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Chapter 9

Real Ocean W aves

9.1. Introduction

Kinsman (1965, p. 386) is careful to point out that a valid specification of real

ocean waves m ust integrate the following three concepts: 1) Fourier and spec

tral analyses of random processes, 2) probability theories applied to stochastic

processes, and 3) hydrody nam ics. Techniques from the first two concepts that

do not depend on the physics of the hydrodynamic processes are available for

analyzing real ocean waves. However, only those techniques from concepts

that may be related rigorously to the physics of the hydrodynamics of real

ocean waves are reviewed.

The theoretical techniques reviewed are applicable to

stationary ergodic

processes and are limited strictly to short term statics. Isaacson and

M acKenzie (1981) give an excellent review of long term statistical and prob

abilistic techniques applied to real ocean waves. The significance of the

stationary ergodic

hypothesis is that the

ensemble

average

E[x(t\)]

at the

same time t\ shown in Fig. 9.1 of an infinite number of finite length time

series x\(t\),X2(t\),x-s(t\), ,*oo(^l) is equivalent to the temporal aver

age over all times shown in Fig. 9.2 of an infinitely long single time series

X\(t\),X\(t2),Xl{tl),....,X\(tooY,i-S;

E[x(t

n

)]=

lim —

I

R

xi(t)dt.

(9.1)

T

R

^OO 1R JO

719

W a v e s a n d W a v e F o r c e s o n

C o a s t a l a n d O c e a n S t r u c t u r e s D o w n l o a d e d f r o m w w w . w o r l d s c i e n t i f i c . c o m

b y M C G

I L L U N I V E R S I T Y o n 0 4 / 0 5 / 1

6 .

F o r p e r s o n a l u s e o n l y .

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720 Waves and Wave Forces on Coastal and O cean Structures

Xjrft)

x

3

(0

-A/l/yAA^JjV/

x

2

(t)

xtf)

-X -

1

' /

*

/ /

Fig. 9.1. Two ensemble averages of finite length records

*,- (t

) at times

t\

and

ti

(Bendat and

Piersol, 1986).

Xrft)

Fig. 9.2. Temporal average of a single time series

x\ it)

of infinite length.

9.2. Fourier Analyses

Definitions of Fourier coefficients

Fourier coefficients are defined separately for

deterministic

and for

non-

deterministic

(or, equivalently,

random)

analyses.

W a v e s a n d W a v e F o r c e s o n C o a s t a l a n d O c e a n S t r u c t u r e s

D o w n l o a d e d f r o m w w w . w o r l d s c i e n t i f i c . c o m

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Deterministic:

Fourier coefficients are those coefficients that provide the

best least squares fit to the data.

Non-Deterministic: Fourier coefficients are those coefficients that explain

the contribution by each frequency to the total variance of

a

random process.

W hen both time and frequency are continuous, the Fourier transform pairs

are given by the following integrals:

i

r

+0

°

r)(t)

=

—f==.

/

F(a))exp±(i(ot)d(D,

(9.2a)

J—

c

'In j — oo

i

r

+

°°

F(a>) = - = t](t)exp^(icot)dt, (9.2b)

V2TT

J—OO

where u>

= 2nf =

radian frequency; where

F(co) = Fourier transform

of the time series

r){t);

and where the plus + sign in Eq. (9.2a) must be

paired with the negative — sign in Eq. (9.2b). Both choices for the ±

signs in the arguments of the exponential functions exp(«) in Eqs. (9.2)

may be found in the literature; and the placement of the normalizing con

stant

2n

is also arbitrary (Ligh thill, 1964). If the frequency / is given

in Hertz, the normalizing constant

2n

does not appear and Eqs. (9.2)

reduce to

/

+ 0 0

F(f)ex

V

±(i2nft)df,

(9.3a)

-00

/ + 0 O

r,(t)exp

T

(i2nft)dt. (9.3b)

-00

Data records that are continuous in time are termed

time series;

and data

records that are digitized to discrete values of time are termed

time sequences.

Modern Fourier analyses employ discrete/inite Fourier transform (FFT)

algorithms that are designed

to

take advantage of high speed digital compu ters.

Discretization of Eqs. (9.3) requires a finite record length TR and discrete


C o a s t a l a n d O c e a n S t r u c t u r e s D o w n l o a d e d f r o m w w w . w o r l d s

c i e n t i f i c . c o m

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722 Waves a nd Wave Forces on Coastal and O cean Structures

values of both time and frequency; i.e.;

t ->• t„ = nAt, f ->• f

m

= mAf, co -» co

m

=

2nf

m

= InmAf,

(9.4 a-c)

where At(=T[t/N) and Af(=l/NAt) are constant fixed temp oral and

frequency intervals, respectively, of N total discrete values of time and fre

quency. A discrete Fourier transform pair may be approximated from

Eqs. (9.2) by

N-l

rj(n) = A / ^ F ( r a )e x p ± {ilizrimAf

At),

n = 0 , 1 , 2 , . . . ,N - 1,

m=0

(9.5a)

N-l

F(m) = At y ^ rj(n)exp ^(Unnm AtAf), m = 0,1,2,.. .,N — 1,

n=0

(9.5b)

where the total number of discrete time and frequency values are equal to ./V;

and where

the

FF T coefficients F(m) are complex-valued quantities.

The

mean

value of the discrete time sequence r\ (n) is given by the real-valued FFT coeffi

cient ^ ( 0 ) ; and the FFT coefficient

F(N/2)

is also real-valued at the Nyquist

or folding frequency

fy

=

(N/2)Af

in Eq. (9.5a). The discrete positive-

definite frequencies f

m

> 0 in Eq s. (9.2 and 9.3) are represented by the

indices 1

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723

Eqs.

(9.5) did not permit zero indices and Eqs. (9.5) were given by only

positive-definite indices m, n > 0 according to

N

rj(n) = A / ^ F ( m ) e x p ± ( / 2 ; r ( M - l)(m - 1 ) A / A 0 , n = l,2,...,N

(9.6a)

N

F(m) = At^t](n)exp =F(/2jr(n - \)(m - \)AtAf), m =

1,2,...

,N.

(9.6b)

The mean value of r)(n) in Eqs. (9.6) is given now by the real-valued FFT

coefficient

F(l);

and the real-valued coefficient

F(N/2

+ 1) is now at the

Nyquist or folding frequency

/N = (N/2

+ 1 ) A / . The discrete positive-

definite frequencies

f

m

> 0 in Eq. (9.5a) now have the indices 2

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Waves and Wa ve Forces on Coastal and Ocean Structures

that may be reduced to the following compact result:

-;\-z

N

SN —

\-z

N

, z # l

, z = l

If z is a complex-valued variable given by

z

= exp ±

{i2n(m

—

m)/N),

where m and m are integer constants; then Eq . (9.7a) becom es

N-l

S

N

= ^ exp[±i2jr(#n - m)/Nf

n=0

and Eq. (9.7b) is

SN =

1

—

exp ± \i2n{m

—

m)\

1

—

exp ± [ilTz{m — m)/N]

N, m

—

m =

0,

0 , 0 < m —

m < N

— 1.

(9.7b)

(9.7c)

(9.8)

(9.9a)

(9.9b)

(9.9c)

The normalizing constant CN may be determined by substituting Eq. (9.5a)

into Eq. (9.5b) with a careful change of dummy indices from

m

to

m

and

obtaining

N-l

F(m) =

AtJ2

n=0

N-l

A / y ^ F(m) exp ± (ilnnmAf At)

x exp

==

{ilnnm Af At)

N-l (N-l

= A ? A / ] T F (m) | ] T exp ± (ilnim - m)nAfAt)

m=0

N-l

l n = 0

N-l

= AtAf J^

F

^)

J ]

{ e x p ±

(

i2n

(™

~

m)AfAt)f.

(9.9d)

m = 0 n = 0

If

AtAf = N

1

; then the term in curly brackets {•}" = N (5^

m

by

Eqs.

(9.9b, c

and 9.5b) reduces to

F(m) = ( A f A / ) F ( m ) { A ^

m

)

(9.9e)



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725

that is possible only if

At • Af • N = I,

so that

Af A / = i

=

CN

(9.10)

and the normalizing constant CN is inversely proportional to N.

For example,

the FFT

algorithm

in the

symbolic software

MATHEMATICA™ employs the finite Fourier transform pair Eqs. (9.6),

pre-multiplies the complex-valued FFT coefficients F(m) by a normalizing

constant

CN = s/N

and selects the minus (—) sign for the exponential func

tion in Eq. (9.6a) and the positive sign (+) for the exponential function in

Eqs. (9.6b). With this convention, Eqs. (9.6) become

N

r](n)= ] [ ^ f l ( m ) e x p - ( i 2 j r ( n -

l)(m -

\)/N);

n = l,2,...,N,

(9.11a)

N

B(m) = */NF(m) =

^ rj(n) exp+(/27r(n - l)(ra - l)/N);

m = l,2,...,N. (9.11b)

Because Eq. (9.11a) applies the minus (—) sign for the argument in the

exponential function exp(«), the com plex-valued FFT coefficients B(m) are

expressed with a negative (—) sign for the phase by

B(m) = \B(m)\

exp -

ia(m),

(9.11c)

where a(m) = phase angle at the discrete frequency mAf. Figure 9.3 illus

trates the frequency domain representation of the amplitudes (m oduli) \B(m)\

in Eq. (9.11c).

To illustrate a simple numerical test that may be applied to determine

where a particular FFT algorithm places the norm alizing constant

CN,

synthe

size the following time sequence that consists of

a

non-zero m ean F(l); first

F(2)

and fourth

F(5)

harmonic components with positive and negative phase




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726

Waves and Wave Forces on Coastal and Ocean Structures

\B(m)\

, I-J I i I U,J_I I .

t i >

i s

1

e • \ i | J « J . « " •

, M ,

-

m H H

" ' ' "

fff

Fig. 9.3. Frequency d oma in representation of the amplitudes of the FFT coefficients

\B{m)\.

1 1

0.5 ,

0

"sT -l

-2 -

1 1 1 .

i 1 j i

i

i i T •

"-• r r r r '

J- J - J- 4_

1 1 •

1 1 1 - 1

. • L L L » L

I • I • I I

1 -

» - H -i i

I

I I I

i * i i

1 4 7 10 13 1

n

Fig. 9.4. Discrete nondim ensional time sequence for

N =

16.

angles

a(2)

and a (5 ), respectively; and a total number of discrete sequences

N

= 16 = 2

4

(NOT E: «/w«ys

select N = 2

2M

where 2M must always be an

even integer so that if the FFT algorithm applies the square root J conven

tion for the normalizing constant

CN

then the normalizing constant CM will

always be a rational num ber). Consider the discrete time sequence

3 (Inn

TT

1 (litAn n

+

2

C

°

S

U ^ ~ 4

(9.12)

that is illustrated in Fig. 9.4. A program for synthesizing the normalized dis

crete time sequence in Eq. (9.12) by the FFT algorithm in

MATHEMATICA™

is listed below. The amplitudes

\B(m)\

of the complex-valued FFT coeffi

cients are illustrated in Fig. 9.5; and the phases aim) of the complex-valued

FFT coefficients are illustrated in Fig. 9.6. Note the change in sign of the

phases a(m) in Fig. 9.6 as a result of the sign convention defined for the



b y M C

G I L L U N I V E R S I T Y o n 0 4 / 0 5 / 1 6 .


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Real Ocean Waves

727

5

4

~^~ 3

1

0

7 10

m

i

i

-r-

i

13

16

Fig. 9.5. Am plitudes of the FFT coefficients for the discrete time sequ ence

r)(ri)

in Fig. 9.4.

1 4

Fig. 9.6. FFT phase angles for the discrete time sequen ce shown in Fig. 9.4.

FFT coefficients in Eq. (9.11c). Table 9.1 lists both the expected amplitudes

\F(m) |

of the Fourier coefficients without regard for the normalizing constant

C

N

and the amplitudes \B(m)\ from the program MATHEMATICA™. By

dividing the values of the amplitudes

\B(m)\

from

MATHEMATICA™

by

the expected amplitudes \F{m)\ of

the

Fourier coefficients in Table 9.1 , it is

easy to obtain the normalizing constant CV = V 1 6 = 4. Note that in Table 9.1

that the negative-definite frequencies in the FFT algorithm are stored in the

discrete frequencies identified by the indices N /2 + 2

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728 Waves and Wave Forces on Coastal and Ocean Structures

Table

9.1.

Determination of normalizing constant Q y and phase angles

a (m)

from MATHEMATICA™ FFT algorithm for N = 16 = 4

2

{terms in (•)

are the amplitudes and phases of the complex-conjugate FFT values}.

a(m)

m

1

2(16 )

3(15)

4 (14 )

5(13)

6(12)

7 (11)

8(10)

9

\F(m)\

1.0

0.75 (0.75)

0(0)

0(0)

0.25 (0.25)

0(0)

0(0)

0(0)

0

\B(m)\ = C

N

\F(m)\

4.0

3.0(3.0)

0(0)

0 (0 )

1.0(1.0)

0(0 )

0 (0 )

0 (0 )

0

C

N

= JN

4.0

4.0 (4.0)

- ( - )

- ( - )

4.0 (4.0)

- ( - )

- ( - )

- ( - )

-

(Radians)

n

-JT/4(+TC/4)

0(0)

0(0)

+ j r / 4 ( - 7 r / 4 )

0(0)

0(0)

0(0)

0(0)

of the positive-definite frequencies with indices 2

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730 Waves

a nd

Wave

Forces on Coastal and O cean Structures

SameQ[finv==f]

(* OUTPUT FILE TO BE IMPORTED TO SPREADSHEET "QP RO " *)

SetDirectory["C:\lfn\"];

z=Table[ { t[[i]],f[[i]],absbn[[i]],phase[[i]]}, {i, 1,16}]

ColumnForm[z];»debugfft.prn

9.3.

Ocean Wave Spectra

Spectral representations of stationary ergodic random seas are required for

engineering analyses of random ocean waves. Several two-parameter theoret

ical spectral models have been applied to engineering applications along with

five-

and six-parameter spectral models. The parameters applied

in the

original

derivations of these spectral models are varied. Among the most commonly

applied parameters are wind speed U

w

, fetch length F , significant wave height

H

s

and period T

s

. For those two-parameter spectral densities, it is convenient

for engineering design

to

replace the original parameters with

the

zeroeth spec

tral

m oment mo (= to the variance

a

2

of the time series or the area under the

spectrum) and the frequency of the spectral peak /o or COQ.

The mean ( = p,\), the mean square value ( = \xi), the variance (=

a%

=

mo) and the standard deviation ( =

9

= lim — /

(x(t) - n\Ydt = 112- p\,

(9.14b)

T

R

^oo

TR

JO




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Real O cean Waves

731

where £[•] = an ensemble averaging operator defined in

Eq.

(9.1);

p(»)

= the

probability density function (pdf) for the random variable (•) ; and 7> = a tem

poral record that is infinitely long in

Eq.

(9.1). Data are frequently normalized

by subtracting the mean \x\ and then dividing by the standard deviation a

x

of

x(t)

in order to obtain the following zero-mean , unit variance data record:

=

* ( O - M I W

( 9 1 4 c )

Ox

The

Wiener-Khinchine Fourier transform pair

is

similar

to the

Fourier series

transform pair from Eqs. (9.2) but relates the covariance function C

xy

(x) to

the two-sided spectral density function G

xy

(co) (Bendat and Piersol, 1986)

according to

1 r+°°

C

xy

(r) = —— / G

xy

(oo)exp ±(ioor)dco, (9.15a)

V2TT

J—

oo

i r+°°

G

xy

(co)

= —= I C

xy

(r)exTp^(ia)T)dr,

(9.15b)

V2TT J-OO

where G

xy

(co) = a complex-valued, two -sided cross-spectral density function

that may be expressed as

G

xy

(co) = C

xy

(co) ± iQ

xy

(a>), (9.15c)

where C

xy

(oo) = coincident spectral density function and Q

xy

(co) = quadra

ture spectral density function. Alternatively, the complex-valued, two-sided

cross-spectral density function

in Eq.

(9.15c) may be expressed

as

an amplitude

and a phase by

G

xy

(co) - \G

xy

(oo)\exp ±ia

xy

(co), (9.15d)

where the amplitude

\G

xy

(oo)\

and phase

a

xy

(co)

are computed from

G

xy

(oo)\ = JC^Jco) + Q lJco), a

xy

(co) =

arctan

Q xy (CO)

C

xy

(00) _

(9.15e,f)



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732

Waves

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Wave

Forces on Coastal and Ocean Structures

A real-valued coherence function may be computed by

2

\G

xy

{co)\

2

YI

y

(w ) = \ )

= area l-valu ed, two-sided spectral density function for the time

series £(/)• A real-valued one-sided spectral density function

S

m

{co)

for the

time series r) (t) may be computed from the two-sided spectral density function

G

xy

{(o) by

S

m

{a>)=2G

m

(a>)U{a))

(9.15h)

where U(a>) = the Heaviside step function in Eq. (2.1) in Sec. 2.2.2. One

sided spectral

density

values

S

nri

(a>

m

) m ay be computed for the discrete radian

frequency

co

m

from two-sided, com plex-valued discrete FFT

amplitudes

| B

m

\

in Eq. (9.11c) by

2\B

m

\

2

\B

m

\

2

S

m

(a )

m

) = ' " ' = —^Ndt, (9.15i)

IjcdfC^i TTCN

where Eq. (9.10) has been substituted for ^f and where

CN

= th e FFT nor

malizing constant defined in Eq. (9.10) for the FFT coefficients computed by

MATHEMATICA™. Similarly, values may be computed from Eq. (9.1 5i)for

the random wave simulations in Sec. 9.6 by

2\B

m

\

2

S^icom) = ^ - ^ (9.15J)

where CN = the FFT normalizing constant defined in Eq. (9.10) for the FFT

coefficients computed by

MATHEMATICA™.

One-sided spectral density functions

S

m

(•) m ay be expressed as functions

of the independen t variables (• ) of radian frequencies a>(=27tf), or of cycles-

per-second (cps) frequencies / , or of wave periods

T,

or of scalar wave

numbers k{=2iz/X) or of vector wave num bers k. In order to determine the

relationship between spectral densities expressed with different independent

variables, equate the differential area under each spectral density in a small

incremental interval of the independent variable according to

-S^mdT =

S

nr}

{(o)da>

= Sr,„(f)df = S

nn

{k)dk = S

nri

{k)dk. (9.16a)



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For co

= 2nf,

the transformation Jacobians required for Eq. (9.16a) may be

determined from

dco

= litdf = -^rdT = -—d T =

-27tf

2

dT; (9.16b)

T

z

2n

so that, accordingly, for the independent variables

T, f

and co

7

T

CO

S

m

(T) =

f

2

S

m

(f)

= —

S

m

(a>),

(9.16c)

lie

S^f) = 2nSr,

n

(eo).

(9.16d)

Transformations between frequency / and wave num ber

k

spectra require the

linear dispersion equations (4.15) in Chapter 4.3 given by

co

2

=

( 2 T T / )

2

=

gktanhkh;

(9.16e)

and the corresponding Jacobian transformation from Eq. (9.16e) is given

by Eq. (4.60c) in Chapter 4.5

where CG = t h e wave group velocity. The Jacobians from Eq. (9.16f) for the

deep-

and shallow-water approximations, respectively, for Eq. (9.16e) are

dco co dco co /—- ,„ , ̂ , s

TT =^r> ^ =

T

= Vgk, (9.16g,h)

UK deep-water ZICQ UK shallow-water K

where the deep-water wave num ber ko

= co

2

/g.

Correlation-covariance definitions

There do not appear to be consistent definitions with respect to the time

series C

xy

{x) (Kinsm an, 1 965). The function C

xy

(r) is defined as a cross-

correlation

function if the times series

x(t)

and

y{t)

are scaled by their

means

[i\

and standard deviations

at

in accordance with Eq. (9.14c) such

that each time series has zero-mean /xi = 0 and unit variance

of

= 1.

If the time series

x{t)

and

y(t)

of record length

TR

are not scaled by

Eq. (9.1 4c), then the function

C

xy

( r ) is defined as a

cross-covariance

function.




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Waves and Wave Forces on Coastal and O cean Structures

The two-sided spectral density function

G

xy

(a>)

is always defined as a cross-

spectral density function regardless of whether or not the time series x(t)

and

y(t)

are scaled in accordance with Eq. (9.14c). If the two time series

x(t)

and

y(t)

are identical, then Eq. (9.15a) is defined as an

auto-covariance

(or

-correlation)

and Eq. (9.15b) the

auto-spectral

density

function.

Historically, the covariance (or correlation) function

was

computed in order

to efface the randomness from a time series of a random process in order to

expose the invariant statistical anatomy of the process (Wiener 1964, p. 6).

The cross-covariance and cross-correlation functions are computed from time

series x(t) and y{t) by

1 fT

R

/2

C

xy

(x) = lim — / x(t)y(t + r)dt, | r | < oo, (9.17a)

TR^OO 1R J-T

R

/2

T

R

-+oo 1

R

J-T

R

/2

T

R

/2

C

xx

(r)= lim — / x(t)x(t + r)dt, (9.18a)

r

f i

^ o o

1

R

J-T

R

/2

c

? x & ( T ) =

T

l m i — / —

2

dt, ( 9 . 1 8 b )




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where

Eqs.

(9.18a, b) are always real-valued functions, symm etric about

x

= 0

and equal to the variance

0%

of x(t) for r = 0; i.e.,

Cxx(0) 1

=

}a

x

2

] (9 8c )

lCfc

&

(0)J 1 1 J (9.18d)

Unless x(t) is a strictly periodic time series, Eq. (9.18b) is also proportional

to the mean ii\ (x) for r

—

> ±00; i.e.,

j

y/C

xx

{±oo)

I _ j ^ j ^ J (9.18e)

I V ^ ( ± ° ° ) J ~ 1 ° J (9.18f)

The first analyses of random data computed the cross- (or auto-) covariance

(or correlation) function from the time series by Eqs. (9.17a, b or 9.18a, b) and

then applied the Wiener-Khinchine Fourier transform Eq. (9.15b) to obtain

the two-sided spectral density function. Modern analyses of digitized dis

crete time sequences (t

n

= ndt, Eq. (9.4a)) employ the discrete finite Fourier

transform (FFT)

(f

m

— mdf,

Eq. (9.4b)) to compute the complex-valued

discrete FFT coefficients

F

x

(m )

and

F

y

{m)

of the discrete time sequences

x(n)

and

y(n);

and then apply these discrete coefficients to com pute either

the discrete cross-covariance

C

xy

{n)

(or cross-correlation

C

xy

{n))

function or

the two-sided

G

xy

(m)

(or one-sided S

xy

(m)) cross-spectral density functions.

A comparison of these two methods for obtaining discrete spectral estimates

from an FFT algorithm is illustrated

in

Fig. 9.7 where the notation F F T implies

the forward transform from Eq. (9.11b); and the notation F F T

- 1

implies the

inverse transform from Eq. (9.11a).

To illustrate how either a two-sided G

xy

(m) or a one-sided

S

xy

(m )

cross-

spectrum may be com puted by either of the two paths in Fig. 9.7 by a discrete

FFT form of the Wiener-Khinchine Fourier transform pair in Eqs. (9.15a, b),

a random time series

rj{t)

of six cosine waves is synthesized from the ampli

tudes and phase angles listed in Table 9.2 and is illustrated in Fig. 9.8a. The

amplitudes

A

m

,

phase angles

a

m

and discrete frequencies

co

m

= lizmdf

for

each of the six cosine wave are summarized in Table 9.2. Continuous time

and frequency are discretized by Eqs. (9.4a, b) for application of an FFT algo

rithm where dt = 0.2 sec,

AT

= 64 and df = 1/Ndt = 0.078125 Hz. The

auto-correlation function C^(x) for the normalized random time series f (/ )

computed from x (t) = rj (t) by Eq. (9.14c) m ay be computed from the discrete




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{x(n),y(n)}

C„

{n) " F

x

(m),F

y

(m)

G

xy

(m),S

xl

,(m)

Fig. 9.7. Com parison of method s for compu ting spectra by FFT and by covariance functions.

Table 9.2. Parameters for the six cosine waves for

the random time sequence in Figs. 9.8a and b (d t =

dx

= 0.2

sec, N =

64,

df = 1/Ndt =

0.078125 Hz, MI (?) = 0 ft and tr

2

= 64ft

2

).

frequency m

5

1

9

11

13

15

a>

m

(rads/ sec)

2.4544

3.4361

4.4179

5.3996

6.3814

7.3631

A

m

(f t )

2.0

4.0

8.0

6.0

2.0

2.0

a

m

(rads)

1.0192

2.0579

5.2495

5.3168

2.6336

0.6556

Fourier coefficients F

f

(m ) by the FFT coefficients B

m

for f (n) following the

horizontal path in the middle of Fig. 9.7 according to

C

ff

(T„) = |F

f

( m ) |

2

= - ^ - , (9.1 9a)

where Civ=the FFT normalizing constant defined in Eq. (9.10) for the

complex-valued FFT coefficients

B

m

computed by

MATHEMATICA™.

The two-sided discrete amplitude spectrum

\G^(m)\

in Eq. (9.15d) may be

computed from discrete FFT coefficients B

m

by

\Gu(m)\ = \F

(

(m)\

2

= ¥^-, (9.19b)




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0 2 4 6 8 10 12 14

t

(sec)

Fig. 9.8a. Random time sequence r){t

n

) of six cosine waves in Table 9.2 where t

n

=

n(0.2) sec.

JJJ..

JJJ..

JJJ..

JJJ..

JJJ.

JJJ..

JJJ..

JJJ,

JJJ..

JJJ,.

JJJ..

JJJ.

_LLL

_LLL

.LLL

-LLL

.LLL

JJJ.

.LLL

.LLL

.LLL

.LLL

.LLL

J.1L

J-LL

J-LL

AIL

J-LL

JJ.L

J 1 L

J 1 L

J L L

AIL

ALLALL

_

L

L

L

L

L

L

_UJ.

JJJ.

JJJ.

JJJ.

JJJ.

JJJ.

JJJ.

JJJ.

JJJ

.LLL

.LLL

.LLL

.LLL

.LLL

.LLL

.LLL

.LLL

.LLL

.LLL

.LLL

.LLL

—J,

AAL

AIL

AIL

AAL

J-LL

J 1 L

J-LL

i l l

J L L

J 1 L

J 1 L

J 1 L

-J—

JJJ.

-UJ.

.UJ.

JJJ.

-UJ.

-UJ,

-UJ-

-UJ_

-UJ-

_UJ_

-UJ.

.LLL

.LLL

LLL

LLL

LLL

LLL

LLL

LLL

LLL

LLL

LLL

-UJ*

0 8 16 24 32 40 48 56 64

m

Fig. 9.8b. Two-sided discrete amplitude spectrum computed from FFT coefficients for six

cosine waves in Table 9.2 for f

m

= m/Ndt Hz.

where

CM

=the FFT normalizing constant defined in Eq. (9.10) for the FFT

coefficients computed by

MATHEMATICA™.

The two-sided discrete amplitude spectrum |G

w

( m ) | for the time series

rj(t)

computed from FFT coefficients

B

m

by Eq. (9.19b) is illustrated in

Fig. 9.8b. The symmetry about the Nyquist or folding frequency

m — N/2 +

1 = 33 in Fig. 9.8b of the two-sided discrete amplitude spectrum

\G

m

(m)\

is

a consequence of the negative-definite frequencies being represented by the

discrete frequency interval

N/2 + 2 < m < N

— 1.

The mean and standard deviation of

the

random time series

r](t)

synthesized

from the parameters in Table 9.2 are ii\{r)) = Oft and o

r]

=

8.0ft,

respec

tively. The random time series r](t) is normalized by this mean ii\(,rf) and

standard deviation

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738


2 4 6 8 10 12 14

t

(sec)

Fig. 9.9a. Normalized random time sequence f (t

n

) of the six cosine waves in Table 9.2 where

0.25

_ 0.2

10.15

to. 0.1

0.05

0

H-4 +

'4 4 +

43±

-1-4 +

lOT

TO

E d :

4-14-

L I J - l L U -

i-l-ti-i—n-t-

TTT44-HI4T

- t -H- + 4-4-4+4-

xrt

+ 4-h

xnr

-4 +

3 1

44 +

trnnTrriTT

uremic 431

B J '

i t t i ± t J i t

1-4 +

331

-U +

111

0 8 16 24 32 40 48 56 64

m

Fig. 9.9b. Two-sided discrete amplitude spectrum computed from FFT coefficients for the

normalized time sequence

r (t)

in Table 9.2 for

f

m

= m/Ndt

Hz.

by

Eq.

(9.19b) and is illustrated in Fig. 9.9b. This illustrates the computational

procedure for computing the two-sided discrete amplitude spectrum \Grr(m)\

following the FFT path on the right side of

Fig.

9.7.

The two paths illustrated in Fig. 9.7 for computing the two-sided discrete

amplitude spectrum \Grr(m)\ for the normalized random time sequence f (t

n

)

in Fig. 9.9a will both be followed in order to demonstrate their differences.

First, the auto-correlation function Crr(t) is computed by Eq. (9.19a) from

the FFT coefficients B

m

for the normalized random time sequence £(?„) and

is illustrated in Fig. 9.10a. Algorithms for constructing covariance or corre

lation functions from FFT coefficients are given by Brigham (1974, p.206,

Fig.

13-6a) or by Bendat and Piersol (1986, Chapter 11.6.2, pp. 406-407).

Note in Fig. 9.10a the symmetry about T = 0 of the auto-correlation function

Crr (T)

in accordance with Eq. (9.17d); and the limiting values of the auto

correlation function Crr (T) ~ 0 = the mean /AI(£) as r ->• ±7>2dt = 6.4

sec in accordance with Eq. (9.18f). Next, the two-sided discrete amplitude



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- 6 - 4 - 2 0 2 4 6

T (sec)

Fig. 9.10a. Au to-correlation function for normalized time sequence f (?„) in Table 9.2.

0.25

-H4-

0.2

FmF -"

fc

444-

; 0 . 1 5 : m

0.1 --H-I-

: B I

0.05

444*^4-

H 4 - H 4 -

^ 4 -

- 1 ^ * 1 .

-|—i*r« -jp r-i -T-T—I"

Q ^ . . i | r 5 y h » | T r T

: P 3 : : c q :

bzfci l t d : t t t

- t - H - - n - i - - n - t - - i - n -

t c t l t c a i t z l i t

-H -

rp:

1 1 1 : 3 1 1 : 3 1 1

-t-i-t--t-t-t--i-t-t-

4 J -H4

41- 1 - 4

iIBIWi

• l -H-

4 H - 4 +

t + t

i n n t t n n i r

4

:

S

-144

-144-

nrx

. . . . . . . J -H -

m c d u m t a r

' - + 4 * 4 4 4

: •

0 8 16 24 32 40 48 56 64

m

Fig. 9.10b. Two-sided discrete amp litude spectra for the auto-correlation function C jf (r ) com

puted from Table 9.2 by the FFT coefficients from f (r){ »»} and by the Wiener-Khinchine

Fourier transform pair {xx}.

function

\G^(m)\

is then computed from the FFT coefficients of the auto

correlation function

C^(t) by Eq.

(9.19b) following the path on the lower left

side in Fig. 9.7. Second, the two-sided discrete am plitude spectrum |

G^

{m) \

is computed from the FFT coefficients of

the

normalized time sequence £ (/)

by Eq. (9.19b) following the path on the lower right in Fig. 9.7. Both of these

two-sided discrete amplitude spectra \G^(m)\ are compared in Fig. 9.10b.

The non-deterministic definition for Fourier coefficients in Sec. 9.2 is

illustrated in Fig. 9.10b. Both of the two-sided discrete amplitude spectra in

Fig. 9.10b have unit variance; but contributions to the variance by the ampli

tudes from each frequency are very different. The discrete spectrum computed

from the FFT coefficients for f (f) by the path on the right side in Fig. 9.7 are

limited

to

only

the

six discrete frequencies in Table 9.2. In contrast, the d iscrete




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then a dimensionless generic four-parameter one-sided wave density spectrum

S( Q)

may be expressed by the product of the two dimensionless functions

F\ (Q ) and F2(fi) that are defined by

m mour

p

'

F i(f l ) = — , F

2

(Q) = exp - « " « ,

and that may be com bined according to the following product

— < oo. . (9.23e)

m

Dim ensionless values of

F \ (Si)/A, F

2

(fi) and S( fi) are illustrated in Fig. 9.11

and demonstrate that

F\{Q )/A

controls the spectral behavior at frequencies

higher than the dim ension less spectral peak frequency £2 > £2o =

a)/mo =

1

and that

F

2

(Q )

controls the spectral behavior at frequencies lower than the

dimensionless spectral peak frequency

Q

< £2o = «/

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742


defined by

mour

n

Jo \m J mo/m

'

/0, (9.25a)

q \ q J

where T(«) = Gam ma function defined by Eq. (2.6a) in Chapter 2.2.5 and

the dimensionless zeroth moment mo = 1. For each dimensionless spectral

moment

n

in Eq. (9.25a), there corresponds a dimensionless characteristic

radian wave frequency defined by (Vanmarcke, 1983, Chapter 4 .1 , Eq. (4.1.4))

On =

(^)

l/

"

=

(-J^-Y"

=

(m,)

1

'-, »>0. ( 9 . 2 5 b )

m I \mozrr"

Spectral peak frequency

a>o

The dimensional spectral peak frequency u>o may be computed from

Eq. (9.23e) by

— ^ = 0, n = «

0

= - = 1 (9.26)

ail coo

so that the dimensional characteristic radian wave frequency parameter

m

may

be defined by COQ as

m =

coo

(-)

(9.27)

where (p/q)

l

^

q

is the multiplicative constant noted following Eq s. (9.20).

The dimensionless generic parameter

A

in Eq. (9.23e) may be replaced by




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Real Ocean Waves

143

Eq. (9 .25a) wi th

n =

0 and wi th the d ime nsionless var iance of the t ime

series mo = 1.0 (or, equivalently, the area und er the dim ensio nless spec trum

5 ( f i ) ) ;

i.e.,

A = - (9 .28)

r « / > - i )A? )

and the d imensional character is t ic radian wave frequency

m

from Eq. (9 .27)

so that Eq . (9 .23e) becomes

S(Q

0

) =

jP/o

q(.(j>/9)-D r ((p - \)/q)

u>

0 <

SIQ =

— < oo .

coo

Q

0

p

exp

—

| — Q

0

q

(9 .29a)

A dimensional form of Eq. (9 .29a) for arbitrary values of the exponents

p

and

q

may be obtained by mul t ip ly ing Eq. (9 .29a) by

mo/m

wi th

m

defined by

Eq. (9 .27) to obtain

S(co,mo,coo,p,q) =

1

(iq + \-p)lq)

m

0

p ( d - p ) / « ) co

0

r((p - \)/q) \ o

x exp

(9 .29b)

_ im

q\co

A number of theoret ical wave spect ra l densi ty funct ions have exponent

values of

p —

5 and

q =

4 . The parameters of several of these theoret ical

wave spectral density functions may be converted to the parameters of mo and

coo

to obtain a generic d imensional two-parameter spect ra l densi ty funct ion

given by

S(co,

m o,

coo,

P = 5 , q = 4) = 5

mo

(coo

COQ \

co

exp

5

/coo

~4\~co~

(9 .29c)

and that are tabula ted in Table 9 .5 . A d im ensio nless plo t of the gene ric spectral

density function Eq. (9 .29c) is shown in Fig. 9 .12.

It is not an easy task to dete rm ine the spe ctral peak freque ncy

coo

f rom me a

sured wave data because of the variabil i ty in real spectral est imates obtained

from dim ensiona l FFT algori thm s. An es t imate of the spectra l pea k frequency

may be computed from FFT coefficients in a best least-squares sense by




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Wave

Forces on Coastal and Ocean Structures

mju>

a

1 . 9

n

K-

n

i

i

[.../.

-4

i

1

\

1

0 0.5 1 1.5 2 2.5 3

G5/W„

Fig. 9.12. Dimensionless two-parameter generic spectral density function for

p

= 5 and

q=4.

applying the linear Taylor differential correction method (Marquardt, 1963).

Because the dimensional FFT coefficients are indexed to discrete frequen

cies m, the method may be defined with discrete dimensional radian wave

frequencies a)

m

= mAco = m2nAf for the dimensional FFT algorithm

from Eq. (9.5b). A dimensional mean-square error e~ between a dimensional

measured

spectral density estimate SM(W) computed from the dimensional

two-sided discrete FFT coefficients at discrete frequencies m by Eq. (9.15i)

and the dimensional theoretical generic spectral density

S(m)

computed by

Eq. (9.29c) may be defined by

M

C

2

J2 [SM("0

-

5(m)]

,

(9.30)

1 =

1

M

c

-M

s

m—Ms

where

Ms =

the starting index for the first significant dimensional FFT

coefficient and Mc = the index of the cut-off frequency above which the

dimensional FFT coefficients are negligible in the measured spectral density

SM(»») -

By restricting the linear Taylor differential correction method to only

those few frequencies in the vicinity of the estimated value of COQ , the algorithm

is very efficient. The generic dimensional theoretical spectral density S(m)

from Eq. (9.29c) may be expanded in a Taylor series about the dimensional

spectral peak frequency u>o by

dS(m)

S(m) = S(m) +

dcoo

-

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gives

SCOQ =

9 < w o

j : (S

M

(m )

- S(m))

m=Ms

(9.32)

An initial estimate for

J

0

+

8 1 .

(9.33)

The iterations are terminated when the corrections 8a)

J

Q

are stable and

acceptably small (10~

6

,say). Note that the theoretical spectrum S(m) from

Eq. (9.29c) must be recomputed after each iteration because of the newly com

puted value of o from data, the dimensional characteristic radian wave frequency m in the

dimensionless generic four-parameter spectrum in Eq . (9.23e) may be replaced

Table 9 .3 . Sum mary of least -squares fit to Hurr icane Car la da ta for

MQ

—

Ms

= 305

and N = 4096 (Hudspe th , 1975) .

R e c o r d N o .

06885 /1

0 6 8 8 6 / 1

06886 /2

06887 /1

Init ial col.

[rad/sec]

0 .52

0.50

0.50

0.50

Final

Q

[rad/sec]

0 .4990

0 .5187

0.4847

0 .5199

Final

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Waves

a nd

Wave

Forces on C oastal and Ocean Structures

by the dim ensional average or mean radian w ave frequency

cb

or period T that

may be computed easily from the spectral moments in Eqs. (9.25) for

n =

1

by (Vanmarcke, 1983)

m\

{mQ Tu)m\ „ \ i J

co

= — =

— — — =

mmi = w—)

—-Y =

mA

2

\(p,q)

(9.34a)

m

0

(m

0

)(m

0

= 1) H ^ )

from Eq. (9.25a) and where

Aijip, q)

=

)

q

.{ .

(9.34b)

A dimensional characteristic radian wave frequency

m

and dimensionless

frequency ratio Q may be replaced by

OJ

m = coAn(p,q), fit = —, (9.35a,b)

OJ

where A12 is the multiplicative constant noted following Eq. (9.20) and

Eq. (9.23e) becomes

qA\

2

{p,q) (A

n

(p,q)\

q

CO

S(Q )= ' " " ' " e x p - "-?'

1

' , 0s

radian wave fre

quency may be defined as

mi

/

(mom

2

)m

2

_ .— .- __ .

a>

z

=Q )s = J— = J - T — — = mjmi (9.37a)

V mo V (

m

o)(mo = 1)

W a v e s a n d W a v e F o r c e s o

n C o a s t a l a n d O c e a n S t r u c t u r e s


b y M C G I L L U N I V E R S I T Y o n 0 4 / 0 5 / 1 6 .


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and the dimensionless seco nd-mo ment m,2 from Eq. (9.25a) with the constant

A defined b y Eq. (9.28) for

mo

= 1 as

m

2

= ) \[ =

A31

(p,q), Vz = ^- (9.37b,c)

and Eq. (9.23e) becomes

•S("z) = —7 rv

TZ v

e x p - '

'(V)

« r

v «?

0 < ^

z

= - ^ < o o . (9.38)

Vanmarcke (1983) defines a generic characteristic wave frequency

Q k

as

i/k

" (

mA

m

0

/

«jfc = — • (9 -3 9 )

9 .3 .2 .

Wave and Spectral Parameters Computed from Spectral

Moments m

n

All o f the variab les in this section Sec. 9 .3 .2 are dimensional variables; and ,

consequently,

th e

tilde

( • )

notation applied

in

Sec. 9.3.1

is not

applied here

to denote dimensional variables. Dimensional spectral density moments

are

computed from

a

dimensional one-sided spectrum

S

m

{a))

b y

/•OO

m

n

= I

co

n

S

nr)

(oo)da). (9.40)

Jo

Many of the wave

a n d

spectral parameters that are computed below are eval

uated

for the

dim ension al generic two- param eter sp ectrum

S

m

{ma,a>o,co,

p

= 5,q =

4) in Eq. (9.29c); consequently, th e first four dim ens ion less

spectral moments computed b y a dimensionless E q . (9.40) are summarized

in Table

9.4

where

th e

Gamma function

F(») is

defined

b y E q .

(2.6a)

in

Chapter 2.2.5.

Spectral shapes

Dimensional spectral moments

m

n

computed from

E q .

(9.40)

are

functions

of the shape o f the dimen sional spectral density S

nn

{co). These dimensional



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748


Table 9.4. Summ ary of dimensionless spectral moments

m

n

for a dimen-

sionless generic two-parameter spectrum S^imo,ci)o,a>,p = 5,q = 4 )

where m

n

=

m

n

/o,5),p = 5,

q = 4) in Eq. (9.29c) for the first five dimensionless spectral mom ents

n =

0 - 4 in Table 9.4.

Spectral bandwidth parameters

e,q and

v

A dimensionless spectral bandwidth parameter e that may be applied to com

pute some extreme value statistics for a Gaussian process (vide., Sec. 9.4) is

the following (Cartwright and Longuet-Higgins, 1956):

e

2

= 1

2

-, (9.41)

where the dimensional spectral density moments m

n

are computed from

Eq. (9.40). For a narrow-banded spectrum, e -> 0 and the maximum values

for a stochastic process that is represented by a narrow-banded spectrum are

Rayleigh distributed (vide., Sec. 9.4.2). For a broad-banded spectrum,

e

-» 1

and the maxim a values for a stochastic process that is represented by a broad-

banded spectrum are Gaussian distributed (vide., Sec. 9.4.1). For the generic

two-parameter dimensional spectral density 5^^(mo,a>0 P = 5,g = 4) in

Eq. (9.29c), Eq. (9.41) is

rHm_

=

r ( i ) r (0)






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where r ( 0 ) = oo in Table 9.4. Consequently, all of the generic two-parameter

dimensional spectra

S

vr)

{m,Q,coQ,CL>,p

— 5, q = 4) that are tabulated in

Table 9.5 may be interpreted as being dimensional broad-banded spectra;

i.e., e —> 1. However, even though the dimensionless spectral bandwidth

parameters e,

q

and

v

discussed here are useful for estimating certain sta

tistical quantities, the effects of the variability in the spectra computed from

measured realizations limit the interpretation that these measured spectra are

broad-banded for engineering applications. The effects of this variability in the

realizations from these spectra are evaluated by the Hilbert transform and the

envelope function for engineering applications to damage estimates for rubble

mound breakwaters in Sec. 9.5. The fourth spectral moment m.4 is undefined

by Eq. (9.40) and in Table 9.4 for the generic two-parameter dimensional

spectra S(mo,coo,co,p = 5,q = 4) in Eq. (9.29c). In addition, m.4 is also

non-converging when computed numerically from data. For this reason, the

spectral bandwidth parameter e may not be computed reliably in engineer

ing applications from data; but it does continue to serve a useful purpose for

evaluating the theoretical distributions of the maximum values in time series

analyses (vide., Cartwright and Longuet-Higgins, 1956, Sec. 4, and Rice,

1954).

Extremal statistics may also be computed from the dispersion of the

wave frequency spectrum about the central spectral wave frequency from the

following alternative dimensionless spectral bandw idth parameter q (Vanmar-

cke,

1972):

2

, mi

q

2

= 1

1

— ,

(9.43a)

mom2

where the bold q in Eq. (9.43a) should not be confused with the italic q

in the exponent in Eq. (9.21). Because the spectral bandwidth parameter q

in Eq. (9.43a) does not require the fourth spectral moment ra4,q

2

may be

computed reliably from data. Vanmarcke (1983, Chapter 4) determines that

extreme values are Poisson distributed. For the generic two-parameter spec

trum

S(mo,(Oo,(o,p = 5,q =

4 ) , q

2

from Eq. (9.43a) with the dimensional

spectral mom ents ra„ from Table 9.4 is given by

7 r

2

( 3 / 4 )

q

2

= 1 - ^ = 0.152787. (9.43b)

Jn




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Table 9.5. Conversion of two-param eter theoretical spectra to generic spectral param eters

m Q , a>o

a n (

l £/(•) = Heaviside step function.

SPECTRUM S(u>)

m

0

a>0

S(mo, C O Q , a>)

S-M-B: {H

s

,co

s

) (Bretschneider 1 958, 1966)

1.618f(^)

5

exp [-1.03(f)

4

]

Neumann {C , U

w

] (1943) (Kinsman, 1965)

(¥)

-)a> "exp '•\u

w

w) ]

P-N-J: Pierson, etal. {a,U

w

} (1955)

(vide., Pierson and M oskowitz, 1 964 )

^

e x

P

- 0 . 7 4

\UuCo) ]

ISSC{H

s

,

0

+ 0.26) -U(co-

co

0

- 1.65)}

WALLOPS

IP,p,coo)

(Huang, etal., 1981)

(^r-p[-f(^)

4

]

0.3932 tf

0

2

0.9528&).

3 C JT

64

\

Q.11U5)

0.71ft>

7

ft)o

ft)o

5^( f)

5

exp[- e)

4

]

24/|^(t)

6

ex

P

[-3(^)

2

]

5^(^)

5

exp[-I(f)

4

]

5^( f)

5

ex

P

[-I(^)

4

]

5 ^ e )

5

e x p [ - f e )

4

]

3.424mo exp

( a ) - a ) p )

[ 0 . 0 6 5 ( a ) - w o -

A l

s

8

£

a

s

a.

55

SPECTRUM S(f)

mo

/o

S(m ) « P -

(ff W a v e s a n d W a

v e F o r c e s o n C o a s t a l a n d O c e a n S t r u c t u r e s D o w n

l o a d e d f r o m w w w . w o r l d s c i e n t i f i c . c o m


F o r

p e r s o n a l u s e o n l y .

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Longuet-Higgins (1975) defines a spectral narrowness parameter v that

may be related to the Vanmarcke param eter q according to

v2 =

mom _

1 =

/momA

q 2

{gM)

m\ \ m\ ]

Goda dimensionless spectral peakedness parameter Q

p

Goda (1970) identifies a parameter Q

p

that correlates with wave groups in

Sec.

9.5 and is defined for dimensional one-sided spectral densities S

m

(f) by

2 r

0 0

Q P = —,\

fSl(f)df.

(9.45)

vv

The Goda spectral peakedness parameter Q

p

appears to be less sensitive to

the cutoff frequency that is required in order to compute spectral moments

m

n

from data and also less sensitive to wave nonlinearities than the spectral

narrowness parameter v in Eq. (9.44). The generic two-parameter spectrum

S

m

(mo, coo, co,p = 5,q

= 4) in Eq. (9.29c) may be converted from radial fre

quencies co = 2n f to Hertzian frequencies / by equating differential spectral

areas from Eq. (9.16d) according to

S(f)df = S(co)dco = S((o)2 7tdf,

S(f) = 2TZ S(CO).

For the generic four-parameter spectrum

S^ (mo, coo,

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Waves

and

Wave


For the dimensional generic two-parameter spectrum S^ (mo, U>Q , CO, 5,4)

in Eq. (9.29c), Q

p

is

Q

P

= h (9.47)

that is approximately equivalent to

Q

p

for a broad-banded Gaussian white

noise spectrum (Goda, 1985).

9.3.3. M ulti-Param eter Theoretical Spectra

All of

the

variables in this section Sec. 9.3.3 are

dimensional

variables; and,

consequently, the tilde (•) notation applied in Sec. 9.3.1 is not applied here

to denote dimensional variables. Multi-parameter theoretical spectra include

both variance-preserving variable shape spectra and multiple peak (bi-modal)

spectra.

Goda-JONSWAP variance-preserving spectrum

The dimensional variance-preserving Goda-JONSWAP one-sided wave spec

trum is (Goda, 1985 or Chakrabarti, 1987)

e x p [ - ( / V o )

2

/ (2 r ,

2

/

0

2

) ]

;

(9.48a)

where

„ _ 0.0624

_

0.230

+ 0 . 0 3 3 6 / - 0.1 85(1 .9 + y ) -

1

'

r

a

= 0.07 i f / < /

0

; r

b

= 0.09 if / > /< ,,

and where 1 ,p = 5,q = 4) in Eq. (9.29c)

Snri(f) = a *

Hi

/o

exp

- 1 . 2 5

(9.48b)

(9.48c,d)



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derived in Sec. 9.3.1 if

Eq.

(9.29c) is converted from radian frequency

co

to

Hertzian frequency / by Eq. (9.16d) and

H

s

= 4^/mo

from Table 9.6. When

the spectral peakedness parameter y = 10, Eq. (9.48a) is a relatively narrow-

banded spectrum. Exam ples of these two extreme values for y are applied to

rubble m ound breakw aters in Sec. 9.5 in order to evaluate the effects of spec

tral shapes on wave groups and the corresponding damage to rubble mound

breakwaters. Goda (1998) modified the variance preserving coefficient a in

Eqs.

(9.48); but these modifications do not appear to be a good as the variance

preserv ing coefficient

a*

in Eqs. (9.48).

OCHI-HUBBLE six-parameter (bi-modal) spectrum

Ochi-Hubble (1976) derive a theoretical six-parameter (bi-modal) wav e spec

trum consisting of both low and high frequency peaks that results in a double

peaked spectrum. Each of these two frequency components require three

param eters: viz., a significant w ave height

H

Sj

,

a modal peak frequency /o

7

,

and a shape (or peakedness) parameter

Xj

where the subscript

j =

1 for the

low frequency components and j = 2 for the high frequency components.

Each of the two three-parameter spectral peaks may be combined into the

following single double peaked, one-sided spectral density function:

'nv

(/) = i E

(^-f

H

j=i

x exp

r(kj)

2TT/O,

f

4A.J + 1

4A./ + 1

f

(9.49a)

Ai = 2.72,

X

2

=

1.82exp(0.0277/,) = 1 .82exp(0 .108v^o) .

(9.49b,c)

The dimensional significant wave heights H

s

. in each of the two frequency

regions j may be replaced by the total variance of the wave spectrum

according to

Hi

H's

HU]

• = i

-«i

+• H ^ =

1 6 m

0

Rfrf]

= 16mo-

(9.50a)

(9.50b,c)



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754 Waves

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Wave


f/foi

Fig. 9.1 3. Hinged wavemaker laboratory simulation of Ochi-Hubble 6 parameter wave spec

trum (| = raw unsmo othed spectral density estimates; = smoothed estimate; and =

theoretical spectral density function).

In order to compare laboratory or field spectra with Eqs. (9.49), it is

convenient to scale Eq. (9.49a) by m o /2 ^ /o , and obtain

Snnif/foj) _

m o / 2 7 T /

0 l

4((4A

1

+ l)/4)*l f, , (

H

\̂

2

\ (X-\

r ( M ) j

i +

U J J \foJ

, 4 /

0 l

«4A

2

+l)/4)*2 J Oh.}

2

]

1

(J-\

+

/o

2

r(x

2

) 1

1 +

\n

n

)

I

\fo

2

)

(4A, + 1)

exp

- (4X2+1)

-*[-m(*r]

(9.51)

An exam ple of a laboratory simulation of random waves by Eq. (9.51 ) in

a 2D w ave channel (vide., Fig. 5.1 in Chapter 5.1) by a hinged wavem aker at

the O. H. Hinsdale-Wave Research Laboratory at Oregon State University in

the USA is shown in Fig. 9.13.

The parameters applied in the laboratory simulation in

Fig.

9.13 w ere Ai

—

2.72,^2 = 1.8,/o

2

//o, - 3 . 5 , /

0 l

= 1/7Hz,

fo

2

=

1/2Hz,

H

S2

/H

S1

=

1.0

and mo = 0.01 1m

2

.

9.3.4. Spectral Directional Spreading Fun ctions

The ocean wave spectra reviewed in Sec. 9.3 and tabulated in Table 9.5 are

unidirectional spectra. The Wave Project I and II hurricane wave force records



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reviewed in Chapter 7.6.6 identify in-line and resultant force coefficients C

m

and

Cd

•

This

distinction

is due

to the directionality of

the

random waves passing

the instrumented offshore platform. The spectral models that incorporate this

directionality of random waves are reviewed briefly (vide., Borgman, 1969a

for an extensive review)

A fundamental method for incorporating wave directionality is to assume

a separation of variables model according to

~S

m

(eo, 9) = S

m

{o))D(e,

n,-), (9.52a)

D (0 ,n . - ) = % ^ , (9.52b)

where

S

m

(a>)

=

the unidirectional one-side spectra models reviewed above

and D(0,Tli) = a directional spreading function that may depend on sev

eral empirical parameters n, and/or radian wave frequencies

w

and that must

satisfy the variance-preserving constraint required by

/ .

71

D(9,Tli)d9 = l

(9.52c)

n

so that the variance of the time series of random w aves may be com puted from

m

0

= tf=

/

S

nJ]

{(o)D{e,Ui)dedo.

(9.52d)

Dirac delta distribution for uni-directional

wave

spectra

For the special situation where the random waves propagate uniformly in the

same direction, the spreading function may be represented by the Dirac delta

distribution in Chapter 2.2.3 that is given by

S(9-9

0

)

D(e,Yli)= \ °\

(9.53)

27T

where

0

= the direction of propagation of the unidirectional wave spectrum.






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756 Waves and Wave Forces on Coastal and Ocea n Structures

Cosine raised to even integer powers

Mitsuyasu (1971) proposed a fetch-limited directional wave spreading

function given by

D(9, n,o

C(s) =

C(s)

cos

2s

o-e

0

-n < 9

<

n

C(s)cos

2

'

s

(0-9o)

2

(2s-l)

r

2

( j +

j )

it r ( 2 s + i ) '

n

<

2

_

< f

C(s) =

1 l\s + l)

(9.54a)

(9.54b)

(9.54c,d)

where 9Q

=

principal direction of propagation of the unidirectional wave spec

trum,

the

param eters n,- = J and 5 = empirically determined integer constants

that control the amount of angular spreading about the principal direction

#o; and the Gam ma function T(«) is defined by Eq. (2.6a) in Chapter 2.2.5 .

The parametric dependency on the empirical parameter s of the w idth of the

spreading function is illustrated in Fig. 9.1 4.

When the integer parameters s = s = 1, then T

2

(2) = 1, T( 3/2 ) =

*Jn72, r(3) = 2 and Eqs. (9.54) reduce to (Borgman 1969a and 1972b)

'9-9

0

'

0(0,n,-)

s>

f

=1

=

C(s =

1) cos"

-it < 9 < n,

n

Cis =

1)

cos

z

(9 - do),

-

-z

<

9 <

TC

1 2

C(s = 1) = - , C(s = l) = -= 2C(s = 1).

it n

(9.55a)

(9.55b)

(9.55c,d)

-180-120 -60 0 60 120 180

(9-e„) (deg)

Fig. 9.14. Parametric dependency of cos

2i

(fl

— 9Q )

spreading function on the shape

parameters .



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Waves and Wave Forces on Coastal and O cean Structures

1.4i

1.2

^ 1

CD

0.8

Q 0 . 6

0.4

H

0.2

0

— j _

— i

_ _ _ u .

—f_

« = 8

—f-

— h a = 4

— h

___;_.

s^^s

— r -

-\—

/

//"

~ ^

~3T^

y

a

- 1

a =

10-i-j--

NT~

M "

- ^v\

afO

__L

a =

6

. J _ l _

- ; - r -

- H -

i

- t

-fi-

i

— J — i —

H ^ S f ^

-90 -60 -30 0 30 60 90

(6-e

0

) (deg)

Fig. 9.1 5. Param etric dependency of circular normal spreading function on the shape

parameter a.

to the shape parameter

5

in Eqs. (9.54) by

2 a r c c o s f l - — } = 4 ar cc os [( 0. 5)

1 /2

*]. (9.58b)

Figure 9.15 illustrates the parametric dependency on the shape parameter a of

the circular normal directional wave spreading function.

SW OP w ave spreading function

A wave directional spreading function derived from stereo photographs of

ocean waves is the SWOP model (Stereo

Wave

Observation Project, Cox

and Munk , 1954a, b) given by

D(6, &) = - •

it

1 + I 0.50 + 0.82 exp

+ 0.32 exp

-*©'

K I

0)'

cos46>

cos 20

(O =

U

5

'

it

(9.59a)

(9.59b)

where Us = wind speed measu red at 5 m (16.4 ft) above the sea surface.

W a v e s a n d W a v e F o r c e s o n C o a s t a l a n d O c e a n S t r u c t u r e

9789812774828_0009

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