A Simple Green Function for Diffraction-Radiationof Time-Harmonic Waves with Forward Speed

FYancis Noblesse, David Taylor Model BasinlChi Yang, George Mason University2

1 Introduction

Prediction of the behavior of ships and offshore structures in time-harmonic ambient waves is a coreissue in free-surface hydrodynamics. For offshore structures, robust and highly-efficient panel methodshave been developed, and are routinely used, to solve the canonical wave diffraction-radiation problemsassociated with the definition of added-mass and wave-damping coefficients, and wave-exciting forcesand moments. These potential-flow methods are based on numerical solution of a boundary-integralequation formulated using the Green function that satisfies the linear free-surface boundary conditionfor difiraction-radiation of time-harmonic waves without forward speed. Application of this classicalapproach, often identified as the free-surface Green-function method, to wave diffractiorr-radiationby ships (i.e. with forward speed) has also led to useful methods see e.g. Diebold (2003), Boi,n etal. (2002,2000), Chen et al. (2000), Gui,lbaud et al. (2000), Fang (2000), Wans et al. (1999), Duet al. (2000,1999), Iwashi,ta and lto (1995), Iwashita (1997) - although not to a comparable degreeof practicality because forward speed introduces major difficulties (not present for wave diffraction-radiation at zero forward speed).

A basic difficulty is that the Green function satisfying the linear free-surface boundary conditionfor diffraction-radiation of time-harmonic waves (frequency u) with forward speed U is considerablymore complicated than the Green functions corresponding to the special cases U - 0 or o : 0, whichcan be evaluated relatively simply and efficiently, at least in deep water, e.g. Poni,zy et al. (1991. Anumber of free-surface Green functions, based on alternative mathematical representations, have beenproposed and used in the literature on wave diffraction-radiation with forward speed. Briefly, two maintypes of free-surface Green functions have been used: (i) Green functions defined by single Fourierintegrals that involve relatively complicated special functions (related to the complex exponentialintegral) of a complex argument, and (ii) Green functions expressed as single Fourier integrals alonga steepest-descent integration-path (that must be determined numerically) in the complex Fourierplane (Bessho's method). These free-surface Green functions, and related singularity distributionsover flat rectangular or triangular panels, have been considered in numerous studies, and relativelyefficient numerical-evaluation methods have been developed, e.g. Maury (2000), Chen (1999), Boinet aI. (1999), Brument and Delhommeau (1997), Ba and Gui,lbaud (1995), Iwashi,ta and Ohkusu(1992), Bougis and Coudray (1991), Jankowski (1990), Hoff (1990), Wu and Eatock Taylor (1987),Gueuel and Bougi,s (1952), Ingli,s and Price (1982), Kobayashi, (1981), Bessho (1977), Wehausen andLai,tone (1960). Nevertheless, Green functions that satisfy the free-surface boundary condition forwave diffraction-radiation with forward speed are relatively complicated building blocks that are notnecessarily best suited for practical applications.

These free-surface Green functions are commonly expressed in the form

4 n G - - T l r + R + W + L ( 1 )

Ilr is the fundamental free-space Green function, R stands for an elementary Rankine source - asgiven by Eq.(9) and the components W and L account for free-surface effects. The component Iztrl,dominant in the farfield, accounts for the waves included in the Green function G, and is definedby one-dimensional Fourier superpositions of elementary waves (i.e. single Fourier integrals). Thecomponent L accounts for nearfield free-surface effects and is defined by a singular double Fourier

tCode 52 NSWC-CD, 9500 MacArthur Blvd, West Bethesda MD 20817-5700, NoblesseFl@nswccd.navy.mil2School of Computational Sciences, Fairfax, VL 22030-4444

Schiffstechnik 8d.51 - 2004/Ship Tech nology Research Vol. 51 - 2004 35

integral, which can be expressed in terms of single Fourier integrals involving the complex exponentialintegral or related special functions. Alternative mathematical expressions for the wave componentW and the local-flow component L are given in the literature; e.g. in the studies of free-surface Greenfunctions listed in the previous paragraph, Noblesse and Yang (1996), Noblesse (1951). The mostcomplicated of the three components in the decomposition (1) is the local-flow component L.

The Green function (1) satisfies the Michell linear free-surface boundary condition, Eq.(3), every-where, i.e. in both the farfield (where the linearized free-surface condition is valid) and the nearfield(where the lineartzed condition is only an approximation, because of nearfield effects). Thus, a naturalalternative to the Green function (1) is a Green function that satisfies the Michell linear free-surfaceboundary condition accurately in the farfield, but only approximately in the nearfield. Such a Greenfunction cannot be obtained by simply ignoring the local-flow component L in the alternative mathe-matical representations (1) given in the literatur€, ffi can be seen e.g. from the representation (82)-(84)of the Green function for steady flows. Specifically, the wave component W and the local-flow com-p-onent L in this representation of the steady-flow Green function C€;d), where fr - (*,A,2) anrl€ - (€,Tl,O respectively stand for the singularity point and the field point (where the flow is ob-served), involve the sign function sign(( - r) and the absolute value lt- " l, respectively. Thus, thewave component W and the local component L do not satisfy the Laplace equation (although the sumW + tr does), and Eq.(t) with L - 0 therefore does not yield a satisfactory Green function.

We obtain here a Green function that satisfies the Laplace equation, more precisely the Poissonequation (2), the radiation condition, and the Michell linear free-surface boundary condition (3) inthe farfield - and approximately in the nearfield - by extending the analysis of Noblesse (2001). TheGreen function given here is expressed as the sum of a local-flow component defined by four elementaryRankine sources and three wave corrponents, which represent distinct wave systems generated by apulsating source advancing at constant speed. These wave components are given by one-dimensionalFourier integrals with limits of integration that are independent of the field point, and continuous inte-grands that only involve elementary functions of real arguments. Thus, the Green function given belowis considerably simpler than the free-surface Green functions that have been used in the literature onwave diffraction-radiation with forward speed. The four elementary Rankine sources in this "simple"

Green function account for the dominant terms in both the nearfield and farfield asyrnptotic approxi-mations to the non-oscillatory local-flow component associated with the Michell free-surface boundarycondition (which thus is satisfied approximately in the nearfield and accurately in the farfield).

2 Basic Rankine-Fourier representation

The Green function G(i;r-) associated with diffraction-radiation of time-harmonic waves with {br-ward speed is considered here. This Green function, which represents the velocity potential of theflow created at a field point d: ({, q,e) by a moving pulsating source located at a singular pointfr: (r,U,z), vanishes in the farfield limit ll d- frll + oo and satisfies the Poisson equation

Gee * Gq, * Gee :6({ - ")a(rt - a)6G - r)

and the Michell linear free-surface boundary condition

Ge r F|G* - f 'G * i ,2rGq - e(F,Gs * i f G) - g

at ( - 0; e.g. Noblesse (2001). Here, 0 < e ( 1, and the Froude number .Fi, the nondimensional wavefrequency f, and r are defined as

Fn:u lJsL f -,1fi1s r : F n f - U w l g

g is the acceleration of gravity, U is the ship speed, L is a reference length (typically the ship length),and c..' is the frequency of the waves encountered by the ship. If the reference length is chosen asL -Ulu, expressions (4) yield F3: r and f' : r, and the free-surface condition (3) becomes

Gelr+ (0e + i lzG - e(4* z)G - o

(2)

(3)

(4)

(5)

36 Schiffstechnik Bd.51 - 2004lshap Technology Research Vol. 51 - 2004

This boundary condition only involves 0 < e ( 1 and the parameter r:Uulg.

The coordinates iand fr arc nondimensional in terms of the reference length tr. The Green functionG is nondimensional with respect to a reference potential U L, where the reference velocity [/ may bechosen as JgL. Alternative choices for (J are aL and the ship speed U. The flow is observed from aCartesian system of coordinates that moves with speed U along the path of the ship. The r-axis ischosen along the ship path and points toward the ship bow. The z-axis is vertical and points upward,and the mean free surface is taken as the plane z - 0.

DefineX - { - r , - t f f a TY - q - y r * : l F l @ #Z : e - z r F : WZ * : C + , r I : Wh - JT'TP rFf :

The horizontal coordinates X and Y. and the Fourier variables a and 6 used further on. may beexpressed in the polar forms

(X,Y) - h(cos 0,s in?) (o, 0) : k(cos 7, s inT)

w i th - r 10 S r and -z - 11 I n .

The Green function can be expressed as

4 n G - - L l r + R - F R ( S )

R is defined in terms of elementary Rankine sources and FR is given by u two-dimensional Fouriersuperposition of elementary waves. In deep water, the Rankine component R in the Rankine-Fourierdecomposition (8) can be chosen as the single Rankine source

R - Ilr*

The Fourier component FR associated with this Rankine component is given by

(6)

(7)

(e)

FR: r io, l T ouT o**y'!e--++o 7r J L tt *'ieD1

-oo -oo

(10)

( 1 1 )

e.g. Noblesse (2001). k - JFT@ is the wave number related to the Fourier variables a and p. Ois defined as

Q - X o - + Y P : h k c o s ( 7 - d )

The dispersion functions D and D1 are given by

D - k - (P,o - f ) ' D t : F " a - f

The elementary wave function expfk( - i("€ + grt)] satisfies the Laplace equation and the free-surfacecondition (3) if the Fourier variables a and p satisfy the dispersion relation D +'ieDr - 0. Eqs.(8)-(12)define the Green function in terms of two elementarv Rankine sources and a two-dimensional Fouriersuperposition of elementary waves.

3 Rankine component

The Rankine-Fourier decomposition (8) may be considered for alternative Rankine components R,notably Rankine components that approximately satisfy the free-surface boundary condition

f r e + F l R u - f t R * i 2 r 0 q - s

SchifFstechnik Bd.5L - 2004lShip Technology Research Vol. 51 - 2004

(r2)

(13)

37

at ( - 0. Here, ii - -Llr -f R. The boundary condition (13), which corresponds to (3), becomes

R<- f 'A t , i 2 r f r .q -o (14)

for wave diffraction-radiation by a ship at low forward speed or an offshore structure in a current, and

€e + r j i lee * i ,2r i4-s

for wave difiraction-radiation by a ship advancing in low-frequency (long)in Noblesse (200,1/ shows that the local Rankine components

(15)

waves. The analysis given

R - I l r *R - - I l r * + 2 l r r

R - L l r * - 2 l r r

(16)

(17)

(18)

(1e)

(20)

(2r)

with r*,rf, rp given by (6), approximately satisfy the conditions (13), (L4), (15), respectively in boththe nearfield limit r* ) 0 and the farfield timit r* -+ oo. Specifically, the foregoing Rankine compo-nents account for the dorninant terms in both the nearfield and farfield asymptotic approximations tothe local-flow courponents associated with the boundary conditions (13)-(15). The more general localRankine component defined by (6) and

R- I l r * -2 l r r *2 l rpy

i s i d e n t i c a l t o ( 1 7 ) i f F , : 0 a n d ( 1 8 ) i f f : 0 . I f F n l 0 a n d f + O , t h e R a n k i n e c o m p o n e n t s ( 1 9 ) a n d(16) are asymptotically equivalent in both the nearfield r* ) 0 and the farfield r* -+ oc. Furthermore,(19) i s iden t i ca l to (16 ) i f Fn :oo o r f _ oo . I f F , - 0and f :0 , theboundarycond i t i on (13)becomes Re :0, and (19) accordingly yields R - -llr*. The Rankine component -If r in (8) satisfiesthe Poisson equation (2), and the Rankine components ,R defined by Eqs.(16)-(19) satisfy the Laplaceequation in the lower half space ( < 0.

4 Fourier cornponent

The Fourier components f'n that correspond to the alternative Rankine components (16)-(19) inthe Rankine-Fourier decomposition (8) of the Green function are given by

- 1 T T A e - i v + z . kF R : l i m r

l a p l a o ^ :e-+*o r J J D+, ieD1

A is a function of the Fourier variables a and 0. Eqs.(9) and (10) yield A - | for the Fourier componentFR related to the Rankine component (16). The amplitude functions A associated with the Rankinecomponents (17) and (18) are given in Noblesse (2001). These amplitude functions correspond to thespecial cases F, :0 and f : 0, respectively, of the amplitude function

A _ r - e-F?*(t _ e-k/ f")Dlk

associated with the Rankine component (19). The dispersion function D in (21) is defined by (12).Eqs.(21) and (12) yield

(22)

(23)

2004

k A l - z k l f , i f f + o l ,l

- t Fik'(i +.or27) ir 'y - s t

as k + o

kA I I l$ - Flkcos2 1) i f F, + 0 It - t f i r ' l k \ r i ; - � o l ' a s k - + o o

38 Schiffstechnik Bd.51 - 2004lship Technology Research Vol. 51 -

Thus, we have AID < Ilk in both the limit k + 0 and the limit k -+ oo. This property, which holds forthe general case F"f + 0 and the special cases f : 0 or Fn: 0, confirms that the Rankine component(19) accounts for the leading term in both the nearfield and farfield asymptotic approxirrtations to thenonoscillatory local-flow components associated with the boundary conditions (13)-(15).

Noblesse (2001)shows that a singular double Fourier integral of the type (20) can be expressed asthe sum of a double Fourier integral L that represents a local-flow component, and a wave componentW that is dominant in the farfield limit h -+ oo and is given by one-dimensional Fourier superpositionsof elementary waves along the dispersion curves associated with the dispersion relation D - 0. Thus,expression (8) for the Green function becomes

4n G - -I lr + /? - W - L (24)

(25)

(26)

Neither the Rarrkine component -Ilr+R nor the local-flow Fourier component L in the decomposition(24) contributes to the waves associated with the Green functionG, which are entirely defined by thewave component W . Accordingly, the amplitude function A in (20) is equal to 1 at a dispersion curveD - 0 for the Fourier components -PE associated with the Rankine components (16)-(19), as canindeed be verified in (2L). A general expression for the wave component W in the Rankine-Fourierdecomposition (2a) is given in the next section.

5 Generic wave component

The singular double Fourier integral (20) is now considered for a generic amplitude function A andgeneric dispersion functions D and D1. This integral, where O is given by (11), can be expressed as

FR : Iime-+*0

A e - i ( @ + i Z - k )

D + ieDl

The analysis of (20) given in Noblesse (2001/ shows that, in the farfield, the singular double Fourierintegral (25) can be approximated by a wave component defined as

1 T ouT o*T r J J

w - -nDo o,[-oo' ffi

e-io+z*k

The summation is performed over all the dispersron curves defined by the dispersion relation D : 0,ds stands for the differential element of arc length of a dispersion curve, and

l l V D l l - J V D . V n w h e r e V D - ( D o , D p )

Do and D p are the derivatives of the dispersion function D with respect to a and p.

The function A in (26) is defined as

A - sign(Dr) * sign(doDo + 60 ng erf(r/tlo)

i - (6o,,5t3) is a field of unit vectors in the Fourier plane (o, 0) that is roughly orthogonaldispersion curve, and erf is the usual error function. Furthermore, o is a positive real functioncontrolsthewidthof thedispersionstr ipsused inNoblesse (2001) for thefarf ieldanalysisof (20),r/ is defined as

{ : k(|opa * 5P pp) with 9 : Q +' iz*k

Eq.(11) yields 9a: X + i,Z.af k and pp - Y + iZ.Plk Thus, 'rl is given by

$ : k(6"x + lPY) + iz*(6"o + tp 0) - xa +Yb * ' iz*c

SchifFstechnik Bd.51 - 2004/Ship Technology Research Vol. 5L - 2004

t o athatand

39

with a - |ok,,b : 6ak and c - 6ou+ |PP. The integrand of the Fourier integral (26) involves afunction of the form

, ( Xa + yb + iZ.c) "_, (x . "+y i l+z_k

\ o /where O stands for the error function, and a,b,c are functions of the Fourier variables a and B. Theforegoing product of two functions, and consequently the wave component (26), satisfies the Laplaceequation if the functions a,b,c satisfy the two conditions a2 + b2 : c2 and aa + Pb - kc. Thesecondi t ionsaresat is f ied i f thef ie ldof uni tvectorsdischosenasd- (o, P) lk . Indeed, th ischoiceyie lds( t r : 2 t r b : / r c - k a n d

4s : Xa +Y p + ' iz*k - Q +, iz*k

byvi r tueof (11) . Fur thermore,wehave 6oDo+6PDp:VD.(" , ,Olk- Dpwhere Dlr is theder ivat iveof the dispersion function D in the radial direction (o, p)lk.

Thus, the function A in (26) becomes

A - sisn(Dr) * sign(Dr) O

with O - erf(O/o * i'Z.klo). The error function erf may be replaced by the simpler hyperbolicfunction tanh. Thus, the function O becomes

(27)

(28)

(2e)

(30)

(31)

O _ tanh (a +iV\ _tanhQ0lo) +is in(zVlo) lcosh(2alo)\ o / 1+cos(2v | " ) lcosh(2Dlo)

with Q - Xa+YP in accordance with (11) and V - Z*k.We have

O = s i s n ( o ) i f 1 < N < l 2 a l o l

This approximation can be used for N = 10. The function O is finite except if

If O : 0, we have

O : 0 a n d c o s ( 2 V f o ) : - 1

O: - ' d tan ) w i th ) . - -Z*k lo

The function f/ : \10+ ^4 lc4) has a maximrm H* - cJ3l414 for ) : \* : clTrla. The functiontanH is finite if H^ l rf 2,, i.e. if C <2nlT3l+. Thus, v in (2g) is modified as

V _z*k

L + (Z.k)n l(C")a

with C <2r l33la x2.756

The error in the Laplace equation associated with expression (30) for V is insignificant if - Z*k <Co. This error is negligible also for sufficiently large values of -Z*k because the exponential flnctionez*k io (26) vanishes as -Z*k + oo, and the wave component W is insignificant in comparison tothe local-flow component -Ilr * R - L in Q\ for large values of - Z*. The value of the exponentialfunction ez*k - e-o\ in (26) for .\ : \^ is smaller than the small positive real number e if 1n(1/e) <oA^ - Co l3tla. This condition yields

Jrl4:o'1le) < co

We then obta in 3 <Co for e -0.1, 6 <Co for e - 0 .01, and 9 lCo for e :0 .001.

If Z* - 0, (28) yields O - tanh(O/o). The variation of the function Qlo that corresponds to oneperiod of the trigonometric function e-iQ in (26) is given by 2rlo. We have

O . e e < l t a n h ( o / " ) l s 1 i f 2 . 6 5 < l a l l "

SchifFstechnik Bd.5 L - 2004lShip Technology Research Vol. 51 - 200440

Thus, if o 12nr12.65 = 2.37n,, the function lOl becomes approximately equal to 1 within n periods

(wave lengths) of the function "-iQ. The choice o - 2.4 ensures that local-flow effects are insignificant

at distances greater than approximately one wave length.

Thus, acceptable choices for o in (28) and Co in (30) are o - 2.5 and Co: 5. The correspondingvalue of C, equal to 2, satisfies the restriction (31).

6 Dispersion functions and dispersion curves

The dispersion functions (12) and the related dispersion curves, defined by the dispersion relation

D - 0, are considered in Noblesse (2001) and lfoblesse and Yang (2003). These studies show that the

dispersion curves, which are symmetric about the axis 0 :0, intersect the axis c - 0 at P - *.f2,

a n d t h e a x i s p - 0 a t a : L k f , a n d a - + k ; w i t h

t " l - t 2 t r l r t , t I , r L 1 / r \ 2h l : -

I l l r l I l tJ l \ v - / - ' t - l ' l k * : ( yq l 4+ "+ r l 2 )2 l F3

The roots kr+ and kj are real for every value of. r. However, the roots ko and k; are real only if

r < I14 . In the spec ia l case r :114 , we have k ; : f lF " - k ; .

If r {I14,, the dispersionrelation D - 0 defines three dispersioncurves located in the regions

(32)

(33)a < - k ; - k ; 1 > o < k { k ! < o

I f r _ I l 4 , t h e d i s p e r s i o n c u r V e S i n t h e r e g i o n s a 1 - k ; a n d - k 1 < �ll4 < T, we have two dispersion curves located in the regions a < kf, anduseful to subdivide these two regions into the three regions

k{ < o. However, itIfis

Thus, the dispersion curves can be decomposed into three branches, identifie<l as the inner branch 1

and the outer branches O- and O+. The outer branch O+ corresponds to the dispersion curve in

the regio" k: ( a in (33) and (34). The outer branch O- corresponds to the dispersion curve in the

reg ion a< -k ; i n (33 ) i f r 1L14 , o r the reg ion o < - f lFn in (34 ) i f L l4<r . F ina l l y , the inner

branch l corresponcls to the dispersion curve in the region -k; I a l kf in (33) if r !l la, or the

region -f lF" I a I kf in (34) if Il4 S r. The dispersion curves in the regions o < -f lFl,., and- f I F" I a 1. k{ in (34) are connected at

a { - f l F " - f l F " < o < k { k I < o

* : - f lF " k : 4 f 2

(34)

(35)

(36)

Tlre wave numbers kr+ and kf defined by (32) and the wave number k : 4f2 in (35) provide reference

wave numbers for the dispersion curves in the five distinct regions (33) and (3a).

The functions sign(D1) and sign(D7r) are constant along the three dispersion curves I and O+

related to the regions (33)-(34), arrd equal to

I s i g n ( D r ) : - l

O - s i g n ( D 1 ) : - 1g+ s ign(D1) : 1

sign(D6) - |sign(D/,) - -1

sign(D6) - -1

The property that sign(D1) and sign(D6) are constant along the three dispersion curves I and O-justifies the subdivisionof the dispersion curves for Il4 ( r into the three branches (34).

At a dispersion curve (defined by D - 0), expressions (12) and the relation k - 6+F define

the Fourier variables a and B in terms of the wave number k as

a: f lF , *s ign(D){E lF" P -+Jk2 - l f lF"*s ign(D){n lP, l 'r v L U(37)

4L

16r ' 2 - I f l F "

SchifFstechnik Bd.51 - 2004lShip Technology Research Vol. 51 - 2004

If the wave number k is expressed in terms of a parameter f, expressions (37) define the dispersioncurves in terrrs of parametric equations

k : k( t) a - a( t ) 0 : 0(t)

Specific parametric representations of the dispersion curves are given below.

7 Wave components

At a dispersion curve, the vectors VD : (Do,Dil and dl : (da,d,p) are orthogonal. Thus,

J= _ )yl, _ kldol _ r lo' , ,1 dtl l v D i l - w - '

v J l - *

l p l * "

w - -oLo I

, * -W( s ignDl * s ignD *@)Ae- iF+z*k

Here, expression (12) for the dispersion function D was used, and at stands for the derivative of thefunction a(f) in the parametric Eqs.(3s). Thus, (20) and (27) yield

(38)

(3e)

(40)

(42)

where O - Xa * Y p in accordance with (11). Furthermore, signDl and signDp are given by (36),and the function O is defined by (28)-(31).

The regions (33)-(34) and the related branches 1 and O+ of the dispersion curves yield distinctcontributions to the Fourier representation (a0) of the wave corrponent W. Specifically, the wavecomponent W may be expressed as

w - i , (w i + w- - w+) @r)

where the components Wi and, Wr are associated with the inner branch .I and the outer branchesO+, respectively, of the dispersion curves. In the special case r - 0, the decomposition (a1) of thewave component W can be expressed as

W - i W ' _ W O

where Wo - i'(W+ - W-) accounts for the contribution of the two outer branches O*.

8 Wave component lV+

The wave component W+ is associated with the outer dispersiorr curve O+ located in theregion k: < a in (33) and (31). The reference wave number for this dispersion curve is takendefined by (32) as

k! : (\FU +, + rl2)2 lFiThus, k[ : I|FS for r - 0 and k[ - flF"as r -+ oo. The waveO+ is expressed as

K : KI JTTP

outer

as kj

(43)

number k along the dispersion curve

with - oop - (sig" t) kIJTTF:E with

a - L *( t + t 2 ) r l 4 - r

J T F + r + r l 2trq.(11) then yields

(sign qYh + t, - "r)a : k [ ( x " +

SchifFstechnik

(44)

a : kfa and

(45)

(46)

Vol. 51 - 200442 8d.51 - 2004lship Technology Research

Eqs.(40) and (36), and the foregoing parametric equations of the outer dispersion curve O+ show that

the wave component W+ in (a1) is defined as

147+_L+ \nJE f o ,AF,i J

(1 - o) l t l Aez.k- ia

The function O is given by (28)-(31) with (46) and

(1 + 121t1+uf 1P -P

(44). Equation (45) yields

Itlltn + t' - "? - r/t * Llttr + 4, as t -+ 0

f ( o o .Thus, the integrand of @7) is continuous for -oo (

9 Wave cornponent W- for r < Ll4

If r I Il4, the wave component W- is associated with the outer dispersion curve O- located in

the outer region a < -k; in (33). The reference wave number for this dispersion curve is taken as k;

defined by (32) as

(47)

(48)

(51)

(53)

(54)

k o : ( \ F + r l D 2 l F y

T h u s , k ; - l l F l f o r r - 0 a n d k ; - I | ( 4 F Z ) : f l F " - 4 f ' f o r r : L 1 4 . T h e w a v ethe dispersion curve O- is expressed as

k: k i \n +t ' � (50)

with -oo { f ( oo . Expressions (37) and (36) then yield the parametric representation a - -ki a

and, B - (sign t) k; \trTF -ffi with

a - I *( t + t 2 ) r l 4 - r

\Frc- +Ll2

Equation (11) then yields

Q: k i ( - t "* (s ign t )Y, 'n +t ' - " '

Eqs.(a0) and (36), and the foregoing parametric equations of the outer dispersion curve O- show

the wave component W- in (41) is defined as

w_ _r+\E= r4F37 , , (1 + o ) l r l 4 "Z"k - iot r t ' t r -

I -" (r + t2\Lt4rETP=i( 1

r4q\\ ̂ " , ,

number k along

(52)

that

+ t2\Lt IETP=;Z)-oo

The function O is given by (28)-(31) with (52) and (50). Equation (51) yields

I t l l t t r + t t - " ' - @ a s r - + o

Thus, the integrand of (53) is continuous for -oo ( I ( oo 1f r 1L14.

10 Wave component W- for 7f 4 < r

If ll4 I r , the wave component W- isthe outer region " 3 - f I Fn ir (34). Thegreater than 4f2,, in accordance with (35)expressed as

associated with the outer dispersion curve O- located inwave number k along this dispersion curve is equal to orThus, 4f2 is chosen as reference wave number, and k is

(55)

43SchifFstechnik Bd.51 - 2004lShap Technology Research Vol. 51 - 2004

a - -4f2a and

Eq.(11) then yields

Q : 4fz (- t"* (s ign DY\n + t2 - a2

Eqs.(a0) and (36), and the foregoing parametric equations of the outer dispersion curve O- show thatthe wave component W- in (a1) is defined as

W- nL n +t2)1t4frTP=;2

(58)

with -oo ( f { oc. trqs.(37) and

B - (sign I af2/1 a[P with(36) then yield the parametric representation

a - [ 2 ( r + t ' ) r l 4 - l l @ r ) (56)

(57)

t-. n

LIl4

ot

+( 1

1 -

t l A e z - k - i odt

The function O is given by (2s)-(31) with (57) and (55). Eq.(56) yields

I tV\n+P - " , - l r l l \F-Ug,y as r + oThus, the integrand of (58) is continuous for -oo <, < oo if Ll4 < r.

(5e)

11 Wave component Wx for If 4 < r

If I l4 ( z, the wave component Wi is associated with the inner dispersion curve 1 located in theinner region -f lF" ( a ( k{ in (34), and we have n{ S k < 4f2. The reference wave number fbrthis dispersion curve is taken as k{ defined by (32) as

k { : f ' t f [ t ++ r+L l2 )2

T h u s , k f - - 4 f ' 1 6 [ 2 + D 2 = 0 . 6 8 6 3 f 2 f o r r - r l | a n d k o + - f l F n a s r - + o o .along the dispersion curve 1 is expressed as

k :k f \ n+ t , (61)

(62)

then yield

w i t h - t t l t l t 6 a n d

T h u s , t o - t l 6 / Z + L ) 4 - 1 = 5 . 7 4 2 f o r r : I l 4 a n d . t i - 4 r a s r - + o o . E q s . ( 3 2 ) a n d ( 3 6 )the parametric representation cy - kf, a and p - (sign t) kf \/TTP -A with

a - I -( t + t 2 ) l l 4 - L

(63)

(64)

that

\ f r I [+r-r l2

(60)

The wave number k

Eq.(11) then yields

o - k{ (*"f (sisn i lY\n +t, - "r)

Eqt.(40) and (36), and the foregoing parametric equations of the inner dispersion curve / showthe wave component Wi in (a1) is defined as

wi: f lF , t i

or 0-" ) l r lo" t . r - .o*1 a 'f aE !,0*" Q + t2)tl4fiT t'fz=;t

The function O is given by (28)-(31) with (6a) and (61). Equation (63) yields

Itll,,/L + p - ", ^, {t - Ll\nTC as / -+ o

(65)

(66)

t t : \ / ( t + t / t + +r)a - t

44 SchifFstechnik Bd.SL - 2004lShip Technology Research Vol. 51 - 2004

Thus, the integrand of (65) is continuous at f : 0. The integrand of (65) is also continuous at t - Ltt

1 f L l4 < r .

L2 'Wave

component Wi for r < Ll4

If r .,.-L14, the wave componentWi is associated with the inner dispersioncurve l located in the

inner region -ki 1a 1kf, in (33), and a can be expressed as a - f2a with

and -zr < t < n. Here, K.

K { :

Thus , K ! - l f o r r : 0 ,

Eqs.(40) and (36), and thethe wave component Wi in

_ _ K d - K ; , K : + K ia _ _!

2____-!_ + _Ln____!_ cos f

kf lf' are defined by (32) as

\@+r lz)2 K, : r l ( \F1+ - "+

equations

- ra)4

of the

( 1 - O ) l s i n l l 4 " Z " k - i Q

(70)

dispersion curve .I show that

- _i -

r l (- - l

K !

rl2)'

Lf 4, and

(67)

(68)

(71)

- 4l U2 + I)' = 0.6863 and Ko - 4 for r -

4 l ( t / r + D ' < K : < 1 < K ; < 4

The dispersion relation D - 0, (12), and the relation a : f2a show that the wave nunrber k alongthe inner dispersion curve ,I is given by

k : f2(r - ra)2 (69)

T h u s , t h i s d i s p e r s i o n c u r V e i s d e f i n e d b y o - f 2 a a n d p _ ( s i g n D f 2 f f i w i t h _ r 3 t 3 n .Equation (11) then yields

/e : f 2 ( X " * ( s i g n" \

foregoing parametric(a1) is defined as

wi : r rK{ tN; i o ,u 2 J-'|f

with (70) and (69). Eq.(67) yields

1 - 0 , ) w i t h { r o : ( r + K ; l K { ) ( L l 2 + ' r [ p ] g 2 )t -+ t,r J L rn: (1 + K{ lK;)(rlz - " lK;) )

Thus, the integrand of (71) is continuous at t - 0, and also at

-)/

inner

The function O is

l s i n l l

Eqs. (68) yieldt - t r i f r <

(28)-(31)

Ll1/fl asIlffl as

-+ Lla.

grvenL b y

f, l

a s 1T n - > 0r l4 -

T

13 Wave component W' for r - 0

The wave conrponent Wi in (42) corresponds to the limit r -+ 0 of the wave component Wi given

by (67)-(7t). This wave component is associated with the inner dispersion curve 1 located in the innerregion in (33). In the special case r - 0, trqs.(67)-(69) yield Kf,:1, a: cosf and k : f2, and thedispersion curve 1is the circle (o,g) - f2(cosf,sint) with -r 3t , . .-r . Eqt.(70) and (71) then yield

,17

wi : f 2 [ a t 6 - o ) 4 "2 *k - i 'Q" ! *

where k : f2 e : f2 (Xcos t+ } ' s in f )

Schiffstechnik Bd.51 - 2004/Ship Technology Research Vol. 51 - 2004

(73)

(74)

45

and O is defined by (28)-(31) with (74). The wave component (73)-(74) corresponds to the limitFn :0 with f + 0, i.e. diffraction-radiation of time-harmonic waves without forward speed.

14 Wave component Wo for r - 0

The wave component Wo : i'(W+ -W-) in @2) is associated with the outer dispersion curves O-and O+ located in the outer regions a < -k; and k[ ( a in (33). Eqs.(43) and (49) show that thereference wave numbers ki for the outer dispersion curves O+ are given bV k* - I lF3 in the specialcase 7 - 0. In accordance with (44) and (50), the wave number k along these dispersion curves isexpressed as

k : r lF l w i t h T_ � \E+ t2and -m < t < oo. -lhe dispersion curves O+ are symmetric about the axes c - 0 and p: 0, and aredefined by o - +'/TlF] andp - (sign t)\tr\/T--I1F3, in agreement wirh (4b) and (51). Eqs.(a7)and (53), (46) and (52), arrd (28) show that the wave component Wo :'i,(W+ -W-) is given by

dt JT! a"z*k rm (1 - o) ,,*TW O :

1 TF3 I

(75)

(76)

(77)

(78)

(7e)

Im stands for the imaginary part, O is given by

. - (t * (sign t)Y\E-) ntr:

and o is the complex conjugate of the function @ defined by (28)-(31) with (77) and (75). Theseexpressions for the wave component wo correspond to the limit f : 0 with Fn * 0, i.e. steady flowabout a ship advancing at constant speed in calm water.

15 Wave filters

The amplitude function A in (a0) and the related Eqs. (47), (53), (bS), (65), (TI),, (23), (26) is equalto 1. The wave components (47), (53), (58), (76) include infinitely short waves, defined in the limitsf -+ *oo. These short waves can be filtered by modifying the function,4. E.g., the function A maybe taken as

. ( 1 i f k < k *a - )" - I " -u(klk--r)2 i f k* < k

The filter function (78) yields A < L0-3 for koo < k with k* lk* equal to 1.10, 1.15 and 1.20 if u istaken approximately equal to 700, 310 and 175, respectively. The amplitude function (78) effectivelyeliminates all waves having a wave length .\ smaller than l* - 2n f k*. The filter function (78) canbe used in (aT), (53), (58) , (76) and also in (65) , (7I), (73). Waves having a wave length ) greaterthan ,\0 - 2n lk0 can be eliminated in a similar fashion.

16 Summary

Eqs.(24), (19) and (41) yield

4 t r G - - L l r * I f r * - Z l r o * 2 f r p y + i ( W + - W - - W i 1 + t

where L - -L- Furthermore, r, r*trFtrpy are given bV (6) and the wave componentsW+ and,Wi aredefined by (+3)-(72) with O given by (28)-(3t). The integrands of the Fourier integrals that define thewave components Wt and W' are continuous and only involve trigonometric and hyperbolic functionsof real arguments. Furthermore, the limits of integration for these Fourier integrals are independentof the coordinates X, Y,, Z*.

The Green function that is obtained if the local-flow component L in (79) is ignored satisfies thePoisson equation (2), the radiation condition, and the Michell linear free-surface boundarv condition

46 SchifFstechnik Bd.Sl - 2004lShip Technology Research Vol. 51 - 2004

(3) in the farfield. This Green function orrly satisfies the free-surface boundary condition (3) approx-

imately in the rrearfield, unlike the related free-surface Green functions used in the literature, which

satisfy the free-surface condition (3) exactly everywhere. The Green function given by (79) with

L - 0 is expressed in terms of four elementary Rankine sources and three Fourier integrals, which

define distinct wave components.

In the special case /:0, i.e. for steady flow, (24), (18) and (a2) yield

4 n G - - I l r * I f r * - 2 l r r + W o + L ( 8 0 )

where the wave component Wo is defined by (75)-(77). In the special case .Flr, : 0, i.e. for diffraction-

radiation without forward speed, (21), (17) and (a2) yield

4 n G - - I l r - I l r * * 2 l r y - i W i + L ( 8 1 )

where the wave component Wi is given by (73)-(74). In (80)-(31), r, r*, rF, ry are given by (6).

TIre Green functions given by (S0)-(S1) with L:0 are defined in terms of three elementary Rankine

sources and a single wave component.

LT Comparison of alternative free-surface Green functions

Alternative free-surface Green functions are now cornpared for the special case / : 0, corresponding

to steady free-surface flow about a ship advancing at constant speed in calm water. The Green function

that satisfies the Michell linear free-surface boundary condition for steady flow everywhere (nearfield

included) is expressed in Noblesse (1981) as

p2

4rFlG - -7 7- signX) / dt ez*k I* "ia +

J

I

? [a,r . l

- l

+ F 3 + 2 OTx

p'2 2.tr2 oo

- n _ - ' , + 2 tr * TF

F : L _ f t * T o r " z . k r ^2 r F J

- x

rm ez 4(z) (82)

(83)

(84)

where k and O are defined as

k : ( r + * ) l F : o : (X +Yt)fi + t2lF2

and r, r*t Xt Y, Z* are given bV (6). Furthermore, Z is defined as

z - Q.Jt - t , +Yt + i lx11/1 - 7z 1Pz

,81 stands for the exponential integral, e.g. Abramowitz and Stegun (1965). The first integral on the

right of (82) is a wave component and the second integral corresponds to a non-oscillatory local flow.

A practical method for evaluating the local-flow component (i.e. the second integral) in (82) is given

in Poni,zy et al. (199/t).

Eq.(80) and a change of variable in (76) yield

pr2

at rFSG - - 'n +r

d,t ez.k Im (1 - 6) "io + FIL

I

( 6 - s i gnx ) " o * + r I a tm"zn11z1n . J- 1

(85)

(86)

47

k and (D are given by (S3), O is the complex conjugate of the function O defined by (28)-(31) with(83), and r, r*, rF , X, Y, Z* are given bV (6). The wave components in (85) and (82) are identical

except that the sign function signX in (82) is replaced by the function 6 in (85). Expression (82)

shows the local-flow component in (85) can be expressed as

SchifFstechnik 8d.51 - 2004/Ship Technology Research Vol. 51 - 2004

The foregoing expressions and (6) show that the Green function flG, and the related. wave andlocal components in (S2) and (85), are functions of the speed-scaled coordinates (X,Y,Z,Z*)lF3 -(€- *,,T-U,e - t,C+ 4lFi Thus, the steady free-surface flow generated by a unit source located attr :0, U :0, z - -5 is a function of the speed-scaled coordinates {lF},rllF3,,e lFi and. subnergencedepth 6lFi .

For purposes of comparison, the steady flow generated by a point source and a point sink advancingat constant speed U in calm water is considered. The source and the sink are slightly submergedbelow the free surface z - 0. The point source is located at (0.5,0, -0.02)tr and the point sink at(-0.5,O, -0.02)L, and the strength q - Ql(UL2) of the source-sink pair is taken equal to 0.001. TheFroude number Fn : U lJgL is chosen equal to 0.3.

The velocity potential - evaluated using the usual Green function (82) and the "simple" Greenfunction (85) - of the flow generated by the source-sink pair at the free-surface plane z - 0 is depictedin the upper and lower halves of Fig.1, respectively. These velocity potentials are compared furtherin Figs-2a,b. Specifically, Fig.2a and Fig.2b depict the velocity potential of the flow generated bythe source-sink pair along four horizontal lirres, with -4 < r I L, in the vertical plane of symmetryU :0 and the free-surface plane z :0, respectively. The lines in Fig.2a are defined by y - 0 andz : 0, -0.05, -0.1, -0.5, and the l ines in Fig.2b by , - 0 and U : 0,0.1,0.5, 1. Figs. l and 2a,b showthat differences between the usual Green function (82) and the simple Green function (85) vanish inthe farfield, as expected, and are relatively moderate in the nearfield.

tlsual Green Function

Simple Green Function

Fig.l: Velocity potential of steady flow generated by u source-sink pair

48 SchifFstechnik Bd.51 - 2004lShip Technology Research Vol. 51 - 2004

0

-0.005

-0.01

0.01

0.005

0

-0.005

-0.01

0.01

0

-0.005

-0.01

0.01

0.005

0

-0.005

-0.01

u s u a l , z - 0 -s imp le ,z=0 " " " - " "

u s u a l , z = - 0 . 1 -s imp le ,z= -0 .1 ' " - " ' ' -

-1

u s u a l , Y = 0 -s i m p l e , y = 0 ' - ' - - - "

u s u a l , Y = 0 . 5 -s imp le ,y=0 .5 " " " " " "

-2

0.01

0.005

0

-0.005

-0.01

0.01

0.00s

0

-0.005

-0.01

0.01

0.005

0

-0.005

-0.01

0.01

0.005

n

-0.m5

-0.01

u s u a l , z = - 0 . 5 -simple, z = - 0.5 ............

-3 -1

Fig.2a: Velocity potential, defined by the usual and simple Green

to a source-s ink pai r a longz _�0, z : -0 .05, z - -0 .L, z :(in vertical plane of symmetry)

functions, of the steady flow due- 0 . 5 ; - 4 < r 1 l , , A : 0

- 2 - ' r 0 1

z - 0 (in free-surface plane)

u s u a l , y = 0 . 1 -s imp le ,y=0 .1 - " " " " "

u s u a l , y = 1 -s imp le ,y=1 " " " " " "

- 4 - 3 - 2 - . 1 0 1 - 4 - 3

Fig.2b: As Fig.2a, but for y : 0, A - 0.1, A - 0.5,, ' ! l : I ; -4 < r I L,

SchifFstechnik Bd.5L - 2004lShip Technology Research Vol. 51 - 2004 49

1-8 Conclusion

The main result obtained in this study is the Green function defined as

4 t r G - - l l r * l f r * - 2 l r r * 2 f r p y + i , ( 1 4 7 + - W - - W o )

where r, r* t rF t rpy are given by (6), and the wave components W+ and, Wi are defined by (43)-(72) and (28)-(31). The Green function (87), associated with a frequency-domain analysis of wavediffraction-radiation by a ship (with forward speed), is expressed as the sum of four elementary Rankinesources and the three wave components Ir7+ and Wi. The four Rankine sources in (87) account for thedominant terms in both the nearfield and farfield asymptotic approximations to the non-oscillatorylocal-flow component associated with the linear free-surface boundary condition (3), which thus issatisfied approximately in the nearfield and accurately in the farfield.

The three wave components in (87) represent distinct wave systems generated by a pulsating sourceadvancing at constant speed. Specifically, the wave component W+ represents a system of inner-Vwaves; the wave componentW- represents a system of outer-V waves if r .,.-I14, or a system of fanwaves if Ll4 I r ; finally, the wave component Wi represents a system of ring waves if r { Il4,or a system of ring-fan waves if Ll4dimensional Fourier superpositions of elementary waves. The integrands of the Fourier integrals thatdefine these wave components are continuous and only involve trigonometric and hyperbolic functionsof real arguments, and the limits of integration for these Fourier integrals are independent of thecoordinates X, Y, Z*- Thus, the Green function (87) is remarkably simple. The Green function (82)becomes

(s7)

4 t r G - - l l r I L f r * -2 l re +Woin the special case / : 0, i.e. for steady free-surface flow about a ship, and

4n G - - I l r - l l r * *2 l r y - iw i

(88)

(se)in the special case Fn: 0, i.e. for wave diffraction-radiation by an offshore structure (without forwardspeed)- The wave components Wo in (8S) and, Wi in (89) are defined by (7b)-(TT) an4 (73) -(74),respectively, and r, T'*, rF t ry are given bV (6). The Green functions (88)-(39) are expresse4 in termsof three elementary Rankine sources and a single wave component.

The Green function (S7) satisfies the Poisson equation (2), the radiation condition, and the Michelllinear free-surface boundary condition (3) in the farfield. Unlike usual free-surface Green functions,which satisfy the free-surface condition (3) exactly everywhere, the Green function (82) only ap-proximately satisfies the free-surface boundary condition (3) in the nearfield. Although the linearfree-surface condition (3) is valid far away from a ship advancing in regular waves (i.e. in the farfield),this boundary condition can only be regarded as an approximation in the vicinity of the ship (i.e.in the nearfield) due to nearfield effects. Thus, it is not a priori evident that a Green function thatexactly satisfies the Michell condition (3) everywhere is necessarily superior to a Green function thatsatisfies the Michell boundary condition only approximately in the nearfield.

The Green function (S7) is considerably simpler than the free-surface Green functions that have beenused in the literature on wave diffraction-radiation by a ship within a frequency-domain analysis. Thefrequency-domain Green function (87) is no more complicated than the Green function associated witha time-domain analysis, which requires a convolution integral to account for time-history (memory)effects, e-g. Bertram (2000), Wehausen and, Lai,tone (1960). The frequency-domain Green function(87) is more complicated, but includes considerably rrore physics, than the elementary Rankine source-Ll, used as Green function in Rankine-source panel methods. These methods require appropriatenumerical-differentiation schemes to enforce the radiation condition (no useful scheme is known forw a V e d i f f r a c t i o n - r a d i a t i o n i n a r e g i m e a p p r o x i m a t e l y d e f i n e d b y t / a l g < � � � � � � � � � � � � � � � � � �Yasulcawa (1996), Bertram (2000).

No distribution of singularities over the free surface is required to compute diffraction-radiation oftime-harmonic waves by u ship within the linear approximation associated with the Michell free-surf'ace

SchifFstechnik Bd.5L - 2004lShip Technology Research Vol. 51 - 2004

boundary condition (3) if a usual free-surface Green function is employed (although a free-surface

distribution of singularities is required to account for nearfield free-surface effects; e.g. if free-surface

linearization about the zero-Froude-nurrber double-body flow is used). However, use of a simple Green

function like (87) requires a distribution of singularities over a nearfield portion of the free surface in

the vicinity of the ship waterline. An important property of the free-surface distribution of singularities

associated with the use of such a simple Green function is ttrat it rapidly vanishes in the farfield (in

practice, at a small distance frorn the ship) because the linearized free-surface boundary condition

becomes exac;t in the farfield. This property is a significant difference with panel methods based on

Rankine sources.

The comparison, reported in the study, between the simple Green function (88) for steady

free-surface flow and the corresponding usual Green function (that satisfies the Michell free-surface

boundary condition everywhere) shows that differences between these alternative Green functions

vanish in the farfield, as expected, and are relatively moderate in the nearfield. A similar comparison

between the simple Green function (87) for wave diffraction-radiation with forward speed and the

corresponding Green functions given in the literature would be interesting and useful. The analysis

used in the present study to obtain the Green functions (87) and (88)-(89) for deep water can be

extended to the more general case of uniform finite water depth, and this extension to finite water

depth will be given elsewhere.

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52 SchifFstechnik Bd.5L - 2004lShip Technology Research Vol. 51 - 2004