Stream function wave theory: Validity and application

CHAPTER 12

STREAM FUNCTION WAVE THEORY; VALIDITY AND APPLICATION

By Robert G. Dean,l A. M. ASCE

SYNOPS1S

The engineer required to calculate theoretical wave characteristics

such as wave profiles and wave forces and moments on piling, is confronted

with a problem which includes (1) selecting one of a number of available

theories and (2) calculating the required information which, for some of

the theories, is a relatively complicated procedure. This paper presents

criteria for assessing the validity of various wave theories; these

criteria are then applied to test the validity of several theories for

two wave conditions and, it is found that for these conditions, the Stream

function numerical wave theory is the most valid of those tested. The

Stream function theory is developed into graphs of dimensionless crest

displacement, and total maximum wave forces and moments on a vertical

piling.

~ormerlY:

Senior Research Engineer, Chevron Research Company,

La

Habra, California. Presently: Acting Associate Professor of Oceanography,

University of Washington, Seattle, Washington.

2 7 0 C O A S T A L E N G I N E E R I N G

INTRODUCTION

The problems o f a n a l y s i s ( t o d e t e r m i n e d r a g and i n e r t i a c o e i

^y,e s u b j e c t s

and c a l c u l a t i o n o f t h e o r e t i c a l waves and wave f o r c e s have been ^' o f c o n s i d e r a b l e i n v e s t i g a t i o n d \ i r i n g t h e p a s t 15 y e a r s . These 3-'

a n a l y s i s t i o n s have r e s u l t e d i n t h e p r e s e n t a t i o n o f a f a i r l y wide range "-^

some r e s u l t s and i n t h e development o f s e v e r a l n o n l i n e a r wave theori® '

. j i e d e s i g n o f ^rtiich a r e n o t i n reasonable a c c o r d . The problem c o n f r o n t i n g

p a r t i c u l a r e n g i n e e r i s t h e r e f o r e a d i f f i c u l t one; t h a t i s t o s e l e c t f o r tïi®

d e s i g n c o n d i t i o n s t h e most v a l i d a n a l y s i s r e s u l t s f o r use w i t h ^ g j , o n o f p r o p e r wave t h e o r y . The p r e s e n t paper w i l l n o t i n c l u d e a d i s c U ^

^ t n e p a p e r the problem o f s e l e c t i o n o f a n a l y s i s r e s u l t s . I h e purposes o i

-vjave a r e t o (1) p r e s e n t a b a s i s f o r a s s e s s i n g t h e v a l i d i t y o f v a r i o H ^ t h e o r i e s and t o a p p l y t h i s t o two cases, and ( 2 ) t o p r e s e n t g r ^ P ' ^

^ « o n p i l i n g ; d i m e n s i o n l e s s c r e s t d i s p l a c e m e n t , t o t a l wave f o r c e s and m o m e n t »

_ v i i c h , on t h e s e graphs a r e developed f r o m a Stream f u n c t i o n wave t h e o r y •*

t l i e b a s i s o f two cases t e s t e d , appears more v a l i d t h a n o t h e r 0-^ wave t h e o r i e s .

CRITERIA FOR ASSESSING THE VALIDITIES OF WAVE THEORIES

^ a f o r The purpose o f t h i s s e c t i o n i s t o d e v e l o p a r a t i o n a l c r i ' t s ^ a s s e s s i n g t h e v a l i d i t i e s o f v a r i o u s a v a i l a b l e wave t h e o r i e s . _ ; n t i a t i o n The t h e o r e t i c a l wave l i t e r a t u r e p r e s e n t s v e r y l i t t l e i n f < ^ - ^ -^3.6 wave p e r t a i n i n g t o t h e r e l a t i v e a p p l i c a b i l i t y o f t h e v a r i o u s a v a i l ^ * - ^

S T R E A M F U N C T I O N 2 7 1 t h e o r i e s . W i l s o n ^ , i n a s h o r t d i s c u s s i o n , has I n d i c a t e d r e g i o n s o f v a l i ' ^ ^ ' ' ^ f o r s e v e r a l o f t h e wave t h e o r i e s , however a l l o f t h e a v a i l a b l e t h e o r i e s were n o t i n c l u d e d i n V/ilson's p r e s e n t a t i o n . F u r t h e r m o r e , W i l s o n d i d n o * a t t e m p t t o d i s c u s s , i n h i s s h o r t p r e s e n t a t i o n , h i s b a s i s f o r assessii>S the v a l i d i t i e s o f t h e v a r i o u s t h e o r i e s .

The e n g i n e e r r e q u i r e d t o c a l c u l a t e t h e o r e t i c a l wave i n f o r m a t i o n trais^ s e l e c t one from a number o f a v a i l a b l e wave t h e o r i e s , i n c l u d i n g : A i i ^ wave t h e o r y ^ . Stokes t h i r d o r d e r t h e o r y ^ . Stokes f i f t h o r d e r t h e o r y ^ > C n o i d a l wave t h e o r y ^ , S o l i t a r y wave theory'^. V e l o c i t y p o t e n t i a l

S j i l s o n , B. W., D i s c u s s i o n o f paper "Long Wave M o d i f i c a t i o n b y Jji-'^®^ T r a n s i t i o n s " , J . Waterways and Harbors D i v i s i o n , P r o c . ASCE, v . 90,

Nov. 1961;.

•^Wiegel, R. L., " G r a v i t y Waves, T a b l e s o f F u n c t i o n s " , P u b l l s h e < i

C o u n c i l on Wave Research, The E n g i n e e r i n g F o u n d a t i o n , Richmond, C a X i ^ ' ^ ' ^ * ' Feb. 19$h.

^ S k j e l b r e i a , L., " G r a v i t y Waves, Stokes T h i r d Order A p p r o x i m a t e ' - ' ^ ' T a b l e s o f F u n c t i o n s " , P u b l i s h e d b y t h e C o u n c i l on Wave Research, T l x ^ E n g i n e e r i n g F o u n d a t i o n , Richmond, C a l i f o r n i a , 1959.

^ S k j e l b r e i a , L., and Hendrickson, J . A., " F i f t h Order G r a v i t y W^'^^ Theory w i t h T a b l e s o f F u n c t i o n s " , N a t i o n a l E n g i n e e r i n g Science C o m s u f ^ ' Pasadena, C a l i f o r n i a , 1962

^Masch, F. D., and W i e g e l , R. L., " C n o i d a l Waves, Tables o f F t * * * * ' * ^ ' " ^ ^ " ' ïchmond P u b l i s h e d b y C o u n c i l on Wave Research, The E n g i n e e r i n g F o u n d a t i o n ,

C a l i f o r n i a , 1 9 6 1 . "^Munk, W. H.

Ann. N. Y. Acad. S o l . , v . 5 l , p . 376-k2k, 19U9.

2 T 2 C O A S T A L E N G I N E E R I N G

n u m e r i c a l wave t h e o r y ^ , Stream f u n c t i o n n u m e r i c a l wave t h e o r y ^ , and perhaps o t h e r s . S e v e r a l o f these t h e o r i e s ^ ' ^ ' ^ ' ^ have been t a b u l a t e d f o r r e l a t i v e l y easy c a l c u l a t i o n o f wave p r o f i l e s and wave f o r c e s and moments on v e r t i c a l p i l i n g . There i s no w e l l - f o t m d e d b a s i s , however, f o r s e l e c t i o n o f t h e most a p p l i c a b l e o f t h e seven, o r more, a v a i l a b l e wave t h e o r i e s . I t i s g e n e r a l l y assumed t h a t t h e h i g h e r o r d e r S t o k l a n t h e o r i e s a r e improvements over t h e l o w e r o r d e r t h e o r i e s ; c e r t a i n l y t h e amount o f e f f o r t t o develop t h e h i g h e r o r d e r t h e o r i e s i s g r e a t e r . Recent q u e s t i o n s have been r a i s e d w h e t h e r o r n o t t h e h i g h e r o r d e r t h e o r i e s a r e u n i f o r m l y more v a l i d t h a n t h e l o w e r o r d e r t h e o r i e s . The C n o i d a l and S o l i t a r y wave t h e o r i e s a r e developed f o r t h e l o n g e r waves, however no p u b l i s h e d i n f o r m a t i o n i s a v a i l -a b l e f o r j u d g i n g t h e r e l -a t i v e m e r i t s o f , s-ay, t h e h i g h e r o r d e r S t o k l -a n t h e o r i e s and t h e C n o i d a l o r S o l i t a r y wave t h e o r i e s .

I n o r d e r t o develop t h e c r i t e r i a f o r a s s e s s i n g t h e v a l i d i t i e s o f t h e v a r i o u s wave t h e o r i e s , i t w i l l be n e c e s s a r y t o d e s c r i b e b r i e f l y t h e non-l i n e a r wave f o r m u non-l a t i o n .

N o n l i n e a r Wave Theory F o r m u l a t i o n ; Two Dimensional Case.—The n o n l i n e a r wave t h e o i y i s f o r m u l a t e d as a boundary v a l u e problem; t h e f o m u l a t i o n v j i l l be p r e s e n t e d h e r e f o r t h e tvro-dimensional case. Boundary v a l u e problems a r e s p e c i f i e d b y ( l ) a d i f f e r e n t i a l e q u a t i o n p r e s c r i b e d on t h e i n t e r i o r

Chappelear, J . H., " D i r e c t N u m e r i c a l C a l c u l a t i o n o f Wave P r o p e r t i e s " , J . Geophys. Res.. 66(2), p. 501-^08, Feb. 1961.

^Dean, R. G., "Stream F u n c t i o n R e p r e s e n t a t i o n o f N o n l i n e a r Ocean Waves", J . Geophys. Res.. 7 0 ( l 8 ) , p . U56l-li572, Sept. 1965.

S T R E A M F U N C T I O N 273 o f t h e r e g i o n o f i n t e r e s t , and ( 2 ) c o n d i t i o n s vdiich must be s a t i s f i e d

on t h e b o u n d a r i e s o f t h e r e g i o n . The number o f r e q u i r e d c o n d i t i o n s on each boundary depends on t h e n a t u r e o f t h e boundary, e.g., whether t h e boundary i s f i x e d o r f r e e t o move under t h e a c t i o n o f f o r c e s .

The t h e o r e t i c a l wave f o r m u l a t i o n has been p r e s e n t e d eiseerdere^ and w i l l t h e r e f o r e be d e s c r i b e d h e r e o n l y i n b r i e f d e t a i l . R e f e r r i n g t o F i g u r e 1 , t h e d i f f e r e n t i a l e q u a t i o n t o be s a t i s f i e d on the wave i n t e r i o r f o r an i n c o m p r e s s i b l e , i r r o t a t l o n a l f l u i d i s L a p l a c e ' s e q u a t i o n w r i t t e n f o r e i t h e r t h e v e l o c i t y p o t e n t i a l $ o r t h e stream f u n c t i o n Y, i . e . , V^^ = V ^ Y = 0 ertiere = — 2 - + -2 9x2 2j,2

The v e l o c i t y p o t e n t i a l and s t r e a m f u n c t i o n a r e d e f i n e d i n terms o f t h e v e l o c i t y components, i . e . ,

^ = - Y z = -4>,

( 2 )

The sea bottcan i s c o n s i d e r e d impermeable and h o r i z o n t a l ; t h e s p e c i f i e d c o n d i t i o n a t t h i s boundary i s t h a t t h e v e r t i c a l component o f v e l o c i t y i s z e r o , i . e . ,

D i r e c t i o n o f M E A N W A T E R L E V E L CO V E L O C I T Y C O M P O N E N T S O CO

>

:^

S T R E A M F U N C T I O N 2 7 5

Because t h e f r e e s u r f a c e i s n o t c o n s t r a i n e d , b u t i s f r e e t o move under the a c t i o n o f f o r c e s , two boundary c o n d i t i o n s must be s p e c i f i e d on t h i s boundary: (1) a k i n e m a t i c boundary c o n d i t i o n w h i c h eocpresses t h a t t h e m o t i o n o f t h e w a t e r p a r t i c l e s a t t h e f r e e s u r f a c e a r e i n a c c o r d w i t h t h e m o t i o n o f t h e f r e e s u r f a c e , and (2) a dynamic boundary c o n d i t i o n s p e c i -f y i n g t h e u n i -f o r m i t y o -f p r e s s u r e on t h e -free s u r -f a c e , i . e . ,

S ' ^ 1 ^ = " , . = 1 ( W

and ^

7 + 2 i b-^^'^ J " i ^ t = ' = ='!

I f t h e wave i s assumed t o p r o p a g a t e w i t h t h e wave c e l e r i t y , C, and w i t h o u t change o f f o r m and i f a r e f e r e n c e c o o r d i n a t e system i s chosen

•atioh moves w i t h t h e wave, t h e n t h e p r o b l e m i s reduced t o one o f s t e a d y

m o t i o n and Eqs. ( l ) and (3) a r e u n a f f e c t e d b u t Eqs. ( i i ) and ( 5 ) a r e s i m p l i f i e d t o t h e f o l l o w i n g f o r m s ,

3 x - ïï^ ( 6 )

7 + [(^ - of + w2] = oonstg = Q (7)

Eq. ( 7 ) i s t h e f a m i l i a r B e r n o u l l i e q u a t i o n f o r s t e a d y s t a t e c o n d i t i o n s . Eqs. (1), ( 3 ) , (6), and (7) now r e p r e s e n t t h e s p e c i f i c a t i o n o f t h e t h e o r e t i c a l wave; i t t h e r e f o r e appears t h a t t h e o n l y way o f a s s e s s i n g t h e r e l a t i v e v a l i d i t y o f v a r i o u s wave t h e o r i e s i s t o compare t h e degree t o w h i c h t h e y s a t i s f y t h e s p e c i f y i n g e q u a t i o n s .

2 7 6 C O A S T A L E N G I N E E R I N G

No a t t e m p t w i l l be made i n t h e p r e s e n t paper t o c a r r y o u t a comprehensive i n v e s t i g a t i o n o f a l l a v a i l a b l e wave t h e o r i e s f o r a l l wave c o n d i t i o n s o f

i n t e r e s t . As examples, two cases w i l l be examined f o r w h i c h t h e wave h e i g h t i s a b o u t ^0% o f t h e b r e a k i n g h e i g h t .

The two cases s e l e c t e d f o r s t u d y and t h e t h e o r i e s f o r w h i c h t h e b o i m d a r y c o n d i t i o n f i t s were c a l c u l a t e d a r e shown i n Table 1.

TABLE 1.—CHARACTERISTICS OF WAVES CHOSEN FOR BOUNDARY CONDITION COMPART SUN

Case

Approximate Wave C h a r a c t e r i s t i c s

Boimdary C o n d i t i o n s Checked f o r Theory Case H ( f t ) T ( s e c ) h ( f t ) A i r y Stokes T h i r d Stokes F i f t h C n o i d a l Stream F u n c t i o n ( O r d e r ) 1 l 5. 0 13.0 30 X X X X X (7) 2 l l l . O 5.80 100 X X X (2)

Because most o f t h e wave t h e o r i e s o f i n t e r e s t ( A i r y , Stokes' T h i r d , S t o k e s ' F i f t h , and Stream F u n c t i o n ) s a t i s f y t h e d i f f e r e n t i a l e q u a t i o n and b o t t o m boundary c o n d i t i o n e x a c t l y , o n l y t h e f i t s t o t h e two f r e e s u r f a c e b o u n d a r y c o n d i t i o n s w i l l b e employed as measures o f t h e r e l a t i v e v a l i d i t i e s

o f t h e v a r i o u s wave t h e o r i e s .

The two " l o c a l " e r r o r components d e f i n e d f o r t h i s s t u d y a r e ( l ) as a measure o f t h e e r r o r i n t h e f r e e s u r f a c e k i n e m a t i c boundary c o n d i t i o n , i . e . .

S T R E A M F U N C T I O N

A x . - u.-C ^ 1 ^ , z = >?. ( 8 )

and ( 2 ) C^, t h e e r r o r i n t h e f r e e s u r f a c e dynamic boundary c o n d i t i o n as d e f i n e d b y

* - + " i ^ ] = % ' 2 = •••• (9)

where t h e s u b s c r i p t , i , i n d i c a t e s v a r i o u s wave phase p o s i t i o n s and Q i s t h e average B e r n o u l l i " c o n s t a n t " . The f i n i t e d i f f e r e n c e a p p r o x i m a t i o n t o ^ as r e p r e s e n t e d i n Eq. (Ö) i s n o t e x a c t and w i l l t h e r e f o r e i n t r o d u c e a c o n t r i b u t i o n t o €-^, T h i s a p p r o x i m a t i o n , however, w i l l p r o b a b l y c o n t r i b u t e a b o u t t h e same t o a l l t h e o r i e s . Because, as w i l l be d e s c r i b e d l a t e r , t h e Stream f u n c t i o n wave t h e o r y f i t s t h e k i n e m a t i c boundary c o n d i t i o n e x a c t l y , t h e € ^ c o n t r i b u t i o n s f o r t h i s t h e o r y w i l l s e r v e as an i n d i c a t i o n o f e r r o r s due t o t h i s a p p r o x i m a t i o n .

To summarize t h e Case 1 and Case 2 l o c a l e r r o r i n f o r m a t i o n , an o v e r - a l l range e r r o r { E ) j j i s d e f i n e d , i . e . ,

( % ) R -= | ( ^ l ) m a x . - ( ^ i W I ( 1 0 )

A s i m i l a r d e f i n i t i o n f o r (E^)!^ a p p l i e s .

F i g u r e 2 p r e s e n t s t h e Case 1 comparisons o f t h e boundary c o n d i t i o n f i t s b y t h e v a r i o u s wave t h e o r i e s . The c o r r e s p o n d i n g i n f o r m a t i o n f o r Case 2 was n o t c a l c u l a t e d i n as complete f o r m . The Case 1 and p a r t i a l Case 2 o v e r - a l l e r r o r i n f o r m a t i o n i s p r e s e n t e d i n T a b l e 2.

278 C O A S T A L E N G I N E E R I N G

L E G E N D

S T R E A M F U N C T I O N 279

TABLE 2.~SUI4MARY OF OVER-ALL BUUNDARI CONDxTION ERRORS

Case Wave C h a r a c t e r i s t i c s O v e r - a l l E r r o r Case H ( f t ) T ( s e c ) h ( f t ) Theory-( % ) R 1 15.0 13.30 30.0 A i r y 0.066 0.1i95 I l i . 53 13.29 30.0 Stokes I I I 0.211i 2.026 13.2 12.25 30.0 Stokes V 0.071 5.105 15.0 13.30 30.0 C n o i d a l 0.176 li.835 111.95 13.30 30.0 Stream Fn. ( 7 ) * 0.059** O.OOli 2 l l i . 0 5.8 100.0 A i r y 0.102 1.762 l l l . O 5.8 100.0 Stokes V Not c a l c u l a t e d 0.027 l l i . 1 5.8 100.0 Stream Fn. ( 2 ) * 0.009** 0.012 Order o f Stream f u n c t i o n t h e o r y . A c t u a l l y z e r o .

DISCUSSION OF VAUDITEES OF VARIOUS WAVE THEORIES

Case I . — F r o m T a b l e 2 i t i s seen t h a t o f t h e Stokes' t h e o r i e s t e s t e d

( t h e A i i y t h e o r y w i l l be r e g a r d e d as "Stokes' 1 " t h e o r y ) , t h e o v e r - a l l

k i n e m a t i c boundary c o n d i t i o n e r r o r s , ( E j ^ ) ^ , a r e l a r g e s t f o r t h e S t o k e s ' ij.1

t h e o r y and a r e a b o u t t h e same f o r t h e A i r y and Stokes' V t h e o r i e s . The

o v e r - a l l dynamic boundary c o n d i t i o n e r r o r s , ( E2 )R, a r e u n i f o r m l y b e t t e r

f o r t h e l o w e r o r d e r S t o k e s ' t h e o r i e s . The reasons t h a t t h e h i g h e r o r d e r

280 C O A S T A L E N G I N E E R I N G

Stokes' t h e o r i e s do n o t p r o v i d e b e t t e r f i t s t h a n t h e l o w e r o r d e r t h e o r i e s ~) i s n o t o b v i o u s . Perhaps t h e h i g h e r o r d e r t h e o r i e s should n o t be employed f o r t h e r e l a t i v e l y s h a l l o w - w a t e r c o n d i t i o n s o f t h i s example ( w a t e r d e p t h / •wave l e n g t h = 0.068). A second, and t h e a u t h o r b e l i e v e s a more p l a u s i b l e i

/ e x p l a n a t i o n , i s t h a t d i f f e r e n t a n a l y t i c a l approaches should be employed ''iin t h e development o f S t o k i a n t h e o r i e s i n s h a l l o w and deepwater c o n d i -('

t i o n s . 'i

The k i n e m a t i c boundary c o n d i t i o n e r r o r s a s s o c i a t e d w i t h t h e C n o i d a l wave t h e o i y a r e s l i g h t l y l e s s t h a n c o r r e s p o n d i n g Stokes' I I I e r r o r s . Th© dynamic boundary c o n d i t i o n e r r o r s , however, a r e over t w i c e as l a r g e f o r t h e C n o i d a l t h e o r y as f o r t h e Stokes' 111 t h e o r y . The C n o i d a l t h e o r y i s g e n e r a l l y r e g a r d e d as b e i n g a p p l i c a b l e f o r s h a l l o w - w a t e r c o n d i t i o n s .

The Stream f u n c t i o n wave t h e o r y p r o v i d e s b y f a r t h e b e s t f i t o f t h e t h e o r i e s t e s t e d , e s p e c i a l l y c o n s i d e r i n g t h a t t h e k i n e m a t i c boundary c o n d - i " t i o n i s a c t u a l l y s a t i s f i e d e x a c t l y .

Case 1 1 . — F o r t h e r e l a t i v e l y deep-water c o n d i t i o n s o f Case I I ( w a t e * * depth/wave l e n g t h = 0.55), t h e i n f o r m a t i o n i n Table 2, a l t h o u g h n o t

complete, i n d i c a t e s t h a t t h e Stokes' V t h e o r y i s S i g n i f i c a n t l y more v a l i d t h a n t h e A i r y ( S t o k e s ' I ) wave t h e o r y .

The Stream f u n c t i o n wave t h e o r y a g a i n p r o v i d e s a b e t t e r f i t t o t h e boundary c o n d i t i o n s t h a n t h e o t h e r t h e o r i e s t e s t e d .

BRIEF DISCUSSION OF STREAM FUNCTION WAVE THEORT

I t i s a p p a r e n t f r o m Table 2 t h a t f o r t h e two cases examined, t h e Stream f u n c t i o n t h e o r y p r o v i d e s a c o n s i s t e n t l y b e t t e r f i t t o t h e b o u n d a . * " ^ c o n d i t i o n s t h a n t h e o t h e r t h e o r i e s . The advantages o f t h e Stream f u n c t - ^ ^

S T R E A M F U N C T I O N 2 8 1

t h e o r y i n c l u d e : ( l ) an e x a c t f i t i s p r o v i d e d t o t h e k i n e m a t i c f r e e s u r f a c e boundary c o n d i t i o n , and ( 2 ) >ri.thin r e a s o n a b l e l i m i t a t i o n s , t h e t h e o r y can be extended t o as h i g h an o r d e r as n e c e s s a r y t o o b t a i n t h e a c c u r a c y r e -q u i r e d f o r t h e p a r t i c u l a r wave c o n d i t i o n s .

I n t h e n e x t s e c t i o n , graphs r e p r e s e n t i n g d i m e n s i o n l e s s c r e s t e l e v a t i o n s and wave f o r c e s , and moments on a v e r t i c a l p i l i n g , vri.11 be p r e s e n t e d based on t h e Stream f u n c t i o n t h e o r y .

CREST ELEVATIONS, WAVE EORCES, AND MOMENTS BÏ STREAM FUNCTTÜD WAVE THEORY

D i m e n s i o n l e s s C r e s t E l e v a t i o n s . — T h e d i m e n s i o n l e s s c r e s t e l e v a t i o n s were c a l c u l a t e d f r o m t h e Stream f u n c t i o n t h e o r y and a r e p r e s e n t e d i n F i g u r e 3 as f u n c t i o n s o f h/T^ and H/T^, where h, H, and T a r e t h e w a t e r d e p t h , wave h e i g h t , and wave p e r i o d , r e s p e c t i v e l y .

D i m e n s i o n l e s s T o t a l Maximum F o r c e s . — T h e t o t a l f o r c e on a s i n g l e v e r t i c a l p i l i n g e x t e n d i n g f r o m t h e ocean b o t t o m t h r o u g h t h e f r e e s u r f a c e can b e w r i t t e n as A

+

. . +

where C^ and Oy^ a r e t h e c o e f f i c i e n t s o f d r a g and i n e r t i a , u and ü a r e t h e h o r i z o n t a l components o f v e l o c i l y and t o t a l a c c e l e r a t i o n , / ' i s tJie mass d e n s i t y o f w a t e r and t h e o t h e r v a r i a b l e s a r e as shown i n F i g u r e 1 . I f CQ and Cj^ a r e c o n s i d e r e d t o b e c o n s t a n t s , t h e n Eq. ( l l ) can be w r i t t e n i n d i m e n s i o n l e s s f o r m as

S T R E A M F U N C T I O N 283 .1 + 7 ( e ) A u ( 9 ) | u ( 9 ) l / h y /S\ ^1 + >?(e)A "l/gH" 1/gH TT ' ^ I D gH U j ' * h

i n w h i c h i i s t h e s p e c i f i c w e i g h t o f w a t e r . The phase a n g l e , 6, can h e v a r i e d t o d e t e r m i n e t h e maximum d i m e n s i o n l e s s f o r c e 4)^^. I t i s a p p a r e n t •fchat depends o n l y on t h r e e p a r a m e t e r s , i . e . ,

^'m = K ( h / T ^ H/T2, ¥ )

where Cj.j p (13)

The Stream f u n c t i o n wave t h e o r y was used t o c a l c u l a t e t h e d i m e n s i ""l®^^ t o t a l maximum f o r c e s , (|)^, f o r a number o f s e t s o f h/T^ and H/T^ and f

f o u r v a l u e s o f W; t h e r e s u l t s o f these c a l c u l a t i o n s have been develop®'* i s o l i n e s o f ( j ) ^ , and a r e p r e s e n t e d i n F i g u r e s U, 5, 6, and 7. I n t h e c a l c u -l a t i o n s , a s u f f i c i e n t -l y h i g h o r d e r Stream f u n c t i o n t h e o r y was used s o '^^'^ i n c r e a s i n g t h e wave t h e o r y o r d e r b y one d i d n o t change t h e maximum •^e'^^^'^'^'S and a c c e l e r a t i o n v r i t h i n t h e vrave b y more t h a n one p e r c e n t . The requi^^®*^ o r d e r , m, t o meet t h i s c r i t e r i o n i s shown as a f u n c t i o n o f h/T^ a n d H / T i n F i g u r e 8. F i g u r e 8 I l l u s t r a t e s one o f t h e advantages o f t h e S t r e s u t n .

f u n c t i o n wave t h e o r y . Because t h e t h e o r y can be developed t o any Tea-^°'^^^^ o r d e r , i t i s p o s s i b l e t o employ a t h e o r y o f an o r d e r c o n s i s t e n t w i t h

288 C O A S T A L E N G I N E E R I N G

O

00

S T R E A M F U N C T I O N 289

i t i s necessary t o use a t e n t h o r d e r t h e o r y t o o b t a i n t h e v e l o c i t y and a c c e l e r a t i o n accuracy r e q u i r e m e n t s s t a t e d above.

Dimensionless T o t a l Maximum Moments.—A development s i m i l a r t o t h a t l e a d i n g t o Eq. (12) f o r t h e d i m e n s i o n l e s s t o t a l moment <^ w o u l d y i e l d M ^ 1 rc^H^Dh 2 ^1 + '?(e)/h 1 + 1 ( e ) A ih\ls\ls\ ,,,,

The c a l c u l a t e d d i m e n s i o n l e s s maximum moments, f o r g i v e n s e t s o f the p a r a m e t e r s , h/T^, H/T^, 3l 2., were developed i n t o i s o l i n e p l o t s and

CQ H

a r e p r e s e n t e d i n F i g u r e s 9, 10, 11, and 12. Spot checks o f F i g u r e s U-7 and 9-12 i n d i c a t e t h a t t h e i s o l i n e s a r e a c c u r a t e t o w i t h i n 5-10^.

The d i m e n s i o n l e s s t o t a l maximum f o r c e and moment asymptotes as d e t e r m i n e d f r o m s m a l l a m p l i t u d e ( A i r y ) deep-water wave t h e o r y a r e i n d i c a t e d i n t h e l o w e r r i g h t c o m e r o f F i g u r e s l i- 7 , 9-12. I n a l l f i g u r e s t h e s e asymptotes a r e c o n s i s t e n t w i t h t h e p l o t t e d i s o l i n e s .

Example.—To i l l u s t r a t e t h e use o f F i g u r e s 3-7, 9-12 suppose t h a t

i t i s r e q u i r e d t o c a l c u l a t e t h e c r e s t e l e v a t i o n , >y^, t h e maximum t o t a l f o r c e , Fj^j and moment,M„i, f o r t h e f o l l o w i n g wave and p i l i n g c o n d i t i o n s :

Wave H e i g h t , H = 65 f t . Water Depth, h = 120 f t . Wave P e r i o d , T = 13 sec. P i l i n g Diameter, D = 6 f t .

294 C O A S T A L E N G I N E E R I N G

v j i l l

F o r purposes o f t h i s example, d r a g and i n t e r t i a c o e f f i c i e n t s t a k e n as t h e average v a l u e s f r o m a summary cranpiled b y W i l s o n The summary i n c l u d e d t h e r e s u l t s f r o m a anuniber o f l a b o r a t o r y ^ wave f o r c e a n a l y s e s ; t h e averages o f t h e d r a g and i n e r t i a o o e f f i " ' ' ' ^ " i n c l u d e d i n t h e summary a r e

= 1.05 c^, = i.Uo

i n t e r p r e t e d Use o f these average c o e f f i c i e n t s i n t h i s example s h o u l d n o t b e

as endorsement o f t h e i r a p p l i c a b i l i t y f o r d e s i g n purposes. A n ^ o f t h e c o m p l e x i t y o f t h e problem o f s e l e c t i n g d e s i g n d r a g a n d c o e f f i c i e n t s i s p r o v i d e d b y t h e f o l l o w i n g ranges o f d r a g and i r ^ e c o e f f i c i e n t s i n c l u d e d i n t h e summary b y W i l s o n and R e i d , O.UO < CD < 1.60 0.93 < C M < 2.30 ^j-ying R e t u r n i n g t o t h e example, t h e d i m e n s i o n l e s s parameters spe*'-^ t h e p r o b l e m a r e : h/T^ = 0.71 H/T^ = 0.385 W = 0.123 e necessary Note t h a t graphs a r e n o t a v a i l a b l e f o r W = 0.123; i t i s t h e r e f ^ f i g u r e s 10 t o i n t e r p o l a t e between F i g u r e s 5 and 6 ( f o r f o r c e s ) and b e t w e ^ " ^

"'•'^Wilson, B. W., and R e i d , R. 0 . , D i s c u s s i o n o f "Wave F c ^ * ^ * ^

- r ^ l v i s i o n ? C o e f f i c i e n t s f o r O f f s h o r e P i p e l i n e s " , J . Waterways and H a r b o r f ~

S T R E A M F U N C T I O N

and 11 ( f o r moments). From t h e d i m e n s i o n l e s s graphs, t h e f o l l o w i n g i s o b t a i n e d r]^/R = 0.69 ( f r o m F i g u r e 3 ) m 0.22, W = 0.1 ( f r o m F i g u r e 5) 0.26, W = 0.5 ( f r o m F i g u r e 6) '0.21, W = 0.1 ( f r o m F i g u r e 1 0 ) p.25, W = 0.5 ( f r o m F i g i i r e 11) U s i n g v a l u e s o f = 0.22 and = 0.21, t h e r e q u i r e d q u a n t i t i e s a r e = 0.69(65.0) = hh.9 f t .

= (0.22)(6U.O)(l.O5)(65.O)2(6.0) = 37ii,000 l b s .

ÏV = (0.21)(6U.0)(1.05)(65.0)2(6.0)(120) = U2,800,000 ft,'"'^^^'

I n a p p l y i n g t h i s I n f o r m a t i o n t o t h e c a l c u l a t i o n o f s t r e s s e s , i t ^ r e a s o n a b l e e n g i n e e r i n g a p p r o x i m a t i o n t o assume t h a t t h e maximur.i wave f ï"'^® and moment o c c u r a t t h e same phase a n g l e .

I t s h o u l d be reemphasized here t h a t ocean waves and wave f o r c e s a J ^ * ^ moments a r e n o t c o m p l e t e l y d e t e r m i n i s t i c . T h a t i s , d i f f e r e n t waves vri-i d e n t vri-i c a l h e vri-i g h t s and p e r vri-i o d s vri-i n t h e same w a t e r d e p t h w vri-i l l e x h vri-i b vri-i t a sï'"'^^* o f wave p r o f i l e s , c r e s t e l e v a t i o n s , and maximum wave f o r c e s and momen.'t'^ ' T h i s a s p e c t o f t h e problem has n o t been t r e a t e d h e r e , b u t must be r e c ' ^ ^ ' ^ ^ ^ ^ ^ and a c c o u n t e d f o r i n d e s i g n .

2 9 6 C O A S T A L E N G I N E E R I N G

CONCISIONS AND RECuMMKNDATiUNS FOR FURTHER TORK

For t h e two s e t s o f wave c o n d i t i o n s examined i n t h i s p a p e r , t h e Stream f u n c t i o n wave t h e o r y p r o v i d e s a s i g n i f i c a n t l y b e t t e r f i t t o t h e s p e c i f i e d boundary c o n d i t i o n s t h a n t h e o t h e r t h e o r i e s t e s t e d . I t v/ould be i n t e r e s t i n g t o i n c r e a s e t h e number and range o f wave c o n d i t i o n s t e s t e d t o e s t a b l i s h more t h o r o u g h l y t h e ranges o f v a l i d i t y o f t h e v a r i o u s wave t h e o r i e s . I n a d d i t i o n , more wave t h e o r i e s s h o u l d be t e s t e d . I t i s n o t c o r r e c t t o assume t h a t t h e h i g h e r o r d e r S t o k i a n wave t h e o r i e s a r e u n i f o r m l y more v a l i d t h a n t h e l o w e r o r d e r t h e o r i e s . A t p r e s e n t (1965), t h e r e does n o t e x i s t an e s t a b l i s h e d r e l a t i v e d e p t h l i m i t , below w h i c h t h e s o - c a l l e d s h a l l o w w a t e r wave t h e o r i e s a r e u n i f o r m l y more a p p l i c a b l e t h a n the S t o k i a n o r o t h e r t h e o r i e s . A vrave t h e o r y w o u l d be e x a c t i f i t i d e n t i c a l l y s a t i s f i e d t h e L a p l a c e e q u a t i o n and t h e s p e c i f i e d boundary c o n d i t i o n s . I t i s n o t n e c e s s a r i l y t r u e , however, t h a t t h e same e r r o r i n e i t h e r boundary c o n d i t i o n f i t o r i n t h e L a p l a c e e q u a t i o n f o r two competing wave t h e o r i e s i m p l i e s t h e same e r r o r s i n a l l wave c h a r a c t e r i s t i c s . F o r example, f o r t h e s h a l l o w i-ra.ter c o n d i t i o n s t e s t e d , t h e A i r y i/ave t h e o r y p r o v i d e d a b e t t e r f i t t o t h e b o u n d a r y c o n d i t i o n s t h a n t h e h i g h e r o r d e r S t o k i a n t h e o r i e s . The S t o k i a n t h e o r i e s , however, w o u l d p r o v i d e a b e t t e r p r e d i c t i o n o f the c r e s t d i s p l a c e m e n t t h a n would t h e A i r y t h e o r y w h i c h does n o t a c c o u n t f o r t h e n o n l i n e a r l t i e s . The r e l a t i o n s h i p between b o u n d a r y c o n d i t i o n and L a p l a c e e q u a t i o n e r r o r s and a s s o c i a t e d e r r o r s i n wave c h a r a c t e r i s t i c s r e q u i r e s f u r t h e r work.

I t would be v e r y w o r t h w h i l e t o determine t h e most v a l i d irave t h e o r i e s f o r a l l wave c o n d i t i o n s o f importance t o c o a s t a l e n g i n e e r i n g ( i . e . , r a n g i n g

S T R E A M F U N C T I O N 2 9 7

f r o m s h o r t i r i n d - g e n e r a t e d vraves t o l o n g waves such as Tsunamis). I n t h e same manner as t h e d i m e n s i o n l e s s c r e s t d i s p l a c e m e n t s and f o r c e s were p r e s e n t e d i n t h i s paper i n g r a p h i c a l f o r m , t h e wave p a r a m e t e r s o f g r e a t e s t use c o u l d t h e n be t a b u l a t e d o r p r e s e n t e d i n g r a p h i c a l f o r m u s i n g t h e most v a l i d t h e o r y f o r each s e t o f wave c o n d i t i o n s .

ACKIWWLEDGMEKTS

The m a j o r i t y o f t h e work p r e s e n t e d here was e a r n e d o u t b y t h e a u t h o r w h i l e employed b y Chevron Research Company; t h e i r a p p r o v a l t o p u b l i s h this paper i s hereby g r a t e f u l l y acknowledged.

2 9 8 C O A S T A L E N G I N E E R I N G

Ai-PENDix. —NUTATION

The f o l l o w i n g symbols have been adopted f o r use i n t h i s paper»

C wave c e l e r i t y ;

Cp d r a g c o e f f i c i e n t ;

CJ,} i n e r t i a c o e f f i c i e n t ;

D p i l i n g d i a m e t e r ;

(E-[^)fj range o f e r r o r a s s o c i a t e d w i t h k i n e m a t i c boundary c o n d i t i o n j

( E ^ ) ^ range o f e r r o r a s s o c i a t e d w i t h dynamic boundary c o n d i t i o n ;

F t o t a l wave f o r c e on s i n g l e v e r t i c a l p i l i n g ; g g r a v i t a t i o n a l a c c e l e r a t i o n ; h s t i l l w a t e r d e p t h ; H wave h e i g h t ; — ^ s u b s c r i p t " i " d e n o t i n g d i s c r e t e wave phase p o s i t i o n s ; s u b s c r i p t "m" d e n o t i n g maximum o f f c r c e o r moment; M t o t a l wave moment, a b o u t b o t t o m o f s i n g l e v e r t i c a l p i l i n g ; Q B e r n o u l l i " c o n s t a n t " ; -y upwards; S v e r t i c a l c o o r d i n a t e , o r i g i n a t sea b o t t o m , o r i e n t e d p o s l t i v ^ - ^ - ' T wave p e r i o d ; u h o r i z o n t a l component o f w a t e r p a r t i c l e v e l o c i t y ; • u h o r i z o n t a l component o f w a t e r p a r t i c l e a c c e l e r a t i o n ; w v e r t i c a l component o f w a t e r p a r t i c l e v e l o c i t y ; W p a r a m e t e r d e f i n e d as ^ - ^ ; D . — a c t i o n X h o r i z o n t a l c o o r d i n a t e , o r i g i n a t wave c r e s t , p o s i t i v e i n d a - ^ o f wave p r o p a g a t i o n ;

S T R E A M F U N C T I O N 2 9 9 v e r t i c a l c o o r d i n a t e , o r i g i n i n s t i l l w a t e r l i n e , p o s i t i v e l y upvrards; d i m e n s i o n l e s s t o t a l moment on s i n g l e v e r t i c a l p i l i n g ; l o c a l e r r o r a s s o c i a t e d v r i t h k i n e m a t i c boundary c o n d i t i o n ; l o c a l e r r o r a s s o c i a t e d w i t h dynamic boundary c o n d i t i o n ; s p e c i f i c w e i g h t o f w a t e r ; wave d i s p l a c e m e n t above s t i l l w a t e r l i n e ; wave c r e s t d i s p l a c e m e n t above s t i l l w a t e r l i n e ;

wave phase a n g l e , o r i g i n a t wave c r e s t , p o s i t i v e i n d i r e c t i o n o f wave p r o p a g a t i o n ; n u m e r i c a l c o n s t a n t , 3.lUl59...; mass d e n s i t y o f w a t e r ; d i m e n s i o n l e s s t o t a l f o r c e on s i n g l e v e r t i c a l p i l i n g ; v e l o c i t y p o t e n t i a l ; stream f u n c t i o n .