The Ultimate Image Singularities for External Spheroidal and Ellipsoidal Harmonics

THE ULTIMATE IMAGE SINGULARITIES

FOR EXTERNAL SPHEROIDAL

AND ELLIPSOIDAL HARMONICS

by

T o i i v i a M i l o l i

sponsored by •Ofries of Naval Research

Fluid Dynamics Branch Contract No. N00014-68-A-0196-0004

I I H R Report No. 146

Iowa Institute of Hydraulic Research The University of lovi'a

Iowa Git)', l o v a

A b s t r a c t

The image system o f s i n g u l a r i t i e s o f an a r b i t r a r y e x t e r i o r p o t e n t i a l f i e l d w i t h i n a t r i ~ a x i a l e l l i p s o i d i s d e r i v e d . I t i s f o u n d t h a t t h e image system c o n s i s t s o f a s o u r c e and d o u b l e t d i s t r i b u t i o n o v e r t h e f u n d a m e n t a l e m p s o i d . The p r e s e n t c o n t r i b u t i o n i s a g e n e r a l i z a t i o n o f p r e v i o u s t h e o r i e s on t h e image system o f an e x t e r i o r p o t e n t i a l f i e l d w i t h

-i n a sphere and s p h e r o -i d . A p r o o f o f Havelock's s p h e r o -i d theorem w h -i c h a p p a r e n t l y i s n o t a v a i l a b l e i n t h e l i t e r a t u r e i s a l s o g i v e n .

The knowledge o f t h e image system i s r e q u i r e d , f o r example, when h y d r o d y n a m i c a l f o r c e s and moments a c t i n g on an e l l i p s o i d iirniiereed i n a p o t e n t i a l f l o w a r e computed by t h e L a g a l l y theorem.

• The two examples g i v e n c o n s i d e r t h e image system o f s i n g u l a r i t i e s o f an e l l i p s o i d i n a u n i f o r m t r a n s l a t o i y m o t i o n and i n p u r e r o t a t i o n .

The a u t h o r i s i n d e b t e d t o P r o f e s s o r L. Landweber f o r s t i m u l a t i n g d i s c u s s i o n s .

T h i s r e p o r t i s based upon r e s e a r c h s p o n s o r e d b y t h e F l u i d Mechanics B r a n c h , O f f i c e o f N a v a l Research under C o n t r a c t NO0Oll+»-68-A.^0196~OOOIK

T a b l e o f C o n t e n t s

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

The Image System f o r S p h e r o i d The Image System f o r E l l i p s o i d Examples

Exa.mple 1: Pure T r a n s l a t i o n Example 2: Pure R o t a t i o n References

THE ULTIMATE IMAGE SINGU.LARITIES FOR EXTERWAL SPHEROIDAL AND ELLIPSOIDAL HARIvIOWICS

l j _ _ _ I n t r o du c t i o n

The L a g a l l y theorem [ l ] t o g e t h e r w i t h i t s r e c e n t g e n e r a l i z a t i o n s [ 2 , 3] y i e l d s e x a c t e x p r e s s i o n s f o r t h e f o r c e s and moments on Rankine b o d i e s immersed i n a r b i t r a r y i n v i s c i d , p o t e n t i a l f l o w s . The main d i f f i c u l t y w i t h t h e a p p l i c a t i o n o f t h e L a g a l l y t h e o r e m i s t h a t i t i s n e c e s s a r y t o know t h e image system o f s i n g u l a r i t i e s a s s o c i a t e d w i t h t h e a n a l y t i c c o n t i n u a t i o n o f t h e e x t e r n a l p o t e n t i a l f l o w i n t o t h e body. The s p h e r e i s t h e o n l y t h r e e - d i m e n s i o n a l shape f o r w h i c h t h e r e e x i s t s a " s p h e r e t h e o r e m " [4,5,6] y i e l d i n g t h e d i s t u r b a n c e p o t e n t i a l due t o t h e p r e s e n c e o f t h e sphere i n terras o f t h e u n d i s t u r b e d p o t e n t i a l . A l s o a t h e o r e m due t o Hobson [7] ( p . 13^+) p r o v i d e s an e x p r e s s i o n f o r any s p h e r i c a l harmonic i n t e r m s o f s i n g u l a r i t i e s ( sources.;doublets o r m a l t i p o l e s ) a t t h e c e n t e r o f t h e s p h e r e , ' ' n Ü ± ! Ü - (-1)1 ill a , . A^ll where ( R , 6, (j)) a r e s p h e r i c a l c o o r d i n a t e s , y = c o s e , P^^^(M) i s t h e Legendre f u n c t i o n o f t h e f i r s t k i n d d e f i n e d b y ( 5 ) , and n and s a r e p o s i t i v e i n t e g e r s . An i n v e r s i o n t h e o r e m s i m i l a r t o t h e sphere t h e o r e m , i s n o t a v a i l -a b l e f o r -a s p h e r o i d . However, -a most u s e f u l r e l -a t i o n w-as g i v e n w i t h o u t p r o o f b y H a v e l o c k [ 8 ] . T h i s r e l a t i o n e x p r e s s e s an e x t e r i o r s p h e r o i d a l h a r -monic i n term.s o f s i n g u l a r i t i e s d i s t r i b u t e d on t h e m a j o r a x i s o f t h e s p h e r o i d between t h e t w o f o c i . I f t h e f o c i a r e chosen t o be a t (±1,0,O) i n C a r t e s i a n n o t a t i o n , t h e H a v e l o c k f o r m u l a may be w r i t t e n as

s

v/here a r e s p h e r o i d a l c o o r d i n a t e s d e f i n e d by {h), and Q^^(?) i s t h e Legendre f u n c t i o n o f t h e second k i n d g i v e n by ( 7 ) .

A p r o o f o f t h e H a v e l o c k f o n n u l a (2) i s n o t a v a i l a b l e i n t h e l i t e r a -t u r e . However p r o o f s o f (2) a r e g i v e n i n an u n p u b l i s h e d -t h e s i s [9] and i n c l a s s n o t e s o f P r o f e s s o r L. Landweber, These p r o o f s a r e based on a s u g g e s t i o n o f H a v e l o c k t o expand t h e f u n c t i o n l/R i n t e r m s o f an i n f i n i t e s e r i e s o f

s p h e r o i d a l . h a r m o n i c s . A b r i e f e r p r o o f o f t h e Havelock f o r m u l a w i l l be p r e s e n t e d i n s e c t i o n 2. T h i s p r o o f i s based on a n a l y t i c c o n t i n u a t i o n 8,pplied t o a g e n e r a l i z a t i o n o f bhe Neumann f o r m u l a f o r t h e Legendre f u n c t i o n o f t h e second k i n d Q^'^(C) •

E q u a t i o n s ( l ) and (2) p r o v i d e e x a c t e x p r e s s i o n s f o r t h e image system w i t h i n a sphere o r s p h e r o i d o f an a r b i t r a r y u n d i s t u r b e d e x t e r n a l p o t e n t i a l f l o w . Once t h e image system o f s i n g u l a r i t i e s i s known w i t h i n t h e b o d yj t h e d i s t u r -bance p o t e n t i a l may be w r i t t e n i r a r a e d i a t e l y i n terms o f t h e s e s i n g u l a r i t i e s .

A c c o r d i n g t o Morse and Feshbach [ l O j t h e most g e n e r a l c o o r d i n a t e s f o r t h e s e p a r a b i l i t y o f t h e L a p l a c e e q u a t i o n a r e t h e e l l i p s o i d a l or t h e f o c a l c o o r d i n a t e s . The e l l i p s o i d a l harmonics a r e a l s o t h e most g e n e r a l h a r m o n i c s w h i c h a r e a s o l u t i o n o f t h e t h r e e - d i m e n s i o n a l L a p l a c e e q u a t i o n . I n f a c t t h e t r i - a x i a l e l l i p s o i d i s a t r u l y t h r e e - d i m e n s i o n a l f o r m , w h i l e b o t h t h e s p h e r e and t h e s p h e r o i d a r e a x i s y n m i e t r i c a l f o r m s .

An e l l i p s o i d t h e o r e m , s i m i l a r t o t h e sphere ( l ) and t h e s p h e r o i d

(2) t h e o r e m s , w o u l d be most i m p o r t a n t i n s h i p hydrodynamics s i n c e s h i p f o r m s can be b e t t e r a p p r o x i m a t e d by a t r i - a x i a l e l l i p s o i d t h a n by a s p h e r o i d . Such a t h e o r e m , w h i c h y i e l d s t h e s i n g u l a r i t y system w i t h i n t h e e l l i p s o i d o f an

-3-a r b i t r -3-a r y e x t e r n -3-a l p o t e n t i -3-a l f u n c t i o n , . w i l l be d e v e l o p e d i n s e c t i o n 3.

2.__JIhe Image System f o r a S p h e r o i d

A s p h e r o i d a l harmonic o f degree n and o r d e r s , w h i c h v a n i s h e s a t i n f i n i t y , i s o f t h e f o r m

H ^ " ( l - i , 4.) = P ^ ' ( y ) Q / ( d e^"'^ • ( 3 )

where ( p , t,, tf)) a r e s p h e r o i d a l c o o r d i n a t e s d e f i n e d by

X =^ y c ; y + i z = >/U^2)(c2„i^ e^* ^ ik)

and t h e two f o c i o f t h e s p h e r o i d a r e a t (± 1 , 0, O) i n C a r t e s i a n r e p r e s e n t a -t i o n ,

s

The L e g e n d r e f u n c t i o n s o f t h e f i r s t k i n d , P _^ , and o f t h e second k i n d , Q^", a r e d e f i n e d by [ l l ] ( p . 1 ^ 2 ) . s_ 2 ,n+s P ""(y) = - ^ 1 7 ( M ^ - I ) " , V<1 ( 5 ) " 2''n! dM^"""^ _s •1 2 -,n+s 2 n! etc" • dC^ I " [ P ^ ( d ] ^ The f o l l o w i n g r e l a t i o n i s g i v e n b y Hobson ( p , 97): s_ s_ 2 , " 2 c ? n\ ,n+s / o T X ^n~s Üi!zi) ( 2 _ i ) n ( i ^ L i ) f i (^2„i)n. /- , ^ I . n+s / N , , n.-s ( n + s ) ! dp ( n - s ) ! dy

(8)

The above r e l a t i o n t o g e t h e r w i t h t h e R o d r i g u e z f o r m u l a (,5) y i e l d t h e f o l l o w i n g e x p r e s s i o n f o r P ^ ( i j ) :

s_ ( 9 ) ïn a d d i t i o n , t h e w e l l - k n o w n Netimann f o r m u l a f o r Q ( x ) , ' n 1 P J C ) x-5

d5

- 1 ( 1 0 ) i s a v a i l a b l e .

When e q u a t i o n (9) i s s u b s t i t u t e d i n t o ( l O ) , and when t h e l a t t e r i s I n t e g r a t e d b y p a r t s s t i m e s , e q u a t i o n ( l O ) becomes s s ! ( n - s ) ! 2 ( n + s ) ! 1 ( l - ? 2 ) 2 p ^ ( 5 ) df.

-1

s+1 ( 1 1 )

The above r e l a t i o n may be c o n s i d e r e d as a g e n e r a l i z a t i o n o f t h e Neumann f o r m u l a ( l O ) , w h i c h c o r r e s p o n d s t o t h e case s - 0. E q u a t i o n s (H), ( 5 ) , and (6) i m p l y t h a t t h e e x t e r i o r s p h e r o i d a l h a r m o n i c ( 3 ) mav a l s o be w r i t t e n i n t h e f o r m ^n ^1^^ . ^ d^^P ( u ) d^Q ( d l y + i ^ ; dy dc On t h e p a r t o f t h e x ~ a x i s where | x | > l , y = 1 and ( 1 2 ) d ^ d ) _{( n + s ) !} dy'' 2''.s!(n~s)! ( 1 3 ) S u b s t i t u t i n g e q u a t i o n s ( l l ) and ( l 3 ) i n t o ( l 2 ) y i e l d s l i m 1 ( l - f . 2 ) 2 p S^^) y->l ( y + i z / ;x-c) 2 s + l

d5

( l U )

A n a l y t i c c o n t i n u a t i o n arguments a p p l i e d t o {ik) i m p l y t h a t f o r p o i n t s o f f t h e x a x i s s_ , 2 s / ^ s, . . _ x s . , ( (o U n ( 1- ^ 2 ) - P^ - ( 0 ( y+ i z ) ° dK " " 2^ . s! (15) The f o l l o w i n g r e l a t i o n i s e a s i l y v e r i f i e d by m a t h e m a t i c a l i n d u c -t i o n , ( 1 . . , i X ) B , y 2 , , 2 ] ^ % . d l ) ! . i 2 s ) ! . „ i j ü ^ : (16) 9y 2 ^ s ! [(x-.5)2+y2+.^]"'^'^ Wlien t h e above r e l a t i o n i s s u b s t i t u t e d i n t o ( l 5 ) , t h e l a t t e r y i e l d s t h e Havelock f o r m u l a ( 2 ) . _3. _ T h e Jmag_e System f o r an E l l i p s o i d L e t t h e e q u a t i o n o f t h e e l l i p s o i d be g i v e n b y r2 ,r2 „2 •1- H -a2 b2 c2 1 i a>b>c (17) The e l l i p s o i d a l c o o r d i n a t e s ( p , u, v ) a r e d e f i n e d by t h e s o l u t i o n o f t h e c u b i c e q u a t i o n i n A, x i + = 1 ( 1 8 ) f o r f i x e d v a l u e s o f ( x , y , z) where k^ = a-'-c The t h r e e r o o t s o f ( l 8 ) a r e chosen so t h a t co>p2>k2 ;, k2>y 2>h2 i h2>_v2>0 (20) The t h r e e s u r f a c e s , p = c o n s t ( e l l i p s o i d s ) , y - c o n s t ( h y p e r b o l o i d s

o f one s h e e t ) and v c o n s t ( h y p e r b o l o i d s o f two s h e e t s ) t h e n f o n a a t r i p l y -o r t h -o g -o n a l c -o -o r d i n a t e system i n space.

The t r a n s f o r i T i a t i o n between t h e C a r t e s i a n and t h e e l l i p s o i d a l c o o r d i n a t e s i s g i v e n by (Hobson p. '+55) 2,,2,,2 9 I . X 2 = p ^ j ^ ' j : ( 2 1 ) and h2k2 = (p2-h2)(p2-.h2)(t,2._^2) ^^^^ h2(k,2-h2) ,2 ^ (p2-k2)(k2-y2)(i,2„^2) ^ ^ k 2 ( k 2 - h 2 ) An e l l i p s o i d a l harmonic w h i c h i s r e g u l a r a t i n f i n i t y i s d e f i n e d as H^'^p,P,v) - F ^ " ( p )

E

;"(

P

)

E^*"(V) {2k)

where n and m a r e j p o s i t i v e i n t e g e r s such t h a t m <^ 2n + 1 . Here E^''^ d e n o t e s

t h e Löiué f u n c t i o n o f t h e f i r s t k i n d w h i c h i s r e g u l a i ' a t t h e o r i g i n , and F^™ i s t h e Lamé f u n c t i o n o f t h e second k i n d w h i c h i s r e g u l a r ad: i n f i n i t y . The Lame f u n c t i o n o f t h e second k i n d F^"\p) i s d e f i n e d i n t e r m s o f E^^'^( p )

(Hobson p. »+72),

F ^ p ) = ( 2 n + l ) E "\p) _ _ _ _ _ • ( 2 5 )

[ [E^™(p)]2

/(PThT(7

^k2)

F o l l o w i n g Hobson ( p . ^6o) , t h e r e e x i s t f o u r d i f f e r e n t c l a s s e s o f Lamë f u n c -t i o n s g i v e n by P ( p ) , / p ^ ^ 2 " p ( p ) ^ v^^r^^"^ p ( p ) ^^^1 y^'p^^-k^H'p"2'-ï?T p ( p ) , where P ( p ) denotes a p o l y n o m i a l i n p. U s i n g Hobson's n o t a t i o n , t h e f o u r c l a s s e s

o f t h e Lame f u n c t i o n s w i l l be d e n o t e d b y K ( p ) , L ( p ) , M(p) and N ( p ) r e s -p e c t i v e l y . The n o r m a l f u n c t i o n s K(-p ) K ( i i ) K ( v ) and L(-p ) L(y ) L ( v ) t h e n y i e l d i n t e r i o r e l l i p s o i d a l harmonics w h i c h a r e even w i t h r e s p e c t t o z, w h i l e

P r o o f o f Theorem 1 .

L e t us assume t h a t t h e image system o f an even e x t e r i o r e l l i ^ D S o i d a l gP+q+2r

harmonic c o n s i s t s o f m i i l t i p o l e s o f t h e o r d e r ~ —~ d i s t r i b x i t e d o v e r 3x^9y^9z2^

t h e f u n d a m e n t a l e l . l i p s o i d where p , q and r a r e p o s i t i v e i n t e g e r s . The aboA^e d i f f e r e n t i a l o p e r a t o r o p e r a t e s on l / R , where R denotes t h e d i s t a n c e between t h e p o i n t ( 5 , n, O) on t h e f u n d a m e n t a l e l l i p s o i d and a f i e l d p o i n t

( x , y , z ) . S i n c e l / R i s an harmonic f m i c t i o n , we have

9xP9y^9z2^^ ^ ~ Sx^Sy^ 9x2 3^2^

= ^

i

-

i

i

-

.

(

3

1

)

The i n t e g r a t i o n o f (27) i s c a , r r i e d o u t over b o t h f; and n hence t h e m u l t i p o l e s g i v e n by ( 3 l ) may be reduced by an. i n t e g r a t i o n by p a r t s t o simpJ.e s o u r c e - s i n k d i s t r i b u t i o n o v e r t h e f u n d a m e n t a l e l l i p s o i d and a. l i n e d i s t r i b u t i o n o f m u l t i p o l e s o v e r t h e c o n t o u r o f t h e e l l i p s e g i v e n b y ( 2 6 ) . The p o t e n t i a l o f a l i n e m u l t i p e l e d i s t r i b u t i o n i s s i n g u l a r a t p o i n t o f t h e d i s t r i b u t i o n . On t h e o t h e r hand, t h e S t i e l t j e s theorem i m p l i e s t h a t (25) i s a c o n v e r g e n t i n t e g r a l f o r p=k f o r a l l p o i n t s on t h e f u n d a m e n t a l e l l i p s o l d . Hence we e x c l u d e t h e p o s s i b i l i t y o f a c o n t o u r d i s t r i b u t i o n o f m u l t i -p o l e s on t h e e l l i -p s e ( 2 6 ) . I n o r d e r t o d e t e r m i n e t h e s o u r c e s t r e n g t h , use w i l l be made o f t h e Gaxiss f l u x t h e o r e m w h i c h , s i n c e we a r e d e a l i n g w i t h a p l a n e d i s t r i b u t i o n , y i e l d s t h e f o l l o w i n g r e l a t i o n f o r t h e s o u r c e d i s t r i b u t i o n ; s ( p > . ) = i l l m ^ _ ^ [ f ^ - H ^ ^ x , y , z ) ] [ i 3 ^ H „ ° ( p . „ , v ) ] (32)

where h^ i s t h e l i n e a r i z i n g f a c t o r i n t h e p d i r e c t i o n g i v e n by P ( p 2 - h 2 ) ( p 2 _ k 2 ) S u b s t i t u t i n g . e g u a t i o n s {2h) , (25) and ( 3 3 ) i n t o ( 3 2 ) y i e l d s ( 2 n+ l ) E " A i ' ) E '""(v-) s ( y ' , v ' ) = - -

-^^^-^

(31,) 2TT E^™(k) >/rk^C,7^2^)TPC7r?y

Por t h e case where p=k eqi.iations ( 2 l ) and (22) raa.y be s o l v e d e x p l i c i t l y f o r i j ' ( x , y ) and v ' ( x , y ) . The r e s u l t i n g expi'essions a r e g i v e n b y (29) and ( 3 0 ) r e s p e c t i v e l y . S i m i l a r l y , t h e d e n o m i n a t o r o f (3!!-) i s g i v e n i n C a r t e s i a n r e p r e s e n t a t i o n 'by (k2.= y ' 2 ) ( k 2 _ v ' 2 ) = k 2 ( k 2 - - h 2 ) ( l - A „ _ïi_ ) ( 3 5 ) k2 k2-h2 S u b s t i t u t i n g (35) i n t o ( 3 ^ ) y i e l d s ( 2 8 ) , and t h e p r o o f o f Theorem 1 i s c o m p l e t e d . Theorem 2

An odd ( i n z ) e x t e r i o r e l l i p s o i d a l harmonic m^ay be g e n e r a t e d b y a n o r m a l d o u b l e t d i s t r i b u t i o n i n t h e z - d i r e c t i o n o v e r t h e f u n d a m e n t a l e l l i p * s o l d , \ ( p ) l y ) \ ( v ) - ^ - ^

d(g ,n)dCdn _„ ^3

/ /(--fp^(7:.-^7pTz2''

s O where ( 2 n + l ) E '""(y-) E "'( v ' n n and d ( x , y ) ^ (37) 2frk E '"(k) y'k^ n E " ^ ( p ) f ( k ) = l i m , . ( 3 8 )

10-P r o o f o f Theorem 2,

By t h e same argrmients as were used t o p r o v e Theorem 1 , one can show t h a t an odd e x t e r i o r e l l i p s o i d a l harmonic may be g e n e r a t e d b y a d i s t r i b u t i o n o f d o u b l e t s o r i e n t e d i n t h e z d i r e c t i o n o v e r t h e f u n d a m e n t a l e l l i p -s o i d

An odd e x t e r i o r e l l i p s o i d a l harmonic may be w r i t t e n as

H "'(p,P,v) = (2n-M) Ë ™(p) E "^(y) E ™(v) ______.___JJ3 _ _ _ _ 1 ( 3 9 ) J [E^^"^(p)]2 ( p 2 - k 2 ) ^ / 2 (p2„h2)l/2 j P The d i s c o n t i n u i t y i n t h e p o t e n t i a l a c r o s s a n o r m a l d o u b l e t d i s t r i b u t i o n t h e n i m p l i e s t h a t t h s n o r m a l d o u b l e t d i s t r i b u t i o n over t h e f u n d a m e n t a l e l l i p s o i d , w h i c h i s t h e image system o f an odd e x t e r i o r e l l i p s o i d a l h a r m o n i c , i s g i v e n b y

. d ( y ' ,v') = ^ l i m H '"(p,y,v) (l+o) "^"^ p->k "

By t h e S t i e l t j e s t h e o r e m , t h e o n l y s i n g u l a r i t y o f t h e i n t e g r a n d o f ( 3 9 ) as p approaches k i s ( p 2 _ k 2 ) . I n t e g r a t i o n b y p a r t s o f ( 3 9 ) y i e l d s

H^"^(p,y,v) = (2nH-l) i ^ " ^ ( p ) E^™(y) E^"^(v)

p [ E ^ " ^ ( p ) ] 2 >/j2r)?- J 'ip I p [ E ^ ^ ^ ( p ) ] 2 ƒƒ

A p p l y i n g t h e l i m i t p->k t o H^"^(p , j i , v ) , e q u a t i o n s {ko) and ( l + l ) y i e l d t h e d o u b l e t d i s t r i b u t i o n g i v e n by ( 3 7 ) , and t h e p r o o f o f Theorem 2 i s c o m p l e t e d .

_it_j____ExaD2]Dles_ L e t (j)(p,y,v) be a g i v e n p o t e n t i a l f u n c t i o n f r e e o f s i n g u l a r i t i e s i n t h e r e g i o n P £ a.' I n t r o d u c i n g an e l l i p s o i d d i s t u r b s t h e f l o w and t h e v e l o c i t y p o t e n t i a l i n t h e r e g i o n p >_ a i s t h e n g i v e n by #^,(p,y,v) = (j) ( p , y , v ) + <i>^(p,y,v) {k2) X Ï J O e

where cf^CpsP^v) i s t h e v e l o c i t y p o t e n t i a l due t o t h e image system w i t h i n t h e e l l i p s o i d .

S i n c e b o t h * and i) a r e harmonic f u n c t i o n s , t h e y may be expanded o e i n t e r m s o f e l l i p s o i d a l harmonics i n t h e f o r m 2n+l cl, ( p , y , v ) = J I A ''(p) E " A ) E "^(v) p < a (^3) n=0 m=l " " and <^ 2 n + l (t, ( p , y , v ) ' ^ 1 1 •^V"'^'^'* ^n"^'"*^ ^n"^^^'^ P 1 ^-^ n=0 m=l

where A and B ^ a r e c o n s t a n t s t o be d e t e r m i n e d . F o r t h e Neumann p r o b l e m , t h e n n n o r m a l d e r i v a t i v e o f must v a n i s h on t h e e l l i p s o i d p = a. S i n c e t h e n o r m a l E derivatiA'^e on t h e e l 3 - i p s o i .d i s t h a t w i t h r e s n e c t t o p , we t h e n o b t a i n from. (i+2), {k3) and {kk) t h e f o l l o w i n g e x p r e s s i o n i n t h e r e g i o n e x t e r i o r t o t h e e l l i p s o i d p = a: «> 2 n + l A , J p , y , v ) ^ - . I I F; ( P) E > ) EJ^V) {k3) n--0 m=l p ^ ^ where . F "'(a) C = - i ^ {k6) ^ Ê ; ( a )

12E q u a t i o n (k^) i s an e x p r e s s i o n f o r t h e d i s t u r b a n c e v e l o c i t y p o t e n -t i a l when an e l l i p s o i d i s i n -t r o d u c e d i n a p o -t e n -t i a l f l o w f i e l d g i v e n b y ( ^ 3 ) . l?§?iple. L u L ' S _ ? L 3 n s l a t i o n _ ^ L e t ( j ) ^ ( x , y , z ) be g i v e n b y ^J^,J,z) = Ux + Vy + Wz ( l l 7 )

w h i c h r e p r e s e n t s a u n i f o r m stream w i t h v e l o c i t i e s U, V and W i n t h e x , y and

z d i r e c t i o n s r e s p e c t i v e l y . As a l r e a d y m e n t i o n e d t h e r e e x i s t s t h r e e Lame f u n c t i o n s o f t h e f i r s t o r d e r ( n = l ) . These t h r e e f u n c t i o n s a r e o f t h e c l a s s K, L and M d e f i n e d e a r l i e r and w i l l be d e n o t e d h e r e i n as , L^ and M_^. I n t e r m s o f t h e s e f u n c t i o n s , e q u a t i o n (hi), m \ i t t e n i n e l l i p s o i d a l c o o r d i n a t e s , i s o f t h e f o r m C o n s i d e r now t h e case-where a s o l i d e l l i p s o i d i s i n t r o d u c e d i n t o t h e s t r e a m . The d i s t u r b a n c e v e l o c i t y p o t e n t i a l a t p o i n t s i n t h e e x t e r i o r r e g i o n may be c o n s i d e r e d t o be g i v e n by a c e r t a i n s i n g u l a r i t y d i s t r i b u t i o n o v e r t h e f u n d a - ' m e n t a l e l l i p s o i d .

E q u a t i o n s (3'-0, ( 3 7 ) and (i+5) t h e n i m p l y t h a t t h e image system c o n s i s t s o f a source d i s t r i b u t i o n o f s t r e n g t h

2ïïhv/(k2rp'-2)(]-Z„^,7y k 2 c ^ ( K ) (k2-h2) C^(L)

Here G ^ ( K ) , C ^ ( L ) and C ^ ( M ) d e n o t e t h e t h r e e v a l u e s o f C^" d e f i n e d i n (^+6) c o r r e s p o n d i n g t o t h e t h r e e p o s s i b l e f o r m s o f E ™ , i,e.,K , L and M .

I t i s more c o n v e n i e n t t o use C a r t e s i a n r e p r e s e n t a t i o n . The source d i s t r i b u t i o n (k^) i s t h e n g i v e n b y 2uk ./p-h^ k2 k2-h2 k2 C^(K) (k2.-h2) C ^ ( L ) and t h e d o u b l e t d i s t r i b u t i o n ( 5 0 ) i s d ( x , y ) --^.^^^-^^^.^^ ( 1 - ^ - -^-1 ( 5 2 ) 2iTk C ^ ( M ) /k^--h^ k2 k2-h2 The above e x p r e s s i o n s a r e a r e d u c e d f o r m o f H a v e l o c k ' s [ l 3 ] r e s u l t s f o r t h e image system o f an e l l i p s o i d i n a u n i f o r m s t r e a m .

The c o e f f i c i e n t s C ^ ( K ) , C ^ ( L ) and C ^ ( M ) a r e g i v e n i n t h e Appendix i n t e r m s o f t a b u l a t e d e l l i p t i c i n t e g r a l s . Example 2: Pure R o t a t i o n L e t us assume t h a t t h e e l l i p s o i d i s r o t a t i n g about i t s p r i n c i p a l axes i n an i n f i n i t e i n v i s c i d f l u i d o t h e r w i s e a t r e s t w i t h a n g u l a r v e l o c i t y , ^ (W 5 w s w ) ( 5 3 ) The boundary c o n d i t i o n t o be s a t i s f i e d on t h e e l l i p s o i d i s - (w X ? ) . n (5^0 where r i s a u n i t v e c t o r f r o m t h e o r i g i n t o a p o i n t ix,j^z) on t h e e l l i p s o i d and n d e n o t e s a u n i t nox-mal v e c t o r t o t h e s u r f a c e o f t h e e l l i p s o i d . F o l l o w i n g Lajnb ( p . ikT) t h e i n t e r i o r v e l o c i t y p o t e n t i a l i s

li2_^Z c2-a2 a2-b2

<!>^(x,y5z) = ™ - w y z + Ui zx + — — w x y ( 5 5 )

-Ik-\Then n=2 t h e r e e x i s t f i v e Lame f u n c t i o n s , two o f c l a s s K, and one f u n c t i o n

each o f c l a s s e s L, M and N (Hobson p. ^t65). These f i v e f u n c t i o n s w i l l be

T 2

d e n o t e d - a s K^, K^, U^, and r e s p e c t i v e l y . The i n t e r i o r v e l o c i t y p o t e n t i a l e x p r e s s e d i n t e r m s o f e l l i p s o i d a l c o o r d i n a t e s i s t h e n g i v e n b y Ü) ( b 2 - c 2 ) N ( p ) N„ (M) N ^ ( V ) CÜ ( c 2 - a 2 ) M„(p) MAM) UJV) <f.^(p,y,v) = 2 _ _ _ _ 2 2 _ _ ^. „ X ^ 2 _ 2 , 2 _ _ hll(b2-l-c2) ( k 2 _ h 2 ) h i 2 ( c 2 + a 2 ) ^^-T'^T-0) ( a 2 - b 2 ) L ( p ) LJU) LAV) + 2 _ _ „ ^ 2 ^^g, h 2 k ( a 2 + b 2 ) v^^h"^' A g a i n e q u a t i o n s ( ^ 5 ) and {k6) y i e l d t h e e x p a n s i o n i n e l l i p s o i d a l harmonics o f t h e e x t e r i o r v e l o c i t y p o t e n t i a l i n terras o f t h e e l l i p s o i d a l harmonic e x p a n s i o n o f t h e i n t e r i o r p o t e n t i a l . The image system o f t h e e x t e r i o r p o t e n

-t i a l i s g i v e n b y Theorems 1 and 2 . T h i s system c o n s i s t s o f s o u r c e d i s t r i b u t i o n

5(.> ( a 2 . - b 2) p ' v ' / ( u° ^ ' 2: d T T ( h ^ ' ' ^ ) s (u ' , V • ) = • • ^^^^-.^^ . ( 5 J) 2ïïk2h2(a2-i-b2)(k2-h2) C^{L) / ( k - ^ I ^ ' " ? ' and normal d o u b l e t d i s t r i b u t i o n d ( y M _

M i E Z i ) i l ? L L l i r

r • V( c 2 - a 2) p ' v ' 2-rThk2(k2-h2) k 2 ( c 2 + a2) C,^{M) a)^(b2-c2) ./(^n^2)-(^7::^T2-) (b2-t-c2)(k2-h2) C^^W ( 5 8 )

Here t h e c o e f f i c i e n t s C^{L) , C^^iu) and C ^ ( H ) a r e t h e t h r e e v a l u e s o f d e f i n e d i n (kS) , w h i c h c o r r e s p o n d t o t h e r e p l a c e m e n t o f E^"^ by L^, and H^ r e s p e c t i v e l y .

The e q u i v a l e n t e x p r e s s i o n s i n C a r t e s i a n r e p r e s e n t 8 , t i o n a r e

501 ( a 2 - b 2 ) x y 2 2

s ( x , y ) = x__ ^_ ^ -2-,Tk3(a2+b2)(k2-h2)-^/2 C ( L ) k2 k2-h2

-15-and 2 , , 2 . % d ( x , y ) = ( 1 ^ -(60) UJ {c^-8,^)x CO ( b 2 . ^ c 2 ) y k 2 ( c 2 +a 2 ) C2(M) (k2-h2)(b2-l-c2) 0^(1^) E x p r e s s i o n s f o r C ^ ( L ) , C,^{M) and C^(w) i n t e r m s o f e l l i p t i c i n t e g r a l s a r e g i v e n i n t h e A p p e n d i x . 6 J

-16-Ref e r e n ces__

[ I ] L a g a l l y , M., "Berechnung d e r Krüfte und Moraente, d i e stromende Flüssigkeiten a u f i h r e Begrenzung Ausiiben," Zeitsalwlft füv Angewandte Mathematik und Meohanik^ Bd. 2 , H e f t 6 , Dec. 1922. [ 2 ] Cuimnins, W.E., "The Forces and Moments A c t i n g on Body Moving i n an

A r b i t r a r y P o t e n t i a l Stream," Jouv. of Ship Research, V o l . 1 , Wo. 1 , A p r . 1957.

[ 3 ] Landweber, L. and Y i h , C.S., " F o r c e s , Momentg, and Added Masses f o r Rankine B o d i e s , " Joia>. of Fluid Mechanics, V o l . 1 , P a r t 3, Sept. 1956.

[h] Vfeiss, P., "On H y d r o d y n a j u i c a l Images; A r b i t r a r y I r r o t a t i o n a l Flow D i s t u r b e d by a Sphere," Pvoo> Camhin-dge Philosopliioal Society, V o l . i+0. P a r t 3 , 19^^!.

[ 5 ] B u t l e r , S.F.T., "A Hote o f Stokes Stream F u n c t i o n f o r M o t i o n w i t h a S p h e r i c a l Boundary," Proo. Cambridge Philosophical Society^ h9, 1953.

[ 6 ] L u d f o r d , O.S.S., M a r t i n e k , J . and Yeh, G.C.K., "The Sphei'e Theorem i n P o t e n t i a l T h e o r y , " Proc, Cambridge Philosovhical Society, 5 1 , 1955.

[ 7 ] Hobson, E., "The T h e o r y o f S p h e r i c a l and E l l i p s o i d a l Harmonics," Chelsea P u b l i s h i n g Co., Hew ïoi^k, 1955.

[ 8 ] H a v e l o c k , T.H., "The Moment on a Submerged S o l i d o f R e v o l u t i o n Moving H o r i z o n t a l l y , " Qu.avt, Joia\ Meah. ajtd Applied Math., V o l . V, P a r t 2 , 1952.

[ 9 ] B o t t a c c i n i , M.R. , "The Added Masses o f P r o l a t e S p h e r o i d s A c c e l e r a t i n g Under a Free S u r f a c e , " Ph.D. D i s s e r t a t i o n , The U n i v e r s i t y o f

I o w a , 1958.

[ 1 0 ] Morse, P.M. and Feshbach. H., "Methods o f T h e o r e t i c a l P h y s i c s , " McGraw H i l l P u b l i s h i n g Co., 1953.

[ I I ] Lamb. H., "Hydrodynamics," Dover P u b l i s h i n g Co., 1932.

[ 1 2 ] W h i t t a k e r , E.T. and Watson, G.ÏÏ., "A Coui'se o f Modern A n a l y s i s , " Cam-b r i d g e U n i v e r s i t y P r e s s , hth Ed., 1935.

[ 1 3 ] H a v e l o c k , T.H., "The Wave R e s i s t a n c e o f an E l l i p s o i d , " Proo. Roy. Soa.,A., V o l , 1 3 1 , p. 275, 1 9 3 1 .

[ih] B y r d , P.P. and F r i e d m a n , M.D., "Handbook o f E l l i p t i c I n t e g r a l s , " 2nd Ed., S p r i n g e r - V e r l a g , New Y o r k , H e i d e l b e r g , B e r l i n , 1 9 7 1 .

I T -Appendix E x p r e s s i o n s f o r t h e c o e f f i c i e n t s i n t e r m s o f e l l i p t i c i n t e g r a l s , S i n c e K ^ ( p ) = Ps e q u a t i o n (25) y i e l d s F_^(p) = 3p Hence (hS) i m p l i e s t h a t dp p - p 2 / ( p 2 - k 2 ) C p 2 i h 2 ) ( 1 ) C^(K) =

3

p 2 ,rp"rpy( p^-b 27 abc

The above e q u a t i o n may be e x p r e s s e d i n teimis o f t a . b u l a t e d e l l i p t i c i n t e -g r a l s [ 1 ^ ] as

C (K) = -3_ [ F ( ( l , , t ) - I E ( ( j . , t ] - (3)

kh2 abc

where IF and IE denote t h e Legendre i n c o m p l e t e e l l i p t i c i n t e g r a l s o f t h e f i r s t and second k i n d r e s p e c t i v e l y , and

h . . - 1 k /), \ t = ; ^ = sm- ^ . ( 4 ; S i m i l a r l y , t h e r e s t o f t h e c o e f f i c i e n t s may be e x p r e s s e d i n t e r m s o f t h e t a b u -l a t e d i n c o m p -l e t e e -l -l i p t i c i n t e g r a -l s [-l^+J. Here we w i -l -l g i v e o n -l y t h e f i n a -l r e s u l t s ; 3k r-n./, h2 , ^ , l£ s i n i . c o s A _ - , 3 C ( L ) = A>^. [ F (<f,,t) - ( 1 - ~ ) E {i>,t) - —

1 -u2\_{h2(k2~h2) k2 k2} 1,2 v 2 Jl _{/ I - i l l 3in-^ abc}

( 5 ) k2

C (M) = [tg<j,

v T T T I

sin2(j, - ]E ( * , t ) ] - - —

(6)

-18-Appendix ( c o n t i n u e d ) C ( L ) = -^-^-5 r j , ( ^ , , ) „ - j j i n l ^ e o g i L ^ ] „ 5 _ _ „ I . h 2 ( k 2 - h 2 ) / T ^ . i n ^ , abc(a2.,-b2) k2 C (M) = [ t g ^ v^lT sin2,j, 2 k 3 ( k 2 - h 2 ) k2 k 2 k 2 - ( ^ " ~ 1 ) F ( < l ) , t ) + 2) 1^ ( < ^ , t ) ] - ( 8 ) h2 h2 atic(a2-)-c'^) h2

tgf|>(l - —y sin2(f)) + sincj) cos<|) C J N ) - — ^ [ _{: ( k 2 - h 2 ) 2 ^}

1 - — s i n ' - i j )

k2

-i- - 2 ) I E ( ^ , t ) - ^-W ( ( } , , t ) ] ^