slide - spectral analysis
TRANSCRIPT
3 Wind Generated Waves 3 .6 Description of wave spectral analysis
Wave Record Analysis (J.M.J. Journee 2001)
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
Surface elevation time series of a regular wave
and its spectrum (Briggs et al. 1993)
Surface elevation time series of an irregular
wave and its spectrum (Briggs et al.1993)
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
• Fast Fourier Transformation technique is employed
• The wave energy spectral density S(f) or S() can be obtained from a continuous
time series of the surface (t) with the aid of the Fourier analysis
• In actual practice, the total data length N has to be a power of 2 for computational
efficiency.
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
Wave energy:
The surface elevation of a small amplitude wave is:
The average wave energy per unit area is:
Variance of surface elevation of a linear wave:
Superposition of linear waves:
Variance spectral density Sη()
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
Variance Diagram
Variance (m2)
1 2 3 4 (rad/s)
212aVariance
ai： Amplitude of waves at different i
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
Variance Diagram
(m2/(rad/s))
1 2 3 4 (rad/s)
212( )a
S
ai： Amplitude of waves at different I
: depends on signal recording frequency.
3 Wind Generated Waves 3 .6 Description of wave spectral analysis Variance Diagram
(m2/(rad/s))
1 2 3 4 (rad/s)
212( )a
S
ai： Amplitude of waves at different I
: depends on signal recording frequency.
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
Energy density of one individual regular wave component
of an irregular wave:
Total energy of an irregular can be written as:
where: S() is the wave energy spectral density
Note： If the wave energy spectral density is a function of the wave frequency
f, it follows as : S() =S(f)/2 or S(f)= S() X 2
If the wave energy spectral density is a function of the wave period T,
S(T)= S(f) /T2 or S(f)= S(T) xT2
2
01 1
1( ) ( )
2
N N
n n n
n n
g a g S g S d
21( )
2n n nga gS
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
Variance Diagram
(m2/Hz) or (m2s)
f1 f2 f3 f4 f(Hz)
f(1/s)
212( )a
S ff
ai： Amplitude of waves at different fi f: depends on signal recording frequency.
3 Wind Generated Waves
3 .6 Description of wave spectral analysis
Variance Diagram
(m2/s)
T1 T2 T3 T4 T(s)
212( )a
S TT
ai： Amplitude of waves at different fi f: depends on signal recording frequency.
3 Wind Generated Waves
3 .7 Wave Spectral Characteristics
The spectral moment mn of general order n
are defined as:
where f is the wave frequency, and n= 0,1,2…
The spectral moment Mn of general order n
are defined as:
The relationship between Mn and mn is:
3 Wind Generated Waves
3 .7 Wave Spectral Characteristics
Sea state parameters
The following sea state parameters can be defined in terms of spectral moments:
• The significant wave height Hs is given by:
• The mean zero-up-crossing period Tz can be estimated by:
• The mean wave period T1 can be estimated by:
3 Wind Generated Waves
3 .7 Wave Spectral Characteristics
Sea state parameters
The following sea state parameters can be defined in terms of spectral moments:
• The mean wave crest period Tc can be estimated by:
• The significant wave steepness Ss can be estimated by:
3 Wind Generated Waves
3 .7 Wave Spectral Characteristics
Spectral Width Parameters
Several parameters may be used for definition of spectral bandwidth:
Note： the use of is not recommended because of its sensitivity to high frequency
due to the higher order moments, in particular m4.
Goda (1974) proposed a spectral peakedness parameter Qp defined as:
where Qp=1 corresponds with 1 broadband spectra
while Qp becomes very large for very narrow spectra (0)
Under natural conditions Qp=1.5-5.
0 2
2
1
1m m
m
2
2
0 4
1m
m m
0 2
2
1
1M M
M
2
2
0 4
1M
M M
2
2 00
2( )pQ fS f df
m
3 Wind Generated Waves
Example 6:
The wave spectrum shown in the figure below is defined by:
S(f)=800f -40 for 0.05 ≤ f < 0.10
S(f)=1000f2-600f+90 for 0.10 ≤ f ≤ 0.30
Assuming wave height and wave period both follow the Rayleigh Distribution,
determine the following parameters:
1. The significant wave height Hs
2. The zero-up-crossing period Tz
3. The mean period T1
3 Wind Generated Waves 3 .8 Wave Spectra
• Pierson-Moskowitz • JONSWAP • Bretschneider • ITTC & DNV • ISSC
Note: • Developed for deep water condition • Developed in the term of a reference wind speed • In the term of wave frequency f or
3 Wind Generated Waves 3 .8 Wave Spectra
The typical wave spectrum form: or
3 Wind Generated Waves
3 .8 Wave Spectra
Pierson-Moskowitz Spectrum
Pierson and Moskowitz （1964 ）assumed
that if the wind blew steadily for a long time
over a large area, the waves would come into
equilibrium with the wind. This is the concept
of a fully developed sea.
Here a long time is roughly ten-thousand
wave periods, and a "large area" is roughly
five-thousand wave-lengths on a side.
Wave spectra of a fully developed sea for
different wind speeds ( Moskowitz 1964)
3 Wind Generated Waves
3 .8 Wave Spectra
Pierson-Moskowitz Spectrum
The PM Spectrum is given by:
where =0.0081， W= wind speed at an elevation of 19.5 m
The following relationship can be developed from the PM spectrum:
The significant wave height:
The peak wave frequency:
3 Wind Generated Waves
3 .8 Wave Spectra
JONSWAP Spectrum
The JONSWAP spectrum for fetch-limited seas was obtained from the Joint North
Sea Wave Project - JONSWAP (Hasselmann et al. 1973) and may be expressed as:
where
In JONSWAP spectrum, typically has values ranging from 1.6 to 6 but the value of 3.3 is
recommended for general usage.
3 Wind Generated Waves
3 .8 Wave Spectra
Bretschneider spectrum
The basic form of the spectrum is:
where W is the wind speed at the 10 m elevation
where H1oo and T100 are the average wave height and period. The parameters
F1 and F2 are non-dimensional wave height and periods.
2
1
4
2
3.44F
F
3 Wind Generated Waves
3 .8 Wave Spectra
Comparison of the PM and JOHNSWAP Spectrum JONSWAP vs Bretschneider Spectrum
3 Wind Generated Waves
3 .9 Directional wave spectrum
Directionality in waves
• In reality, waves are three-dimensional
in nature and different components travel
in different directions.
• Measurements of waves are difficult and
thus spectra are made for “uni-directional”
waves and corrected for three dimensionalities.
where G(f,) is a non-dimensional directional spreading function
A schematic for a two-dimensional
wave spectrum E(f,)
( , ) ( , ) ( ) ( , )E f S f S f G f
3 Wind Generated Waves
3 .9 Directional wave spectrum
A directional spectrum and its frequency and direction spectrum (Briggs et al 1993)
3 Wind Generated Waves
3 .10 Short-Term Statistics Summary
• Short term statistics are valid only over a period of time
up to a few days, while a storm retains its basic features
• During this period the sea is described as a stationary
and ergodic random process
• Wave spreading and swell are two additional parameters
of importance. Fetch also plays an important role.