The added mass of bodies heaving at low frequency in
water of finite depth
p. M c I V E R
Department ofMathematics and Statistics, Brunei University, Uxbridge,Middx UB83PH, UK* C . M . L I N T O N
.Sclwol of Matliematics, University of Bristol, Bristol BS8 ITW, UK
N u m e r i c a l results f o r the added mass o f a heaving t w o - d i m e n s i o n a l b o d y i n f i n i t e - d e p t h water show t h a t the zero-frequency l i m i t is d i f f e r e n t f o r the cases of a surface-piercing b o d y a n d a b o t t o m - m o u n t e d b o d y of the same wetted shape. A t f i r s t sight, this m a y appear s u r p r i s i n g as the h m i t i n g b o u n d a r y - v a l u e problems f o r the potentials i n v o l v e d are identical. H o w e v e r the solutions are indeterminate to w i t h i n a n a d d i t i v e constant and it is the value o f this constant t h a t accounts f o r the difference. I n the present w o r k , the m e t h o d of m a t c h e d a s y m p t o t i c expansions is used t o show t h a t the added mass l i m i t s f o r the t w o cases differ by a n a m o u n t t h a t depends very s i m p l y o n the geometry o f the bodies.
1. I N T R O D U C T I O N
T h e l o w - f r e q u e n c y behaviour o f the heave added mass f o r a t w o - d i m e n s i o n a l cylinder i n the water surface has a t t r a c t e d interest over a n u m b e r o f years. F o r i n f i n i t e water d e p t h UrseU"^ showed t h a t the heave added mass o f a half-immersed circular cyhnder becomes i n f i n i t e i n the l i m i t o f zero frequency. N u m e r i c a l calculations of Y u a n d U r s e l P seemed to suggest that the same m i g h t be true f o r f i n i t e water depth. H o w e v e r , these calculations p r o v e d t o be i n e r r o r and subsequent calculations b y Bai a n d Yeung^ gave a finite value f o r the zero frequency l i m i t o f the added mass i n f i n i t e depth water a n d this b e h a v i o u r was c o n f i r m e d a n a l y t i c a l l y by Urseh*.
Recently, L i n t o n ^ has made calculations o f the heave added mass f o r a b o t t o m - m o u n t e d rectangular cylinder i n water of f i n i t e depth. H e f o u n d that his values f o r the zero-frequency h m i t differ f r o m those f o r s i m i l a r cyhnders i n the free surface, w i t h the same wetted shape a n d size, as calculated by Bai*^. A t first sight, this m a y appear s u r p r i s i n g as the b o u n d a r y - v a l u e problems f o r the t w o potentials i n v o l v e d become identical i n the zero-frequency l i m i t . H o w e v e r , the b o u n d a r y c o n d i t i o n s of the l i m i t i n g p r o b l e m s i n v o l v e o n l y derivatives of the p o t e n t i a l so t h a t their solutions are indeterminate to w i t h i n a n a d d i t i v e constant w h i c h m a y o n l y be determined by consideration of the non-zero frequency behaviour. Here, i t is s h o w n that i n the b o t t o m - m o u n t e d case the constant, and hence the added-mass h m i t , is different f r o m the surface-piercing case. ( M a r t i n a n d D a l r y m p l e ^ have f o u n d previously unsuspected differences i n the reflection a n d transmission coefficients of i d e n t i c a l bodies o n the bed a n d i n the free surface w h e n they scatter l o n g waves.)
T h e constant i n the p o t e n t i a l that p a r t l y determines the l i m i t i n g value of the added mass has been calculated
Paper accepted January 1990. Discussion closes August 1991. * Present address: Dept. of Mathematical Sciences, Loughborough University of Technology, Loughborough, Leics L E l l 3 T U , U K .
analytically f o r surface-piercing cylinders by Bai*". However, there appears to be d o u b t a b o u t the v a l i d i t y o f his expansion procedure a n d so here a scheme of m a t c h e d asymptotic expansions is used a l o n g the hnes suggested b y Y e u n g a n d N e w m a n ^ i n a discussion to Sayer a n d U r s e l P . I t s h o u l d be n o t e d t h a t i n the latter author's reply to the discussion they p o i n t o u t some inconsistencies i n the w o r k of Y e u n g a n d N e w m a n , thus there are a n u m b e r of differences f r o m the present w o r k . S a y e r ^ ° has also considered this p r o b l e m using m a t c h e d a s y m p t o t i c expansions. However, his s o l u t i o n o n l y gives p a r t o f the l o w - f r e q u e n c y expansion whereas here the complete expansion is considered. T h e results given here show t h a t the l i m i t i n g values of the added masses f o r the surface-p i e r c i n g and b o t t o m - m o u n t e d cases differ b y an a m o u n t t h a t depends very s i m p l y o n the geometry of the b o d y concerned.
The K r a m e r s - K r o n i g relations m a y be used t o o b t a i n i n f o r m a t i o n about the type of expansion considered here. A n interesting p o i n t t h a t arose d u r i n g discussion o f the present w o r k is t h a t the f a m i h a r f o r m of the K r a m e r s -K r o n i g relations requires m o d i f i c a t i o n for two-dimensional problems i n f i n i t e - d e p t h water. T h i s is described b y M . G r e e n h o w i n a n appendix t o the present paper.
I n three dimensions the l o w - f r e q u e n c y l i m i t o f the heave added mass is f i n i t e f o r a surface-piercing b o d y i n i n f i n i t e d e p t h water. I n the f i n a l section o f the present paper, i t is s h o w n that f o r f i n i t e d e p t h f l u i d the added mass becomes i n f i n i t e i n the l i m i t a n d the l e a d i n g order b e h a v i o u r is the same f o r b o t h surface-piercing a n d b o t t o m - m o u n t e d bodies. T h i s c o n f i r m s , f o r example, the a n a l y t i c a l results f o r l o w frequency f o u n d by Yeung^^ a n d M c l v e r a n d Evans^^ f o r a t r u n c a t e d v e r t i c a l cylinder.
F o r a symmetric b o d y i n t w o dimensions a n d a n a x i s y m m e t r i c b o d y i n three dimensions the l i m i t i n g behaviour of the sway added mass is the same f o r i d e n t i c a l bodies o n the bed a n d i n the free surface. T h i s is because an " a n t i s y m m e t r y " c o n d i t i o n m a y be i m p o s e d o n the
r- -1 I I
SI
/ / / / / / / / / / / /
Fig. 1. Defimtion sketch and tiie contour S (see equation (11)) for a surface-piercing body
p o t e n t i a l so that the s o l u t i o n of the l i m i t i n g p r o b l e m is f u l l y determined w i t h o u t reference to the non-zero f r e q u e n c y behaviour.
2. T W O D I M E N S I O N S
Consider the t w o - d i m e n s i o n a l p r o b l e m of the heaving of a b o d y o n the surface of water o f constant depth h. Cartesian coordinates [x, y) are chosen w i t h y directed vertically upwards and w i t h the o r i g i n i n the mean free surface (see F i g . 1). T h e body, symmetric a b o u t x = 0 and intersecting the free surface at x = + a, performs vertical oscillations of r a d i a n frequency co a n d velocity amphtude U. As the p r o b l e m is symmetric a b o u t x = 0 o n l y the s o l u t i o n f o r x > 0 need be considered. T h e wetted surface of the b o d y i n x > 0 w i l l be denoted by S^. U n d e r the usual assumptions o f t h e linearised t h e o r y of water waves, t h e m o t i o n m a y be described using a velocity p o t e n t i a l
$(x,3',O = R e ( W ! 0 ( x , y ) e - ' " ' ) (1) where 4) satisfies Laplace's equation,
dx^'^dy' '
i n the f l u i d region, the linearised free-surface c o n d i t i o n (2)
dy
K4>; y = Q,x>a (3)
where K = a)^/g, a n d velocity c o n t i n u i t y c o n d i t i o n s o n the b o d y a n d bed, namely
dé 1 dn h and dy - = 0 ; on Sf y = - / 7 (4)1 (5)
Here n is a coordinate measured n o r m a l t o a n d directed o u t of the f l u i d a n d = cos(fi, y). I n a d d i t i o n the s o l u t i o n must satisfy a s y m m e t r y c o n d i t i o n o n x = 0 a n d the r a d i a t i o n c o n d i t i o n
8x
(6)
where the wavenumber /( is the s o l u t i o n o f the usual -dispersion r e l a t i o n
K = /ctanh/c/i. (7) I n the low-frequency Hmit, as X ^ O , a l l of the b o u n d a r y
c o n d i t i o n s i n v o l v e only derivatives o f <f> and so the s o l u t i o n o f the l i m i t i n g p r o b l e m is indeterminate to w i t h i n an additive constant. H o w e v e r this constant m a y be f o u n d f r o m the non-zero frequency b e h a v i o u r o f t h e p o t e n t i a l .
The a i m is t o develop an expansion f o r 4> v a l i d f o r l o w frequencies i n terms of the parameter e = kh, assumed to be small, using the m e t h o d of m a t c h e d a s y m p t o t i c expansions. I t w i l l be assumed t h a t a a n d h are o f the same order of m a g n i t u d e . T h e f l o w d o m a i n is d i v i d e d i n t o t w o f l o w regions, an inner r e g i o n w i t h i n a distance 0{li) of the b o d y , where wave effects are n o t discernible, and an outer r e g i o n f a r f r o m the b o d y where the s o l u t i o n consists o f o u t w a r d p r o p a g a t i n g plane waves.
F o r each r e g i o n a p p r o p r i a t e l y scaled coordinates w i l l be used. I n the inner region the m o t i o n has a l e n g t h scale h and inner variables x = x / / i a n d y = y/lt w i l l be used. W i t h this change of variables i n equations (2)-{5), i t can be seen that the p o t e n t i a l ^(ic,}!) s 0(x, y) i n the i n n e r region is a s o l u t i o n of Laplace's e q u a t i o n satisfying the b o u n d a r y c o n d i t i o n s 8y £*-hO(e« y = 0, x > -dn dtp = 0; on S B y = - I . (8) (9) (10)
T h e r i g h t - h a n d side of e q u a t i o n (8) results f r o m expanding the dispersion r e l a t i o n (7) i n terms of s. T h e r a d i a t i o n 'Condition is not a p p r o p r i a t e to the inner region, the large .X behaviour of the inner s o l u t i o n must m a t c h with the outer s o l u t i o n . F o r later use, i t is n o t e d t h a t as </> is a h a r m o n i c f u n c t i o n then, f o r any c o n t o u r S b o u n d i n g a f l u i d region,
• dn
-ds = 0, (11)
where iï is the o u t w a r d n o r m a l coordinate t o S. I n particular, S is chosen as the c o n t o u r i n x > 0 , i l l u s t r a t e d i n F i g . 1, consisting of the free-surface f o r a/h<x<X, where X is a r b i t r a r y , the wetted surface o f the b o d y Sg, t h a t p a r t o f x = 0 w i t h i n the f l u i d , the bed ïor 0<x<X and a vertical line at x = Z j o i n i n g the bed a n d free surface.
I n the outer region, because of the a s s u m p t i o n kh«l, the flow is essentially t h a t of linear shallow-water t h e o r y a n d there are different l e n g t h scales f o r the h o r i z o n t a l and vertical m o t i o n , respectively k~ ^ and T h u s suitable sealed coordinates f o r the outer region are x = kx and 37 = 3;//! i = y)so t h a t f r o m equations (2)-(3) a n d (5)-(6) the o u t e r p o t e n t i a l 0(x, y) = (pix, y) satisfies
i n the fluid and the b o u n d a r y c o n d i t i o n s
dy V 3
i ^ = 0 ;
y = - l , dy dx -j(^->0; x-*oo. (12) (13) (14) (15)T h e b o d y b o u n d a r y c o n d i t i o n does n o t appear i n the outer region, rather the smah x behaviour of the outer s o l u t i o n must m a t c h w i t h the inner s o h i t i o n .
A leading order s o l u t i o n f o r the outer region is posed i n the f o r m
< ^ « / Q ( S ) 0 O (16) where / ^ ( E ) is a gauge f u n c t i o n t o be determined by
m a t c h i n g w i t h the inner s o l u t i o n . R e t a i n i n g o n l y the lowest order terms i n e q u a t i o n (12)-(15) shows t h a t cpQ m u s t satisfy
8f = 0 (17)
i n the f l u i d , homogeneous c o n d i t i o n s o n the free-surface a n d bed a n d the r a d i a t i o n c o n d i t i o n . T h u s
= c/.e (18)
an o u t w a r d p r o p a g a t i n g plane wave w i t h a p o t e n t i a l a m p h t u d e a w h i c h w i h be f o u n d f r o m m a t c h i n g w i t h the inner s o l u t i o n . The m a t c h i n g p r i n c i p l e to be used is that described i n Chapter V of the text by V a n D y k e ^ ^ , t h a t is, the m-term inner expansion o f t h e n-term outer s o l u t i o n m u s t correspond t e r m by t e r m w i t h the n-term outer expansion o f the ;n-term i n n e r s o l u t i o n . R e w r i t i n g (18) i n terms o f t h e inner coordinates a n d expanding i n powers o f 8 gives the t w o - t e r m inner expansion o f the leading-order outer s o l u t i o n
/ Q ( £ ) ^ e = / e ( £ ) « ( l + + ^ ( s ' ) ) . (19) T h e f o r m o f the b o d y b o u n d a r y c o n d i t i o n , e q u a t i o n (9),
suggests t h a t the inner s o l u t i o n w i l l have an expansion of the f o r m
--fp{e)4)p + 4>o + o{l) (20)
where the subscript P is used t o denote a possible t e r m larger t h a n the 0(1) t e r m f o r c e d b y the b o d y b o u n d a r y c o n d i t i o n . T h e f o r m o f the gauge f u n c t i o n fp{e) w i h be f o u n d by m a t c h i n g w i t h the outer s o l u t i o n . S u b s t i t u t i n g the expansion (20) i n t o equations (8)-(10) and equating like coefficients of e shows t h a t b o t h (pp a n d 0o satisfy homogeneous c o n d i t i o n s o n the free-surface a n d bed, w h i l e o n the b o d y surface 4>p satisfies a homogeneous c o n d i t i o n b u t 4>o satisfies e q u a t i o n (9). As 4)p satisfies homogeneous b o u n d a r y c o n d i t i o n s a n d also e q u a t i o n (11) t h e n 0p can generate n o net " f l o w " o u t of the inner region. Conversely, because o f the non-homogeneous b o d y b o u n d a r y c o n d i t i o n 4>o will generate a flow o u t the i n n e r region. Thus the leading-order constant t e r m i n ejijuation (19) must m a t c h w i t h the outer expansion of 4>p a n d t h e j t e r m i n x w h l m a t c h w i t h the outer expansion of (pQ. As 4>p tends t o a constant at large distances and satisfies homogeneous c o n d i t i o n s elsewhere, the o n l y possible s o l u t i o n is
(21)
where Cp is a constant. F u r t h e r m o r e , the m a t c h i n g demands t h a t / P ( £ ) =/<2(£) = e"^ (so t h a t f r o m n o w o n P = Ö = — 1), Cp = a a n d 4>o ~ io'-^ as oo. Thus, the inner s o l u t i o n m a y be w r i t t e n as
4) = -a + Co + 'i'o{x,y) + oil) (22)
where CQ is a constant, separated o u t f r o m (f>o f o r
iconvenience, and is a h a r m o n i c f u n c t i o n satisfying
dy 8y y = 0, x> o n Su h y=-l and >0; • 0 0 . (23) (24) (25) (26) turns out to i(The constant CQ i n the inner p o t e n t i a l (pQ
c o n t r i b u t e t o the l o w - f r e q u e n c y l i m i t of the added mass a n d its value differs a c c o r d i n g t o whether a b o d y is isurface-piercing or b o t t o m - m o u n t e d . ) T h e constant a can be calculated b y a p p l y i n g (11) t o and t a k i n g the h m i t as X->oo, thus
i a s l i m 8 ^ dx
dy
nyds= - J ,
(27)t h a t is a = ia/h and as a consequence *Fo is real-valued. T h e inner expansion of 0 suggests the need f o r a t e r m at order e i n the inner s o l u t i o n , thus
e h
•scpi+ois). (28)
S u b s t i t u t i n g e q u a t i o n (28) i n t o (8)-(10) a n d e q u a t i n g l i k e powers of s shows that 0^ is a h a r m o n i c f u n c t i o n satisfying homogeneous c o n d i t i o n s o n the b o d y surface and bed b u t o n the free surface
h ^ I
dy (29)
The c o n d i t i o n (11) applied to can o n l y be satisfied i f its expansion f o r large x contains terms p r o p o r t i o n a l t o
X a n d x^. W h e n r e w r i t t e n i n outer coordinates, this i n
t u r n suggests the need f o r at t e r m at 0 ( 1 ) i n the outer expansion so that
+ ^ 0 + 0(1). e h
(30)
F r o m equations (12)-{15), (po satisifes the same equations as <p_^ a n d so
(31) where /? is a constant. T h e inner expansion of (30) is f o u n d by w r i t i n g i n inner coordinates and e x p a n d i n g to
0(B) to o b t a i n • 1 a > = ! -£ /; 1 f £!X -1 + P(l+eix) + Ois^). (32)
I f the outer expansion o f e q u a t i o n (28) is t o m a t c h w i t h (32), then i t is r e q u i r e d t h a t /? = Co and (p^~ (Hx-^ifx^ as x—KX). T h e constant CQ is determined b y a p p l i c a t i o n of (11) to 4>v T h e o n l y non-zero c o n t r i b u t i o n s t o the surface i n t e g r a l are f r o m the free surface a n d the v e r t i c a l hne at x = X so t h a t ' ^ d s = l i m dn x~>co rx , dy dx + .J dx x = X dy} = 0 (33)
and, o n substituting f r o m e q u a t i o n (29) a n d f o r the a s y m p t o t i c behaviour of i t is f o u n d that ^ = C O = F. T h u s tlie inner p o t e n t i a l has the f o r m
4>{x, y)=-ij + "^^ + ^o(x, y) + s4>,{x,y) + o(e). (34)
N o t e that is a realvalued f u n c t i o n b u t 0 i is i m a g i n a r y -valued.
T h e added-mass and d a m p i n g coefficients, A a n d B respectively, are given by
A + i~=^2pli'
CO
bn^, ds (35)
where, as previously, the arc-length s has been made n o n - d i m e n s i o n a l w i t h respect to b. So, f r o m the expansion 1(34) and e q u a t i o n (27) iand A a h^ + -Ipa" h a^ B ^ B i -»Pon,,rfs + o(e) 0i??,,rfs + o(e). (36) (37) 2copa^ T h e l o w - f r e q u e n c y h m i t of the n o n - d i m e n s i o n a l added mass as is therefore given by the terms given ^explicitly i n equation (36) where is the h a r m o n i c f u n c t i o n satisfying equations (23>-(26). B y proceeding f u r t h e r i n the expansions the integrals o n the r i g h t - h a n d ;sides of equations (36)-(37) m a y be related a n d this is done i n A p p e n d i x A .
T h e behaviour w i t h £ = fe/j i n equations (36H37) is d e a r l y very different f r o m the m o r e f a m i l i a r i n f i n i t e d e p t h icase w h e n the zero-frequency l i m i t s (c£)->0) of the added mass and d a m p i n g are respectively i n f i n i t e a n d finite. K o t i k and Manguhs^'^ have used the K r a m e r s - K r o n i g relations t o derive general relations between the l o w -frequency behaviours of the added mass and d a m p i n g f o r heaving bodies i n i n f i n i t e depth water. A similar calculation for finite depth has been made b y M . Greenhow and is described i n A p p e n d i x B . T h e c a l c u l a t i o n shows t h a t the f o r m o f one of the K r a m e r s - K r o n i g relations f a m h i a r i n h y d r o d y n a m i c s requires m o d i f i c a t i o n f o r t w o - d i m e n s i o n a l m o t i o n s i n f i n i t e d e p t h water and also c o n f i r m s the consistency o f the expansions (36)-(37).
T h e case of a heaving b o d y m o u n t e d o n the bed, rather t h a n piercing the free surface, m a y be treated similarly. As f a r as the zero-frequency l i m i t of the added mass is concerned, the essential difference is i n the c a l c u l a t i o n of the constant CQ appearing i n the inner s o l u t i o n , e q u a t i o n (28). T h e free-surface secfion o f the c o n t o u r S n o w occupies 0<d<X so t h a t the l o w e r l i m i t o f i n t e g r a t i o n a/h i n e q u a t i o n (33) is replaced by zero w h i c h gives Co = 0. T h e f u n c t i o n satisfies the same b o u n d a r y -value p r o b l e m as previously so t h a t the leading order b e h a v i o u r of the added mass is given b y e q u a f i o n (36) w i t h o u t the first t e r m . T h u s the zero-frequency l i m i t of the added mass, made n o n - d i m e n s i o n a l as i n e q u a t i o n (36), w i h differ f r o m the result f o r the surface-piercing
case b y a/h.
B a i ^ has calculated upper and l o w e r bounds f o r the zero-frequency h m i t o f the added mass f o r a surface-piercing rectangular cyhnder and some of his results, resclaed t o give the present n o n d i m e n s i o n a h s a t i o n , are reproduced i n T a b l e 1. Values f o r the b o t t o m - m o u n t e d
Table L Zero frequency limit of added mass for a surface-piercing cylinder (^is) and a bottom-mounted cylinder (jig), eacli cylinder lias a rectangular cross-section of half-width a and height b and the water is of depth h. See text for further explanation
alb an,
lower bound upper bound
0.5 0.333 1.022 1.048 0.699 1.032 0.25 0.792 0.818 0.551 0.801 0.167 0798 0.830 0.643 0.809 1 0.5 O.806 0.823 0.313 0.813 0.333 0.680 0.695 0.352 0.685 0.125 0.978 0.999 0.861 0.986 2 1.333 1.775 1.899 0.540 1.873 1.033 1.056 0.042 1.042 0.25 0761 0773 0.515 0765
case (denoted b y calculated by the e i g e n f u n c t i o n m e t h o d described i n L i n t o n ^ are also given. T h e final c o l u m n shows that these n u m e r i c a l results are consistent w i t h the t h e o r y given i n the present w o r k , t h a t is i n the zero-frequency l i m i t = / i ^ -|- a/h.
3. T H R E E D I M E N S I O N S
A s i m i l a r c a l c u l a t i o n to t h a t of the previous section m a y be p e r f o r m e d f o r a heaving a x i s y m m e t r i c b o d y . H o w e v e r , the leading-order behaviours f o r l o w frequency o f the added mass a n d d a m p i n g m a y be o b t a i n e d m o r e q u i c k l y as f o h o w s . T h e leading-order c o n t r i b u t i o n t o the heave exciting force X i n i n c i d e n t l o n g waves is t h a t due t o the wave elevation at the b o d y , t h a t is
X'^pgSo, (38)
where SQ is the water-plane area f o r a surface-piercing b o d y a n d the area of intersection w i t h the bed f o r a b o t t o m - m o u n t e d b o d y (disregarding a difference i n sign f o r the t w o problems). N o w , the d a m p i n g c o e f f i c i e n t B (defined by the three-dimensional equivalent o f e q u a t i o n (35)) f o r an axisymmetric b o d y is related t o the exciting force b y
B = - (39)
Anpgcg
(see Ref. 15, p. 304) where the g r o u p velocity Cg~{ghY'^ f o r l o n g waves. Thus, c o m b i n i n g equations (38) and (39),
(opSl B'
Ah
(40)
t o leading order. T h i s l o w - f r e q u e n c y b e h a v i o u r o f the d a m p i n g has been examined b y K o t i k a n d M a n g u h s ^ * w h o show f r o m the K r a m e r s - K r o n i g relations t h a t the added mass
Inh'
I n kh. (41)
T h u s , the added mass is l o g a r i t h m i c a l l y i n f i n i t e as /c/j->0 f o r the three-dimensional case a n d the l e a d i n g - o r d e r b e h a v i o u r is the same f o r b o t h the surface-piercing a n d b o t t o m - m o u n t e d cases.
A C K N O W L E D G E M E N T S
The authors are g r a t e f u l to Prof. D . V . Evans and D r . M . G r e e n h o w f o r useful discussions. C . M . L . is ;supported b y S E R C g r a n t no. G R / E / 0 6 9 7 . 8 ( M T D L t d ) .
R E F E R E N C E S
1 Ursell, F . On the heaving motion of a circular cylinder on the surface of a fluid, Quart. J. Mech. Appl. Math., 1 9 4 9 , 2 , 2 1 8 - 2 3 1 2 Yu, Y. S. and Ursell, F . Surface waves generated by an oscillating
circular cylinder on water of finite depth: Theory and experiment, J. Fluid Mech., 1961, 11, 529-551
3 Bai, K . J. and Yeung, R. W. Numerical solutions to free-surface flow problems, Proc. 10th Symp. on Naval Hydrodynamics, M.I.T., 1974, 609-647
4 Ursell, F. On the virtual-mass and damping coefficients for long waves in water of finite depth, J. Fluid Mech., 1976, 76, 17-28 5 Linton, C. M . Wave Reflection by Submerged Bodies in Water of
Finite Depth, Ph. D . thesis, University of Bristol, 1988 6 Bai, K . J. The added mass of two-dimensional cylinders heaving
in water of finite depth, J. Fluid Mech., 1977, 81, 85-105 7 Martin, P. A. and Dalrymple, R. A. Scattering of long waves by
cylindrical obstacles and gratings using matched asymptotic expansions, J. Fluid Mech., 1988, 188, 465-490
8 Yeung, R. W. and Newman, J. N . Discussion paper on Ref. 9, Proc. 11th Symp. on Naval Hydrodynamics, London, 1976,560-561 9 Sayer, P. and Ursell, F. On the virtual mass, at long wavelengths, of a half-immersed circular cyhnder heaving on water of finite depth, Proc. 11th Symp. on Naval Hydrodynamics, London, 1976, 529-564
10 Sayer, P. The long-wave behaviour of the virtual mass in water of finite depth, Proc R. Soc. Lond. A 1980, 372, 65-91 11 Yeung, R. W. Added mass and damping of a vertical cyhnder
in finite-depth waters, ^pp/iedOcea/ii?esearc/!, 1981,3,119-133 12 Mclver, P. and Evans, D . V. The occurrence of negative added
mass in free-surface problems involving submerged oscillating bodies, J. Eng. Math., 1984, 18, 7-22
13 Van Dyke, M . Perturbation Methods in Fluid Mechanics, Parabohc Press, 1975
14 K o f i l f , J. and Mangulis, V. On the Kramers-Kronig relations for ship motions. Int. Shipbiulding Progress, 1962, 9, 3-10 15 Newman, J. N . Marine Hydrodynamics, M . I . T . Press, 1977 16 Cummins, W. E. The impulse response function and ship
motions, Symp. on Ship Theory, Inst, für Shiffbau, Hamburg, 1962 17 Landau, L. D. and Lifshitz, E. M . Statistical Physics, Addison
Wesley, 1980
18 Greenhow, M . High- and low-frequency asymptotic consequences of the Kramers-Kronig relations, J. Eng. Math., 1986, 20, 293-306
A P P E N D I X A
I n this a p p e n d i x the expansions derived i n the m a i n b o d y of the paper are extended to higher order i n o r d e r to relate the integrals o f the potentials a p p e a r i n g i n equations (36) a n d (37).
The i n n e r expansion ofcj) calculated f r o m equations (30) and (31) indicates the need f o r a t e r m at O(e^) i n the i n n e r s o l u t i o n . P u t
^ = - ' 7 + + * ^ o ( ^ ' + 3~) + ^^^2{x, y) + o{s^) £ h Ir
( A l )
where ^ 2 is a h a r m o n i c f u n c t i o n h a v i n g zero n o r m a l derivative o n the b o d y a n d the bed and
dy h^ 0>
y = 0 (A2)
T h e f l u x c o n d i t i o n (11) apphed to 4>2 m a y o n l y be satisfied if the large x e x p a n s i o n c o n t a i n s terms p r o p o r t i o n a l to
p o s i t i v e powers of x up to and i n c l u d i n g x^. M a t c h i n g is possible o n l y i f the next t e r m i n the outer s o l u t i o n is at 0{e) so t h a t £ /; Ii^ where tj),^ satisfies a f h i n the f l u i d region, dy h y = 0 (A3) (A4) (A5)
together w i t h the usual homogeneous c o n d i t i o n o n the bed and the r a d i a t i o n c o n d i t i o n . The s o l u t i o n f o r (j)^ is
a 1 ., -1 = 1 ) ' + ' 7 y + - y M l e "
/! V 2
(A6)
where y is a constant to be f o u n d by m a t c h i n g w i t h the inner s o l u t i o n . W r i t i n g (A3), w i t h (A6), i n terms of inner •coordinates gives the inner expansion
<b=-i~ l+etx — £^x^ — £^;x^ - I - - - l - t - £ ( x — s^x^
s h \ 2 6
J
/ J H 2af 1
+ s{y + i- y+-y^] ( 1 + £jx)-l-0(e^). h \ 2
(A7)
The outer expansion o f the inner s o l u t i o n can m a t c h w i t h this o n l y i f ^2 (pi-y-i-r y+-y h\ 2 1 a -i^x-\—/'-X / i ^ 2 h •-.2_ (A8) and a 1 . 2 \ \ ~ I f l ' 2 l a 3 h \ 2 JJ 2lr 6h as x ^ c o . The value o f y is f o u n d by a p p l i c a t i o n o f (11) to (j>2 and is y = / l i m dx+^-^X^ a^ la 2/t ,- + -3/1 (AlO)
The outer f o r m _ o f 4>i given by (A8) can be used to relate the i n t e g r a l of 4>i over Sg to integrals o f over Sg.
I f Green's t h e o r e m is a p p l i e d over S to *Po and each of (pi, y and y'^—x^ i n t u r n and all integrals i n v o l v i n g a n d 4>i, excepting those over Sg, are e l i m i n a t e d , i t is f o u n d t h a t >iny ds^ ^ ' , . ^ i f - x ' ) d s - 2 i j s„ dn h "fou, ds
/A
2 7 3 /;2 It* where ( A l l ) yds = 0,S^2= i f - x ^ ) d s = ^,V2= ydV, (A 12)the b o d y a n d SQ is the i n t e r v a l — - < j c < - i n > ' = 0 (i.e.
h h
the w a t e r p l a n e 'area'). N o t e t h a t ah quantities are made n o n - d i m e n s i o n a l w i t h /; as the l e n g t h scale as i n the i n n e r region described i n Section 2.
T h e d a m p i n g m a y n o w be expressed i n terms o f the zero-frequency added mass jUs(O) as
4 ( £ ) = - - 6 - j 2 ^ / i s ( 0 ) +
£ a I
Ir
s, Sn(A13)
A P P E N D I X B"*
F r o m the t h e o r y o f C u m m i n s ' ^ we can i d e n t i f y the complex force coefficient
q(aj) = {A{o})~A{co)) + i B{co)
Ü)
( B I )
w i t h the F o u r i e r t r a n s f o r m o f the m e m o r y f u n c t i o n Ö ( T ) f o r w h i c h causahty implies Q(T) S 0 f o r r < 0. Consequently
q{a)) is a n a n a l y t i c f u n c t i o n i n the upper h a l f m plane
h a v i n g the symmetric p r o p e r t y q*{—o}} = q{a}), where * denotes complex conjugate (see L a n d a u and Lifshitz^''). Such a f u n c t i o n satisfies the K r a m e r s - K r o n i g relations, a f o r m o f w h i c h was first given i n this context b y K o t i k a n d Mangulis^"^. H o e v e r , the present e x a m p l e (see equations (36) a n d (37)) shows that q{aji)'^i^ as a;->0 and this pole m o d i f i e s one o f the relations given b y K o t i k a n d M a n g u l i s ^ * . B y c o n t o u r i n t e g r a t i o n a l o n g the real axis i n d e n t e d at Q = 0 a n d Q. = co and a c i r c u l a r c o n t o u r at i n f i n i t y , i t is easy t o show that
q{co) =
1
^ d ü + i^
, Q —CO CÜ
(B2)
where j denotes p r i n c i p l e value i n t e g r a t i o n (see L a n d a u and L i f s h i t z ^ ' , p . 383). E q u a t i n g real a n d i m a g i n a r y parts .and using the s y m m e t r y p r o p e r t y o f q{co) gives
R e [ g ( c o ) ] = -' Ü lm[q{Q.)-'] dD. -co a n d I m [ g ( c ü ) ] -CÜ (B3) (B4)
I n terms o f dimensionless parameters p,„(/?) = HS{E) a n d
Pa{P) = 2.s{£) as used b y K o t i k a n d Manguhs^*, where 'P==Ka = m^a/g = liatanhkh^e^f,-\-Ois'^) as e = kh-^0,
e q u a t i o n (37) is
p^iP)^'-l^.^b,,J''' + oW^'')
( e x p l i c i t l y b.^/^^ia/hy^). W e then have
dt
(B5)
(B6)
* By M . Greenhow, Brunei University.
as i n K o t i k and M a n g u l i s b u t
p.m
- ^1/2 •1/2 'P,„i'X))-p,„it) t ' l \ t - p )
dt.
(B7)w h i c h differs f r o m the r e l a t i o n given by K o t i k and Mangulis^"* b y the a d d i t i o n o f the first t e r m .
W e can n o w use the M e l l i n t r a n s f o r m a p p r o a c h used by G r e e n h o w ' ^ to show that the l o w - f r e q u e n c y expansions o f equations (36) and (37) are consistent w i t h e q u a t i o n s (B5) and (B6) above. Specifically, we need the b e h a v i o u r as x - > 0 o f f i x ) g{t) dt w i t h g{t)=- .1/2 (B8) (B9) U s i n g M e l l i n t r a n s f o r m s (defined b y G{s) = \Qg{t)f ^ dt) we t h e n have f i x ] 1 " 2 ^ X ''G{s)n cot ns ds, i < R e ( s ) < l (BIO) w h i c h , w i t h the assumed f o r m f o r g{t), can be s h o w n t o have simple poles at x = j , 0, — — 1 , . . . . M o v i n g the v e r t i c a l c o n t o u r o f i n t e g r a t i o n t o the left across these poles gives / ( x ) = Ci-|-C2X-t- x ^ O ( B l l ) where
rv
=
u
f git) 0^ 2 — « - i 39(0-
•1/2 t'" I t -dt-h 'ait) dt-2a. 1/2 (B12) a,„t^l^ i , d t + .1/2 'git) dt 1/2 -2a 1/2- (B13) A p p l y i n g ( B l l ) t o e q u a t i o n (B6) w i t h the d a m p i n g given by e q u a t i o n (B5), shows t h a t the added mass as ^ ^ 0 is a p o w e r series i n p , at least as f a r as the hnear t e r m . T h e coefficients o f the series are given b y integrals over the d a m p i n g . S i m i l a r l y , w i t hgit)-Lp,„i<x>)-p,„it)-]/t''\ (B14)
e q u a t i o n ( B l l ) m a y be a p p l i e d t o (B7). T h i s recovers
the series (B5) w i t h an a d d i t i o n a l t e r m where i i / 2 a n d ^3/2 are integrals over the added mass t e r m
p,„(co) —p,„(f). W e c o u l d t h e n a p p l y e q u a t i o n (B6) again to give a q u a d r a t i c t e r m i n the added mass expansion, w h i c h via e q u a t i o n (B7), implies a t e r m i n ^ ^ ' ^ i n the d a m p i n g . C o n t i n u i n g to ' l o o p r o u n d ' i n this f a s h i o n generates f u r t h e r terms i n b o t h expansions. T h i s procedure o f itself does n o t , however, preclude other p o w e r o r l o g a r i t h m i c terms e n t e r i n g the expansions at a higher order, unless the f o r m o f the complete series e x p a n s i o n is k n o w n f o r either the d a m p i n g o r added mass.
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