Evaluation of the wave-resistance green function: Part 3-The single integral near the singular

Journal of Ship Research, Vol. 38, No. 1, IVIarch 1994, pp. 1 - 8

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M&Mmg 2, 2628 CD Dai .

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E¥alys]tion of the Wave-Resistance Green Function: Part 3—

Tlie SiogI© Sntegral Near the Singular Axis

J.-M. Clarisse^ and J . N. Newman^

The single integral part of the waveresistance Green function is considered near its singular axis. C o m -pleting the work of Ursell (1988) an asymptotic expansion is proposed that is valid in the vicinity of the source when this lies on or near the free surface. For the complementary region away from the singular axis an Improvement of Bessho's (1964) convergent series is also obtained. Numerically, both expres-sions along with previous results guarantee fast and accurate evaluations of the single Integral within two wavelengths downstream of the source. For larger distances downstream, the result of Ursell (1988) Is shown to be still valid within bounded distances of the singular axis. Hence the single integral can be evaluated analytically or numerically in a whole neighborhood of its singular axis.

1. I n t r o d u c t i o n

T H E P O T E N T I A L of a submerged source m o v i n g w i t h con-s t a n t h o r i z o n t a l velocity beneath the free con-surface i n a f l u i d of i n f i n i t e depth is f u n d a m e n t a l to the analysis of the flow past a ship. T h i s p o t e n t i a l has been decomposed i n P a r t 1 ( N e w m a n 1987a) as the s u m of (a) the R a n k i n e source and its image above the free surface, (b) a double i n t e g r a l w h i c h represents a s y m m e t r i c a l n o n r a d i a t i n g f i e l d , and (c) a single i n t e g r a l w h i c h is included only f o r field points downstream o f t h e source. E f f i c i e n t n u m e r i c a l schemes f o r the e v a l u a t i o n of t h e double i n t e g r a l were proposed i n N e w m a n (1987a) w h i c h reduce the computational b u r d e n of t h i s component to a r e l a t i v e l y simple n u m e r i c a l task. A comparable r e s u l t for the single i n t e g r a l p a r t is, however, more d i f f i c u l t to ob-t a i n due ob-to iob-ts singular behavior. I n P a r ob-t 2 ( N e w m a n 1987b) t h i s single i n t e g r a l was treated w h e n both the source and the field points are i n the same l o n g i t u d i n a l "centerplane." I n t h i s s i t u a t i o n the single i n t e g r a l is regular f o r a l l v e r t i c a l positions of the source and the field point, i n c l u d i n g the l i m i t on t h e free surface, thus r e n d e r i n g the analysis m u c h s i m -pler. I n the present w o r k we do not consider such restric-tions.

F o r any position of the p a i r source-field points, the use of N e u m a n n series as proposed by Bessho [1964, equations (4.2) and (4.3)1 was shown by B a a r & Price (1988) to be compu-t a compu-t i o n a l l y more e f f i c i e n compu-t compu-t h a n direccompu-t n u m e r i c a l i n compu-t e g r a compu-t i o n . U n f o r t u n a t e l y , one of Bessho's [1964, equation (4.3)] se-r i e s — t h e asymptotic sese-ries—is incomplete and does not ac-count f o r the singular behavior of the single i n t e g r a l along the t r a c k of the source p o i n t w h e n t h i s is on the free surface. T h i s defect was corrected by U r s e l l (1988) who proved t h a t by adding an i n t e g r a l t e r m , the o r i g i n a l series of Bessho was, indeed, asymptotic i n a bounded region away fi-om the source. Subsequently N e w m a n proposed a n a l g o r i t h m f o r the eval-u a t i o n of t h i s a d d i t i o n a l t e r m w h i c h proved to be effective

' C E A , Centre d'Etudes de Limeil-Valenpon, Villeneuve St. Georges, France.

^Department of Ocean Engineering, M . I . T . , Cambridge, Mass. Manuscript received at S N A M E headquarters December 8, 1992.

i n t h i s region (see N e w m a n 1988). However, the r e s u l t of U r s e l l (1988) cannot be used w h e n the field p o i n t approaches t h e source p o i n t , the l a t t e r being on the free surface. F u r -t h e r m o r e , i n i -t s s-ta-temen-t, -t h i s resul-t is also l i m i -t e d -to a bounded r e g i o n downstream of t h e source.

The purpose of t h i s paper is to present a n a l y t i c a l expres-sions f o r the single i n t e g r a l w h i c h :

— I n the same s p i r i t as P a r t 1 and P a r t 2, f u r n i s h robust and efficient means for i t s n u m e r i c a l e v a l u a t i o n . ( I n practice t h i s translates to the same demand of a u n i f o r m absolute accuracy of six decimals.)

—Complete or extend the r e s u l t of U r s e l l (1988) so t h a t t h e single i n t e g r a l can be evaluated anywhere i n the v i c i n -i t y o f -i t s s -i n g u l a r ax-is.

— F o l l o w as m u c h as possible the w o r k s of Bessho (1964) a n d U r s e l l (1988) i n p r o v i d i n g expressions w h i c h can be generalized to evaluate derivatives or i n t e g r a l s of the single i n t e g r a l .

For the t r e a t m e n t of the single i n t e g r a l i n the v i c i n i t y of t h e source w h e n t h i s lies on or near the free surface, we gen-eralize the idea of u s i n g Dawson's i n t e g r a l ( N e w m a n 1987b) to any position of the source and field points. T h u s a new asymptotic expansion w h i c h extends the v a l i d i t y of UrseU's r e s u l t to the neighborhood of t h e source is proposed, along w i t h a modification of Bessfio's convergent series f o r s m a l l distances downstream o f t h e source. The proof of the v a l i d i t y of t h i s new asymptotic expansion relies o n two results sim-i l a r to t h a t of U r s e l l (1988) b u t f o r : (1) a f u n c t sim-i o n d d sim-i r e c t l y r e l a t e d to t h e complex Dawson i n t e g r a l , (2) another f u n c t i o n specifically chosen i n r e l a t i o n w i t h the d e f i n i t i o n s of d and t h e single i n t e g r a l . The i n t r o d u c t i o n of t h i s a d d i t i o n a l func-t i o n is j u s func-t i f i e d by func-the a b i l i func-t y , c o n func-t r a r y func-to simpler expressions, to define recursions w h i c h are w e l l behaved i n the v i -c i n i t y of the sour-ce. The m o d i f i -c a t i o n of Bessho's -convergent series is a direct extension of the r e s u l t of P a r t 2 [ N e w m a n 1987b, equations (13) and (14)]. A g a i n t h e use of t h i s new series allows t h e d e f i n i t i o n of w e l l behaved sequences, a fea-t u r e nofea-t p e r m i fea-t fea-t e d by fea-the direcfea-t use of Bessho's convergenfea-t series f o r s m a l l distances downstream o f t h e o r i g i n . N u m e r -i c a l compar-isons of these new express-ions w -i t h UrseU's

asymptotic series ( U r s e l l 1988) and Bessho's convergent se-ries show agi-eement w i t h i n six decimals as w e l l as good ef-ficiency. Hence we c o n f i r m t h e above a n a l y t i c a l results and show t h a t the proposed i m p l e m e n t a t i o n s f u r n i s h robust and e f f i c i e n t means f o r e v a l u a t i n g the single i n t e g r a l up to t w o wavelengths downstream o f t h e source. [ U r s e l l (1989) has proposed a d i f f e r e n t t r e a t m e n t of t h i s r e g i m e based on a con-vergent series expansion of t h e single i n t e g r a l ]

I n the context o f unbounded distances downstream o f the source, we reexamine the proofs given by U r s e l l (1988). W h e n i t appears t h a t w i t h i n a bounded distance f r o m t h e t r a c k of the source, UrseU's r e s u l t s t i l l holds u n i f o r m l y d o w n s t r e a m of t h e source. N u m e r i c a l c o n f i r m a t i o n of t h i s r e s u l t w i l l be presented w h e n d e a l i n g w i t h t h e single i n t e g r a l i n t h e f a r field—the r e m a i n i n g d o m a i n .

2. T h e s i n g u l a r b e h a v i o r

T h e conventions we use f o r t h e d e f i n i t i o n of t h e wave-re-sistance Green f u n c t i o n are t h e same as i n P a r t 1 ( N e w m a n 1987a). I n p a r t i c u l a r , we i-ecall t h a t t h e o r i g i n of t h e Carte-sian coordinate system (x,y,z) is t a k e n at t h e position of the image of the source p o i n t above the f r e e surface, t h e x-axis b e i n g i n the d i r e c t i o n of m o t i o n a n d the z-axis positive downward. (x,y;z) designates t h e coordinates of the field point, n o r m a l i z e d i n t e r m s o f t h e wave n u m b e r g/lP. Here g'is the acceleration of gi-avity, U t h e v e l o c i t y of f o r w a r d m o t i o n . I n a d d i t i o n , R denotes the distance between the image of t h e source and the field points w h i l e p a n d a are defined hy y = p s i n a, z = p cos a such t h a t (x-,p,a) is a system cylindrical coordinates based on t h e »;-axis. W i t h these conventions, t h e Green f u n c t i o n , or t h e p o t e n t i a l due to a source o f s t r e n g t h -4-77, is: G = 1 1 R f 2 + i l i i i -[ I T n / 2 cos (f) ,T/2

-kz+ili\x\ sec <^-i-ky t a n <!, k - cos^ 6 + ie

•dli d^,

- 8m~x) f{x,y,z) (1)

where RQ is the n o r m a l i z e d distance between the source and the field points, a n d - 8 H{~x) fi.x,y,z) is the single i n t e g r a l part. H e r e a f t e r we w i l l c a l l ƒ t h e single integral, f o r w h i c h a d e f i n i t i o n is [Clarisse 1989, e q u a t i o n (1.4)]:

- 2 / 2 f<,x,p,c

exp •<3i

• cosh (2 M - ia) s\n{x cosh u) cosh u du \ (2)

i n t e r m s of the c y l i n d r i c a l coordinates (x,p,a). I n w h a t f o l -lows we w i l l use t h e f a c t t h a t ƒ is a f u n c t i o n even i n a a n d

odd i n X i n considering o n l y positive values of x a n d a.

The a n a l y t i c properties of t h e single i n t e g r a l have been studied by m a n y authors. A m o n g t h e m , U r s e l l (1960) was the first to i d e n t i f y — a t l e a d i n g order—the s i n g u l a r behav-ior of t h i s f u n c t i o n w h e n the source is on t h e free surface and t h e f i e l d p o i n t near i t s t r a c k . U r s e l l also noted t h a t the asymptotic expansion g i v i n g t h i s behavior d i d not r e q u i r e R to be large i n order to be v a l i d , b u t r a t h e r t h a t a c e r t a i n parameter, M = .T^/4p, should be large. I n fact, he gave t h e complete f o r m of t h i s expansion i n his l a t e r proof of Bessho's asymptotic series (Ursell 1988). There, his result (Ursell 1988,

Theorem 2) is g i v e n i n terms o f a c e r t a i n f u n c t i o n F satis-f y i n g the r e l a t i o n :

- z / 2 f{x,p,a) = dF

dx

(.r,p,a)

w h i c h is i n v o l v e d i n the d e f i n i t i o n of the K e l v i n wave-source p o t e n t i a l . W h e n i n t e r m s of ƒ, the same t h e o r e m becomes:

T h e o r e m 1 — S u p p o s e x and p strictly positive and bounded, and 0 < a < T T / 2 . Wiien M = x^/4p is large we liave:

y 2 „ , - i ( % ) ] +

+ ix cosh u

f ;a/2-:-ir

exp

= + i c i / 2

- cosh {2u - ia)

cosh udu \ + 0

e x p [ - M ]

(3) The s u m over m corresponds to Bessho's o r i g i n a l series [Bessho 1964, e q u a t i o n (4.3)] w h i l e the i n t e g r a l is the sup-p l e m e n t a l t e r m i n t r o d u c e d by U r s e l l .

I n n u m e r i c a l computations the Bessel f u n c t i o n s / „ and F „ m a y be computed f o l l o w i n g the principles described i n Baar and Price (1988), or as follows:

—The sequence [7„(p/2)] is evaluated by means of

bacii-ward r e c u r s i o n . T h e s t a r t i n g v a l u e of n = No is chosen such

t h a t /Af„(p/2), obtained f r o m the l e a d i n g asymptotic behavior f o r large orders, is equal to the tolerance o f t h e computer. -'wo+iCp/Z) is t h e n t a k e n to be 0 a n d the r e s u l t i n g q u a n t i t i e s are n o r m a l i z e d by the s u m ZQ - 2/2 + 2/4 — . . . f o r consis-tency.

— [ ^ „ ( x ) ] is computed by means of forward recursion, Yoix) a n d Yi(.x) b e i n g g i v e n by Chebyshev p o l y n o m i a l approxi-m a t i o n s .

The i n t e g r a l t e r m of (3) is evaluated u s i n g N e w m a n ' s a l -g o r i t h m w h i c h i m p l e m e n t s Laplace's m e t h o d as described i n N e w m a n (1988).

The c o m b i n a t i o n of the above a l g o r i t h m s p r o v i d e e f f i c i e n t n u m e r i c a l procedures. I n double a r i t h m e t i c precision, these e v a l u a t i o n s a r e c o m p l e m e n t a r y to t h e r e s u l t s g i v e n b y Bessho's convergent series for t h e range 1 < x £ 16, 0 s a s Tr/2. F o r six decimals absolute accuracy, t h e p a r t i t i o n sur-face between these t w o expressions is comprised between M = 17 a n d M = 2 1 . T h i s accuracy is achieved by s u m m i n g t h e series up to m = 8, or m = 1 i f p is s m a l l e r t h a n lO"^", a n d by t r u n c a t i n g t h e asymptotic expansion of t h e i n t e g r a l at i t s "smallest t e r m " (see the A p p e n d i x , Section F ) , or a f t e r its s i x t e e n t h t e r m , whichever occurs first. W h e n x < 1 or w h e n X > 16, expression (3) is no longer complementary to Bessho's convergent series for t h e desired accuracy. (For a more d e t a i l e d account f o r x < 1, see Clarisse 1989.) B y ex-ample, f r o m (3) i t appears clearly t h a t f o r s m a l l values o f x t h e neglected t e r m o f order 0 ( e x p [ - M ] / V p ) is no more neg-l i g i b neg-l e : t h e c o n s t r a i n t of keeping M c o n s t a n t — o r at neg-least b o u n d e d — i n order to ensure the c o m p l e m e n t a r i t y w i t h Bes-sho's convergent series requires t h a t p 0. We are thus compelled to a special t r e a t m e n t of t h e s m a l l x region.

8. T h e n e i g h b o r h o o d of t h e o r i g i n

As noted i n N e w m a n (1987b) t h e decomposition o f the sin-gle i n t e g r a l u s i n g Dawson's i n t e g r a l is of i n t e r e s t whether for a n a l y t i c a l or n u m e r i c a l considerations. H e r e we extend to the w h o l e range 0 s a < T T / 2 , the idea of e x t r a c t i n g t h e d o m i n a n t singulai-ity at the o r i g i n by s u b t r a c t i n g now a t e r m i n v o l v i n g the complex Dawson integi-al. I n particular, we show

t h a t once t h i s d o m i n a n t s i n g u l a r i t y is subtracted, the re-m a i n d e r is continuous a t the o r i g i n . We proceed i n t w o steps by considering at f i r s t the asymptotics of the single integi-al i n the large M regime, t h e n i t s expansion i n convergent se-ries f o r the complementary region of bounded M.

A new asymptotic expansion

The consideration of f i n the v i c i n i t y of the x-axis, i.e., i n the large M regime, close to the o r i g i n requires two

inter-mediate results. The f i r s t is concerned w i t h the complex

Dawson i n t e g r a l f o r w h i c h we prove a r e s u l t s i m i l a r to t h a t of U r s e l l (1988). L e t d(x,p,a) be the i n t e g r a l : -2/2 I _{exp ^ - e ^ ' " cosh 2u} 2 sm{x s i n h u) cosh u duj (4)

w i t h ^ = pe"'", w h i c h is r e l a t e d to the complex Dawson i n -t e g r a l F -t h r o u g h -t h e r e l a -t i o n :

d e^-F{Q 2i. X

(5)

where ^ = V M e'"^^. W e have t h e n t h e f o l l o w i n g r e s u l t f o r d:

T h e o r e m 2 — S u p p o s e x and p strictly positive and bounded, and 0 < a £ 'n/2. Wiien M is large, we tiave:

exp cos 2u - X s i n u cos u du \ M'-l/2 h \ l e - ' A C,{x) + 2 Im(-e-''^][C^,„^,{x) 2 / m = 1 \2 — ix s i n h u m=l =+ici/2 exp + l a/ 2 - i T r cosh u du + ^ / e x p [ -m cosh 2u -z/2 ₍₆₎

witfi M' = - x^/i and wliere tiie sequence (CJ is de-fined as in Section A of tiie Appendix.

The f o r m of t h i s r e s u l t is v e r y s i m i l a r to T h e o r e m 1 i n the presence of the Bessel f u n c t i o n s ƒ„,, and of neglected t e r m s of the same order. H o w e v e r differences appear i n the depen-dencies of the series i n t h e variables (p,a), and i n t h e func-tions C2m±i r e p l a c i n g t h e Bessel f u n c t i o n s Y2m±i- T h i s re-m a r k leads us to our second result, a g a i n v e r y s i re-m i l a r to

T h e o r e m 1 and T h e o r e m 2. We now deal w i t h t h e i n t e g r a l : h{x,p,a) -2/2 I .Jo exp — e P -2 cosh 2u

sinix cosh u) cosh u du\ (7)

w h i c h has been chosen i n r e l a t i o n w i t h the d e f i n i t i o n s of f and d. (Note t h a t t h i s choice is n o t unique.) Indeed, b y i n -specting the integrands of equations (2), (4), and (7) i t ap-pears clearly t h a t h and d have the same dependence on (p,a), w h i l e h and f have the same dependence i n x. Therefore we

should expect, under the same assumptions as i n T h e o r e m 2, the existence of a n asymptotic series i n v o l v i n g the prod-ucts / , „ ( p e " ' " / 2 ) Y2m±i(.x) f o r h. This is indeed t h e case and we have the f o l l o w i n g result:

T h e o r e m 3—Suppose x and p strictly positive and bounded, and 0 < a < 17/2. When M is large, we have:

h / \ + ia/2 2 Jx + ia exp , / 2 - i u - e"'" cosh 2u

: cosh u cosh udu + O

e x p [ - M ]

(8)

[Proofs o f b o t h theorems can be f o u n d i n Clarisse (1989) ref-erenced as T h e o r e m I I f o r T h e o r e m 2, and as T h e o r e m I I I f o r T h e o r e m 3, a l t h o u g h i n a d i f f e r e n t f o r m . Those proofs are too l e n g t h y to be presented here b u t are i n a n y case sim-i l a r to t h e proof g sim-i v e n sim-i n U r s e l l (1988).]

W e m a y n o w subtract d f r o m / b u t w i t h t h e help of h, t h a t is:

f = { f - h) + { h - d ) + ?Ii\ e-''l^ — FiQ (9)

T h e n f r o m t h e above theorems we o b t a i n the f o l l o w i n g re-sult:

T h e o r e m 4 — S u p p o s e x and p strictly positive and bounded, and 0 s a < -77/2. When M is large, we have:

{ f - h ) + { h - d )

= -e^''^ 'i!i{A+B + C + D} + OiVp e x p [ - M ] ) (10) where:

1. A is the finite integral:

A = ê exp - - e cos 2u- X s i n u 2

cos u du (11)

2. B is the sum corresponding to ( f — h) or:

TT B = -2 Yyix) 7 „ ( - ) c o s ma-lJ^e^'" , 2 / \ 2 [Y^m.iM - Y2,n-i(x)] (12)

3. C is the sum corresponding to (h - d) or:

M / p

Y i ( x ) + e« Ci(x)

- y 2 „ , + i W + C2„,+i(x) - - Y2,n-i(.x) - C2„,-i(x) 2 ^

(13)

4. D is tlie difference between the two infinite bound

in-tegrals present in (3) and (6) or.

D i e x p [ - M e ~ ' ° ] 4 exp[-MT^] exp / X 1 - exp pe imO-- ie'^^'Sf 2 ( l - i e ' " / \ ) pe 1 6 M ( 1 - ie"'"'xY 2(1 - ié°-"T)\ X 1 p^e'-x-'(l - ie"'"iYI p

T h e m a g n i t u d e of the last t e r m i n (10) guarantees the va-l i d i t y of the decomposition (9) i n the v i c i n i t y of t h e o r i g i n . N a t u r a l l y the n u m e r i c a l interest of t h i s last expression relies o n the a b i l i t y of e v a l u a t i n g , w i t h h i g h accuracy, the d i f -f e r e n t t e r m s o-f (10) as w e l l as the t e r m i n v o l v i n g the com-plex Dawson i n t e g r a l . The c o m p u t a t i o n of these q u a n t i t i e s is done as follows:

—The c o m p u t a t i o n of the finite i n t e g r a l A does n o t raise any p a r t i c u l a r d i f f i c u l t y and is carried out t h r o u g h a T a y l o r series expansion of the integi-and [see the A p p e n d i x (B)].

—The t e r m 2^ F ( 9 / x is computed f r o m a l g o r i t h m s f o r t h e complex Dawson integi-al based on ascending series a n d con-t i n u e d f r a c con-t i o n represencon-tacon-tions [see con-the A p p e n d i x (C)].

— T h e i n t e g r a l D is evaluated by means of Laplace's method, the r e s u l t i n g asymptotic expansion being computed f o l l o w i n g N e w m a n (1988). T h i s expansion of l e a d i n g order V p e x p [ - M e ~ " ' ] proves, i n p a r t i c u l a r , the c o n t i n u i t y of D at the o r i g i n [see Clarisse (1989)].

— T h e e v a l u a t i o n of the two sums B and C requires some precautions as t h e y i n v o l v e products of f u n c t i o n s w h i c h are s i n g u l a r at the o r i g i n — F „ and C „ — b y f u n c t i o n s of value 0 there. I n p a r t i c u l a r , the c o m p u t a t i o n of these sums necessitate t h e i r r e w r i t i n g i n t e r m s of new sequences [see the A p -pendix ( D ) ] . I n the process i t appears t h a t these sequences are r e g u l a r at the o r i g i n and along the x-axis. Hence t h e l e a d i n g s i n g u l a r behavior of the single i n t e g r a l is e n t i r e l y accounted f o r by t h e t e r m ^ F ( ^ ) / x , the r e m a i n i n g t e r m s i n (10) b e i n g continuous on the x-axis i n c l u d i n g the o r i g i n . However, t h i s decomposition p r o p e r t y does n o t apply to t h e h i g h l y oscillatory behavior of f . Indeed, i f t h e t e r m g ¥(Q/x accounts f o r the h i g h l y oscillatory and s i n g u l a r behavior of

f along the x-axis, the i n t e g r a l D does also c o n t a i n a h i g h l y

oscillatory (but continuous) component of f . I n particular, the l e a d i n g order of t h i s i n t e g r a l and the order of the neglected t e r m s i n (10) indicate t h a t derivatives of (9) w i l l n o t be suf-ficient for e v a l u a t i n g accurately the corresponding deriva-tives of f i n the v i c i n i t y of the o r i g i n . I n such cases a more r e f i n e d analysis of the bounds i n T h e o r e m 4 w o u l d be nec-essary.

Nevertheless, expression (9) is s t i l l of interest i n numer-ical computations. T h i s is c o n f i r m e d by n u m e r i c a l experi-ments i n the region 0 < x < 1. I n double precision, expres-sion (9) and Bessho's convergent series agree along a surface comprised between M = 10 a n d M = 18 w i t h at least six decimals absolute accuracy. For such results b o t h summa-tions i n B and C are carried o u t up to m = 8, w h i l e t h e asymptotic series f o r the i n t e g r a l D is t r u n c a t e d near i t s smallest t e r m , as i n Section 2. A n i d e n t i c a l accuracy is also achieved between (3) and (9) i n the d i s k x = 1 and M > 17. Such an accuracy is obtained at the expense of a h i g h e r com-p u t a t i o n a l burden: an e v a l u a t i o n of (9) is u com-p to five times more expensive t h a n the corresponding evaluation of (3). The

m a i n c o n t r i b u t i o n to t h i s a d d i t i o n a l cost is due to the eval-u a t i o n of two series instead of one. T h i s inconvenience is, however, necessary as f o r m u l a t i o n (10) is c r u c i a l f o r the def-i n def-i t def-i o n of well-behaved recursdef-ions def-i n the v def-i c def-i n def-i t y of the or-i g or-i n [see the A p p e n d or-i x (D)].

Improvement of convergent series

O f interest i n practice i n the region M < 22, Bessho's [1964, equation (4.2)] convergent series m a y be i m p r o v e d by the same m a n i p u l a t i o n of s u b t r a c t i n g the complex Dawson i n -tegi-al. The corresponding r e s u l t is as follows:

T h e o r e m 5 — f can be written as:

e"'"/VT (14) f='3l\e 2£ ( - 1 ) " K,, cos n a

cos(n + l ) a X J „ + i ( x )

, - z / 2

(15)

where Kn is the modified Bessel function and (A„) is defined through the recursion:

^Ao = 0

(16) A„

An

„ y 2

the series being uniformly convergent in any compact set in-cluded in the region x > 0 , p > 0 , 0 < a < -77/2.

[Proof of t h i s theorem is g i v e n i n Clarisse (1989) as Theorem I . ]

N a t u r a l l y , the r e s u l t of N e w m a n [1987b, equations (13) a n d (14)] is recovered for a = 0.

A n u m e r i c a l i m p l e m e n t a t i o n of (15) requires c e r t a i n pre-cautions [see the Appendix (E)]. N u m e r i c a l l y , i t appeal's t h a t t h i s i m p r o v e m e n t of Bessho's convergent series is not nec-essary to o b t a i n an agreement (of six decimals absolute ac-cm-acy) w i t h expression (9) as long as double precision is used. Nevertheless, (15) is more suitable i n the v i c i n i t y of the or-i g or-i n because of reduced cancellator-ion errors.

T h i s concludes the t r e a t m e n t of the neighborhood of the o r i g i n . Expression (9) and Bessho's convergent series, or i t s i m p r o v e d f o r m (15), are complementary i n the region 0 :£ x < 1. T h u s the e n t i r e p o r t i o n of h a l f space z > 0, p 7^ 0 com-prised between the t w o planes x = 0 and x = 16 is t r e a t e d as i l l u s t r a t e d i n F i g . 1, w i t h at least six decimals absolute accuracy. The question r e m a i n s of e v a l u a t i n g the single i n -t e g r a l , s -t i l l i n -the v i c i n i -t y of -the x-axis b u -t now f o r large values of x, i.e., i n practice f o r x & 16.

4. L a r g e d i s t a n c e s d o w n s t r e a m o f the s o u r c e I n v i e w of the efficiency p r o v i d e d by t h e n u m e r i c a l i m p l e m e n t a t i o n of (3) i t is n a t u r a l to seek a n extension of U r -seU's r e s u l t to large values o f x. I n a d d i t i o n , i t is of interest to understand the restrictions imposed on the v a l i d i t y of (3) f o r t h i s regime. T h i s requires a fine analysis of the proof of

T h e o r e m 1 as g i v e n i n U r s e l l (1988) t h a t we summarize

here.

T h e assumption t h a t p is hounded is c r u c i a l i n establish-i n g the bounds establish-i n L e m m a 1.1, and establish-i n obtaestablish-inestablish-ing equatestablish-ion (2.15) and i n e q u a l i t i e s (2.19) and (2.21) of U r s e l l (1988). However, i t i s n o t necessary to assume t h a t x and p are small i n order to o b t a i n these results. I f x and p were s i m p l y assumed to

4 8 12 16

(la)

p _{Bessho's convergenl series ^}

p 0.01 _ 0 (15) X (3) ^ ^ ^ ^ (9)-(lQ) 0 _

1

0 0.5 I ( l b )

Fig. 1 Sections at constant a stiowing regions ot validity of various expressions for 0 s x s 16 (a), and for 0 =s x s 1 (ö). In both figures portions of parabola p = are shown as possible locations of traces of

partition surface

be positive, the bounds i n v o l v i n g e x p [ - A f | w o u l d have t o be modified to include functions of p only. For example, the most demanding restriction would then produce a bound i n expE-Af + 2p + p^/16], instead of e x p [ - M ] , w h i c h w o u l d be f o r equa-t i o n (2.15). The a d d i equa-t i o n a l assumpequa-tion o f large x / 2 p w o u l d t h e n be necessary f o r the proofs to r e m a i n unchanged. B u t i f we w i s h to keep expression (3) as is (i.e., w i t h e x p [ - M ] f o r t h e order of the neglected terms), i t is necessary to as-sume t h a t p is bounded. However, t h e assumption t h a t x is bounded is i n no w a y r e s t r i c t i v e a n d T h e o r e m 1 (or Theor e m 1 of U Theor s e l l 1988) still holds f o Theor laTheorge b u t f i n i t e x. S i m -i l a r l y , the proof of T h e o r e m 3 o f U r s e l l (1988) necess-itates some m i n o r changes ( i n the behavior o f the H a n k e l f u n c t i o n

If^^ f o r large order) w h i c h leave i t s r e s u l t unchanged. Hence

we can state a m o d i f i e d version of T h e o r e m 1:

T h e o r e m 6 — S u p p o s e that p is positive and bounded, x real and positive, 0 < a < 'n/2. Then for large values of M and x/2p, expression (3) holds.

We have t h u s means f o r e v a l u a t i n g the single i n t e g r a l i n the v i c i n i t y of the x-axis and t h i s u n i f o r m l y f o r any dis-tances downstream of t h e source. I n p a r t i c u l a r , the decom-position i m p l i e d i n w r i t i n g (3) is s t i l l v a l i d i n t h i s region: there, the i n t e g r a l i n (3) accounts entirely f o r the h i g h l y os-c i l l a t o r y and s i n g u l a r behavior of f . I n the large x r e g i m e we m a y therefore associate the series i n (3) to the transverse wave system w h i l e the i n t e g i ' a l represents the r a p i d l y osc i l l a t i n g d i v e r g i n g waves near the t r a osc k of the sourosce. N a t -u r a l l y the n -u m e r i c a l properties of t h i s e v a l -u a t i o n can o n l y

be assessed b y comparison w i t h other expressions dedicated to the f a r - f i e l d regime, b u t we reserve t h i s m a t t e r f o r a sub-sequent paper.

5. C o n c l u s i o n

W e have proposed an a n a l y t i c a l t r e a t m e n t of the single integi-al i n t h e v i c i n i t y of the o r i g i n under the f o r m of two complementary results:

1. The asymptotic expansion (10) w h i c h ensures a n eval-u a t i o n w i t h e x p o n e n t i a l l y s m a l l errors i n the regime of s m a l l X and large M, and

2. The convergent series (15) w h i c h provides a u n i f o r m l y converging series i n the regime of s m a l l x and s m a l l M .

N u m e r i c a l implementations of these expressions c o n f i r m their c o m p l e m e n t a r i t y i n the whole r e g i o n 0 < x < 1, - I T / 2 < a

< T T / 2 , p 7^ 0, w i t h a u n i f o r m absolute accuracy of six

dec-i m a l s . T h e dec-i r p r dec-i n c dec-i p a l characterdec-istdec-ic dec-is the use o f sequences w h i c h are wellbehaved i n the v i c i n i t y of the o r i g i n , i n c l u d -i n g the o r -i g -i n -i t s e l f Comb-ined w -i t h -i m p l e m e n t a t -i o n s of pre-viously derived expressions, n a m e l y UrseU's asymptotic se-ries (3) and Bessho's convergent sese-ries, expressions (10) and (15) provide e f f i c i e n t means for c o m p u t i n g the single inte-gi-al i n the w h o l e r e g i o n 0 < x < 16, - T r / 2 < a < T r/ 2 , p ^ 0. S t i l l i n the v i c i n i t y of the x-axis b u t f o r unbounded x, we have seen t h a t UrseU's r e s u l t ( T h e o r e m 1) is s t i l l vaUd. Hence, i n c o m b i n a t i o n w i t h the above results, the single i n -t e g r a l can e f f i c i e n -t l y be compu-ted w i -t h a u n i f o r m error i n the v i c i n i t y o f the x-axis by means of (3) a n d ( 9 - 1 0 ) . I n the prospect of a u n i f o r m evaluation of the wave-resistance Green f u n c t i o n i n i t s e n t i r e d o m a i n o f d e f i n i t i o n , i t r e m a i n s to evaluate the single i n t e g r a l i n t h e f a r f i e l d away f r o m the

X-axis.

U n t i l now we have emphasized the interest of the v a r i o u s expressions (3), (9), (15) and of Bessho's convergent series i n d e v i s i n g a l g o r i t h m s f o r f a s t and accurate evaluations of the single integi-al. However, t h i s should not hide the equally i m p o r t a n t characteristic of these expressions w h i c h d i s t i n -g u i s h t h e m f r o m n u m e r i c a l q u a d r a t u r e techniques, n a m e l y t h e i r power as a n a l y t i c a l tools. T h e most obvious advantage of u s i n g asymptotic expansions f o r the v i c i n i t y o f the x-axis is to i d e n t i f y and account f o r the complex s i n g u l a r behavior o f t h e single i n t e g r a l there. I n p a r t i c u l a r , we m a y decompose t h i s p a r t of t h e Green f u n c t i o n as the s u m of:

• a continuous and slowly v a r y i n g component r e l a t e d to the transverse wave system. T h i s corresponds to the se-ries i n (3), or the s u m -e'"'^ 2/l{A + B + C} i n ( 9 - 1 0 ) . • a h i g h l y oscillatory and s i n g u l a r component r e l a t e d to

the d i v e r g i n g wave system. T h i s corresponds to the i n -t e g r a l i n (3), or -the s u m d - e'''"^ gi{D} i n ( 9 - 1 0 ) . B u t above a l l , a l g o r i t h m s such as the ones we derived are c e r t a i n l y n o t the panacea f o r p r a c t i c a l applications. The simple f a c t t h a t the wave-resistance Green f u n c t i o n presents a nonisolated s i n g u l a r i t y should call f o r extreme care i n i t s use. A too-famous i l l u s t r a t i o n o f the d i f f i c u l t i e s associated w i t h t h i s p e c u l i a r i t y is the controversial N e u m a n n - K e l v i n p r o b l e m . T h e r e the boundary i n t e g r a l f o r m u l a t i o n of t h i s p r o b l e m f o r a surface-piercing body involves, i n p a r t i c u l a r , a n i n t e g r a l over the w a t e r l i n e ( B r a r d 1972). The single i n -tegi-al b e i n g the k e r n e l of t h i s l i n e i n t e g r a l , the u s u a l ap-p r o x i m a t i o n s made i n n u m e r i c a l methods u s i n g constant source s t r e n g t h panels are h i g h l y suspicious. The a n a l y t i c a l properties of expressions (3) and (10) are u s e f u l f o r c l a r i f y i n g such questions. For example, the consideration o f the poten-t i a l induced b y a segmenpoten-t or a f l a poten-t panel indicapoten-tes poten-t h a poten-t p o i n poten-t evaluations o f the single i n t e g r a l near t h e free surface are

meaningless (see Clarisse 1991). Instead, contributions f r o m integrals over segments or f l a t panels of the single i n t e g r a l should be considered. For t h i s purpose, the derivations o f a l l the above asymptotic expansions and convergent series are i n t u r n useful i n t h a t t h e y can be extended to such integi'als. The r e s u l t i n g expressions w o u l d t h e n present s i m i l a r char-acteristics of y i e l d i n g fast a n d accurate algorithms. B u t practical applications other t h a n the N e u m a n n - K e l v i n prob-l e m m a y have d i f f e r e n t requirements. I t is t h e n advisabprob-le t h a t each p a r t i c u l a r case be treated w i t h equal care.

A c k n o w l e d g m e n t s

Support for t h i s w o r k was provided by the Office of N a v a l Research.

R e f e r e n c e s

A B R A M O V I T Z , M . A N D S T E G U N , I . A . 1964 Handbook of Mathematical

Functions with Formulas, Graphs, and Mathematical Tables, Dover, 9th printing, New York.

B A A R , J . J . M . A N D P R I C E , W . G . 1988 E v a l u a t i o n of the wavelike

disturbance i n the K e l v i n wave source potential. J O U R N A L OF S H I P R E

-SEARCH, 32, 1, 4 4 - 5 3 .

BESSHO, M . 1964 On the fundamental function i n the theory of the wave-making resistance of ships. Memoirs of the Defense Academy of Japan, 4, 2, 99-119.

B R A R D , R . 1972 The representation of a given ship f o r m by singular-ity distributions when the boundary condition on the free surface is

linearized. J O U R N A L OF S H I P R E S E A R C H , 16, 1, 7 9 - 9 2 .

CLARISSE, J . - M . 1989 On the numerical evaluation of the Neumann-K e l v i n Green function. MS thesis, M.I.T., Cambridge, Mass. CLARISSE, J . - M . 1991 H i g h l y oscillatory behaviors i n the

Neumann-K e l v i n problem. 6 t h I n t e r n a t i o n a l Workshop on Water Waves and F l o a t i n g Bodies, Woods Hole, Mass., A p r i l 14-17, 1991.

N E W M A N , J . N . 1987a Evaluation of the wave-resistance Green func-tion: part 1—the double integral. J O U R N A L OF S H I P RESEARCH, 3 1 , 2, 7 9 - 9 0 .

N E W M A N , J . N . 1987b Evaluation of the wave-resistance Green func-tion: part 2—the single integral on the centerplane. J O U R N A L OF S H I P

R E S E A R C H , 3 1 , 3, 145-150.

N E W M A N , J . N . 1988 E v a l u a t i o n of the wave-resistance Green func-tion near the singular axis. 3rd Internafunc-tional Workshop on Water Waves and Floating Bodies, Woods Hole, Mass., A p r i l 17-20, 1988. U R S E L L , F . 1960 On Kelvin's ship-wave pattern. Journal of Fluid

Me-chanics, 8, 4 1 8 - 4 3 1 .

U R S E L L , F . 1988 O n the t h e o r y of the K e l v i n ship-wave source: asymptotic expansion of an integral. Proceedings, Royal Society of London, A 418, 8 1 - 9 3 .

U R S E L L , F . 1989 The near-field convergent expansion of an integral. 4th International Workshop on Water Waves and Floating Bodies, Oystese, Nonvay, May 7-10, 1989, and unpublished manuscript, March 1989.

W A T S O N , G . N . 1948 A Treatise on the Theory of Bessel Functions, Cambridge U n i v e r s i t y Press, 2nd edition, Cambridge, U . K .

Appendix

A T h e s e q u e n c e (C„)

The functions C„ present i n (6) can be found i n Watson (1948, § 10.13) and appear to be related to Bessel functions but not i n a simple manner, Nevertheless, they may be defined by the integral representation:

c„a-) • e x p [ - . ï cosh u] sinh u du for x > 0 or by the recursion, for .r > 0:

' C o ^ O ) C,(x) = — ^

.X-2n 2 C„+i(.ï) C„{x) - C„_i(.ï) = - e-'

X X

B C o m p u t a t i o n of i n t e g r a l A

The integi-al (11) is computed by expanding its integr-and i n a T a y l o r series. This results i n a convergent series:

A = e»"

2

where the sequence (a„) is given by the recursion: 1 - e-' X M n o„ 2 / 1 x'e-" 4 n ! 4 M • (2n + .%•)

This recursion being well behaved and defined even for x = 0, i t can be used up to any order so as to ensure the desired accuracy.

C C o m p l e x D a w s o n i n t e g r a l

The complex Dawson integr-al F , or Dawson's i n t e g r a l of a complex argument, is related to the error f u n c t i o n of a complex argument erf, through the relation:

(

2

F ( 2 ) ^ • e r f ( - i ' 2 )

When we deduce the ascending series for F :

( - 1 ) " 2"0'"*'

% 1 X a X . . . X (2n 4- 1)

(17)

after the ascending series for erf [Abramovitz & Stegun 1964, equation (7.1.6)].

The continued f r a c t i o n representation for F can be obtained f r o m the continued fraction representation of the complementary error f u n c t i o n erfc [Abramovitz & Stegun, 1964, equations (7.1.4) and (7.1.15)]:

1/2 1/2 1 3/2 2

(18)

For computations, the ascending series (17), w i t h a m a x i m u m of one hundred terms, is used i n the region x > 0, 2.x- -1- j / s 9, 0 £ =^ 3, w h i l e the continued f r a c t i o n (18), w i t h sixteen terms, is used i n the comple-mentary region i n the first quadrant. Values i n the f o u r t h quadrant are obtained by using the conjugation relation: F(z) = F(z). I n F O R T R A N double precision, the accuracy ranges f r o m nine to t e n decimal places.

D C o m p u t a t i o n of S u m s B a n d C

As already seen (see New asymptotic expansion section) the numerical computation of the sums B and C require some precautions. I n this context, expression (9) reveals its importance: a simpler f o r m u l a t i o n i n -v o l -v i n g only /-and d as i n Newman (1987b), does not allow the definition of regular sequences through well-behaved recursions as permitted by (10). The new sequences required by the computation of B and C are based on the behaviors of the various f u n c t i o n s — Y „ , I„, C„—for small values of their argument. These sequences are associated to sequences of m u l t i p l i e r s i n order to recover the original sums. Hence we have:

ysm+i 2m(2m - 1) \ 2 y 2 , „ - i 1 2m(2m - 1) \ 2 for B, and M C = 7o5'oAyi + 2 -^m^m I A>'2,„ + 1 m=l for C.

Herein above, the sequences (i,„) and (AJ-^) are related to the Bessel functions ƒ„ through the relations:

3'„, = m\(-) i j e

-and

A.*„ = (m + 1)!

S i m i l a r l y to (/„) both sequences are computed w i t h the help of backward recursions, namely:

m(m _ P 1)

u

and m + 1 - e'" A3-„ + • (OT + Dim + 2) \ 4 ;n + 1 which are well-behaved for small values of p including p = 0. For the initializations of these recursions, we take advantage of the fact t h a t p is small. Namely, the ascending series for Jf,„ w i t h five terms is used to determine the starting order A''o. This order corresponds to the index m £ 30 f o r which the modulus of the fifth t e r m i n the series exceeds the tolerance of the computer. Then S'^f^+i and .J^o ai'e given by t h e i r five t e r m series, while A i « ( , + i , AI^v,, are given by their three term series.

The sequences and (A^^) are related to the Bessel functions Y,„ and the functions C„, (see Appendix A ) through the relations, for m. > 0: 1 /xY'K y,n = - - Y j x ) (OT - 1)! V2/ 2 and Ay„, fx) ml -(OT - D!

_K2)

- Y,„(x) + C,„(.ï) 2 Both sequences are computed using forward recursions, namely

ir IX y„,+i = y, mim - 1) \2, y„.-i and Ay,=-Y,(x)+ — Z X

^y2 = -\-Y2ix) + e'C,{x)

_J /xY X

Ay„,+i - Ay„, -I- _ A y „ ^ , ^^^^^ _

m! V2

I n i t i a l values are obtained from the Chebyshev approximations for Y„(x), OT = 0, 1 and by means of the recursion for (Y„,) i n order to obtain Y^ix).

F i n a l l y , the m u l t i p l i e r sequences (p„) and (7„) are given by 7o 1

2m 2 M

xp 2M(m + 1) 16 7„,

I t must be noted t h a t a l l of the terms i „ „ A.$,„, y,,, and Ay„ stay of order 1. Hence the asymptotic properties of the series and, i n particular, the fact t h a t the truncation is optimal for m — M are highlighted i n the definitions of the m u l t i p l i e r sequences (p^) and (y„).

E M o d i f i e d convergent series

The problems raised by the numerical computation near the o r i g i n of the series (15), or even of Bessho's convergent series, are somewhat simi l a r to those encountered when dealsiming w simi t h the asymptotsimic sersimies. I n -deed, both convergent series involve products of functions singular at the o r i g i n — K „ of A„—by continuous functions t a k i n g the value 0 there— namely t/„. I n addition, the opposite behaviors of these quantities as functions of their indices must be taken into account: K„ is exponentially increasing w i t h n while J „ is exponentially decreasing. This peculiarity may be taken care of by means of scaling factors as i n N e w m a n (1987b) or by defining new sequences as here.

Hence, we define the sequence (S'C,,) f r o m (K„) by the relation, for re > 0

4 / V2

in - 1)! In (p/4)

the sequence (2i„) f r o m (A„) by the relation, for n > 0 2 1 / p

%, =

in - D! In (p/4) V4

and the sequence (^v) from (J„) by the relation

^ „ = (« + 1)! l^-j J,.(.r)

Consequently the various recurrence relations satisfied by the original sequences yield the new recursions

3Co 3C, 2 ip I n (p/4) \ 2 I n (p/4) 4 ' \ 2 3C„,. = -e'°3C„ + nin - 1) \ 4 for ÖCJ

r

% = o I n (p/4) I n (p/4) 1 3 „ « = - e ' ° 2 ) „ + , nin - 1) \ 4 for (Si)„), w i t h ^ = pe^'", and

n\ In (p/4) \ 4

n + l ' in + 2)(n -I- 1) \ 2

for (^„).

I n order to take f u l l advantage of the form of (15), another sequence

(A3C„) is introduced for ; i > 0

A3C„ - (3f„ + S),.)

w h i c h satisfies the recurrence relation

A3Ci A3C 2e" I n (p/4) 2 I n (p/4) 4 A3f„+i = -e''"A3C„ + e''» - 1 p P -P e^'^/fal - U A, .2y nin - 1) \ 4 P \ 2 1 A3f„_i n\ In (p/4) \ 4 nin - 1 ) 4

A l l of the above sequences present the advantage of being continuous at the o r i g i n and of satisfying stable recursions. I n i t i a l values for the f o r w a r d recursions are determined f r o m the Chebyshev approximations for Koip/2) and i f i ( p / 2 ) , and the expressions for 3C2 and A3f2. The s t a r t i n g index A^o for the backward recursion for (^„) is given by m i n [40 + 6 £ ( l n x) 70] and the corresponding values of ,Jivo+i(^) ai^d $NO^X) are,

respectively, 1 and 1 + 1 0 " ' " (or whatever the tolerance of the computer is).

F i n a l l y , f r o m the above definitions, the sum i n (15) may be r e w r i t t e n as

( - 1 ) " K„{ - ] cos n a + i f „ + i ( - ) cos (n + l)a

2/ V2/

\o(A3(i + 3{o - % ) , w i t h (k„) given by ^ 0 ^ \„A3C.+i + — (3f„ - a „ ) 4(n + D M (2n + 4)(2n + 3)

Here (\„) highlights the convergence properties of the series as well as its l i m i t a t i o n for large values of M . I t must be noted t h a t i n the region of interest, i.e., for bounded M, ko is continuous and of l i m i t 0 at the origin.

F T r u n c a t i o n o f a s y m p t o t i c s e r i e s

The use of asymptotic series i n computations raises the practical prob-lem of their truncation. I t is a profound result t h a t the generic t e r m of

order n of an asymptotic series i n the large parameter N is of the form n\/N". O p t i m a l t r u n c a t i o n of such series as we i m p l y by w r i t i n g expres-sions w i t h exponentially small terms as i n (3), (10), and others, corre-^, spends to a t r u n c a t i o n index Ug ~ N. I n practice, however, such optimal use is not possible for a l l values of N: the number of terms required would then increase lil^e N. For low values of A'' i t is then recommend-able to truncate a series optimally, i.e., at its smallest term. The crite-r i o n we choose to apply is:

The truncation index no is the smallest index n such that:

\a„\ < |a„+i| < |a„+2|

ifa„ designates the generic term ofthe series we consider. A l t h o u g h quite demanding (two extra terms i n the series are required), t h i s three-term criterion is more robust t h a n one based on a two-term inequality, or on a reversed inequality, e.g., | a „ - i | > \a„\.

I n order to l i m i t the computations performed for large values of iV, an additional constraint should be applied along w i t h the above criterion. This constraint can take the form of either an actual bound on the num-ber of terms, either a m i n i m a l accuracy criterion to be satisfied by

Kol