Wave effects on large floating structures with air cushions

ABSTRACT

An analysis is made of a VLFS which is par-tially supported by an air cushion. The interface between the air cushion and the water is considered to be a free surface. The elevation of this surface is represented by an appropriate set of Fourier gen-eralized modes, and extended equations of motion are derived for the rigid-body motions and gener-alized modes. The dynamic effect of the air is rep-resented by appropriate acoustic added-mass coef-ficients. The hydrodynamic coefficients are evalu-ated using the B-spline based panel code HTPAN. Illustrative computations are presented which show significant resonant effects.

1. INTRODUCTION

One of the most important challenges in the de-sign of very large floating structures (VLFS) is to achieve the required overall length without incur-ring large wave-induced loads. Some designs pro-posed for floating airports have included massive breakwaters to protect the structure from incident waves. Most concepts for mobile offshore bases con-sist of several relatively short modules with flexi-ble connections. A design proposed by Norwegian Contractors uses a single hull with long cantilever extensions to achieve the required deck length.

Pinlcster et al. (1997, 1998) describe an inter-esting alternative where the structure is supported partially by air cushions. This is achieved by the use of side and end walls extending vertically to a sufficient depth to retain the interior air, which has

a positive pressure relative to the atmosphere to

provide static support. Assuming a uniform pres-sure in the air chamber, the wave-induced moments and structural loads will be substantially reduced by comparison to a conventional hull of the same

dimensions. The results presented by Pinkster et

al. include computations of the body motions and air pressure, based on a low-order panel method,

Cambridge, MA 02139, USA E-Mail:

chleeArainbow.mitedu and jnn@mitedu

as well as supporting experiments. However their results are limited to cases where the length of the vessel is less than 3-4 wavelengths (L/) <3 4).

For VLFS applications it is necessary to consider the regime where L/A > 10. It is not practical to use the low-order panel method, due to the large number of panels required in this regime, and the assumption of a uniform pressure distribution is questionable.

In this paper we consider the example of a rect-angular barge with one pressurized air chamber. The hydrodynamic analysis is performed using the

higher-order panel program HIPAN described by Lee (1997). The motion in the air chamber is rep-resented by an eigenfunction expansion. The

el-evation of the interface is represented by an

ap-propriate set of Fourier generalized modes, with unknown amplitudes. Imposing the condition of pressure continuity across the interface leads to an extended set of 'generalized equations of motion' including the conventional rigid-body motions and the interface Fourier modes.

Since a set of specific generalized modes are pre-scribed for the elevation of the interface, conven-tional Neumann conditions specify the normal ve-locity on the interface. Thus the interface and the submerged body surface can be combined to form

one 'global body surface' with the same type of

boundary condition. Green's theorem is used to

solve for the velocity potential on this global sur-face. This approach is somewhat indirect from the physical standpoint, and can be contrasted with an approach where the pneumatic pressure is specified directly on the interface. That 'direct' approach leads to a modified integral equation for the

un-known potential on the submerged structure sur-face only

These two approaches are described further by Lee, Newman & Nielsen (1996) in their analysis of an oscillating water column device for wave-power

conversion. The advantages of the indirect

ap-proach are (1) it can be implemented numerically without modification of the panel program, which

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WAVE EFFECTS ON LARGE FLOATING

STRUCTURES WITH AIR CUSHIONS

C.-H. Lee and J. N. Newman

accepts arbitrary generalized modes; (2) the use

of a relatively small number of orthogonal Fourier modes is more efficient computationally than the need to integrate the pressure accurately over the domain of the interface; and (3) poor condition-ing of the linear system correspondcondition-ing to the dis-cretized integral equation, in the vicinity of physi-cally relevant resonances, is transferred to the much smaller system of equations of motion. On the

other hand, the direct approach restricts the com-putational domain of the unknown potential to a smaller boundary surface, and it is more easily un-derstood.

Our method is first applied to the model used by Pinkster et at (1998), shown in Figure 1 (left), to permit comparison and verification of the results. We then analyze the prototype barge shown in Fig-ure 1 (right), with a length of 1500m and other di-mensions selected to be representative of a mobile offshore base. For this prototype the air cushion

supports 55% of the total displacement, and the

remainder is provided by the buoyancy of the ver-tical walls which contain the air chamber

Two obvious sources of resonance exist for the motions of the air/water interface, sloshing due to standing waves in the water and acoustic standing waves in the air chamber. The resonant frequen-cies can be estimated easily for a rectangular air chamber with vertical side walls. Since the struc-ture is much longer than the hydrodynamic wave-length, sloshing can be expected for several adja-cent wave periods of spectral interest. Acoustic res-onance may also occur within the range of relevant wave periods for a structure as large as a VLFS.

Figure 1: Perspective views of the submerged portions of the model used by Pinkster et at (left) and the prototype VLFS (right). The view is from below, with patches outlined by heavy lines and panels by light lines. Part of the air/water interface is visible inside the vertical walls. Pertinent dimensions of the model are length 2.5m, beam 0.78m, and draft 0.15m to the bottom of the walls. The wall thickness is 0.02m at the ends and 0.06m at the sides. The air chamber extends from 0.05m below the exterior free surface to 0.13m above. The corresponding dimensions for the prototype are length 1500m, beam 150m, draft 32m, wall thickness 20m, with the air chamber extending from 16m below the free surface to 20m above.

For periods of 6-9 seconds the acoustic wavelength is 2000-3000m, and the first half-wave resonance will occur in a chamber of length 1000-1500m.

The possible importance of acoustic resonance was one of the motivations for this work, suggested by the analogous 'cobblestone effect' experienced by surface-effect ships. This phenomenon is de-scribed by Nakos et at (1991) and Ulstein & Faltin-sen (1996). The preFaltin-sent computational results for the prototype VLFS indicate that the most impor-tant resonant response in heave can be predicted in a similar manner as in the simplified analysis of Nakos et al, with the inertia force balanced by the sum of the hydrostatic and aerodynamic stiffness coefficients, which are of comparable orders of mag-nitude. For pitch the most significant resonance is due only to the moment of inertia of the body and

the hydrostatic restoring moment, as in the case

of a conventional slender ship. The heave resonant frequency is substantially larger than for pitch.

The analysis which follows is based on linearized potential theory. Plane progressive waves of fre-quency ca move in the longitudinal direction. The unsteady motions of the fluid, structure, and air are oscillatory at the same frequency. The fluid depth is assumed to be infinite. For simplicity we con-sider only head seas. Thus the rigid-body motions

include surge, heave, and pitch, and the Fourier modes are restricted to be symmetrical in the trans-verse direction. After outlining the method of anal-ysis, computational results will be presented and discussed.

2. FORMULATION

We consider a rigid body floating on the free sur-face, partially supported by the pressure in a

rect-angular air chamber (a <x < a,b <y < b,zi <

z < z2). Here 2a is the length, 2b the width, and

z2 z1 the height of the air chamber, z 0 is

the plane of the undisturbed free surface outside the body, and z = z1 <0 is the mean position of the interface. The static pressure in the chamber

is po =

Pw9Z1 > 0, where pu, is the density of water and g denotes gravity. The subscripts a and w are used to designate the air/water densities, re-spectively. For other parameters and variables up-per/lower case symbols are used. The sides, ends,

and top of the chamber are fixed with respect to

the body.

The fixed surface of the air chamber is denoted

by Sc and the wetted surface of the body by Sb

The air/water interface is Si. The complete closed

surface bounding the air chamber is S. = Sc +

S. The complete boundary surface of the water is = Sb+ Si, and the free surface outside the body. The motions of the air and water, are represented by the velocity potentials

Re {0 (x, y, z) et} and Re {cb(x, y, z) et}

(1)

The time-dependent factor et is assumed implic-itly hereafter. These potentials are governed by the Helmholtz and Laplace equations,

v2,1, + K2 _{= 0}

_{and V20 = 0}

₍₂₎ which are valid throughout the corresponding phys-ical domains. Here K wk.° is the acoustic wavenumber and co is the velocity of linearized

sound waves. From the linearized Bernoulli equa-tion the pressures are

P(x,Y,z) = ipaw4)(x,y,Z)

(3)

p(x, y, z) = ip,wd)(x, y, z)

pwg(z +c d)

(4)

The aerostatic pressure pag(z+cd) is neglected

on the assumption that co >> glw.

The elevation of the interface can be represented by the superposition of the following modes: (a) uniform vertical motion with amplitude equal to

the heaving motion of the body, (b) the vertical

component of the rigid-body rotations in roll and pitch, and (c) a complete set of orthogonal Fourier modes

Cmn(r, Y) = cos urn,/ (cos vnY) (5)

(sin unix sin vnY

where

M7r

=

Vn = (6)

2a 2b

and the integers m and n are even or odd, respec-tively, for the modes proportional to the cosine or sine. Each Fourier mode is associated with a cor-responding pair of velocity potentials q5j and (Di

Li > 7) which are appended to the six radiation potentials (j = 1 6) for the rigid-body motions. The explicit relationship between (m, n) and

j is

not important, as long as the expansion is com-plete. Thus, for 7 <

j < co

we require that all combinations of m and n are represented in the

ranges 0 < < 00, 0 < n < oo. (In the numerical implementation to follow, the upper limits of these ranges are truncated at sufficiently large finite val-ues to demonstrate convergence of the results.) It is convenient to use the notation (i in place of (inn. The elevation of the interface, relative to its static level, is

00

c=Ev.;

(7)

j=1

The potentials can be expanded in the forms CO

=

e, 4)3 ₍₈₎

j=1

co

0 = OD + icuE&.;

(9)

Here ei (j = 1

6) are the amplitudes of the six rigid-body motions and ei (j > '7) are the ampli-tudes of the Fourier modes for the interface eleva-tion. The kinematic boundary conditions on S. for

j= 1

6 are

(Din = Nj on Sa (10)

where (Ni, N2, N3) = N is the unit normal vector pointing out of the air chamber, and (N4,N5,N6)= x x N. For the generalized modes (j > 7) we define

= (3(x,y) on Si, N3=0 on Sc

(11)

Similarly for the fluid domain,

q5jh = on (12) where

(ni,n2,n3) = n

(13) (n4, n5,726) = x x n (14) and, for j > 7,

ri3=(3(x,y) on Si,

= 0 on 5b. (15)

The potential OD is the solution of the diffraction problem where incident waves are present and there is no motion of the body or interface. Thus

Note that this is not the complete solution of the physical diffraction problem, where the interface is free to move; that solution will include appropriate contributions from the modes j > 7.

Equating the air and water pressures on the in-terface gives the dynamic boundary condition

ipwcaik

pg(

ipacoto (17)

After using the kinematic condition Oz = icd( to eliminate C, in the same manner as for a conven-tional free surface, it follows that

Pw(w20 90z) = Ai(d2'D011 Si (18)

This is analogous to the modified free-surface con-dition which is applicable when an oscillatory pres-sure is imposed on the free surface (cf. Wehausen & Laitone, 1960, equation 21.2).

On the free surface outside the body the conven-tional free-surface condition is applicable, equiva-lent to the homogeneous form of (18). In addition to these boundary conditions, each potential Oi and the scattering component of OD satisfy the radia-tion condiradia-tion of outgoing waves in the far field, and vanish at large depths.

3. EXTENDED EQUATIONS OF MOTION

The conventional equations of motion stipulate that the six components of the force and moment due to the air and water pressures should be equal to the inertial force and moment of the body. The integrated pressure force and moment acting on the body will include contributions from the hydrody-namic pressure acting onSb, and from the air pres-sure acting on Sc. It is convenient to add the total hydrodynamic pressure force and moment acting on Si, and the corresponding air pressure force or moment on the same surface, since these are equal and opposite. This permits us to define force co-efficients which are consistent with the extended normal vectors ni and

The six hydrodynamic contributions are

fi=

f

nipdS = Xi+ E ;(w2. _ iwk; _ C.ij)

i=1

(19)

where i = 1, 2, 3, 4, 5,6 and the coefficients in this equation are defined as follows:

X, = ihopw

f

no5DdS (20)

(i/w)bii = p,,,

fs.ndS

(21)

and

cii = pwg

f

ninjdS (i = 3,4, 5) (22)

These are, respectively, the coefficients of the ex-citing force, added mass, damping, and hydrostatic restoring force (and moment).

The analogous expressions for the aerodynamic force and moment are

=-

f

NiPdS =i2E ajAij

(23)

j=1

where

j = pa

f

NiPidS

s.

is the acoustic added-mass coefficient. (As will be shown in the Appendix, the acoustic potentials are real and there is no analogous damping coefficient. This can be anticipated phySically by noting that the air chamber is closed, with no energy radiation or dissipation.

Assuming the body is free to respond in each mode of rigid-body motion, with the body mass

distribution represented by an inertia matrix six conventional equations of motion follow in the form

co

E[w2

C.ii)] =

i= 1

for (i = 1 6).

An extended set of equations of motion including the generalized modes is derived from the dynamic boundary condition on the interface. If this con-dition is multiplied by Gkin = n2 (i > 7), and inte-grated over the interface, a set of linear equations for the modal amplitudes e; follow in the form

Ej [ca2(a22 + Aii)+icabi; + C.ii )1 = Xi (26) j=1

for (i > 7). The definitions of these coefficients are unchanged, except for the extended range of the

index ti.

The generalized equations of motion, including (27) and (28), can be solved by truncation and stan-dard linear algebra to obtain the modal amplitudes j. The hydrodynamicforce coefficients in this lin-ear system can be evaluated by a panel method such as WAMIT or HIPAN. The only non-standard co-efficients required are the added-mass coco-efficients 44,3 . These are derived in the Appendix.

a 18 16 14 12 mY10

4. COMPUTATIONAL RESULTS

Our computations are based on the higher-order B-spline code HIPAN, described by Lee (1997). One quadrant of Su, is represented by six rectangu-lar patches on Sb and one patch on Si. The geome-try is described exactly. Cubic B-splines are used to represent the velocity potential on each patch. The patches are subdivided into panels, with B-spline knots at the intersections of adjacent panels.

We first consider the model with one air

cham-ber used by Pinkster et al (1998) for their

free-floating model tests, with the dimensions indicated

in the caption of Figure 1. The water depth is

2.5m, the center of gravity is at (0, 0, 0.15) and the pitch radius of gyration about this point is 0.751 meters. The total number of panels used in these computations is 150 and the corresponding num-ber of unknowns (i.e. the number of lEl-spline co-efficients) is 441. The elevation of the interface is represented by 16 Fourier modes in the longitudi-nal direction and 2 symmetric Fourier modes in the transverse direction. The computational results are estimated to converge to two significant digits. Fig-ure 2 shows the heave and pitch response computed by the present method, at 100 closely-spaced fre-quencies, and comparisons with the numerical and

experimental results of Pinkster et al. The

com-putational results agree quite well, except for rel-atively small differences in the vicinity of the

res-10 0 2 4 6 8 10

(1)

Figure 2: Normalized heave (left) and pitch (right) response amplitude operators of the model used by Pinlcster et at (1998). The solid lines are the computational results from the present analysis. The dotted lines and square symbols are the computational and experimental results of Pinkster et al (1998). 6 is normalized by the incident wave amplitude A, and 6 is normalized by AIL.

onant peaks. These are assumed to be caused by the limitations of the low-order panel method used by Pinlcster et al (1998).

Next we consider the prototype barge described in Figure 1. The center of gravity is located at

(0, 0, 0) and the radius of gyration for pitch is

as-sumed to be 400m. The patch definition and B-spline order are the same as in the paragraph above, but the number of panels is increased to 182 and the corresponding number of unknowns is 596. The number of Fourier modes in the longitudinal

direc-tion is increased to 64. A total of 225 wave

fre-quencies are used to define the oscillatory features of the results.

The heave and pitch RAO's are shown in Figure 3, and compared with a conventional barge with the same horizontal dimensions and displacement. For the low-frequency (long-wave) limit, the air-cushion barge maintains static equilibrium relative to the incident wave and it behaves as if the body and the air Chamber are a single rigid structure without mo-tion of the interior free surface relative to the body. Thus the heave and pitch RAO's approach to the amplitude and the slope, respectively, of the inci-dent wave. For intermediate frequencies the RAO's are oscillatory in a similar manner for both vessels, but the peaks are amplified by sloshing in the case of the air-cushion barge.

co

Figure 3: Heave (left) and pitch (right) response amplitude operators for two barges of length 1500m and beam 150m, normalized as in Figure 1. The solid lines are for the prototype with air chamber, and the dashed lines are for a conventional rigid barge with the same displacement.

02 0.4 0.6 08 0 02 0.4 0. 0.8

02 0.4 0.6 0.8 02 0.4 0.6 0.8

(0

Figure 4: Heave exciting force (left) and pitch moment (right) for the two barges in Figure 3. These results are normalized by the hydrostatic limit pgALB and by pgAL2B112' where L =length and B =beam.

Figure 3, for heave at w = 0.595 (period 10.6 onds) and for pitch at w = 0.336 (period 18.7 sec-onds). These resonances can be explained by noting that, since the walls are relatively thin and deep, the dominant pitch moments are due to the body inertia and to the hydrostatic restoring coefficient

associated with the walls. This is essentially the

same as for a conventional slender floating body, except that the body inertia is about 2.2 times the displaced volume of the walls (due to the

hydro-static support of the air cushion). Thus the

natu-ral frequency can be approximated from the simple equation con

(02.2r1/2

) = 0.37 where T is the

draft of the walls. The error in this estimate can

be attributed primarily to neglect of the added mo-ment of inertia due to the walls, and to the small motions of the interface. For heave the hydrostatic restoring coefficient is augmented by the effective

Stiffness of the air chamber, and the natural

fre-quency is increased relative to pitch. This expla-nation is not complete, however, since there is sub-stantial vertical motion of the interface relative to the body, and it is necessary to consider the cou-pled two-degree-of-freedom system j =3 and j =7

in order to confirm the heave resonance. (At the

heave resonant condition, the RAO for the mode

j = 7 is almost twice as large as for j = 3, and

with opposite phase.)

Figure 4 shows the heave exciting forces and pitch moments for the both vessels. The low-frequency limits of the exciting force and moment are reduced by the air cushion, relative to the

con-ventional barge. At intermediate frequencies the

force and moment are oscillatory in a similar

man-ner for both vessels, due to diffraction. The os-cillations of the RAO's in Figure 3 are correlated with these diffraction effects. A small local peak is noticeable in the pitch exciting moment at the half-wave acoustic resonant frequency w = 0.74, but this does not have a significant effect on the pitch RAO. The elevation of the interface is shown in Figures 5 and 6, plotted as a function of w at six fixed points in Figure 5 and conversely in Figure 6. The first two frequencies shown in Figure 6 correspond to the resonant peaks of heave and pitch in Figure 3, and the others correspond to the three highest peaks of Figure 5. Only the symmetric modes are relevant at x = 0, as indicated in the bottom curve of Figure 5 which includes only half of the peaks shown in the other curves. The normalized elevation approaches 1 for long waves, and peaks are obvious at the reso-nant frequencies for sloshing modes. The elevations for the three higher frequencies shown in Figure 6 are clearly similar to standing waves.

5. CONCLUSIONS

We have analyzed the motions of a prototype barge with dimensions similar to a VLFS, and with partial support from an air cushion. The method is based on matching of acoustic modes in the air chamber with the hydrodynamic solution below the air/water interface. The acoustic problem is solved

in the rectangular air chamber by the method of

separation of variables. In the fluid, the

three-dimensional higher-order panel program HIPAN is

used. The displacement at the interface between the air and water is expanded in Fourier modes

whose coefficients are obtained from the kinematic and dynamic conditions at the interface.

For the relatively small vessel analyzed by

Pinkster et al (1998) our computational results are consistent. This favorable comparison, together with their experimental results, support the valid-ity of the present computational methodology. For the prototype vessel with dimensions more relevant to a VLFS convergent results have been obtained which account for both the short-wavelength effects of the water wave diffraction and radiation, and for nonuniform acoustic pressures in the air chamber. The former effects are qualitatively similar to those associated with a conventional rigid barge of the same horizontal dimensions and displacement, but the occurrence of sloshing modes in the water below the air chamber amplify the frequency-dependent variations in the RAO's and exciting forces.

The initial motivation for this work was the possi-ble importance of the half-wave acoustic resonance, which occurs for the very long VLFS within the

practical range periods of ocean waves. The

com-putational results do show this effect as a small resonant peak of the pitch exciting moment, but

there is no significant effect on the pitch amplitude due to the relatively high frequency. The lack of a strong resonance in this mode can be attributed to the large discrepancy between the wavelengths of the acoustic and wave waves.

The most significant resonant peaks in heave and pitch are primarily due to the balance between the

body inertia, hydrostatic restoring, and the

stiff-ness associated with the change of volume in the air chamber. The oscillatory air pressure only af-fects the heave response, in a manner analogous to that of cobblestone oscillations for air-cushion ve-hicles. At these resonant frequencies the hydrody-namic damping is small, and the large amplitudes of response are cause for practical concern. Viscous damping may help to attenuate these peaks, but probably not sufficiently to overcome their practical consequences. Further developments including the

5 4 3 2 1 0 4 3 2 1 5 4 3 2 1 0 5 4 3 2 5 7 4 3 2 0-5 4 3 2 0 x=-584 x=-292 0 0:2 0.4 - 0.6

Figure 5: Amplitude of the air/water interface ele-vation normalized by the incident wave amplitude

A. Six positions along the centerline are shown, with values of the longitudinal coordinate x in me-ters. 0 -5 10 -10 og.693 op0.735 %."

Figure 6: Real (solid lines) and imaginary (dashed) parts of the elevation along the centerline of the interface at the indicated wave frequencies w.

0.8

considerations of passive and active damping may be useful in this context. Except for these peaks the motions of the air-cushion barge are substantially the same as for a conventional rigid monohull, and

the structural loads are expected to be relatively

small. It remains to compare the structural loads themselves, which requires a straightforward exten-sion of our analysis.

From the practical design point of view, it may be necessary to compartmentalize the air chamber to increase the static stability in pitch and roll. The extension to two compartments with a longitudinal subdivision is included in the work of Pinkster et al (1998). It is straightforward to extend the present method to this configuration. With increased sub-division the structural wave loading will be greater, unless some type of pneumatic control system is in-troduced to partially equalize the pressures in dif-ferent chambers. An important consequence of sub-division is that pitch motions will be accompanied by (antisymmetric) changes in the volume of the air chambers, and thus the pitch response will be more similar to that of heave.

Another practical concern is the large amplitude of resonant sloshing modes at the air/water inter-face. An amplification factor of 10 is indicated by

the bottom curves in Figure 6, at a wave period

of 8.5 seconds where significant wave energy can be expected. Viscous effects may attenuate this

resonance somewhat, but it may be necessary to use special hydrodynamic or pneumatic dampers to keep the resonant modes within acceptable levels.

The concept of a VLFS which is partially sup-ported by air cushions is a very interesting alterna-tive to others which have been considered more ex-tensively. Extensions of the present analysis would

be appropriate to provide a more complete and

practical appraisal of this concept.

ACKNOWLEDGMENT

This work was supported by the Office of Naval Research, Grant N00014-97-1-0827, under the di-rection of the Naval Facilities Engineering Service Center. The development of HIPAN was supported by a Consortium including the Chevron Petroleum Technology Company, David Taylor Research Cen-ter, Exxon Production Research, Mobil Oil Com-pany, Norsk Hydro, Offshore Technology Research Center, Petrobras, Saga Petroleum, Shell Develop-ment Company, Statoil, and Det Norske Veritas. Professor Pinkster kindly provided data files for the results shown in Figure 1.

REFERENCES

Lee C.-H. 1997, 'Wave interactions with huge float-ing structures,' 8th International Conference on the Behaviour of Offshore Structures, Delft, The Netherlands, 2, 253-265.

Nakos, D. E., Nestegard, A., ULstein, T, and

Sclavounos, P. D. 1991, `Seakeeping analysis of surface effect ships,' FAST`91 Conference, Trond-heim, Norway.

Lee, C.-H., Newman, J. N. & Nielsen, F. G. 1996. 'Wave interactions with an oscillating water col-umn,' International Offshore and Polar Engineer-ing Conference (ISOPE-96), Los Angeles. Pinkster, J. A., 1997. 'The effect of air cushions

under floating offshore structures,' 8th Interna-tional Conference on the Behaviour of Offshore Structures, Delft, The Netherlands, 2 143-158.

Pinkster, J. A., Fauzi, A., Inoue, Y. & Tabeta, S., 1998. 'The behaviour of large air cushion supported structures in waves,' 2nd International Conference on Hydroelasticity in Marine Technol-ogy, Fukuoka, Japan, pp. 497-505.

Ulstein, T., & Faltinsen, 0. M., 1996. 'Cobble-stone effect on surface effect ships,' Schiffstechnik / Ship Technology Research 43, 78-90.

Wehausen, J. V., and Laitone, E. V., 1960. 'Surface waves', Encyclopedia of Physics 9, 446-778.

APPENDIX

Added-mass coefficients A,,j The acoustic added-mass coefficients (26) are

derived from the appropriate solutions of the

Helmholtz equation which satisfy the boundary

conditions (10). Since the domain is rectangular, the method of separation of variables can be used.

For the rigid-body modes (i < 6, j <6) it is con-venient to use an indicial notation in conjunction with a coordinate system where the origin is at the centroid of the air chamber. Thus

= (x, y, z z2 + c) (27) where z2 z1 = 2c is the height of the chamber. In addition we define the parameters

= (a, b, c) nir cti(n) =

ki(n) = V K2 - a(n)2 (30)

where (i = 1, 2,3) in all cases and the cyclic con-vention applies. The explicit dependence of ai and ki on n will be implied hereafter.

For i = j = 1, 2,3 the potentials and added-ass coefficients are easily derived in the forms

sin Kxi K cos Kai

tan Kai Aii = 8Pciai+rai-r

The potentials for the rotational modes are more complicated. To satisfy the boundary condition (10) it is helpful to replace j in (12) by aiq, where the normalized variable q is in (-1,1), and to use the Fourier series

co

. Sirig

ll

h2 2

)

71=1 (n odd)

where sa sin(nr/2). The potentials (4)4, d's, (Ds) then can be derived in the form

8 sr,

=

_.L

n=1 (nodd)

ai+i sin ai+iii.+1 sin ki+1±.2,-1

ki+i cos ki+-iaz-i

aa,..4 sin 1 sin kt _ it +1 ) ki_1 COS

and the corresponding added-moment coefficients in the I, reference frame are

A = ypnabC64i+3,i+3

E

-1-n=i

(nodd)

[

1 1 1 ) tan ki+i ch-i

k2 a? 2 ) i+1 ki+rai--1 1 1 1 tan ki-lai+1 , ,_ - _ 35 + lq_i a ?_/ I C1 ISi= r a i +-1.

For the generalized modes (j > 7) the potentials satisfy the boundary conditions (I)j N = 0 on Sc and (1)3. = (3(x,y) on Si, where ci is defined by (5).

The appropriate solutions are

8 = it2 (33) (34) where 0, C S Wj (Z2 Z) (1):7 (X Z) (XI Y) wi sin 2w3c w V K2 v2 n (36) (37)

For the 'pumping mode' (j = 7), m = n = 0 and the acoustic pressure is independent of the horizon-tal coordinates. If Kc << 1 this pressure is equiva-lent to the time-varying air pressure in the analysis of Pinkster et al. The solutions for j > 8 represent acoustic standing waves in the chamber.

For (i > 7) and (j > 7) the only contributions to the added-mass coefficients are front the interface Si. Since the functions ci are orthogonal, the only nonzero coefficients are

Aii = -paca, = -p

cot 2w1c f ,2 ,

_em

cot 2wic

apen

wi S, Wi

(38)

Here eo = 2 and em = 1 (m > 1), and the indices m, n correspond to the Fourier modes (5).

For (i < 6) and (j > 7) the added-mass coeffi-cients can be evaluated by integration of the poten-tials (38) over the surface Sa For the surge force (i = 1) the only contributions are from the modes where m is odd and n = 0:

4pab r = 2 S /un0

wj

where 5mn is the Kroenecker delta function. Simi-larly for the sway force i = 2 the only contributions are from the modes where m = 0 and n is odd:

4Pa a

A.2i = rSnum0

Wi

The only contribution to the heave force is from the pumping mode:

A37 = 4pnaotanKc (41)

K 2

The moments are

(1

-

4w )1 8,a0 Vn A4 = 4paa _n [Z1 + tan w3c W 4pa 1 A63 = T SmSn_W u2_m v2 n

The remaining elements (i > 7) and (j < 6) can be evaluated directly, but it is simpler to use the symmetry relations Aii Aji. Except for the co-efficients Aii Which are explicitly evaluated above, and the reciprocal coefficients Aii, the off-diagonal elements A3 (i j) are equal to Zero when i > 7 and/or j > 7. In all Cases the potentials and added-mass coefficients are real.

(39) (40) 4pab

A53,=--sm

w4 1 tan wic 2

)]

(42) nO Zi [ wi (1