Abstract

The peak to trough distributions of nonlinear high sea waves in bimodal sea states in deep water are investigated. The statistical distribution of wave height is first analyzed by considering the Boccotti’s expression, where the parameters of the distribution are calculated for some bimodal spectra of sea states recorded in the Atlantic Ocean. The nonlinear crest and trough distributions are then obtained, particularizing for two peaked spectra the second-order Fedele and Arena expression, which depends on two parameters. The results have been finally validated by means of Monte Carlo simulations of second-order random waves with bimodal spectra.

1.
Guedes Soares
,
C.
, 1984, “
Representation of Double-Peaked Sea Wave Spectra
,”
Ocean Eng.
0029-8018,
11
, pp.
185
207
.
2.
Guedes Soares
,
C.
, 1991, “
On the Occurrence of Double Peaked Wave Spectra
,”
Ocean Eng.
0029-8018,
18
, pp.
167
171
.
3.
Aranuvachapun
,
S.
, 1987, “
Parameters of the JONSWAP Spectral Model for Surface Gravity Waves-II Predictability From Real Data
,”
Ocean Eng.
0029-8018,
14
, pp.
101
115
.
4.
Rodriguez
,
G. R.
, and
Guedes Soares
,
C.
, 1999, “
The Bivariate Distribution of Wave Heights and Periods in Mixed Sea States
,”
ASME J. Offshore Mech. Arct. Eng.
0892-7219,
121
(
2
), pp.
102
108
.
5.
Rodriguez
,
G. R.
, and
Guedes Soares
,
C.
, 2001, “
Correlation Between Successive Wave Heights and Periods in Mixed Sea States
,”
Ocean Eng.
0029-8018,
28
, pp.
1009
1030
.
6.
Rodriguez
,
G.
,
Guedes Soares
,
C.
,
Pacheco
,
M.
, and
Pérez-Martell
,
E.
, 2002, “
Wave Height Distribution in Mixed Sea States
,”
ASME J. Offshore Mech. Arct. Eng.
0892-7219,
124
(
1
), pp.
34
40
.
7.
Ochi
,
M.
, and
Hubble
,
E.
, 1976, “
On Six-Parameter Wave Spectra
,”
Proceedings of the 15th International Conference on Coastal Engineering
, pp.
301
328
.
8.
Pierson
,
W. J.
, and
Moskowitz
,
L.
, 1964, “
A Proposed Spectral Form for Fully Developed Waves Based on the Similarity Theory of S. A. Kitaigorodskii
,”
J. Geophys. Res.
0148-0227,
69
, pp.
5181
5190
.
9.
Hasselmann
,
K.
,
Barnett
,
T. P.
,
Bouws
,
E.
,
Carlson
,
H.
,
Cartwright
,
D. E.
,
Enke
,
K.
,
Ewing
,
J. A.
,
Gienapp
,
H.
,
Hasselmann
,
D. E.
,
Kruseman
,
P.
,
Meerburg
,
A.
,
Miller
,
P.
,
Olbers
,
D. J.
,
Richter
,
K.
,
Sell
,
W.
, and
Walden
,
H.
, 1973, “
Measurements of Wind Wave Growth and Swell Decay During the Joint North Sea Wave Project (JONSWAP)
,”
Dtsch. Hydrogr. Z.
0012-0308,
A8
, pp.
1
95
.
10.
Torsethaugen
,
K.
, 1993, “
A Two Peak Wave Spectrum Model
,”
Proceedings of the 12th International Conference on Offshore Mechanics and Arctic Engineering (OMAE)
, Vol.
II
, pp.
175
180
.
11.
Longuet-Higgins
,
M. S.
, 1952, “
On the Statistical Distribution of the Heights of Sea Waves
,”
J. Mar. Res.
0022-2402,
11
, pp.
245
266
.
12.
Earle
,
M. D.
, 1975, “
Extreme Wave Conditions During Hurricane Camille
,”
J. Geophys. Res.
0148-0227,
80
, pp.
377
379
.
13.
Forristall
,
G. Z.
, 1978, “
On the Statistical Distribution of Wave Heights in a Storm
,”
J. Geophys. Res.
0148-0227,
83
, pp.
2353
2358
.
14.
Forristall
,
G. Z.
, 1984, “
The Distribution of Measured and Simulated Wave Heights as a Function of Spectral Shape
,”
J. Geophys. Res.
0148-0227,
89
, pp.
10547
10552
.
15.
Longuet-Higgins
,
M. S.
, 1980, “
On the Distribution of the Heights of Sea Waves: Some Effects of Nonlinearity and Finite Band Width
,”
J. Geophys. Res.
0148-0227,
85
, pp.
1519
1523
.
16.
Tayfun
,
M. A.
, 1980, “
Narrow-Band Nonlinear Sea Waves
,”
J. Geophys. Res.
0148-0227,
85
, pp.
1548
1552
.
17.
Boccotti
,
P.
, 1982, “
Relations Between Characteristic Sea Wave Parameters
,”
J. Geophys. Res.
0148-0227,
87
, pp.
4267
4268
.
18.
Prevosto
,
M.
,
Krogstad
,
H. E.
, and
Robin
,
A.
, 2000, “
Probability Distributions for Maximum Wave and Crest Heights
,”
Coastal Eng.
0378-3839,
40
, pp.
329
360
.
19.
Boccotti
,
P.
, 1981, “
On the Highest Waves in a Stationary Gaussian Process
,”
Atti Accad. Ligure Sci. Lett., Genoa
0365-0278,
38
, pp.
271
302
.
20.
Boccotti
,
P.
, 1982, “
On Ocean Waves With High Crests
,”
Meccanica
0025-6455,
17
, pp.
16
19
.
21.
Boccotti
,
P.
, 2000,
Wave Mechanics for Ocean Engineering
,
Elsevier Science
,
Oxford
.
22.
Tayfun
,
A.
, 1990, “
Distribution of Large Wave Heights
,”
J. Waterway, Port, Coastal, Ocean Eng.
0733-950X,
116
, pp.
686
707
.
23.
Naess
,
A.
, 1985, “
On the Statistical Distribution of Crest to Trough Wave Heights
,”
Ocean Eng.
0029-8018,
12
, pp.
221
234
.
24.
Forristall
,
G. Z.
, 2005. “
Understanding Rogue Waves: Are New Physics Really Necessary?
,”
Proceedings of the 14th ‘Aha Holiko’a Hawaiian Winter Workshop
, University of Hawaii, pp.
29
35
.
25.
Tayfun
,
A.
, and
Fedele
,
F.
, 2007, “
Wave-Height Distributions and Nonlinear Effects
,”
Ocean Eng.
0029-8018,
3
4, pp.
1631
1649
.
26.
Tayfun
,
A.
, 1986, “
On Narrow-Band Representation of Ocean Waves
,”
J. Geophys. Res.
0148-0227,
91
, pp.
7743
7752
.
27.
Tung
,
C. C.
, and
Huang
,
N. E.
, 1985, “
Peak and Trough Distributions of Nonlinear Waves
,”
Ocean Eng.
0029-8018,
12
, pp.
201
209
.
28.
Arena
,
F.
, and
Fedele
,
F.
, 2002, “
A Family of Narrow-Band Non-Linear Stochastic Processes for the Mechanics of Sea Waves
,”
Eur. J. Mech. B/Fluids
0997-7546,
21
(
1
), pp.
125
137
.
29.
Forristall
,
G. Z.
, 2000, “
Wave Crest Distributions: Observations and Second-Order Theory
,”
J. Phys. Oceanogr.
1520-0485,
30
(
8
), pp.
1931
1943
.
30.
Al-Humoud
,
J.
,
Tayfun
,
M. A.
, and
Askar
,
H.
, 2002, “
Distribution of Nonlinear Wave Crests
,”
Ocean Eng.
0029-8018,
29
, pp.
1929
1943
.
31.
Song
,
J. B.
, and
Wu
,
Y. H.
, 1999, “
Statistical Distribution of Maximum Surface Displacement in Weakly Nonlinear Random Waves
,”
J. Waterway, Port, Coastal, Ocean Eng.
0733-950X,
125
(
2
), pp.
59
65
.
32.
Tayfun
,
M. A.
, 2006, “
Statistics of Nonlinear Wave Crests and Groups
,”
Ocean Eng.
0029-8018,
33
(
11–12
), pp.
1589
1622
.
33.
Fedele
,
F.
, and
Arena
,
F.
, 2005, “
Weakly Nonlinear Statistics of High Random Waves
,”
Phys. Fluids
1070-6631,
17
, p.
026601
.
34.
Arena
,
F.
, 2005, “
On Non-Linear Very Large Sea Wave Groups
,”
Ocean Eng.
0029-8018,
32
(
11–12
), pp.
1311
1331
.
35.
Arena
,
F.
,
Ascanelli
,
A.
,
Nava
,
V.
,
Pavone
,
D.
, and
Romolo
,
A.
, 2008, “
Three-Dimensional Nonlinear Random Wave Groups in Intermediate Water Depth
,”
Coastal Eng.
0378-3839,
55
, pp.
1052
1061
.
36.
Boccotti
,
P.
, 1989, “
On Mechanics of Irregular Gravity Waves
Atti Accademia Nazionale dei Lincei. Memorie, Rome, Italy
,
8
, pp.
111
170
.
37.
Boccotti
,
P.
, 1997, “
A General Theory of Three-Dimensional Wave Groups
,”
Ocean Eng.
0029-8018,
24
, pp.
265
300
.
38.
Boccotti
,
P.
,
Barbaro
,
G.
, and
Mannino
,
L.
, 1993, “
A Field Experiment on the Mechanics of Irregular Gravity Waves
,”
J. Fluid Mech.
0022-1120,
252
, pp.
173
186
.
39.
Tromans
,
P. S.
,
Anaturk
,
A. R.
, and
Hagemeijer
,
P.
, 1991,
A New Model for the Kinematics of Large Ocean Waves—Application as a Design Wave
,” Vol.
1042
,
Shell International Research
,
Rijswij, The Netherlands
.
40.
Arena
,
F.
, and
Fedele
,
F.
, 2005, “
Non-Linear Space-Time Evolution of Wave Groups With a High Crest
,”
ASME J. Offshore Mech. Arct. Eng.
0892-7219,
127
(
1
), pp.
46
51
.
You do not currently have access to this content.