MEASUREMENTS OF WIND WAVE KINEMATICS
J.A. B a t t j e s J . van H e t e r e n
MEASUREMENTS OF WIND WAVE KINEMATICS J.A. B a t t j e s , D e l f t U n i v e r s i t y o f Technology J. van H e t e r e n , R i j k s w a t e r s t a a t R i j k s w a t e r s t a a t R e p o r t WWKZ-G007 J u l y 1983
MEASUREMENTS OF WIND WAVE KINEMATICS CONTENTS page INTRODUCTION 1 OBSERVATION PLATFORM 6 INSTRUMENTATION 7 3.1 I n t r o d u c t i o n 7 3,2 V e l o c i t y m e t e r 7 3,3 Wave gauge 9 3.4- Anemometer 12 3,5 Data a c q u i s i t i o n system 12
DATA ANALYSIS PROCEDURE 14
4.1 I n t r o d u c t i o n 14 4.2 E s t i m a t i o n o f a u t o - and c r o s s - s p e c t r a 14 4.3 E s t i m a t i o n o f coherence s p e c t r a 15 4.4 E s t i m a t i o n o f phase s p e c t r a 15 4.5 E s t i m a t i o n o f g a i n s p e c t r a 17 4.5 E s t i m a t i o n o f p r o b a b i l i t y d e n s i t i e s 17 OBSERVATIONS • 19 5.1 Data sequences 19
5.2 Wind and sea c o n d i t i o n s 20
RESULTS AND DISCUSSIONS 2 1
6.1 I n t r o d u c t i o n 2 1
6,2 Coherence s p e c t r a 2 1
6,3 Phase s p e c t r a 22
6.4 Gain s p e c t r a 24
6.5 P r o b a b i l i t y d e n s i t i e s . 26
SUMMARY AND CONCLUSIONS 27
ACKNOWLEDGEMENTS REFERENCES
- 1
-1. INTRODUCTION
The realistic modelling of the internal kinematics of wind-generated waves, in relation to the motions of the surface, is a topic of scientific interest and practical importance. The linear Gaussian model of'the wave motion provides a manageable approximation with many features which are at least qualitatively realistic. Its quantitative perfor-mance under field conditions has been checked in a number of studies, with varying and in some respects conflicting -result;:;.
In the following, some of the results published previously will be mentioned concerning spectral transfer functions between surface eleva-tion (1;;) and horizontal
(li
=
u,v) and vertical (w)velocity.The theoretical gain and phase spectra between two processes x(t) and yet) will be designated as H (f) and
¢
(f), in which H (f) representsxy xy xy
the amplitude of y per unit amplitude of x, and
¢
(f) represents the xyphase lead of y with respect, to x, both as function of frequency f.
The theoretical gain f;unction_ for the horizontal velocity in the propa-gation direction (denote~ by
u),
for given surface elevation, isH (f)
=
2'1Tf cosh k(h+
z) 1;;u sinh ,-kband for the vertical velocity it is
Hrw( f)
=
2'1Tf sinh k(h+
,z)'" sinh kh
in which h is the mean depth, z is a vertical coordinate measured posi-tive upward from mean water level, and k is the wavenumber which varies with f according to the linear dispers ion equation:\>
(2'1Tf)2
=
gk tanh kh } I,' (f,
f lr
I
,
I
2
-The l i n e a r - t h e o r y p r e d i c t i o n s f o r t h e phases a r e ( j ) ^ ^ = 0 - , = 0 , " T ' t h e r e f o r e a l s o (}.^^ = | , c j ) ^ = | .
The d a t a by T h o r n t o n and K r a p o h l ( 1 9 7 4 ) a r e by and l a r g e i n agreement w i t h t h e l i n e a r t h e o r y p r e d i c t i o n s as g i v e n above, e x c e p t f o r an u n d e r e s t i m a t e o f t h e h o r i z o n t a l v e l o c i t i e s f o r g i v e n s u r f a c e e l e v a t i o n , t o an amount o f about 6% a t t h e s p e c t r a l peak f r e q u e n c y ( f j ^ , ) . However, o n l y one s e r i e s o f r e c o r d s was a n a l y s e d , i n a l o n g , l o w s w e l l (rms wave h e i g h t - 0.27 m, i n a
depth o f about 19 m). The r e s x i l t s a r e t h e r e f o r e n o t i n d i c a t i v e o f t h e ap-p l i c a b i l i t y o f t h e l i n e a r t h e o r y i n c o n d i t i o n s o f a s t e e ap-p , young sea. F o r r i s t a l l e t a l . ( 1 9 7 8 ) have e s t i m a t e d H ^ ~ ( f ) f r o m measurements i n c o n d i -t i o n s o f a c -t i v e wave g e n e r a -t i o n , i n -t r o p i c a l s -t o r m D e l i a . They a l s o a p p l i e d v a r i o u s n o n l i n e a r t h e o r i e s . They f o u n d t h a t t h e i n t e r n a l v e l o c i t i e s were b e t t e r m o d e l l e d by t h e l i n e a r t h e o r y , w i t h a l l o w a n c e f o r . d i r e c t i o n a l and f r e q u e n c y s p r e a d i n g o f t h e e n e r g y , t h a n by any o f t h e n o n l i n e a r t h e o r i e s f o r u n i d i r e c t i o n a l , monochromatic waves. A l t h o u g h t h i s i s a v e r y v a l u a b l e r e s u l t , t h e q u a n t i t a t i v e i n f o r m a t i o n about t h e e s t i m a t e d t r a n s f e r f u n c t i o n s p r e s e n t e d by F o r r i s t a l l e t a l . i s l i m i t e d t o one f m o t i o n H ^ ~ ( f ) / H ^ ~ ( f ) f o r each o f t h r e e d i f f e r e n t e l e v a t i o n s , i n w h i c h H i s t h e v a l u e o f H e s t i m a t e d f r o m t h e d a t a . (No phase i n f o r m a t i o n i s g i v e n . ) These show a t r e n d w i t h f , f r o m a 10% u n d e r p r e d i c t i o n o f t h e v e l o c i t i e s by t h e l i n e a r t h e o r y n e a r f = f t o a r o u g h l y 20% o v e r p r e d i c t i o n n e a r f r e q u e n -c i e s o f about 2 f . The l a t t e r ( o v e r p r e d i e t i o n ) i s s t a t e d t o be " p r o b a b l y m
due t o n o n l i n e a r phase l o c k i n g e f f e c t s between harmonics"-.- I f t h i s "phase l o c k i n g " r e f e r s t o t h e o c c u r r e n c e o f b o i m d h i g h e r h a r m o n i c s , t h e n t h i s h y p o t h e s i s w o u l d seem t o be u n t e n a b l e , s i n c e t h e wavenumber a t a f r e q u e n c y 2 f ^ i s s m a l l e r i n t h e n o n l i n e a r a p p r o x i m a t i o n ('^JJQJJ3_;^JJ " ^ k ^ , where k = (2TTf )^/g i n deep w a t e r ) t h a n i t i s i n t h e l i n e a r a p p r o x i m a t i o n m m ° ^ (k . = (4TTf ) /g 4k ) , w h i c h r e s u l t s i n a s l o w e r decay o f t h e v e l o c i t i e s I m m m ' w i t h d e p t h b e n e a t h t h e s u r f a c e i n t h e n o n l i n e a r case. Thus, i n c l \ a s i o n o f t h i s n o n l i n e a r e f f e c t w o u l d g i v e an even g r e a t e r o v e r p r e d i c t i o n o f t h e v e l o c i t i e s i n t h e f r e q u e n t y range o f about 2 f ^ , f r o m a g i v e n s p e c t r u m o f s u r f a c e e l e v a t i o n .
3
-C a v a l e r i e t a l . ( 1 9 7 8 ) and -C a v a l e r i ( 1 9 8 2 ) p r e s e n t g a i n f u n c t i o n s w h i c h show an o v e r p r e d i c t i o n o f t h e - v e l o c i t i e s ( u and w) by about 10% i n t h e f r e q u e n c y range o f h i g h coherence between v e l o c i t i e s and e l e v a t i o n . These r e s u l t s , a l t h o u g h a l s o t a k e n i n c o n d i t i o n s o f a c t i v e wind-wave g e n e r a t i o n , c o n t r a s t w i t h t h o s e o f F o r r i s t a l l e t a l . ( 1 9 7 8 ) i n . t h e e n e r -g e t i c band a r o u n d t h e s p e c t r a l peak. The d i f f e r e n c e s w i t h t h e l i n e a r t h e o r y exceed s i g n i f i c a n t l y t h e p o s s i b l e c a l i b r a t i o n e r r o r s w h i c h C a v a l e r i e t a l . e s t i m a t e a t a few % o n l y . ( I t i s n o t e d t h a t t h e u n d e r -o r - -o v e r - p r e d i c t i -o n s r e f e r r e d t -o ab-ove appeared t -o -o c c u r s y s t e m a t i c a l l y , w h i c h makes i t v e r y u n l i k e l y t h a t t h e y 'T^ould be due t o random s a m p l i n g e r r o r s . )
The r e s u l t s w i t h r e s p e c t t o t h e phases a r e more c o n t r o v e r s i a l t h a n t h o s e f o r t h e g a i n f u n c t i o n s . S h o n t i n g ( 1 9 6 4 ) , Y e f i m o v and K h r i s t o f o r o v ( 1 9 6 9 ) , C a v a l e r i e t a l . ( 1 9 7 8 ) and C a v a l e r i ( 1 9 8 2 ) a l l r e p o r t e d s i g n i f i c a n t o u t -o f - q u a d r a t u r e r e l a t i -o n s between h -o r i z -o n t a l and v e r t i c a l v e l -o c i t y , w h i c h w o u l d i m p l y a mean v e r t i c a l f l u x o f h o r i z o n t a l momentrum. The m a g n i t u d e o f t h i s e s t i m a t e d f l u x d e n s i t y i s t y p i c a l l y o r d e r s o f magnitude l a r g e r t h a n t h e w i n d - i n d u c e d mean momentum t r a n s f e r a t t h e s u r f a c e . , - . -.. . V a r i o u s . a d hoc h y p o t h e s e s have been advanced t o " e x p l a i n " t h e a p p a r e n t anomaly. S h o n t i n g ( 1 9 6 4 ) s t a t e s somewhat v a g u e l y t h a t i t " p r o b a b l y r e f l e c t s an anomalous s h o r e c o n d i t i o n " . Y e f i m o v and K h r i s t o f o r o v ( 1 9 6 9 ) , C a v a l e r i e t a l . ( 1 9 7 8 ) and C a v e l e r i ( 1 9 8 2 ) a s c r i b e i t ( t e n t a t i v e l y ) t o t u r b u l e n c e a s s o c i a t e d w i t h t h e wave b r e a k i n g o c c u r r i n g i n a. w i n d - d r i v e n sea.
C a v a l e r i (19 82) f i n d s s p p o r t f o r t h i s view i n t h e f a c t t h a t t h e s t r o n g phase anomaly (- 25°) was o b s e r v e d o n l y i n c o n d i t i o n s o f a c t i v e wave gen-e r a t i o n ; i t v i r t u a l l y d i s a p p gen-e a r gen-e d ( t o - 5°) whgen-en t h gen-e wavgen-e b r gen-e a k i n g d i m i n - . i s h e d as t h e sea t r a n s f o r m e d i n t o s w e l l . D e s p i t e t h i s i n d i c a t i o n , t h e sug-g e s t i o n t h a t t h e o b s e r v e d phase anomaly''.would be due t o t u r b t i l e n c e l e a v e s many imanswered q u e s t i o n s . To b e g i n w i t h , i t i s n o t c l e a r how t u r b u l e n c e w o i i l d cause such a d r a s t i c change i n t h e phase r e l a t i o n between t h e wave-i n d u c e d v e l o c wave-i t wave-i e s . (The e s t wave-i m a t e d coherence between s u r f a c e e l e v a t wave-i o n and v e l o c i t i e s was q u i t e h i g h ( y > 0.98) i n t h e e n e r g e t i c f r e q u e n c y r a n g e , i n d i c a t i n g t h a t t h e o b s e r v e d v e l o c i t i e s were v i r t u a l l y f u l l y wave i n d u c e d . )
•Secondly, t h e r e i s a s u b s t a n t i a l u n b a l a n c e d momentum f l u x i n t h e upper l a y e r s , s i n c e t h e f l u x c a l c u l a t e d f r o m t h e measured v e l o c i t i e s ( i n a range w h i c h i n t h e measurements b y C a v a l e r i e t a l . goes up t o 0.3 Hz) i s o r d e r s o f magnitude l a r g e r t h a n t h e w i n d s t r e s s . C a l l i n g upon t u r b u -l e n c e o u t s i d e t h e measured range t o compensate f o r t h i s excess does n o t seem r e a l i s t i c s i n c e i t i s h a r d t o see how such t u r b u l e n c e , w h i c h u l t i -m a t e l y i s due t o t h e w i n d a c t i o n on t h e s u r f a c e , c o u l d r e s u l t i n a l a r g e upward f l u x o f downwind momentum. I n t h e o p i n i o n o f t h e p r e s e n t a u t h o r s , t h e anomalous phase d i f f e r e n c e s and a s s o c i a t e d momentum f l u x e s r e f e r r e d t o above s h o u l d be h e l d i n d o u b t as l o n g as no p h y s i c a l p r o c e s s has been i d e n t i f i e d ( n o t j u s t c o n j e c t u r e d ) w h i c h c o u l d e x p l a i n t h e a p p a r e n t i m -b a l a n c e .
I n t h e c o n t e x t o f phase e s t i m a t e s , r e f e r e n c e i s made t o some p r e l i m i n a r y -r e s u l t s o f t h e p -r e s e n t s t u d y ( B a t t j e s and van H e t e -r e n , 1980) w h i c h a l s o gave a p p a r e n t phase d i f f e r e n c e s w h i c h seemed t o be m r e a l i s t i c . These p r e -l i m i n a r y f i n d i n g s m o t i v a t e d a s c r u t i n o u s i n v e s t i g a t i o n o f t h e e n t i r e system o f measurement and d a t a h a n d l i n g , w h i c h b r o u g h t t o l i g h t numerous t i m e d e l a y s w h i c h h a d p r e v i o u s l y been o v e r l o o k e d , and w h i c h b y and l a r g e an-n u l l e d t h e p r e v i o u s l y s u s p e c t e d aan-nomalies (see b e l o w ) . Whereas t h i s does n o t i m p l y s i m i l a r e r r o r s i n t h e o t h e r r e f e r e n c e s q u o t e d above, i t does i n
-d i c a t e t h e g r e a t c a r e nee-de-d i n t h i s t y p e o f a n a l y s i s .
I t appears f r o m t h e p r e c e d i n g summary o f pre-vious-Tres.ülts- th&t""there--.are u n r e s o l v e d q u e s t i o n s about t h e a p p l i c a b i l i t y o f l i n e a r wave t h e o r y f o r t h e p r e d i c t i o n o f v e l o c i t i e s f r o m s u r f a c e e l e v a t i o n s , w i t h r e s p e c t t o g a i n as w e l l as p h a s e , p a r t i c u l a r l y f o r c o n d i t i o n s o f a c t i v e wind-wave i n t e r a c t i o n . The p r e s e n t s t u d y was u n d e r t a k e n t o i n v e s t i g a t e t h i s p r o b l e m f u r t h e r . To t h a t e n d , measurements were p e r f o r m e d f r o m a p l a t f o r m i n t h e s o u t h e r n N o r t h Sea, and a n a l y z e d p r i m a r i l y w i t h r e s p e c t t o s p e c t r a l t r a n s f e r f u n c t i o n s . A s e c o n d a r y i t e m o f a t t e n t i o n was t h e p r o b a b i l i t y d i s t r i b u -t i o n o f -t h e v e l o c i -t y componen-tè., w h i c h i n -t h e l i n e a r -t h e o r y s h o u l d be Gaussian. T h i r d l y , t h e d a t a have been a n a l y z e d t o e s t i m a t e d i r e c t i o n a l s p e c t r u m p a r a m e t e r s , b u t t h e s e r e s t i l t s a r e r e p o r t e d e l s e w h e r e .
5
The c o n t e n t s o f t h e p r e s e n t r e p o r t a r e as f o l l o w s . The p l a t f o r m and i n s t r u m e n t a t i o n a r e d e s c r i b e d f i r s t ( c h a p t e r s 2 and 3 ) . P a r t i c u l a r a t t e n -t i o n i s -t h e r e b y g i v e n -t o , c a l i b r a -t i o n s o f s e n s o r s and o -t h e r sys-tem com-p o n e n t s . The a n a l y s i s com-p r o c e d u r e s a r e d e s c r i b e d i n c h a com-p t e r 4. ' . . T h i s i s f o l l o w e d i n c h a p t e r 5 by a d e s c r i p t i o n o f . t h e e n v i r o n m e n t a l con-d i t i o n s con-d u r i n g t h e measurements. The r e s u l t s a r e p r e s e n t e con-d ancon-d con-d i s c u s s e con-d i n c h a p t e r 6. F i n a l l y , t h e c o n c l u s i o n s a r e s t a t e d i n c h a p t e r 7.
6
-2. OBSERVATION PLATFORM
The meastirenients were made f r o m t h e p l a t f o r m N o o r d w i j k (MNP), w h i c h i s l o c a t e d i n t h e s o u t h e r n N o r t h Sea, n e a r N o o r d w i j k , about 10 km o f f t h e Dutch c o a s t ( s e e f i g u r e s 1 and 2 ) . The b o t t o m i n t h e n e i g h b o u r h o o d o f t h e p l a t f o r m i s r a t h e r f l a t . The l o c a l depth i s about 17.3 m b e l o w MSL, and t h e l o c a l t i d a l range i s about 1.8 m.
The p l a t f o r m r e s t s on a j a c k e t s t r u c t u r e c o n s i s t i n g o f t u b u l a r e l e m e n t s . Four o f t h e s i x j n a i n p i l e s , w h i c h have a d i a m e t e r o f 0.8 m, a r e p o s i -t i o n e d i n -t h e c o m e r s o f a r e c -t a n g M o f a p p r o x i m a -t e l y 11.15 m x 18.30 m a t Mean Sea L e v e l (MSL) ( s e e f i g u r e 3 ) . They have a 6° b a t t e r .
The o r i e n t a t i o n o f t h e p l a t f o r m ' s l o n g e s t a x i s i s 85° e a s t f r o m t r u e N o r t h .
Sensor s u p p o r t s have been mounted v e r t i c a l l y a t t h r e e o f t h e f o u r c o m e r p i l e s o f t h e j a c k e t s t r u c t u r e , e x t e n d i n g f r o m t h e p l a t f o r m deck t o a p o i n t c l o s e t o t h e sea b e d . They a r e f a i r l y t r a n s p a r e n t space t r u s s e s , w i t h f o u r .11m d i a m e t e r t u b e s i n t h e c o m e r s o f a 1 m x 1 m s q u a r e . They can be l i f t e d o u t o f t h e w a t e r f o r a t t a c h m e n t , i n s p e c t i o n and c l e a -n i -n g o f t h e s e -n s o r s . F o r t h e measureme-nts d e s c r i b e d h e r e , s e -n s o r s were
used f i x e d t o t h e s u p p o r t a t t h e s o u t h w e s t c o m e r o f t h e p l a t f o r m ( s e e f i g u r e 3 ) .
7
-3. INSTRUMENTATION
3.1 I n t r o d u c t i o n '
The p r i n c i p a l i n s t r u m e n t s used i n t h i s s t u d y were a v e l o c i t y m e t e r a l l o w i n g measurements o f t h r e e o r t h o g o n a l v e l o c i t y components, anda-wave gauge f o r t h e measurement o f t h e s u r f a c e e l e v a t i o n . These a r e
d e s c r i b e d b e l o w , t o g e t h e r w i t h c a l i b r a t i o n p r o c e d u r e s . I n t h i s c o n t e x t , t h e p r o b l e m o f t i m e d e l a y s i n t r o d u c e d b y t h e s e n s o r s and b y t h e d a t a l o g g i n g system r e c e i v e s c a r e f u l a t t e n t i o n , because o f t h e a p p a r e n t phase e r r o r s i n t h e r e s u l t s o f p r e l i m i n a r y d a t a a n a l y s e s ( B a t t j e s and van Heteren»1980). T h i s s t i m u l a t e d a d e t a i l e d i n v e s t i g a t i o n o f p o s s i b l e t i m e d e l a y s , based on n o m i n a l p r o p e r t i e s o f system components (Botma,
19 8 1 ) . The r e s u l t s were l a t e r v e r i f i e d w i t h s i n u s o i d a l t e s t s i g n a l s . I n t h e f o l l o w i n g p r e s e n t a t i o n , t h e r e s u l t s o f t h e s e i n v e s t i g a t i o n s w i l l be m e n t i o n e d .
3.2 V e l o c i t y m e t e r
The v e l o c i t y m e t e r i s b a s e d on t h e p r i n c i p l e o f t r a v e l t i m e o f a c o u s t i c p u l s e s . A m e t e r o f t h i s t y p e , d e s i g n e d f o r use i n t i d a l r i v e r s , has been d e s c r i b e d b y Botma ( 1 9 7 8 ) , who a l s o d e s i g n e d t h e m e t e r used f o r t h e mea-surements r e p o r t e d h e r e .
Three p a i r s o f t r a n s d u c e r s were mounted i n a f r a m e , i n t h r e e m u t u a l l y o r t h o g o n a l l i n e s , a t ' d i s t a n c e s o f 1.71 m (see f i g u r e 4 ) . The f r a m e - c o n s i s t s o f copper p i p e s ( o u t e r d i a m e t e r 50.4 mm) t o m i n i m i z e m a r i n e f o u l i n g . F o r t h e same r e a s o n , t h e e l e c t r i c w i r i n g was mounted i n s i d e t h e p i p e s . The j o i n t s o f t h e p i p e s were g l u e d t o p r e v e n t c o r r o s i o n .
8
-The v e l o c i t y m e t e r was mo\mted o u t s i d e t h e s e n s o r s u p p o r t , w i t h one o f i t s axes a l i g n e d v e r t i c a l l y . The o r i e n t a t i o n o f t h e o t h e r s was a p p r o x i -m a t e l y f r o -m s o u t h - w e s t t o n o r t h - e a s t ( x - a x i s ) and f r o -m s o u t h - e a s t t o n o r t h - w e s t ( y - a x i s ) , see f i g u r e 3. The d i s t a n c e f r o m t h e v e r t i c a l a x i s
o f t h e v e l o c i t y m e t e r t o t h e c e n t r e o f t h e s e n s o r s u p p o r t was 1.75 m. I n one s e t o f measurements, t h e e l e v a t i o n o f t h e c e n t e r o f t h e v e l o c i t y m e t e r was 4.91 m b e l o w MSL, and i n a second s e t i s was 7.83 m b e l o w MSL.
The c a l i b r a t i o n f a c t o r o f t h e v e l o c i t y m e t e r was c a l c x i L a t e d u s i n g t h e n o m i n a l speed of__sound i n sea w a t e r . (Temperature and s a l i n i t y v a r i a t i o n s a r e a u t o m a t i c a l l y c o r r e c t e d f o r . ) An. e f f e c t i v e t r a v e l d i s t a n c e o f 1.61 m was used. T h i s i s 10 cm l e s s t h a n t h e d i s t a n c e between t h e s e n s o r s , t o
a l l o w f o r wake e f f e c t s . The c a l c u l a t e d c a l i b r a t i o n f a c t o r was checked e x p e r i m e n t a l l y by o s c i l l a t i n g t h e frame r e c t i l i n e a r l y i n a l a b o r a t o r y b a s i n . Two o r i e n t a t i o n s were used. I n one o f t h e s e , one p a i r o f s e n s o r s was a l i g n e d w i t h t h e d i r e c t i o n o f t h e b a c k - a n d - f o r t h m o t i o n . I n t h e o t h e r , t h e y e n c l o s e d an a n g l e o f 45° w i t h t h i s d i r e c t i o n . The c a l i b r a t i o n f a c t o r c a l c u l a t e d b e f o r e h a n d t u r n e d o u t t o t m d e r e s t i m a t e t h e v e l o c i t y by about 13% i n t h e p a r a l l e l a l i g n m e n t , w h i c h can be c o n s i d e r e d as an u n d e r e s t i m a t e o f t h e e f f e c t i v e w a k e l e n g t h , and t o o v e r e s t i m a t e i t by about 4% i n t h e case o f t h e 45° a n g l e , w h i c h c o r r e s p o n d s t o an o v e r e s t i m a t e o f t h e e f f e c t i v e w a k e l e n g t h . D e v i a t i o n s as l a r g e as were o b s e r v e d i n t h e case o f p a r -a l l e l -a l i g n m e n t -a r e n o t e x p e c t e d t o o c c u r i n t h e -a c t u -a l me-asurements, because r e c t i l i n e a r f l o w does n o t o c c u r a t sea. I n t h e a n a l y s i s , t h e c a l i -b r a t i o n f a c t o r c a l c u l a t e d -b e f o r e h a n d -b a s e d on n o m i n a l v a l u e s was u s e d , w i t h p o s s i b l e e r r o r s e s t i m a t e d n o t t o exceed 5%.
As t o t h e phase r e s p o n s e o f t h e v e l o c i t y m e t e r has i t been assumed t h a t the m e t e r i n t r o d u c e s no d i s t u r b a n c e t o t h e f l o w f i e l d , because i t i s v e r y t r a n s p a r a n t and i t i s mo\mted o u t s i d e t h e , s e n s o r s u p p o r t o&-tthè. ;pla€fOEm^x-• The v e l o c i t y m e t e r measures t h e v e l o c i t y components a l o n g t h e s u r v e y l i n e s
c o n n e c t i n g t h e t h r e e p a i r s o f t r a n s d u c e r s , a v e r a g e d o v e r t h e s e l i n e s . T h i s a v e r a g i n g p r o c e s s a c t s l i k e a low-pass f i l t e r on t h e v e l o c i t i e s i n t h e c e n t e r . I t i n t r o d u c e s no- phase s h i f t . For t h e h o r i z o n t a l v e l o c i t i e s ; . i t s modulus i s { s i n ( J k . r ) / 2 k . r ) } , i n w h i c h ï< i s t h e wavenumber v e c t o r and
9
-r t h e h o -r i z o n t a l s e p a -r a t i o n v e c t o -r between t h e t -r a n d u c e -r s . The -r e l a t i v e e r r o r i n t r o d u c e d t h e r e b y : a s l e s s t h a n 3 x 1 0 i n t h e f r e q u e n c y r a n g e
c a r r y i n g most o f t h e wave e n e r g y i n o u r measxirements ( 0 . 1 H z ^ f < 0 . 2 H z ) . -2
Even f o r f = 0.3 Hz i t s t i l l i s o n l y 1 . 6 x 1 0 a t most. F o r t h e v e r t i c a l v e l o c i t y component comparable numbers a p p l y .
The o u t p u t s i g n a l s o f t h e v e l o c i t y m e t e r l a g b e h i n d t h e i n s t a n t a n e o u s mean v e l o c i t i e s between t h e s e n s o r p a i r s , due t o a w a i t i n g t i m e i n an i n t e r n a l s c a n n i n g c y c l e p l u s a f i r s t - o r d e r s m o o t h i n g f i l t e r . The e f f e c - : t i v e d e l a y i s 15 ms. The a c t u a l mean v e l o c i t y between s e n s o r p a i r s i n t u r n may be s l i g h t l y o u t o f phase w i t h t h e u n d i s t u r b e d one, due t o t h e p r e s e n c e o f t h e v e l o c i t y m e t e r . However, t h i s e f f e c t i s c o n s i d e r e d n e g -l i g -l i b -l e because t h e frame o f t h e v e -l o c i t y m e t e r i s h i g h -l y t r a n s p a r e n t .
3.3 Wave gauge
The wave gauge used f o r t h e measurements o f t h e s u r f a c e e l e v a t i o n was a 15 m l o n g r e s i s t a n c e s t e p gauge ("TPD"-type) w i t h e l e c t r o d e s .spaced a t 7.5 cm i n t e r v a l s . The gauge was mounted n e a r t h e c e n t e r o f t h e s e n s o r s u p p o r t , a t a d i s t a n c e o f 1.90 m t o t h e v e r t i c a l a x i s o f t h e c u r r e n t me-t e r ( p r o j e c me-t e d h o r i z o n me-t a l l y ) .
The c a l i b r a t i o n o f t h e gauge i t s e l f was t a k e n a t i t s n o m i n a l v a l u e , de-t e r m i n e d b y de-t h e d i s de-t a n c e bede-tween de-t h e e l e c de-t r o d e s .
The wave gauge o u t p u t s i g n a l l a g s b e h i n d t h e i n s t a n t a n e o u s s u r f a c e e l e v a -t i o n , due -t o w a i -t i n g -t i m e s i n an i n -t e r n a l s c a n n i n g c y c l e and a f r e q u e n c y s m o o t h i n g . The e f f e c t i v e d e l a y i s 52 ms. T h i s i s Z\t^ =-52 ms - 15 ms = 37 ms ' more t h a n i t i s f o r t h e v e l o c i t y s i g n a l s .
10
-P o s s i b l e e f f e c t s due t o s c a t t e r i n g o f t h e i n c i d e n t waves by t h e s e n s o r s u p p o r t and b y t h e p l a t f o r m p i l e s a r e c o n s i d e r e d i n t h e f o l l o w i n g .
To i n v e s t i g a t e w h e t h e r t h e f l u c t u a t i o n s o f t h e s u r f a c e e l e v a t i o n i n s i d e t h e s e n s o r s u p p o r t d i f f e r e d s i g n i f i c a n t l y f r o m t h o s e o u t s i d e , t e s t s were
c a r r i e d o u t i n a l a b o r a t o r y f l u m e u s i n g a 1:5 model and p e r i o d i c waves. Due t o l i m i t a t i o n s o f t h e f l u m e and t h e wave g e n e r a t o r , t h e model waves had a s h o r t e r w a v e l e n g t h , compared t o t h e dimensions o f t h e s e n s o r sup-p o r t , t h a n t h o s e i n t h e f i e l d ( r o u g h l y a f a c t o r 3 ) . T h i s i s e x sup-p e c t e d t o g i v e r i s e t o an^_exaggerated s c a t t e r i n g o f t h e waves i n t h e model.
S u r f a c e e l e v a t i o n s were measured i n s i d e and o u t s i d e t h e model s e n s o r sup-p o r t , i n t h e same fl\jme c r o s s - s e c t i o n , f o r a v a r i e t y o f i n c i d e n t waves. The measurements were made w i t h t h e c r o s s - s e c t i o n a l d i a g o n a l o f t h e supp o r t e i t h e r supp a r a l l e l t o t h e f l u m e a x i s o r a t 45°. No s i g n i f i c a n t d i f f e r -ence c o u l d be o b s e r v e d due t o t h e o r i e n t a t i o n . The measured r a t i o s H. /H were a l s o f o u n d t o be i n d e p e n d e n t o f wave s t e e p n e s s ( i n t h e m o u t range 0.016 < H/L < 0.066); t h e i r average v a l u e i s 0.98, w i t h a s t a n d a r d d e v i a t i o n o f 0.0 3. Phase s h i f t s between t h e u n d i s t u r b e d s u r f a c e e l e v a -t i o n s and -t h o s e i n s i d e -t h e s e n s o r s u p p o r -t were f o u n d -t o be l e s s -t h a n 1 % c o f t h e w a v e p e r i o d , o r l e s s t h a n about 4 ° . C o n s i d e r i n g t h e s e r e s u l t s , and t h e f a c t t h a t i n t h e model t h e i n f l u e n c e o f t h e s u p p o r t i s e x p e c t e d t o be e x a g g e r a t e d (see a b o v e ) , no c o r r e c t i o n s were made t o t h e s u r f a c e eleva¬ t i o n measurements t o a l l o w f o r such i n f l u e n c e .
The p o s s i b l e d i s t u r b a n c e s t o t h e i n c i d e n t waves due t o t h e p l a t f o r m sup-p o r t s t r u c t u r e were e s t i m a t e d t h e o r e t i c a l l y . To t h i s e n d , t h e s c a t t e r i n g o f t h e waves by t h e n e a r e s t c o m e r p i l e ( d i a m e t e r 0.80 m) was c o n s i d e r e d . T h i s p i l e was t a k e n t o be v e r t i c a l , a t a d i s t a n c e o f r = 3 m f r o m t h e wave gauge. The l i n e a r , p o t e n t i a l - f l o w s o l u t i o n t o t h e p r o b l e m o f t h e s c a t t e r i n g o f a p l a n e , p e r i o d i c wavesystem b y a r i g i d impermeable c i r c u l a r c y l i n d e r , p o s i t i o n e d v e r t i c a l l y , has been g i v e n by Lamb ( 1 9 3 2 , a r t . 3 0 4 ) . We w i l l o n l y n e e d t h e a p p r o x i m a t i o n t o t h e e x a c t s o l u t i o n f o r t h e case o f a s l e n d e r p i l e , ak « 1 , i n w h i c h a i s t h e p i l e r a d i u s and k i s t h e wave-number.
11 -I f t h e s u r f a c e e l e v a t i o n o f t h e i n c i d e n t waves i s g i v e n by , i ( k x + a t ) n ( x , y , t ) = Re e t h e n t h e e l e v a t i o n o f t h e s c a t t e r e d wave i s a p p r o x i m a t e d as f i ( r , 0 , t ) = R e J ( k a ) ^ { i H [ j ^ ^ : ( - k r ) + 2 H ^ ^ \ k r ) c o s e } e ^ ^ * + OCka)"^
where ( x , y ) = ( r cos 0 , r s i n 9 ) , and t h e p i l e i s c e n t e r e d a t
( x = 0, y = 0 ) . H^^-* i s t h e H a n k e l f u n c t i o n o f t h e second k i n d o f o r d e r ^ n / 2 )
n. (Lamb uses t h e n o t a t i o n , where D ^ ( k r ) = - i H ^ ( k r ) . ) U s i n g
H ^ ^ ^ ( k r ) = J ^ ( k r ) - i Y . ( k r ) , n u u
and n e g l e c t i n g t h e t e r m s o f 0 ( k a ) ^ , t h e a m p l i t u d e o f t h e s c a t t e r e d waves r e l a t i v e t o t h a t o f t h e i n c i d e n t waves i s e x p r e s s e d as •„
^ = • J - ( k a ) 2 { j Q + - Y j + 4 ( J ^ Y Q - j Q Y ^ ) c o s e + 4(4 + Y^) cos^ e } *
where = j Q ( k r ) , e t c . The maximum v a l u e o f n c o n s i d e r e d as a f u n c t i o n o f 9 a t r = c o n s t . , i s For n . , a s i m i l a r e x p r e s s i o n h o l d s , w h i c h i s n o t shown h e r e . U s i n g mm a = 0.4 m , r = 3.0 m and h = 17.5 m ( t h e ' l a t t e r i s needed i n t h e d i s p e r -s i o n e q u a t i o n ) t h e f o l l o w i n g v a l u e -s r e -s u l t f o r 3 d i f f e r e n t f r e q u e n c i e -s : f [Hz] k a k r n max ^min 0.1' .022 .163 ,0038 ,0005 0,2 .066 .500 ,013 .002 0,3 .145 1,09 ,037 .009
12
-I t appears t h a t i n t h e f r e q u e n c y range f r o m 0.1 Hz t o Ó.2 Hz,irwhich i s t h e most e n e r g e t i c range i n o u r measurements, t h e s c a t t e r i n g o f t h e waves by t h e n e a r e s t c o m e r p i l e a f f e c t s t h e measurements by r e l a t i v e
-3 -2
amounts o f t h e o r d e r o f 10 t o 10 a t most, w h i c h i s n e g l i g i b l e f o r o u r p u r p o s e s .
3.H Anemometer
Measurements o f ^ w i n d speed and d i r e c t i o n were made w i t h an anemometer ( t y p e Munro, Mark 4 ) and a vane, b o t h mounted on a 9.5 m h i g h mast a t an e l e v a t i o n o f 28 m above MSL.
3.5 Data a c q u i s i t i o n s y s t e m
A 32-channel d a t a l o g g i n g and t r a n s m i s s i o n s y s t e m has been used t o c o l l e c t t h e d a t a and t r a n s m i t them by an F M - r a d i o - l i n k t o a s h o r e - b a s e d r e c e i v i n g s t a t i o n where t h e y were r e c o r d e d on m a g n e t i c t a p e .
B e f o r e s c a n n i n g f o r t r a n s m i s s i o n , a l l d a t a had t o be c o n v e r t e d t o a s u i t - ^ a b l e f o r m a t . For t h e v e l o c i t y m e t e r , w h i c h has a DC o u t p u t , t h e d e l a y caused h e r e b y i s o n l y 1 ms, b u t t h e wave gauge s i g n a l r e q u i r e s a f r e q u e n c y c o u n t i n g , r e s u l t i n g i n a t o t a l i n t e r f a c e d e l a y o f 367 ms, o r
A t 2 = 367 ms - 1 ms = 366 ms more t h a n f o r t h e v e l o c i t y s i g n a l s .
The s c a n n i n g p r o c e d u r e i t s e l f and t h e s u b s e q u e n t r e a d - o u t f o r d a t a e n t r y i n t h e a n a l y s i s p r o c e d u r e s cause a d d i t i o n a l l a g s . The v e l o c i t y s i g n a l s (u,V5w) and t h e e l e v a t i o n s i g n a l (<;) were f e d t o a d j a c e n t c h a n n e l s , w h i c h were scanned a t 6.25 ms i n t e r v a l s , i n t h e o r d e r c i t e d . However, i n t h e
d a t a r e a d - o u t t h e e l e v a t i o n v a l u e o f one s c a n n i n g c y c l e was combined w i t h t h e v e l o c i t y v a l u e s o f t h e f o l l o w i n g c y c l e , c a u s i n g a l a g o f 250 ms. Thus, t h e p r o c e d u r e o f s c a n n i n g and r e a d o u t i n t r o d u c e s a l a g o f t h e e l e -v a t i o n s i g n a l b e h i n d t h e -v e l o c i t y s i g n a l s g i -v e n by A t ^ ^ - 250 ms - 6 ms - 244 ms, A t o - 250 ms - 12 ms 2 38 ms. and A t _ „ - 250 ms - 19 ms o j V o,U 231 ms.
13
-A siJinmary o f t h e t i m e l a g s m e n t i o n e d above i s g i v e n i n t h e f o l l o w i n g t a b l e , m d e r t h e h e a d i n g " n o m i n a l " , because t h e s e v a l u e s have been c a l c u l a t e d f r o m n o m i n a l s y s t e m p r o p e r t i e s . Lag o f
'X
, N o m i n a l Meas'd b e h i n d ^ , A t 3 3 E A t . i = l " 3 E A t . i = 2 . 3 E A t . .i=2_ " ;u 37 366 231 634 597 589 V 37 366 238 6 4 1 604 595 w 37- 366 244 647 610 601Table A - Summary o f system d e l a y t i m e s , e x p r e s s e d i n ms.
A p a r t i a l check on t h e n o m i n a l v a l u e s has been c a r r i e d o u t by s i m u l a t i n g s e n s o r o u t p u t s w i t h s i n u s o i d a l t e s t s i g n a l s a t d i f f e r e n t f r e q u e n c i e s , and by s u b j e c t i n g t h e s e t o t h e s t a n d a r d p r o c e d u r e o f i n t e r f a c e c o n v e r s i o n ( A t ^ ) , s c a n n i n g and r e a d o u t ( A t ^ ) , and subsequent c r o s s s p e c t r a l a n a l y -s i -s . The r e -s u l t a n t pha-se l a g -s between -s i m u l a t e d v e l o c i t i e -s and -s i m u l a t e d e l e v a t i o n showed v e r y n e a r l y a l i n e a r v a r i a t i o n w i t h f r e q u e n c y , - a s expec-t e d f r o m a c o n s expec-t a n expec-t expec-t i m e s h i f expec-t (see f i g . 5 f o r an e x a m p l e ) . The v a l u e s o f t h e t i m e s h i f t c a l c u l a t e d on t h i s b a s i s a r e g i v e n i n t h e l a s t column o f t a b l e A g i v e n above. They s e r v e as a check on ( A t 2 + ^ ^ 3 ) °f w h i c h t h e n o m i n a l v a l u e i s g i v e n i n t h e n e x t - t o - l a s t column. There i s a d i f f e r e n c e
o f l e s s t h a n 10 ms between t h e two s e t s , w h i c h i s n e g l i g i b l e i n c o m p a r i s o n t o t h e wave p e r i o d s i n v o l v e d .
14
-4. DATA ANALYSIS PROCEDURE
4.1 I n t r o d u c t i o n .
The d a t a a n a l y s i s t o be r e p o r t e d i n t h i s c h a p t e r was aimed p r i m a r i l y a t e s t i m a t i n g s p e c t r a l t r a n s f e r f u n c t i o n s between v e l o c i t y components and between t h e s e and t h e s i i r f a c e e l e v a t i o n . A secondary o b j e c t i v e was t h e e s t i m a t i o n o f p r o b a b i l i t y d e n s i t i e s o f i n s t a n t a n e o u s v a l u e s o f t h e measured q u a n t i t i e s . I n a d d i t i o n t o t h i s , d i r e c t i o n a l s p e c t r a l d e n s i t i e s were e s t i m a t e d , b u t t h a t a s p e c t i s n o t d i s c u s s e d h e r e . We w i l l use o n l y one r e s u l t o f t h e d i r e c t i o n a l a n a l y s i s , v i z . t h e p r i n c i p a l d i r e c t i o n o f p r o p a g a t i o n ( 0 ^ ) o f t h e sea o r s w e l l as f u n c -t i o n o f f . 4.2 E s t i m a t i o n o f a u t o - and c r o s s - s p e c t r a
The a n a l y s e s were p e r f o r m e d on 30-minute l o n g records,, d i g i t i z e d w i t h At = 0.25 s. No d i l u t i o n o f t h e d a t a was a p p l i e d , so t h a t t h e N y q u i s t f r e q u e n c y i s a t f = 2.0 Hz. T h i s was deemed h i g h enough t o p r e v e n t s i g n i f i c a n t a l i a s i n g .
S t a n d a r d s p e c t r a l a n a l y s i s p r o c e d u r e s as d e s c r i b e d e.g. by Bendat and P i e r s o l ( 1 9 7 2 ) and J e n k i n s and W a t t s .(1968) were used. These i n c l u d e d
c o r r e c t i o n s f o r m i s s i n g d a t a o r u n a c c o u n t a b l e v a l u e s and r e m o v a l o f sample mean and l i n e a r t r e n d . S i n g l e t o n ' s (1969) FFT a l g o r i t h m was ap-p l i e d f o r t h e s ap-p e c t r a l c a l c u l a t i o n s . .
The r e c o r d s . w e r e d i v i d e d i n t o 30 segments o f 60 s each. No d a t a o r s p e c t r a l window was a p p l i e d , so t h a t t h e r e s \ i l t i n g s p e c t r a l e s t i m a t e s each had 60 degrees o f f r e e d o m , e x c e p t t h o s e a t z e r o f r e q u e n c y and a t t h e N y q u i s t f r e q u e n c y , w h i c h had 30. U s i n g t h e x^-<ü-Stribution, t h e l o w e r - and u p p e r bounds o f t h e 90%- c o n f i d e n c e bands o f t h e e s t i m a t e s o f t h e a u t o - s p e c t r a a r e 0.72 and 1.32 t i m e s t h e sample e s t i m a t e i n case o f 60 degrees o f freedom.
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Each e s t i m a t e d s p e c t r u m i s h e r e d e n o t e d as S ^ ^ ( f ) . , w i t h s u b s c r i p t s i n -d i c a t i n g t h e p r o c e s s e s o f w h i c h i t i s t h e a u t o - o r c r o s s - s p e c t r u m . The a u t o - s p e c t r a have been made o n e - s i d e d , so t h a t t h e i r i n t e g r a l o v e r a l l p o s i t i v e f r e q u e n c i e s e q u a l s t h e v a r i a n c e o f t h e p r o c e s s .
4.3 E s t i m a t i o n o f coherence- s p e c t r a
The c r o s s - s p e c t r a have been n o r m a l i z e d t o s p e c t r a o f coherence. I n case o f C ( t ) and w(tr)- t h i s i s w r i t t e n as
L,t, WW
9 0 % . c o n f i d e n c e bands f o r t h e coherence were c a l c u l a t e d u s i n g t h e e x p r e s -s i o n -s p r e -s e n t e d by Bendat and P i e r -s o l ( 1 9 7 2 , p a r . 6 . 6 ) . The-se a u t h o r -s l i m i t t h e v a l i d i t y o f t h e i r e x p r e s s i o n s f o r t h e c o n f i d e n c e i n t e r v a l o f
t o t h e range 0.35 < < 0.95. O u t s i d e t h i s range no c o n f i d e n c e bands were c a l c u l a t e d .
I n t h e f o l l o w i n g , coherence v a l u e s a r e used o n l y f o r t h e v e r t i c a l v e l o c i t y i n r e l a t i o n t o t h e s i r r f a c e e l e v a t i o n , i n w h i c h case i t s t h e o r e t i c a l v a l u e i n absence o f n o i s e i s 1 ( a s s t i m i n g l i n e a r i t y ) . The coherences between a h o r i z o n t a l v e l o c i t y component ( u o r v ) and s u r f a c e e l e v a t i o n ( ? ) o r v e r -t i c a l v e l o c i -t y (w) depend on -t h e d i r e c -t i o n a l wave energy d i s -t r i b u -t i o n . I n f a c t , t h e i r v a l u e s a r e used i n e s t i m a t i n g t h a t d i s t r i b u t i o n . S i n c e t h i s s u b j e c t i s n o t d e a l t w i t h h e r e no coherence v a l u e s i n v o l v i n g u o r v a r e p r e s e n t e d .
4.4 E s t i m a t i o n o f phase s p e c t r a
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-Confidence bands on t h i s e s t i m a t e were c a l c u l a t e d u s i n g t h e e q u a t i o n s p r e s e n t e d i n Bendat & P i e r s o l ( 1 9 7 2 , p a r . 6 . 7 ) .
The s p e c t r a l c a l c u l a t i o n s were made w i t h o u t t a k i n g i n t o , a c c o u n t t h e t i m e d e l a y s m e n t i o n e d i n c h a p t e r 3, w h i c h became known a t a l a t e r d a t e . The phase a n g l e s i j i , ^ ^ ( f ) , c a l c u l a t e d as i n d i c a t e d above, t h e r e f o r e had t o be c o r r e c t e d t o a c c o u n t f o r a t i m e s h i f t , w h i c h was a p p r o x i m a t e d as a c o n s t a n t A t = .64 s (see t a b l e A ) . The c o r r e s p o n d i n g phase c o r r e c t i o n i s Atjj = -2ïïfAt. (The d e f i n i t i o n s a d o p t e d i m p l y t h a t a p o s i t i v e v a l u e o f ^ i n d i c a t e s a l a g o f I, b e h i n d u , v o r w.) A f i n a l phase c o r r e c t i o n was a p p l i e d t o .account f o r t h e f a c t t h a t t h e c e n t e r o f t h e v e l o c i t y m e t e r was 1.90 m f r o m t h e wave gauge ( p r o j e c t e d h o r i z o n t a l l y ; see f i g u r e 3 ) ^ c a u s i n g an a d d i t i o n a l phase l a g o f r, b e h i n d w g i v e n b y k r cos ©, i n w h i c h r = 1.90 m i s t h e h o r i z o n t a l d i s t a n c e between t h e s e n s o r s , k i s t h e wave-number and 9 i s t h e p r o p a g a t i o n d i r e c t i o n r e l a t i v e t o t h e d i r e c t i o n f r o m
v e l o c i t y m e t e r t o wave gauge. The m a g n i t u d e o f t h i s c o r r e c t i o n was c a l c u -l a t e d p e r f r e q u e n c y , u s i n g t h e -l i n e a r d i s p e r s i o n e q u a t i o n . The p r i n c i p a -l p r o p a g a t i o n d i r e c t i o n p e r f r e q u e n c y , 0^( f ) , as d e t e r m i n e d i n t h e d i r e c ¬ t i o n a l a n a l y s i s ( v a n H e t e r e n and K e i j s e r , 1 9 8 1 ) , was used as an e s t i m a t e o f 9. The maximum v a l u e o f t h i s c o r r e c t i o n (assixming 9 = 0 ° ) c o r r e s p o n d s t o a t i m e s h i f t o f about 0.2 s i n t h e most e n e r g e t i c f r e q u e n c y range ( f between 0.1 Hz and 0.2 H z ) . A l t o g e t h e r , t h e c o r r e c t e d e s t i m a t e d phase v a l u e s become
$ ^ ^ ( f ) = - 2TTfAt - k r cos 9 ^ ( f )
Cross s p e c t r a between w and e i t h e r u o r v were a l s o c a l c u l a t e d . Phase a n g l e s were c a l c u l a t e d as a r g S^^ o r a r g S ^ . The o n l y c o r r e c t i o n w h i c h i s needed h e r e i s due t o t h e s c a n n i n g sequence, w h i c h causes a l a g o f 6.25 ms i n case o f (v,w) and 12.50 ms i n case o f (-u,w).
I n t h e d a t a g i v e n b e l o w , e i t h e r ^^^j^Cf) ^vw^^'^ °'^ ^ r e c o r d have been p r e s e n t e d . The c h o i c e between u and v was d e t e r m i n e d by t h e c r i t e r i o n
t h a t t h e component s h o u l d be used w h i c h most n e a r l y c o i n c i d e d w i t h t h e p r i n c i p a l wave p r o p a g a t i o n d i r e c t i o n , i n o r d e r t o m a i n t a i n a r e a s o n a b l y h i g h coherence.
17
-4.5 E s t i m a t i o n o f g a i n s p e c t r a
I n order t o e l i m i n a t e p o s s i b l e i n f l u e n c e s o f n o i s e , the gain f u n c t i o n H ( f ) was e s t i m a t e d a c c o r d i n g t o Cw
|L
( f ) s ( f ) WW C o n f i d e n c e bands f o r ( f ) were c a l c u l a t e d , u s i n g t h e p r o c e d u r e d e s c r i b e d QW by Bendat and P i e r s o l ( 1 9 7 2 , p a r . 6 . 7 ) .The h o r i z o n t a l v e l o c i t y components r e q u i r e a s e p a r a t e t r e a t m e n t , because o f t h e d i r e c t i o n a l d i s t r i b u t i o n o f t h e wave energy. Gain f u n c t i o n s were c a l c u l a t e d f o r t h e r e s u l t a n t v e l o c i t y i n t h e h o r i z o n t a l p l a n e ( u ) a c c o r d i n g t o S ( f ) + S ( f ) T H ( f ) = { ^ ^ }i S ^ , ( f ) and S ( f ) + S ( f ) T i ^ ( f ) = i ( f ) WW T h i s p r o c e d u r e does n o t p e r m i t t h e e s t i m a t i o n o f c o n f i d e n c e bands. I n s t e a d , t h e s e g a i n f u n c t i o n v a l u e s were c a l c u l a t e d i n a f r e q u e n c y b a n d w h i c h was c o n s i d e r e d t o be v i r t u a l l y f r e e o f n o i s e , as i n d i c a t e d by a v a l u e o f a t l e a s t 0.9 o f ( f ) * QW 4.6 E s t i m a t i o n - o f p r o b a b i l i t y d e n s i t i e s
A p a r t i a l check on t h e v a l i d i t y o f t h e Gaussian model f o r sea waves was c a r r i e d o u t by c o n s i d e r i n g , 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 o f t h e i n s t a n t a -neous v a l u e s o f C , u , V and W-, and b y comparing i t t o t h e b e s t f i t t i n g Gaussian d i s t r i b u t i o n . - •
18
-To t h i s e n d , a h i s t o g r a m was c a l c u l a t e d f r o m t h e 7200 d a t a p o i n t s o f each sample f u n c t i o n , w i t h a c l a s s i n t e r v a l o f 0.10 m f o r C and o f 0.02 ms""*" f o r u , v and w. No d i l u t i o n o f t h e d a t a was a p p l i e d , as w o u l d be needed t o have i n d e p e n d e n t samples o n l y . T h e r e f o r e , c o n v e n t i o n a l q u a n t i t a t i v e t e s t s o f g o o d n e s s - o f - f i t do n o t a p p l y . A v i s u a l i n s p e c t i o n was c a r r i e d o u t i n s t e a d .
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-5. OBSERVATIONS
5.1 Data sequences
I n t h e p e r i o d o f t h e measurements, s e r i e s o f 30-minute r e c o r d s were made o f t h e s u r f a c e e l e v a t i o n and o f t h e v e l o c i t y components... I n a d d i t i o n , t h e f o l l o w i n g p a r a m e t e r s were c a l c u l a t e d and p r i n t e d e v e r y h a l f h o u r : 1 0 - m i n u t e averages o f w i n d speed and d i r e c t i o n , t i d a l e l e v a t i o n , m a g n i t u d e and d i r e c t i o n o f t i d a l c u r r e n t , and s i g n i f i c a n t wave h e i g h t .
I n o r d e r t o m i n i m i z e p o s s i b l e d i s t u r b a n c e s due t o t u r b u l e n c e i n d u c e d b y t i d a l c u r r e n t s , p a r t i c u l a r l y ..in t h e wake o f t h e p l a t f o r m s u p p o r t s t r u c -t u r e , o n l y r e c o r d i n g s made a r o u n d -t h e -t i m e s o f s l a c k w a -t e r were used i n t h e a n a l y s e s r e p o r t e d h e r e . The c r i t e r i o n a d o p t e d was t h a t t h e c u r r e n t v e l o c i t y s h o u l d n o t exceed 0.25 ms ^, An a d d i t i o n a l consequence o f t h i s s e l e c t i o n i s t h e v i r t u a l absence o f D o p p l e r f r e q u e n c y s h i f t s i n t h e ob-s e r v a t i o n ob-s ( c u r r e n t v e l o c i t y l e ob-s ob-s t h a n 0.03 t i m e ob-s t h e phaob-se ob-speed i n t h e e n e r g e t i c f r e q u e n c y range f < 0.2 H z ) .
Two s e t s o f measurements were made. I n t h e f i r s t s e t , s i x s e r i e s (number 4 t h r o u g h 9 ) were r e c o r d e d , w i t h t h e c e n t e r o f t h e v e l o c i t y m e t e r a t 4.91 m b e l o w MSL. Of t h i s s e r i e s , 23 r e c o r d s were used i n t h e a n a l y s i s ( r e c o r d 4 0 1 t h r o u g h 9 0 5 ) . U n f o r t u n a t e l y , no s u r f a c e e l e v a t i o n d a t a were o b t a i n e d i n t h i s e n t i r e s e t i n o r n e a r t h e same v e r t i c a l i n w h i c h t h e v e l o c i t i e s were measured, so t h a t t h e d a t a o f t h i s s e t c o u l d be a n a l y z e d o n l y w i t h r e s p e c t t o t h e v e l o c i t y s i g n a l s and t h e i r i n t e r r e l a t i o n s h i p s .
A second s e t o f measurements was made w i t h t h e c e n t e r o f t h e v e l o c i t y m e t e r a t 7.83 m b e l o w MSL. Of t h i s s e t , 37 r e c o r d i n g s were a n a l y z e d w i t h r e s p e c t t o t h e v e l o c i t i e s , t h e s u r f a c e e l e v a t i o n , and t h e i r i n t e r r e l a t i o n -s h i p -s ( r e c o r d -s 1501 t h r o u g h 2 6 0 5 ) .
20
-5.2 Wind and sea c o n d i t i o n s
H o u r l y v a l u e s o f t h e p r i n c i p a l p a r a m e t e r s o f w i n d and waves on t h e days o f t h e r e c o r d i n g s a r e p r e s e n t e d g r a p h i c a l l y i n A p p e n d i x 1 . A t a b u l a r summary o f t h e v a l u e s a t t h e t i m e s o f t h e r e c o r d i n g s w h i c h were a n a l y z e d i s g i v e n i n Appendix 2. The f o l l o w i n g q u a n t i t i e s p e r t a i n i n g , t o t i d e s and w i n d a r e l i s t e d i n Appendix 2: r e c o r d number, d a t e , t i m e o f r e c o r d , mean w a t e r e l e v a t i o n above MSL, x- and y-óomponents o f mean v e l o c i t y ( u , v ) ,
10-minute average w i n d v e l o c i t y ( V ^ ) , and w i n d d i r e c t i o n w i t h r e s p e c t t o t r u e N o r t h ( 9 ^ ) . - A p p e n d i x 3 g i v e s t h e p r i n c i p a l d i r e c t i o n o f t h e waves w i t h r e s p e c t t o N o r t h ( 0 ^ ) a t t h e peak f r e q u e n c y o f t h e v a r i a n c e s p e c t r u m
( f ^ ) , s i g n i f i c a n t wave h e i g h t (Hm^^ , c a l c u l a t e d as ^/m^^ , i n w h i c h m^ i s the v a r i a n c e o f C ) , peak f r e q u e n c y if^) and f i n a l l y wave age 3 and s t e e p -ness Y, d e f i n e d as c H B E
J i
and Y E ^ w m i n w h i c h °m g ^ -u 27Th m 27Tf m m The c o n d i t i o n s r a n g e d f r o m a m i l d e a s t e r l y ( i . e . , o f f s h o r e ) wind('^^==6 ms ) and a low s w e l l (H ^ 0.8 m, wave s t e e p n e s s y ^ 0.01, wave age 3 - 2) i nthe s e r i e s 15, t o a w e s t e r l y g a l e (maximumVi^ 24 ms ) w i t h a p\ire sea (H 4 m, y - 0.04, 3 0.6) i n s e r i e s 26. ( A c t u a l l y , t h e maxim\im w i n d
^ - 1
v e l o c i t y o f 24 ms o c c u r r e d about i j h o u r p r i o r t o t h e f i r s t r e c o r d i n t h i s s e r i e s , n r . 2601; a t t h e t i m e o f t h i s r e c o r d i n g t h e w i n d v e l o c i t y had dropped t o about 20 ms •''.) Less s e v e r e b u t s t i l l p u r e sea c o n d i t i o n s p r e -v a i l e d i n t h e s e r i e s 6, 9 ( f i r s t p a r t o n l y ) , 16/17, 22/23, 24 and 25. The wave age 3 i n t h e s e s e r i e s i s i n a l m o s t a l l r e c o r d s l e s s t h a n 0.8. S e r i e s 4, 7 and 18 a r e cases o f m i x e d sea and s w e l l , w i t h two peak f r e -quenciès, o f w h i c h however o n l y one i s l i s t e d i n A p p e n d i x 2, v i z . t h e one w i t h t h e h i g h e s t s p e c t r a l d e n s i t y .
2 1
-6. RESULTS AND DISCUSSIONS
6.1 I n t r o d u c t i o n
I n A p p e n d i x 4- t h r o u g h Appendix 10, a number o f s p e c t r a o f v a r i a n c e , c o h e r e n c e , phase and g a i n have been p l o t t e d as examples. I n most c a s e s , t h e f i r s t r e c o r d s o f each s e r i e s o f t h e second s e t o f measurements have been used f o r t h i s purpose ( r e c o r d s 1 5 0 1 , 1 6 0 1 , e t c . ) . T h i s s e t i n c l u d e s v e l o c i t y and e l e v a t i o n d a t a .
The s p e c t r a l e s t i m a t e s have been p l o t t e d a t a f r e q u e n c y i n t e r v a l o f ~ Hz
1 2 0
f r o m Hz t o — r Hz. A t h i g h e r f r e q u e n c i e s t h e energy l e v e l s become q u i t e 60 60
l o w .
The r e s u l t s w i l l be d i s c u s s e d i n terms o f s p e c t r a l , r e l a t i o n s h i p s between measured v a r i a b l e s , as e x p r e s s e d i n t e r m s o f c o h e r e n c e , phase and g a i n . The v a r i a n c e s p e c t r a o f t h e s u r f a c e e l e v a t i o n and t h e i r v a r i a t i o n w i t h w i n d speed e t c . a r e n o t d i s c u s s e d s i n c e t h i s s t u d y i s n o t aimed a t wave g r o w t h o r wave f o r e c a s t i n g . F o r o v e r a l l measures o f t h e sea s t a t e p e r r e c o r d , r e f e r e n c e i s made t o A p p e n d i x 3.
6.2 Coherence s p e c t r a
The coherence between e l e v a t i o n and v e r t i c a l v e l o c i t y ( A p p e n d i x ft) i s i n "2
a l l measurements c l o s e t o 1 (y^^>0.95) i n t h e c e n t r a l , e n e r g e t i c f r e quency band. T h i s i s s i g n i f i c a n t s i n c e i t i m p l i e s a v i r t u a l l y l i n e a r r e -l a t i o n between C ( t ) and w ( t ) . I t i s p o i n t e d o u t t h a t t h i s c o n c -l u s i o n h o l d s f o r a l l t h e r e c o r d s w h i c h were a n a l y z e d , r e g a r d l e s s o f t h e w i n d -speed, t h e wave age o r t h e wave s t e e p n e s s .
A f u r t h e r consequence o f t h e h i g h o b s e r v e d c o h e r e n c e - v a l u e s i s t h a t t h e s a m p l i n g v a r i a b i l i t y i n t h e e s t i m a t e s o f phase and g a i n s p e c t r a i s l o w . The t o t a l w i d t h s o f t h e 90% c o n f i d e n c e i n t e r v a l s f o r t h e s e q u a n t i t i e s i n t h e e n e r g e t i c f r e q u e n c y band a r e t y p i c a l l y l e s s t h a n 5° f o r t h e phases and l e s s t h a n 10% o f t h e t h e o r e t i c a l v a l u e s . f o r t h e g a i n s .
_ 22
-The l a t t e r v a l u e does not. i n e l u d e an u n c e r t a i n t y i n t h e c a l i b r a t i o n o f t h e v e l o c i t y m e t e r , w h i c h however was e s t i m a t e d n o t t o exceed 5% ( s e e p a r . 3 . 2 ) .
6, 3 Phase s p e c t r a
The e s t i m a t e d phase l e a d o f w w i t h r e s p e c t t o ^ ( A p p e n d i x 4) i s i n most cases i n agreement w i t h t h e t h e o r e t i c a l l y e x p e c t e d v a l u e o f 90°, i n t h e sense t h a t t h e l a t t e r i s u s u a l l y w i t h i n t h e e s t i m a t e d 9 0 % c o n f i d e n c e i n -t e r v a l o f -t h e e s -t i m a -t e . The average o f ( f ) i n -t h e e n e r g e -t i c f r e q u e n c y band o f each s p e c t r u m d e v i a t e s g e n e r a l l y l e s s t h a n 2 i n a b s o l u t e v a l u e f r o m t h e t h e o r e t i c a l l y e x p e c t e d v a l u e o f 90°.
The m a j o r i t y o f t h e e s t i m a t e d phase s p e c t r a ^-^^^^^ show a weak t r e n d w i t h f r e q u e n c y , o f about 6°/(0.3 Hz) o r 20°/Hz, c o r r e s p o n d i n g t o a d e l a y o f X, b e h i n d w by about 0.05 s.
Note t h a t d e l a y s i n t h e s i g n a l p r o c e s s i n g a r e a l r e a d y a c c o u n t e d f o r , t o an e s t i m a t e d a c c u r a c y o f 10 ms ( p a r . 3 . 5 ) . The o b s e r v e d d e l a y i s p o s s i b l y due t o t h e f a c t t h a t t h e s h o r t - o r e s t e d n e s s o f t h e waves was n o t t a k e n i n t o a c c o u n t i n t h e c o r r e c t i o n ' k r cos 0 ^ ( f ) ( s e e p a r . 4 . 4 ) . The use o f t h e p r i n c i p a l d i r e c t i o n 0^ o n l y l e a d s t o an o v e r e s t i m a t i o n o f t h e d e l a y o f C b e h i n d w.whioh i n c r e a s e s w i t h i n c r e a s i n g f r e q u e n c y . T h i s i s q u a l i -t a -t i v e l y i n agreemen-t w i -t h -t h e o b s e r v e d -t r e n d . F u r -t h e r m o r e , i f -t h i s e x r p l a n a t i o n i s c o r r e c t , no t r e n d i s e x p e c t e d i n 4 ' ^ ^ ( f ) and < j ) ^ ( f ) > s i n c e u ,
V and w have been measured i n t h e same v e r t i c a l s . T h i s agrees w i t h t h e
r e s u l t s o f t h e measurements ( s e e b e l o w ) .
The phase s p e c t r a ^^„(f) o f some r e c o r d s ( 1 6 0 1 , 2 2 0 1 , 2 3 0 1 , 2501) d e v i a t e f r o m t h e g e n e r a l p a t t e r n i n t h e sense t h a t f o r l o w f r e q u e n c i e s , where t h e l e v e l s o f energy and coherence a r e l o w , t h e e s t i m a t e d phases d e v i a t e s t r o n g l y f r o m t h e e x p e c t e d v a l u e , t o such e x t e n t t h a t t h e l a t t e r i s w e l l o u t s i d e t h e 90% c o n f i d e n c e i n t e r v a l . T h e o r e t i c a l l y , t h e f a c t t h a t t h e l o w f r e q u e n c y energy l e v e l i s l o w r e l a t i v e t o t h a t i n t h e peak s h o u l d n o t m a t t e r . Even t h e f a c t t h a t the- coherence drops t o l o w v a l u e s s h o u l d n o t m a t t e r i n t h e sense t h a t t h i s i s r e f l e c t e d i n - t h e d i v e r g e n c e , o f t h e u p p e r and l o w e r l i m i t s - o f t h e c o n f i d e n c e b a n d on é .
23
-The f a c t t h a t i n t h i s l o w - f r e q u e n c y band we majr s t i l l be m e a s u r i n g p h y s i c a l l y m e a n i n g f u l v a l u e s i s s u g g e s t e d by t h e b e h a v i o u r o f t h e e s t i m a t e d g a i n f u n c t i o n . Y e t t h e i n d i c a t e d phase r e l a t i o n i n t h e
^ Cw
l o w f r e q u e n c y range does n o t seem r e a l i s t i c . A s a t i s f a c t o r y e x p l a n a -t i o n has n o -t been f o u n d . However, -t h e p r o b l e m i s n o -t o f much p r a c -t i c a l i m p o r t a n c e i n view o f t h e l o w energy l e v e l s f o r t h o s e f r e q u e n c i e s where i t o c c u r s .
I n t h e case o f r e c o r d 2 6 0 1 , t h e e s t i m a t e d phase l a g o f X, b e h i n d w ap-p e a r s t o . d i f f e r . - s i g n i f i c a n t l y .(about 8°) f r o m t h e t h e o r e t i c a l , v a l u e i n t h e c e n t r a l , e n e r g e t i c f r e q u e n c y r a n g e . I n t h i s r a n g e , t h e coherence i s n e a r l y 1 ( a s i n a l l o t h e r r e c o r d s ) , w h i c h i n d i c a t e s n e a r - l i n e a r i t y , so t h a t t h e a p p a r e n t d i s c r e p a n c y i n t h e phase cannot be a s c r i b e d t o n o n l i n e a r b e h a v i o u r . No s a t i s f a c t o r y e x p l a n a t i o n f o r t h i s e r r a t i c r e s v i l t has been f o u n d . W i t h r e s p e c t t o t h e phase s p e c t r a $ ^ ^ ( f ) and ^ . ^ ( f ) ( A p p e n d i x 7 ) , w h i c h a r e o f p a r t i c u l a r i n t e r e s t i n v i e w o f t h e c o n t r o v e r s y m e n t i o n e d i n t h e I n t r o d u c t i o n , s i m i l a r comments a p p l y as i n t h e case o f $ ^ ^ ( f ) . Here again''-t h e again''-t h e o r e again''-t i c a l l y e x p e c again''-t e d v a l u e o f 90° i s g e n e r a l l y w i again''-t h i n again''-t h e 90% con-f i d e n c e i n t e r v a l o con-f t h e e s t i m a t e . The average p e r r e c o r d o con-f t h e e s t i m a t e s i n t h e e n e r g e t i c f r e q u e n c y band d e v i a t e s g e n e r a l l y l e s s t h a n 2° f r o m 90°, r e c o r d 2601° a g a i n b e i n g t h e e x c e p t i o n , w i t h an average d e v i a t i o n o f about 8 ° , t h e same as was f o u n d f o r The s p e c t r a $^^7^^^ ™ ^ ^vw^"^"* g e n e r a l l y have no t r e n d w i t h f r e q u e n c y , i n c o n t r a s t t o T h i s f a c t , t o g e t h e r w i t h t h e absence o f any v i s i b l e i n f l u e n c e o f t h e p r i n c i p a l ' wave p r o p a g a t i o n d i r e c t i o n , i s e v i d e n c e t h a t t h e r e i s no s i g n i f i c a n t t i m e d e l a y i n t h e v e l o c i t y measurements i n d u c e d b y t h e v e l o c i t y s e n s o r s o r t h e i r sup-p o r t s t r u c t u r e . (The e l e c t r o n i c d e l a y s have been a c c o i m t e d f o r - see sup-p a r .
3 . 5 ) .
The r e s u l t s c o n c e r n i n g è ( f ) and (f) c o n f i r m t h e t h e o r e t i c a l l y p r e d i c t e d
uw -VW
q u a d r a t u r e r e l a t i o n between h o r i z o n t a l and v e r t i c a l v e l o c i t i e s . I t i s emphasized t h a t t h i s c o n c l u s i o n i s based on d a t a o b t a i n e d i n a w i d e range o f w i n d - and sea c o n d i t i o n s , i n c l u d i n g n\amerous cases o f a c t i v e wave gene-r a t i o n f o gene-r w h i c h some p gene-r e v i o u s s t u d i e s had g i v e n l a gene-r g e d e v i a t i o n s f gene-r o m q u a d r a t u r e , w i t h r e s u l t a n t e s t i m a t e d momentum f l u x e s w h i c h c o u l d n o t be b a l a n c e d by i d e n t i f i e d : : p h y s i c a l p r o c e s s e s .
_ 24
-C a v a l e r i ( 1 9 8 2 ) f o u n d a d i f f e r e n t phase r e l a t i o n between h o r i z o n t a l and v e r t i c a l v e l o c i t i e s .for s w e l l t t h a n f o r a w i n d d r i v e n s e a , b u t -- as n o t e d above -- t h e p r e s e n t d a t a do n o t show such b e h a v i o u r . An i l l \ i s t r a t i o n o f t h a t i s a f f o r d e d by a comparison o f t h e r e s u l t s o f r e c o r d 1501 ( s w e l l ; w i n d s p e e d =; 4 ms ; wave age 2.3) t o t h o s e o f r e c o r d 1601 (young sea; w i n d s p e e d - 17 ms ; wave age - 0 . 5 ) . T h e i r phase s p e c t r a ( A p p e n d i x 7) a r e n o t e s s e n t i a l l y d i f f e r e n t .
6.4 Gain s p e c t r a
The g a i n s p e c t r a E^^(f) and H ^ ~ ( f ) have been compared t o t h e l i n e a r -t h e o r e -t i c a l v a l u e s i n v a r i o -t i s ways. Values o f H ( f ) and H ( f ) have been p l o t t e d i n one g r a p h . These d a t a have been r e p l o t t e d as t h e r a t i o
R ( f ) E H ( f ) / H ( f ) . The same r a t i o s , t a k e n a t f = f ^ , have f o r a l l a n a l y z e d r e c o r d s been t a b u l a t e d . F i n a l l y , l o g a r i t h m i c p l o t s a r e g i v e n o f v a r i a n c e s p e c t r a o f v e r t i c a l v e l o c i t y and h o r i z o n t a l v e l o c i t y , as c a l c u l a t e d d i -r e c t l y f -r o m t h e v e l o c i t y measu-rements, compa-red t o t h o s e c a l c u l a t e d f -r o m t h e e s t i m a t e d s u r f a c e e l e v a t i o n s p e c t r u m u s i n g l i n e a r t h e o r y .
I n s p e c t i o n o f t h e f i g u r e s i n Appendix 4 and 8 shows t h a t i n most cases t h e t h e o r e t i c a l v a l u e s o f ( f ) a r e w i t h i n t h e 90% c o n f i d e n c e i n t e r v a l o f t h e e s t i m a t e , and t h a t H ^ ~ ( f ) ( f o r w h i c h no c o n f i d e n c e i n t e r v a l can be s p e c i f i e d ) i s g e n e r a l l y w i t h i n _+ 10% o f H^-.(f) i n t h e f r e q u e n c y i n t e r ¬ v a l on w h i c h Y > 0.9. The r e c o r d s 2420 and 2501 f o r m an e x c e p t i o n t o t h i s , i n p a r t i c i i l a r r e c o r d 2 5 0 1 . I t appears f r o m t h e f i g u r e s i n A p p e n d i x 5 and 9 t h a t t h e r a t i o R^^(f) shows no t r e n d w i t h f r e q u e n c y , whereas R^^^''^' ^'^ g e n e r a l shows some i n crease w i t h i n c r e a s i n g f . I t i s l i k e l y t h a t t h i s t r e n d i s due t o an i n -c r e a s e i n n o i s e / s i g n a l r a t i o i n t h e v e l o -c i t i e s w i t h i n -c r e a s i n g f r e q u e n -c y , because o f t u r b u l e n c e . I n t h i s c o n t e x t i t i s i m p o r t a n t t o n o t e t h a t H^-.(f) was c a l c v i l a t e d f r o m a u t o s p e c t r a l d e n s i t i e s . T h i s w o t i l d e x p l a i n why t h e t r e n d i s n o t p r e s e n t i n H ^ ^ ( f ) , w h i c h has been c a l c u l a t e d by c r o s s -s p e c t r a l a n a l y -s i -s , t h e r e b y e l i m i n a t i n g p o -s -s i b l e n o i -s e . F u r t h e r e v i d e n c e f o r t h i s v i e w i s a v a i l a b l e i n t h e v a r i a n c e S p e c t r a , shown i n A p p e n d i x 6 and 10. " ' . • ' ' : /
25
-Those f o r ü ( s t r i c t l y s p e a k i n g , S^^ + S ^ ) behave q u i t e s i m i l a r l y t o t h o s e f o r w, v i z . i n t h e e n e r g e t i c : ^ f r e q u e n c y band t h e r e i s by and l a r g e agreement between d i r e c t l y measured v e l o c i t y - s p e c t r a and t h o s e c a l c u l a t e d f r o m t h e s u r f a c e e l e v a t i o n , and an i n c r e a s i n g u n d e r e s t i m a -t i o n o f -t h e measured v e l o c i -t i e s w i -t h i n c r e a s i n g f r e q u e n c y . M o r e o v e r , t h i s u n d e r e s t i m a t i o n i s s i m i l a r between ü and w, as w o u l d be e x p e c t e d i f t h e d i f f e r e n c e w o i i l d be due t o t u r b u l e n c e . The t r e n d i n ^'^ t h e p r e s e n t d a t a i s o p p o s i t e t o t h e one f o u n d by F o r r i s t a l l e t a l . ( 1 9 7 8 ) . I t i s p o i n t e d o u t t h a t t h e t r e n d i n t h e i r d a t a cannot be a s c r i b e d t o h i g h - f r e q u e n c y t u r b u l e n c e ( n o r t o n o n l i n e a r bound h i g h e r h a r m o n i c s see t h e comments about t h i s i n t h e I n t r o d u c
-t i o n ) .
S i n c e t h e r a t i o ( f ) shows no t r e n d w i t h f , and R _ ~ ( f ) o n l y a weak
5W L,U one, t h e v a l u e s o f t h e s e r a t i o n s a t t h e peak f r e q u e n c y ('f = f ^ ) a r e more o r l e s s r e p r e s e n t a t i v e f o r each r e c o r d , a p a r t f r o m s a m p l i n g v a r i a -b i l i t y w i t h i n each r e c o r d . T h e i r v a l u e s a r e l i s t e d i n Appendix 1 1 f o r a l l t h e r e c o r d s . P l o t s o f t h e s e v a l u e s a g a i n s t wave s t e e p n e s s ( n o t shown h e r e ) showed t h a t t h e s e v a r i a b l e s were n o t c o r r e l a t e d , a l t h o u g h on t h e o -r e t i c a l g-rounds a c o -r -r e l a t i o n c o u l d be e x p e c t e d . The ave-rage and s t a n d a -r d d e v i a t i o n o f R ( f ) , R ^ ~ ( f ) and R ~ ( f ) , as d e t e r m i n e d f r o m a l l a n a l y z e d ?w m m wu m ' r e c o r d s ( a t o t a l number i n d i c a t e d by n ) , a r e l i s t e d i n T a b l e B. n Av. S t . d e v . R^ ( f ) 37 1.00 0.08 R ^ ~ ( f ) Cu m 37 0.95 0.05 R ~ ( f ) wu m 60 0.96 0.05 T a b l e B. These v a l u e s s u g g e s t e x c e l l e n t agreement w i t h t h e l i n e a r t h e o r y f o r t h e v e r t i c a l v e l o c i t i e s and a s y s t e m a t i c o v e r p r e d i c t i o n o f t h e h o r i z o n t a l v e l o c i t i e s by an e s t i m a t e d 5%. However, i t s h o u l d be remembered t h a t t h e
26
-c a l i b r a t i o t i may be s y s t e m a t i -c a l l y + S l T i j f f . , so t h a t i t ' w o u l d be b e t t e r , t o s a y - t h a t on t h e average t h e d a t a a r e cons i s t e n t " w i t h t h e t h e o r y t o ' w i t h i n a m a r g i n o f +'5%. ' .
-6.5 P r o b a b i l i t y d e n s i t i e s o f i n s t a n t a n e o u s v a l u e s
Sample h i s t o g r a m s o f u , v , w and C o f r e c o r d s 1501 and 1601 a r e p r e s e n t e d i n A p p e n d i x 12. These r e c o r d s are o f a l o w s w e l l i n a v e r y l i g h t w i n d (wave age 3 = 2.26, steepness y - 0.017, V 4 ms , see A p p e n d i x 3) and o f a young sea i n a s t r o n g w i n d (3 - 0.54, y - 0.0 37, 17 ms r e s p e c t i v e l y . The c o r r e s p o n d i n g l e a s t s q u a r e s b e s t f i t t e d Gaussian p r o b -a b i l i t y d e n s i t y f u n c t i o n s h-ave been p l o t t e d f o r comp-arison..
I t has been n o t e d above ( p a r . 4.6) t h a t a l l sampled v a l u e s o f each r e -c o r d a t i n t e r v a l s o f 0.25 s have been i n -c l u d e d i n t h e h i s t o g r a m s , so t h a t t h e samples a r e h i g h l y dependent. For such cases, no m e a n i n g f u l , q u a n t i t a t i v e t e s t o f g o o d n e s s - o f - f i t seems t o have been d e v e l o p e d . Only a v i s u a l i n s p e c t i o n has been c a r r i e d o u t . T h i s i n d i c a t e s an a c c e p t a b l e f i t i n a l l cases. I t i s n o t e w o r t h y t h a t t h e r e i s no o b v i o u s d i f f e r e n c e i n t h i s r e s p e c t between t h e s w e l l r e c o r d s (1501) and t h o s e t a k e n i n con-d i t i o n s o f a c t i v e wave g e n e r a t i o n ( 1 6 0 1 ) .
27
-7. SUMMARY AND CONCLUSIONS
Measurements have been p e r f o r m e d o f s u r f a c e e l e v a t i o n and i n t e r n a l v e l o c i t i e s i n t h e f r e q u e n c y range o f w i n d g e n e r a t e d waves a t a l o c a -t i o n i n -t h e s o u -t h e r n N o r -t h Sea, 10 km o f f -t h e Du-tch c o a s -t , where -t h e depth i s about 17 m. V e l o c i t i e s were measured a t two d i f f e r e n t e l e v a -t i o n s , v i z . a p p r o x i m a -t e l y 9 m and 12 m above -t h e s e a bed. The d a -t a c o v e r a w i d e range o f c o n d i t i o n s o f w i n d and s e a , f r o m young sea t o o l d s w e l l , i n c l u d i n g m i x e d cases. S i g n i f i c a n t wave h e i g h t s r e a c h e d -v a l u e s up t o 3.7m, whereas t h e most e n e r g e t i c f r e q u e n c y band t y p i c a l l y e x t e n d e d f r o m 0.1 Hz t o 0.2 Hz.
The d a t a have been a n a l y z e d m a i n l y w i t h r e s p e c t t o s p e c t r a l t r a n s f e r f u n c t i o n s , i n an e f f o r t t o v e r i f y l i n e a r t h e o r y p r e d i c t i o n s .
The p r e c e d i n g c h a p t e r has d e a l t w i t h r e s u l t s f o r v a r i o u s a s p e c t s sepa-r a t e l y , and i n some cases w i t h sepa-r e s u l t s o f i n d i v i d u a l sepa-r e c o sepa-r d s i f t h e y were seen t o d e v i a t e f r o m t h e g e n e r a l p a t t e r n . T h i s p r o c e d u r e g i v e s p e r h a p s undue w e i g h t t o e x c e p t i o n a l cases. A more g l o b a l p r e s e n t a t i o n
i s g i v e n h e r e , w h i c h f o c u s s e s on t h e o v e r a l l p i c t u r e e m e r g i n g f r o m t h e r e s u l t s .
I n t h e summary o f t h e r e s u l t s , a d i s t i n c t i o n i s made between r e l a t i o n s o f t h e v e r t i c a l v e l o c i t y t o s u r f a c e e l e v a t i o n , t h o s e o f t h e h o r i z o n t a l v e l o c i t y t o s u r f a c e e l e v a t i o n , and t h o s e between t h e h o r i z o n t a l v e l o c i t y and t h e v e r t i c a l v e l o c i t y . These cases a r e a f f e c t e d d i f f e r e n t l y b y
d i r e c t i o n a l s p r e a d i n g , n o i s e and c a l i b r a t i o n e r r o r s .
R e l a t i o n between v e r t i c a l v e l o c i t y and s u r f a c e e l e v a t i o n
The r e s u l t s i n t h i s c a t e g o r y can be a f f e c t e d b y c a l i b r a t i o n e r r o r s i n t h e wave gauge and i n t h e v e l o c i t y m e t e r s , w i t h a p o s s i b l e , s y s t e m a t i c e r r o r i n t h e e s t i m a t e d t r a n s f e r f u n c t i o n w h i c h i s e s t i m a t e d t o be l e s s t h a n 4° i n phase and 5% i n g a i n . There i s t h e o r e t i c a l l y no b i a s due t o d i r e c t i o n a l s p r e a d i n g o f wave energy o r n o i s e , E s t i m a t i o n o f s a m p l i n g v a r i a b i l i t y i s p o s s i b l e b y s t a n d a r d methods.
