G E N E R A T I O N O F W A V E S BY W I N D S T A T E O F T H E A R T by C h a r l e s L . B r e t s c h n e i d e r P r e s e n t e d 3,t I n t e r n a t i o n a l S u m m e r C o u r s e L u n t e r e n , The N e t h e r l a n d s September 1-18, 1964 C o n f e r e n c e sponsored by N e t h e r l a n d s U n i v e r s i t y I n t e r n a t i o n a l C o o p e r a t i o n and N o r t h A t l a n t i c T r e a t y O r g a n i z a t i o n P r e p a r a t i o n of Notes sponsored by O f f i c e of N a v a l R e s e a r c h D e p a r t m e n t of the Navy W a s h i n g t o n , D . C , , 20360 C o n t r a c t Noo Nonr-4177(00) NESCO R e p o r t S N - 1 3 4 - 6 J a n u a r y 15., ^1965 N A T I O N A L E N G I N E E R I N G SCIENCE C O M P A N Y . 1001 C o n n e c t i c u t A v e n u e , N . W , W a s h i n g t o n , D . C. , 20036 R E P R O D U C T I O N I N W H O L E OR I N P A R T IS P E R M I T T E D FOR A N Y PURPOSE O F T H E U N I T E D S T A T E S G O V E R N M E N T
T A B L E O F C O N T E N T S Page 111 R E F E R E N C E S V ix. 1 T A B L E O F C O N T E N T S LIST O F FIGURES P R E F A C E I , I N T R O D U C T I O N I I . P R A C T I C A L A P P L I C A T I O N S - - D E E P W A T E R 17 A . S I G N I F I C A N T W A V E CONCEPT 1'^ B . C O M P L E X N A T U R E OF SEA S U R F A C E 20 1. Wave V a r i a b i l i t y C. W A V E S P E C T R U M CONCEPTS 39 D . F R O U D E S C A L I N G O F T H E W A V E S P E C T R U M 46 I I L P R O P A G A T I O N O F W A V E S A N D S W E L L S I N T O 50 S H A L L O W W A T E R I V . G E N E R A T I O N OF W I N D WAVES I N S H A L L O W W A T E ^ 6Z A . G E N E R A T I O N OF W I N D WAVES OVER A B O T T O M 62 OF CONSTANT D E P T H V . D E C A Y OF W A V E S I N D E E P W A T E R V I . W A V E S T A T I S T I C S v n . W I N D S P E E D VERSUS W I N D S P E E D 79 68 76 84 111
L I S T O F FIGURES
7. H - 1 - F- T D i a g r a m f o r F o r e c a s t i n g W i n d - G e n e r a t e d Waves
1. Wave M o t i o n at I n t e r f a c e of T w o D i f f e r e n t F l u i d s 2 2. N o r m a l and T a n g e n t i a l E n e r g y T r a n s f e r , A i r to W a t e r 5
3. F e t c h Graph f o r Deep Water 21 4. Deep W a t e r Wave F o r e c a s t i n g C u r v e s as a F u n c t i o n of 22 W i n d Speed, F e t c h L e n g t h and W i n d D u r a t i o n 5. R e l a t i o n of E f f e c t i v e F e t c h to W i d t h - L e n g t h R a t i o f o r 23 R e c t a n g u l a r Fetches 6. Methods of Wave R e c o r d A n a l y s i s 24 26
8. Graph R e l a t i n g Wave Height t o W i n d Speed and D u r a t i o n , 27 and to F e t c h ; f o r Oceanic W a t e r s
9. Graph R e l a t i n g Wave Height to W i n d Speed and D u r a t i o n , 28 and to F e t c h ; f o r C o a s t a l W a t e r s
10, Graph R e l a t i n g Wave P e r i o d to W i n d Speed and D u r a t i o n , 29 and to F e t c h ; f o r Oceanic W a t e r s
11. Graph R e l a t i n g Wave P e r i o d to W i n d Speed and D u r a t i o n , 29 and to F e t c h ; f o r Coastal W a t e r s 30 30 12. S t a t i s t i c a l D i s t r i b u t i o n of Heights 13. P e r i o d S p e c t r u m 14. D i s t r i b u t i o n F u n c t i o n s f o r P e r i o d V a r i a b i l i t y and 32 Height V a r i a b i l i t y 15. Sample W e i b u l l D i s t r i b u t i o n D e t e r m i n a t i o n f o r Wave 35 Height 16. Sample W e i b u l l D i s t r i b u t i o n D e t e r m i n a t i o n f o r Wave 36 P e r i o d
17. Scatter D i a g r a m of 1 and A f o r 400 Consecutive Waves 37 f r o m the Gulf of M e x i c o
18. R a t i o of Wave H e i g h t s to the Square of A p p a r e n t Wave 41 P e r i o d s ; H / T ^
Page_
19. D u r a t i o n Graph: C o - C u m u l a t i v e Spectra f o r W i n d 43 Speeds f r o m ZO to 36 Knots as a F u n c t i o n of D u r a t i o n
ZO. F e t c h Graph: C o - C u m u l a t i v e Spectra f o r W i n d Speeds 44 f r o m ZO to 36 Knots as a F u n c t i o n of F e t c h
Z l . C o m p a r i s o n of P e r i o d S p e c t r u m f o r 6 Z - f o o t S i g n i f i c a n t 49 Wave w i t h Station " J " Data
ZZ. R e f r a c t i o n E f f e c t 51 Z3. D i v e r g i n g O r t h o g o n a l s 5 2 Z4. C o n v e r g i n g O r t h o g o n a l s 5 3 Z5. E x p e r i m e n t a l L e n g t h ^ X of R i s i n g Sea B o t t o m i n 55 D i r e c t i o n of M o t i o n Z6. R e l a t i o n s h i p f o r F r i c t i o n L o s s over a B o t t o m of Constant 60 Depth Z7. Kg v e r s u s T ^ / d ^ 61 Z,8. G e n e r a t i o n of W i n d O v e r a B o t t o m of Constant Depth 63 f o r U n l i m i t e d W i n d D u r a t i o n Represented as D i m e n s i o n -less P a r a m e t e r s 29. Wave F o r e c a s t i n g R e l a t i o n s h i p s f o r Shallow W a t e r of 64 Constant Depth 30. G r o w t h of Waves i n a L i m i t e d Depth 65 3 1 . Wave Spectra f o r A t l a n t i c C i t y , N . J . 67 32. F o r e c a s t i n g C u r v e s f o r Wave Decay 69 33. Decay of Wave Spectra w i t h Distance 70 34. Decay of S i g n i f i c a n t Waves w i t h Distance 71 35. T y p i c a l Change of Wave E n e r g y S p e c t r u m i n the 7 3
B u i l d - u p and Decay of Waves
36. E x a m p l e of P e r i o d Spectra of C o m b i n e d L o c a l S t o r m 74 and Swell 37. E x a m p l e of F r e q u e n c y S p e c t r u m of C o m b i n e d L o c a l 75 S t o r m and S w e l l 38. C o m p a r i s o n of Wave H e i g h t D i s t r i b u t i o n s D e r i v e d f r o m 78 V i s u a l O b s e r v a t i o n s and f r o m M e a s u r e m e n t s of Wave
Heights at A t l a n t i c Ocean Stations I and J v i
39. Geostrophic W i n d Scale 80 40. S u r f a c e W i n d Scale 81 4 1 . Computed v s . O b s e r v e d Surface W i n d Speed f o r 43 83
R a n d o m l y Selected P o i n t s
P R E F A C E
T h i s r e p o r t was p r e p a r e d o r i g i n a l l y as a s e r i e s of l e c t u r e s given at the I n t e r n a t i o n a l S u m m e r Course on "Some A s p e c t s of Shallow W a t e r Oceanology" h e l d at L u n t e r e n , the N e t h e r l a n d s , The subject of t h i s phase of the l e c t u r e s , " G e n e r a t i o n of Waves by W i n d , " i n c l u d e d both deep and shallow w a t e r c o n d i t i o n s . The decay of s w e l l i n b o t h deep and
shallow water was also d i s c u s s e d . I n a d d i t i o n to the o r i g i n a l p r e p a r e d m a n u s c r i p t , t h i s r e p o r t i n c l u d e s some m a t e r i a l r e s u l t i n g f r o m the d i s c u s s i o n s d u r i n g and f o l l o w i n g the l e c t u r e s .
G E N E R A T I O N O F WAVES BY WIND S T A T E O F T H E A R T
I , I N T R O D U C T I O N
When a i r f l o w s over a w a t e r s u r f a c e waves are f o r m e d . T h i s is an observable phenomenon. Just why waves f o r m when a i r f l o w s over the w a t e r i s a question about n a t u r e w h i c h has not yet been a n s w e r e d c o m p l e t e l y or s a t i s f a c t o r i l y by t h e o r e t i c a l m e a n s , F u r t h e r m o r e , why do waves have the heights and p e r i o d s t h a t a r e observed? A l l t h e o r i e s
begin e i t h e r w i t h the a s s u m p t i o n that waves do f o r m when w i n d b l o w s over the w a t e r s u r f a c e , or else that waves m u s t a l r e a d y e x i s t by the t i m e the w i n d begins to b l o w , or else the t h e o r y i s i m m o b i l e .
The b r o t h e r s E r n s t H e i n r i c h and W i l h e l m Weber (1825) w e r e the f i r s t known to r e p o r t e x p e r i m e n t s on waves, and A . P a r i s (18TI) made a c t u a l wave o b s e r v a t i o n s on the state of the sea. These o b s e r v a t i o n s w e r e made a b o a r d the D U P L I E X and the M I N I E R V A . A l t h o u g h A i r y (1848) d i d t h e o r y on waves and t i d e s , w i n d f o r c e s w e r e not i n c l u d e d . Other e a r l y c o n t r i b u t i o n s on wave o b s e r v a t i o n s at sea i n c l u d e d A b e r c r o m b y (1888), Schott (1893), and Gassenmayr (1896). H o w e v e r , the best e a r l y d o c u m e n t a t i o n on wave o b s e r v a t i o n s was perhaps that p r e p a r e d by C o r n i s h (1904, 1910 and 1934). C o r n i s h also attempted to r e l a t e wave conditions to m e t e o r o l o g i c a l and g e o g r a p h i c a l c o n d i t i o n s .
Rather than t h i n k i n g of the s c i e n t i s t i n the r o l e of a n s w e r i n g the question " W h y do waves f o r m , " one m i g h t r a t h e r t h i n k of h i m as a p r a c t i c a l engineer who knows that the phenomenon does o c c u r and who
can then r e c o m m e n d what should be done about i t . P r o g r e s s i s made only by e n g i n e e r i n g a p p l i c a t i o n of s c i e n t i f i c t h e o r y . T h e o r y o f f e r s no p r o g r e s s , except when i m p l e m e n t e d ; o t h e r w i s e i t i s d o r m a n t .
Since a l i t t l e t h e o r y has n e v e r h u r t a p r a c t i c a l engineer, an oceanographer, or an a p p l i e d s c i e n t i s t , i t seems quite a p p r o p r i a t e to m e n t i o n v a r i o u s t h e o r i e s w h i c h have been p r o p o s e d t h r o u g h v a r i o u s
stages i n the advancement of the state of the a r t .
A l t h o u g h Stevenson (1864) e s t a b l i s h e d the f i r s t known e m p i r i c a l f o r m u l a f o r wave g e n e r a t i o n , the c l a s s i c a l w o r k on w i n d wave t h e o r y was due to L o r d K e l v i n (1887) and H e l m h o l t z (1888). The K e l v i n - H e l m h o l t z t h e o r y , w h i c h can be found i n H y d r o d y n a m i c s , by L a m b (1945), p e r t a i n s to the study of the o s c i l l a t i o n s set up at the i n t e r f a c e of two f l u i d m e d i a of d i f f e r e n t d e n s i t i e s - - w a t e r and a i r , f o r e x a m p l e .
H e l m h o l t z c o n s i d e r e d the m e d i a of m a s s densities f ^ and f l o w i n g w i t h v e l o c i t i e s U ^ and U ^ w i t h r e s p e c t to each other as i l l u s t r a t e d i n F i g u r e 1.
F i g u r e 1
The i n t e r f a c e i s a wave surface., f ^ and U^^ the naas? densitA' and v e l o c i t y of the upper f l u i d , and f ^ and the m a s s d e n s i t y ana v e l o c i t y of the l o w e r f l u i d , r e s p e c t i v e l y . The p r o p a g a t i o n a l v e l o c i t y C i s the speed at w h i c h the i n t e r f a c e t r a v e l s i n a f o r w a r d d i r e c t i o n , a r d L is the distance between two successive peaks of the i n t e r f a c e .
Helmholtz. (1888) showed that the induced o s c i l l a t i o n , i f s m a l l c o m p a r e d w i t h the distance L - took the f o r m of a wave t r a i n at the i n t e r f a c e t r a v e l i n g at the v e l o c i t y C such that
w h e r e g i s the a c c e l e r a t i o n of g r a v i t y and k i s the wave number g i v e -as k = 2 - r r / L .
K e l v i n (1887) d e r i v e d the same r e s u l t i n a d i f f e r e n t m a n n e r and made some v e r y i n t e r e s t i n g c o n c l u s i o n s , F o r e x a m p l e , when = = 0 i t can be shown that
\ 9 . P \ k
I f one c o n s i d e r s the upper f l u i d to be a i r and the l o w e r f l u i d w a t e r , f o r w h i c h / I / i s equal to about 1, 29 x lO"-^, then E q . (2) reduce? v e r y n e a r l y to the s i m p l e f o r m
w h i c h i s the c l a s s i c a l equation f c r wave c e l e r i t y obtained f r o m line.^.r wave t h e o r y .
By use of the quadratic f o r m u l a , E q . (1) can be s o l v e d fc?, C na one obtains:
c
^ 1 ^ 2 ( A - ^ 2 V Now one can w r i t e E q . (4) as f o l l o w s :
C = Ü + C 1/2 (5) (4) w h e r e U = ^ ^ 2 " 2 /^2 and = C ƒ ) ƒ , ^ 1 ^ 2 2 - U ^ ) ' (6) (7)
I n the above U and C r e p r e s e n t an average o f t h e c o r r e s p o n d i n g values of U and C, and i s the e x p r e s s i o n i d e n t i c a l to E q . (2).
I f C o 2 is l e s s than the t e r m i n v o l v i n g U , i n E q . (7), i t w i l l be found that C becomes i m a g i n a r y , w h i c h i m p l i e s a c o n d i t i o n of m - ^
s t a b i l i t y i n the d e v e l o p m e n t of the waves, and t h i s leads to a p r o g r e s s i v e i n c r e a s e i n a m p l i t u d e . Under these c o n d i t i o n s the w i n d i s t r a v e l i n g f a s t e r than the waves and t h e r e w i l l be a continuous t r a n s f e r of energy to the waves, w h i c h i n t u r n goes into the f o r m of i n c r e a s e i n wave h e i g h t and
i n c r e a s e i n wave c e l e r i t y . The t e r m U ^ - U^ represents^the^wind v e l o c i t y
U1 i n c r e a s e i n w c L v e t ; e i e x i u y . ^xi.'^ 1,^x1^.^ ^ 2 ^ j ^ " x - * r e l a t i v e to the w a t e r and i s u s u a l l y e x p r e s s e d s i m p l y by U = U2 The c o n d i t i o n of i n s t a b i l i t y is d e f i n e d when 1 1 + (8) Since
f^lf^.^ 1.29X 10'
one m a y o b t a i n U > 28 C (9) C o O tr < ^ ~ 2 Fhas been d e f i n e d as the wave age, and waves a r e The r a t i o
unstable when t h e i r wave age i s l e s s than 1/28; t h i s i n s t a b i l i t y m a n i f e s t s i t s e l f as a p r o g r e s s i v e i n c r e a s e i n wave a m p l i t u d e .
F r o m o t h e r c o n s i d e r a t i o n s i t can be shown that the s m a l l e s t v e l o c i t y that a c a p i l l a r y wave (such as a r i p p l e ) can have i s 23. 2 c m / s e c , c o r r e s p o n d i n g to a wave l e n g t h of 1. 7 c m . and a p e r i o d of . 073 second?. W o r k by C r a p p e r ( 1957), Schooley ( I 9 6 0 ) and P i e r sen (1961) show? thar because of n o n l i n e a r e f f e c t s , the 1.7 c m . is somewhat l o w . A c c o r d i n g t o K e l v i n (1887) the waves w i l l always be unstable i f U > 28 x 23, 2 c m / sec. T h a t i s , i f U > 6. 5 m / s e c (12. 5 k n o t s ) , then the waves are u n s t a b l e . U = 6. 5 m / s e c i s also c a l l e d the c r i t i c a l w i n d speed r e q m r e d f o r g r a v i t y wave g e n e r a t i o n . A c c o r d i n g to M u n k (1947) t h e r e i s a c r i t i c a l w i n d speed below w h i c h waves do not f o r m . T h e r e have been n u m e r o u s a r t i c l e s on the subject of c r i t i c a l w i n d speed, some s u p p o r t i n g and o t h e r s o b j e c t i n g to the existence of a c r i t i c a l w i n d speed. R e f e r e n c e i s made to the w o r k of Cox and M u n k (1956), L a t e r M u n k (1957) appears t o be dubious as t o whether or not a c r i t i c a l w i n d speed e x i s t s , c i t i n g the w o r k of M e n d e l -b a u m (1956) and L a w f o r d and V e l e y (1956). I f a c r i t i c a l w i n d speed e x i s t s , i t appears f r o m a l l l i t e r a t u r e sources that i t e x i s t s appro-ximately between
2 and 6 m e t e r s per second.
The concept of c r i t i c a l w i n d speed is indeed a c o n t r o v e r s i a l sub-j e c t at p r e s e n t . N e v e r t h e l e s s , t h e r e i s s t i l l b e l i e f that t h e r e is a c r i t i c a l w i n d speed s o m e w h e r e between 4 and 6 m / s e c . , and that below the c r i t i c a l w i n d speed the f l u i d f l o w i s h y d r o d y n a m i c a l l y smooth or l a m i n a r and
above the c r i t i c a l w i n d speed t u r b u l e n c e develops and the i n t e r f a c e becomes h y d r o d y n a m i c a l l y r o u g h , f o r w h i c h wave a m p l i t u d e s i n c r e a s e w i t h t i m e and d i s t a n c e .
The d i f f i c u l t y w i t h t h i s t h e o r y i s that t h e r e i s a d e n s i t y d i f f e r e n c e between a i r and w a t e r even when the w i n d does not b l o w , and yet no waves a r e f o r m e d . Thus d e n s i t y d i f f e r e n c e alone i s i n s u f f i c i e n t to s t a r t wave g e n e r a t i o n ; the waves m u s t a l r e a d y have e x i s t e d by some other means,^ then they can propagate as f r e e g r a v i t y waves. I f t h e r e i s a c r i t i c a l w i n d
speed the w i n d i s a l r e a d y b l o w i n g , but t h e r e a r e no waves. W i t h an i n c r e a s e i n w i n d speeds, waves do f o r m . Why?
I t was not u n t i l 1925 t h a t J e f f r e y s i n t r o d u c e d the t h e o r y of " s h e l f e r i n p h y p o t h e s i s , " based on the concept of a h y d r o d y n a m i c a l l y rough sea r e q u i r e d f o r wave g e n e r a t i o n . I n h i s paper " O n the F o r m a t i o n o f Waves by W i a a , J e f f r e y s (1925) p r o p o s e d that eddies on the l e e w a r d side of the waves r e s u l t e d i n a r e d u c t i o n of n o r m a l p r e s s u r e as c o m p a r e d w i t h the w i n d -w a r d face and i n a consequent t r a n s f e r of energy f r o m -w i n d to -w a v e s . H i s r e s u l t s suggested t h a t the w i n d c o u l d add energy to waves only so long as the w i n d speed was equal to o r g r e a t e r than the wave c e l e r i t y , and that when the wave c e l e r i t y became equal to the w i n d speed the wavers reached m a x i m u m h e i g h t and the sea was one of steady state. A l s o , the l o w e s t w i n d speed r e q u i r e d f o r wave g e n e r a t i o n was cn the order- o^ two k n o t s , or about one m e t e r per second. I t appears that the value 'i-i two knots i s the b e s t accepted value f o r the c r i t i c a l w i n d speed.
D u r i n g the t e n y e a r s f o l l o w i n g the w o r k of J e f f r e y s , somewhat m o r e d e t a i l e d o b s e r v a t i o n s w e r e made of the sea surfa.ce c o n d i t i o n s .
F o r example, Schumacher (1928) r e p o r t e d the f i r s t known study of "Stereophotography of Waves" f r o m the G e r m a n A t l a n t i c E x p e d i t i o n , and W e i n b l u m and B l o c k (1936) also r e p o r t e d r e s u l t s on s t e r e o p h o t o -g r a m m e t r i c wave r e c o r d s . T h i s l a t t e r c o n t r i b u t i o n -gave r e s u l t s of m e a s u r e m e n t s c a r r i e d out on b o a r d the m o t o r ship SAN F R A N C I S C O , under o b s e r v a t i o n of V . C o r n i s h . Other data on waves d u r i n g t h i s p e r i o d i n c l u d e d that of W i l l i a m s (1934), who r e p o r t e d on sea and s w e l l o b s e r v a -t i o n s , i n c l u d i n g e a r l y me-thods of o b -t a i n i n g da-ta, and W h i -t e m a r s h (1935) who r e v i e w e d data on unusual sea c o n d i t i o n s as r e p o r t e d by m a r i n e r s , and d i s c u s s e d the cause of h i g h waves at 'sea and the e f f e c t of these waves on s h i p p i n g .
A f t e r the founding of the Beach E r o s i o n B o a r d i n the War D e p a r t m e n t i n the e a r l y 1930's, s e r i o u s r e s e a r c h began on t h e o r y and f o r m a -t i o n of g r a v i -t y waves. The f i r s -t i m p o r -t a n -t r e p o r -t was c o m p l e -t e d i n 1941 and p u b l i s h e d i n 1948, " A Study of P r o g r e s s i v e O s c i l l a t o r y Waves i n W a t e r , " by M a r t i n A . M a s o n (1948). T h i s r e p o r t updated the state of the a r t to about 1940.
The next great advance i n the t h e o r y of wave g e n e r a t i o n i n deep w a t e r was that by S v e r d r u p and M u n k (1947), although Suthons (1945) had a l r e a d y p r e p a r e d f o r e c a s t i n g methods f o r sea and s w e l l waves. Whereas J e f f r e y s (1925) t o o k into account only the t r a n s f e r of energy by n o r m a l s t r e s s e s , S v e r d r u p and M u n k c o n s i d e r e d both n o r m a l and t a n -g e n t i a l s t r e s s e s . (See F i -g u r e 2 . )
U
F i g u r e 2
The average r a t e at w h i c h energy i s t r a n s f e r r e d to a wave by n o r m a l p r e s s u r e i s equal to
^ N " 1 7 p w dx o (10)
w h e r e w = - k A C cos k (x - Ct) i s the v e r t i c a l component of the p a r t i c l e o
ve l o c i t y at the s u r f a c e , and p is the n o r m a l p r e s s u r e a c t i n g on the sea s u r f a c e . L i s the wave l e n g t h .
The average r a t e at w h i c h e n e r g y i s t r a n s n a i t t e d to the waves by t a n g e n t i a l s t r e s s i s equal to
R = _ - I (L u ax (11) ^ T L I
'o
u denotes the h o r i z o n t a l component of p a r t i c l e v e l o c i t y at the sea s u r f a c e , o
and i s the w i n d s t r e s s .
u = k A C s i n k (x - Ct) U 2 ) o
w h e r e is the r e s i s t a n c e c o e f f i c i e n t . V a r i o u s e x p e r i m e n t s and obser^-vations have been made l e a d i n g to c o n t r o v e r s i a l values of f as a
Iunct?on of w i n d speed. H o w e v e r , a n u m b e r of d i f f e r e n t a u t h o r i t i e s appear to have advocated a value of c l o s e to 2. 6 x 10" , and i t is t h i s value u t i l i z e d by S v e r d r u p and M u n k (1947).
A c c o r d i n g to the above a r g u m e n t s , the e n e r g y of waves ^an i n -c r e a s e only i f ( R j ^ + R ^ ^ ' ''^^^ ^^^"""^ ^""^""^^
n o r m a l and t a n g e n t i a l s t r e s s e s of the w i n d , exceeds R ^ . the r a t e at w h i c h energy i s d i s s i p a t e d by v i s c o s i t y . The energy added by the w i n d
goes i n t o b u i l d i n g the wave height and i n c r e a s i n g the wave speed. T h a t i s , R ° R = ' ^^^"^^ ^^^^ p o r t i o n of energy t r a n s -f o r m e d i n t o wave heights and R ^ i s that p o r t i o n o-f energy t r a n s -f o r m e d into wave speed.
D u r i n g the e a r l y stages of wave development^moBt of t ^
i s t r a n s m i t t e d b y n o r m a l s t r e s s e s , but when C / b > 0 37 the t r a n s m i s s i o n by t a n g e n t i a l s t r e s s i s d o m i n a n t . The e f f e c t of t h e n o r m a l
S t r e s s e s l o m i n a t e s f o r a s h o r t t i m e o n l y . D u r i n g the t i m e that t h e waves a r e g r o w i n g , the e f f e c t of the t a n g e n t i a l s t r e s s i s m.ost i m p o r t a n t . Whe... a r e g r o w i g _ ^ ^ ^ ^ ^ ^ t a n g e n t i a l s t r e s s , but t h e r e i s a sma 1 amount l o s t due to n o r m a l p r e s s u r e , and, f o r t h i s r e a s o n ^ ^ ^ ^ ^ w r i t t e n w i t h + R^^ . When R ^ = R ^ t R ^ ' the wavea-are said to have
r e a c h e d m a x i m u m height and c e l e r i t y f o r a P f . ^ ^ ^ ^ - ^ ^ ^ ^ ^ ^ ^ ^ J ^ ^ ^ f f f . o m e » a r e independent of f e t c h l e n g t h and w i n d d u r a t i o n . T h i s c o n d i t i o n ... . o m e t i m e s c a l l e d the f u l l y developed sea.
A c c o r d i n g to the w o r k of S v e r d r u p and M u n k (1947), the s o l u t i o n of the h y d r o d y n a m i c equation d e s c r i b i n g wave g e n e r a t i o n ^^^ailed a
knowledge of c e r t a i n c o e f f i c i e n t s or constants r e s u l t i n g f r o m ma.thematxcal f n t e l r a t i o n w h i c h , of c o u r s e , c o u l d not be d e t e r m i n e d b y t.heory alone. The a p p r o p r i a t e constants w e r e d e t e r m i n e d by use of e m p i r i c a l data.
Hence, there was no way of knowing whether or not the theory was c o r r e c t . In order to evaluate these constants it was n e c e s s a r y to r e s o r t to
e m p i r i c a l wind and wave data, which at that time was very limited. Wind speeds, fetch lengths and wind durations were estimated f r o m m e t e o r -ological situations, the data of which were also based on v e r y meager coverage. The waves were estimated by visible means. Out of this theoretical investigation grew the concept of the significant wave. The
significant wave height was estimated as the average wave height of the waves in the higher group of waves, which later became identified v e r y closely as the average of the highest one-third of the waves in a r e c o r d of about 20 minutes duration. The significant wave period was the c o r -responding average period of these waves.
According to t h è theory as evaluated with "ancient" data for the significant wave, the fully developed sea resulted in the foUowing relations:
g H / U ^ = 0, 2 6
and
| I = = C / U = 1, 37
where H and T are the significant wave height and period r e s p e c t i v e l y , and U i s the wind speed. It then became quite apparent for any situation, either wind waves or swell, that a whole spectrum of waves was present, including a probability distribution of wave heights and a probability d i s t r i -bution of wave periods. Much of the above work was performed during the days of World War I I . Otherwise e a r l i e r publications would have appeared in the l i t e r a t u r e . In fact, as e a r l y as 1935, the I m p e r i a l Japanese Navy encountered a typhoon in the P a c i f i c Ocean and many observations were taken but were not published until much later by A r a k a w a and Suda (1953).
The time had then a r r i v e d when no further advance in wave gen-eration theory could be made without reliable recorded data and an advance in s t a t i s t i c a l theory and data reduction and a n a l y s i s . T h e r e i s no necessity to d i s c u s s wave recording here since this subject is w e l l covered by T u c k e r (1964).
B a r b e r and U r s e l l (1948) w e r e perhaps the next to present a v e r y important paper. The r e s u l t s of their investigation proved the existence of a spectrum of waves. A completely new field of theory and r e s e a r c h had been initiated, but it should be noted that oceanographers were slow to take advantage of this concept. T h i s r e s e a r c h had laid the foundation upon which many advances have been made in the "state of the a r t , " and it i s because of this r e s e a r c h that the Sverdrup-Munk (1947) works a r e con-sidered "ancient. "
Although much r e s e a r c h was c a r r i e d out during the next few y e a r s , no great advances in the state of the art were published until T h i j s se and Schijf (1949) presented wave relationships for both deep and shallow water
based on wave data and some c o n s i d e r a t i o n s of the S v e r d r u p - M u n k t h e o r y . E x p e r i m e n t s on a p a r a f f i n m o d e l of w i n d - g e n e r a t e d waves by T h i j s se and S c h i j f show a h i g h negative p r e s s u r e at the c r e s t of the wave, w h i c h is i n c o n f l i c t w i t h the S v e r d r u p - M u n k concept of a constant w i n d along the f r e e
s u r f a c e , but w h i c h i s i n a c c o r d a n c e w i t h B e r n o u l l i ' s equation f o r an i n c r e a s e i n w i n d speed at the c r e s t and a decrease at the t r o u g h . T h i s
e x p e r i m e n t was c e r t a i n l y a great c o n t r i b u t i o n .
Johnson (1950) a p p l i e d the P i - t h e o r e m concept f o r d i m e n s i o n a l a n a l y s i s and p r e s e n t e d wave r e l a t i o n s h i p s f o r deep w a t e r based on a c o l l e c t i o n of n u m e r o u s data f r o m A b b o t s Lagoon, C a l i f o r n i a . A t the same t i m e the U . S. A r m y C o r p s of E n g i n e e r s (1950) had p r e s e n t e d wave data generated under h u r r i c a n e w i n d c o n d i t i o n s f o r shallow L a k e O k e e -chobee, F l o r i d a . The C o r p s of E n g i n e e r s also p r e s e n t e d w i n d and wave data f o r i n l a n d r e s e r v o i r s . F o r t P e c k , Montana (1951), and l a t e r f o r L a k e T e x o m a , T e x a s (1953). B r e t s c h n e i d e r (1951) p r e s e n t e d r e v i s e d wave f o r e c a s t i n g r e l a t i o n s h i p s of S v e r d r u p and M u n k ( 1947), based on the f i e l d data of Johnson (1950), l a b o r a t o r y data of B r e t s c h n e i d e r and R i c e (1951), and n u m e r o u s other data c o l l e c t e d by v a r i o u s a u t h o r s , B r a c e l m (1952) p r e s e n t e d an u n p u b l i s h e d r e p o r t on o b s e r v i n g , f o r e c a s t i n g , and r e p o r t i n g ocean waves and s u r f . A n e x c e l l e n t s u m m a r y of wave r e c o r d i n g s was p r e s e n t e d by W i e g e l (1962).
I t seems t h a t the year 1952 w i t n e s s e d the f i r s t a c c e l e r a t i o n i n the " s t a t e of the a r t . " L o n g u e t - H i g g i n s (1952) p r e s e n t e d the R a y l e i g h d i s t r i b u t i o n f o r wave height v a r i a b i l i t y based upon a n a r r o w s p e c t r u m . P u t z (1952) p r e s e n t e d a G a m m a - t y p e d i s t r i b u t i o n f o r wave height and wave p e r i o d v a r i a b i l i t y based ü p o n a n a l y s i s of 25 t w e n t y - m i n u t e ocean wave r e c o r d s . D a r b y s h i r e (1952) and Neumann (1952) each p r e s e n t e d wave
s p e c t r a concepts and r e l a t i o n s f o r wave g e n e r a t i o n based on c o l l e c t i o n of wave data. The m e t h o d of d e r i v a t i o n used by Neumann (1952) i s c o n t r o -v e r s i a l ; i t l e a d to the i n t r o d u c t i o n of a d i m e n s i o n a l constant, and f o r h i g h f r e q u e n c y , the energy was f o u n d to be p r o p o r t i o n a l to f , w h e r e f = ^ wave f r e q u e n c y . W a t t e r s (1953) d e r i v e d the R a y l e i g h d i s t r i b u t i o n of wave h e i g h t v a r i a b i l i t y i n a l e s s s o p h i s t i c a t e d m a n n e r than L o n g u e t H i g g i n s , and the data of D a r l i n g t o n (1954) supported the R a y l e i g h d i s t r i -b u t i o n . I n f a c t , the G a m m a - t y p e d i s t r i -b u t i o n f o r wave heights of P u t z (1952) was r e p r e s e n t e d v e r y c l o s e l y by the R a y l e i g h d i s t r i b u t i o n . I c h i y e (1953) studied the e f f e c t s of w a t e r t e m p e r a t u r e on wave g e n e r a t i o n .
Homada, M i t s u y a s u and Hase (1953) p e r f o r m e d l a b o r a t o r y t e s t s on w i n d and w a t e r .
The f i r s t a c c e l e r a t i o n of the "state of the a r t " d i d not mean m u c h i n r e g a r d to the development of the t h e o r y of wave g e n e r a t i o n , since t h i s was p u r e l y e m p i r i c a l , i n c l u d i n g b o t h f i e l d and l a b o r a t o r y data c o l l e c t i o n , except f o r the w o r k of L o n g u e t - H i g g i n s (1952) and W a t t e r s (1953).
U r s e l l (1956) had s u r v e y e d the p r o b l e m of w i n d wave g e n e r a t i o n and opened w i t h the statement that " w i n d b l o w i n g over a w a t e r s u r f a c e
generates waves i n the w a t e r by a p h y s i c a l p r o c e s s w h i c h cannot be
r e g a r d e d as k n o w n ; " he concluded that "the p r e s e n t state of our knowledge i s p r o f o u n d l y u n s a t i s f a c t o r y . "
B r e t s c h n e i d e r and R e i d (1954) p r e s e n t e d a t h e o r e t i c a l d e v e l o p -naent f o r the "Change i n Wave Height due to B o t t o m F r i c t i o n , P e r c o l a t i o n , and R e f r a c t i o n ; " B r e t s c h n e i d e r (1954) c o m b i n e d these r e l a t i o n s h i p s w i t h the wave g e n e r a t i o n r e l a t i o n s h i p s g i v e n by S v e r d r u p and M u n k (1947), as r e v i s e d by B r e t s c h n e i d e r (1951), to o b t a i n shallow w a t e r wave g e n e r a t i o n r e l a t i o n s h i p s f o r wave h e i g h t and wave p e r i o d as a f u n c t i o n of w i n d speed, f e t c h length and water depth. Sibul (1955) i n v e s t i g a t e d i n the l a b o r a t o r y the g e n e r a t i o n of w i n d waves i n s h a l l o w w a t e r . A s i d e f r o m the above r e f e r ¬
ences and the c o n t r i b u t i o n s of T h i j s s e and S c h i j f (1949) and the U . S. A r m y C o r p s of E n g i n e e r s , J a c k s o n v i l l e D i s t r i c t (1950), t h e r e i s s t i l l another c o n t r i b u t i o n on w i n d - g e n e r a t e d waves i n shallow w a t e r . I n 1953 and 19!34 Keulegan p e r f o r m e d e x p e r i m e n t s at the N a t i o n a l B u r e a u of Standards;
but as f a r as i s known, t h i s i n f o r m a t i o n has not been p u b l i s h e d . H o w e v e r , i t has been a s c e r t a i n e d that the data of Keulegan i s i n a g r e e m e n t w i t h that obtained f o r L a k e Okeechobee and i s also i n a g r e e m e n t w i t h the r e l a t i o n -ships p r e s e n t e d by B r e t s c h n e i d e r (1954). S a v i l l e (1954) p u b l i s h e d a u s e f u l r e p o r t on the e f f e c t of f e t c h w i d t h on wave g e n e r a t i o n .
Data c o l l e c t i o n , although l i m i t e d i n quantity and q u a l i t y , also
p e r s i s t e d d u r i n g the p e r i o d f r o m 1950 to about 1955. F o r e x a m p l e , U n o k i and Nankano (1955) i n Japan p u b l i s h e d wave data f o r H a c h i j o I s l a n d ;
T i t o v (1955) p r e s e n t e d w o r k s i n R u s s i a n ; and B r e t s c h n e i d e r (1954) d i d w o r k on shallow w a t e r of the Gulf of M e x i c o .
T h e r e seems to be a t r a n s i t i o n p e r i o d d u r i n g the y e a r s 1955 to 1960 H i s ideas based on t h e o r y and v e r i f i e d w i t h data, K r y l o v (1956 and 1958) p r e s e n t e d the R a y l e i g h d i s t r i b u t i o n f o r wave height v a r i a b i l i t y and p o s t u l a t e d also that the R a y l e i g h d i s t r i b u t i o n a p p l i e d to the wave l e n g t h v a r i a b i l i t y , w h i c h c o u l d be t r a n s f o r m e d i n t o a p e r i o d d i s t r i b u t i o n f u n c t i o n .
B r e t s c h n e i d e r (1957 and 1959) also v e r i f i e d the R a y l e i g h d i s t r i b u t i o n f o r wave height and wave l e n g t h v a r i a b i l i t y . He developed a d i s t r i b u t i o n f u n c -t i o n f o r wave p e r i o d v a r i a b i l i -t y w h i c h was i n a g r e e m e n -t w i -t h -tha-t p o s -t u l a -t e d by K r y l o v (1958) and was i n v e r y close a g r e e m e n t w i t h the G a m m a - t y p e d i s t r i b u t i o n f u n c t i o n f o r wave p e r i o d p r e s e n t e d by P u t z (1952). R o l l and _ F i s c h e r (1956) made a r e v i s i o n of the s p e c t r u m by Neumann (1952), e l i m i n -a t i n g the d i m e n s i o n -a l const-ant, -and f o u n d th-at the energy -at h i g h f r e q u e n c y wks p r o p o r t i o n a l to f " ^ i n s t e a d of f ' ^ a c c o r d i n g to the s p e c t r u m of Neumann (1952).
I t then became apparent that f o r a n a r r o w s p e c t r u m i t was safe to assume that the R a y l e i g h d i s t r i b u t i o n a p p l i e d equally w e l l f o r both wave height and wave length v a r i a b i l i t y , the l a t t e r r e a d i l y t r a n s f o r m e d i n t o a wave p e r i o d d i s t r i b u t i o n f u n c t i o n .
S t a t i s t i c a l r e p r e s e n t a t i o n of the sea by wave s p e c t r a concepts t h r o u g h the w o r k of T u k e y (1959) and B l a c k m a n and T u k e y (1958) was
g r e a t l y p r o m o t e d by the s t a f f at New Y o r k U n i v e r s i t y , p a r t i c u l a r l y P i e r s o n and M a r k s (1952).
Under the a s s u m p t i o n that the j o i n t p r o b a b i l i t y d i s t r i b u t i o n of wave height and p e r i o d was u n c o r r e l a t e d , B r e t s c h n e i d e r (1957, 1958, 1959) p r o p o s e d a development of a wave s p e c t r u m concept. T h e r e seemed to be some s i m i l a r i t y between the B r e t s c h n e i d e r s p e c t r u m and that proposed by Neumann (1952), and f i n a l l y the f o r m of B r e t s c h n e i d e r ' s s p e c t r u m r e s o l v e d as the proposed s p e c t r u m of P i e r s o n (1964) based on the s i m i -l a r i t y t h e o r y of K i t a i g o r o d s k i (1961).
A t t h i s t i m e also B r e t s c h n e i d e r (1958) again r e v i s e d the wave f o r e c a s t i n g r e l a t i o n s h i p s f o r both deep and shallow water.- The p r a c t i c a l graphs f o r wave f o r e c a s t i n g a r e g i v e n i n the r e v i s e d v e r s i o n of Beach E r o s i o n B o a r d T e c h n i c a l R e p o r t No. 4 (1961). These r e l a t i o n s h i p s p r e s e n t l y a r e undergoing a f u r t h e r r e v i s i o n w h i c h should i n c r e a s e the a c c u r a c y of wave f o r e c a s t s .
H o w e v e r , d u r i n g the ten y e a r s p r e c e d i n g about 1955, m o s t of the e f f o r t was devoted to a n a l y t i c a l e x p r e s s i o n s and l i t t l e to t h e o r y of wave g e n e r a t i o n . D u r i n g the days of J e f f r e y s (1925), S v e r d r u p and M u n k (1947) among o t h e r s , the concept of wave s p e c t r a was not p r o m o t e d . However, because of the wave s p e c t r a enlightenments of B a r b e r and U r s e l l (1948), S e i w e l l (1948), Neumann (1952), D a r b y s h i r e (1952), B r e t s c h n e i d e r
(1957), B u r l i n g (1959), and P i e r s o n (1964), among o t h e r s , a new channel was opened f o r wave g e n e r a t i o n t h e o r y .
A t t h i s p o i n t m e n t i o n should be made of the great c o n t r i b u t i o n s on wave t h e o r y , wave p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n s , and wave s p e c t r a proposed by M i c h e (1954). I n p a r t i c u l a r , M i c h e p r o p o s e d the R a y l e i g h d i s t r i b u t i o n to wave steepness, a t h e o r y not p r e v i o u s l y p r o p o s e d . T h i s d i s t r i b u t i o n f u n c t i o n should have a wide a p p l i c a t i o n f o r e n g i n e e r i n g
studies.
I t was not u n t i l P h i l l i p s (1957) and M i l e s (1957) that a d d i t i o n a l t h e o r e t i c a l concepts w e r e developed. S v e r d r u p and M u n k (1947) c o n
-s i d e r e d that the w i n d wa-s con-stant i n v e l o c i t y i n o r d e r to develop t h e i r t h e o r y , but t h i s was p r o v e n o t h e r w i s e by T h i j s s e and S c h i j f (1949). H o w e v e r , P h i l l i p s ( 1957) c o n s i d e r e d the f a c t that the w i n d was r a p i d l y f l u c t u a t i n g about some m e a n v a l u e . I t i s v e r y t r u e that w i n d s b l o w i n g over water do not c o n s i s t of s t r e a m s of a i r i n steady and u n i f o r m m o t i o n but, r a t h e r , of an i r r e g u l a r s e r i e s of " p u f f s " and " l u l l s " c a r r y i n g eddies and s w i r l s d i s t r i b u t e d i n a d i s o r d e r e d m a n n e r . The a t m o s p h e r i c eddies, or r a n d o m v e l o c i t y f l u c t u a t i o n s i n the a i r , a r e a s s o c i a t e d w i t h r a n d o m s t r e s s f l u c t u a t i o n s on the s u r f a c e , both p r e s s u r e s ( i . e. n o r m a l s t r e s s e s ) and t a n g e n t i a l s t r e s s e s . The eddies a r e borne f o r w a r d by the m e a n v e l o c i t y of the w i n d and, at the same t i m e , they develop, i n t e r a c t , and decay, so that the a s s o c i a t e d s t r e s s d i s t r i b u t i o n moves a c r o s s the s u r f a c e w i t h a c e r t a i n c o n v e c t i o n v e l o c i t y dependent upon the v e l o c i t y of the w i n d and also evolves i n t i m e as i t moves along.
I t is these p r e s s u r e f l u c t u a t i o n s upon the w a t e r s u r f a c e that a r e r e s p o n s i b l e f o r the e a r l y g e n e r a t i o n of waves. The t a n g e n t i a l s t r e s s i s not c o n s i d e r e d , but P h i l l i p s (1957) states i n some cases that the shear
s t r e s s a c t i o n m i g h t not be n e g l i g i b l e . The t h e o r y i s i n a g r e e m e n t w i t h wave o b s e r v a t i o n s d u r i n g the e a r l y stages of g e n e r a t i o n , but as C / U approaches u n i t y t h e r e a r e other wave generating p r o c e s s e s to take into account, such as s h e l t e r i n g and the e f f e c t s of v a r i a t i o n i n shear s t r e s s e s . A l t h o u g h t h i s t h e o r y tends to an u n d e r - e s t i m a t i o n of wave heights f o r
C / U close to u n i t y , i t m a y be c o n s i d e r e d as a g r e a t advance i n wave generation t h e o r y i n s o f a r as the i n i t i a l b i r t h and g r o w t h of waves a r e concerned. A v e r y i m p o r t a n t aspect r e s u l t s f r o m P h i l l i p s (1957) fcased on d i m e n s i o n a l c o n s i d e r a t i o n s i ' - i . e, , tcfor hi,gh f r e q u e n c y oomponehtsi ithe
.-5 energy v a r i e s as i
M i l e s ' t h e o r e t i c a l m o d e l f o r the generation of w a t e r waves is based on the i n s t a b i l i t y of the i n t e r f a c e between the a i r f l o w and the w a t e r . The t h e o r y of P h i l l i p s p r e d i c t s a r a t e of g r o w t h of the sea p r o -p o r t i o n a l to t i m e , w h e r e a s a f t e r the i n s t a b i l i t y m e c h a n i s m of M i l e s takes o v e r , the r a t e of g r o w t h becomes exponential. The P h i l l i p s m o d e l i s an uncoupled m o d e l i n the sense that e x c i t a t i o n ( a i r f l o w ) i s a s s u m e d to be independent of response (sea m o t i o n ) . The M i l e s t h e o r y r e p r e s e n t s a coupled m o d e l i n w h i c h the c o u p l i n g can lead to i n s t a b i l i t y and consequent r a p i d g r o w t h . T h e r e can be l i t t l e doubt that both m e c h a n i s m s occur m any p r a c t i c a l s i t u a t i o n . A t some f r e q u e n c i e s i n the s p e c t r u m the uncoupled m o d e l w i l l g o v e r n and at others the i n s t a b i l i t y m o d e l w i l l g o v e r n . The w o r k of M i l e s (I960) i s a r e c o g n i z e d c o n t r i b u t i o n on wave g e n e r a t i o n t h e o r y .
I j i m a (1957) p r e s e n t e d an exceltent paper on the p r o p e r t i e s of
ocean waves f o r tljie .Japanese a r e a of i n t e r e s t . T h i s study i n c l u d e d v a l u a b l e i n f o r m a t i o n on wave s p e c t r a obtained under typhoon c o n d i t i o n s . A l s o a decided d i f f e r e n c e e x i s t e d between wave s p e c t r a obtained on the open P a c i f i c Coast and that obtained f o r the coast of the Sea of Japan.
A n o t h e r i m p o r t a n t e f f o r t f o r obtaining wave s p e c t r a was that c o n -ducted by m e m b e r s of the New Y o r k U n i v e r s i t y : Chase, Cote, M a r k s , M e h r . P i e r s o n , Ronne, Stephenson, V e t t e r and Walden (1957). A U
e m b a r k e d upon a great task of obtaining the f i r s t d i r e c t i o n a l s p e c t r u m of a w i n d - g e n e r a t e d sea by stereophotographic techniques, although W e m b l u m and B l o c k (1936), about 20 y e a r s e a r l i e r , c a r r i e d out m e a s u r e m e n t s
e n t a i l i n g s t e r e o p h o t o g r a m m e t r i c r e p r o d u c t i o n of ocean waves on b o a r d the m o t o r ship SAN F R A N C I S C O .
S u l k e i k i n (1959) p r e s e n t e d the R u s s i a n methods of f o r e c a s t i n g w i n d waves over w a t e r , w h i c h f o l l o w e d f r o m h i s e a r l i e r w o r k s on the t h e o r y of sea waves (1956). B u r l i n g (1959) p r e s e n t e d a s p e c t r u m of waves at s h o r t fetches and f o u n d a range i n values of m i n the h i g h f r e q u e n c y f ' " ^ of the wave s p e c t r u m . K n r v i n - K r o u k o v s k v (1961), i n h i s T h e o r y of Seakeeping, s u m m a r i z e s m u c h of the e a r l y w o r k on w i n d wave g e n e r a t i o n .
I n p a r t i c u l a r , t h i s book i n c l u d e s a v e r y conaprehensive b i b l i o g r a p h y on waves and.wave t h e o r y .
I n 1961 an I n t e r n a t i o n a l Conference on Ocean Wave Spectra was h e l d at E a s t o n , M a r y l a n d , the proceedings of w h i c h w e r e p u b l i s h e d i r i
1963.* T h i s c o n f e r e n c e i n c l u d e d about 30 p r e s e n t a t i o n s , plus d i s c u s s i o n s , and had an attendance of l e s s than 100 p a r t i c i p a n t s , r e p r e s e n t i n g a v e r y l a r g e percentage of the s c i e n t i s t s and engineers i n the w o r l d who have been c o n t r i b u t i n g to the advancement of the science of ocean wave s p e c t r a . I t can be said t h a t t h i s c o n f e r e n c e b r o u g h t the "state of the a r t " up to date. Known and unknown p r o p e r t i e s of the f r e q u e n c y s p e c t r u m of a w i n d
-generated sea, by P i e r s o n and Neumann (1961, 1963) was the m o s t l o g i c a l paper to lead o f f the p r o g r a m . Unless one understands the concept of a f u l l y developed sea, the w o r k of Walden (1961, 1963) m i g h t be m i s i n t e r -p r e t e d since h i s data w e r e f o r v e r y s h o r t e f f e c t i v e f e t c h e s . f^°w®7f^' t h i s d i f f i c u l t y was c l a r i f i e d i n the d i s c u s s i o n of B r e t s c h n e i d e r ( 1961, 1963). N u m e r o u s d i s c u s s i o n s f o l l o w e d and the p r o g r a m continued w e l l on i t s way t h r o u g h o u t the f o u r - d a y p e r i o d . The c o n f e r e n c e p r o d u c e d two sources of d i r e c t i o n a l s p e c t r u m : L o n g u e t - H i g g i n s , C a r t w r i g h t and S m i t h (196iL, 1963) and M u n k (1961,1963). On the l a s t day, w i t h heads s t i l l s p i n n i n g , i t
became an accepted f a c t that the o n e - d i m e n s i o n a l l i n e a r concepts w e r e not always s u f f i c i e n t to d e s c r i b e the state of the sea.
H o w e v e r , the c o n f e r e n c e was not intended to b r i n g f o r t h new t h e o r y on how waves f o r m when w i n d blows ever the w a t e r , except f o r d i s c u s s i o n of the w o r k of P h i l l i p s (1957) and M i l e s (1957), and an i n t r o d u c t i o n " O n the N o n l i n e a r E n e r g y T r a n s f e r i n a Wave S p e c t r u m " by H a s s e l m a n n
(1961, 1963). O t h e r w i s e the t h e o r y was l i m i t e d to t h a t r e q u i r e d f o r data c o l l e c t i o n , data r e d u c t i o n and a n a l y s i s , and data p r e s e n t a t i o n and a p p l i -c a t i o n s . The w o r k of H a s s e l m a n n (1961,1963) i s indeed a -c l a s s i -c a l
c o n t r i b u t i o n , but i t s t i l l does not t e l l us why waves a r e f o r m e d a c c o r d i n g to p r e - d e s c r i b e d elevations and f r e q u e n c i e s .
D u r i n g the f i n a l stages of development of a w i n d - g e n e r a t e d sea, two n o n l i n e a r p r o c e s s e s could become s i g n i f i c a n t . T h e r e i s a d i s s i p a t i o n of wave energy due to b r e a k i n g ( w h i t e c a p s ) , and a t r a n s f e r of energy f l u x between f r e q u e n c y bands m a y take place. The t h e o r y of H a s s e l m a n n concerns the l a t t e r p r o c e s s . The t h e o r y shows that by the f i f t h o r d e r i n t e r a c t i o n s energy can be t r a n s f e r r e d between f r e q u e n c y bands i n the
s p e c t r u m . I n f a c t , P h i l l i p s (I960 and 1961, 1963) has shown that t h i s t h e o r e t i c a l energy t r a n s f e r can occur at c e r t a i n t h i r d o r d e r resonant i n t e r a c t i o n s . So f a r s p e c t r a l energy t r a n s f e r is a p u r e l y theqre};ical c o n -j e c t u r e and has not been v e r i f i e d by o b s e r v a t i o n or e x p e r i m e n t .
I n the f o l l o w i n g m a t e r i a l r e f e r e n c e s t o the above c o n f e r e n c e and proceedings a r e shown by (1961, 1963),
C a r t w r i g h t (1961) and P i e r s o n (1961) gave b r i e f r e p o r t s on the papers presented at the 1961 c o n f e r e n c e , p r i o r to the p u b l i c a t i o n of the p r o c e e d i n g s .
A f t e r the conference was o v e r , the p a r t i c i p a n t s went honne to w o r k again, hoping to advance the state of the a r t . M o r e data w e r e to be c o l l e c t e d , and t h i s r e q u i r e d f u r t h e r development of i n s t r u m e n t a t i o n , and an advancement of s t a t i s t i c a l t h e o r y and c o m p u t a t i o n p r o c e d u r e s .
A f t e r the conference s e v e r a l i m p o r t a n t p a p e r s appeared, although they had p r o b a b l y been w o r k e d on f o r s e v e r a l y e a r s . These i n c l u d e d a paper by K o r n e v a (1961) on wave v a r i a b ü i t y w h i c h tended to v e r i f y the p r e v i o u s l y m e n t i o n e d p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n s ; a j o i n t paper on "Data f o r H i g h Wave Conditions O b s e r v e d by the OWS 'Weather R e p o r t e r ' i n December 1959" by B r e t s c h n e i d e r , C r u t c h e r , D a r b y s h i r e , Neumann, P i e r s o n , W a l d e n and W i l s o n (1962); and a paper by Schellenberger (1962) on "undersuchungen uber W i n d w e l l e n auf l i n e m Binnensee. " P i e r s o n and M o s k o w i t z (1963) c o n t r i b u t e d a new f o r m of the o n e - d i m e n s i o n a l wave
s p e c t r u m based on the s i m i l a r i t y t h e o r y of K i t a i g o r o d s k i (1961) and found out that t h i s s p e c t r u m f e l l s o m e w h e r e among the other past _ p r o p o s e d s p e c t r a , c o n s i d e r i n g the i n h e r e n t e r r o r s a r i s i n g f r o m d i f f i -c u l t i e s i n d e t e r m i n i n g and d e f i n i n g w i n d speeds.
Walden and P i e s t (1961) p r e s e n t e d data and a n a l y s i s on wave s p e c t r a obtained near the M e l l u m P l a t e L i g h t h o u s e , l o c a t e d i n the s o m e -what s h e l t e r e d w a t e r o f f the N o r t h Sea coast b f G e r m a n y ,
A good s u m m a r y of wave t h e o r y and wave g e n e r a t i o n i s p r e s e n t e d i n V o l u m e I of The Sea, edited by H i l l (1962), p a r t i c u l a r l y Chapter 19, " W i n d Waves, " by B a r b e r and T u c k e r ( 1962),
N u m e r o u s data r e p o r t s on wave spectra a r e now b e c o m i n g a v a i l a b l e . F o r e x a m p l e , M o s k o w i t z , P i e r s o n and M e h r (1962, 1963) p r e p a r e d r e p o r t s on wave s p e c t r a e s t i m a t e d f r o m wave r e c o r d s obtained by the OWS
"Weather R e p o r t e r I , I I and I I I " and the OWS "Weather E x p l o r e r , " and P i c k e t t (1962) p r e s e n t e d wave s p e c t r a f o r the A r g u s I s l a n d t o w e r o f f
B e r m u d a ,
B r e t s c h n e i d e r (1962) p r e s e n t e d a concept on m o d i f i c a t i o n of wave s p e c t r a over the c o n t i n e n t a l shelf, and I j i m a (1962) p r e s e n t e d an i n t e r e s t i n g development of the c o r r e l a t i o n between wave heights and wave p e r i o d s f o r shallow w a t e r , K i t a i g o r o d s k i and S t r e k a l o v (1962, 1963) p r e s e n t e d c o n -t r i b u -t i o n s -to an a n a l y s i s of -the s p e c -t r a of w i n d wave m o -t i o n based on_
e x p e r i m e n t a l data. T h i s w o r k was a c o n t i n u a t i o n of the w o r k of K i t a i -g o r o d s k i (1961),
Goodknight and R u s s e l l (1964) presented the f i r s t data on l a r g e w i n d waves i n shallow w a t e r of the Gulf of M e x i c o , These waves w e r e generated under h u r r i c a n e w i n d c o n d i t i o n s . The s t a t i s t i c a l a n a l y s i s of the data showed that, f o r a l l p r a c t i c a l p u r p o s e s , the R a y l e i g h d i s t r i b u t i o n
was quite s a t i s f a c t o r y f o r r e p r e s e n t i n g wave height v a r i a b i l i t y f o r l a r g e h u r r i c a n e waves i n shallow w a t e r . The d i s t r i b u t i o n of wave p e r i o d s d i d not f o l l o w the d i s t r i b u t i o n f u n c t i o n of B r e t s c h n e i d e r (1957) or that of P u t z (1952) but f e l l somewhere between these d i s t r i b u t i o n f u n c t i o n s and the R a y l e i g h d i s t r i b u t i o n . The data seem to c o n s i s t of long p e r i o d waves a r r i v i n g f r o m deep w a t e r c o m b i n e d w i t h l o c a l w i n d waves generated at a l a r g e angle to the s w e l l s .
Hamada (1963, 1964) p r e s e n t e d two v e r y i n t e r e s t i n g r e p o r t s based on l a b o r a t o r y e x p e r i m e n t s of w i n d wave g e n e r a t i o n . The law o r i g i n a l l y p r o p o s e d by P h i l l i p s (1957) was stated to be a p p l i c a b l e to the l i m i t i n g boundary of the i n s t a b i l i t y . F o r v e r y s h o r t fetches and h i g h w i n d
speeds, Hamada (1964) f i n d s the high f r e q u e n c y r e l a t i o n s of f " " and £-8,94_
A c c o r d i n g to the w o r k of B r e t s c h n e i d e r (1959), the h i g h f r e q u e n c y en er g y v a r i e s w i t h f " " " w h e r e m = 9 f o r v e r y l o w gF^/U , and
d e c r e a s e s i n m a g n i t u d e to m = 5 f o r v e r y l a r g e g F / U , c o r r e s p o n d i n g to f u l l y developed seas. Thus t h e r e i s a g r e e m e n t at i n i t i a l wave g e n e r a -t i o n f o r f""^ be-tween Hamada (1963) and B r e -t s c h n e i d e r (1959), and also a g r e e m e n t at f u l l y developed wave g e n e r a t i o n f " ^ among P h i l l i p s (1957), B r e t s c h n e i d e r (1959), P i e r son (1963), and Hamada (1964). T h e r e are s t i l l e f f o r t s r e q u i r e d f o r the exponential p a r t of the s p e c t r a l equation, i e e'-^^''^ w h e r e B i s a constant. A c c o r d i n g to B r e t s c h n e i d e r (1959) and P i e r s o n (1963), n = 4 , but the f a c t o r B i s s t i l l i n some d i s a g r e e -m e n t . The w o r k of B u r l i n g (1959) i n d i c a t e s that n -m i g h t be l a r g e r t h a n n = 4. B r e t s c h n e i d e r ( 1 9 6 1 , 1963) states that n m i g h t v a r y between 4 and 8 or 9.
T o date, a l l t h e o r i e s a r e u s e f u l i n a t t e m p t i n g to u n d e r s t a n d the m e c h a n i s m s i n v o l v e d i n the g e n e r a t i o n of waves. None of the theories_ t e l l s us why waves are f o r m e d , l e t alone why the wave heights and p e r i o d s a r e as o b s e r v e d or why t h e r e is a s p e c t r u m of w a v e s . H o w e v e r , t h e r e i s enough i n f o r m a t i o n to f o r m u l a t e v a r i o u s e m p i r i c a l wave f o r e c a s t i n g r e l a t i o n s h i p s f o r c e r t a i n p r a c t i c a l a p p l i c a t i o n s . The a c c u r a c y of such wave f o r e c a s t i n g r e l a t i o n s h i p s depends on the a c c u r a c y of the w i n d and wave data c o l l e c t e d and used f o r the e m p i r i c a l r e l a t i o n s h i p s . The a c c u r a c y of the wave f o r e c a s t s then also depends upon the a c c u r a c v of the w i n d f o r e c a s t s .
A t t h i s point of the d i s c u s s i o n i t appears i n o r d e r to i n t r o d u c e an equation w h i c h d e s c r i b e s the sea state wave s p e c t r u m , i n c l u d i n g the v a r i a b i l i t y of wave d i r e c t i o n . The equation can be w r i t t e n as f o l l o w s :
CO t e - 1 m t , m t • oo TT (k,CO ) c o s ^ (14) w h e r e m = CO U COS ^ CJu,,, cos - 1 d T g CJ ^ I n E q . (14) TT ( k , ) i s the t h r e e - d i m e n s i o n a l p r e s s u r e s p e c t r u m as a f u n c t i o n of the v e c t o r wave n u m b e r k and t i m e T ; U i s the c o n v e c t i o n v e l o c i t y of the p r e s s u r e s y s t e m s , and u>;< i s the f r i c t i o n v e l o c i t y of the
shear f l o w . /Ö i s the c o e f f i c i e n t c a l c u l a t e d by M i l e s ( I 9 6 0 ) , and and 7 ° a r e w a t e r and a i r d e n s i t i e s r e s p e c t i v e l y .
T h e r e i s l i t t l e need to extend the above r e v i e w any f u r t h e r since the d i r e c t i o n a l s p e c t r u m has been discussed quite adequately by T u c k e r (1964). I t i s hoped that m o s t of the i m p o r t a n t c o n t r i b u t i o n s on wave gen-e r a t i o n havgen-e bgen-egen-en m gen-e n t i o n gen-e d . E q . (14) r gen-e p r gen-e s gen-e n t s thgen-e p r gen-e s gen-e n t statgen-e of the a r t on wave s p e c t r u m g e n e r a t i o n t h e o r y , but a d d i t i o n a l e m p i r i c a l data a r e r e q u i r e d .
I n r e g a r d to p r a c t i c a l m e t h o d s f o r wave h i n d c a s t i n g , B r e t s c h n e i d e r (1964) p r e s e n t e d a paper w h i c h takes i n t o account the c o m p l e t e p r o b l e m of deep w a t e r waves, s t o r m surge and waves over the c o n t i n e n t a l shelf, the b r e a k i n g wave zone, the wave r u n - u p on the beach and dunes f o r the M a r c h 5-8, 1962, E a s t Coast S t o r m . T h i s paper shows the r e s u l t s based on p r e s e n t methods of wave h i n d c a s t i n g and also e m p h a s i z e s the a r e a s of need f o r f u r t h e r r e s e a r c h . A v e r y i m p o r t a n t c o n s i d e r a t i o n of w i n d wave g e n e r a t i o n over the s h a l l o w w a t e r of the c o n t i n e n t a l shelf i s that of the t o t a l w a t e r depth. The t o t a l w a t e r depth includes the c o m b i n e d e f f e c t of o r d i n a r y t i d e and s t o r m surge, The v a r i o u s p r o b l e m s of w i n d set-up and s t o r m surge have been d i s c u s s e d by B r e t s c h n e i d e r (1958). No f u r t h e r d i s c u s s i o n on t i d e s and s t o r m surge is given h e r e since the sub-j e c t s w e r e w e l l d i s c u s s e d by v a r i o u s l e c t u r e r s at L u n t e r e n , e. g. D r s , J . R. R o s s i t e r , W, Hansen, P , G r o e n , J. T h . T h i j s s e , and J . C. Shonfeld. A d d i t i o n a l w o r k on s t o r m surge p r o b l e m s i s planned f o r subsequent r e p o r t s .
I I . P R A C T I C A L A P P L I C A T I O N S - - D E E P W A T E R A . S I G N I F I C A N T W A V E C O N C E P T
The s i g n i f i c a n t wave naethod of wave f o r e c a s t i n g was t h a t o r i g i n a l l y i n t r o d u c e d by S v e r d r u p and M u n k (1947) and is s o m e t i m e s c o n s i d e r e d the "ancient" m e t h o d . The wave f o r e c a s t i n g p a r a m e t e r s p r e s e n t e d by t h e m e v o l v e d f r o m t h e o r e t i c a l c o n s i d e r a t i o n s , but the a c t u a l r e l a t i o n s h i p s r e q u i r e d c e r t a i n basic data f o r the d e t e r m i n a t i o n of v a r i o u s constants and c o e f f i c i e n t s . Hence, t h i s m e t h o d can be c a l l e d s e m i - t h e o r e t i c a l or s e m i - e m p i r i c a l .
The s i g n i f i c a n t wave m e t h o d e n t a i l s c e r t a i n d e f i n i t i o n s . The s i g -n i f i c a -n t wave height i s the m e a -n or average wave height of the highest
1/ 3 of a l l the waves p r e s e n t i n a given wave t r a i n . The s i g n i f i c a n t wave p e r i o d r e p r e s e n t s the m e a n p e r i o d of the s i g n i f i c a n t wave h e i g h t . I t was f o u n d f r o m the a n a l y s i s of wave r e c o r d s that the s i g n i f i c a n t height is n e a r l y equal to that h e i g h t r e p o r t e d f r o m v i s u a l o b s e r v a t i o n s , and f o r t h i s r e a s o n t h e r e was s o m e t i m e s a c e r t a i n amount of a g r e e m e n t between v a r i o u s e m p i r i c a l f o r m u l a s used p r i o r to the development of the t h e o r y .
I t m i g h t be m e n t i o n e d that the s i g n i f i c a n t wave p e r i o d r e p r e s e n t s a p e r i o d a r o u n d w h i c h is c o n c e n t r a t e d the m a x i m u m wave energy. F r o m the w o r k of P u t z (1952), L o n g u e t - H i g g i n s (1952), and B r e t s c h n e i d e r (1959), the d i s t r i b u t i o n of the v a r i o u s wave heights can be d e t e r m i n e d by use of the s i g n i f i c a n t wave h e i g h t .
The wave p a r a m e t e r s obtained f r o m the t h e o r e t i c a l w o r k of S v e r d r u p and M u n k (1947) can also be obtained f r o m d i m e n s i o n a l c o n
-s i d e r a t i o n -s . T h i -s ha-s been done by John-son (1950), among o t h e r -s , u t i l i z i n g the B u c k i n g h a m P i - t h e o r e m (1914).
The f a c t o r s on w h i c h the w i n d wave p a r a m e t e r s f o r deep w a t e r depend are the w i n d v e l o c i t y U , the f e t c h length F , and the w i n d d u r a t i o n t . Of the wave p a r a m e t e r s only wave height and wave p e r i o d need to be c o n s i d e r e d since, i n deep w a t e r , the wave length L
-(g/2TT)T2 and the wave c e l e r i t y C = ( g / 2 T T ) T . The waves w i l l s u r e l y be subject to the i n f l u e n c e of g r a v i t y and then i t m a y be supposed that:
C = f ^ ( U , F , t , g) (15) and
H = f^ ( U , F , t , g) (16)
E q s . (15) and (16) state that C and H r e s p e c t i v e l y a r e f u n c t i o n s f J and f^ of U , F , t , g, but apply f o r deep w a t e r only. *
F r o m the s y m b o l s appearing i n E q s . (15) and (16) one m a y w r i t e the d i m e n s i o n s f o r deep w a t e r as f o l l o w s ; S y m b o l D i m e n s i o n s C H L U F L t T g
A d d i t i o n a l r e l a t i o n s h i p s can be w r i t t e n i f the w a t e r depth i s t a k e n i n t o account.
F o r each of the above equations t h e r e a r e f i v e v a r i a b l e s and two d i m e n s i o n a l u n i t s , whence f r o m the B u c k i n g h a m P i - t h e o r e m the s o l u t i o n s w i l l each be f u n c t i o n s of 5 - Z = 3 d i m e n s i o n l e s s p r o d u c t s , w i t h 2 + 1 = 3 v a r i a b l e s to each p r o d u c t . I n r e s p e c t to E q . (15) one can w r i t e
O = F J ( C U ^ g ^ ) , (FU^g"^)., tU^g^) (17)
W i t h r e s p e c t to equation (17) one obtains the d i m e n s i o n s
1 r a . -,b and T 1 r e T LT^ J
* I f one c o n s i d e r s wave g e n e r a t i o n i n shallow w a t e r , E q s . (15) and (16) become (15a) and C = f j ( U , F , t , d , g ) H = f^ (U, F , t , d , g) (16a) w h e r e d i s the w a t e r depth. 18
E q u a t i n g to u n i t y the sum of the exponents f o r the c o r r e s p o n d i n g d i m e n s i o n s , one obtains the f o l l o w i n g equations:
1 + a + b = 0 1 - a - 2b = 0 1 + c + d = 0 • c - 2d = 0 e + f = 0 1 - e - 2f = 0
The simultaneous s o l u t i o n of the above r e s u l t s i n a = - 1 b = 0 c = -2 d - 1 e = - 1 f = 1
Using values of the above exponents, E q . (17) becomes
O - F . c ' u
1 ^ 1 1
\ 1 or (18) 4 I n a s i m i l a r m a n n e r the c o r r e s p o n d i n g e x p r e s s i o n f r o m E q . (16) becomes J l u (19) I t m i g h t be m e n t i o n e d that the P i - t h e o r e m i s a m o s t p o w e r f u l t o o l i f p r o p e r l y u s e d . I t i s e x t r e m e l y i m p o r t a n t to r e a l i z e that the e x p r e s s i o n s f o r p h y s i c a l f a c t m u s t be d i m e n s i o n a l l y homogeneous; o t h e r -w i s e t h e r e a r e some s c i e n t i f i c f a c t o r s m i s s i n g .Equations (18) and (19) r e p r e s e n t the wave g e n e r a t i o n p a r a m e t e r s f o r deep w a t e r , based on d i m e n s i o n a l c o n s i d e r a t i o n s , and y/ ^ a r e f u n c t i o n a l r e l a t i o n s t h a t m u s t be d e t e r m i n e d by use of wave data. g H / U ^ , C / U , g F / U ^ and g t / U a r e d e f i n e d r e s p e c t i v e l y as the wave height p a r a m e t e r , the wave speed p a r a m e t e r , the f e t c h p a r a m e t e r , and the w i n d d u r a t i o n p a r a m e t e r . The wave speed p a r a m e t e r can also be w r i t t e n ^ , w h i c h i s a b e t t e r f o r m because the wave p e r i o d i s m o r e
e a s i l y m e a s u r e d than the wave speed. 19
Using the above p a r a m e t e r s B r e t s c h n e i d e r (1951) r e v i s e d the o r i g i n a l f o r e c a s t i n g r e l a t i o n s of S v e r d r u p and M u n k (1947), u t i U z i n g m u c h a d d i t i o n a l wave data. B e f o r e 1951, h o w e v e r , A r t h u r (1947) also
r e v i s e d the same r e l a t i o n s , but d i d not have the data that w e r e a v a i l a b l e i n 1951. F u r t h e r r e v i s i o n s of these r e l a t i o n s h i p s w e r e made again by B r e t s c h n e i d e r (1958). These f o r e c a s t i n g r e l a t i o n s h i p s have a c q u i r e d the name S - M - B m e t h o d f o r S v e r d r u p , M u n k and B r e t s c h n e i d e r ,
The f i n a l f o r m of the d i m e n s i o n l e s s wave g e n e r a t i o n p a r a m e t e r s appears i n f i g u r e 3.
The c u r v e t U / F i s the r e l a t i o n s h i p between w i n d speed, m i n i m u m d u r a t i o n and f e t c h l e n g t h , and was d e t e r m i n e d b y n u m e r i c a l i n t e
-g r a t i o n of the f o l l o w i n -g r e l a t i o n s h i p s :
gt f U r gF \ , _ C g ^ _ L ^
(20) t U _ gt gF
— - U • ^ 2
Wave f o r e c a s t i n g r e l a t i o n s h i p s given i n F i g u r e 4 are based on the d i m e n -sionless p a r a m e t e r s of f i g u r e 3.
F o r long n a r r o w bodies of w a t e r such as m a n - m a d e r e s e r v o i r s , r i v e r s , canals, or n a r r o w i n l e t s , c o r r e c t i o n s need to be made f o r the f e t c h l e n g t h . F i g u r e 5, based cn the w o r k of S a v i l l e (1954), can be used to c a l c u l a t e an e f f e c t i v e f e t c h l e n g t h , based on a c t u a l f e t c h l e n g t h and f e t c h w i d t h . The e f f e c t i v e f e t c h l e n g t h F ^ 5 F ^ - ^ should be used w i t h F i g u r e 4 to d e t e r m i n e s i g n i f i c a n t waves f o r long n a r r o w bodies of w a t e r .
B . C O M P L E X N A T U R E OF SEA S U R F A C E
The s i g n i f i c a n t wave d e s c r i p t i o n i s a s i m p l e and p r a c t i c a l m e a n s of d e a l i n g w i t h p r o b l e m s i n wave f o r e c a s t i n g . H o w e v e r , i t i s i m p o r t a n t to r e c o g n i z e that the sea i s v e r y c o m p l e x , made up of many v a r i a b l e heights and p e r i o d s . F i g u r e 6 shows a schematic i n t e r p r e t a t i o n of a
t y p i c a l wave r e c o r d w h i c h m i g h t be obtained f r o m a wave r e c o r d e r . T h e s i g n i f i c a n t wave height i s the a v e r a g e of the h i g h e s t 1/3 of the waves m a given wave t r a i n , of at l e a s t 100 consecutive waves, and is t h e r e f o r e a s t a t i s t i c a l p a r a m e t e r . The s i g n i f i c a n t wave p e r i o d i s the average p e r i o d of the highest 1/3 of the wave h e i g h t s , and i s c o m m o n only to the S - M - B f o r e c a s t i n g m e t h o d . T h e P i e r s o n - N e u m a n n - J a m e s ( P - N - J ) f o r e c a s t i n g m e t h o d c o n s i d e r s a m e a n apparent wave p e r i o d and ranges m wave p e r i o d .
B o t h S - M - B and P - N - J consider the p r o b a b i l i t y d i s t r i b u t i o n of wave heights and i n both methods the R a y l e i g h d i s t r i b u t i o n i s used. The S - M - B
F I G U R E 4 DEEP WATER WAVE FORECASTING CURVES
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O O I U J i»7 UJ a. ü O t¬ I X O 3 O cc O O I U J s to z V) </) O o: O O. n I O o: U J U J I UJ cr D O Q: UJ ü. O. 3 a O X O O CC I ¬ I O h¬ I UI O. O UI cc cr UI O _i CO </)
