A note on the linearized deep water theory of wave profile and wave resistance calculations

(1)

BY

J.K.LUNDE

A NOTE ON THE LINEARIZED DEEP WATER

THEORY OF WAVE PROFILE AND WAVE

RESISTANCE CALCULATIONS

NORWEGIAN SHIP MODEL EXPERIMENT TANK PUBLICATION N°48

REPRINT JANUARY 1963

ARCHib-

Lab. V.

S4w.,-zr...1s!)on4wnd

NORWEGIAN SHIP MODEL EXPERIMENT TAM

P"':1-"""1

(2)

INSTITUTE OF EN

BERKELE

A NOTE ON THE LINEARIZED DEEP WATER THEORY OF WAVE PROFILE AND WAVE RESISTANCE CALCULATIONS

by

J. K. LUNDE

ERING RESEARCH

LIFORNIA

SERIES NO.

a2

ISSUE NO

DATE

Dezenitez,. 195.7

(3)

A NOTE ON THE LINEARIZED DEEP WATER THEORY OF WAVE PROFILE AND WAVE RESISTANCE CALCULATIONS

by

J. K. Lunde

Fourth Technical Report

submitted to the

Office of Naval Research on

Contract Nonr. 222(30)

Hydrodynamics of Naval Architecture

Series 82, Issue 4

December, 1957

Institute of Engineering Research University of California

(4)

by J. K. LUNDE

§ 1. Introduction. It appears that Wigley was the first to show that it

was practically possible to calculate the wave profile for certain fairly ship shaped bodies of small beam and of infinite drafts, moving at a constant speed of advance, and to obtain a fair agreement with the corresponding measured wave profile

[13*.

Immediately afterwards

Havelock gave a general linearized solution which is applicable to any ship shaped body of infinite draft, moving at a constant speed of advance and satisfying the linearized conditions [2). Since then some additional papers have appeared in the literature, but their number has always been considerably less than those dealing with the corresponding linearized

theory of wave resistance [3)

M.

In this report we shall return to the general linearized theory of wave profile and wave resistance.

§2. Free surface condition. Let the x-, y- axis of a right hand,

orthogonal reference frame coincide with the undisturbed free surface of a body of inviced and incompressible fluid with the z-axis directed

vertically upwards. Suppose the fluid fills the space

z

0

. If a body

is at rest in a uniform stream whose velocity is -c in the x-direction, the velocity potential 4t can be written in the form

cip, (2-1)

where cx is the velocity potential of the uniform stream and where is the

velocity potential of the perturbation caused by the body.

We shall assume that the (1) produces negligible effect on cx at sufficiently great distance upstream from the disturbance. We assume

(5)

also that the deviation of any fluid particle from the state of uniform flow is resisted by a fictitious dissipative farce proportional to the perturbation velocity. This force does not interfere with the irrotational character of the motion. Hence the components

X

Z.

of the external force are given by

x ) ₉

)(==-

_/Li

\,(==

wherejut is a positive coefficient which is retained in the intermediate

analysis to a degree sufficient to attain its chief purpose and is made zero in the final result.

Since the velocity components u, v, w of the perturbation are

given by (1)3

(2-2) also as

dOrespectively, we can write

_z.

7ti

) --/41q5y

z= 9-/

Multiplying the first equation by dx, the second by dy, the third by dz and adding them together, we have

9 dz.

Integrating this equation we find that .. 3

z

where S2 is a force potential.

When the fluid motion is steady and irrotational and the external force is conservative, the pressure equation reduces to Bernoulli's equation

9,2

=

C

where q is the resultant fluid motion, p is the pressure,

f

the constant density and C is a constant which has the same value throughout the fluid

at all times.

2.

(2-2)

-

(6)

Equating Bernoulli's equation for a point within the disturbance produced by the body with the same equation for a point far in front of the disturbance where

_(I)-4-0

we have

(cP,12,-

qz)z]

₊₃

_(-z-)

_{74 4 Ci)}

c2÷3

7-)

f

where '>(x

- Y, 0) is the free surface elevation. But

(1=02"-t- (cV-+ (U2-=

2c

(4>J+

(kr;

and so

P-

Pi

+

C c1D + (c1)),)14((k3)1+

(41)1+3

71qb

0.

Assuming

_{(ci))11-- ((i) )1+}

₍ (12

to be negligible compared with the other terms in this equation, we get at the free surface where F>

p

--riqb= 0)

(z=0).

(2-3)

When the fluid is in contact with a rigid surface, or with another fluid with which it does not mix, the general kinematic condition which must be satisfied if contact is to be preserved is that the fluid and the surface with which contact is maintained must have the same normal velocity at the surface. If the surface

S

is given by

relative to a reference frame fixed in space, the general kinematic boundary condition Which must be satisfied at every point on

S

is

(2-0

=

(7)

Since the free surface is given by Z= ;(x, y, 0), we put

z

y, 0):= 0 and (2-4) reduces to

ck)A,

=

0

Neglecting small quantities of second order, i.e., the non-linear terms

d)xx

and

i0:3

we have

ci)= C.-. =---. - \./ CZ = 0)

2- k )

)

which is the kinematic boundary condition which must be satisfied at

the free surface. From (2-3) and (2-5) we get

v

ciD

=

cAP_x

Which is the kinematic boundary condition Which must be satisfied at the wetted surface of a fine ship, i.e., at the vertical centre line plan

of the ship. 4. (2-5) (3-1) cipt qb),

0 (z= o)

_(2-6) where 9/d-z and

/4)-§3.

The velocity potential. If 9 =-- -±.11) (X.) _{is the equation of}

the wetted surface of a ship at rest in a uniform stream moving with a velocity -c in the x-direction, we put

F=9:p4)(,,z)

and

(24)

reduces to

(C Ck)(1)),

-Ci)zlilt.=

0

Keeping the same order of approximation as in (2-3) and (2-5) we have to assume a fine ship, i.e., we neglect the non-linear terms

Ci)), 4Ik (1), 4)-z. and

F7==

=

)

(8)

Subtracting the second limit from the first we have

We now make use of the

following

theorem from. potential theory [5].

TheOrem I. If the density (S(G) Of A,

surface,

distribution on

the

regular

surface

8

is continuous at on

S

the normal derivative of the potential

LI

at, P approaches limits as1D approaches along the normal to s5* at

Q

On either side of . These limits are

where U(G1) br14 Q)

+-SiS WaS

-6.U(OL)

2

_G-(Q) _+ifs.01.70.

U

₁₁₄

dS

Subtracting (3-4) from (3-3),

_

Let U be a velocity potential

d)

and let

c.

be a point

the surface 'yr-=10' 1 then becomes

respectively, and (3-6) can be written

_

-2)4) -4'y+

4X600,3j,

we have (3-2) (3-3), (3-4) (3-5) (3-6) z), _on

Vay4_,

a/Dy-(3-7)

)d.S ,

S

(9)

where from (3-5)

)4)0<)tJ)-2-)=

fl'Cr(k')

_ciS3

(

3-8a.)

and r=

-k)a-1-91+

S

Equation (3-8a)can be intreperted as the velocity potential due to a continuous distribution of sources over the portion $ of the plane

, the source strength per unit area or source density at (K)

O)

being

d(0)-f)

. Hence, the significance of (3-7) is that it enables

us to determine the density of this source distribution once normal derivatives of the velocity potential are known.

Comparing (3-2) and (3-7) we have on the surface

_

(

cr(x,0,z)=- 2c4x (>4,0)7-).

-by't "e3-

(3-8)

Hence the boundary condition of the wetted surface of a fine ship can, in the linearized theory, be satisfied by assuming a continuous distribution of sources over the vertical centre line plane

S

of the ship so that the velocity potential CI) due to the perturbation can be written

(t)

_ds

(3-9)

where _k)1,4- _z-i- _{z. -01",} and where,

from (3-8), the source density ( (

0,f)

is given by

6.

(1)1 0, )= 25t (-Ph (h) f

(3-10)

(10)

Source Of

Stre-nen

rn tat (0,0,-1)

FT5. ,

uniform stream moving with a velocity -c in

the x- direction (Fig. 3-1).

It will be shown that

the velocity potentialof

suCh a source can be

determined in a form which This last result is, however, obvious when considering what happens at the elementary area

SS

of the distribution

8

away from the boundaries of

the source distribution. As cc' is the source density, the total strength

over is (58S and the corresponding flux is equal to 4TEC

8SL

Owing to the neighboring source distribution and assuming a uniform distri-bution, the fluid does not flow radially out from as in the case of an isolate point source but is forced to flow out along the normal to the distribution at SS and at both sides of it. Thus the positive normal velocity V is given by

v.=

4itc:585/2aS-=--

27Cd , but from

(3-1), V=- and hence c5

,

the source density, is given by (3-10).

In the analysis which follows we shall assume that the density of the source distribution always is given by (3-10), even in the presence of

a free surface. The velocity potential 4) due to the perturbation caused

by a fine Ship and given by (3-9) for an infinite fluid has, however, to be modified in order to satisfy the free surface condition. However,

whatever form this modification takes d':) must tend towards (3-9) when

the free surface is moved further and further from the body. Hence, in the analysis which follows, we consider a source of strength

nn

situated

(11)

satisfies the linearized free surface condition and the equation of continuity for an incompressible fluid. The velocity potential 16

,

due to perturbation caused by a fine ship, is then obtained by integrating Y over the

vertical center line plane of the ship, assuming the source density 1 to be given by (3-10) since this expression of ot) will satisfy all

linearized boundary conditions as well as the equation of continuity.

Assuming that the fluid extends to infinity in every direction we have for a source in a uniform stream (Fig. 3-1)

where

(I)

C 22

r,

z

r ,z-t-8. (3-11)

If c is sufficiently small we have from (2-5) that the kinematic boundary condition of the free surface reduces tof4s1S*))(7))

₀₎

i.e.,

the surface condition when

0

approximates to the condition of a rigid wall given by

z=.0

. This condition can be satisfied by adding to (3-11) a term nn/ral

cl1==

x2.+ya-1- (z-f)2"/

which is the velocity potential due to a source at the image point (0)01*n

and of strengthrn .

Assuming that ct>5 , the velocity potential of the perturbation

caused by the presence of the source of strength rn , vanish at

infinite distance upstream from the source and is bounded at every point on the free surface, we have from (2-3)

(12)

If the integral

jci)(x,9,0)clx

C.0 exists, then

P,S

(f)N y 0) d

C.

We have stated in §2 that/tijor/mi is to be made zero in the final result, hence

dPs(x) 9) o)

==

y, 0)d .

Since the integral will presumably be finite we have that the free surface pressure condition for sufficient large values of c may be taken as

4)5(x)Y,0)

=

, which corresponds to motion under a free surface

neglecting gravity. This condition can be satisfied by adding to (3-11)

a term

c-=

which is the velocity potential of a sink at the image point

(0,0)-4).

Having established the image system of a source below the free surface in a uniform stream for these two limiting cases, namely when

0 and oo

,

we now assume that the image system for intermediate velocities can be built up of a certain

distribution of sources above the free surface and that one of these with strength nns is situated at the point ) s .

The velocity potential of the image system therefore may be written

Hence the velocity potential of a source

rn

at (0)0,-f)

and at rest in a uniform stream of velocity -c is given by

(1)

x ci)s )

4)s=

2-1 rn s 2. where

(c)

(9-9s)1--4-(3-12)

0

(13)

We do not yet know the exact form of the image system, but this will be determined when we satisfy the free surface condition given by

(2-6). First of all we have to write the terms in

(3-12)

in integral

forms. For this purpose we make use of the known integrals

ic

it(tal-z)

27.(x24-91-i--zzi-

id& e

_K

_-ferroxlci

J

-7f 0

=jd9JeK)8

$trrZ'.O

-7C

where xco-s -t- Thus

(3-12)

now becomes

-it .c,

1

r_n

89

.eKU(7.)

(z

4-f)3

2T

eik ...-r 0

Kii.gx-xs)c.,yyr,,,,4-2.2.4

_I_Lrn4c19je

79 (-)s 9](-q

2.7c s --it 0

where in the first integral term we require

2-44>0

and the second

-4

7-s

0

Hence, in the form c4),s is written in equation

10.

(3-13)

(3-13) it applies to

the

region --+ Z. Z-s ) and in particular to

the free surface where

Z=.0

Moreover,

(3-13)

satisfies the

equation of continuity for an incompressible fluid. Writing

(3-13)

in the form

Pc'

-(z+f

(iL77.-1-z) EIL\

j88

-k-

-1-69

Zit1

a

.

-'it

(14)

and substituting in (2-6) we have

acosze

[rn

r(e) e).]

Cef.rn-C(G, k).]

14 Cos

&['m

k)]

0)

or

me.

(-1Kcos260-

4L& Cu e) - r(e,)

(i< tue".0

,

0+

M.Cos (4 This equation will be satisfied if

f

r. ors 13, "4 NO 4' le CASS

9

zer_ 9

CA-s?-9 -4 o yAcet k-ke,st-c.1 sex, 9

write (3-15) in the forth.

jci

ejeK t:

-cz+f)]

-7t

71 00 cz.a3

!I IdJe

-It

Icso, .rt I/1 K.[ Era (7. -

f

rnV-41se.2.9

de

e? 7C--

)

(7.-f)]

d.

(3-10

(3-16)

nd (3-13) can

be written

as

CI)c

- ez+-r)]

d

foo 2,7t

rn

fciek+

#14 b=3, +

f 2,

ice.c."60 Z'itA se,8

-71 0

(3-35)

where

> 0

F

O

CArs(901- Ild'n

9

and real parts-have to

be taken We may also

--it

(15)

or

cso -t-

(z-f33

rr,

sele

d

e.

se4..9 0 % p

where q

+

t

and where real parts have to be taken. From the discussion in 44.2 it is to be understood that,- is to be made zero whenever its chief purpose is

attained. It will readily be seen that each one of the terms of (3-15),

(3-16)

and (3-17) satisfies the equation of continuity for an incompressible

fluid.

It has been assumed that the velocity potential CI:is of the source

vanish at infinite distance upstreat frim the source. It

appears that we are justified to make this, ,assumption, at least as,

far as

the wave motion is concerned, since the group velocity of gravity waves is never greater than the speed of propogation of the waves and hence gravity waves cannot precede the disturbance When the speed of advance is

constant.

We now have to show that

(3-17)

satisfies this condition.

Since each of the two first terms behaves as

0(11)*

a.s

it i the last term which has to be considered. Let ot,

Va

it

_{r.s-+ (z-f )3}

1. \)s#20daje

lk--=_. te..c./9 4. ?... se. a 0 0

12. (3-17)

where

CO=

XCA,-Sta -I,- . Substitute 9 ..

+1

and ID becomes

Irk 00

_{14 c(z_o Z}

6c:cats, y,s%14.19)3,

61-, ,i'secle

°tele

o o

e2&

iiA See. e:;

...---*

x-4.x2'=ii

0

(x2). when A --i. C=.40 ) k *- ke/z.._-- ON) when --i.

)

. o (x29 when *---0,.,

x when x--10. c o * 3--..:-_-, c.)(*)

but

C

(16)

Substitute S

and c becomes

C

1Ce-2'6)

In the last integral, we substitute G+ 7t and it becomes

7

AL 0.0 [(1... j>cCArva S4-;" 0)) .1

St-39de?

Hence,,

0. j

r

=...fs,42.6cla ft-t-fc);.,

(,,c4-1 e + d' 9). .e.Z:

ic.(.NeLod. ,,,

s,;._ e)

o

0

r'"--- 4 Str'"& -k

C-7l s.Qc G Tr/2_

We want, however, the real parts,

-rtia

0

write the double

Sec_ 19

-t-AseG

Lie.; 10.a C-17'Se 2;e%

9 _d.

.2.

os#"- e-ty4^ &IL

9 -s,;,g)

-1-e

ktr...,276)

c-45s12-9 dejr

e-

_d

ste-9.) Las (km co-se) GArs

s.;.te)

\t-osect6.)%_t_/..s.,36).

-iy4A sec e Cc.* ta-sta)co-r(ik.s..., G.).]

Since it is the limiting value when /A-4O we have to consider we shall integral as bO

39- e

set./

cos

dk.

co (K-it seee) 7-4.. 'Lk,.

e.

_{(1kc Loqe)c.0-+(knrir% 9)1C:kj} 0

(3-18)

(3-19)

!I koStAltii

ci4

L +

-i.e.,

(17)

it ms consider the first integral term and in particular the part which is integrated with respect to

K.

This can be written

le-6.--.1)fc..s[x(icei)c-rs 93

(4k

co:

9)

I

0 s'untx N-q1c4fse],y,(kktcose)} urs (

44) (3-20)

where

i"

te,J,je

and

Are now make use of the following theorem which may be obtained

from the theory of Fourier integrals.

Theorem II. If fcs) is sectionally continuous in the interval (a, b)

and has derivatives from the left and right at a point

s=s0

,

then

Irfcset-o)

I.

rvin, irm

ds..

_{i f(a.+o)}

> -t-i-.... _a when

fcs. 0))

a c

s.4

Sc,

b

so,

s

If

Ifcs)ds

is absolutely convergent, i.e.,_

PM8s

is

o

convergent, the limits a and b can be taken as 0 and oo respectively.

Under similar conditions,

rvrn cs) _c4s(s-s°)

t

Applying these results to (3-20) and noting that

(i==koseje>0

we have

li_..._-ke.

t(zf)s:81

(14.1x co-s (3-21) 14.

--

0

ks--b.

(18)

We now consider the second integral term of (3-19),

Ic(x-f)

'T Urn e_ _{&`".' (ex} ce-s

e) Los(

cik

a

7-b-01

00

-=---

rt..4.0

Urn

j

(

_{4-v112-+/-sec.le}

(3-22) where

_{14i= K:, e164> 0}

and where

r (lc)

is a continuous function.

As 7-0,- 0,

,

only values of K for which K- ,i--0' 0 makes the integrand

different from zero. Let at therefore write -- Ki.--=-- n/c where

the limits of fl are _ cx) and -4- c=e3 since

1:1> 0

Assuming that tyt. --- 0 _{whatever the value of n} v.re now have

2cf

P(te )

k...rn

_/-

t

dr)

(iki) c-os

AA% (n1-:+ste-9)

C0-5

e.,kikcctsfio)

COI(

SINN9)

ki

(7.-f

(3-23) Substituting

(3-21)

and

(3-23)

into (3-19), we find that the two integral terns which (3,19) consists of cancel each other when k'

Hence the velocity potential of the perturbation cauSed by

the

,source, vanish at infinite distance upstream. Since

kf

114.11 F(9, ryle

the velocity potential of a source as given, by

(3-15)

reduces to

KLLC,) (M14 )1 Ct

+

f

=

G

e

cik+

Crajci4

cik rc) rt.)

7I

when

0 ,

Hence

(3-15)

satisfies the ,free surface condition

C1)4z(x, 9,0) -a-

0

when i.e.,

-0

,

(19)

-Similarly, since

-10

(11 -Re,;k)== )

kd'i°0

the velocity potential given by (3-15) reduces to

it .6. -rt

1;.(Z(z+f).3

, C

ct-03

Era

e

rn

2.1L r r

-7t

when C-.A.c="Q Hence (3-15) satisfies also the free surface condition

(CO)) ---

0

When

Since we have assumed that 4, is bounded on the free surface,

0 when/A-a0

and -2-=.0.

From (2-3) we then have that the wave profile (*.,J)0) is given by

*We can.also use contour integration in connection with (3-17) and the question of the condition at large distances upstream and downstream, but we have then to make what may be called arbitrary assumptions as regards the indentation

of the contour at the pole

c-vVostje

. By making use of Theorem II we avoid

this and the result determines uniquely the contour to be used in connection

with (3-31).

16.

(3-24) If the fictitious dissipative coefficient/m- is not introduced

41) , the velocity potential of a source, apparently can still be assumed

to be given by (3-17), but the denominator of the last integral term will be ie-esejt9 instead of

k-k see

-4- (74" St-C. (3.

By making use of (3-24) we find from (3-17) and (3-21) that we then have waves upstream from the source and that these approximate to

TfiL

_ 4

rne

c.-1,1c,Ksec.9)cas(V.e',69r4c(ii)sec-ecie

3

(3-25)

0

when C.104

*

This is, however, contrary to one of our assumptions, namely that the perturbation vanish at infinite distance upstream. It is,

therefore, necessary to superpose a freely moving wave pattern in such a way as to cancel out the waves at a very large distance upstream from the

(20)

source., We notice from (3-23) that the required term is

r

4rnleafe.;(z-f)smae

t;en(kik

co-4G)

c4r: (qb

9)

sejg

cIff

which satisfies the equation of continuity by itself.

Therefore the velocity potential CV of a source, wilen the

fictitious dissipative force is neglected but otherwiSe undet the same

conditions which, apply to (3-17), is gival by

Io

1c/2

k(2-f)

cps=

_rc

4 rrl

Self; de e-

Led oc,,.c.s.rt

et)ci.rs(kys,:,,9) dk.

ri

7

k-

secl-0

--im

leo o (k dc sec G ) cas ( _sec..1-9 ref:"

6? Li ,e)

lcCe-{ )ser...19

Where

r--=

(z+

f

and .Q=.=

a/de

It may be conceivable that the as given by (3-17), which tontains the fictitious dissipative forcer is not identical to _CI)i as giveh

by (3-27). Indeed it is difficult to prove in .a rigorous nianner the equality

of these two expressions at all points in the fluid when transforming the

integral terms by contour integrations just, as they appear in (3-17) 4ftd.

(3-27).

From the discussion of the free surface conditions It will readily be seen that whilst

( y, 0)

=

3

_4)._oc. _tV _(3-28)

gives, the free surface elevation, the same expression, but with

0

(3,26)

(3-27)

gives the elevation of a surface of constant pressure above its position

0

(21)

-'when there is no perturbation of the stream,. the elevation-of the

isobars.

From (3-27) and,

(3-28)

we have

i

-Friz. c2.0

is

s (k,1,2):= Cm** (1 - 1 ).=.

4 n-1

secadti .42-

i

H e-(v.4)'

s;.(k w c,:ps a co-s N

c:,,,9)1<clic ItAL

'k-vosig_to

o

009 =

Im

We Consider -t-

4rY,K.f.(/-f

)re2:19

c,34 (k,,k sA4 a)c4sfr,tdslos, lec-69

d

₍₃₌₂₉₎

Ignoring constants and substituting A

=

9) CIA cei-t 9 d9 where A 4_- we find, that the dOuble integral tern of

(3-29)

can be

written "r/t, 0,3

tx--=-1Sec,,&c19

e-k-ikostc.ze

(k), etrs e) cfrs(c;, t4) 'k

OcV

kt

A)

tc, )4

W:7

i671 = Az

7

c17 18. (3-30)

(3-31)

where 17 is a complex variable equal. to

in

)

F-7 -

0

and where

Irn

stands for the imaginary part.. The integrand has two Simple

poles at _ik6J,&K0z4.- and

Ki-=

At

on

the teal

exit.

The

contour C

we choose iflhen

x

0

consists of a

circular arc of a radius which tends to infinity and the positive half of the real and imaginary axes. Since the integrand has a. pole at,4 on the positive part of the real axis we cut out, the part of the path within a

i.e.,

Ste'La)

(22)

Ak-Cauchy's integral theorem,

fci

=-.7.

0.

Consider first the integration along the arc

r'

. We substitute and ultimately let

2-4.

c..0. , thus

7*- ;.oe zid

A OViz)2'

21e(1

I jf(

:Re

+2i0c+ilx1Re

I

Res) 12 e;:c4i. d

ticI ..., 1

<r: _1ZdoC

1a

KoRe.ze4s Al

'Re-44

F. 3-L.

0 rr? -

?lone

V2.-Rlx

I s,,,

<

e

---..1 1.2. i3O,R__ A2

R do(

0

since P4

0 and

El- Wej2L°(3112.=

Pi7

4

1

,

when

Tz-a. ...

_.

Noting that S'vrice (2170oC we have that

small circle nr

of radius

r

about the pole,

and then make

r

tend to zero thus taking the princi-pal value in

the usual sense

(Fig. 3-2).*

Hence, by

for 0 oe-...4 7c/2.

*As the contour integrations which follow are round similar contours, we

will, for short, call them first, second, third and fourth quadrant when going round the origin in the counter-clockwise direction, the contour in Fig. 3-2 being the first quandrant. The direction of integration is

(23)

1..f(Re4()Riecz d

p

9 .

3-3.

7C/2

j

Vlxioc

Ko-R -Az

-Rix II

(R2- k tg

e

0.0

Consider now the integration along the semicircle (Fig.

3-3).

Substituting 17= kit_{t'e_4°C}

so that

pe

Kr2

rl'ic-ka-1-42() we find that

0

Irn

tif(Pel:.°() cle( T

(04*

rdA()

-

ix

i(kI4- re`.°C) 2_ At

irr,

be

E+ re c

(lei+ re )

doc

F

[KA VA2+

(40/2)1) ___4 7C

e

ViA2+

(4/2)2

[K12+02+

ka/2)ticeskko2/24 (14/2)a When

v.-4.a

Integrating along the real and imaginary

axes,

and collecting the results we have

Se"

i< 2- -,.4.)(kile"-A7-)leo

- Az

led

A

F [4/2 -01A1-1- (/2J

re.

_{iecia+VAt+ Wel}

21M1-1-

W2),-i

x rikr-E-At

e Unti- A2 ) Li, n F lKon, us n F

+

nd 0

(nt+A2)2 4- Ka r)2.

0 k0y2+,)Az-(4/2)'j

where x <0 ) (z-f) <0 and noting that the result from the

20.,

(3-32)

(24)

(i--integration along the real axis from

0

to

A

is real and accordingly does not contribute to the value of the integral.

Similarly, integrating round the fourth quadrant, we have

tencs..n At) _{d K}

=--J Kt KoKA%

FE At+04./211

pee1'24_4ft+

(e.k),-jecksrxik.2/2+ of./2 aV (ke,/2)z 00

r)F

ri c../T

nF]

je

ndn) (3 -3 3) where .x>

0

and F.,

(z-{)

0

.

Substituting

=

0wec29

in the first term on

the right hand side of

(3-32)

and

(3-33)

and

fl=pstr,e) A=

I:1 cos 6

in the second term of

(3-32)

and

(3-33)

we find that

(3-29)

can be

written 00 -r/a j4->tp,A4p2szn[ct-f)pl:infEidkop.sr16 Cesat-4)Ps'n 93

9

ri

rt

7C. _1='2 Koz sirz

e

c,(pticas9*ede

o (3-3)4) where

x>0

and 7112. cc_i_mx )+ItrleKpop

0jpBz-f)pE)]-0SW,9casaz--OPsnq

P2*-1-ic'as-vn2

9

cc s (py cos )(;

e de

712. C.

° e

cirs(kstc.93-s(ktet.1-6Str364) set-39 c16,

0

(3-35)

where x40,7.4 LO and where rit= (z )1:, (-2,q

)2-in both

(3-3)4)

and (3-35). Substituting

Z 0 in

ts(x.,y,7-) and

expression (3-3)4) gives the free surface profile in front of the source and

(25)

22.

(3-35) behind the source. Since the terms of

(3-34)

and the corresponding

terms of (3-35), all with

z=.0

representing the non-oscillating part of the wave profile, dies out quickly with increasing distance from the source, they have in the past been called the "local disturbance" in order to distinguish them from the last term in

(3-35)

which represents the oscillating part of the wave profile.

From

(3-17)

we have

cm

-

r-'r--'19.4.9 de,r

e

k

--- a

_"11.

rt./

C7

k<---c,SeC.,69 +VA-Se.C.9

K

CI

-x

where real part has to be taken. This expression of AS corresponds to

that given

by (3-29)

and in fact the two should be equal.

We consider the double integral term of (3-36). Ignoring the

constants, reducing the range of integration with respect to

a

from

- )t

to

o,

7C/2 and substituting A.== CSLr

9

we have

KFLZ-t (1-÷)]

Inn 19fte

Gie

x set-2* 4. (;tA. sec a

{ZO

tio

21Arj--TVL

k171:=---A"-1,-,-, 2ICCS(b /\)d)\j) (2.-C)

e

_]c11(

,,K- At+ t i.k te*--

e-K.e.--2(2.-0 A

We consider the two terms

Im

7,9.4,t:x62:-XL

4c0)6,9 I

e

7?--14012 Ctr2)

C.

In,

9(.9)67 "

Irr

Vta

-At_<;,..fri_p_

76 7)

(3-37)

where the first integrand has a simple pole in the second and fourth

quadrant and the second integrand has a simple pole in the first and third

quadrant.

(3-36)

(7.-= 4)3

e

(26)

The contour C we choose when

x>,0

in the first and x4.0

Sex crrui

in the integrand is the first quadrant (Fig.

3-2).

When >C4. in

the first

and

X>0

in the second integrand we choose the fOurth

quadrant.

Hence, by Cauchy' s theorem of residues

ferdthi

'C.

where

ZAR

is the sum of the residues of the integrand at its poles within

C.

With the contour chosen and with the Corresponding Conditions on the sign of x as stated, we get. no contribution when integrating along

the circular arcs of these two contours. Carrying out the integration and taking the limiting value when /tA. --4- '0 we find that

firm

f e.F'` slit-4 i

iA-.1.0 i=2-koK-Az--ILY VL-A2.

Ft4A-4- VA% + (44)2-3

i

V At+ (k./2

)1. -

[K4I+VAC4-04/2.)7j1cairdia2-4-kdAt+

(v0/2)2

Ta

_ex V

r

nt---T-2--A ,

Vril*X9 tin n;--

.:3n,C-1)-S

n

;'3

an,

(n4.

z

A2)2- .4 14:lzrI2-

n

0

`

where X 4-Oil 7= tt- f ) and that

oo ₀₆

n .--1.7-1Unt+At) *00 n F - i<on c 0-S nd

ci

.inn

(3-39)

,u-s()

Kt- V,,K-Az + :it r- ' 2 -_- - -- - Tv 1 `.44"

-

le-xi

(flt 4 Anz 4.

tc2-n2-S'

lien gt,n(x ie--i---1- A '1'

4

where ><>0j F= (z-f).

Making the same substituting in

(3-38)

and

(3-39) as

in the

oase of

(3=32)

and

(3-33)

We find that (3-36), when 14.-4. reduces to (3-3/4:

and (3-35). We therefore conclude that the two velocity potentials, gives

identical free surface profile and identical isobars in the fluid.

(3-38)

(27)

Moreover, since neither velocity potential contains terms which are independent of x the two expressions of

V

as give n by (3-17) and (3-27) are

identical as will also be readily seen by contour integration.*

It may look as if given by (3-34) and

(3-35) is

discontinuousalong the plane given by

cO,

but this is not

so (away

from

the source) since, the value of the single integral term is

positive and twice the. value of the double integral term Which is

negative for x---0)(3-35)),and positive f Qr. :>( 4.10 ( 3-34).

In particular,

.5( o, p, zj

2 for 't

4-c

may be, expressed in

term S of known functions,.

It will be found that the isobars on the plane %.11.t.r.0 can be

written as

s(x, 0, z.)

_

rn x r

9 Vki+ (2+01-

01+ (x-f

it]

irk

4 rn

encrsseec

fleas (ri - f1)-kost4.2'e

(n17.- fj)

c

Ste.

& 4 nt

flan

(3-40)

0

where

=01.10%...11

The use

of

the fictitious /A' as a means of suppressing waves in advance

of a disturbance has been critized from time to time by mathematicians, and, even recently, a paper by Green has been quoted as evidence that the

"-, method may give incorrect or incomplete result. Whatever faults ,AA-may have, that particular paper cannot be used as evidence. Indeed, no result can be quoted in which the solution obtained by the,- method is in

conflict with the result from other methods. Even if capillarity is taken

into account ,Ax operates the right way with capillary waves in front

and gravity waves to the rear of the advancing disturbance. and

(28)

sc)()C rn x

1

9 1/x+

(Z1-)(z-f)2-3

where

x<

Continuity of 5 ( , x 00

4. Ann_r

4r61.

9 43 g

n ars (rdt-f

ec..219.sin-,(n tz -f I) ,

J

k,;:lsec, G -4- t+

non

o o icia -r-, arnk,1-4:41z-f isee..2.9

C

ctrs (kox sec-6)sec- e

1

de

0

tx,

inces(nlz- f I)- euseje

(r)

(1-)ndn=

_K,

es-k. "1-"C I

I"-C.

sE.c464 +.

St29

n2-Substituting

ezfisec.26)

um= nlz -ft

of the known results

CerS0-1.1

_u

a.°-r

_{> o,}

o,

) 0 and

_oo

usn,o_u ,..i

i

1 -o. p

vu --:----

2e

0,.Di p>0)

1:::?--- ut

J

0

we find that the value of the integral term is

at the origin requires

__S00

-0(

_f

LA Ctr1 U S.1%.r) it

e.

(te

1A1)1z. - fi

f I =

_Kle'

Seje

0

Hence, the suggested equality of the two terms of (3-)2) is true.

and making use

(3-41)

(3-42)

d

du

-

I

t

0

(29)

c"-9i-M-4-14+01 N/k

tz--i ₁

n

being an integer (see Appendix I). Making use of (8) Appendix I we have finally

me.

_{olt-ft rAr}

_{okolz-f 0+}

L

(!6c

Substituting ZmO0 in (3-1460we have of' course the surface elevation

gtt:Lle4

origin above the source.

It is of interest to determine also, at least roughly, how' behaves for small, intermediate and large values of x and y. We driall, however, limit our investigation to the simpler

case Of

e 0,z)

as given by (3-40) and (3-41).

Since, for every value, of x)

oo

E(-I) ij

---

A

r).0

Care

26,

(ec,12.--410

.

We have also that

7/L

s(00)z)=

1+44-Yikt. ""x°11.6fisejesec.3#9

(3-)43Y Y ("E-44)2' k2.4. (z.- C,

e

de

.

The integral term is, however,

kdl-z- Ostc:49

sec.309

d

4ack0htt -f I))

(3-44)

where the A-functions are defined as 1/2

An 6,0

jors2n+

16 oe.seJ 98 (3-)45)

9

0

Sec. 7c/a

(30)

we have, for the most interesting term,

/2-k (;0,-4,

8m ko

cArs(k,,,se,e)se.J9 69.

0

7/a

2rr`k''

r(_0()

NA" -eolzflsec2e

2n4.3

C L.`

sec..

de

n=0

(2,-4

6_00 No421

A

iiKs..z_cij

C.

r1=0

(211)! "-n

(3-46)

where the /4.41P) is given by (7) to (14) of Appendix I, and where

stands for the oscillating part of the isobars, i.e.,

(v)

0,

oi

is the oscillating part of the wave profile along the negative part of the x-axis. The expansion given by (346) is suitable for

small values of

Since, for every x,

-e

_$

ri=o

-rift

w )4' L)

GCS (1e0C Se-C.ei s.ec=36

k.lzfl-toan29

ars

(les,xsec.e) sec..39 d

gmifQ2n9 3

_L=0

0°17-""firics.,s(iQ(sec.e)tv)

se 6 de.

(3-47)

nT

n=0

₀

we have also

j

G

(31)

The integral may, for different values of r-1 be expressed in terms of

the Bessel function of the second kind. From (2) and

(4)

of Appendix II

we find that

7/2

n=

0

Icors(e.0(sic.9) se.,...19.

d. =

d No()

0

-Tc

Yi

040')

(Kox)

4 ceok)

From (2) and (

6)

of Appendix II,

n =_- 1

(e,),

e9 f(sp_45G

-

Sec.30) s cec. cl

=

rvil

lit (x)

Yi (0)1_

T

r

3

yo (v..><)+

_,-/k)

_kxg.

a a ()1

(e.x)

a *At

()i

sec 9) sec_2-6 (AG

From (2) and (9) of Appendix II,

Ith

"R/2_

n

ta

j

ors

Not

sc4 icur,40 se.t.,143ci (scc.79

-

2 secs& -1-sec! 61)cos (1c,,* Sec (9)

de

o 9

=_LicZydLox) +2c1311(.k)

d X (+Q()

L

(e.x)

_(Kx)3

_{d (ke,x)}

=

r r 3

Co

2 L\ (koe

0<004) 210

(4,,()

_0e001

y (eA]

28.

( 3-148) ( 3-1490) (3-149)

:

.a

(32)

From (2) and (12) of Appendix II,

-rck

I

3 l

c,,,., ,et 9

j

+.,(2,9 .e.,..16

de

=

J(-3e&

+ 3secse -se

2

(9)cys sec

G)de

0

1--,37Kce,)

,ds-yotido+

_d-,A(,k)

I

(

ciCkok)

((.>63

ljj' 7 YDS

2520

( IS

G60 _L_

5o 4o vy-(,,a

aL\ (lesk)1+ 6 04.43

(K)()3

(k.k)1

_-1/

(3-50)

and so on. The asymptotic expansion of

X(e)

is given by (14) of

Appendix III. This expression together with (3-48), (3-449) and

(3-50)

gives a suitable formulae from which(x 0

w Z.) can be obtained

for large values of

We have found an expression for the double integral term of

(3-40

or

(3-41)

at K.=0

_O)

(2- which is the

line along which the term has its maximum absolute value for a given value of z To show how the absolute value of this term falls off with increasing value of

ti

we can use the method used by Havelock

in his discussion of the wave profile of a doublet in a stream

[1.

Thus the result of the integration with respect to

n

can be expressed in

terms of the it- or function. Writing down the expansion of this

function for large value of argument and integrating term by term we

obtain, without too much difficulty, the first few terms in the asymptotic

expansion, A more direct method is to write the integral in

n

as the

sum of

two integral termsone, a, in

which

n>

and one,

b, in which Se,19 Carrying out the double integration it

(33)

will be found that the contribution from the O. is 0 (Z.

k°'/.x)

for large values of In the case of

b

we expand the denominator in a series of ascending powers of

n/(osec.264)

integrating first with respect to In and then with respect to ED we

find the first few terms in the asymptotic expansion,

7/z

4 rn (.39 Tc.

n cc,(rdz--11)

flLfl

sec_4(4 ina c2.rv: [121 1/-f1(6k4.+Sx2.17.-f12-212-f(4)] 1)(11[0-.4.

(3-S1)

If,

If, in addition, we assume that

lzfl

is small in comparison with additional terms may readily be obtained in the asymptotic expansion. Wigley has published a table of values of the integral in

(3-51)

and the

Mathematical Division of the National Physical Laboratory has prepared a table of values of the single integral term of

(341)

PI,

-We shall in the following make use of the velocity potential as given by

(3-17)

or the corresponding velocity potential of a

horizontal doublet. According with the discussion in the beginning of

the present article, the velocity potential due to the perturbation caused by a ship is then given by

C°

qi(hi-f)dL,L4;jsej96a

e

27(.2%,

kj

-+ Sec.G

(3-52)

0

where r-1.7-.= (.)c (Z - _(7.3 )Cv;

s,

Sis the vertical centre line plane of the ship below the water line and

where

y==-Lp(4,7)

is the equation of the wetted surface of a fine ship.

30

k fl)

-( -{)cih.d

(34)

It should be remarked, however, that although the method of obtaining the velocity potential of a fine ship by introducing the idea of source and sinks is convenient and descriptive, especially from an engineer's point of view, it is by no means necessary.

§4.

Wave profiles. Let us assume that we have a doublet at (0,0,-f)

and that its axis is parallel to the )(- axis. In an infinite fluid the

velocity potential of the doublet can be obtained by taking

(l)k)4)5

(the velocity potential of a source) and substituting NI , the moment of the

doublet, for nn , the strength of the source. Assuming that

Ifl

is sufficiently large we get from (3-17)

clpci

Mx_ Mx

,1

or, from (3-15),

kCL(Z (1+4)3

cz 't'licer.s(a6a

e-271: -7r k-C(LLM +(2 )3

-+ Yfr,:et &

Cert

<14

Z1

00

4-'11'141secOde

7C

1<,,,se,7-(:).-recti 4'34.)

Using (3-28) we find from (4-2) that the wave profile due to

a doublet at and of moment M is

(4-1)

(4-2)

where "z--+ >0, z-f- 0 and c C.4

and where real parts have to be taken,

(35)

where .it 0,Q rk(67,-f) S\d(x, 0)

= -

co-4'6 J I e 2T9

7

icctlz-f) k(Lc7J- A + k,SeCIS 640. M I

-e

e... 4. --

de

zrzq

e-

ct"iii ;fr,

7tt

-

Lfr, ) -7C o (14-3)

it being understood that it is the limiting real value

when/A

is made to vanish which is to be considered.

In this form , the surface elevation, is finite and continuous and the expression can therefore be generalized by

integration over any distribution of doublets. We consider a distribution over a finite length of the vertical plane y=0 and extending downwards from to If

M(h4)

is the moment per unit area at

the point (h)0,-f)

,

then, from (4-3), the surface elevation along the

plane

y.0

is given by

71 0,,

1

()(,, 0,0)

=

71c,,fOlkIM

(kr

)3 fid6

_sec"'

0

(k-.1)C.crS

is independent of the

f-

ordinate thus

()(,),J)=7-cic+1(b) dkidej k-"%Sec1.6

+

Itc

o

32.

We shall, however, assume that Ni

'

.

(36)

Let the graph of Nei to a base of

h

between the stern el and the

M(6) bow -4-t consists of

N1(1)

straight lines and

//

\\N

ranges of continuous changing gradient (Fig. 4-1). Let In V) 1)

2 1""

let

h'

i., 2"

h 1 gradient

Oh/10,VA

rcg. 4-i.

be junctions at which the straight lines meet and be junctions at which the continuous Changing

changes abruptly.

kt

fekciimeskok)c.-se ( Err, Gik k, 1! ki t4,

But NI (4t)

m(4)= 0)

and all the other terms for which the integration

has been carried out cancel in pairs. Hence, integrating once more with respect to

k

and noting that c1N1/(c)kl is zero along the

straight lines whilst

cit'l/dt-,

is constant we have formally (Fig. 4-2)

1 kt

(011

1 +11i -0,11

41.14,..+,0 4...4.

) C-GrN3 1,1 LI.

_I

2 kr,

Integrating by parts with respect to

h

we have formally

.itvi(o.ei:k(*-1,)co-s 9 mcin) (!k(c--k)c.0-s

e

ki

kt

kn4t

MNie

(37)

If Li - ni.1 It

I

. co,_

e

( j

atm

ik (X -11)CcrS Oci

i

---ir

'

)

l(tCOVe kit V1r, kt+0, _1,0.0 1

f

1 ocOciJi)ce,-19

-

d m

ikOt -InOcers 6,

n--t

tu,stat oh

dh

1 .1.

d m

V%

fro

4,+0

-k

)co-,se

e.

can t'k(se-k)

cs-1 far

dkl-

e

ode% -4- .,i, ,

vs,.

4 -dk

E((-1,;,),cfrs

V 11,7- 0 e-f

dm

1,2"..:kbc-I.,

ti.0

&

34. Gin

6,40

itc(x-t, )co-ze

4

I.

dk

1,4() o .*

(38)

The integration is from stern to bow so that

16.71-0

(dm/c16)

tain oc1i+0

ock

0

in terms of the

slope of the adjacent parts of the M(k) graph (Fig.

4-2)

The surface elevation ().-4) now becomes

b= 1134'0 7C (111-0) (k, 0) -- 1 stje-de 7t c ks- 0

-o)

k=1; -0

1--+1 Tc.

4--Where the first summation covers all points of sudden change of slope

of the M()-graph, including the bow and stern, and where the second

cit,i(11)/dt,.

summation covers all ranges of continuous changing gradient

Since

t

c1-17:

(04+0)di

k((- )ca-ye oc) ci ( (7... sece) bQ

(),-1A)cb-se

Sec.2:9 39

d

(L_

stc's,

_/4.A.P.cC9) ki+41-o

Sak).0

0

when the summations cover all points of sudden changes of slope and all ranges of continuous changing slope, equation

(44)

can be written

(39)

and

6r°

-(111-o) ct,tcrt C.rn PR_ . /"'10 k (4- VoSe-L-e Ltc,19)

zk(x-

Ips),c..0-3 L.

e

(I)

9.

Secede

t((e.-ke,see 4.k?,itee)

obz, p

_{(xhtce.re 9}

i

1- e.

stjede

1

cl

., ,

k (k.

I4bS1.e9, -t-

ysee 8)

0 _A °yr( qe-cy dtrs Ea )

scje)

tqnC4-s(kg.co-sG)

(;01 cy co-1Lsi;v1( ct-co-.09).]

k(

ems,e020)

-

Secte

----_

0,

36. (4-7)

There appears

to be

a singularity

at

; this is not

really so

as will be seen by taking

the real part of the integrand when 71/4

0 Thus

-1

(40)

If the distribution of doublets considered is submerged in an infinite stream, the velocity potential is given by

oo

'cif

I

Wh).(x-h)

_c4 .

J

(z* Pnr

0

Integrating by parts we have

j)

MN). (k-h) _M(k)

Uk-

V01"

(z+

ni-32/2

c191

EN- 14' -+ yt+

(z, 0%

V2-h 1 -4-

4---II nfl

chi/dh

2-+

WY"'

d

k

since

(t) = 0)

and since all the other terms for which

the integration has been carried out cancel in pairs.

Hence,

dfrlid

ff91fr2.

d

QC, 00

MU).(-J)

0 m/c1

_dk

J

j[6,-

k)14 162-4

_{(z- f)%"]2}

EN-

lir-

+

(2- f

nvz

L(*-6)z-+

1

(41)

/

-0

JT (114(i))Jk sec.td kt.1+0 )c.e-s -e. K( - Kt, sett& -4 ki ko-D 4-1-1 1,1+0)

I

7c4,.(6)

c.4,"(k)Pck-k)db/

2.7Ca L'"" t 111-° 1-1°

(1-0)

38.

The velocity potential for a corresponding distribution of sources is given by

jciff

c(k)ZtVtd

where G(h) , the source strength per unit area at k--- h2

td= 0,

is assumed to be independent of the

f.-

ordinate.

Comparing these two velocity potentials we have that they are

equal if

cim(b)

(k).

(4-8) Making use of (3-10) we finally have that

61'1(1) (.1)

V(h)

(14-9)

since in this case Lid(x) is the equation of the wetted

surface of the ship. Hence, (4-7) can be written in the form )ter-Set

o,

(kr1+°)

_J

(ss)

S4c--

(ekt,

SECS

+it

Stce) C1K 2.70" ins-0 ₀

M-0)

(4-10

+(z-f)t3A-=

(42)

Note that the magnitude of the contribution due to an angular point on the ship is directly proportional to the change of slope that occurs there.

Example 1. Consider now a model Whose half water lines for positivalues

of Y are given by e -ihAs

91

6 )-y- 6

_

tz

ihAs

A-a.

I Also

41=

, and

7 klki4

6

6 6.1-0 = t- 0-

I

From _(14-10)(4-10) and (14-11) we have

-rt CNC>

e6r,o,o)=

srrIL

clef

or - -e)cu..t

9)

2X2a-exj i(e-K,ser-ae [(I

-7c 0

)ce.s

e)- (

-

(i-R_

"-0-)ctrIG

Substituting 9,17-

t),

9s., 0( -)

.t.,2.)

)3.

(4-11)

where

+4.

are measured from bow, stem, forward shoulder and after shoulder respectively and reducing the range of integration with respect to e from -7t, 71.

fo

7t/2 _{we have}

0 +

(43)

ith

korictafit

,

c.6-1 Nk.d. co-se)

6 aisa!eciej[0-e

(I- e,

64..Le")

(1-4 Dra 74 )1

2.-Kt(e-a) ,

K(

eL"te)

6)

0 _ Coq

9cas

fi-e

1-1-(1-e-

2 ) 4-( k - Iko se_c_t9 If we therefore write A(q.:1

2.72(.t...e.t)

454 clef{

K(K-Voseje

+t AA'

e.

0 7

b

-F-Co()

then the total wave profile becomes

(co

t)

(k4-

4(,

-4)-4 (x+. a-))

it being 'understood that real parts of (4-13) hate to be taken. We consider R4.4){("9>G17 tb-S fa -.Aar Crt

e

)- (1-

e.

)]

eg)

404 (4-12)

(4-13)

(4-34)

(4-15)

where?? is a complex variable and equal to k+in Note that we are going,

-bcckcos

+

v..

-

sec_

e)

0, 0)

e

.

(44)

z.oc

r e..

Consider first the integration along the arc

r

We substitute and ultimately let

,f()Lao<

2.

7c

where

Consider the first integral on the right hand side

712.

"Vz

7/2. (..C10( I

S

doe,

I

1Z-C11.( _{Tc /a}

0

0_s 12e"(-0... I o I ei7-44_ 0 0

Consider now the second integral on the right hand side. _{We shall assume}

le4-0

i.e.,

,kO.

Noting that St,noC (2.1n.) _for _71Z _,

we have , thus 7r(2 - co< z./kee. Z.-d do: a. to put tk=0 wherever this

can be done with-out introducing indeterminateness. As contour we we choose the fourth quadrant (Fig. 4-3).

e

0

(45)

Ia t.12. _ AI

eec

-C zclod

Jdo< 6

sea."'_ 01/4. 04, I 0

0

te.

-4,\IE

₀

_{04 ';.?}

e

cld ) (e-to1/41) 0

The integrand

or

0-15), has a simple pole at sq

which is within the contour. Since, the residue of

4cv

is

at a simple pole,

we have

residue at 4setQ f;A&ice)

C%0

where we have put pt.T.--0 Bence, using Cauchy' s residue theorem,

ceps G

. L'eda

irrt S

it -

,

o,k ()4-kose.,..ze Sec

el

'taX

Ski

.N.001. Sec

)1-cfrd.0- ,ei.

e_

n

kg.stc4

k0 Ct2-(5 dL9'

.0

for

Cie- O..

When c.>0 we

integrate ()4-15) tOund d contour consisting of the

*hole of the 14- axis and A large semicircle in the upper half. 12 -plane.

It will readily be seen that the integral round the semicircle tends to

Zci.kesee.. 9

e

_- _{S):(%Corkosec.G.,)} sec.2e9

42.

zero as the radius , provided 00,0

Hence.

I1.

(46)

ilccr-coce

e

e_

k(K-e.ser.!a+zps,ec.e

_S14(k

_{lc ostc_19} _ZitA.S44.e)

de

0

for

cif> 0,

Integrating round the first quadrant we have

d<ti,arse .1' 0 I --7-.. Z. Kci, c4,-s!e

d

,'

/,..-0

k-Ce- 1/4,sAct-e-f- itasec e )

-Collecting the results0,

it= S2t,

ti,rn eL=4-e-crse 27c 4NI

(14

ste a)

z.Co+zrTic.0 kc, ses2-9

oo,

co& (rn1(00:,-.ree. 0) ,

0

,

0.e

r9 rn (14m)

where we have put.

Or =---icir'Y

and

Or- >

.

Also

co-1 (rn Ko

(+ate)

"

Alexc. of,...> 0

sec% rr, (1* rn)

` )

1 urcATT Sa.t 43- cfin

n(r)2

0 (14-16)

(1-17)

for Or> 0.

Substitute e---rnk se.2.9 _I

hence

_cl -- 1/4,3

caLts d

oo

(rn koorSet

Sec.,E9 rn (It

-rr, ) 0 )

(47)

quadrant that

'12 =

Kn.

j

814, 2-ics.1:4,1k0 arstc0 Kosee.ta --itete.) oo

not-col a

I

-(n2-t. kjse.219)1 -43 oc, _z Greets ta

f

Re_ Vrn ,17

-&Lee _

-00

Uvrvt

e

' at lc. tA--*.0

1;4(K

s.jia

__Cap

In similar

manner.

we find when integrating round the first

-

*COS (2, I Oct

-jI

Cp-Car e n(n2....1-k,otsam,49) kostc.I.6

de

sAc,63ir

for 0( 4 0

.

By integrating

round á

Contour consisting of the whole of

the al-cis and a large semicircle in the lower - plane we have

oo

e

crart

k(t-k,seJe-ty.tstc.e.)

sejta

P

for

Integrating, round the fourth quadrant we have

44.

for

c- >o

0

(48)

Also by substituting

\c3 sej

"Re_ rp

-Collecting the results,

I a=

k(e-lcrcec.1-ci-civs,,9)

rac

zec-409

oo

1 -

La-1(n',ko

(14 rn )

0

tJ").-eirt. Or = (),' airtAA 0,1) 7, 0) ov,-..6

I ccrsfry k,Or

st,

ci

ry) ( -4..

where Or> 0

We consider the integrals in (4-20) and (4-21). From

(4-18)

(4-21) where we have put Cy...--CA and

Gli>

c".0

k cc-s (rn koCir Sec 9) _wifuLr

0 Also

e_ C'>o.

(4-19)

I

61, ₎

k0st,116) re, 04 r, )

ca

Substituting these results into (4-13) we have

7t/t

7C /2.

,)

00

0

R, lir, rao ___

.!:11.cs14,(e,,ci.l.stce) d G - LI".

/A-) 0

ro

k'oy

de

... 0 (..) -"'`erIcrn

°()'''' ''''9)

clryi,

_(4-20)

( I -+ rn )

-

s(

_k.or.St.e)

sej-9

1-+ rr, )

e

0

(49)

(3) Appendix II

Yo(x)=

Gc.,-sk, 4-)C.4.

where

x>o

, and so

itjz Yo(k)

The first integral term of (4-20) notation introduced by Havelock [9)p

1CIZ )eo

ars

11,,A(.0(4%st.c...e)d

-

V0( kc,01%) d(

where the

7-

functions are defined as

1V2

?2,q) -=

On' CArS'M e S,...1(p 5,e

and 7c/2 szn+1

?

(I))

)

JC0-r

5 Cc.t.T(t

ta) d

'2.n*t

(-70('')=

t-) Sec Eii-; scc, GJ(5., 7172. 0

can now be written, in a

being an integer (see Appendix III).

then [63

If 0 and '.e(In) 4

jr,(x)

(k in-it) ccrst,

where Jr,(k) is Bessel function of the first kind. Hence

7/2.

st;,A(k Ste_S)2e.c6.i6,

Tr

46. (4-22)

(4-23)

(4-24)

(4-25)

2 - 77- Sect3 c..(n(k a)

k

-2 _(k _t9) 7C 0 )

0 9)

0 1

(50)

and roct!

Ici(ifoc)ISec.9de

xn

Jr) (c)-')

k

0 7q2.

Jo(ttArrn

nn

= -T7

2 ccseo9da

S'in(Qcvirnsr"-t4_

rn

1-1-But for 4/?..4-Re(r)) 1/2- we have 1,61

where

H()

Struve function,, Hence

zit

24sec,9

de

("in'

se' t3)

II+ 're\ 0 7/.

r H

-2 c

L no-zoiTC

21[K,(4) -Y0(1404)1,

172 oo

(ico9rfrnSec-e) (4. 1- co-sc aterSeC,

e)

,

onn

r)-

1-v-t (114 Inn) 2.

r

'

-47 0 0

(/CoCit) d

001,

a

0

"))

0

(4-26)

'using a notation introduded by Wigley [1] (see Appendix III).

Substituting' (4-24) and (4-26) into (4-20) and (4-21) we have

1F-ccy)

L4P,

-

OD( k.:,c41)

(4-27)

for 0)t.>

and. 0 is 0

(51)

-Re, kixy

POO,

ZasItQ (

0

for c+(> 0

I

where 'Pcov) is defined by ap--13).

We ,Can now

write

down the disturbance associated with the bowl shoulders and stern separately as indicated by (14-13)1, or

(V).0)

4 f

A ,yr d 1.(.460,)-{, Q. [ Ck-.1

> .ej

(14-,29)

0.4)

G e-4413)

>

(4-30 (x,o)))(f.s. iksk(11.

6 _Q

()(-0,) -7*t--0,)e, (4=28) U.

(4-31)

kwc\s.to,

oss:41,0 ; 04t)

744 iiirPork,3(cL-k

(14-32)

0_{0)(eLt.',,fs.e f; 41'1.40-rot)} Dto(ia.+4..4j

44-0. >0)

(14-33)

aft)

_{j4.?Ck, (-etc-4k)]}

-

_x_-a5)3'- > 0, (4-314)

)

f

itse

(52)

bc,o,o)

(S4t/rrlLd

a)._

__-.b

Q EK,(.t-hkij

t-1-)e

>0

°

that

(4-35)

14..0 0.--00V0

(4-36)

Where C.s. and a's, stands for forward and after shoulders respectively and

Where we have sibstituted for +x' in order to conform to the reference frame Chosen. In these equations stands for the local disturbance or non-oscillating part of the wave profile and is expressible in terms of the

-functions. w stands for the oscillating part of the wave profile,

and is expressible in terms of the -functions. Lastly, _kr,t4--C

stands for the sum of the oscillating part and the local disturbance of

the wave profile.

Substituting (4-29), ,

(4-36) in (4-14)

we have that total

wave profile becomes

Z7r

n -1

0;10,0) 1,1Y,,(L'" IcLi

Lk.(0.k)i

tic(-',-")3

7(.{aivo

_

go [k/

_?)]

Q.[K. (-k-

- Q01--00c-c..).)

(4-37)

where we have introduced the convention that ?z,(-io)

0

and

o)(sA.eirn [k (

0J

(53)

The

Or function,

by which the local disturbance is expressed, is zero at its origin and increases indefinitely with distance from it.

However, the local disturbance as a whole decreases with increasing distance. The total wave profile is given by

(4-37)

is in finite and

continuous throughout.

Consider the wave profile at a very long distance to the rear

of the model. Since the terms in

(4-37)

tend to cancel each other,

i.e., (k) 0,

0 0) ()K, W

values, or from

(4-37)

when >, °J, it follows that

when x takes large negative

4

oc,0;o)

kpc,

0, 0) E - x)J

Iv. (-

- %Ike

(4-38)

when

In (19) Appendix III it is shown that the predominant term in the asymptotic expansion for

(I')

is

(7/2e

)VLco-1 (T -7/4)

Therefore, ultimately when , each U,0) of the four

wave systems becomes a cosine curve of continuously diminishing amplitude and of wave length 27/K, 27Z

c9

which is the natural wave length of

a

free gravity wave travelling at the speed of the ship.

Substituting for 9. Q-1) its integral, i.e.,

712.

(e)

ga)

we have

(0)0)

46)e

(-(--k)s-ec aj ep]

s;,,D-S(-Jc-0,-)s.kc

50 .

(4-39)

(54)

1

77-The "velocity potential, 4)' of the perturbation due icy a Slender

ship is given by (3-52). Making use of Theorem II,

gi.3, and

in particular

of (3-19),

(3-21)

ani

(3-23) we

find that

4--C--f1p.1.7f)

tiffe11.6,,4

-tR,ejc0.s( S.C. 9) Sec:L9

d

.0 (.14,14.0

when

From

(3-28) we

have that the wave profile at any point far behind

the ship is given by

7th

kof sec%)

o) -1-=1144,01,-f)

dffe

c

sic

cal( te.

s 6 .sej.G).14%1

6

0

when. eaQ

Let as consider the mcdel given by ()-11). Integrating

(441)

with respect to

and lb

we find that

4 2- fit

S

7,3 (e-)t)

ISvrV

-

Set t9..) 4

DC0 4

ej

0 prja

Ty. c-

0.)z. ,

fs]

(K, yisc, 9 ;I:La)

cis

(4-4?)

when

k

-We. notice that

(4-39)

and (4-)42)

ate

eqUal-Whet Substituting

in the latter., From the

manner by

which

_{("4-39) was}

Obtained we have that

the first term of

(4-39) and (442),

focused at the

bow, is

due to the

(55)

70.,

ki(xii0)(L..,7-:.

-fit ;4')

4 6'7

fi

Xs (4- o,) 0

6

i.k Cot)

Fcth

as

k

,

0..

ko 'Sec C433(

Z-e

se

efe) d e

772

260_2'

_{Evosec29(xtos}

kO.

73 (4-a)

-

Triz,

(4-43)

Since we are not to much interested in the absolute value of the integral

as in the values of -the direction angle

0

which contribute most to the

value of .(4-443) we shall make use of the principle Of stationary phase

in the form. stated. :in Theorem III below

[10]

'Theorem III. If 4.(2) and F-(2) are analytic functions of a complex variable Z. regular. in a domain :containing

C

a segment of the real

axis on which both

fet)

and

F(z)

are real, then:

(a) if

coo

has no stationary point in

a

4.- 4

6

_(contai.ned

C )D

Z1c4(k)

via)

ikko..) / I

-4-10 (17)

Zk (14.44)

52.

finite angle at the bow, and the second term, focussed at the stern, is clue to the

finite angle at the stern. The two last terms focused at the forward and -J

after Shoulders respectively are due to the finite discontinuity of the

water line ,angle at the forward and after shoulder respectively.

We shall discuss briefly any one of the component patterns of

(4-42), say the first one.. Moving: the origin to the bow we have

,

in

F(b)

(56)

rl

(b) if To() has

one

stationary value at x=r7c,C in

o'--E

(contained in

C ),

and if -C\ )> 0) as

0(41

k-ccx),

Ia

e

F7x)

r

-4/4-yr

PK)e

_(+;)

04E

however, -1"(3() C. 0) c(-41 t:leftx; 14. k-fc.c)

(i/toz

C`cx)

r

_Pc.c)ea. _-L c"64) E

(contained in

C )

and if fu(c4)"---' Oj

4""64)*

0;

li

ilec(x), e- FF/3)( 6 )1/1

r

k-c-(.4)

(1-

W3,)' - kif"661, at t=1.3 _'k

(4-45)

0446)

(4-47)

as --->

(c) if

fo)

has

one

stationary

point

at

5c=oc

in E

k

.

If,

(57)

cbr)

I=je

F-0,)

(1

into a finite number of segments in any one of which .c-o,o has either

no stationary point or has only one stationary, Theorem III shows that the most dominant part of

I

..s cx) arises from those

segments of

C

which contain the stationary points, assuming of course

that f(() and F-N satisfies the conditions of the theorem.

Returning to

(4-43)

we substitute rccaet and rs.,Jr193

Where 7r/1 4=

n)

i.e.,

we consider

(4-43)

only for

()

positive values of 4, since the profile is by tri

-\

cal with respect

to the X -axis

(Fig. 4-4).

Ignoring the constants, the

Hence, dividing the range of integration of an integral of type

integral of

(4-43)

becomes

54.

(11-48)

(x

co-s

e

CA, 9)..]

dp

i(or can (9es)s.t...9]

(4-49)

-7/z.

(58)

Let or..- be the angle the radius vector makes with the negative

axis of (Fig.

44.4).

Then

of=

and by substituting

co+04.,

=

we have

1++ZA

Cos (

e

stc,

e)

=

i=

- {(4.,+)

(14-50) C.erse c -c:46 )--tc::1 so that

fo.14._7.1&,fror(to_t)i-i

I

cit

(e,t2]

I4tb (i+tt)

j

where and

r

are positive quantities, r being the distance from

the origin to the point on the free surface at which we consider the

wave pattern.

We shall only consider

(4-51)

when

r

is a very large quantity.

Hence the method of the stationary phase as stated in Theorem III may be used to find the first term in the asymptotic expansion of (4-51). The