Comments on note by P. McIver

Wave drift enhancement effects in multi column structures." R. Eatock Taylor and S. M. Hung As r/2 is a constant for the incident wave there is no contribution to any component of the second order force from the integral over the waterline in equation (6). The remaining terms have no dependence upon r and 0 and so contribute an equal amount to the mean vertical force on each body. That is, the total force on the N bodies is N times the force on a single body. In particular, the ratio of the force on N bodies to the force on an isolated body tends to N as ka tends to zero, in agreement with Eatock Taylor and Hung. For low frequencies when the scattered field is neglected there is no contribution to the horizontal components of the mean second order force from any of the terms in equation (6). Therefore it is necessary to include the scattered wave field when considering the limiting behaviour. This is perhaps most easily done by considering the 'far-field' expressions for the second order force. As kr/. tends to infinity (i.e. in the far field) the expression (7) for the wave field scattered by a single body may be approximated using the large argument asymptotic forms of the Hankel functions (Abramowitz and Stegun, p. 364) 4 as

/

2 ,~12 +O'~II](O/.)~-r] ) exp [i(kr/. -- rr/4)]

(10)

Expanding in powers of si/r it is easily shown that

and

sj

cos O/= cos 0 + 0 -and therefore ~)/.(1) ~

Ao--iAx cos 0 + O (ka) 2si' (/ca)4 r

\~r]

exp[i(/o'--n/4)]

11(0) = NIIi(0) + O (ka) 2sl (ka) 4 r

271

& (11)

n=0

By considering a m o m e n t u m balance Maruo s and Newman 6 derived the expression

(18)

The ratio of the total mean horizontal second order force on the N identical bodies to the force on an isolated body is therefore

N2 I ( 1 - - c o s 0 ) ~ An(-- i)n cos nO/.

1+O(~)(17)

and so by linear superposition and comparison with (13)

where, 11i(0/.) :

(16)

r

Fix

II/(0)+ O

(ka)2Sl(ka)4r

2 dO

0 2rr

f(1

cos 0) Hi(0 ) dO

o

(19)

2rr

Fix - ogAZ kcTr

cg

f (1 -- cos 0) 111i(0)12 dO

(12)

o

for the component of the mean second order force on an isolated body in the direction of wave advance. Here c is the phase velocity and cg is the group velocity of the linearised incident wave. The formula (12) is equally applicable to a group of bodies provided II1(0 ) is replaced by a function II(0) such that the total far field scattered potential is / 2 ~/2 q~(1) ~ 11(0)/_2_ / exp [i(kr -\rr~/

rr/4)l

(14)

The relationship between 11(0) and Hi(0 ) is sought in the limit of low frequency. Note that the array of bodies must be finite, that is it cannot extend to infinity in any direction. As a first step consider the scattering of the incident wave within the array. As pointed out previously, the largest scattering coefficients (A o and A 1) are each O((ka)2). Thus if the wave field scattered from the incident wave by one body is further scattered by a second body this will produce a modification to the leading order scattering coefficients of O((ka)4). Hence, as a first approximation multiple scatterings within the array may be neglected and only the scattering of the incident was considered. The contribution to the far field potential due to scattering by t h e / t h body is therefore given by equation (10) with

Ili(01) = Ao -- iAx cos 01 + O((ka) 4)

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Applied Ocean Research, 1985, Vol. 7, No. 3

(14)

which tends to N 2 as ka tends to zero, again in agreement with Eatock Taylor and Hung. This result is easily extended to the component of the drift force perpendicular to the direction of wave advance.

REFERENCES 1 Pinkster, J. A. Mean and low frequency wave drifting forces on floating structures. Ocean Engineering 1979, 6,593 2 Longuet-Higgins, M. A. and Stewart, R. W. Radiation stress and mass transport in gravity waves, with application to surf beats. Journal of Fluid Mechanics 1962, 13,481 3 Mei, C. C. The Applied Dynamics of Ocean Surface Waves, Wiley-Interscience, New York, 1983 4 Abramowitz, M. J. and Stegun, I. A. Handbook of Mathematical Functions, Dover, New York, 1972 4 Maruo, H. The drift of a body floating in waves. Journal of Ship Research 1960, 4, 1 5 Newman, J. N. The drift force and moment on ships in waves. Journal of Ship Research 1967, 11, 51

COMMENTS ON NOTE BY P. MeIVER We are most grateful to Dr McIver for his interest in our work. His analysis investigates the theoretical basis for our conjecture regarding drift enhancement effects, for groups of general axisymmetric bodies. It is always satisfactory if a simple explanation can be found for some heuristic results obtained by numerical analysis. The verification of our conjecture 1 regarding vertical forces seems to be based on an approach in which the scattered wave field is neglected. One must be careful in

Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung Mean vertical drift force on a cylinder (radius 9 m, draught 32 m, water depth 148 m) to/rad s-1

fzi/Nm -2

fz/Nm -2

0.125 0.150 0.250 0.350 0.450 0.550 0.650

--4 862 --5 116 --6 280 --7 260 - 7 033 --5 478 --3 497

--9 393 --9 627 --10 810 --11 800 --11 380 --9 071 --5 864

incident potential alone. For a single cylinder as shown in our Fig. 3a this leads directly to the results given in the second column of the following Table. The third column gives corresponding results based on incident plus scattered potentials (as in Table 3 of the original paper)) These suggest that, even at low frequencies, the scattered wave contribution is as important as that from the incident wave. But this need not affect the conclusions regarding the influence of interactions, for cylinders in a group.

REFERENCE the interpretation of this assumption. One may deduce immediately from Mclver's equations (6) and (9) the exact component of vertical drift force due to the first order

Eatock Taylor, R. and Hung, S. M. Wave drift enhancement effects in multi column structure~ Applied Ocean Research 1985, 7

Applied Ocean Research, 1985, Vol. 7, No. 3

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