Variational principles associated with ship motions in waves

(1)

18 OKIIS72

ARCHIEF

..,...

Lab.

y.

Scheepsbouwkunde

Technische Hogeschool

VARIATIONAL PRINCIPLES ASSOCIATED WIH SHIP NOTIONS IN WAVES.1.

H. ISSHIK.I

Dr.

0.L

ABSTRACT

General variational principles for linear water waves are

obtained. _{Reasonable algorit1ms to calculate hydrodynamic}

quantities, which are important _{in discussing ship motiDne}

in

waves, are obtained from the standpoint _{of thé variatio:2a].}

principles. _{A general form of the equation of ship motions in}

ocean waves is discussed. _{A general theory of}

onedimnsjonal

tidal waves is also discussed, _{and the corresponding variational} principle is obtained. _{Using this principle,}

a simple numerical example is given to show the _{character and importance}

of -the

(2)

M. Bessho has presented an important variational principle to deal with linear surface waves associated with ship motions in waves [i], and the utilities are shown by him

_[13

and T. Nizuno.

(2,33.

Recently, added mass and damping coefficients concerned

with heaving oscillations of an axis symmetric coluizin were

calculated based on this principle by K. Sao, H. _and

J. P. Hwang _4]. In this paper, the same principle is derived from another point of view.

General variational principles associated with linear water waves are obtained. _{Physical systems considered here are those}

of continuous spectrum, and the mechanical energy dissipates from the system without changing its form. _{Mechanical energy dissipates}

as mechanical enerv. _{A process analogous to the transition}

from

E (

in discrete spectrum ) to _Ç C ïn continuous

spectrum ) _{is an essential of variational principles in, continuous}

7

spectrum [7lO).

i In §1, general linear 'water wave theory is developed.. _Simple

examples of the deep water wave and general tydal wave theory are dealt with in §1.1 and §1.2, respectively. _{Radiation and.}

diffraction problems associated with a surface -ship are discussed

in §2.1, and variational expressions for coupling forces and diverging wave amplitude are also obtained. _{Since these}

quantities are the basic ones in the equation of ship motions in waves, these discussions are important'concerned. with the

approximations of the equation. _{1n §2.2, a general form of} the equation of ship motions in ocean waves is derived. _The

(3)

physical meanings of the quantities defined in §2.1 _{may be olarifie'd} by the discussions given in this section.

Numerical example given in §1.3 shows the òharac-er of the

variational principles in continuous spectrum, and the results are satisfactory.

The Bessho variational principle _{and, variational approach}

(4)

r.

§1. General Linear Water Wave Theory

Linear water waves ( gravitational waves ) may be classified as so-called surface waves, tidal ( surface ) waves and general surface waves (5). At the present theory, we &istingiiish these waves by the character of the wave velocity. The velocity of

so-called surface waves depends on the frequency, and is independent of the coordinates. On the contrary, that of tidal ( surface )

waves is independent of the frequency, and. depends on the coordinates. We cal]. any other type of waves as general surface waves

In this paper, we exclude the last ones.

§1.1. Surface Waves associated with Radiation and Diffaotion of Waves by a Circular Cylindrical Shell

If we consider the special cases of the surface wave problems concerned with a circular cylindrical shell as shown in Fig. 1.1.1,

Pig. 1.1.1 we may find very simple examples of radiation and dïffraction of

waves by a three-dimensional body. _{Prom the considerations of thig} problem, we can get much knowledge.

§1.1.1. Water Waves generated by Forced Vibration of a Circular

(5)

2 2 2 2 2 2 2 O

influid

p

yap

pa

az

Ks

_m

+=o

8z

an

J

27piK

e_iP

exponentially zero z -+ oo ( m = 0,1,2,...

..,

00

where : velocity potential =

w2

/ g : wave number c...): circular frequency g :

gravitational

_acceleration

n : normal to the

surface of

the cylindrical shell to fluid (x,y,z) : Cartesian

coordinates

(p,Q,z)

: cylindrical coordinates

ai _{= (upuQ9uz) = -}

grad.

: velocity of fluid motion a : radius of the cylindrical shell

S : surface of the cylindrical shell'

PS : free surface i : imaginary unit

: normal veloÒity distribution given on the surface of the

cylindrïcaJ. shell under free surface.

on PS - e cos mO 1. _{sin mO}

onS

cos mO

-+00

sin mO

(6)

dm

can be written as d

-(

p)

dp dp

=-1

dp

i2

=0

rn

p'=a

Ix

_eP

as p.-+oo.

2m' hm()A\J27r1,i

his problem has the exact solution:

H(Kp)

= i

_F(2)(Ka)

-

E,(Ka))

Ka hm(K) = K2

2)(Ia)

-Ka a

(1.1.3)

where 2) is the Hakel function of the second kind, of order rn.

Prom the assymptotic expansion of the Hankel function, we obtain

- 2

eim+1'2

(1.1.5)

Let _m(K) , K)

and. Ç(K)

( scattering phase, cotangent

of scattering phase multiplied by K and generalized hyd.rô&ynamic

forcel) _{respectively )} _{be defined as follows:}

i) The hydrodynamic pressure

m corresponding to the velocity

potential

m

can be written as

=

P

m =

fi

Ret " m3

_P

Imtm)

Hence, the added mass and:wave ieleî'g damping are proportional. to

the real and imaginary parts of

m

respectively. cos mQ

Çl .1.2)

sin mQ.

(7)

(Vt.

(0

_ma

Î_mm(E) ₌ a d dz (a,9,z) f (9,z)

= (a).

(1.1.6)

-i

We then. obtain the exact values of these quantities from (1.1.4)

and (1.1.5):

%

S' (x)

₍ m

+ ].

) = J. - ( 2m +1 )

m

= K cot _i: ₍ 2m

.

1 )

J

₌ -pta

H2)(Ka)

f 2K2 _c H(2)(Ka)

H(Ea)

_f

Kam

Let us consider variational principles corresponding to the

boundary value problem (1.1.3). _{We consider, at first, a variational} principle which is a variational expression. for the _genexalized

bydrodnamio

force f(K)

. Let us consider a following

variational probleml): 2K

ma

21

n

_t(a)

fao

d )_a _dp _dp

)+p(K2

tI

C

n

= 0,2,4,... )

-K _{( m =1,3,5,...} m dp (1.1.8)

1) Prime

dose not

mean

derivatives. It is used, to

distingnish

the

boundary

value

problem from the corresponding variational problem.

. _(1.1.7)

( ) = stationary,

(8)

1K

h' (K)

,/

eP

p

-+c'o.

m m

_V22i

It is easily verified that the solution of the

boundary value

problem (1.1.3)

and

that of the variational problem (1.1.8)

are

equivalent. The stationary value of P

_{equals Ç(K)}

A complex number _h'm is

also

subjected to variation. _{To show}

the correspondence between the boundary value problem and the

variational principle, let us consider the variation of

r at

kl

nm-' m m

-P

.=.-S'

(a)

oo

_d

_'m _m2

+i

_2XJa

_L(p

_dp

p

(00 d

d.I'

2 m2

t-(

2K

a

dp

)+f(K_)tm

ira

d'm(a)

+ -

'm(a)

₊

_"m

+ 'rn(a) ). 2K

dp

_dp

(1.1.9)

Prom the generalized Green's

integral

_{formulae in one dimension}

rn2

(oo

d

J

_a

[(p

d2

(00

d

2

[(p

)+p(K

- J

a dp

dp

'

:i

'

rn

mr

m

dp

m2 tøo cl m

i) The integral

\

[ - ( p

) ₊

p (

i2

-

) _'m3 _'m df)

Ja

dp

can be defined under the condition (1.1.8a).

(9)

V

₌

_a

_{6m(a) + a}

_'

_(a)

m

d'm

+ hm

(

_p

_'

).

p.b.00

d dp K

Substituting (l.1.8a) and

_'m-

_h'm

eKP

_{, we icnow}

2i

that the second term of the right-hand side of

(1.1.10)

is zero.

We, therefore, obtain the expression of

_variation

_{as follows:}

Ir.

oP

mm

_Ja

fo

nio##

_I

h(K)

dp dp

d'

ta

nl'

+

-f.

+

iJ

'(a).

K

dp

Hence, the boundary value problem (1.1.3)

_{is identical}

with

the

variational principle (1.1.8), and

m

h)

=

Z1(K)

Let

be

i

.

We then obtain eua;ioas:.

nl2

(p. )+?(K2_.)=O

_ap

a<p<oc

dp

_p

at p=a

dp

sin

cos (Kp-.Ç)

)+

(K2

nl2

8 (1.1.11)

ç

a

00$

Xj ₎ )

ø,

(.1.1.12)

(10)

d

dms

)+(2..)

in2

=0

ap<co

dp

dms

at p=a

do

Sin

(Kj.&) cos 1p - - sin Kp ) p -

00.

z: (1.1.13)

quation(1.1.13) defines eigen-value problem of ßtruxlL-Liouville type in continuous spectrum

_C7-loJ:

d ms in2

-(p

)+p(K

_---!

=0

dp dp

_p

o -( co s Kp - - sin. Ko )

p

-z: (1 .1.14)

It is shown that this eigen-value problemi is

identical with a

variational principle which is a variational expression of

One of

the

variational

principles is given

as ío11ows

rrt

.T

_'t

in' ins '

m- -

in

1) One of other type of the variational principles is applied to

the scattering. o gravity waves by a circular dock by J. Miles and P. Gilbert (ii).

dp

(11)

under

I

ms ( 008

Kp -

sin 1p )

p

(l.1.15a) where

_tm(K)

is also subjected to variation. The variation

of Vmt1m8 ;

can be

written as o 2

Ç=2f[__(p

_d

)p(

ms

'msd?

+ 2a ms ' (a),

and

ÇE

_ms

Ç)

=

Let the approximate value of

&()

be *m(X) The

real and imaginary parts

of f(K)

can

be calculaed. using this

value.

We

define

4.)

and

as follows:

. = +

m(+) mc ms

= mc ms

and

satisfy (1.1.12)

and (1.1.13).

Hence,

m( is

the: solution of the following

bomidary

value problem:

-°

d.

d'ms

m2

dp

)(2__)]

&i'ms(a) + a

'(a)

= stationary, (

d.

-(

dp ni2 + p ( 2 - ;..) m(+) ° a <

p

co

i)

*

denotes the numerical or approximate value for the corresponding, quantity. 1 15) (1.1.15b) (1.1.16) C dp

(12)

di,

at

Ihm(K)I

sin

2ltp

t(

i

+ )cos

Kj,

( 1

sin

lpJ

_p

(1.1.17)

The corresponding variational principle can be written as Zollows:

I'mI

=_'

(

\(a)

21

øo d

'm(±)

2 ni2

p

) +p (

_'n(±)

'()

dp

211a

di,

_p

2K di,

d' ())

(l1.i8)

ni +

i )

'm(+)

=

stationary,

under

'm(+)'' Ihlm(K)1 sin

'

₂₁₉ s m.

+

i

+ - )

cos Ii, + ( i

- ) sin

Ii,)

_p

(l.l.l8a)

where

_111v(K)

sin

is subjected to variation, and

*(K)

is substituted Thr

(K)

.

The above variational principle is

the variational expression o

Re

_(K)J

+ Iin

K))

:

(13)

Imay be calculated from the energy dissipation Im Ç(K)]

Let us introduce an unknown Íunction uÇp) defined as

= u(p) /j7

a<poo,

(1.1.20)

and describe the boundary value problem (1.1.3 in ter'msof u(p)

We can easily obtain

the

ol1owing equations:

du u

e =

-dp 2a

=0

a<p<oo

at p=a

u. " h)

'

_2ii

eP

_P+oo.

m=O

The general treatments of this

type of

equations are discussed in §1.2.

§1.1.2. Diffraction of Waves by a

Circular Cylindrical Shell

Let us consider diffraction of an incident wave

which is the regular plane wave coming from

the

direction of cc

radian with respect to the xaxis :

=

+ 1K ( x 00

+ y sin. cc,)

=

Ç

_e

_{003 m(Q-}

_,

(1.1.22)

(1.1.21)

in2

4

2

P.

(14)

d(cx) ]. d(cx) P d(c) +

=0

3z d(oc)' / e -lip v2lrpi - exponentially zero on PS

i

ad(cx

Bd()

'=0

influid

+_T

_az2

Poo

(1.1.24)

(1

for m = O

Em

= il

L2

for m = 1,2,3,...

3m = Bessel function of the first kind of order m. . (l.l.22a)

e

J(Kp) cos m() is a cylindrical wave, and

haE

the assymptotic

behaviour according to that of _Jm(Kp) :

r].

_C

2n1+l)

+ e

"p

J,

-+00.

(1 i 2 2b)

let the velocity potential (x,y,z;oc) be the solution of the dif!raotion problem as follows:

= ₊ (1.1.23)

We obtain the following equations:

d(o) o(cc)

=--

on S

(15)

$a(«)(x,Yz)=

i

We can write

and. according to the expansion (1.1.22)

oZ the

plane

wave

as Lollows:

(p)

cas

m(-oc)

hd((Q,K)

=

__

Ç ih(K) cas m(Q-o

(l.1.2)

We, then, obtain equations Lor

and

h(K) :

2 _d ₂ md 1 tad 2 2

+(K_)=o

a<p<oo

dp

pdp

_P

d2md

dJ(Kp)

dp

at p=a

md 2irpi

Also, in this case, we

can

obtain exact solution:

mdf

=

ta

J(Ka)

-

Ka Jia)

u112)_(Kp)

(1.1.26)

C

phase shift ), the cotangent

o

the1 scattering phase multiplied

by K and the generalized exciting Íorce, reapectiveLy, defined as o11ows: hmd.(K)

Let

= ta

H(Ka)

ta

J(Ka)

--

Ka H(Ka)

Ka J+i(Ka)

_2'fT

-m

H2)(Ka)

à(K)

_md

- Ka H(Ka)

K

(1.i.27)

(16)

=

argument

o

h(K)

+

argument o

i

/1JT

md = K ct S'md

ed(I)

= a d dz

E

m e cos m(Q.-o m

dp

?ca

(Em)2(_1)m

dJm(Ka) =

2K dp

Ç1.1.28)

The exact

value

o these

can

be calculated from the exact solution

(1.1.27): md =

om(K)

=

ed(K)

=

+a

2m1

+,

.it

o)

4

a roo _à

+t

E-p

Ja

d m

= 0,2,4,...

m

= 1,3,5,...

(

m Jm(Xa)

Ka m+l )2

2Xa ni H(2)(Ka) -

Ka

H(Ka)

+ ni2

-'mà

J

'md dp

2)

2!

(1.i.29)

The boundary

value

problems (1.1.3) in §1.1.1

and

(1.1.26)

are oÍ

the same ones except the

boundary

conditions at

_p

=

a .

ThereThrs,

we consider a variational principle:

mmdC _'md h'màj = a 2K L df) md

dJ(Ka)

+ )

'(x:a)J=

stationary,

(1.1.30) dp

dp

dJ (Ka)

jme_Kz

cos m(Q-cx)

(17)

' d? #-'In

_2ti

eP-h'

(K) , (l.l.30a)

where h'(K) is also subjected to variation.

The Euler condition

of this variational principle lead to the boundary value problem,

(1.1.26), and this principle is a variational expression of ed.(K) :

md md ; h

) = ed(K)

(1.1.31)

]..2. Genera]. Theory of Wave Motions in a Canal of a Variable

Section

Let us consider wave motions in a canal of a variable section 5) as shown in Fig. 1.2.1. We assume that b _{( the}

Fig. 1.2.1 breadth of the canal ), S ( the cross sectional area ) and

h ( the mean depth: S = bh ) vary along x-axis 4_n approximate

treatment of this problem is studied by Green [5). We extend. the

theory to have an example of general wave theory [6-10).

Let

1(x)

e be the surface elevation. The governing equations are as follows:

g d

dx)

b(x)

( h(x) b(x) -) + w2 1,(x) = O

o (x <oo

dx

at x=O,

(1.2.1)

where g is the gravitational acce1exation, and. is the

amplitude of the ocean wave. The boundary condition at infinity is discussed in the followings. If we introduce a variable ,Z defined by

(18)

dx

(X

-=/

or dx

JO

'gh(x)

in place oZ x , the equation (1.2.1) transZorrns into

d2t

d log bd

₂

= o

Z<°,

_(1.2.3)

let us introduce :

b4)/2

We, then, obtain equations written in Ç :

+[2

1 d log bV 23 = 0

0< Z<

0°

c

.--(

₎

4

Hence, i 1

d log br

2

-('

)

-+0

4 dZ'

it is, then,

verified that has the assymptotic Zorm:

+ W 'C

Ç- Ae

(-+oô)

or

in

the term of

bZ. , where

-

'1og bj

dx

(z

g(x) ='=l

/gh(x) (1.2.2) (1.2.4)

at '= O

. (1.2.5) (1.2.6) (1.2.7)

( x-+oo)

_(1.2.8) (1.2.8a)

(19)

We, then, obtain the basic equations for the transmission problem of waves in a canal of a variable section:

a2

_0Ct[J

-C-.t[3E

=;

' A e'

_+oo,

where i

dlogb

4

w2v(t) )J.

=0

0<"<oo

at

Z'=O

log

b(0)Vh(0)

,

L0 . Cr

-+00

(1.2.9) (l.2.9a)

A=

IAte'.

_(1.2.11)

We, then, have equations for

(1.2.9)

can be. written in terms of 12v, as follows:

gd

d.

b(x) dic dx

O <z Zoo

C-R3E

ki, _=kl

at x=0

A f(x) e

x -oo,

_(1.2.10)

We introduce two variational principles _{or the phase shift} correspo"tding to the boundary value problem (1.2.9) C7-10).

(20)

a2

-+

(2-V()

_)J

=o

o

<'<oo

az2

at

''=O

= stationary, under

-= o

_{= i}

corresponds to the eigen-value

problem (1.2.12), and is a

variational

expression

Thr woot

(ca1 +

6' ),

that is

(]..2.16ä)

rGc;)

=

wcot

(wa1

+ S' ) . (1.2.1Gb)

-

IAl 8m

(c.ix+ 6')

r,co.

(1.2.12)

(1.2.12) defines eigen-value problem

_Lor

the phase sh.ft. _1Í

Zor _(1.2.13)

the

relations hold:

=-IAj sin(c+8')

or >

a

,

_(1.2.14)

and

d5(a1)

/

= &cot (wax + S')

_1.2.15)

íor arbitrary

a1

(

a )

.

Because o

the relation (1.2.15),

a variational principle: ra

rG'5

= i

E C

'8

)2 -

v(t) )

, 2

at

(21)

If in general

we, then, consider an eigen-value problem

_cCtC..5]

=

o

C[3J

D

-'

cosç+sinWt,.

where

=Wcot

s'

can

be calculated by a variational principle:

-.rc5 ;

= ₊

_J;)°°S

£[']

= stationary, under a 20

(1.2.17)

(1.2.18)

(1.2.lSa)

(1.2.19)

o

_'t=O

cos)Z - .n")Z

(]..2.19a)

where is also subjected to variation.

_The

_{above variational} principle is a variational expression Lor , that is

(22)

§1.3. Numerical Example

Let us consider to calculate the phase shift for the eigen-value proble.m (1.2.18) in §1.2 based on the variational principle (1.2.19). Por the sake of simplicity, we consider the case:

h(x) = h = constant, = x / './E

C

b(x) = - exp(

'g'-45

e' /« d)

'Ji

Jo

This shape of the canal leads to

1

dlog(

v()=(

4

This type of v(') is called Yukawa totentia1. L. Hu1thn [7,81

has studied this problem for positive value of

_fi

. We calculate according to his ideas for negative value of 9 . Let us assume

o = cosc

+ - sinw

w

N

- [ e+

L1

0m

em

_{( i - e)}

_J

(1.3.1) (1.3.2)

and calculate by Rayleigh-Ritz procedure based on the variational

principle (1.219). The results are shown in Tab. 1.3.1. We can Tab. 1.3.1

see that the integral Ç0

C'8Ci4'8]d'

in

(1.2.19) cancels the error

invalid

in.

(23)

22

2.1. General Variational Principles Associated with Ship Notions

Let us consider floating _{cylindrical shéll ( two-dimensional} model of a ship ) _{and take the origin at the}

center of the water plane of the ship, the _x-axis

horizontally and y-axis _vertically downwards, as shown in Fig. _2.1.1. _{For the sake of simplicity,}

Fig. 2.1.1 we assume that the ship form is symmetrical with respect to _y-axis The velocity potential of steady state of harmonic motion of

frequency c can be represented as

Re[

«x,y,Ca.)]1C

_.

Then,

the velocity potential _{must satisfy the free surface}

(PS) condition: e -

cscIp

J

C - +

)

(x,y,w) = o

ay,

where

K=wavenumber=,2/g

on PS , _(2.1.1) (2.1.2)

Ship motions can be _{decomposed into fo11owings2:}

The oscillation parallel to x-axis _{( swaying )...'.,} _{suffix 1.} The oscillation parallel t.o _{y-axis ( heaving )}

suffix 2. The, rotational oscillation _{around the axis of}

the cylindrical shell _{( rolling )}

. suffix 3.

The first mode ship hull vibration

suffix 4.

1): Three-dimensional _{theory can be obtained without any difficulties}

by the direct extension of the present _'results.

2) Accordin.gto the theory of natural _{vibration of a elastic}

structure without damping, the ship motions can be decomposed into _{rigid body} motions (or oscillatjons _{in this case )}

(24)

The second mode ship hull vibration ... .. . . . ... . . .suffix 5.

e. .

. . . ...

.S ...'

Let us use the symbols

Q(+1) and. to express incident,

regii1ai waves coming from x-positive and x-negative directions, respectively. Hence

-.K.y+ i.x

0(1)(XYI) = e.

-The diffraction potential due to the incident wave _c(+l)(X,Y,K) is expressed by _$d(+l)(xy,K) . The velocity potential of ship

motions in regu.lar waves can be obtained by superposing the incident regular waves, the radiation potentials by ship motions and the

diffraction potentials induced by the regular waves. The general problems in (irregu.lar) ocean waves are discussed in the next

section. In the followings, we use the sufl'ixès defined above

to denote each motions, incident waves and diffraction _potentials. We consider to normalize the motionU _{as follows:}

= jwX1 _{j = 1,2,3,...} _{, 00}

iga

..

=A.

'Q :J

j

= O(c), d(o, cx.= +1 , (2.1.4)

where _{a means the amplitude of the incident wave, and.} _the amplitude of the ship motion . The boundary condition. on the

ship surface is given as

on,

(2.1.5)

(2.1.3)

(25)

where f

i

an ax2

f.=x

x

-for j=1,21

for

j

₌ 3 = w for

j=

4,5,6,....

2)

The radiation

and

_{diffraction potentials have the diverging waves}

in xpositive

and

in xnegative directions,,

_and

tend to exponentially zero as

y -

00,

that is

- i e

iKx

h(K+1)

X -+ ±00

- exponentiallyr zero

y+.00

C

j

= 1,2,3,...

...,00 ;

d(x,),.

.=

1 )

_, _(2.1.6)

where

h1(K,o

=5

(

Let us consider the quantity f defined as

fjj=JifjdS=_5.j___..d

Ci,

_i

= _1,2,3,....

00

_d(c),

oi= ±1 ) . (2.1.8)

may be realized as the appropriately normalized j-th componen

of the

_{hydrodynaxnic}

_{force by the i-th motion,.} _We

_cal.

f the

1)x1=x,

x2=y.

2) w :

normal

displacement mode of the ship hull vibration.

t'o(o) a .

-OIL 'J._.I

)dS

. (2.1.7) fi = an

(26)

generalized force component. In the sain.e way, we normalize the generalized f ree component excited by the incident wave

o()

and express this by

e(Kcx.)

:

e(K) =58

o()

+ 6d(o() f dZ

=

[s1

si

an an a

(K)

_ay

si

:i dS on FS

1) X means to take the complex conjugate of X

çj

= 1,2,3,...

s s o(.= -i-1 ) .. (2.1.9)

Promi(2.l.7) and (2.1.9), we obtain the Haskind theorem:

e1(K,c) = h1(K,x)

(

j

= 1,2,3,... ...,00 ; d(oQ,

.=

_{±1 )} . (2.1.10)

Using the assymptotic form (2.1.6), the frea surface condition.

(2.1..l) and reezi's integral formula, the following relations hold:

fi1 =

= .

)

(K,c() h1(K,c)

1)

(

i

= 1,2,3,...

...,00 ; d(oc), cx.= 1 ) . (2.1.11.)

The boundary value problem associated with. the ship motion of the type j , or the diffraction of wave by the hip orresponding

to j =

d()

_{is formulated as follows:}

(27)

r': :;1;::i

where 2 2 0x ay (

j

= 1,2,3,.... ...,00 ;

d(),

_{OC.= ±1 )} . (2.1.12a)

Let us consider a variational principle E7-1O]:

Flic h'1

/

5'

iKx

_h(K+1)

X

-e- exponentially zero

$fluid $'

£E$'

-

ç-,

Js

on S y-.,p.00

s'10E s

1) '

is used to distinguish the variational problem from the corresponding

boundary

_{value problem.}

= stationary,

(2.1.13)

under

S'i

-' i _h'1(K,l) _X

5'

exponentia1ly zero

y__poo

(2.l.l3a)

where h'(X1)

is also subjected to variation, _and

(

j

= 1,23...,

...,00 ;

d(),

_.=

_{±1 ) .}

_(2.1.1»).

(28)

Fjj

=

£[

_J

tj

fluid

-2

_C'

r' J

J

F5 FS

j

_2SC[$tj

/ f]

as

Hence, the solution of the variational problem (2.1.13) ïs the solution of the boundary value problem

'(2.1.12),

_and _{is a} variational expression. of the generalized hydrodynamic _force

that is

; h / = _(2.1.15)

Let an approximate solution

J :1)

=)

c*, w(x,y)

)11

w

i e t»(Ic+i) X

w -' exponentially zero

_{y 'oe}

_(2.1.16)

be obtained by Rayleigh-Ritz procedure _{based on. the variational} principle (2.1.13). We,, then, obtain

= _; *, / f..]

=J

f. dß

(2.1.17)

Let us consider a variational expression for f

_{( 1j )}

This is important to discuss ship motions _{in waves.}

_For

_an example, the rolling motion and the swaying motion., are not

independent. _{They effect each other through the quantity} _f13

dxdy

(2.i.14)

(29)

or f31 . We notice a relation:

ir

= =

_{-\ (..+$.) (f.+f. )as}

4JS

- -

\ C

-

) ( f1 - ) as.. (2.i.18)

4)S

From this, we

obtain

=

f1

ijC'i(+)j

_;

/

_f ₊

-

±

j_j[1j(_)j

_{; h_}

_/

_f

-

Í]

j

(2.1.19)

where

s.v.E

.f.,.

_...]]

means

the stationary value of the variational problem:

P +E$'j(+)

;

/

±

£)

± ) dS

Sfluid

'i(+)j £E'1(+)]

dxdy

'1()j

_'1(±)

under

-L

'i(+)j

C't'1(±)

/

± as

(30)

= exact solution of (2.1.20)

=

Let. a Rayleigh-Ritz approximation. of the variationaj. _rroblem.

(2.1.20) be

15*()j

:

0*

=

___

(±)ì

w31(x,y)

)11

We easily obtain the equation:

=

£

15*(), (

fi _±

and, fr9m (2.1.20) and (2.1.22), the equations:

c*i(+)j y. = 31 ± 31 (

P

= 1,2,...

Prom (2.1.24)

15* 15*

15*

i-

_j

..,J ) s

Hence,, we obtai. an important result &om the stand point of

variational calculus: (2.1.21) (2.1.22) (2.1.23) (2.I.24) (2,1.24e.)

-

i e7

iKx

hl()(K±l)

_{X -+ ±00}

15'.

-+

exponentially zero y

-400

(2.l.20a)

It is obvious that the relation hold.

= exact solution. of (2.1.20)

(31)

f

= f f* =

;p-

i. ji

ir

it

= -

\ $*. f. dS

+ -.\

$* f. dS

2J5

'

;j _2.JS

i

( i,

j = 12,3,...

...,00 ;

d(c),

.=

_{±1 )} .. (2.1.25)

A variational epression for the amplitude of the .iverging

wave h(K+1)

, (

j, = 1,2,3,...

...,00

_; d(cx,

o.=

+1 )

can

be obtained in a following way. We notice the relation: r'

=$S

SS

a,s o(+l)

d.S

-0(±l) f d + f

d(l)

a

dS (2.1.26)

The first term

$

o(l)

f. dS ( Froude-Kria.lof force ) is

exactly calculated, and the variational expression for the second

Rayleigh-Ritz approximation of the variational problem (2.1.13) ( to solve We obtain ) be N

=)

c*d(±l)

»

w(x,y)

h(K+l)

_h*(Kl)

°(±') f dS

.11

'r

+

.js

d(l)

dS +

*(1)f d

A variational expression for the _{generalized exciting :eorce is} obvious from the relation (2.i.10).

(2.1.27)

(2.1.28)

(32)

r

(

' -,

If we discuss the coupled ship motions in _{waves:. (2.1.25)}

Pictorial represemtations of these waves _$

given in Pig. 2.1.2. _{The diffraction. potential}

obtained by superposing the potential _$

à syin

and

à (+1)

We notice the fact that ₍ ₌

o sym

/

Sn. ) .nd 'd ant

=

ant

/

Bn ) are real functions defined on S , that is

8$

_{O sym}

dsym

en ax _'ay = -K ₍ sin Kx + ) ,

_(2..l.3].A)

Bn.

31 ant are can be Pig. 2.1.2

8dant 3O

ant + ₌ O .

(2.1.3oB)

Bn

and (2.1.28) may be the basis for the numerical calculations. It seems to be interesting to the author to make a

variational principle to determine the reflexion and transmission rates of regalar incident waves. _{For this purpose, we decompose} the incident wave

O+l)

= e + into two waves $,.

and _$

ant e

'O(1)

= sym + O ant (2.1.29)

where

$=±(e+e1Y_iKX)

ant

= (

e + iKx e iKx _(2.1.29a)

and the potential

ant

C-rI

1. -

exit ' d ant

8$

A rrn '.. J.LL

j

S à srm à sym Sn Sn

(2.1.3oA)

(33)

where

ant

=

O

ant

an

It shouid be noticed that

d

a.nt

are

complex valued

functions in general. _{We, then., assume that}

d and.

ant

have the assymptotic form:

s

_d e h (K)

x

- co

Ç '. '1d

ant ..

.&)

X + +00

s

_dant

-i

e3T

+

iXx

_lid

ant

X +

o 5

_{d. srm}

h

exponentially zero o 1) 5d _syin OC Im C _5d _sym

:i

ax

cosKx -

sinKx )

an 8n

Let us consider an eigen-value problem corresponding to the

imaginary part of

5d : 1) sym = in fluid e on _PS ay Q

=0

on s on

dsymi

5

_{d sym}

--

e ₍ +

sin x

+ cos Kx )

_x

+ +

co K 32 (2.l.31B) (2.l.32A) (2 .1. 32B)

=K ey

( y

+. co

(2. 1. 33A)

(34)

'd sym

sym = argwent of ha (2.1.3Aa)

is distinguished by the stationary value of.a 'rariaional principle:

o

rd

SIfl'

_a

_sym _d

sym3 ='

_sym

r o K I

5'

_LES'

_dxdy Jfluid d sym r )

dsym PS

dsym

r.

o T

/

Kis

5'

_symCE

_{d sym} _s

à.

J

dS = stationary,. _(2.1.34A) under where ' d sym o

-

exponentially zero d ant3 ( - + ay o

1)

5

antd ant

o

_dsym

oosKx)

K

can be obtained with respect to the imaginary part of Let us, now, consider an eigen-value problem:

o o

(ri

1Ai

01

"d ant - ' ''d ant - in fluid

:c -+

_{± 00}

à ant

(2 .1. 34Aa)

is also subjected to variation. _{Similar results}

(35)

'd.

ant

-s a

ant-a ant o

-.e3T(

sillKx±

dantK

e - exponentially zero

d ant

ant

=0

on S

dx

J

(2.1.3 3B)

where

ant = K cot S'd ant

ant = argument

of

hd

ant

(2.1.33Ba)

d

ant

is distinguished

by

the stationary value of a variational

principle: o

rdantcTd

_ant

dant

-(

-K

t

$?dantE$tdant

dxdy

J

fluid.

_KÇ

_dantPS

r

_dant

Jrs

KS0

'd ant

tr°ì

SLP

d ant

/

s d ant

= stationary, _(2.i..34B)

under

d.

ant

o - e C

sin Kx

+ co s Kx )

x

+oo

dan

_K

o

ant

exponentially zero

) 00

(2.1.34Ba)

where

'd ant is also subjected to variation. The above

variational principles are variational expressions for

cos Kx ) X

_{± 00}

(36)

dsym

₌ O(+1) s3m

(ant

{ant

+d58Q(l)13n)s.

o(i)

/

'

d sym

/

'òn

and

d. ant

/

n

are

given

explicitely. as follows:

8

_8dSYn

_8dant

=-

i

that is

=-Ke(

sinKx

+cosKx )

Bn Bn

ex 8y

ix e ( co s Kx, -

sin

Kx )

Bn Bn

dSsym = c

dçsym

+ _{'s dçsym}

laut

lamt

_lant

syin = h _sym + Í h5

d çsym

laut

iant

(2.1.36)

(2.1.37)

(2.1.38)

=

r symE

ant

=

rd ant

_d

_ant

_d

_ant

_J

(2.1.35)'

Let the approximate

value

of and. be

and

. Using these approximations, w-e consIder to obtain variational expressions to calculate _{ha sym(L)}

_and

ha ant(K)

From

the expression of the diverging wave amplitude

( the Koohin function ),

8n en

Let us decompose

d , h sym

ant

and

ha

ant

into

the rea]. and. imaginary parts:

(37)

Prom (2.1.36), (2.1.37) and (2.1.38),

h0

d sym and h ant

he5ym=$eY

_{cosKx f}

_dS d sym

à symd sym

ha d ant e sin x ant dS + c à ant d ant dS

The second terms o± the right-hand side of (2.1.39) can be determined as the stationary values of thè following variational problems:

{syin

d{sYmC'Pc:

_dsym

h'

dçsyni / àçs3rm J

ant ant laut tant tant

=

Ç,3T.

J

S ° dSsym d{sYrn

dS

ant ant

fluid

'P

d{sE?o dçsym] dxdy

ant ant

dçsyiz

'P

_dçsym .2 clx

-ant Liit

d1sym

'P

dçsym _àçsym)

Lant _laut = stationary, (2.1.40) under h' e

_{( Cos Kx}

± Kx )

odsym

_sdsym

X 4 +

dS (2.l.39A) (2.l.39B) SS

P

C

(38)

a.ua a zero

y )-+Oo

à ant

h' ( cos Xx + sin Xx )

odant

sdarLt

K

X

'C d a.nt - exponentially zero y ). +oo (2..l..4OBa) h? _and _h'

where rea]. parameters

d -m d are also subjected

to variations, and

ant are approximate values

determined from the variational _{problems (2.l.34A) and (2.l.34B).}

Let h* and h*5

à ant be approximate values of h0

a

cdsyin

and

's d ant Calculated from (2.1.39) and (2.1.40). The imaginary

part of hà

and

the real

_{part of}

hà _ant are determined from.

o d

sym '

_d _{sym '} _{s à ant} and

à ant as

follows:

h5

d syxn h*3

= (

IC / _*d

sym ) h*0 _sym

h0

d ant h*0 _{à ant} _ant / K ) h*5

. ant

The diffraatjo _potentia1

d(+l) due to the incident wave

is obtained

by superposing

_and

ant that is

'd(+l)

= _{d sym} + d axt

hence, we have

hd(+l)(K,+J.) _{ha sym}

ant

(2.1.4OAa) (2.1.42) (2.1.43)

The reflexion rate _{J hd(+l)(K,+1)I}

and the transmission _rate

I hd(l)(K,_l) _I can be approximated as follows:

i _I

h*d1 (K,±')

)2 2 = d\j h* sym h _{à ant}

+ (

h* à sym + h*0 _{à ant} ) (2.1.44)

(39)

(+oo

= _m(&.,)

et

_th4.,

l.j-oo

+ +W=o

on S.

8ii n

From (2.2.2), the velocity potential _{ca.n be written as}

(x,y,t)

_{= dy,t) +}

where

(2.2.1)

where m(c)) _{is the spectrum density of the ocean} _wave. _{We, then,}

consider ship motions in ocean wave (2.2.1). Lét W(x,y,t) and

(x,y,t) ₊

_e(xyt)

be the normal velocity distribution on the

ship surface S and the velocity potential of the water,

respectively. _{The mutual interactions between these are based on} the continuity equation on

8

(2.2.2)

2.2.3)

1) Por the sake of convenience, _{we use somewhat different notations} and definitions from those in the preceding section without any references. This may not cause any confusions.

IIIPIIIFT

-38

§2.2. equations of Ship Notions in Waves

Let us consider a general form of hydio-elastio equations of ship motions in ocean waves. Prom the investigations in. this

section, the physical pieanings of the generalized hydrodynamic

forces defined in the preceding section may be clarified. _Por

the sake of simplicity, it is assumed that the ship is completely submerged, and the structural damping is ignored.

Let

_{e3Tt) be the velocity potential of the ocean}

_wave. We assume that the ocean wave can be written as

(40)

d e +

=0

on S 8n

8r

+w=0

on 3.

8n

Let _d(X,y,W)

e)t

_{be the diffraction potential due to the} regular wave

eY+1 et

_, _{that is}

¿hgd=O

influid

+

=0

on S

8 _{i e.} + _{/ On} _{= O} _on _S

ç i e

iKx

hd(U),+1)

X -+

for

'iV>

O

L-

e3' ±

iKx

_hd(,+l)

_{x -}

_±00

_for _CII)< _O

Ad - exponentially zero

y -

_(2.2.6).

Then, the diffraction potential (x,y,t.)

due to the ocean wave can be written as

+00

d,y,t)

i

Ç in(&)

AdY

et

_dc.

We, then, consider to obtain an expression o± the radiation potential (x,.y,t) _{, which is induced by the normal Telocity}

distribution W on the ship hull S . The differential equation

for this problem can be written as

for C&J< O

(2.2.4)

2.2.5)

(2.2.7)

i) _$d(x,Y,C) ₌ _and. _ha(w,+l)

= hd(+l)(I,±i)

for cJ>0 ,and

(41)

influid

1

ar

r

- _

on PS g t2 Oy

+w=o

on. S On

The same notation i is used for the suffix of the series and for the imaginary unit, but any serious confusions may not

occur.

(t

J

A(t) dt

is the generalized coordinate of the normal

displacement.

(2.2.9a)

- diverging wave

X ab. ± 00

- exponentially zero (2.2.8)

According to the theory of natural vibration of a elastic structure,

the normal velocity distribution W may be decomposed

into2)

w(Q,t) ₌

)A(t) w(Q)

Q E S, (2.2.9)

where

w1(Q) = the rigid body oscillation mode (. swaying mode )

w2(Q) = the rigid body oscillation mode ( heaving mode )

w3(Q) = the, rigid body oscillation mode ( rolling. mode )

w4(Q) = the first mode of the ship hull vibration w5(Q) = the second mode of the ship hull vibration

(42)

From the theory of the Fourier

transform,

_A.(t)

can. be written, as 1

(+00

A (t)

a.(w)

et

dcv,

V2ICJ-oo.1

where a1(UJ) =

k(t)

e_t

_dt

Substituting (2.2.10) into (2.2.9), we obtain

00 1 W(Q,t)= ,.._..) J

a(w)

_dtuw(Q) i=].

Let ,(x,y,t)

et

be the solution of the radiation problem due

to the normal velocity distribution w(Q)

e1t

on S , that is

influid

+

8,31/ay =0

on PS

+

Wi =0

on S

ji

e_KY _h1(w,±1)

_{x +oo}

_{for w0}

-

a-Ky ±

x

(2.2.10)

(2.2.lOa)

(2.2.11)

1)

=

_Á(x,y,K)

and

_h(w,l)

₌

h1(,+1)

for w > O

,

and

= _6(x,y,X)

and

h(w,L)

= (K,+l)

for W< O

Then, the radiation potential

lr(xyt)

_can

be written as

-J

a(w)

1(x,y,u) e" dw,

(2.2.13)

i=l

(43)

-- ,.

-mce,

_r(xyt)

satisfies (2.2.8).

Let P(x,y,t) be the hydrodynamic pressure:

a

d P(x,y,t)

_{= P}

( i = 1,2,3,4,...

Ss

?swi

-={

at (2,.2..14)

where _p is the density of the fluid. _{We assume that the equation..}

of the motion of the sbip hull can be written as

a

W(Q,t) (t

p_s + IC W(Q,t) dt _J + P(Q,t) = o

)

where _p is the mass density of the ship hull, _and _L _{is the}

appropriate differential operator corresponding to the deformation.

of the ship hull. Since w is the mode of the natural vibration of the ship hull, w _{is the solution of the eigen-val.ie problem:}

-wj ?swiLCwiJ =0

on S

where _{is the circular frequency of the natural vibration} corresponding to the vibration mode _w . The first tiree are

the rigid body oscillations,, _and

W1

= "2 = W3 = O

Ia[w] = O

i = 1,2,3. .

. (2.2.16a)

In general, the operator L satisfies the relation.:

iw.

LCw j

=

w LCw1 J

. (2.2.l6b)

s

We,, therefore, choose _{a systen which satisfies}

1

i=j

O

ij.

on , (2.2.15)

(2.2.16)

(44)

p'

+

(

)-ii

m( dL'.)

+

,Ii

Jo°

I(A) a(cù) t

dw.

(2.2.17) i=1 Prom (2.2.].1) P(x,y,t) = QQ ₊₀₀ Ps

[LOO

and from (2.2.11) and. (2.2.16),

(t

LE)

Wdt3 =

00 (+00

ir-)

iw m()

e

dW

IC'.) a1(i,)

et

5

(w2

w2

) a() w(Q)

u2 m(w) e-Ky+iKx _m(w) _(Q,c'.,) a1(ci.) 1(Q,w) .. i=J. ,

Then, we rewrite the equation of the motion. (2.2.15). _{Por this} purpose, we substitute (2.2.1), (2.2.7) and. (2.2.13) into _(2.2.14),

and we obtain n (2. 2 .17a)

a1(w) etdww.(),

_(2.2.17b) (2.2.18) i=1

Substituting these into (2.2.15), _{we obtain an equation in the} term of a1(w) :

(45)

Using the ie1ation (2.2.16e), we obtain the result: where e .() = Ç C +

a(Q,w) ]

Js

1

P(t)

= r )_

iw f(w)

et

d,

w(Q) dB

and use the following formulae of the Fourier _tran.sÍor:

(2.2.20)

=

J

m(w) e() -

a(c) fo

(2.2.19)

=

5 $(Q,w) w(Q) dS

= (w) (2.2.19a)

The equation (2.2.19) is the equation of the motion written in

the term of a(c)

. Prom this equation, we recoiize that the

quantities

e(w)

and f(w)

are the basic ones to discuss ship motions in ocean waves. _{The variational consideratio.s for these}

quantities are given in the previous section.

The equation of the motion writtén in the term of A(t) _is

obtained in the followings. Por this purpose, we define _M(t) ,

and P1(t)

(+oa M( t) = m(w) e

it

1

E(t)

=

,rj_00

iwe(w) et d

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s(,) = R(t)

et

dt

(+00

r() s()

R(t-) S() d

dt

,9Ùoo

J-00

(2.2.21) where i ç+oo

_{R(t) e)t dt}

s(t.)

eLt

dt . (2.22].a)

From (2.2.19), (2.2.20) and. (2.2.21), we o:btain the result:

d A.(t) 2 + ci. A.(t) dt dt

_p

= N() E(t-t) dtC

00

+

%E)

A() P(t-) dC

. (2.2.22) i=l

We may äall the equation (2.2.22) as a general equation of the ship motion in ocean waves. _{Let us study the physical meanings}

of the quantities E(t)

and F(t)

. From (2.2.20)

00

p

E(t)

[i) t yiwe(w) 3

e dc _(2.2.23A)

'V 2?C)-o0

p

P(t)

=

_{ic oo}

Ei] [iwí (c) J

dcv.

Prom (2.2.23A), _y

E(t)

is the j-th component of the wave

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exciting force by the incident wave which has the spetrum density

m(aJ) = i . Prom (2.2.23B),

F(t)

is the jth component of

the hydrodynamic force by the ith motion which has the spectrum density

a(c) = i

. Let us use the symbols M(t) and _(t)

for N(t) and

A(t)

when m(u) = i and a(c&i) = i , respectively.

Then, we have

N(t) =

i

ç+0Ò

[lJe1)t

dw=

iE(t)

(2.2.24A)

I«t) =

1 (+00

=

iF)

El]et

dw= ((t)

, (2.2.24B)

where (t) is the Dirac delta function, On the ot:ierhand, M(t) and. can be written as the superposition of the

impulse, that is

5+00

-00

+00

(t) =_ AY) S'(t) dZ

. _(2.2..25B)

The jth component. of the wave exciting force corresponding to

N() 6'(t) d'

is

M(t) E(t) dZ

'1ir.

and. the jth component of the hydrodynamic force by _{t:ae i-4h motion}

A1('r) _{(tz) d'} _is

.P

A.(t) F1.(t')

d'C

i

The above discussions may clarify the physical meanings of the equation (2.2.22).

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Let us consider a special case when

=

S'(c% )

where is an arbitrary cons-tant.

corresponds to a regular incident wave:

i

L

eYt) =

d A(t)

dt +00

(c4c0 )

e +

i -Key + iK0x iLiJ0t

e e

where

K0 2 _{/ g}

And, in this case, M(t) becomes as follows:

i Ç+0o M(t) _{= ____} jC)t i iC*)0t e

d)=

e

iwt

=

'_iCQ

e.(&J) e

i.flj

O 00

A(V) P(t_t')

1KX d (2.2.26)

This spectr density

(2.2.26a)

(2. 2.

2.2.26e)

Substituting (2.2.26c) into the first. integral on the rig:at-hand side of (2.2.22), we obtain (+00 i

iWZ

P

e0t.

d=

e(c0)

(2.2.26a)

If we substitute (2.2.26d) into (2.2.22), the following equation of the ship motion in a regular incident wave (2.2.26a) is obtained:

(2.2.27)

st

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he solution of (2.2.27) may have the form:

i(A) t

A(t) =

e _(2.2.28)

Hence, the second integral on the right-hand side of (2.2.27) becomes as follows: 00 P

r-

)

e

F(t-'r) dZ

= y ) A 14)Q f(w0) e 00 _{d. Â(t)} Therefore,

?

Re[ f()] and

- ,pw0 Im[ f1(w0)) may be

called as the added mass and damping coefficient matrixes corresponding to the ship motion (2.2.28), respectively.

=

___

[ Re[ f1(w0) :i

dt

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one lu si ons

Geñeral variational principles are obtained to discuss linear water wave problems.

We should pay special attention to the facts that Variational principles obtained are stationar

principles, and.

The stationary values of these variational principles corresponds to required physical quantities, l'or eg., hydrodynamic forces.

Hence, we may expect promissing results, even when aai. approximate potential is obtained by other methods than the Rayleiph-Ritz procedure ( for eg., by collocation method ), and the required physical quantities are calculated from the correspondi

variational expressions. _{This is a fundamental phylosophy in} variational calculus presented by J. W. S. Rayleigh, and is called as the Rayleigh principle. _{A variational approach to strip method} of calculation is discussed from the standpoint of the Rayleigh prïnciple in reférence [12).

Reasonable and sound basis for approximations in discussing ship motions in waves may be expected also from the variational approaches (4).

Ac]mowle dgement s

The author wishes to express his aciQiowledgements _to Professors H. Naeda, J. F. Hwang, M. Bessho and. T. _{Mizuno for} their continuous encouragements. _{He i's deeply indebted to}

Professor Y, Yamamoto, University of Tokyo, for his valuable advices in writing this paper.

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References

[i] Bessho, "On Boundary Value Problems of an Oscillating Body

Floating on Waterti, Meni. Defence Academy, Vol. VIII, No. 1,

(1968),

pp. 183-200.

T. iviizuno, *tQ _{Swaying Motion of Some Surface-Piercing Bodies",}

em. Defence Academy, Vol. IX, No. 1,

(1969), pp. 221-237.

-T. Mizuno, 'tOn Swayr and

hou

Motion of Some Surface-Piercing

Bodies",. J. of Soc. of Ìav. Architects of Japan, Vol..

127, (1970).

i. Sao, H. Naeda and J. P. Hwang, On the Heaving Oscillation.

of a Circular Dock", J. of Soc. o± 1av. Architects of Japan,

Vol.

130, (1971).

H. Lamb, Hydrodynamics, Cambridge University Press, 6-th ed.,

(1932)

chap. VIII-IX.

G. B. Whithum, "A General Approach to Linear and Non-1inear. Dispersive Waves using a Lagrangian", J. Fluid Nech., Vol. 22, part 2, (1965),

_pp.

_273-283.

L. Hulthn, "Variational Problem for the Continuot.s Spectrum of a Schrödinger EQuations11, Ku.ngl. Fysiografiska SäliskapetsI Lund Förhandlinger, Bd. 14, Nr. 21,

(1944), pp. 257-29.

L. Hu1thn, 110n the trum-Liouvi11e Problem Connec:ted with'

a Còntinuous Spectrum", Arkiv Ír Natematik, Astronomi Och Fysik,

Band. 35A, No. 25, (1948), pp. 1-14.

T.

Kato, "0. Henbun-Ho&', lu SHIZEN KAGAKUSHA NO T.ANi NO SUG-AKU

GAIRON ( Ooyohen ), ed., by K. Terazawa. ₍

'o.

_{Variational Calculus",} in Mathematics for Scientist ( Advanced Course ) ), IWANANi SHOTEN,

Publishers,

(1963), pp. 447-455.

[io]

P. M. Morse and H. J'eshbach, Met?iod of Theoretica.l Physics, part 2, NcGrawHu11,

_{(1953), pp.}

_1123-1131.

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t

[ii J. Miles and F. Gilbert, Scattering of Gravity Waves by a

Circular Dock", J. Fluid Mech., Vol. 34, part

4, (1968), pp. 783-793.

f12] H. lashiki, Variational Principles Associated witì Surface Ship Motions ( A Variational Approach to Strip Method )tt, Korea-Japan Seminoi' on Ship Hydrodynamics, Proceedings, Soc. of Nay. Architects of Korea, (1970).