OCE421 Marine Structure Designs

Lecture #6 (Short-term Wave Statistics)

Reading Material

• Coastal Engineering Manual – Part II– Chapter 1: pp. 59-76

Nonbreaking Design Wave

• If breaking in shallow water does not limit wave height, a non-breaking wave condition exists.

• For non-breaking waves, the design height is selected from a statistical height distribution.

Wave Statistics

• Long-term wave statistics (a few years, 20 years, etc) – Fisher-Tippet II distribution, etc.

• Short-term wave statistics (20 minutes, 3 hours, etc.) – Rayleigh (wave height) distribution etc.

Long-term vs. Short-term

A 20-min record may have been recorded (and statistics of each record computed) every 3 hr for 10 years (about 29,000 records) and the statistics of the set of 29,000 significant wave height compiled.

If we have measured the waves for 20 min and found that the significant wave height is 2 m, what is the chance that a wave of 4 m may occur?

a common short-term question:

If the mean significant wave height may be 2 m with a standard deviation of 0.75 m, what is the chance that once in 10 years the significant wave height will exceed 4 m?

a common long-term question:

Recording Period and Interval

recording interval

recorder on

20 minutes

3 hours

t

(t)

recorder on

20 minutes recording

period

Wave Identification: zero-crossing technique

H1

H2

H3

T1 T2 T3

zero-upcrossing technique

individual wave height

H1, H2, H3, …

T1, T2, T3, …

individual wave period

Zero-crossing Wave Height Identification

2 blue waves, 1 red wave

Matlab Code: zerocrs.m (I)

function [H,T]=zerocrs(t,eta);%-----------------------------------------------------------%% function [H,T]=zerocrs(t,eta)%% Perform zerocrossing method to identify individual wave % height and wave period%% H = Wave heights of individual waves% T = Wave periods of individual waves%% H is a 1 by (number of waves) array% T is a 1 by (number of waves) array%%-----------------------------------------------------------

Matlab Code: zerocrs.m (II)

nstep=length(eta);eta1=[eta(2:nstep),0]; % a shift of eta by 1 steptem=eta.*eta1; % negative when a zero-crossing takes placecrs_ind=find(tem<0); % index of wave elevation at zerocrossingnum_crs=length(crs_ind); % number of zerocrossings%num_wave=fix(num_crs/2); % number of waves%H=zeros(1,num_wave-1); % initialization, % for simplicity, drop the last waveT=H;%for n=1:(num_wave-1), start=crs_ind(2*n-1); % starting index for the n-th wave endd=crs_ind(2*n+1); % ending index for the n-th wave peak=max(eta(start:endd)); valley=min(eta(start:endd)); H(n)=peak-valley; T(n)=t(endd)-t(start);end;

Representative Wave Heights

The 1/nth wave height, denoted as H1/n is defined as theaverage wave height of the highest 1/nth waves.

For n=3, H1/3 termed as the significant wave height, Hs.

For n=1, it represents the mean wave height,

H1/10 , H1/100 and H1/250 are defined accordingly

Hrms is defined as: Hrms=

vuut 1N

NX

i=1

H2i

1

if N individual wave heights are given.

Fundamental Probability Functions

PH (h) =Prob[H <h]

2

_

Cumulative distribution function (cdf):

PH (h) =Z h

0pH (x)dx

1

pH (h) =dPH (h)dh

3

Probability density function (pdf):

Probability of exceedance (poe):

QH (h) =Prob[H >h]

4

P +Q =1

1

H = random variable

h = fixed number

Relationships among pdf/cdf/poe

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

pdf

cdf

poe

area

Theoretical Models

• Wave elevation : Gaussian distribution (due to central limit theorem)

• Wave height : Rayleigh distribution (narrow band assumption)

Gaussian

Normal (Gaussian) Distribution

¡ 1 < x <1

1

f (x) =1

¾p2¼exp

(

¡12

µx ¡ ¹¾

¶2)

1

Probability density function (pdf)

Cumulative distribution function (cdf)

F (x) =1

¾p2¼

Z x

¡ 1exp½¡12

µv¡ ¹¾

¶¾2dv

2

2

Rayleigh

Rayleigh Distribution

f (h) =2hµ2exp

"

¡µhµ

¶2#

; h ¸ 0

1

F (h) =1¡ exp

"

¡µhµ

¶2#

; h ¸ 0

2

Q(h) =1¡ F (h)

3

=exp

"

¡µhµ

¶2#

; h ¸ 0

4

f (h) =dF (h)dh

1

F (0) =0, F (1 ) =1

2

a monotonically increasing function

cdf

pdf

poe

Root-Mean-Square Value

E[H2]=Z 1

0h2f (h)dh

1

=µ2

2

f (h) =2hµ2exp

"

¡µhµ

¶2#

; h ¸ 0

1

=Hrms

2

(mean-square value)

root-mean-square (rms) value

second moment

pdf

µ=qE[H2]

1

substitute in

notation for the rms of H

Rayleigh Distribution in RMS Value

f (h) =2hµ2exp

"

¡µhµ

¶2#

; h ¸ 0

1

f (h) =2hH2rms

exp

"

¡µh

Hrms

¶2#

; h ¸ 0

F (h) =1¡ exp

"

¡µh

Hrms

¶2#

; h ¸ 0

Q(h) =exp

"

¡µh

Hrms

¶2#

; h ¸ 0

1

f (h) =2hH2rms

exp

"

¡µh

Hrms

¶2#

; h ¸ 0

F (h) =1¡ exp

"

¡µh

Hrms

¶2#

; h ¸ 0

Q(h) =exp

"

¡µh

Hrms

¶2#

; h ¸ 0

1

Matlab Display: Rayleigh

hrms=1; h=0:0.01:4;psd = 2*h/hrms^2 .* exp(-(h/hrms).^2);cdf = 1 - exp(-(h/hrms).^2);subplot(211); plot(h,psd); grid;subplot(212); plot(h,cdf,'r-'); grid

pdf

cdf

Rayleigh Distribution in Mean Value

E[H]=¹H =H =Z 1

0hf (h)dh

1

=p¼2Hrms

2

f (h) =2hH2rms

exp

"

¡µh

Hrms

¶2#

; h ¸ 0

3

Hrms=2¹p¼

1

(drop subscript H from H for simplicity)

f (h) =¼h2¹ 2

exp

"

¡¼4

µh¹

¶2#

; h ¸ 0

2

pdf in terms of mean value (change of parameter)

Histogram to pdf

average probability “density” for H