Chap 2. Statistical Properties and Spectra of Sea Waves

Chap 2. Statistical Properties and Spectra of Sea Waves

2.1 Random Wave Profiles and Definitions of Representative waves

2.1.1 Spatial Surface Forms of Sea Waves

• Long-crested waves: Wave crests have a long extent (swell, especially in shallow water)

• Short-crested waves: Wave crests do not have a long extent, but instead consists of short segments (wind waves in deep water)

2.1.2 Definition of Representative Wave Parameters In nature, no sinusoidal wave exists

→ Wave forms are irregular (or random)

→ Difficult to define individual waves

→ Zero-crossing method is used.

Assume that we measured η(t) at a point.

Zero-upcrossing:

Zero-downcrossing:

Statistically, zero-upcrossing and zero-downcrossing are equivalent if the wave record is long enough. But zero-upcrossing is more commonly used.

Arrange the wave heights in descending order.

(a) Highest wave: H_max, T_max

Note: T_max is not the maximum wave period in the record, but the wave period corresponding to H_max.

(b) Highest one-tenth wave: H_1/10, T_1/10 Average of the highest N/10 waves

∑

=

= ^/10

1 10

/

1 /10

1 ^N

i

Hi

H N

10 /

T1 = average of the wave periods corresponding to the highest N/10 waves (c) Significant wave, or highest one-third wave: H_1/3, T_1/3 (or H_s, T_s)

∑

=

= ^/3

1 3

/

1 /3

1 ^N

i

Hi

H N

(d) Mean wave: H (or H₁), T

∑

=

= ^N

i

Hi

H N

1

1

2.2 Distribution of Individual Wave Heights and Periods

2.2.1 Wave Height Distribution

Gaussian stochastic (linear) theory for narrow-band spectra (range of periods is small) suggests Rayleigh distribution of H_i (discrete) = H (continuous) for large N→∞. Probability density function:

⎟⎠

⎜ ⎞

⎝⎛−

= ²

exp 4 ) 2

(x x x

p π π

with H x= H

satisfying ( ) 1

0 =

∫

^{∞ p x dx}

Probability of exceedance:

) given

~ (arbitrary~ y

probabilit

exp 4 exp 4

) 2

( ^{2 2}

H x x H

x d

x

P x

>

=

=

⎟⎠

⎜ ⎞

⎝⎛−

⎟ =

⎠

⎜ ⎞

⎝⎛−

=

∫

^{∞π ξ π ξ ξ π}

Field data indicate that Rayleigh distribution based on restricted assumptions (linear + narrow-band) yields good agreement with data.

2.2.2 Relations between Representative Wave Heights Exceedance probability based on Rayleigh distribution

x N x

P _{N N} 1

exp 4 )

( ² ⎟=

⎠

⎜ ⎞

⎝⎛−

= π

2 /

)1

2 (ln N x_N

= π for given N (e.g., N = 3, 10) By definition

∫ ∫

∫

^∞

∞

∞

=

=

=

N N

N

x x

N x

N xp x dx

dx N x p

dx x xp H

x H ( )

/ 1

1 )

( ) (

/ 1 /

1

Using p(x) dx

dP =− and P x ^{= ⎜}_⎝^⎛−π x ^⎟_⎠^⎞=

∫

x^∞ p ξ dξ ) 4 (

exp )

( ²

{ }

{ }

⎭⎬

⎫

⎩⎨

⎧ ⎟

⎠

⎜ ⎞

⎝⎛−

⎟+

⎠

⎜ ⎞

⎝⎛−

=

+

=

−

−

−

=

⎟⎠

⎜ ⎞

⎝⎛−

=

∫

∫

∫

∫

∞

∞

∞ ∞

∞

N N

N N N

N x N

N x N

x x N x

dx x x

x N

Pdx x

P x N

dx P xP

N

dx dx xdP N

x

2 2

/ 1

exp 4 exp 4

) (

) ( ]

[

π π

using complementary error function, ⁼

∫

x^{∞ −} ,

t dt e

x) ²

( erfc

⎭⎬

⎫

⎩⎨

⎧ ⎟+ −

⎠

⎜ ⎞

⎝⎛−

= N N

∫

^∞ xN

N N x x t dt

x

2

2 2

/ 1

) 2 4 exp(

exp _π

π π

⎟⎟

⎠

⎞

⎜⎜

⎝ + ⎛

= _{N N}

N x N x

x 2 erfc 2

/ 1

π

π with 2 (ln )^1/2

N x_N

= π

See Table 9.1 (p 263) for H_1/N /H vs N=100,50,20,10,L

H H

H_s = _1/3 =1.6 , H_1/10 =1.27H_s, H_1/100 =1.67H_s

2.2.3 Distribution of Wave Periods Not well established.

Local wind waves (~10 s) + Swell (~15 s) → two peaks (bi-modal) or two main direction Typically, T_max ≅T_1/10 ≅T_1/3 ≅(1.1~1.3)T

2.3 Spectra of Sea Waves

2.3.1 Frequency Spectra

Free surface oscillation at a point:

∑

^∞

=

+

=

1

) 2

cos(

) (

n

n n

n f t

a

t π ε

η

where

an = amplitude of wave with frequency

n

n T

f 1

= εn = phase of wave with frequency f_n

Energy of wave with frequency f_n = ² 2 1

gan

ρ

Real sea waves → infinite number of frequency components → (continuous) frequency spectrum

Note: a_n =S(f_n)Δf 2

1 2

a_n = 2S(f_n)Δf

Standard spectra

↑

ensemble average of large number of wave records (1) Bretschneider-Mitsuyasu spectrum

Fully developed wind waves in deep water

(energy input from wind = energy dissipation due to breaking)

] ) ( 03 . 1 exp[

257 . 0 )

(f = H²T⁻⁴f ⁻⁵ − T f ⁻⁴

S _{s s s} (2.10)

for given H_s and T_s.

Modified by Goda (1988): 0.257→0.205, 1.03→0.75 as in Eq. (2.11).

(2) JONSWAP (JOint North Sea WAve Project) spectrum Growing wind seas in deep water

] 2 / ) 1 ( 4 exp[

5 4

2 ^{2 2}

] ) ( 25 . 1 exp[

)

(f = β_JH_sT_p⁻ f ⁻ − T_pf ⁻ γ ^{−Tpf− σ} S

where

) (γ β

β_J = _J (2.13) )

, ( _s γ

p

p T T

T = (2.14)

⎩⎨

⎧

>

≈

≤

= ≈

p b

p a

f f

f f for 09 . 0

for 07 . 0 σ σ σ

peak enhancement factor γ =1~7 (typically 3.3)

Need to specify H_s, T_s, σ_a, σ_b, and γ (sharper spectral peak as γ ↑).

(3) TMA spectrum (Bouws et al., 1985, J. Geophys. Res., 90, C1) ↓

Includes effects of finite water depth )

, (f h S

S_TMA = _Jφ_k ↑

Kitaigordskii shape function (depth effect)

⎪⎩

⎪⎨

⎧

>

≤

≤

−

−

<

=

2 for

1

2 1

for ) 2 ( 5 . 0 1

1 for 5

. 0 )

,

( ²

2

h h h

h h

k f h

ω ω ω

ω ω

φ

2 /

)1

/ ( 2 f h g

h π

ω =

2.3.2 Directional Wave Spectra (1) General

frequency spectrum → assumes waves with many different frequencies but single direction.

However, real sea waves consist of many component waves with different frequencies and direction. Therefore, we need directional wave spectrum:

)

; ( ) ( ) ,

(f θ S f G f θ

S =

↑

directional spreading function ↓

directional distribution of wave energy ↓

varies with frequency f . 1

)

;

( =

∫

−

↑ π

πG f θ dθ 43 42 1

represents relative magnitude of directional spreading of wave energy

∴ Total energy =

∫ ∫

0^∞ −^π S(f, )d df

π θ θ

=

∫ ∫

0^∞ −^π S(f)G(f; )d df

π θ θ

=

∫

0^∞S(f)df

(2) Mitsutasu-type directional spreading function Based on field measurements,

⎟⎠

⎜ ⎞

⎝

= ⎛

cos 2 )

;

(f θ G₀ ^2s θ

G ← symmetric about θ =0

θ = wave angle from principal direction (θ =0) See Fig. 2.11 for the variation of G versus θ.

Intuitively, G=0 for −180°≤θ ≤−90° and 90°≤θ ≤180°. But, G is very small as long as s is large.

Must satisfy 1

cos² 2

0 ⎟ =

⎠

⎜ ⎞

⎝

∫

− ⎛

π

π θ θ

d

G ^s

Symmetric about θ =0: 1

cos 2

2 0

2

0 ⎟ =

⎠

⎜ ⎞

⎝

∫

^π ⎛^{θ dθ}

G ^s

) 1 2 (

)]

1 ( 2 [

1 cos 2

2

1 _{2 1} ²

0 2

0 Γ +

+

= Γ

⎟⎠

⎜ ⎞

⎝

= ⎛ ⁻

∫

^s

s d

G ^s

s θ θ π

π

where Gamma function Γ(n)=(n−1)! for integer . n

s depends on frequency f :

⎪⎩

⎪⎨

⎧

>

= ₋ ≤

p p

p p

f f f

f s

f f f

f s s

for )

/ (

for ) / (

5 . 2 max

5 max

where

p

p T

f = 1 = peak frequency, and roughly T_p ≈1.05T_s.

smax

s= at f = f_p, and s decreases as f − f_p increases.

Hence, directional spreading is the narrowest near f = f_p. See Fig. 2.12 for where (=1 at peak frequency).

max =20 s

fp

f f*= /

(3) Estimation of the spreading parameter s_max As s_max increases, more long-crested.

Tentatively, in deep water.

⎭⎬

⎫

⎩⎨

⎧

=

=

swell for 75

~ 25

waves for wind 10

max max

s s

smax increases as H₀/L₀ decreases (see Fig. 2.13).

As waves propagate to shallow water, they become long-crested due to refraction. In other words, increases as h decreases. On the other hand, increases more rapidly with decreasing for a larger incident angle because of more refraction (see Fig. 2.14).

smax s_max

h

(4) Cumulative distribution curve of wave energy Read text.

(5) Other directional spreading functions Simplest:

⎪⎩

⎪⎨

⎧

≤

= ≤

≡

for 2 0

for 2 2cos

) ( )

; (

2

θ π θ π π θ

θ

θ G

f

G (2.29)

which is independent of f .

SWOP (Stereo Wave Observation Project):

)

; ( )

;

(f θ Gω θ

G = ; ω =2πf (2.30) Eqs. (2.29) and (2.30) are similar to Mitsuyasu-type with s_max =10 except energy spread with f .

Wrapped normal function (Borgman, 1984):

∑

=

⎭ −

⎬⎫

⎩⎨

⎧− +

= ^N

n

m

m n

f n G

1

2

) (

2 cos ) exp (

1 2 ) 1

;

( σ θ θ

π θ π

θm = mean wave direction

σm = directional spreading parameter (broad directional spreading as σ_m↑)

2.4 Relationship between Wave Spectra and Characteristic Wave Dimensions

⎥⎦

⎤

→

→

domain frequency

spectrum Wave

domain time

analysis wave

- by -

Wave ← relates each other.

∑

^∞

=

+

=

1

) 2

cos(

) (

n

n n

n f t

a

t π ε

η

2

1

2 cos(2 )⎥

⎦

⎢ ⎤

⎣

⎡ +

=

∑

^∞

= n

n n

n f t

a π ε

η

∫ ∑

^⎢_⎣^{⎡ + ⎥}_⎦^⎤

= ^∞

= T

n

n n

n f t dt

T ⁰ a

2

1

2 1 cos(2π ε )

η

Using orthogonality, 1 cos( )cos( ) 0

0 =

∫

^{T m t n t dt}

T σ σ if m≠n,

0 0 1

2

0 1

2 2 2

) ( 2 1

) 2

( 1 cos

m df f S

a

dt t

f T a

n n T

n

n n n

≡

=

=

⎥⎦

⎢ ⎤

⎣

⎡ +

=

∫

∑

∫ ∑

∞

∞

=

∞

=

ε π η

where m₀ = zeroth moment of S(f) η2 = variance of η(t)

Defining η_rms = η² = root-mean-squared value of η(t), we get

0

2 m

rms = η =

η

For Rayleigh distribution of H,

4 0

4 m

H_s ≅ η_rms =

Since H_s and 4 m₀ are not exactly the same, use

3 /

H1

H_s = = average height of highest 1/3 waves from zero-crossing method

0

0 4 m

H_m = = spectral estimate of significant wave height

As for wave periods,

p

s T

T ≅0.95

2 0/m m

T ≅ ; ⁼

∫

0^∞ = 2

2

2 f S(f)df

m ^nd moment of S(f)