Stochastic characteristics of wave groups in random seas, Part 1: Time duration of and number of waves in a wave group and Part 2: Frequency of occurence of wave groups

Stochastic characteristics of wave groups in random

seas

Part 1: Time duration of and number of waves in a

wave group

MICHEL K. OCHI AND IORDANIS I. SAHINOGLOU

Coastal and Oceanographic Engineering Department, University of Florida, 336 Weil Hall, Gainesville, Florida, 32611, USA

This paper presents the results of a study in which a method for evaluating various stochastic characteristics of wave groups is developed. Here, wave groups are defined as a sequence of at least two high waves exceeding a specified level. As the first part of the study, the probability density function applicable for the length of time the wave group persists is analytically developed. Formula to evaluate the probability of number of wave crests in a group is also derived based on the probability density function of time duration. The average time duration associated with wave groups computed by the present theory agrees reasonably well with the data observed in North Sea, while the average time duration computed by applying the formula currently used is substantially smaller than observed data.

KEY WORDS: Wave group, Time duration, Number of waves in a group

INTRODUCTION

A unique phenomenon associated with wind-generated waves in the ocean is a sequence of high waves having nearly equal periods commonly known as wave groups. Two examples of wave groups observed in the ocean are shown in Figure I. Figure 1(a) is taken from Rye in which he shows group waves recorded in the North Sea during a storm of significant wave height of 10 meters, while Figure 1(b) shows wave groups recorded in severe

seas by a weather ship at Station K in the North

Atlantic.

It has been known that wave groups often cause

serious problems for the safety of marine systems when the period of the individual waves in the group is close to the marine system's natural motion period. This is not because wave heights are exceptionally large but occurs primarily because of motion augmentation due to resonance with the waves which, in turn, can induce capsizing. As another example, a moored marine system tends to respond to successive high waves which induce a slow drift oscillation of the system resulting in large forces on the mooring lines.

Since the wave group phenomenon is extremely

interesting and important for design consideration of marine systems, many studies have been carried out to date on various phases of the phenomenon. One area of considerable interest is the stochastic analysis of wave groups in random seas. Several studies made along this Accepted June 1988. Discussion closes July 1989.

TECHNISCHE UNNERSITEIT taboratorlum voor Seheepshydromechantoa Makelweg 2- 2628 CD DELFT

ARCHIEF

line may be categorized according to the following two approaches.

One approach is to consider wave groups as a level-crossing problem associated with the envelope of a random process as shown in the explanatory sketch given in Figure 2. The principle of this approach is credited to Rice" and Longuet-Higgins4.5; although they did not discuss wave groups specifically; instead, basic properties of group phenomena of a Gaussian ran-dom process were clarified in their work. Following this approach, Nolte and Hsu 6, Ewing', and Goda8 have developed methods to evaluate the mean values of the length of time the wave group persists (7, in Figure 2) and the mean number of wave crests in group, etc.

Another approach is to treat a sequence of wave

heights as a Markov chain wherein the derivation of the transition probability is based on the two-dimensional Rayleigh distribution (Kimura 9).

In this approach,

however, the probability distribution of wave period -and thereby the time duration associated with wave groups is assumed somewhat arbitrarily. Rye 1.16 and Arhan and Ezraty 11 evaluated the correlation between successive wave heights from data observed in North

Sea and compared with that evaluated by theory

developed by Arhan & Ezraty.

Recently, Longuet-Higgins 12 has shown that two

dif-ferent approaches (wave envelope function approach and Markov-Chain approach) to the analysis of wave groups in random sea lead to almost identical results, and that these approaches may be applied only for suffi-ciently narrow-band random processes.

1-,77

Stochastic characteristics of wave groups in random sects:it/fiche K. Ochi et at_,

of wave groups is developed. In the first phase of the study, the probability density function applicable for

the length of time the wave group persists, rci+, is

analytically developed. In the study, a sequence of (at least two) high waves is considered as a wave group so that more accurate probabilistic information on time duration as well as the number of waves involved in a

, group can be predicted.

The average time duration associated with the

envelope exceeding a specified level computed by the

newly developed theory agrees reasonably well with data

observed in North Sea for high crossing level, while the

average time duration computed by applying the

formula in current use is substantially smaller than observed data. The average time duration associated with wave groups computed by the present theory also agrees reasonably well with the data observed in North

Sea.

Figure I. Examples of wave groups observed at sea (Example (a) from Rye 1974)

Figure 2. Level crossing of the envelope of a random process

Although stochastic analysis of wave groups follow-ing the approach based on the level-crossfollow-ing problem of the envelope of a random process has analytical founda-tion, the mathematical derivation of the probability density functions applicable for the time duration (T.. in Figure 2) and the time interval between successive groups (ra in Figure 2) is extremely difficult. Hence, in almost all studies, the number of waves involved in a group and the length of time the wave group persists, etc., are all mean (average) values, and no precise prob-abilistic information on these individual quantities has been provided.

Extreme care must be taken in applying the currently available methods for evaluating the mean number of waves and mean time duration of wave groups. It has been customary, to date, to consider the exceedance of the envelope above a certain level to identify a wave group. However, this is not correct; if the time duration Ta + shown in Figure 2 is relatively short, there may be only one wave crest (or no wave crest) in time ra+ which obviously does not constitute a wave group even though the wave envelope exceeds the specified level. Therefore, consideration of (a) exceedance, and (b) the number of wave crests must be incorporated in the

development

of the

probability density function associated with wave groups.

This paper presents the results of a study in which a method for evaluating various stochastic characteristics

40 Applied Ocean Research, 1989, Vol. 11, No. 1

PROBABILITY DISTRIBUTION OF TIME DURATION OF ENVELOPE EXCEEDING A CERTAIN LEVEL

As stated in the Introduction, an exceedance of the envelope above a specified level a does not necessarily mean a wave group unless two or more wave crests are compromised in the exceedance. Therefore, we first derive the probability density function of the time

dura-tion, re,+, associated with the envelope exceeding a level

a; then this probability density function will be trun-cated taking into consideration the condition required for the existence of a wave group.

In the development of the probability density

func-tion of time durafunc-tion of the envelope exceeding a

specified level a, waves are assumed to be a Gaussian random process with a narrow-band spectrum. Then, following Rice's approach (Rice2), the probability that

the envelope of a random process x(t) exceeds a

specified level a at time ti with velocity R; during an

upward crossing, and then crosses that same level

downward at time tz with velocity RI is given approx-imately by the following formula:

{

Upward crossing of a level a at time tl with Pr velocity R; followed by a downward

crossing at time 12 with velocity R2'

R;Rif(a, R a, Ri) dR I dRi

+ rce +

Na+

Rfficr, dRI

O<Ri<co

co < < 0 (1) Here, the numerator, 17.4.(1-.4.), represents the

expected number (per unit time) of envelope upward crossings with velocity R; followed by the downward

crossings with velocity RI; while the denominator, Kra+,

represents the expected number (per unit time) of

envelope upward crossings of the level a with velocity RI. f(a, RI, a, Ri) is the joint probability density func-tion of displacement R and velocity R' at times ti and

-'where' .?. ; x,(1)= cn cos I (,,,_c72)t --e,,],

.;(1) 7 >c,,

I(Con, C-01 en) -n=1

7 mean frequency of wave Spectrum

Let the -random Variable. Xci represent the value- of x(t) at time th and the random variable Xsi represent the value of x5(t) at time t, etc. $irice We take sine and cosine components for both displacement. and velocity at times t1 and t2., we have a set Of. eight 'random

variables (X,,, X;1/ -/t2) which_

composes a random vector X..Kere, each element 'Of obeys the normal probability distribution, and We may write the joint normal 'probability density, function in

the following form: :

.RX)=""-4

.1EI1/2 C (Z7r) where,

X =

Xc2, 4.s2, -X12) i and

PO, PI

0 0 112 PI P2 0 PO PI AO no( PI P2 0 112 0 .0' 711 0 0 ni o' no, v-ni 0 0 , Po ni

712 00

. Stochastic characteristics of wave groups 'in random seas: Michel K.- Ochi elat

,

-12 with /?1,=, /22 = a in this .case, while- fig, R() is the .

joint Probability density of displacement R and velocity R' of the envelope at time it with R1,-=

a.-The probability given inequatiOn (1) may be inter-preted as the time that the -envelope spends above the

level a. In other words, by letting ti =7-12.=704,equation (1) is equivalent to the probability density function of

for a Specified level ,a.

Integration of the joint probability, densitY function involved in the numerator of tiination,(1)dannot be car-ried out directly : It is necessaryto first obtain the joint

probability density Of the wave profile i(t) and velocity x'(t).at times

ti

and 12, and the to transform this joint probability density function of displacement

and velocity to that, of the wave envelope and its

velocity. :

Assuming a harrow-hand Gaussian random process, the wave profile x(t) can:be:written as,

-.x(t) = c,, COi(o;nt .

-.

= xc(i)cos'41 - x(t)sin iet_{_} (2)

(4)

\

6'. 770 ni ni n2 ni 4.: -0v _0 0,-'.:, po

-Pi

112 r - .p-2 "Pi 74.0" ..._ 0 P2' 11 (5) where =Lt2 ti . wave spectrum , .

-At'Can be seeifinequatiOns (4) thrOugh (6), the joint ; _{probability density function f(X ) Can he evaluated}

,from a given wave spectrum In practice, however, the

,inverse operation of the covariance matrix involved in equation (4)- is extremely difficult to perform. Even thoUgh, the inverse Can be---a_Chieired,-the follow up -ttantforniation to the joint prObabilitY density function

Of the envelope and its velocityii not feasible.

One ki,,ay to OVertome theii difficulty it to decompose

a given speatrurninto two parts, , eadh part being sym-medic abotit its mean frequency. One contains

primar-Ily,thelOwer frequency_ energycomponents of the wave ..spectrum, 'while the other -contains the higher

frequen-cies -Of the spectrum as illustrated in Fittire-3. Each part Of the sPeCtrtirn can he' expret-ted in the" form' of,

tiomo =

-S(01) do) -112 (0) (.7))2S(0.0- d

-

d ) d Po(co)cos(o)-(7))7.-4.r.o-- 0

vi S(4.004 w)stn(o) - CO)7 CICO 0

= S(i.d)Sin(W,,7*-Ca.1)7003

- = c)cas(o) C1.07

712 - S(0))(co - &,)2 sin(w - (.0)7 d(.0

' ,FFIC-01.010

Figure DeOrnpoSitiort-of waive spectrum into - symmetric-shape spectra

-two

E is_the coVariance matrix given by

Applied Ocean Research, 1989, Vol: 11, No. 1 41

- Stochastic characteristics of wave groups in random seas: Michel K. Ochi et al.,

where,

i= 1, 2

co.; = modal frequency of Si(w)

al= value of SAO at the modal frequency

moi:= area under the spectrum SAW)?

The sum of the two areas m'ol and minis equal to the area under the original spectrum S(w) which, in turn is equal to the wave variance, ,to. Since the variance of the

velocity, 1L2; in the covariance matrix also plays a

signif-icant role in developing the probability density functiOn of time duration 7,.+, the sum of the variances of the two decomposed spectra is Maintained constant and is equal to the variance 1'2 of the original spectrum. Under these conditions, the parameters ai, Wirth and mph are determined numerically- through a, nonlinear least squares

fitting technique such that

the difference

between the shape of the original spectrum and the

sum of the two symmetrid spectra, expressed, by

1S(w)= iSi(W)j2 iS 'minimal. A detailed discussion

on the decomposition of the spectrum is .given in

Appen-dixl.

Figure 4 shows an example of wave spectrum

expressed by the stint of tvio. syniinetric spectra. The original spectrum represents the Avid-parameter Wave spectral formulation for a 'significant -Wave, height of 10.0 m with a modal 'frequency _of 0.42 rps. Although there is some discrepancy in the niagnitude of spectral -densities between the original and The combined Sym-metric spectra at frequencies ranging from 0.60 and 0.85 rps, the values of 'moments which are significant

for the analytical scilution of vvaye groups are very close.

Mat is, PO = 6.25 m2. and 142 -t 0317 m2/sec2 of the combined spectra are chosen to be equal to those of the

original spectrum. The computed moments m1 and 1712 are 3.36 in?Isec and 2.22 in z/Sec2, -respectively, for the

combined: spectra as compared with 3.40 m2/sec and 2.19 M2jsecz, respectively, for the original spectrum.

The significance of representing the shape of a given wave spectrum by the sum of two symmetric spectra'is that the covariance matrix given in.equation (5) can be drastically simplified for each spectrum such that all covariances between x and: become zero (see

Appen-dix n). That is,-the covariance matrix of waves for each symmetric spectrum becomes,

Figure 4.. Comparison of (significant wave height 0.42 rps) and sum of two

0.8 12

.FREOLIENCY 0 IN

two-parameter'.wave ,spectra

10.0 in, modal frequency syrninetric-shape spectra

42 Applied Ocean Research, 1989, Vol. 11, No. I

where, 0. P21

O0 0

0 imat POi 0 0 .112i P11 P01 PI POi P '1,2i 0 (8)

where, i = 1 and-2, and elements in the above covariance

matrix are evaluated for each decomposed spectrum. As can be seen in the covariance matrix, the sets of random variables'' (xri, 412 Xc2, 42) and (xsi, xs2, 312) are

uncorrelatecl and this makes further analytical devel-Opment: of the distribution function feasible:

We now consider two 'sets of random variables

Xcf1;Xc2,-212) and (zsi, xs2, 42) both normally distributed with

Zero mean and assumed

to be

statistically independent. Hence, the sum of these two sets of random variables has a normal distribution with zero mean and a covariance matrix which is equal to the sum of the'two covariance matrices evaluated for the symmetric spectra. Hereafter, each element of the covariance matrix given in equation (5) represents the sum of the two elements in equation (8) Computed for each symmetric spectrum. All n in the covariance matrix given in equation (5) are zero.

Since the inverse of the covariance matrix can be ,evaluated, the -joint probability density function of

...,2d2) can now be written as follows:

f(xci, 4i; 42,-42, 4i; 41, xsi, x;i)

e - (9) co < all, x, x' < co A= ( (p2 + p2)(go ) 1 (A2 .P2)040 + PO (10) A, 2 ' 2 2 2

K=

A1

1(xci

xe2.+ xsi + X 52)

+ M22 (x,? +"x12÷ )

+ 21412 (Xt I Xc2.42 X51X;1 Xs2.42) 211/13(XclXc2 X51X52)

+ 2 M24 (4142 +

42)

+ 2M14 (*el Xe2-41+ xs, xax;1)1 (11)

and

MI I = 140(/41 vi) Pi/42

MI2 P2/10)

M13 "=.

yo(id

+ PtP2

MI4 !I(0142' Pi - POP2) M22 " .142(140 Pi) P1140

M24 =, P2(14 .4)+ PO!i

The detailed derivation of equation (9)

is given in Appendix 4-of Reference 13.

(12)

1

Stochastic characteristics of wave groups in random seas: Michel K. Ochi et al. Next, by applying the polar coordinates

xi = R1 cos 01 xi =R1 sin 01 Xt2 = R2 cos 02 Xs2 = R2 sin 02,

the random variables in equation (9) are transformed to a set of new random variables

RI, R2, Ri, 01, 01, 02, 0i).

Since R1 and R2 represent the envelope of a random

process x(t) at times t1 and t2, respectively, we set

121= R2 = a for the problem at issue. Then, the joint probability density function of R(, RI, 01, 01, 02, and 01 for a specified a can be written by,

fin R1,01,01,02,01; a) a4 1 _-(L)/2IAI (14) (27r) IA I 0 --5. Ri < co, co < Rl .<.. 0, 0 .<... Oh 02 ..'5 2r, CO < Of , Oi < co where, L = 2a21 M11 + M13 COS(01 ₀₂₎₎

+ 2a[ M12 M14 COS(0I _{02)1 (RI Ri)}

M22(RI 2 + R2 2) + 2M24R1R1 cos(01 02)

+ M22a2(0I 2 + 022) + 2M24a20101 cos(01 02)

+ 2M24cx (R191 RIO( )sin(01 02)

+ 2A/i4a 2(01+ 01)sin(01 02) (15)

It is noted that M11, M22, M13, etc. in equation (15) are functions of Po vi and P2 as shown in equation (12), which in turn are functions of the time duration rc,+ associated with the envelope exceeding the level

a.

Next, equation (14) is integrated with respect to 01 and 02. The exponential part of equation (14) consists of many terms containing cos(01 02) and sin(01 02) as

shown in equation (15); hence the integration is carried out with respect to 01 and (01 02). The results yield,

4 2.2r

f(R (, 121,01,191; a) = , e P dio (16)

(27) _Al 0

where,

1

P=

[ 2(MII + M13 cos co)a2

2 1 A I

+ 2(M12 Mpscos co )cr (RI

Ri)

M22(121 2 + RI 2) + 2M24R1R1 cos 40 m22a2(01 2 + _{) 2M24a 20(0/ cos so} + 2M24a sin v,(RfOi RA)

+ 2A/114a2(01+01)sin sol

After much lengthy mathematical rnaniputalion, the details of which are given in Appendix 5 of Reference

13, the joint probability density function

f(R1, R1,0 ( , 01; a) is integrated with respect to 01 and 01,

and then, the joint probability density function (17)

f(12(,121; a) can be written as,

(

\ 2 r 1 f(R 1, 121;a).

24

30 M2241 (13) a 2 (14° Po cos ,p) x exp[

[

where, 2 2 PO X (Rf 2 + 2pRIRI + RI 2) 2M22 (1 P2) 2k(1 p)(R;

k))]]

dos (18)

ka

0.to COS 40 PO)

2 2

ILO PO

M24

_{P2(IL pa)-Fvovi}

p =

cos 40 =2 1,

2 COS SO

M22 MULTI Po) IlL0

(19) In order to carry out further integration_{necessary for} the derivation of the probability density function of the time duration, f(ra+), the following transformation of the random variables is made:

2 D , M ,T I r 1'4 = 2 222 =..\11L2 POP 2 2 ''-'

,I

M22 P-OP I2

v

Ri= 2--

V= ;42 2 2

'

AO PO N /40 PO

Then, the joint probability density function of U and V for a specified a becomes,

f(u,v;a)=

(a

1 ____,

) 2 .f 27

2r

po Po 0 ,Il _p`

x exp{ [a2(1.40 Po COS fp) 1

Po2 PO2 + 2(1 p 2 ) x (u2

2puv + V2)

2 X

{(u v)(k

t.

'°)

M22

cjia

M22v1) 2}]] clio (21) 0 u < co 0 v < co

By applying the property associated with the bi-variate normal probability, distribution (see

Cramer"),

the second

term of

the exponential

expression in equation (21) can approximately be writ-ten as follows: 1

ex{

'

2(12)

(u 2 2puv + v2)} ,11 exp +--2 1.72} 2 (22)

This approximation method was used by Tikhonov in the derivation of the probability distribution of time duration associated with excursion of Gaussian noise. The joint probability density function f(u, v; a) then

1

1 +p

(20)

.

Stochastic characteristics of *rye groups in random seas: Michel Ochi et irt.

approximately 'beeomes, f(u, v; a) = 271-

- Po o

(a ) 2 ' By letting -[ 2 a2Gio Po COS or 1O 'O. ' xexp{ (k.1. 2

}]]

(23) . .

By transformation of the random variables Rf,andRi

- to U and V, respectively, the numerator Ofequation, (1)

- can be expressed in terms of U and V as follows:

. . . ,

:

.,

Ra. 0'6:0 =-, -_

--. RfRif(Ri, Ri; a) d/21 dRi

-

-. o

-M22 _{: f-' 7 1:1,pfoi-, v";:co du du (24)}

Ao T-po 9, 0 -' ' ; - ..,

Then, from equations (23) and (24), We have, -iCra+fra+, 1

a 2'

--M22. (2w

rr

- ,././,i

- 27r (IA - FO)2 Jo_. o.-_.-o , =

xexp- 2 .(u +..y...1

- {

-

1u2+-v2 .\I ..x... AIL vl '

)

, ! M22: , -: du

la

2 (

-

COS 9)4. 1 -

-x--(k) ]}.clio.

--',. (25) 2 .2 2

Since the terms associated with u and v in equation (25) ate syminetric, 'e ean simply write,

- 2 -uu

expt

[

. 0' -di/ dv =-2 ''''." X exp[ u _ :..-' . , . A AZ 7: iii-....,2 :-I + p, M22' . 7

the integration 7giVen In equation

(26)an. be obtained

asi_

, ..\Iii(2)-+ pO

1-...+P..;,-4122

clu'F.- .1 F;,--/ry. F2) 2 [(1 + (JiY))

where. -Jva+

+ =

(4)

(ii- Po)

"13 a2(Ati_t' Po Os ) x e?c13 4(1 + + v2 po,- Po 2 x (1 + e.2Y2,( + 4:( -y)) )2 do 2- 7

(to

P0)2 0 4:2

-+p

. .(26) (27)" (28)

Hence, equation (25) can be eValuated by,

44 Applied Ocean Researck1989, Vol. 11, No. 1

xexp{- r(p4j - vo cOS it?)+ 4p 2]}

.PO

.

.,. -X (e-i72 "I' .;:27i 1 1 + 0(,27)1 )2-' do .

\I

-,m2: ,:- f 2ir :0177-1'121).%. 30

- 1 xexP{

Ao - Po

[a2Uto. 2.1/0 COS ) ' 2 2 -I- 4p-y

(.e -:2",i!' -.7 ., 2iry ( 1 _{ck 0y)) )2}

--. .

. e .,270(.,2-y))

" (29) .

It can be piiived that the first term in the brackets Of the above equation represents the expected number (per unit tithe) of the envelope downward crossing with - veldeity RI to:Aid-wed by the upward crossing with Velocity RI, namely Ra...(7a_.) (see Appendix 7 of Reference 13). Since ) is much smaller than Rd+ (rat. ) for a large a, the first term in the brackets of

. equatiOn (29) can be neglected. It is also noted that the

_ uppetlimit of integration of equation (29) is 42 in

real-ity in. order to satisfy the condition that R.. (T0.,) be a

;

positive nurnber. Thus, we can write

-

m

r(iso-- Po)

o

x exp[

-

[a2 kiL0,- Po cos. jp + 4" 2]}

.

x 1 e +

,/70(7)} clio

(36)

. Oh the other hand, the denoininatOr of equation (1)

becomes,

= 1?1.11R(; a) dRi= e-(a2)/2" (31)

o - p.o

As stated in regard to equation (1), the equation is equivalent to the probability density function of the time - duration, 'T.+,

associated with the envelope

exceeding a specified level a. Thus, from equations (30) and (31), the probability density function of roc+ can be obtained as,

..

J lTo +.1 --- , 2 2 2

-A-; . ' 2410 _Mix i712

... '..._. :7r/L2 who =-- I, o )- o . ( ra2(k) -- Po COS (p) -- ---. exP t -- L ---- - 2 2 - + 4P72 . - . AO 2u0 ,-} .-- 7 772--4-.j-i-ici)(J-7) ch0 (32)

It is assumed that co is relatively small for large a, and

thereby we have the approximation cos 1,0'7 1 92/2.

Then, equation (32) may be written as, T/2

B

)= C2Cip.0 M22 .71.A2 (AO P0)- SO ra+

---

2 2 2 x exp[ "2(/-43

P"I

_{2 2}9212)1 ILO PO , 2 2 4p7 I

_{e_22 +}

' 0 L7r where,

a

PI(A0(1 02/2) POI ₍₃₄₎ 2(1 + p), M22 %AO PO

C= normalization factor to make the area under the density function unity

f

_rTa+i

= 1

I

(Ta + ... 0

PROBABILITY DISTRIBUTION OF TIME

DURATION ASSOCIATED WITH WAVE GROUP In this section the probability density function of the time duration associated with wave groups will be de-rived. The probability density function derived in the previous section is that applicable to the time duration that the envelope of a narrow-band random process exceeds a specified level. If the time duration is not

sufficiently long then there may be no wave crest or only

one wave crest occurring during the time the envelope

exceeds the specified level, and this cannot be considered

as a wave group. Therefore, the probability density function of the time interval associated with the envelope

exceeding a specified level should be modified so that it is only concerned with the time interval during which two or more wave crests are present. For this, let usfirst

examine the relationship between the time duration rai., and the number of wave crests involved in Ta+.

From the assumption of a narrow-band random pro-cess, the time interval between two successive wave crests associated with wave groups is considered to be

equal to the average zero-crossing period, To, which can

be evaluated by the following formula:

To = 27,./no/m2 (35)

where,

mi = jth moment of wave spectrum.

Then, we can evaluate the number of waves during

the time interval To+ based on the average zero-crossing

period as follows:

(i) tcy+ To: If the time duration of the envelope,

ra4-, is less than To, there is either no wave crest or only

one wave crest in T.+. Hence, this situation cannot be

considered as a wave group although the envelope

exceeds the specified level a. The probabilities of occur-rence of this situation can be evaluated from equation

Stochastic characteristics of wave groups in random seas: Michel K. Ochi et al.

dio (33)

(34) as,

I.

Po =

f(

+) dr.+

(36)

(ii) To < Ta+ 4 2T0- : If the time duration T.+ is

bet-ween To and 2T0, then there is either one wave crest or two wave crests during The probabilities of one and two wave crests during can be evaluated by,

Pr ( One wave crestI Ta+ _{= I}

T

_I)

To (37)

a+

Pr ( Two wave crests Ir.+ = (2:1=-1; 1) (38)

To

Since one wave crest in rix., cannot be considered as a wave group, this situation should be eliminated from the probability density function for the time duration associated with wave groups. The probability of

occur-rence of one wave crest in To <

Ta+ 27-0 can be

evaluated by, 2f0

pi = {1 1)]f(ree+) (39)

fo To

(iii) 2T0 4 T.+ 4 3 To- : Under this condition, there are

either two or three wave crests in time duration T.+ hence, this situation can be considered a wave group. Thus, in summary, an exceedance of the envelope

above the specified level does not necessarily mean wave

groups, since there may be no wave crest or only one crest if the exceeding time duration is not sufficiently long. The probability of occurrence of this situation is given by (po + pi) as shown in the above. Therefore, the probability of wave group when envelope exceeds the specified level a is given by the following formula:

Pr Wave groups when envelope] = 1 (po + pl) exceeds a level a 2f.

= 1

[

Ara+) dra+ +

.f 0 x {i-7 1)]

f(ra+)

drc,+] (40) To

The probability density function of time duration associated with wave groups can be obtained from the probability density function f(7.+) by eliminating the portion in which either no wave crest or one crest occurs

f(ro,)

itto)

.(%

21-0

(741± -1) ftra+)_To

Figure 5. Sketch showing the relationship between prob-ability density function associated with wave envelope,

"4, and that associated with wave groups, ffrg)

Stochastic characteristics of wave groups in random seas: Michel K. Ochi et al. in r4. That is, the entire probability density function

for ra.,. To is discarded, and the magnitude of the density function fOr .To 4 '7'a+ 42TO -is modified by multiplying by the probability given in equation (38). This is shown in the piCtorial sketch given in Figure 5.

Then, the modified probability density function is

normalized so that the area under the density func-tion becomes' unity.. The probability density funcfunc-tion thus derived, derioted by f(ro); which is applicable to the time duration associated with wave groups is sum-tharized as follows:

Aro

for 2 To G (41)

The expected (average) time duration associated with wave groups, dehoted by fq, can thereby be evaluated

as, for 0 < 4.. To (TaZ,L =7,

(po+ pi)

To . . for To -1 T.G 4 ;To IAra + I. T., = (po + pi) TGf(TG) drc. . s. T., drG (42)

46 Applied Ocean Research, 1989, Vol. 11, No. 1

- The probability of r (m + I) waves for a given TG

becomes,

.

Pr ( (m 1)waves I ra ) = ---m 0

Frorri the conditional probability given above, the probability of occurrence of m waves in TG, where

inn 4 TO 4 (m + 1)To, can be evaluated by,

Pr ( m waves ) = (m+ 1) J-(7G) drG cinm, of. {

71

fo To Similarly, we have, . 1)fo Pr (m + 1)waves -

m f (ro)

mfo tTo -

-The Under6ring concept for equations (45) through

(47) is the same as that demonstrated by

Longuet-Higgins in his derivation .

of wave group

length

(L.onguet-Higgins 12 ). '

By applying the formulae derived in equations (46) and (47), we can evaluate from the wave spectrum the

probability of occurrence of a specified number of

waves in a group as follows:

rr

.=

r

j(7G) AuTG

(in in a group iro' 'To

wives

(ro

,)

Fir{Three waves

in a group -404' +Ara + ) .Cfra,+ L

i2to3

RTG) clr'G -2n to

3''(;. 2)/(TG)

dra

7: 4 fo Tr: :4- .f ' -=-7)f(TG) drc To . 4; -

,

(

1

7.,,,,

i 3 f(r9).1:1TG,

1:: (

5 ..77711- fOG) dT. (48) , :EXAMPLE OF APPLICATION .

As an example of the applicati6U Of the prediction method for evaluating the time duration and number of

- waves associated with wave groups, numerical

computa-tions were carried Oht Using the two-parameter wave spectrum shown in Figure 4 'having a significant wave -height of 10.0 m and a modal frequency of 0.42 rps.

The probability density function of the time duration of the envelope exceeding levels of 5.0, 6.0, 7.0, and 8.0 m above the mean water level was computed by equation (33), and the results are shown in Figure 6. The

'average values of the time duration fa+ were then

evaluated from equation (33) by

(45)

(49)

1

and the results were compared with those computed

by applying the following formula which has been

1 _(PO

+ pl)t

-To TG , -1. . 27;,,

(

I JITTG . . +

.c.1 Bra +

2T, = Ti; d:TG}

NUMBER OF WAVE CRESTS _IN 'A GROUP

Once the probability function of' the time duration

associated with- wave groups is derived, statistical

.infor-mation on the number of waves involved-in a group can be obtained. therefrom. For example, the probability of 'occurrence of a specified .number of waves in a group can be evaluated from the probability density function

wave crests in a wave group for a given wave'Spetfrum. developed in the Previous section.

Let us first 'evaluate the average (thean)- number of

Asshming that the time interval between tvvO StIccessive

wave crests is equal to the average zero-crossing period, the average number of wave crests can be obtained from equations (35) and (42), as

Average. number Of wave crests = 'To/To '(43)

,

Next, let us evaluate the probability Of two wave

crests, three.Wave crests,- etc., in a group. In general, for the time duration m To 4 7G 4 (77:71-1)To, where m is an

integer, 1, 2,1, there are either m waves or (m + 1)

waves in -7.G. The probability of m-wa.ves in a specified

Tc.is given by,

Pr m-waves ITG ) = 1

(1-To = (m + 1) TGTo (44)

where,

inn

ro 4 (m + 1)To

commonly used to date: ,

f(R) dR

7a+ = /43 - - (50) R, .1101,Rij dRs 1L2 0 ' .

where, f(a; is the-joint:probability density function of amplitude and velocity of the envelope, ' and ii0-_and #2 are as defined in equation

s As can be seen in Table 1,the avetaie'tinie.duiatiOn for a given 'level evaluated by the probability density function, derived in the present study -(equation 49) is substantially greater than that conitinted.,by the formula

-given in equatiOn:(50).

Additional coMputations. Were carried Out using wave

data measured it-seg.:the data-Were'tien in North Sea off Norway - the Norwegian Marine' Technology

Research Institute. The water depth at the measured site

"-it 230 meters.- Two sets Of data (signifidant'Waye-heights 3

of 8.13 m and 7.44 m) Were -analyzed : Figure '7- shows

the ,measufed,spectruth as well as the sum Of the two symmetric shaped spectra used in the -computations for . each record.

Since the Observation tithe- for each. recoil:11°i which

' the seas were considered to be steady-state is not

suffi-ciently long,- the sample size Of the _envelope exceeding

a specified level is rather,smalli:Therefore; theaverage_ values obtained from the measured data may not be accurate Nevertheless, 'a consistent:trerid Can be seen in

Figure 6. .Probability density function oftienedUration 'asiOciatedifvith wive envelope at various levels

Table 1 --Comparison of average time durations associated

with thaanyaiOpaeaceeding.AnieCiiied

- - .

(Two-parameter wave spectrum)

24 213

Stochastic characteristics of _wave groups in random seas: Michel K. Ochi et al.

12

Olt 12

FREQUENCY ca 04 RFS

10

2

21.INOF ii,/0 240.161ETRIC SPECTRA

-MEASURED SPECTRUM

\

tjJ

s--...i... .' OAT' t.. 02 12 .12

'..-FREQUENCY to IN FIPS

. (b) Wave spectrum B, significant .wave height 7.44 M

Figure 7.- Measured wave spectra used in :analysis.

. _ .

-,

Tab'fa. 2 Comparison betvaan obseroad.and essiated tint durations

assosiatad with.the suwa1opieeicseding'a specif lanai '

...

-A

Table 2 Where the Measured and computed average time

durations- above- various levelg..are tabulated. That is, although the:average Yalues obtained from the records are greater than:thcise computed by 'equation (49), the tWo'are very closefor high crossing level. On the other

hand,- the computed 'values by equation (50) are smaller

than the values obtained from the records by

con-siderable amount.

As Stated' in connection with ;equation (40), an

exceedanCe of the envelope above A specified level does

not necessarily mean al.wave group. The probability of

' occurrence of wave group . when the ;envelope exceeds

the specified level was calculated' by equation (40) using

the twO=Paratneter-Wave spectrum and the results are

,tabulated in Table 3. . ,

As can be.Seen in the table, the majoritji of the level Crossings cannot be considered as indicative of the

(a)' Wave 'spectrum A, signifiCant wave' height'8.13 m

-Applied Ocean Research, 1989, Vol. 11,No. 1 47

"

Observed

- -- -at/erase ties: duration

- . SaM.39 tine durationAverf.li i Eq.(49) 39:_(393

o al g r. ... 4.0 a 5.0 9 6 15.9see 10.8 -set 7.9 - _ 3.999c 3.1 Z 6.0- 7. _{- -} 2.6 4.0 a V 7 10.7 sec' -, 7.6 'lee 3.7See .1. ., . 5.0 4 71"

'

6.3

l'

Z.. rom _ ' Level

Average time duration computed by

Eq.(49) Eq.(50) .. 5.0 m

8.2 sec

5.6 sec 6.0 7.3 4.6 7.0 6.4 4.0

8.0,

..! 5.5 3.5 ...

3

Stochastic characteristics of wave groups in random seas: Michel K. Ochi et al.

Table 3 Probability of occurrence of wave group when

envelope exceeds a specified level

(Two-parameter wave spectrum)

TO.128 DURAT2261712.42 tGIN.SEC

Figure 8. Probability density function of time duration associated with wave groups at various levels

' rabic 4 Comparison between observed and computed average time

durations associated with wave group (Spectrum A, 118+ 8.13 m)

occurrence of wave groups since there is either nci wave crest or only one crest during the crossing time interval. Figure 8 shows the probability density function of the time duration associated with wave groups exceeding levels of 5.0, 6.0, 7.0, and 8.0 m above the mean water

level computed by equation

(41) using

the

two-parameter wave spectrum. It can be seen from a com-parison of Figures 6 and 8 that there is a substantial dif-ference in these probability density functions.

-Table 4 shows the comparison between observed and computed average time durations associated with wave group for the measured wave spectrum A shown in Figure 7(a). Agreement between them appears to be satisfactory.

CONCLUSIONS

This paper presents the results of a study on stochastic analysis of wave group phenomenon observed in the ocean; in particular, the derivation of the probability function applicable for the time duration associated with wave groups.

It has been customary, to date, to consider the

exceedance of the wave envelope above a certain level to

identify a wave group. However, if the time duration

48 Applied Ocean Research, 1989, Vol. 11, No. 1

above a certain level is relatively short, there may be only one wave crest (or no wave crest) in the duration which obviously does not constitute a wave group even though the wave envelope exceeds the specified level. Hence, in the present study, a sequence of (at least two)

high waves exceeding a specified level is considered as a

wave group so that more accurate information on time duration as well as the number of waves involved in a group can be predicted.

Formula to evaluate the probability of number of wave crests in a group is also derived based on the prob-ability density function of time duration associated with wave groups.

The average time duration associated with the

envelope exceeding a specified level computed by the

newly developed theory agrees reasonably well with data

observed in North Sea for high crossing level, while the average time duration computed by applying the formula in current use is substantially smaller than observed data (see Table 2).

The average time duration associated with wave

groups

computed by

the present theory agrees reasonably well with the data observed in North Sea (see Table 4).

ACKNOWLEDGEMENTS

This research was sponsored by the Office of Naval Research, Ocean Technology Program, through con-tracts to the University of Florida. The authors would like to express their appreciation to Dr. E. A. Silva of the Office of Naval Research, Dr. T. Dawson and Dr.

D. Kriebel of the U.S. Naval Academy for their

valuable discussions. The authors are sincerely grateful to Professor Longuet-Higgins who kindly provided his valuable comments on the paper. The authors would like to acknowledge Dr. A. Terum and Dr. H. Rye of the Norwegian Marine Technology Research Institute for their kind help in providing wave records used in the present study. Many thanks are also due to Mrs. Kathy Winstead for typing the manuscript.

APPENDIX I: WAVE SPECTRUM COMPOSED OF TWO SPECTRA HAVING SYMMETRIC SHAPE

In order to simplify the covariance matrix given in equa-tion (5), a given wave spectrum is decomposed into two parts; each being symmetric about its modal frequency as illustrated in Figure 3. One part contains primarily the lower frequency energy components of the wave spectrum, while the other contains the higher frequen-cies of the spectrum. For this, express each decomposed spectrum in the form of a normal probability distribu-tion with mean comi and variance a?. That is,

.51(co) exp{ (51)

2crt

i = 1, 2

where, moi is the area under the spectrum so that the integration of equation (51) over the frequency domain is equal to moi. The mean value cumi is the modal fre-quency of Si(co). By letting the value of SAO at the modal frequency be a, the unknown variance a? can be

Probability of

Level Wave group

No wave group For T <Y CC+ 0 For Y ( T

o'

ct+'<2.fo 5.0 m 0.15 0.71 0.14 6.0 0.12 0.76 0.12 7.0 0.08 0.83 0.09 8.0 0.05 0.88 0.07 0.12 008 0.0. Alb. s.om )

r:Ager

_

IN

.4.%7,;// 2...11M 6.7% 8.0 4ttt-S____ 412 18 Level Observed Computed average time duration . Sample: : size Average time duration 4.0 m 5.0 s 1 19.0 sec 13.7 -17.9 sec 17.0

obtained as

2_

21

(moiy

7r ai

Thus, the symmetric shape can be expressed by

a. )2

Si (co) =

(co

,fl,)2 (53)

moi

The second moment about the mean, 112, of the spec-trum can be evaluated as

mo; ImoA2

= (co _{Wmi)2S1(0) do' =} (54)

0

Two conditions are considered in representing the shape of an arbitrarily given spectrum by the sum of two

spectra, S1(w) and S2(0.1), given in equation (53). These

are,

The area under the original spectrum, mo, is equal to the sum of mot and moi, each pertaining to the area under the spectrum Si(co) and S2(w),

respectively.

Since the parameter 1.42 defined in equation (6) plays a significant role in developing the prob-ability density function of time duration To+, the sum of 1121 and A22 evaluated for SI(w) and S2 (c0), respectively, is maintained constant and is equal to /12 of the original spectrum.

The procedure to determine the parameters of the two decomposed spectra is as follows:

Divide the given spectrum S(co) into two parts and evaluate the areas mot and m02 (where mot + m02= mo). Choose the modal frequencies W1 and om2, and assume

al as the value of the spectrum at the modal frequency

CLIm . All these values are appropriately chosen and will

be used as the initial values for the computations.

Evaluate /121 from equation (54) and obtain ic22 by subtracting /22 I from tc2 which is calculated for the given

spectrum.

Evalute a2 from equation (54) for subsequently obtaining /.122.

By applying a non-linear least squares fitting technique, repeat the above computation procedure for various values ofmoi, al, 1 and com2, such that the

difference between the shape of the original spectrum and the sum of two symmetric spectra, expressed by

.S(0) EL-i S(o) 12, is minimal.

APPENDIX II: ELEMENTS OF COVARIANCE MATRIX FOR WAVES WITH SPECTRUM OF SYMMETRIC SHAPE

The covariance matrix of a random vector X associated with wave profile is given in equation (5). If the shape of a given wave spectrum is represented by the sum of two symmetric spectra, then the covariance matrix can be drastically simplified since all elements, no, m and n2 of the covariance matrix become zero. The proof is given in the below:

Since the shape of a given wave spectrum

is represented by the sum of two symmetric spectra, the element of the covariance matrix can be written as the

sum of two elements of the covariance matrices

evaluated for each decomposed symmetric spectrum.

Stochastic characteristics of wave groups in random seas: Michel K. Ochi el al.

(52)

Applied Ocean Research, 1989, Vol. 11, No. 1 49 For example, 770 = noi + n02, where no; (i= 1 and 2) is given by

1101=

o

Si(w)sin(co do)

where

Si(cu)= spectrum with symmetric shape given in

Equation (7)

= me-Ti al)2kCa Wm/ )2 MOi

= modal frequency of Si(co)

ai= value of S,((.0) at the modal frequency mw = area under the spectrum Sa(w)

By letting 71"

H2

ai = b, and co Wm; = a moi we have,

Si(a)=

b'°2 and thereby 7701 = Si(a) sin ra do a;

e0 sin ra do

(58)

-Since it is assumed that the symmetric spectrum S1(a) is narrow-banded and its energy is concentrated in the

neighborhood of its mean frequency, we may write 1701 as

noi= S1(a) sin to do

= a,

e° sin ra do

- b 2 (59) The integrand is the product of a symmetric and an

unsymmetric functions, hence no; is equal to zero. Thus,

it can be proved that

nol + = 0.

In a similar fashion, n2; defined as follows can be proved to be zero. That is,

n21= Si(w)- (w (.4,02 sin(co con,i)r do,

a, e- "2 a2 sin TO do = 0 (60) Thus, 772 = 1721+ 7722 becomes zero.

For nuwe may write I. oa

711; = Si(4)) ' (CO Wird) COS(CO Woli)T dco

0 b a,

a e-

42 cos to do =

ai{[1

_b02 TO sin 717 do}

2b; ,

-6122 (61)

: Stochastic characteristicsof wavegroupsin 'random'seas:-Michel K. Oehiet.

The first term of equation (61) becomes Zero and the second term is also equal to zero as'shown.in equation. (19). Thus, and thereby ni becomes

; '

.1 REFERENCES

-i - Rye, H., 'Wave Group Formation Among Storm Waves ,' Proc.: 14th int. CoaletairCe On Coastal Eng.-, 1914, 1, 164-183, 2 'Rice, S. 0., 'The Mathematical Analysis-of Random iioise,! Bell

System Tech. Journal, 1945, 24,,46i56

'3 Rice, S. 'of ihe;buration of- RandOm-NOiie,'

Bell 'SystemTech.Journi; 1958', 31; 581-635a

Longuet-Higgins; '.14:-S:;.-The:StatisticalAnalysis of A- Random Moving Surface ;!-- Ado's Trins Roy Soc .,:London; -Set: A,

1957, 249, 3217381 ,

Longuet Higgins M S The DistribuliOn,of Intervalitier:Yeen

.,- A Stationary Random Function,' Phila. Trans. Roy,-,..Stic.

London, Sir. A, 1962, 254, -551599" -

-6 Nolte, K. G. and Hsty,- F. 1-1.,''Statistics of Ocean Wave Groups,' Proc. 4th Offshore Tech. Conf.', 2;' paper !Id16.88,:637-64

- . , ,

Applied Ocean ResearCh,..: LOA' Vol.'11, No.' 1

7 Ewing...1. A:, 'Mean Length of Runs or High Waves,' Jouthril deophy: 19737 78, 1933-1936

G_oda, Y., Wave Groups,' Proc. Con!Behavior of Offshore ; StructuteS, 1976, 1; 1 _14

91 Kinnifa; A., 'Statist-kat Properties of Itandoin Wave Groups', _Proc.' 17th Int. Conf. on Coastal Eng., 1980, 3, 2955-2973

10 Rye, H., 'Wave parameter Studies and Wave Groups,' Prot. Mt. Sea airizatologY , 1919, 89-123

11 .Arhan; M. and Ezraty, R., 'Statistical Relations Between Suc-cessive Wave Heights,' Ocenologica Ada, 1978; 1, 2, 151=,15-11

12 Longuet4iiggitis, M. S., "Statistical Properties of Wave Groups in a Random Sea State,' Philos. Trans. Ray.Soc.London, Ser. . Ai 1984, 312. 219-250

13;'. Ochi, M. K. and Sahinogloti, I. 1., 'Stochastic Characteristics Of

Wave droiipsin Random Seal: Pail 1, Time Duration of and -.Number of Waves in A Wave Group,' Univ. of Florida ;Report

UFL/COEL.T12/058, 1986

14:L. trainer, a., .Matherhatjeal Moho& of Statistics ,' Princeton

Press, 1946 .

.l5. V. 1., 'The Distribution of the Duration of Excursions

,of.- Not-Mal fiuctiraliont," Non-Linear transformations of

Stochastic Proces.ses_pergamon Press,, 1956, 354,-367

`

Stochastic characteristics of wave groups

in random seas

Part 2: Frequency of Occurrence of Wave Groups

MICHEL K. OCHI AND IORDANIS I. SAHINOGLOU

Coastal and Oceanographic Engineering Department, University of Florida, 336 Weil Hall Gainesville, Florida, 32611, USA

This paper presents a method to statistically predict the frequency of occurrence of wave groups for a given wave spectrum as the second part of a study on stochastic characteristics of wave groups in random seas. The probability function of the time interval applicable for wave groups is newly derived, and then a method to evaluate the frequency of occurrence in a specified time is developed by applying renewal theory. The results of numerical computations carried out using wave data measured in the North Sea show that the predicted frequency of occurrence of wave groups agrees well with those observed.

Key Words: wave group, frequency of wave groups, time interval of wave groups

INTRODUCTION

As the second part of a study to evaluate various

stochastic characteristics of wave groups in random seas, this paper presents the results of a study on predic-tion of frequency of occurrence of wave groups for a given wave spectrum. For this, the probability density function applicable for the time interval between two successive wave groups is analytically derived.

The time interval between wave groups is usually

con-sidered to be that between two successive up-crossings of the envelope of a random process. The derivation of the probability density function of the time interval

be-tween successive up-crossings is difficult, in general, and

only a few studies have been carried out.

Longuet-Higgins derived the distribution of the time interval

assuming the envelope up-crossing phenomenon obeys a

Poisson random process. On the other hand, the

average (mean) number of waves involved in the time in-terval between successive up-crossings was evaluated by

Longuet-Higeins 2 and Ewing. The average frequency

of envelope crossing of a specified level was obtained by

Nolte and Hse.

It should be noted that a wave group is defined as a

sequence of at least two high waves exceeding a specified

level as stated in Patt 1 of the present study.3, This im-plies that the probability density function of the time in-terval between two successive envelope up-crossings does not necessarily represent that associated with wave groups.

In order to elaborate on the above statement, Fig. 1 shows a pictorial sketch of the time interval between successive up-crossings of the envelope of a random

Accepted 1988. Discussion closes September 1989.

process. Suppose the up-crossing of the envelope at

Point B in the figure is not associated with a wave

group, then the time interval AC (instead of AB and BC) should be considered as the time interval between wave groups.

In the present study, the probability density function of the time interval applicable for wave groups is newly developed, and then a formula for evaluating the fre-quency of wave groups in a specified time period is derived. The results of numerical computations carried out using wave data measured in the North Sea show that the predicted frequency of occurrence of wave groups agrees well with the observed number of wave groups.

PROBABILITY DISTRIBUTION OF TIME

INTERVAL BETWEEN SUCCESSIVE CROSSINGS OF SPECIFIED LEVEL BY ENVELOPE

We first consider the level crossing of the envelope of a narrow band random process, and derive the proba-bility distribution of the time interval between suc-cessive positive crossings of the envelope, R(t), at a specified level a which is denoted by 7. as shown in 'Fig. I.

The distribution function of T. can be evaluated by applying the concept presented in Part 1 of the present study. That is, the probability distribution function of

time duration of the envelope exceeding a specified level

a, denoted by rt., was developed in Part 1. In a similar fashion, the probability distribution function of time

duration of the envelope below a specified

level,

denoted by re,., may be developed. Since the time

inter-val between successive positive crossings of the envelope, rft, is the sum of 70 + and T.-, the probability

Stochastic characteristics of wave groups: Michel I. Ochi and lordanis i. Sahinoglou

Figure

distribution function:of- re,. carrbe derived by the con-volution integral of two probability density functions

f(r+) and f(r). This apprOach, however, has a

drawback in the accuracy of the probability density function

f(r_.).

To elaborate, the following . explanation is given:. ;

The method for deriving the probability densityfunc-tion:developed in.Part r is valid for a relatively small

lime duration.. Since a fairly high level of envelope crossings is considered for wave groups, the time

dur-ation To+ is small, but the time durdur-ation of the envelope crossing below the level, is essentially large. ,Hence,

there is no assurance as to the accuracy. of the proba-bility density functionArc, ) if it is derived following

the approach developed in Part I.

'One way to overcome the difficulty. involved in the derivation of the probability density function of the time interval between succeSsive positive crossings of the envelope is to assume that the probability density

function is approximately equal to that of the time

inter-val between successive maxima of the envelope above a specified level a, denoted by in, in fig. I. This:assUmp-tion is permissible because the crossings fake place at a

fairly large , distance above.. the zero-line for wave groups; and this results in the time interval between positive crossings, being' nearly equal to that between successive maxima of the envelope,

Following Ride's method developed for evaluating the

average number-of Maxima per unit' time (kice6), we assume waves to' be a narrow band random Process whose profile' x(t) can be written as,

'.y(1)=. 'E 'en cos(cont.4 en) = Xi(1)

n=1-.cos (Tit Sin (et (1)

where,

) = cos1(c.0 Li)t

n =

90 _{Applied Ocean Research, '1989, Vol. 11, No. 2}

Since we are interested in the maxima of the envelope shown in Fig: 1, we consider a random vector X which is composed of a set of six random variables

_4,

_4,

x,. .%1 representing the cosine and sine components

of displacement, velocity, and acceleration.

Each element of X obeys the normal probability

distribution and we can write the probability density

function of X in the following form:

1 1

fix).-where,

X ' = Xc",Xs,

E is the covariance .matrix given by,

.7 I/2X'E."'X where 1.4= (Ce7 4))r. S(0)) do) 'S(o-) = waveipectruM (3) (5)

As can be seen in equation (4), it is difficult to invert

the covariance matrix E as needed in equaiton (3).

xs (1 )

E

I (-Of (2)

A=1

cl; = mean frequency of wave spectrum

= S(co) d(w)I1 .3( )

0 0

However, by interchanging the positions of 4 and .4, the matrix given in equation (4) can be written as

.

and thereby the inversion of the matrix is feasible.

The joint probability density function of (xc, x,4, x5, .4, Al) can now be written as follows:

, 1 1 e-1/2 s;, , _{) =} (27)3 I A I 03 < all x, x', x" < co where, I Ai = P-0(iL2P-1 AZ

K =1A-1_I (MI 1(4+ xi) + M22 (x2 + xr2)

+ .1'133 (.4.'2

x2)

2 M12 (XcX; XCX5) 2M13(-Y4 X5X;) + 2 M23 (.4 X;

xxfl

and M11 =-1L2A-3 M22 = AWLS 11433 = P43P2 M12 = P2P3 M13 = M23 = 120143

The detailed derivation of equation (7)

is given in Appendix 2 of ref. 7.

Next, by applying the following polar coordinates,

the joint probability density function given in

equation (7) will be transformed to that of R. R', R", 0, 0', 0". That is, by letting

.vc = R cos

= R sin 0

xcf = R' cos 0 RO' sin 0

= R' sin 0 + RO' cos 0 (10)

x; = R" cos 0 2R'0' sin 0 R0'2 cos 0

RO" sin 0

= R" sin 0+2R'0' cos 0

R0'2 sin 0

+ RO" cos 0;

the joint probability density function can be written by,

R3 1

IIe-v,,.A,

f(R,R',R",0,01,0")=

3 "

(271-) A

0 R < co,

< R' < co,

< R"

0

ø<O' <co, co<r<co,

Stochastic characteristics of Wave groups: Michel!. Ochi and lordanis Sahinoglou

(6) and L = M11R2 + M22(R '2 + R20'2) + M33(R"2 + 4R '20r2 + R20'4 R20n2

2RR"0'2 +4RR'0'0")+ 2M12R20'

+2M13(RR" R20'2) + 2M23(RR"O' 2R'20'

R20'3 RR'0")

By integrating equation (11) with respect to 0 and 0", we have,

R2 _{e- Pi(2 I A}I)

f(R, R',R",0')=

(27)312(IA IM33) I/2

0 R < co ,

co < R' <

co < R"

0, 0' < c0 where, P= MI1R2 + M22(R '2 + R20'2) + M33 (R "2 + 4R '20 '2 + R20'4 2RR"O'2) + 2/14'12R20' + 2M13(RR" R20'2) + 2M23(RR"O' 2R '20 R20'3) +(R'21M33)(2M338' + M23 )2. (14) Next, the probability density function of positive maxima of the envelope and phase velocity will be

derived from equation (13). Let 2.7, be a random variable

representing the local positive maxima of the envelope. Then, the probability that F. will exceed a level E with a phase velocity 0' is equivalent to the expected value of the ratio of the number of positive maxima above the level

per unit time with velocity 0', denoted by

N(,0'), to the total number of positive maxima per

unit time, denoted by N(E.4). That is,

'

PrtZ> E,O'l = 1 F(,0')=

E[N '0

(15) N(E+ )

where E[ ] stands for the expected value and F(E,0') is the joint cumulative distribution function of E and 0'.

If we assume that N(E,0')IN(E4.) and N(E.) are

statistically independent, then equation (15) can be written as

PO: > t,01 = 1

F(Z,6")E[NM8')]

(16)

E[N(E+11

Taking R' =0 at positive maxima into consideration, the numerator and denominator of equation (16) can be evaluated as follows: E[N(E,0')] =RE.6. = so. .f E[N(E+)] = RE+ =

-x R"f(R, 0, R" ,r) dR" dR dr .

(17) x R "f( R , 0, R", 0') dR" dR d0'. (18)

From equaitons (16), (17), and (18), the joint pro-bability density function of the envelope maxima and

Applied Ocean Research, 1989, Vol. 11, No. 2 91

"

0

A2

0 0

0 \

0 1L2 A3 0 0 0 1 A2 P3 P4 0 0 0 0 0 0 /10 0 A2 0 0 0 0 112 P3 , 0 0 0 A2 A3 A4/

/

E

StOchastic characteristics okwave:groups: Michel 1. .0chi and Iordanis I Sahinoglou,

Phase velocity can be obtained as

M+:

,

'

-Where

< co,

<0<a.

-:.The, numerator Of,ecitiatiOn (19) can be evaluated analytically, however tbe:,ititegralion Witti.respectlo R

.4.641 iriyolvedr-in. the denominator 'carr, be evafuated,

' in

Appendix.:2The resulting : joifit: probability density

= fUrictiOn'Of t and V'''hee'orges.

i

0')

-1 + ireqv? 4, (Ey) H where,

it=

, ( 2.1116)0' 2 Z. I vY: ;

L.2Mi30'3.Mj0-Yt1'

From the joint probability density (unction of E and 'W-e-ean derive the conditional probability density

Rljti,'0;jrAi't0') di?"'- function of

- - 'giVen.E. That is.

-, (19) -.4"f(R,s0,:R1,64.1) dir3.7dR dO' 1.110! I E) .c -e=-R3u " e(Ry)! (Ri))11 dR dB' 0 < O' <:co (20) 21-A1)433W. 24

02

Figure 2.

92 Applied Ocean Research, 1989, Vol. 11, No. 2

; ;,...-rr.; M230 M330'" )

r

(21)-- I Acp(x)= . e

-VAE;

VP.: -(22)

Nextah_e, conditioPal- tirobability density function ivnin equation (22) will be converted to the condi-tional probability density function of the time interval --betWeen successive :peaks above a specified level a by

lettipg;t-,rt

-7

0' =

(23)

,

Fierei;the Jiii-arrieter deferrnined by the iteration method front, a. -to-Aside-ration of the probability of en elope crossing at the level ck. The derivation of

equa-_tiati,(23) is -2N'en in Appendix II along with a method to

determine the parameter ,/c._ThuS, froth equations (20) through (23), the ,conditional probability density

func-ticih- of r,. given =.cy, can be written as follows: ,

2irk- , x (24).,-; :=

+ 4000'

where,_ . u' = 1

{Pfil

+2M12.(27k) (M22 2M13) - . -:---, ,e7"": (.1.717 (XV' ,77-17 OFIF12 I1

0)} 1

'

-

. _ _ .. .. _ . --. 7

.

.3.

1 + atk.Tr e(a"( 1, -F-I(oiv)] ] -dr:

, " - . - .. :-., 0.030 0.020

< 0.010

.iti23(2-irky

)43,(2iky-t

.-k*.,),-,

k ia I

k

I i

, f C ,. . - - ... ,. ;T., 1 (.:; IA i M33 ) "2 -.7 ' . ' - - -1 -0P.3

)

.

±(

.A0A2 ---7.' - - . - , _ 1

0

271:;!C .

.271

. . ....,, -T. . .. . . _:;.. ' - A;:.: '' '' A, . 17t.k 27. k 2 . ' -.. '. . '.-- (25), . A, X{In 13 + in 23 :,.., 33 .4-7- r Tor 3+ A4igatiA. "-TT 1.1 Al-, .1-21L Tfr + (pow 3;c1)(3-7k) '2

-Equation (24) can also, be.eicpressed. ,terms of the

central moments defined in eqUation (5')'..i.ith the 'aid of

- , .

eqtiation (9). In this case, the parameters

u', v and

v' become

, .;

C= dohstitit fort normalization of., the ;probability density function

I

(r 1a) dr

x. (2.7tk)

(-727k)2}-' ' .

Numerical computations are carried out for the two

parameter wave speCtrurwiwith -significant Wave height of 10..0 rh and 'n .modal frequed-Cy:Of-.042 rps 'Showti

, Fig!: 2.- The firobability.--deniity,,function of the time

interval between-. tub SOcCessive.-envelope up-crossings for level Of 5.0, 6.0,-and,7.0 rn-above the still water level

. arecomputed by equation (24) and the ,results are shown

- -

.-111-171S:-.3-,-;--LEVEL,

_ 5.0m

.

30a-;-.1111;1E,1NTEE.NAL Ta_IN SEd.

Figure 3.

Applied Ocean Research, 1989,. Vol. II, No. 2 93 _ - )1 + 0 t ... e .r2. ..v

"

1/2 Ilz :. 1,p3

;

21:A . .. . . 1/2 2l'A 1

Sioehastiechargeterigics.9 wave groups: Michel -L Ochi and lordanis L Sahinoilazi . '

We mayrecall the assumption that-the time intervAl ,-.

si*cifie4 )00a

;-eclual.I6.ihe.: time interval' between successive ,ufi=dicis8-ings''ofr_the' leer by the envelope.

betweenuCceSsives,peaks. of the envelope above- a

Hence the conditional- probability density.: function

'l

given in equation (24) re-presenti;-askwell; the condi-conal probability density fiOtion-..bf,the time interval

bet-tkeeri $ucdes-Weup`-diiiisitts a'specified-le41 a by

the-envelope.

Stochastic characteristics

PROBABILITY:DISTRIRLIflON OFTIME INTERVAL BETWEEN WAVE GROUPS

At disCussed-in7the IntroduCtion referring- tec Fig. 1, an

uperossing of the envelope does not necessarily mean r., the occurrence of a wave group, The _probability of

--occurrence of wave groups when the envelope exceeds a

specified level a was Obtained in equation (40) of Part l': Let thisprobability- be denoted by p then, it can be said thatp-fraction-Of f(7c, I-a) derived in equation (24) represents the probability density ;function .bf the time

interval .between two,,wave groups.

However, the (1,7 -1:7)rftaqiOrl Of Bra I

a) can still

become the probability density Of the time interval between two wave groups by'suniniing two, three or more of the succesSive time intervals between envelope crossings, T.. Hence, the' (1 p)-fraction Of Ara] a)

should be modified so that it is .associated With the time interval between two wive grOU0S.- The 'modification of

the probability density function may be made as

follows:

First, let us consider three up-crossing i

of the

envelope.

If the

first

and third .uP-crossines are

. associated with.-waVe groups but the time duration of - - the second up-crossing is so short that it does-. not

con-stitute a- wave group then the time duration between two wave groups is that between the first and third

up-- rcrossines.,ThisiS the situation of the up-crossings A, B,

and.0 showh- in Fig., 1.. In this case the time interval

be-tWeen Points AándC is considered ;as thesurn of two :independent erandom _variables, each having- the pro-bability density function derived in equation (24) BY

riting the =,sbm' of two indePenderitAirne- intervals as .ria, its probability density.funetioh- for a specified level

0.030

Figure 4.

94 Applied Ocean Research, 1989, Vol. II, No. 2

,

of wave groups: Michel I. Ochi and lordcinis Sahilloglou

a,canbe,obtaitied -b-j, the following convolution integral:

-Isir a)-* f(0:2F. Tc,)1a) dri, (27)

where,

I a) 7-.." probability density .furiction of time inteival

betWeen successive up-crossings of envelope

abOve a Specified level cc (see equation 24).

The Same concept can beipplied to four up-crossings

of the _envelope in- which the first and fourth

up-crossings are associated with Wave groups. In this case the time interval_ between successive wave groups, denoted by r3.,.ean be considered as the sum of three independent random, variables each having the bability density function given in equation (24). Its pro-bability density function can be Obtained from equa,

lions '(24)-and (27)- as .

I ei)-- Aria, = 72,JI ay (472a (28)

The, same procedure canbe eRtended to any number

Of uperOssitigs of the envelope in which the first and the

last -up-crossings are associated with wave groups: An -:ekarnple of-the-probability density function f(ri,, a) -.Jot n front 2 to 5; Computed for the two-parameter wave

.- spectrum, is shown in Fie. 4. The crossing level a is 5 m in:thrs example: Included also in the figure, for cornR

parison is the probability density function f(7,,

.6)1

shown.jin Pig 3-

-From the results of analysis of wave records obtained at-,sea; it-Was found that, in 'practice, an upper limit Of

75n. is Sufficient for consideration of the longest tirfie in

500

100 200 300 ₄₀₉

Stochastic characteristics of terVal betWeen'up-crossings for wave groups, and that.

cbances" -at

occurrence for highilevel crosiings. Here, implies the .

time interval between the .first and the last up-crossings of a total 4.0.3ccros'sings of the envelope. Therefore; in the present analysis, the probability density functions

I a') for-ii. from2 to 3 alieValiiateilthibugh -solution integrals and then these probability density functions are accumulated with equal 'Weight p)/4 'along with the 'probability density (unction f(7-91 a) whiCh has a\Yeight p Thus, in summary, the proba-bility density-functibn of the time interval between

.suc-

cessive wave groups for a specified leYel- a, denoted by =

..f(rc I a); Can be written as, '

I

p

f:Cre,G I a) = P " fir. I a) + -4 X 1./(7.2. I a) +/(73. +f(T4.I a) + /0'5.1 (29) . Where,

p = probability of occurrence of waye_arbiiPt

'hentheenvelope exCeeds a.rtpecified Ci (ke

equa-tion-, (40) of Part 1).

1(7-. cr) =

f((rn.

Tot- I lu ) I a) 0

f(7(n-ii.

n= 2, 3, 4, and 5

As an example, Fig. 5 shows the 'probability density

function .1.(r00 a) computed for the to

Avaye spectrum for the levet:cc,

The average time interval;:beiWeen.sliccessive 'Wave groups, denoted by r can be "eValuated from equation

(29)-; as

V' "'

' ' ict'G AroG I '01) dT6G :IFigure 5.

::.

. 0.002

0

cr 0.001 :.VV 200 _

wave groups: Michel I. Ochi and lordanis L Sahindglou EVALUATION' OF OCCURRENCE OF WAVE. :diti)VP:$7

-It of considerable interest to evaluate for a given waVe

-spectrum how many,times,a wave group will occurin a

specified time period, say 30thinuts Or one fiohr; as long

as the sea severity is ;unchanged..-Foi-,:thit, renewal theoll, it apPlied to the probability density function of -the time interval between wave groups;

It has Often been said : that the ,,wave group 'phenothenoh,obeyslthe Poisson random process This :assumption may betitle if we consider the wave group phenoMenonto-be a rare event.' In thisFate, the time

in-terval_betWeeh successive occurrence of the event (wave

groups 1 fOr the present problem)' -Mtist follow the `exponential probability lavir:! However, as shown

- Fig. 5,, the probability density-function,of the time

inter-val ibetWeen wave groups, f(df a), cannot be

con-sidered:to be an exponential distribution, in general._ Hence,- in the Present 'analysis, ' the 'probability density .-,function f(ral:ei) is approximated by the following

gamma - probabilit)%:distribution whic_h is a generalized

forth of the. exponential

"distribittion:'-1

r(ii)

e-krnG 0

ra

(32)

_ , Note, that,f(raq).:Jeduces to the tipOnential.

:=7 Iv.:.-The..paratheters in and N. of the

gamma .distribution given in equation (32) can be deter-.

mine-04'6Y equating. the mean and variance computed from the probability density-function given in equation

(29) to the-tridan'irid...varialice.Of the gamma distribution.

An example of ,acbttiparison between the probability

cie.pot)., functions computed frbri); equation _{(29) and the}

-44inina'dit'tiibution'foithe iwb=paraineter Wave-

spec-. . '.frutrtite shbWn'

Based n the probabi ity den-sit;fu.nction given in equation (32), the probability of ii-Pdcurrences' of_, wave

.

0.0006.

-.

60&,crn: Research, _1989, Vol..!!,. No _2 95 Are,G)'-,=

1000

600., 800::

Stochastic_ _ of wave groups.- MicheiZ c1 anis 1.2:Sah1;ioglent

Q '0.003

0.004

Figure-6:

_

grciiipt in -a specified time ,p'eriOd can 6e 'derived as

' follows:- ' .r

Let T bethiyiiting".time tip to the i-th'WaVe grouP. Then, we may write

"-_= (Tao)! + (T ao)2 + (Tao)3+ + (7"&,0n.:: (33)

'where all:rardikeqUation (33) are statistically independ-ent 'and:follov'the.Piobabiliiy density function given in -equation:(32).-,Since the characteristic the

. .

gamma pro-liability distribution :gigeh in equation (32) can. be:written as

_

n

(1' (34)

X

the characteristic 'function of Ta becomes_ . _ . _

-inn

T,;(U) = 1 (35)

. .

This is also the characteristic -:function- of the gamma .probability distribution with the parameters X- and_ inn..

"-Thus, the probability, function Of_ can be obtained as

- 1

--2

Atn)

= X mr.Tgin- e-xT". (36)

r (mn)

.;'SirailatlY; the probability density function foT

:the--Waiting time up, to the (n.-1- 1)th.wave.irmip, denoted by T,, .1,-, can be writted,as

f(Tn+i)_

Xm(nti)T:(n+1)-1

r(hz(n -44_1)),

-0.002

0;001

Then, we have the following relationship:

.,: ..,i. n,occurrences of wave : --,

P

{ .

= PrIT, 4 II'

--, groups in thrie t . ' , PriT..1

BTrdTn-- 0

6 Applied Ocean Research:4989,'Ir/Ol. 11, No. 2

200_

_:400

600 -,

1.

TIME INTERVAL IN SEC

XT. -(37)

800

- .

-:^;,(171ii ; )(i11(n A-, .1),: NE)

_ .

r 0170

r(,n(n*.-1))i-- 1000

-n-= 2, 3, (3-8)

where,'

_ -y( )= incomplete einima function: ihpprarntipCii9la,tcrurfro_e r n = 0, we have.f,.:

[

Wave eroup in-:time i. group. . '-',.

-I (m.Xt)

17.(n)-

--As an . example of application Of the prediction

method derived in above, numerical computations are Carried out using wave data measured in the North Sea

off NOrwaY. The recording time Was 17 minutes, and the

significant wave height was 8.13 m. The spectra density function. obtained from data is shown as Wave Spec-trum A in Fig. 7.(a) in Part 1 Of the present study:

The probability density function of the time interval between successive wave group; fitad), computed for a

-- level Of ii = 4.0 m is shown in Fig. 7 along with the

gamma distribution computed by equation (32).

, The probabilities of wave groups in 17 minutes for a

level of a =" 4.0'M are computed for various number of

occurrences by equatibn. (38). and the

results are tabulated in Table 1. As can be seen in the table, the

' probability of Occurrence Of 4, 5, and 6 wave groups is

high; specifically the highest possibility is 0.204 for 5 oc-currences . This predicted result agrees very well with the Measurements in Nv hich five wave groups were observed

in ,17 minutes at the level of 4 meters.

' Currently available methods for evaluating the time interval -between successive wave groups consider the

..aveiaie time interval between two successive envelope

crossings ofa specified level. This results in the average number of occurrences of wave groups being extremely (39)