Stochastic modelling of quadratic wave components

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STOCHASTIC MODELLING OF

QUADRATIC WAVE COMPONENTS

by

M. DOGLIANI

Technical Bulletin N. 103 Genova, April 1989

This work has been completed with a grant by Italian National Research Council (CNR) under the

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Direttore responsabile: ALFREDO DELAVIGNE Stampato da: CENTRO STAMPA R.I.NA.

Genova

Editore: REGISTRO FFALIANO NAVALE

Via Corsica 12 - 16128 GENOVA

Tel. (010) 53851

Autorizzazione dei Tribunale di Genova: N° 27/73 dell'li aprile 1973

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tJ?i O / ftBST1Cr

fri

questo bof Iettino si oresenta un model lo idrodinarnco de/ /ednde

di suoecficie in cui i'elvazicrie ondosa e' trattata come un nrocsso

stoeastco debolmente non ),aussiano. Lo studio, effettuato nel dom.j'r'iio dei le frecuenze. conta al fa: formulazione del Io sDettro quadratico e del

b i st tro del I e onde,

r

-Attraverso un'analisi del/e relazione di dispersione si mostra

i no i tre come, a secondo ord i ne le soorae / evaz i one d 'onda vada model/ata come un carnoo stocastico e come la sua funzione densita'

soettrale di ootenza sia definita nel dominio frequenza - numero d'onda

(ùj, ir),

Le formulazioni fornite oer soettro quacratico, bisoettro e sttro

(i,k) acciono della caratteristica di Doter essere ottenute. piuttosto

se'n.ol icemente, noti i I tradizionale soettro. del mare ed i I fonda/e.

A quadratic hydr'odynamic model of surface waves is presented which

treats the wave elevation as a weakly non aaussian random process. The

procedure is f u I y developed i ri the frequency

dma n

and the

formuiaticn o-1- the spec'tr'urnas wel as the bispectrum of quadratic wave elevation are theoretical lderive,

A detaiied analysis -of the dispersion relation o-F quadratic

components t, locked waves) is made show inq that the wave elevation; at second order, should be treated as a random field, Its spectral denity

-1-unct ion is _{ca i ned in 'the frequency - wave number (u, k) dma in,}

The formulations Qiven are such that the quadratic wave spectrum.

the wave bispectrum and th(w,k) spectrum can be obtained in a simple

way from the knowledge of the usual linear spectral density function

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C'TENTS

I NTROE)UCT ON pag

- TERM I NOLOGY pao. 5

- SECTION 1 - LITERARY REVIEW pag. 11

Chapter 1,1 - EVCLUT I ON CF WAVES pag 11

1,1 1 - Evolutionary properties of unidirectional

wave train

_P9.

11

1,1.2 - Evolution properties of the wave train in

two dimensions pag. 1.5

Chapter 1 2 - STAT I ST CAL CHARACTER ZAT ON CF RANDG"1 WAVES pag 17

12,1 - Wave elevation pag. 17

122 - Wave envelope pag. 22

1, 2. 3 - B i soectrum pag. 23

Chapter 1 3 - CONCLUDI NG REMARKS paq. 24

- SECTION 2 - SECOND CDEP DETERMINISTIC MODELLING pac. 27

Chapter 2 1 - GOVERN I NG EJAT I ONS _pac. 27

2,1,1 - Equations of motion paq. 27

2 1 2 - L i near approach paa. 32

2.1., 3 - Higher a-dm expansion paa. 33

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Ohapter 2.2 - D SPERS ON RELAT ON pa 37

2 2. 1 - Genera pag 37

2.2.2 - Thea'etical resu Its _paa. ₃₈

22.3 - Experimental results paa. 46

2.2,4 - Canments pea. 4.9

Chaoter 2 3 - OSBORNE MODEL paq 51

2.3.1 - Theaetical backqrjnd paq 51

2.3 1 - En i neer i ng aspects of the mode

...,,,,,,

_p-an ₅₃

2.3,2 - Possible appi ications paa. 62

2.3.3 - Summary o-F characteristics _paa, ₆₄

Chapter 2,4 - CONCLUDING REMARKS paa. 65

- SECTION 3 - SECOND ORDER STOCHASTIC MODELLING

...paa.

69

Chapter 3. 1 _{- CHARAC raR I ZAT i CN 0F OSBORNE S MODEL AS A}

RANDC11 PROCESS _paa. ₆₉

3,1.1 - Genera.I _pap. ₆

3,1 2 - Intermediate water depth

,..,...,,.,.,,,,

_oa.

₇₀

3.1.3 - Shallow water limit _pag. ₇₂

3,1,4 - Canparisa- with Tick's model ïaa. 72

Chapter 3.2 - FREQUENCY SPECTRA CF WAVE ELEVAT i CN _paa. ₇₃

3,2,1 - General _pag. ₇₃

3.2.2 - Mean values and cörreiation functirns _paa. ₇₅

3,2.3 - Covariance functias and spectral density

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Chate 3 3 -

PECTRA pac. 84

j j I - paa, '34

33,2 - Third arder corre!atian functians rjaa, 88

3,3,3 - hrd arder dovanian functions 94

- i?,

nctnai ddnity

oaa. 99

:3 4 RANUC! FRCX2ESSES AND RANL1 F ELLIS aaa', I 00

3,4.1 --

cH

on

nndan fields ,,,.,...,

oac. 100

:j,4,z - Wave elevation n

he (w) pane

rac. 103

:3.4,3 - Sj:;tr

in t;hc (i,i<) p'ane pac. 10E

::ner :

- cc(:buL G REMARKS rac. I 08

hC C0 4 .A'PL.C.A CS pac. 1:3

c:h '.í NR(tuL:[;CN ac, 113

Chat-' 4,2 - T ME SER!ES Cf THE LiNEAR ANL QJALRAi !L

WAVE. ELEVATiCX'J pac,

l4

'4..?.

-

Near'kr per jic wave trains pea. i

4,2..? Inreouiar ave trains a, lib

Lhapt.er 4 :j - SPEC. RA Cf SE4TC'JD CRLER coRE:CT C D1C. i2A

4,3, 1

-

Lip

dé

fr'

sjtn'jm .oa:1, 124

- Iwo Lirac delta in the mear sr..x,ct:rum pac. 129 ChaDter

4,4 -

BISPECTRA Cf WAVE ELEVAI ICN

....,...,..

ac :32

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Chapter 4,5 - SPECTRA IN THE (wyk) PLM'iE pag. 139

4 5.1 - Che Dirac delta in the I mear spectrum pag 139

4 5 2 - P i erscn Moskowi tz i near spectrum pag 145

Chapter 4.6 - CONCLUD I NG REMARKS pag 148

- SECT I ON 5 - CONCLUS IONS peg 151

B BL OGRAPHY pag 1 55

- APPEND i X A - MATHEMAT CAL BACKGROJND

pag, 161

A, i - Four i er transform peg 162

A. 2 -

m.i

I se f unct on peg. 163

A. 3 -

Autoca-re I at i on f unct ions for a stat i a,ary

gauss Ian randaii 'ocess pag. 164

A,4 -

Gram Char! 1er series expansion peg. 165

A.5 - Danain of definition

...,...

pag 16?

- APPENDIX B - TICK'S GIJADRATIC

WAVE ML

pan. 169

- APPEND I X C - YUEN 'S APPROACH TO THE ZACHAROV'S EQJATI ON . pa< 177

- APPENDIX D - DISPERSION RELATION IN THE (w,k) PLANE .., pag, 181

- APPENDIX E - DERIVATION OF EQ (2,8) AND (2.9) .,,... paq 186

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I NTRcDJCT I (N

As is well - knn, sea waves and their effects

ai

the structure

d be treated i n a stochast i c way t i s canmai p-act i ce fa'

eng i neers to cais i der then as ha'nogenews, ergod ic and gauss i an

processes. This is a. very limiting way to look at the real ity of the

phenc4iena, wh i ch can genera I I y be subd ivi ded into gauss i an and nai

-gauss i an

The sea in real ity is a nor - gaussian p-ocessj a well

-example is the wave el evat i ai i n sha I I c water which shows sharp crests

and wide trghs.

However, a I arge number of mar i ne structures is ab I e to f i I ter any

nor - gaussian energetic caitribution of the input; in this case both

the sea and the structure respaise can be mode I I ed as gauss i an Sh i ps are a typical example of this kind, aIthogh they show also p-ctlems of

nor - gaussianity, such as the roll mot lai and the hogging and sagging

bend Ing manent

When the respaise is nor - gaussian, it is in general influenced by

both the non - I i near i ty of the system and the nor - gauss i an i ty of the

sea. These prcb I ems are frequent in offshore structures, such as the

resporse o-f caiipl iant structures (single point mooring, tens ion l

pl atform, and so on).

For the c ¡ ass of nor - I i near systems subjected to a nor gauss i an

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- the respmse cannot be ca-s i dered a p-' i

a-

i gauss i an

- i t i s necessary to take into account the effects due to both the

nm -

I inearity of the system and the na- - gaussianity of the inp.it

however, these effects can be treated separately

- and i t i s necessary to f i nd a caiìp-'aî i se between the rea i ty and the

models, which should be as simple as possible.

The effects due to the na- - inearity of the system can be dealt

with by fol low ing the Volterra app-' ach canputatia-al ly, that has been

shown to

be

feas i b I e i f I i near and secmd a-den ca-tr i but i a-s a- y are'

taken into account /60/,

C the othe nd, the prcb I ems of non - gauss i an i ty of the inp.it

(the sea mot ion, which leads back to the study of the non - i mear sea

surface elevation) are the main concern of this wa-k,

The sea surface elevat ion can be described by the. usual dynamic

differential equations, which, also within

the

incornF'essible fluid

hypothesis, are stra-g 1 y na- I i near (Eu ¡ en' s equat i a-) and. can be

solved only with a number of simpl if icatims,

The app-'oadhes wh ich are fol I owed to sol ve Eu I er' s equat i

a-

can be split into two main groups: the so - called evolutionary mes, which are used to study the shape mod i f i cat ia-s (evo ut ions) of non stat f mary wave trains during their p-'opagatim in time and space.: and

the stochastic ones, fa- the statistical characterization of a sha-t

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The two approaches are different in their theoretical bases, the

methodologies and eve-i in the results cbtained, In particular, the evoluticnary approaches are ma-e refined in the definitirn o-F the wave

nm -- linearities but are basically deterministic and aimed at the

study of na- - stat imary effects, These characteristics drastical y

reduce their applicability to engineering prlems however, they have

bee-i used i n tn i s work i n orde- to show the importance o-f the non -i near phenomena wh -i ch shou I d be taken -into account i n the stochast i c

representat im o-f the sea,

(Yi. the other , the stoehast i c approaches are ca-s i derab ly

:irnp!r fran

th

onqineering point of view but seen less able to take into account the physical aspects shown by the evolutimary approaches,

A model recently developed by Osborne /3/ ca'nb ines some features of

both approaches and, i n part i cu I an, i t seems to be ab I e to app I y some

results typical of the evolutimary models, As an example, it al ows the numerical simulatims of the quadratic wave elevation to be made

fairly simply, However, it has mly been developed in the time domain and it is substantial y deterministic,

The scope o-F this work is to extend this quadratic model to the

frequency domain where the wave elevaticn is treated as a stochastic

process, Part i cu I an attent im is focused on the def i n i t im of spectra

and bispectra and

a-

the influence o-F the dispersion relation,

Sect im i is a dota i led literary review of the two ma in approaches,

I n Sect im 2, the evo I ut i mary Zacharov approach is presented i n dotai with particular attention to the d ispersim relat im which

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governs the secaid mder i ocked waves. The character- i sti Cs and poss ib e

applicatfais of Osbcrn&s model are also reviewed in this sectiai. Sect im 3 i s devoted to the extent lai of Osbaine' s app'oadi and to the

themet i ca fa-mu I at i

ai

of spectra and b i spectra. Maeover, the

i nf uence of the d i spers. i ai rei at im i s h i gh i i ghted by the f req.ency wave numbe' (k) spectrum, tained by model I ing the wave elevatim as

a randan f i eid Severa i si mp i e exarnp I es are developed i n Sect i ai 4

A mathematical backgrcund is p-'ovided in Appendix A Appendix B

presents T i cl<' s approach, wh i ch has been the f i rst me

ai

this subject.

Sane mathernat i cs rei ev ant to Zacharov' s method are deve I oped i n

Appendix C Appendix D deals with the dispersiai relatiai in the (u,k)

pl ane Sane f crmu I as are der i ved i n ApØe-d ix E and;fTna I I y, Append ix F i s the I st of symbol s

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rEJ

I NY

Throghaut the paper extens i ve use w i I I be made of the term i noi ogy na-ma I y adopted i n techn i ca i i i teratur'e when dea I i na w i

th

stochastic hydrodynamics. Fa- an explanation of these terms the reader

shad Id refers to the basic wa-ks on this subject (/1 / and /2/), Less

common terms used are def i ned bel ow.

wave tra in: a t i me (a- space) ser i es of waves wh i ch, i f not d i f-f erent i y spec i f i ed, are intended to be regu I ar

irreau lar wave train: a wave train compod by irregular waves

homogeneous wave tra i n a wave tra in, either rg..i I ar a- i rrequ I ar,

whose power spectra I dens i ty funct i on does not change neither i n t i me

na- in.space

stationary wave train: a wave train which is homogeneous in time only

periodic wave train: an infinitely long, stationary wave train, either

repu i ar

a-

irregl aï-, Which repeats after a -f ixed (time

a-

space) period

wave packet: an irregular, not stationary wave train ccmposed of a

f in i te. nunber of waves travel i ing ai an otherwise calm surface

enve I ope: gi ven a stat i onary random process X (t) w i th zero mean, its

envelope A(t) may be though as a relatively smoothly varying funçt ion

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a-near the peaks of IX(t)I. In

the case of

a wave train, either

stat imary or not, i t may be thwgh as the I i ne wh i ch connects a I

wave peaks

modulates wave train: a wave train whose envelope exhibits a regular

beháv i ojr

weakly modulated wave train: a modulated wave train whose envelope has

a smal ampI itude

slowly moduiated wave train: a modulated wave train whose enveloce not

necessari y has a small ampi itude but varies very slowly

I mear wave: regu ar sinusoidal wave (Airy wave)

weakly non linear wave: regular Stokes wave

sol itary wave: highly na- linear surface wave which 'opaqates withwt

chang i nq form for a F ong t i me

envelope sol itm: wave packet

whose envelope has the shape of

a

sol itary wave

internai wave: wave which creates at the interface between two layers

of fluid having different density, Usuai ly they fOl low Soi iton

dynamics

evolution: all the mod if icat ions which may occur to a nor stationary wave train while propagating in time or space, Such modifications may

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topograiy,

current and

so on) and fran non -

mear

effects

(such as

wave

break i ng and wave-wave a- wave-current interact i

ais)

=

T2

j_T,

2

Fourier

transform: given a function ri(t)

fa- which the

I imit

j_T

'

dt

(0,1)

exists, its representat

lai in

the

fa-m

C

= _{.1 _W}

(t)e

dt

is

it-s Fourier- trans-form

Genera

i zed Four i er transform: the representat ion of a

i ven f unct i on

r1(t) as:

-iwt

-iut

Nr(U) -

N(0) =J

q(t)

e

-

e

°

dt

-the

ca-responding

inversion integral o-F which is the Stielt les

2nit

1(t) exp(i

T dt

Four i er ser es

the

representat

ion

of a

g i ver

funct

ion r (t) as

a

2niit 2nut 2niit

--- ¿

-n=1 cos n _T

+bsin

T = E n

=-cexp(i

_T

T/2

T2

w i th

an =

2

jri(t)

cos

271t

dt b- =

j

r)(t)

sin dt

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inte ra I

=

dN(w)

i f the Fo.jr er transfa-'m of r () ex i sts, then

(J)

N,,,,() - N(w0)

_=J

j(y)

dy (J)0

-d(w) Note that:

the envelope of a regular wave train is a straight ¡ne (frg, n

the enve ope of a s I ow y modu ated wave tra i n i s a very smooth y varying functim (see fig 2)

the enve i ope of an i rregu I ar wave tra i n i s a funct ¡ cri more regu I ar

than the train itself but it is still quite irreguIar..:

the enve ope of a wave packet i s zero everywhere except for a sha't

t ¡me a-' space ag (see f ig 4 where an envelope sol iton i s shown)

a periodic either' regular' or irreaular wave train has a a-o-File r

wh i ch may be represented i n t i me (a- space) as a Four i

er

ser i es, but is

not Fourier transformable since the limit (0.1) is not bounded in this

case

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representat ion as a Fax' 1er sen ¡ es over an i nf i n i te ly ong i nterva i is

zero

the prof i I e of sea waves is norma i y. expressed by means of an hanogenedus and ergod Ic stochastic f leid If, fa- instance, ae of its

realizations in time (or space) ¡s considered, this is stationary but

i nf ¡ n i te ly I cng. As a consequence it i s neither a period Ic nor a

Four i

er

trans-fa-mab I e f uict ion i n th i s case i t may be expressed in

terms of the genera I i zed Four i er transform by means of a St i el tjes

integral enve lop i oo Û ÖO

-o-5

Fig I

smoothly varying function

ÍÍATÏTI AÏTíÏïì

LYl

Y

IYtI1

Fige 2 regu I ar wave sily modulated wave train

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-04-0.12 0.04

/

frci,t

/

Fiq 3 7. Fig 4 enve ope.

-I

j

/

wave packet irregular. wave train

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cria'i i

LITERARY REVIEW

Qiapt' I I - EVQJJT I (Z G WAVES

1.1.1 - Evo I ut ¡ mary p'opa't i e of a u-i i d i rect 1mal wave tra in

A wave

brain can be characterized by three parameters: its ampI itude a, wave number k (= 2n/X) and frequency w General y, its

evoluton is described by equaticns fon the conservation of mass wave

number, wave action (*) and by the dispersion relation which are

qeneral ly non - linearly coupled and this fact leads to sane remarkable

phenomena occurr- i ng duri ng the evo ut im

In i 965, L iahthi /4/ studied the evo lut ion (in this case sane

non - mear effects only were taken into account disreqardinq the environmental effects) of weakly non - I mear' me dimensional waves in

deep water and examined two sets of ihitial conditions: a wave packet

with a Gaussian envelope and a weakly modulated Stokes wave train, Both

of them developed into a cusp and the time taken to reach this

si tuat im was found to be inverse I y pnoper't i ma I to the wave steepness

( a/X; fon the Stokes like wave tra in this is not true fon the wave

packet even if, in that case, the time was f inite,

(X) _{In the simple case of a regular, stationary wave train travel I}

on a stra i ght bottaii w i thout current, they I ead to the usua I mass

(19)

A deta i I ed pe-turbat i a-i ana i ys i s of the wave tra i n was perfa-'med,

starting frau Eule- equatias, by Benjamin and Feir /5/ in 1967 and it

was fojnd that a unifcrm wave train with ampi itude wave nunbe- k0

and frequency u0 is unstable to perturbations with wave number K in the range O < K < 2

fl

k a0 (K refers to the envelope of the wave train

there-f cre i s a measure of the perturbat ion) they a I sÖ reperted expe- ¡ menta data which f i tted the p'ed i Ct ions we i I An example øf such

a modulation gring on an

initially Lnifcrfn wave train is shn in fia, 1,1 x= f t

il

r ' Fig. 1.1 C TAlN FF?Ct.1 FERNC

f7

)

The resuits of Benjamin and Feir were confined to the mit ial

growth per i od, eav Ing the quest ion of the subsequent evo I ut i on o-f the

unstable non - I mear wave train unresolved

Hasimoto and Co /6/

= i 5+t

=20 ft

=30 ft

(20)

showed

n 1971

that the evo lut ion of a non -

i near crie-d i mens i ona I

wave tra i n obeys the non -

I

near SchrExi i nger (NLS) equation, wh ch was

given this name Fa- its analogy with the equation used

in particle

phys i Cs

In part i cul ar, the resu I ts of Benjam i n and Fe i r were cbta i ned

by a stability analysis of the unifa-m solution of NLS.

NLS is obtained by writing the wave train as:

q(xt) = Re [ a(xt) exp [iO(x1t)] J

where a(x,t)

is slowly varying

If the wave train is narrow banded the

phase may be written as O(x,t) = &(x,t) + (k0x - u0t) where

and k0

are the peak frequency and wave number and e (x, t)

i s s I ow ly vary i ng

The wave train thus may be exp'essed in termof the caîiplex envelope

A(xt) = a(x,t) expL fe(xt) J:

1(xt) = Re f A(x,t) exp Ei(k0x - w_t)] L

An

i mpa-'tant character' i st i c

of. .NLS

is that

i t can be exact ly

solved when a wave packet is dealt with

lt was shown by NLS that any

initial

wave packet

eventual ly evolves

into a

number of envelope

sol i tons and a ta i

I

of d i spers i ve waves and that the bu I k of the energy

is conta i ned

i n the enve.l ope sol i tons WhiCh do not change the i r fa-m

wh

I e propagat ¡ng (thus are not of a d i spers i ve natLre),

Thea-et i ca i

resu I ts were conf i rrried by exper i ments; fa- instance

i n f i g 1

2 an evo I y i ng wave packet ¡s shown together with the enve I ope,

abta i ned by appl y i ng NLS

In the case of wave packets and sol i tons, the

descr i pt ion o-f non -

I

i near deep water wave dynam i cs prov i ded by NLS

was fajnd to be quai itatively and quantitatively correct and this

is

very

i mportant because the theory pred i cts the ex i stence of packets of

deep water waves with sol ¡tai properties (envelope sol itons),

(21)

S ft z

--I

20 ft

Fia, 1.2

(TAJCN FC1s1

EFEENCE f.f

)

Mo-'eover long time evolution o-f an unstable wave train was studied

exper irnental ly and a Ferm 1-Fasta--U am recurrence phenienon detected

was conf i rmed by numen i ca I canputat ion The 'esence o-f th i s k i nd of

recurrence means that, in a weakly non - I mear wave train, in the

absence of viscws dissipation, there are no steady states but a series

of modulation and demodulation cycles (see fig, 1.3)

JI i. .-

i'

L..

I L'

rnrnrnorn

Nrn1Arn

FMI0

i &iittrn ...

rnlir rnrnrnrnrn IIrnII*fiI ,rntw

IMtf

trnk!I M

iio'ii

Laauarnr .

i.

4IIuIø3IrnMMr rna..

rnIirnarMïrn MIrS ...E 'E IwimlIIMaLi.

A LI

IItl

...IUYM rniuthm

MIIHWAIIIl.iIIW rnIII0'rnII L'IiIWVÍ iDiIrn '11rnilW11J E1IUL IE MUIIIIWIRa1rn a L

arnHlriIH i

irnrnnrnrnrnmi i rnai . W I'il III 'i - J-[

FiQ

1,3

(TN FCZ'1 REFEENC /7/

i x=Sft Oft _15f-t - 25ft 3Oft WiI'W .... m

Iíirnttarn

IIILlrnt ç

't ww

r WiWll Will tllV

rnwi

v

--ernrnrnrnrnrnM

IMWllllMlllXIll

ItY1I

flMllffl

llWWillllhI iW&mll ' Ill

iIllW ,WIlllllIW llrn IrIIII0

aim' a

wwrnw'

-2Oft Illllh1ll B0ffi 0lW MUMll

aRllWlluInI

rnmIntLul$lrnll Lll0W X 25 ft

(22)

Ma-e deta i I ed numer i ca invest ¡ at ions showed that the on t i me evoiut ion of an unstable wave train is daninated by the unstable modes

contained in the initial cchditicns: if there are none. the wave train

is stable and the envelope renia ins constant or periodic; i f there is

one,

à recurrce sequence

i s set up When the i n i ti a cord i t i on conta ins more than ore unstable mcde the evolution process is highy irregular and its develop'nent turns out to be extremely sensitive to

the number of stable modes also included, These stable modes contain a

sma I I amount of energy, thus the evo I ut ion of the system, be i ng extremely sens i t i ve to the sma I I energet i c perturb at i ors represented by the stab I e riiodes of thé enve I ope, shows a chaot i c behav i our.

1 .1 .2 - EvolUti a-i popert les of the

wave tra in

¡ n t d i mms lais

So f ar we have rev i ewed sane interest i ng enanena which were

thecr'et i ca I ly and exper i manta I ly demonstrated to occur; i n part i cu I ar,

we have highlighted the naiienoñof modulaticnal instability, the

ex i stence of enve I ope sol i tons and of recurrence phenaiiena i n the I ong crested case

Simi lar -iencmena are I ikaly to occur in the two- dimensional case

where a weak two-d i mens i ora I <pans ion of NLS may be obta i ned by letting the wave envelope also depend on the y axis, A(x,y,t) while

the carr i

er

wave, st i I I propagates i n the x d i rect ion. Therefore, the

free surface is given by:

fl(x,y,t) = Re [A(x,y,t) exp [i(k0x - o0t)]}

wh i ch i s not a model of a tru ly short - crested sea since cannot account fa- the spread i ng of energy around the dan i nant d i rect ion of

propagat ion typ i ca I of a shor"t - crested s9a,

(23)

neither for what cmcerns envelope sol itms (which turn ait to be

unstable

and

do not exhibit true

sol itcn

behaviour)

na-

fer' the

recurrence phenomena wh i ch

i s detected together w ¡ th a terma

i zat i eri

effect not present

i n the me-d i mens i ma I

ease and wh idi makes the

equat Fai inval ld

(this effect causes energy to be gradual y spread out

to h i gher and h i gher modes at every recurrence so that after saiìe t i me

the bu k of the enerqy

i s not caif i ned around the carr i er wave nuliber

k0 as i n the hypothes i s)

A sat F Sf acta-y

tool

was

found

i n

a nm -

I i near

integro

-d i fferent i a I

equat im due to Zacharov wh i ch descr i bes the evo I ut im of

the campi ex envel ope spectral funct im A(k, t) and which takes care of

h i gher order effects and degenerates into NLS inaie-d i mens i ma I

case.

Th is equat io-i w i

I be exam i ndd

in sane deta i

I

i n sect im 2.

Fer'

now

it

is

sufficient

to

say

that

al I

the

phenomena

described

prey i ous ly are present, w i th sane m i nor d i fferences, i n th is 2D case

To end th is rev Few o-f the evo! ut im propert i es of water waves, we

sha I I

br i ef I y out! i ne the trends of deve! oneno th is k md of study.

lt is possible to identify four major directims

further thecret i ca I

numer i ca I a- _{exper i menta I}

_{¡ nvest i gat i ais}

ai

the Zacharov equat im. Arnaig others

we quote the theoretical wa-k

of Shemer and St i assn i e /9/ who extended the

equat im to a h i gher

order and to the f i n i te depth case.

investigatim of chaotic behaviour Fri water waves. A paper by

Vuen

/10/ has dealt with this matter.

(24)

thea-et i ca stud i es w i th the a im to asses the i r i nf uence on

surf ace waves and structures,

d) expe'- i merita study of non I i near nteract ais between waves fa-' a better understand i nc of the d i spers ion re I at ion, To

th

s c I ass

be cngs a wa-k by Harnrnack et a I, /11/ who stud i ed r ¡ pp I es ( wh i ch

have. quite an easy d ispers ion rélat ion),

thapte" 1 .2 - STAT I ST I C 0-IARACTER I ZAT I C1'4 CF RANDCN WAVES

1 .2.1 - Wave eIe,atia

W i rid generated wave p-'ope-"t i es such as hfht and per i cxl are not

readily definable man individual basis but may be characterized by a

probabilistic approach enabling engineers to apply stochastic analysis.

in almost al I studies, the wave elevation is assumed to be a stat irnary randrn process, subject to ergod ic propert ¡es, Moreover, i t i s often assumed to be a gauss i an d i str i buted process.,

The wave field described by the gaussian model can be visual ized as

consisting of a simple superposition of infinitely many independent

Airy waves (which are pure sinusoids), Although this simplified model

offers a good approx i mat im to a great number of prob I ems, there are

sane difficulties, To begin with, independent pure sinusoidal

canponents

ai

ly sat i sfy the f i rst crde- equat im of mot ion, theref me

they ho d true when the s I ope of the waves approaches zero, Even fran

an en i ner i ng point of y i ew, they sh I d be used aily f or I i ght seaways that are either wind driven and uncontaminated by swel I a-' that

(25)

Moreover, sorne pA-encmena observed i n ocean and coasta I engineer i ng

appear to be non gauss i an randan processes because quadrat ic a-' h i gher

order interact ions between caiiponent waves affect the wave marqinal

distribution, For instance, heavy stormy waves show a trend o-F skewness

i n the i r p-of i I e w i th an excess of h I gh crest points and a I ack of I ow

tragh points

and a s im i ar tendency has been observed near the

shoreline, as illustrated in figure 14.

Fig 1,4

C TAicN FC*'l

FPENCE / I 3/

>

Höwever, even i f they are often taken as si non i ma i t i s not the

same to speak o-F non - I mear

or

nor - gaussian behaviojr o-F sea waves. and two approaches wh i ch are se dan recogn i zed as be i ng d i fferent are

used, The first, which arises fran the hydradynamic model I mo of waves

and which w i be ca I ed the hydrodynam i c approach and the second wh i ch

fol lows fran the study o-F the prcbabi I istic structure of the wave

elevation process which will be indicated as the probabili ic method,

As far as the wave elevation q(x,t) is concerned, it is pössible to

assess the gaussianity of a wave time series simply by evaluating its

h i gher order expectat i ors (prcbab i I i st i c approach) and i t is

stra i ghtforward to see, by us i ng the centra I im i t theorem / 2/ that

r(xt) foi lowing fran. I mear dynamics has a gauss Ian marginal

d i str i but ion Thus, the two approaches g i ve the same resu I ts in the

(26)

i terature

A second reason fa-' the frequent m i sunderstand ng

i s that

both approaches use the perturbat lai

analysis1

which gives at

the

f Inst order a gauss i an d istribut ion,

in the prabi ist Ic approach

and

the

mear wave dynamics,

in the hydnodynamic aie

Histcrical ly,

the

first

attempt,

at

least

to

the

authors'

know ledge,

repa-'ted

in

I iterature to develop a hydrodynamic

second

crder aprx'oach

is due to Tick /14/ who (as summarized

in Appendix A)

af ter mak i ng a

Tay Ior

expans ion up to second order

of

free surface

eqat ions,

cbtained the second order

potential

2)

_{(superscniDtim}

within bracket-s indicates the arder) as a function of the first order

one < ( 1) and

(2)

as a f unct Ian of both

1 and

(2? then, under the

hypothesis that

rì(i)

is gaussian, found the spectniJn

S(2)(u

o-f

ri(2

Uw i ng to the non-

i near interact ions wh ich occur between waves, there

is a spread of energy to

I ow and h i gh frequency.

I f the

I

i near spectrum

is a discreet one containing only two ccniponents u1 and u2, energy at

second order ww I d be found at frequenc i es 2u1,

2u2,

- u2 and w1 +

u

roughiy speaking this is what happens also in the continuous case

and s 2) (u) reLjresents both I ow and h i qh frequency correct ions to the

i

i near spectrum S

(u)

These non -

I iriear contributions may be very

important and they have been investigated by many researchers such as,

among othcrs

Hudspeth and Chen IlS!, Sharma and Dean /16/, Masuda et

al

/17/

and

Pinkster

/18/

for

their

formulations

based on a

"frequency

ap'oach similar to Ticks.

n the probabilistic approach Longuet Higgins /19/ derived a ser les

expansion of the probability density function based

on

Edaeworth's form

of the type A Gram-Char! ier (see append ix A) ser les truncated to the

fourth term thus abie t-o take care of the third and fourth cumu!ants

(27)

eva uated how we I the Q'am-Char I i er ser i es rep-'esents the marq i na I

d i str i but i on of I abcratary w i nd generated waves i ncreas i ng the

truncat ion order. The surface el evat i on d i str i but ¡ on was we I

approx i mated by a f i f th order ex pans ion but the same resu I t d ¡d not

hold true for wave acceleration (see fig 15 and 1,6 ).

l0 _Linar N

aaItjon

N ,00

io

N

-I

a

u

a

Liasur us.otto. z .0

a

N

a

I-u

a

Fig, 1.5

( TAkN F=c.l

, I 3J ) N Nonlinear Fig, 1,6 .0 0.0 P.O O

T.A1N FCI'I EFENCE ,1 3./

n 1980-81 Bietner /201,121/ made comparisons between measx'ed at

sea and Gràm-Char i er d i str i but ions ¿f - wave amp Ii tudes for both deep and shal ow water reaching the conclusions that they may be treated as

a weakly non gaussian mocess.

.StL)n is /22/ suggested describing non-gaussian seas as a random

z

0.0 7.0 '1.0

(28)

sequence of Std<es' th i rd a-d er waves but there i s a phys ca I

imitatiì in this model owing to the fact that two waves placed side

by s i de present a d i scont iiiu ity i n the wave part ic e dynamics at the

cmjunct i a- point, Na-e recent y, Tayfun /23/ p'esented, i n the deep water and narrow band I im it, a model based on second a-der Stdces

thea-'y dta in i ng the marg i na d i str i but i on i n the hypothesis of

Rayleigh distributed wave heights. Rang et al /24/ were able, taking

i nto account a constant (not osc i I I at i ng) term often neg I ected i n

Stôkes expansions, to cbta in the i n i t i al d i str i but i on by- third a-der

Stckes' series, reaching good results (fig. 1.7)

Fig, 1,7

C TAkN F1Ct EFENC

2A./

Fa-' what concerns wave height distributia-i, _{it must be said that it}

is usua I y assumed to be a Ray I e i gh one even i -f i n thea-y th i s i s true an y in the i near case. In the non - I i near case, i t is suggested /25/

that as a parameter of the distribution the va fue (0,925)2 a2 is used

instead of the a2 of the I mear case (a2 is the variance of the

if.

(29)

marginal distribution), The result was confirmed /21/ for deeo water

but not in shallow water /20/, /21/ where a non negi iaible sensitivity

of the distribution to the spectral width was assessed.

Ccnsideratia-s similar to the last are not, however, very useful

because their goal is not to go to the prabi list Ic nature of the

phenomenon

but to try

to assess the gauss ianity of its marginal distribution fran the fact that its peaks are Rayleigh distributed, This is not correct, even if a gaussian distributed narrow band process

has heiohts which are Rayleigh distributed, the oppösite is not always

true

1 .2.2 - Wave ve I ope

Fol lowing Vanmarcke /61/ we can define the sea surfce cievatim

as the rea I part of the comp I ex process

c(t) = A(t) exp(iut)

where is a midband frequency which is taken as reprsentat ive of the

sur-f ace e evat i an process, f the process i s suf f i c i ent I y narrow banded

and If is closed to its peak frequency, A(t) turns cut to be slowly

aryin and the complex part of c(t) is read i

y dta mable fran

knowledge of its real part only via Hubert transform,

In this case A(t) can be defined as A(t) =

E R(c(t))

+ I(c(t))

12

which is the wave envelope,

Naess /26/. /27/ has developed a method to evaluate extreme wave

height-. assuming that the extremes of a narrow-banded random r'ocess are

(30)

The wave enveiooe is strictly related to the eno-nenon of wave

qrcupinc cmsist;inq o-F a sequence of hiqh waves) which, in turn, olav

an important role in t.he enomenon of

slow drift osciHatims of

moored obiects /28/. Ce of the suqqest:ed ways of ap'oachinc this

-nanchan makes use of the wave envelope statistics /29/.

1.2.3 - Bispectr.sn

Within this general context, an important topic to consider is

hiQher order soect.ral analysis which is strongly linked to both the stat i st i ca I and the frequency apr'oaches H i qher order spectra may be

important i n the assessmt of the response character i st i cs of a

non-I mear system in a randan sea as wel non-I as in civinq a lot, of infcrmat ion

on the statistica.l pr'opert les of sea waves themselves,

The second order spectrum (ca I I ed b i spectrum) of a stat i mary

erqodic random p-'ocess, which was first introduced by Brillinqer /30/,

is theoretically examined in /31/ and /32/ whilst a sta'-t encineermnq

introduction may be found in /33/. At- this staqe, however, it is

sufficient t:o state that, in the simplest cased the bispectrum is the (two

dimensional) Fourier transform of the third

order

correlation function

r

'i"

(-t ,-c ) = E. {x(t;) x(t + -'r ) x(t, + )), Therefore, it is

1 2 1 2

related to the third moment. o-f the initial distribution o-f _{waves, and}

is identical ly zero for gaussian processes.

A first attempt to obtain more information about non - I mear

interactions in waves by using the bispectrum was made in 1963 /34/ and

relatively qood agreement between theory and experimental data was

(31)

bispectral estimate (which has high statistical variabi I ity) we had to

wait ti I 1985 to find bispectral servatias of sea waves /35/,

However, both these wa-'ks refer to sha I low (11 m deep i n /34/) and very

sha I ow water depth (O to 8 m i n /35/) so that nctody has tr i ed to study an engineering model of the bispectrum as was done in the case of

spectra.

C the other hand cross-b i spectra I ana lys is (which is based a the crossc'rel at ion f unct ion of inpjt, inpjt, output) is quite well estab I i shed i n the assessment of non - I i near i t i es of the response of wave exc i ted non - I i near systems. I t I s suf f i c i ent here to ment i on the

basic works of Yamanouchi /36/, /37/, DaIzel I /38/, /39/ and Pihkstgr

/ I l/ on th i s subject.

thata' i 3 - CX1ID ING REMAR

rwo main different topics are related to the 'opagatim of waves:

the eime -

spatial evolution of a wave train and the statistical characterization of a sea state in a point (random 'ocess) or over an

area (random field),

When linear dynamic (Airy waves) is used, the two dDlems are

easily solved When non - linearity is taken into consideration, sane

remark-able effects arise in both cases.

Dur ing the I ast twenty years a number of enomena have been

high i ighted Which mear theories cannot take into account. Among them:

(32)

Fermi Pasta and Ulam recurrence

ex i stence of enve I ope so i tons

are relevant to the space - time evolution of a wave train and

displacements of the initial distribution fran gaussianity

existence of a non zero bispectral density

are important when deal Ing with the statistical characterization of a

sea state,

Final ly it must be underlined that evolutionary type apçr'oaches do

not need the hypothesis of stationariety o-F the wave train, lt is not

clear if such apy-'oaches are a I ways canpat ib le w i th the stochast ic one

which requires stat lanar iety, At second arder seems to be thus, in

the fol lowing chapters, stat lanar iety will be hypothesized a-, at

least, _{the period of observation needed to evaluate the statistical}

pror3ert es of the sea state w i I be cons i dered suf f ic ¡ ent y short and

the time evOlution of the train sufficiently slow to allow

(33)

(34)

CTIa4 2

cc1'JD CRR CETEFd'i i NI ST IC i'iaxu.. I NG

Qiap- 2.1

G/

I NG EJAT I

S

2.1 i - Eqiat ¡

i of rrKt im

The dynamics o-f surface water waves is governed by inertial,

resta-' i ng ana camp i ng forces and _by _{f a-ces due to the act im of the}

externa env i ronment,

The two major restoring fa-ces are due to gravity _{and surface} tension and their contributions are approximately equal fcr a wave

about 17 cm, _{mg. Waves longer- than 17 cm. are gravity daninated and}

are therefore known as gray i ty waves whereas those shorter than 1 7 cm

are cal leci capiHary,

Damping arisefrcm viscosity which is expected to affect caQiilarv waves cniy, having a reduced effect or gravity _{waves, at} _{east over a}

short term extent

i f the e-f fects o-f the interact ions _{such as w i nd and current are} d sregar-ded, _{the externa i} _{act i}

_or

i s due to the atmosi±er i_{c pressure}

and, Since the fluid is assned to be incanpressibIe _{its density can}

be na-mal ized to ore and the external _{pressure. can. be taken to be a}

constant WhICh can be set to zero without loss _{of generai ity,}

(35)

inccrn-essible inviscid fluid is based on the Euler equations which, in

the infinite water depth case are written as fol s:

= 0 _(2,1)

i i

+

(V)

+ gî = p

fOE' z =

q (x,y,t) (2.2)

rlt + - = O for z = T)

(x,y,t)

(2.3)

4

0 z

-

_(2.4)

where ' is the ve oc i ty potent i a q is the e evat i on of the -free

surface. g is the gravitational accele-'at:on p is the fluid density and p i s ari externa _{x-essure exerted on the surface o-F the -F I u i d. The}

horizontal coordinates are x and y and the vertical z is pointinq

upward, The apiadan operator

a2, a2 a2 _a _a _a

y2 j ₍ + + .) and V is the OPerator

(

-ax2 ay2 az2 - ax ay az

wn le subscripts denote part i al der ivat ives

-After Yuen and

Lake /1 the

surface velocity potential is

introduced as 8(x)t) (where x (xy)) and is defined as:

4S ix.,t) = I' (x. r x,t),t) _(2.5)

-fran whicri it foi lows that:

(36)

V> 5t = V> (xr(x,t),t) + ,z,tt V< Qt>

(2.1>

where V, i s U ha- i za-ta I grad i ent ( ).

Using (26) and (2,7) the surface baindary caditia-'s (2,2) and (2,3) may be written as(cfr, Appendix E):

+ gr + _x _- .

{i+

(yx

j.)2] ₌ ₀ _(2,8)

[i +

(VXq)2J

=0

(2.9)

¡n the foHowing, the time evolutia-i in _{the period} _T _of _a

deterministic wave packet will be dealt with, This is not the most

genera case; however, sane resu I ts are app I idab I e to the case of randan f ieids as well,

Due to the na- statimariety of the wave packet, it is useful to

treat the çrd i n terms of the FOEr i er transfa-'m, w i th respect to the space var i ab I es, I eav i ng the t ime as a parameter,

The two-d imens i aia I Fair i er transfa-m of the velocity

potential is defined as (see Appendix A):

4 (k,zt) = I 4'(x,z,t) exp (-i k X) dx

J

-he Fair i er transfcr'm of t-he Lap I ace equat i a- (2,1) g i ves

(k,z,t)

which, together with the bcundary ca-id itian at the bottan (2,4) yields

A

(37)

(k,z,t) ± 3(,t) exp(-IkI±) (2.11)

A

where the notation +(kt) means (k,O,t

4ntitrans-form o-f (2,11) is highly ccrnpl icated and wwld aive a

strange function which moreover, ww!d not be very useful because what

is needed is not the ootentia; at a fixed quote z but the potential at

the free surface x,t) that

is, the antitrans-onm of 3(krx,t),t)

which is a more complicated p'oblen, The optimal solution is to find

fon (kq(xt),t) an ex-ession deDending on _{(kt) and q(k.t) which} ar-e readily usable, The meehed to. obtain this will be discussed in

some deta i rì the next paragraph, for now et' s suppose one has such

an exp-essicn which will be indicated as fi _{t)] =} t>

and its use

wI! I

be dnonstr'ated,

he f Inst step is to transform with Fanier the tec-ms in eq (2,8)

and (2,) this can be made by notin that (App. A):

(

S )

=

3(,q(x,t)t) =(kt)

) = i k (k,t) , (V<

q) =

i k (it) ç s i

A,

Vq) =

k.1f(,1,t) (K,-

t)qK.-

_1c.t)

dk1

J-= _JJ

k2(k2t)

- , - k2>

dk, dk2

= -

jJ

j,2q

,t)qt) 6k - k.1 - k) dk

dk2

AI

A

(38)

where ( )

denotes the Fojrier transfcrm

of

Fran

(2M)

we have

rr oe A A A A 1 A k2 k.1tf(k2,t)ô(k -

k1 -

k2) dk.1 dk2 + -JJ Ik IIK2If(k.1,t)f(k21t)

l2

dk,dk2 + (2,12)

-JJjjJ

oe1

t) (k2,t) exp[- 1k.1 I +

Jk2I) ri(xt)]

-6(k5Hc-k2) k3 k4 I(k3)I(kq)6(k-k5-ka---K4)] dk1dk2dk3dkqdk5 = O

recai

Ing the properties

of

the impjise functfa

s5( )

(Appendix A) the

last term of (2.12) may be written as:

jjjj

oe

k1 I

1k21 k3 k,1 3(k.1,t)(k2,t)(k3,t)(k4t) -oe r -1

e)(Pj-(k.1I+Ii2I) 1(.t)j ók-k1-k2-<3-k4) dk,1

dk2

dk3 dk4

and fran (2,9) rr oe A A A _r rn-- _JJ _c1!s2fQ1 it)r(k2t)S(k-ki--k2)dk1dK2-IkI(kt)exPj--IkIr(xt)J +

(213)

=0

Equatia-ìs

(2,12) and (213)

toqether with the Laplace equatim

(39)

2.1 2 - L inr app-'oadi

If only linear' terms are retained fran (2.12) and (2,13) it is

possible to write:

A A

)t) + q rl(k.,t) = 0 _(2,14)

A . _r

i

t11(k,t) - JI (k,t) exp[IkI r(xt)J O (2,15)

Moreover, within mear assumptions, 4(x,r),t) (xO,t) is taken,

tnus

(k,t) (kt) (2.16)

which ai lows (.2,15) to be re-written in the f01 lowing form

(k,t) - k (k,t) O . (2,17)

Taking. 'from (215), and substituting it in (2,17) me cets:

kJt) + II

3(k,t)

= O . (2,18)

whicn is the usual harmonic equation for _{which can be solved, by the}

A A

function (kt) = (k) cos(wt) _{p"ovided that w fol lows the} _linear dispersion relation:

()2 =

g Iki _(2.19)

The dispersion relation (2,19) contains a! I the information needed

(40)

k I inearly indicates that the wave celerity (so-net imes cal led wave

phase velocity) c(w) = ¡s a function of k, The result is that

unequa I y ong waves travel at d i fferent speed s and two waves whose crests are located at x at time t will have shifted crests at time t+t

(see fig. 2,0, taken fran /62/),

?±-t

±

I

\\

Fig. 2.0

Propagation of plane progressive waves. Each wave crest is

connected in successive photographs by grey diäonal lines which advance in time with the celerity which reduces when wavelength is reducing.

2.1 .3 - Hi q a'd pans i ai

I n the non I ¡ near case, we sha I I start aga i n with eq. (2.12) and

A _I. _A

(2.13),

4'(k,l(x,t)t) being expressed in terms of 4' and

T) (X),

After Zacharov (as cited in /8/), the canplex variable b(k,t) is

¡ ntroduced and def i ned as

( ùi(k) A r IkI 11/2 A b(k,t) = I. 21k1 )1/2 fl(kt) + ¡ I 2u(k) _J f(k,t) (2.20)

(X) _{see Appendix C fo'a derivation o-f}

(kt) as dDtained by VuGn /8/

(41)

r

Bearina in mind the de-Fir,itim of the ceie-'itv c(u), it is

straightforward to write (2,20) as

(

?I<I 11/2

A 1 A

I b(k,,t) = r(k,t) + i f(kt)

w(,cz) )

-

c(w)

-which cives b(kt) the foi owinq physical meaning;

A A

« r(kt) and lm[b(k,t)] a

Substitutior of (2,21) ih the free surface boundary cdndition

al ls a sincie complex equatim for b(kt) to be obtained from the two

free surface boundary ccnd it ions,

This gkxal

equation. which is omitted here because of its

complexity but which can be f and in /8/ iS an exact solution u to

the second order' in kIr, Because of its complexity it. is usual lv

treated by means of a multiple scale approach assumin that the wave

field can be divided into a slowly time vary inq component B and a smal ¡

but rapidly varying bound component B' and that most of the ener'ay is

contained in the slowly varying component B,

Since we are dealing with waves and we are intested in their

evolution, it is reasonable to assume that both r and

can be

represented as a modulated snusoidal siqnal, the modulational

amplitude of which is slowly varying, cooether with rapidiv varvina

h i qher order connect ion

(2,21)

r

b(k,ti IcB(k't)

-fe2

B'(k:,t')I exp(-iu(k:)t)

L J

where B(k.'r) is the moduiational ampi itude

(42)

B' (k ,t') i s the h i gher aid er cairect i ai

u(k) i s the frequency of the carr i er

i s the sma I I parameter of the 'db I em

't

= c2t is the slow time scale whereas the fast one is t',

SLbst i tut ion of equat ian (2,22) into the above ment i a-ed q da I

equat i an, which i s obta i ned f ram the ba.ind any cand i t i ais, and

separat ia-i of terms acca-d ¡ng to their aider in c leads to the so lut ian

of the prd Iem

lt is to be noted that: aider E is satisfied identical y

Cder

c2

g ives an equat ¡an which rei ates B to B wh ii e aider

C3 g i

ves an

equat ian wh i ch descr i bes the t i me evo I ut i an of B ¡ n f unct i on of B

itself and B', Substitutiai of C2 results into C3 aies leads to an

equat ian descr i b i ng the s I ow evo I ut ian of the dan i nant campaient B of a

weakly non-1 mear wave field,

This equatian is a very useful tool fai the study of the evolution

of a nan statianary wave train and it is known as the Zacharov equation

by the researchers involved in this kind of study, The whole method

has, however, sane characteristics which may be very useful when deal ¡ng with quadratic effects in a stationary wave tra in in this case

some equat ¡ais cbta i ned i n der iv i ng the Zacharov equat ion w i I I be used

instead of the equation itself, An example is the equation which arises

from E2 terms,

2.1 4 - Example of appi icat im of Zacharcw' s rnethcxi

(43)

elevat ion f

or

a stat imary wave train canposed of a single harmonic

(Airy wave)

of

frequency w,

wave number k0 and ampi itude a,

The

so ut ion

i s ve-y s i mp i e and

is descr i bed by Stckes' second crder wave

theory.

Appi icat ion of the Zacharov method

is very calipi icated but

stil

possible withcxjt the use of computer facilities, In this case the

ihear' wave term B is:

I

_w(k)

21k1 )

1exP[I(k_jt]k_k0)+exp[i(k+w0)t]k+3(0)

Iki

i,,-

li

,

t 2u(k))

' exp[i(k+u0)t]k+k0_exp[i(uCk_0)t]&k_k01j

corresponding to a wave elevation q(x,t) = acos(k0x

-Second order wave el evat i on

i s, accord ing to the method:

(2> a2 a2k0

q

(xt) =

f(k)

(k)

+ 2

cos[2(k0x - w0t)]

where f(k) is a function which app'oaches zero,

The first term in this relation is the low frequency term'which is

zero for a second order Stokes wave in

f nf i n i te depth

The second, term

is exact I y the second order

I ocked wave el evat i on of the Stokes wave.

The analytical details of this example are not reported here due to

their canpexity but may be f cx.'nd

in /64/. With further calculations,

information related to the velocity potential

and to the dispersion

reiat ion cojid be

tained but this is not done here sir-ce the

rpose

(44)

to study the mg t i me evo I ut im of a non stat imary wave tra in, can

give very useful information about the quadratic characteristics of a

stationary wave train, Unfortunately to do this requires considerable

analytical effort,

So far we have tried

to show, quai itativeIy how Zacharov's equat ions may be dta i ned from the free surface wave equat ions, De to

the considerable mathematical compl icat ion,

we have shown only the

initial passages, but the whole theory is developed in /8/. It is

important to underline that, despite its complexity, the theory gives a

deep ins i ght into the phys i cs of the ocean (i t turns out to be usefu I

in the study of envelope sol itons, wave groups, modulated wave trains

and so on) and it is al so used to invest igat& nm-I mear effects in

water waves spec i a I I y from the point o-f y i ew of I ong t i me evo I ut i on of

a two dimensional wave train,

Another f i el d i n wh i ch Zacharòv' s equat ion was used, wh i ch is far

more interesting for engineers, is the short term time evo lut ion, which

allows us to relate u to k and to study the dispersion relation and its

modification due to non-1 inearities,

This will be the subject of chapter 2.2.

Qapter 2.2 - DI 9ERS I N RELAT I ci

2.2.1 - GaI

When dealing with the dispersion relation of a wave field, two

(45)

components in the power spectrum travel at different wave speeds which

are qi ven by the (linear) d ispers ion rei at ion (2 19) Second, in a

weakly nm-i mear steady and uniform wave train (Std<es waves), - al the

harmonics have the same speed which is also the speed of the

fundamental one. The fundamental one obeys the mear dispersion

reat ion. therefore higher harmonics do not.

[his quite different behaviour- indicates that in a system composed

of finite waves both free waves and their harmonics (often cal ed bound

on i ocked waves) ex i st and an i mpontartt quest i on to be answered is how

ncr-I inearities and bandwidth of the wave system affect the balance

between these two d i f f erent components,

1 he coex i stence of both free and bound waves has been shown to be

rea I i n open ocean as wei as i n I aba- atcry measurements, Moreover,

some detailed studies on topics related to this problem have been

undertaken: for instance Sand /40/ and Sand .& Mansard /41/ deal with a

rather secial one which is the generation of real I mear waves in a laboratory tank

A variet' of explanations have been proposed includinq drift

current effects, directional ffects and various kinds of nm-i mear

interactions, The experimental results obtained by Crawford et, ai,

/42/, with mechanical ly generated wave spectra without wind show phase

speed anomal ies similar to those observed in wind waves and, for

instance, may be accounted for by Zacharov's equation.

2.2.2 - The'icaI reiIts

(46)

i n-if I uence of ncn- ¡ i near i ty and bandw i dth cn the behav i or of a wave

component. The theoretical analysis performed was limited to the me

d i mens i ma I case (I mg-crested sea) for two ma i n reasons, the

complexity o-f the topic and the absence of experimental data concerning

wave speed of individual components of a wave field.

The wave phase speed is, ¡n fact, no longer univocal y determined

for a given wave number or wave frequency as in the I mear case but it

has different values depending on the component (linear or slowly varying) to which it refers.

To make a comparison between experiments and theory, it is possible

to define a sort of mean celerity C(k) which averages the phase sDeed

o-f the waves wh i ch have wave number k be long i ng to the two d i fferent

components. This can be done either in a theoretical way, by using the

Zacharov equation, or by fi ter ing the experimental results in an

app-'opriate way, This kind of comparison,

made by

Vuen /8/, is

summarized in the fol owing while experiments are dealt with in the

next paragraph.

First of ail, it is important to say that, when considering the

I mear component only, the well-known linear results C(k) = w/k can

be dDta i ned regard I ess of the structure of the spectrum.

A slightly modulated wave train corresponds to a dominant component

of strength S0 at k = k0 and a pair of "sidebands' with strength s.

sma I I compared to S0, I ocated. at k0 away fran the carr i er component

and centered around it (see fig. 2,1), which generate the long wave modu at im of the enve I ope The degree of modu I at im is proport i ma I to the sidebands strenth and the wave length of modulation is inversely

(47)

propa-t i ma I to the s i debands separat ¡ m

r ak Ing th s ystenl to be the p- ¡ mary system B (k), secmd a-'der effects pred i cted by the thecry yi e d fcr'ced caîlpments at k = ¿k0,

2&, 2k0-

2k0- ¿ka, 2k0, 2k0+ ¿k0, ad 2k0. Time-lo_I k Ca,'Do...nt

4

345 lO 50 Frequency f (Hz) Fig, 2,1 i TAicN FC*.l REFERENCE / 8/ )

In fig, 2.2 the actual ase speed C(k), ncrmal ized with respect to

the speed cor'respond ing to the daninant wave number CL(k0), is shown, A

sol Id me indicates the mear dispersim relatim (deep water case) and ¿k0= 0.2 k0.

From the f i gure, the component at which phys i ca I y describes

the wave enve I ope, seems to trave w i th the grwp vet cc i ty wh i ch

cor-respmds to half the carrier pkase speed and the same happens fcr

the component at 2k0 which does not possess any fami liar physical meaning, The primary system ( k0, k0 ¿k0) closely follows linear

thecry with a very slight difference due to Stokes non linear

Wave amplitude

r

(48)

correct i on proport i ma to (ka ) . The second harmon i c at 2k0 trave I s

at the same speed as the p-' i mary and its ne i ghbm i ng components behave

i n a manner s im i ar to that of the s i debands of the pr mary systr.

Thus the forced components

do not

fol low the I mear dispersion relation but bchave in a locked nm-I mear way

('J

I'

o-o

o 0.0 k 2.0 30

ç

Fig, 22

C TAIEN FC1"

// )

The ex i stence of Iocked waves i s independent of the degree of

non-inearity but their intensity is not, In f acta takina as a measure of

non--I ineanity the root mean square of the I mear wave steepress,

the study o-f

the two cases (ka)rs= 0.10 and (ka)r=

0.01

g ves the same qra for the actua I wave speed pl otted aqa inst wave

number (-F i o 2 2)

wh'eas the

spectra corrected w i th second order contribueims are substantial y different, n fact, nm-I mear

(49)

F(k

contr ibut ¡ ans are too sma I to be detected in the second case (see f io

2,3),

t

Fig, 2,3

(TA)cN FP'.1

FENCE /8/

Most of the present concI usions can be extraoolated to harmonics

higher than the secondi al the harmonics generated by the primary

system crave at the same speed as the dorn i nant wave, As a crnseauence

of modulation, an increasing number of forced caliponents would be f curd

around higher harmonics and their behaviour would be, at east

quai it.atively, similar to that of the second harmonic's torced

components but a weaker and weaker dependence on wave number i s

expected 1h ¡ s f act is h i qh i ghted by the I ess pendent s I ope of second

order components w i th respect to the I i rear curve shown i n f i q, 2.2

o

O

00 05 '0 '5 20 25 30

e

(50)

At this staqe, it is important to renember that when deal inn with a

continuous spectrum3 high frequency cantr i but ions are due to both free

and bound waves and the resut.ing wave speed wI reflect a mixed

behav i our eventua y more s i mi I ar tö one of the two depend I ng on the range of frequency exam i ned,

To tain more information.. Yuen and Lake /8/ assumed the leadIng

order wave number spectrum 8(k)

to take the form of

a I orentz i an

spectrum (which hasP a decay rate of k(4) and perf armed same numerical

experiments fixing the value of the bandwidth c to 0.05 corresponding

to a narrow spectrum, and making the non-1 mear parameter (ka)rmg

assume the values of 0,01, 010 and 0,20.

Íhe total power spectra cbtained by adding the non-Hnear

contribution 8(k) are shown in fin. 2.4 which confirms what could be

expected: the non- I i near contr i but i an i ncrea ses as (ka )_rms OflOWS

The component phase speed 'normal ized by its value at spectra peak

i s shown i n f i a. 2,5 pl otted aga inst wave number. Departures f ran

i near i ty seen to be con-f i ned to qu i te a h i qh frequency zone that cou Id

be cal led 'the second harmonic" range for the quasi-1 mear sQctrum

ut become less localized as non-linearity is increased. C 0,10,

0,20).

Figures 2,6 and 2.7 show phase speed plotted against frequency and

the dispersion relation plots. Yuen /8/ notes that3 while far karma 0.01

(51)

cJ o o. o o q b o 00.0

Fig 24

C TAJN FC'tI EFERNC

ei

o

Fiq, 25

(TAJCN FCS'1 EFERNCE / 8f

The same values of the nm-I inearity parameter were adopted wher

analyzing a relatively broad spectrum (c=O2), The results, shown in

fig 2. are simHar to the narrow band case but departures fran

ineanity are more uniform,

0.0 0.5 1.0 I.5 2.0 2.5 30

k k0

(52)

w, w,. flIk0) . . C(fl01 e o o 0. o 0 2.0 30 000 ko Fiq. 2.6 .TAicN

C11 EFENC / ./ )

k0 Fig. 2,8

(TAIN FC3.1

EFRENCE J wJ )

Moreover, another set of ca ICu at ions were performed keep i

(I.

rms constant and letting assume the fol lowinq values: 0,05 (+)

0.10 (o), 0.20 (A) and 0,30 (e), Fiqure 2,9 refers to the case

ka)rms= 0.01 while fig, 2,10 to _rms

0,20,

Fig. 2.7

30

20

i.0

(53)

Fia. 2,9 Fia, 210

(TAicN RC*l

/9/)

[bese analyses show that /8/, at least theonetical ly, a parameter

which can oe thought to better characterize the non- mear procert i es

of a wave system is r' _c/('<a)rms. When r » 1, free wave canpments

dan i nate the systems and i near themy wcrks very we I when r « 1 the

opposite holds true,

2.2.3 - Exp i mta I ri Its

Lxperirnentai results have been analyzed in or'der to find a

val idatim of

the themetical mes and

they are dealt with au it-e

canprehensiveiy in the cited wa-k o-f Vuen and Lake /8/ from which we

have taken the toi owing figures and to which the reader is referred to

obta i n deta i I ed i nfarmat i on about experiments and the correspond i nq bibi ioqraphy

(54)

The first callpar'ison (fi'c. 211) relates to a sinole modulated wave

train qenerated in a tank by a wave paddle. Theoretical data (solid symbois) ca-respond to values abLained for k0a0 = 001 and 010 whereas

exrjerimentai mes (open symbols) are relevant to laboratory results with ranaina Fr-orn 0,02 to 0,10, T'ne resuts seem to be insensitive

to the value of k0a0 and to fol ow the theory very closely,

he case of a d i screet spectrum o-F waves n absence of w nd was

also investiaated and -FIQQ. 212 - 2.14 show a case with e 0,13 and

k0a0 ranainq fr-rn 0.01 to 0,09i3: the experimental results are contained

i n the naded area whereas the thearet i ca i resu I ts obta i ned by means of a bretschne i der spectrum of character i st i cs equa I to the mea sured one

are rely-esented by squares., The theory fits the experimental results quite weil Cpite L'ne difficulty which can arise in the evaluation of

the spect:na I bndw idth of the I mear part 8(k) of the measured spectrum. 2.0 1.5 c/co 1.0 0.5- e- o Fi-c, 2,11

n

n0

CTAIcN RRcti

_{EFEENCE //}

(55)

n no 20 30

TAN FC*"1

FEPENCE / /)

Fiq 214

TAN FCl EFENCE / /

'r mal y, in f ic 2 15 canpar i sas were made between theoretical

dar-a (solid

(1)

curve: Vefimov spectrum kz0a0 = 0,03, solid (2) curve;

'0 20 30

n

n0

(56)

Bretschneider k0a0 = 007, sol id (3) curve: Bretschneider k0a0 = 010,

dashed curve: mear thea-y), I abcratmy data (shaded area) and ocean

measurements (hatching area), Again, the results are satisfacta-y,

2.5 2.0 1.5 c/co 1.0 0.5 o m tOm/sec

i

n n0 Fig, 2,15

(TAN FC'l RFEPENCE

't/

2.2.4 - (aiiiits

At this stage, sane considerations are useful, Frst, it must be

under-I med that. the thecry which leads to Zacharov's equation is hiqhly

i dea i zed, -spec i a I y because _{i t does not account fm w i na current and} other nm-I mear effects

Keeping this in mind, the results are still val Id for demmstratinQ

that the non-I i near coup I i ng between d i fferent harmon i cs and the subsequent violation of dispersive equation (2,19) is important, at

(57)

idnored by the thea-v are not important

A second consideration is to be made about the method used to

evaluate wave phase speed, Essential y, it consists of a band pass

f i ter, centered around

a wave numbe- k, Which selects a

_{f i} _tened signal flp(X..t) containing only certain wave lenaths which is used to evaluate a sort of averaaed wave phase speed, There is no doubt. that if

every wave contained in q i s free, the result ing averàged speed w i fol ow t mear theory and if af the waves are bound the speed will not

fol ow it The ob len is that when there are both free and bound waves

the averaged speed will

reach an intmediate vaiu

between the two

bounds and the relative distance frän then will only quai ìtatively show

that free

or

bound waves are dom i nant w i thout quant i fy i nq the bound waves and w i t:hout i nformat i an about the i r frequency

1he correct way to proceed would be to plot the phase speed ,aqainst

wave number and frequency and the same shou I d be. made far wave spectra

¡n this way.. the results wuid also b quant itatively interestinq (App,

D), The reasons why this was not done is mainly that such data are not

readily available fran <periments, specially because soohisticat:ed instrumtation is needed, However, something is under way and the