G E N E R A T I O N OF WAVES BY WIND
STATE OF T H E ART
by
Charles L .
Bretschneider
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 Course
L u n t e r e n , The Netherlands
September 1-18,
1964
Conference sponsored by
Netherlands U n i v e r s i t y I n t e r n a t i o n a l Cooperation and
North Atlantic Treaty Organization
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
Washington, D . C , , 20360
C o n t r a c t Noo Nonr-4177(00)
NESCO R e p o r t SN-134-6
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 Connecticut Avenue, N . W ,
Washington, 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 STATES G O V E R N M E N T
T A B L E OF CONTENTS
111
T A B L E OF CONTENTS
LIST OF
Page
V
FIGURES
ix.
PREFACE
1
I,
INTRODUCTION
II.
P R A C T I C A L APPLICATIONS - - DEEP WATER
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 OF T H E W A V E S P E C T R U M
46
IIL
P R O P A G A T I O N O F WAVES A N D SWELLS I N T O
SHALLOW WATER
50
IV.
G E N E R A T I O N OF WIND WAVES I N S H A L L O W W A T E ^
6Z
A.
62
G E N E R A T I O N OF W I N D WAVES OVER A B O T T O M
OF CONSTANT D E P T H
V.
D E C A Y OF WAVES I N D E E P W A T E R
VI.
W A V E STATISTICS
vn.
WIND SPEED VERSUS WIND S P E E D
68
76
79
84
REFERENCES
111
LIST OF FIGURES
1.
Wave M o t i o n at I n t e r f a c e of Two 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 Water
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 Curves as a F u n c t i o n of
Wind Speed, F e t c h Length and W i n d D u r a t i o n
22
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 Ratio f o r
Rectangular Fetches
23
6.
Methods of Wave R e c o r d A n a l y s i s
24
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
26
8.
Graph R e l a t i n g Wave Height to Wind Speed and D u r a t i o n ,
and to Fetch; f o r Oceanic W a t e r s
27
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 ,
and to F e t c h ; f o r Coastal Waters
28
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 ,
and to Fetch; f o r Oceanic W a t e r s
29
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 ,
and to Fetch; f o r Coastal W a t e r s
29
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
30
13.
Period Spectrum
30
14.
D i s t r i b u t i o n Functions f o r P e r i o d V a r i a b i l i t y and
Height V a r i a b i l i t y
32
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
Height
35
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
Period
36
17.
Scatter D i a g r a m of 1 and
f r o m the Gulf of M e x i c o
37
18.
Ratio of Wave Heights to the Square of A p p a r e n t Wave
Periods;
A
H/T^
'.V
f o r 400 Consecutive Waves
41
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
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
43
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
f r o m ZO to 36 Knots as a F u n c t i o n of F e t c h
44
Zl.
C o m p a r i s o n of P e r i o d Spectrum f o r 6Z-foot S i g n i f i c a n t
Wave w i t h Station " J " Data
49
ZZ.
Refraction Effect
51
Z3.
D i v e r g i n g Orthogonals
52
Z4.
Converging Orthogonals
53
Z5.
E x p e r i m e n t a l Length ^ X
D i r e c t i o n of M o t i o n
Z6.
Relationship 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
Depth
60
Z7.
Kg v e r s u s
61
Z,8.
Generation of W i n d O v e r a B o t t o m of Constant Depth
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
63
29.
Wave F o r e c a s t i n g Relationships f o r Shallow W a t e r of
Constant Depth
64
30.
G r o w t h of Waves i n a L i m i t e d Depth
65
31.
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 Curves 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
B u i l d - u p and Decay of Waves
73
36.
Example of P e r i o d Spectra of Combined L o c a l S t o r m
and Swell
74
37.
Example of F r e q u e n c y Spectrum of Combined L o c a l
S t o r m and Swell
75
38.
C o m p a r i s o n of Wave Height D i s t r i b u t i o n s D e r i v e d f r o m
V i s u a l Observations 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
78
of R i s i n g Sea B o t t o m i n
T^/d^
vi
55
39.
Geostrophic W i n d Scale
80
40.
Surface Wind Scale
81
41.
Computed vs. Observed Surface W i n d Speed f o r 43
Randomly Selected P o i n t s
83
vii
PREFACE
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 Summer Course on "Some Aspects of Shallow Water
Oceanology" h e l d at L u n t e r e n , the Netherlands, 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 water c o n d i t i o n s . The decay of s w e l l i n both deep and
shallow water was also discussed. 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 includes some m a t e r i a l r e s u l t i n g f r o m the
discussions 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 .
ix
G E N E R A T I O N OF WAVES BY WIND
STATE OF T H E ART
I,
INTRODUCTION
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 water is a question about nature w h i c h has not yet been answered
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 means, 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 that are observed? A l l t h e o r i e s
begin either 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 blows over
the water s u r f a c e , or else that waves m u s t a l r e a d y exist by the t i m e the
w i n d begins to blow, or else the t h e o r y is 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
actual wave observations on the state of the sea. These observations
w e r e made aboard 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)
did t h e o r y on waves and t i d e s , w i n d f o r c e s were 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 observations at sea included A b e r c r o m b y
(1888), Schott (1893), and Gassenmayr (1896). However, the best e a r l y
documentation on wave observations 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 geographical conditions.
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 "Why 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 occur 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 is made
only by engineering 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 is d o r m a n t .
Since a l i t t l e t h e o r y has never h u r t a p r a c t i c a l engineer, an
oceanographer, or an applied 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 proposed 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 .
Although Stevenson (1864) established 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 generation, 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 densities - - w a t e r and a i r , f o r example.
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 mass densities
flowing with velocities U^
and U ^
i l l u s t r a t e d i n F i g u r e 1.
1
f ^ and
w i t h respect to each other as
Figure 1
The i n t e r f a c e i s a wave surface.,
and v e l o c i t y of the upper f l u i d , and
f ^ and
f ^ and
U^^ the naas? densitA'
the mass density 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 propagational v e l o c i t y C
is 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
where 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 is the wave number giveas 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 manner and
made some v e r y i n t e r e s t i n g conclusions, F o r example, when
=
= 0
i t can be shown that
\
9
.
P \
k
If one considers 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 ,
for which / I /
is equal to about 1, 29 x lO"-^, then Eq. (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 is 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 , Eq. (1) can be solved fc?,
one obtains:
2
C
na
1/2
^ 1
^2
(4)
c
( A - ^ 2 V
Now one can w r i t e Eq. (4) as f o l l o w s :
C
= Ü+
(5)
C
where
U
^
=
^2"2
(6)
/^2
and
ƒ) ƒ,
^1^2
= C
I n the above
values of U and
U and
2
(7)
- U^)'
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
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 n E q . (7), i t w i l l
I f C o is less than the t e r m i n v o l v i n g U ,2
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 development 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 conditions the w i n d is 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 height 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
i n c r e a s e
in
wcLve
t;eiexiuy.
1,^x1^.^
^xi.'^
^ 2
^ j
^ "x-*
r e l a t i v e to the water and i s u s u a l l y expressed s i m p l y by U = U2
The condition of i n s t a b i l i t y is defined when
1+
Since
f^lf^.^
1.29X 10'
U1
1
(8)
one m a y obtain
U > 28 C
(9)
C
o <^~2F
O
has been defined as the wave age, and waves are
The r a t i o tr
unstable when t h e i r wave age i s less 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 .
3
F r o m other considerations 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 is 23. 2 c m / s e c ,
c o r r e s p o n d i n g to a wave length 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 (I960) 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.
That i s , i f U > 6. 5 m / s e c (12. 5 k n o t s ) , then the waves are unstable.
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 generation. A c c o r d i n g to M u n k (1947) t h e r e is 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 numerous 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 others objecting
to the existence of a c r i t i c a l wind speed. R e f e r e n c e is 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 baum (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 exists 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 subj 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 there is a c r i t i c a l
w i n d speed somewhere 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 rough, f o r w h i c h wave amplitudes i n c r e a s e
w i t h t i m e and distance.
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 is a density d i f f e r e n c e
between a i r and water even when the w i n d does not blow, and yet no waves
are f o r m e d . Thus density d i f f e r e n c e alone is i n s u f f i c i e n t to s t a r t wave
generation; the waves m u s t a l r e a d y have existed by some other means,^
then they can propagate as f r e e g r a v i t y waves. I f there is a c r i t i c a l wind
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 are 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 that 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
hypothesis, " 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 generation. 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) proposed 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 wind to waves.
H i s r e s u l t s suggested that the w i n d could add energy to waves only so
long as the w i n d speed was equal to or greater 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 wind speed the wavers
reached m a x i m u m height and the sea was one of steady state. A l s o , the
lowest w i n d speed r e q u i r e d f o r wave generation was cn the order- o^ two
knots, or about one m e t e r per second. I t appears that the value 'i-i two
knots i s the best 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 ten 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 detailed observations were made of the sea surfa.ce c o n d i t i o n s .
4
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 stereophotog 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 FRANCISCO,
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
included 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 methods of obtaining data, 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 conditions as r e p o r t e d by m a r i n e r s ,
and discussed the cause of high waves at 'sea and the e f f e c t of these waves
on shipping.
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, serious 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 completed i n 1941
and published 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 generation i n deep
water 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) took 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 gential s t r e s s e s . (See F i g u r e 2.)
U
Figure 2
The average r a t e at which 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 is equal to
^N
where
"
17
w
p
w
o
dx
(10)
= - k A C cos k (x - Ct) is the v e r t i c a l component of the p a r t i c l e
o
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 acting on the sea
ve
surface.
L
i s the wave length.
5
The average r a t e at w h i c h energy i s transnaitted to the waves by
tangential s t r e s s i s equal to
R
^T
u
= _L
I
I
'o
(L u
ax
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
o
and
(11)
surface,
is the w i n d s t r e s s .
u
= k A C s i n k (x - Ct)
U2)
o
where
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 leading 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 number 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
close to 2. 6 x 10" , and i t is this 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 energy of waves ^an i n -
crease 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 into 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
is, R ° R
=
' ^^^"^^
^^^^ p o r t i o n of energy t r a n s f o r m e d into wave heights and R ^ is that p o r t i o n of 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 ^
is t r a n s m i t t e d by 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 is dominant. The e f f e c t of the n o r m a l
Stresses l o m i n a t e s f o r a s h o r t t i m e only. D u r i n g the t i m e that t h e waves
are g r o w i n g , the e f f e c t of the tangential s t r e s s i s m.ost i m p o r t a n t . Whe...
are 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 reason
^ ^ ^ ^ ^
w r i t t e n w i t h + R^^ . When R ^ = R ^ t R ^ ' the wavea-are said to have
reached 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 »
are independent of f e t c h length 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 generation ^^^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 , could not be d e t e r m i n e d by t.heory alone.
The a p p r o p r i a t e constants were d e t e r m i n e d by use of e m p i r i c a l data.
6
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
empirical wind and wave data, which at that time was very limited. Wind
speeds, fetch lengths and wind durations were estimated from meteorological 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 very
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:
gH/U^ = 0,
26
and
|I=
= C / U = 1, 37
where H and T are the significant wave height and period respectively,
and U is 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 II. Otherwise e a r l i e r publications would have appeared
in the literature. In fact, as early as 1935, the Imperial 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 Arakawa and Suda (1953).
The time had then a r r i v e d when no further advance in wave generation theory could be made without reliable recorded data and an
advance in statistical theory and data reduction and analysis. T h e r e is
no necessity to discuss wave recording here since this subject is well
covered by Tucker (1964).
B a r b e r and U r s e l l (1948) were perhaps the next to present a v e r y
important paper. The results 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
is because of this r e s e a r c h that the Sverdrup-Munk (1947) works are considered "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
7
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 high 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 is i n accordance 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) applied 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
analysis and presented wave r e l a t i o n s h i p s f o r deep water based on a
c o l l e c t i o n of numerous data f r o m Abbots 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 Corps of Engineers (1950) had presented wave
data generated under h u r r i c a n e wind conditions f o r shallow Lake Okeechobee, F l o r i d a . The Corps of Engineers also presented 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 Lake
T e x o m a , Texas (1953). B r e t s c h n e i d e r (1951) presented 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 Rice (1951),
and numerous other data c o l l e c t e d by v a r i o u s authors, B r a c e l m (1952)
presented an unpublished 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 excellent s u m m a r y of wave r e c o r d i n g s was
presented by W i e g e l (1962).
I t seems that 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 "state of the a r t . " L o n g u e t - H i g g i n s (1952) presented 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) presented 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 analysis 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
spectra concepts and r e l a t i o n s f o r wave generation based on c o l l e c t i o n of
wave data. The method of d e r i v a t i o n used by Neumann (1952) is c o n t r o v e r s i a l ; i t lead 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 found to be p r o p o r t i o n a l to f
f =^
wave f r e q u e n c y .
, where
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 height v a r i a b i l i t y i n a less sophisticated 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 generation.
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 tests 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 generation, 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 both 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 surveyed the p r o b l e m of w i n d wave generation
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
8
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 known;" he concluded that "the p r e s e n t state of our knowledge
is 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) presented a t h e o r e t i c a l developnaent 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) combined these r e l a t i o n s h i p s w i t h
the wave generation r e l a t i o n s h i p s given by Sverdrup 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 obtain shallow water wave generation
r e l a t i o n s h i p s f o r wave height 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) investigated i n the l a b o r a t o r y
the generation of wind waves i n shallow 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 Schijf (1949) and the U . S. A r m y
Corps 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 published. H o w e v e r ,
i t has been ascertained that the data of Keulegan i s i n agreement w i t h that
obtained f o r Lake Okeechobee and is also i n agreement w i t h the r e l a t i o n ships presented by B r e t s c h n e i d e r (1954). Saville (1954) published 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 generation.
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 example, Unoki
and Nankano (1955) i n Japan published wave data f o r H a c h i j o I s l a n d ;
T i t o v (1955) presented w o r k s i n Russian; and B r e t s c h n e i d e r (1954) did
w o r k on shallow water 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) presented 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
postulated also that the R a y l e i g h d i s t r i b u t i o n applied to the wave length
v a r i a b i l i t y , w h i c h could be t r a n s f o r m e d into 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 length 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 agreement w i t h that postulated
by K r y l o v (1958) and was i n v e r y close agreement 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 presented 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 ating the d i m e n s i o n a l constant, and found that 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"^ instead 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 applied 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 into
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 spectra 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
9
g r e a t l y p r o m o t e d by the staff 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 assumption 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)
proposed 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 this 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 are given 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 are 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
accuracy of wave f o r e c a s t s .
However, d u r i n g the ten y e a r s preceding 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 expressions and l i t t l e to t h e o r y of wave
generation. D u r i n g the days of J e f f r e y s (1925), Sverdrup and M u n k (1947)
among others, the concept of wave spectra was not p r o m o t e d . However,
because of the wave spectra enlightenments of B a r b e r and U r s e l l (1948),
Seiwell (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 generation t h e o r y .
A t this point 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 spectra
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 proposed 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 proposed. 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 engineering
studies.
It 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 were developed. Sverdrup and M u n k (1947) considered that the wind was constant 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) considered the f a c t that the wind was r a p i d l y
f l u c t u a t i n g about some mean value. I t i s v e r y t r u e that winds blowing
over water do not consist 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 manner. 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 , are associated 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 stresses)
and tangential s t r e s s e s . The eddies are borne f o r w a r d by the mean v e l o c i t y
of the wind and, at the same t i m e , they develop, i n t e r a c t , and decay, so
that the associated s t r e s s d i s t r i b u t i o n moves across the s u r f a c e w i t h a
c e r t a i n convection v e l o c i t y dependent upon the v e l o c i t y of the wind and
also evolves i n t i m e as i t moves along.
10
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 water s u r f a c e that are
responsible f o r the e a r l y generation of waves. The tangential s t r e s s is
not considered, 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 is i n agreement w i t h
wave observations d u r i n g the e a r l y stages of generation, but as C / U
approaches unity there a r e other wave generating processes to take into
account, such as s h e l t e r i n g and the effects of v a r i a t i o n i n shear s t r e s s e s .
Although 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 unity, i t m a y be considered as a great 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 are
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 considerationsi'-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 water 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 , whereas 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 is 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 flow) is assumed 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 which the coupling 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 frequencies i n the s p e c t r u m the uncoupled
m o d e l w i l l govern and at others the i n s t a b i l i t y m o d e l w i l l govern. The
w o r k of M i l e s (I960) is a recognized c o n t r i b u t i o n on wave generation
theory.
I j i m a (1957) presented an exceltent paper on the p r o p e r t i e s of
ocean waves f o r tljie .Japanese area of i n t e r e s t . T h i s study included valuable
i n f o r m a t i o n on wave spectra obtained under typhoon conditions. A l s o a
decided d i f f e r e n c e existed between wave spectra 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.
Another i m p o r t a n t e f f o r t f o r obtaining wave spectra 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) presented the Russian 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) presented a s p e c t r u m of waves
at short fetches and found 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 his 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 generation.
11
I n p a r t i c u l a r , t h i s book includes 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
held at Easton, M a r y l a n d , the proceedings of which were published iri
1963.* T h i s conference included about 30 presentations, plus d i s c u s s i o n s ,
and had an attendance of less 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 scientists 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 spectra.
It can be said that t h i s conference brought 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 off 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 his data w e r e f o r v e r y short e f f e c t i v e fetches. 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).
Numerous discussions 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
throughout the f o u r - d a y p e r i o d . The conference produced 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 Smith (196iL, 1963)
and M u n k (1961,1963). On the last day, w i t h heads s t i l l spinning, 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.
However, the conference 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 wind 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 "On
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 Hasselmann
(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 that 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 cations. The w o r k of Hasselmann (1961,1963) is 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 are 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 nonlinear processes could become s i g n i f i c a n t . T h e r e is 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 (whitecaps), 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 Hasselmann
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 conference and
proceedings are shown by (1961, 1963),
12
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 conference, p r i o r to the p u b l i c a t i o n of the
proceedings.
A f t e r the conference was over, 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 this 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 computation 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 papers 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 included 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 mentioned 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 Observed 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 , Walden 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 somewhere among the other past _
proposed spectra, 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) presented data and analysis on wave
spectra obtained near the M e l l u m P l a t e Lighthouse, located i n the s o m e what s h e l t e r e d water o f f the N o r t h Sea coast b f Germany,
A good s u m m a r y of wave t h e o r y and wave generation is presented
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),
Numerous data r e p o r t s on wave spectra are now becoming a v a i l a b l e .
For example, 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 spectra 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) presented wave spectra f o r the A r g u s I s l a n d tower o f f
Bermuda,
B r e t s c h n e i d e r (1962) presented a concept on m o d i f i c a t i o n of wave
spectra over the continental shelf, and I j i m a (1962) presented 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) presented c o n t r i b u t i o n s to an analysis of the spectra of wind 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 continuation 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 water 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 wind c o n d i t i o n s . The s t a t i s t i c a l analysis of
the data showed that, f o r a l l p r a c t i c a l purposes, the R a y l e i g h d i s t r i b u t i o n
13
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 consist of long p e r i o d waves
a r r i v i n g f r o m deep water combined 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) presented 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 wind wave generation. The
law
o r i g i n a l l y proposed by P h i l l i p s (1957) was stated to be applicable 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 " "
£-8,94_
and
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
energy v a r i e s w i t h f""" where m = 9 f o r v e r y low gF^/U , and
decreases i n magnitude 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 there is agreement at i n i t i a l wave generat i o n f o r f""^ between Hamada (1963) and B r e t s c h n e i d e r (1959), and also
agreement at f u l l y developed wave generation 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,
ie
e'-^^''^ where 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) indicates that n m i g h t be l a r g e r than
n = 4. B r e t s c h n e i d e r (1961, 1963) states that n m i g h t v a r y between 4
and 8 or 9.
To date, a l l t h e o r i e s are u s e f u l i n a t t e m p t i n g to understand the
m e c h a n i s m s i n v o l v e d i n the generation of waves. None of the theories_
t e l l s us why waves are f o r m e d , let alone why the wave heights and p e r i o d s
are as observed or why t h e r e is a s p e c t r u m of waves. 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 accuracy 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 wind f o r e c a s t s .
14
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 :
e
CO t
mt
,
- 1
mt
• oo
TT (k,CO ) c o s
^
(14)
CO U COS ^
dT
- 1
where
CJu,,, cos
m
=
CJ ^
g
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 number k and t i m e T ; U is the convection
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>;< is 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 is the c o e f f i c i e n t calculated by M i l e s ( I 9 6 0 ) , and
and 7 ° are water and a i r densities 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 gene r a t i o n have been mentioned. E q . (14) r e p r e s e n t s the p r e s e n t state of
the a r t on wave s p e c t r u m generation 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 are 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 methods f o r wave hindcasting, B r e t s c h n e i d e r
(1964) presented a paper w h i c h takes into account the complete p r o b l e m of
deep water waves, s t o r m surge and waves over the continental 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, East Coast S t o r m . T h i s paper shows the r e s u l t s
based on present methods of wave hindcasting and also emphasizes the
areas 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 generation over the shallow water of the continental shelf is
that of the t o t a l water depth. The t o t a l water depth includes the combined
e f f e c t of o r d i n a r y tide 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 discussed 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 tides and s t o r m surge is given here since the subjects were w e l l discussed 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 , Groen, 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 is planned f o r
subsequent r e p o r t s .
15
II.
P R A C T I C A L APPLICATIONS - - DEEP WATER
A.
SIGNIFICANT W A V E CONCEPT
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 that
o r i g i n a l l y i n t r o d u c e d by Sverdrup and M u n k (1947) and is s o m e t i m e s
considered 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
presented by t h e m evolved 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
actual 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, this method 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 entails 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 is 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 height. I t
was found f r o m the analysis 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 height 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 reason there was s o m e t i m e s a c e r t a i n amount of agreement 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 around which is concentrated 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 height.
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 has been done by Johnson (1950), among o t h e r s ,
u t i l i z i n g the Buckingham 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 water
depend are the wind v e l o c i t y U, the f e t c h length F , and the wind
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/2TT)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)
H
(16)
and
= f^ (U, F , t , g)
17
Eqs. (15) and (16) state that C and H r e s p e c t i v e l y are 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 symbols appearing i n Eqs. (15) and (16) one m a y w r i t e
the dimensions f o r deep w a t e r as f o l l o w s ;
Dimensions
Symbol
C
L
H
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 is taken into
account.
F o r each of the above equations there are 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 solutions
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 respect to E q . (15) one can w r i t e
O = F J (CU^g^),
(FU^g"^).,
tU^g^)
(17)
W i t h r e s p e c t to equation (17) one obtains the dimensions
1 r
a .
-,b
and
1 r
e
T
T
LT^J
* I f one considers wave generation i n shallow w a t e r , Eqs. (15) and (16)
become
(15a)
C = f j (U, F , t , d , g )
and
H
where
= f^ (U, F , t , d , g)
d i s the water depth.
18
(16a)
Equating to unity 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
d - 1
b = 0
e = -1
c = -2
f = 1
Using values of the above exponents, E q . (17) becomes
O - F .
c'
u
\ 1
1^11
(18)
or
4
I n a s i m i l a r manner 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
(19)
u
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 is 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 used. 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
expressions 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 wise t h e r e are 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 generation 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/ ^
are f u n c t i o n a l r e l a t i o n s that 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 are 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
written ^
, w h i c h is a better f o r m because the wave p e r i o d is 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 Sverdrup 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, however, 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 available
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 method f o r Sverdrup, 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 generation 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 is 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 length, and was d e t e r m i n e d by 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)
tU
—
_
-
gt
U
•
gF
^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 water 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 length. F i g u r e 5, based cn the w o r k of Saville (1954), can be used
to calculate an e f f e c t i v e f e t c h length, based on actual f e t c h length 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 length 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
water.
B.
C O M P L E X N A T U R E OF SEA SURFACE
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 means
of dealing w i t h p r o b l e m s in 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 recognize that the sea is v e r y complex, 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 .
The
s i g n i f i c a n t wave height i s the average of the highest 1/3 of the waves m a
given wave t r a i n , of at least 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 heights, and is 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 . The P i e r son-Neumann-James ( P - N - J ) f o r e c a s t i n g
method considers a m e a n apparent wave p e r i o d and ranges m wave p e r i o d .
Both 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
20
FIGURE
3
FETCH
GRAPH
FOR
DEEP
WATER
FIGURE
4
DEEP WATER WAVE FORECASTING
CURVES
AS A F U N C T I O N OF WIND S P E E D ,
FETCH
L E N G T H , AND WIND D U R A T I O N
( FOR
TFTCHP?;
I rn
i oor»
n'it
<t>
WIND
WIND
1.0
OF
EFFECTIVE
FETCH:
OF
WIND
i.e.
OVER
30°
ONLY
FETCH••
OF
WIND
i.e.
0 V {: R
45°
ONLY
ElT H E R
90°
SIDE
DIRECTION^
60°
EITHER
DIRECTION.
EFFECTIVE
OF
SIDE
- V
0.9
w
0.8
<-
0.7
0.6
«
//
/
£
//
//
0.4
^
-
WIND
EFFECTIVE
OF
FETCH :
OF
WIND
i.e.
OVER
90°
ENTIRE
EITHER
180°
SIDE
DIRECTION.
y
^^CHANGE
IN
HORIZONTAL
SCALE
0.3
0.2
0.1
//
DASHED
/'
f
OVER
FULL
AS
r -
LINES
EACH
LINES
THt
INDICATE
(EQUAL
INDICATE
COSINt
WIND
SIZED)
WIND
I Hh
OP
CONSIDERED
SEGMENT
OF
EQUALLY
EFFECTIVENESS
I-XNüLfc
EFFECTIVE
FETCH.
CONSIDERED
WIND
TO
COMPONtN r
VARY
CONS I D E R E D .
1
O
0
0.1
02
0.3
0.4
0.5
0.6
0.7
0.8
RATIO . OF
FIGURE
5
RELATION
WIDTH-LENGTH
OF
0.9
1.0
FETCH
1.1
WIDTH
EFFECTIVE
RATIO
FOR
1.2
TO
1.3
LENGTH
FETCH
1.4
L5
2.0
2.5
^
TO
RECTANGULAR
FETCHES
3.0
3.5
4.0
O
O
I
UJ
O
O
I
UJ
a.
ü
CO
</)
O
t¬
I
<
X
O
3
O
cc
s a
UJ
V) OX
</)
ir
O
O
LJ
QSL
n
Lü
to O
z
O
o: O
O CC
O. I ¬
I
O
I
O
h¬
o: I
UJ
UJ
I
UI
O.
O
O
UI
CO
O
O
X
H
UJ
UJ cc
cr
D
O
i»7
Q:
UJ
cr
ü. UI
O.
3
O
_i
CD
UJ
ID
CS
24
m e t h o d uses a p r o b a b i l i t y d i s t r i b u t i o n also f o r wave p e r i o d . W i l s o n
(1955) i n t r o d u c e d t h é s p a c e - t i m e concept f o r f o r e c a s t i n g waves m m o v i n g
fetches and i n 1963 extended the w o r k f o r use on a h i g h speed c o m p u t e r .
F i g u r e 7 is r e p r o d u c e d f r o m W i l s o n (1955),
D r a p e r and D a r b y s h i r e (1963) presented r e l a t i o n s h i p s f o r f o r e casting m a x i m u m wave heights. F i g u r e s 8, 9, 10 and 11 are r e p r o d u c e d
f r o m D r a p e r and D a r b y s h i r e (1963) where i t should be kept i n m m d that
the m a x i m u m wave is equal to 1. 6 t i m e s the s i g n i f i c a n t wave height.
F i g u r e s 8 and 10 are f o r deep w a t e r and F i g u r e s 9 and 11 are f o r
coastal w a t e r s .
1.
Wave V a r i a b i l i t y
a. S i g n i f i c a n t Wave H e i g h t . The s i g n i f i c a n t wave height,
as mentioned above, is a t e r m c o m m o n to both the S - M - B and P - N - J
methods of wave f o r e c a s t i n g . Just how the s i g n i f i c a n t wave height i s
r e l a t e d to the p r o b a b i l i t y d i s t r i b u t i o n and also the wave s p e c t r u m can
best be i l l u s t r a t e d by f i g u r e s 12 and 13. F i g u r e 12 shows the d i s t r i b u t i o n of wave heights as v i s u a l i z e d by a h i s t o g r a p h , i . e. the number N
(or percent P of the t o t a l number) of waves i n each wave height range.
F i g u r e 13 i s a schematic d i a g r a m of the wave s p e c t r u m and m t h i s f o r m
i s c a l l e d the p e r i o d s p e c t r u m .
b
D i s t r i b u t i o n of Wave Heights. The s i g n i f i c a n t wave
height i s a s t a t i s t i c a l p a r a m e t e r , and about i b or 17 p e r c e n t of the waves
w i l l be h i g h e r than the s i g n i f i c a n t wave height. I t is shown by LonguetHiggins (1952), W a t t e r s (1953), K r y l o v (1956 and 1958), and D a r l i n g t o n
(1954) and v e r i f i e d by B r e t s c h n e i d e r (1959) that the d i s t r i b u t i o n of wave
heights f o r a n a r r o w s p e c t r u m i s given by the R a y l e i g h d i s t r i b u t i o n . The
p r o b a b i l i t y density f o r wave heights i s given by:
.
p (H) dH = S-
and the c u m u l a t i v e d i s t r i b u t i o n is
TT
P (H) =
where
TT
H_
e
1 - e
^
dH
(21)
'H^2
H
(22)
H is the i n d i v i d u a l wave height and H i s the average wave height.
The average wave height ïï , the s i g n i f i c a n t wave height H ^ j ,
and the average of the highest ten percent of the waves
as f o l l o w s :
25
H ^ Q , are r e l a t e d
9Z
00 g
m
m
—
O
O
O
I
O
* i
O
/AVE
HE I G H T
- H-(FE ET)
ETCH -
-n
1
z
\UTICAl.
MILE:
I
c
:^
z
O
•
WIND
VELOCITY
(KNOTS)
-J
Duration
Maximum
in
Hours
V/avo Height
/n feet
4
i
FETCH, NAUTICAL
MILES
(AFTER
FIGURE
8
GRAPH
TO
AND
RELATING
WIND
TO
SPEED
FETCH;
WAVE
AND
FOR
DRAPER
AND
HEIGHT
DURATION,
COASTAL
WATERS
DARBYSHIRE
1963)
Ourfltion in Hours
Maximum W a « Height
In Feet
40 50
MILES
FETCH. NAUTICAL
FIGURE
9
GRAPH
TO
AND
RELATING
WIND
TO
WAVE
SPEED
FETCH:
TOP
AND
^ ^^^^^
HEIGHT
DURATION,
a . . ^ - ' -
oARBVSH.RE
FETCH, NAUTICAL
MILES
(AFTER
FIGURE
10
GRAPH
WAVE
FOR
FETCH,
TO
WIND
OCEANIC
NAUTICAL
WATERS
DRAPER AND DARBYSHIRE 1963)
GRAPH
RELATING
PERIOD
TO
WAVE
AND
FETCH;
SPEED
MILES
(AFTER
M
1963)
DURATION, AND TO
FETCH;
FIGURE
AND DARBYSHIRE
RELATING
PERIOD
AND
DRAPER
WIND
SPEED
D U R A T I O N , A N D TO
FOR, COASTAL
WATERS
MOST PROBABLE H
•MEAN
P=33.3
H
FIGURE
H
12
.
STATISTICAL
DISTRIBUTION
OF
HEIGHTS
| > - 2 . ^ SIGNIFICANT
FIGURE
PERCENT
13
PERIOD
SPECTRUM
30
PERIOD
H
= .625 H
33
(23)
= 2. 03 H = 1. 27
The c u m u l a t i v e d i s t r i b u t i o n P(H)
is given i n F i g u r e 14,
c. D i s t r i b u t i o n of Wave P e r i o d s . The s i g n i f i c a n t wave
p e r i o d i s not a s t a t i s t i c a l p a r a m e t e r of the wave p e r i o d d i s t r i b u t i o n funC'
t i o n . The s i g n i f i c a n t wave p e r i o d has s t a t i s t i c a l significance only when
both wave heights and wave periods are c o n s i d e r e d . Putz (1952) obtains
a 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 v a r i a b i l i t y , K r y l o v
(1956 and 1958) postulates the R a y l e i g h d i s t r i b u t i o n also f o r the wave
lengths, and B r e t s c h n e i d e r (1959) shows that the d i s t r i b u t i o n of the
square of the wave p e r i o d s ( p r o p o r t i o n a l to the deep water wave lengths)
can be r e p r e s e n t e d a p p r o x i m a t 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 , a t r a n s f o r m a t i o n of w h i c h leads to the f o l l o w i n g d i s t r i b u t i o n f u n c t i o n f o r wave
periods:
p (T) dT
3
^
= 2.7
-0,675'-'
dT
e
(24)
and the c u m u l a t i v e d i s t r i b u t i o n is
-0.675
P (T)
1 - e
^
IT
(25)
The c u m u l a t i v e d i s t r i b u t i o n P ( T ) i s also given i n F i g u r e 14. K o r n e v a
(1961) gives r e s u l t s w h i c h v e r i f y the 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 of
wave height and p e r i o d .
d. The W e i b u l l D i s t r i b u t i o n F u n c t i o n . I n m a n y cases i t
m a y be d e s i r a b l e to r e p r e s e n t e m p i r i c a l data by means of a s i m p l e
d i s t r i b u t i o n f u n c t i o n . W e i b u l l (1951) proposes a simple a n a l y t i c a l
d i s t r i b u t i o n f u n c t i o n f o r use i n c e r t a i n c i v i l engineering p r o b l e m s as
follows:
(26)
where P
B
i s the c u m u l a t i v e d i s t r i b u t i o n
and
X ^
m
are constants
0 is the v a r i a t e ,
i . e. H , L , or
The p r o b a b i l i t y density i s given by
31
T
as the case m a y be,
Percent
FIGURE
14
DISTRIBUTION
accumulative
FUNCTIONS
FOR
PERIOD
.m
(27
I t can be seen that when m = 2, the W e i b u l l d i s t r i b u t i o n f u n c t i o n i s the
same as the R a y l e i g h d i s t r i b u t i o n ( E q . 21 or 22) and when m = 4 the
W e i b u l l d i s t r i b u t i o n f u n c t i o n is the same as E q . 24 or 25. Thus the
W e i b u l l d i s t r i b u t i o n f u n c t i o n m i g h t be suitable f o r i n v e s t i g a t i n g wave
v a r i a b i l i t y when the data deviate somewhat f r o m the R a y l e i g h d i s t r i b u t i o n .
The m o m e n t s can be generated f r o m
m
where
P
n
ü
m
p (X) dX
=
— I
1+Ü
m
(28)
r e p r e s e n t s the Gamma f u n c t i o n .
If the t e r m s of E q . (26) a r e r e a r r a n g e d and the l o g a r i t h m i s taken
t w i c e , one obtains
i
in in
E q . (29) is that of a s t r a i g h t l i n e .
(29)
n B +m i n X
When i n i n y ^ p "
:-plotted against
i n X , the i n t e r c e p t becomes i n B and the slope óf the l i n e i s given by
m . P is the c u m u l a t i v e p r o b a b i l i t y d i s t r i b u t i o n , c a l c u l a t e d f r o m the
data and B i s r e l a t e d to the m e a n wave height or m e a n wave p e r i o d as
the case m a y be.
—
~2
F r o m E q . (28), m ^ = 0 , m ^ = X ,
:= X , etc. I t then
f o l l o w s that
m
(30)
1 + —
X s H = T =
m
or
1 +
B
m
^
m
X
Thus Eqs. (26) and (27) become
m
P
= 1
_x
33
-
(31)
m-1
P (X) =: m
2L
X
1
m
r
2£
X
_1_
X
r
1
1+ m
m
(32)
By use of E q , ( 3 i ) , E q , (29) becomes
in
in
m in
1 - P
r(i + A )
m /
+ m i n
(33)
I t m u s t be emphasized that the W e i b u l l d i s t r i b u t i o n should be used
only where the t r e n d of the data shows a n e a r l y lineair r e l a t i o n s h i p a c c o r d ing to Eq, (29) or E q . (33), I f such a l i n e a r r e l a t i o n s h i p becomes apparent
f r o m the data, then a g r a p h i c a l s o l u t i o n i s possible, and a m o r e accurate
solution can be obtained by the s t a t i s t i c a l method of least squares.
A l t h o u g h the W e i b u l l d i s t r i b u t i o n has no t h e o r e t i c a l b a s i s , the f u n c t i o n
does have a wide range f o r p r a c t i c a l a p p l i c a t i o n s . F o r example. F i g u r e s
15 and 16 r e p r e s e n t an analysis of wave data a c c o r d i n g to the W e i b u l l
d i s t r i b u t i o n . These data a r e based on long wave r e c o r d s obtained at
L a k e T e x o m a by U . S, A r m y engineers (1953) and are s u m m a r i z e d i n a
r e p o r t by B r e t s c h n e i d e r (1959). The s t r a i g h t l i n e s given i n F i g u r e s 15
and 16 w e r e f i t t e d v i s u a l l y , although a least squares f i t m i g h t have been
made to obtain better a c c u r a c y ,
e. Joint D i s t r i b u t i o h of Wave Heights and P e r i o d s . The
j o i n t d i s t r i b u t i o n of wave heights and p e r i o d s is quite c o m p l e x , except
f o r the special case of zero c o r r e l a t i o n between wave height and wave
p e r i o d . F i g u r e 17 is a scatter d i a g r a m of
ï j = H / Ï T and A = T ^ / ( T ) ^
f o r a case v e r y n e a r l y zero c o r r e l a t i o n . I f the number of wave heights is
summed independently of the wave p e r i o d s , one obtains the m a r g i n a l
d i s t r i b u t i o n of wave heights, and i f the number of wave p e r i o d s squared
is summed independently of the wave heights, one obtains the m a r g i n a l
d i s t r i b u t i o n of wave p e r i o d s squared. Both of the above d i s t r i b u t i o n s
are r e p r e s e n t e d a p p r o x i m a t e l y by the Rayleigh d i s t r i b u t i o n . F o r the case
of zero c o r r e l a t i o n between wave height and wave p e r i o d , the j o i n t d i s t r i bution i s given d i r e c t l y by
p ( H , T ) = p (H) • p (T)
(34)
Based on Eqs, (21), (24), and (34), the j o i n t pr_obability d i s t r i ¬
butions have been c a l c u l a t e d f o r v a r i o u s ranges of H / H and T / T and
are s u m m a r i z e d i n Table 1. T h i s table assumes zero c o r r e l a t i o n between
H and T ^ . Gumbel (I960) discusses the b i v a r i a t e exponential d i s t r i butions, t a k i n g into account c o r r e l a t i o n c o e f f i c i e n t s d i f f e r e n t f r o m zero,
Eq. (34) can be c o n s i d e r e d a p p r o x i m a t e l y c o r r e c t f o r p r a c t i c a l
engineering uses, except that the l i m i t of b r e a k i n g waves should always
be checked. The b r e a k i n g wave l i m i t i s given t h e o r e t i c a l l y by M i c h e
(1954) as f o l l o w s :
H_
tanh
L
34
2-rrd
L
(35)
3
« /
2
9
/a
/
/
TH
In l n ( j r p ) = - C(.075+-J-In H
X 15
us:
P= l" e
VH
Y
9 /
-2
•
/
-3
-2.0
-1.0
0
I
2
In H
FIGURE
15
S A M P L E W E I B U L L DISTRIBUTION
DETERMINATION FOR WAVE HEIGHT
(DATA FROM U.S. ARMY ENGINEERS, 1953)
35
3
/
2
//
In In ( T r p ) = - 3 . 3 5 + 3 . 4 In T
THUS!
PM-e-°.e3(i)O
/•
-2
-3
*/
-4
•/
-
2
-
1
0
1
2
In T
FIGURE
16
SAMPLE
WEIBULL
DETERMINATION
(DATA
FOR
DISTRIBUTION
WAVE' PERIOD
FROM U.S. ARMY ENGINEERS, 1953)
36
3.0
1
r
2.8
2.6
2.4
2.2
TL
2.0
e»
e
e «
ee
e
^
..4
•• •
e
•
e
O 1.2
•.Vv •
•.:->v.
.a
V|»
i.o
SI
«
.8
• •
•
•
e
c:-.:: •
6
•
•
• • e « «fe
.4
.
.2-
I
V .
•«
.2
FIGURE
_L
4
<
•
.6
IZ
e
.8
±
1.0
1.2 1.4 1.6
1.8
2.0
Relative Wave L e n g t h , X
2.2
2.4
SCATTER
DIAGRAM OF
V
AND A FOR
400
CONSECUTIVE
WAVES FROM
THE
GULF OF M E X I C O
17
37
2.6
2.8
3.0
TABLE 1
JOINT D I S T R I B U T I O N O F H AND T F O R Z E R O C O R R E L A T I O N
Number of Waves P e r I , 000 Consecutive Waves for Various Ranges in Height and P e r i o d
Range in
Relative
Heig_ht
H/H
RANGE
00.2
0.20.4
0.40.6
O.-O. 2
2-0. 4
4-0. 6
6-0. 8
8-1. 0
0-1. 2
2-1. 4
4-1. 6
6-1. 8
8-2. 0
0-2. 2
2-2. 4
4-2. 6
6-2. 8
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
1.
2.
2.
2.
2.
1.
1.
0.
0.
0.
0.
0.
0.
2.
5.
8.
9.
9.
8.
7.
5.
3.
1.
1.
0.
0.
0.
0-3. 0
1. 09
16. 06
66. 44
157. 46
248. 65
Cumulative
1.09
17.15
83.59
241.05
489.70
0.
0.
0.
0.
1.
1.
1.
1.
1.
2.
2.
2.
2.
03
10
14
16
16
15
12
09
06
03
03
01
01
50
41
06
40
40
14
74
30
90
48
42
18
09
04
05
81
54
91
92
87
21
37
72
99
72
76
39
18
0.60.8
TN R E L A T I V E
4.
13..
20.
23.
23.
21.
17.
12.
8.
4.
4.
1.
0.
0
86
78
23
48
51
02
07
72
82
72
09
80
93
43
0.81.0
7.
21.
31.
37.
37.
33.
26.
20.
13.
7.
6.
2.
1.
0.
68
76
95
08
13
19
96
09
93
45
45
84
47
67
1.01.2
PERIOD
1.41.6
1.61.8
1.82.0
31
05
10
65
69
96
65
90
64
15
47
97
02
47
1. 92
5. 44
7. 99
9. 27
9. 28
8. 30
6. 74
5. 02
3. 48
1. 86
1.61
0.71
0. 37
0. 17
0.
0.
1.
1.
1.
1.
1.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
0.
262. 93
172. 03
62. 16
752.63
924.66
8.
23.
33.
39.
39.
34.
28.
21.
14.
7.
6.
2.
1.
0.
09
92
65
06
11
97
40
16
67
85
80 99
55
71
1.21.4
T / T
5.
15.
22.
25.
25.
22.
18.
13.
95.
-4,
1.
1.
0.
34
98
44
67
67
49
21
90
63
33
29
13
07
03
03
07
11
12
12
11
09
07
05
03
02
01
11. 18
0. 83
986.82 998.00
99& 83
02.0
30. 81
88. 32
128.21
148.80
148.99
133.20
108.19
80. 62
55. 90
29- 89
25. 90
11. 40
5. 90
2. 70
Cumulative
30. 81
119.13
247.34
396.14
545.13
678.33
786.52
867.14
923. 04
952. 93
978. 83
990.23
996. 13
998.83
w h e r e the l i m i t i n shallow water i s given by the s o l i t a r y wave t h e o r y
J L
= 0.78 4 -
(36)
Based i n p a r t on t h e o r y and i n p a r t on e m p i r i c a l data, B r e t s c h n e i d e r
(1960) p r e s e n t s a r e l a t i o n s h i p of the b r e a k i n g wave l i m i t w h i c h is as
follows:
4L
1 + 0. 152 tanh
= 0.124 tanh
(37)
Values of H and T obtained by use of T a b l e I should always be
checked by use of the above equations.
C.
W A V E S P E C T R U M CONCEPTS
The o r i g i n a l wave s p e c t r u m concept f o r f o r e c a s t i n g waves is that
due to Neumann (1952). The wave s p e c t r u m m e t h o d r e s u l t i n g f r o m this
concept is that due to P i e r s o n , Neumann and James (1955). M u c h w o r k
has also been done w i t h r e g a r d to a wave s p e c t r u m m e t h o d by D a r b y s h i r e
(1952 and 1955). B r e t s c h n e i d e r (1959) has added a wave s p e c t r u m
approach to the s i g n i f i c a n t wave m e t h o d . A m o r e r e c e n t s p e c t r u m concept
has been proposed by P i e r s o n (1964), the f o r m of w h i c h is i n agreement
w i t h that d e r i v e d by B r e t s c h n e i d e r ( 1959). Whereas the s i g n i f i c a n t wave
m e t h o d is c o m m o n l y c a l l e d the S - M - B method, the P i e r s o n , Neumann
and James method i s c o m m o n l y c a l l e d the P - N - J m e t h o d .
Just as the S - M - B m e t h o d i s based on e m p i r i c a l wave data, so i t
i s w i t h the P - N - J m e t h o d . As m e n t i o n e d b e f o r e , any suitable wave f o r e ¬
c a s t i n g method m u s t be c a l i b r a t e d by use of wave data. The P - N - J m e t h o d
can be used to p r e d i c t the s p e c t r u m of waves f r o m w h i c h one m a y obtain
the s i g n i f i c a n t wave height as w e l l as the s t a t i s t i c a l d i s t r i b u t i o n of the
waves! The S - M - B m e t h o d i s used to p r e d i c t the s i g n i f i c a n t wave height,
f r o m w h i c h i t is possible to obtain the wave s p e c t r u m and the s t a t i s t i c a l
d i s t r i b u t i o n of the waves. Both methods u t i l i z e the d i s t r i b u t i o n f u n c t i o n
d e r i v e d t h e o r e t i c a l l y by L o n g u e t - H i g g i n s (1952). T h i s d i s t r i b u t i o n f u n c t i o n
i s i n v e r y close agreement w i t h the e m p i r i c a l r e l a t i o n s h i p s given by
P u t z (1952) based on the analysis of 25 ocean wave r e c o r d s .
Consequently,
when both S - M - B and P - N - J methods p r e d i c t exactly the same s i g n i f i c a n t
wave height, then the two methods r e s u l t i n exactly the same d i s t r i b u t i o n
of waves. I t m u s t be r e m e m b e r e d that both methods are based on actual
wave data. However, the wave data are not the same u t i l i z e d m the
two methods, and the m e t h o d of analysis of the data i s d i f f e r e n t . I t i e
analysis of the data f o r the S - M - B m e t h o d consists of d e t e r m i n i n g the
s i g n i f i c a n t wave height and p e r i o d , w h i c h i n t u r n are r e l a t e d to w i n d speed,
f e t c h length, and w i n d d u r a t i o n .
P r i o r to the development of the P - N - J m e t h o d , Neumann (1952)
p r o p o s e d a t h e o r e t i c a l wave s p e c t r u m based on a great abundance of
39
i n d i v i d u a l wave observations. T h i s s p e c t r u m should be c o n s i d e r e d as a
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 s p e c t r u m i n the s t r i c t sense of the
d e f i n i t i o n . F r o m his wind and wave observations Neumann (1952) computed
the p a r a m e t e r s
H/T
and T / V , and a r r i v e d at the f o l l o w i n g r e l a t i o n ;
y
-
const
e
(3ö)
(See f i g u r e 18..) The p a r a m e t e r H / T i s d i r e c t l y r e l a t e d to the^wave
steepness and T / V the wave age. H i s the i n d i v i d u a l height. T i s the
i n d i v i d u a l wave p e r i o d , w h i c h he c a l l e d the apparent wave p e r i o d , d e f i n e d
as the t i m e between two successive u p - c r o s s i n g s of the S t i l l w a t e r s u r f a c e
by the wave f o r m .
F r o m t h i s e m p i r i c a l r e l a t i o n s h i p (Eq. 38) and other c o n s i d e r a t i o n s ,
Neumann (1952) d e r i v e d his t h e o r e t i c a l wave s p e c t r u m of energy. F r o m
these r e s u l t s he concluded f o r u n l i m i t e d f e t c h length and d u r a t i o n ( f u l l y
developed sea) that the wave height was equal to some constant t i m e s the
w i n d v e l o c i t y to the 5/2 power, and the energy at h i g h f r e q u e n c y was f
Because of the method of s o l u t i o n , a d i m e n s i o n a l constant was i n t r o d u c e d ,
but the equation has been evaluated by c o n s i d e r i n g that the t o t a l energy
under the curve of the s p e c t r u m was p r o p o r t i o n a l to the energy.XDJ-ithe equivalent
r o o t m e a n square wave height. The d i m e n s i o n a l constant has c r e a t e d
objections by v a r i o u s i n d i v i d u a l s i n the f i e l d of wave f o r e c a s t i n g , p a r t i c u l a r l y those who adhere to d i m e n s i o n a l homogeneity. However, i f one
accepts the r e l a t i o n s as e m p i r i c a l r a t h e r than t h e o r e t i c a l , then these
r e l a t i o n s ought to be suitable f o r wave f o r e c a s t i n g , at least over the range
of p a r a m e t e r s f r o m which the data w e r e analyzed. The i m p o r t a n t t h i n g
i n his development of the s p e c t r u m is the f a c t that t o t a l energy of the
s p e c t r u m i s c o r r e c t , a s s u m i n g that the w o r k of L o n g u e t - H i g g i n s is
c o r r e c t , and t h i s appears to be the case. The actual d i s t r i b u t i o n of the
energy w i t h r e s p e c t to wave p e r i o d or f r e q u e n c y m i g h t not be e x a c t l y
c o r r e c t , but perhaps is i n close agreement w i t h the actual data f r o m
w h i c h i t was d e r i v e d . Not u n t i l m u c h a d d i t i o n a l data become available
w i l l i t be possible to r e c o n c i l e any d i f f e r e n c e s between NeumannLs wave
s p e c t r u m and a m o r e c o r r e c t s p e c t r u m .
R o l l and F i s c h e r (1956) suggested a d i f f e r e n t d e r i v a t i o n of the
Neumann s p e c t r u m . I n t h i s d e r i v a t i o n there r e s u l t e d no d i m e n s i o n a l
constant, the wave height f o r f u l l y developed sea beca,me p r o p o r t i o n a l
2
-5
to U , and the energy at h i g h f r e q u e n c y was f
The concept of the wave s p e c t r u m i s c e r t a i n l y a great advance i n
t r y i n g to understand the n a t u r e of wave generation. I t i s shown, f o r
example, i n spite of the objections to the Neumann s p e c t r u m , that the
waives generate f r o m the high f r e q u e n c y end of the s p e c t r u m , and w i t h
t h i s concept Neumann (1952) d e r i v e d f r o m his s p e c t r u m r e l a t i o n s h i p s
40
i
FIGURE
18
RATIO
SQUARE
OF
OF
WAVE
APPARENT
41
HEIGHTS
WAVE
TO
THE
PERIODS,
% 2
f o r the s o - c a l l e d young or t r a n s i e n t state of sea. That i s , r e l a t i o n s c a l l e d
CO-cumulative power s p e c t r a are developed f r o m w h i c h i t is possible to
p r e d i c t E - v a l u e s , where E is r e l a t e d to the generated wave energy.
The t h e o r e t i c a l wave d i s t r i b u t i o n d e r i v e d by L o n g u e t - H i g g i n s
(1952) i s d i r e c t l y r e l a t e d to the E - v a l u e s , as t h i s c o n d i t i o n was u t i l i z e d
i n the development of the wave s p e c t r u m . The s i g n i f i c a n t wave height,
mean wave height, etc. are r e l a t e d to E as f o l l o w s :
H
1.772
ave
Hj/3
= 2.832
fË,
(39)
JT"
^ 1 / 1 0 - ^-600 . / E
T y p i c a l examples of 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 based upon the above:
concept are given i n F i g u r e s 19 and 20.
The s p e c t r u m proposed by B r e t s c h n e i d e r (1959) was obtained by
use of a t h e o r e t i c a l f u n c t i o n f o r 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
heights and p e r i o d s . The p e r i o d s p e c t r u m was obtained as f o l l o w s :
H
(40)
p ( H , T) dH
where the i n t e g r a t i o n is over a l l wave heights as a f u n c t i o n of wave p e r i o d .
A s s u m i n g no c o r r e l a t i o n between H and T :
p(H,T)
p(T)
= p(H) • p(T) , whence
ƒ H^p(H)dH
-
H^P(T)
= ±.(H)^p(T)
(41)
(42)
By use of E q , (24) one obtains the p e r i o d s p e c t r u m :
.3
-.675
3. 43 (H)^
(43)
(T)
F o r the f r e q u e n c y s p e c t r u m :
T =
1
and dT =
T
df
f r o m w h i c h the f r e q u e n c y s p e c t r u m s becomes:
Sj,(f)
-
3.43
2
(H)' £-5 ^ - : 6 7 5 ( f T ) - ^
(T)'
42
(44)
FIGURE
19
DURATION
GRAPH
CO-CUMULATIVE
SPECTRA
FOR
WIND
SPEEDS FROM
2 0 TO 3 6
KNOTS
AS A FUNCTION
OF
DURATION
43
I I I I I
I
28.0
I
I
.06
•OA
I
ZOO
.08
I
I
16.0 14.0
.26 f=J4
.10
I
I
120
IQO
I
9.0
I
4.0 T
80
( H.0. PUB 6 0 3 , 1957)
FIGURE
20
FETCH GRAPH
CO-CUMULATIVE
SPECTRA
FOR
WIND S P E E D S FROM 2 0 TO 3 6
KNOTS
AS A F U N C T I O N OF
FETCH
44
0
If one lets
gH
_
625 gH33
(45)
U
and
(46)
2 TTU
i t then f o l l o w s that
675
gF.
Sj,(f) =
27T Uf
g
f-^e
3.43
(47)
(2rrFj'
F o r the h i g h f r e q u e n c y end of the s p e c t r u m S^(f) i s p r o p o r t i o n a l
to f " ^ T h i s is i n agreement w i t h the r e s u l t s of R o l l and F i s c h e r (1956)
and P h i l l i p s (1957) a ! w e l l as the n e w l y proposed s p e c t r u m of P i e r s o n and
M o s k o w i t z (1963).
Other f o r m s of the wave spectra have been suggested by B r e t schneider i n the Proceedings of the ConLerence on Ocean Wave Spectra^
(1963). The c o r r e s p o n d i n g T ^ T C T ^ Ï Ï H T r e q u e n c y s p e c t r a are as f o l l o w s .
S(T)
= a T ^ e-^T^
S(f)
-
or
(48)
•n
(49)
a f - " ^ - ^ e-^^
F o r the W e i b u l l d i s t r i b u t i o n f u n c t i o n ( m = n - 1) one can propose
the f o l l o w i n g f o r m of wave spectra:
m+l
- m •1
m
(50)
S(f/g = (f/f„)""'"^e
where the peak of the s p e c t r u m occurs at f = f ^ .
f o r m the above equation becomes
ill+i
m
S(f) = - E
where
S(f) df
E =
F o r the c u m u l a t i v e
( f / f )-«^
o
(51)
(52)
•'oo
P i e r s o n and M o s k o w i t z (1963), 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) propose the f o l l o w i n g f o r m of wave spectra f o r a
f u l l y developed sea:
45
(53)
S(CJ) dCO =
CO
where
CO
=
2TT£
.5
COo '
= 8. 1 X 10 -3
ai
B
= . 74
Uj^g g was the m e a s u r e d w i n d speed at 19. 5 m e t e r s above m e a n
sea l e v e l , and was used d i r e c t l y instead of m a k i n g an a d j u s t m e n t to the
standard anemometer l e v e l of 10 m e t e r s . T h i s was to avoid selecting
someone elses drag c o e f f i c i e n t . A c c o r d i n g to the Weather B u r e a u
c r i t e r i a , the wind speed over water at elevation 19. 5 m e t e r s is about
1. 1 t i m e s that at e l e v a t i o n 10 m e t e r s .
If E q . (53) i s i n t e g r a t e d to obtain the t o t a l energy and r e l a t e d to
the s i g n i f i c a n t wave height, one obtains
gH/uJg_5 =
.21
or
gH/U^Q
w h i c h is v e r y n e a r l y equal to
(1947).
=
0.254
g H / U ^ = . 26 used by Sverdrup and M u n k
Eqs. (47) and (53) are d i r e c t l y r e l a t e d , any d i f f e r e n c e s r e s u l t i n g
only f r o m the e m p i r i c a l r e l a t i o n s f o r CO, 60^,
and
as f u n c t i o n s
of wind speed, and the c o r r e s p o n d i n g i n t e r p r e t a t i o n s of the w i n d speed,
D.
FROUDE SCALING OF T H E WAVE SPECTRUM
A s s u m i n g that the Froude l a w applies f o r wave spectra, i t i s
possible to estimate design wave s p e c t r u m by the concept of F r o u d e
scaling. The design wave p a r a m e t e r , f o r example, the s i g n i f i c a n t wave
height, can be obtained f r o m the analysis of the c o m p i l e d wave s t a t i s t i c s
(wave s t a t i s t i c s can be c o m p i l e d by long t e r m m e a s u r e m e n t s or by use
of wave hindcasts). M e a s u r e d wave spectra are available f o r c e r t a i n
s t o r m c o n d i t i o n s , f o r example, B r e t s c h n e i d e r , et. a l . (1962). F o r t h e
design s t o r m , once i n 100 y e a r s , f o r example, the s i g n i f i c a n t wave height
can be c o n s i d e r a b l y l a r g e r than that c o r r e s p o n d i n g to a m e a s u r e d wave
spectrum.
46
I n a r e p o r t by B r e t s c h n e i d e r and C o l l i n s (1964), the nnost severe
h u r r i c a n e w h i c h m i g h t occur i n the A t l a n t i c Ocean can generate a 62-foot
s i g n i f i c a n t wave height. The m e a s u r e d wave s p e c t r u m of B r e t s c h n e i d e r ,
et. a l . (1962), was used to estimate the c o r r e s p o n d i n g wave s p e c t r u m f o r
the 6 2 - f o o t s i g n i f i c a n t wave height. The f o l l o w i n g table gives s i g n i f i c a n t
wave heights based on f i v e m e a s u r e d wave spectra. F o r c o n v e r s i o n to
the 6 2 - f o o t s i g n i f i c a n t wave height the Froude length scale is obtained
from
A = 62 (H measured) and the c o r r e s p o n d i n g t i m e scale i s
-r =jT.
Table I I
A = 62 -;- Hg
Time
Date
17 Dec 1959
11
0600
35. 2
1. 76
1. 33
0900
34. 4
1. 80
1. 34
11
1500
33. 9
1. 83
1. 35
11
1800
39. 7
1. 56
1. 25
18 Dec 1959
OOOO
35. 3
1. 76
1. 33
To c o n v e r t the m e a s u r e d f r e q u e n c y s p e c t r u m to design s p e c t r u m ^ ^
the o r d i n a t e S(f) , having d i m e n s i o n f t ^ s e c , , m u s t be m u l t i p l i e d by
and the abscissa f , having d i m e n s i o n
A
s e c " ^ m u s t be m u l t i p l i e d by
A
To c o n v e r t the c o r r e s p o n d i n g p e r i o d s p e c t r u m to design s p e c t r u m
the o r d i n a t e
S(T), having d i m e n s i o n f t ^ s e c " \ m u s t be m u l t i p l i e d by
X •^''^ and the abscissa
by
T , having d i m e n s i o n sec,
m u s t be m u l t i p l i e d
A
I f the design s i g n i f i c a n t wave w e r e d i f f e r e n t f r o m H = 62 feet, then
A and X would change a c c o r d i n g l y .
(Data tabulated by f r e q u e n c y analysis is o f t e n given f o r i n t e r v a l s
of A f- I n o r d e r to c o n v e r t such data to a design spectrurn^^one m u s t
make an a d d i t i o n a l c o n v e r s i o n by m u l t i p l y i n g A £ by
A
• )
F o r a number of reasons the p e r i o d s p e c t r u m has a m o r e p r a c t i c a l
a p p l i c a t i o n than the f r e q u e n c y s p e c t r u m . A s i m p l e o p e r a t i o n c o n v e r t s the
f r e q u e n c y s p e c t r u m to the p e r i o d s p e c t r u m as f o l l o w s ;
S(T) dT
= - S(f) df
47
(54)
where
f
df
=
l/T
= - 1/T^ dT
Thus
S(T)
= f^S(f)
(55)
F i g u r e Z l shows the F r o u d e scaling of the p e r i o d s p e c t r u m f o r
Hg = 6Z feet as given i n T a b l e I I , The Froude scaled spectra i n F i g u r e
Z l are shown s u p e r i m p o s e d on the t h e o r e t i c a l s p e c t r u m given by E q . (43)
The s i g n i f i c a n t v a r i a b i l i t y at some f r e q u e n c i e s is apparent. The theor e t i c a l s p e c t r u m by E q . (43) can be considered as y i e l d i n g the mean
s p e c t r a l energy density i n any f r e q u e n c y band, but w i t h i n such a band an
i n d i v i d u a l ordinate w i l l v a r y a c c o r d i n g to some skew-type ' d i s t r i bution. The areas under a l l spectra given i n F i g u r e Z l are i d e n t i c a l and
a r e r e l a t e d to the same value of the s i g n i f i c a n t wave height, but only the
i n d i v i d u a l ordinates v a r y . F o r example, i f a given ordinate v a r i e s
a c c o r d i n g to the R a y l e i g h d i s t r i b u t i o n , then the upper and l o w e r 95%
confidence l e v e l s would be r e s p e c t i v e l y a l m o s t t w i c e and half of the
mean values.
In F i g u r e Z l the l a r g e i n s t a b i l i t i e s i n the scaled observed spectra
at l o w p e r i o d s (high f r e q u e n c i e s ) are not quite so apparent i n the frequenc
s p e c t r u m . These i n s t a b i l i t i e s are p r o b a b l y caused by the i n t r o d u c t i o n
of noise i n the o r i g i n a l r e c o r d , e i t h e r by sample rate or i n s t r u m e n t i n s e n s i t i v i t y , o r , m o s t l i k e l y a c o m b i n a t i o n of both. Since t h e r e is a
smoothing p r o c e s s ( r e l a t e d to A f) f o r the f r e q u e n c y s p e c t r u m , i t
becomes apparent that a s i m i l a r smoothing process ( r e l a t e d to A T)
should also be applied to the p e r i o d s p e c t r u m . T h i s is because A T
is not l i n e a r l y r e l a t e d by a constant to l / A f - A d d i t i o n a l studies are
r e q u i r e d to i n v e s t i g a t e the d i s p a r i t y observed between spectra i n t e r m s
of A f and those i n t e r m s of ^ T .
48
6f
SPECTRAL
Ci
C
m
ro
o
O
Tl
O
-D
>
cn
ro
o
I
o
"Tl •n
O
O
m
cn
O
o
D
05
o
>
H
S
O
H
C
•D
m
o
o
</)
m
o
o
z
O
ENERGY DENSITY
IN
FT/^/SEC.
III.
P R O P A G A T I O N OF WAVES A N D SWELLS I N T O
SHALLOW WATER
When waves or swells propagate into shallow w a t e r , a number of
m o d i f i c a t i o n s take place: r e f r a c t i o n , shoaling and energy l o s s e s . These
f a c t o r s w i l l be discussed subsequently, c o n s i d e r i n g only a s i m p l e s i n u soidal wave. The f a c t that waves t r a v e l i n groups of v a r i a b l e a m p l i t u d e
and f r e q u e n c i e s - - the wave s p e c t r u m - - entails additional c o n s i d e r a t i o n s
discussed l a t e r .
I f the waves are l o n g - c r e s t e d and a r e m o v i n g obliquely t o w a r d s a
s t r a i g h t s h o r e l i n e whose depth contours are also s t r a i g h t and p a r a l l e l to
the shore, those p o r t i o n s of the wave f r o n t which e f f e c t i v e l y f e e l b o t t o m
f i r s t are r e t a r d e d f i r s t ; t h e r e f o r e , the wave becomes subject to a p r o gressive c u r v i n g or r e f r a c t i o n w h i c h , i n i t s o v e r a l l e f f e c t , tends to a l i g n
the wave f r o n t to the depth c o n t o u r s . T h i s is i n analogy to simple l i g h t
rays of a p a r t i c u l a r wave length when t r a v e l i n g f r o m one m e d i a to a
m e d i a of d i f f e r e n t density. The l i g h t is bent or r e f r a c t e d . When a l l the
wave lengths f o r the s p e c t r u m of l i g h t t r a v e l f r o m one m e d i a to a m e d i a of
greater density, each wave length is r e f r a c t e d to a d i f f e r e n t degree, w h i c h
r e s u l t s i n s o r t i n g of the c o l o r s . S i m i l a r l y , when the s p e c t r u m of ocean
waves enters a r e f r a c t i n g a r e a , the d i f f e r e n t f r e q u e n c i e s or wave lengths
are sorted, and the r e s u l t i n g s p e c t r u m is changed a c c o r d i n g l y . F i g u r e
22 i l l u s t r a t e s the e f f e c t of r e f r a c t i o n f o r a s i m p l e wave.
The orthogonals (wa.ve rays) r e p r e s e n t the d i r e c t i o n that the wave
f r o n t s are t a k i n g . They become c u r v e d i n the p r o c e s s of r e f r a c t i o n and
i n general m a y tend to d i v e r g e cr converge. Over a submarine canyon
( F i g u r e 23) the waves w i l l always tend to diverge., and c o n v e r s e l y , over a
submarine r i d g e ( F i g u r e 24), the waves w i l l always tend to converge.
I t is g e n e r a l l y assumed that the wave energy contained between
orthogonals r e m a i n s constant as the wave f r o n t p r o g r e s s e s ; t h i s supposes
that there is no d i s p e r s i o n of energy l a t e r a l l y along the f r o n t , no
r e f l e c t i o n of energy f r o m the r i s i n g b o t t o m , and none l o s t by other
p r o c e s s e s . I f b^ r e p r e s e n t s the distance between orthogonals i n deep
w a t e r , and b
the distance between orthogonals somewhere i n shallow
water (where the c o r r e s p o n d i n g wave heights are
and H ) , since
O
X
the energy of the wave is p i o p o r t i o n a l to the square of the wave height,
i t f o l l o w s that
bH^Cg
= b H ^ C g
or
H
= H
X
Ih Ih ' jcg
/Cg '
O v/ O X vy ®0 ^X
60
H
(56)
COAST
FIGURE 2 2
REFRACTION
51
EFFECT
FIGURE
23
DIVERGING
52
ORTHOGONALS
COAST
WAVE
FRONTS
ORTHOGONALS
BOTTOM CONTOURS
FIGURE
24
CONVERGING
53
ORTHOGONALS
where
/ b /b
= K
, the r e f r a c t i o n c o e f f i c i e n t . The above is a
s i m p l i f i e d example and the general mechanics of t r a n s f o r m a t i o n f o r a
s i m p l e wave f o l l o w .
A s s u m i n g that the gradient of the sea bed is not so steep that
wave energy i s r e f l e c t e d , and that t h e r e is no gain or loss of energy due
to l a t e r a l d i f f r a c t i o n or d i s p e r s i o n , then the wave energy t r a n s m i t t e d
between orthogonals r e m a i n s the same, except f o r loss due to b o t t o m
f r i c t i o n and p e r c o l a t i o n i n the p e r m e a b l e bed. F r i c t i o n a l losses r e p r e s e n t
the energy expended i n o v e r c o m i n g r e s i s t a n c e , shea,r s t r e s s e s or d r a g
f o r c e s on the sea b o t t o m . P e r c o l a t i o n losses r e p r e s e n t energy l o s t
t h r o u g h actual i n f i l t r a t i o n of the wave m o t i o n into the s e m i - f l u i d m a s s of
c e r t a i n f i n e l y suspended sediments of the permeable sea. b o t t o m .
If
P^ and Pp r e p r e s e n t the f r i c t i o n a l and p e r c o l a t i o n energy
losses per u n i t a r e a of b o t t o m per unit time^ r e s p e c t i v e l y ; then f r o m
F i g u r e 25, w h i c h shows the e l e m e n t a l length
^ x o£ r i s i n g sea b o t t o m
i n the d i r e c t i o n of m o t i o n , the energy loss ( t i m e r a t e ) , i n t e r m s of power
(P) t r a n s m i t t e d a c r o s s the v e r t i c a l section bh between the orthogonals
w i l l be:
(P^+PJb^x
=
[pb -
( 5 ( P b ) ] - Pb
(58)
whence
d(Pb)
dx
The wave energy
_ _(p
+ p )b
f '
p'
(59)
E per unit s u r f a c e a r e a is given by
E
= -1 / g H ^
(60)
w h i c h is t r a n s m i t t e d a c r o s s the unit a r e a w i t h the group v e l o c i t y
P
= E C
C
; i . e.
ê
The group v e l o c i t y changes f r o m deep water into shallow
.
ê
w a t e r a c c o r d i n g to
C
- nC
(61)
g
where C is the wave c e l e r i t y and n is the t r a n s m i s s i o n c o e f f i c i e n t
and, a c c o r d i n g to l i n e a r wave t h e o r y , is given by
1
where
k
2 77/L,
1+
sinh 2 kh
(62)
I t then f o l l o w s that the power of t r a n s m i s s i o n i s
Pb
= Eb C
54
=
ƒ g
b (nC)
(63)
FIGURE
25
RISING
EXPERIMENTAL
SEA
BOTTOM
LENGTH
IN DIRECTION
B%
OF
OF
MOTION
I n deep water, j u s t b e f o r e the waves are due to f e e l the e f f e c t s of
depth, E q . (63) becomes
Pb
=
OO
(64)
~ / ° g H ^ b n C
ö y ° o
c o c
The r a t i o of Eqs. (63) and (64) r e s u l t s i n
/ b \ ^ ' ^ i n C.
o o
nC
H
H
1/2 ,
1/2
(65)
w h i c h ma,y be expressed as
H
H~
where
K
, K and
r '
s
fp
= K K
r
s
(66)
fp
are known as the r e f r a c t i o n c o e f f i c i e n t , the
,
shoaling c o e f f i c i e n t , and the f r i c t i o n - p e r c o l a t i o n c o e f f i c i e n t , respectively^
The r e f r a c t i o n c o e f f i c i e n t is separately d e f i n e d as
(67)
K
the square r o o t of the r a t i o of the widths between orthogonals i n deep and
shallow w a t e r . A c c o r d i n g to linea.r t h e o r y ,
depends only on wave
p e r i o d , water depth contours and i n i t i a l deep water wave d i r e c t i o n . I n
p r a c t i c a l coastal engineering p r o b l e m s , r e f r a c t i o n c o e f f i c i e n t s can be
d e t e r m i n e d by one of two methods: wave f r o n t and d i r e c t ray or orthogonal
methods, both of w h i c h are d i s c u s s e d i n B, E . B , T . R. 4 ( 1 9 6 1 ) .
The shoaling c o e f f i c i e n t is given c o m p l e t e l y , a c c o r d i n g to l i n e a r
wave t h e o r y , by:
(68)
K
The wave c e l e r i t y C i n shallow water is r e l a t e d to
C
and i n deep water
C
o
i n deep water by
tanh kh
(69)
n^ = 1/2, whence f r o m Eqs, (63), (68), and (69)
-1/2
K
2 kh
1+
sinh 2 kh
tanh kh
is then given e x p l i c i t l y as a f u n c t i o n of h / L or h / L ^
K
obtained f r o m tables by W i e g e l (1954) where K,, = H / H
56
(70)
and m a y be
The f r i c t i o n - p e r c o l a t i o n c o e f f i c i e n t K^^ i s delined as
1/2
^
>
fp
(71)
o o
R e t u r n i n g to E q . (59) and s u b s t i t u t i n g Pb
d ( K . ^)
J ^
dx
p b
o o
=
frona Eq. (71), one obtains
- (D. + D ) b
i
p
(72)
or
A c c o r d i n g to P u t n a m and Johnson (1949)
*
^
T
(sinh kh)-^
and according to P u t n a m (1949)
D
p
.
±Jll. ,rr,2
^T
.gPH^^
smh 2 k h
(75)
In Eqs. (74) and (75)
= m a s s density of water
f
= dimensionless f r i c t i o n f a c t o r
U
- k i n e m a t i c v i s c o s i t y of sea water
p
= p e r m e a b i l i t y c o e f f i c i e n t of D a r c y ' s L a w having
the dimensions of (length)
E q . (75) is applicable f o r h / L > 0. 3 but becomes m o r e c o m p l i c a t e d f o r h / L < 0. 3.
R e t u r n i n g to the d i f f e r e n t i a l E q . (73) and i n view of E q s . (74) and
(75), one r e c o g n i z e s , a f t e r c a r e f u l examination, that this is of the same
general f o r m as B e r n o u l l i ' s nonlinear d i f f e r e n t i a l equation of the f i r s t
o r d e r ; i . e.
+ My
57
= Ny'"
(76)
Without going into the algebra, one m a y w r i t e E q . (73) as f o l l o w s :
F
(77)
F,
K
fp
and the s o l u t i o n i s :
1
K
-/^pdx
ƒ
F
P
dx
F^ dx + Const
fp
78)
In g e n e r a l , n u m e r i c a l i n t e g r a t i o n of E q . (78) is r e q u i r e d , but a
number of special cases have been investigated by B r e t s c h n e i d e r and
Reid (1954). Of m o s t i n t e r e s t at p r e s e n t a r e two s i m p l e cases: (1) no
b o t t o m f r i c t i o n , and (2) no p e r c o l a t i o n .
The f i r s t case f o r F , = 0 r e s u l t s m
f
rx
F dx
K
= • e
fp
^
K
P
(79)
The second case f o r F ^ = 0 r e s u l t s i n K^^ =
^f
1+
=
'X
where
where
-1
F^ dx
(80)
F o r a b o t t o m of constant depth E q s . (79) and (80) become
respectively
K
- F X
P
(81)
1 + F
(82)
= e
and
E q s . (81) and (82) can be used f o r p r a c t i c a l engineering solutions f o r a
b o t t o m of v a r i a b l e depth where s m a l l i n c r e m e n t s of A x a r e selected,
each i n c r e m e n t having a b o t t o m of constant depth equal to the m e a n depth
over the i n c r e m e n t ; i . e.
K fp = l A K f
• A K
(83)
where
-FpAx
AK
:
58
(84)
and
1+
Ax
(85)
Graphs presented by B r e t s c h n e i d e r and Reid (1954) can be used
to f a c i l i t a t e computations by the n u m e r i c a l method.
B o t t o m f r i c t i o n and shoaling m o d i f i c a t i o n s can be obtained by use
of F i g u r e s 26,and 27. F i g u r e 26 i s based on a bottom of constant depth,
but can be used f o r a v a r i a b l e b o t t o m by r e p r e s e n t i n g the b o t t o m by a
s e r i e s of sections, each having an average bottom depth.
/
59
WITH
.999.998
.995 .99
.98
.95
.90
.80
.70
.60 .50
. 40
.30
.20
MAY
FIGURE
26
RELATIONSHIP
FRICTION
BOTTOM
OF
LOSS
FOR
OVER
CONSTANT
60
A
DEPTH
.10
1961
1.2
in
1.0
0.91
0.1
!
\
0.2
l
1
0.3
i
!
i
0.4
0.5
J
0.6
L_J
L_l
0.7 0.8 0.9 1.0
I
L
1.5
2
dj[ sec?/ft.)
FIGURE
27
Kg
VERSUS
T^/dj
9 10
MAY
1961
G E N E R A T I O N OF WIND WAVES I N S H A L L O W W A T E R
IV.
Less i n f o r m a t i o n is available on wind waves i n shallow water than
i n deep w a t e r . T h i s is t r u e i n r e g a r d to both t h e o r y and available data.
The f i r s t i n f o r m a t i o n on t h i s subject was given by T h i j s s e (1949), based
on l i m i t e d data. A d d i t i o n a l data and r e l a t i o n s h i p s w e r e brought f o r t h
by D r . Garbis Keulegan at the National Bureau of Standards, although
never published to the knowledge of the author. The U. S. A r m y Corps
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 (1955), p e r f o r m e d an extensive f i e l d
i n v e s t i g a t i o n on w i n d , waves and tides i n Lake Okeechobee, F l o r i d a .
Based on the h u r r i c a n e w i n d and wave data f r o m L a k e Okeechobee, and
some o r d i n a r y w i n d wave data f r o m the shallow regions of the Gulf of
M e x i c o , B r e t s c h n e i d e r ( 1954) was able to e s t a b l i s h a n u m e r i c a l p r o c e d u r e
f o r computing w i n d waves i n shallow w a t e r t a k i n g b o t t o m f r i c t i o n into
account. A f r i c t i o n f a c t o r of f = . 0 1 appears s a t i s f a c t o r y . P r e s e n t l y
these techniques are used f o r the continental shelf, but m a y r e q u i r e f u r t h e r
c a l i b r a t i o n when m o r e w i n d and wave data are a v a i l a b l e .
A.
G E N E R A T I O N O F WIND WAVES OVER A B O T T O M OF
CONSTANT D E P T H
I f d / T ^ < 2. 5 f t / s e c ^ , then the waves e f f e c t i v e l y " f e e l b o t t o m "
and the depth and b o t t o m conditions enter as a d d i t i o n a l f a c t o r s w i t h
r e s p e c t to the heights and p e r i o d s of waves which can be generated. The
e f f e c t of f r i c t i o n a l d i s s i p a t i o n of energy at the b o t t o m f o r such waves
l i m i t s the rate of wave generation and also places an upper l i m i t on the
wave heights w h i c h can be generated by a given w i n d speed and f e t c h
length.
F i g u r e 4, the d i m e n s i o n l e s s deep water wave f o r e c a s t i n g r e l a t i o n ships, i n e f f e c t r e p r e s e n t s the generation of wave energy i n deep w a t e r
as a f u n c t i o n of F , U and t , since the energy i.s p r o p o r t i o n a l to H ^ ,
whereas F i g u r e 26 r e p r e s e n t s the d i s s i p a t i o n cf wave energy due to
b o t t o m f r i c t i o n . F i g u r e s 4 and 26 w e r e combined by a n u m e r i c a l method
of successive a p p r o x i m a t i o n to obtain r e l a t i o n s h i p s f o r the generation of
waves over an i m p e r m e a b l e b o t t o m of constant depth. Best agreement
between wave data and the n u m e r i c a l method was obtained when a b o t t o m
f r i c t i o n f a c t o r of f = . 01 was selected. Perhaps a " c a l i b r a t i o n ' f r i c t i o n
f a c t o r " is a m o r e a p p r o p r i a t e t e r m since i t would take into account other
i n f l u e n t i a l f a c t o r s not n o r m a l l y i n c l u d e d in the f r i c t i o n f a c t o r t e r m such
as energy loss by " w h i t e c a p s , " The c u r v e s of f i g u r e 28 are the r e s u l t s
of these computations.
The c u r v e of g T / U v e r s u s g d / U is based on
2
2
2
the wave data, whereas the curves of g H / U v e r s u s g d / U and g F / U
are based on the n u m e r i c a l computations. The curves c>f these f i g u r e s
are not too m u c h d i f f e r e n t f r o m those presented by T h i j s s e and S c h i j f ,
(1949), r e p r o d u c e d i n F i g u r e 30, F i g u r e 29, based on F i g u r e 28, gives
wave f o r e c a s t i n g curves f o r shallow water of constant depth and u n l i m i t e d
w i n d d u r a t i o n and f e t c h l e n g t h . F i g u r e 28 m a y be used when both the
f e t c h length as w e l l as the depth are r e s t r i c t e d .
62
Ui
FIGURE 2 8
GENERATION OF WIND OVER A B O T T O M OF C O N S T A N T DEPTH
FOR U N L I M I T E D WIND DURATION REPRESENTED AS DIMENSIONLESS PARAMETERS
4
5
6
7
8 9 10 II 1213 14 15 I6I7I8I920
MEAN WATER DEPTH IN FEET
25
FIGURE 2 9
WAVE FORECASTING RELATIONSHIPS
SHALLOW WATER OF CONSTANT
DEPTH
64
30
35
FOR
40
FIGURE
30
GROWTH
OF
WAVES
IN
A
LIMITED
DEPTH
The i m p o r t a n t f a c t f r o m the above m a t e r i a l , however, i s the
establishment of a n u m e r i c a l procedure f o r computing w i n d waves m
shallow water of constant depth which can be v e r i f i e d by use of wave
data
T h i s p r o c e d u r e can be extended to a b o t t o m of constant slope
w h e r e i n the b o t t o m is segmented into elements, each element h a v m g
a mean depth assumed to be constant. Sample computations f o r a t y p i c a l
continental shelf are given by B r e t s c h n e i d e r (1957).
P r e s e n t l y , the system of calculations is being p r a c t i c e d on gene r a t i o n of w i n d wave spectra over the continental shelf. I t w i l l be some
t i m e i n the f u t u r e b e f o r e the system is p e r f e c t e d . However, F i g u r e 3 1 ,
r e p r o d u c e d f r o m B r e t s c h n e i d e r (1964), i s a t y p i c a l example r e s u l t i n g
f r o m c a l c u l a t i o n of wave spectra i n shallow w a t e r , t a k i n g into account the
combined e f f e c t of b o t t o m f r i c t i o n and w i n d a c t i o n . The hindcast deep
water wave spectra is based on the m e t h o d of B r e t s c h n e i d e r (1957) and
b o t t o m f r i c t i o n i s taken into account a c c o r d i n g to the method of
B r e t s c h n e i d e r (1963) w h e r e i n the wind e f f e c t is added.
66
600
f sec-'
FIGURE 31
WAVE SPECTRA
FOR A T L A N T I C C I T Y , N . J .
f sea
V.
DECAY O F WAVES I N D E E P W A T E R
When waves leave a generating area and t r a v e l through a c a l m
or an area o£ l i g h t w i n d s , a t r a n s f o r m a t i o n takes place. A t any p a r t i c u l a r
t i m e and w i t h respect to decay distance, the s i g n i f i c a n t wave height
decreases and the m o d a l p e r i o d s h i f t s to the longer p e r i o d waves,
r e s u l t i n g i n an apparent increase i n s i g n i f i c a n t wave p e r i o d . H o w e v e r ,
at a p a r t i c u l a r decay distance the m o d a l p e r i o d decreases. To u n d e r stand the s i g n i f i c a n c e of a p e r i o d s h i f t , one m u s t r e f e r to the spectrum,
of waves and the o r i g i n a l w o r k of B a r b e r and U r s e l l (1948). Recent
w o r k of Munk (1964) shows that m o s t of t h i s t r a n s f o r m a t i o n takes place
in the f i r s t 1000 m i l e s of decay. Based on f i e i d tests of waves a r r i v i n g
f r o m the southern h e m i s p h e r e and t r a v e r s i n g a great c i r c l e to A l a s k a ,
M u n k (1964) shows that t h e r e i s a v e r y r a p i d loss or t r a n s f o r m a t i o n of
energy d u r i n g the f i r s t 1000 m i l e s and a s h i f t of peak f r e q u e n c y to about
0. 6 to 0. 7 sec" ^ a f t e r w h i c h l i t t l e change o c c u r s .
The w o r k of W i e g e l and K i m b e r l y (1950) and B r e t s c h n e i d e r ( 1950)
was included i n the r e v i s i o n s i n f o r e c a s t i n g deca.y of s w e l l . E m p i r i c a l
r e l a t i o n s h i p s w e r e developed, based on the c o r r e l a t i o n of wave data
obtained o f f Southern C a l i f o r n i a and w i n d and f e t c h c h a r a c t e r i s t i c s obtained f r o m weather c h a r t s of the Southern H e m i s p h e r e , f r o m where the
waves had o r i g i n a t e d . The r a t h e r meager data w e r e r e c o r d e d i n shallow
water and had to be t r a n s f o r m e d f o r deep water a c c o r d i n g to the shoaling
c o e f f i c i e n t s given by F i g u r e 27. I n spite of the above d i f f i c u l t i e s ,
e m p i r i c a l f o r e c a s t i n g c u r v e s f o r wave decay were d e t e r m i n e d by B r e t schneider (1951) and are presented i n F i g u r e 32. The scatter of data
on w h i c h these curves are based indicates that the above r e l a t i o n s cs.n
deviate by as m u c h as t 50 percent. A l t h o u g h these curves are not c o m p l e t e l y s a t i s f a c t o r y , they are the best available to date. F u r t h e r
refinements are certainly necessary.
A s s u m i n g average conditions f o r a severe s t o r m , a wave spect;ra
hindcast was made f o r the end of the f e t c h , and assuming that F i g u r e 32
applied f o r H ^ ^ / H ^ as a f u n c t i o n of T f o r decay of the wave s p e c t r a ,
computations w e r e made f o r v a r i o u s elements of the p e r i o d s p e c t r u m .
The r e s u l t s of these c a l c u l a t i o n s are shown i n F i g u r e 33. The c u r v e f o r
D = 0 i s zero decay distance or the end of the f e t c h . Other values of the
decay distance D are given i n n a u t i c a l m i l e s . The short p e r i o d waves
disappear r a p i d l y , leaving only the longer p e r i o d waves to decay m o r e
slowly.
The areas under each s p e c t r u m are r e l a t e d to the square of the
c o r r e s p o n d i n g s i g n i f i c a n t wave height, and the m o d a l p e r i o d is close
to the s i g n i f i c a n t wave p e r i o d . F i g u r e 34 shows the change i n sign..£icant
wave height and p e r i o d as a f u n c t i o n of decay distance. F r o m t h i s f i g u r e
i t is seen that m o s t of the decay takes place d u r i n g the f i r s t 1000 m i l e s
68
T f . S E C O N D S , SlGNrFICANT
WflVE
PERIOD
AT
EMO
OF
F,^,
TD/T,.
0.9
Hf,
FEET,
SIGNIFICANT
WAVE
HEIGHT
AT END O F F„
RELATIVE
0.8
HQ/MP
WAVE
0.7
RELATIVE
PERIOD
AT
0.6
0.5
WAVE
HEISHT
END
OF
DECAY
0.4
AT
END
0.3
OF
DECAY
DISTANCE
0.2
0.1
DISTANCE
( B r e t s c h n e i d e r , 1952)
FIGURE
32
FORECASTING
CURVES FOR WAVE DECAY
T
FIGURE
33
DECAY
SPECTRA
SEC.
OF
WITH
70
WAVE
DISTANCE
SEC.
16
O
1000
2000
DECAY
3000
DISTANCE -
4000
NAUTICAL
DECAY
3000
4000
DISTANCE— NAUTICAL
5000
6000
7000
6000
7000
MILES
35
0
RGURE
1000
34
2000
DECAY
OF
SIGNIFICANT
WAVES
5000
MILES
WITH
DISTANCE
a f t e r which l i t t l e change o c c u r s , w h i c h is i n agreement w i t h the idea
proposed by M u n k (1964). The r e l a t i o n s h i p s of F i g u r e 34, having been
based on e m p i r i c a l data, do not indicate how the energy is d i s p e r s e d ,
dissipated or t r a n s f o r m e d f r o m one f r e q u e n c y to another.
No r e l a t i o n s h i p s have been developed to show how the wave
s p e c t r u m decays w i t h r e s p e c t to t i m e at a p a r t i c u l a r l o c a t i o n . The
wave spectra data of I j i m a ( 1957) show that the s h i f t i n m o d a l p e r i o d is
opposite to that given i n F i g u r e 33, as should be expected. F i g u r e 35
r e p r e s e n t s f r e q u e n c y s p e c t r u m f o r i n c r e a s e of waves and f o r decrease
of waves, r e p r o d u c e d f r o m I j i m a (1957). The g r o w t h of wave energy i s
f r o m h i g h f r e q u e n c y (low period) and the decay of wave energy i s f r o m
low f r e q u e n c y (long p e r i o d ) .
The c o m b i n a t i o n of w i n d waves and s w e l l can lead to a m u l t i peaked s p e c t r u m . T h i s i s p a r t i c u l a r l y t r u e f o r h u r r i c a n e conditions
where the h u r r i c a n e winds act at n e a r l y a r i g h t angle to the d i r e c t i o n
of s w e l l propagated ahead of the s t o r m as the s t o r m moves over the s w e l l .
A t y p i c a l i l l u s t r a t i o n of these conditions is 'shown f o r H u r r i c a n e Donna
by B r e t s c h n e i d e r (1961) iri; anvliippendixito the-paper by G a l d w é l l ah^
W i l l i a m s (1961) as published i h Ocean Wave Spectra (1963). A d e m o n s t r a t i o n of t h i s phenomenon is shown i l l u s t r a t i v e l y i n F i g u r e 36 f o r the
p e r i o d s p e c t r u m and w h i c h , when t r a n s f o r m e d to the f r e q u e n c y s p e c t r u m ,
i s shown i n F i g u r e 37, r e m e m b e r i n g that S(f) = T ^ S(T), I t m i g h t be
noticed that a c a r e f u l i n s p e c t i o n of F i g u r e s 36 and 37 reveals the f a c t that
f o r c e r t a i n applications the p e r i o d s p e c t r u m has c e r t a i n advantages over
the f r e q u e n c y s p e c t r u m . I t appears that i f the f r e q u e n c y s p e c t r u m w e r e
plotted on a l o g scale, t h e r e would be a c l a r i f i c a t i o n of the apparent
difficulties.
F i g u r e 32 is based e n t i r e l y on m e a g e r e m p i r i c a l data. These
r e l a t i o n s h i p s take into account the generating f e t c h length, and as a
r e s u l t give a f a m i l y of r e l a t i o n s h i p s f o r wave decay, whereas the o r i g i n a l
w o r k of S v e r d r u p and M u n k (1947) gives a single r e l a t i o n s h i p f o r wave
decay. I t should be m e n t i o n e d that the w o r k of P i e r s o n , Neumann and
James (1955), as outlined i n H y d r o g r a p h i c O f f i c e P u b l i c a t i o n 603, o f f e r s
the best s c i e n t i f i c means f o r d e s c r i b i n g the decay of waves. F u r t h e r
r e s e a r c h and c o l l e c t i o n of f i e l d data is n e c e s s a r y b e f o r e the f i l t e r
techniques proposed i n H . O. p u b l i c a t i o n 603 can be i m p l e m e n t e d f o r
practical applications.
72
T
FIGURE
35
WAVE
TYPICAL
ENERGY
BUILD-UP
( AFTER
CHANGE
OF
SPECTRUM
AND
DECAY
73
OF
IN
THE
WAVES
IJIMA )
30
28
26
/
^ '1000
^
24
22
20
18
16
2(T)
H
"•"/sec. 14
\
\
12
( / \
\
10
LOCAL
STORM
V \
\ N
\
\
\
\
\
\
\
\
4
SWELL
2
O
0
2
FIGURE
4
36
6
8
10
12
14
T
SEC.
EXAMPLE
OF
SPECTRA
LOCAL
OF
STORM
16
18
PERIOD
COMBINED
AND
SWELL
20
22
24
26
75
VI.
W A V E STATISTICS
Wave s t a t i s t i c s are d e f i n e d i n t e r m s of p r o b a b i l i t y of o c c u r r e n c e
or r e c u r r e n c e i n t e r v a l s , f o r example the average number of y e a r s
r e q u i r e d f o r a p a r t i c u l a r value of the s i g n i f i c a n t wave height to be
equalled or exceeded. T h i s d e f i n i t i o n does not n e c e s s a r i l y have the same
s t a t i s t i c a l m e a n i n g as wave v a r i a b i l i t y . Wave v a r i a b i l i t y i s r e s e r v e d f o r
wave height or p e r i o d d i s t r i b u t i o n f o r a p a r t i c u l a r continuous wave r e c o r d .
I n this section p r o b a b i l i t y is d e f i n e d as the percent of t i m e a
p a r t i c u l a r event i s expected to o c c u r . The c u m u l a t i v e p r o b a b i l i t y is
the percent of t i m e a p a r t i c u l a r event is equalled or exceeded. The
r e c u r r e n c e i n t e r v a l is the t i m e r e q u i r e d f o r a p a r t i c u l a r event to r e c u r .
I n o r d e r to d e t e r m i n e c u m u l a t i v e p r o b a b i l i t y and r e c u r r e n c e i n t e r v a l s
of wave heights f r o m severe s t o r m s w h i c h m i g h t be p r o g n o s t i c a t e d f r o m
c l i m a t o l o g i c a l events, s e v e r a l methods of approach m i g h t be u t i l i z e d .
I f data w e r e m e a s u r e d over a s u f f i c i e n t length of t i m e , t h i s would be
u s e f u l . O t h e r w i s e , wave hindcasts f r o m past m e t e o r o l o g i c a l weather
maps can be made f o r p r e s e n t a t i o n of the data. The m e t h o d of B e a r d
(1952) has been used s u c c e s s f u l l y f o r s m a l l samples - - 20 to 40 y e a r s
of r e c o r d s - - b y h y d r o l o g i s t s and h y d r a u l i c engineers to p r e d i c t r e c u r rence i n t e r v a l s f o r peak f l o o d s i n engineering studies of watersheds,
r e s e r v o i r capacity and dam c o n s t r u c t i o n .
The equations f r o m B e a r d (1952) f o r d e t e r m i n i n g p r o b a b i l i t i e s
and r e c u r r e n c e i n t e r v a l s a r e as f o l l o w s :
P
=
100
^-=-^
(86)
and
I ^
where
lOOY
SP
S = t o t a l number of o c c u r r e n c e s on r e c o r d ,
s
= the s u m m a t i o n of o c c u r r e n c e s , beginning w i t h the l o w e s t
value to any successive h i g h e r value u n t i l s = S.
P
= the c u m u l a t i v e p r o b a b i l i t y that an event is equal to or
less than a p a r t i c u l a r value.
Y = the number of y e a r s of r e c o r d ,
I
- the r e c u r r e n c e i n t e r v a l i n y e a r s c o r r e s p o n d i n g to the
p r o b a b i l i t y P and the number of y e a r s of r e c o r d Y ,
(87\
G e n e r a l l y , the p r o b a b i l i t y P versus the magnitude of the event
is aplotted on p r o b a b i l i t y paper and a smooth curve is c o n s t r u c t e d . F r o n
the smooth c u r v e values of the event and the c o r r e s p o n d i n g p r o b a b i l i t y
are d e t e r m i n e d , and the r e c u r r e n c e i n t e r v a l I is then d e t e r m i n e d f r o m
E q . (87).
Instead of using the g r a p h i c a l plot on p r o b a b i l i t y paper, one could
use to a better degree of a c c u r a c y the W e i b u l l d i s t r i b u t i o n f u n c t i o n , E q .
(33):
(88)
I f the data f o l l o w a n e a r l y l i n e a r r e l a t i o n s h i p a c c o r d i n g to the
above equations, then a g r a p h i c a l d e t e r n i i n a t i o n can be made. I f
r e q u i r e d , a s t a t i s t i c a l least squares technique can be used to obtain the
best f i t p a r a m e t e r s
A and M . Once the p r o p e r a n a l y t i c a l equation
has been d e t e r m i n e d f o r the p r o b a b i l i t y P , then the r e c u r r e n c e i n t e r v a l
I can be d e t e r m i n e d as b e f o r e .
Jasper (1956) presented s t a t i s t i c a l d i s t r i b u t i o n s of s i g n i f i c a n t
wave heights based on about 5 - 1 / 2 or 6 years of Weather B u r e a u data
f o r c e r t a i n stations of the N o r t h A t l a n t i c , These data r e s u l t e d i n a
l i n e a r r e l a t i o n s h i p on l o g n o r m a l p r o b a b i l i t y paper. F i g u r e 38 is
r e p r o d u c e d f r o m Jasper (1956),
M o r e r e c e n t data on wave s t a t i s t i c s have been presented i n t a b u l a r
f o r m by M . D a r b y s h i r e (1963). These data are i n t e r m s of the m a x i m u m
wave m e a s u r e d and the c o r r e s p o n d i n g wave p e r i o d .
0.01
99.99
NOTE: THE STRAIGHT LINES ARE THE LOG-NORMAL
CUMULATIVE DISTRIBUTIONS COMPUTED
DIRECTLY FROM THE E X P E R I M E N T A L DATA.
0.05
0.1
999
STATION I
> OBSERVATIONS
OF SIGNIFICANT WAVE •
HEIGHT (FROM U.S. WEATHER
BUREAU DATA 1 9 4 7 - 1 9 5 3 )
0.2
0.5
I
99.8
99.5
99
98
STATION J
OBSERVATIONS
OF SIGNIFICANT WAVE
HEIGHT (FROM U.S WEATHER
BUREAU DATA 1 9 4 7 - 1 9 5 3 )
2
•
5
10
95
90
20
80
30
70
40
60
50
50
60
40
70
30
80
20
90
95
I
STATIONS I AND J
MEASUREMENTS OF MAXIMUM WAVE
HEIGHT OBTAINED BY DARBYSHIRE
(REFERENCE 9) DURING PERIOD F E B .
1953 JAN 1954
98
99
99.5
99.8
2
I
0.5
0.2
0.1
99.9
0.05
0.01
9a99
4
5
6 7 8 9 10
20
30
40
( AFTER
WAVE
FIGURE
38
OF
WAVE
HEIGHT
DISTRIBUTIONS
FROM
VISUAL
OBSERVATIONS
ATLANTIC
JASPER )
HEIGHT-FEET
COMPARISON
MEASUREMENTS
50 60 7 0 8 0 9 0
OF
WAVE
OCEAN
AND
HEIGHTS
STATIONS
78
DERIVED
FROM
AT
I AND
J
CD
<
CQ
O
OC
a.
VII.
WIND SPEED VERSUS WIND SPEED
T h i s section on wind speed v e r s u s w i n d speed is indeed a subject
about w h i c h m u c h d i s c u s s i o n can be p r o m o t e d . Although there are c o n t r o v e r s i e s about use of the p r o p e r wind speed i n wave f o r e c a s t i n g , t h e r e
can be no d i s a g r e e m e n t that m u c h r e s e a r c h i s r e q u i r e d to standardize
wind speed as r e l a t e d to wave generation. I t appears that each author
of a p a r t i c u l a r wave f o r e c a s t i n g technique has his own d e f i n i t i o n of w i n d
speed, whereas the user of the wave f o r e c a s t i n g techniques sometimes
uses a m i x e d d e f i n i t i o n . I t is no wonder that the v a r i o u s methods have
been c i r i t c i z e d by the u s e r s .
P i e r s o n (1963) f o r the f i r s t t i m e gave a r e m i n d e r that the w i n d
data obtained by Neumann (1948 and 1952) was a c t u a l l y m e a s u r e d at 7. 5
m e t e r s above the sea, aboard a ship, w h i c h i t s e l f was bobbing i n the sea.
A hand anemometer extended on a pole away f r o m the ship was used to
r e c o r d the w i n d speed, and, at the same t i m e , the waves w e r e o b s e r v e d .
The type of average wind speed d e t e r m i n e d is d i f f i c u l t to know. P e r h a p s
some of the scatter of the data given i n F i g u r e 18, r e p r o d u c e d f r o m
Neumann (1952), is due to conditions under w h i c h the m e a s u r e m e n t s w e r e
made. H o w e v e r , the averages can s t i l l be of i m p o r t a n c e .
A n a t t e m p t by P i e r s o n (1963) was made to i n t e r p r e t wave s p e c t r a
i n t e r m s of the w i n d p r o f i l e instead of w i n d m e a s u r e d at a constant
elevation, and P i e r s o n and M o s k o w i t z (1963) proposed a wave spectra
based on w i n d data m e a s u r e d at a constant height of 19, 5 m e t e r s above
mean sea l e v e l . T h i s was done to avoid using d r a g c o e f f i c i e n t s by other
authors f o r r e d u c i n g the wind speed to some other standard elevation,
P i e r s o n (1963) also attempted to r e c o n c i l e the d i f f e r e n c e s between the
v a r i o u s spectra on the basis of the e l e v a t i o n at w h i c h winds were mea.sured,
and he advocated longer averages f o r w i n d speed i n o r d e r to obtain b e t t e r
c o r r e l a t i o n w i t h the averaged wave data,
D a r b y s h i r e (1952, 1955) r e l a t e d wave data w i t h the geostrophic
wind speed c a l c u l a t e d f r o m the s u r f a c e p r e s s u r e gradient. Since the
i s o b a r s w e r e smoothed t h r o u g h the r e c o r d e d p r e s s u r e data, the geostrophic w i n d r e p r e s e n t e d some type of a longer t i m e average.
B r e t s c h n e i d e r (1951, 1959) c a l c u l a t e d a surface wind speed f r o m
the geostrophic w i n d speed, t a k i n g into account c u r v a t u r e and a i r - s e a
t e m p e r a t u r e d i f f e r e n c e s . The c o r r e c t i o n s due to c u r v a t u r e and a i r - s e a
t e m p e r a t u r e d i f f e r e n c e s w e r e based on the o r i g i n a l e m p i r i c a l S c r i p p s
data given by A r t h u r (1947). These data w e r e c o r r e l a t e d w i t h the observed or r e c o r d e d o n e - m i n u t e average w i n d speeds. F i g u r e 39,
based on the geostrophic w i n d d u r a t i o n , and f i g u r e 40, based on the
Scripps data of A r t h u r (1947), can be used to calculate a mean s u r f a c e
wind speed, p r e s u m a b l y f o r an elevation of 10 m e t e r s above the m e a n
\
79
30
35
40
45
50 55 6 0 6 5 70
DEGREES LATITUDE
( B R E T S C H N E I D E R , 1952)
FIGURE
39
"
GEOSTROPHIC WIND
Züpsmtf,
FOR
Ap = 5mb a
An
p
T
o
=
:
=
=
80
3mb
DEGREES LATITUDE
1013.3 mb
I0-C
1.2 xlO"'=gm/cnfi'=>
SCALE
. 4 5 1
15
-10
DIFFERENCE
FIGURE
40
-5
0
+5
+10
SEA-AIR TEMPERATURE (TgTs-TaTa')
SURFACE WIND
(BRETSCHNEIDER
11
SCALE
1952)
+15
sea s u r f a c e . Sometimes the p r e s s u r e gradients are not too w e l l defined
p a r t i c u l a r l y f o r l i g h t e r winds, but i n any case the calculated w i n d speed,
should agree w i t h the m e a s u r e d wind speeds to w i t h i n t 15%.
The data of Goodyear (1963) show a considerable amount of
scatter between the c a l c u l a t e d wind speed and the observed wind speed,
and states that the " c a l c u l a t e d wind speeds r e p r e s e n t a 10 t o 15 m i n u t e
average" whereas the m e a s u r e d wind speeds are at the m o s t s e v e r a l
m i n u t e s d u r a t i o n . F i g u r e 4 1 , r e p r o d u c e d f r o m Goodyear (1963), showthe deviations between observed one^ or t w o - m i n u t e a.verage w i n d speed
and the calculated 10 to 15 m i n u t e average wind speed f r o m p r e s s u r e
g r a d i e n t s . I t is no wonder that W i l s o n (1963) could not calculate w i n d
speeds f r o m the p r e s s u r e gradients and obtain a c o r r e l a t i o n w i t h the
r e p o r t e d w i n d speeds. Instead W i l s o n (1963) uses the r e p o r t e d wind sp^
to calculate i s o l i n e s of constant w i n d speed, but this m u s t also e n t a i l an
averaging p r o c e s s , an average w i t h r e s p e c t to distance. F u r t h e r , when
using the s p a c e - t i m e wind f i e l d concept t h e r e is s t i l l another average:
a. t i m e average f r o m one weather map to the next. Some of the ship
r e p o r t s of winds are winds m e a s u r e d at elevations d i f f e r e n t f r o m , 19. 5
m e t e r s above the sea. One can h a r d l y help wonder what wind speed
is being t a l k e d about.
C e r t a i n l y t h e r e are d i r e c t i o n a l spectra of winds (which v a r y w i t h
elev"i,tion), j u s t as there are d i r e c t i o n a l spectra of waves. H o r i z o n t a l
va.riation i n w i n d speed and d i r e c t i o n m u s t be considered of great i m portance i n wave generation, p a r t i c u l a r l y when one discusses e f f e c t i v e
fetches and d u r a t i o n s as w e l l as f u l l y - d e v e l o p e d seas. U n t i l m o r e c a r e
is ta.ken i n r e p o r t i n g winds and d e f i n i n g w i n d v a r i a b i l i t y , one cannot
expect a,ny novel advances i n wave f o r e c a s t i n g . Even then i t w i l l be
r e q u i r e d to c o r r e l a t e new wave spectra data w i t h new wind speed d e t e r minations .
P r e s e n t l y the f o r e c a s t e r must use the e x i s t i n g tools f o r calc -:'.:-: ".i
waves. He m u s t use the type of wind analysis f o r m a k i n g the forecas'.
which is c o m p a t i b l e to the methods used to develop the f o r e c a s t i n g
r e l a t i o n s h i p s and p r o c e d u r e s . A change i n the procedures on which the
tools are o r i g i n a l l y based should also r e q u i r e a change i n tne t o o l s , or
else the a c c u r a c y of the f o r e c a s t s w i l l become w o r s e instead of b e t t e r .
82
O
10
20
30
COMPUTED
FIGURE
41
COMPUTED
SURFACE
RANDOMLY
40
WIND
vs.
WIND
SPEED
(Ms)
(AFTER
60
SPEED
FOR
70
GOODYEAR, 1963 )
OBSERVED
SELECTED
83
50
43
POINTS
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B r e t s c h n e i d e r , , g . L- (1957). " H u r r i c a n e Design Wave P r a c t i c e s . " P r o c . ,
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Chase, J. , L . J. Cote, W. M a r k s , E. M e h r , W. J. P i e r s o n , J r . , F . C.
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"The 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 as D e t e r m i n e d
f r o m Data obtained f r o m the Stereo Wave G b s e r v a t i o n P r o j e c t . "
T e c h n i c a l R e p o r t , R e s e a r c h D i v . , College of E n g i n e e r i n g , New
York University.
C r a p p e r , G. H . (1957). " A n E x a c t Solution f o r P r o g r e s s i v e C a p i l l a r y
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È(6)'532.
I j i m a , T a k e s h i (1957). "The P r o p e r t i e s of Ocean Waves bn the P a c i f i c
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B l a c k m a n , R. B . and j . W. T u k e y (1958). The M e a s u r e m e n t of P o w e r
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D o v e r , New Y o r k
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~
"
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Spectra. ' by Walden. " D i s c u s s i o n presented at Conference on Ocean
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C a l d w e l l , J. M . a n d ! , . C. W i l l i a m s (1961). "The Beach E r o s i o n B o a r d ' s
Wave Spectriom A n a l y z e r and I t s P u r p o s e . " P a p e r presented at
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C a r t w r i g h t , D. E . (1961). " P h y s i c a l Oceanography. " Science P r o g r e s s ,
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'
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K o r n e v a , L . A . (1961). " S t a t i s t i c a l C h a r a c t e r i s t i c s of the V a r i a b i l i t y
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M a r i n e H y d r o p h y s i c a l I n s t . , USSR A c a d e m y of Science, V o l .
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J o u r n a l of Geophysical Research, A G U , V o L 66, No, 1,
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and D e s i g n . " Beach E r o s i o n B o a r d T e c h . R e p o r t No. 4 ( B E B
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Deuts. W e t t . Seewetteramt N r . 30, H a m b u r g , G e r m a n y .
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B r e t s c h n e i d e r , C. L . , H . L . C r u t c h e r , J . D a r b y s h i r e , G. Neumann,
W. J. P i e r s o n , J r . , H . Walden and B . W. W i l s o n (1962).
"Data
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B r e t s c h n e i d e r , C. L . and J . I . C o l l i n s (1963). " P r e d i c t i o n of H u r r i c a n e
Surge: A n I n v e s t i g a t i o n f o r Corpus C h r i s t i , Texas, and V i c i n i t y . "
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I n s t i t u t e of Oceanography, " P r o c . , 1961 Conf. on Ocean
Wave Spectra, P r e n t i c e - H a l l , pp. 111-132.
Hamada, T o k u i c h i (1963). " A n E x p e r i m e n t a l Study of Development of
Wind Waves. " R e p o r t No. 2, P o r t and H a r b o u r T e c h n i c a l R e s e a r c h
I n s t i t u t e , Yokosuka, Japan.
Hasselmann, K . (1961). " 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
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W a v e s . " P r o c . , V l l l t h Conference on Coastal E n g i n e e r i n g ,
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the M o t i o n s of a F l o a t i n g Buoy. " P r o c . , 1961 Conference on
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M o s k o w i t z , L . , W. J . P i e r s o n , J r . , and E . M e h r (1963). "Wave Spectra
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F o r m f o r F u l l y Developed W i n d Seas Based on the S i m i l a r i t y
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~
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96
D I S T R I B U T I O N LIST
ADDRESSEE
NO. OF COPIES
Chief of Naval R e s e a r c h
A T T N : Geography B r a n c h
O f f i c e of N a v a l R e s e a r c h
Washington, D . C. , 20360
2
Chief of N a v a l R e s e a r c h
A T T N : Geophysics B r a n c h
O f f i c e of N a v a l R e s e a r c h
Washington, D . C. , 20360
2
Defense Documentation Center
C a m e r o n Station
A l e x a n d r i a , V i r g i n i a , 22314
20
D i r e c t o r , Naval R e s e a r c h L a b o r a t o r y
ATTN: Technical Information Officer
Washington, D . C. , 20390
6
Commanding O f f i c e r
O f f i c e of N a v a l R e s e a r c h B r a n c h O f f i c e
1030 East Green Street
Pasadena 1, C a l i f o r n i a
1
Commanding O f f i c e r
O f f i c e of N a v a l R e s e a r c h
Navy #100
F l e e t Post O f f i c e
New Y o r k , New Y o r k
5
Chief of Naval R e s e a r c h
A T T N : Code 416
O f f i c e of N a v a l R e s e a r c h
Washington, D . C. , 20360
1
Defense I n t e l l i g e n c e Agency
DIAAP-1E4
Department of Defense
Washington 25, D. C.
1
The Oceanographer
U. S. N a v a l Oceanographic O f f i c e
Washington, D. C. , 20390
1
Director
Coastal E n g i n e e r i n g R e s e a r c h Center
U. S. A r m y Corps of E n g i n e e r s
5201 L i t t l e F a l l s Road, N . W .
Washington, D . C. , 20016
2
NQ. QF CQPIES
ADDRESSEE
D r . R i c h a r d J.
Coastal Studies
L o u i s i a n a State
Baton Rouge 3,
Russell
Institute
University
Louisiana
2
D r . Charles B . H i t c h c o c k
A m e r i c a n Geographical Society
Broadway at 156th Street
New Y o r k 32, New Y o r k
1
Assistant D i r e c t o r for Research
and Development ,
U. S. Coast and Geodetic Survey
D e p a r t m e n t of C o m m e r c e
Washington 25, D. C.
1
Director
N a t i o n a l Oceanographic Data Center
Washington 25, D; G. .
Robert L . M i l l e r
Department of Geology
U n i v e r s i t y of Chicago
Chicago 37, I l l i n o i s
Director
M a r i n e Sciences D e p a r t m e n t
U. S. N a v a l Oceanographic O f f i c e
Washington, D . C. , 20390
P r o f , F r a n c i s P , Shepard
Scripps I n s t i t u t e of Oceanography
La Jolla, California
Dr. Maurice Ewing
L a m o n t Geological O b s e r v a t o r y
P a l i s a d e s , New Y o r k
1
Prof. Filip Hjulstrom
Geografiska Institutionen
Uppsala U n i v e r s i t e t
Uppsala, Sweden
1
M i s s C. A . M . K i n g
L e c t u r e r i n Geography
U n i v e r s i t y of N o t t i n g h a m
N o t t i n g h a m , England
1
ADDRESSEE
NO. O F COPIES
D r . G i f f o r d C. E w i n g
Scripps I n s t i t u t e of OceanographyLa Jolla, California
D r . V . HenryMarine Institute
Universit-y- of Georgia
Sapelo I s l a n d , Georgia
Coastal Studies I n s t i t u t e
L o u i s i a n a State Universit-yBaton Rouge 3, L o u i s i a n a
2
M r . M„ P. O ' B r i e n
Wave R e s e a r c h Laborator-y
Universit-y- of C a l i f o r n i a
Berkele-y 4, C a l i f o r n i a
1
M r . Joseph M . C a l d w e l l
Coastal E n g i n e e r i n g R e s e a r c h Center
U. S. A r m y Corps of E n g i n e e r s
5201 L i t t l e F a l l s Road, N . W ,
Washington, D. C. , 20016
Oceanographic P r e d i c t i o n D i v i s i o n
U. S. Naval Oceanographic O f f i c e
Washington, D . C , , 20390
U. S. Naval A i r Development Center
Johnsville, Pennsylvania
C h a i r m a n , D e p a r t m e n t of M e t e o r o l o g y
and Oceanography
New Y o r k U n i v e r s i t y
U n i v e r s i t y Heights
New Y o r k 53, New Y o r k
Director, Arctic Research Laboratory
Barrow- Alaska
Dr. H. Lundgren
Coastal E n g i n e e r i n g L a b o r a t o r y
T e c h n i c a l U n i v e r s i t y of D e n m a r k
(Öster Voldgade 10
Copenhagen K , D e n m a r k
Dr. Per Bruun
Coastal E n g i n e e r i n g L a b o r a t o r y
U n i v e r s i t y of F l o r i d a
Gainesville, F l o r i d a
ADDRESSEE
NO. O F C O P I E S
P h y s i c a l Oceanography Section
U. S. Navy E l e c t r o n i c s L a b o r a t o r y
San Diego 52, C a l i f o r n i a
F l u i d Mechanics Division
N a t i o n a l B u r e a u of Standards
Washington, D . C.
Chief, B u r e a u of Y a r d s and Docks
A T T N : R e s e a r c h D i v i s i o n , Code 70
D e p a r t m e n t o f t h e Navy
Washington, D. C. , 20390
Commanding O f f i c e r and D i r e c t o r
David T a y l o r M o d e l B a s i n
Washington, D . C, , 20390
ATTN: Library
U. S. N a v a l Oceanographic O f f i c e
Washington, D. C. , 20390
A T T N : L i b r a r y (Code 1640)
D i r e c t o r , Waterways E x p e r i m e n t Station
U. S. A r m y Corps of E n g i n e e r s
Vicksburg, Mississippi
D r . Garbis H . Keulegan
Waterways E x p e r i m e n t Station
U. S. A r m y C o r p s of E n g i n e e r s
Vicksburg, Mississippi
Commanding O f f i c e r and D i r e c t o r
U. S. Naval C i v i l E n g i n e e r i n g L a b o r a t o r y
P o r t Hueneme, C a l i f o r n i a
Commanding O f f i c e r
Naval R a d i o l o g i c a l Defense L a b o r a t o r y
San F r a n c i s c o , C a l i f o r n i a
Chief, B u r e a u of Ships
D e p a r t m e n t of the Navy
Washington, D. C, , 20360
A T T N : Code 373
Director
Woods Hole Oceanographic I n s t i t u t e
Woods H o l e , M a s s a c h u s e t t s
1
NO. O F COPIES
ADDRESSEE
D r . W a r r e n C. Thompson
Dept. of M e t e o r o l o g y and Oceanography
U. S, Naval P o s t - g r a d u a t e School
Monterey, California
1
Commanding O f f i c e r
U. S. Navy M i n e Defense L a b o r a t o r y
Panama C i t y , F l o r i d a
2
Wave R e s e a r c h L a b o r a t o r y
U n i v e r s i t y of C a l i f o r n i a
B e r k e l e y 4, C a l i f o r n i a
1
Director, Hydrodynamics Laboratory
Massachusetts I n s t i t u t e of Technology
C a m b r i d g e , Massachusetts
1
Head, Department of Oceanography
U n i v e r s i t y of Washington
Seattle 5, Washington
1
Dept. of Oceanography and M e t e o r o l o g y
Texas A & M U n i v e r s i t y
College Station, Texas
2
O f f i c e , Chief of E n g i n e e r s
U. S. A r m y Corps of E n g i n e e r s
R m 2031, G r a v e l l y P o i n t
Washington, D. C.
A T T N : M r . C h a r l e s E . Lee
1
O f f i c e , Chief of E n g i n e e r s
U. S. A r m y Corps of E n g i n e e r s
R m 1306, G r a v e l l y P o i n t
Washington, D. C.
A T T N : M r . A l b e r t L . Cochran
1
Registrar
NUFFIC
27 M o l e n s t r a a t
The Hague, the
2
Netherlands