Basic theory of wave analysis

Basic Theory of Wave Analysis

(State of the Art 1975)

K. Eggers

- jUI.I

1976

ARCHIEF

Lab

v. Scheepsbouwkunde

Technische Hogeschool

H(k,O) H h J(u,w) J Ic

Basic Theory of Wave

Analysis

(State of

the Art 1975)

Hauptvortrag auf dem

International

Seminar

on WaveResistance

in

Tokio und Kansai

3. bis 9. Februar 1976

ABSTRACTL

The. paper provides an evaluative sur vey on mathematical mc4els which are

under-lying the current methods for determining

wave resistance through wave pattern mea-surernents. £mphasis i on elucidating im-plicit a3aurnptiofls rather than analytical

derivations. Special weight is given to

such problems hich are still open for fu-ture research. Some comments are made to questions related to the decay of' the

'local' wave, field.

NOMENCLATURE

A,B,CD = control surface for

energy flux wave amplitude

btankwidth

C waterlin& along a ship

C(w),S(w) Fourier transforms for

longitudinal cut method

C2(w),S2(w) Fourier transforms for

-longitudinal cut method F(u),G(u) 2 sine, cosine component

of free wave spectrum f.(0),g(8) alternative definition

of free wave spectrum

O(x,y,z;x;y;s') : Green's Function àf

point source of unit strength

acceleration due to gravity

complex Kochin function = complex conjugate of H water depth alternative form of Kochin function = complex conjugate of J dimensionless circular wave number in

direc-tionG

fundamental

wave number

2 :'Ccos = cosine component of

wave number C

rn = 2s-inO : sine component of wave

- number e

: unit vector normal to the ship

P(w),Q(w) p(x),q(x)

wave resistance

R,CL polar coordinates with respect to origin r s distance of flow point

from source point r1 = distance of' flow point

from image point of source

S control surface

a sCu)

= function of' u time

U ship speed

uv,w

flow component in horiz.1

lateral and vertival dir.

u kainG = dimensionless circular

wave number induced in y-direction

u,: discrete value of u in tank of finite width

w

_iccoa3

= dimensionless circular

wave number induced in

x-direction

X longitudinal component of

horizontal wave force transverse component of horizontal wave force x = coordinate in direc,tio

of notion of ship

y horizontal coordinate

po-sitive to portside

vertical coordinate posi-tive upwards

flow point Kelvin angle wave elevation

partial derivatives of direction of wave propa-gation = k k wave number O fluid density wave frequency perturbation potential 2 components of perturba-tion' velocty Fourier transform of p(x) q(x), respectively

-As -long- as- man has -used. naval

trans-portation,, he must have observed that moving floating bodies make waves, which reflect in some way part of the effort needed to keep a vessel advancing, in the desired direction. ilut it was not-earlier than 25 years ago that interest awakened to. determine the power Supplied to the wave pattern from the ship quantitatively through evaluating the wave pattern geometr

Exploratory enterprise came up in. this country (1] and in France (2) around

1950,

but it was the truly pioneering work of Prof. Inui which cleared the soil for the second, more rational phase of research. lie presented his achievements at the - I

may now say historca1 - 18th meeting of

the "li-5 panel" - that is the ship-wave committee of the American Society of Naval Architects and this worked like an igni--tiom spark for world wide simultaneous research.

The timing for the first international seminar on wave resistance theory in 1963 was just appropriate to have the attendance confronted with about ten different methods of wave analysis already. This set signal for the next phase, which meant synoptical analysis and generalisation together with

systersatic.assess.nent of practical expe-rience. Thus the variety of different approaches could be tied to and re-derived. from common mathematical models.

It would be premature if I claim that this phase is facing its completion alread but I feel that meanwhile so much progress is evident that we are under an almost compulsary need to have the state of the art presented here and to be discussed cooperatively by potential experts. This is in particular so with respect to rapidly increasing demands facing the naval archi-tect for most accurate prediction of power-; ing requirement of ships.

My role to give an overall report on wave analysis theory within this seminar is

made easier in so far as four well-arranged

surveys on current theoretical development

arealready available: One (3) by Eggers, Snarma and Ward

_(1967,

referred to as ESW in the sequel)., one

₍₁₉₆₉₎

by Ikehata [4)., one

₍₁₉₇₃₎

by Wehausen

_[5J

and finally one

(1975)

by Gadd [6] within the report of the ITTC Resistance Committee. I take the liberty to assume that you had opportunity to study at least one of above papers so

that I can concentrate on certain

partcu-lar aspects which - I feel - have not been treated with adequate weight so- far.

Apart fron reporting what has been achieved I intend to compile material which still has to be evaluated and completed, n ordex to find cross-correlations and to show gaps where further research is-needed. For-this purpose, I have built up a bibliographical -appendix. Its aim ia to update the

litera-ture given in ESW to cover all papers somehow related to problems of wave analy-sis since

1967.

References to this appendix are made in standard form, i.e. by author and year of publication as is used in the bibliographical work of Wehausen and in

ESW.

One basic assumption of all wave ana-lysis methods is that viscosity is

neglec-ted iñthe following sense:

Ci)

There should be no vorticity where measurements are taken.

(ii) Any attenuation of waves off the ship due to viscosity is neglected

- (this is justified through

investi-gations of Nikitin

(1965),

Cunber-batch

(1965)

and Brard

(1970) ).

Another tacit implication is that wave-brea-king resistance, though originating from gravity effects, can neither be measured nor

detected

by wave analysis. This questi-on together with interactiquesti-on of resistance

components has been analysed by Wehausen

(5]

and in more special works of L3aba (7),

Brard [8) Landweber [9), Sharma (io] and

Weinblum [ii]

1. MODELS OF WAVE. FLOW

Any deviation of the water surface from its state-of-rest position represents some positive amount of potential energy, which may be evaluated directly from the wave -pattern geometry. Due to the associated

fluid m o t i o n ,however, there must

be also kinetic energy. And the wave re-sistance, in which we are finally inter-ested, results from the lengthwise compo-nent of total energy transport. The group velocity, which governs this process, depends on wave length of individual corn-ponents. Hence it can not be expected as

suggested by Xorvin-Kroukovsky [12) that there is a direct relation- between poten- -tial energy of some area under obseration and the resistance of the ship creating it,

We can consider some modes for-the flow manifested through the wave pattern -and derive wave resistance as function of

the flow components. Inserting then rela- -tions between flow and wave elevation, we are aiming to express wave resistance

through characteristics of the wave geome-try only. This is actually possible fot several models of wave flow, wbich:in ttrn give reasonable approximations to the actual flow related to the wave pattern, at least away from certain domains close to the ship, where "local waves" can not be disregarded. This restriction comes up because these flow models cam be accepted only under special assumptions. We

menti-oned already that there should be no vorticity, hence a velocity potential C'

can be introduced such' that flow compo-

-nents

UJ -

may be expressed as -its

necessary to require the smallness of wave

elevation and flow components Cf,C

in such a sense tha liriearised free sur-face condition can be assumed to. hold the undisturbed free surface.

l.i Expression for the Total Wave Pattern (includinp Near-Field ffects)

Let 'us take a standard coordinate system to describe waves and the flow. The plane z-O is the undisturbed free surface, the positive x axis is in the direction wnere the ship advances steadily with speed U, and a is positive upwards. We shall not commit ourselves

to,

take a specific choic.e of origin regarding y, unless a symmetry plane

yO

can be found. The position of the coordinate origin 'regarding x is arbitrary in so far as 'it will not influence in General the quality of approximations used; there is a special situation, however, in connection with

Kelvin

wave patterns 1.3.

Any arbitrary wave pattern : _(x,y)

with sufficient decay at

infinity

will admit then a global double Fourier integral representation regarding x and y. Antjci-pating later insight1 we 'shall separate some

particular denominator from the Fourier

spectral function and set

(i,,)

_!.

u,w) e' k.uy

with k0

_{- 91U2}

Thus the wave field is governed by some "Ko,cnin function" J(u,w) which is in general complex. It is convenient or more

instructive to express t through polar.

coordinates in the x-y and u-w plane equi-valently as

-I,

(i,.)

_{1" {J}

14(k,0) kcu,O

h- cw'O k do]

(ib) Cia) ãnd(lb) are then interlinked

through the system of relations,

x Rcoscd w kcos0 dudw .: kdkdO

y Rsin u ksinO J(u,w) }i(k,

0 ).

(ib) is in close conformity with the notation introduced by iavelock [13] and by'

Xochir. [I4] . We shall write down our

results in both notations simultaneously if consiuered helpful for better understanding.

The evaluation or (Ia) near the line

w = _{+ s(u)}

_((1liuZ)/2)

_{(or the}

-2

evaluation of (Ib) near the line k cos 9)

must., be performcu in terms of complex inte-r.ration 'in such a manner that no far-field waves appear for x positive. No special con-siucr&.ions would be required if J(u,z(u))

rcsp.

i(coa0, 0)

were identically zero. £r. this caae l(x,y) degenerates to a "wave tree" wave pattcrno In fact, it is only the

one-variable "degenerated Kochin function" .J(u,s(u)) which determines the far-field

and thus the wave resistance . But there

isno method available to determine R, froma wave field described through (1).

It is a fortunate circumstance that al-ready not too far from the ship (1) may be

fairly well replaced by muchaimpler

expressions found from (1).

1..2 Wave Patterns reresented..through

Sinple-interal Expressions or

through a Series

For regions sufficiently faz behind the ship (i.e. for x = x$< 0) we may deduce from '(1) up to terms of order 0(xb '

(x,,,y)

- { Flu)

Mtk (0uiuy)

+ G(w)

with G)+F(u) (u,s(u?)

In polar notation this leads to

+

fj f

(a). o;r% (kocoi'GRbcoo (e-))

* g(e)

ccs(k,coieR5cosCe4 } 48 'with (8) iifLO) - , cost OH(cos'ø,O) (2b)

(2b)rnakes clear that this is a

repre-sentation, of through a continuous, system of plane waves ("free waves") with wave number k ranging from k0 to infinity and ang1e of propagation 0 against x axis

between t/2 and t/2.

If the ship is moving in a canal .with. vertical walls along y = + b/2, the expres-sion analogue to (2a)- as well as to Cia) can be derived by evaluating the u-iñtegra-tiori in the sense of atrapézoidal rule

with. a stepwidth u

21/k0b. We then

obtain (up.toO(x))

(z,,,y)

E { F(U1)5(3(Lip)Xtwuy)

P.G(u)_{cQ9ko(51IAaX+M9Y)1J} Au with ' (3a) Inserting G(w,) +

(v)

_{{' 9 (o r if(O,) } ;} up Sin3, coi'O we have the polar representation

(e,..)

-( f-(o,)s.'i (k.co'O.r,co,(e..4

+ 9(0.) cc3(coO,R,cos,-3) n,co,.0.

(3b)

We obtain (2a), (2b) from

_(3a),

_(3b) reversely with b tending to infinity1 hence u.-.du. .

and minus infinity so does in this case. An asymptotic analysis shows that

for x positive,

_c2

finally cancells

-the range of value C2(w) and S,(w),for w

less than 1 is irrelevant - wheruas for.x

tending. to minus infinity, duplicatcs1.

If, however, y iz:nonsero, no far-field

contributions ?rom can be found with

_(xl

approachin infinity (Stoke's phenon'.enon of mathematical physict).

1.3 Slowly Var_yinr Wave Trains, Kelvin Patterns in carticular

If to (2) or to (5) in case y$0-a

second asymptotic evaluation

is.

performed,

a representation of' is found with no

waves outside sorn wedge-shaped region x ( O,(y/x( i/TB, whereas for each point of this region two systems of wave

("trans-verse" and "divergent") are present,, whose

characteristics 'e and 0 depend on space.

coordinates x and y, or R and

if

we

'use polar notation. Explicitly we obtain

CeJ'Oj,O) 5°ar

degenerating to

(R,) H (i,o) '

'(6b)

, =

-R,

with

os,;

as roots.

foz

of

{w(o)(o.)

-0

(7)

This is the condition of stationary

phase. In

case of deep water, we have

'(9)-k0cài'O'

(8)

so that

0,,0

are roots of cot+ 2

tanO

+

cot 0

0, i.e.'

tan

-[

zfl,t'

_8}

(9)

and thereby tan0,L1/1/L.tan02. In case

that (8) does not hold, e.g. for

finite

depth h, factors of (6a), (6b) will be

different.

It should be observed that in case

depends on 0

solely -

in particular

under (8) - for each of above wave, systems the locus of constant

0

(and thus con-stant 2C ), the so called "characteristic curves",comes out as straight lines, radi-ating from the origin within the wedge domain.. A system of curved "wave crests" may be constructed by' eliminating

0

from

two equations (7) and

Ree(9) cos (o...) - nE _; n - 1,2.5...

(10)

as shown by

Hogner

f15] in the special

(6a)

t2a) is the basis for, the so. called

"transverse

cut"

wave analysis methods.

(3a) is the starting point for the matrix method of Hogben (1972) and for the

"multiple longitudinal cut" method of Moran and Landweber (1972).We have for simplicity assumed here that the wave pattern is sym-metric to the plane y=O, accordingly we

-must take F(Vu ) =

F(-uLu), G()

0(-u).:The velocity potential associated

to (2a) is

cx,y,z)

UT

4 6(u) SIn k0(e(ii)x+uy)

5(u) (14) du

from which the expression for finite tank width can be derived following above rules.

ónsiderinS now 'unrestricted water

we m9 approximate

.1+uptO order

again, but taking yryC sufficiently large,

O(y ) by . .

-

.,1

Wrcr

f C

(w) cesk,w ( 'p i'

y)

''l (5a)1

+ 5(w)

eàk.w(x+'

y)}dw

with C(w)+iS(w).

(&).

IF(u)) u; U.

-

Im C,(w)iS:(w) _dwj

We shall

give,

a corresponding polar

representation only for , as it displays

an analogy to (2b), though with different upper limit of,integration,

r

f

(f(0)Sin(k.co5'ORccuo(O.c))

-I

(5c)

+9(ø)eO(k.CO'ORcCOS(O..i))} _dO

It was outlined in ESW that for each

method of wave analysis just

oneparticu-'lar wave flow rnodel.is pertinent. The ap-proximate basis for "longitudinal cut" wave analysis is (5a).. It is certainly true that (2a) could serve as an approximation as well for x sufficiently large, and Havlock's "variable integration limit" model (ESW p. 1143) is valid in a formal

sense under roughly the same restrictions

as (5a).ut only (a)leads to a consistent

result, perhaps due to being "uniformly .varid" in a certain sense along a cut

It needs some pondering to understand the Joint action of the two components ,

and . Whereas ' . again is a system of

free waves, the componentsof display

non-oscillatory decay with increasing LyI. With (1) given, C(w), S(w), C2(w), S2(w) depen clearly on the choice of y-coordi-nate 'origin, as does the value of

Y0.

But

for y 0, - though convergent - can

not be considered part of some "local dis-turbance" in so far as it contributes to far-field waves with x tending both to plua

case (8).

Wave crests are thus envelopes to

a parametricmanifold of plane waves.

It is easily seen that the approxima-tion (6) to the wave pattern depends essen-. tially on the coordinate origin through which R and. are determined. If we

re-place x by :x+x, y by :y+y and thus R

by : c by :arctan i/x, we can

expect that the function (6a) may be of quite different character if(Ax'+nyt is not

small compared to H - i.e. far away - and that this change cannot be compensated

through selecting another function H(coiO). Only intuitive arguments are at hand for optimal selection -of x (and of y if no symmetry arguments can be used). For most.practical applications1 at least when

considering waves generated along the fore body, the fore perpendicular is taken as oric.n. This may be justified through the fact that for a discrete pressure point 10-cated in the plane z:O the peak of the wedge, wherein waves occurs coincides

ac-tually with the location of the pressure point. This is substantiated.through nume-rical calculation (Ursell (1960a)-). How-ever1 for submerged sources it is knownthat

their wave pattern may be imagined as cre-ated by some distribution of pressure points over the entire plane z:O with

rnaxi-above the singularity!

If we have to analyse a (symmetric) wave pattern which was obtained through ex-periment, it may be tempting-starting from.

a transverse profile at xx say to find

values .yas limits of an interval _LyLy*

such that (xb,y) vanishes outside this range; then x could be defined in such way

tnat

Iy/xbXI1//3.

But such a procedu-re is not only inaccurate for a measuprocedu-red profile: There is ample evidence that in general the inclination of boundaries to the area where waves are observed is

cx-essive close to the fore body (Hogben (1972), Standing (197A4)) with regard to thO Value t19°28":arctan

1/Vs'

predicted by theory, which is actually observed behind the ship (Newman (1971)). This is an indi-cation of obviously nonlinear mechanisms invalidating result8 of the classical ii-nearised ship wave theory.

An analytical tool for explaining this phenomenon is available since Lighthill (1967) and Witham provided the theory of slowly varyinç wave trains. It applies to monocnronatic wave fields - this term means that only one plane wave is felt in the vi-cinity of each point - where in general the relation between wave number SC and angle of

propagation

0

may be effected by inhomoge-neities of the environme1t. In order that the wave train may be called "slowly

vary-ng", it is required that variations e

0 and the wave amplitude 0. can be detected: only on a scale comparable to several

dis-tances between wave crests. Both the trans-verse and the divergent components of a Kelvin wave pattern (6) may be considered as special cases, where

(8)

holds. In this case no influence of the envir9rnnont j considered, which otherwise may result frpm nonuniformities

_Ci)

of average wave ampli-tude (ii) of basic flow (for example due to the presence of wake) or (iii) of water depth h

There is a direct formal analogy (Eg-gers (1971)) between stationary ship wave patterns and unsteady disperse wave system in one dimension; so that mutually cbrres-ponding sets of symbols can be freely cx-changed. If unsteady wave systems are

cha-racteiised through (x,t,Ge) with x as

space coordinate, t as tiñie, G as

frequen-cy, C as wave number,and C.3G/3cas

appro-priate definition of group velocity, the corresponding parameter set is (x,y,m,Z) for a stationary ship wave pattern

(Light-hill (1967)), with m:SCsinO, L:eecosO The "dispersion relation' m:m(L) -is

given through ni':k'(rn2+12) if (8) holds. Using this analogy, we may then say that regions of constant "frequency" rn "travel" with "group velocity" along curved charac-teristic line in the x-y plane if the re-lation between rn and 2 is not dependent on

x and y.

-Both causes (1) arid (ii) willbe ef-fective near the ship. The influence of

finite wave amplitude is treated in

Lighthill's paper and in the work of Howe (1968) together with experiments. The ac-tion of basic flow non-uniformities has been investigated by Longuet Higgins (1961) by Ursell (1960b) and by Wijngaarden (1969). Their point is that small amplitude wave perturbation - and thus linearisation in particular - should be applied to the.basio flow (arround the ship) rather than to pa- -rallel uniform flow relative to the ship. As far as non-uniformities -of basic flow

must be considered.small themselves dueto

a thin ship assumption, this argument may be inconsistent.

Relation between SC and

0

generalising (8) are subject to two requirements:

The wave pattern must be stationary with regard to a system fixed to the

ship

Waves travel with phase velocity c against the basic flow component normal to the wave front.

-With basic flo)j.omponents(-U+L4,U,L)3 and

with c(ae,c) =Vgflc(1i-(aea)')up to higher order

terms in 0. , we then must require that

co 0 0 - r::' . (see)')

11)

instead of-(8).

-- It is beyond the scope of this lecture to touch the question how the chance Of wave pattern can be found once a profile along

the ship is given (see however the approach made by lnui, Kajitani and Okamura (1975)) From he viewpoint.of wave

analysis,refer-ring to results deriyea

later,.thetolloW-ing

insight is important.:

ven if (8) is not valid close to the ship, A,V and a. wil]. be small

enough for sufficiently far away that this relation can b taken as a starting point for wave analysis.

utfunct'ionsF(u), G(u),S(w), C(w.)

need not to be derivable from some

global spectral furction J(u,w) It i Only in case

of

one monochro-matic wave train that the local flux of energy (or momentum) can be de-

-termined from the associated wave elevation directly. In general, only integral effects as the wave

resi-stan9

., or the wàvé spectrum

IL(co 0,0) can be determined. For the latter purpose, measurements for one single longitudinal wave profile will be found sufficient (see 2.3)

if (8) holds.. :

t will be shown that the wave ana-lysis method of Roy and Millard

(1971) is justified under assumption of one monochromatic wave train,sub-. ject to (8) rather than the more general wave model (5).

If h(cos 0 ,0) is not changing

rapid-ly with 0 for 0 close to

0,

then.

sufficiently far behind the ship,

is

a slowly varying function of.x

fory constant and the truncation

correction proposed by Tanaka

and

Adachi (1967)can beapplied.

1.4

Otner

Wave Flow Models

HethOda for

predicting ship wave

patterns based on ideas of Gui].loton [163 and refined recently by Gadd _f17) ,

Standing

(1974); Dagan

[18)

and Noblesse [19] are

not explicity mentioed wit1in the pro-.iram

of

this seminar. But it seems that

the distortion of;wave patterns found

through this type of approach -

compared

tO our models (1)

(2),

(3) -

has

some-thing

in commoh w.t.h: the

distortion of

characteristic lines due to non-uniform

basic flow as considerea under 1.3. We

should, howcvcr,'observe that so far we

have rather

artificially split

up the flow

into a.basic component

and a wavy one.

esulting from the Guillotoh's method

as

.c1l

as:-froñ Wehausen's [20] treatment of the ship wave problem by. Lagrangian

formulation - there id a distortion Of the total flow field from time-integrated action of total flOw.. And no use is made of semi-heuristic concepts of wave spread-ing over non-uniform flow. A quantitative comparison of both types of.approach

seems

ieairable, but has môt been achieved

so far.

It' has been found out experimentally by Adachi (19714) that.át least for a

ship

with very

long parallel middle body waves

generated near the bow fade of f much faster with increasing x distance than predicted by the Kelvin wave pattern model (6). Using the mathematical technique of rnatched.asyn-.totic,expansions,. Adachi derived a flew

model which explains this behavior. But it. has obviously still to be clarified under what coñditiorts and

in

which. region his

model has to be appliOd.nd.it is not

clear

how any

type

of wave analysis

can

be performed for determinimg resistance under these circumstances.

. SOME METHODS OF WAVE ANALYSIS

2.1 Wave Resistance défined.via Momentum Flux Intecrals .

There is a very general formula to

express R..,

in

terms of flow components

and

wave profile at some control surface 5 en-closing the ship

and cutting

the undistur -. bed free surface vertically along a closed

curve :0

p'4yi.

Jf(c.q.+)nds

-5/f

x1fn dC ₍₁₂₎ The orientation should betaken such that

both dy along C and n on .5 is positive behind the ship,' i.e. the normal vector

?=[nx,ny, nz} should be directed towards

the

ship in.our case. Thus i seems to be desirable to express qA,c,q'a on S in terms of , , and y along C . Even if 5 is

a vertical cylinder1 this is.possible only in case of a single monochromatic wave field under (8); for the other flow models

Oonaidered so far even the contribution to

(12) from vertical integration cannot be found that way. .

Let us take S as a rectangular cylin-dOr generated through the intersection

of

tour

lines

'

A: x=xa>O, B: X=Xb<O) C: 'y=y>O, D:

Equation (12) simplifies then to

Kw.

_-

_{_fJ '}

f/f (' '

_-

_T)

sff

(13)

where in

the double integrals vertical

in-tegration ranges from lower: fluid

r

;nn (2,h)

(z Ih)

2 .$iiPl (1,h)

where is the root of

ratio of group : velocity to phase

'velocity ae9s'Au):k

From (p16.) it can be concluded that the wave resistance associated with some wave pattern (3) is amaller if the tank has ahallow depth! Expressions for sidewise unrestricted water follow through analogy between Fourier integrals and Fourier series. It is possible. to find F(u).and

0(u) from and wave slope along B. In praxi, several cuts are taken and only wave height is measured. Redundant data serve to smooth out errors in the sense of least square fit.

tanh(

2.3 Lonpitudinal. Cut Methods

If in (13) we let x tend to minus in-finity, assuming symmetr? of flow regarding the plane y:0, we obtaIn

-25

fjcTy

where the cOntribution of the wave

eleva-tionto z-integration may be also deleted.

We shall show that (17.) can be evaluated in

terms of or or along C,.

If a global representation (1) holds for the wave pattern, it would do in prim-.ciple to use (15) and determine F(u) and

0(u) from the profile via (5a). However, though, a longitudinal cut wave analysis should lead to the same R1,as (is), the Fourier transforms S(w) and C(w) must not necessarily be connected to F(u), 0(u) ob-tained from a transverse cut if non-linear wave distortion effects iear the Ship must be expected. Discrepancies were actually found in ESW. We shall therefore rather start from (17). It is remarkable that (17) can be expres8ed through wave characteris tics along C under tib simple assumptions

for the flow in the plane

y=y0,

z(0 namely

C? _+k0?5 0 (heat flow differential

XX

equation) _CiBa)

and

?x)c'yy'fzz 0 (Laplace equation)

(18b)

Due to (18), under very general conditions p and its derivatives may be expressed

through values along z0, namely

--

v'

J.(*',c.o)

dii'

-?

I

(i4y,,o) J c''' ca.wk.(z-ic) iwx'

(17)

(19).

This means that the x-Fourier transforms of f _{and its derJvatives depend on z Just}

via

a

factor Z If we define

(20)

and through Parsevall's theorem we find from

:(17) -

_[ (r'(i.) s

61(u)) 2S'(w)-I (15a)

-E

{Pt(u.)+G1()} 25'(w,)-t (15b)

withu:2t/kb in case of a ship amidst

_a

tank. (15) may be writen in sligtly dif-ferent notation, valid then even for finite water depth h. It reflects then a genera-lisation to three dimensions of Lathb's [21j, explanation of wave resistance as increase of wave energy ahead of B minuS energy transport with group velocity component normal to B:

R,,

-,

(4+ b) (i- r9ce1'O)

₍₁₆₎

with a9=F(u.u)u/ic0; b,:G(vAu)u/k0; cosO9:1I8('.tIu)

2.2 Transverse Cut Methods

Let us now in (13) either have and

Then there is no contributiors from C and

yd_b/2 in case of a tank with°width b.

tend to infinity - or let y :b/?,

L) tO (13). and as that from A must vanish for x sufficiently large1 it must be zero

for aY.Xa ahead of the ship. We are left

with

R1

-1! J

_(Xb,y)

+ *ff(q;x..,.z

4 c-.q) dyctz

(1')

To our degree of approximation, it is pertinent to drop the contribution of the wave elevation tO the s-integral. It is then possible for flow models (2) and (3) to express R.,in -tez'ms of the functions F(u), 0(u) which in turn can be determined from the wave profile along . Inserting

(2) or(3)

into

(ia)

we obtain

X(w,z)

k

J q', (x.),,z)

C"

Y(w,z) -

_J

_'fi (y,,z) dii'

We must accordingly have

X(w,z) :X(w,O)ekoZ,

R

2gk.1e {f/ X(w,O) Y(-o) c""

dw}

-

f.1

X(w.Q) Y(w0)/w' dw} (21)

k0uJ' (x,y0) -. eko dx:

out uositive) we may deduce that Y(w,0.) :

have to determine X(w,O) and Y(w,O) in order to evaluate (21). However from (18b) (and from reasoning that _{R.shou1d cOme}

Vw-1'X(wO), or more general, as X(w,0): At first glance it may appear that we

uJ

XX(w,O) Ii _{,y,) c'' dx}

- iwk,X(w,O).

XY(w,O)

k.uf

_{,(X,yc)c".'' th - (wkf7 X(w.0)}

(22)

It can be further seen that there is

no contribution to (20) from the _range

-1wj. From (5) it is apparent that

X(w,z) should_have a steep rise due _{to a} factor 1/

yw2_1

near w:1; this has been verified experimentally by Ikehata and -Nowaza-(196.7). -Chen (1973) observed the

analogue in case of finite depth. Now for. optimum evaluation of (20)

one may ask if. Or or . should be

subject to measurement. - This question was investigated by Hichelsen and Uberoi ₍₁₉₇₁₎ who transformed (20) to include aut000rre-lation functions between these quantities

(see Gadd 16]).

The requirement that the quantity to be measured should decay fast with x (to keep records short and to avoid tank wall.

reflection effects) is counteracted by the desire to have the Fourier transform intensity Concentrated near w=1 (i.e. long waves). Lee (1969) -concluded from

numeri-cal investigation that should have be

preferred.

-2.4 Longitudinal Cut Methods for Tank _Wave

Systems

-Transverse cut -methods are permissible

in quite narrow tanks in principle,

-provided there is ideal reflection along the walls. But there are two shortcomings

Of these methodes in compense:

-(.i) If not taken simultaneously, data - --. must be collected in a reference

system moving with the ship.

-(ii) Part of the wave profiles extends Over domain of the viscous wake where _the basic, asswnption of no vorticity is invalided;-- furthermore _the

length-wise basic flow component relative to the ship will.nárkodly deviate from

-u there. -.

For this reasons methods have been devised for- determining the coefficients

a%,

F(u)u/k0 and

b C(u)u/k0 -from

a wave pattern (3a) up to some adequate limit

I'

umax from one or more

longi-tudinal wave records. of finite length. (We should observe that even if a cut is nowhere crossed by wake flow, the wave

profile will contain componentswhjch have

passed the wake after being reflected at. the opposite tank wallif- the record is not short (Maruo and Iiayasak-i (1972)).

- For one cut only, having a proper-aidewise location y:y _{such that for}

cos uy5 (u9 - 21r/k, is not too small (in-particular for y :b/6 I) Eggera (1962)

suggested a straghtrorward determination

through -, -a Lm .7.

'

C9U,7 - turn cós kw.,x CO3Mv dx

if the record begins at xx . Such

proce-dure should even eliminate local wave components of monotonje decay.

-Some exploratory numerical experiments -made clear that the above limiting process

shows poor convergence if applied to some mathematical wave profiles. This can be

-understood from the fact that the set of -wave lengths ? = 2ltIs(u,) is too "dense"

asymptotically as to allow only one unique

expanion of a given comtimuos profile in

-terms; of free wave components (3a) over a semi-infinite domain [22].

Landwebez. and Moran (1972) took this

problm up again and succeded in deter--mining the set of a, b by the method of

least. square fit to FOurier-integrals over finite records. Their approach was ;modified by Tsai (1972) with the aim of reducing the sensibility of results re-garding the choice of longitudinal cut location. He studied the influence of cut location, of record length and of

ax withip his numerical and experimentaT

work. - -

-An independent path has been followed -by Hogben (1975) in refining his matrix

method (1972). With 'mx given, the set

of- as,, b, is determinea now by least -square fit to a set of pointwise- wave measurements along fOur longitudinal cuts. Both approaches have require _{of -solving a} systeffl of linear equations so that thear

efficiency depends on numerical techniques as well as instrumentation. Toai proposed a mode of combining iteration and

elimina-tion.

One may feel disturbedby one special

feature of above methods. For a given set

of more than 2)

a +1 data, i.e. for measu-rements with reuIdancy, needed for a

least square fit, any variation of

will in principle affect the determiPion

of all ac, b .inc.uding the long wave components, - this does not occur in case of transverse cut methods, where the data set is selectea "orthogonal". This point has been raised by Sharma in a discussion to Mogben and Standing (1974). - We should admit, nowever, that the situation is. not basically different in case of

longitudi-nal cut methods described under 2.3 for

sidewise unrestricted water. Any extension of the record length for improved

approxi-mation of long wave components will in

reciprocity nere affect the high frequency spectral range as well.

2.5

Ward's X-Y-Method and related Topics If we could find two measurable quail-tities p(x) and q(x) along the cut y:y

with Fourier transforms C

P(w)

i/2itf p(x) e1Co dx and

Q(w) = I/2;J(x) eo

dx such that we have

X(w,O) Y(-w,O) w2 P(w)Q(-.,) (23)

then it is possible to express R.,., in terms of p(x), q(x) directly without any use of Fourier transforms! Using Parsevall's

theorem in opposite direction, we obtain rom (21) under (23)

-

9/

pi)

q() dx (24)

One particular pair of such functions s

p(') -! (x,y,z) dx

-

J

P(w) - .wX(w.0)/w'

0(w) iwjT Y(w.0)/wt

(2)

This is the rationale of Ward's X-Y-method,

based on theapproximation that forces

acting on a vertical circular cylinder have

components proportional to cf(x,y,z) and

in x- and in y-directioñ. It ray be shown (private communications by

harna) that in some average sense we even have

-

J/q'dxdx/k

(26)

if

-X

is.sufficiently large - however, this does not imply

da _xx

xy dz/k

under wave flow model

₍₅₎

and (19). Such equivalence does, howover, hold in a

mono-chromatic wave field in the area behind the ship where

cosO (x,y)

is close tO 1.

Sharmä found that by one- and two-fold integration of the wave elevation

: U/g and transverse wave slope

U/CCixy another pair of functions aatia fing (22) can be found, namely

p(x) =

kJXfCfx(x)

dx dx

q(x) = koJ.

xy dx

certainly without 8uggesting practical application.

But there is one practical conclusion which we may draw from our reasoning: Ir we want to determine wave resistance

through flow measurements - and using. Laser' techniques this may be as simple as measu-ring wave profiles - there is no need to.

measure Cf or at z=O, where we. must expect disturbances through wave troughs where no fluid will be present. We can

meaaure at one arbitrary depth zz0<O and. still find Fourier transforms for zO

through (20); analogue reasoning holds for transverse cut analysis.

If the water depth h is finite, most of above conclusions remain valid, but in spite of (20) the influence of z on Fourier transform is through.a factor.

coshse(z+h) / cosh(reh) instead of eIoZ where eis the positive root of

ctani (ach).

But it is obvious that Ward's X-Y-method is no longer justified if h is small. Lon-. gitudinal cut wave analysis for finite

water:

depth has been investigated in

theory and experiment by Chen (1973). As mentioned earlier, the quantity

-

f

Lf (?ydZ - i.e. the flux ofmomentuin

through a vertical line cannot be

ex-pressed in terms and

,

in case

of a flow mOdel (19). Butthisis possible

in case of one single slowly varying wave

train under (8)Z If we have

: k0 coø

assuming

-

f

A(x.y,)Co32c(xCO3Oy5øQ) +

where A, B and are slowly varying func-tions of x and y, we may approximate R.was

- 9

J

'(x,y,) sit(2O(x.yc)) d

₍₂₇₎

This result ma' serve as a basis for. the wave analysis method proposed by Roy and Millard

(1971).

It is, however, not evident how thi3' formular could be justi-' fied fpr more general wave patterns

(5)

or at least for two monochromatic systems 000uring simultaneously, as we must expect in case of a Kelvin pattern. Even if diver': gent waves could be disregarded at low Froude numbers, an interaction of waves from bow and from stern already invalidates the underlying assumption. (It is certain-ly true that sufficientcertain-ly behind the ship Xelvin wave patterns trots bow and from 'stern add up approximately to one single Xelvin pattern, for which the origin can-not be defind precisely).

On the other hand,

₍₂₁₎

may be genera-'used to the case that the relation

between ze and

0

is affected through local inhomogeneities as 'presented in

(1l),'provi-ded the wavelength and thus can be

measured sinultaneously with and 0 We

Obtain the.more general formula

dx

(28)

t we still can assume 'that the relation

between and

(x,y0) = U/g'(x.y01O).

. CONSIDERATIONS AOUT SOME SPECIAL TOPICS

.1 On Decay of the Local Components in Wave PatternS

The error in wave analysis due to con--tamiriation of records through local-wave

components has been investigated along two lines, both reported upon in ESW:

(i) tiumerical experiments (recently ex-tended'by Lee

_(19691).

(ii) by comparison of resistance obtained from analysis along different cute (see also Ikehata and Nowaza

_(1967)).

The numerical evaluation of local rIow components is admittedly tedeous, different Froude numbers require separate

calculations. Not' much has been'done along another line, i.e. calculation of the

amoupt by which the differential operator applied tO

_.f

leads'to non-zero values.

Apljed to potentials (2),

₍₃₎

and (5),

10

-this functional iS a mapping on zero even

for z unequal to 0, thus

_xcf'z

clearly

isa measure for local-flow intensity.

-Calculations are easy for that rcncratod

by source distribution, as fora single

source potential 0, singular like inverse distance 1/r, this operator leads to a rational function, dependent on speed only

through a linear factor. We have _L23]

G+kG

= (hr

_h/ri) +. k0(1/r-h/r1)5

where r1 is the distance from the image point with regard to z:O.

The right hand side falls off like y. (the first term even like y) sidewise,

and in. ]engthwise direction the decay is

like x3 . For distibuted sources, inte-.

grating (29) through some quadrature for-'mula will display the same rates of decay.

A very simple check on. local-wave decay can be performed just' by inspection, at least if the ship is symmetric to the main section. In this case linear theory predicts a fore and aft symmetry of the

local wave elevation. This means that the local wave profile along a transverse cut must equal the total wave elevation alone the image cut ahead of the ship, where no

free wvos occur. And the sidewise decay

along'the cut must be as well equal that of the total waves on the image cut.

-It jtt that for real ships part of

their form is distributed in antimetry to the main seàtion and the influence' of Froudé number on source distribution will spoil such symmetry anyway - but this

meansonly a slight modification to above

'rule.

More elaborate numerical investiga-tions of local-wave influence to trans-verse cut wave analysis were reported by Landweber and Tsou

(1968).

3.2 Determination of "Eauivalent

Sinulari-ty Systems" from Wave Analysis Mesults Classical methods for deriving wave flow, and-wave resistance for a given hull form have one common feature: As a first step at least implicitely - a system of singularities (sources and sinks in gene-ral) has to be determined which has to stand-on the ship's stead with regard to her wave making. Considering 'shOrtcomings of wave resistance theories, it may be 'found tempting to try determining these

singularities from observed patterns rather than from the ship geometry data. Under thin-ship assumptions1 such singularities depend on speed only -through a linear fac'

11

-tor and are not affected by 'change in water depth.

One merit of such investiGations is that tney show up limits of the range where

the underlying hypotheses are sound (Eve-rest and Hogben (1968)) and where not. In particular. they may display invalidity of

linear wave models (without distortion) it they lead to singularities outside (ahead of) the volume displaced by the ship

(Hog-ben (1970)). Hog(Hog-ben evaluated free waves behind the ship for determining of equiva-lent source arrays along some line in lengthwise direction. In' a broader sense we should also mention here measurements of waves (and flow) alongside the ship using more-general' wave model (1) and allowing greater freedom for the source distribution, namely determining sources all' over the ship's center plane. This is the approach of Zion, Inui and Kajitani

(1972) and' Inui (j974)..

Another motive for determining source distributiOns from wave pattern is to by-pass- the need to avoid local-flow effects in the region where measurements are taken (Sabuncu (1969) for transverse cuts, Zaba (1973) for longitudinal cuts), and thus

measure waves closer to the ship. For

truricated,longitudinal' cuts this may lead to some analytic continuation of the wave 'profile to infinity (Ikehata and Nowaza '(1968)) or of the spectral function towards

o = o (Bessho (1969) derived therefrom).

With locus of dingularities prescribed

along a given line, their intensities may certainly be determined 'uniquely by

postulating least square fit to a given set of data. Thus Sabuncu found that the wave profile of 'one single transverse cut is

needed for determining equivalent (anti-symmetric) sources', along the axis 'of a submerged slender body Of revolution.

However, we have seen earlier that even with local waves disregarded there is

no redundancy left if we have measured two

transverse 'cuts or one longitudinal cut.

II waves are Polluted by local flow

compo-merits, one should therefore expect that more data must be at hand, at.least if.we

cou).d not' anticipate that wave spectra

should be smooth and welibehaved.

In case of a ship wave pattern, there appears some arbitrariness in selecting

z0,

the depth of submergence for singularities.

}iowever, a governs the rate of exponential decay of F?u). 0(u) with large u, and from 'linear ship wave theory', no such decay can be expected for a floating ship. It may be argued, nevertheless-, that the ultimate range of divergent waves affected is

un-portant'fOr, wave analysis anyway.

A more fundamental caveat against such

equivalent source arrays along one line nay' be based on postulates of one-to-one con-respondance. 3essho £2 haa shown that one and the same far-field wave pattern may

re-sult from very different systems of 3ingu laritics. This is already obvious if we

'recall that a wave pattern (1) already

under the weak restriction J(u, s(u))=O does not display .any far-field waves, but:

by no means needs to be small. - It is evi-dent that promoters of 'equivalent source array' concepts are as well aware of all 'limitationS mentioned so far, we want' to

make sure that their followers do.

ACKNOWLEDGEMENT

This paper was completed within the research programm of the Sonderforschungs-bereich 98, Schiftstechnik urid Schiffbau, sponsored by the Deutsche ?orschungsgemein-schaft. The author feels happy to let know that Mr. Hang Shoori Choi gave valuable assistance in composing and editing this

.paper. ' '

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APPENDIX

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Determina-.tion of Wave-making Resistance or a Ship by

the Method of Wave Analysis, I, II.

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1969, pp35-43

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Jubilee Meeting, NSMB, Wageningen, 1972,

pp32-45

. . .

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Of Eggers' Method for the.Determination of

Wavernaking Resistance,

J.Ship Res., Vol.

12, 1968, pp213-230

.

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Longitudi-nal Wave Profiles and the Wave Resistance

of a Shipwith A Proposed Local-Disturbance

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Non-linear .Theory of Wave Propagation in

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