Frederic R. Harris, Inc., Ne1V York, N. Y.; no", \lith
Van Houten, SchHartz and Nurphy, Ne", York, N. Y. Jeppe J. Edens
Mueser, Rutledge, HentlIorth & Johnston, Nell York, N. Y. ABSTRACT
Drag and mass coefficients corresponding to
sto~
es
'b
fifth-order theory have been determined from previouslY pUt .-ta ~s -lished wave and wave-force data. Comparisons between S t tical distributions for fifth-order and corresponding £i r s -es
pness . order coefficients have been made for varying wave stee co-For other existing wave data, from which the fifth orde r 'b efficients could not be determined, the statistical
di
st
r~o~
tions of first-order coefficients have been investigated varying wave steepnesses.drag Statistically based fifth-order and first-order
coefficient design values for varying Reynolds numberS are presented. Statistically based fifth and first-order mass coefficient design values are suggested.
Comparisons between drag-coefficient distri but i ops .
~:d
corresponding mass-coefficient distributions seem to
P
:rO
;~i_
some insight into the reasons for the scatter on the c C)ecients.
INTRODUCTION
Notation. - The letter symbols adopted for use paper are defined where they first appear.
:i-:P this
p and
The method presented by Morison, O'Brien, JohnS C7 . des
Schaaf (1950) for determining wave forces on piles, d:i--v:l- r o-the total force into a drag force, which is numerical L
~
~s
of portional to the second power of the horizontal compon. e ! Pthe water particle velocities, and a mass or inertia f
c>
:r
~;ti_
proportional to the horizontal components of the water P cylin
-cle accelerations. The wave force per unit length of ~
drical pile is expressed as:
J'V D2tJ
.9
U2F = CD D 2 + C ...,l'c...::-....:M 4 J=.-
U
::i-cient, in which CD is the drag coefficient, C
M the mass coeff: ~locity, D the pile diameter, U the horizontal water particle ~
220 C O A S T A L E N G I N E E R I N G
U t h e h o r i z o n t a l 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 , and 9 t h e mass d e n s i t y f o r w a t e r .
The c o e f f i c i e n t s C and C a r e d e t e r m i n e d f r o m t h e above r e l a t i o n s h i p by u s i n g measured v a l u e s f o r F and t h e o r e t i c a l v a l u e s f o r U and U. A l r y ' s f i r s t - o r d e r wave t h e o r y ( w h i c h i s s u p p o s e d l y o n l y v a l i d f o r r e l a t i v e l y low waves o f low s t e e p -n e s s ) a-nd S t o k e s ' seco-nd-, t h i r d - a-nd f i f t h - o r d e r t h e o r i e s
( w h i c h s u p p o s e d l y r e p r e s e n t storm-wave a c t i o n more c l o s e l y ) have been employed.
M o r i s o n , Johnson and O ' B r i e n (1954) used A i r y ' s f i r s t -o r d e r t h e -o r y t -o d e t e r m i n e C and C v a l u e s f -o r m-odel p i l e s -o f v a r i o u s c r o s s s e c t i o n s i n a wave c h a n n e l and a 3.5 i n . d i a m -e t -e r c y l i n d r i c a l p i l -e i n p r o t o t y p -e wav-es. Th-e D -e p t . o f Oc-ean- Ocean-o g r a p h y Ocean-o f Texas A & M (1954, 1955, 1956 and 1957) used l i n e a r wave t h e o r y on d a t a f r o m t h e G u l f o f M e x i c o .
W i e g e l , Beebe and Moon (1959) used A i r y ' s t h e o r y t o de-t e r m i n e C and C v a l u e s f o r ocean-wave f o r c e s on c y l i n d r i c a l p i l e s . ™
On some o f t h e above d a t a t h e Beach E r o s i o n B o a r d ( 1 9 6 1 ) a p p l i e d S t o k e s ' s e c o n d - and t h i r d - o r d e r t h e o r i e s .
C o n s i d e r a b l e v a r i a t i o n s i n mean v a l u e s f o r C and C» as w e l l as l a r g e s c a t t e r and s t a n d a r d d e v i a t i o n s have r e s u l t e d f r o m t h e above i n v e s t i g a t i o n s . The r e a s o n s f o r t h e d i s c r e p -a n c i e s m-ay be m-any. Among t h e more i m p o r t -a n t -a r e :
a. I r r e g u l a r wave s p e c t r a c o n s i s t i n g o f s e v e r a l p e r i o d s and a m p l i t u d e s . I n t h e a n a l y s e s t h e wave s p e c t r a a r e u s u a l l y r e d u c e d t o one wave p e r i o d and one wave h e i g h t i n a r a t h e r p r i m i t i v e manner. b. T u r b u l e n c e a r o u n d t h e s t r u c t u r e s by w h i c h t h e t e s t p i l e s a r e s u p p o r t e d . c. V i b r a t i o n s o f t e s t p i l e s . d. I n a c c u r a c i e s on w a v e - f o r c e r e c o r d i n g s . e. I n a b i l i t y o f wave t h e o r i e s t o d e s c r i b e a c t u a l w a t e r -p a r t i c l e m o t i o n . I n model t e s t s w i t h r e g u l a r wave t r a i n s , i t s h o u l d be p o s s i b l e t o e l i m i n a t e a, ||bj and c and r e d u c e d c o n s i d e r a b l y . ? X
i t i s p o s s i b l e t o e l i m i n a t e e. T h i s was done by L a i r d , John-son and Walker (1960) who r e p o r t e d t h a t t h e i r CQ v a l u e s f o r a c c e l e r a t e d and u n i f o r m m o t i o n a g r e e d w i t h a c c e p t e d v a l u e s f o r u n i f o r m m o t i o n , whereas d e v i a t i o n s o c c u r r e d f o r d e c e l e r -a t e d m o t i o n . T h e i r v -a l u e s f o r n o n - u n i f o r m m o t i o n were how-e v how-e r d how-e t how-e r m i n how-e d undhow-er t h how-e a s s u m p t i o n t h a t Cj^j was how-e q u a l t o u n i t y . T h i s paper p r e s e n t s Cp and v a l u e s d e t e r m i n e d by t h e w r i t e r s by a p p l y i n g S t o k e s ' f i f t h - o r d e r t h e o r y a d a p t e d f o r
t h i s purpose by S k j e l b r e i a , H e n d r i c k s o n , Gragg and Webb (1960) on wave and w a v e f o r c e d a t a by W i e g e l e t a l ( 1 9 5 9 ) . S t a t i s -t i c a l a n a l y s e s and c o m p a r i s o n s w i -t h c o r r e s p o n d i n g f i r s -t - o r d e r v a l u e s f o r CD and Cj^j a r e I n c l u d e d . The f i f t h - o r d e r t h e o r y p r e s e n t e d by S k j e l b r e i a e t a l ( 1 9 6 0 ) as t a b u l a t i o n s o f computer s o l u t i o n s i s s p e c i f i c a l l y d e s i g n e d f o r c o n v e n i e n t d e t e r m i n a t i o n o f wave p r e s s u r e s , f o r c e s and moments on p i l e s . Volume I o f t h e above r e p o r t , w h i c h i s c l a s s i f i e d "For O f f i c i a l Use O n l y , " c o n t a i n s s t a t i s
t i c a l l y based d r a g and mass c o e f f i c i e n t s f o r d e s i g n use t o -g e t h e r w i t h t h e t a b l e s , w h i c h a r e n o t i n any way c l a s s i f i e d .
DETERMINATION OF FIFTH-ORDER C^ AND C, VALUES D M
T a b l e I by W i e g e l e t a l (1959) c o n t a i n s wave h e i g h t s H and p e r i o d s T, t o g e t h e r w i t h wave f o r c e s FQ a t t h e passage o f a wave c r e s t (phase a n g l e 0 = 0 ° ) . The f o r c e i s f o r one l i n e a r f t . o f a 12.75 i n . d i a m e t e r p i l e a t a d i s t a n c e S o f 42.5 f t . above t h e b o t t o m . The w a t e r d e p t h d i s r e p o r t e d t o be between 49.3 f t . and 46.0 f t . An a v e r a g e v a l u e o f 47.65 f t . was used h e r e . T a b l e I I c o n t a i n s t h e same i n f o r m a t i o n f o r a 24 i n . d i a m e t e r p i l e a t a d i s t a n c e o f 33.0 f t . above t h e b o t t o m w i t h t h e w a t e r d e p t h r e p o r t e d t o be between 48.0 f t . and 46.0 f t . An average v a l u e o f 47.0 f t . was used h e r e .
For w i d e ranges o f d/T^, H/d, 9 =0° t o 180° and S/d v a l u e s f r o m 0.1 t o 1.5, t h e t a b l e s by S k j e l b r e i a e t a l (1960) g i v e v a l u e s f o r t h e p r e s s u r e f u n c t i o n s Pi and P2 f r o m w h i c h d r a g p r e s s u r e i s d e t e r m i n e d as H CQ and i n e r t i a p r e s s u r e as P2 D Cj^j. The t o t a l wave f o r c e on t h e p i l e and i t s moment a b o u t t h e b o t t o m a r e f o u n d e q u a l l y c o n v e n i e n t f r o m t h e t a b u -l a t e d f o r c e and moment f u n c t i o n s Fi, F2, M]^, and M2 .
L i n e a r i n t e r p o l a t i o n s were used where n e c e s s a r y . The t a b u l a t e d v a l u e s f o r d/T2 have a l o w e r l i m i t o f 0.20. Of t h e
222 C O A S T A L E N G I N E E R I N G
1 9 5 waves i n T a b l e s I and I I , 2 5 had d / T 2 v a l u e s below 0 . 2 0 ' Those w i t h d/T^ v a l u e s e q u a l t o o r e x c e e d i n g 0 . 1 7 5 were i n -c l u d e d by l i n e a r e x t r a p o l a t i o n f r o m 0 . 2 0 and 0 . 2 5 . For e = 0 ° 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 s and a c c o r d i n g l y i n e r t i a f o r c e s a r e z e r o . The d r a g c o e f f i c i e n t f o r w h i c h * ® symbol Cjjs i s employed i s f o u n d f r o m t h e e q u a t i o n : C ^ D 5 - p — Ï T D T a b l e I I I by W i e g e l e t a l ( 1 9 5 9 ) c o n t a i n s t h e w a ^ e f o r c e Fg^yj^ c o r r e s p o n d i n g t o t h e passage o f t h e s t i l l - ^ * * ^ i ^ j ^ g l e v e l p o i n t on t h e wave p r o f i l e , w h i c h p r e c e d e s a c r e s t -o t h e r d a t a i n T a b l e I I I a r e t h e same as i n T a b l e I I .
The f i f t h - o r d e r mass c o e f f i c i e n t f o r w h i c h t h e ey'"^°''" % 5 i s used may be d e t e r m i n e d f o r a t o t a l o f 7 4 waves
w h i c h Cj)5 a t t h e passage o f t h e wave c r e s t i s d e t e r m i n e d
-One f e a t u r e o f t h e computer s o l u t i o n s t a b u l a t e d ^ ^ ^ g S k j e l b r e i a e t a l ( 1 9 6 0 ) i s t h e n o n - d i m e n s i o n a l wave s u ï " * ^ o r d i n a t e Y. The s t i l l - w a t e r l e v e l i s c h a r a c t e r i z e d b y
The c o r r e s p o n d i n g v a l u e s f o r and P2 a r e f o u n d by lin®^''^ i n t e r p o l a t i o n s . The e x p r e s s i o n f o r Cjj5 i s :
^ ^SWL - ^ " ^ ^ D 5 CMS = T,
T a b l e s 1 and 2 a r e s i m i l a r t o T a b l e s I and I I t>5^ j j _ W i e g e l e t a l ( 1 9 5 9 ) . Columns w i t h C-Q^ v a l u e s and c o r r ^ ^ ^
i n g R e y n o l d s numbers Res have been added. The symbol ^^\±x-has been used i n s t e a d o f CQ f o r t h e f i r s t - o r d e r d r a g < ^ * ^ ^ - j ^ i i c i e n t s and Re 1 f o r c o r r e s p o n d i n g Reynolds numbers. I t ^ ^ n be n o t i c e d t h a t C^s ( i n a l l b u t two c a s e s ) i s s m a l l e r ^ ^ \ ^ the c o r r e s p o n d i n g u s u a l l y c o n s i d e r a b l y s m a l l e r . ' ^ ^ ^ b v was e x p e c t e d as t h e w a t e r - p a r t i c l e v e l o c i t i e s d e t e r m i n ^ T ' the h i g h e r - o r d e r t h e o r y a r e u s u a l l y h i g h e r . The 0^5 ^ ^ have been t a b u l a t e d w i t h two d e c i m a l p o i n t s , w h i c h i s m*^-^^ t h a n s t r i c t l y n e c e s s a r y .
T a b l e 3 i s s i m i l a r t o T a b l e I I I by W i e g e l e t aO- ^ However, a column f o r 0^,5 has been added and t h e s y m b c ^ ^ i has been used i n s t e a d o f f o r t h e mass c o e f f i c i e n t t»
on t h e f i r s t - o r d e r t h e o r y . T a b l e 3 does o n l y c o n t a i n ^ waves o u t o f t h e t o t a l o f 7 4 waves f o r w h i c h 0^5 i s k n - ''^^oüld The r e a s o n f o r t h i s i s t h a t t h e s e 2 8 Cjj5 v a l u e s , w h i c l i
-224 C O A S T A L E N G I N E E R I N G
be d e t e r m i n e d w i t h o u t e x c e s s i v e t i m e - c o n s u m i n g two-way o r f o u r way i n t e r p o l a t i o n s , i n no case d e v i a t e d more t h a n + 0 . 1 f r o m t h e c o r r e s p o n d i n g C^^ v a l u e s . A p o s s i b l e e x p l a n a t i o n f o r t h i s may be t h a t t h e a c t u a l s t i l l - w a t e r phase a n g l e s were n o t known t o t h e w r i t e r s . The t h e o r e t i c a l f i f t h - o r d e r s t i l l - w a t e r a n g l e s were employed. The p r o d u c t H D C^s d i d n o t exceed 2% o f FsWL e x c e p t i n one case. Thus t h e i n a c c u r a c y on C^s v a l u e s was n o t i n c o r p o r a t e d i n t h e Cj^g v a l u e s . I t was n o t c o n s i d e r e d w o r t h w h i l e t o compute t h e r e m a i n i n g 46 Cjj5 v a l u e s p a r t l y f o r t h e r e a s o n s t a t e d above and p a r t -l y because t h e Cj^jj v a -l u e s a r e much h i g h e r t h a n t h o s e f r o m o t h e r s o u r c e s . DRAG COEFFICIENTS STATISTICAL ANALYSES OF CD VALUES
As t h e s c a t t e r on t h e CQ v a l u e s i s n o t s m a l l compared t o t h e mean v a l u e s , t h e s t a t i s t i c a l d i s t r i b u t i o n c a n n o t be assumed t o f o l l o w t h e normal law o f e r r o r . T h i s may e a s i l y l e a d t o e r r o n e o u s c o n c l u s i o n s . A s i m p l e f u n c t i o n o f CD w h i c h may be assumed t o f o l l o w t h e n o r m a l law o f e r r o r i s l o g CD The s c a t t e r and t h e s t a n d a r d d e v i a t i o n f o r l o g CD a r e n o t changed i f a l l t h e Cp v a l u e s and t h u s t h e mean a r e m u l t i p l i e d by a c o n s t a n t . T h i s f a c i l i t a t e s s t a t i s t i c a l c o m p a r i s o n o f d i s t r i b u t i o n s w i t h d i f f e r e n t mean v a l u e s and p r o v e d t o be v e r y u s e f u l . P a r a l l e l l i n e s s i m p l y i n d i c a t e t h a t t h e s t a n d a r d d e v i a t i o n on t h e l o g CD d i s t r i b u t i o n i s t h e same. S t a t i s t i c a l d i s t r i b u t i o n s f o r CD v a l u e s a r e shown i n F i g s . 1 , 2 and 3. The a b s c i s s a s c a l e i s l o g a r i t h m i c , w h i l e t h e o r d i n a t e s c a l e i s a s o - c a l l e d p r o b a b i l i t y s c a l e . I f t h e n o r m a l l a w o f e r r o r i s v a l i d f o r t h e l o g CD d i s t r i b u t i o n , a s t r a i g h t l i n e w i l l r e s u l t , when t h e numbers o f CD v a l u e s ( i n p e r c e n t o f t h e t o t a l number) s m a l l e r t h a n o r e q u a l t o c e r t a i n CD v a l u e s a r e p l o t t e d as a f u n c t i o n o f t h e s e CD v a l u e s . F i g . 1 c o n t a i n s t h e d a t a f r o m T a b l e 1 f o r a 12.75 i n . d i a m e t e r p i l e . Two s t a t i s t i c a l d i s t r i b u t i o n s f o r Cjjg as w e l l as C D I v a l u e s a r e shown. One i n c l u d e s a l l t h e v a l u e s , t h e o t h e r o n l y t h o s e f o r w h i c h t h e c o r r e s p o n d i n g wave s t e e p n e s s H/L e q u a l s o r exceeds 2.0%. The wave s t e e p n e s s v a r i e d f r o m 0.7% t o 3.6%. The w r i t e r s a r e aware o f t h e l i m i t e d r e l i a b i l -i t y o f a s t a t -i s t -i c a l a n a l y s -i s o f o n l y 20 Cn v a l u e s f o r w h -i c h H/L ^ 2 . 0 % .
-•CD5 A L L V A L U E S ® CD5 - f t 2 . 0 % xC 01 + C 01 I I M I I I I 99.9r 5 (» y-2 : I .
0.5 '
1 1 1 1 1 — I I I I 1 1 \ 1 1 I N I
0.1 0.2 | 0 . 3 0.4 0.5 1.0 1.5 2.0 3.0 4.0 5.0 10.0 Co(LOGARITHMIC S C A L E )Figure 2. Statistical Distributions of C^^ Values for D = 24.0 in. Based on Data by Wiegel, Beebe and Moon
226 C O A S T A L E N G I N E E R I N G t i o n s f o r l o g CQ c l o s e l y a p p r o x i m a t e s t r a i g h t l i n e s f o r t h e p o i n t s above 5%. S t e e p e r d i s t r i b u t i o n s and t h u s s m a l l e r s t a n d a r d d e v i a t i o n s r e s u l t when t h e lower l i m i t f o r H/L i s r a i s e d t o 2.0%. The s t a n d a r d d e v i a t i o n i s d e f i n e d as t h e d i f f e r e n c e between l o g Cj) v a l u e s c o r r e s p o n d i n g t o 83.5% and 16.5%. I n t h e f o l l o w i n g s e c t i o n t h e r e s u l t s a r e d i s c u s s e d f u r t h e r and compared w i t h t h o s e o b t a i n e d f o r a 24 i n . d i a m -e t -e r p i l -e i n F i g . 2 and a 16 i n . d i a m -e t -e r p i l -e f r o m B r e t s c h n e i d e r ' s (1957) d a t a . F i g . 2 c o n t a i n s t h e d a t a f r o m T a b l e 2. Two d i s t r i b u -t i o n s f o r each o f Cps and C D I shown. One f o r a l l v a l u e s and one f o r 40 v a l u e s w i t h c o r r e s p o n d i n g H/L = 2.0%. H/L v a r i e d f r o m 1 . 1 % t o 3.4%. The s t a t i s t i c a l d i s t r i b u t i o n s a p p r o a c h s t r a i g h t l i n e s f o r v a l u e s above 20%. Only t h e p l o t t e d p o i n t s a r e shown f o r t h e d i s t r i b u t i o n o f Cj^i v a l u e s f o r H/L ^ 2.0% w h i c h i s v e r y c l o s e t o t h e 0^5 d i s t r i b u t i o n f o r a l l v a l u e s . An i n c r e a s e o f t h e l o w e r l i m i t o f H/L seems t o i n f l u e n c e t h e mean v a l u e s b u t n o t t h e s t a n d a r d d e v i a t i o n -F i g . 3 shows f o u r C^-^ d i s t r i b u t i o n s f r o m B r e t s c h n e i d e : ( 1 9 5 7 ) d a t a . Only waves w i t h a p e r i o d o f 6.0 s e e s , o r mo^e were i n c l u d e d , as t h e e r r o r on t h e measured wave h e i g h t s exceed + 2 0 % f o r l o w e r wave p e r i o d s . Out o f a t o t a l o f l ^ ^ waves, 95 had a s t e e p n e s s o f 2% o r more, 38 had a s t e e p n e s s o f 3% o r more and 15 had a s t e e p n e s s o f 3.5% o r more. So^"® p o i n t s a t t h e l o w and h i g h ends s t r a y away f r o m t h e s t r a i g l ^ * l i n e d i s t r i b u t i o n s .
ANALYSES OF DISTRIBUTIONS
For t h e i n d i v i d u a l p i l e i n a c o n s t a n t w a t e r d e p t h , w i t h t h e f o r c e s measured f o r o n l y one e l e v a t i o n i n t e r v a l ? o n l y v a r i a b l e s a r e t h e wave c h a r a c t e r i s t i c s . The r e c o r d e d v a r i a b l e s a r e t h e wave h e i g h t H and t h e wave p e r i o d T. T h e wave l e n g t h , L, i s computed f r o m T and t h e w a t e r d e p t h d -n o -n - d i m e -n s i o -n a l wave p a r a m e t e r s , t h e s t e e p -n e s s H/L a-nd 1 1 ^ ^ r e l a t i v e h e i g h t H/d, a r e commonly u s e d . As d i s c o n s t a n ' t and t h e l a r g e r v a l u e s f o r H/L g e n e r a l l y c o r r e s p o n d t o t h e la.ï'ëer v a l u e s f o r H, i t was d e c i d e d t o use o n l y H/L as a p a r a m e ' t e r f o r t h e i n d i v i d u a l p i l e . F i g . 1 shows t h a t t h e f i f t h - o r d e r and f i r s t - o r d e r ^T) d i s t r i b u t i o n s f o r a l l waves a r e v e r y c l o s e t o b e i n g p a r a. 1 i® •'•' t h a t i s t h e s t a n d a r d d e v i a t i o n on l o g Cp i s t h e same. A-S t h e a v e r a g e wave s t e e p n e s s i s l e s s t h a n 2%, we c o u l d n o t exp>^*^''' t o f i n d any s i g n i f i c a n t d i f f e r e n c e s . The f i r s t - o r d e r ttx&°^y may be assumed t o be adequate f o r such g e n t l e s w e l l s . X* ï^^*
X CDI A L L V A L U E S ® C D I ^= 3.0%
+ CDI f = 2.0%
ffl CDI
^ = 3.5%99.9| I 1 I 1 1 — I I I I 1 1
0.1 0.2 0.3 0.4 0.5 1.0 1.5 2.0 3.0 4.0 5.0 10.0 Co(LOGARITHMIC S C A L E )
Figure 3. Statistical Distributions of C^^^ Values f o r D = 16.0 in. Based on Texas A & M Data (1957)
228 C O A S T A L E N G I N E E R I N G t h e f i f t h o r d e r d i s t r i b u t i o n l i e s b e l o w t h e f i r s t o r d e r d i s -t r i b u -t i o n i s -t o be e x p e c -t e d , as -t h e f i f -t h - o r d e r w a -t e r - p a r -t i c l e v e l o c i t i e s a r e g e n e r a l l y c o n s i d e r a b l y h i g h e r t h a n t h e c o r r e s -p o n d i n g f i r s t - o r d e r v e l o c i t i e s . When waves w i t h a s t e e p n e s s o f l e s s t h a n 2% a r e d i s -r e g a -r d e d b o t h Cp d i s t -r i b u t i o n s become c o n s i d e -r a b l y s t e e p e -r , w h i c h means t h a t t h e s t a n d a r d d e v i a t i o n s d e c r e a s e d . T h a t t h e f i r s t - o r d e r d i s t r i b u t i o n became s t e e p e r t h a n t h e f i f t h - o r d e r d i s t r i b u t i o n i s perhaps n o t s i g n i f i c a n t as t h e s t a t i s t i c a l d i s t r i b u t i o n s a r e based on o n l y 20 Cp v a l u e s . However, Dean
( 1 9 6 5 ) has shown t h a t convergence p r o b l e m s e x i s t f o r t h e
h i g h e r - o r d e r t h e o r i e s a p p l i e d t o s h a l l o w w a t e r waves, and t h a t t h e f i r s t - o r d e r t h e o r y i s s u p e r i o r t o t h e f i f t h - o r d e r t h e o r y w i t h i n c e r t a i n r a n g e s o f t h e p e r t i n e n t p a r a m e t e r s . I t was n o t t o be e x p e c t e d t h a t t h e s t a n d a r d d e v i a t i o n w o u l d d e c r e a s e f o r waves o f h i g h e r s t e e p n e s s u s i n g e i t h e r f i f t h - o r d e r o r f i r s t - o r d e r t h e o r y . N e i t h e r o f t h e t h e o r i e s s h o u l d be a b l e t o d e s c r i b e s t e e p waves b e t t e r t h a n l e s s s t e e p waves . The r e a s o n f o r t h e decrease i n s t a n d a r d d e v i a t i o n may be t h a t t h e r e l a t i v e a c c u r a c y on t h e measured wave f o r c e s i s b e t t e r , a s h i g h e r f o r c e s g e n e r a l l y c o r r e s p o n d t o h i g h e r s t e e p -nesses and h i g h e r a b s o l u t e wave h e i g h t s . A l s o , a g a i n i t s h o u l d be c o n s i d e r e d t h a t t h e d i s t r i b u t i o n s a r e based on o n l y 20 Cp v a l u e s .
F i g . 2 shows Cp d i s t r i b u t i o n s f o r a 24 i n . d i a m e t e r p i l e . Here t h e l i n e s f o r a l l waves and t h e l i n e s f o r s t e e p -nesses of 2% o r more a r e v e r y c l o s e t o b e i n g p a r a l l e l . The s t a n d a r d d e v i a t i o n on l o g Cp i s t h u s about t h e same. A g a i n , the f i r s t - o r d e r d i s t r i b u t i o n f o r waves o f h i g h e r s t e e p n e s s has a s l i g h t l y l o w e r s t a n d a r d d e v i a t i o n t h a n t h e c o r r e s p o n d i n g f i f t h - o r d e r d i s t r i b u t i o n . 40 Cp v a l u e s c o r r e s p o n d e d t o a s t e e p n e s s o f 2% o r more. F i g . 3 shows C p j d i s t r i b u t i o n s f o r a 16 i n . d i a m e t e r p i l e f r o m B r e t s c h n e i d e r ' s (1957) d a t a . As m e n t i o n e d p r e v i o u s -l y , t h e wave f o r c e d a t a , f r o m w h i c h f i f t h - o r d e r c o e f f i c i e n t s c o u l d have been d e t e r m i n e d , were n o t a v a i l a b l e . The wave h e i g h t v a r i e d f r o m 3.5 t o 13.3 f t . and t h e wave p e r i o d f r o m 6.0 t o 8.8 s e e s . The s t e e p n e s s v a r i e d f r o m 1.3% t o 8.0%. The p i l e e x t e n d e d 13 f t . below s t i l l - w a t e r i n a w a t e r d e p t h o f 40 f t . The t o t a l wave f o r c e s on t h e p i l e were measured. The R e y n o l d s number v a l u e s were g e n e r a l l y low, o n l y two exceeded a v a l u e o f a b o u t 4 x 10^. From F i g . 3 i t i s once more f o u n d t h a t t h e s t a n d a r d d e v i a t i o n on f i r s t - o r d e r l o g Cp v a l u e s does
s t r a i g h t l i n e s c l o s e l y . Only a few p l o t t e d p o i n t s a t t h e l o w e r and upper ends d e v i a t e c o n s i d e r a b l y f r o m t h e l i n e s .
For t h e r a n g e s o f v a r i a b l e s i n v e s t i g a t e d above, i t i s c o n c l u d e d t h a t t h e f i f t h - o r d e r a p p r o a c h i s n o t s u p e r i o r t o t h e f i r s t - o r d e r a p p r o a c h . A s l i g h t i n d i c a t i o n o f t h e o p p o s i t e i s f o u n d . F o r l a r g e r d/L r a t i o s t h e f i f t h - o r d e r t h e o r y may be s u p e r i o r . See Dean ( 1 9 6 5 ) . DISCUSSION For s t e a d y - s t a t e f l o w a r o u n d c y l i n d e r s , a d e f i n i t e r e l a t i o n s h i p between Cp and Reynolds number has been e s t a b -l i s h e d e x p e r i m e n t a -l -l y . The c u r v e shown i n F i g s . 4 and 5 f o r an i n f i n i t e l y l o n g c y l i n d e r was c o p i e d f r o m P r a n d t l and
T i e t j e n s ( 1 9 3 4 ) . When f i r s t - o r d e r Cp v a l u e s f o r waves a r e p l o t t e d as a f u n c t i o n o f Re, c l o u d s o f p o i n t s appear. See F i g s . 11 and 12 by W i e g e l e t a l ( 1 9 5 9 ) . No c o n c l u s i o n s may be d r a w n , o t h e r t h a n t h a t Cj) v a r i e s v e r y c o n s i d e r a b l y w i t h i n r e l a t i v e l y n a r r o w r a n g e s o f Re, e.g. on t h e above m e n t i o n e d F i g . 12 Cp v a r i e s f r o m 0.11 t o 0.9 r i g h t a r o u n d R e = 7 x l 0 ^ and CD v a r i e s f r o m 0.4 t o 2.0 f o r Re v a r y i n g f r o m 3 t o 4x10^. S i m i l a r c l o u d s o f p l o t t e d p o i n t s r e s u l t when t h e f i f t h - o r d e r t h e o r y i s a p p l i e d .
G e n e r a l l y , i t i s n o t d e s i r a b l e t o use Reynolds number as a p a r a m e t e r f o r w a v e a c t i o n phenomenon, when t h e w a t e r -p a r t i c l e v e l o c i t i e s c o n t a i n e d i n Re a r e n o t measured ( w h i c h t h e y a r e g e n e r a l l y n o t ) , b u t d e t e r m i n e d a c c o r d i n g t o t h e a p -p l i e d wave t h e o r y . One o f t h e m a j o r r e a s o n s f o r t h e s c a t t e r on CQ may be t h e d i f f e r e n c e s between a c t u a l and t h e o r e t i c a l p a r t i c l e v e l o c i t i e s . K e u l e g a n and C a r p e n t e r (1958) s t u d i e d f o r c e s on a c y l i n d r i c a l p i l e exposed t o s t a n d i n g waves i n t h e l a b o r a t o r y . I t a p p e a r e d as i f Cp was a f u n c t i o n o f t h e p a r a m e t e r U^^^^ T/D w h i c h a c c o r d i n g t o f i r s t - o r d e r t h e o r y i s p r o p o r t i o n a l t o t h e r a t i o between t h e h o r i z o n t a l p a r t i c l e a m p l i t u d e and p i l e d i a m -e t -e r . I n t h -e i r s t u d i -e s t h -e R-eynolds numb-er r a n g -e f r o m 4 x 10^ t o 3 X 10^ i s c o n s i d e r a b l y below c r i t i c a l v a l u e s . The p e r i o d f o r t h e s t a n d i n g wave i s c o n s t a n t c o r r e s p o n d i n g t o t h e con-s t a n t b a con-s i n l e n g t h and d e p t h . The a c t u a l p a r a m e t e r i con-s t h u con-s n o t Unjax T/D b u t Umax/D. W i e g e l (1964) i n h i s F i g . 11.9
shows t h e r e l a t i o n s h i p f o u n d by Keulegan and C a r p e n t e r (1958) t o g e t h e r w i t h p o i n t s p l o t t e d f r o m t h e d a t a by W i e g e l e t a l
CO O
Figure 5. Coefficient of Drag as a Function of Reynolds Number. First Order Theory
232 C O A S T A L E N G I N E E R I N G
( 1 9 5 9 ) . Large s c a t t e r on Cp i s f o u n d w i t h i n r e l a t i v e l y narrow l i m i t s o f t h e p a r a m e t e r . The p l o t t e d p o i n t s go f a r above and b e l o w t h e c u r v e f o r s t a n d i n g waves.
For h i g h v a l u e s o f t h e h o r i z o n t a l - p a r t i c l e a m p l i t u d e t o p i l e - d i a m e t e r r a t i o combined w i t h r e l a t i v e l y l o w maximum-p a r t i c l e v e l o c i t i e s ( l o n g waves o f l o w s t e e maximum-p n e s s ) one m i g h t e x p e c t t h a t c o n d i t i o n s s i m i l a r t o s t e a d y - s t a t e f l o w f o r lower R e y n o l d s numbers w o u l d be approached under t h e wave c r e s t . The d a t a by W i e g e l e t a l (1959) g e n e r a l l y c o n f o r m t o t h e s e r e q u i r e m e n t s . However, t h e c o r r e s p o n d i n g v a l u e s do n o t a p p r o a c h t h o s e f o r s t e a d y s t a t e f l o w . I t seems t o t h e w r i t e r s , t h a t i t c a n n o t be i n s i g n i f i -c a n t f o r a -c o n s t a n t v a l u e o f D and Umax T/D w h e t h e r a l o w v a l u e f o r Uj^^x i s combined w i t h a h i g h v a l u e f o r T, o r a h i g h v a l u e f o r Umax i s combined w i t h a low v a l u e f o r T. As Umax T/D=ReVT/D2 t h e f o r m e r case w o u l d c o r r e s p o n d t o l o w e r v a l u e s o f R e y n o l d s numbers and s l o w l y c h a n g i n g p a r t i c l e v e l o c i t i e s , t h a t i s , i t w o u l d a p p r o a c h s t e a d y - s t a t e f l o w f o r l o w e r v a l u e s o f R e y n o l d s number. The l a t t e r case w o u l d c o r -r e s p o n d t o h i g h e -r and mo-re -r a p i d l y c h a n g i n g v a l u e s o f Rey-n o l d s Rey-number. The w r i t e r s f i Rey-n d i t h a r d t o u Rey-n d e r s t a Rey-n d t h a t t h e s e w i d e l y d i f f e r e n t c o n d i t i o n s s h o u l d have no i n f l u e n c e on t h e d r a g c o e f f i c i e n t . I t seems t o t h e w r i t e r s t h a t R e y n o l d s number, as w e l l as i t s g e n e r a l r a t e o f change, w h i c h may be e x p r e s s e d by t h e wave p e r i o d T, w o u l d be s i g n i f i c a n t p a r a m e t e r s . T h i s d i s r e -g a r d s t h e above s t a t e d o b j e c t i o n t o u s i n -g t h e non-measured p a r t i c l e v e l o c i t y as p a r a m e t e r . For n a r r o w r a n g e s o f Rey-n o l d s Rey-number, i t was a t t e m p t e d t o p l o t CQ as a f u Rey-n c t i o Rey-n o f t h e wave p e r i o d . The p l o t s were i n c o n c l u s i v e .
The measured p e r t i n e n t p a r a m e t e r s a r e t h e wave h e i g h t and p e r i o d , t h e w a t e r d e p t h , t h e e l e v a t i o n i n t e r v a l s f o r w h i c h t h e wave f o r c e s a r e measured, t h e p i l e d i a m e t e r , and
t h e sea w a t e r v i s c o s i t y . A number o f n o n - d i m e n s i o n a l r a t i o s between t h e s i x d l m e n t i o n a l p a r a m e t e r s w o u l d be n e c e s s a r y t o d e f i n e t h e p h y s i c a l e n v i r o n m e n t . However, even f o r c o n s t a n t v a l u e s o f a l l t h e v a r i a b l e s Cp v a r i e s v e r y c o n s i d e r a b l y . Thus a s t a t i s t i c a l a p p r o a c h seems i n d i s p e n s a b l e . I n l a c k o f a b e t t e r p a r a m e t e r , i t was d e c i d e d t o use R e y n o l d s number.
For r e l a t i v e l y n a r r o w ranges o f R e y n o l d s number as compared t o t h e c o r r e s p o n d i n g CQ r a n g e s , s t a t i s t i c a l d i s t r i -b u t i o n s f o r CD v a l u e s were d e t e r m i n e d . The Re r a n g e s were chosen based on t h e need f o r a r e l a t i v e l y l a r g e number o f CD
t h e number o f CD v a l u e s u s u a l l y v a r i e d between 40 and 80. W i t h i n each Re range t h e Re d i s t r i b u t i o n was n o r m a l o r c l o s e t o n o r m a l . The CD d i s t r i b u t i o n was c o n s i d e r e d r e p r e s e n t a t i v e o f t h e mean v a l u e f o r Re.
DESIGN VALUES FOR CD
F i g . 4 shows s t a t i s t i c a l l y based p l o t t e d p o i n t s f o r CD5 v a l u e s and s t r a i g h t l i n e s , w h i c h f a l l w i t h + 10% o f t h e c o r r e s p o n d i n g p l o t t e d p o i n t s . Three s e t s o f p l o t t e d p o i n t s and t h e c o r r e s p o n d i n g s t r a i g h t l i n e s a r e shown. The upper one i s f o r CD5 v a l u e s c o r r e s p o n d i n g t o 98% v a l u e s on t h e s t a t i s t i c a l d i s t r i b u t i o n , or i n o t h e r w o r d s , t h e CD v a l u e s f o r w h i c h t h e p r o b a b i l i t y o f b e i n g exceeded i s o n l y 2%. The m i d d l e and l o w e r CD5 v a l u e s a r e t h o s e f o r w h i c h t h e p r o b a b i l
i t y o f b e i n g exceeded i s 10% and 50% (mean v a l u e s ) r e s p e c t i v e l y . The s t r a i g h t l i n e s used a r e m a i n l y a m a t t e r o f c o n -v e n i e n c e , a l t h o u g h f o r s t e a d y s t a t e f l o w t h e r e l a t i o n s h i p between CD and Re i s q u i t e c l o s e t o b e i n g l i n e a r between Re X 10-5 e q u a l t o 1 and 5. I t s h o u l d be n o t i c e d t h a t t h e r e i s no v i s i b l e t r e n d f o r t h e d r a g c o e f f i c i e n t s t o become con-s t a n t f o r h i g h e r Re v a l u e con-s acon-s t h e y do f o r con-s t e a d y con-s t a t e f l o w p a s t a Re v a l u e o f about 5x10^. i t i s b e l i e v e d t h a t d a t a e x i s t f o r Re v a l u e s above t h o s e c o v e r e d h e r e . These d a t a have n o t been p u b l i s h e d , however.
The d e s i g n v a l u e f o r CD5 t o be used w i t h c e r t a i n v a l u e s o f Re w o u l d depend upon w h i c h s a f e t y f a c t o r s a r e used; on how r e l i a b l e t h e d e s i g n wave c r i t e r i a a r e ( w h e t h e r t h e y a r e based on r e c o r d i n g s o r d e t e r m i n e d by f o r e c a s t i n g f r o m w i n d d a t a ) ; and how r e l i a b l e o t h e r d e s i g n c r i t e r i a ( s u c h as s o i l s p a r a m e t e r s ) a r e . I f mean v a l u e s a r e u s e d , i t i s known t h e r e i s a f i f t y - f i f t y chance t h a t t h e y w i l l be exceeded, and t h a t t h e y m i g h t be exceeded by a f a c t o r o f 2 or more. T h i s c o u l d c o m p l e t e l y a b s o r b commonly used s a f e t y f a c t o r s , l e a v i n g no m a r g i n f o r t h e o t h e r u n c e r t a i n t i e s i n v o l v e d . I f 90% v a l u e s a r e u s e d , t h e p r o b a b i l i t y t h a t t h e y w i l l be exceeded i s o n l y 10% and t h e p r o b a b i l i t y t h a t t h e y w i l l be exceeded by 25 t o 3 3 % o r more i s o n l y a b o u t 2%. The 98% v a l u e s w i l l r a r e l y be exceeded and n o t by v e r y much. For t h e h i g h e r end o f Re v a l u e s c o v e r e d i n F i g . 4, i t i s recommended t h a t CD5 d e s i g n v a l u e s o f 0.8 t o 1.0 be u s e d .
F i g . 5 shows s t a t i s t i c a l l y based f i r s t o r d e r d r a g c o -e f f i c i -e n t s as a f u n c t i o n o f R-eynolds numb-er. Th-e t r -e n d i s
234 C O A S T A L E N G I N E E R I N G
Figure 6. Statistical Distributions of 0^^^ and Cj^^^ V a l u ' ^ D = 24.0 in. Based on Data by Wiegel, Beebe and M c ^
as t h e f i r s t - o r d e r v e l o c i t i e s a r e lower t h a n t h e f i f t h - o r d e r v e l o c i t i e s . F o r t h e h i g h e r end o f Re v a l u e s c o v e r e d by F i g - ^ C D I d e s i g n v a l u e s o f 1.0 t o 1.4 a r e recommended, based on s i " " " l i a r r e a s o n i n g as f o r t h e CD5 v a l u e s i n F i g . 4.
MASS COEFFICIENTS STATISTICAL ANALYSES OF Cji VALUES
I n F i g . 6, CM d i s t r i b u t i o n s f o r a 2 f t . d i a m e t e r pil® c o r r e s p o n d i n g t o t h e CQ d i s t r i b u t i o n s i n F i g . 2 a r e p l o t t e d i n a l i n e a r p r o b a b i l i t y g r a p h . The CM d i s t r i b u t i o n s f o l l o w t h e n o r m a l law o f e r r o r ( s t r a i g h t l i n e ) . The s t a n d a r d d e v i a ' t i o n t o mean-value r a t i o f o r CM i s much s m a l l e r t h a n f o r Cp • Three d i s t r i b u t i o n s a r e shown. One f o r t h e 28 CMS v a l u e s , one f o r t h e c o r r e s p o n d i n g C M I v a l u e s , and one f o r a l l t h e 149 C M I v a l u e s i n T a b l e I I I by W i e g e l e t a l ( 1 9 5 9 ) . The l a * " t e r f o l l o w s f a i r l y c l o s e l y t h e s t r a i g h t l i n e shown. The fv/*^ o t h e r d i s t r i b u t i o n s s t r a y a l i t t l e f u r t h e r f r o m straight-li^^® d i s t r i b u t i o n s . They a r e v e r y c l o s e t o each o t h e r , h a v i n g a number o f p o i n t s i n common. A s t a t i s t i c a l a n a l y s i s o f t h e 59 C M I v a l u e s ( f o r whi'^*'' t h e c o r r e s p o n d i n g H/L = 2.0%) f r o m T a b l e I I I by W i e g e l e t a-^
(1959) has a l s o been c a r r i e d o u t . I t i s n o t shown i n F i g . w h i c h w o u l d have become t o o crowded. I t f o l l o w s t h e d i s t r i - ^ ^
t i o n f o r 149 C M I v a l u e s v e r y c l o s e l y , b u t i s perhaps a t r i f s t e e p e r . F i g . 7 shows C M I d i s t r i b u t i o n s f o r a 1 6 - i n . d i a m e t e a r p i l e c o r r e s p o n d i n g t o t h e C D I d i s t r i b u t i o n s i n F i g . 3. FoX" i n c r e a s i n g wave s t e e p n e s s e s , t h e s t r a i g h t - l i n e d i s t r i b u t i o n ^ r e m a i n a l m o s t p a r a l l e l . P a r a l l e l l i n e s do n o t , however, ^ r e p r e s e n t e q u a l s t a n d a r d d e v i a t i o n s . When t h e a b s c i s s a s c ^ ^ ^ i s l i n e a r , t h e s t a n d a r d d e v i a t i o n (as w e l l as t h e mean val'«-J-^ ^ ^ ^ ^ j i s m u l t i p l i e d by a c o n s t a n t i f a l l t h e CM v a l u e s a r e m u l t i
by t h i s c o n s t a n t . Thus t o compare d i s t r i b u t i o n s w i t h d i f f ^ - ' f ^^^^ e n t mean v a l u e s , t h e s t a n d a r d d e v i a t i o n a l o n e i s n o t s u f f i < ^ I t i s n e c e s s a r y t o use t h e r a t i o between s t a n d a r d d e v i a t i o s ^ and mean v a l u e . As t h e mean v a l u e s i n F i g . 7 v a r y o n l y f r * ^ * ^ 2.0 t o 1.7, t h e s t a n d a r d d e v i a t i o n t o mean v a l u e r a t i o i n
-c r e a s e s o n l y s l i g h t l y w i t h i n -c r e a s i n g wave s t e e p n e s s . I n F i g . 8 i s shown C M I d i s t r i b u t i o n s f r o m B r e t s c h n e
(1955) f o r a 2.5 f t . d i a m e t e r p i l e e x t e n d i n g 13 f t . below ^ s t i l l - w a t e r l e v e l i n a w a t e r d e p t h o f 40 f t . Out o f a tot
236
C O A S T A L E N G I N E E R I N G
XCM, A L L V A L U E S ®CM, f > 3 . 0 %
+ CM, 2 . 0 % = C„| 3 . 5 %
238 C O A S T A L E N G I N E E R I N G
o f 203 waves, 134 had a s t e e p n e s s o f 3% o r more, 77 had a s t e e p n e s s o f 4% o r more and 44 had a s t e e p n e s s o f 5% o r more. The wave h e i g h t v a r i e d f r o m 3.1 t o 8.4 f t . and t h e p e r i o d f r o m 3.6 t o 8.2 s e e s . The c o r r e s p o n d i n g C D I v a l u e s were n o t a v a i l a b l e . When t h e l o w e r l i m i t f o r wave s t e e p n e s s i s r a i s e d f r o m 3% t o 4% t h e s t a n d a r d d e v i a t i o n t o meanvalue r a t i o d e c r e a s e s s l i g h t l y and when t h e s t e e p n e s s l i m i t i s r a i s e d f u r -t h e r -t o 5%, -t h i s r a -t i o r e m a i n s c o n s -t a n -t . For t h e r a n g e s o f v a r i a b l e s i n v e s t i g a t e d h e r e , f i r s t - o r d e r a p p r o a c h does n o t show s i g n i f i c a n t l y i n c r e a s i n g s c a t t e r on C ^ i v a l u e s f o r i n c r e a s i n g wave s t e e p n e s s .
DESIGN VALUES FOR MASS COEFFICIENTS
As t h e mass c o e f f i c i e n t s a r e r e l a t e d t o 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 s i n t h e f i r s t power and t h e d r a g c o e f f i c i e n t s a r e r e l a t e d t o p a r t i c l e v e l o c i t i e s i n t h e second power» t h e r e i s r e a s o n t o b e l i e v e t h a t t h e f i f t h - o r d e r mass c o e f f i c i e n t s a r e c l o s e r t o t h e f i r s t - o r d e r mass c o e f f i c i e n t s t h a n vi^as t h e case f o r d r a g c o e f f i c i e n t s . W i e g e l (1959) and (1964) ^ a s a t t e m p t e d t o c o r r e l a t e C j j i w i t h R e y n o l d s number, water—P^^*-*--c l e a water—P^^*-*--c water—P^^*-*--c e l e r a t i o n , wave p e r i o d and U^j^x T/D. O n l y f o r
C M I v e r s u s w a v e - p e r i o d p l o t was a s l i g h t c o r r e l a t i o n f o ^ i ^ d . CM showed a t e n d e n c y t o i n c r e a s e w i t h i n c r e a s i n g wave p e r i o d . I n F i g s . 7 and 8, t h e 98% C M I v a l u e s f o r H/L ^ 3 . 0 % v a r y f r o m 1.8 t o 2.5 and t h e 90% C M I v a l u e s f r o m 1.6 t o 2.2. Based on t h e same c o n s i d e r a t i o n s as f o r d r a g c o e f f i c i e r i t s , i t i s recommended t h a t a d e s i g n v a l u e f o r C M I o f 2.0 ( e q u a l t o t h e t h e o r e t i c a l v a l u e ) be u s e d . For CMS, t h e same d e s i g n v a l u e o f 2.0 ( o r perhaps a s l i g h t l y l o w e r v a l u e ) i s r e c o m -mended u n t i l f u r t h e r knowledge i s a v a i l a b l e .
COMPARISONS OF DRAG AND MASS-COEFFICIENT D I S T R I B U T I O N As p r e v i o u s l y s t a t e d , t h e s t a n d a r d d e v i a t i o n t o mean-v a l u e r a t i o s a r e much s m a l l e r f o r m a s s - c o e f f i c i e n t d i s ' ^ ^ - ' - ^ " " t i o n s t h a n f o r t h e c o r r e s p o n d i n g d r a g - c o e f f i c i e n t d i s ' t ^ ^ ^ i ^ ^ " t i o n s . T h i s i s o b v i o u s f r o m c o m p a r i s o n s between F i g s - ^ 6 as w e l l as F i g s . 3 and 7.
When t h e r a t i o between any p e r c e n t a g e v a l u e fo:x^ CM and t h e mean v a l u e i s compared t o t h e c o r r e s p o n d i n g CD r a " C i o i t i s f o u n d t h a t t h e CD r a t i o i s e q u a l t o t h e CM r a t i o i J ^ ^
power w h i c h v a r i e s between 1.7 and 2 . 1 . T h i s may p e r t ^ ^ P ^ be e x p l a i n e d by assuming t h a t t h e r e l a t i v e d e v i a t i o n s b e " t " ^ ^ ^ "
f a c t t h a t CM i s r e l a t e d t o t h e a c c e l e r a t i o n i n t h e f i r s t power and Cp t o t h e v e l o c i t y i n t h e second power. I f t h i s e x p l a n a t i o n i s v a l i d , t h e p r o s p e c t s f o r d r a s t i c a l l y r e d u c i n g t h e s c a t t e r on CQ v a l u e s do n o t seem t o o p r o m i s i n g , even i f c o n s i d e r a b l e p r o g r e s s i s a c h i e v e d i n a p p r o a c h i n g wave t h e o -r i e s t o a c t u a l wave a c t i o n . I f i t , f o -r example, i s -r e q u i -r e d t h a t a l l CD v a l u e s s h o u l d be w i t h i n t h e mean v a l u e m u l t i p l i e d or d i v i d e d by a f a c t o r o f 1 . 5 , i t w o u l d mean t h a t t h e r a t i o s between a c t u a l and t h e o r e t i c a l v e l o c i t i e s (and a c c e l e r a t i o n s ) c o u l d n o t exceed a v a l u e o f 1 . 2 2 ,
CONCLUSIONS AND RECOMMENDATIONS
From t h e i n v e s t i g a t i o n s d e s c r i b e d above i t was c o n -c l u d e d t h a t :
a. F o r t h e ranges o f v a r i a b l e s c o v e r e d , t h e f i f t h -o r d e r a p p r -o a c h i s n -o t s u p e r i -o r t -o t h e f i r s t - -o r d e r a p p r o a c h . A s l i g h t i n d i c a t i o n o f t h e o p p o s i t e i s f o u n d .
b. No d e f i n i t e r e l a t i o n s h i p s between t h e d r a g and mass c o e f f i c i e n t s ( r e s p e c t i v e l y ) and t h e measured p a r -a m e t e r s seem t o e x i s t . Even f o r c o n s t -a n t v -a l u e s o f a l l measured p a r a m e t e r s , t h e s c a t t e r on t h e c o e f f i c i e n t s i s v e r y c o n s i d e r a b l e . c. A s t a t i s t i c a l a p p r o a c h was c o n s i d e r e d i n d i s p e n s a -b l e f o r d e t e r m i n i n g d e s i g n v a l u e s f o r t h e d r a g a n d mass c o e f f i c i e n t s . d. Perhaps t h e p r i m a r y r e a s o n f o r t h e s c a t t e r on t h e c o e f f i c i e n t s i s f o u n d i n d i f f e r e n c e s between a c -t u a l and -t h e o r e -t i c a l p a r -t i c l e v e l o c i -t i e s a n d a c c e l e r a t i o n s . For d e s i g n p u r p o s e s t h e f o l l o w i n g v a l u e s a r e r e c o m m e n d e d : a. C D 5 = 0 . 8 t o 1 . 0 f o r Res ^ 7 x 1 0 ^ . b. C D I = i - 0 t o 1 . 4 f o r Re^ ^ 6 . 5 x 1 0 ^ .
c. F o r s m a l l e r v a l u e s o f Re, see F i g s . 4 and 5 . d. CMS and C M I = 2 . 0 f o r s t o r m waves.
240 C O A S T A L E N G I N E E R I N G ACKNOWLEDGEMENTS The w r i t e r s w i s h t o t h a n k F r e d e r i c R. H a r r i s , I n c . f o r p r o v i d i n g a s s i s t a n c e i n t y p i n g o f t h e m a n u s c r i p t and p r e p a r a -t i o n o f -t h e i l l u s -t r a -t i o n s . REFERENCES
A «5 M C o l l e g e o f Texas, The, ( 1 9 5 4 ) . Wave F o r c e E x p e r i m e n t s a t A t c h a f a l a y a Bay, L o u i s i a n a : Techn. R e p o r t No. 3 8 - 1 , D e p t . o f Oceanography. (Not p u b l i s h e d ) .
Beach E r o s i o n Board ( 1 9 6 1 ) . Shore P r o t e c t i o n P l a n n i n g and D e s i g n : Techn. R e p o r t No. 4, U.S. Army Corps o f E n g i n e e r s B r e t s c h n e i d e r , C.L. ( 1 9 5 5 ) . An E v a l u a t i o n o f I n e r t i a l C o e f f i
c i e n t s i n Wave Force E x p e r i m e n t s , I n v e s t i g a t i o n o f Wave F o r c e s on S t e e l P i l e s : D e p t . o f Oceanography, The A & M C o l l e g e o f Texas, Techn. R e p o r t No. 55-3. (Not p u b l i s h e d ) B r e t s c h n e i d e r , C.L. ( 1 9 5 7 ) . E v a l u a t i o n o f Drag a n d I n e r t i a l
C o e f f i c i e n t s f r o m Maximum Range o f T o t a l Wave F o r c e : D e p t . o f Oceanography, The A & M C o l l e g e o f Texas, Techn. R e p o r t No. 55-5 (Not p u b l i s h e d ) .
Dean, E.G. ( 1 9 6 5 ) . A C r i t i q u e o f Wave F o r c e A n a l y s i s and C a l c u l a t i o n Methods: P r o c e e d i n g s ASCE S p e c i a l t y C o n f e r -ence on C o a s t a l E n g i n e e r i n g .
K e u l e g a n , G.H. and C a r p e n t e r , L.H. ( 1 9 5 8 ) . F o r c e s on C y l i n -d e r s an-d P l a t e s i n an O s c i l l a t i n g F l u i -d : J . Res. N a t i o n a l Bureau o f S t a n d a r d s , 60,5.
L a i r d , A.D.K., Johnson, C A . and W a l k e r , R.W. ( 1 9 6 0 ) . Water F o r c e s on A c c e l e r a t e d C y l i n d e r s : T r a n s a c t i o n s o f t h e ASCE V o l . 125.
M o r i s o n , J.R., O ' B r i e n , M.P., Johnson, J.W. and S c h a a f , S.A. ( 1 9 5 0 ) . The Force E x e r t e d by S u r f a c e Waves on P i l e s : P e t r o l e u m T r a n s a c t i o n s , AIME, V o l . 189.
M o r i s o n , J.R., Johnson, J.W. and O ' B r i e n , M.P. ( 1 9 5 4 ) . Exper i m e n t a l S t u d i e s o f F o r c e s on P i l e s : P r o c e e d i n g s F o u r t h C o n f e r e n c e on C o a s t a l E n g i n e e r i n g , C o u n c i l on Wave
R e s e a r c h .
P r a n d t l , L. and T i e t j e n s , O.G. ( 1 9 3 4 ) . A p p l i e d H y d r o - and A e r o m e c h a n i c s : M c G r a w - H i l l Book Co.
o g r a p h y , The A & M C o l l e g e o f Texas. (Not p u b l i s h e d ) . S k j e l b r e i a , L., H e n d r i c k s o n , J.A., Gragg, W. and Webb, L.M.
( 1 9 6 0 ) . L o a d i n g on C y l i n d r i c a l P i l i n g Due t o t h e A c t i o n o f Ocean Waves. T h e o r e t i c a l R e s u l t s : An i n v e s t i g a t i o n c o n d u c t e d a t N a t i o n a l E n g i n e e r i n g S c i e n c e C o . f o r U.S. N a v a l C i v i l E n g i n e e r i n g Lab., V o l s . I I , I I I and I V . W i e g e l , R.L., Beebe, K.E. and Moon, J . ( 1 9 5 9 ) . Ocean Wave
F o r c e s on C i r c u l a r C y l i n d r i c a l P i l e s : T r a n s a c t i o n s o f t h e ASCE, V o l . 124.
W i e g e l , R.L. ( 1 9 6 4 ) . O c e a n o g r a p h i c a l E n g i n e e r i n g : P r e n t i c e H a l l .
W i l s o n , B.W. ( 1 9 5 7 ) . R e s u l t s o f A n a l y s i s o f Wave F o r c e D a t a - Confused Sea C o n d i t i o n s Around a 3 0 - I n c h D i a m e t e r T e s t P i l e , G u l f o f M e x i c o : D e p t . o f Oceanography, The A & M C o l l e g e o f Texas, F i n a l Techn. R e p o r t No. 55-7.
242 C O A S T A L E N G I N E E R I N G
Table 1. Drag Coefficients - D = 12.75 in.
D = 12. 75 i n . , S = 42.5 f t , d = 49 .3 t o 46 .0 f t WAVE T H ^05 Re 1 Re^ NO. sec . f t . l b s / f t x l O - 5 x l O - 5 228 16. 5 7 . 0 6.8 0. 8 0 . 50 2. 14 2,73 229 16. 8 9. 6 14.4 0. 9 2. 95
-230 17. 1 7. 6 14.0 1. 5 2. 28 -2 3 1 16. 1 6. 6 6.0 0. 8 0. 50 2. 02 2.58 232 15. 5 8. 8 18. 0 1. 3 0. 84 2. 77 3 .49 233 15. 0 8. 8 7.0 0. 5 0. 32 2. 75 3.47 234 14. 6 6. 0 9.2 1. 4 1 . 01 1 . 87 2 .23 235 15. 3 9. 0 34 2. 3 1 . 44 2. 78 3 .60 236 15. 9 13 . 6 40 1. 2 0. 65 4. 17 5.80 237 15. 4 17. 6 56 1. 0 0. 50 5 . 44 7 .81 238 15. ,2 20.5 58 0. ,8 0. 35 6. 30 9.58 239 15. ,6 17. 8 62 1. , 1 0 . 54 6. 34 7.92 240 16 10. 1 21.0 0. ,5 0. 68 3 . 12 4.11 2 4 1 16 8. 0 10.0 0 . ,9 0. 54 2. 47 3,18 242 15, ,5 5. 4 9.2 1. ,3 1 . 25 2, 00 2.01 243 13 , ,8 8, ,5 12.4 0 , ,9 0. 63 2. 67 3 .27 244 14, ,8 9. ,2 21.0 1, .4 0 . .87 2. 85 3 .63 245 10, .5 7. ,1 12.0 1, .3 1. ,01 2. ,25 2.55 246 14, .3 9, ,3 15.2 1, .0 0 , ,63 2. ,86 3 .64 247 14, .0 8, ,4 22.8 1, .8 1. ,21 2, ,64 3,21 248 14, .0 6, ,3 21.2 3 , .0 2, ,19 1, ,96 2.30 249 13 .0 6, .6 10.2 1 .0 0, ,94 2, ,07 2.44 250 12, .7 8, .6 9 0 .7 0, ,48 2, ,65 3 .22 2 5 1 12 .8 8, .9 12 0 .8 0 , .59 2, .78 3 ,35 252 12 .8 9, .6 14 0 .9 0, ,58 3 , ,01 3 .63 253 12 .4 6 .6 6.6 0 .9 0, .61 2, .08 2.43 254 16 .1 9 .2 15.2 1 .1 0 .60 2, .82 3.71 255 14 .6 8 .6 22 1 .6 1 .08 2 .68 3 .34 256 14 .8 8 .6 22 1 .6 1 .07 2 .66 3.36 257 14 .8 5 .6 8.8 1 .5 1 .09 1.76 2,10 258 14 .8 8 .0 8.6 0 .7 0 .48 2 .48 3 .12 259 16 .1 8 .3 7.8 0 .7 0 .40 2 .52 3 ,29 260 14 .0 8 .2 7.2 0 .6 0 .41 2 .56 3 .09 2 6 1 14 .0 5 .3 10 2 .0 1 .49 1 .65 1,91 262 15 .4 7 .0 13 .2 1 .5 0 .99 2 .18 2.70 263 15 .8 4 .4 6.6 1 .7 1 .47 1 .37 1.57 264 15 .0 7 .8 8 0 .7 0 .48 2 .45 3.02 265 15 .0 10 .2 25 .2 0 .9 0 .82 3 .18 4.09
WAVE T H ^C NO. sec . f t . I b s / f 1 266 15 12 .6 40 267 15.5 13 .3 54 268 15 13 .6 36 269 14.5 11 .5 40 270 13 .7 9, .4 30 271 14.5 11 .6 28 272 15 5, .8 10 273 15 9 .0 10 274 15 9, .2 16 275 15.4 7, .0 6 276 15.4 8, .3 12 277 15.1 9, ,2 6.4 278 14.6 7 , .9 10 279 15 9, ,6 15 280 14.4 7 , ,6 4 281 14.0 7 , ,8 10.4 282 14 7 . ,4 8.6 283 15.3 8. .6 12 284 12.5 9, ,2 24 285 15.6 9. ,0 22 .6 286 16.3 9, ,0 25 287 16.5 7 , ,7 7.2 288 16.3 9, ,0 13 .4 289 15.6 8, ,9 14.6 290 14.3 12, ,5 22 291 18.6 7 , ,8 12.4 292 14.8 7 , ,0 20 293 14.5 9, ,8 28 294 14.5 9, ,2 26 295 13 9, .3 24 296 15.3 9, ,0 12 297 15 6 , ,6 10 298 14 7 , ,8 16 299 15.4 9. ,0 17 300 15.4 14, ,5 52 3 0 1 14.8 17, ,0 54 302 15.3 13 , ,5 46 303 13.8 8, ,0 8 304 15.0 7 . ,6 10 305 15 .3 11. ,5 26 3 06 15.0 19. ,0 58 C D I R e i Res xlO-5 xlO-5 1.4 0 .81 3 .89 5 .22 1.7 0 .94 4 .08 5 .61 1.1 0 .61 4 .22 5 .68 1.5 1 .02 3 .58 4 .63 1.9 1 .24 2 .94 3 .64 - 0 .70 3 .00 4 .68 1.6 1 .19 1 .82 2 .14 0.7 0 .43 2 .83 3 .56 1.0 0 .66 2, .89 3 .64 0.7 0 .45 2, .66 2 .69 1.0 0 .62 2, .56 3 .27 0.4 0, .26 2, .93 3 .68 0.9 0, .59 2, .44 3 .05 0.9 0.57 2, .98 3 .81 0.4 0, .26 2, .34 2 .90 0.9 0, .65 2, .42 2 .95 0.9 0, .61 2, .31 2 .77 0.9 0, .57 2, .68 3 .40 1.6 1, .09 1.65 3 .47 1.6 0, .96 2, .79 3 .60 1.8 1, .03 2, .75 3 .64 0.7 0, .43 2.38 3 .02 1.0 0, .55 2, .75 3 .64 1.0 0 , .64 2.77 3 .54 0.8 0, .47 3 , ,90 5 ,05 1.1 2.48 2.2 1, .54 2, .17 2 .66 1.6 1, .02 3 , .04 3 .88 1.7 1, .09 2, .84 3 ,61 1.6 1, .05 2, ,92 3 .54 0.8 0, .51 2, .78 3 .60 1.3 0, ,86 2, .07 2 .52 1.5 1, ,00 2 , .42 2 .96 1.2 0 , ,72 2 , .80 3 .60 1.4 0, ,74 4.46 6 .21 1.0 0, ,55 5, .28 7 .34 1.4 0, ,78 4, ,14 5 .70 0.7 0 , .47 2, ,50 3 .06 1.0 0, ,63 2,38 2 .94 1.1 0, ,64 3, ,56 4 .71 0.9 0, ,44 5 , ,88 8 .50
244 C O A S T A L E N G I N E E R I N G T A B L E 1 - ( C o n t i n u e d ) WAVE NO. T sec . H f t . Fc l b s / f t C D I CD5 R e i x l O - 5 Res xlO^' 3 0 7 1 4 .5 1 5 , ,0 4 8 1.2 0 . 6 7 4 . 6 4 6 . 2 8 3 0 8 1 3 .5 1 3 , ,0 4 4 1.4 0 , 9 0 4 . 0 9 5 . 1 6 3 0 9 1 3 .7 6. ,6 1 1 1.4 0 . 9 9 2 . 0 8 2 . 4 6 3 1 0 1 3 .0 8. ,0 1 4 1.2 0 . 8 4 2 . 5 3 3 , 0 2 3 1 1 1 3 .0 9. ,3 1 7 1 . 1 0 . 7 4 2 . 9 2 3 , 5 4 3 1 2 1 4 .0 8. ,4 1 5 1 . 2 0 . 8 0 2 . 6 2 3 , 2 1 3 1 3 1 2 .0 8. .6 1 6 1 . 2 0 . 8 7 2 . 7 5 3 , 1 8 3 1 4 1 4 .0 7 . ,0 8 0 . 8 0 . 6 4 2 . 3 0 2 , 6 1 3 1 5 1 4 .3 8. ,0 1 0 0 . 8 0 . 5 8 2 . 5 4 3 , 0 8 3 1 6 1 3 .8 9. ,7 1 8 1 . 0 0 , 7 0 3 . 0 7 3 , 7 6 3 1 7 1 3 .2 8. 5 1 3 .6 1 . 0 0 , 7 1 2 . 7 0 3 . 2 3 3 1 8 1 4 .7 7. 0 1 0 1 . 1 0 , 7 8 2 . 2 3 2 , 6 6 3 1 9 1 3 .7 6. 5 8 1 . 0 0 . 7 5 2 , 0 8 2 , 4 1 3 2 0 1 4 . 1 8, 0 1 0 0 . 9 0 . 5 8 1 , 7 8 3 . 0 6 3 2 1 1 4 .8 8. 2 1 7 1.3 0 . 9 3 2 , 6 3 3 . 1 5 3 2 2 1 4 .2 8. 9 1 2 . 4 0 , 8 0 . 5 7 2 . 8 5 3 . 4 6 3 2 3 1 5 . 1 6. 0 9.2 1 . 1 0 . 9 9 1 , 9 2 2 , 2 5 3 2 4 1 4 .5 6. 0 7 , 6 1 , 1 0 . 8 4 1 . 9 2 2 , 2 3 3 2 5 1 4 .0 5 . 5 1 2 2.0 1 . 6 4 1 . 7 6 2 . 0 0
D = 24 i n . , S = 33.0 f t . , d = 48.0 t o 46.0 f t .
WAVE T H Fc C D l C D5 Re 5 NO.
Re 5 NO.
sec . f t . l b s / f t xlO-5 xlO-' 23 15.0 9.8 25,5 0 .8 0 .53 5,75 7.00 24 15.2 12.0 29.0 0 .6 0.39 7.05 8.75 25 12.8 10.1 11.6 0 .3 0 .26 5.87 5.53 26 12.1 8.6 41.6 1 .7 1 .39 5.03 5.55 27 13.8 9.0 11.6 0 .4 0 .33 5.29 6.06 28 12.2 9.3 11.6 0 .4 0 .32 5.45 6,10 29 14.3 10.8 29.0 0 .7 0 ,52 6.37 7,58 30 13.2 11.4 5.8 0 .1 0 .10 7.21 7,75 31 13.5 9.0 11.6 0 .4 0 .33 5.20 6,06 32 14.6 7.0 11.6 0 .7 0 .53 4.11 4,74 33 14.3 7.8 29 1 .5 1 ,07 4.46 5,30 34 13 .5 8.9 26.1 1 .0 0 ,74 5.10 6.03 35 14.4 9.0 23 .2 0 .9 0 ,63 5.19 6.18 36 13.6 12.2 5.8 0 .1 0,08 7,05 8.50 37 11.6 8.5 11.6 0 .5 0.41 4.84 5.39 38 11.2 10.0 11.6 0 .4 0 .29 5.65 6.36 39 13.0 7.3 11.6 0 .7 0 .53 4.17 4.75 40 12.9 9.0 11.6 0 .4 0 .34 5,20 5 .94 41 14.4 6.1 11.6 1 .0 0.73 3 ,51 4.04 42 14.0 11.5 11.6 0 .3 0, .19 6.80 7 .96 43 13.7 8.0 40.6 2 .0 1, .46 4,60 5.34 44 15.2 9.5 20.3 0 .7 0, .47 5,45 6.70 45 14.7 8.6 17.4 0, .7 0. ,51 4.98 5.95 46 15.4 8.1 11.6 0, .5 0, ,38 4 .70 5.58 47 13.2 8.0 11.6 0, .6 0. ,43 4.75 5.28 48 12.0 8.5 11,6 0, ,5 0. .40 4.83 5.45 49 13 .5 8.5 29.0 1. ,2 0. 92 4.85 5.70 51 15.3 7.5 23 .2 1, ,3 0. 89 4.33 5,19 52 12.2 10.2 31.9 1, ,0 0. 73 5.84 6 .70 53 13 .0 10.0 29.0 0. 9 0. 66 5.81 6,70 54 12.0 6.0 11.6 1. 0 0. 82 3 ,42 3 .80 55 14.2 11.5 11.6 0. 3 0. 18 6,86 8.10 56 14.4 11.0 40.6 1. 0 0. 70 6,29 7,75 57 14.3 12.5 23 .2 0. 5 0. 30 7,29 8.91 58 13.5 11.5 34 .8 0 . 8 0. 56 6,60 7.96 59 15.0 12.2 31.9 0. 7 0,41 7.06 8.91 60 14.4 8.0 23 .2 1. 1 0. 81 4.62 5.44 61 12.0 8.0 11.6 0. 6 0, 46 4.62 5.12
246 C O A S T A L E N G I N E E R I N G TABLE 2 - ( C o n t i n u e d ) WAVE T H NO. sec . f t . 6 2 10.0 5.2 6 3 12.2 9.0 64 14.3 8.5 6 5 15.7 6.2 6 6 11.0 9.5 6 7 11.0 8.0 6 8 11.8 8.5 6 9 12.5 6.0 7 0 13 .9 9.0 71 16.2 8.5 7 2 15.0 9.2 7 3 13 .2 9.5 7 4 15 .4 10.0 7 5 12 .3 9.0 7 6 13 .9 8.2 7 7 11.9 6.6 7 8 12.0 6.0 7 9 13 .2 9.2 80 13.5 10.3 8 1 11.8 10.6 84 12.6 7.5 8 5 9.1 10.5 86 13 .1 10.0 8 7 13 .5 7.5 88 13.7 11.5 90 13.8 7.8 9 1 12.0 8.1 9 2 13.6 6.5 93 9.6 8.0 94 14.4 10.2 9 6 12.3 9.5 9 7 15.2 9.1 98 14.7 10.6 9 9 14.8 7.3 101 13.6 9.1 1 0 2 14.0 8.2 1 0 3 13.5 8.0 1 0 4 15.2 10.5 1 0 5 11.7 9.2 106 15.0 10.0 Fc CD5 l b s / f t 20 .3 2.4 2.21 23 .2 0.9 0.71 23 .2 1.0 0.71 17 ,4 1.4 1.04 40 .6 1.4 1.18 17 .4 0.9 0.72 17 .4 0.7 0.61 17 .4 1.5 1.21 11 .6 0.4 0.33 17 .4 0.7 0.49 8 .7 0.3 0.21 23 .2 0.8 0.60 29 .0 0.9 0.59 5 .8 0.2 0.17 17 .4 0.8 0.59 17 .4 1.2 1.04 11 .6 1.0 0.82 17 .4 0.6 0.47 29 .0 0.8 0.60 29 .0 0.8 0.63 17 .4 0.9 0.76 40 .6 1.2 1.08 34 .8 1.1 0.79 17 .4 0.9 0.72 26 .2 0.6 0.42 14 .5 0.7 0.55 14 .5 0.7 0.56 11 .6 0.8 0.65 17 .4 0.9 0.78 5 .8 0.2 0.12 17 .4 0.6 0.46 11 .6 0.4 0.29 17 .4 0.5 0.32 34 .8 1.9 1.46 17 .4 0.6 0.47 17 .4 0.8 0.59 5 .8 0.3 0.21 29 .0 0.8 0.53 29 .0 1.0 0.86 34 .8 1.0 0.70 R e i xlO-5 X l ^ 3 .08 2.94 p ; 82 5 .20 6 78 4.87 r-5 9^ 3 .55 r-5 9^ 5.41 A.9& 4.52 5 . 4 3 4.99 3 . 0^ 3 .46 « 0 6 5.25 6 . 0 3 5.00 6 . 0 3 5.30 6 5.51 7 .15 5.65 5.22 4.81 5 .86 3 .79 3 . 80 3 .44 6 Y . 08 5.27 6 Y . 08 5.99 e . 8 9 6.08 4.33 6 .24 5 .88 & .73 5.73 4 .99 4.40 -7 .98 6.75 5 .20 4.54 5 .15 4.75 4 .29 3 .83 4 .79 4.54 -7 .14 5.99 e .21 5 .55 6 . 4 4 5.32 V .51 6.30 4.95 4.28 6.16 5.28 5.49 4.80 5 .34 4 .70 7 .54 6.32 5.90 5.32 7 .14 5.88
WAVE T H NO * s e c . f t . l b s / f t 107 14 .2 7 .5 23 .6 1 0 8 14.5 10 .3 34 .8 110 17.0 9 .3 11 .6 111 11.2 8 .5 5 .8 112 9.8 7 .0 8 .7 113 12 .7 7 .3 17 .4 1 1 4 15.2 9 .0 34 .8 116 13.1 8 .3 5 .8 117 14.1 10 .8 40 .6 118 11.0 11 .5 20 .3 119 11.4 7 .5 17 .4 120 12.0 11, .0 17 .4 121 13.7 7, ,0 26 .2 122 12.8 9, .6 23 .2 123 14.8 7, ,5 29 .0 124 13.6 8, , 1 34 .8 125 15.3 10, ,0 29 .0 126 14.5 1 1 . ,0 20 .3 127 15.3 10, ,0 31 .9 % 1 % 5 Re 2^ Reg x l O - 5 x l O j : ^
—=
1.2 1 .03 4, ,36 5.05 1.8 0 .68 6, ,06 7.22 0.4 5, ,47-0.2 0, .21 4, ,98 5.34 0.6 0, .51 3, ,99 4.19 1.0 0, .81 4, ,30 4.69 1.3 0, .90 5, ,32 6.30 0.3 0, .20 4, ,86 5.50 1.0 0, .74 6, ,30 7.56 0.5 0, .39 6, .58 7 .34 0.9 0, .81 4.36 4 . 7 0 0.4 0, .34 6, ,39 7 . 2 1 1 .6 1, ,25 4 , ,07 4 . 6 4 0.8 0, ,59 4, ,61 6 . 3 6 1.5 1, ,13 5. ,13 5 . 1 2 1.6 1, ,24 4. ,76 5 . 3 8 0.9 0, ,59 5. .89 7 . 1 5 0.7 0, ,35 6. 44 7 . 7 0 1.0 0, ,64 5. 89 7 . 1 Ö
248 COASTAL ENGINEERING
Table 3. Mass Coefficients
D = 24 i n . , S = 33.0 f t . , d = 48 t o 46 f t . ^SWL s e c . f t . l b s / f t 49 13 .5 8 .5 23 .2 2 .3 2 .3 51 15 .3 7 .5 23 .2 3 .0 2 . 9 59 15 .0 12 .2 40 .6 3 .0 3 . 1 65 15 .7 6 .2 29 .6 4 .7 4 .6 66 11 . 0 9 .5 37 .7 2 .8 2 . 8 67 11 .0 8 .0 17 .4 1 . 5 1 .5 68 11 .8 8 . 5 31 .9 2 .8 2 .7 69 12 . 5 6 .0 13 .3 1 .8 1 .7 70 13 .9 9 ,0 34 .8 3 .3 3 .3 74 15 .4 10, .0 40 .6 3 .9 3 .9 75 12 .3 9, .0 11 .6 1 .0 1 .0 80 13 .5 10, ,3 37 .7 3 , 1 3 . 1 84 12 .6 7 , ,5 29 .0 3 . 1 3, , 0 87 13 .5 7, ,5 23, ,2 2 , .5 2, ,6 92 13 .6 6, ,5 26, , 1 3 , ,4 3 , ,4 93 9 .6 8. ,0 29, ,0 2 , ,2 2, ,3 96 12 .3 9, ,5 40, ,6 3 , ,3 3 , ,4 103 13 .5 8, ,0 23 , ,2 2 , ,4 2, ,5 104 15 .2 10, ,5 23, ,2 2. , 1 2, , 1 106 15 .0 10. 0 3 1 , ,9 2 . ,9 3 , ,0 112 9 .8 7 . 0 23 , ,2 2 . 1 2, 1 114 15 .2 9. 0 34 . 8 3 . 6 3 . 6 117 14 .1 10. 8 12. 8 1. 0 1. 0 119 11 .4 7 . 5 11. 6 1. 1 1. 1 121 13 .7 7 . 0 26. 1 3 . 2 3 . 1 124 13 .6 8. 1 45. 3 4 . 8 4. 8 125 15 .3 10. 0 29. 0 2 . 7 2. 7 127 15 .3 10. 0 36. 0 3 . 4 3. 4
