An axis-symmetrical dock in waves

ARCHIEF

SEOUL NATIONAL UN W. RSITY

COLLEGE OF ENGINEERING

DEPARTMENT OF NAVAL ARCHIÌECTURE

AN 'AXIS-SYMMETRICAL DOCK

IN WAVE

by

H. Isshiki and J.H. Hwang

SHIP HYDRODYNAMICS LABORATORY

Report No. 73-1

DeIfL

by

H.

Isehiki and J.

H. Ifwang

Report No. 73-1

SHIP HYDRODYNAMICS LABORATORY

Department of Naval Architecture

College of Engineering

Seoul National University

Seoul, Korea

Abstract

Introduction ....

Linearized theory of motion of an axis-symmetrical

dock 3

Variational formulation of the boundary value

problem

lo

k. Rayleigh-Ritz procedure for the variational

formulation 1k

5. Numerical calculations and discussions 22

RefQrences 23

Appendix A 25

Appendix B ₂₇

Tables ₂₉

1. Introduction

Motions of an axis-symmetrical dock freely floating on a free surface of water are important for designs of

artificial islands or ocean platforms. _{The hydrodynamic}

properties of a floating semi-sphere were discussed by

Havelock (1955) and Barakat (1962), and those of'

semi-spheroias by Kim (1965) and Sao, lVlaeda & Hwang (1971). By H. ISSHIKI AND J. H. HWANC

Department of Naval Architecture, College of Engineering,

Seoul National University, Seoul, Korea

Linearized motions of an axis-symmetrical dock freely

floating in a regular plane wave are discussed. An

extension of the Bessho variational principle (Bessho (1968)) is derived to obtain a numerical procedure for a solution of the boundary value problem associated with

the fluid motion.

The added mass and the damping coefficients for a circular dock in vertical (heave) and horizontal (surge) oscillations are evaluated numerically, and the results

Miles & Gilbert (1968), Garrett (1971), Miles

₍₁₉₇₁₎

and

Black, Nei & Bray (1971) discussed wave forces on a circular

dock. In the present work, a general variational formulation

for radiation and scattering of water waves by an

axis-symmetrical dock is obtainedas an extension of the Bessho.

variational principle (Bessho

_(1968),

Isshiki

(1970, 1972),

Mizuno

(1970),

Sao, Maeda & Hwang

(1971)).

Sao et al. also discussed heave oscillations of _a

circular dock on the basis of the Bessho variational _principle

and slender body assumptions. _{But their method seems to be}

out of their assumptions for this problem.' _{In this paper,}

the same problem is dealt with from a little different _point

of view. _{An extension of the Besaho variational principle}

is obtained, and the Rayleigh-Ritz _{procedure is derived based}

on the variational formulation and an admissible potential

similar to that by Miles & Gilbert (1968).

The added mass and the damping coefficients _are.

calculated for a circular dock in _{heave and surge oscillations.} The numerical results seem to be reasonable, and are

compatible with the results for _{a sphere and epheroids.}

Near the end of the _{preparation of this paper, the}

authors received an.interesting papBr from Dr. K. J. Bai

(Bai (1972)J which also _{discusses linear water wave problems} from the standpoint' of variational calculus. _{He regards}

the Sommerfeld radiation condition _{as a boundary condition'} on a circular cylindrical surface which is assumed to be

located far from 'a body, _{and approximates the}

boundary valu

problem in an infinite region _{by a problem in a finite}

fluid region within the cylindrical _surface.

Let an axis-symmetrical dock be freely floating on a free surface of water as shown in figure 2.1. Q(x,y,z)

Fig.21

is a right-handed Cartesian coordinate system fixed in the

space. x- and y-axes are taken on the calm water surface, and z-axis is directed vertically upwards. (r.,,.z) refers to the space-fixed cylindrical öoordinate system Buch that

= r cos Q

(

y=r sin Q.

Let G _{be the center of mss of the dock, and be on the} axis of the dock. _{At an initial instant of time, the dock}

is assumed to be in its hydrostatic equilibrium with _z-axis coincident with the axis of the dock. Furthermore, the equilibrium is assumed to be stable.

Let the free surface elevation %(x,y)e_it

_{of a}

regular plane wave, which travela.in water of the depth d

along x-axis..from left to right, _{be approximated by}

= 0 e

where

Jm(1C coB mQ,

i = imaginary unit = circular frequency

ampiitude of the free surface _elevation

for m=O

for _{m = 1,2,3,}

= Bessel function of the fist kind of order

2.1):

YO X,y, z

. coshk(z+d) jm (kr) cos 1n, 'O d cosh kd . m m

whére g is the gravitational acceleration.

Since is on the axis of the axissymmetrical dock, themotiön of the dock ,de to the regular plane 'wave (2.2). is assumèd 'to be confined to xz plane. If is

sufficiéntiy small compared with a' principal dimension'

of the doc:k, the. motion of the dock may be assumed tobe small from the assumption of the stable hydrostatic

equilibrium of the dock Then, the motion of the fluid

also bécornes small. The linearized theory Is assumed in

the followings. Let

ÍGe_t

, ZGe_'* and

be the stationary harmonic parts of .the horizontal, the vertical and the angular displacements of G respectively (figure 2.2), and 9(x,y,z) be the velocitr potential .

Fig.2.

of the fluid motion. Then, theunknowns X _,

and Y are the, solutionsof the following linearized

boundary value problem (Wehausen & Laitone (1960)):

(.+

2+)0

òx ày òz

nx(idXG)

+

nz(_iZG)

-

_'

cosh. k(z+d) eC O 'cósh lcd in

The velocity potentia.10(x,y,z)

e_t

of the regular incident wave (2.;2) may be approximated by

2

( )5=O

on _(2.5è)

y=o

at z = -d (2.5d) 4J ( ik )(',) «-'

o()

as

r +,

(2.5e) and 2N

_{XG = °?}

Ss dS (2.6a) + S% _{) ZG} =

1P Ss

n2 dS (2.6b)

= iclp J5{xn2. _{- (z_zG)nXJ.ffas, (2.6e)} where

p = density of the fli4d

D = mean region occupied by.the fluid S = man wetted surface of the dock

F5 = mean free surface (i.e., calm water surface) V = mean displaced volume of the dock

S _{= mean water plane area of the dock}

=

(n)

= unit normal of S into the fluid = height of G above calm water surface

M = mass of the dock

'G = moment of inertia o _{the dock corresponding} to the angular displacement

I

= J (Z_2G) dV

=

Js

2

dsw.

(2.5e) is the Sommerfeld _{radiation condition..} Since (2.5b) can be written as

= - zn =

Substituting (2.8) into (2.6), the following equations are obtained: 2M X ₌

_iøpçoFox

_-

_idpO1DI

d2pxaFxx _-

d 1OX+ zGFxx)

(-d2M + pgS)Z.

_{= 4poFoz -}

_PODZ

-(_2

+ pg I + pg

)O

=

-ip0(F

_{+ ZGFOX)} - + ZGPJJX) - d2pXG(PxO + ZGFXX) - + ZGP + ZGFOX +

¼2

(2.8)

(2.9)

Qn S (2.10) (2.11)

y

₌

- Z

+ nZ(icZG) + (xn

- zn)(_i

,5 may be decomposed as = o

o +

+ (_iÖx

- ZGIÓG)X

+ (_iccZG)$Z + where - j cosh k(,z+d eu1 cosh lcd

00

-

ii cosh k(z+d) '-" cos mO, cosh kd m=O m and

D and are harmonics in

D which

satisfy the free surface condition, the bottom condition, the Sommerfeld radiation condition and

ô A

ônF'D

A

F1j=j1fjdS=jiidS

(I = o,D,X,Z,e; J =

It must benoticed that and F1 (I,J = D,X,Z,)

are determined by th geometrical characteristics of 3

Because of Green's integral theorem, the hydrodynamic

force (I,J = D,X,Z,e) is equal to . i

-=

Js[$I

_{_.J}

-] dS = O

(i,J =D,X,Z,).

Let X , Z and be the displacements of the

point fixed to the dock which has the mean coordinates (x=y=z=0)...±t' then follows that

XG=X_zG

ZG=Z

From (2.10), .-iX, $ and -i.

are the.

velocity, potentials due to the simple harmonic oscillations of the dock in still water corresonding to the

displacements X , Z and respectively. From the

pressure equation and (2.12),

_-2p{x,z,OJF1

_{(I,J = X,}

z,e) is the J-th component o± the hydrodynamic reaction due to:the I-th motion (I = X,Z,). Let

be the component _{f the hyd.rodynamlc reaction with repeet}

tothe center of mass G _{coxresponding to the dispiaòement} (XG,ZG,©G). Then, the relations between F1 and

can easily be obtained as . , . .

(2.12)

(2.13)

GZZ = ZZ

9 = PlrI + ZF)( + z&F + ZG' xx

= F

_{+ ZGXX}

GX = + ZGFXX,

where other components are equal to zero from the geometrical and the kinematical syminetricities.

Re[-pF13]

and Im{-pF1:l

1) (I,J = x,Z,) are

called the added mass and the damping matrices corresponding to the motion (X,Z,), since

'j

= {-d2X,-d2Z,-d} Re [-pP1 _I

+ {icX,id'Z,ic$O} Im[-dpF1]. (2.16)

Let F (I,J = 'X,Z,) be the complex conjugate of P1

It then follows that

2k im[F1j] _{= PIJ}

-*

*ò$I

= (O (21L _* = 11m dz t _{rdQ [}

-rJ-d

JO

i âr J'

since, from (2.10),

$1/n

(I = x,z,) is a real

valued function on S . Hence,

Im[F1]

(I,J = X,Z,e)

can be determined by the assymptotic behaviour of the corresponding potentials. Let the assymptotic form of

i) Re[ J and Im[ J mean to take the real and the

imaginary parts of the quantity in the square bracket.

(2.17) (2.15)

(I

= x,z,ÇJ)

, vhen r tends to infinity, be given as

A1()

_{cash k(z+d) eiOE}

r

d cash k

substituting (2.18) into (2.17), 2i Im[P1j] can be

derived as

2ig2 d cosh kd sinh kd 2i Im[P1j3

2(cosh k5' + 2k (27

j

4(Q)

(I =

x,z,)

represents the wave

excitation force to the dock fixed in the space,

and

_iccp0P01 is called the FroudeKrylOV force. Prom Greèns

integral theorem,

_OIDI

can be written as

o$ DI =

505D

1

dS

J3[$o

- i (0 (27t sI =

hm

dz \ rdQ[ 5

-r+DoJd

JO

Substitution of (2.9) and (2.18) into (2.20) leads to the following expression of the wave excitation force:

2j/3E. g e2 ilt/4

d cash kd sinh kd

'0IDI

2(cosh kd)2

(

2k

A(1L).

This relation is called the HaskindNewman relation

(Newman (1962)).

(2.18)

(2.19)

(2.20)

3. Variational formulation of the boundary value problem

Prom discussions in 2, it can be deduced that. a

boundary value problem of the following type must be solved to determine the wave forces and the hydrodynamic reactions:

2 2 2

mD

(3.la) ay öz on S (3.,lb)

(Ô2)

on

at z=-d

(3.ld) jf ( - ik ) $ -..-0() as

r -oo

(3.,le)

Bessho (1968) and leshiki (1970, 1972) obtained an variationál expression for the hydrodynamic force associated with the

boundaryvalue. problem (3.1). In the followings, an extension of their theory will be discussed.

Devide the region D into two subregions D and DU

as shown in figure 3.1, that is

I

U

D=D +D.

I

I

The intersection of S and D is denoted by S , and

U U

I

U

that of S and D by S . Let $ and $ refer to

I

U

the velocity potential _$ in D

and D

respectively. Then, the boundary value problem (3.1) may be formulated

F1S3.1

(3.2)

I

in terms of $ and a2 U $ + o2 as follows: r in in on on

I

D U D

I

S (3.3a) (3. 3b) Ox2 + dy2 Oz2

The boundary value problem (3.3) can be transformed into a

I

variational problem. Let L{ ,$ ;f] be a functional of

z ir

the argument functions _$ and $ such that

L [$'

's;

±' = (

4+

$ dD +

7=I,TIJS

-f (

'_ dF8 F8 Oz g ,. =1 ! ôz dxdy

+_1z

:ir( JD,-D Or d(DtDR)

It then follows that

'L=2

A=1,xJD

ôx2 Oy + 2

t: I

₍

'-

_f)

_dS On - 2

₁

₍

'-

$'

_dF Oz g dD a 2 = I

= o

C ik )

3' o()

on at as

I

D on z = -d r -,co, and

IiI

D,-D D : (3 3c) (3.3d) (3. 3e) (3. 3f) (3.:g) A,Í

and the conditions

a

.;p

-on the intersecti-on of =

a

1

=p

:. z (O (27t.

- uni

I dz _I rd@ ₍

Á1I)

p

rooJ-d

Jo

+ 2

JD',.D

Á

à(D'i-J?). (3.5)

It is easily seen that (3.3a-d) and (3.3g) are the stationary conditions of the functional L under the subsidiary

conditions (3.3e) and (3.3f). Hence, the following variational problem is equivalent to the boundary value

problem (3.3): L[,3',?;f] = stationary, (3.6a) under .ff ( -

ik )

t'-, O() as r

-tao

(36b)

I

I

Z = on D,-D . (3.6e) X

It must be noticed that _L[

_,$

_;f] is a variational

expression for a quantity -Çf dB , that is

iS

= -

dS, _(3.7)

,=I ,fl

JS

when $ (¼=x,1!) is the solution of the variational problem (3.6).

If the natural conditions (3.3a), (3.3c) and (3.3d) of the variational problem (3.6) are also considered as the subsidiary ones, the following variational problem is

then obtained:

+ 2

L1

dxdy

undé r

M

-.1

.

t

E ,

;f] =.L[

,

;f]

2f) t

dS

J'cD

Ç?-stationary, a2

ay

az

2 =

I

a 1

l$1r

= O

.,,ç

-. ¡k )

'....'o()

.1

E

mD

E in. D on

as

r

-too,

I

If on D,-..D

The natural cönditions of this variational

problem are

as

follows:

,Ön

(3.8g)

= on

D',-D.

(3.8h)

The

variational problem (3.8). is an extension of the

Bessho

variational formulation (Bessho 1968).

-..

I

U'

If the continuity of the normal velocity. on

_D,-D

is assumed, a complementary variational fprmulation of.

(3.6); oai be ôbtained as follows:

E. ,on S (3..8a) (3.8b) (.3.8e) (3.8d) (3.8e) (3,.8f)

-I

:ii

-

'-i

I

-

r $fl) d(D',-.D)

Jii ,-j

= stationary, _(3.9)

under (3.3e) and (3.3g). (3.3a-d) and (3.3f) then become the natural conditions of this variational problem.

Corresponding to (3.8), it then follows the following complementary formulation: * i t M 1 ,$ ;f] = M[ , _;f] - 2

Ç

₍

''_

$) d(D'r?)

JD,.D

= stationary,

under (3.3a), (3.3e-e) and (3.3g). _{Then, (3.3b) and} (3.3f) are given as the natural conditions _{of (3.10).}

4. Rayleigh-Ritz procedure for the variational formulation

An admissible function for the variational problem (3.8) will be obtained _{in the followings, and a} Rayleigh-Ritz approximation based on the admissible function will

be discussed. Let _$

00

= Em jm cos inQ (3.10)

I(cXr)F J(ftr) H(1) _(pr)

3t'(r,z)

= E

eos mQ

t

in D K in D 1

[e' + em+/2)71]

1/27cjr 1

[ei{_12m)/41

+

eiß«2m+1)/4}

s/2JLr

i i{r-7i(2m+]j/4}

Vmre

(4.4) (4.5) 4.1)

be a solution of the Laplace equation:

r2

(4,2)

It then follows that

j; j; + ) 3b1

= o.

The solution of this equation by the separation of the variables are as follows:

Km()1

(cos cXz Im(atr)) sin

dz,

where

= real parameters

'm' Km = modified Bessel functions of order m

H1)

_{= Hankel function of the first kind of order}

m

I(cfr)

,

Jm(r)

are regalar at r = O , and _Km(cfr)

unbounded at r = O . When r tends to

infinity, these functions have the following assymptotic

expansione:

e0

J(r)

1r

f

e

H(r)J

I

e,

PrOm the propertis at r 0, cO.

and

the condition at

d ,it may be assumed as follows:

)bX(r,z)

-)(r,z)

K(cr) cos c«z+d

I(cI'r) cos '(z+d)

The free surfacé cond tio. (3.3e) then yields the following eigen-value equations for and.

(4.7a) has an infinite number of roots

and the solutiön of (4.7b) is equal to the wave number of shallow water k (see (2.3)). Hence, 3b1(r,z) _{may be}

assumed as

X'

_(r,z)

= A

K(r)

cos 1(z+d)

B' H(kr) eoshk(z+d)

m m

coshkd

where': = linear combination of o(

tan. cd

= -d2/g

ß tanhpd=

where A1

(i=l,2,

. )

and

Bm

are integral constants,

and r0 is the radius of the

cylinder

DK.

(figure

.1).

From (3.8f),

3C(r,z) must be,a function such that

. . )b(r0,z).= 3b'(r Let '(r,z) be decomposed as z, _Ku

(-r,z)

= 'th (r,z) + 'm (r,z), = r) cosh ß(z+d)

and-i

(46)

cosh 1ß'(z+d).

i:

ce.>Q

ß>O.

.1 K

on D,-D

I

On

DpD

4,8)

4.9)

(.4 .lOa) LiJ ,z) = C)

Then, it may be assumed that rio

I(c1r)

'"m (r,z) = A I

r)

i=1 where B mi 00 _{cosh k1(z+d)} "m (r,z) =

B, J(k1r)

cosh k d 1=1 _mi

are integral constants,

_and

=

m,i1O

Hence, 1//(r,z) _{may be assumed as}

00

_I(.r)

= Ami

_{i (OEr)}

cos c1(z+d)

+ B' H1)(kr0)

J(kr)

cosh k(zd) m J (kr0) _{cosh kd} in ori _{cosh k (z+d)}

+ T

B. Jrn(kmjr)

coshk.d

In the followings, _{an approximation based}

on the

variational problem (3.8) _{and the admissible functions}

(4.8) and (4.14) _{will be considered.}

For the sake of simplicity, a numerical procedure will _{be discussed for} a circular dock of the _{radius a}

(figure 4.1). _And r0 is assumed to be _{equal to a} (i = cosh k(z+d) + B J (lcr) in in

coshkd

where

r

A . = A . (i = 1,2, ₎ mi ini (i B J (kr0) = B H (kr0). _(4.:llc) in in m m

Let j . be the ith zero point of the Bessel function

:

in, i

= o

_0<i1<i

< . (4.12)

m _m,2

Then, (r,z) may be written as

1,2, )

cos

c(z+d)

(4.14).

Substituting (4.1) and (4.15) into the functional

I

M[5 ,,S ;f]

M

_-

2f'(z,) ) _Jr=a

adQdz

N[$',?;f]

1-(d-h)JO

ta(27t + J

[(

-

-

2f (r,e) ) _]z=-(d-h rdrd9

joJo

1-(d-h)

_12n[M'

₁

J-d

j

r är ir=a ad9dz, (4.16)

the following expression for M is obtained:

z 00 z u z ut.,

M[5 ,5

;f] = _{Em(_i) Mm[3'.t'm;fm,fm.J,} where

Mm[XE!

Y'm,Y'm;fm,fm]E

Io

[

-

a ( - - 2f(z) ) b1 ]r=a dz - J-(d-h) u 2f(r) ) _{rn]z=-(d-h) rdr} (-(d-h) _Y' + a I

[]r=a dz

J-d

1_(d_h)

J-d

In r=a dz.

-a

Substitution of (4.8) and (4.14) into (4.17b) leads to the following expression of _{Mm :} (4.17a) (4.l7b) Let f f =J be expanded on S

I

(z,Q)

as

I

(fm(z) cos m on on z S S (4.15) = E jm m

1f (r,)

-U

i

n

-

IA .,B ,Bmi;fm,fm]z z n

r

it

in'- mi in

00 00 _{z r}

_r

00

zr

=)-

[A,A]

A

i1 j=1

ni ni ij miAmj +

>-

i=1

[A,B]1 AB

r1

I

12

00

I

It

I

U + 'B ,B

i

(B '- ni

m'

ru +

y-

[B,BJ. B

B

i

m ini i=1 00 00 _{r ir}

_I

_it 00 00 +

E:

AmiBmj +

E

i=1 j=1 _{1=1 1=1}

[BB] BBJ

00 _X

_I

_I

_I

U Ir

-

2

y-

_{[AJ± Ami - 2} [Bm] B

-

2 >

[B1i

, (4.18) i=1 ₁₌₁

where [AA]

, etc. are given in appendix A.

Because of Green's integral theorem, the following relations

hold:

[AA]

=

=

[BB]1

(4.19)

From (4.17a)

and (4.18),

_{the stationary condition of}

the functional M is given as

Mm[Ai,3,Bi;fX,fU

_]=

stationary

_(4.20)

Hence, the following linear

X

I

algebraic equation for the

_unknowns

Ami B

and

_Bmj is obtained:

r

r-

x

r

r

r

rit

_{E x}

'T

2[Am,A ijAmj + [A ,B ] B +

y-

[A,B:li

_.B

= 2[Am]i in

mirn

[t

AB

I-

Z

I

X

I

I

_{E it}

_I

. +

2[a,

]B +

[B ,B

J B

_{= 2}

mjm

_mm

=i ru in

j

mj [B'] (4.21) for _{in =} 0,1,2, 00

YT [A»].iAT.

j

=1 +

[I

U

I

_{It Z ir E}

BmBmiiBm

+ = 2[Bm]i

I,

I

U where i _{= 1,2,} . IA I. ,

[b]

and IB

i.

are

rni

'-determined according to the conditions on S . These conditions are given iii table 4.1, where "heave", "surge" and "pitch" refer to the vertical (Z) motion, the

horizontal (X) notion and the angular () thotion about y-axisrespec±ively ((2.10)). By using the formulae in appendices A-B and table 4.1, ,

[B]

and

for these motions can be written exlicitely as shown in

table 4.2. In numerical calculations, the infinite series

in (4.21) are truncaed byfinte tèms according to an

approximation:

I

=.O for

i'P

E (4.22)

= O for

where P and Q are appropriate integers determined by the

numerical convergence.

The added mass and damping can be approximated as

follows. From (2.12), (2.16) and (3.8), these quantities

p

e-= j.[ M[ _{,$ ;f] ],.}

pIm)

-

I

E

where (.$ ,$ ) is the exact solution of the variational

problem (3.8). i'rom (4.17a), it then follows that added mass damping. where

m=1

P

Re}[

],

apIm

for. heavé

for surge or pitch.

4.23a)

(4. 2 3b)

can be written as

added mass

-pRe

_(.

j=.

f. dS _E $

_dB]

I.

Let _Ami (i. = 1,2,...P) B and _Bmj (ï 1,2,....Q)

be ail approximate solutionof (4.21) based on the assumption

I.

n

(4.22), and ,)" ) the corresponding approximation of'

(i,,y) . Then, may be approxImated by

Mm[31 ,3

;f,f]

, that is

ME,;f] M[',4

,3bm

From. (4.18) and (4.20), the following expression for

Mm[)I ,Y'

,f,f]

is obtained.

i z ir z n

z x n z

n

mrn

m'

Ç]

=

M[A

Bm ,Bmj;fm,fm]

___ T

T

Î

= T

[A1]i Ami -

[Bj

The óoupling force such as the surge (pitch) component in

itch (surge), is also easily obtained (,Issh±ki (1970)).

If the numerical convergence is satisfactory, the radiation

V _V.

z.

wave amplitude may be calculated from _Bm. by using the

assymptotic expansion of H(kr)

((4.5)) and the linearized dynamical condition bn the free surface:

_jV

. -idt V

=0

at z=0,

where is the free surface elevatiom. _{From the}

Haskind-Newman. relation (.2.21), the waver excitation force

will bé easily' caicu.lated.l). V

i) From .a standpoint.6f thé _{variational calculus,, the}

radiation wave amplitude br the wave excitation force should be. approximated by using the _{radiation and the}

catteringpötentja1s (Isshjki. (1970))..

(4.24)

[B] B:

. (4.25)

5. Numerical calculations and discussions

The added mass and damping of a circular dock (figure 41)

were calculated on the basis of the numerical procedure

developed in

54.

The numerical calculations were carried out by the computer IBM 1130 at Seoul National University,

and the single precision (six significant figures) were adopted.

The added mass and the damping coefficients are defined

as follows i

added mass coefft. = added, mass/Ep2(d-h)]

2 (5.1)

damping coefft. = damping/['p7a (d-h)].

In tables

5.1-2,

the numerical tendancies of the added mass

T.Lb.

5.1

and the damping coefficients with increasing P and Q are

Tab. 5 . 2 shown, and seém to be satisfactory. From tables

_5.3-4,

Tab. 5. 3

the convergence of _Bm seems also to be reasonable.

Tab .5. 4

In figures 5.1-2, the frequency dependencies of the

Fig. 5.1 added mass and the damping coefficients are shown. _The

Fig. .5. 2 free surface effects seem to be compatible with the results

of a semi-sphere by Havelock

_(1955),

Barakat

₍₁₉₆₂₎

and spheroids by Kim

(1965),

Sao, Maeda & Hwang

_(1971).

The

horizontal force on the dock ( d/a =

0.75,

h/a =

0.25, 0.5

₎ due to wave ( k =

1.32

₎ was compared with the correporiding results by Garrett

_(1971),

and the agreement was satisfactory.

The authors are deeply indebted to _{Mr. 1. H. Kim}

and Mr. K. P. Rhee for their unlimited cooperations _{in the}

R}FERENCES

BAI, K. J.

₁₉₇₂

A variational method in potential flows

with a free surface. Dissertation, Univ. of Calif.,

Berkeley.

BARAKAT, R.

₁₉₆₂

Vertical motion of a floating sphere

in a sine-wave sea. J. Fluid Mech.

13, 540-56.

BESSHO, N.

₁₉₆₈

Boundary value problems of an oscillating body floating on water. Memoirs Defense Acad. Japan

8, 183-200.

BLACK, J. L., NEI, C. C. & BRAY, N. C. G.

₁₉₇₁

Radiation and scattering of water waves by rigid bodies. J. Fluid

Mech.

46, 151-64.

GARRETT, C. J. R.

₁₉₇₁

Wave forces on a circular dock.

J. Fluid Mech.

46, 129-39.

HAVELOCK, T. H.

₁₉₅₅

_{Waves due to a floating sphere} making periodic heaving oscillations. _{Proc. Roy. Soc.} Lonion, Ser. A

251, 1-7.

ISSHIKI, H.

1970

Variational principles associated with

surface ship motions. _{Korea-Japan Seminor on Ship}

Hydrodynamics, Soc. Naval Arch. Korea.

ISSHIKI, H.

₁₉₇₂

_{(under publication) Variational principles}

associated with ship motions in _waves.

KIN, W. D.

1965

_{On the harmonic oscillations of a rigid}

body on a free surface. _{J. Fluid Mech.}

_{21, 427-51.}

MILES, J. & GILBBRT, F.

₁₉₆₈

_{Scattering of gravity waves}

MIZÌJNO, T. 1970 On sway and roll motion of some surface-piercing bodies. J. Soc. Naval Arch. Japan 127,

55-70.

NEWMAN, J. N. 1962 The exciting forces on fixed bodies in

waves. J. Ship Res. 6(3),

10-7.

SAO, K., MAEDA, H. & HWANG, J. H.

₁₉₇₁

On the heaving oscillation for a circular dock. J. Soc. Naval Arch. Japan _{130, 121-30.}

WFHAUSEN, J. V. & LAIT0NF, E. V.

₁₉₆₀

Surface waves.

Encyclopedia of Physics vol.

9, 446-778,

Berlin: Springer-Verlag.

Appendix A.

K(.5)

O cos c(1(z+d) cos c(z+d) dz [AmAm]ij =

ca

_Km(ctia)

_d

+ Im(cia)Im(ci.ja)

r Im(ctjr)

i(cr) dr sin

cth

cos d.h

I(ca) ((dh)

cos d(z+d) cos

oz+d) dz

- dia

Im(ja)

Jd

K(ca)

_i

=

{ka H'(ka)

+

dja H1(ka)

_K(da)

_{}coèh kd}

(O .

cosh k(z+d) cos c(z+d) dz

Jd

(

k81h

{_

I(1a)

Ç

r I(cX1r) dr _{cosh kd} COS cXh

cosh kh - °i cosh kd

i

((dh)

_{cosh k(z+d) cos ci(z+d) dz}}

H1(ka)

J(ka)

cosh kd

(O i (cosh k(zd))2 dz

[B,B1

= ka

H1)'(ka) H1)(ka)

(cosh kd)2 )d

ra

r

(J

(kr))2dr

sinh kh cash kh

)o

cosh kd cash kd.

i

_{(cosh k(z+d))}

ka J(ka)

(cosh kd)2

Jd

2dzJ.

H1»(ka)

2 )

sin cX1h ₎

( ka

J(ka)

+

o(ja J(1c5

a _{cosh kh}

sinh k.h

((dh)

H1)(ka)

.1 cash k(z+d) cash k(z+d) dz}

J(ka)

)d

=

-

Im(ja)

: r

mmj

Im(çr) dr

sinh kmih cosh k h

'(k

mj cosh kmjd cas

ch -

csin

ch

cosh krnd

I'(ca)

- (

;5

J(kmja)

+ cX.1a )'

r(dh)

i _I cosh kmj(Z+d) cas ç(zd) dz cosh kmjd

_)d

r a sinh k .h cosh k r J (k .r) J (k .r) - Ifli m

LBmBmJij =

_kmi _{m mi m m} ar cash kmjd cosh kmjd

- kmia

J(k1a) JmCmj

_{cosh kmid}

_Cosh

kd

((dh)

cosh k (z+d) cosh k .(-z+d) dz

d

m

(o

[A].

= a I f' (z) cas c(z+d) dz

)(dh)

m +

r f(r) I(t1r) dr cas

cth

[Bt,BU]. sinh r

_mmi

kh

cash kh

)

dr ( kmi

_{cosh kd cosh kd}

- (

ka

J(kmja)

i

+ k

cash kd cosh kmjd

(O

I

[B ] = a

HW(ka)

i )_(d_h cosh k(z+d) dz m m cosh kd

cosh ich

H»(ka)

(a. + )

r f(

dr cosh kd r (a cosh k .h_mi

LB ]

= r f (r) Jm(kmjr) dr còsh k .d

mi

m mi

Formulae for the integrals fcos 1(z+d) cos

(zd) dz

,

r Im(c:ir) Im(cjr) dr , etc. are given in appendix B.

Appendix B.

cos cos _cXz dz

i

( sin cXz cos cz

- Cos cXZ Sin Z )

{

-z i

+

sin 2cz

for c=.c(.

Jcosh kz cos cxz dz

( k sinh kz cos jZ _{+ cXcosh kz sin}

1z )

Çcash kz cash

kz

dz i

( k .cosh k .z sinh k .z - k .cosh

k .z sinh k . )

2 2 mi mi mj mi

kmi_kmj

_{for k1 kmj}

z i

+ 2k cosh kmjZ sinh kmjZ

for km= k.

I

(cx.r)

Im(jr) dr

j

mi

{_cXjlm(cxjr) Im_i(jr) jI_1(cxjr) Im(cxjr)}

for

r Jm(kr) I(cx.r) dr

mit1)

- I_1(1r) 'm+l

r

- k2+

{jJm

Imi(jr) - k Jm_i(1C1

Im(jr)}

r Jm(kmjr) Jm(1Cmj1 dr

rk1_k

kinjJm(1cmi

_mlmj

- kmiJmi(kmir) Jm(kmjr)

2

for kmi4: kmj

t{ mmiY2

- m-1mi

m+1miJ

for _kmj=

m+1

rm+i

I(c.r)

_dr r

_I1(c.r)

rn+i

(rm+1

_{j (kr) dr}

-)

TABLE 4.1. Kinematical condition on

S

ft(z,)

?(r,Q)

m

f(z)

f(r)

heave(Z)

0

-1

0 0

-1

surge(X)

cos

O

i

-1/2

0

TAL$L

4.2.

I

[Bu] and _[Bm]1 heaveZ); m = O surge(X); m = i jAm] i a 11(d.1a) -( s in

a -

sin ci1h )

-

EÇ ₍Ç a) co s [B - cosh kh

H,1)(ka)

la

H(ka)

_i sinh kd_{cosh kd} sinh Ich J1(ka) cosh kd J0(ka) cosh kd r LB

J.

a

coshk0h

o J1(k01a)

- ç

cosh

k0d

pitch(s);

rn = i z [Am]i cos cL d cos cL h 1a ci. + _cj21 - sin

cth

₎

I

a

12(cL1a)

+

Ç

I1(ci.a) 005 IBm]

i

H(ka)

d sinh kd i h sinh kh i cosh Ich

cosh kh

H(ka)

coshkd;2-1COShkd;2COhkd)

a + osh kd J1(ka)

J2(k1a)

cosh k11h 2k1 _{.cosh k1d}

TABLE 5.1. ConvergenCe of theadded mass and the damping coefficients (heave);

(2a/g

= 1, d/a =6, h/a = 4).

TABLE 5.2. Convergence of the added mass and thè damping cbefficints (surge.);

= 1, d/a 6, h/a. 4. ). 5.. added mass .coefft. 0.2462 0.2609 damping coefft. 0.00422

0.00393

6 added mass coefît.

0.2519

. 0.2665 - -a-... damping coefft. 0.00412 0.90383 i 5 -adde.d mass coefft. .

0.5411

0.5415

-damping coefft.

0.-5346

.

0.5351

6 added mass coefft.

0.549

0.5433

damping òoefft.

0.5354

.

.

.0.5359.

TABLE 5.3. convergence of B

/

cosh kd (heave);

(2a/g

= 1, dIa = 6, h/a = 4).

5 6 i P (O.2821O.4959i)

.io

(O.2787+0.4900i) .10 2 (0.2720+0.47861) (O.2686+O.4727i) .iO

TABLE 5.4. Convergence of B

/

cosh kd (surge);

(2a/g

= 1,

d/a = 6, h/a = 4).

i.

2

5 _(o.4273+0.1541i).102 _(o.4275+0.1543i).102

01

7

d

I

region D

I I

I.

t regionu

I I I t

'reg jDi

I

J

FIGURE 3.1. Division

of D into

subregions

D1

and

DU.

region

region D

z

4 I

region D

region

D11.

h

y

rjIs,'

region

/

FIGURE 4.1. Circular dock,

1.0

08

.0.6.

0.4

.02

o

i

2

3

4.

o'4a/g.

FIGURE 5.1.

_{Added mass and damping}

coeff. for a circular dock in heaveZ).

0.5

0.

02

o'2cig

FIGURE. 5.2

_{Addéd mass and damping}

.cöeff for a circular dock in

surge (X).