The linearized velocity potential round a Michell ship

i

TFE _{LI.AP.DED VLUY PONTIAL rOUD A MICLL - SIll?}

Prof .dr.R.Tinman and G.Vossers,

SUL3ÇT.

Mich1ts express ion for tl _{velocity potential of the f 1o17 round a}

sLer ship, travelling

_{over infinitely deep water is derived by}

_ue

of complex Fourier integral trarìsfom _{.The solution is uniquely deterndnsd}

by the condition that at infinity ahead of the thip the dHurbance potential and velocities rust vanish. Results _{are compared with other}

thods (Havelock),introducing an artificial

_{disipation factor in order}

to det'ttLne the solution uniquely, and with _{Michell' original expression.}

Colete agreer'ent with all former results _{is obtained, tlmus sheing the}

equivalence cf these f orner results with Micheil's.

In a sirriplified case ( _{o dinnsional flow ) the equivalence of this} nethod with Lr1b ts arrt Peters' results are shom,

IÌTOD: CT ION

-2

During the last decade a cons iderahie amount of papers has been

published on the way -res is tance of ship-forr ,using the class ical integral formula of Michell.This forriula is derived from the expression

for the velocity poteritil round a ship of known shape,which determines

the pressure fie li .Several methods have been given to ded'.ce this

velocity potential for a ship moving with constant velocity,hcwever

it din not seem possible to prove the equivalence of these rthods as the

solution was not unique]y determined by the boundary conditions introduced The problem in two dinnsions was discussed by L&B.[J Calculat5jig the wave forni in a moving stream,due

to a steady pressure on the surfae,it

appeared to be necessary to introduce a small dissipative

force, to

obtain an uni.quely determined solution of the problem.This law of

dissipation isn't a viscous one,f or in the final result the dissipation

factor is made infiitesinal sinall.The introduction of this force is only

a mathematical device to avoid some infinite integrals ,but for a phys ical

understanding this factor is confusing;even in instationary phenorena it

loada to the suporess ion of a part of the solution,as has been sbovm by

GPEEN [2.]

HOGR [3] has treated the three-dimensional case oí' a moving pressure

system on the water surface,without the use of the dissipation factor; he was able to give a more deta±led discussion of the wave form,but in

or:Ier t.o determine the

cl:tion uniquely, it was rcessary to î'ke an

arbitrary decision about the way of integration,vthich was only

justified

by the final result.

DTAN [L4] has given in l917 the solution ol' a two-dinnsionai case by using a ccìplex va±able.Asswning that far ahead of the pressure system

the disturbance vanishes ,it wasn't neces sarj to introduce a diesipation

factor to determine the solution.This method however can't be generalized

to a three-dimerisiona]. cas.

-In l9t9 FTi

[5] has given a nxre general deduction of the

three-d3.zns iona]. velocity potential under the influence of an arbitrary pres

-sure distribution on the surface ;he reduced the determination of the

potential to the deduction of a pressure funct.on in the considered fluijì domain0

since it is impossible to give a direct

relation between the pressure distribution on the water surface, end the corrcsponding ship form,these methods coula not give any contribution to a systematic knowledge about

the relation between ship form and. wave-resistance (planing vessels

excepted).

In 1898 MIC}LL6]deduced the velocity potential of a stationary

moving slender shi .To make the problem deternin ed,he chose the soletion which (in his own words) : '1riakes the elemantary diverging

-3-waves trail aft" It is pos siblo to deduce fron this potential a f crrla for the wave resistance,and although the approxination of the theory

(linearized conditions on the water surface, and a srnafl anGle between s hip surface and longitudinally

plane of syrtry)

0are rather crude

,this

forIra2la was succossfufly appited by HAVL(CK ], 'TIQTJY [9) and

especially EINB!JJM [g) to analyse the wave resistance of ordinary ship

ferns.

TITJIOTO!T 1o) [u) used the Michel' s pot.enti to calculate the pressure disturbance of a siriolified eoretric body, "elerxrrtary coin1t,

and b adding these disturbances he was able to trace the streamlines or.

fine hufls,and b2 intrating the pressure on the ship

surface to calculate

the wave rcsistance.In ¶t) he Gives a substantial sun1rrr of hi.e calcu1.ted

tables.

HkVELOCK [tlapproxirated the hull of a ship in an other vïay.1r using

the conception of sources and sin,be represented a ship feriti by a

con-tinuous source distribution in the longitudinal plane of syrrunetry, and

even approximately by a distinct rruraber of sources

[3I .It is possible to

give the potential of a source,rnoving in a straight course under the 'rater surface,but in the coniron deductIon it is necessary to assure once agai.n

a small frictional f oroe;it is true that in

¶1L)HAVFL1CK has given a rrtethod to deduce the potentl without the use of a

frictional force in

cons ide ring the special case of an ins tat ionary ixvement,but a direct

deduction of the velocit: potential of a stationary moving source isn't

known.

flT!1flE 1S)deter!riines the sOlution uniquely by introducing arbitrary

assumptions ,based on the final result.

The express ions for the wave res istance, as calculated by HAVE 100K and MICFELL are equal,as shown by WIGLLY tI 6] .For the wave dis turbance in the longitudinal plane of synmetry both expressions give the sari

nunrical results. [iji) ,but the equivalence of the expression for the velocity potentials in the complete domain of the fluid coukl not be proved because the boundary conditions underlying HAVELCCK'S and

MIC1ELL.s approach were different.Y.IGTEY made a first attempt to prov-e the equivalence,but he wasn't able to conclude his investigation.

Aplicating Four±er double integral transform

nthods,it is possible

to give a direct determination of the velocity potential,without using

the artificial device of the friction coefficient; in applying this

method round a slander ship, called MicheU ahip,it is possible to prov-e

the complete equivalence of both poentiTh.The only assumption to

be made is that far ahead of the ship the disturbance of the fluid

vanishes.

In l the boundary conditions of the general problem are discussed, 92 gives a summary of the Fourier integra]._{- -} transforns.

In 33 the method is illustrated with some two dinensional examples.

In 14 the Michel-problem is treated with a double integral transform.

'

In is shown that the results of 14 are in agreement with the }Javelock's

i.

BOIThTDARY corinrrIor.

We consider a ship,mc'ing with constant velocity on the surface of an infinitely deep,incouipressible and inviscous fluld.The siip gives

rise to a pertubation velocity field in the fluid thich is asuned

to

be irrotational,an consequently deternined by a velocity potential p

The velocity coruponents U,V,W, with respect to a system of rectangular coordinates with the XY-plane along the undisturbed water surface arì

ti-E Z- axis verticafly dnwards then are given by

u= -'

; V- _% ;

As u,v,w rmLst satisfy the equation of continuity U,i- \f), 4 t.Jz. O

is a solution of Laplace 's equation

Fron Euler's equation of notion for iristationary flow

U 4 U tL V Uy + _z -ç, V 4-U-V, 4 \IVy + WV1 4, i- ;.&-J, k 4-f'

Ernouili's law for ins tational unsteady rx'tion is f oLnd

t+

- (t) _(1.2)

Fixing the origin C of the coordinate-system ori the ship,rnoving

with a cortant velocity c in the positive X-dfrection,we assuma that the "disturbing" velocities round the ship are sn11 as coipared with the

velocity

e; viz.,putting

u---

; VV ;

'ge as siae c ; V < C w' « C

Introducing the disturbance potential 'tI by

V'_4y ;

then

cxi-ji

As the rrtion is steady with respect to these ccordnates,rnouii's

law becornes,neglecting srnafl quantities of second order, i. (cii- zc i- - z Con

J _P

The condition that the pressure is a constant everywhere on the

surface gives - !L5. p'

o z

where is the surface olevation. (i' ig.l)

The condition that the disturbance vanishes for ,fixes the constant

2 _p

and the condition of constant pres sure on the

surface becors

i

and in connection with our assuition that the disturbing velocities are saall, this condition is replaced by

_{he similar one for z0,}

In the second place we 1-ave to apply the kinematical _{boundary condition}

(l.L) to the surface of the ship.It

_{is cuctorrry to represent the ship}

form b3ran equation: y (%z) _Y>0

ye

and because of the syxnzetry of the ship

y= Ç(x)

_y(o

(fig,2)

In this case F(x)y- f(xz)

and

X

(i.t)

_{leads to}

- =0

The assumption of 1icheU for a slender

shi.p is,that the tangent plane of the

ship surface everywhere 1-as a small angle with the XZ- plane,so that

Ç, «t

and Ç2((! ,and the surface condition

:Yo

5-A kinematical boundary condition to be satisfied by an invis cous

fluis,is that the velocity relative to the surface of

_{a particle lying in it}

riust be tangent ial, or

(l.t.)

by

wherein F (%yz')is the equation of the surface.

In the first place wo have to apply this condition to the free surface

of the f].uid,

F(x3'z) z + (xy)a O _{is the equation of the free surfaoe,and (i.1.)}

becomes:

_{(_ct&')-+.v'y+wo}

t+40 I

Eliminating from (1.3)

_{and (1) gives the free surface corrliticn:}

The boundary conditions in infinity can be stated as:

sislifies to

V

-

z

fig .2 - C

J=CÇX= &)

and similarly as f o the froc surface

ony f(xz),

z):frz)

j (1.7) on y o.

a. the dsturbance dies out at great depth b. the disturbance also

_{dies out athwartsbs}

c _{the ship is advancing in still vTater,which is a physically sound}

asuinption,because of the f_{sct that for vrater waves the group velocity}

is less tnari the _{wave velocity}

-6

4

y -

t

X -)+to

o o

Fourier sin.transform:

_e(,)

Cop1ex Fourier transform:

Ç(x)

7T

I

--L'o

For a proof of these _{irrversion-farmulae, reference has}

to SEDDON _{and Th.ÄNTER} be nade

So the ais of _{our investigation can be stated as follows:}

To obtain a so:lntion

_{of the Ip}

_{ace equation}

- I/yY t

which satisfies the _{bouncial7 coniitions}

10 Z _{- '<}

y -O

_'/i=

(')

z

j'.y 3 o

,to

X - -j. DO

20 SLTBY OF TI-TE F(XRR _{fl1'EGFAL TRMF(flÇ}

The Fourier integral transform is a functional _{transîorn,deí'ined}

by or _{of the following relations:}

Furier cos.trarisform:

t) J&

,x _b) x

By reans of these _{transforms a differential}

equation in the

variable x

_{can be trarisforried into an algebraio equation in}

tho

variable p. To obtain the solution as

_{a function of the orina1}

coordinate,it is necessarr to invert the integral _{equations 2.l) and}

(2 .2) ,which is _{done with the f orrlae:}

The existerioe of ?(p) as given by (22) poses severe roctrictions

upon the forni of f(x) ;ard it ray be poas ible,that app]ying the corlex

Fairier transform in the X direction of the Michell problein,the integral in (2.2) does riot exist,t.ecause of the fact that *lxyz.)does not necessarily

tend to zero for X--x7.

it even if ?(p) 1oes not exist,the functions defined by

00 f

Ç(p)

j'

()

cix 00

(2J)

() :) e

y exist for suitable ranes of

p

u+ iv ; thus ?(p) exist for all

real values of p,irh11ì

f _p) Iiy exist for a sufficiently large negative

value of y (say b).

In that casC

_)( LL&

Ç()

e ctx.

,so that

?_(p)

is the

transform of a function

and this fuctJ.on would tend to zero for x

- - of or sufficiently 1are

values of

b.It Is

rthìt in our inves tiation, 'vherc

& 'enains

bounded, b is a very small quanti.y,which tonds in the liriit to zero.

The inversion £ormu) in this case is given by:

I.

f

- 1;:- -LX» _f

-) 4_(r)e (2.$)

¿J,-uo

b

is related to the singularities of

_{f...(p), and it is}

neces-sary that b shall be less than the inginary parts of the

al&rit±eE

of f_(p) .

cf. (t8] pio

The advantage of the lifte aral transforns Ls, that we can exprese the

integral transfori of the derivatives ci' a function, in terns cf the

function itself,

bO ¿rx

f ()

x = -

_p f(°)

_{Ç (&}

o o eQ

-7

í?DX cL

if

f' (x) and £(x)

tends to zero for X ) o

In the s arie way we can deduce

g,

-(x

f()x

flh(x)\ ₌ o _o

for x(o

,

o

f

fe

= 1C'(o) -pf(o)

-8--

_fío)

- Çx) x

fo)

-Applying an integral transforri to a partial diÍferent4i7 equation it is transThrd into an ordinaxy dffferenta1 equation,which _{can be}

solved by e eentary rar.

3 APPLICATION TO S( T7O - DDNCIONAL OBLC

We conîder first the tro-üiìrnsional problem of a pressure

distribution ving with constant velocity c o-ver the surface of

a twodnensional fluid of

if mite depth.Fizing the coordinate

syte

to the pressure distribution, the undisturbed stream is rving

with constant velocity o in the negative X- direction. The pressure coiìition on the s-urface give3

=

Co',S( -'-_

0 _p

p0 atm. ressure

p-)= arbitrary pressure dis tr ibutisn.

If for x -

o the dis turbance tends to zcro, (3,1) beoorre

i CSxf Z

and Uiinating

Z -3 bO +

4-o

_- P-3 Ç

The ìdrt5.ca1 boundary corition remains the sa

_{as in}

(3.3)

( I

\_/ si-4

We have already assud that the uave heights are suail, and this condition is a3zo true for z O, in stead of

z .

ec ides it is clear tLat the disturbance -rh die out at groat

depth

S mce the X - c oornate extends from - o-o o + p.o _, dof ins

a function:

(-s_.5.---3

/

¿X

?ífr)

_J e

and the boundary condition

9

bd f ic(X ! _1b,(x)ac t>

z: j w

e can

rite down a similar rslation if X<O:

* (ca) = - t1Ç(oz) - ¿vc.

z=o -Qc71(cco) -koco)z 77 7oc) - + ¿

Z -3 o

I

¿.oc

77_±

je

_/'(xickx

(,o

o

(3G) k ,Ç(xz) -o o o (3 6)

(

e ./11xz)X

In that case we still

expect that both inte[a].s ril1 exist, although it is propable that SL(x2) #0 for X9

-Tt

integral transforns of th derivatives of th potential are

related to ths trazoforn of th potontial

sLf with

g

f L

Je yÇ(sz) dix.: 'Ç(oz) LOE i/.(oz)

-ptx

1L» frz)c.ts= Ç(oz) - ¿Pc (02.) - t7(ccz)

-rhe laplace equation becomes, if ?c)o

_Ìctz) *

1ccz) - ¿oi (oz:) _(3.7)

with the bouary condition (3 .li):

zo

(0)[acO)..-

lit1,1) _oo)-toct(oo)

ID

The general oktion of the reduced equation of (3 7)

i:

¡z

e

t

e _{(3 1o)}

wherein )' represents the square root of c ,f or whic)i J) .9

Th3.iecessary to ap1y 'uniquely the condition

j zoo

ITitb the thod of the variable paraìeters in (3 alO),W8 deduce

the general soltiticn

_{of (3.7):}

-d CedZ c, e

ì

Fe_lZo £/4!/2. _Cta.e'

Iit:

t

f _'

7tz'

c/2e1*C2)tei. C,,j.

- C7je

7t

-

'J.2

, _d2 -J'; . (oz)-Lc,t(OZ) FroTn (3.11) and (3,12):

.f _{L,ioz)_}Lai. oz).J?

ç

e. ta. z

,:

L

- L _{+ R} cz

f()(o/

+ (3.11) (3.12)

and the o1ution of (3°?) is:

z z

z

fN-)

flea- è _{( )}

_B

ç

o o

The condition

if zoo,givcs

-a.

fl?L

oi.)e

c(

z'-J

o

The constant B can be deterrnined by substitution

boundary condition

(3.):

()

(o)(oo)

le-J-

'f)

o

and the general so)aition l'or bd u -

(

-.j- - i

j( z)

Ç(o)-

_L.(o3)

o of (343) in the

T+oo)-to(oo)

(3'11) In the way we deduce

f ig.

U

°

_.ç+-

_»iaï

7ç(cz) =1

_lZ(oo)*u/7o0)

-e _(3.:5) D(

To vrite down the required solution of 4xz)

_{,it is necessary tò}

apply the i!rers ion f orru1a (2 .):

-* o

t(xz) 1['7.i)

_{* (3.16)}

In the cecorid intogre.1 the way of _{integration rur below ail}

singular points of -(.cz).

Coi1ering

iz) ,the first integrai

_{of (3.16) does not. eern to}

exist,since the denosthzator _{vanishes if * ±a.Bat we have} require:1. in

_{(3.6) the integral:}

LX

to exlst,even for Li -o ,cinoe L(xz) -o,ir

_x-

o'.

Hence ti mine rators in (3 Th) z

and ?7'7.() 4foo)

_oL(o)

nust go to zero at the sane tire with

Therefore we can integrate the first intsal

_{in (3.16) along the}

real axis,and even divert the contour in

_{a snll se- circle unger}

the axis ,since tITE integrand is regular in this _{d»iin. ( f Lg.3)}

,wl (. r

opLan

_t(u)

(o

£ig.3

In the 50002x1 integral of (3.16) _{ve have to integrato along a}

contour,which ruiis under ail sthgulartias _{of ?( (ocz). (f i0L)}

S mcc 7Ç 7ciz) _{goes to zero if .c}

'

_{we are allowed to nov-o}

the contaurFto the'e

_{aais,only if}

_{re interrate round the joles}

c ± K

along a sii1 seni-circie under the

_{axis. so we lot coincide}

the contours of both integrals of _{(3.15) ,integrating along the}

contour _{H ,indioated in}

_{ig 5.}

Freni this anal3' is it is obvious that the boundary e onditiori

far

uptreani unbigious1y deterTnin tLo contoxrs of integration, and.

already in this phe o! t

calculation it is possible to fix the

contours. Equation (3.16) becore:

(

X - e(Ps

.L

j aQ (3.17)

-'

₊ -H _wherein:

Hr

_-! J'p)e.

c

Workir out this epress.on,it is necessary to discrininate between

ReJ)ancftkCo;for, in the first

case - -s-

,and in the latter

r-oc

_{(ci'. p.19),} and we re'iì'ite (J17) o -(SIX

-!

e _{e duc} i e

()

zar

I

-

f

_i

---J

Introducing the coi1

variatle

¿ ie vrite dovrn: o

_L

(

e

iii) e

dIc

j.

Tî() -&

_e

ZIT _{j 2rT J}

H1

and this cxprecsion is equal to (3.13),

if

is real,

Therefore e cor1ote the oontour H by an arc of a circle of radixs

R ,and a path along the negative inaginary azis. (f 1g.6)

-K 'k

If x)o, the integral round the circular arcs

_{iU tend to zcro,is the}

radii of the arcs tend to ini inity,and (3.13) can be written as:

o ¿Iz - I j%x

_i

_r)

((j)e

_{df_ _L (}

e 2ff )

-o IJ)

Q fr, L _(3.19) VT _ZrT) o

If x<o ,vre have to colete the oontir H

by an arc of a circle

of radins R,. ,snd a path along the positive

inery axisif the

integral round the circular arcs will tend to zero. (fig7)

x)o

aj

Case a:

1dx= P

Rut by coleting t oontr now,rre are passing a polo, an ic

tve

to take account Ó± t residue,

/ , iJz

(

I

cj5+ _rTJ

x(o

:

o

o we have

f

ow _{a general e:p ess ion f}

the potiiftal of a

stationary two - diier iial motion Trith free surface, umer th

influence of an arbitrary pressure diìtribution on tho surface. We aly -bhi result to three s thinle cases:

a .A concentrated pres6ure P

in ti

origin.

b. A uniform pronire acting on a band of finite breadth.

e A uniform pressure ,applied to t1ì ur2ace £ro:i the origin th

infinity.

K dx pLx) C? -L

'

- -b 11LC)

L_'

t(._Lj2)_

!

C? C17 C) (3.20)

o

K(o

o

LaJE (i) deduces only an expression foi' the wave distirbanee

by uíng the dissipation factort1

_{result is in agreerient with the} formulae

(3.23) aI (321).

OQ

p

1 (Q. o rrp J o

+ !J

_(.6(X cf? with

This result is in areee''the

_{result cl' ETE.S (S) .The} express ions

7ith the integral are syetrical

_{with resp3ct to the}

and disapear at great dLstance,so _{that indeed I1.(xZ)oif} X

It is stnj1e to calculate the

_{ïave disturbance}D

ZO _-

_Lo)

04 -j3x

je

° Ç3x 2PJ(

___

1(2Pccc

; 04

X)o

zpc

Í$X

(13 i-ff)

cß

ri o (3.21) (322) (324) X<O : g -f ₂ -kZ 'Ç origin, (3.23)

zPc (eßX,ßaßßß)

L

1-1f;

p(I3Xç

d53

ri- ) K+j3t

,ç(o

xJßecLf

ciPct- II,.. .11.,. iC' n_ )

The wave diirbnce far dowrtream is in

_{a reent rìth the}

results of DEAN Cts). C-.se c:

xo :

cl,

-

(X(zJz

-'Cz rrcf

-f 2P'e

Po'C

(f3x

(3,2f;)

'TP')

_o

i

ti

Equation (3,2f;) is in agreerrrt with t,he result

_{of L. Ci)}

4. AP?LICATICW OF THE F(JT.LEkL TFRPL TASF

CALcTJtATION OF T} VLÜCTTY PONTL.L RCTD A CIL

TTTP

We ìapitu1ate the boundary conditions of 2:

Required the soi2tion of the Laplace equation

,which satisfies the comiitions:

'Q 2j

L -L0

<--2DD -<<

)c(--

?'-)

_L>,o

Because of the syrtry of the problen with respect to the plane

z

Ve have _{discussed in the single Fourier trarìsforn,but for the}

soition of the problem of this paragraph _{we need a double Fourier}

transform.Since the X _{coordinate extends from - towards -} o , _and

t

Y coordinate _from O to ,and since we have to satisi) _the

condition

(x)

_{g(xz), we define the f oflcing transform:}

i

¿ x o oq (Pc X faL

_Jy

_xa)

-"q

Just as in

_{we expect both integrals to exist under these}

conditions .We can v'rite the Laplace equation,

_if

_{x) o:}

00 f00

f Jí

(za)°

The partial 'ierivatives of

_{'xy)are to be transforrd in:}

oyz) ¿

(oe)

c.cç X

Z

xz)x]

-

(oz)

*/oz)

t'

₀

with

_(ojz)

c- if;y

oia)

,_ _o., &

Çf

xrz) °h

o =

t() J32(z)

¿'-'X o)çQ

t(xoz

And so the Laplace equation _{is tran$.fornd into:}

x)o -octp;) yj°'z)

t

_ofz)

7z)

-i. (oj3) <o

_(°d)

(l.l)

and the boundary condition for z O :

-ßo)-Jo):

yÇ(ojo) _oc(oo) (1.2)

and

if

z -

r(/Cß.)

zo (3)

If

x <O we can deduce stailar relations:

X<O _{' 7a J,2) -t »}

-

7o)

91 (oj3) L1Lf'ßa) th. .h)

Z;:,:,

c7j3o).

- K1jo)

_-

_lojo)

_{+ ¿}

_'ojo)

₍₁₎

o

-/

c'fl'x

2 - '->°

a'- ).LX.

_(x).

'

_{ith this trferni we have reduced}

_{a part1 differential equation}

with three boundary conditions

_{to an dinary differential equation}

(L.i) and (h.L.) rith to boundary conditions

(..X

,Çdx e y(xz)

z

-

I

53)

*

neIl

x)

II " _{in which:}

-

17

-T1 general solution of the reduced equation

(,l)

is:

41jz)

e*

Cz'

square root of

(2)

The solution of the cori1ete equation

, in

which-is the positive

(iol) is:

We can obtain this solution,as in 3, by using the thod of the variable parameter.

A and B are to be deterthd from the bou±Iary conditions.

The condition

¿Loz):off Z-'

gives:

f2)_

-- L

I $/oj'3)e

c

zo)

o

B f oflcws from the condition :

*

f

-t( e A (oßo)

-

¿ 21

and the solution of the equations (L.l) , (1.2) ard _(b.3)

is:

,D

-

e f e - i 3 / - o) d

-i

o

(-dst(ft)

ç-)

4(cj;3) _&(3

z)

2)

- (ojeo)

fc)

-

(.6)

(

j3) _j(*

_e

-K0-In the same way we deduce an expression, if

017)

?c <c

i

(e

cÇ

_._,..__

L)

-(oj

(Wz) is found fron the inversion fo .. -

(2,)

, as vie have dor

in In the first place ire perform this for the X transform :

-t ¿Q

2(XZ)=

f()E

-

-t-x

¿2(eÇ (Ii. 8)

-o

f2

(aß _{/oo)e (e7)}

-t-

z-(5a/

0 ₅ 3)

-*

8e

+ oß (14.S) - ¿

7/o)

a)

l3

-In t

first int3gra1

oc.

is to be real, but in the second integral

o&

has to be a rgative igirry part

b , that is sra1ler

t!a

all

singularIties of X7ß2).

The singularitie.s oJ are givon by

i c'c?-

«r

VK<2J7'

Since we have chosen the positive root of

oiiß

, we can

ûy

us e

*

z

The singularities of y7oßz)are given by ± c

z

These singularities are situated on the real axis (

j3 Is a)a:!s real).

With the sai arguient as in 3, we let como i both contours of

(i48), onJr if we integrate round the 'oies ±.

a. along a snail

senil - circle under the Wec xis

i'ig.9).

'e

(ct)

f ig.9

And so the condition at infinity uniquely gives _{the direction in} which to indent the contours round the poles.

Equation (1.0) becorins:

2'jZ)

_Aft)

e e

/2

j./+)

/ o po -tQ

f[(ou

de

J/e:()

I

e H

Using the inversionThri.ìla far the Y -direction, we get:

Xy2-

_)(2):

I

-e(z-3) _t)

-+3))

(z. 2.

I

I-f o 1 3)

¿(;J

d o

PC

_(x-,)

j:

-&. .-, o

.1

-t- e + tck I-II f

(

cU\

-j

_-ir

-)+

.:i C4C) y Zi7

,"

2

-í

ç,17) J t)

_tir

'I Cr-(1.9)

In (1.9) we have found a general express ion for the velocity potential round a .IicheU - ship, and in the sul equent paragrap we will shag th3 equivalonce with the potential of a continuous

source distribution in the XZ - plane of the !hip, as bas been derived by HP.VLOC! Çi23; alzo we will shov that (1i.9)

is equal to the original

exession of MIC1IL C"].

5. EQUTÄINCF TITH TLWELOCK 'S POTENTIPL We consider the integral:

(

4fx-)

.í

(f3)

û FI _Si2 (5.1)

2j

o

This integral does not have ar s ingulzrities, and ie can also

write: _{-t 9}

I

('.()(-fft dß) e

o

/ (yI\

_{1dç e}

-I2-)

2) - _ and introducing L

("

dt)

(

di:) <,112 L

1z41 t(Xc4>-- y.)

-

CTt

z-1 ¿4cc

Patting çzç in

fig.

10. - 2iT 20

t4; ¿?g

olé 1?-3(&'+1) .-(41&'.-l) (xya)

V z-)

7y2 z

In the same waywe prove

¿ç 2,72 We re putting f ig.10o ':t. Lc) r

,

L

d

c9f

-

-

_(.3)

arr

(2)

C 217

1Iç

()

o

inthich:

and whereby one bas to integrate with a srl será. circle under the

Zn

More difficult to trnforri in the third terLa of

(.l)

2 Zn-? J o

(f

--fly-d()

- H # We put - , thon:

¿D('(4)

_{-iij;y _c(?3)}

e

pole

L(J-,jf

, and vrith a sinai]. semi c:Lrcle above

the pOle

i"?

j -

OE

This requirement f oflows l'rom the2condition that ¿«'J >0 at the piace

of the poles.

By treating the wave - disturbance of a moving pressure distribution

on a water - surface, HOGR (-J corres across integrals of the sm

type as (.b), but he has to make an arbitrary assuition about the inte

gration round the poles, to get the required solution.

Taking the three terns

to-gether, we get in stead of (l):

¿IT e (x,z) ' I k

-

-t.

2ii,

Z)

Í

o

This expression is equivalent with the potential of a source of

strength "= --L

,situated at (o3) ,as has been derived by

HVELOK 6(xi2J: kb ¿<lAi

n-- 21

-(E)

( .6)

In timis expression /4 represents the vanishing small diiration

-factor, and. c1ear1 the only purpose of this factor is the location of

the pole, or if 1L - o

,the way tc evade the pole .The place of the pole is given by

and if - <7 < the integration along the real axis runs above the

pole, aîi in case of <1'< under the pole.

It is clear that

ve1ocks potential,calu1ated for a continuous

source distribution in the lonoilii.dinal p1ar of syimnetr, with strength:

a- - 3)

is equivalent with the potential, as has been derived in pararaph li.9)

This follows from a comparison of the equations (IL.9), (g.!) and (.6).

Aleo a simple kinematical crans ide ration proves the equation: V <XOZ)

-

c'j [is[.

6 EEDIJCTION TO T L4ICFIELL P

rrIL

Je consider the result cf 4,wherc we have found an oxì,ress ion for

the velocity potential round a .cheU - shin: (p. ¡g)

,.,I 0

(xz)

Jdoc.

--

_{_J(2-)}

4k»

-In the first case we consider the expression:

-'()(.

«JftYftf

(6.1)

I-1

In order to proof the equivalence with the Michell

potential, it is

necessary to change the orLer of integration,aw therefore we exanine

the singularities of

(6.1) .These are given by:

(2-±

tJ0ç_?

6 J i /<

-(cfo f ig.11)

£i4.li.

For every ß constant, we Lind two poles, and we know frein the foregoing analysis the way to evade t1se poles: viz in the negative

¿*) _plane0

If cX = constant, and vie firs t integrate the / - coordinate, we only

encounter two poles if ° <-' and o') ¿Ç _{, and the requirenent we have}

found to encircle the poles in the o( - plane, gives now uniquely a

prescription about the inte'ation pathe in the ß - plane0

}itting _{'-- ¿oc"}

_{'. ¿je,,}

we draw the (acc") , (J3'fi'5 _and (a'ß')

piarEs ( fi.12)

çft

:

I

,t _:1:

fig.12 .e.

In the first quadrant of the (a'f%')

- plane, J) a .If we give

_¡3

at the place of the pole I a sll positive inaginary part

, it is seen

f ror

(-tt thc ole I sets a positive

iragir1ary part in the (° ) ,- plane, and is shifting c.hove the * - axis; so the integration along the" - axis renairis under

the singularities,

which was necessary to satisfy the corctitior

at infinity.

Sirnilrr

we calculate the integration paths for the other poles, and so we have to

integrate along P

if #'<-'(, and along

_{Q if}

*)

sc (f ig.13).

p -z- (

(z)

3/

- 23

-1' ig.l3. ¿ - _{- , cfiy} _/(7# 3)

c'-k

-L0X-,)

- -F -°(1'x-,)

'Zi')

* (6.2)

3,

After changing the sequence of integration, (6.1) can

_{written a:}

p<. J), e

a(()

We introduce

_{a colex variable}

_V

'\ , anri cor iìer ti

exprsss ion:

7

_:ac()

-(6.3)

K K

T! Y _{real, the integrand}

of (6.3) is equal to the integrath of (6.2).

Ttrefore

we _{comolete t contour with two}

arcs of radius R _{, as shown}

in fig.Th.Considering _{only the case}

y.)o _{the integrals taken rour the}

arc ter1 to zero as R tends _to

infinity.

Further the ntegrard has two

points of rari! ication

'-which we connect _{with a cut}

through infinity, _{separating the}

parts of the

-v _{plane, where} > O _{,froni the parts}

wlere _{e Vc,v2' <o 0EviLently}

vie have to choose alonn T the

fig.lL.

_{positive square}

_{root VÇi'}

and along S the neat1ve

square root -\[cs'i _{The integral along the infinit.e1j s!nafl}

circle

round

_{V-toi vanishes,}

Coileting the

_{contours in this way, tI}

only contributions to (6.2)

are given by the integration _{along the itnaginary}

axis rnd the reiiues

_of

the poles,

In case of _{c<-'c and °Q'-'c}

_t

_{poles are situated on the real}

axis, as we have _{seen; and if - K c <ç}

the poles _{are situated on the}

imaginary axis An so we calculate: j.c 2 2ìT.! 'C z<J

b

bç *

z)

-'ç oc. 2h iocf-) -./f+ 5''?*) o(f a(?« _J,7-fÇl

21TL ±. 6< -(x--)-/\y-- v° i..

±f

/ot/\

e

p?« Lz2

ji

_fc, o( - 2

;

i'z,) o V.«'

«e. J «

/ot

e Q

/

7 _J K 2rr,. (QL2t. ₊

_V2?

4< e f.

ífccc

_o('x) '\'

-e -i< V,_'z ¿Ç

zL

£

_(?)

VK?'

2«)

C, -t -A

t

1 ¿4'i-Ç %j,L,1?

9-4

fig .160 f ig.l7. + o

Addition of the

_{£oLr tr}

K

(doe

- -* !Ti

'<L

Lt 'ç-2fl 26 1f -'ç e -+ .

-'--Ay

Í/

((/1Y,) -'iy-

7)

+

--Jd.aç/'

(z)

-Q ¿Q e e a ?t

£r:. ( e

¿1-)- ¿

v' -'i'-) _ÇY Vci4i)

e

-e

'<I

I-a

Í

K

\[i

Dt7

«Jff

_2v?'

C-) -1-

/

bf

ojoç / o(ì\ e

csx).

/

cL< VÇ?

) d( ttiiig I 2 oSf\ _7Z

citi:

1a1 '2 2 G( vie get :

(

,(e

r

K ¿

(2* ) y (x

₎

_± k

k)

û

.i;-s-i

íÇo'x-)

_'-.Y

e ?(21.3) y

vIa

a'

s

o gives: tcC

_(I(('%) ()_a

J79j1-7 e _e

.0 K

\f-çZ I'

<-)

jJJ)

t)

2Jf

i

-__ I

-,l,

L) L'

(x) t

(z)

3Q

We cone ìer now the other terr of the express ion for the velocity potential round a !iche1l BiIp (page 22) :

It is clear that this integral alzo can he ritten as:

because both express ions are equal tc

IT 2,(yz)

ff

o( Adding

¿

H t-o -f -4-L-o 7 VDS -1 t, 27

_x) Íy

-

?fZ 'J-/

Vocj

-1-L'o

d

(a(5c)

J2-j;7;;

>,

V-i-

y-We f ith a sinilir expiss ion for

-t- " -p. L)

(

t

¿ ß V( L4

f

ç (

'(-)

-V'

E

20)

f _e¿Il(?-) Li{2-f)

(i2--1

-l?-?,I

-+-

Calculating _*

I _¿

3C

Í

wfoi

°

_/i,f

_{2k(z).-2I7S&i(2-)}

pS'1-t-t?-$-) -l1(-)))

ì)-) z Us lug ,,¼

the express ion _{b? corfls}

z

-2

3':

Adding aU _{thgethr, we write:}

y

fÍ)

+Vo c,4 cE4

(ct le%)

+

28

-'1 , We get :

c6)OV'x-) ¿c5(i.i-f) 45)("1-'i)

y

_a(6.(I.i?-) 4S)(Li7,-)

O _¿<fr1 xC

yI

y aÇC Q (6.)4) L

which le in _{corriplete agreernt with the}

exprgssion,whjchwas published

by _{1!ICLL in l898}

_[6}.

Attention riiist b? _{cafled to the factor in the}

second terii of (6,L):

which was chosen _{by Michell,}

"to r.ke the

_{elerntazy diverging}

waves

trail aft",

Tb? other solution of the problen _{in given by (6i14),}

in wtiich

replace the factor

_{(6.) by}

This solution

_{is not in}

_{agreer-nt with the}

boundary condition

Lyz)off x.Ö,

_{bit with the condition}

_L(z)o

if

We can derive this solution by changing the integration paths P and Q in fig.130This would be the solution with the "eleintary diverging waves advancing to the front

From the foregoing analysis, it is clear that the solution is not

inerTninate, nor is it

necessary to make a distinction between a

practical, stable solution and an urtable solution, as bas beeb dcne

by soi authors; bit the right interpretation of the conditions

at

infinity wuely givoS the required solution ol' the problema

So we have proved the coirplete equivalence of MïC}ELL 'S potential with PAVLLOCK' S potential for a continucue source distribution in

the longitudinal plane of synni?try.

Mathematical Departnt

Deift University, April 19S3

Ship Model Basin

Shipb.iilding Departrnt

[6j MICHELL -" The wave - resistance of a &dp ". Phil.1bg.l8980

L71 HAViLOC -" Studios in wave - resistance : Influence of the

form of the waterplane section of the ship' Prcc,.Royal Soc. A 103, 1923.

" Effect of parallel ittddle body G

Proc.Roa1 Soc.A l06,].92.

[8 WIGEY

[911

-

29

--

_{Ship wave resistance '. Trans of the north East}

Coast Int.of Eng.and Shipb. l930/31 " ave res is tance of uxy!.. trical f orr ".

T,I.NA.191.l

- TI

Anvendungen :1er 1ichelschen Werstat1oi&'0

J.S ,T,G.1930.

RltLNCS

[1] IMB

-" Hydrodynanics ", 6th edition, 1932.

{2] GRN

Vaves in deep water, due to a concentrated surface

-

pressure". Phil.!ag.19h8.

[3] H0GR '" A contrIbution to the theory of ship waves «,

Ark.for t.,Astr. och Physik. a. D1. 17/12 1923. b. Bd. 10/10 1923.

[1] DFÄN -" Notes on waves on the surface of running water". Proc.Canib.Phil.5oc,l9L7.

[e;] TFS -" A new treatrnt of the ship wave probleif.

1i) GUILICTO! [u GUILLOTON [izi HA7LOCK [!3j1 HA.VLOGK 11611 WIGrEY Li?] SEDDON [1811

30.

der tar1suntersuchurLger. an SchiTen

ZAOM.L 193S0

" chiffsforiii und Werstand" chfffbau 1939.

-" Stres1ines on fine hulls ",

T,I,N.A. 19I.8

- _{Fbtential theory of wave resistance of}

with tables for its calculation."

Tr,S 0NJiM,E 019S1.

-" The theory of wave res is tance " Proc3Boy.SocA 138. 1932,

- " The approxirate calculation of wave resistance at

placement ships in steady arx. acc1erated rrotion"

ThNÇAMÇEO 19S1. cÇ p!9

- "

_{l'Etat actuel des calcula de résistance de au".}

A OT.LPA .19t9.

- " Fourier transforms ". l9i.

Integral trn.for

in matheiiiatical phys ics " 193..

high speed". Tr0North - East Coast Lrst.of Eng.nd Shipb. 19L3/L,

[i HAVELK wave resistance thecry

proble - Tr . and its application to ship_19S10

and discussion to ('lj.