E L S E V I E R
0 1 4 1 - 1 1 8 7 ( 9 4 ) 0 0 0 2 0 - 4
Printed in Great Britain. A l l rights reserved 0141-1187/94/507.00
Analysis of wave induced drift forces acting on a
submerged sphere in finite water depth
G. X . Wu, J . A. Witz, Q. Ma & D. T . Brown
Department of Mechanical Engineering, University College London, Torrington Place, London, UK, WCIE 7JE (Received 20 June 1994)
A solution is presented for the wave induced drift forces acting on a submerged sphere in a finite water depth based on linearised velocity potential theory. In order to obtain the velocity potential, use has been made of multipole expansions in terms of an infinite series of Legendre functions with unknown coefficients. The series expression for the second order mean forces (drift forces) is provided by integrating the fluid pressure over the body surface. The horizontal drift force is also expressed by a series solution obtained using the far-field method.
1 I N T R O D U C T I O N
D r i f t forces acting on a body i n waves represent an important source of excitation which has significant impUcations in a variety of apphcations. Wave induced d r i f t forces have been considered by a number of researchers. Considerable effort has been devoted towards the numerical calculation o f the drift forces based on the far-field and near-field methods proposed by N e w m a n / and Pinkster and Van Oortmerssen^ respectively. Newman and Lee^ have shown that numerical procedures for calculating the d r i f t forces are more time consuming than for hnear forces, even for a simple body shape, such as a box. I t is advantageous therefore to establish analytical solutions to the problem.
Extensive work has been reported over a number of years on analytical solutions for the linear or first order force acting on a floating or submerged body of hemispherical, spherical or spheroidal shape i n waves. Havelock's'* pioneering work investigating the vertical motion of a floating hemisphere in infinite water depth was extended by Hulme^ to consider both heave and surge degrees of freedom and to investigate the added mass and radiation damping. I n the work of Gray,^ the scattering problem for a submerged sphere was solved by expanding the Green's function and the associated velocity potential in spherical harmonics. Srokosz^ investigated a submerged spherical wave power absorber solving the corresponding radiation problem by the method of multipoles. Wang^ considered the problem o f linear forces acting on a submerged sphere. Wang solved the governing equations by employing a special series solution to represent the associated
velocity potential in infinite water depth and by imposing the zero normal flow body boundary condition at specified points on the sphere's surface. W u and Eatock Taylor^''° considered a submerged spheroid and provided analytical solutions f o r the linear forces. More recently, L i n t o n ' ' developed a solution to the radiation and diffraction of a submerged sphere i n a flnite water depth by expanding the multipole potential given by Thorne'^ into a series o f Legendre functions. However, these publications present resuhs for the linear forces only.
This paper presents an analytical procedure f o r the calculation of wave d r i f t forces acting on a submerged sphere i n a finite water depth. The work of Thorne'^ is used as a starting point to represent the hnear velocity potential by a multipole potential for a finite water depth. The hnear velocity potential is formulated by the method similar to that used by L i n t o n . " Expanding the multipole potential into a series o f Legendre functions and using the body boundary condition, allows a linear system of equations with unknown constants to be set up. By solving the hnear system, the constants and therefore the linear velocity potential are found. As a result new expressions for the wave induced d r i f t forces are derived by both the far-field and near-field methods (direct integration .of fluid pressure over the body surface method). I n this paper, only the diffraction contribution to the d r i f t force is considered. However i t is believed that the radiation contribution can be derived in a simflar manner.
2 M A T H E M A T I C A L F O R M U L A T I O N
The formulation is defined with respect to two i3
co-ordinate systems: one is a right-handed Cartesian co-ordinate system {x,y,z), in which the x-y plane coincides with the undistributed free surface and the z-axis is orientated vertically downwards; the second is a spherical co-ordinate system {r, 0, tp) with the origin at the geometric centre o f the sphere. The axes systems are shown in Fig. 1 which shows a sphere of radius a in water of depth d with its geometric centre located at
(0,0, h) with respect to the Cartesian co-ordinate system. The relationship between the co-ordinate systems is given by R = + ;;2 r = V R ^ + { Z - hf R t a n ö t a n ^ z - h X for 0 < Ö < TT f o r — T F < l j ) < - K
Based on the assumptions of a homogeneous, inviscid and incompressible fluid and irrotational flow, the fluid motion can be expressed by the velocity potential Hx,y,z,t) as
$ ( x , 3 ; , z , 0 = - i t ^ ^ [ < ^ i + < ^ D ] e - ' " ' (1) where w and A are the wave frequency and amplitude
respectively. The motion is assumed harmonic.
The term (t)i{x,y,z) represents an incident wave approaching f r o m the negative x direction and is given by ^ _ 1 C O S h / C o ( / c - r f ) ^ i M n c o s ^ ' K c o s h IcQd (2) wave direction free surface (r,e,v) sea bed
Fig. 1. Reference co-ordinate system.
w i t h K = üj^/g and /CQ is the finite depth wave number defined by
/co sinh/cQf^ - i^cosh/co<i = 0 (3)
The term (f>o{x, y, z) i n eqn (1) is the diffraction potential that satisfies the Laplace equation subject to boundary conditions on the free surface, sea bed and body surface in addition to the radiation condition. These conditions are mathematically described by
(4) (5) (6) (7) (8) = 0 9n dn dz 0 lim V in the fluid on z = 0 on /• = a on z = d
where n denotes the normal vector f r o m body surface to fluid.
The incident wave potential in eqn (2) is expanded i n terms of associated Legendre functions i n the fluid region near the sphere as
^ ( / j j , , cos mip m=0 where K Xs = cosh/co(rf — h) cosh/co<i sinh ICQ {d - h) pill . cos( {s + 2my. f o r j = 0,2,4,. (9) (10) (11) f o r s 1,3,5,. cosh k^d
w i t h e,„ = 1 f o r m = 0 and e,„ = 2 f o r m> I.
The term Pj"(cosö) is defined as the associated Legendre function given by
,d"'Pnix)\
p;;'icosi
sm"'9-dx" (12)
using the Legendre polynomial P„{x).
Therefore the boundary condition on the body surface in eqn (6) can be rewritten as
dr
_ ^ d ^
cos m>p (13)
Equation (13) suggests that the diffraction potential can be constructed as
^D=Y '"^ (14)
m=0
where ip„, is a function of polar variables /• and 9 only; that is, it is independent o f the term ip. I t can be
expressed as
(15) where a is the radius of the sphere and A^l, are the unknown complex coefficients. The terms G,'" are multipole potentials similar to those derived by Thorne''^ but are modified here by introducing the second term i n the expression
„,„ P,'"(cosö) P'"(cosa) 1 (_cos(
,.n+l ,.H+1 {n-m)\
( i ^ + /c)[e-'^(''+^' + ( - l )"+"'< -kin
(16) Jo k sinh kd - i f cosh kd
X /c" co&hk{z - d)J,„{kR) dk
where k is the integration variable. The term J,n{kR) i n eqn (16) is the /nth order Bessel function and a and /'i are defined using
= ^ R ^ i d + H - z f R
t a n a = - — — d + H - z
where H is defined i n Fig. 1.
The hne integration in eqn (16) passes under the singular point of the integrand at k = /CQ. The potentials G'" and therefore 0 d satisfy the Laplace equation together with conditions (5), (7) and (8) but not the body boundary condition. I n the fohowing the body boundary condition wih be imposed on (f>Y) to find the unknown coefficients A'," involved in eqn (15).
The second and third terms i n eqn (16) are therefore expanded in the region near the body surface into a series o f Legendre functions in order to separate the variable ;• f r o m 9. This gives
. . « + 1 s=0 2H ,(cos( (17) and 1 0 k sinh kd - K cosh kd m)!jo xk!'co5\ik{z - d)J„,{kR) dk 2H {cos 9) = 2 ^ C,{",>n
The terms B"l, and Cs{n,ni) are given by 1 {s + n + m)\ B:: ( 2 j ï ) " + ' {s + 2f72)\{n - m)\ i2Hy+'" (18) (19) +ii+m ^ ^ ( " ' ' " ) = ( . - » 0 ! ( . + 2 m ) ! j " ( j f + /c)[e-^'^+^' + ( - l ) -m g-Wi
j
k sinh kd • X u,{kH)dk K cosh led (20)where iis{kH) is given by
cosh kH f o r s •• - sinh kH f o r ^ :
iisikH) 0,2,4,.,
1,3,5,., (21)
Finahy, G"' can be expressed in the region near the body surface as P - ( C O S Ö ) , , « + 1 s=0 r r is+iii 2H (cos 6») i = 0 p ; % , ( c o s ö )
N o w employing eqn (13) gives dG"
dr dn
(22)
(23) By multiplying both sides of this equation by Rfl (cos Ö) and integrating with respect to ii over - 1 < n < 1, where fi = cos 6, one derives the linear system of equations n=m T"' for s = m,m +\,m+2, where E;;i = -{n + m„ + D';:{s D:{S) = c m) "+'(^ + ' « ) ( ^ f " ' [ Q ( « , m ) and is the Kronecker delta function.
The term T"' is given by -e„,i'"(/cofl) s-\ {s^ni)\ K A.S-II (24) (25) (26) (27) Solving this linear set of equations allow the coefficients, A'^, and, therefore, the diffraction velocity potential to be determined.
Equations (24) are derived using the orthogonality property of Legendre functions. They are decoupled f o r different m and hence it is possible to solve independ-ently f o r each value of m. The selected maximum value for in depends on the required level of accuracy. Furthermore T"' given by eqn (27) is of order I/nil and it is shown that only the first few terms i n the expansion need to be considered. This is covered i n further detail i n the discussion.
3 E X C I T I N G FORCES A N D D R I F T FORCES The hydrodynamic forces and moment acting on a body can be evaluated by carrying out direct integration o f fluid pressure over the body surface. The pressure is given by
Focusing only on the first order forces and the d r i f t forces, the wave induced pressure may be written as
p = Rc{p,m')+P2 (29)
where p\ is the amplitude of the linear pressure given by
P\ = p(^^A<j)iY) (30) and p2 in eqn (29) is the time independent second order
pressure which can be written as
Pl (31)
where (^iijo = + (/^d and the symbol * denotes the complex conjugate.
On the body surface 0 i d can be reduced to a very simple expression i f use is made o f the body surface boundary condition'^'''*
° ° CO 2 « - I - 1
- ^ I D =
« E E
<P'^ ('^"s ^) ' " ^ (32)where the condition n
dv 0 (33)
has been apphed.
Using eqn (32) ahows the velocity components to be expressed as I D ( 2 / 2 + l ) ( 7 7 - « 7 ) ( « + «7 + 1) 1 ( ' de 1 oo oo I T ,
E
xA';;+^p;;'{cos9) cos ( « 1 + 1)7/; 1 ~ 2 1 ^ 277 + 1 EE^'^-i^T^ =1 n=in " x ^ ; r ' P ™ ( c o s 6 i ) c o s ( ; 7 7 - 1)^ (34) V,: 1 d(l)_ I D sin 9 dtp 1 CO 00 sinö ^ ^ 7 7 7 ( 2 7 7 + 1 ; xA';,'P;;'{cos9) smmtp (35)where )i'„, = 2 f o r 777 = 0 and it'„, = 1 for 777 > 1.
The second order pressure coefficient on a body surface is given by
Pl
pgA'K (36)
Representative distributions of the pressure coefficient are presented in the following section.
The forces acting on the sphere can be calculated by integrating the pressure (eqn (28)) over the body surface
5 b
piij és (37)
Here the term itj represents the components o f the normal out o f t h e body surface which can be expressed as
77y = P{(cos0) cos jlj) (38)
with indices y = 0 and /' = 1 corresponding to heave and surge modes respectively.
When the second order time dependent forces are ignored, the total forces then reduce to
Fj = Kc{fjC-'-')+fj (39)
Substituting eqn (28) into eqn (37), the first order exciting forces and drift forces are r.rspectively given by
f j and f j -pu?A \poj'A' bo)njds (40) (41)
I n the following, f j and f j are expressed i n terms of constants A',". T o do so, the method proposed by W u and Eatock T a y l o r ' ° is employed.
3.1 Exciting forces
Using the orthogonality property o f cosine functions, the exciting forces (eqn (40)) can be expressed as
2
pTTu?Aa^ P/(cos 9) (0] + ( p f ) sin 9 dö (42)
with ej = 1 for 7' = 0 and ej = 2 for 7 > 1. Use is made o f eqn (32) to give
(43) Finally by using the orthogonahty property of the Legendre function, the exciting force coefficient (42) can be expressed as
f j 1 p-naJ'uP-A 3.2 Drift forces
-2>A{ f o r 7 = 0,1 (44)
Once the linear velocity potential is established, the expression for heave and surge d r i f t forces can be derived by the near-field method,^ and that f o r the surge force can also be derived by the far-field method.' Results for both these methods are now stated.
Near-field method
The d r i f t forces given by eqn (41) have been obtained by integrating the fluid pressure over the submerged sphere surface and then taking the temporal average. In order to express the force in terms of coefficients A'", the fohowing derivation is carried out.
Introducing
W = V(j>io and Q
dn dz
9 d(l> ID
and making use of the identity
dn dx 15 W-VQnjds = YjQds (45) (46) (47)
allows for an alternative representation of eqn (41) given by
u?A
(48)
In addition, Yj can be derived by considering the body boundary condition to give'^''''
oo oo Fo =
- ^ E ("+!)("-
« 0 ^ ; ; - i ^ » ' ( c o s ö ) c o s mtl, ^ »ï=0 n=m-\-\ 1 n(« + 2 ) ( « + m + 1) (49) m=0 « = / » J ' " X ^;;ViP"'(cos0) cos mt}} and 1 OQ 1 oo CO+0;; E E +
Hi=0 n=m x < ' + / P " ' ( c o s 0 ) cosm^ 1 2^ _^ ^ ^ n{n + 2){n + m + 1)(« + 7 ) 7 + 2 ) » i = 0 H = » i 7 7 + 1 x 4 j / p ; ' ( c o s 6 i ) cos 7777/; 1 ' 2 ^ 1 ^ 77(77 + 2) m=\ n=m jin-l T>mi 77 + 1 Xyi;;[;-l'P™(cOSÖ)cOS7J77/j (50)Substituting eqns (32), (49) and (50) into eqn (48) gives /o 1^ = — Z T T I A f l 'A'" / l . , r . " + 2 (« + /" + ! ) ! - 2 . ( i f f l ) E X : u ' . ; ^ ^ 7)7 (51) = 7r(.S:a)
E E ' " ' - i „ („_,„)!
Hl=l « = " 1 77 + 1 ( 7 7 - W ) ! (52) where n'„ 2 for 7 7 1 = 0 and H ' „ , = 1 for 777 > 1.Equations (51) and (52) give the heave and surge d r i f t f o r c e s ^ and f .
Far-field metiiod
I n order to obtain the drift force by the far-held method acting on a submerged sphere in a finite water depth, the velocity potential in the region far f r o m the body given by
cosh/co(z-^)
i f c o s h M ""^^^^
16 7r/cnP 7/co1?-i(vr/4) (53) is introduced.Following the methodology of Section 2, the diffrac-tion potential (\>Q for the problem can be rewritten as
= 2cos77iV'5^fl"+^4;'Gr
(54)H;=0
I t is noted that the expression for G"' in eqn (22) cannot be used for the far-field method as the series is nonconvergent for large R.
To obtain the expression for H{t\)), it is necessary to estabhsh the asymptotic properties of G,'" i n eqn (16) for large R. Using the method discussed i n Wehausen and Laitone,''' gives
ITT , i [ i : o « - ( 7 r / 4 ) - m ( , r / 2 ) | (55) " ( 7 1 - 777)!^(fco)
as R tends to infinity where /(/co) = (if + /co)[e-'^°(''+^)
+ (_!)"+'" e-'^o"]/c{j cosh/Co(z - d) (56) and ^(/co) is given by
g(fco)
= sinh A:o(i + k^dcosh/co<i - Kdsinh/cQf/ (57) Substituting eqn (55) into eqn (54), and comparing w i t h eqn (53), the expression for H{t\)) is given by^ ^ ^ ^ ^ i ^ ( X f l ) W o ^ g<,„,,„^,-(.W2) where Gl : G, = 00 n—m ( V ) " ( G i + (-i)"+"'G2)^;r ( 7 7 - 7 7 7 ) ! (58) 1 + -hh
I t is noted that this analytical expression for Hitp) is a direct result of the representation of the diffraction potential using associated Legendre polynomials.
written as 16 1.5 •2-K 0 cos ii>\H{'il)) p d^) + 2 Re H{Q) | (59) p ^ C g f l ' /Co C (60)
dl/-'dH*{tpy ] dtp (61) with q _ 1 / ^ 2/coJ \ C 2 \ sinh2/co(i?/which is the wave group velocity to phase velocity ratio. Terms F^. and Fy give the surge and sway drift forces respectively with the moment about the vertical axis.
Substituting eqn (58) into eqns (59)-(61) gives Fy = 0, M, = 0 as expected and Trd / Ikpd a \ sinh2/co(i {Kaf [{\-Kd)Kd+{k,df\ CO CO C f R e ^ , v „ , C , „ C ; , + i + 2 R e ^ C „ /;i=0
where Cf and C,„ are defined as Tr{Kafkod n:=0 C [{\-Kd)Kd+{k,df] C,„ = (fl/fo)"'e-*('""')'^/2
{k,d)\U, + {-\r^"'UM';.
H=0 (62) (63) (64)Thus the surge d r i f t force established by the far-field method is found in terms of coefficients involved in the diffraction potential.
4 N U M E R I C A L RESULTS A N D D I S C U S S I O N Exciting force coefficients (moduli) for a submerged sphere are presented i n Fig. 2 at a fixed submergence {h/a= 1-25) for a range of water depths. The results have been calculated using eqn (44). Also included are the results of Wang^ f o r infinite water depth. There is agreement between both sets o f results for deep water to
1.0
0.5 h d/a=2.5
d/a=3.0 d/a=5.0 d/a=\l.O
d/a=20.0 and Wang
0.25 0.50 0.75
Ka
1.00 1.25
Fig. 2(a). Heave exciting force {li/a = 1-25).
3.5, .
d/a=2.5 d/a=3.0 d/a=5.0 d/a=n.O
(i'a=20.0 and Wang
0.25 0.50 0.75
Ka
1.00 1.25
Fig. 2(b). Surge exciting force {h/a = 1-25).
within three significant figures although different expansions of the velocity potential have been used. I n long waves {Ka < O'l) the shallow water heave exciting force at this submergence reduces significantly f r o m that in deep water. The converse is true for the surge exciting force where the values in water of depth 2-50 are more than double those in depth 20a.
Figure 3 gives exciting force coefficients on the sphere i n fixed water depth {d/a = 6-0) for a range of submergence values. The results indicate, as expected, that the force coefficients decrease for increased submergence values.
Figure 4 shows the variation with water depth of the time independent pressure distribution over the centre-line cross-section of the sphere f o r a wave frequency o f Ka = 0-2 and a submergence h = 1 -25(7. The large difference in the pressure coefficient between the top and the bottom of the sphere reflects the significant heave d r i f t force f o r these particular conditions. The
1.5
Fig. 3(a). Heave exciting force {d/a = 6).
2.5
Ka
Fig. 3(b). Surge exciting force {d/a = 6).
(15) (10) (5) 0 5 10 15
Pressure coefficient Pressure coefficient (wave side) (eqn 36) (lee side) (eqn 36)
Fig. 4. Note: 0 degrees represents bottom of sphere.
3,0
ol I I 1— i
0 0.3 0.6 0.9 1.2
Ka
Fig. 5(a). Heave drift force.
2.0 I
Ka
Fig. 5(b). Surge drift force.
small differences between the wave- and the lee-side pressure coefficients reflect on the relatively small surge drift force coefficient f o r the above set o f parameters.
Figure 5 and Table 1 give d r i f t force results on the sphere at a fixed submergence o f h = \ -25a for a range of water depth values. The results show that the heave drift force coefficient is relatively insensitive to changes in water depth (from 15a to 11a) for all but the lowest frequencies. The surge drift force coefficient exhibits the same insensitivity over the higher wave frequency range. However i n shorter waves there is up to 40% variation over the analysed water depths.
D r i f t forces on the sphere at a range o f submergence values i n a fixed water depth of d = 6a are now investigated. The results shown i n Fig. 6 confirm that both heave and surge drift force coefhcients are of increasing magnitude as the sphere is brought closer to the water surface.
3.0 I 2 . 0 ,
Table 1. Heave and drift forces {hja = 1-25)
Ka Heave Surge
d/a = 2-5 d/a = 3-0 d/a= 11-0 d/a = 2-5 ( / / f l = 3-0 d/a = 11-0
Near field Near field Near field Near and Near and Near and
far field far field far field
0-10 0-3535 0-4060 0-2436 0-0065 0-0032 0-0001 0-20 0-7204 0-7606 0-5827 0-0154 0-0076 0-0015 0-30 1-0979 1-1349 1-0178 0-0238 0-0126 0-0080 0-40 1-4791 1-5246 1-4794 0-0315 0-0202 0-0232 0-50 1-8503 1-9081 1-9033 0-0412 0-0343 0-0477 0-60 2-1850 2-2481 2-2424 0-0565 0-0577 0-0781 0-70 2-4512 2-5037 2-4743 0-0794 0-0891 0-1093 0-80 2-6211 2-6497 2-5991 0-1076 0-1220 0-1359 0'90 2-6894 2-6930 2-6356 0-1352 0-1490 0-1552 1-00 2-6721 2-6589 2-6069 0-1563 0-1662 0-1664 1-20 2-4897 2-4693 2-4391 0-1714 0-1734 0-1692 1-40 2-2428 2-2291 2-2147 0-1606 0-1595 0-1559 1-60 2-0023 1-9948 1-9885 0-1398 0-1385 0-1363 1-80 1-7865 1-7828 1-7800 0-1178 0-1169 0-1157 2-00 1-5973 1-5954 1-5942 0-0978 0-0973 0-0966 3-00 0-9471 0-9470 0-9469 0-0368 0-0367 0-0367 4-00 0-5851 0-5850 0-5850 0-0139 0-0139 0-0139 5-00 0-3684 0-3684 0-3684 0-0053 0-0053 0-0053
of first order wave exciting forces and second order drift forces is instructive. For comparison purposes identical sphere submergence values of h = \-25a are taken. Inspection of Fig. 2 indicates that exciting force values are influenced by water depths d < 1 la. However Fig. 5 shows that d r i f t forces are water depth dependent f o r values d < 5a. Consequently exciting forces are more sensitive to water depth variation than d r i f t forces f o r the case considered.
8 C O N C L U S I O N S
The work is motivated by the need f o r an analytical
solution to the evaluation of second order forces acting on a body in waves. The influence of body submergence and water depth on the drift forces is demonstrated. Many of the numerical methods that are currently in use depend on the summation o f wave pressure induced forces over the wetted body surface. The results f r o m these methods are sensitive to the number of elements used to discretise the body. Furthermore calculations involving large numbers of elements are computationally time consuming. I t is therefore con-sidered advantageous to provide analytical solutions to the second order forcing as a basis f o r validating the numerical results when applied to a body of spherical shape.
REFERENCES
1. Newman, J. A., Drift force and moment on ships in waves. J. Sliip Res., 11 (1967) 51-60.
2. Pinkster, J. A. & Van Oortmerssen, G., Computation of the first and second order wave forces on bodies oscillating in regular waves. 2nd Int. Conf. Numerical Slup Hydrodynamics, Berkeley, CA, 1977, pp. 136-56.
3. Newman, J. A. & Lee, C. H., Sensitivity of wave loads to the discretisation of bodies. Proc. Beliaviour of Offslwre Structures Conf {BOSS 92), May, London, Vol. 1, 1992, pp. 50-64.
4. Havelock, T. H., Waves due to a floating sphere making periodic heaving oscillations. Proc. R. Soc, A231 (1955) 275-85.
5. Hulme, A., The wave forces acting on a floating hemi-sphere undergoing forced periodic heaving oscillations. Fluid Mecli., 121 (1982) 443-63.
6. Gray, E. P., Scattering o f a surface wave by a submerged sphere. / . Engng Math., 12 (1978) 15-41.
7. Srokosz, M . A., The submerged sphere as an absorber of wave power. J. Fluid Mech., 95 (1979) 717-41.
8. Wang, S., Motions of a spherical submarine in waves. Ocean Engng, 13(3) (1986) 249-71.
9. Wu, G. X. & Eatock Taylor, R., The exciting force on a
submerged spheroid in regular waves. J. Fluid Mech., 182 (1987) 411-26.
10. Wu, G. X. & Eatock Taylor, R., On radiation and diffraction of surface waves by submerged spheroids. J. Ship Res., 33 (1989) 84-92.
11. Linton, C. M . , Radiation and diffraction ofwater waves by a submerged sphere in finite depth. Ocean Engng, 18(1/2) (1991) 61-74.
12. Thorne, R. C , Multipole expansions in the theory of surface waves. Proc. Camh. Phil. Soc, 49 (1953) 709-16. 13. Wu, G. X. & Eatock Taylor, R., Radiation and diff'raction of water waves by a submerged sphere at forward speed. Proc. R. Soc, A413 (1988) 433-61.
14. Wu, G. X. & Eatock Taylor, R., The hydrodynamic forces on a submerged sphere moving on a circular path. Proc R. Soc, A428 (1990) 215-27.
15. Ogilvie, T. F. & Tuck, E. O., A rational ship theory for ship motions: Part 1. Department of Naval Architecture and Marine Engineering, University of Michigan, Ref No. 013, 1969.
16. Mei, C. C, The Applied Dynamics of Ocean Surface Waves. Worid Scientific, London, 1989.
17. Wehausen, J. V. & Laitone, E. V., Handbuch der Physik, Vol. 9, ed. W. Flugge, Springer, Beriin, 1960, 446-778.
