Radiation and diffraction of water waves by a submerged sphere in finite depth

Ocean Engng, Vol. 18, No 1/2, pp. 61-74, 1991. Printed in Great Britain.

0029--8018/91 $3.00 + .00 Pergamon Press plc

RADIATION AND DIFFRACTION OF WATER WAVES A SUBMERGED SPHERE IN FINITE DEPTH

BY

C. M. LINTON School of Mathematics, University of Bristol, Bristol BS8 1TW, U.K. Abstract--The problems of radiation (both heave and sway) and diffraction of water waves by a submerged sphere in finite depth are formulated using the multipole method. In each case this leads to an infinite system of linear equations which are easily solved numerically. Simple expressions are derived for the hydrodynamic characteristics of such a body. A far-field approximation for the scattering problem is also presented. Results showing the effect of varying the depth on both the hydrodynamic characteristics and the diffraction forces are given and it is shown how the method can be used to predict free-surface elevations in the vicinity of the sphere.

1. INTRODUCTION PROBLEMS concerning the radiation and scattering of waves by spherical objects have received extensive study, beginning with Havelock (1955) who solved the heave radiation problem for a half-immersed sphere in deep water. This work was later improved and extended to the case of sway by Hulme (1982). The method of solution used was to express the velocity potential as a sum of a three-dimensional wave source and a linear combination of so called "wave-free potentials", harmonic potentials that satisfy the free surface and bottom boundary conditions but which radiate no energy to infinity, a method pioneered by Ursell (1949) for the case of a half-immersed horizontal circular cylinder. Numerical calculations of forces on a fixed sphere, both half-immersed and totally submerged, due to an incident wave in deep water, were made by Milgram and Halkyard (1971), and Gray (1978) solved the scattering problem for the sumberged sphere, again in deep water, by formulating the problem as an integral equation and then expanding the Green's function and the velocity potential in spherical harmonics in order to reduce the problem to the solution of an infinite set of linear algebraic equations. This method is very complicated and it is much simpler to use the method of multipoles first used by Urseil (1950). Srokosz (1979), in considering a submerged sphere as a wave power absorber, solved the heave and sway radiation problems for a submerged sphere in deep water using this method. More recently, Wang (1986) has used the method of Havelock (1955) to examine the radiation and diffraction problems for a submerged sphere in deep water. In the field of wave-body interactions, comparatively little attention has been given to problems where the water depth is finite, mainly because they are more complicated and involve an extra parameter. Accurate determination of the hydrodynamic characteristics of bodies in water of finite depth has until recently relied almost entirely on purely numerical work; see for example Naftzger and Chakrabarti (1979). Evans and Linton (1989) used the multiple method to solve the two-dimensional problems of radiation and scattering of water waves by a submerged horizontal circular cylinder in finite water depths as part of a need to determine accurately the natural frequencies 61

62

C.M.

LINfON

of oscillation of a highly buoyant tethered cylinder, and in this paper the same method is used to examine the problems of radiation and scattering by a submerged sphere in finite depth. 2.

FORMULATION

Cartesian coordinates (x,y,z) are chosen with z = 0 being the undisturbed free surface and z increasing with depth. A rigid sphere of radius a is centred at (0,(),f) where f > a. Cylindrical and spherical polar coordinates, (R,cx,z) and (r,0,c~) respectively, will also be used (see Fig. 1). The fluid is assumed to be both inviscid and incompressible and the fluid motion is assumed to be irrotational and of sufficiently small amplitude so that the equations of motion can be linearized. Under these assumptions we know that there exists a velocity potential dO(x,y,z,t). We will further assume that all motions are time harmonic with angular frequency to and so we can write

• (x,y,z,t) = Re {+(x,y,z) e-i'°'}. 3.

(1)

THE H E A V E AND SWAY RADIATION PROBLEMS

The solutions to the problems of heave and sway will be denoted by ~o and OL, respectively. In the case of heave the body velocity is given by U° = Re { Ue ~'} k, where k is a unit vector in the z-direction, whereas in sway the body velocity is U 1 = Re { Ue -"°'} i, where i is a unit vector in the x-direction. The boundary value problems for + " , m = 0,1 are then given by V2+' ' = 0 K~b" + 0+-m = 0

Oy

in the fluid

m = 0,1

on z = 0

m = 0,1 (K

R

(O,O,f)

z

Fm. 1.

(2) = toz/g)

(3)

Radiationand diffractionof waterwaves

63

o~ m

0y

-

0

onz

= h

m = 0,1

(4)

o~ ° Or Or

- Ucos0

onr=a,

0---0-<~r, 0-
(5a)

- Usin 0 cos a

on r = a, 0-<0-rr, 0-
(5b)

lim

KR---.~

/K0m] 0

o 01

where K is the real positive solution of K = K tanh Kh.

(7)

It will be more convenient to write the body boundary conditions (5a) and (5b) in terms of Associated Legendre Functions. Note that P~(cos 0) = cos 0 and P~(cos 0) = sin 0, and so we have

Or

- UPf'(cosO)cosmot

onr=a,O<-O<-rr, O<-ot<-2rr,m=O,1.

(8)

In order to solve these problems the velocity potential is expanded in multipole potentials. Multipole potentials are singular solutions of Laplace's equation which satisfy the free surface and bottom boundary conditions and behave like outgoing waves far from the singular point, which in this case will be the centre of the sphere. Expressions for these multipoles in spherical polar coordinates are given in the Appendix. In particular, ~ represents a multipole potential which is singular like (l/r) n+l and has an azimuthal angle dependence of cos ma, and is given by :¢

[( a)n+l P~'(cos 0) + ~ ~bm = a cos met

A,~

(r) s P~(cos 0 )1

(9)

S=I

with A,~ given by (A5). From the boundary condition on the sphere, Equation (8), it is clear that the potentials for heave and sway can be written solely in terms of ~b° and ~b~, respectively. Let ~bm = U X b ~ ~bn"* n=l

m = 0,1

(10)

for some unknown coefficients bm. Note that the n = 0 term which could appear in the expansion for $o has been omitted. This term corresponds to a r -1 singularity at the centre of the sphere which is physically unacceptable, as it would imply an instantaneous flux of fluid across the surface of the sphere. With this expansion ~bm satisfies all the conditions of the problem except (8), application of which gives

r

"1

bm l - ( n + 1)P~'(cos 0) + ~ s Z ~ n=l

L

s=l

P~(cos 0)/ : Pf'(cos 0) J

rn : 0,1. (11)

64

C . M . LINTON

Use of the orthogonality relations

f7

/2/(2r+1) Pg'( c°s O) PT( c°s O) sin O dO = Sr*12r( r + l )/ ( 2r + l )

m=rn=O 1

(12)

enables this to be reduced to [-__3

bT-

[ ~ ] ~ A , , ~ b , , = - 8m, j 2 m

r 1 , 2 = ....

re=O,1

(13)

k''lJ

which is an infinite system of linear algebraic equations in an infinite number of unknowns. For the purposes of computation the system is truncated to an N x N system and approximations to the finite set of coefficients b~, r = 1,2 ..... N calculated. By increasing the value of N the convergence of the method can be checked and provided the method does converge (i.e. the values obtained for the coefficients b" tend to some limiting value as N --~ ~), greater accuracy can be achieved at the expense of computing time by using larger values of N. The system (13) is in fact extremely well behaved and very good approximations can be obtained with small values of N. A value of N = 4 was sufficient in virtually all cases, giving results accurate to three significant figures. The hydrodynamic force on the sphere in the direction of motion is given by Fm Re{fme -i't} where the time-independent part, f " , is given by =

f m = ptoi

+re(a, O, cx) P~'(cos 0) cos m s a 2 sin 0 d0da )

m =0,1.

)

(14) Using (9), (10), and (12), this reduces to

fm = ~,rra3Upo~i b'~ + ~ A,3 b'2 .

(15)

n=l

This can be simplified by using (13) with r = 1. If the force is then non-dimensionalized with respect to the mass of fluid displaced by the sphere and the maximum acceleration of the sphere (Uo~), we get expressions for the non-dimensional added mass and damping coefficients p m and v m. These are

p m + iv . . . .

(1 + 3b'~n)

m = 0,1.

(16)

The damping coefficient v m is related to the energy radiated to infinity; see for example Mei (1983, ch. 7), and this provides an alternative method for its calculation or can be used as a check on the numerical results. The identities that are obtained are (Ka)n

vO = 3~r (2Kh + sinh 2Kh)-' Ka

[e~Cr-m + ( - 1 ) ~e~{h-o] 2

'

b~--n-i .... I

(17a)

Radiation and diffraction of water waves

65

L 3,rr 1)1 : 2~a (2Kh + sinh 2Kh) -1 n=)" b~ (Ka)"÷l (n--l)[

[ eK
(17b)

Curves of added mass and damping coefficients for four spheres are shown in Figs 2-5. In all the curves the immersion depth to radius ratio is the same, namely 1.5. The different curves correspond to radius to depth ratios in the range 0 < a/h < 0.4. As a/h --> 0 the problem tends to that of infinite depth, whilst as a/h ---->0.4 the bottom approaches the sphere. Thus, as a/h increases from 0 to 0.4 the effects of a finite water depth should become more apparent. The multipole method discussed above is a particularly efficient method for solving the radiation problem for a sphere in finite depth, provided that Kh is small enough. Results are difficult to obtain if Kh > 8, but this is a region of little practical interest as the waves involved are so short. Also, the effect of finite depth is negligible in this region and so infinite depth results can be used. In the region Kh < 8 the convergence of the method is excellent with a truncation size of 1 giving accuracy within 2% for most parameter ranges. The results for a sphere with a/h = 0.1 are extremely close to those shown in Srokosz (1979) for the infinite depth case, and as can be seen in the figures unless the sphere is very close to the bottom the effects of finite depth are quite small. Figures 2 and 3 show the added mass coefficients for heave and sway, respectively. It can be seen that the nearer the sphere is to the bottom the greater its added mass. The curves shown in these two figures can all be considered as deviations from 0.5, the added mass of a sphere which is infinitely submerged in infinitely deep water.

1.0.

,,, a/,h=.39 a/.h=.3 a/h=.2 a/h=.1

0.9.

0.8'

............ ........... ", ....

¢/1 0 . 7 " 0'~ -0.6" "10

"10 0 , 5 " "lO rD

• 0.4. > 0 ¢"' 0 . 3 -

0.2

0.1

0.0 0.0

0'2

o',

0'6

0'8

1'o Ko

FIG. 2.

1'2

,',

;6

1'8

2.0

66

C, M. LINTON 1.0

•

0.9

aZh=.39

a~h=.3 aZh=.2 a/h=.1

0.8 0.7

O E 0.6 "O ~D "I~ 0.5 "O O

0,4 O 0.3

0.2 0.1 0.0

o.o

o:2

o:,

o:6

0'.8

~'.o

1'.2

,'.,

1'.6

1:8

20

Ka FIG. 3.

0.30

,-a/h=.39 a/h=.3 -'- a / h = . Z a/h=.'" 1 -

0.25 ~,/..-'"-~'-.,,"-/ ,. V ;'

ED 0.20

-

-

-

-

"",,',

;~ /

%,

E

"5. E

!

'.~ 0.15> 0

i

,I

'%'¢,; .,,,,

,'

",,

i i

¢" 0.10-

i I i I i /

".. ".. -,.

// 0.05 "

",.

;' /

0.00

0.0

0.2

0.4

016

0.8

110

112

114

116

118

2.0

Ka Fro. 4.

Figures 4 and 5 show that the effect of finite depth is to increase the damping coefficient for sway motion and to lower that for heave motion. Computations based on the analysis given in Evans and Linton (1989) show that this is the same as in the case of a submerged cylinder. Note that unlike the two-dimensional problem of radiation by a submerged cylinder, the sphere in deep water does not exhibit the property that the added mass and damping coefficients in sway are the same as those in heave.

Radiation and diffractionof water waves

67

0,30

,-, 0{h=.39 aZh=.3 -,-aZh=.2 --a/h=.1

0.25

I ~ 0,20 • ¢.(D,.

E 0.15"

>,, o 3: ¢/.1

/.--'72T'."~r~:.

0.10

J "

0.05 '

•

0.00

0.0

/~

" .

/

~-2,

z'

012

i

0.4

0.6

,

0.8

1.0

i

1.2

1.4

1.6

1.8

2.0

Ko Fro. 5~

4.

THE SCATTERING PROBLEM

The total scattering potential can be decomposed into two parts as follows: 4' = 4', + *

(18)

where 4'1 is the potential due to the incident plane wave. The potential ~ therefore must satisfy (2)-(4), (6) and the body boundary condition

0,

_

o4',

Or

on r = a.

Or

(19)

An incident plane wave of amplitude A making an angle ~, with the negative x-axis is described by 4', = _ igA cosh ~ h ) tO

_

cosn

exp[iKRcos(a-e~,)]

(20)

~ emimJm(KR)cosm(a-a,)

(21)

Kn

igA cosh K(Z--h) cosh Kh

m=O

where eo = 1, Em (A3), giving

=

2 for m --- 1. This can now be expanded in spherical polars using

-igA

4'1-2tOcoshKh ~

EmimCosm(ot--OQ)Z {(--1) m+~eK(h-f)

m=0

+ e "0~-h)} ~

(Kr) s

s=m

P?(cos 0).

(22)

6~

C.M. LINTON

For the radiation problems considered in the previous section the dependence on thc azimuthal angle c~ was known, but here it is not and so we must use a more general multipole expansion. We write

= -igA.

~,,.~,,r("~('+

ao.)

i

c,",'+m,, •

(23)

rn = 1 n~=m

It is convenient to redefine the unknown constants by putting d~ . . . ca~,,, . ,

k = 1,2 ....

m>l

(24)

so that all the summations start from 1. Thus

0

.

.

a(.o

[

.[

.

n := I

. + .

.

m- I n = I

d,, (b...... t

(25)

The boundary condition on the sphere is now applied and we can assume without loss of generality that c~, = 0 (i.e. the incident wave is along the x-axis from x = -~c). Using the orthogonality of the functions cos m(~, m = 0,1 .... on (0,2~r) and that of the Legendre functions, (12), yields an infinite system of equations for each of the sets of coefficients c~ and d,m, m = 1,2 ..... They are

c°-

[r

1

J_

r+i

,a,r

=I--,~C,,:2coshKh[(--1ye~(h

J ) + e ~(f " ) ] r ( r + l ) i

(26a)

and *"

d/'

L r+m- J ,,= ~ , A;7,,, 1, r~,,, ld;~' r + m - 1 1 (Ka) . . . . -1 r+m -/(r+2mzi)i

=

cosh

Kh [eK(r-") - (-1)reK("-f)]

m = 1,2 .....

(26b)

These systems can again be solved by truncation with an additional truncation p a r a m e t e r being the number of systems that are solved. The time-independent part of the vertical and horizontal forces on the sphere can be calculated from

fv = p~oi

)

+(a,O,~) P~(cos O) a ~ sin 0 dOd~

(27a)

+ ( a , 0 , a ) PI(cos 0) cos oLa2 sin 0 d0da.

(27b)

)

and

fH = ptoi I2-n)

f~ )

Performing the integration, simplifying and non-dimensionalizing leads to the results

f v = [(v/(pga2A )l = 4=fc?l f , = [f,,/(pga :'A )l --

4
(28a)

(28b)

Figures 6 and 7 show curves o f f v and ]eH plotted against Ka for the same four spheres used in Figs 2-5. Thus the immersion depth to radius ratio is kept constant at 1.5,

Radiation and diffraction of water waves

69

2.2

'" a/h=.39 a/,h=.3

2.0

-'-

1.8

a/.h=.2 a/h=.1

1.6 1.4

///

1.2

1.0

..

%,

"--..

/'v"./

0.8

i/

..

0.6 /~i21f] i/ I'" ,'"'" O!~af ." 0.4" o" 0.2 //," 0.0

0.0

0.2

0.4

0.6

018

1~0

112

114

1.6

Ko Fic. 6.

2.2

•............... ..

2.0

... a/h=.39

. . . . . .

•/

•"" . "'.. •~ ,................ .,.,,, "'..

1.8

a/h=.3

-'" a/.h=.2

~ a/h=.1

1.6 .

/

,-.:¢

,,,

1.4

,.._=

/ / /~ " / i ./ J

1.2

~-'~' ,,%,~..,.. "',

1.0 0.8 0.6 0.4 0.2 0.0 0.0

o7=

oi,

0'.8

0.8 Ka

1.o

,;2

,;,

~.6

FiG. 7.

while the depth to radius ratio varies. The curves therefore show the effect of reducing the water depth on the vertical and horizontal forces. It is noteworthy that shallower water increases the horizontal force but decreases the vertical force. This phenomenon has been noticed before by Naftzger and Chakrabarti (1979), for the case of a submerged circular cylinder in finite water depths.

70

C, M. LINTON

The Haskind relations [Haskind (1959)], which relate the exciting forces on a fixed body to that wave field radiated by the same body in motion imply that the coefficients c~ and a"~ must satisfy the identities (Ka) ''~'

c'? = (2Ka cosh Kh) ~] - n ! tl-

[eK(/-h) + (-l)'~eK(" /)] b',',

(29a)

] ze

(,,a),,~'

d~ =i(2KacOshKh) -I ,,:~, (n_l).t

[e KCj "~ - (-1)"e "(h /~] b],

(29b)

where the numbers bE are the solutions of (13). These identities were used as a numerical check on the results obtained from the radiation and scattering problems and in all cases good agreement was obtained. The total velocity potential in the scattering problem is given by

+

o~ [

cosh Kh

n=l

c"q'" +

m=l

n=l

d,, ~b,,,-,,, l J

(30)

and the free-surface elevation can be computed, using (A8), from I

.......

~ra

+ a -I

c,,+,,(R,(x,O) II =

1

+

d , , + ...... I?rl ~ ; I

It ~

,(R,(x,O)

.

I

(31) An example of the results that can be obtained using (31) is given in Fig. 8, which shows the effect that a submerged sphere ( a / h = 0.3, f/h = 0.45) has oll a unit amplitude incident wave with non-dimensional frequency K a = 1.

3¸

2

y/a

0

Above

1.15 1.10- 1.15 1.05- 1.10

-1 -2

!

-3

1.00- 1.05 0.95- 1.00 0.90- 0.95 0.85- 0.90

U~

-4

-4

-2

0

x/a FIG. 8.

Below 0.85

Radiation and diffraction of water waves 5.

71

A FAR-FIELD SOLUTION TO THE SCATTERING PROBLEM

In this section a far-field approximation to the scattering problem is developed based on results presented by Davis (1976). The velocity potential is decomposed as in (18), the incident wave is given by (21) and the scattered wave has the far-field behaviour t~ ~ - igA cosh K(z-h) o~ 2 cosh Kh ~

~,~i"a,,, H~ ) (KR) cos m ( a - a t )

as R ~

trl=0

(32) for some complex constants am. This is a general form for the scattered part of the potential for any axi-symmetric body. The unknown coefficients am are independent of aj. Thus as R ~ ~ the total potential satisfies zc

igA cosh K(z-h) ~- 2 cosh Kh

qb -

%,i"[(l +a,,)H~) (KR) + H~) (KR)] m=O

cos m ( a - e t t ) .

(33)

In order to evaluate the coefficients am we construct a radiation potential as follows ~bR = ~

+"

(34)

m=O

where +m is the solution to a radiation problem as discussed in Section 3, but with the body boundary condition

o(~m Or

P~(COS 0) = cOS met Pm( cos 0)

r=a

m = 0 m - 1.

(35)

Thus +o is the solution to the heave radiation problem and +1 that for sway. The remaining potentials ~bm, m = 2,3 . . . . . represent the radiation potentials resulting from some non-physical body forcing. The problem for the potentials qb", m -> 1 can be solved as in Section 3 by putting ~b" = ~] bm~bm = ~ n=n~

where e~'

=

e'a''n s-n+,,,-~

(36)

n = [

bkm~m_l k = 1,2 ..... The equivalent of (13) in this case is

Ir+m-ll C~ ] ,,=IEemAnm+m-1,r+m--1 ------ ~rl/(m+l) ~c

e~ -

r = 1,2,...

(37)

which reduces to the Equation (13) when m = 1. A careful analysis of the far-field behaviour of the potentials qb" shows that qbR

igA cosh K(z-h) 2coshKh ~, e,,B,, H~ ) ( KR ) cos mot rn=O

(38)

72

C.M. L~N[ON

where _

Bm

2ito 2"rriacosh Kh gAem 2Kh + cosh 2Kh n = m i n~][ I , m I c,~ [e Kct-h)

(Ka) ',-~l + ( - 1),,~ me~(h-29] i.nLm)!.

(39)

Applying Green's theorem to the total scattered potential + and the potential +R - d~R gives rise to the equation dS = 0

(40)

where S~ is a vertical cylinder far enough away from the body for Equations (33) and (38) to be valid. (Note that since +R satisfies (35) the potential ~n - (bn has zero normal derivative on r = a.) Substituting the asymptotic forms for ~b and ¢bR into (40), noting that the angle of incidence az is arbitrary, leads to the simple result am =

m = 0,1 .....

--1 + B,,,/Bm

(41)

Thus the far-field wave amplitude in the scattering problem is given by ~e,~l ( - 1 + B m / i L . ) H ~ I ) ( ~ R ) c o s m ( a - a I )

(42)

/rt =[)

which is expressed solely in terms of the solutions to radiation problems. If only farfield properties are sought, therefore, the need to solve the full scattering problem is eliminated. On the face of it this may not appear to be much of a saving since both problems require the solution of a number of systems of equations (4 was found to be an adequate number). However, the far-field approach eliminates the need to compute the velocity potential from (A8) which can be very time-consuming, especially if R / h is large. Numerical comparisons between (31) and (42) show that the far-field approximation, (42), is a good approximation if KR is large enough with KR > 5 giving agreement within about 1%. 6.

CONCLUSION

The method of multipoles has been shown to be an extremely powerful method for solving radiation and scattering problems involving submerged spheres, thus eliminating the need to use large and cumbersome numerical packages for the solution of such problems. Clearly the method discussed above is only applicable due to the simple nature of the geometry, but would provide an approximation to problems involving almost spherical bodies and also could be used as a check on numerical codes suitable to more elaborate geometries. AcknowledgementlThc author would like to thank ProfessorD. V. Evans for the suggestionof the problem

and useful comments. CML is supported by SERC (MTD Ltd) grant No. GR/F/32226. REFERENCES DAVIS, A.M.J. 1976. A relationl~etween the radiation and scattering of surface waves by axisymmetric bodies. J. Fluid Mech. 76, 85-88.

Radiation and diffraction of water waves

73

EVANS, D.V. and LlNTOrq, C.M. 1989. Active devices for the reduction of wave intensity. Appl. Ocean Res. 11, 26-32. GRAY, E.P. 1978. Scattering of a surface wave by a submerged sphere. J. Engng Math. 12, 15-41. HASKlrqD, M.D. 1959. The exciting forces and wetting of ships in waves (in Russian). lzv. Akad. Nauk. SSSR, Otd. Tekh. Nauk. 7, 65-79. English translation available as David Taylor Model Basin Translation (No. 307). HAWLOCK, T.H. 1955. Waves due to a floating sphere making periodic heaving oscillations. Proc. Roy. Soc. Lond. A231, 1-7. HuI.ME, A. 1982. The wave forces on a floating hemisphere undergoing forced periodic oscillations. J. Fluid Mech. 121,443-463. MEI, C.C. 1983. The Applied Dynamics of Ocean Surface Waves. Wiley Interscience, New York. MILGRAt~,J.H. and HALKYARO,J.E. 1971. Wave forces on large objects in the sea. J. Ship Res. 15, 115-124. NAVrZrER, R.A. and CnAKRABARXLS.K. 1979. Scattering of waves by two-dimensional circular obstacles in finite water depths. J. Ship Res. 21, 32-42. SRO~COSZ,M.A. 1979. The submerged sphere as an absorber of wave power. J. Fluid Mech. 95, 717-741. THORNE, R.C. 1953. Multipole expansions in the theory of surface waves. Proc. Camb. Phil. Soc. 49, 709-716. URSELL, F. 1949. On the heaving motion of a circular cylinder on the surface of a fluid. Q. J. Mech. appl. Math. 2, 218-231. URSELL, F. 1950. Surface waves on deep water in the presence of a submerged circular cylinder--I. Proc. Camb. Phil. Soc. 46, 141-152. WANG, S. 1986. Motions of a spherical submarine in waves. Ocean Engng 13, 249-271.

APPENDIX:

SPHERICAL

MULTIPOLE

EXPANSIONS

Thorne (1953) gives expressions for three-dimensional multipoles in finite depth. F r o m his paper, multiplying by a "+2 for convenience and factoring out the time dependence gives

qb~ = a cos rna

[i:],+,

P'2~(cos 0) + (n~m).V Jo Kh cosh kh - kh sinh kh

[e k0~-h) (K sinh ky - k cosh kz ) - ( - 1 ) . . . . ( K + k ) e - k l c o s h k( h - z ) ] dk 2"tri (Ka) "+l cosh K ( h - z ) ] + ( n - m ) ! (2Kh + sinh 2Kh) [eK0~ m + ( - 1 ) ..... eK~h-f)] J,, (KR)

]

n = 0,1 ...

m-< n.

(A1)

Writing q ( u ) = - ( K h + u) ( e - " + ( - 1 ) ' + m e "(l-2i/h))

(A2a)

c2(u) = (Kh - u) e "(2r/h-I) - (Kh + u ) ( - 1 ) " + ' ~ e - "

(A2b)

c3(u) = 1 + ( - 1 ) . . . . e 2"~l-r/h)

(A2c)

c4(u )

(A2d)

=

e2,(y/h ,) + ( - 1 ) " + "

and using ( +_kr).~ e -+k(z+I) Jm(kR) = ( +-1)m *--,nZ(8~m)5 Pro(cos O)

(A3)

shows that Equation (A1) can be reduced to n+

+~ = a cos ma

1

r

P',"(cos 0) + ~ .s=m

s

A'~ ~"(cos 0)

(A4)

74

C. M. LINION

where

(a/h) ...... '

[(~ ( - 1 ) ...... c, (u) + <~ ( . )

A:,", = 2 i n ~ m i i ( s + m i i [ ~

( - 1 ) ...... c3(Kh) + c,(~h)] + 2wi(Kh) ...... 1

....

1/h cosh i i : u s i n h u

2Kh + sinh 2Kh

J

u

du

n>-O'm<--n

(A5)

This expansion for 6;;' is valid in the region r < 2)" (see Thorne, 1953). On the free surface (z = 0) the region of validity of this expansion will be insufficient unless we are very close to the sphere and so (A1) must be used. On z = 0 the spherical and radial coordinates are related by r = (R: +F-) l'~-

(A6)

cos 0 = -fir.

(A7)

and

Substituting these formulae into (A1) gives a slightly simpler form for the velocity potential on z = 0, namely +',;'(R,a,0) = a cos m a [[(RZ+fe),/~ j

P~' (R ~

ii/5

(a/h)"+' I ~ u"g,(U)Jm(uR/h) du + 2 (n -n~)! Kh cosh u - u sinh u + 2rri

+' g2( h) 2Kh + sinh 2Kh

JJ

(AS)

where

gl(u) = - 2 ( u e ''(s~h i ) + ( - 1 ) ..... e 'e/h(Kh + u)cosh u)

(A9a)

gz(u) = e'¢/h(1 + e - : " ) + ( - 1 ) ...... e-'¢~h(l+eZ").

(A9b)

It can be shown that (Thorne, 1953), "m cosh K(h--z) ,,,I, ~b',7- a A,, cosh K h - rt~,,, (KR) COSmc~

(AIO)

where 2"n'i (Ka) "+' cosh Kh

A;;' = (n-m)! (2Kh + sinh 2Kh) [eK~/-hl + ( - 1 ) ...... e K(h "3]

(All)

and thus that (bnm Iz=(,

~ a m~ n --m/4(1) (KR) c o s m c ~ .

(AI2)