Dual pontoon floating breakwater

~ ) Pergamon

S0029-8018(96)00024-8

Ocean Engng, Vol. 24, No. 5, pp. 465~-78, 1997 © 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0029-8018/97 $17.00 + 0.00

D U A L P O N T O O N F L O A T I N G B R E A K W A T E R

A. N. Williams* and A. G. Abul-Azm# *Department of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4791,

U.S.A. tlrrigation and Hydraulics Department, Faculty of Engineering, Cairo University, Giza, Egypt

(Received 15 December 1995; accepted in final form 31 January 1996)

Abstract The hydrodynamic properties of a dual pontoon floating breakwater consisting of a pair of floating cylinders of rectangular section, connected by a rigid deck, is investigated theoretically. The structure is partially restrained by linear symmetric moorings fore and aft. The fluid motion is idealized as linearized, two-dimensional potential flow and the equation of motion of the breakwater is taken to be that of a two-dimensional rigid body undergoing surge, heave and pitch motions. The solution for the fluid motion is obtained by the boundary integral equation method using an appropriate Green's function. Numerical results are presented which illustrate the effects of the various wave and structural parameters on the efficiency of the breakwater as a barrier to wave action. It is found that the wave reflection properties of the structure depend strongly on the width, draft and spacing of the pontoons and the mooring line stiffness, while the excess buoyancy of the system is of lesser importance. Copyright © 1997 Elsevier Science Ltd

1. I N T R O D U C T I O N

Floating breakwaters offer an alternative to conventional fixed breakwaters and may be preferred in relatively low wave energy environments or where water depth or foundation considerations preclude the use of a bottom-founded structure. Furthermore, in certain applications, aesthetic or water circulation considerations may require that the breakwater does not pierce the free-surface and/or extend down to the sea-bed. In the present paper, the behavior of a dual pontoon floating breakwater is studied.

Various aspects of the two-dimensional problem of wave interaction with long submerged, bottom-founded or floating, surface-piercing structures of rectangular cross-section have been studied previously by several investigators. Both the diffraction (waves incident on fixed structure) and radiation (structure oscillating in otherwise calm fluid) problems have been treated. A variational formulation approach for rectangular bodies either on the free surface or on the sea bed was presented by Mei and Black (1969) and Black et al. (1971). For bottom-founded rectangular bodies, approximate solutions for long waves have been developed by Ogilvie (1960), for long obstacles by Newman (1965), and for low-draft structures by Mei (1969). Drimer et al. (1992) presented a simplified approach for a floating breakwater where the breakwater width and incident wavelength are taken to be much larger than the gap between the breakwater and the sea-bed. Mullarkey et

al. (1992) utilized an eigenfunction expansion approach to calculate the hydrodynamic coefficients for rectangular TLP pontoons. An integral equation formulation for the calculation of hydrodynamic coefficients for long, horizontal cylinders of arbitrary section has been presented by Naftgzer and Chakrabarti (1979) and Andersen and Wuzhou (1985). Mclver (1986) investigated hydrodynamic interference effects between a pair of semi-

465

466 A.N. Williams and A. G. Abul-Azm

immersed bodies of rectangular section using both the method of matched eigenfunction expansions and a wide spacing approximation.

In the present paper, a boundary element technique is utilized to calculate the wave transmission and reflection characteristics of a dual pontoon floating breakwater consisting of a pair of floating cylinders of rectangular section, connected by a rigid deck. The structure is restrained by linear symmetric moorings fore and aft. The fluid motion is idealized as linearized, two-dimensional potential flow and the equation of motion of the breakwater is taken to be that of a two-dimensional rigid body undergoing surge, heave and pitch motions. The fluid domain is divided into three regions and the boundary integral equation method is applied on each domain using an appropriate Green's function. The integration contours are discretized into small elements and the fluid velocity potential and its normal derivative are assumed to vary linearly in each. The resulting discrete algebraic systems are then solved simultaneously by standard matrix techniques. Numerical results are presented which illustrate the effects of the various wave and structural parameters on the efficiency of the breakwater as a barrier to wave action. It is found that the wave reflection properties of the structure depend strongly on the width, draft and spacing of the pontoons and the mooring line stiffness, while the excess buoyancy of the system is of lesser importance.

2. THEORETICAL DEVELOPMENT

The geometry of the problem is shown in Fig. 1. The system is idealized as two-dimensional, Cartesian coordinates are employed, the breakwater is located symmetrically about x=0 and the z-axis is directed vertically upwards from an origin on the still-water level. Each pontoon is of draft b and width 2a, the edge-to-edge spacing between pontoons is 2h and the uniform water depth is denoted by d, The structure is subjected to a train of regular, small-amplitude waves of height H and frequency oJ travelling in the positive x- direction and is assumed to respond linearly in surge, heave and pitch to this excitation.

Assuming that the fluid is inviscid and incompressible, the irrotational fluid motion may

Z

/ - " - - . .

v

2a 2h 2a d

Oo

Fig. 1. Definition sketch.

Dual pontoon floating breakwater 467

be described in terms of a velocity potential ~b(x, z; t)=Re[~(x, z) e-i"~], where Re[ ] denotes the real part of a complex expression. It follows that this potential must satisfy the Laplace equation

V2~ = 0 (1)

everywhere in the region of flow. It is subject to the following boundary conditions on the quiescent fluid free-surface and the sea-bed, respectively,

g ~ z - t °2~ = 0 o n z = 0 (2)

-0 onz=-d (3) bz

The boundary condition on the structure is best written in terms of the various components which make up the total velocity poential. The total potential is taken to consist of incident, scattered and radiated wave components, i.e.,

3 -: (I)I "[- liftS + E "~mtff~ Rm (4)

m=l where 7/m is the displacement amplitude in the mth mode of oscillation. In Equation (4), re=l, 2, and 3 correspond to the surge, heave, and pitch modes of oscillation, respectively.

The structural boundary conditions may then be written as

3~s 0~1 3x - 3x on x = +_h, +[2a + h], -b<-z<-O (5a)

~ s ~eP l - on z = - b , h<-[x[<-[2a + h] (5b)

~z bz

~(I)R l / Ox - - i to

~(I)R2 on x = +_h, +[2a + h], -b<-z<-O ~x - 0 ]

o3~R3 ~ x - i t o ( z - z c )

(6a)

(6b)

(6c)

b~RI / OZ - 0 (7a)

~(I)R2 3z - - i to on z = - b , h<-Ixl<-[2a + h] (7b)

~q~R3 Oz - itox (7c)

in which zc is the vertical coordinate of the center of mass G of the structure. The structure is assumed to possess symmetric mass properties about the z-axis (i.e., x~-0).


Finally, a radiation condition must be specified at large distances from the structure. This condition may be written as

lim ~ -7-ik}(Cb-~,) = 0 (8)

In Equations (4) and (8), the complex amplitude of the incident wave potential, ~ , is defined by

gHi cosh k(z + d) ~ / - 2oJ cosh kd exp(ikx) (9)

where k is the incident wave number which is related to the angular frequency through the linear dispersion relation o.,2=gk tanh kd.

The structural response will be analyzed by assuming that the breakwater behaves as a two-dimensional rigid body undergoing small-amplitude surge, heave and pitch motions. The equations of motion of the breakwater acted upon by fluid pressure may be written

) m~l + To sin20o + 2K coS20o ~ L

+ (K sin 20o(2a + h) -2K cosZOo(b + zc) - 2To sinZOo(b + zG)

L

To sin 20o(2a + h)) - L ~3 = F , ( 1 0 )

/

( o os 0o ) m~2 + L + 2K sin20o + 40ga ~2 = F~ (11)

+ (K sin 20o(2a + h) - 2K cos2Oo(b + za) 2To sin2Oo(b + za)

Ic~3 L

sin 20°(2a+h)]~l+(2To cos Oo(2a+h)+2To sin Oo(b+ZG) L / \

+ 2K sin2Oo(2a + h) 2 + 2K cos2Oo(b + zc) 2 (12)

2To sin 20o(2a + h)(b + zG) - 2K sin 20o(2a + h)(b + zc) +

L

2To cos2Oo(2a + h) 2 2To sin2Oo(b + z~) 2 2 + L + L + 3 pg { (2a + h) 3 - h 3 }

+ 4pgab(zs-z~))~3 = F3

where m is the mass of the structure, Ic the mass moment of inertia about G, K the mooring line stiffness, p the fluid density, L the mooring line length at equilibrium, zn the location of the center of buoyancy, B, To the mooring line pretension, and 0o the mooring line


angle in the static position, as defined in Fig. 1. For simplicity, in the present case the mooring system has been taken to be symmetric fore and aft of the breakwater. It is noted that the mass, inertia, mooring stiffness and pretension are defined per unit length of pontoon. Also, in Equations (10) and (12), the ~m m=l, 2, 3 are the surge, heave and pitch displacements respectively, ~m(t)=Re{ ~m e-i~°t}, and Fro(t) m=l, 2, 3 are the corresponding total hydrodynamic forces and moment in these directions, including both excitation and reaction components. The rotational displacement and loads are defined about G. Finally, the pretension To may be obtained from static equilibrium as To={4pgab -mg}/2sinOo and the mooring line length L=(d-b)/sinOo.

3. SOLUTION BY INTEGRAL E Q U A T I O N APPROACH

The above problem for the fluid velocity potential is solved numerically utilizing the boundary integral equation method. First, by introducing a pair of fictional boundaries at x=-L-_[a+h],-d <-- z <--b, the fluid domain is divided into three regions, region 1 (x -< -[a+h]), region 2 ( - [a+h] -< x --- [a+h]) and region 3 (x --- [a+h]). Denoting the complex velocity potentials in the three regions by q~j=l , 2, 3, continuity of mass flux and pressure across the fluid interfaces between the regions implies the following matching conditions:

I[I)l = lffl D2 on x = --[a + h], -d<-z<--b (13a)

~(~D I ~(I )2

bx bx

(I D2 = (I)3

O~(I )2 ~(I )3

3x ~x

on x = - [ a + h], - d < z < - b (13b)

on x = [a + h], -d<-z <--b (14a)

on x = [a + h], -d<-z <--b (14b)

Two auxiliary vertical boundaries x=+_x = are also introduced (Fig. 2), these boundaries are located sufficiently far from the breakwater such that the radiation condition, Equation (8), is valid on each. Applying Green's theorem in each of the regions to either the dif-

D E K J N 0

X ---- "Xoo F

x = -(a+h)

A A H H

L I X = X ~ M

x = (a+h)

C 8 8 G G P

Fig. 2. Sketch of three fluid sub-domains showing integration directions, matching boundaries at x=!_(a+h), and auxilliary boundaries at x=+_x=.


fracted (incident plus scattered) velocity potential, ~d=~;+~s/, or the unit radiation potential, q~R2, m=l, 2, 3, in region j, and the free-space Green's function of the Laplace equation,

G(G;_r) = lnlr-r~[ (15)

leads to the following integral equations,

ENJ(r ~) = ~j(r) On (r~;_r)-G(~;r) N-~ (0 dr j = 1, 2, 3 (16) rj

in which ge=q~j or q~Rff, Fi is the boundary of each subregion and Ep is the interior angle at the source point r_p=(Xp, Zp) which is restricted to lie on the boundary Fj.

With reference to Fig. 2, after imposing the free-surface, radiation and bottom boundary conditions, the integral equations in region 1 may be written,

~ - G bx j + * ' ~zz dX AB BC

[ Ox + ikG dz- G dx + J [~z g

CD DE

f Ox dz- xltl = 2ik ~tG dz for ~1 = ~ol (17a) Oz

EF FA CD

f odi)Rml f ~f~Rml ~t~l - G ~ - - x d Z - G ~ z z d X for =(~.)Rm I (17b)

EF FA

for m = l , 2, 3. Similarly, in region 2,

%~2(r~)+f{xt,2bG O'kI/2 ] f ~x - G ~ x l d Z + ~2 Oz AB BG

ffx - G Ox j *2 Oz J ffxx dz GH HI IJ

JK KL

Oz = 0 for ~I f2 = ( I ) o 2 ( 1 8 a )

LA

f~gI]Rm 2 ( ~ f ~ R m 2 f ~ d l k R m 2 f ~ f ~ 2 m = G ~ x dz -JG~z-z dx - G ~ z dx- G~x-x dz (18b) KL LA HI lJ


f o r ~,I f2 = ~ ) R m 2

for m=l, 2, 3. Finally, for region 3,

ax - G Ox J Oz ~ GH H M

az- j taz g dx

M N NO

- f * 3 1 a G - i k G } d z + f * 3 a G W 3 [ o3x a z dx = 0 for = *03 (19a)

OP PG

f O m3 foo 3 .3 = - G ~ z dx+ G~--x dz for = ~ m (19b)

HM M N

for m=l, 2, 3. The numerical solution of the above sets of integral equations proceeds by discretizing the integration contours into a large finite number of segments, the interzonal points are numbered twice. In each segment, the velocity potential is taken to vary linearly between its nodal values. It is noted that in the present formulation the nodes in regions 1 and 3 are numbered clockwise for convenience in applying the matching boundary conditions later. The discretized system may then be solved by matrix techniques.

Once the scattered and unit radiation potentials in each of the fluid domains have been calculated, various quantities of engineering interest may be determined. The t e r m s Fq,

q=l, 2, 3 on the right hand side of Equations (10) and (12) contain both hydrodynamic excitation and reaction effects. The complex-valued amplitudes of the hydrodynamic excit- ing forces in the x and z-directions (El, E2) and the moment (E3) about rc=zck may be expressed as

Eq = oi~o~ ~jnqJdS q = 1, 2, 3 (20) j = l s,,j

in which nl j = n2 and n~ = n~ the components of the unit normal to Soj, the equilibrium wetted surface of the breakwater in region j, and n3J=([z-z~ ] n2-x ril).

In a similar way, the complex-valued amplitudes of the hydrodynamic reaction force or moment R m, defined as the force or moment in the q~ direction due to breakwater oscillation in the pth mode, may be expressed as

3 I gpq = pioJ~ dPR/nqJ dS (21)

j = l So j

for q=l, 2, 3, where p=l, 2, 3 denotes the surge, heave or pitch oscillation, respectively. These reaction loads may be decomposed into components in phase with the acceleration and velocity of the forced oscillation and give rise to the added-mass and added (radiation)


damping matrices respectively. The hydrodynamic loading terms on the right-hand side of Equations (10)-(12) may now be written as

F q ( l ) = + "OeRpq e - i°Jt q = 1, 2, 3 (22) p = l

The structural displacements in the surge, heave and pitch modes may now be determined from Equations (10)-(12).

The effectiveness of the structure as a wave barrier may be measured in terms of the reflection and transmission coefficients R and T. On the auxiliary boundary x=-x~, the velocity potential may be written

qb3(x~, z) - --igHT cosh k(z + d) e i ~ + io, + (23) 2to cosh kd

where Hr is the transmitted wave height and a ÷ a phase angle, similarly on x=--x~,

- i g ~ l ( - - X ~ , Z) = ~ (nRe i ~ + i,~- + n t e - i k~ ) cOShcoshk(Zkd + d) (24)

P

o ( 3

m

q m

G) 0 0

C 0 i m

t O o ql,,,

t r

1 . 2 I . , , . I , ' ' , I , , ' ' I , ' ' ' I ' ' ' ' I ' ' ' ' I ~ ' ' ' I ' ' ' '

O m m m ' m ' m m m - - ~ ~ m, , , - - I

- , ~ s |

I , J - 0 . 8 i

| , I I OG'm~ I ~ -

0 . 6 ,, • * , . . n I

i I I I

• I I - 0 4 , S '

I I ' I

0 . 2 " ~ I I -

0 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4

D i m e n s i o n l e s s W a v e n u m b e r k d

Fig. 3. Influence of pontoon draft on reflection coefficient for d/a=5, hla=l, K/pgd--~. Notations: bla=0.5; b/a= 1; . . . . . . b/a=2.


where HR is the reflected wave height and a - its associated phase. The reflection and transmission coefficients are then given by

R -InRe'°-I (25) H,

T - Iare/~ + J (26) H,

4. RESULTS A N D D I S C U S S I O N

A parametric study was carried out to investigate the sensitivity of the breakwater performance to the various wave and structural parameters. In obtaining the numerical results presented herein, the domain boundaries were each typically discretized into approximately 300 line segments, in several cases a more refined element mesh was utilized in order to ensure that numerical convergence had been achieved. The auxiliary boundaries at x=#_x= were placed 3-4 water depths away from the structure, sufficiently far to ensure that the evanescent wave modes could be neglected (Dean and Dalrymple, 1984).

The performance of the dual pontoon structure as a wave barrier depends upon several parameters, the width, draft, and spacing of the pontoons, denoted by 2a, b, and 2h,

1 , 2 ~

C 0 iM

U m

Q 0

C 0

m m

0 m qb~

0

1

0.8

0.6

0.4

0.2

0

i m l m

0 ° •

. : ' / . I

, / i~ VI i II

, f ' I

' I !

' I |

' I I

' I |

I I I

|

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4

Dimensionless Wavenumber kd

Fig. 4. Influence of pontoon width on reflection coefficient for d/b=5, h/b=l, K/pgd--oo. Notations: . . . . . b/a=0.5; b/a= 1; b/a=2.

4 7 4 A, N. W i l l i a m s a n d A. G. A b u I - A z m

,I,,11

C: Q

i m

cJ , , i q m q , m

O) o CJ

I=

2 cJ q)

m q u

O) IZ:

1.2

0 .8

0 . 6

0 . 4

0 .2

O Ill

0

, , , , I , ~ , , l + , , , i , , , , i , , , , i r , , ~ I , ] , ,

• -~a~,.- I - O @O • u, ##

t • I u i : i

/ - : I

/",. / V i! , " t ' |

0 . 5 1 1 .5 2 2 .5 3 3 .5 4

D i m e n s i o n l e s s W a v e n u m b e r kd

Fig. 5. Inf luence o f p o n t o o n spac ing o n ref lect ion coef f ic ien t for d/a=5, b / a = l , K / p g d --~. Nota t ions : - - - - - - - - h /a=0.5; h /a=l ; . . . . . . h/a=2.

respectively, the mooring line stiffness and angle of inclination, K and 0o, and the excess buoyancy of the system. This latter quantity may be expressed in terms of the relative density of the pontoon material (assumed uniform), p*. In all calculations, for simplicity, the mass center, G, was taken to be at the still-water level, and the inertia of the structure was characterized by specifying a radius of gyration, kc.

First, the influence on reflection coefficient of the pontoon draft was investigated. Figure 3 shows the variation of reflection coefficient with dimensionless wavenumber kd for three example structures of relative draft, b/a--0.5, l, and 2. All other wave and structural parameters were held constant. The mooring line stiffness, K, is taken to be infinite, i.e., only wave scattering effects are considered in this figure. It can be seen that although the general forms for all three curves are the same, the deeper draft structure is, as intuitively expected, a more efficient wave barrier. It is noted that, in short waves the reflection coefficient approaches unity due to the infinite stiffness of the mooring system. Also, there exists, for each dataset, a sharp minimum of the reflection coefficient in the frequency range of interest. This minimum does not occur for the single pontoon case; in that case, for infinite mooring line stiffness, the reflection coefficient is a monotonic function of dimensionless wavenumber. Therefore, this minimum value must be associated with hydrodynamic inter- eference effects between the two pontoons.


e, o

m m

o m m

o o

c o

i m

o e

m

o IIC

1 . 2 t . . . . i . . . . I . . . . i . . . . i . . . . i . . . . m . . . . i . . . . 4

0 . 8

0 . 6

0 . 4

0 . 2

0

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4

Dimension less W a v e n u m b e r kd

Fig. 6. Influence of pontoon excess buoyancy on reflection coefficient for d/a=5, h/a=l, b/a=l, kc,/a=l, Klpgd--O.O16, and 00--45 °. Notations: p'---0.1; p*--0.25; . . . . . . p*=0.5.

Figure 4 presents the influence of pontoon width on the reflection coefficient as a function of dimensionless wavenumber, kd. Results are shown for three example structures corresponding to b/a=0.5, 1, and 2, with all other wave and structural parameters held constant. Again the mooring stiffness is assumed infinite (wave scattering only). The forms of the curves are generally the same as those in Fig. 3. The reflection coefficients each exhibit a sharp minimum in the mid-frequency range, and tend to unity in short waves (high kd). It is noted that, in general, the efficiency of the breakwater increases as the pontoon width increases.

The effect of pontoon spacing on the reflection coefficient is shown in Fig. 5 for three example structures corresponding to h/a=0.5, 1, and 2. All other wave and structural variables are kept constant. Again, in each case, the reflection coefficients exhibit a minimum in the mid-frequency range and tend to unity at high frequencies. At low frequencies, it can be seen that the smallest spacing, h/a--0.5, leads to the maximum reflection coefficient, while at higher frequencies (kd > 1.7), it is the largest pontoon separation which results in the most efficient wave barrier. This is because in long waves the small spacing allows the two pontoons to act like a continuous barrier, while in short waves, the larger spacing allows the pontoons to act independently, in essence as two single pontoon breakwaters in series.


r- Q

, m ¢J i i q) , , , q m

Q O (J

e- O

i i

¢J Q

m q , , , ,

o re

1 . 2 t ' ' ' ' l f ' ' ' l ' ' ~ ' l ' ' ' ' [ ' ' ' ' l . . . . I . . . . I ' ' ' ' t

0 . 8

0 . 6

0 . 4

0 . 2

0

0 0 .5 1 1 .5 2 2 .5 3 3 .5 4

Dimensionless Wavenumber kd

Fig. 7. Influence of mooring stiffness on reflection coefficient for d/a=5, h/a=l, b/a=l, kc,/a=l, p*--0.25, and 00--45 °. Notations: Klpgd--O.O16; Klpgd=O.08; K/pgd=O.16;

. . . . . . KIpgd=0.32.

Figure 6 shows the influence of the excess buoyancy on the reflection coefficient as a function of dimensionless wavenumber, kd. In the present case the dual pontoon structure is idealized as a homogeneous solid with an equivalent relative density, p*, from which the breakwater mass is computed as m=p*(pV), where V is the volume per unit length of the barrier. Therefore, varying the relative mass density is equivalent to varying the excess buoyancy of the structure. This quantity may also be thought of as determining the pretension in the mooting system. In Fig. 6, three example structures are shown corresponding to p*=0.1, 0.25, and 0.5. It can be seen that all three curves exhibit a sharp maximum in the range 0.5< kd <0.75, and peak again when kd is approximately 2.0. For 0 .75< kd <2.0, the lightest breakwater, corresponding to p*--0.1, is the most efficient breakwater, while for 2 .0< kd <3.5, the heaviest breakwater, corresponding to p*=0.5, is most efficient. The rapid drop in reflection coefficient for kd >3.5 in the case of p*=0.5, is also noted.

The influence of mooting line stiffness on the reflection coefficient is shown in Fig. 7. Results are presented for four example structures corresponding to a dimensionless stiffness Klpgd=O.O16, 0.08, 0.16, and 0.32. All other variables were kept constant. It is noted that at low frequencies, the structures with the lower mooting stiffnesses provide more


effective wave barriers due to their resonant peaks in this region. The location of the resonant peak (reflection coefficient maximum) can be adjusted by varying the mooring stiffness, although this may result in a very narrow frequency band of acceptable breakwater performance in this region. The structures with lower mooring stiffnesses (Klpgd=O.O16, 0.08), being more dynamically sensitive, exhibit sharp maxima and minima over the frequency range of interest while the higher stiffness structure (K/pgd---O.16) exhi- bits a more uniform reflection coefficient with frequency. The highest stiffness structure, corresponding to K/pgd--0.32, is seen to be inferior to the lower stiffness structures for kd > 1.75. However, as the stiffness increases further, the reflection coefficient minimum at kd=2.5 becomes more pronounced and the reflection coefficients at the higher frequencies (kd >3) increase and eventually approach those for Klpgd=~ seen in Fig. 3.

Figure 8 presents a comparison between the reflection coefficient for a dual pontoon breakwater structure and that of an equivalent single pontoon structure of draft b and width equal to (4a+2h). Results corresponding to dimensionless mooring line stiffnesses K/pgd=O.Ol6 and 0.08 are presented. As intuitively expected, the dual pontoon breakwaters possess higher natural frequencies than the corresponding single pontoon structures. Indeed, for a given mooring stiffness, the dual pontoon structures can be seen to be more

e- G) ,m

c,) ii

G) 0 0

t-

O i m

o o q,.

e cC

1 . 2

t , , , , I , , , 1 1 , , , 1 1 , , , , I , , , , I , , , , I , , , , I , I , , t

1 | , ,to o o ~ Q ~

L °I 0.8

0.6

0.4

0.2

0

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4

D i m e n s i o n l e s s W a v e n u m b e r kd

Fig. 8. Comparison of reflection coefficient for dual pontoon structure(lines) and single pontoon structure (symbols) of draft b and width (4a+2h) for d/a=5, b /a=l , h la=l , kc,/a=l, p*---0.25, and 0°=45 °.

Notations: (O) Klpgd=O.O16; - - - - - - ( e ) Klpgd=O.08.


efficient wave barr iers in both the lower- f requency (kd < 0 . 7 5 or kd < 1 . 0 ) and mid-frequency ( 1 . 5 < kd < 3 . 0 or 1 .75< kd <2 .75 ) ranges.

5. CONCLUSIONS

The hydrodynamic proper t ies o f a dual pontoon floating breakwater consis t ing of a pair o f f loating cyl inders of rectangular section, connected by a r igid deck have been investigated theoret ical ly . The structure is restrained by l inear symmetr ic moor ings . The fluid mot ion has been ideal ized as l inearized, two-d imens iona l potent ia l flow and the equat ion of mot ion o f the breakwater was taken to be that o f a two-d imens iona l r igid body moving in surge, heave and pitch. The solut ion for the fluid mot ion was obta ined by the boundary integral equation method using an appropr ia te G re e n ' s function. Resul ts have been presented which i l lustrate the effects o f the different wave and structural parameters on the eff iciency of the breakwater . It has been found that the wave reflection propert ies o f the structure depend s trongly on the draft and spacing of the pontoons and the moor ing line stiffness, while the excess buoyancy o f the sys tem is of lesser importance.

Acknowledgements--Partial support for this study was provided by the University of Houston, Limited-Grant in Aid Program 1994-95. This support is gratefully acknowledged.

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Dual pontoon floating breakwater