Slow drift motions of a floating body in narrow-banded seas

(1)

SUMMARY

In an irregular sea, waves of different

wavenurnbers interact nonlinearly and give

rise to second order forces at the sum and

difference frequencies.

A moored or

dynarn-ically positioned vessel (ship or platform)

can be induced to perform slow drift

oscil-lations at the difference frequencies.

To

study the slow motion in a narrow-banded

sea, we combine the methods of multiple

scales and matched asyniptotics.

It is first

shown in general terms that slow drift

motion is accompanied by long waves.

The

range of applicability of a formula for the

wave force by Neiian [7] is discussed.

An

exception to the formula is a long body in

beam seas with a small clearance under its

keel.

Some recent results for this case are

presented, exhibiting resonant motion.

1. INTRODUCTION

Ships and offshore platforms are often

subject to seas with narrow banded spectra.

Usually the natural frequencies of the

moor-ing system are at the order of 0.01 Hz,

which is much below the peak frequency of

the incident sea.

However, at these low

frequencies, long period resonance can be

induced by a second order force associated

with the difference frequencies of short

waves.

_{The low frequency force can also be}

sufficiently large to affect the design

or

operation of dynamic positioning devices.

Therefore, during the last fifteen years

there has been increasing interest in the

low frequency effects of irregular waves

interacting with floating bodies.

Most authors have approached the slow

drift problem by a straight forward second

order theory (cf. [2], [9]).

For a

compre-hensive review the reader is refered to [8],

ihere a few papers on viscous effects are

also cited.

In the special case of a

regu-lar (monochromatic) wavetrain, the steady

irift force, which is second order in

wave

slope, can be found from the solution of the

first order (linearized) problem ([5], [6]).

Newman [7] has suggested that the drift force

can be written in terms of quadratic transfer

functions of the wave components.

The terms

contributing to the slow drift force

are

functions of the frequency difference.

_If

the spectrum is narrow-banded, these terms

can be approximated by their values at zero

difference.

Thus the slowly varying drift

force is formally the same as the steady

drift force, except that the incident

wave

amplitude now varies slowly in time.

Newman's result has been widely used in

recent studies, but its range of

applicabil-ity appears not to have been fully examined.

Since oscillations of long periods should

imply the presence of long waves, the

com-plete hydrodynamic problem must involve both

long and short length scales.

_{Thus the}

method of multiple scales, recently employed

by us [1] for an idealized problem, should

be

useful.

In this paper we shall first deduce

some general results for the case where the

slow displacement of the body is small.

It

will be shown that while Newman's result is

quite adequate for horizontal drift forces in

most cases, an additional contribution to the

vertical drift force should be accounted for.

This new force is associated with

a long wave

field.

The mechanics of the long wave

gener-ation is explained.

_{Sample results on the}

drift motion of a rectangular block

are

discussed.

2.

PERTURBATION EQUATIONS FOR THE FAR FIELD

Uith reference to Figure 1,

we divide the

horizontal plane into near and far fields.

In the near field which is the 0(1/k)

neigh-borhood of the body, the characteristic

length is the wavelength 2./k or the body

dimension a, with the assumption that ka

=

0(1).

(An additional characteristic length

arises in a later example where the

gap

between the sea bed and the keel

is small.)

In the far field, slow modulation in

r9q_

TECHNISCHE UNiVERSJTEF

Laboratorium voor

SLOt! DRIFT 1OTIONS OF A FLOATING BODY IN NARROL-EANDED SEAS

Scheepshydromochan Loa.

Archiet

Mek&weg 2, 2628 D Deift

YEHUDA AGNON1 AND CHIANG C. MEl2 t1L ß15-veee?5.. Fein 015-181836

1Ioods Hole Oceanographic Institution and Massachussetts Institute of Technology, MA (USA)

2Massachussetts Institute of Technology, 77 Massassachusetts

_{Avenue, Cambridge MA 02139 (USA)}

(2)

The omitted ternis are functions of the short time t and are unimportant in later discus-sions.

The condition on the seabed yields simply

- O

j =

1, 2, 3 (2.13) j must satisfy the radiation condition

a _{infinity for both short and long scales.}

The far field potentials are not directly responsible for the boundary conditions on

the body.

FIELD

EQUATION FOR THE NEAR

In the near field the slow spatial

coor-dinates x1 and y1 are irrelevant but the

dependence on t1 must be allowed. The

per-turbation equations can be obtained from those in Section 2 simply by letting

_a/x1

₌

= O.

For the sake of distinction, we shall

denote the near field potential by _{p and} write its perturbation series as

= +

2

2 +

=

(3.2)

On the body the kinematic and dynamic

conditions are quite lengthy. _Fortunately

Ogilvie [8] has given a detailed review; his results will be cited here with minor

modifications. Let = (X, Y, Z) denote the

linear translation of the point O' in the

body whose rest position coincides with O,

and ₌ _, -y) denote the angular

dis-placement of the body about O'. If 1' is

the position vector of a material point measured from O' and its position vector

measured from the fixed origin O, then

=

+ (

+ ''

+ H

fi'

+ O(e3) (3.3)

cix

where H is the second order quantity

(2+2)

0 0

H

= - --

_-2a5 ( +- ) o

(3.4)

22

-2 (

Furthermore, let

fi'

denote the unit normal fixed at a point P on the body surface, directed into the body;

fi'

is a function of

fi'

only. In the fixed coordinate system,

the sane unit normal is denoted by , where

. =

fi'

+ +

a X

fi1 + H ' (3.5) The kinematic condition states that the normal velocity of the fluid next to the body must equal the normal velocity of the body at

the sane point.

dfi =

2T

Upon introducing the expansions

(,

, fi

fi)

= (o,

O ;' _{') +}

+

1'

fi1)

+

+ 2( OE2' fi2, fi2) +...(3.7)

where _eri' _n'

. n=1,2,...

depend on t and t1, we get from (3.6), + =

'U1

+ (3.1)

t

(3.6) (3.8) + fi'.{ + + + fi +

X

_j1

altix

+ Ht _{'} +} x

i').(t1

+ x

fi')

t t

(3.9)

We now turn to the dynamic boundary

con-ditions on the body, which are just the

con-servation laws of linear and angular momenta

of the body. Let _{G and} fi6' be the

posi-tion vectors of the center of gravity of the

body, measured from O and O' respectively.

fiG = + (X +

fib)

+ H

_{+ O()}

(3.10)

After introducing multiple scale

expan-sions the linear inertia becomes

c1 +

tt

+

al

tt

x

fi)+2{(2

_tt

+ 2

_tt

+ 2(X1 + x

fib)

+ Htt fi J + ttl ttl (3.11) where = + 1

')i -

x

(3)

We shall assume weak mooring so that the constraining force is second order in ;

further details will be specified later.

Equating this to the hydrodynamic force and

the reacting force from physical constraints, we get

0(c):

+ X

= -

f!

tt tt so

-

pA(Z1 + Yfa1 - xf1)e

(3.12)

O(e2): + +

')+2(i

+ X 2

x r0

t tt = 2 + 2 - MHtt

where is the 0(2) mooring force and

= -p 5f

ds + 2 So ti t + X +

ix

+

f

d

I[(i)2

21(z1+ Yf1

_Xf1)]

+ gA (Z +

wpL 2

Yf2

XfB2+

11(xfa1+

fi]z

(3.14)

which is esentially an expression deduced by Ogilvie ([8] Equation 73) except for the

additional term

aq/at1.

A0 is the area of the water plane at rest whose center of gravity is located at the point (xf,yf,

0), and CQ is the water line. From the

conservation law of angular momentum, the perturbation equations by Ogilvie can be similarly modified, and moment due to the weight of the body accounted for. The

expressions are similar to Equations (3.12

-3.14) but are lengthier, hence are omitted. The solution at each order will be

decomposed into harmonics with respect to t.

Specifically: n n' n' n' = nm' nm' anm' nm m=-n e

imt

(3.15) (3.13) A(t1) ch k(z+h) ikx -ikx ig 2(A ch kh (e + Re ) x<0

At each order the perturbation problem

leads to boundary value problems for each

harmonic, which are discussed below

4. THE FIRST HAR0NIC - THE FAST £ûTION The first order, first harmonic

represents the short scale moEion af the leading order

_O(e).

In the near

field the governing equations on includ-ing the radiation condition are well known.

Since the mathematical problem for the first order floating body motion and the

potential is routine, l).

,

and will be regarded as known. In particular for a two-dimensional floating cylinder in beam seas the potential involves only the

propa-gating modes at the outer limit of the near

field -igA(t1) ch k(z+h) ikx Te x>0 - ch kh (4.1)

where R and T are constant reflection and

transmission coefficients. In three dimen-sions -igA(t1) ch k(Z+h)( ikx

_y(0)

kr = 2 ch kh 'kr kr»1 (4.2)

where (r,) are polar coordinates in the

horizontal plane and is the scattering coefficient.

In the outer field (kx1, ky1) = O(i),

only propagating waves are present. In two dimensions we have -ig ch k(z+h) 11 ch kh [(x1,t1)e1kX+ + 0(x1 ,t1 )e kx] x1<O -

ta

ch k(z+h) ikx 2 ch kh [t(x1,t1)e ] x1>O (4.3)

The slow evolution of the progressive wave

can be found by demanding the solvability of with the well-known result that

= (t. -x,/00),

T =

_{T(ti-xl/Cn) and}

= ci(t1+xj?Cg3.

Ey matching (4.3) to

(4)

and define the far field where k(x1 ,y1) = 0(1). in the far field (x,y,z,t,x1,y1,t1) are all pertinent.

Our general plan is first to deduce the approximate equations for the near and far

fields, then to solve the perturbation

equa-tions by separating into harmonics with

respect to the carrier wave frequency. It

will be shown that the slow motion of both the body and the fluid in near and far field can be found without solving the entire

second order problem.

Consider a general geometry as depicted

in Figure 1. The origin of the fixed

cartesian system is in the mean water plane of the body, with the z axis pointing

upward. The depth of the sea bed below the

calm sea surface is constant, h. Incident

waves of amplitude A, mean frequency (wave

number k) arrive from x + . We assume

small amplitude and finite depth relative to the mean wavelength, i.e., kA = O(e) « i

and kh = 0(1). The wave amplitude is

modu-lated slowly over time scale 2/ç which is much longer than the carrier wave period

i.e.,

where * denotes the complex conjugate,

= gk tanh kh (2.7)

and A is a slowly varying function of space

and time, i.e.,

A = A(t1

-

x1/C)

(2.8)

We now introduce the following expansior: for the far field

= 1 + 2 + 2 (2.9) where = x1, y,, t,) n=1,2,.. (2.10)

Denoting by y1 the horizontal gradient

with respect to x1 and y1, we get from the

Laplace Equation

2

O(e) « 1

(2.2) V i

= O (2.ila)

Viscous effects will be ignored. Within 2

the fluid bounded above by the free surface V 2

h1

+ Vh.Vl)l

(2.11b)

z = (x,y,t) Laplace Equation holds for the

velocity potential 2 2

2

2 2 2 y

hl

+

1h2

(2.11c)

:__+4+$=

O -h < z <

y z From the free surface condition (2.5) we get

(2.3)

2

Keeping nonlinear terms to second order in

amplitude, the free surface height is

rela-ted to the potential by

g

t-)ti

= O

2

1 1 2

= -[et - +

(v)2],

z=O (2.4) +

2tt1 -

()

t

The combination of kinematic and dynamic conditions on the free surface gives

1

21

tt

+ 9z =

- 2t +

tztt

-hth'

z=0 (2.5)

where _h _(B/ax, 0) denotes the orizontal gradient. 2

1 ,1

-zzJt

2 2 1 2 L 2

+ 93

- Bt dL i 1

-(r

(2.12a) (2.12b) (2.12c)

ency implies a long croup length O(1/k). The incident wave has the following

paten-We therefore introduce slow variables tial at the leading order

(x,,y1,t1) =

(X,

y, t) (2.1)

iqA cosh k(z+h) i(kx-t)

e ± * (2.6)

(5)

the outer limit of the near field, (4.1) we

find

= 11(ti_Xi'0g)

=

R A(t ±x ¡C

11g

(4.4)

= T

A(t1X1/Cg)

In three dimensions the outer solution is

s imply -ig ch k(z+h) {A(t X1/C ikx ii 2 ch kh - )e + A(t1-

r1lcg)

eikr] ./kr

5. THE ZEROTH HAR!ONIC - THE SLOW DODY

1OTION AND LONG WAVES

To the leading order the slow fluid motion has the potentials p10(x,y,z,t1) and

10(x,y,z,x1,y1,t1) in the near and far

fields respectively. It is easily shown

that both the variations of and with

respect (x,y,z) are governed by Laplace Equation and homogeneous Neumann conditions

on Sf ,S ,SB and S (see Figure 2),

therefore

= 10(t1) and = 10(x1,y1,t1) (5.1)

To match and in the intermediate

region we simply require

= c1i(O O, t1) (5.2)

The fact that *10(t) is independent of position in the near field makes the calcu-lation cf the slow body motion a simple

mat-ter, as will be shown.

We now assume that the horizontal mooring force to be linear in the slow displacement,

with K being the elastic tensor.

We give the dynamic boundary condition for the body translation only

i 2 K ₌ -p ₊

iviiI

S0

_Li

*

+ +1

+ [+ 11X

r ).iwV411+ *}Ids

*

+ pg

_f

_{c11

_ii

CO

*

-

[11(Z11+ y X +

*J1d

-

gA (Z + wp 20 (4.5)

Yf20- xfB2o)z

(5.3)

LJith the exception of the tern

and the terms (Z0 yfa90-xfß20), a?l

other terms on te righf hand side are form-ally the same as those in the steady drift

force or torque formulas for regular (unnod-ulated) waves. Since is spatially

con-stant on the wetted surlace of the body, it contributes no net horizontal force or

moment.

All

other terms on the right of (5.3) are

proportional to

A2

and can be written in the

form A2F20 where A is a function of

(-00t1) and F20 is the same constant

factor known in steady drift force theory. Thus the horizontal (sway, surge and yaw)

components of the slow drift displacements

and can be found from the steady drift force theory using the first order

result alone.

Once and are solved, we can

examine te radiation of long waves. The vertical slow motions (heave, roll and

pitch) can be shown to be small (of second

order in e) since the buoyancy restoring

force inhibit larger vertical notion unless

the body has a special shape (such as a

bottle neck with a very small water plane). As a special case of the general theory

of this paper we consider a horizontal cyl Inder with a wide gap between the keel and the seabed H=O(h). In this case and

are both continuous across x1=Ö. is constant along the hull hence does no _{contribute to slow sway which is the} dominant slow mode. Therefore, the slow sway

of the vessel is immediately given by

X10 =

pgA2R2C9/CK

(5.4) with

A=A(t1),

_{C is the phase velocity K the}

mooring stiffness and R the reflection

coefficient (cf. [1]). Formally this result is the same as that for a

regular

(unmodulated) wave train [4], and would be obtained by [2], [7], [8], [9].

In the far field the two continuity conditions on , provide the boundary

conditions. To rind the governing equation forc10(x1,y1,t), we must proceed to higher orders.

At O(2)

2O satisfies the same

homogeneous equation with respect to x,y,z and can be discarded. At the third order is governed by 2 2 3O =

1iO

2 3O a

*

g -

2t1

[v11.v11 +

(5.5)

(6)

*

+

± *Y1

_-gv1.[v11+

]

11

z

i

z= 0 (5.6)

We remark that this condition is related to nass conservation (see [1]).

Being in the far field, only the

propa-gating parts of and matter to the forcing terms on fFe righE hand side of

(5.8). The solution to can be expressed

as the sum of inhomogeneous and homogeneous

solutions. In two dimensions we have

I io - 10 (x -1 C t )+

gi

0(x1+ C t

gi

+ j0(x1+ jiT t1), x1<0

_T(

- o x1-C t )+

gi

0(x1-/ t1), x1>O (5.9)

where the inhomogeneous parts, which are locked to the wave envelopes, are given in

[1].

In three dimensions, we have

I y(o) s

Ct)

lO = 10(x1- Cgt1) + 10(r1- g i

(5.10) where r1 cr.

We now illustrate the steps to determine

the free long waves and 4 in two dimensional cases. To the far field

= h +

fJ

-h x1=O

axiIo+

l

observer the near field shrinks to a line at

x1=0 across which the potential is

connected via In view of (5.2) we simply have

(-o, t1) = (+O, t1) (5.11)

which implies

4(-/

t1)-

(/i

t1) =

ogti) ±

0(Cgt1)

iogti)

(5.12) We further need a jump condition on

which is related to the horizontal flux. To

s+

+ {v11.v11

-+ which is equal to (5.13)

(5.8)

x1=0 z= O (5.14) The overbar stands for averaging over a

short period Since the quadratic

terms, which correspond to Stokes' Drift, are equal on x=O by energy conservation of

the short waves, we have

h

O

dz (5.15)

ax1

x1=0 Lh ax

There is a similar equation on the left side. It follows that

X++co

x1=0+

2Ol

h____

₌

_f

dz 3X1 x1=0- -h

x

We now examine the governing equations for _2O: 2 2

+ L)

20 = -h<z<0 (5.17) x az az - O z= -h (5.18) (5.16)

order O(2), the flux across S where

kx

»

1, x1 « 1) is O = dzdz _____ S+ z=0 Lh ax Lh X 30 z= -h (5.7)

The omitted terms in (5.6) are important only on the short scale x and y. By

averag-ing over the short wave, only the long scale

part of the forcing in (5.6) is left. The corresponding response in must be

inde-pendent of x and y. Thus the long scale part of is governed by a boundary value

problem in z only. The solvability

condi-tion of this problem yields the following

equation for

{ghv1 -2

2

(7)

2O

*

a11

_*1 = -h z=O 3Z - [ ₁₁ _3X

ix -

ax

*

hU

[ü11

+ * i

z=O

On the body the kinematic boundary condition

requires that

2O_

+1

*

an'

11X

r

).v)v+

_[(ix

11x '

-+

*1

+

1ot

} 1

*20 which satisfies the boundary value

problem of Equations (5.17-5.21) can be

decomposed into 20 + 20' where *20

satisfies Laplace Equation with homogeneous boundary conditions on the free surface and

the seabed, and

-an' -

fi'.

_1Oti

on the body. Therefore,

_*20

is the

solution to the regular wave train problem. It can be shown by mass conservation that

O

+

dz = O (5.23)

Since in two dimensions, involves only

sway, the total volume in e near field is not changed. We must have

o

X

]dz

= o

(5.24)

Adding (5.23) and (5.24), we find from

(5.16) X 0+ ax

I

= O 1

x1=O-(5.25)

With (5.2) and (5.25), and can be solved fron (5.9). Because of energy

con-servation of the short waves, we can show t ha t I R T - = at

x1 =

O (5.26) (5.19) where (5.20) (5.21) (5.22)

as in [1] (Equation 4.12). Using this we find

+ R

(f

'

io'

i

=

10(t1)

(5.27)

Hence, the free long wave is equal in ampli-tude and opposite in phase to the locked reflected long waves. Substituting (5.27) into (5.9) we get

= *10(t1) =

0(t1), x1 =

±

0 (5.28)

so that

is the same as if the body were

absent.

is similar tc the

Fraude-Krylov approximation in the linearized

theory of long waves past a small body, and

can be useful

in calculating the vertical

drift.

The fact that the slow displacement of

the body can be found before the long wave

is due to the constancy of *10(t1) in

space. If, however, * varies appreciably

on the wetted body surface, the

surface

integrals involving

a*10/at1 need not

vanish and long waves and the driff motion are then

coupled.

An exarnole of this kind is the two

dimensional block in bean seas where the

block slides on the sea bottom [1].

In that

case a*10/at1 takes on two different values

on two vertical sides of the block.

In the

next section we shall examine the case where the bottom of the swaying block is slightly

higher than sea bed. The narrow gap creates strong blockage and an appreciable variation

of 1j

along the gap.

We stress that the second order poten-tials etc. need not be solved at all in order to solve for the leading

order slow drift and bong waves.

To study long wave radiation in three

dimensions, we must decompose

and _*20

into Fourier series in cos m0 and sin mc.

By

matching the Fourier coefficients of the

horizontal flux we also get a boundary

cork-dition for the Fourier coefficients of 27r1

.a1/ar1.

Free long waves and the slow body

dispTacement can also be determined.

Details

appear straightforward and will not be

pursued here.

6.

A TWO DIMENSIONAL BLOCK IN BEAM SEAS In this section we discuss briefly a simple geometry to point out an exceptional

circumstance under which a general result

must be modified.

The geometry is a two

dimensional

cylind-er of rectangular cross section with beam 2B

and draft D. The keel to seabed h-D is des-ignated as H (Figure 3). To the leading

order 0(e) the fast heave and roll can be

(8)

showr.tc be negligible; the first harmonic

near field is the same as that for a

block in sliàing contact with the bottom

(i.e., H=0) which has been treated in [1]. In the far field we must still maintain

the continuity of

,10/x1 across x1=0

because of mass conservation.

However

is no longer continuous because of strong

blocking by the narrow gap. This is caused by the potential jump in the near field

(x

+

-

p10(x

+

-)]

2Vc1 (6.1)

where c1 the blockage constant

c1 = Eh/H (6.2)

(cf. [2]) and V is the apparent current

passing a fixed body by an arguement similar to Equations (4.20-4.21) in [1] and a

Calilean Transformation, we find V to be:

V x1 10 lO

-u +

-

4-rn 1

(6.4)

where U is given by (5.20).

In conjunction with the dynamic boundary condition on the body, we finally obtain a different equation for the slow sway X10

C K X - - g 1

2h/

10l

2phc1XlO lot 1 c s 2 gC 1R12A2 g A + g + _{1- 1R2} ) pg(gh-Cg2) ¿hLc1 6.5)

where

(6.6)

A variety of inputs A(t1) can be studied with this simple differential equation.

Figure 4 gives the norTalized amplitude

of the slow displacement X10 X10 h/A2 of a

block for a sinusoidally modulated incidt wave, for kh=l, M=h and c1/h=1. Damped

resonance occurs near the modulation frequency (K/2phc1)'/2.2 K is the mooring line stiffness (KEK/pgh takes the values 1/2, 1, 2). From(6.5), 2hc1 may be

regarded as the apparent mass.

In the limit of H=0, c1, the case of

the sliding

block

1 is recovered. Figure 5

gives sample results of fast

(X11EXh/A2)

and slow displacements for sinEsoidaT

envelopes. X0 is the negative of the mean

slow sway.

Thr comparison, X, the mean of

(5.4) in which the jump in is not accounted for, is also shown.

Figure 6 gives the long wave due to an incident wave packet on a sliding block. Note that on both the incidence (x1<0) and

transmission sides, there are two outgoing wave packets travelling at ;C0 and ;/

respectively. Here we have chosen kh=1.25,

M=h2 and K=gh.

In the limit of a wide gap

c11 Equation 6.5 reduces to Equation 5.4.

7. CONCLUSIONS

The response of a floating offshore

structure in slowly varying waves depends on

body geometry, mooring stiffness as well as wave characteristics. In this paper

atten-tion is focussed on the cases where fast and

slow displacements of the body are

compar-ably small. By the notion of near and far fields and multiple scales, we have shown (i) that the slow drift problem can be

solved without the full treatment of the

second order problem and (ii) that the slow

motion in the near field is accompanied by

the propagation of long waves in the far

field. In many practical cases the

horizon-tal slow motion of the body only affects but

is not affected by, the long waves. The low frequency vertical force is however coupled to long wave radiation; this coupling can

induce significant vertical drift if the body

has a small water plane area. For such a

case the vertical drift and long wave

propagation must be solved together.

For a

horizontal cylinder in beam seas, the slow

motion is related to the so called blockage

coefficient which measures the obstruction by

the body to the long period current arround

it. It has the effect of an added mass. In

particular, if blocking is complete (c1=, nO

gap), the added mass is infinite. The drift

force is negative; the equation of body notion is first order and there is no

resonance. tihen blocking is large but finite (c1=0(1)) as for a narrow gap between the keel and the seabed, the added mass dominates the nass of the body. The equation of body

motion is second order; there is both

resonance and radiation damping. Lastly when blocking is weak, (c1«l) damping and added mass are both negligible. The moored system

then reacts in a quasi-static manner.

When the mooring stiffness is 0(c2) the

slow drift motion becomes, in general 0(1). Extension of ideas of this paper to large amplitude drift motion is in progress and will be reported elsewhere.

2

dz 11

+

h 2

x=B

(9)

a

ACKNOWLEDGEMENT

This research has been supported by the Office of Naval Research

(N00014-8O-C-0531)and the National Science Foundation (Grant MEA 77-17817-A04).

RE FERENCES

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Faltinsen, O. M. and Loken, A. E., 1979, Slow drift oscillations of a Ship in irre-gular waves, Appl . Ocean Res., 1, 21-31.

Flagg, C. N. and Newman J. N., 1971,

way added-mass coefficients for rectangular

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Berkel ey.

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