k
k k

1 2 an E ( k , )k 2
2
( 2.6.8) ( 2. 6. 9)
2

k 0 0

1 2 an 2

0

2
E ( k , )dkd
0
Next we can define the wave number spectrum
Significant wave, or highest one-third wave: H1/3, T1/3
Highest one-tenth wave: H1/10, T1/10 Mean wave: H T ,
The other is the wave height exceeded by a given percentage of waves in the irregular sea state such as (H10%, H1%).
p(T ) 2.7 T3
4
T Where p(T) is the wave period density distribu ( ) T
(2.6.6)
The above relation means that the distribution of T, which is proportional to the wave length, is expressed by a Rayleigh distribution curve.
Methods of Threshold definition
•Running mean •Nominal wave height •Function of peak or absolute maximum •Recalculated zero •Function of noise
It is also necessary to define the statistically representative wave by using the height and period data. Statistical representation of the individual wave heights in a sea state is a useful tool to describe random waves. In it, there are two types of statistics.
P( H ) 1
H
0
H 2 p( H )dH exp ( ) 4 H
(2.6.2)
and is shown in Fig. 2.6.3. 注意不同的表示方法
2-6-3 Wave Period Distribution
It is known that wave period distributions differ corresponding to the wave development stags. But it is accepted that the wave period distribution of a fully developed sea is given by
When the frequency spectrum is given, the nth Moment矩 at the origin原点, is defined by
(2.6.11 )
f
2 n

g 4
2
kn
( 2.6.12)
showing that the wave number and the frequency are uniquely correlated.
Therefore we can introduce the quantity

2


E (k )dk
Random Waves
重要内容提要
Ocean Wave Characteristics Wave Height Distribution Wave Period Distribution Directional Wave Spectra
frequency spectrum， wave number spectrum directional distribution function
2-6-4 Ocean Wave Energy Spectra 能量谱
Generally speaking, wave energy spectra are classified into : frequency spectra 频谱 wave number spectra波数谱. The former spectra express the wave energy density distribution as a function of wave frequency, while the latter express it as function of wave number.
0


E ( f )df
(2.6.13)
0
where E(f) is defined as the frequency spectrum.
Based on wave data analysis as well as theoretical considerations, numerous expressions for E(f) have been presented. Some of the representative spectra are listed below:
Pierson and Moskowitz Spectrum (2) Bretschneider Spectrum (3) Mitsuyasu Spectrum (4) JONSWAP (Joint North Sea Wave Project) Spectrum 我国海港水文规范采用的谱
(1)
Fig. 2.6.6 Definition of parameters of JONSWAP spectrum
E (k ) E (k , )d
0

2
(2.6.10)
Frequency spectrum ( E(f) )
wave number spectrum
For deep water wave conditions, we have the dispersion relation.
2 2 n 4 2 f n gkn
( x, y , t )
a
n 1

n
cos(xk n cos n yk n sin n nt n )
(2.6.7)
Either one of the following equations can be used to define the directional wave energy spectrum.
2-6-1 Statistical characteristics of Ocean Waves
It is necessary to define the wave height and period from a series of wave records. The zero-up crossing method (上跨 零点法) is the one generally accepted.
One is the average of waves with heights above a certain level. The following parameters are frequently used: Highest wave: Hmax, Tmax 1/n highest waves：H1/n，T1/n
2-6-4 Ocean Wave Energy Spectra 能量谱
Based on the idea that the wave field might be well described by a sum of sinusoidal waves with different frequencies, the term “ wave energy spectrum” is defined.
H 2 p( H )dH exp ( ) dH 2 2 H 4 H
H
(2.6.1)
where
H
is the mean wave height. 累积频率 The probability of an event occurring in which the wave height exceeds the value of H is given by
Definition of Threshold
A threshold阀值 is used in the analysis of the force ( or pressure) time series for several purposes:
*To define „zero‟ crossings between events and therefore define these events
2-6 Ocean Wave Characteristics
In the foregoing sections, regular waves 规 则 波 of constant height, period, and direction were described. However, actual wind-generated waves are very irregular.不规则的 There are two ways of treating such irregular waves: (1) to express them by representative wave characteristics in the statistical sense (2) to describe them via the energy spectrum能量谱