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

Ocean (Irregular) Waves Definitions of Zero-Upcrossing & Downcrossing

Root-mean-Square (RMS), Skewness and Kurtosis Ochi (1998) Ocean Waves

Wave Pattern Combining Four Regular Waves

FFT & IFFT – (Inverse) Fast Fourier Transform. Irregular wave Regular Waves (Frequency Domain Analysis)

Ocean Wave Spectra: P-M & JONSWAP Types

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

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)

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).

Actual Versus Design Seas

Discretization of a continuous wave spectrum

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)

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

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.

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