Spectral Analysis and non-linear high wave groups in wind generated sea states

4 
Introduction 
Engineering works are varied and complex. Particularly, maritime and ocean 
structures can be of different types: breakwaters for ports and harbors, seawalls for 
shore protection, platform for exploitation of oil beneath the seabed are some 
examples. Earthquakes, currents, effects of wind, waves are just some of the 
actions generated in the natural environmental subjected to the listed structures 
must perform their functions. In order to ensure their designated performance, deep 
investigation must be carried out, to be able to correctly understand the 
environmental conditions. 
Moving in the specific, waves are the most important phenomenon to be considered 
among the environmental conditions affecting maritime structures; since they 
exercise the greatest influence  the presence of waves makes the design procedure 
for maritime structures quite different from that of structures on land. In this view, 
description and assessment of wind generated ocean waves provide vital 
information for the design and operation of marine systems such as ships and ocean 
and coastal structures. However, it must be said that waves are one of the most 
complex and changeable phenomena in nature and it is not easy to achieve a full 
understanding of their fundamental character and behavior. Wind-generated seas 
continuously vary over a wide range of severity depending on geographical 
location, season, presence of tropical cyclones etc. Furthermore, the wave profile in 
a given sea state is extremely irregular in time and space and thus any sense of 
apparent regularity is totally absent. Having said this, the random wave behavior of 
sea waves can be interpreted by mean of stochastic prediction approach that allows 
to estimate the response of marine systems with reasonable accuracy. Such 
probabilistic approach has become an integrated part of modern design technology 
in naval, ocean and coastal engineering. 
In this view, the main objective of this work is to investigate the random behavior 
of the wave motion by means of stochastic approach in order to interpret the nature
5 
of sea in terms of wave mechanics. To this aim wave surface displacement data are 
analyzed, those data made available thanks to the hospitality of the research center 
of naval engineering CENTEC of Lisbon. The analysis is focused first on the 
spectral analysis in order to identify the nature of sea states, subsequently the 
statistical properties of waves is computed. Finally the first formulation of the 
Quasi-Determinism Theory,  or new wave  theory, is applied up to the first and 
second order, with the purpose of detecting the non-linear deterministic profile in 
high wave groups related to the identified unimodal sea states.  
.
6 
Chapter 1  
Waves in the sea 
 
 
1.1 Random sea waves behavior 
A prerequisite for the reliable estimation of waves on maritime structures is a 
detailed understanding of how waves transform during their propagation toward the 
shore, after they have been generated and developed by the wind in the offshore 
region. 
Considering a point at sea, at a certain instant of time waves begin forming at this 
point. These waves may be under the influence of the wind in a generating area, or 
may also be waves out of their generation area. In the first case they are called wind 
waves, in the second case they are called swells. 
The flow of the transformation and actions of sea waves may be described starting 
from a wind wave, that becomes swell when it moves out of the generating area. 
The height of the swell gradually decreases with distance as it propagates; when the 
wave enter an area of depth less than about one-half of its wave-length, it stars to 
be influenced by the sea bottom’s topography. 
Having propagated into a shallow region, the wave undergo refraction by which the 
direction of wave propagation, as well as the wave height, varies according to the 
sea bottom’s topography. Concerning waves inside a harbor, the phenomen on of 
diffraction by breakwaters is the governing process. 
 
The propagation of a wave in a shallow region gradually change in height as a 
result of the change in the rate of energy flux due to the reduction in water depth,
7 
even if no refraction takes place. When the wave reaches an area of depth less than 
a few times the significant wave height, waves of greater height in a group begin to 
break one by one and the overall wave height decreases as the wave energy 
dissipates. 
When the wave arrives at the site of a proposed structure, it will experience the 
above transformations and deformations of wave refraction, diffraction, shoaling, 
breaking, etc. 
 
1.2 The sea state 
In figure 1.1 it is showed a record of the surface displacement h at a fixed 
location at sea. The function ( ) t h represents the free surface elevation above the 
mean water level. A single wave is a piece of ( ) t h between two consecutive zero 
up-crossing; the wave period is the interval between the two consecutive zero up-
crossing; the wave crest is the highest local maximum and the wave trough is the 
lowest local minimum of the wave; the wave height is measured from trough to 
crest. 
 
Fig. 1.1 Record of sea waves at a fixed point - data coming from North Alwyn platform ( see 
Chapter 5)
8 
The ideal sea state means an infinitely long stationary time series of wind 
generated waves. 
To understand this definition, a number of sets of N consecutive waves  should be 
considered; the mean height and period of each of these sets will be: 
1
H ,
1
T for the 
first set, 
2
H ,
2
T for the second set, and so on. For a small N, the pairs 
1
H ,
1
T , 
2
H ,
2
T … will generally be very different from one another. However, as N grows, the 
differences between these pairs will tend to vanish, and as N ޴ all the pairs will 
become equal to each other.  
The  real sea state is a sequence of a few hundred wind-generated waves (typically 
100-300 waves).  Such a sequence is sufficiently short to be nearly stationary, and 
it is long enough for its statistical properties to be meaningful. In other world , it 
can be thought of as a sequence drown from an ideal sea state, and it can be 
assumed that its mean wave height and period are very close to the mean wave 
height and period of this ideal state. Considering a sequence of a few waves from 
an ideal sea state (i.e. from a stationary random process), the mean wave height and 
period of this sequence could be very different from the mean wave height and 
period of the ideal sea state. On the contrary, if we recorded a sequence of a few 
thousand sea waves, generally it could not be thought of as a sequence from an 
ideal sea state, in that the assumption of stationary time series might be grossly 
unsatisfied (i.e. the first half of the sequence might belong to a stage of nearly calm 
sea, while the second half might include a sea storm).  
 
1.3 The theory of sea states 
Considering a point at sea, at a certain time instant waves begin forming at this 
point, the function ( ) t h being recorded at the fixed point after some time from the 
beginning of the wave motion, from instant 
inf
t to instant 
sup
t , with 
sup inf
t t - 
defined as the duration of a real sea state. If the same storm is repeated many times
9 
with the same speed direction and duration of the wind , the surface displacement 
( ) t h at the fixed point can be recorded, each time starting at instant 
inf
t to instant 
sup
t . 
The obtained records are:  
1 2
( ), ( ), ... ( )
n
t t t h h h 
According to the theory of the sea states to the first order in a Stokes’ expansion, 
each of the n time series 
1 2
( ), ( ), ... ( )
n
t t t h h h is a piece of a new realization of a 
stationary Gaussian process. Each realization of this process has an infinite 
duration, and thus it represents the ideal sea state. 
The analytical form of the process is: 
 
1
( ) cos( t )
N
i i i
i
t a h w e
=
= +
  (0.1) 
Where it assumed that the frequencies 
i
w are different from each other, number N 
is infinitely large, phase angles 
i
e are uniformly distributed on (0,2 ) p and are 
stochastically independent of each other, and all amplitudes 
i
a are of the same 
order.  
The last hypothesis regards the spectrum, which is assumed to be continuous and to 
be the same in each realization; it is defined as follows: 
 
2
1
( )
2
i
i
E a w dw =
  (0.2) 
for i such that: 
2 2
i
dw dw
w w w - £ £ + 
Under this hypothesis, the function 1.1 represents a stationary Gaussian random 
process [Boccotti 2000].
10 
1.4 Basic relations in the theory of sea states 
In order to have a direct idea of the strength of the wave motion, the first 
analysis shall regard the standard deviation, given by: 
 
2
( ) t s h ” < >  (0.3) 
Where the temporal mean must cover the whole duration of the sea state. The 
larger thes the higher the waves. 
The most common parameter used for representing the severity of sea condition is 
the significant wave height. It may be defined as the average of the one-third 
highest observed or measured wave heights 
1 3
H . For this representative wave, the 
waves in the record are counted and selected in descending order of wave height 
from the highest wave, until one-third of the total number of waves is reached. The 
means of theirs height are then calculated and denotes 
1 3
H . However, the 
significant wave height is rarely evaluated following this definition; instead it is 
commonly evaluated by using the variance computed from the spectrum: 
 4
s
H s =  (0.4) 
The peak frequency 
p
w is the frequency of the highest peak of the spectrum. The 
peak period is the wave period associated with the peak frequency: 
 
2
p
p
T
p
w
”  (0.5) 
The wavelength relevant to the peak period, on deep water, is: 
 
0
2
p
p
gT
L
p
”  (0.6) 
The jth moment 
j
m is given by: 
 
0
( )
j
j
m E d w w w
¥
”
  (0.7)
11 
In particular, the zero-th moment of the spectrum coincides with the integral of the 
spectrum over (0, ) ¥ . 
The autocovariance function is: 
 ( ) (t) (t T) T y h h ”< + >  (0.8) 
and thus it is the mean value of the product of the surface displacement at time t 
and the surface displacement at the later time t+T. This mean value depends on T, 
that is the autocovariance is a function of T. 
 
The main properties of the autocovariance are: 
 ( ) ( ) T T y y - =  
 | (T) | (0) 0 for T y y < „  
 
| |
lim ( ) 0
T
T y
޴
=  
 
Starting from the definitions 1.3, 1.4, 1.7 and considering the following relations: 
 
2
0
( ) ( ) t E d h w w
¥
< >=
  (0.9) 
 
0
( ) ( )cos( T)d T E y w w w
¥
=
  (0.10) 
the following equalities can be derived: 
 
2
2 2
0
0
( ) ( ) (0)
16
s
H
t m E d h s w w y
¥
< >= = = = =
  (0.11) 
 
1.4.1. The relation between Tp and Hs 
Considering the JONSWAP spectrum form, which will be defined in Chapter 2  
( Paragraph 2.4.2), the following relation between 
p
T
 and 
s
H
 can be derived: