n s o f ' t l t c - h u r g e a r e L = B = 9 0 r t r . l ) = . 1 0 r n . 1 1 = l ( X ) n r ( / - :leng th .B: bearn .D: d ra t t .H : t lu id t lep th) . Thc dashec l l ine g ives the to ta lscconc l -o rc le ' r lb rce . u 'h i le the so l id l ine s ives the spcc i f i c par t o l - thcscconc l -o rder lb rce due to the t iee sur lhce in l romogenc i t r (Par t IV) .Fi-gure 4(c) shou's thc' specit ic part of ' the secorrd-orcler di l ' l -ere l ice-l ' r ' equcncr hor izon ta l l i r l ce cor respond ing to the e xc i ta t io r r o f ' thc ' bargcb1 the f - rec sur lhce inhornogene i t r on lv anc l lb r the case o l -a b ichrorna t ici r rc ic len t \ \ ' i l ve u ' i th t , , l - = 3 l anc l vh l - : I ( r ,= ,u ) /g l . J 'hc \ \ 'aveampl i tuc les a re here in assunred equa l to I n r . ' Ih is l igure shorvs thc lo rccca lcu la ted as a lunc t ic ln o l - the rac l ius o f the c i rc lc separa t ing the i r re ls o l 'numer ica l and ana lv t i ca l in tegra t ion on the l ' ree sur f i rce . Thc so l i c l l i ncg ives the resu l ts lb r the lb rce u 'hen the ana l l ' t i ca l i l l tegra t ion is inc luc lec l .u 'h i le the c lashec l l ine s ives the resu l ts fo r the case o f - onr iss ion o f ' thc 'i tna lv t i ca l in tegra t ion . From th is l igure . the ra thcr qu ick convL ' rucr . rc r ' o l 'the ana lv t i ca l in tc 'g ra t ion can he conc ludec l .

In F igs ,5 (a) . - i (b ) anc l -5 (c ) . resu l ts lo r thc ' l l r s t - anc l sc .conc i - r t rc le rnro t ions in surge . hcave and p i tch o f a ver t i ca l l loa t in_q c r , l in r le r . sub jec tto a n lonochromat ic inc ident \ \ ' aYe are 1 ' l resen lec l . Thc 'cv l incL ' r has i rc l ra f t - to - rad ius ra t io equa l to 7 .9 : - l and the ra t io o l - the depth o l ' the l lu id tothe rac l ius o f thc c r ' l i nder i s equa l to 4 -5 :4 . The r lashed anr l so l i c l l i nc -srepresent the (numc ' r i ca l ) resu l ts o f Mo l in anc l Mar ion . r i wh i le thesvmbols cor respond to the resu l ts o f the presen l rne thoc l . The c lashc 'd l inca n d t h e s v m b o l * q i v e t h e l i r s t - o r d e r m o t i o n s o l - t h e c v l i n d e r . g h i l c t h cso l id l ine and svmbc l l C g ive the seconc l -o rde r r r ro t iuns . As can be sccr rl -rom these f igures. the agreement hetween thc' two theorics is ven goocllo r t l r s t - t t rdcr n to ( io t ts . no l lha t good fo r lhc seconr l -o rder surge nro l ion .bu1 n tuc l t be t tc r fb r the second-order heavc 'anc l p i tch n to t ions . l t i sobv i r t l t s l ' ronr l l te the ore t ica l po i r r l o l ' r ieu ' . lha l lhe se conc l -o rc lc r mot ionrcspot rscs r . r i l l exh ih i t n ran i lo ld resonances t lue t< t v r r r ious rno t lcin te n rc t ions in thc f i r s t - i rnd sccond-orc le r n to t io l ts .

In T i rb lc I r rc t lna l l r p rcscn t rL -su l l s o l ' ( to t i r l ) scconc l -< l r r l c r l i t r ces i r n r ln ronrcn ls l i r r -u h< l ( ton t - l l ( )L l r . r lL ' r l rc ' r t i c t r l c r l i r t r - l c r o l 'n r r l ius R, , : l (X) n tI t t t t l c le p l l t H = l0 l l r t t . s t rh . iec t to [ r i chr t l r r r i r t i c rv , i r , " ' cs o l ' i r r r rp l i tu r leu , , = I rn .

- [ ' l r c ' r ' csu l ts i r rc g iv r -n t i l r thc c l i l ' f c rc r rcc - l l ' c r lL l r . l t c \ i rn r l sunr -

0 2 ,

t 8

Sa<ond-onlL'r thon. fitr -lD hodics in v.uyt,.s

R

l 8 r

0 0

^ f J t t' " 1 r : . J

;r;t-- i . o

P o d ( l V )

Totol Forcc

I

( a )

4.0, f - ( 2 ) t^ I t z ' J

;sL;I-3.0

Port (rV)

Totol Force

t 00.0

- ( 2 ) , - ,f z i l r ( K'

t0.0

EO.0

70.0

60.0

50.0t 50.0 200.0 2g.o Joo.o J50.0 100.0 ,150.0

( c )

F ig . 4 . Sccont l -o r t l c r hor izon ta l $ 'avc lb rc r 'anrp l i lu ( l c on i t l l oa t ing bursc . / - = B = 99 n t .D = 10 n l : (a ) rca l par t : (b ) rn rag inan p i l r l . (c ) Scconc l -o rc lc r t l i f l t r cncc l - rcqucncvhur izon ta l \ r 'avc l i ) rcc unrp l i tudc on a l loa t inu bargc in b ichromat ic \ . \ ,avcs as a l i rnc t ion

o l in t r -g r i l t ion rac l ius R, , . L = B = 90 nr . D = :10 nr .

( h )

la @ 2 t

t 8

l s l

0 . o !

0.00

0.o0.00

0 0 0

0.50

( h )

1 _ 5

' { 2 )R o ( t

;a1 . 0

0 . 0

R o ( , 1 I- I r t

1 l

1 l/ \

I . f\A''lI

\ ^ k R

0 . 2 5 0 . 5 0 0 . 7 5

( c )

2 o( . mot ion

- 1 o ic . mot ion

F i g . 5 . I : i r s t - a n d s c c o n ( l - o r ( l c r : ( a ) s u r u c : ( b ) h c u v c i l n d : ( c ) p i t c h n l ( ) t i ( ) n a r l l p l i t u ( l c s ( ) l i l

l l o i t t i n ! c \ l i n ( l c r i n n r o n o c h r o n l i l l i L \ \ ' i l \ c \

l 'ABt . l - . I

l l i ch ro r r r i r t i c Wuvcs o l l Jn i t A rnn l i t u t l c

IJollotrr-ttrorurtul c.t'lintler. R. = H = lO0 nt. ir,, = I nt

u L t

l . ol ol 0l oI ol 0t lt ll ll ll lt il {

l . ll . lt . 4

l 6

t 6l 6I 8l sl o

, , r ,_

l ol ll . ll 6l ll 0l l1 . 1l 6I . l {

2 0l 4l 6l l l2 0l 6l l l2 0I l t2 ( )l 0

I ' I

-.iti'i -,ril;;"

- ( ) l l . l5 l ] .1 . -0 5] l l : . - l-0 7 l0 l : .1 . * ( ) 9 l0 l r . l- ( ) 57.1L. ] . - ( ) l12E4-o - l t l t l . - ( l 14,11: .1-( ) 2448-r . -0 15684-0.1.126h.1. 0 000E0-0.751{El . -0 425tr - l- {) 667 F..r. - 0. 766 t-_-l-0.549E3. -0. 103 E4-0 .4 t 283 . -0 . 123 E4-0 76981. () 000F.0-0 719E1 . -0 . t 6983-0 6-16F..1. -0 66tiE-3-0 515F .3 . - 0 9 t lE l-0 740Ir-.1. 0 0mtr0- 0. 69-s t:-1. - 0. l-13 E3-0.614t- .3. -0.60283-0 7 t7E3 . 0 00080-0 66783. -0 l0-sE4-0 6li4F.-1. 0.(X)0h0

l ,

-0 9 l l t r5 . 0. (X)0f :o-0 .S7 .1 I l 5 . - 0 | 771 ]5-0 l {071: .s . -0 l93l :5- 0 7 l 7 l : 5 . - ' o . l 8 l l : 5- - ( ) ( ) ( ) f i l r 5 . - ( ) 4 l l t l . 5-0 4 l i l t : 5 . - 0 461F .5-0 ti70h-s. () (XX)trO-0 l i l-sE-s. -0 l l0F--s-0 7tt0tr5. -0 226F-.s-0 '70l t r -s . -0 195E5-() 601 E5. - ( ) - t42E5-0 846E5. 0 (XnF.0-0 t t20E5. -0 l68E' l-0 767E5. -0 lr, i.rE-s-o 691F.5. -() l. l9F.-s-0lt. lt{t:.5. 0 (XnL.0-0 t i l2 I :5 . -0.1 i9-sE4-0 755F .5 . *o t - s lE5-0 !i l7F:-5. 0 (XnE0-{). 794F.-s. -() 746F.,1-0.802F.-s. 0 (xx)t..0

[) i lfr'rcrr c c .f rtq r t o t ( _t' I r( ) l) I utl . \ tu t t f i t ' t1 t t t , r t t t

1,,

- ( ) 1 7 ) l : l 0 . ( X X ) [ : l-0 -1761: r . 0 ( i l s1 . .1- ' () 6f i" l l : .1. 0 S16l : .1- 0 l i6. l I : .1. 0. -1901 :.1-0 ( ) ( \ . l l . l .1 ) | ( ) .11 . l- 0 9 6 7 | - 1 . 0 9 l I I r l-0 67581. 0 4 - l0 t r - l- () f i (r-11:.1. 0 2l i0l : .1- 0 l 0 l [ ] 4 . 0 l T l t : . ]-0 l (x )E4.0 902E]* ( ) t 0 4 E 4 . 0 7 8 5 E 1-0 l ( ) I E .1 . 0 lgsF. l- 0 l l l F . 4 . 0 l s 2 t r l- 0 I l 7 t : - 1 . 0 l 7 5 F . . l-0. 1 I l tr . l . 0. l0l i l : . ]- 0 l l l t r 4 . 0 l . l l F - l-0 l22 t r4 . ( ) .107E1-0 104F.4 . 0 5 l l i t r l-0 I l l t r ' l .0 6 l . lF . l- 0 l 0 . l F 4 . 0 t { l 7 I : . 1- 0 9 1 1 8 . 1 . 0 l l l I ] . 1

t ,

O . l . l 7 l 5 . 0 l O ( ) [ ( r

0 . 1 { ) . 1 l 15 . 0 f i i i I l : 5( ) l J 6 l : 5 . 0 6 ( ) 0 1 : . 5

0 l f i l l 1 5 . 0 . 5 : 6 [ : s

0 l - l l ) l : 5 . { ) . 1 i ( ) l 5

0 | l 9 [ 5 . 0 ] 5 l l : 5( ) l . l ( ) t r .S .0 68 .1 f : 5( ) l 7 . t t r 5 . 0 5 l l t 5{ ) | l . l l r 5 . 0 . 1 6 6 [ : 50 ()l(6[:'1. 0 ].1.r tr-s0 7t.r F-.1. 0 l.l4F.50 I I ] F 5 . 0 . 1 6 4 F 50 5t)91:.:1. () 16Str-s0 l7 lE .1 . 0 l69 t : i0 I . l4 l : -1 . 0 975 l . ' l

-0 l99 l - . .1 . { ) l l {31 :5- 0 1 . l ] F . . 1 . o I l 6 l : 5

0 l ( ,9E.1 . 0 lR lF50 7 l r ) F . l . 0 l l s t : 5

- 0 . 1 6 S I : 1 . 0 I | ( ) l 1 50 l96 l : -1 . 0 l ( r .11 :s

-.

)

].

18.1 (i t t ryio.s,\'. Z o ru p h o n ilr,r. .{7rr.t.r/o/os D. I'u p tt n i k t t I t t r t u

f requc'ncv problr 'nr. A conrparison ol ' thesr- results wit l - r those bl Kirn anclYuea s l tows sa t is tac ton ' agreenrent .

7 CONCLLJSIONS

A rne thod fo r the eva lua t ion o t ' the seconc l -o rc le r po te n t ia l . p ressures .fbrcc-s and motions for boci ies of -seneral shape. subject to monochromaticor bichromatic w'aves in f lu id ol ' f in i te clc-pth is presented. Typical rcsultsof second-order forces and motions for three di f l -erent body shapres.including a rect i tngular barge. are given. Comparisons with otheravai lable results show. in general" verv good agreement.

t .

l

l l

l - )

l 0

l 1

REFERENCES

Mol in. B. . Second-order d i l f ract ion loads upon three c l i r rensional bc ld ic 's .Appl. Ot'ean Res., I (1979) 197-101.Eatock Taylor . R. & Hun-e. S. M. . Second order c l i f t iact ion forces on avertical cylinder in regular waves. Appl. Ocean Res.. 9 (1987) l9-10.Kim. M. H. & Yue. D. K. P. . The complete second-order d i f f ract ion solut ionf icr an ax isvmme(r ic bodl ' . Par t I . Monochromat ic inc ident waves. J F lu idMech.. 200 ( 1989) ns-e.Kim. M. H. & Yue. D. K. P.. The complete second-order diffraction solutionfor an ax isvmrnetr ic body. Pan IL Bichromat ic inc ident waves and bodrmotions. J. Fluid Mech.. 2l I (1990) 557-93.Loken. A. E. . Three d imensional second order hydrodynamic ef fects onocean structures in waves. Report UR-86--54. Dept. of Marine Technologl'.Univers i fv of Trondheim. Noru 'ay 1986.Wehausen. J. V. & Laitone. E.. Surface Waves. ln Handhuch der Ph.t,sik.Spr inger Ver lag. Ber l in . 1960. pp.447-718.Papanikolaou. A. . On integra l -equat ion-methods for the evaluat ion o l 'mot ions and loads of arb i t ran 'bodies in waves. Ingenieur-Arcf i i r ' . 55 (198-st11 -29 .John. F.. On the motion of f loating bodies: l. Conrntuni('etions on Pure undApplied Mathentatics, 2 ( 1949) ll-57.John. F. . On the mot ion of f loat ing bodies: I I . S imple Harmonic Mot iorrs .Contnrunication.s on Pure and Applied Mathematics. 3 (1950) 45-101.Abramowitz. M. & Stegun. l. A.. Handbook of Mathematircl FuttL't itttt.t.Government Pr int ing Of f ice. Washington DC. November 1970.Zaraphoni t is . G. . Second-order theory of mot ions and loads for thrc 'c ' -d inrensional bocl ies of ,ueneral shape in l \ 'aves. PhD Thesis. NTLI . Atherrs.Grcece. 199() .Borvers. E. C. . Long pe r iod osci la t ions o l ' r roored ships suh. lect to shor l \ \ 'avcseas. frnr.s. Ro.t' ln.st. of Naval . lrchitecturc. I lft (1976) l lJl-9 l.Pap l rn i ko laou . A . & Za raphon i t i s . C i . . On an i n rp rovec l r r r e th t l d l i r r t hc

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