chapter 7. wave statistics & spectra wave statistics rayleigh distribution (narrow-banded...
Post on 19-Dec-2015
238 views
TRANSCRIPT
![Page 1: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/1.jpg)
Chapter 7. Wave Statistics & Spectra
•Wave Statistics
•Rayleigh Distribution (Narrow-banded spectrum)
•Wave Spectra (P-M & JONSWAP)
•FFT & IFFT
•Cross Spectra & Directional Wave Spreading
•Wave Simulation
![Page 2: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/2.jpg)
Ocean (Irregular) Waves Definitions of Zero-Upcrossing & Downcrossing
Root-mean-Square (RMS), Skewness and Kurtosis Ochi (1998) Ocean Waves
![Page 3: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/3.jpg)
Wave Pattern Combining Four Regular Waves
FFT & IFFT – (Inverse) Fast Fourier Transform. Irregular wave Regular Waves (Frequency Domain Analysis)
![Page 4: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/4.jpg)
Ocean Wave Spectra: P-M & JONSWAP Types
![Page 5: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/5.jpg)
Pierson-Moskowitz Spectrum
42
4 5
5( ) exp
42
where --- constant depending on wind
PMp
g fE f
ff
JONSWAP Spectrum
22
4 exp2 2
4 5
5( ) exp
42
where --- constant depending on wind
sharp factor =1 - 7 (average 3.3)
p
p
f f
f
PMp
a p
b p
g fE f
ff
f f
f f
![Page 6: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/6.jpg)
JONSWAP Spectra & H1/3 and Tp
2 4 5 41/3
1
2
2
5( ) exp[ ( ) ]
40.06238
where [1.094 0.01915 In ]0.230 0.0336 0.185(1.9 )
( / 1) exp[ ]
2
dJ p p
J
p
S f H T f T f
f fd
1, (sharp factor) 1 7(mean 3.3),
0.07
0.09
pp
p
p
fT
f f
f f
Goda (1987)
![Page 7: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/7.jpg)
Wave Directionality & Directional Waves
•Wave components do not travel in the same direction.
•Single Summation Model: Wave components of different freq. travel at different directions but at the same freq., they travel at the same direction.
•Double Summation: At the same freq. wave components travel at different directions. (Energy spreading).
![Page 8: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/8.jpg)
Actual Versus Design Seas
![Page 9: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/9.jpg)
Discretization of a continuous wave spectrum
![Page 10: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/10.jpg)
1
cos ,
where , * ,
2* * ( ) ,
and is randomly selected.
N
n nn
n n n n n
n n
n
a
k x t n
a S
Simulation of Irregular waves
•Uni-directional waves (long-crested)
![Page 11: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/11.jpg)
0.05
0.1
0.15
0.2
-50
0
50
0
50
100
150
f (Hz)
Direction ( )
S(f
,)
( m
2 sec)
Directional wave energy density spectrum
![Page 12: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/12.jpg)
1 1 1 1
,
2
cosh[ ( )]cos , sin ,
cosh
where (cos sin ) ,
2 / , ( 1) , 2 ( , ) ,
tanh
N M N Mnm n
nm nm nmn nn m n m
nm n m m n nm n
m n m n m
n n n
a g k z ha
k h
k x y t n
M m a S
gk hk
Directional Waves:Double Summation Model
The above directional waves may form a partial standing wave pattern and consequently the related resultant wave amplitude at this frequency is no longer uniform in the x-y plane.
![Page 13: Chapter 7. Wave Statistics & Spectra Wave Statistics Rayleigh Distribution (Narrow-banded spectrum) Wave Spectra (P-M & JONSWAP) FFT & IFFT Cross Spectra](https://reader036.vdocuments.net/reader036/viewer/2022081503/56649d375503460f94a0f1aa/html5/thumbnails/13.jpg)
To avoid non-uniformity, it was suggested that at each discrete frequency the wave component is in one direction although the directions of waves at different frequencies are different. Hence, inner summation be eliminated and the representation of irregular wave elevation reduces to,
1
cos ,
where (cos sin ) ,
N
n nn
n n n n n n
a
k x y t
Directional Waves: Single Summation Model