Statistical description of ocean waves

>J1 O ^ O O c» S T A T I S T I C A L D E S C R I P T I O N • F • C E A N W A V E S u s e r s n o t e s c o m p i l e d by I r . E. A l l e r s m a W.W. Massie. P.E. f o r l e c t u r e s g i v e n by P r o f . d r . i r . E.W. B i j k e r U) OD O Ul • e l f t U n i v e r s i t y o f Technology D e l f t , The N e t h e r l a n d s F e b r u a r y , 1973 BIBLIOTHEEK TU Delft P 1855 1350 C 580574

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

1 . I n t r o d u c t i o n Page 1

1 . 1 The Phenomena 1

1 . 2 S t a t i s t i c a l Models 3

1 . 3 Observations,- t h e i r P r o c e s s i n g and Use 5

2 . P o p u l a t i o n s o f Water L e v e l s , 7

3 . Time S e r i e s o f Water L e v e l s g

4 . Energy and Energy D e n s i t y Spectrum 1 3

4 . 1 A p p l i c a t i o n t o Ocean Waves 1 6

5 . P o p u l a t i o n s o f Wave H e i g h t s 1 7

6 . P o p u l a t i o n s o f Wave P e r i o d s 3 2

7. Wave L e n g t h 34

8. Wave D i r e c t i o n and Wave C r e s t s 35

8.1 D i r e c t i o n 35 8.2 C r e s t s 35 9. P o p u l a t i o n s o f Sea C o n d i t i o n s 3B 9.1 H e i g h t 37 9 . 2 P e r i o d 4Q 9 . 3 H e i g h t and P e r i o d 4 0 9 . 4 D i r e c t i o n 43 9 . 5 D i r e c t i o n , H e i g h t and P e r i o d 43 9 . 6 Sources o f I n f o r m a t i o n 43 1 0 . L i t e r a t u r e 44 1 0 . 1 G e n e r a l 44 1 0 . 2 S t a t i s t i c s 44 1 0 . 3 Time S e r i e s 45 1 0 . 4 Energy and S p e c t r a 45 1 0 . 5 Wave H e i g h t s 4 6

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s t a t i s t i c a l D e s c r i p t i o n o f Ocean Waves

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

T h i s s e c t i o n i s i n t e n d e d t o p r o v i d e an o v e r - v i e w o f t h e s t a t i s t i c a l methods used t o d e s c r i b e and c l a s s i f y ocean waves. S i n c e t h i s o v e r - v i e w i s i n t e n d e d p r i m a r i l y f o r t h e c o a s t a l e n g i n e e r i n g u s e r s , most d e r i v a t i o n s a r e s k i p p e d ; t h e s e may be f o u n d i n o t h e r l i t e r a t u r e . We r e s t r i c t o u r s e l v e s , h e r e , more t o s t a t i s t i c a l models, t h e i r i n t e r r e l a t i o n s h i p s and t h e i r use.

1.1 The phenomenon •" An o b s e r v e r v i e w i n g t h e ocean w i l l f i n d t h a t i t u s u a l l y has a v e r y i r r e g u l a r wavy s u r f a c e . T h i s s u r f a c e i s n e v e r s t a t i o n a r y ! i t v a r i e s w i t h t i m e . Waves can be o b s e r v e s t r a v e l l i n g a c r o s s i t i n , p e r h a p s , s e v e r a l d i f f e r e n t d i r e c t i o n s . I t w o u l d , o f c o u r s e , be p o s s i b l e t o p r e c i s e l y r e c o r d t h i s s u r f a c e by u s i n g m o t i o n p i c t u r e methods. Each frame o f f i l m ( o r s e t o f s t e r e o frames made a t t h e same i n s t a n t ) w o u l d c o m p l e t e l y d e s c r i b e t h e s i t u a t i o n a t t h a t i n s t a n t i n a p a r t i c u l a r a r e a . W h i l e such a method w o u l d p r o v i d e a l o t o f I n f o r m a t i o n , we w o u l d q u i c k l y be b u r i e d i n an a v a l a n c h e o f p h o t o s . We must seek a more u s a b l e , b u t p e r h a p s more l i m i t e d s o l u t i o n .

A much more p r a c t i c a l method m i g h t be t o make a c o n t i n u o u s r e c o r d of t h e w a t e r s u r f a c e e l e v a t i o n a t one f i x e d p o i n t . We w o u l d now have a r e c o r d

( g r a p h , m a g n e t i c t a p e , e t c . ) o f t h e w a t e r s u r f a c e e l e v a t i o n a t o u r p o i n t as a f u n c t i o n o f t i m e . We have s a c r i f i c e d t h e s p a t i a l i n f o r m a t i o n o f t h e p h o t o s , b u t have o b t a i n e d a r e c o r d w i t h w h i c h we can s t i l l work. We have, however, l o s t d i r e c t o i n a l i n f o r m a t i o n . Even so, we can s t i l l d e r i v e much i n f o r m a t i o n f r o m t h i s r e c o r d . V i a known p h y s i c a l models, we can g e t an i n d i c a t i o n o f wave c o n d i t i o n s a t a n o t h e r nearby p o i n t .

t o t h e d i r e c t i o n o f wave p r o p a g a t i o n . I n t h e d i r e c t i o n o f p r o p a g a t i o n , t h e r e i s a f i n i t e d i s t a n c e between wave c r e s t s , t h e w a v e l e n g t h , L.

. ^ U n f o r t u n a t e l y , o u r p o i n t r e c o r d c a n n o t r e c o r d t h e s e s p a t i a l p r o p e r -t i e s . I n s -t e a d , we do g e -t a measure o f -t h e i n d i v i d u a l wave h e i g h -t s and -t h e i r p e r i o d s , t h e t i m e between passage o f s u c c e s s i v e c r e s t s . L u c k i l y , we have a s i m p l e p h y s i c a l r e l a t i o n s h i p between wave l e n g t h and wave p e r i o d .

I f we w i s h t o g e t i n f o r m a t i o n about t h e wave d i r e c t i o n and t h e l e n g t h o f wave c r e s t s , t h e n we must use s e v e r a l wave gauges s i m u l t a n e o u s l y . A w e l l c h o s e n c o n f i g u r a t i o n f o t h r e e m e a s u r i n g u n i t s can y i e l d t h i s i n f o r m a -t i o n .

F u r t h e r i n v e s t i g a t i o n o f t h e c h a r a c t e r o f t h e sea s u r f a c e shows t h a t t h e waves come i n g r o u p s . T h i s shows up i n a wave r e c o r d as a slow v a r i a t i o n i n a m p l i t u d e ( b e a t ) as i f caused by t h e m u t u a l i n t e r f e r e n c e o f two waves h a v i n g s l i g h t l y d i f f e r e n t p e r i o d . Qne can o b s e r v e t h i s phenomena on a beach. There seems t o be a m o r e - o r - l e s s r e g u l a r v a r i a t i o n i n wave h e i g h t s . A common s a y i n g i s t h a t e v e r y s e v e n t h wave i s t h e l a r g e s t . T h i s may g i v e a c r u d e i n d i c a t i o n o f t h e s i z e o f a g r o u p , b u t one who r e c k o n s on t h i s w i l l g e t wet s o o n e r o r l a t e r . '

Sea waves a r e caused by w i n d s . Wind f i e l d s o f low p r e s s u r e a r e a s , monsoons, h u r r i c a n e s , e t c . , a r e o f v a r i a b l e s t r e n g t h , d u r a t i o n and d i r e c t i o n . They a r e a l s o l i m i t e d i n e x t e n t ; t h e y c o v e r o n l y a l i m i t e d area o f t h e sea. Because o f t h e s e i r r e g u l a r i t i e s i n t h e w i n d , t h e waves g e n e r a t e d and s e n t o u t

( r a d i a t e d ) a r e a l s o i r r e g u l a r . Each w i n d f i e l d g e n e r a t e s and r a d i a t e s a p a t -t e r n o f waves w h i c h , a -t a d i s -t a n c e f r o m -t h e s o u r c e , appear as -t r a i n s o f waves. The f i r s t waves t o a r r i v e a r e t h e l o n g ( f a s t t r a v e l l i n g ) low components. The

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wave p e r i o d ( l e n g t h ) g r a d u a l l y decreases w h i l e t h e wave h e i g h t f i r s t shows an i n c r e a s e t o w a r d s a maximum a f t e r w h i c h t h e wave a c t i o n d i e s out g r a d u a l l y

(Thompson, 1 9 7 0 ) . The p a s s i n g o f a wave t r a i n can t a k e a few hours t o s e v e r a l days depending on t h e s i z e and t h e d i s t a n c e t o t h e g e n e r a t i n g w i n d f i e l d .

I t i s p o s s i b l e t h a t one o b s e r v e s a t some p o i n t t h e s u p e r - p o s i t i o n o f i n d e p e n d e n t wave p a t t e r n s g e n e r a t e d by w i d e l y s e p a r a t e d s t o r m s . These over-l a p p i n g p a t t e r n s , o f t e n c o n s i s t o f over-l o n g s w e over-l over-l s f r o m an o over-l d , d i s t a n t s t o r m , w i t h a s u p e r i m p o s e d t r a i n o f y o u n g e r , s h o r t e r , s t e e p e r waves f r o m a l o c a l s t o r m . The l i f e t i m e o f wave t r a i n s can v a r y f r o m a few h o u r s t o s e v e r a l days. Wave t r a i n s o r i g i n a t i n g i n t h e A n t a r c t i c Ocean have been o b s e r v e d t o t r a v e l more t h a n h a l f way around t h e e a r t h .

L a s t l y , on a s t i l l g r e a t e r t i m e s c a l e , s e a s o n a l v a r i a t i o n s i n t h e sea a r e common.

1.2 S t a t i s t i c a l Models

There a r e two d e c i d e d l y d i f f e r e n t a s p e c t s t o c h a r a c t e r i z i n g waves i n a s t a t i s t i c a l way. These two i n d e p e n d e n t a s p e c t s a r e :

a. The d e t e r m i n a t i o n o f t h e v a l u e s o f t h e s t a t i s t i c a l v a r i a b l e s n e c e s s a r y t o d e s c r i b e t h e wave s i t u a t i o n a t some t i m e , and

b. The d e t e r m i n a t i o n o f t h e f r e q u e n c y w i t h w h i c h t h e s e c o n d i t i o n s o c c u r . T h i s second may be viewed as t h e d e t e r m i n a t i o n o f t h e p r o b a b i l i t y w i t h w h i c h a g i v e n sea c o n d i t i o n o c c u r s .

G e n e r a l l y , knowledge o f t h e above i n f o r m a t i o n a l o n g w i t h a p h y s i c a l model o f t h e waves i s n e c e s s a r y i f one i s t o make a t e c h n i c a l and economic e v a l u a t i o n o f a g i v e n d e s i g n .

B 3 f o r e c o n t i n u i n g t h i s d i s c u s s i o n i t w i l l be p r a c t i c a l t o i n t r o d u c e an i m p o r t a n t d e f i n i t i o n o f t h e t e r m p o i n t o f t i m e . A " p o i n t o f timd' i n t h e sense i n w h i c h i t i s used h e r e i n f a c t i s a t i m e i n t e r v a l s a t i s f y i n g t h e f o i

-l o w i n g two c o n t r a d i c t o r y c o n d i t i o n s :

1. The t i m e i n t e r v a l i s s u f f i c i e n t l y l o n g t h a t a r e p r e s e n t a t i v e sample o f t h e measured waves (wave c o n d i t i o n s ) can be o b t a i n e d i n a s i n g l e i n t e r v a l . Thus, t h i s t i m e i n t e r v a l must c o n t a i n s e v e r a l wave groups and must a l l o w s u f f i c i e n t l y a c c u r a t e s t a t i s t i c a l p r o p e r t i e s t o be d e t e r m i n e d .

2. The t i m e i n t e r v a l must be s u f f i c i e n t l y s h o r t t h a t t h e waves w i t h i n t h e i n t e r v a l may be c o n s i d e r e d t o be a s t a t i s t i c a l l y s t a t i o n a r y p r o c e s s . T h i s means t h a t t h e s t a t i s t i c a l c h a r a c t e r i z a t i o n s remain c o n s t a n t w i t h i n t h e i n t e r v a l . Thus, a t i m e i n t e r v a l must be s h o r t e r t h a n t h e d u r a t i o n o f a wave t r a i n .

I n p r a c t i c e , a t i m e i n t e r v a l d u r i n g w h i c h t h e waves a r e r e c o r d e d c o n t a i n s a few hundred waves and l a s t s somewhere between f i f t e e n m i n u t e s and one hour. From t h i s r e c o r d t h e c h a r a c t e r i s t i c p r o p e r t i e s o f t h e wave c o n d i t i o n s a t a c e r t a i n " p o i n t o f t i m e " ( t h e m i d d l e o f t h e t i m e I n t e r v a l ) can be d e r i v e d .

There a r e v a r i o u s s t a t i s t i c a l models a v a i l a b l e t o us. The c h o i c e among them depends more o r l e s s on t h e t i m e s c a l e o f o u r e n t i r e s e t o f ob-s e r v a t i o n ob-s . The f o l l o w i n g f o u r m o d e l l i n g mehtodob-s a r e f o u n d :

a. P o p u l a t i o n o f Water L e v e l s w i t h i n a r e c o r d .

These a r e s e t s o f i n s t a n t a n e o u s w a t e r l e v e l r e a d i n g u n r e l a t e d t o one a n o t h e r . From t h e s e we can d e t e r m i n e t h e a v e r a g e , s t a n d a r d d e v i a t i o n , e t c . b. Time S e r i e s o f a r e c o r d .

From t h e s e more d e t a i l e d r e c o r d s , one can o b t a i n t h e a u t o - c o r r e l a t i o n f u n c t i o n , t h e e n e r g y d e n s i t y s p e c t r u m , t h e t o t a l e n e r g y , a t e .

c. P o p u l a t i o n o f Waves w i t h i n a r e c o r d .

The h e i g h t s and p e r i o d s o f t h e waves i n a s e r i e s may be d e t e r m i n e d and r a n k e d . Here, a l s o , we can compute t h e a v e r a g e , s t a n d a r d d e v i a t i o n , o r s i g n i f i c a n t wave h e i g h t .

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d. P o p u l a t i o n o f wave c o n d i t i o n s .

Each o f t h e t h r e e f o r e g o i n g models d e s c r i b e s t h e wave c o n d i t i o n s a t a " p o i n t o f t i m e " w i t h t h e v a l u e o f one o r more p a r a m e t e r s . T h i s can be r e p e a t e d a t r e g u l a r i n t e r v a l s (a few h o u r s ) d u r i n g l o n g p e r i o d s ; p r e f e r -a b l y one o r more y e -a r s . Thus -a p o p u l -a t i o n o f c h -a r -a c t e r i s t i c p -a r -a m e t e r s o f wave c o n d i t i n n s i s o b t a i n e d w h i c h , i n t u r n , can be s u b j e c t t o s t a t i s t i c a l e l a b o r a t i o n s y i e l d i n g an a v e r a g e , f r e q u e n c i e s o f exceedance, s e a s o n a l f l u c t u a t i o n s , e t c .

The c h o i c e o f w h i c h model t o use depends upon t h e d a t a a v a i l a b l e and t h e d a t a needed.

1•3 O b s e r v a t i o n s ; t h e i r p r o c e s s i n g and use

Much o f t h e a v a i l a b l e i n f o r m a t i o n on sea waves has been d e r i v e d f r o m o b s e r v a t i o n s made f r o m s h i p s t o g e t h e r w i t h t h e w e a t h e r r e p o r t . These o b s e r v a -t i o n s u s u a l l y i n c l u d e e s -t i m a -t e s o f -t h e wave h e i g h -t , p e r i o d and d i r e c -t i o n as w e l l as a n o t a t i o n o f t h e l o c a t i o n and t i m e . They a r e c o l l e c t e d , s t a t i s t i -c a l l y p r o -c e s s e d and p u b l i s h e d by v a r i o u s m e t e o r o l o g i -c a l o r h y d r o g r a p h i -c a g e n c i e s . P u b l i c a t i o n s a r e u s u a l l y i n t h e f o r m o f a t l a s e s . O f t e n , more det a i l e d and more r e c e n det d a det a i s a v a i l a b l e on s p e c i f i c r e q u e s det f r o m det h e c o l l e c -t i n g agency ( e . g . -t h e K.N.M.I.). T h i s method o f v i s u a l o b s e r v a t i o n s i s a l s o used a l o n g c o a s t s . A more dependable wave h e i g h t r e c o r d can be o b t a i n e d i f t h e wave h e i g h t s a r e measured u s i n g a f i x e d s c a l e , however.

Measurements u s i n g more s o p h i s t i c a t e d i n s t r u m e n t s a r e , as y e t , o n l y made i n l i m i t e d areas. These r e s u l t s a r e o f t e n o n l y p u b l i s h e d t o a l i m i t e d e x t e n t .

Most o f t h e s p e c i a l wavemeasuring i n s t r u m e n t s r e c o r d t h e i r i n f o r -m a t i o n i n a f o r -m -most s u i t a b l e f o r t h e chosen p r o c e s s i n g -method w h i c h i s t o

f o l l o w . T h i s p r o c e s s i n g may be done by hand o r can be a c c o m p l i s h e d w i t h t h e a i d o f e i t h e r a n a l o g o r d i g i t a l computers.

Now t h a t we have t h e o b s e r v a t i o n s ( o r s t a t i s t i c a l i n f o r m a t i o n d e r i v e d f r o m t h e m ) , how a r e we g o i n g t o use t h i s data? The d e s i g n e r o f some o b j e c t t o be exposed t o waves i s u s u a l l y n o t i n t e r e s t e d i n t h e waves t h e m s e l ves. I n s t e a d , he i s i n t e r e s t e d on t h e e f f e c t s o f waves on h i s d e s i g n e d p r o -j e c t . T h e r e f o r e , he must f i r s t use some m a t h e m a t i c a l o r p h y s i c a l model t o t r a n s l a t e t h e wave i n f o r m a t i o n i n t o i n f o r m a t i o n about t h e e f f e c t s t h e s e waves have on h i s c o n s t r u c t i o n . Then, s e c o n d l y , t h e s t a t i s t i c s g i v e s h i m an i n d i c a t i o n o f how o f t e n a c e r t a i n wave c o n d i t i o n has o c c u r r e d a t h i s l o c a t i o n i n t h e p a s t . He must hope t h a t c o n d i t i o n s w i l l remain e s s e n t i a l l y t h e same i n t h e f u t u r e .

W i t h t h e s e s t e p s , t h e d e s i g n e r can e s t i m a t e a q u a n t i t a t i v e d e s i g n c r i t e r i a and t h e chance ( p r o b a b i l i t y ) t h a t t h i s c r i t e r i a w i l l be exceeded. Examples o f t h e s e d e s l g h c r e t i e r a i n c l u d e :

wave run-up on shores and s l o p e s o v e r t o p p i n g o f s t r u c t u r e s by waves l o n g s h o r e c u r r e n t s

l o n g s h o r e s e d i m e n t t r a n s p o r t

areas and i n t e n s i t y o f c o a s t a l e r o s i o n and d e p o s i t i o n i n t e r n a l s t r e s s e s i n s t r u c t u r e s

damage t o s t r u c t u r e s

d e l a y s t o s h i p p i n g t r a n s p o r t v i b r a t i o n s o f s t r u c t u r e s .

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2, P o p u l a t i o n s o f Water L e v e l s

I n t h i s s e c t i o n we s h a l l d i s c u s s t h e f i r s t o f t h e s t a t i s t i c a l models p r e s e n t e d e a r l i e r . Water l e v e l s may be t a b u l a t e d by s i m p l y n o t i n g t h e i n s t a n t a n e o u s w a t e r l e v e l a t t h e end o f r e g u l a r l y - r e c u r r i n g t i m e i n t e r v a l s (sample i n t e r v a l s ) A t . We can d e n o t e such an i n d i v i d u a l o b s e r v a t i o n by r\^. We c o l l e c t a t o t a l o f N o b s e r v a t i o n s d u r i n g o u r o b s e r v a t i o n p e r i o d ( r e c o r d i n t e r v a l ) l a s t i n g N A t .

As a b e g i n n i n g s t e p , we can compute t h e average w a t e r l e v e l a t t h e " t i m e p o i n t " : ^

n = 1

The waves cause t h e a c t u a l w a t e r l e v e l o b s e r v a t i o n s t o v a r y about t h i s a v e r -age, o r mean v a l u e .

Over a l o n g e r t i m e , t h e v a l u e o f ri w i l l v a r y as a r e s u l t o f t i d a l o r m e t e o r o l o g i c a l e f f e c t s . O f t e n , we assume t h a t o u r o b s e r v a t i o n p e r i o d , N A t , i s s t i l l s h o r t enough so t h a t t h e s e e f f e c t s may be i g n o r e d . Thus, t h e t i m e i n t e r v a l . A t , used here w i l l p r o b a b l y be q u i t e s h o r t and on t h e o r d e r o f a second. I f t i d a l e f f e c t s may n o t be i g n o r e d because our r e c o r d e x t e n d s o v e r t o o l o n g a t i m e , t h e n we may, i n s t e a d , d e t e r m i n e n as a f u n c t i o n o f t i m e , n ( t ) , u s i n g a n o t h e r source o f d a t a such as a t i d e gauge.

We a r e now a b l e t o d e t e r m i n e t h e i n s t a n t a n e o u s v a r i a t i o n i n t h e w a t e r l e v e l caused by t h e waves. T h i s i s

y^ = - n ( 2 )

From t h i s , we can compute t h e s t a n d a r d d e v i a t i o n

' n = 1

I t has been f o u n d f r o m o b s e r v a t i o n t h a t t h e d i s t r i b u t i o n o f t h e v a l u e s o f t h e s e i n d i v i d u a l Water l e v e l s , y^, measured f r o m t h e mean, seem t o

s a t i s f y t h e normal p r o b a b i l i t y d i s t r i b u t i o n . We have a l s o assumed, now t h a t we have made o u r measurements i n deep w a t e r . T h i s normal d i s t r i b u t i o n i s

/ \2 g i v e n by: _ V .

,,,, = l F i y l = _ _ l _ e ( 4 )

y

From t h i s 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 , we can compute t h e p r o b a b i l i t y t h a t a g i v e n w a t e r l e v e l , y, i s exceeded: F ( y ) = f ( y ) dy ( 5 ) ' y R e p r e s e n t a t i v e v a l u e s a r e shown i n t a b l e 1. y F ( y ] a 0.16 y 20 0.025 y 3a 0.0025 y

T a b l e 1: Values o f F ( y ) f o r a normal d i s t r i b u t i o n computed u s i n g e q u a t i o n s 4 and 5. S i m i l a r l y , we c o u l d compute t h e p r o b a b i l i t y t h a t t h e l e v e l i s l o w e r t h a n a c e r t a i n w a t e r l e v e l , e t c . S i n c e we chose o u r t i m e i n t e r v a l A t t o be c o n s t a n t , we can d i r e c t l y e s t i m a t e t h e t o t a l t i m e d u r i n g w h i c h a c o n d i t i o n w i l l o c c u r . T h i s t o t a l w i l l most l i k e l y be made up o f a s e r i e s o f s h o r t i n t e r v a l s . T h e o r e t i c a l l y , we must p e r f o r m t h e s e c o m p u t a t i o n s w i t h an i n f i n i t e number o f v a l u e s y ( N -> » ) . We can g e t an i m p r e s s i o n o f t h e a c c u r a c y o f •'n o u r computed v a l u e s o f and a , r e s u l t i n g f r o m t h e f a c t t h a t N i s f i n i t e , ^ a f r o m t h e f a c t s t h a t t h e e r r o r o f 7) i s e q u a l t o and t h a t t h e e r r o r o f a a i s e q u a l t o T^ÏT '

9

3. Time S e r i e s o f w a t e r l e v e l s

I n t h e a n a l y s i s o f t h e p r e v i o u s s e c t i o n , we have i g n o r e d t h e f a c t t h a t t h e sequence o f w a t e r l e v e l s more o r l e s s f o l l o w s some p a t t e r n . We mean, h e r e , t h a t i t i s v e r y l i k e l y t h a t V^^^^ i s n o t t o o d r a s t i c l y d i f f e r e n t f r o m y^. T h i s i s c e r t a i n l y t r u e i f o u r t i m e i n t e r v a l . A t , i s s m a l l r e l a t i v e t o t h e

wave p e r i o d , T.

A b e t t e r method, w h i c h does n o t I g n o r e t h i s e s s e n t i a l I n f o r m a t i o n m i g h t be t o make a c o n t i n u o u s r e c o r d o f t h e w a t e r l e v e l as a f u n c t i o n o f t i m e , n C t ) . F i g u r e 1 shows examples o f such r e c o r d s . We can s t i l l sample t h i s

r e c o r d a t t i m e i n t e r v a l s At i n o r d e r t o , a g a i n , g e n e r a t e n^. T h i s t i m e , how-e v how-e r , whow-e s h a l l bhow-e c a r how-e f u l t o khow-ehow-ep o u r N v a l u e s i n t h e i r p r o p e r o r d e r w i t h r e s p e c t t o t i m e . We have b o t h a c o n t i n u o u s r e c o r d and a s e t o f d i s t i n c t v a l u e s . E i t h e r may be used i n f u r t h e r c o m p u t a t i o n s . The a v e r a g e w a t e r l e v e l comes f r o m : N 1 ^ = 0 n Nn e t ] dt = ^ E n„ , n ( 6 ) 0 n = 1 where 0 = N A t , t h e d u r a t i o n o f t h e r e c o r d . A l s o , as b e f o r e : y ( t ) = n ( t ] - n C7a) and: y^ = - n ' (7b)

The f a c t t h a t t h e w a t e r l e v e l v a r i e s i n a c o n t i n u o u s way ,has a l -ready been i n d i c a t e d . I n d e e d , t h e r e i s a c o r r e l a t i o n between s e c c e s s i v e o c c u r r e n c e s , t h e waves have a c e r t a i n degree o f p e r i o d i c i t y . A measure o f t h i s p r o p e r t y can be o b t a i n e d f r o m t h e a u t o c o v a r i a n c e f u n c t i o n . To g e n e r a t e t h i s f u n c t i o n we m u l t i p l y two v a l u e s o f w a t e r s u r f a c e e l e v a t i o n s e p a r a t e d by a t i m e d i f f e r e n c e T and t h e n t a k e an a v e r a g e o f a l l o f t h e s e p r o d u c t s . I n e q u a t i o n f o r m : y C t ) y ( t + T ) d t ( 8 a ) 1 ^® R(x) = 1 0

o r : R ( T : _N

n=1 + v

(8b)

F i g u r e 1 EXAMPLES OF CHARACTERISTIC RECORDS WITH THEIR AUTOCOVARIANCE

FUNCTIONS AND SPECTRA.

Comparison o f e q u a t i o n s ( 3 ) and ( 8 b ] i n d i c a t s t h a t <?y = 7 R ( • ] ' ( 9 ) The a u t o c o v a r i a n c e f u n c t i o n g i v e s an i n s i g h t t o t h e c h a r a c t e r o f o u r t i m e s e r i e s . Two l i m i t i n g cases s h o u l d be d i s c u s s e d : a. I f o u r t i m e s e r i e s c o n s i s t s o f a c o m p l e t e l y random p r o c e s s , sometimes c a l l e d w h i t e n o i s e , t h e n t h e a u t o c o v a r i a n c e f u n c t i o n reduces t o a d e l t a f u n c t i o n Ö ( T ) a t T = 0. ( R ( T ] H O f o r T 0, and R ( T ] -> « f o r X = 0 ) . b. I f , a t t h e o t h e r e x t r e m e , o u r t i m e s e r i e s i s a p u r e , p e r i o d i c , h a r m o n i c f u n c t i o n t h e n t h e a u t o c o v a r i a n c e f u n c t i o n has a s i m i l a r n i c e f o r m . ( I f 2 y = a s i n tot, t h e n R ( T ] = | - cos UT ] .

Real waves on t h e ocean s u r f a c e w i l l have an a u t o c o v a r i a n c e f u n c t i o n w h i c h f a l l s between t h e s e e x t r e m e s . R ( T ] o s c i l l a t e s about t h e x a x i s w i t h d e c r e a s -i n g a m p l -i t u d e as we move away f r o m x = G. ( s e e f -i g u r e 1 ] . T h -i s -i n d -i c a t e s t h a t t h e r e g u l a r i t y o f o u r ocean wave r e c o r d o n l y e x t e n d s o v e r a few wave p e r i o d s . The p e r i o d o f t h e a u t o c o v a r i a n c e f u n c t i o n agrees w i t h t h a t o f t h e waves.

Some p e o p l e use t h e a u t o c o r r e l a t i o n f u n c t i o n , R' ( X ] , i n p l a c e o f t h e a u t o c o v a r i a n c e f u n c t i o n . These f u n c t i o n s a r e r e l a t e d by:

R'^r^ - R ( T )

1 3

4. Energy and Energy D e n s i t y Spectrum

The energy c o n t a i n e d I n a wave o f wave l e n g t h L has been computed I n o t h e r n o t e s . For r e g u l a r waves h a v i n g a u n i t w i d t h ( c r e s t l e n g t h ) we g e t :

E^ = I p g L = ^ p g a' L - (11)

where a = ^ is t h e wave a m p l i t u d e (see t h e f i r s t s e c t i o n o f t h e n o t e s

" I n t r o d u c t i o n t o C o a s t a l E n g i n e e r i n g " ) . E q u a t i o n 1 1 , e x p r e s s e d , i n s t e a d , as energy p e r u n i t o f w a t e r s u r f a c e a r e a becomes:

E = I p g = I p g a^ (12) o 2

We remember t h a t t h i s e n e r g y , E, i s e q u a l l y d i v i d e d between K i n e t i c and p o t e n t i a l energy.

The above two r e l a t i o n s h i p s a r e w e l l known; t h e m s e l v e s , t h e y have n o t h i n g t o do w i t h s t a t i s t i c s . However, i f we n e g l e c t t h e p h y s i c a l c o n s t a n t s l n v o l v e d ( p and g) we g e t s o m e t h i n g t h a t l o o k s r a t h e r l i k e what t h e s t a t i s t i c s p e o p l e c a l l t h e t o t a l energy o f a s t a t i o n a r y p r o c e s s . B e i n g more s p e c i f i c , t h e y d e f i n e t h i s t o t a l energy as: rT 1 m = lim Y T-X» y ( t ) ] ' ^ d t (13) 0 w i t h ( 8 a ) and ( 9 ) we see t h a t : m = R(0) = (14) For a w e l l - b e h a v e d harmonic v a r i a b l e w i t h a m p l i t u d e a: r2Tr 1 = 2^ 2 . 2 , 1 2 a s i n q dq = •2 a 0 (15) I n d e e d , ( 1 5 ) l o o k s much l i k e (12) We can a s s o c i a t e a s p e c t r u m w i t h any s t a t i o n a r y s t a t i s t i c a l p r o c e s s j u s t as we do w i t h sound and l i g h t . We can t h i n k o f t h e phenomenon

t h e sum o f a l a r g e ( t h e o r e t i c a l l y i n f i n i t e ) number o f harmonic v a r i a b l e s . Each o f t h e s e v a r i a b l e s has i t s own a m p l i t u d e a^, f r e q u e n c y , w^, and an a r b i t r a r y phase a n g l e , (j)^. Expressed i n a f o r m u l a : N y ( t ) = V a cos (w t - (f) ) ( 1 6 ) ^ ^ n n ^n n = 1 w h i c h has a t o t a l e n e r g y : 1 ^ 2 m = 1 E a^ ( 1 7 ) n=1

Because o f d i f f e r e n c e s i n f r e q u e n c y and phase, i t i s i n c o r r e c t t o make a

t o t a l o f t h e a m p l i t u d e s , b u t s i n c e t h e energy o f a wave remains c o n s t a n t , t h e e n e r g i e s can be summed as i n ( 1 7 ) .

The energy d e n s i t y s p e c t r u m , S(a)), i s d i f l n e d as a f u n c t i o n o f t h e f r e q u e n c y , o), h a v i n g t h e p r o p e r t y t h a t :

1 a^ = S((o ) Au ( 1 8 )

2 n n

Thus, t h e energy d e n s i t y s p e c t r u m g i v e s an I m p r e s s i o n o f t h e d i s t r i b u t i o n o f t h e energy o v e r t h e v a r i o u s components w h i c h make up o u r wave p a t t e r n .

Example s p e c t r a a r e g i v e n i n f i g u r e 1 .

As t h e t o t a l number o f wave components, N, approaches i n f i n i t y , t h e n t h e i n t e r v a l AÜJ approaches z e r o . E q u a t i o n 1 8 now becomes:

m = 3( 0 ) ) dw ( 1 9 )

0

The a r e a under t h e s p e c t r u m g r a p h i s t h e t o t a l energy.

I t can be shown ( R i c e , 1 9 4 4 , 1 9 4 5 ) t h a t t h e energy d e n s i t y s p e c t r u m i s t h e same as t h e F o u r i e r t r a n s f o r m o f t h e a u t o c o v a r i a n c e f u n c t i o n .

3( 0 ) ) = I ^ R (T ) COS O)T d T ( 2 0 a )

/oo

and t h u s c o n v e r s l y : R( T ) 3( 0 ) ) cos ü)T dü3 (2üb)

1 5

T h i s r e l a t i o n s h i p can be handy when we must a c t u a l l y d e t e r m i n e t h e s p e c t r u m f r o m a wave r e c o r d . I t s h o u l d be p o i n t e d o u t t h a t o t h e r t e c h n i q u e s u s i n g s p e c i a l i z e d F o u r i e r t r a n s f o r m s can a l s o be used. One s h o u l d c o n s u l t r e c e n t l i t e r a t u r e f o r t h e l a t e s t d e v e l o p m e n t s .

I n r e a l i t y , no s p e c t r u m o f i n t e r e s t t o us w i l l c o n t a i n a l l f r e q u e n c i e s f r o m ÜÜ = 0 t o o) = ~. The w i d t h o f a s p e c t r u m , u s u a l l y c a l l e d s p e c t r a l w i d t h , o r band w i d t h , i s d e f i n e d as t h e range o f v a l u e s o f omega i n w h i c h most o f t h e energy i s c o n c e n t r a t e d . T h i s s p e c t r a l w i d t h , a l o n e , can i n d i c a t e much about t h e a c t u a l wave p a t t e r n . Compare t h e p a t t e r n s and s p e c t r a l w i d t h s i n f i g u r e 1 . I f o u r s p e c t r u m c o n s i s t s o f a s i n g l e s p i k e ( d e l t a ) f u n c -t i o n , -t h e n -t h e wave p a -t -t e r n i s p u r e h a r m o n i c . Our s p e c -t r u m c o u l d a l s o c o n s i s -t o f a f i n i t e number o f t h e s e s p i k e s . T h i s s p e c t r u m w o u l d l o o k l i k e t h e l i g h t s p e c t r u m o f , f o r example, a mercury v a p o r lamp. The p h y s i c a l wave p a t t e r n w o u l d c o n s i s t o f a f i n i t e number o f s u p e r i m p o s e d harmonic waves. The t i d e s

are an example o f t h i s s o r t o f s p e c t r u m w h i c h can be i m p o r t a n t t o us. A n a r r o w s p e c t r a l w i d t h ( n a r r o w band) c o r r e s p o n d s t o a n e a r l y p e r i o d i c o c c u r e n c e . A t t h e o t h e r e x t r e m e , w h i t e n o i s e has a u n i f o r m e n e r g y d i s t r i b u t i o n o v e r a t h e o r e t i c a l l y i n f i n i t e band w i d t h .

The band w i d t h can be d e s c r i b e d i n a m a t h e m a t i c a l way v i a t h e moment o f t h e s p e c t r u m a b o u t t h e a x i s u = 0 ,

^00 i

m. ü)-^ 3( 0 ) ) dü) (21 )

O

E q u a t i o n ( 2 1 ) y i e l d s t h e t o t a l energy when i = 0. From ( 2 1 ) we d e f i n e a measure f o r t h e s p e c t r u m w i d t h as: m.^ m - m^ g = 0 4 2 ( 2 2 ) m „ nn „ 0 4 T h i s p a r a m e t e r v a r i e s between 0 end 1 , i n c r e a s i n g w i t h t h e s p e c t r a l w i d t h (see C a r t w r i g h t and L o n g u e t - H l g g i n s , 1 9 5 6 ) .

A p p l i c a t i o n t o Ocean Waves

Wind g e n e r a t e d waves g r a d u a l l y d e v e l o p and i n c r e a s e t h e i r h e i g h t and l e n g t h as t h e y p r o g r e s s . O b v i o u s l y , t h e a s s o c i a t e d s p e c t r u m ( t o t a l e n e r g y ) a l s o I n c r e a s e s . The c e n t e r o f g r a v i t y o f t h e s p e c t r u m s h i f t s t o w a r d t h e

l o w e r f r e q u e n c i e s as t h e s p e c t r u m d e v e l o p e s ( s e e f i g u r e 2 ) . Neumann ( 1 9 5 3 ) , P i e r s o n and Moskowltz ( 1 9 6 4 ) , and Kinsman (1964) d e s c r i b e t h i s p r o c e s s i n d e t a i l .

F r e q u e n c y (a)/2Ti:)

F i g u r e 2 E N E R G Y D E N S I T Y S P E C T R A A C C O R D I N G TO N E U M A N N T H R E E D I F F E R E N T W I N D S P E E D S .

1 7

A f u l l y d e v e l o p e d sea i s one w h i c h i s i n e q u i l i b r i u m w i t h t h e w i n d ; t h e energy o f t h e waves remains c o n s t a n t . I n such a case, t h e s p e c t r u m , S((ü), g e n e r a t e d by a w i n d o f v e l o c i t y , W i s g i v e n by: 2 3 ( 0 ) ) = C " I 0 ) " ^ exp - ( 2 3 ) W Ü) 2 5 where C = 3.05 m /sec From t h i s , we d e r i v e : m = 3 4- j 2 ^ ^ j ( 2 4 )

E q u a t i o n s ( 2 3 ) and ( 2 4 ) a r e v a l i d o n l y w h i l e t h e wave remains w i t h i n t h e w i n d f i e l d . The s p e c t r u m o f s w e l l (waves o u t s i d e t h e i r g e n e r a t i n g w i n d f i e l d ) i s o f t e n q u i t e n a r r o w . T h i s i s shown i n f i g u r e 3.

5. P o p u l a t i o n s o f wave h e i g h t s

The wave h e i g h t i s d e f i n e d as t h e v e r t i c a l d i s t a n c e between suc-c e s s i v e suc-c r e s t s and t r o u g h s o f t h e wave. I n d i v i d u a l v a l v e s o f t h e wave h e i g h t may be d e n o t e d by H and a r e a l w a y s p o s i t i v e (see f i g u r e 4 ) .

I . I . I 1 = n

50 25 20 15 10 8 6

Figure 3 SPECTRA OF WAVES OBSERVED ON 9 SUCCESSIVE DAYS

AT ABOUT 6000 KM FROM THE SOURCE A R E A .

3}

I

m

m

m

Given a s e t o f v a l v e s o f we can make a f r e q u e n c y d i s t r i b u t i o n , f ( H ) o r even a c u m u l a t i v e d i s t r i b u t i o n . F ( H ) . U s u a l l y , we d e t e r m i n e two r e l a t e d s o r t s o f p a r a m e t e r s f r o m t h e s e d i s t r i b u t i o n s . - I n a g e n e r a l n o t a t i o n t h e s e a r e : Hp = t h e wave h e i g h t exceeded by a f r a c t i o n p(Q^p^1) o f t h e t o t a l number o f waves, N. H = t h e average h e i g h t o f a l l waves h i g h e r t h a n H . P p C e r t a i n s p e c i f i c v a l u e s o f p have s p e c i a l s i g n i f i c a n c e f o r t h e c h a r a c t e r i z a t i o n o f waves. Some examples a r e :

'^0 5 ~ '^1 0 ~ ^ ~ average wave h e i g h t .

^0.33 '^0.135 ^ s i g n i f i c a n t wave h e i g h t , denoted H^.

H^ i s o f i m p o r t a n c e , s i n c e i t has been f o u n d t o c o r r e s p o n d w e l l w i t h t h e v a l u e e s t i m a t e d by an e x p e r i e n c e d o b s e r v e r who must e s t i m a t e a wave

h e i g h t .

A n o t h e r p a r a m e t e r , i n d e p e n d e n t l y d e f i n e d , i s :

1 X ' n = 1

T h i s l a s t p a r a m e t e r can be i m p o r t a n t f o r energy r e l a t i o n s w h i c h depend upon H^.

As was done p r e v i o u s l y f o r w a t e r l e v e l s , we can compute a s t a n d a r d d e v i a t i o n about t h e average:

N

^H

4

\

n=1

E q u a t i o n ( 2 6 ) g i v e s a measure o f t h e v a r i a t i o n i n wave h e i g h t s around t h e average wave h e i g h t .

2 1

Rice and L o n g u e t - H i g g i n s have t h e o r e t i c a l l y i n v e s t i g a t e d t h e d i s t r i b u t i o n o f maximum v a l u e s i n s t a t i s t i c a l l y s t a t i o n a r y p r o c e s s e s . When t h e s p e c t r u m (band w i d t h ) i s s m a l l , t h e y f o u n d t h a t t h e s e d i s t r i b t u i o n s s a t i s -f y a R a ^ ^ i g h D i s t r i b u t i o n . T h i s d i s t r i b u t i o n i s g i v e n m a t h e m a t i c a l l y by: , a 2m f ( a ) = - e 2 (27) 3 o r : F ( a ) = e '^^ (28) E q u a t i o n (27) i s shown g r a p h i c a l l y i n f i g u r e 5.

2 3

C a r t w r i g h t and L o n g u e t - H i g g i n s d e m o n s t r a t e d t h a t t h e w i d t h o f t h e s p e c t r u m (e3 i n f l u e n c e s t h e f o r m o f t h e R a y l e i g h D i s t r i b u t i o n . As t h e s p e c t r u m becomes b r o a d e r , t h e n an i n c r e a s i n g " number o f t h e maxima a r e f o u n d t o be below t h e mean w a t e r l e v e l , feee f i g u r e 4 ) . The f r a c t i o n , r , o f t h e s e maxima w h i c h a r e below t h i s mean can be shown t o come f r o m :

1 r = ^ 1 T 1 - (1 - e^) C29) (30} o r , i n o t h e r f o r m : = 1 - d " 2 r ) ^ C31) T h i s l a s t e q u a t i o n a l l o w s us t o compute e, g i v e n an a c t u a l wave r e c o r d . I f e i s s m a l l , we g e t a R a y l e i g h D i s t r i b u t i o n f o r t h e maxima, a. As e i n c r e a s e s , t h e d i s t r i b u t i o n approaches a Gaussian D i s t r i b u t i o n . T h i s i s shown a l s o i n f i g u r e 5.

S i n c e wave minima a r e more o r l e s s s i m i l a r t o maxima, by symmetry then, we c o u l d go t h r o u g h t h i s same d i s c u s s i o n a g a i n f o r t h e s e . T h i s i s l e f t t o t h e r e a d e r .

R e t u r n i n g t o o u r p r a c t i c a l p r o b l e m o f wave h e i g h t s , we f i n d t h a t a wave h e i g h t r e s u l t s f r o m t h e c o m b i n a t i o n o f a maximum w i t h a minimum. F u r t h e r , we see a c e r t a i n degree o f c o r r e l a t i o n between t h e maxima and t h e i r a s s o c i a t e d minima. T h i s c o r r e l a t i o n i s dependent upon t h e band w i d t h o f t h e s p e c t r u m .

When we have a n a r r o w s p e c t r u m , we have a r e l a t i v e l y g r e a t symmetry i n t h e wave r e c o r d ; we a p p r o a c h a harmonic p a t t e r n . T h i s f a c t i s what a l l o w s t h e c o n c l u s i o n t h a t ( C a r t w r i g h t and L o n g u e t - H i g g i n s , 1964)

H = 2a t 3 2 ) / 2 \

F(H) . sxp - ( ^ ^

1 2 1 9

I n a way analogous t o a harmonic wave, f o r w h i c h m = - a = H , we can

Z 8 show t h a t : ,,2 ^ m s = ( 3 4 ) Y i e l d i n g : f ( H ) = exp - ( 3 5 ) rms rms /u2 \ F(H) = exp - ( 3 6 ) / rms f r o m w h i c h , we can d e t e r m i n e : / r H = — H^^ = 0.886 H ( 3 7 ) 2 rms rms ' " (1 " H = 0.453 H ( 3 8 ) H 4 rms rms _ L o n g u e t - H i g g i n s (1952) c o n t i n u e s w i t h e q u a t i o n ( 3 6 ) t o g e t v a l v e s H o f Pj . Some r e p r e s e n t a t i v e v a l u e s a r e g i v e n i n t a b l e I I . rms

2 5 T a b l e I I I Values o f rms and rms f o r v a r i o u s v a l u e s o f p. 10 -5 2x10 5x10' -I 10 2x10 -5 -4 rms 3.39 3.29 3.14 3.04 2.92 rms s i g n i f i c a n c e 5x10 2x10~' 5x10~' 0.01 0.02 0.05 0.10 0.135 0.125 2.76 2.63 2.50 2.30 2.14 1 .98 1 .73 1.52 1 .42 1 .44 2.359 1 .989 1 .800 rms 0.20 0.333 0.40 0.5 0.6 1 .27 0.83 1. 591 1 .416 1.347 1.256 1 .176 rms 0.7 0.8 0.9 1 .00 1 .102 1.031 0.961 0.886 0.453 rms rms 0.353 rms

F i e l d wave measurements, i f t h e y a r e c o r r e c t , seem t o f i t t h e t h e o r y s a t i s f a c t o r i l y . T h i s i s shown i n f i g u r e s 6 t h r o u g h 9.

F i g u r e s EXAMPLES OF DISTRIBUTIONS OF WAVE HEIGHTS AT

CERTAIN " P O I N T S O F T I M E " .

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F i g u r e 8 COMPARISON OF C H A R A C T E R I S T I C D A T A

OBTAINED FROM RECORDS

3 1

H i g h e s t wave i n a r e c o r d .

T h e o r e t i c a l l y , i f we w a i t l o n g enough, we can e n c o u n t e r a wave as h i g h as any v a l u e we choose. I n d e e d , i n p r a c t i c e , t h e h e i g h t , H^, o f t h e

h i g h e s t wave i n a group o f N waves depends b o t h on chance and t h e v a l u e o f N. L o n g u e t - H i g g i n s d e r i v e d an e x p r e s s i o n f o r t h e s t a t i s t i c a l e x p e c t a t i o n o f a

H^ / H^ \

r e l a t i v e wave h e i g h t ^ ^ , E ,_, i n 1952. These r e s u l t s a r e dependent upon li Vn /

rms \ rms'

N and a r e shown i n t a b l e I I I . The t a b l e shows b o t h e x a c t v a l u e s and t h e r e -s u i t -s o f an a -s y m t o t i c a p p r o x i m a t i o n .

The v a l u e s shown i n t a b l e I I I agree more o r l e s s w i t h measurements. T h i s i s shown i n f i g u r e 8A. As a r e s u l t , some p e o p l e e s t i m a t e t h e c h a r a c t e r i s -t i c s o f -t h e waves based s o l e l y upon a measuremen-t o f -t h e h i g h e s -t wave d u r i n g a measurement p e r i o d ( C a r t w r i g h t 1958; Draper 1963; T u c k e r 1 9 5 8 ) . T a b l e I I I E x p e c t a t i o n o f t h e h i g h e s t wave i n a s e r i e s o f N waves. N e x a c t s o l . ^ rms a p p r o x . s o l . 1 0.886 2 1.146 5 1.462 10 1.676 1.708 20 1.870 1.898 50 2.124 100 2.280 200 2.426 500 2.609 1,000 2.738 2,000 2.862 5,000 3.017 10,000 3.130 20,000 3.239 50,000 3.377 100,000 3.478

6. P o p u l a t i o n s o f wave p e r i o d s

The p e r i o d s o f m o r e - o r - l e s s r e g u l a r waves can be d e t e r m i n e d as t h e t i m e between passage o f s u c c e s s i v e wave c r e s t s (maxima] as shown i n f i g u r e 4. These p e r i o d s ( T ^ ] can, o f c o u r s e , be t r e a t e d s t a t i s t i c a l l y t o o b t a i n an average p e r i o d ( T ^ ) , a s i g n i f i c a n t p e r i o d ( T ^ ^ ^ g ^ ] , e t c .

As t h e waves become more i r r e g u l a r , t h i s p r o c e d u r e becomes l e s s adequate. I n between t h e wave groups t h e wave p a t t e r n becomes i l l - d e f i n e d . I t i s , however, s t i l l p o s s i b l e t o d e t e r m i n e t h e p e r i o d o f t h e h i g h e r waves i n a group ( T ^ ) and t h e a v e r a g e o f them ( T ^ ] as shown i n f i g u r e 4. ( I n some cases t h i s i s a l s o c a l l e d t h e s i g n i f i c a n t p e r i o d ] . The wave p e r i o d so deduced f r o m a r e c o r d appears t o c o r r e s p o n d r a t h e r c l o s e l y w i t h t h e p e r i o d r e l a t e d t o t h e peak o f t h e energy d e n s i t y s p e c t r u m ( f ) . A c c o r d i n g t o D a r b y s h i r e (1959]

( 3 9 ] T = 1.14 T^ O.BB T^

and f u r t h e r a f t e r S i b u l (1955] : T^ = 0.89 T^^^g ( 4 0 ]

I t has been shown t h a t t h e band w i d t h o f t h e s p e c t r u m i n f l u e n c e s t h e t i m e I n t e r v a l s between wave c r e s t s . Rice (1944, 1945] g i v e s

m \m^/

The f r e q u e n c y o f exceedance o f a wave h e i g h t T^ i n a p o p u l a t i o n i s g i v e n , a c c o r d i n g t o B r e t s c h n e i d e r (1959, 1966], b y : F(T ] = exp - 0.675 m T \4 ^ \ ( 4 2 ] m y ^ We m i g h t p o i n t o u t t h a t t h e d e f i n i t i o n s o f wave p e r i o d g i v e n above a r e r a t h e r d i f f i c u l t t o h a n d l e i n t h e case o f a u t o m a t i c p r o c e s s i n g o f wave d a t a . A more s u i t a b l e d e f i n i t i o n sometimes used i s t h e t i m e ( T ^ ] between

s e c c e s s i v e c r o s s i n g s o f t h e mean w a t e r l e v e l ( g o i n g upward, f o r e x a m p l e ] . The a v e r a g e o f t h e s e p e r i o d s (Tg) can be d e r i v e d f r o m t h e d u r a t i o n o f t h e r e c o r d

3 3 (SJ and t h e number (N^^) o f t h e z e r o c r o s s i n g s . T „ . 9 / N „ ( 4 3 , R i c e (1944, 1945) r e l a t e d t h i s p e r i o d t o t h e wave s p e c t r u m by 'm \k ^^2 T = 2Tr — (44) The r a t i o T /T i s r e l a t e d t o t h e w i d t h o f t h e s p e c t r u m . The m 0

w i d e r t h e s p e c t r u m , t h e more i r r e g u l a r t h e waves and t h e more maxima o c c u r below t h e mean w a t e r l e v e l (and minima above i t ) . A c c o r d i n g t o C a r t w r i g h t and L o n g u e t - H i g g i n s (1956) : 2 T \2 m \ e- = 1 - 1 = - (45) T h i s r e l a t i o n s h i p can be used t o d e t e r m i n e t h e v a l u e o f e ( T u c k e r , 1 9 6 3 ) .

7. Wave l e n g t h

The wave l e n g t h i s v e r y d i f f i c u l t t o o b s e r v e i n n a t u r e . Very l i t t l e i n f o r m a t i o n i s a v a i l a b l e .

G e n e r a l l y t h e t h e o r e t i c a l r e l a t i o n s h i p

L = |-: T^ = 1. 56 T^ ( i n m e t r i c u n i t s ) ( 4 6 )

i s used t o d e r i v e t h e wave l e n g t h f r o m i n f o r m a t i o n about t h e wave p e r i o d . T h i s , however, h o l d s o n l y f o r r e g u l a r waves ( i n v e r y deep w a t e r ) . P i e r s o n , Neumannand James (1955) d e r i v e d f r o m o b s e r v a t i o n s t h a t t h e " a p p a r e n t wave l e n g t h " i s r e l a t e d t o t h e mean wave p e r i o d a c c o r d i n g t o L = I - ^ T^ = 1.04 ( i n m e t r i c u n i t s ) ( 4 7 ) a 3 2ir m m i n t h e case o f i r r e g u l a r waves. I f a s u f f i c i e n t number o f o b s e r v a t i o n s o f wave l e n g t h ( L ) i s a v a i l a b l e t h e i n f o r m a t i o n can be s t a t i s t i c a l l y t r e a t e d ( a s i n t h e cases o f wave h e i g t h s and p e r i o d s ) t o o b t a i n an a v e r a g e wave l e n g t h ( L ) , a r o o t mean s q u a r e wave l e n g t h C-jp^j^g^' e t c . A c c o r d i n g t o B r e t s c h n e i d e r (1959) t h e wave l e n g t h s a r e d i s t r i b u t e d a c c o r d i n g t o R a y l e i g h : h\ u / ] \2 / i \2 F Z = exp - J = exp - 0.785 ^ ] ( 4 8 ) \ L / \ L / \ L / o r : / \

I

V

The r e l a t i o n s h i p between t h e l e n g t h and t h e p e r i o d e x p l a i n s why t h e wave p e r i o d i s n o t d i s t r i b u t e d a c c o r d i n g t o t h e R a y l e i g h d i s t r i b u t i o n .

3 5

8. Wave d i r e c t i o n and wave c r e s t s D i r e c t i o n

G e n e r a l l y waves p r o p a g a t e i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e c r e s t s . T h i s d i r e c t i o n can be o b s e r v e d v i s u a l l y o r by means o f i n s t r u m e n t s . I n t h e l a t t e r case a c o m p l i c a t e d s e t - u p w i t h t h r e e s e n s o r s can be used and t h e r e c o r d s a r e p r o c e s s e d w i t h a c o m p l i c a t e d computer program. I t i s a l s o p o s s i b l e t o use a f l o a t i n g buoy o r a v e l o c i t y / d i r e c t i o n meter. Longuet-H i g g i n s . C a r t w r i g h t and S m i t h (1963) u t i l i z e d such a method. Chase, Joseph, e t c . (1957) made s u c c e s s i v e s t e r e o p h o t o g r a p h s o f t h e sea s u r f a c e .

The d i r e c t i o n o f p r o p a g a t i o n o f t h e wave energy ahows a d i s t r i b u

-t i o n : C r ^ o 1 . . r^r.r^'^^ f ' J s i

F(X ,s) : : cos ( 5 (pJ

i n w h i c h : ^ = d i r e c t i o n w i t h r e s p e c t t o t h e average

s = exponent depending on t h e r a t i o o f w i n d v e l o c i t y (W) t o wave c e l e r i t y ( c ) o f t h e wave component w i t h l e n g t h ( L ) under c o n s i d e r a t i o n :

W/c = 1 l e a d s t o s = 7 W/c = 5 " " s = 0.25 w i t h t h e c e l e r i t y d e f i n e d as c =/ g L / 2 i T I n t h e g e n e r a t i n g a r e a w i t h s t r o n g w i n d s , t h e d i s p e r s i o n i s c o n s i d e r a b l e . I t i s much l e s s i n s w e l l . C r e s t s

I n accordance w i t h t h i s p i c t u r e o f i r r e g u l a r i t y , waves have s h o r t c r e s t s i n t h e i r g e n e r a t i n g a r e a . The r a t i o o f c r e s t l e n g t h t o wave l e n g t h i s about 3 i n t h e s e w i n d waves. The r a t i o i s l a r g e r i n s w e l l .

Waves t r a v e l l i n g i n t o a s h o a l i n g a r e a g r a d u a l l y d e c r e a s e i n l e n g t h . Conse-q u e n t l y , t h e i r c r e s t s become more and more p a r a l l e l t o t h e d e p t h c o n t o u r l i n e s . T h e r e f o r e t h e c r e s t t o wave l e n g t h r a t i o i s much g r e a t e r i n s h a l l o w w a t e r ;

u p t o a and 10. The s c a t t e r i n t h e d i r e c t i o n s i s a l s o much l e s s i n t h i s case.

9. P o p u l a t i o n s o f sea c o n d i t i o n s

I n t h e p r e v i o u s p a r a g r a p h s i t has been shown t h a t t h e c o n d i t i o n s o f t h e sea a t a c e r t a i n " p o i n t o f t i m e " can be d e s c r i b e d by a c o m b i n a t i o n o f c h a r a c t e r i s t i c p r o p e r t i e s such as:

- wave h e i g h t (H^^gg, H^^^. H, e t c . ) - wave p e r i o d ( T ^ , T^, e t c . )

- wave d i r e c t i o n i<i>)

S t i l l more i n f o r m a t i o n can be condensed i n a energy d e n s i t y s p e c t r u m o r even a group o f energy d e n s i t y s p e c t r a ; one f o r each d i r e c t i o n .

I n t h i s way. an o b s e r v a t i o n o f t h e s t a t e o f t h e sea y i e l d s a s e t o f c h a r a c t e r i s t i c d a t a . The s t a t e o f t h e w a t e r s u r f a c e i n t h e ocean i s o f t e n g i v e n by two o f t h e s e s e t s ; one f o r t h e sea and a second one f o r t h e s w e l l .

I f t h e s e o b s e r v a t i o n s are r e p e a t e d a t r e g u l a r i n t e r v a l s ( e . g . 6 h o u r s ) o v e r l o n g t i m e s (one o r more y e a r s ) a p o p u l a t i o n o f sea c o n d i t i o n s i s o b t a i n e d . T h i s s e t o f d a t a , i n t u r n , can be s u b j e c t e d t o s t a t i s t i c a l e l a b o r a -t i o n -t o o b -t a i n a u s e f u l g e n e r a l p i c -t u r e . The k i n d o f s -t a -t i s -t i c a l -t r e a -t m e n -t depends upon t h e p u r p o s e , such as:

- t h e d e s i g n s o f s t r u c t u r e s w h i c h r e q u i r e i n f o r m a t i o n about extreme sea c o n d i t i o n s ;

- o p e r a t i o n s a t sea f o r w h i c h i n f o r m a t i o n about w o r k a b l e c o n d i t i o n s i s r e q u i r e d , and

- c o m p u t a t i o n s a b o u t l i t t o r a l d r i f t o f s e d i m e n t s t o w h i c h a l l c o n d i t i o n s c o n t r i b u t e .

The v a r i o u s methods t h a t a r e used t o p r o c e s s t h e i n f o r m a t i o n i n t o t h e r e q u i r e d d e s i g n c r i t e r i a w i l l now be b r i e f l y d i s c u s s e d .

3 7

H e i g h t

The o b t a i n e d p o p u l a t i o n o f c h a r a c t e r i s t i c wave h e i g h t s can be g r o u p e d p e r month t o compute t h e average v a l u e p e r month and t h u s t o o b t a i n i n f o r m a t i o n about t h e s e a s o n a l f l u c t u a t i o n s o f t h e wave h e i g h t as i s done i n f i g u r e 1DA. A l s o t h e a v e r a g e c o n d i t i o n s o v e r a l o n g e r p e r i o d can be c a l c u l a t e d .

The c u m u l a t i v e f r e q u e n c y o f o c c u r r e n c e o f wave h e i g h t s can be c a l c u l a t e d f o r a season, f o r a y e a r (see f i g u r e 11] o r even f o r l o n g e r

p e r i o d s . From such graphs t h e average c o n d i t i o n s , t h e v a r i a t i o n o f t h e c o n d i t i o n s and t h e p r o b a b i l i t y o f extreme c o n d i t i o n s can be d e r i v e d .

1.6 1.4 1.2 1.0 0.8 tt

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F i g u r e 10 S E M I - M O N T H L Y A V E R A G E S OF Hs

A N D ^ FROM ONE YEAR OF

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80 70 60 50 40 30 20 10 2 1

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180 100 50 20 —r-_{10 2 , 1 05 0.2 01 005} Days per year

F i g u r e 11 C U M U L A T I V E FREQUENCY D I S T R I B U T I O N OF H

FROM OBSERVATION DURING ONE Y E A R

The c u m u l a t i v e f r e q u e n c y g r a p h s can be e x t r a p o l a t e d t o o b t a i n e s t i m a t e s o f t h e p r o b a b i l i t y o f o c c u r r e n c e o f v e r y r a r e c o n d i t i o n s . The most s u i t a b l e m a t h e m a t i c a l model f o r t h i s t a s k must be s e l e c t e d by t r i a l and e r r o r . Good r e s u l t s a r e o f t e n o b t a i n e d u s i n g graphs h a v i n g a p r o b a b i l i t y d i s t r i b u t i o n f o r t h e c u m u l a t i v e f r e q u e n c y and a l i n e a r o r l o g a r i t h m i c ( D r a p e r , 1963) s c a l e f o r t h e wave h e i g h t . I n some cases i t may be b e t t e r t o use t h e methods o f Gumbel o r G o o d r i c h .

P e r i o d

The p o p u l a t i o n o f p e r i o d s can be t r e a t e d i n a s i m i l a r way as i s done w i t h t h e h e i g h t s .

As an example, f i g u r e 12 shows t h e average p r o b a b i l i t y o f o c c u r r e n -ce i n i n t e r v a l o f t h e wave p e r i o d s ( T ^ ) o b s e r v e d i n t h e c o u r s e o f one y e a r . I t p r o v i d e s i n f o r m a t i o n a b o u t t h e s c a t t e r a j D o u t t h e a v e r a g e . — • 0 2 U 6 8 10 12 IA 16 18 20

Period (Ts) in seconds

F i g u r e 12 AVERAGE DISTRIBUTION OF Ts FROM

ONE YEAR OF OBSERVATION.

H e i g h t s and p e r i o d

4 1

p e r i o d ( l e n g t h ) o f waves. The r a t i o H/L, t h e wave steepness v a r i e s between

z e r o and t h e maximum s t e e p n e s s a t w h i c h t h e waves b r e a k ; t h e o r e t i c a l l y 1/7 'x'o^-^^i'i i n r e g u l a r waves.

The o n l y o c c u r r e n c e s o f i r r e g u l a r waves w i t h a c l e a r r e l a t i o n s h i p between t h e i r h e i g h t and p e r i o d i s i n t h e case o f w i n d waves i n t h e p r o c e s s o f g e n e r a t i o n . S i b u l (1955) and W i e g e l (1961) o b t a i n e d t h e r e l a t i o n s h i p : H_ „„ = 0.135 T^ ( 5 1 ) 0.33 m w h i c h w i t h ( 4 7 ) l e a d s t o : M = 0.13 L ( 5 2 ) U. dd a T h i s i s v e r y c l o s e t o t h e t h e o r e t i c a l l i m i t . S w e l l t h a t has t r a v e l l e d away f r o m i t s g e n e r a t i n g a r e a d e c r e a s e s i n h e i g h t . C o n s e q u e n t l y t h e H / L - r a t i o i s much s m a l l e r as i s shown i n t h e s c a t t e r d i a g r a m o f s w e l l i n f i g u r e 13.

I n t h i s example t h e maximum s i g n i f i c a n t wave h e i g h t o c c u r s w i t h about t h e average wave p e r i o d . There i s no c l e a r r e l a t i o n s h i p between t h e wave h e i g h t and t h e p e r i o d .

I n t h i s way a s c a t t e r d i a g r a m p r o v i d e s i n f o r m a t i o n about t h e n a t u -r e o f t h e waves.

The s c a t t e r d i a g r a m can a l s o be p u t i n t h e f o r m o f a t a b l e showing t h e f r e q u e n c y o f o c c u r r e n c e i n combined i n t e r v a l s o f h e i g h t s and p e r i o d s i n a s i m i l a r arrangement t o f i g u r e 13.

4 3

D i r e c t i o n

The d a t a about d i r e c t i o n s can be t r e a t e d s t a t i s t i c a l l y t o o b t a i n e.g. t h e v a r i a t i o n s o v e r t h e seasons ( f i g u r e 10B] and o t h e r i n f o r m a t i o n .

O f t e n t h e i n f o r m a t i o n i s v i s u a l i z e d i n t h e f o r m o f w i n d ïïoses o r o t h e r c i r c u l a r diagrams w h i c h i n d i c a t e t h e f r e q u e n c y o f o c c u r r e n c e p e r i n t e r -v a l o f t h e d i r e c t i o n (decade, o c t a n t ) . D i r e c t i o n , h e i g h t and p e r i o d The d i s t r i b u t i o n o f t h e c h a r a c t e r i s t i c i n f o r m a t i o n o v e r t h e d i r e c t i o n s i s o f t e n o f g r e a t i n t e r e s t e.g. f o r t h e d e s i g n o f t h e mouth o f a h a r b o r .

Much i n g e n u i t y has been p u t i n t h e v i s u a l i z a t i o n o f t h e d i s t r i b u -t i o n s o f h e i g h -t s and p e r i o d s i n d i r e c -t i o n a l diagrams. Examples can be f o u n d i n o c e a n o g r a p h i c a t l a s e s . I n o t h e r cases t h e i n f o r m a t i o n has been p u b l i s h e d i n t h e f o r m o f s e t s o f t a b l e s showing t h e f r e q u e n c y o f o c c u r r e n c e o f combi-n a t i o combi-n s o f v a l u e s o f t h e c h a r a c t e r i s t i c p r o p e r t i e s .

Sources o f i n f o r m a t i o n

Data o f t h i s n a t u r e , c o l l e c t e d a l l o v e r t h e oceans by merchant v e s s e l s and by o t h e r means, a r e p u b l i s h e d by h y d r o g r a p h i c s e r v i c e s i n v a r i o u s c o u n t r i e s (U.S.A., E n g l a n d , Germany). O f t e n , i t i s p o s s i b l e t o o b t a i n s p e c i f i c d a t a , however, i t - m u s t be kept i n mind t h a t t h e i n f o r m a t i o n o r i g i n a t e d f r o m v i s u a l o b s e r v a t i o n .

10. L i t e r a t u r e 10.1 G e n e r a l P. Groen en R. D o r r e s t e i n Z e e g o l v e n S t a a t s d r u k k e r i j en U i t g e v e r i j , 1958. B. Kinsman Wind waves P r e n t i c e H a l l , London, 1964. G. Neumann en W.J. P i e r s o n P r i n c i p l e s o f p h y s i c a l oceanography P r e n t i c e H a l l , London, 1966. R.L. W i e g e l • 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 , London, 1964. 10.2 S t a t i s t i c s M.J. Norovey F a c t s f r o m f i g u r e s P e l i c a n Books, 1951. A. H a l d S t a t i s t i c a l t h e o r y w i t h e n g i n e e r i n g a p p l i c a t i o n s John W i l e y , New York/London, 1952.

IQ 3 Time s e r i e s J.A. B a t t j e s S t a t i s t i s c h e eigenschappen van s t a t i o n a i r e g a u s s i s c h e p r o c e s s e n De I n g e n i e u r , 7 J u j y 1972, V o l . 84, No. 27. Pages B44 - B51. S.D. R i c e M a t h e m a t i c a l a n a l y s i s o f random n o i s e

B e l l System Tech. J o u r n a l 23_. 1944 and 24, 1945. A l s o i n : N. Wax

S e l e c t e d papers on n o i s e and s t o c h a s t i c p r o c e s s e s Dover P u b l i c a t i o n s , New York.

10.4 Energy and s p e c t r a

D.E. C a r t w r i g h t and M.S. L o n g u e t - H i g g i n s

The 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 maxima o f a random f u n c t i o n Proc. Roy. S o c , A, 237, 1956. Pages 212 - 232.

G. Neumann

On ocean wave s p e c t r a and a new method o f f o r e c a s t i n g w i n d - g e n e r a t e d sea Techn. Mem. No. 43, Beach E r o s i o n Board, 1953.

W.J. P i e r s o n and L. Moskowitz

A p r o p o s e d s p e c t r a l f o r m f o r f u l l y d e v e l o p e d w i n d seas based on t h e s i m i l a r i t y t h e o r y o f S.A. K i t a i g o r o d s k i i

1D.5 Wave h e i g h t s D.E. C a r t w r i g h t

On e s t i m a t i n g t h e mean energy o f sea waves f r o m t h e h i g h e s t wave i n a r e c o r d Proc. Roy. R o c , A, 247, 1958. Pages 22 - 48.

M.S. L o n g u e t - H i g g i n 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 o f t h e h e i g h t s o f sea waves J o u r n a l o f Mar. Res. V o l . X I , 3_' 1952. M.J. T u c k e r A n a l y s i s o f r e c o r d s o f sea waves P r o c I n s t . o f C i v i l Eng., 26. 1963. No. 6 6 9 1 . 10.6 P e r i o d s C L . B r e t s c h n e i d e r

Wave v a r i a b i l i t y and wave s p e c t r a f o r w i n d - g e n e r a t e d g r a v i t y waves Techn. Mem No. 118. Beach E r o s i o n Board. 1959.

C L . B r e t s c h n e i d e r

Wave g e n e r a t i o n by w i n d , deep and s h a l l o w w a t e r C h a p i t e r 3 o f : A.T. I p p e n E s t u a r y and c o a s t l i n e hydrodynamics Mc G r a w - H i l l . New York, 1966. J. D a r b y s h i r e A f u r t h e r i n v e s t i g a t i o n o f w i n d g e n e r a t e d waves Deutsche H y d r o g r . 7, 12, 1959.

4 7

M.S. L o n g u e t - H i g g i n s

The d i s t r i b u t i o n o f i n t e r v a l s between z e r o s o f a s t a t i o n a r y random f u n c t i o n P h i l . T r a n s . Roy. S o c , A, 254, 19B2. Pages 557 - 599.

R.R. P u t z

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 ocean waves

T r a n s . Am. Geophys. U n i o n , 33, 5, 1952. Pages 685 - 692.

10.7 D i r e c t i o n and c r e s t s J. Chase e t a l

The d i r e c t i o n a l s p e c t r u m o f a w i n d g e n e r a t e d sea as d e t e r m i n e d f r o m d a t a o b t a i n e d by t h e s t e r e o wave o b s e r v a t i o n p r o j e c t

New York Univ. C o l l . o f E n g i n e e r i n g , J u l y , 1957.

M.S. L o n g u e t - H i g g i n s , D.E. C a r t w r i g h t and N.D. Smith

O b s e r v a t i o n s o f t h e d i r e c t i o n a l s p e c t r u m o f sea waves u s i n g m o t i o n s o f a f l o a t i n g buoy

Proc. Conf. Ocean Wave S p e c t r a P r e n t i c e H a l l , 1963. 10.8 Sea c o n d i t i o n s S.H.A. Begemann T o e p a s s i n g van de w a a r s c h i j n l i j k h e i d s l e e r op h y d r o l o g i s c h e waarnemingen De W a t e r s t a a t s i n g e n i e u r , 1931, L. D r a p e r

D e r i v a t i o n o f a " d e s i g n wave" f r o m i n s t r u m e n t a l r e c o r d s o f sea waves Proc. I n s t . o f C i v . Eng., 26, 1963. No. 6690.

E.J. Gumbel

S t a t i s t i c s o f Extremes

Columbia Univ. P r e s s , New Y o r k , 1958.

U.S. N a v a l Oceanographic O f f i c e

Oceanographic A t l a s o f t h e N o r t h A t l a n t i c Ocean S e c t i o n I V : Sea and S w e l l , 1963.

A i r M i n i s t r y

M o n t h l y m e t e o r o l o g i c a l c h a r t s o f t h e A t l a n t i c Ocean H.M.S.O., London, R e p r i n t 1959, No. 483.

Deutsches H y d r o g r a f i s c h e s I n s t i t u t und D e u t s c h e r W e t t e r d i e n s t M o n a t s k a r t e n für d i e Südatlantischen Ozean

Hamburg, 1954.

N. Hogben and F.E. Lumb Ocean wave s t a t i s t i c s H.M.S.O., London, 1967.

4 9

L i s t o f Symbols

The f o l l o w i n g i s a l i s t o f t h e most i m p o r t a n t symbols used.

Symbol C E E, rms L L L C L I m rms m. 1 D e f i n i t i o n wave a m p l i t u d e 2 5 c o n s t a n t = 3.05 m /sec T o t a l energy p e r u n i t s u r f a c e a r e a T o t a l energy i n a wave base o f n a t u r a l l o g a r i t h m s = 2.71828 2 a c c e l e r a t i o n due t o g r a v i t y = 9.81 m/sec = 32.17 f t / s e c ^ maximum wave h e i g h t wave h e i g h t o b s e r v a t i o n wave h e i g h t exceed by f r a c t i o n p o f t o t a l waves average o f waves h i g h e r t h a n H P r o o t - m e a n - s q u a r e wave h e i g h t s i g n i f i c a n t wave h e i g h t = H 0.33 i n d e x o f s p e c t r a l moment wave l e n g t h average wave l e n g t h a p p a r e n t wave l e n g t h r o o t - m e a n - s q u a r e wave l e n g t h T o t a l ( s t a t i s t i c a l ) energy s p e c t r a l moment about Ü) = 0 t o t a l number o f o b s e r v a t i o n s Dimensions M L T 2 - 2 M L T L T L L L L L L L L L L

p p a r a m e t e r O < p < 1

r f r a c t i o n o f r e l a t i v e maxima below t h e mean s wave s c a t t e r i n g p a r a m e t e r

T wave p e r i o d T T p e r i o d c o r r e s p o n d i n g t o peak o f energy

s p e c t r u m T T wave p e r i o d measured between c r e s t s T

m

T a v e r a g e c r e s t - t o - c r e s t wave p e r i o d T m

1^ wave p e r i o d measured between s u c c e s s i v e

upward ( o r downward) z e r o c r o s s i n g s T s i g n i f i c a n t wave p e r i o d T A t sample i n t e r v a l T W w i n d speed L T y^ I n s t a n t a n e o u s w a t e r l e v e l w i t h r e s p e c t t o t h e mean L E s p e c t r u m w i d t h p a r a m e t e r n w a t e r l e v e l w i t h r e s p e c t t o some datum L n average w a t e r l e v e l L 0 l e n g t h o f r e c o r d = N A t T V p a r a m e t e r o f a u t o c o v a r i a n c e f u n c t i o n = = A t / x p mass d e n s i t y o f w a t e r M L s t a n d a r d d e v i a t i o n w i t h r e s p e c t t o a v e r a g e L rl a s t a n d a r d d e v i a t i o n o f w a t e r l e v e l w i t h y r e s p e c t t o mean L T parameter o f a u t o c o v a r i a n c e f u n c t i o n T (j) phase a n g l e

wave d i r e c t i o n c i r c u l a r f r e q u e n c y