Calculation of hydrodyanmic forces for cylinders oscillating in shallow water

AND MARINE ENGINEERING

GOTHENBURG - SWEDEN

CALCULATION OF HYDRODYNAMIC FORCES

FOR CYLINDERS OSCILLATING

IN SHALLOW WATER

by

CHEUNG H. KIM

DIVISION OF SHIP HYDROMECHANICS REPORT NO. 36

GotIienburg. February 1967

8. Discussio

Appendix- Ç.

n Not Circular Cylinders s . . s s

5 5 7 9 li 13 16 18 18 18 19 20 21 22 23 s s . . s 30 Nomenc ature 33

Appendix-A. The Components of the Potential and

Stream Function for Deep Water Heaving . 24

Appendix-B. Summary of the Component Potentials and

Stream Functions for Shállow Water Heave. 26

The transformed Formulae of Singular

Integrals

G(vh)

and F(vh) 28

Appendix-D. Wahlström's Procedure for Solution of

Linear Equation System. .

Figures l-56

References . . . 31

3. Component Potentials and Stream Functions

for Shallow Water Heaving . . .

Wave Source Potential

Higher Order Source Potential Calculation of Singular Functions

4. Hydrodynamic Pressures and Forces . .

5. Generated Waves and Energy DissÌpation

6 Method of Numerical Calculation

7. Discussion Circular Cylinder

Wave Length in Shallow Water

Source Intensities AorT/B A0/B

. s ö

. .

.

Hydrodynamic I¼Iass Coêfficient C

d Ratio of Wave and Heave Amplitude

e Damping Parameter 5

Pressure Distribution on the Cylinder

Pressure Distribution on the Bottom of the Water 22

Convergence 22

Page

Introduction . . . i

1. Introduction,

W consider Lewis cy].,inders heaving harmonically _{in shallow}

water. _{The problem is to find the effects due to water depth}

on the hydrodynanic mass, damping parameter and amplitude

ratio of wave and heave.

Yu and Ursell

_Li.]

_{and Porter [.2] have reported on a two}

dimensional theory of heaving in shallow _{water, Wang [3] has}

recently reported on an oscillating sphere _{in a fluid of}

finite depth.

The author extended Grims potential _{for heave in dèep water}

[d.] to the case of finite, depth according 'to _{[i]. and [2].}

The numerical calculation has been _{carried Out for Lewis}

cylinders by Grim's method.

It is assumed that the water is nOn-viscouS _{and moves}

irrotationally, _{Linearised boundary conditions are applied}

in. solving the boundary valué problem.

At first calculations were made for the _{circular cylinder to}

grasp general influences of.depth effects _{on the hydrodynaniic}

characteristics.

Asymptotic formulae of the hydrodynamic _{characteristics i.e.}

hydrodynamic coefficient, hydrodynamic _{damping parameter and}

amplitude ratio of wave and heave for _{very low frequency were}

useful in analyzing the computed results,

In order to obtain the limit of depth parameter (water depth!

ships draft) to which the calculation _{converge, several Lewis}

cylin4ers with extremely deep and shallow _{or extremely fine}

and full forms were computed.

The computations were made _{on the SAAB D21-computer at the}

University of Gothenburg.

The author expresses sincerely his graditude to the head of

the institute, Prof...Falkemo, who has supported this work.

Above all, the author is deeply indebted to Prof Grim

furnishing his tihpublished coriputer program for heave and

pitch of ships in regular wav and also to Prof. Porter, who

allowed me to copy the work of Yu and Ursell from his own library and willingly helped me by sending the rescent report on a heave theory of a sphere in shallow water by his student

Dr. Wang.

Ha also takes this opportunity to acknowiedgc Dipl.-Ing. Claussen

of the waterline in calm water and the centerline of the

profile. _{The x-axis lies horizontally to the right and the}

y-axis is vertical pointing downward.

The complete complex potential for deep. water heaving used by Grim [4] is as follows: OD = U

ewt[Aò um

dK +

110

KV+ill 00 + n=l A L (K+v) e (x+i.y ¿K]

where U = the amplitude of heave velocity,

w = the circular frequency of oscillation,

y = w2/g the wave number of the generated surface waves

A0 and A are complex source strengths, which are

determined by the boundary condition on the surface

of the cylinder.

The velocity potential and the stream function * are

C0 = Re

[00]

= Re

respectively. Both of them are written also as sums of the

components of the potential functiön and the stream function

in the following form:

00

= u

no

[(A

nr

A1

cos wt

-- (A.

- (A

_nr

Änr flj

Sin

wt . . . (2)

where the subscripts r and i represent the real and imaginary

parts of the components and means that the functions apply

to deep water.

The detailed expressions of the comporents are to be found

in Appendix A

=U

E _Anr

'nr

Ani eii) cos wt

The complex potentials for deep water heaving [4] are changed to fit the case of shallow water according to Thorn's theory

[] as has been done by the authors of [i] and [2].

We begin with the wave source potential and higher order

potential for infinitely deep water i.e.

or

_{= f}

-y

dK Oboe

= - î'e'cos

\)X and nroe

(nlj

_eYcos

KX dK,

fljoe = for

ni.

Each of, the above component potentials is extended to the

corresponding shallow water potential in the. following

ub-article s.

3.a. Wave Source Potential. Since the wave potential

Or

fulfils the condition of constant

pressure on the free surface and the bottom condition for infinite water depth, one assumes an additional adjusting

potential

or add.

J [a()

sinh

y +

K).cosh

K(h-y)]cos Kx dK

The coefficients

(Ic)

and ß(K) are determined by the free

surface 'condition

4,

=0

y

where

=

or

+ 4,or add.

We thus obtain the wave source potential for shallow water

corresponding to the potential

4,ör as follows:

=;

d +

I:

(X dK -

¡ cosh K(h-Jos

(X or -

f \)°COSh

KìihKh

dK o

To obtain t]e potential

4,oih corresponding to one considers

the condition of radiation i.e. that the wave generated by the

heaving body nust appear as a progressing wave at great distaflce

fron the body. Since the potential satisfies only the free

surface and bottom conditions one must at first obtain the

asymptotic behaviour of 4,orh at lxi That is

2ir cash y h

4,orh -

2Çi +

---

cosh v0(h-y) sin v0lxl at xl

where is the wave iumber in shallow water defined by

y = tanh

v0h

for the given frequency

y and the depth of water h.

Therefore by introducing the following new potential

Zir cosh y h

4,oih

= - + cosh v0(h-y) cos y0 lxi

it can be easily observed that

- .

- Re

[eWt(4,0

oih]

at y O and

at yh,

Thus the required

oih and consequently the complete wave

potential

oh orh + oih is determined. The

corres-ponding components of the stream functions

oh = orh + tPoih

are expressed as the conjugate parts of the above potentials:

7

esin cx

_{+ L}

e''

v.sinhKyKeCO5h

-) sin KX

= _dK

VCOSh K11K.Slflh Kh

-

K V

27r cosh y h

oih = - 2v0h + sinh Zv0h cosh y0 (h-y) sin v0xI

.b. Higher Order Source Potential.

To obtain the potential corresponding to

nr

we proceeds

similarly as in article 2.a Assume that the additional

potential is of the form

nr ad

f[y()

S1 y + 6(K) cosh

K(h-y)] COS

X dK

o

The constants y(K) and 6(K) are determined by fulfilling the conditions of the free surface and the bottom as in 2.a.

Consequently the potential

_(flT

+ add) corresponding to is obtained as follovs.. 2(n-l) -Ky K (K+v) e cos x i.

f

_K2(n-l) _(K+v) -Kh v.sinh

KV -

c.cosh KV e

vcosh

Kh

- Ksiflh

cos (X

dK

The asymptotic behaviour of the above potential is nrh =

nrh

-or

nrh

4'nh =

2n 2ir cosh v0(h-y)

Vo _{zv...h+Sí1i2 coShv0h.} sin

2m

Vo ₂

I;Si1ii1....2v0&

at lxi +

at xl + and

By introducing a new potential

-i 2m 2T .cosh v0(h-y)

nih = V0

2v0hsin

2v0h

cósh

v0li

CDs

One may also observe that

T

Re [et(flrh +

nih at i and y + ()

represents a surfaçe progressing wave. Noté that

nih approaches

zerO, when h goes to infinity.

Thus one obtains the complete higher order source potential

nh

nrh + 'nih

and further the corresponding stream

functions _{*nh =}

nrh + as the conjugate of them;

it is

-Ky

_{!lfl K}

_th+

2(n-l) _{+K) e_}

(ShKyK'CO5h

IC)

Sfl KX

dK]+

v'cosh icfl-Ks1nh K

2n 2ir cosh v0(h-y)

I

e'" e1

4K

KV

nr add. where

G25l(v)

and e o

are given in detail in Appendix - A, and the aditive source

potentials

or add, and nr add, for shallow water are developed

in infinite series [i] as follows:

=

ecos

cx

(V'sinh K) .cosh _d

or add.

v'cosh hi.1iihKh

K

o

KV

so

(2)

G251(vh)()25.cos

2s

-vh

_(2s+l) ;

e11 .

u21

(v-u)(v cosh u - u sinh u)

r

\Ç2+2

= tan (

IK7

eX

dic

2(n-1) -idi v°sinh cy-ic.cosh

K

(ic+v)

e cos icx

v.osh

ichicejnh

dI 'C

-

I

e(u+v u22'.

F2n+2s..l(V) - _{cos u - u sin u} o -2n (2s)Í

F225_1 (vh)(j)2cos 2S8 +

n e -+ (_].)

_h2"

z F (2s+l) 2n+2s- ( r 2s+l cos(2s+l)

In a manner similar to [i j,. the, additive stream functions

'ör add, and 'nr add, are obtained and represented by

= j _<h5 V..SÎflh.KY-K'COSh K or add. v.cosh KhK.Slflh Kh o G2 2(v) '2n+2S 2s+2 - I 2(n-l) -Kh . v.sinh KY-K.COSh KY

nr add. K (K+v) e sin KX(

cosh

1-K.S1flh h)dK

fl (-1)

_h2

i

_F225(vh)()

r 2s+l sin(2s+1)ß + + i

vh2z

t2s+2)

F225(vh) ()2sin(2s+2)8

where E s=o (25+2J G25

e'

u22

2s+l

sin(2s+l)8.-(vh)Ç,)252 sin(2S+2)ß;

vuj.(v;:co:sh u-u sinhiÏ) du

)

e(u+v)

d

y cosh u .ü sinh u U

A summary of the formulae for component potentials and stream

functions are given in Appendix - B. The transformed formulae

of the singular integrals

G(vh)

and

F('vh)

are given in

Now the component potentials and stream functions for heaving

in shallow water are at our disposal.

Replacing the component

potentials and stream functions in eq. (l).and (2) by the

cörresponding components for finite depth, one obtains the

potential:

'h

By fulfilling the stream condition on the surface of the

cylinder

= Ux cc,s

(Lit o s s I I

(3)

the unknowii complex intensities A0, A

are determined.

SubstitutinÉ these numbers in eq.

(1), one obtains the required

potential

_h

_{which, can give the velocity or pressure field in}

the water.

In what foilowis we omit the Subscript h iñ the

synbols

_h'

h

etc.

The hydrodynamic pressure on the surface of the cylinder is

represented by

p

= -Uw sin wt {p Re(EA

)}

wherö p is the density of water.

The first and second term

f the right hand side of the aove

equation represent the hydrodynainic pressure in phase with the

acceleration and velócity respectively.

As the next step the hydrodynamic pressures are integrated along

the surface of the cylinder and thus the hydrodynamic force is

calculated:

U cas wt {-p w I( E A

_n

_n

)}

¡

Lt

surf

= - U w Sin wt {p J

surf.

Pe( E A

n=o

n dx } +

t

+ U cos wt f-p w

In( E A

)dx}

Thus the force. consists of two components, the so-called

hydrodynamic inertia force in phase with the acceleration and hydrodynamic damping force in phase with the velocity of the

body's motion, We define tht.hydrodynamic mass m" and damping

coefficient N (force/velocity) as follows

r

in" = p ¡ Re( Z

A

)

J surf: _n=o n n

i

For practical use we nondimensionalize the above physical

quantities in the usual way and define thejyn

coefficient

= -p w

rn" C =

irB2

8

and damping pramet.er.

N

6

irB

p w

-g--s i s s- s s s s

where B = breadth of the cylinder

Hydro4ynamic Pressure on the Bottom.

The hydrodynamic pressure on the bottom of the water is represented by

-pUw Re(E AnqnI ) sin wt +

yh

y=h

00

+

_{UtA Im(_EAnqnI}

_{) COS}

wt

(6)

yh

(7)

f

surf.

_n=o

InC Z 00

whe r'

and a = tan

2v0h + sinh 2v0h

-1

________

n ni

S. Generated Waves

and

Ener Dissipation.

The vertical oscillating defonnation of the water surface, at

an infinite distance

from

the body is

nw = aJv

E{A

[nrSi1

ut

+ qcos

wt

A1 [nrCO5 ut - q.sin wt]}

at

lxi

y

-+0

where _ah is the heave amplitude of the body. On the other hand

the component potentials at infinity on the calm water level

are represented by

2ir .cosh2 v0h

iv0

-

7h + sin12v0h

e

j_1)fl

_2n

iv0

lxi

(2n-1).' '2FF+ sinh Zv0h e

Therefore is expressed as the progressing wave in the

following forn: 2ir

cosh2

y h

(1)fl

o fi =

for n1

(2n-1) 2v0h + sinh 2v0h

Consequently the amplitude of the generated wave is determined by the fonnula

noHnh12 + (zHIbA1)

Note that sinçe H + 71 and H + o for h

+ e,

the amplitude

ratio for deep water heaving is = 71v \/A0.2 +

A02'

We shall calculate the energy needed _{to form this wave system.}

The mean power used in forming plane _{progressive waves on the}

surface of watêr of finite depth is given by the _formula

4

p g

a2

Vg where p is the density of water and _{Vg is the}

group velocity of wave propagation in the shallow water. In our

case

2v h

iw

o

V =.---J+

g

z y0

sinh 2v0h)

Therefore the mean power extended is given by the expression

2vh

'mean =

4

g a2 (J?..) (i

+ sinh2v0h

On the other hand the mean power dissipated by the damping force

in the mOtion of the body is

.1 2 2

N'w

_sah

Assuming a conservative system, both _{energy dissipations are}

equal and one thus obtains, a relation between _{the damping}

coefficient N and the amplitude ratio of _{wavó and heave 7 as}

follows: 2v h

(1+

wv0

sinh

2v0h

(8a)

a

= ah.vs +

(zH.A)2

and the amplitude ratio of wave and heave _{is represented by}

Note. that if h +

then

6. Method of Numerical Calculation.

By Lewis transformation i.e. x + iy = + a/rh + b/c3,

where c = ele, the functions of boundary condition (eq. (3))

are expressed by a parameter e and the constants a and b. They are expanded on the boundary of the. cylinder in the form of trigonometric series:

(a,b,e) = C0 (.-

-N-1

+

EC

s'in2me

m=1 nm

The boundary condition is then reduced 'to 2xN simultaneous

linear equations having 2xN unknowns. In 'this calculation

N = 5 was used. In fact N = 10 was tried to improve the

convergence for low depth parameters. However the improvement

was perceived negligibly small at the practically important

low frequency range = O - 1,5 and for the assumed minimum

depth parameter 1,5. To save computing time, N = S was thus

choosen, (see also 7.h).

The principal value integrals

G251(v)

and

F251(v) have

singular points on the u-axis. To avoid these they were

trans-formed by moving the line of integration to arg(u) = rr in the

complex u-plane [i]. ù the same manner the integrals G252(v)

and F252(v) were transformed. The transformed formulae of

these integrals are given in detail in Appendix-C. In the

numerical, integration the limits are taken from 2 = O to 30

and the increment of (2) is 1,0. Simpson's rule was applied.

The computed results of G251(vh) and G252(vh) for s=o and i

are plotted in figure 2 for our reference in the discussion.

The additive potentials and stream functions

nr add, and

add, are represented by infinite series as mentioned

before (see Appendix-B). In the numerical calculation the

Prom for Solving Linear Equation System.

The computing centers of SAAB D21 have not yet prepared all

purpöse standard procedures for customers. After trials to

use several private procedures, one made by Wahlströni (see

Appendix-.D) was chosen. _{The reasonability of this procedure}

was checked by comparing Grim's Program for computing ship motions in waves [7] as translated into ALGOL of IBM 7090 by

Norrbin [12], and the same program [7] translated into ALGOL

SAAB D21 by the author, using Wahlströms procedure.

The hydrodynamic section coefficients for _{a series 60}

ship were computed with both programs and compared. It was

shown that they were in good agreement with only slight

7, Discussion - Circular Cylinder.

Wave Length in Shallow Water.

Shallow wavenuinber y0 for a given depth h and frequency y are calculated and illustrated in figure 1.

Observe that shallow water wave number _{increases according}

to the formular

J7Ìi'

at very low frequency, while the

corresponding deepwater wave number obeys the formula v=v

;

h+.

_{As the frequency increases further the shallow water}

wave number curve gradually approaches the deept water wave

number curve.

From the relation -. = _{it is consequently stated that the}

wave length in shallow water is at low 'frequency generally

shorter than the corresponding wavelength in deep water.

Source intensities. _AOr1T/B A01rr/B

The source intensities Aorlr/B and A011T/B (dimensionless) play

a significant role in explaining the hydrodynamic behaviour

for heaving in Cep water at low frequency. _{Our computed}

intensities are therefore plotted and compared with those of

the deep water potential, see figure 3 and 4. The. figures show

that the intensities are dependent not only on frequency and geometry of cylinder as those of deep water potential but also very much on the depth parameter h/T, especially at very low

frequency. As the depth parameter increases the intensity

curves approach the line of heaving in infinite water. The humps of the curves at very low frequency are judged to be caused by the behaviour of the additive shallow water stream function:

X

or add.

G252(vh)

for

"0 ;

5=0

iifx(

oi add. 2h

(See fig.2 and Appendix-B)

Zc Hdrodnamic Mass Coefficient C.

We consider first the hydrodynainic mass coefficient C at very

low frequency: v-o. It is expressed by

Ch - 2 {Aor1T/B[_lfl() 0,577

+ G2s+l(vh)l]

v+o 1T/B(IT/2Vh)} . . . (9) C, -

.4 {-ln(-) - 0,577}

, v+o

where h and mean finite and infinite depth respectively..

G2s+i(h)I

_s=o represents asymptotically the function or add '

see Appendix-B and figure 2. Also the function

ir/2v0h

represents

asymptotically the additive potential

add.' see figure l

These potentials may give significant influence on the behaviour

of Ch at very low frequency. In other words, they are the

amplitudes of standing waves in water of finite depths and

consequently may contribute to the high increase of hydrodynamic

masS. These coefficients are plotted and compared in figure 6.

It is.concluded that as the frequency approaches zero the values

of Ch are infinite and higher than the values of Ca,.. As the

frequency increases further Ch values suddenly become smaller

than those of Ca, and rise again at higher frequencies. This

behaviour is due to the characteristics of the standing waves. In the following we consider the behaviour of Ch at low and

medium frequency range. It is hardly possible to study this

behaviour by analysis but it is observed in the computed results

given in figure 7 and 8.

The values of Ch are generally lower than those of Ca, at low

frequency. As the frequency increases, they increase and

be-come higher than those of Ca,, for depth parameter

1.2, 1.5,

1.7,

while they remain below the Ca,-curve and gradually approach it

2.d. Amplitude Ratio of Wave and Heave

L

The asymptotic expressions for are

VO

vB

r

+

(A01tr/B)2

₍₁₀₎

;

e

e'

. _{f S' S}

(l1)

First of all the behaviour of the source intensities

+

(A0ir/B)2

_{are illustrated in figure 5. Also}

the above formulae _{are plotted and compared in figure 9. The}

remarkable facts are:

The slope of the _{Ah-curve is infinite at v=o, while that}

of the _{-curve is 2,}

The value _{is higher than those of at very low}

frequency.

Recalling the discussion on the Wave Length in Shallow

Water _{(7.a) it is stated that the shallow water wave}

generated by heave at _{very low frequency has shorter}

length and larger amplitude than those of the corresnònding

deep water wave.

As the frequency gradually increases _{the 'ratios become}

sliht1y lower than those of _{, see figure 10. If the}

frequency increase further they become higher than those of deep Waterwave, see figure 11.

Finally a comparison between the _{values of our computation and}

the experimental results by _{and Urseil [i] are presented in}

figure 12. _{They are in a fairly good agreement at the}

7.e.

Damping Parameter 5.

At very low frequency v-'o, the damping parameter

h is represented by 4 h

vO

cS "r g. ah (A

ti/B)2

_..

_(A07r/3)2.

while the parameter Ç is exactly written as

= _.

(A/B)2

₊

_(A../B)2

(13)

From the above formulae. it is concluded, that the damping

para-meters for shallOw water heave

6h approaches infinity, while

the damping parameters in deep _{water reach a finite value}

as o.

This behaviour is also illustrated _in

_figure

_{13. From figure}

14 and 15 it is seen that, _{if the frequency increases further}

from this very low range _{vo, the values of damping}

_for

_low

depth parameter h/T1.2, 1.5, 1.7 and 2.0 become higher than

those of heave in deep water, while thôse in shallow water fOr

depth parameters 4, 6, 10 become lower than _{those of deep water}

heave and gradually.approach the

_Ç-curve.

7.f.. Pressure Distribution on the C1inder..

The hydrodynamic pressure on thé cylinder was calculated for _a

number of frequencies and 8=00 _to 9Q0

_by

_{step of} 180. _The

pressure in phase. with acceleration

pgsa =

v.Re(zA.)

and the. pressure in phase with velocity

(12)

v.Im(- z

for hIT = 1.5, 2.0, 10 are plotted in figures 16-20. By a

comparison of the figures it is bserved that he influence

of depth is slightly larger on

pg'a than on

pga

This

fact explains that the shallow water effect on damping is more

significant than on ine1

7.g. Pressure Distribution on the Bottom of the Water.

The total dynamic pressure on the bottom

p g ah

was calculated near the bottom of the cylinder, for depth

parameter h/T2.0, figure 21.

The maximum pressure is to be

found at the point nearest to the center line of the cylinder

and decreases along the surface y=h. It is alsó evident that

the pressure is directly proportional to the frequency of os

oscillation. The maximum pressures àt the point (x=O, y=h)

for depth parameters h/T=l.5, 2.0 and 10 are represented as

functions of frequency in figure 22. The comparison illustrates

that the influence of depth is dominant in the whole frequency

range.

7.h. Convergence,

i.e. the amplitude of

1.i

g.a at x=x

y h

The convergence of the calculations were considered by observing the behaviour Qf the source intensity curves and by checking the condition that the amplitude ratió of wave and heave computed from the wave potentials, see eq. (8a) and that

computed from damping, see eq. (8b) must be the same.

Generally the converging for heave in shallow water is worse

than in deep water. As the depth parameter h/T decreases and

frequency increases the convergence becomes not desirable.

However the important fact is that the convergence is secured

at practically important low frequency range, say = O - 1.5

8. Discussion

Not Circular Cylinders.

To find the inininiuni allowable depth parameter for which our

calculation converges and to get a general idea of the

influence of depth in connection with the geometry of the

cylinder, several widely varying forms of cylinders were

chosèn,

he particulars of the cylinders are given in the

following table:

= fulness coefficient of section.

Some of the calculated results are plotted in figures 23-56

where the lowest values of depth parameter represent the

required minimum allowable depths up to which our solution

converge.

It

is concluded from a comparison that:

For thé deep draft cylinders (1 A, i. B) the influence

of dépth parameter is small and therefore the minimuri

depth parameter is very lòw.

For the shallow draft cylinders (3 A, 3 B, 4 A, 4 B,

5 A,

5 B) the influence of depth parameter is remarkablè and

the 'values of minimum parameters are high..

3.' The influence of fullness ß on the minimum allowable

depth is also significant when the shallow draft cylinders

are considered.

Model

H

lA

0,2

0,6

i B

0,2

1,0

2 A

1,0

0,5

2 B

1,0

1,0

3 A

1,8

0,5

3B

1,8

1,0

4 A

2,0

0,5

4B

2,0

1,0

5 A

2,2

0,5

5 B

2,2

1,0

is represented as follows:

is small

Appendix-A

The C'oxnponents of the Potential and Stream Function, for Deep

Water Heaving.

1. The Wave Source Complex Potential i.e.

4o +

o =

oroico +

oreo1oio) = um

_io

or

= e'cos vX

[-ln(I,781'v\1x2+y

n1

o

-Ky

_e1

dK K V

cos nB]

-

e''sin vx [z

S]fl ¡IB

+ arctan

_]

n=l

n.n

y

Oi

-

lT e'

COS V'X

*or

=

cos vx [

cV\Ç24.2)n

Si.fl nB

_{+ arctan}

n1

nn

y

r

J2

2tn

e

''

sin vx [-ln(l,781 v\Jx2+y2) +

(vVx4y )

cas nß1

n=1

n.n

= - * e

sin vx

is represented by

nr

= (_j)fl

[COS2fl

+

v.cbS(2n-9ß

r (2n-l)r2 = (_1)fl

[Sifl

2n8 + vsin(2n-l) j r (2n-l)r2"

ni

= if

vx2+y2

is large n . n

= i (n-l)

{(YiX)

4yix)

_-

_{ie'sign(x)'sin vJxJ}

,

or

7n=l (x2+y

)fl

-

rre'cos

vX ,

oie

-

e

(n-l)

{(Y+1X)

-(y-ix) + ¶e''sign(x)'cos vixi ore

-n4

y" (x2+y2)T1

oi =

-

ire'sjn

vx

2. Higher Order Conpiex Source Potential i.e.

fle

+

nr+ini)+

_nr1'ni)

J

K2 (n- eK (X+ly)d

o

nl

Appendix-B

Simimary of the Component Potentials and Stream Functions. for Shallow Water Heave

eY.cos

KX dic + E (2sjT

G251(vh)()2cos

2s8

-K - V

S0

o rh nih = 2ff cosh v0h s=o

2s+1J1

2s+ 21T cosh y h

oih = 2v0h + sinh2y0h cosh v0(h-y) cos

'orh = dic

+ (2s+i)

G252(vh)()2sin(2s+I)8

-vhE

(2s+2J

+ sinh cash v0(h-y) sin

2(n-l) K (ic+v)eYcbs KX dK -2n (2s)!

2n+2s--2nl

vh z (2s+l) s=o

h)()2cos

2s8 + r 2s+l a cos(2s+i)ß cosh v0(h-y)

o

_2vh

_{+ sinh 2v0h' cosh 'u0h} - cos

'oih =

nrh nih

=

-G251(v)

=

G2+2(v)

=

F2+2(v)

= 2(n-1 K V vh, u =

F225_1(v)

=

v2.G225_1(v)

_-

G221(V)

--'n 1 2 +1 (2s+lfl

F225(vh)()

sin(2s+1)ß + n + t-1) -2n+1 1 2 + (2n-1)

_i2s+2TJT

F22(vh)()

s

sin(2s2)8,

where 8

tan(x/y)

KYi

_{KX dK} 2n 2 cosh v0(h-y) V0

20h

coshv0h

sin IxI

-u 2s+1

e u

edu

-u)(v

cosh u - u sinh u)

u22.du

v-u)(v costi u - u sinh i)

F

Je(v+u).u12ldU

2n+2s-1 _{T y cosh Ù U sinh u} ô

_U(v+u) 'u2'2du

y cosh u - u siiifi'ti

F22(v)

v2.G225(v)

-where 2s+2. N2 = - (2p) - {v2[sin( d(2p) + cos Appendix- C

The Tranformed Formulae of Singular Integrals G('h) and F(vh).

G251(v)

= d(2p) where =

z2±

_{v2[sin( - +

eT2esin

+ v(2p)[cos(. - 2p) - sin(.- _{- 2p)]} +

4(2p)2[e2cos

- cos(.L 2p)]} = 4(v2

-

2pv + 2p2) [v2(cosh 2p + cas 2p) - y 2p(sinh 2p - sin 2p) + 2 +

(2)

(còsh22 - cas 2g)] 2p) +

e2

sin + + v(2p)[-2sin(.. - 2p)] + (2p)2[e2cos(.!L)

- cos(-

2p) + + sin(!L -

- esin(.)]}

F25 (y) where ç N., I

';

d(2p) J _D o i 2s+i (Zp) +2 v [sin(

+ t(2p)[cos(. - 2p) + esin ]

+

4(2p)2 Íe2sin 2L

- sin(.iL - 2p)]}

= 4v2..(c.osh Zp + cos 2p - ..v(2p'sinh 2p - 2p.sin 2p) +

+ p''(cosh 2p - cos 2p).

F252(v) =

d(2p) where

(2p)22

2 cos(.L - 2p) +

e2cos

+ + sin(.- - 2p) +

e2sin

+ + v(2p)[cos(.L - 2p) +

e2cos(.!L)

sin( 2p) +

e2sin

_{-] +} - 2p) +

e2sin

-2p . sit . ,S1T 2 sin sin.2 -- cos( - 2p) +

e2cos

+ (2p

Wahlström9s Procedure for So1ution of Linear .EquatioxiSystei,ì.

bool lwoc gaussinv(n,m,a,banthl);

i n,rn;int n,m;arr a,b;bool anthi;

fl

Proceduren beräknar (a)(-1)(b) och

lagrar i b. a förstöres.a[1:n,].:n)

b[1:n,1:mJ orn tri>]. eijest b[1:nJ. Proceduren

är faisk orn a är ilikonditionerad,

annars sann. Pivâtering pâ

diagonal-elementen ut.föres. ('rn nâgot pivâelernent

är

mindre,n 1OL30 anses a Ilikonditionerad.

Orn antl=true ärb eri 2-dim array;

i.,j,k,p;reaj ill,piv,s;

bool beh[].:n);

ill:=101.(-3O);

Thr i:=1 stp i unt n

beh(i)

:=tru.e;

1.:=i step.]. n

.kg& piv:=O;j j:=i step i unt ti

beh(i) _{abs(a[ j, j])> piv then}

:. p:=j; piv:=abs(a[j,jJ):

end; j pivill.then

beg, gauss mv :=false;gotq gaussinvslut;

eTld,;. piv:=a[p,p];beh[p] ;=fals

£Q

j:=]. step i

unt n

j, j4p ithen

beg s:=a[jpJ/piv.!

Çor k.: = i step;]. J n

j.

_beh[k]

tirien aEj,k):=a(j,kJ-s'a[p,k);

j, anthl then k=i stej i m

b[j,k]:=b[j,k]_s*b[p,k] else

b[ j] t=b( b[ p];

ejrad;end; j i:=i step i

.g.p1v:=i/afjjjJ;j

a't1ü

£

j=i

ste.p.

1. m b[i,j]:=b(i.,j]*plv

else b[ i]

:=b[ i]tpiv;

gauss

mv

:=true

+

f1aftctia of frequency y

td

depth h

RepoTt 36

-

I

Fr

--

I - I

---Tr

:,

:

-I-

...:.

.:I _;:I

,

: : :-

..

-4 J:: .

IiII.-1-,

LIII

I

---.

; '.1L ir::1

,.

:

'::'E.

L : : .. L. _ . -t _r

I1IIIL

_i_

_L:

ij

I -

'

L -II

t1_

I

rj

-Hf

? --

PL,T

4

L

-1

L

L

_i_

t:

r

-T..i

1

-:-

-I -r-

i

i I

r'

4-4. _H I

:J

- .-,-- I I

j

I i

I,

/

H

T

I

:

I

t

_

t

L4?!1

-4

t

T r.

'dB

:

,. I,

.

i4gØr4bh1

. , , ,-'!

_-:

L-

i

b..-:.

::

Il

...

! .

iitQ6

EL1J::

ri _L I . .

'

Í'I-I j I t :1 f I I I -T:: :

j

Il 1j 1 -I

i

J :

'I

I j I

Source intensities as fuctios of

isia'

for depth parameter

36

hIT.

LL

L t

I

₎

i L :i

::::.;:::

_Li:...

:;2

. :: . LT7

;

, 'T T - ,- 4i-.- - T. rt

_A11-

'-r î' j

_:

li

:' ;.r_

.fi

_{'i!'i_4r_}

L

:i1

T '

'k

::.

_'

- iLi _ h-

'jrj _!h

t -1 1

4dlL1.

:'.iJ.i i::i

'!::

ti fir _{_LL i} ' _{_ b} -

._Wlinii_

t ... . II

. r'' ...

_{_.1} . ... .

I ...

L

'ii

T

₄₉₄

-r-

--

*r

. r ' -- L E E ! Eir i E E E E

::

_..::.

.?:

::::. ::j,::4.:;;A:J

E:::::J. :

,. LE

111

i .L J E 1

---I!

-H

:

I E

-y

T

j:-'

____I t I .-L

j

t

- 1IlWA..

_T r Lr

_f1.

r

jI

.. I _ -____1 I ,: ._. E E L __..._L_..J_ T....-r- _L.-f L.. l .- E2 ...

i,I

l..-t r r I I :1 E 1 T4 I EI I I E

.-1L.-

_/

I I E I ... ... _{.:' ...1 ...} ., ...

..;::I.. ,

_... ..I ' :- _;-___a

j.

._II!

I _EEr E

fi

I IL -

-I -J 4 - ... E

... :::.:::::

IT - .. I

.

E

-E _,f__ I

L4vu

i_4 -:-

4i

- _{;II IJI:} Ii _i

:.J:: i.'i:

_4.i.i

freqasacy

hIT.

for depth parwtei

_{Report 36}

-2 j f - i i I lt

-1

il

L_ ' _

l_

f -, .-}

1I

.-r

.

jt

I ,-t t -I L r . _. j4Ti

L!L' L

_

I

I: : r _¡ :' I . i -r t. _j :: ::t.. - L-:r :: f : Ir

::: :: :

, 1':; L ¡_ -:t: I :

,

I t -f ,::;

. 1::r;.j::.

r

__

: .

--

:

-t

L

,::!T T7:

TL

L p_

:11

A,

I L if r i

-_--LI4_

-

.-hT:1:

tr

it=p

r . L L. -t t t

j-i

-.

j

:L

:

-A

Alllrt

4 1

L'\

t

t4.4!V4

A

.1_ e I .1:__ : r I ._i__

L2'

J

...

::

r J

---1H

_t t : t J: . t

*'

r_r

I

)

r .i 'j

tt

*i..

__

-:L

_.H'

i:::L:

It i

:'t

:t'L'.-:.H

: _1_

_-

i

-t

_t I r

L::L'

1.r:!!

..

i

r

----4-

r

r

':iLj::::

t

:H

r :

---

' I r

----: : t ' t :r, J r 4 r

';.

r'

'_ r-t

-H

::':,

r :

h----

4

-t r L ' r r J

F

-r441i4H

f L i1

:L.r:,j,:

I r

jI.:1::H'

r r

L:

r t t r H t r J t rr ¡ r -I _i I L r tj

r4

Q2 r

rÎ

r- rß

t 'r: t:::: 1, _:r-'t :

H

rl r:: : :: tr:;.. ::;f1:: t t t .rr:. , ,r : r- t i; _'::,r:::

t

;' .: ':;:' ' rl ,; 4 ::_4

fictioas of frequency !j!

for

depth paraaeter hIT according to

.

--tfunctions of

t

_{depth p&raeter}

frequeucy

.,TTH,

hIT..

_ J

---f-!j!

,:Hr'

for

t 1 i

1:j'T!:11:1:M;,

t--. ' I L ,

:2-4

. H i ! .1__

-:c:__ I

:IH

:LL1

Lf

---1f

-+

L_L

JI

T

t

-r

r 4111 t __ 4 .1

'AV

t

4iLc

I

VV

.. _ -

L-1_L_1125111I111

il

Ii

tL

I I :

- _

I I

.':

kÌ

i ,

-T

H

Jlll

ITll

'

J j

b' r4..

:1::i:..',

I

49

+

I H

'.ÇtH

L4t

T

'''' .1 H. I

-20

'ja:

JI

I

lIP

1TliAL

jIII

i 11111 "h1

t_Ilj1;

"

I

__

r

e

I ' I -:-I

II

I _-I

.

j d H

i

t : H Ii.:i_t.: t

j::

I I r I...

IDI

JT iI I

H.,

f H: I.. H:. I

IIJ

I I I i

r4

1I

t

P9L N

(3e;)

OÇÇL SS v CG :- Lr_____ 1:;

rH1

T _4_. L___

oc

'

Y

! I -

'i.L

11

'

o-

r

rJ4r

:1z:

::,:

:tpjr 1 L.i

Hr

tT I

j

:

J :E 4 H

-i

G

..,-.

i .___L L i -

i:

i - _.:t_ -F

:r:.T:;!1

____:__ H .

::;;L:

- I 4 _ I r r r 1 - + i L

Fr

'"-

4 r -

::

:

' : . I» J

':.r

« :F I , 4 1 .:. . .: I:. , . - .L:___ : ; - - . J _1 ::: :: ; ::;r_ : r __.._ ' ,- r ; ---_-1-..4-.,-

;1:

L;: - 4 t . L r

J-

I .

;r

H

I - t r

4L

I I _i 14 j L Tr --

1

T-t

F

iI

ITl

t1!

:';: r : r ;: : : p ;:.. :: : .. r . : :

...

. . . .. I 41 t -

-

_i

':: ., . . . . : :;.t_4

i::::

_T I

-r

n

i

...

_

r

I-

r r r H I L - ;

-l--t

- 1 . i... :

:..

r r _! r ' r L . .

l:IL

... -- ,;I.. r- : 1 . . . .

...

F f .:. ,: ,- .t

:rttl

i1-

I F j r

L--

., ; : r r t I ....j r 1 r

-

_. L

-

r

t ._t r

4

fJ._ J I ;4 _ .-

.::;I;,

¡ r

1Hr

-r-t

;_ø-

:.it

... r H . j.-. i4 r ..[. .:',L:..t:. . ::. ». . .. . .:;- ; _., ti - ri '. _i

r:itÀi

1.1

..l..r F rr

44P1

1 ;:,:;J:

_II-t-

t1!

4 t: . ;__...LI! .r.Fi 1: F

'1

; r .

;r:

:ij: Y1 I

:

.: i1 jE : r r

::j'l

;:

... Fr 1 r « r .4_L._ ..____fL r __t!__, I r r r

:

r r TJT F _L r

':.

. r 1 r T

"TF

Ii ._ L:t:.

r

rJt F r r 1 I LI. r r [' ri r r r r ::ïi,._r 4 .

'yHI

r I 1F . r .-;4 ,i :: ft: _ I r s . r I FL L

t;:

: .. ::,: ; r . F 'T r

'-

. :»4fl : :: F rL : ; t' ;

Tf'}

TrTr.. r i . i .. r . 1.. j ,.. .r1 tLF r J _ j r _- . IT , 1j r .4 r- : r _ ...

HL

r r ' ...

ri

r L r

rrIIlt.,..

r -F

1I4I!j1

r :. t-r I r1 i r

;.t;.Hrtil:..f;

;:. r r rr ....

J1_

1 î

4

rr

r {i +rtì : L4 F :l1:, r , I r rI r r r Li , r r t.t .1j . : t: r} ... r Lj1 d- Lt r' f L

iM-

j 1; c 14 4 f t r-- . j.r.SîjIIIIIIOI4 lr 4. _ :.ri1i;.t , jr r

*r

jrrfLr L r i4 1jj Ir Ffli r! J 1ti'1 Trit 4 1 r

-iL

f:4

u

parameter h/T according to the

-il

I

!-

T -

_as

dpth

T

p1itude

fmctions

paratsr

L ,

ratto

of

.1r_

of

wave

frequsncy

h/T

at

10 __:_.îL i -

aitd

heays

fer

uii

freq.ncy

!j!

1'-r , i rr t1

III

i - ;_4:

f

-

::::

-

_r---

j F L : J._ I i .J ;

;i;:,t:i;ç:;l,

:

;I:

i'jJ1i

- i' :.4;J j -

-

I :v :, J

i

r

::r

_ i . ::st: :::,

j:

I i _r - .L._1 __ I .__.i I 4 ._4- J r f'

-

i 4

t

7

E r T 1 Tt!L _!' L L _i .1 I ; l:: Ì; I t I i 0..:øi1ølPI ,:.:, j;.; . : , T ,-- . H I -f t

H

J '«H,

L

_ H ' : H: i

:,,.

. I r -i-. I

__,ø7

,.. L r _f I

--

I t

-t,-

r

e L .-

,.

r

L i ' i t 4

i-

I i:

.

Iii I J _ii I IL

I_

II

I Jj

j

_d

_{._}

L . _II I I I T : 'i I 1

L_Vr

L

Ltf

-4 _J

iYiti!:

;-

-H

'--

,

,"

_:

Ii

T' I

i

I f I I _ I I

!L'!.

' I L 1 . _-I.--

HII

I

i;

ï

j-

;j!

i- :1

J'A

III _L I _'-1 I

ILL

J

j

J

I

JI'

.::-:

I'

ri l ri' .::.r:: :;r;1,.r:.:. j tr:rI f

J

.II

I

-j

_{r ;' I' rr:} ' ...

t-

I :::t I r: :: ri :: : .,..:-i I "

r.t.!

L

If

i r r, r

-

r

--

_r: r r I

23 A4 732501 rg,,p111 r 1634

II'L'JL

':1 - I r

depth parameter hIT.

'i. 1.5

25

;H

H

30

Ql Ir _IJi1 :r.H

:_r,

1:rt

r

_jIi

__

ir

'I I

t

r I L

n'

i L :;I:j._ r

bd:rirT

NIIJII1II1

IIII1OR ...n

inn. aflflt -.

tflflflhitfl -¡t*flhÍISflhSfl

,

II

IIEIIIIIi

i

j

;;;

!!Lllll

OIIIIIIIOIIIIJ

gjjjij jIjillI

Jli;1i:iILJb1iirrffi

JIHi:1:

r

4t L n n q p L i

.-foTUia QQ. (12).

.

*t

IOIiOi

;;

tt

L4 : i '1 T41i i ..f i _; iiLiLkT1 ):r ' 4' _1J2J, In : ' r i f1I; tF

!I!

h

-r't-ti

, '

!:

J

ri

II L

»

F1±

:;1

I YHPLh ,

1t

r

r

i1li-

i

kihL

I

r:;

i ] 1 ' I;r _L . '

i1-I-

'j .," .i:

J __ I .- i ¡±+ .L_l; f _t:_1 L.H .

II.t.if'

I »

'I

II ITti tu

:..

'.L*i;r. Nf1j , r :

-j if1H,_

}I;;

. ± ' j

iI1 ÎL

.._

...,

f_j ._ I 4

I1I

I ;-i ...I ;T çjf-11-;1-: iI i

I

'li-ìJ

ii,

I , ' {

It

I j 111 _j__

...,..

i

'iJ

r ' It ri-1 i... I.

f'i

4-i ' h i J 11L I 1j-..-iI .. I T1 Iji I ,; ' ' i r!

It '1

+

I

I j , ji I Llt'., l_ i It-j-

L

-t--r

}:i'

I I 1 i -.-._ j

\!It

:_:Ï,_ : i i r I I t , I - i i- ... J I ' - FF.I .:

T:.l:T

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'

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t:- :...

_[JI

-: . - _{,, ;:.}

:i.4:' ...

': i:i

r-

I;:;'

. ::Ji! I ; :

.7H.

t.i: : . : r.

::I1..

jj

....I :T I :

H;

: ; J!1 . L: tTL

III

41 I ,j-!

....

I I

4 J 'I r I b I

,;-,

J: L '- I _II ,,

...

I I I

fi

_I:

'7Ï P1

I ' fl__i_ i

I'-J... I II I LLtr L I

I!,1

i-, , I ,

...

Ij J FL' I J J ¿ J I

,..

_I,. II I ...

_-z

iif!tII

IIrI

îï

.,, : ,h I _.j .

....i::;.ij :Ji

..--.- I .: _;._ ;r.: s.:_r J I:;1 :: ir.: r

..I..::L;i

. ::;;.

it _{t_-i.',} L, _j .:

;i

; :;: i: 'H-' . .Ï1f ' I -.. -r-ï1

i

j_.i ti : _f. j r 11L I J -i iL I , , r __Ji _L,1j I r I::: : ;Lj.r j--- -'- i' rr r I-ht_f _ f ' J r

':i.J;

:h!

I _ : :: : ,, :. r ' ' 1r 1 I r 'H»i... ,. ... I I r L... r

.:,

r r r J , I. j Jj rJ . r ::;.r.fl I: W . 'T1 _ :. , ' ::: I } I ; ILt I . , I

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523 A4

; 7325O

4LjpN Nr 634

depth parameter h/T1lO,O.

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