Second-order oscillatory forces on a body in waves

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AppUed Ocean Research 14 (1992) 325-331

econd-order oseiltetory forces on a body in waves

M. Mclver

Department of Mathematical Sciences, Loughborough University of Teclmoiogy, Loughborough, Leicestershire, LEll 3TU, UK (Received 10 June 1991; accepted 10 January 1992)

When a small amplitude, water-wave train is incident upon a fixed body, a second-order analysis predicts that the body experiences a steady force and a force at twice the frequency o f the incident wave. The double-frequency force is comprised of integrals of products o f linear quantities over the surface of the body and the mean vi^aterline and a term due to the second-order potential. A n application of Green's theorem to the first-order potential and its horizontal derivative shows that the integral of the first order terms over the body is related in a simple way to the waterline integral and the far-field representation of the linear, diffraction potential. A minor modification o f the analysis yields the far-field formulae for the drift force.

1 I N T R O D U C T I O N

T h e i m p o r t a n c e o f i n c l u d i n g n o n l i n e a r effects w h e n c a l c u l a t i n g the wave l o a d i n g o n o f f s h o r e structures is w e l l established. T h e r e are m a n y effects w h i c h c a n n o t be p r e d i c t e d b y linear t h e o r y a n d yet have p r o f o u n d consequences f o r the m o t i o n o f a b o d y . F o r example, M a r u o ' predicted t h a t a floating b o d y is subject t o wave d r i f t i n g , and N e w m a n ^ showed t h a t the i n t e r a c t i o n between wave trains o f d i f f e r e n t frequencies produces forces o n a b o d y at the s u m a n d difference frequencies w h i c h , i n c e r t a i n circumstances, can lead to resonant oscillations o f the b o d y . IMore recently, N e w m a n ^ d e m o n s t r a t e d t h a t the v e r t i c a l f o r c e o n a b o d y w h i c h extends a large distance b e l o w the water surface m a y be d o m i n a t e d b y a c o m p o n e n t a c t i n g at twice the f r e q u e n c y o f the i n c i d e n t wave. F u l l y n o n l i n e a r p r e d i c t i o n s o f the wave l o a d i n g o n a b o d y r e m a i n scarce (see, e.g., B r e v i g et alf"), b u t considerable e f f o r t has been expended i n e x t e n d i n g linear t h e o r y t o second order. A l t h o u g h this a p p r o a c h is l i m i t e d t o small a m p l i t u d e waves a n d m o t i o n s , each o f the effects described above is success-f u l l y predicted b y a second-order analysis. I n p a r t i c u l a r , secondorder t h e o r y predicts t h a t a b o d y i n a m o n o -c h r o m a t i -c wave t r a i n experien-ces b o t h a steady f o r -c e a n d a f o r c e at twice the f r e q u e n c y o f the i n c i d e n t wave.

T h e c a l c u l a t i o n o f the steady f o r c e requires k n o w l -edge o n l y o f the first-order p o t e n t i a l a n d the f o r c e is m o s t d i r e c t l y expressed i n terms o f integrals o f p r o d u c t s o f first-order q u a n t i t i e s over the m e a n w e t t e d surface o f

the b o d y a n d the water line. T h i s a p p r o a c h was used by Ogilvie^ to calculate the mean, v e r t i c a l f o r c e o n a submerged, c i r c u l a r c y l i n d e r . H o w e v e r , M a r u o ^ s h o w e d t h a t the steady, h o r i z o n t a l force o n a b o d y (the d r i f t f o r c e ) m a y be w r i t t e n entirely i n terms o f the f a r - f i e l d ampHtude o f the firstorder d i f f r a c t e d wave. N u m e r i -cally, this has the advantage t h a t i t is n o t necessary t o calculate the g r a d i e n t o f the first-order p o t e n t i a l over the b o d y surface, s o m e t h i n g w h i c h is r e q u i r e d i n the direct a p p r o a c h .

T h e c a l c u l a t i o n o f the d o u b l e - f r e q u e n c y f o r c e is m u c h m o r e c o m p l i c a t e d because i t depends i n p a r t o n the second-order p o t e n t i a l . E v e n the specification o f the correct r a d i a t i o n c o n d i t i o n w h i c h this p o t e n t i a l satisfies has caused m u c h c o n t r o v e r s y a m o n g s t a u t h o r s . H o w -ever, i t is n o w w i d e l y accepted t h a t the m o s t complete d e s c r i p t i o n o f the second-order w a v e field is t h a t given by M o U n . ^ B y d i v i d i n g the second-order p o t e n t i a l i n t o a p a r t w h i c h is l o c k e d t o the first-order system a n d a p a r t w h i c h satisfies the homogeneous f r e e surface c o n d i t i o n , he was able t o a p p l y suitable r a d i a t i o n c o n d i t i o n s to each c o m p o n e n t . These c o n d i t i o n s have been v e r i f i e d b y W a n g , ' w h o l o o k e d at the l o n g - t i m e l i m i t o f the i n i t i a l value p r o b l e m . Despite this agreement, the actual c a l c u l a t i o n o f the f o r c e due t o the second-order p o t e n t i a l remains a n o n t r i v i a l task. B y i n t r o d u c i n g a fictitious, d o u b l e f r e q u e n c y r a d i a t i o n p o t e n t i a l , M o h n ^ and L i g h t h i l l ^ i n d e p e n d e n t l y showed t h a t t h i s f o r c e c o u l d be expressed i n terms o f first-order q u a n t i t i e s . H o w e v e r , the c a l c u l a t i o n requires an i n t e g r a l o f p r o d u c t s o f first-order quantities over the entire f r e e surface, w h i c h is n u m e r i c a l l y t i m e c o n s u m i n g . T h e m o r e direct a p p r o a c h o f finding the second-order p o t e n t i a l

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326 _{M. Mclver}

was u n d e r t a k e n i n t w o dimensions b y V a d a ' ' a n d i n three dimensions by ICim and Y u e / ° but the resulting calcu-lations still r e q u i r e a significant c o m p u t a t i o n a l e f f o r t .

C o n t r i b u t i o n s t o the d o u b l e - f r e q u e n c y f o r c e also arise f r o m the q u a d r a t i c t e r m i n B e r n o u l l i ' s e q u a t i o n a n d f r o m the f a c t t h a t the w e t t e d surface o f the b o d y can change d u r i n g a wave p e r i o d . These c o n t r i b u t i o n s m a y be w r i t t e n i n terms o f integrals o f p r o d u c t s o f first-order quantities over the m e a n w e t t e d surface o f the b o d y a n d the w a t e r Hne. H o w e v e r , the purpose o f this w o r k is to show t h a t this p a r t o f the f o r c e m a y a l t e r n a t i v e l y be expressed i n terms o f the f a r field a m p l i t u d e o f the Hnear, scattered wave, i n a d d i t i o n to a water-line i n t e g r a l . T h e r e s u l t i n g f o r m u l a e are s i m i l a r to those d e r i v e d b y M a r u o ' f o r the d r i f t f o r c e a n d a simple m o d i f i c a t i o n o f the m e t h o d used here p r o v i d e s an alternative d e r i v a t i o n o f his results.

T h e f a r - f i e l d f o r m u l a e f o r the h o r i z o n t a l c o m p o n e n t s o f the d o u b l e - f r e q u e n c y f o r c e are o b t a i n e d b y a p p l y i n g Green's t h e o r e m t o the linear, d i f f r a c t i o n p o t e n t i a l a n d its h o r i z o n t a l derivative. M a n y other i n t e g r a l identides i n h y d r o d y n a m i c s have been derived using the s i m i l a r a p p r o a c h o f a p p l y i n g Green's theorem t o t w o p o t e n -tials, a n d a review o f the existing identities is given by N e w m a n . " F u l l details o f the d e r i v a t i o n o f t h e f o r m u l a f o r a t w o - d i m e n s i o n a l b o d y are presented i n the next secdon. The a p p r o p r i a t e m o d i f i c a t i o n s f o r a three-d i m e n s i o n a l b o three-d y are given i n section 3, a n three-d i n the final section, a p p l i c a d o n s t o specific b o d y shapes are given.

2 T W O - D I M E N S I O N A L A N A L Y S I S

A fixed, t w o - d i m e n s i o n a l b o d y is either p a r t i a l l y or t o t a l l y immersed i n an i n f i n i t e l y deep fluid, as i l l u s t r a t e d i n F i g . 1. F o r convenience, i t is assumed t h a t i f the b o d y intersects the u n d i s t u r b e d free surface then i t does so at r i g h t angles. Rectangular Cartesian coordinates x a n d z are chosen so that the x-axis p o i n t s a l o n g the free surface a n d the z-axis p o i n t s v e r t i c a l l y u p w a r d s . T h e line

L

z = 0 coincides w i t h the rest p o s i t i o n o f the free surface. T h e fluid is assumed to be i n v i s c i d a n d incompressible a n d the m o t i o n is i r r o t a t i o n a l . T h u s , the flow is described b y a velocity p o t e n t i a l , w h i c h satisfles

V ' $ = 0 ₍₁₎

i n the fluid. T h e wave steepness e is assumed t o be smaU a n d so the v e l o c i t y p o t e n t i a l , $ , a n d wave elevation, C, m a y be expanded as

a n d

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(3) T h e equations and b o u n d a r y c o n d i t i o n s g o v e r n i n g the first-order p o t e n t i a l $ i are a v a ü a b l e i n m a n y s t a n d a r d texts (e.g. M e i ' ^ ) . F o r convenience, these equations are summarised below. T h u s , e $ i is w r i t t e n as

= Re -\gA ₍₄₎

where A is the a m p l i t u d e o f the first-order i n c i d e n t wave, LÜ is its f r e q u e n c y and g is the acceleration due t o g r a v i t y . T h e time independent p o t e n t i a l satisfies eqn (1) a n d the free surface c o n d i t i o n

/C^i - ^ := 0 o n z = 0

dz (5)

where

K=^''/g (6) There is n o flow t h r o u g h the surface o f the b o d y a n d so

d4>i/d>i = 0 o n (7) where is the instantaneous wetted surface o f the b o d y

a n d d/dn is a derivative i n the d i r e c t i o n o f the i n w a r d n o r m a l to S^. T h e fluid is a t rest at large depths, thus

V 0 , 0 as z ^ - o o

₍₈₎

I n a d d i t i o n , i f the incident wave is t r a v e f l i n g i n the p o s i t i v e X d i r e c t i o n , (pi satisfies the f a r - f i e l d c o n d i t i o n

i ( x , z ) (e iKx Re-'^-^l e^^ iKx+Kz as X as X -oo ( 9 )

Fig. 1. Definition sketch.

where R a n d T are the first-order r e f l e c t i o n and transmission coefficients respectively.

N u m e r i c a l procedures f o r finding (p^ a n d the associated first-order forces o n a b o d y are w e l l estabhshed a n d described i n M e i . ' ^ T h e second-order f o r c e m a y be spHt i n t o a steady p a r t a n d a p a r t a c t i n g at the f r e q u e n c y 2w. T h e steady force is d e t e r m i n e d solely f r o m (f>i whereas the d o u b l e - f r e q u e n c y f o r c e depends o n b o t h 01 a n d the second-order p o t e n t i a l $ 2 . B y i n t e g r a t i n g the pressure over the w e t t e d surface o f the b o d y a n d e x p a n d i n g the r e s u l t i n g i n t e g r a l t o O(e^), i t m a y be s h o w n t h a t the c o n t r i b u t i o n t o the second-order

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Second-order oscillatory forces on a body in waves 321

h o r i z o n t a l f o r c e f r o m the first-order p o t e n t i a l is given b y T h e f a r - f i e l d b e h a v i o u r o f ip is given b y e^F^l\ where .§,2 P 2g [V^^fn^dS {a,Q) FJ. + Re [ƒ; 2.V(1) „-2it^'l e J (10) ( T h i s is the t w o - d i m e n s i o n a l analogue o f the three-d i m e n s i o n a l f o r m u l a o b t a i n e three-d by M e i ' ^ ( p . 662, eqn 9.34b).) Here, is the m e a n w e t t e d surface o f t h e b o d y ,

is the x c o m p o n e n t o f the u n i t i n w a r d n o r m a l to a n d the b o d y is assumed t o intersect the u n d i s t u r b e d free surface at r i g h t angles, at the p o i n t s ( - a , 0) a n d [a, 0 ) . S u b s t i t u t i o n o f eqn (4) i n t o (10) gives, a f t e r some m a n i p u l a t i o n ,

pgA^ AK ( V 0 , ) ' " v d 5 + - [ < / . ! ( - « , 0 ) - < / . ? ( « , 0)] (11) A representation f o r the i n t e g r a l i n eqn (11) i n terms o f the f a r - f i e l d f o r m o f and its value at the intersection p o i n t s is derived below.

I t is convenient t o i n t r o d u c e the f u n c t i o n s

^ = d4>i/dx (12) a n d

X = - d c P y / d z (13) B o t h f u n c t i o n s satisfy Laplace's e q u a t i o n i n the fiuid

a n d , m o r e o v e r , x is the h a r m o n i c c o n j u g a t e t o ip. C l e a r l y , ip decays w i t h d e p t h and, u n d e r the a s s u m p t i o n t h a t the tangendal derivatives o f tpi a n d d^\/dz are c o n t i n u o u s o n the b o u n d a r y z = 0, ijj satisfies the free surface c o n d i t i o n , eqn ( 5 ) . I f the b o d y intersects the free surface, care needs to be taken t o d e t e r m i n e the b e h a v i o u r o f •0 near the intersection p o i n t s . Wigley''* d e r i v e d expressions f o r the a s y m p t o t i c expansion o f a h a r m o n i c f u n c t i o n near a corner i n t w o dimensions. T h e precise b e h a v i o u r o f the f u n c t i o n depends o n the b o u n d a r y c o n d i d o n s t o be satisfied o n the Hnes f o r m i n g the corner a n d the v a l u e o f the i n c l u d e d angle, w h i c h is here assumed to be 7r/2. T h e b o u n d a r y value p r o b l e m f o r (pi near one o f the intersection p o i n t s , say t h a t at

(a, 0 ) , is a p a r t i c u l a r example o f the b o u n d a r y value p r o b l e m I I I described b y Wigley.''* H e showed t h a t as cpi is b o u n d e d at the corner, i t has an a s y m p t o t i c expansion there w h i c h is a p o l y n o m i a l i n ( Z , Z , Z ^ In Z , Z ^ I n Z ) where Z is the c o m p l e x variable x — a - f jz a n d the o v e r b a r denotes c o m p l e x conjugate. F u r t h e r m o r e , expansions f o r the derivatives o f <pi m a y be o b t a i n e d b y d i f f e r e n t i a t i n g f o r m a l l y . T h u s , the first derivatives o f (pi are b o u n d e d at the corner, b u t there is the p o s s i b ü i t y o f a l o g a r i t h m i c s i n g u l a r i t y i n the second derivatives o f (pi, t h e r e f o r e i n the first derivatives o f -0, at this p o i n t .

• i i i : ( e ' ^ - ^ ' - R e - ' ^ - ^ ) e ^ ^ i i s : T e ' ^ ^ + ^ ip{x,z) as X —> —oo as X —> oo (14) A p p l i c a t i o n o f Green's t h e o r e m t o the t w o h a r m o n i c f u n c t i o n s (pi and ip, a r o u n d the c o n t o u r C i ü u s t r a t e d i n F i g . 1, gives

dn dn (15)

where d/dn is the derivative i n the d i r e c t i o n o f the o u t w a r d n o r m a l t o the fluid. T h e possible singularities i n dip/dn at the intersection p o i n t s o f the b o d y a n d the f r e e surface are n o t s u f f i c i e n t l y s t r o n g t o p r o d u c e c o n t r i -b u t i o n s t o the i n t e g r a l f r o m these p o i n t s . T h u s , as -b o t h (pi and Ip satisfy eqn ( 5 ) , c o n t r i b u t i o n s t o the i n t e g r a l arise o n l y f r o m the b o d y surface a n d the closing lines as X —> ± o o . A f t e r some m a n i p u l a t i o n , using eqns ( 7 ) - ( 9 ) a n d (14), the i n t e g r a l i n eqn (15) m a y be r e w r i t t e n as

dip

2KR=0 ; i 6 )

o n (17)

B y d e f i n i t i o n ,

dip dip dip dn dx dz

where n^ a n d n^ ate the x a n d z c o m p o n e n t s respectively o f the u n i t i n w a r d n o r m a l t o the b o d y . T h e u n i t vector p o i n t i n g i n the a n t i c l o c k w i s e , t a n g e n t i a l d i r e c t i o n to the b o d y is given b y {n^, n f ) and so, using the C a u c h y -R i e m a n n equations,

dipldn = -dxlds o n (18) where d/ds is the a n t i c l o c k w i s e , t a n g e n t i a l derivative.

S u b s t i t u t i o n o f e q n (18) i n t o eqn (16), i n t e g r a t i o n b y parts a n d use o f eqn (5) gives

K

d(p\ d^i

s„ dz ds ÓS=^]{a,0) a , 0 ) + 27? (19)

B y expressing drpi/dn a n d d(p\/ds o n S-g i n terms o f the X a n d z derivatives of cpi a n d u s i n g eqn ( 7 ) , i t is possible t o s h o w t h a t d(px/dz — -n^d<px/ds o n 5'b T h u s , eqn (19) m a y be r e w r i t t e n as (20)

i f / ^ ^ V

K ] s X d s ) "-^ d 5 = - [ 2 7 ? + 0 ? ( ö , O ) - 0 ? ( - « , O ) ] (21) ( T h i s expression m a y a l t e r n a t i v e l y be d e r i v e d f r o m eqn (16) using the i n t e g r a l i d e n t i t y given b y B r a n d ' ^ (p. 220, eqn 4 ) . ) A s <p\ satisfies the b o d y b o u n d a r y c o n d i t i o n ( 7 ) , {y(p\Y = {d<p\/ds)^ o n a n d so s u b s t i t u t i n g eqn (21) i n t o eqn (11) gives

ML

pgA^ 1 \R+<p\{a,Q)-cp\{-aM (22)

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3 2 8 M. Mclver

I f the b o d y is c o m p l e t e l y submerged, eqn ( 2 2 ) is replaced by

2.(1)

pgA^

R

2 ( 2 3 )

A s t r a i g h t f o r w a r d m o d i f i c a t i o n to the analysis yields a representation f o r /^^^ w h e n the fluid is n o t i n f i n i t e l y deep b u t has a d e p t h h. T h e details are o m i t t e d b u t the r e s u l t i n g expressions f o r ƒ ^J^-* are

pgA^

(sinh kh cosh kh + kh)R sinh kh cosh kh

+ </.?(«, 0 ) - , ^ ? ( - f l , 0 )

i f the b o d y intersects the free surface and

eYjl

_ (sinh kh cosh kh + kh)R

pgA^ 2 s i n h kh cosh kh

i f the b o d y is submerged, where

K=k t a n h kh

( 2 4 )

( 2 5 )

( 2 6 )

A f u r t h e r m o d i f i c a t i o n to the analysis whereby Green's t h e o r e m is a p p l i e d to t ^ i , the c o m p l e x c o n j u g a t e io (pi, a n d produces the f a r - f i e l d f o r m u l a e o b t a i n e d by M a r u o ^ f o r the h o r i z o n t a l d r i f t f o r c e .

3 T H R E E - D I M E N S I O N A L A N A L Y S I S

I n three dimensions, the first-order p o t e n t i a l (j)y{x,y,z) is w r i t t e n as

, JKx+Kz I /

( 2 7 )

where cp^ satisfies eqns ( 1 ) , ( 5 ) , ( 7 ) a n d ( 8 ) a n d

\-nKR

T h e p o l a r coordinates R a n d 9 are d e f i n e d b y

X = R cos 9 a n d y = R sin 9 a n d the K o c h i n f u n c t i o n H{9) describes the angular v a r i a t i o n o f the scattered

wave. T h e o s c i l l a t o r y , second order, h o r i z o n t a l f o r c e due to the first-order p o t e n t i a l is split i n t o c o m p o n e n t s i n the x and y d i r e c t i o n s , ƒ a n d / V / respectively, where 0 )

2.(1) pgA^a 1 ( V 0 , 2y

!>f;;,dr

( 2 9 ) where j = x o r y, a is a t y p i c a l d i m e n s i o n o f the b o d y a n d r is the m e a n w a t e r l i n e . F o l l o w i n g the m e t h o d o f the p r e v i o u s section, an expression f o r /^J^' is o b t a i n e d by a p p l y i n g Green's t h e o r e m t o 0 i a n d = d(p]/dx. T h e i n t e g r a l f o r m u l a ( 1 5 ) is o b t a i n e d , where C is n o w the surface consisting o f the u n d i s t u r b e d free surface, the m e a n w e t t e d surface o f the b o d y and c o n t r o l surfaces at large d e p t h a n d as 7? ^ oo. (The a s s u m p t i o n is m a d e t h a t the b e h a v i o u r o f Vtjj at the w a t e r l i n e is n o t

s u f f i c i e n t l y singular t o c o n t r i b u t e t o the i n t e g r a l . H o w e v e r , to the a u t h o r ' s k n o w l e d g e this has n o t been p r o v e d a n d there are no general results w h i c h determine the b e h a v i o u r o f a h a r m o n i c f u n c t i o n near a b o u n d a r y c o r n e r i n three dimensions.) B o t h cpi a n d ijj s a t i s f y the linearised, f r e e surface c o n d i t i o n and decay w i t h d e p t h a n d so c o n t r i b u t i o n s t o eqn ( 1 5 ) arise o n l y f r o m the b o d y surface a n d the surface as i? ^ oo. T h e m e t h o d o f s t a t i o n a r y phase is used t o evaluate the i n t e g r a l over the c o n t r o l surface at i n f i n i t y a n d , a f t e r some m a n i p u l a t i o n eqn ( 1 5 ) becomes h ^ d S = -4H{IT) ( 3 0 ) T h e i n t e g r a l i n eqn ( 3 0 ) m a y be r e w r i t t e n b y o b s e r v i n g t h a t V t / ) = V A ( 0 , d(t>x/dz, -dc^i/dy) = V A B say. T h u s , V A ((7iiB).ndS {V(pi/\B).ndS ( 3 1 )

T h e first i n t e g r a l on the r i g h t - h a n d side o f eqn ( 3 1 ) m a y be evaluated using Stokes' t h e o r e m , a n d direct substi-t u substi-t i o n o f 4&gsubsti-t;i a n d B i n substi-t o substi-the second i n substi-t e g r a l gives

an

-K bin^ d P - f _{( V 0 i ) X - d 5 ( 3 2 )}

where the free surface c o n d i t i o n (eqn ( 5 ) ) a n d the b o d y b o u n d a r y c o n d i t i o n (eqn ( 7 ) ) have been used. ( A s i n the p r e v i o u s section, eqn ( 3 2 ) m a y also be d e r i v e d f r o m eqn ( 3 0 ) b y u s i n g the i n t e g r a l i d e n t i t y given b y B r a n d ' ^ (p. 2 2 0 , eqn 4 ) . ) S u b s t i t u t i o n o f eqn ( 3 0 ) i n t o eqn ( 3 2 ) a n d t h e n i n t o eqn ( 3 1 ) gives 2Al) pgA^a 1 2a b\n^

dr -

Ka I f the b o d y is c o m p l e t e l y submerged, replaced by

ML

pgA^a Ka ( 3 3 ) eqn ( 3 3 ) is ( 3 4 )

T h e expression f o r the y c o m p o n e n t o f the f o r c e m a y be o b t a i n e d b y a p p l y i n g Green's t h e o r e m t o pi a n d

d(pi/dy. A s i m i l a r analysis yields

ML

pgA^a ]_ 2a b\nyAV f o r a surface-piercing b o d y a n d 2.(1) pgA^a

0

( 3 5 ) ( 3 6 )

f o r a submerged b o d y . I t is interesting t o observe t h a t the first-order p o t e n t i a l does n o t c o n t r i b u t e to the d o u b l e f r e q u e n c y f o r c e o n a submerged b o d y , i n the

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Second-order oscUlatory forces on a body in waves 329

d i r e c t i o n p e r p e n d i c u l a r to the i n c i d e n t wave advance, irrespective o f any s y m m e t r y i n the b o d y shape.

E a c h o f the expressions m a y be m o d i f i e d t o a l l o w f o r the effect o f finite depth. The resulting f o r m u l a e are equiv-alent t o those i n eqns ( 3 3 ) - ( 3 6 ) w i t h H{-ïï)/Ka replaced by H{TX) (sinh Ich cosh Ich + Jdi)/{Ica sinh kh cosh Icli). A s i n the t w o - d i m e n s i o n a l case, the f a r - f i e l d f o r m u l a e f o r the h o r i z o n t a l d r i f t f o r c e derived b y M a r u o ' m a y be o b t a i n e d by a p p l y i n g Green's t h e o r e m to ip^ and d4>\/dx o r dpi/dy. F u r t h e r m o r e , the expression f o r the steady v e r d c a l m o m e n t o n a three-dimensional b o d y , derived by N e w m a n , ' ^ m a y be o b t a i n e d b y a p p l y i n g Green's t h e o r e m to cpi a n d xd(pi/dy -ydcpi/dx,

4 R E S U L T S A N D D I S C U S S I O N

E q u a t i o n s ( 2 2 ) - ( 2 5 ) a n d ( 3 3 ) - ( 3 6 ) are expressions f o r the h o r i z o n t a l c o m p o n e n t s o f the o s c i l l a t o r y , second-order f o r c e due to the f i r s t - o r d e r p o t e n t i a l w h i c h obviate the need to evaluate Vcpi o n the b o d y surface. N u m e r i c a l l y this is m o r e efficient, b u t the f o r m o f the expressions also a l l o w s deductions t o be m a d e m o r e easily a b o u t the force o n certain special bodies.

D e a n " first p r o v e d t h a t there is n o r e f l e c t i o n f r o m a submerged, c i r c u l a r c y l i n d e r at any frequency. F r o m eqn ( 2 3 ) , an i m m e d i a t e consequence is t h a t the first-order p o t e n t i a l does n o t c o n t r i b u t e t o the second-first-order, h o r i z o n t a l , o s c i f l a t o r y f o r c e . T h i s result was observed b y W u a n d E a t o c k T a y l o r ' ^ w h o integrated {Vcpifn^ over the surface o f the cyhnder d i r e c t l y . T h e y f o u n d , however, that there is a non-zero c o n t r i b u t i o n to the d o u b l e - f r e q u e n c y f o r c e f r o m the second-order p o t e n t i a l T h i s supports the observations m a d e b y K i m a n d Y u e a n d N e w m a n ^ t h a t i t is n o t , i n general, correct to assume t h a t the d o m i n a n t c o n t r i b u t i o n to the second-order o s c i l l a t o r y f o r c e arises f r o m the first-second-order p o t e n t i a l . One b o d y f o r w h i c h the firstorder c o n t r i -b u t i o n is d o m i n a n t , however, is a su-bmerged, h o r i z o n t a l plate. T h e second-order p o t e n t i a l is finite at the plate tips a n d so does n o t c o n t r i b u t e to the f o r c e b u t the first-order v e l o c i t y is i n f i n i t e there, resulting i n a non-zero h o r i z o n t a l f o r c e o n the plate, given b y eqn ( 2 3 ) . T h u s , the v a r i a t i o n o f this f o r c e m a y be derived i m m e d i a t e l y f r o m the graphs o f the r e f l e c t i o n coefiicient associated w i t h a plate, presented b y M c l v e r . I t is also reasonable t o suppose t h a t the first-order c o n t r i b u t i o n t o the f o r c e m a y be d o m i n a n t w h e n the v e r t i c a l extent o f the b o d y is m u c h less t h a n its w i d t h a n d large velocities occur near the ends o f the b o d y .

C a l c u l a t i o n s o f the f o r c e o n a semicircular c y l i n d e r i n the f r e e surface were made f r o m eqn (22), using the m u l t i p o l e f o r m u l a t i o n o f M a r t i n a n d D i x o n . I t was observed n u m e r i c a l l y t h a t 10 AR (37) at a l l frequencies a n d so eqn (22) m a y be r e w r i t t e n as

ML

pgA' 3R 2 (38)

f o r this b o d y . T h u s , n u m e r i c a l values o f this c o m p o n e n t o f the force at d i f f e r e n t frequencies m a y be o b t a i n e d i m m e d i a t e l y f r o m the table o f r e f l e c t i o n coefficients p r o v i d e d by M a r t i n a n d D i x o n . T h e results coincide w i t h the c o m p u t a t i o n s o f W u and E a t o c k T a y l o r ' ^ a n d o f Isaacson and Cheung,^' w h o integrated the pressure d i r e c t l y over the b o d y . T h e a u t h o r was, however, unable t o p r o v e the result i n eqn (38) and i t is c e r t a i n l y n o t t r u e f o r a general shaped b o d y because, as is d e m o n -strated b e l o w , i t f a i l s f o r the surface-piercing, v e r t i c a l b a r r i e r .

Ursell^^ derived a n e x p l i c i t expression f o r the first-order p o t e n t i a l associated w i t h the b a r r i e r , w h i c h is given b y JKx+Kz Re AKx+Kz Te [Kx+Kz C(A-,Z) + C{x, z) i f X < 0 i f A' > 0 where R -mf{bK) T: C Kx{bK)-nf{bK) KiibK)-mf (bK) Ki{bK)-iTxh{bK)] -k\x\ •^JijbK) 0 K^ + k^ X e [k cos /cz 4- K sin /cz] d/c (39) (40) (41) (42) T h e p a r a m e t e r b is the l e n g t h o f the b a r r i e r , Ji is a Bessel f u n c t i o n a n d f a n d Ki are m o d i f i e d Bessel f u n c t i o n s . S u b s t i t u t i o n o f eqn (39) i n t o eqn (22) gives

ML

3R 2 2i Ki{bK) - m f i b K ) ' k f j b k ) - f /c2 d/c (43) is n o t zero i n general eqn 6.566.5) a n d so T h e i n t e g r a l i n eqn (43) ( G r a d s h t e y n & Ryzhik,^^

^^fil/PS^^ evidentiy does n o t equal 3 i ? / 2 i n this case.

I n three dimensions, the first-order p o t e n t i a l associ-ated w i t h a v e r t i c a l , c i r c u l a r c y l i n d e r extending t h r o u g h o u t the d e p t h m a y be w r i t t e n d o w n e x p l i c i t l y (see, e.g., M e i , ' ^ p . 313), as cosh /c(z - f h) cosh kh 11=0 J„{lcR)

H'„{ka) HnikR) cos nO

where

(44)

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330 _{M. Mclver}

J„ is a Bessel f u n c t i o n , H„ is a H a n k e l f u n c t i o n o f the tirst k i n d a n d a is the radius o f the cylinder. T h e K o c h i n f u n c t i o n , H{9), is o b t a i n e d by e x a m i n i n g the large argument behaviour o f the H a n k e l f u n c t i o n . T h i s a n d the expression f o r 0 i a r o u n d the waterline are substituted i n t o the version o f eqn (33) m o d i f i e d to a l l o w f o r finite depth, t o give

Ml

8i _-1)"

PgA^a nika)' H'„{ka)H'„^, {ka)

ka V sinh 2kh

f^e„{-iyj;xka)

h H'„{ka) (46) where the i n t e g r a t i o n i n eqn (33) has been p e r f o r m e d explicitly. C o m p u t a t i o n s o f this expression were made f o r several values o f a/h a n d a c o m p a r i s o n w i t h the predicdons o f ICim a n d Y u e ' " ( T a b l e 4) show exact agreement w i t h the n u m e r i c a l values calculated f r o m the f o r m u l a

M

2i pgA'-a 7r(/ca) ,1=0 " " ( - 1 ) " ;,{ka)H'„+y{ka) 2kh

+

« ( « + ! ) sinh 2kh {k 2kh sinh 2kh (47) f o r a cyhnder f o r w h i c h a/h = 1. T h i s latter f o r m u l a was o b t a i n e d by direct i n t e g r a d o n o f the pressure over the body. K i m a n d Y u e ' ° also calculated the c o n t r i b u d o n t o the second-order force f r o m the second-order p o t e n t i a l and, b y c o m p a r i n g the m a g n i t u d e o f the t w o c o n t r i -b u t i o n s f o r various wave frequencies, concluded that there is no range o f frequencies i n w h i c h the neglect o f the c o n t r i b u t i o n f r o m the second-order p o t e n t i a l c o u l d be j u s t i f i e d . Despite the n u m e r i c a l agreement o f the t w o series i n eqns (46) a n d (47), i t is n o t o b v i o u s h o w one series is t o be t r a n s f o r m e d i n t o the other a n a l y t i c a l l y . C o m p u t a t i o n a l l y , neither series offers any great advan-tage over the other, b u t this is p r i m a r i l y because the latter is o b t a i n e d by e v a l u a t i n g V ^ i j o n the cylinder analytically. T h i s w o u l d n o t be possible f o r a general shaped b o d y , i n w h i c h case the f a r - f i e l d f o r m u l a t i o n s h o u l d be n u m e r i c a l l y m o r e efficient.

5 C O N C L U S I O N

F o r m u l a e have been derived w h i c h relate the h o r i z o n t a l , oscillatory second-order f o r c e due t o the first-order p o t e n t i a l to the f a r - f i e l d a m p l i t u d e o f the linear, d i f f r a c t e d wave and the value o f the f i r s t - o r d e r p o t e n t i a i a r o u n d the waterhne. T h e resulting expressions e l i m i -nate the need t o evaluate the gradient o f the first-order p o t e n t i a l over the surface o f t h e b o d y a n d enable simple observations t o be made a b o u t the f o r c e o n certain

special bodies. A n e x a m i n a t i o n o f the f o r m u l a e f o r p a r t i c u l a r bodies shows that the first-order p o t e n t i a l does n o t , i n general, f o r m the d o m i n a n t c o n t r i b u t i o n t o the h o r i z o n t a l , double frequency f o r c e , a l t h o u g h an exception t o this is that the f o r c e o n a submerged, h o r i -z o n t a l plate is comprised entirely o f this c o n t r i b u t i o n .

A l t e r n a t i v e expressions f o r the v e r t i c a l d o u b l e f r e q u e n c y force o n a b o d y m a y also be o b t a i n e d , by a p p l y i n g Green's theorem to a n d dcjii/dz. H o w e v e r , the s i m p l i c i t y o f the expressions f o r the h o r i z o n t a l f o r c e is a consequence o f the f a c t that the h o r i z o n t a l d e r i v a t i v e o f the p o t e n t i a l satisfies the free surface c o n d i t i o n . This is n o t true o f the v e r t i c a l derivative o f the p o t e n t i a l a n d so the resulting f o r m u l a f o r the v e r t i c a l f o r c e involves an integral over the free surface, w h i c h provides no s i m p l i f i c a t i o n o f the o r i g i n a l expression. S i m i l a r l y , alternative expressions f o r the sum a n d difference frequency forces o n a b o d y i n b i c h r o m a t i c waves m a y be obtained, b y a p p l y i n g Green's t h e o r e m to the first-order p o t e n t i a l associated w i t h one wave a n d a d e r i v a t i v e o f the other first-order p o t e n t i a l . H o w e v e r , this also leads t o f o r m u l a e i n v o l v i n g integrals over the free surface as the t w o potentials satisfy d i f f e r e n t free surface b o u n d a r y c o n d i t i o n s . T h u s , a l t h o u g h the t h e o r y presented here can be extended t o p r o v i d e expressions f o r v e r t i c a l forces a n d sum a n d difference frequency forces, i t is expected that the p r a c t i c a l a p p l i c a t i o n o f the f o r m u l a e w i l l be l i m i t e d t o the c a l c u l a t i o n o f the d o u b l e f r e q u e n c y h o r i z o n t a l f o r c e .

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