E L S E V I E R P I I : 8 0 1 4 1 - 1 1 8 7 ( 9 7 ) 0 0 0 2 9 - 1
All rights reserved. Printed in Great Britain 0141-1187/97/$17.00 + 0.00
Comparison of methods for computing
hydrodynamic characteristics of arrays of wave
power devices
S„ A. Mavrakos'' & P. Mclver'''*
'^Department of Naval Architecture and Marine Enghieering, National Tecimical University of Athens, 9 Heroon Polytecimiou Ave, 15773 Zografos, Atliens, Greece
^Department of Malitematical Sciences, Lougliborough Universiry, Lougiiborougii, Leics LEll 3TU, UK (Received 30 June 1997; accepted 10 October 1997)
A comparison of methods for the calculation of the hydrodynamic characteristics of arrays of wave power devices is presented. In particular, the plane-wave approximation and an exact multiple scattering formulation have been used to compute exciting wave forces, hydrodynamic coefficients and q factors for arrays of interacting wave power devices. The results obtained are compared with each other, and accuracy aspects of the computations are stressed and critically assessed. © 1998 Elsevier Science Limited. All rights reserved.
1 I N T R O D U C T I O N
Within the context of the linearised theory of water waves, a variety of methods have been devised for the calculation of hydrodynamic interactions within arrays of floating offshore structures. These methods have found application in many areas including the dynamics of offshore oil platforms and the design of floating airports. The application of most con-cern in the present paper is power absorption by an atTay of wave-power devices.
The techniques may be broadly classified into two classes. In the first are those methods that attempt to find an accurate solution without approximations other than those involved in the truncation of a numerical scheme. The second class use simplifying approximations, for exam-ple based on the dimensions of the array relative to the incident wave length, before a numerical solution is attempted. The use of an approximate method can give sig-nificant savings in programming and/or run time. Although a number of comparisons between methods have been made in the literature [1-4], a systematic evaluation does not appear to have been made for geometries appropriate to the design of a wave-power station. It is the purpose of the present paper to summarise the findings of a recent study made by the authors as part of 'The Offshore Wave *To whom correspondence should be addressed.
Energy Converter Project' funded by the European Union. The larger part of this study is concerned with a comparison between the 'plane-wave' approximation and a multiple-scattering method which can accurately account for hydro-dynamic interactions; some comparisons are also made between these methods and the 'point-absorber' approxima-tion. The aim is to identify the limitations of the two approx-imate methods.
The point-absorber method [5,6] is based on a weak-scat-tering approximation in which it is assumed that the wave-length is much greater than a typical device dimension. This assumption allows the gain in the maximum power that may be absorbed by an array, relative to the power that may be absorbed by an isolated device, to be expressed entirely in terms of the array geometry. The method might be used for calculating added-damping coefficients but, in general, it is not suitable for calculating other hydrodynamic parameters that might be needed to determine the detailed performance of individual devices.
The plane-wave method [2,3,7] is based on a wide-spacing approximation which assumes that the device spa-cing is many wavelengths. Hydrodynamic interactions are calculated using plane-wave approximations to scattered and radiated waves within the array and the method may used to examine all hydrodynamic characteristics of the array including exciting forces and non-optimal power absorption. These two approximate methods are compared
284 S. A. Mavrakos, P. Mclver with results from a multiple-scattering approach [8,9] in
which the interactions are calculated as a cascade of scat-tering events within the array. In principle, arbitrary accuracy may be achieved by including sufficient scatter-ings of the incident and radiated waves.
2 F O R M U L A T I O N
A device array consists of identical, wave-power devices each able to absorb power in M modes of oscillation. An oscillator index may be defined by
s = M(n-l) + m, n = \,2,---,N, m=\,2,---,M (1) where n is the number of the device and m is the mode of oscillation. For convenience, the oscillator index is refen'ed to as an array mode of oscillation; there are a total of M X A^ such modes.
The results are based on the linearised theory of water waves with all motions assumed to be time-harmonic with angular frequency co. A velocity potential $ exists and the time dependence may be extracted, in a standard way, by writing
* = Re< (2)
where Re indicates that the real part is to be taken. Here, the complex-valued potentials <^o ^nd describe the inci-dent and diffracted waves, while 0^ describes the radiated waves due to the oscillations in airay mode s. The complex quantities U^, s — 1,2,-••,M X N, are the velocity ampli-tudes for each array mode. The boundary value problems that must be solved to determine the potentials </)d and cj), are described in 7.3.1 of Mei [10].
Cylindrical polar coordinates {r,d,z) are used with origin O in the mean free surface and the z axis directed vertically upwards. The fluid is talcen to be of constant depth /). For an incident wave of amplitude A and wavenumber k propagat-ing in the direction d = P, the potential is
-L ' g ^ Jkr(S-l3)
"0- wcoshkh cosh^;(z + h) (3) The hydrodynamic forces on the devices may be calculated by integrating the pressure over the wetted surface of the device. These forces are time-dependent quantities but it is convenient to extract the time dependence in the same manner as was done for the velocity potential in eqn (2). For example, a time-dependent force is written in the form Re{Fe"'"'} where F is a complex quantity containing infor-mation about the magnitude and phase of the force. A l l forces discussed below have the time dependence removed in this way.
The exciting force in mode s due to the incident waves is
= mp (4)
of device n and Vm is the component of the generalised normal. The total hydrodynamic force on device n in direc-tion m due to the forced modirec-tion of the devices is
say, where
MXN
1=1
v.,.dS — ioi
(5)
4>,v„idS and b^t = wplm <i),v,„AS (6) are the conventional added mass and damping coefficients describing the force in the direction of array mode s due to unit velocity amplitude forced oscillations of in mode t. Here Im indicates that the imaginary part is to be taken. 2.1 General results on power absorption
A general theory for power absorption by a system of A' interacting wave-power devices was derived independently by Evans [5] and Falnes [6]. The total mean power absorbed by the system is found by calculating the mean rate of work-ing of the hydrodynamic forces and summwork-ing over the devices to get
^(U'X-F-X*U) -
\\]*B\]
il) where U = ([/,„} is the column vector of complex velocity amplitudes, X = {Z,„} is the column vector of exciting forces, B — {/?„„,) is the matrix of (real) damping coeffi-cients and an asterisk denotes complex conjugate transpose. In general, U is determined from the equations of motion for the individual devices. The first term on the right hand side of eqn (7) is the power taken from the incident waves while the second term is the power lost due to wave radia-tion by the oscillating bodies. This expression may be maximised in a straightforward way ([5], p. 236) and i t is found that the maximum power which may be absorbed isand that this is achieved when u =
IB^^X
(8)
(9)
where p is the fluid density, T„ denotes the wetted surface
It is assumed here that B ' exists. A mean gain factor q(|S) for the an-ay is defined as
q(/3) =
Maximum power that may be absorbed by the array
N X Maximum power that may be absorbed by an isolated device
(10)
I f there were no hydrodynamic interactions then the maxi-mum power that could be absorbed by each device would
be the same as i f it were i n isolation and q would be identically equal to unity for all incident wavelengths. In fact, for a given array of devices, the hydrodynamic inter-actions often result in a strong dependence of q upon wave-length.
3 S O L U T I O N M E T H O D S
Two principle methods, one exact in principle and one approximate, have been used to calculate the exciting forces and associated hydrodynamic parameters of the devices so that eqn (10) may be used to calculate the q factor for the array. These methods are fully described elsewhere, but for the benefit of the reader unfamiliar with the techniques involved a brief description of the solution for the scattering problem is given here. In addition, comparisons are made with calculations of the q factor using the point-absorber approximation and a brief description of this is also given.
3.1 The multiple scattering method
The first method, which accurately accounts for interference effects between the devices, relies on single-body hydro-dynamic characteristics, and describes the hydrohydro-dynamic interactions through the physical idea of multiple scattering. By superposing the incident wave potential and various orders of propagating and evanescent modes that are scat-tered and radiated by the array elements, exact representa-tions of the total wave field around each body may be obtained. As the boundary conditions on each body in the array are satisfied successively, there is no need to retain simultaneously the partial wave amplitudes around all the structures in the array as in direct matrix inversion methods [11,12]. As a result, a considerable reduction of the storage requirement in computer applications can be achieved. The multiple-scattering method was introduced by Twersky [13] to study acoustic scattering and radiation by an array of parallel circular cylinders, and was first applied to free-sur-face wave interactions with floating bodies by Ohkusu [14] who investigated the case of three adjacent, floating, vertical truncated cylinders. The method was extended by Mavrakos and Koumoutsakos [8] and Mavrakos [9] by introducing evanescent modes into the solution of the diffraction and radiation problems for an arbitrary number of vertically axisymmetric bodies that may be in any geometrical arrangement and have any individual body geometry. Since the method has been extensively described in pre-vious works [8,9], only a short oudine will be provided here. The velocity potential of the undisturbed incident wave field, given by eqn (3), can be recast in the form
<^o('„ö,,z;/3)= - / w A ^ ^ ^ ^ ^ | ± ^ 0 o ( r , , ö , , z ; / 3 ) (11)
where
im(B„-0)
(12) when referred to a coordinate frame with origin at body q. Here /,„ denotes the /?2th order Bessel function of the first kind and Q^qfioq) are the polar coordinates of body q rela-tive to the origin O.
In response to the excitation 0o> the body q £ {1,2,-• •,N} scatters its 'first-order' of scattering
,z)= — /coA
(13) where K„, is the ml\i order modified Bessel function of the second land and (Zj(z)^' = 0,1,---) are orthonormal func-tions in [ - /z,0] defined by / I " 1 - f sinlajh'
\
1 2 2ajh/
1/2 COSQ;^(Z-|-/7) (14) (15) Also, {a/J = 0,1,---} are the roots of0? + gajtanajh = 0
with «0 = — ik being purely imaginary. The first-order scattering coefficient 'f,,^^ in eqn (13) can be obtained through the solution of the diffraction problem around body q using one of the well-lcnown methods for this prob-lem. In the present work, the method of matched axisym-metric eigenfunction expansions has been used to obtain the unknown coefficients.
Next, all scattered waves of first-order emanating from the remaining cylinders of the array represent a 'second-order' of excitation for the initially considered body q. That is
p=i
(16) in response to which body q radiates its second-order of scattering such that the total second-order potential
(17) satisfies the imposed boundary conditions in the coordinate system of body q. The potential ^«^j*' is expressed formally through eqn (13) where the second-order scattering coeffi-cients should be introduced accordingly.
To solve the second-order scattering problem, it is neces-sary to express the scattered wave components ' </)^^ in the coordinate system of body q. Apphcation of a Bessel func-tion addifunc-tion theorem (Abramowitz and Stegun [15], p.363) gives
'<^^\r„ z) = - i.A _ _ X I ^ Z , ( z ) e ' I m ( a / ' > ) . n(a,ör/)
286 S. A. Mavrakos, P. Mclver with
N
p = l
X nioijlpq)
is the radius of body q and /^^ denotes the inter-body spacing. Proceeding in a similar way, the ^th order of inter-action may be obtained and then summing over all orders gives the total wave field, including both incident and scat-tered components, around the body as
CO 00
<l>^''\r„d„z)^-io^A X ' " ' I
HI = - CO ^ = 0
X
""JUaja^)^ '"^K^iaja,) Zj(z)e" where
s=l s=l
and the alternative notation
giWopCos(9„^)g-im
(20)
(21)
(22) has been introduced, in accordance with eqn (11), to express the incident wave in the coordinate system of body q. The potential in eqn (20) may now be used to calculate hydrodynamic forces.
3.2 The plane-wave method
The second main method used in this paper is essentially a wide-spacing approximation in which, when calculating the hydrodynanuc interactions, evanescent modes are neglected and non-planar outgoing waves are approximated by plane waves. These approximations mean that once the scattering and radiation problems for an isolated device are solved the hydrodynamic interactions may be accounted for in a very straightforward way. The method was devised [7] to analyse the performance of arrays of wave power devices able to absorb power in heave and then extended in a non-trivial way to allow consideration of horizontal motions and forces [2,3].
The essence of the approximation can be explained by considering a plane wave incident on device q from a direc-tion dq = X with potential
Mrq, dq,z) ^Rcoshkiz + /ï)e''^'-«(''' (23) where Si is an arbitrary complex number. I f evanescent
modes are neglected, this w i l l produce a diffracted field
'cl>f=Rcoshk(z + h)
X
fI^^"H„,(^r,)e""(^'
-X)--Jicoshk{z + hy4>f (24)
say. (For an axisymmetric device, the coefficients F^^l are independent of the angle of wave incidence and should be calculated by an accurate numerical method.) This out-going wave from device q may be expressed in terms of the coordinates with origin at device p using an addition theorem for Bessel functions (Abramowitz and Stegun [15], equation 9.1.79) which gives
CO
B^(kr,W""^ = ^""^'' X H„+„,(«,,)J„(fo-,)e-'"^^'-^-'
« = - co
(25) The plane-wave method assumes that the devices are widely spaced relative to the wavelength so that kl^p > 1 for q i= p. This assumption allows the Hankel function on the right-hand side of eqn (25) to be approximated by its asymptotic expansion for large argument (Abramowitz and Stegun [15], equation 9.2.7) so that
H„ + m(W,^) = Ho(W,^) X , i ^ ( 4 ( m + n ) ^ - l ) / ^ Q ^ , ^ ^ - 2 ; | ^ - , M . + ,o/2 8/c/, IP where ' TTWi 1/2 (26) (27) A combination of these results, together with the Bessel function series representation of a plane wave given in eqn (11), gives an approximation to the wave impinging on device p in the form
' - Spq(x)e*^^^°^^^'' - + Dpq(/>, dp X) (28)
where the first term is a plane wave with amplitude propor-tional to
Spq(x) = Ho(W,p) X F\$,& i'n(.e,„-x)
( W - l)i IP
(29) and Dpij is a non-planar correction term. Note that 5„„ = 0(klp-"^) while D^, = 0(ldp"q^'^) so that, in the vicinity of device p, the leading approximation to an out-going wave from device ^ is a plane wave travelling along the line dp = d^p.
The scattering problem is solved as follows. The depth-independent part of the incident wave (eqn (12)) from out-side the array is written in terms of the coordinates with origin at device q as
Ur,, e,, z; (3) = e"°'=°^('»'" «e'^^'^"'" ^> = L.e^'^^^^^'" (30) say. Also incident on device q are the waves scattered from the other devices in the array. A l l waves incident on device 5 as a result of scattering from device p are approximated
by the plane wave
Cgpe^'^cosie^-dpg) (31)
where the amplitude Cgp is to be found. In turn, this wave arises from the scattering of all plane waves incident on device p, that is c^p is the sum of the amplitudes of the approximating plane waves scattered from device p and propagating in the direction of device q. Thus, using eqn (28),
N
Cgp = LpSqpiP) + X CpjSgpidjp) (32) j*p
where the first term on the right-hand side arises from the incident wave from outside the array. This equation is valid for all values of p and q,p ¥= q, and so a set of simultaneous equations is given for the plane wave amplitudes. It is consistent to neglect the correction term Dp^ here, but it should be included when calculating wave forces [3]. 3.3 The point-absorber approximation
The main assumption behind the point-absorber approxima-tion is that each device is sufficiently small, relative to the wavelength of the incident waves, for scattered waves to be neglected when calculating hydrodynamic interactions. From eqns (8) and (10) it can be seen that the q factor is determined by the exciting forces and the damping coeffi-cients. Further, the damping coefficients may be directly related to the exciting forces (Mei [10], section 7.6) so that the q factor may be calculated from the solution of the radiation problem. In terms of the local device coordi-nates, the field radiated due to the forced oscillations of device q has the form
^^r^
~ Tirwf
(^'?)^°^h^:(z-f
h) as kr^ - co (33)Under the assumptions of the point-absorber approxima-tion, f(Ög) is unaffected by scattering within the array and so is taken to be the same for all devices; a simple expres-sion for the q factor in terms of Bessel functions then follows. Full details of the theory are given by Evans [5] and Falnes [6] and a discussion of the assumptions is given by Mclver [16].
4 R E S U L T S
Based on the two solution methods outiined in the previous section, a systematic comparison of the obtained results was undertaken conceming the hydrodynanuc characteristics (exciting wave forces, hydrodynamic parameters, ^-factors) of an array of wave power devices. The geometry chosen (illustrated in plan in Fig. 1) consists of five equally-spaced truncated vertical cylinders of radius a placed in a line and results are presented for inter-body spacings d = 5a and d — 8a. The draught of each cylinder is equal to its radius a and the water has depth h = Sa.
mode 2
mode 1
0^© © © ©
Fig. 1. Plan view of row of devices.The infinite series eqns (20) and (21), which describe the velocity potential around each cylinder of the array in the case of the multiple scattering method, were truncated only at the stage of its numerical implementation. The number of modes in the vertical and circumferential directions involved in eqn (20), as well as the number of interactions considered (see eqn (21)), were selected in such a way so that convergence of the obtained results is achieved with an estimated accuracy of 0.5%. The implementation of the plane-wave method does not involve any further truncations for the numerical calculation of the hydrodynamic interactions.
The results presented in this paper have been selected to illustrate the largest discrepancies that are likely to occur when using the approximate methods. Note that a number of different scalings of the vertical axis are used in the figures. The first results presented concern the exciting forces on individual cylinders within the array under normal wave incidence. On the relevant figures the notation [m,n] is used to denote the force on cylinder m in the direction n, with the surge, sway and heave directions denoted respec-tively by 1,2 and 3 (see Fig. 1). The modulus of the exciting wave force on the second cylinder of the array is plotted against the wave number parameter ka in Figs 2 and 3 for d/ a —5 and d/a = 8, respectively; the results have been made non-dimensional using the factor 2pgaA. For both d/a values, the two methods give almost identical results for the forces in sway and heave (modes 2 and 3, respectively). As far as the horizontal wave force in the surge direction (mode 1) is concerned, small discrepancies are found for the
Table 1. Modulus of exciting force on device 2 in mode 1 versus ka: angle of incidence /3 = 7r/2, spacing d/a = 8. Comparison of
multiple scattering (MS) and plane wave (PW) methods
ka MS PW 0.1 0.00454 0.00453 0.2 0.02008 0.02002 0.3 0.03023 0.03013 0.4 0.01831 0.01881 0.5 0.08091 0.08384 0.6 0.18565 0.18931 0.7 0.29537 0,30447 0.8 0.34711 0.35615 0.9 0.40808 0.42615 1.0 0.35320 0.36334 1.1 0.20011 0.20351 1.2 0.14497 0.15007
288 S. A. Mavrakos, P. Mclver
ka
Fig. 2. Modulus of exciting force versus ka: angle of incidence /3 = 7r/2, spacing d/a = 5. Comparison of multiple scattering ( X )
and plane wave ( • ) methods.
central cylinder in the neighbourhood of the maximum value, that is around ka = 1.3 for d/a = 5 and ka = 0.9 for d/a — 8. It is typical that, for quantities that oscillate strongly with ka, the largest discrepancies between the two methods occur near the peaks in the oscillation. It is also typical that forces directed along the line of the cylinders are less accurately computed using the plane-wave method. It has been observed, however, that smaller discrepancies between the two theories occur i f the inter-body spacing is increased. This is because the plane-wave method is derived under the assumption that the spacing between the inter-acting devices is much greater than the wave length, that is M > 1, and so is expected to perform better for more widely-spaced bodies. Despite this assumption it is
ka
Fig. 3. Modulus of exciting force versus ka: angle of incidence /? = 7r/2, spacing d/a = 8. Comparison of multiple scattering ( X )
and plane wave ( • ) methods.
remarkable how well the plane-wave method performs for values of kd less than one. The graph conceming the surge (mode 1) exciting force in Fig. 3, is supplemented by Table 1, which gives in tabular form the corresponding results for the d/a — 8 case and so allows a numerical estimate of the accuracy to be made. The biggest difference between the results obtained by the two methods is 4.4% for ka = 0.9.
Returning to the results obtained through the multiple scattering method, thirteen terms in the circumferential and seventy in the vertical direction, that is - 6 < m < 6 and j = 70 respectively, were necessary in the series expan-sion of the velocity potential eqn (20) to achieve conver-gence of the results with the above mentioned accuracy of
7r/2, spacing dla = 5. Comparison of multiple scattering ( X ) and parison of multiple scattering ( X ) and plane wave ( • ) methods, plane wave ( • ) methods.
0.5%. The number of interactions ^ to be retained i n eqn (21) is dependent on the spacing dla and the range of wave frequency parameter ka. For dla = 8 it was found that five {s = 5) and seven (s = 7) interactions were necessary for ka ^ 1 and ka > I, respectively. The corresponding number of interactions for dla = 5 ars s — 10 and s = 15, respectively.
Fig. 4 shows comparisons between the multiple scattering formulation and the plane-wave method for the phases of the exciting wave forces in the case of dla = 5. The results compare favourably with each other.
The next set of figures is concerned with the radiation forces. The notation [m,n;p,q] is used to indicate the force on cylinder p in mode q due to the forced oscillations of
cylinder m in mode n. Figs 5 and 6 are concerned with selected results for added mass coefficients for the dla = 5 array. The coefficients are plotted against the wave-number parameter ka and made non-dimensional by pa^. The results from the two methods compare generally very well with other. It is typical of the results for both added mass and damping coefficients that the curves oscillate more strongly with ka for cylinders at the end of the line and least strongly for the central cylinder. The biggest differences between the two calculation methods are found when calcu-lating the hydrodynamic interaction coefficient in surge on the end cylinder, denoted by [4,1,5,1]. The corresponding results from the two methods dealing with the vertical hydrodynamic reaction forces exerted on the first, the third and the fifth cylinder of the array due to forced vertical
290 S. A. Mavrakos, P. Mclver
0.5 1 1.5 ka
Fig. 6. Added mass coefficients versus ka: spacing d/a = 5. Com-parison of multiple scattering ( X ) and plane wave ( • ) methods. motion of the middle cylinder exhibit a very good correla-tion with each other (Fig. 6).
The results presented here have concentrated on the added mass coefficient but broadly similar features have been observed for the damping coefficient. Further, results for the wider spacing d/a = 8 show excellent agreement.
Finally, Fig. 7 gives the ^-factors for devices absorbing in heave for both inter-body spacings. The results obtained by the multiple scattering (MS) and the plane-wave (PW) methods are supplemented by those derived through the point-absorber (PA) approximation. For all practical pur-poses, the point-absorber theory predictions are identical to those of the multiple scattering formulation for ka < 0.8. I n contrast, the plane-wave method shows substantial deviations from the other two solutions in the low-frequency
1— 1 —1 — n 1 1 O • > 1 1 1 1 1 1 1 1 ' 1 1 1 d / a = 5 MS PW 1 PA • - -1 ^ \ . \ v. 3 I I I I I I I I I I I I I I I I I I I I I I I
Fig. 7. q factor versus ka: angle of incidence 13 = Tr/2, spacing d/ a = 5,8. Comparison of multiple scattering (MS), plane wave
(PW) and point absorber (PA) methods.
range. This is because the evaluation of the q factor involves the inversion of the damping matrix, see eqn (8), which becomes nearly singular in the low-frequency range. As a result, any small errors in the calculation of the damping-matrix-elements, which are in apparent good agreement with the ones of the multiple scattering formulation, are magnified by the inversion leading to the erratic behaviour observed in Fig. 7. As far as the ^-factor is concerned, the plane-wave approximation will only give reasonable results for non-dimensional wave numbers greater than about kd = 2. This contrasts with its good performance down to about kd = 0.4 for the hydrodynamic forces as shown previously.
For ka values higher than 0.8, however, substantial dif-ferences between the multiple scattering and the point-absorber theory can be obtained, particularly for closely-spaced devices. This is because the point-absorber theory is based on a weak-scatterer assumption and for large ka values the devices will strongly scatter waves. The plane wave and the multiple scattering methods give comparable results which converge as ka increases.
5 CONCLUSIONS
A number of calculation methods have been used for the calculation of the hydrodynamic properties of a row of ver-tical circular cylinders which may be used to model wave power devices. The accuracy of two approximate methods.
the plane-wave approximation and the point-absorber the-ory, has been assessed by comparison with calculations made using a multiple scattering technique which can, in principle, achieve arbitrary accuracy. In agreement with previous work, it has been found that in most circumstances of practical interest hydrodynamic forces can be accurately calculated using the plane-wave approximation. The only significant errors that may occur are for forces measured along the row, and these errors are likely to be amplified for rows longer than the five cylinders considered here.
The so-called ^-factor has been used by many authors to estimate the power absorption characteristics of a device array. Calculations of the ^-factor using the plane-wave method break down in the long-wave regime, as the near-singular damping matrix cannot be calculated with suffi-cient accuracy for the successful inversion of the matrix. However, in this regime, accurate estimates of the q factor can be obtained using the point-absorber theory. For short waves the point-absorber theory becomes inaccurate, but the plane-wave method is accurate in this regime.
R E F E R E N C E S
1. Linton, C. M. and Evans, D. V., The interaction of waves with arrays of vertical circular cylinders. /. Fluid Meclianics, 1990, 215, 549-569.
2. Mclver, P. and Evans, D. V., Approximation of wave forces on cylinder arrays. Appiied Ocean Research, 1984, 6, 101¬ 107.
3. Mclver, P., Wave forces on arrays of bodies. J. Engineering Mathematics, 1984, 18, 273-285.
4. Williams, A. N. and Demirbilek, Z., Hydrodynamic
interactions in floating cylinder arrays—I: Wave scattering. Ocean Engineering, 1988, 15, 549-583.
5. Evans, D. V., Some analytic results for two- and three-dimensional wave-energy absorbers. In Power from Sea Waves, ed. B. M. Count. Academic Press, 1980, pp. 213¬ 249.
6. Falnes, J., Radiation impedance matrix and optimum power absorption for interacting oscillators in surface waves. Applied Ocean Research, 1980, 2, 75-80.
7. Simon, M. J., Multiple scattering in arrays of axisymmetric wave-energy devices. Part 1. A matrix method using a plane-wave approximation. / Fluid Meclianics, 1982, 120, 1-25. 8. Mavrakos, S. A. and Koumoutsakos, P., Hydrodynamic
inter-actions among vertical axisymmetric bodies restrained in waves. Applied Ocean Research, 1987, 9, 128-140. 9. Mavrakos, S.A., Hydrodynamic coefficients for groups of
interacting vertical axisymmetric bodies. Ocean Engineering, 1991, 18, 485-515.
10. Mei, C. C, The Applied Dynamics of Ocean Surface Waves. Wiley, New York, 1983.
11. Spring, B. H. and Monkmeyer, P. L., Interaction of plane waves with vertical cylinders. In Proceedings 14th Confer-ence on Coastal Engineering, Copenhagen, Denmark, 1974. 12. Kagemoto, H. and Yue, D. K. P., Interactions among
multi-ple three-dimensional bodies in water waves: an exact alge-braic method. / Fluid Mechanics, 1986, 166, 189-209. 13. Twersky, V., Multiple scattering and radiation by an arbitrary
configuration of parallel cylinders. Journal ofthe Acoustical Society of America, 1952, 24, 42-46.
14. Ohkusu, M., Hydrodynamic forces on multiple cylinders in waves. In Proceedings of the International Symposium on Dynamics of Marine Vehicles and Structures in Waves. Insti-tute of Mechanical Engineers, London, 1974.
15. Abramowitz, M. and Stegun, I . A., Handboolc of Mathema-tical Functions. National Bureau of Standards, Washington, 1964.
16. Mclver, P., Some hydrodynamic aspects of arrays of wave-energy devices. Applied Ocean Research, 1994, 16, 61-69.
