Sea Waves on Deep Water

41

SEA WAVES ON DEEP WATER

PAOLO BOCCOTTI

Department of Fluid Mechanics and Offshore Engineering University of Reggio-Calabria

Reggio Calabria - Italy

1. Introduction ...•...•.••...•.•...••••••..•.•....•••••••..•• 3-1 2. Difficulties of an analytical solution for the wave period

and the wave height probability•.••••••••••...•.•••••..•.•••• 3-3 3. The closed solution for the wave height probability

in the limit of the band-width approaching zero ...••..••....••3-5 4. The closed solution for the wave height probability

in the limit H/~400 ••••••••••••••••••••••••••••••••••••••••• 3-6 5. The wave groups •••...•.•••.••••••••.••••••••...••••••••.••••••• 3-12 5.1 A recent theory ..•...•••.••••.••••••.•••...•••.•••..••• 3-12 5.2 How a high wave forms in an open sea .•••••••.•..••••...••..••. 3-17 5.3 How a high wave forms at a wall •.•••••••...••••....••••••....• 3-18 5.4 How a high wave forms off a wall ....•.••..••••.•.•.••••••...•. 3-19 5.5 How a high wave forms in the shadow cone behind a wall ••..•... 3-22 Appendices ...•...••••...•.•••..•.•.••••..•• 3-23 Symbols .••••...•...•.••.•.••...••.•....•••.••..•.• 3-27

1. Introduction

A person who observes a sea storm in an open sea, say from an offshore platform for a time-span of about half an hour, sees an irregu1ar and nearly steady homogeneous wave field which is called ' sea state '.

The wave field is irregu1ar because the waves at any fixed point have dimensions and shapes continuously variable in time, and at any fixed instant have dimensions and shapes different from one point to another. The wave field is nearly steady because the per cent variation of the energy content that wind is able to produce in half an hour within a sea storm is normally small. Finally, the wave field is nearly homogeneous because differences in the mean energy within the range of vision are typically negligible.

If the observation point, rather than in open sea, is near the coast, say from a wall-breakwater, wh at is seen is an irregular steady wave field being no longer homogeneous . Indeed the wave energy changes from one point to another due to the interaction with the structure. The phenomenon is that of reflection-diffraction of a sea state.

A theory of sea states in an open sea was developed from the SOs and 60s with important contributions by Longuet-Higgins (1963) and Phillips (1967). The first hypothesis is that a sea state is the sum of an infinitely large number N of regular periodic waves with infinitesima1 amplitudes, and with frequencies, directions and phases being different from one another. The

42 PAOLO BOCCOITI

relevant analytical form for the wave elevation and the velocity potential to Stokes'a first order is

N

Tlt(X. Y.1)=

L

a,cos(k,sin

e,x

+ klcose,Y-wi' + E,), 1·1 ( La)

.ti:-

.Icoshk,(h+z), , +t(x.y.z.I)=gLalw, hkh sIn(klsmelx+klcosely-wl'+E,), '_I cos, W~ kItanhk Ih-- g (1.b)

where h is the bottom depth, x,y are orthogonal axes in the horizontal plane, z is the vertical upward axis with origin at the mean water level, and

e,

is the angle that the propagation axia of the ith small wave makea with y-axis.

The second hypothesia is that, for the same wind generation being repeated many timea on the aame baain, aea states all with the same frequency spectrum E(w)liw-

L

!a: lor i such Ihal w<w,<w+liw (2)

I 2

and the same directional '" 1 2 S(w.e)liwbe-<;-2a, lor

spectrum

i such Ihal w<w,<w+liw and e<e,<e+lie (3)

would be produced. Moreover, the valuea of the phaae angleE,would be diatributed uniformly and purely at random in (0.2n).

On those aaaumptions wave elevation (l.a) and velocity potential (l.b) represent random atationary GauBsian procesaea of time.

A very large number of frequency apectra ECw) have been measured both on sea and lake surface. A characteriatic spectrum of the wind waves on deep water is

ECw) - ag2w

'Sex

p[-

~C:d

r}x

p{ Iny

ex

p[ -C;O~:;)2J} (4) It is the JONSWAP frequency spectrum (Hasselmann & al., 1973). Here Wd is the dominant (peak) angular frequency and two uaeful parameters are related to it:Td - ~ - dominant (peak) period and Ld=2•• = dominant wave length on deep

Wd

w!

water. The value of a depends on gFlu2 (F=fetch, u=wind velocity); for large gF / u? , a approachea Phillips' value 0.008, and it grows"for decreasing gF lu2

, so that also the wave steepness grows, aa was pointed out since the work by Bretschneider (1959). The apectrum shape ia governed by parameters yand a whoae typical values during the JONSWAP project proved to be Y"3 and0Ol0.08. A number of other proposed mathematical forma of the frequency spectrum can be found in the litterature (e.g. the Pierson & Moskowitz spectrum is given by eq. (5) without the second exponential function, and, clearly, it repreaented the baaia for the JONSWAP apectrum).

The directional apectrumS(w.e) ia equal to the product of frequency spectrum ECw) and apreading direction function Dce). Typical forma of Dce) which have been propoaed to fit data, are

SEA WAVES ON DEEP WATER

43

D(O): 0 elsewhere

D(O)-N(n) ICOS[~(O-OO)}2'

(5) (6)

where N(n) is simply a normalizing factor. The first form was applied in

particular by Forristall (1980), the second was applied by Mitsuyasu (1975)

who suggested that exponent n depends on frequency: n takes its maximum value

no at the dominant frequency and

n-nO(w/wd)s ij w<wd• n-no(wd/w)2.S ij w>wd (7)

then no grows as gFlu2 decreases. Clearly the greater is n, the smaller is

the directional spread.

2. Difficulties of an analytical solution for the wave period and the wave

height probability

As it has been recalled in sect. 1, wave elevation D.(I)at any fixed point

within a sea state, to Stokes's first order, represents a stationary Gaussian

random process: the most classic in the family of the continuous random

processes.

Only a small number of probabilistic problems dealing with waves in the

Gaussian random processes have been resolved. Most of the wave problems, as,

for example, the problem of the distribution of the wave·periods, are still

unresolved.

At present we are able to get solutions for those problems which require

to answer questions of the type: "what is the probability that some fixed

conditions dealing with ~.and its derivatives take place in one (or even more

than one) fixed small intervals dt? ". For example, the problem of the average

wave period <T> requires to extimate the probability P.(dl)that a given small

interval dt contains a zero of Th with a positive value of derivative 11••

Indeeed <T> is equal to the ratio between time interval 1:(-c --+ co) and number

N.(-c) of zero-upcrossings in 1:, that in its turns is equal to the number

P.(dl)-cldlof intervals dt keeping zero-upcrossings

-c dl

<T>= P.(dl)-Cldl: P.(dl) (8)

dl

H

t

I I I

I

I I I I I I 1

·

1I I I I

I

I I

I I

I I

I

I I

I

I I

1·1I

I I I

I

I >

Fig.l Relerenee sketch lor eq.(B): each point represents a zero-upcrossing

of D,(I).

44 PAOLOBOCCOTTI

The solution of the specific <T>-problem as well as the general way of solution of th is class of problems was given by Rice -see his works of 1944 and 1958-.

We are not yet able to get the analytica I solution for those problems which require to answer questions of the type "what is the probability that a fixed condition takes place within a fixed finite interval t::..p". This is the case for example, of the problem of the wave period distribution. Indeed, for arriving at the solution, we should be able to answer the question "what is the probability that the following three conditions jointly outcome

(i) a zero-upcrossing of 'lR falIe in a fixed small interval (1).I)+dl));

(ii)a zero-upcrossing of.'lR falls in a fixed small interval 12.12+dI2 (12)1)); (iii) no one else zero-upcrossing of 'lR falls in the interval (1).12)",

Here the third condition stops us from getting the closed solution. The situation looks even more critical if we examine the problem of the wave height distribution. Indeed we should answer the question "what is the probability that conditions (i), (ii), (iii) -the same as before- jointly outcome and in addition the following 4 conditions also outcome

(iv) a maximum of 'lR, with an elevation in a fixed small interval b,b+db, falls in a fixed small interval 13.13+dI3 (1)<13<12);

(v) a minimum of 'lR, with an elevation in a fixed small interval b-H, b-H-dH (H>b), falls in a fixed small interval 1•.I.+dl. (13<1.<12);

(vi) no maximum higher than b falIe in (1).12); (vii) no minimum lower than b-H falls in (1).12),"

Conditions (vi),(vii) beside condition (iii) already cited, deal with a finite time interval and that is why the general solution for the wave height distribution is beyond our actual possibilities. It should also be noted that, even in the case that we would arrive at the solution for the joint probability of conditions (i) to (vii), there would remain a considerable body of work to be done. Indeed the probability PH(H)dH that a wave has a crest-to-trough height within Hand H+dH should proceed from the integral of the aforesaid joint probability for 12 in (0.00). for '3 in (1).12), for I. in (13.12), for b in (O,H), which (the integral) should be divided by the probability of condition

(i) •

Clearly, approximate solutions and also exact solutions are possible under some restrictive hypothesis. In particular a number of authors (cf. Longuet-Higgings, 1962) proposed approximate numerical solutions for the wave period distribution, which are effective for small va lues of the wave period. The closed solution for PH(H) in the limit of the band-width approaching

o

is known since Rice (1944). The closed solution for PH(H) in the limit

H

.[rn.-+co (m.being the variance of the wave e1evation) was obtained by Boccotti

(1981, 84, 89), without restrictions on the spectrum.

SEA WAVES ON DEEP WATER

45

3. The closed solution for the wave height probability in tbe liait of th.

band-width approaching zero

To fix the ideas, let us suppose that the spectrum has a finite energy

content (area) and it has an infinitesimal band (Wd-bw/2,wd+bw/2).

Frequency w,of the ith small regular wave forming the sea state can be written

as Wd + bw" where Ibw,l:$; bw/2.

Let us consider random function Th (Le. the wave elevation) at two different

instants tand 1+ n2n/wd (n entire). We can write

N

IJ,(I) =

_,.

L

_,

a,cos(wd' + bw,' +E,), (9)

N

IJ,(' +n2n/wd) =

L

a,cos(wd' +liw,' +E,+ n2n+ n2nbw,/wd)·

,

-

,

(10)

The difference between the two Th values takes on generally finite values only

if n has an order equal to or greater than wd/bw. In practice, this means

that if we take two waves relatively close to each other (being separated by

a number of waves of an order smaller than wd/bw) we shall find only some

infinitesimal difference from each other. But if we take two waves very distant

from each ot her (being separated by a number of waves of an order equal to

or greater than Wd/bUl) we shall find that the second wave generally differs

from the first wave. More in details, each wave approaches a sine with period

Td=2n/wd and the heights of consecutive waves change very slowly, so that

an infinite number of waves is needed to see some finite differences of the

crest-to-trough height.

Hence a simplification follows, and in particular the probability density

function of the wave periods simply approaches a Dirac function. As to the

distribution of the wave heights, it can be deduced straightforwardly from

the Rice solution for the distribution of the elevation of the maxima. It is

an exact and genera1 solution which proceeds from the two following

probabilities:

(i) the probability that a fixed small interval dt includes a maximum of D,

(ii) the probability that a fixed small interval dt includes a maximum of

DR whose elevation falls within a small fixed interval y,y+dy.

The probability Pm(y)dy that a maximum has an elevation within y and y+dy

is equal to the ratio of probabilities (ii) and (i). The result was due to

Rice (1944) and was analyzed in details also by Cartwright and Longuet-Higgins

(1956). It is

Pm(y) = keXP(-ly2/mo){h+m

k

exp( m

kr

·[

I+er/( mk)]> (11)

with k-~(,~q), 1-1/[2(I-q)], m-Jq/[2(I-q)] and q-m~/(mOm2m.)

where mJ is the jth moment of the spectrum with respect to axis w -

o

.

If the spectrum is infinitely narrow, q-+ I so that m-+00 and

PmCy) =L..exp(-

L

)

C12)

rn , Zrn ,

46

PAOLO BOCCOTII

then, if the spectrum is infinitely narrow, the waves tend to sinusoids so that each maximum is a wave crest and the crest-to-trough height is equal to two times the elevation of the maximum, which implies

(H)dH

PH(H)dH-p".

"2

""2

(13)

where PH(H)dH, as already said, is the probability that a wave has a crest-to-trough height within Hand H+dH. Hence the equation of the probability deneity function is

PH(H) - (H/4~)exp(-H2/8mo)' (14)

4, Th. clo.ad .01ution for th. wave haight probability in tha 1i.it H

/..fi1lo

-+<0 Let us consider the wave elevation Tl",(t) at a fixed point Xo,Yo within random wave field (1). The sarnegeneral assumptions by Rice (1944) on autocovariance

~(T) are made, and in addition it is assumed that~(T) has an absolute minimum

which is also the first minimum after T-O. The abscissa of that minimum is

called T' • The jth moment of the frequency spectrum with respect to axis

w - 0 is called mrThen momentsmo and m2 are taken both equal to one, without

loss of generality and for slmplicity only, This implies~(O) - I, since~(O) - mo'

Finally, minimum ~(T')of the autocovariance is called simply a, and for the

assumptionmo - I, a takes a value in the range (-1,0) depending on the spectrum

shape: the narrower is the frequency spectrum the smaller isa. On summarizing

( 15)

Now we summarize the steps of the solution (Boccotti 1981, 84, 89) for the

probability of the wave height. The solution is based on 3 theorems,

Thaora. A : given the condition

Tl",(O)-Il, Tl.(T') - ~Il ( 16)

where 0 is an arbitrary time instant that is taken as time origin,

13

-+<0

and ~ is an arbitrary real number, the probability approaches 1 that the

wave elevation at any time instant T is equal to determinist ic function

D(T) _ [-a1jl(T- T') + lj/(T)] + [:,,(T - T')- alj/(T)J~1l (17)

I-a

plus a random noise of order

ilO,

that is, of a smaller order.

This follows from an analysis of the probability of Tl", at any fixed time

instant T, given condition (16) -see Appendix A-. That conditiona1 probability

is Gaussian, its mean value is D(T) which, as appears from eq.(17), depends

on (3; its root mean square depends on T but does not depend on B. Thus, if

13

tends to infinity, the random function Tl",(T) approaches the deterministic

function D(T).

The result is not related to the particular choice of instant T'; if we take

a different instant in the 2nd of conditions (16), we arrive at a quite similar result, with the only difference of the equation of D(T). The peculiarity of

T' is relevant to the next theorem which deals just with determinist ic function

D(T) •

SEA WAVES ONDEEP WA TER

47

Theorelll B: funetion D(T) -eq. (17)-, for -I ~~~a, represents a wave group

whosehighest wavehas its erest ~high at time instant 0 and its trough ~~

at time instant T' -see Fig,2-.

P

sP

Fig.2 General appearance of function D(T) -eq,(17)- provided that parameter ~ has a value in [-1,a], The ca1eulation was done from the autocovariance

pertaining to a characteristic spectrum of sea states, The theorem consists of two items

(i) funetion D(T) has its absolute maximum,whieh is equal to ~,at instant 0, and has its absolute minimum, whieh is equal to ~~, at instant T';

(ii) the derivative of funetion D(T)is smaller than zero in the range 0 <T<T',

Both (i) and (ii) ean be proved for ~--I and for ~-a, Then, beeause funetion D(T) depends linearly on parameter~, it straightforwardly follows that items

(i) and (ii) hold also for any value of ~in [ l,a), The proof of the two items for the cases ~- -I and ~-a is based simplyon the general properties of the autoeovarianee funetion: 1jJ(T) <1jJ(0)if T..0, 1jJ(-T) -1jJ(T); as well as naturallyon the definition of T': 1jJ(T»1jJ(T') if T~T' and "F-T', ,p(T)<O for 0< T<T',

Theorelll C: given at time instant 0 a wave erest with elevation

Jl--+ "",

the probability that the wave elevation is equal to a fixed ~Jl(-I ~~~a) at any fixed time instant T>Ois infinitesimal of an order exp(-nf32) (n>O) with respect to the probability that the wave elevation is equal to ~~ at instant

T'.

The eonditional probability P(~Jl;T)dy that Th at a fixed time instant T falls in a fixed small interval ~~,

~f3 ..

dy, given that at time instant 0 there is a wave erest of elevation

f3

-400, is equal to the ratio between

(i) the probability that a fixed small interval t,t+dt keeps a maximumof DM

48 PAOLO BOCCOTII

whose elevation falls in a fixed small interval «(3, (3+d(3), and jointly that the wave elevation llRat instant t+T falls in a fixed small intervaln(3,~(3+dy)

(ii) the probability that a fixed small interval t,t+dt keeps a maximum of llRwhose elevation falls in a fixed small interval «(3,(3+d(3),

Both probability (i) and (ii) belong to the Rice (1944) class of solvable problems, and the expression of P(~(3;T) can be simplified because of(3--+ 00':'see Appendix B-. The simplified form is

{ [

I

2]}

T _ [1jI(T)-

~]2

P(~(3;T)O exp -2j(T,~)(3 j(

.

o

[1-1jI2(T)-,j,2(T)] jor (3--+00 ( 18)

where O{ ) means 'has the same order as ' and ~ is the first derivative of

1jI.

To show theorem C, j(T, 0 is proved to be greater than j( T' ,0 for any positive

T'" T' and for -I :Ç~:Ça -see Appendix C-.

T T

Fig.3 Givena maximumwith elevation (3--+00 at instant 0, and given that the

wave elevation af ter instant 0 falls downto a negative level f;(3 (-1 :Ç~Sa);

the probability that the wave is anyone of the type of B,C,D,E is negligible with re.spect to the probability that it is like A.

Let us see the 3 theorems in their actual meaning. provided that, at time instant 0 there is a maximum with elevation (3--+00, and provided that the wave elevation after instant 0 falls down to a negative level ~(3(-1 S ~ Sa), theorem C says that the probability that the wave is anyone of the type of C, D, or

E in Fig.3 is negligible with respect to the probability that the wave is like A or in case B, because in C, D and E the wave elevation is equal to f;(3 at instants different from T' (for a further insight of this topic see also the following Fig.4). Theorems A and B add that the probability of waves like

B,

where the trough at instant

T'

does not follow the crest at instant 0, is negligible with respect to the probability of waves like A; moreover, they specify that the probability that the wave approaches a weIl defined deterministic function approaches 1.

The consequences are two .

consequence(a): the conditional probabilityPTIC(~(3I(3) that the trough elevation is~(3( -I :Ç~ Sa), given that the crest elevation is[3 «(3--+ 00), has the same order

SEA WAVES ONDEEP WATER

49

of the probability that the wave elevation at time instant T" is ~~ given that at instant 0 there is a wave crest of elevation ~, that is,

PT,c(~I3II3)o{exp[- (a-

o

:

I3z]} Jor 13..., and -I S~Sa (19)

2( 1- a )

consequence (b): a wave with given crest elevation ~ and trough elevation ~13(13....

=.

-I S~S a) with probability approaching 1, approaches determinist ic profile D(T) -eq.(17)-.

Then the demonstration cango on from eq. (19) of the conditional probability. The flow is

{ [

(a_~)z+(I_az) z]}

PT.c(~I3·I3)O l3exp - 2(I-az) 13

~

Jor 13....ec and -I S~Sa (20)

{ [

a2-2(I-a)al3+ 2(I-a)132]}

Pil c(a.I3)O ~exp - 2

. 2(I-a)

~

Jor a .... co and-SI3a S-- a

2 I-a (21 )

Pil_.c(a.~+t.)o{aexp[-_{2 4(I-a)}a2 JexP[-~J}_I+a

[or a .... ee and -KaSt.Ska wilhk=~___:=-1+a

2(1-a) (22)

Eq.(20) of the joint probability of the trough and the crest elevation is

obtained by multiplying the conditional probability (19) by the marginal

probability p,C(3) of crest elevation. The probability p,C(3)dl3 that a wave crest has an elevation within ~ and ~+dl3 for ~....co is equal to the ratio of

(i) the probability that a fixed small interval dt keeps a maximumwhose

elevation falls within ~ and ~+dl3 ;

(ii) the probability that the fixed small interval dt keeps a zero-upcrossing (indeed the number of waves is equal to the number of zero-upcrossings ).

Both probabilities (i) and (ii) belong to the Rice class of solvable problems and the result for ~"""', is p,C(3)=l3exp(-~132).

The step from eq~(20) to eq.(21) of the joint probability of the wave height and the crest elevation simply involves the definition

wave heighl - cresl elevalion - Irough elevalion

Finally, the step from form (21) to form (22) of the joint probability deals only with the validity range. Indeed from eq. (21) the validity range of eq. (22) strictly should be 0 S/).Ska. It becomes -ka S /). S ka because of the statistical symmetry of the Gaussian process, that implies

PII.c(a.~+/).) - Pllc(a.~-/).) (23)

50 _{PAOLO BOCCOlTI}

(for the statistical symmetry, the probability of a wave with height a and crest elevation

P

is equal to the probability of a wave with height a and trough elevation-~, which is equal to the probability of a wave with height a and crest elevation a-~, that is, PH.C(a,~) = PH.c(a,a-l3) from which eq ,(23)

follows straightforwardly),

The conclusions proceed from eq, (22) showing that the joint probability has

its maximum order for 6 of order aO For ~ of the same order as a, the joint

probability takes some smaller orders (it can be shown also that PH.C, for 6

outside the validity range, has an order smaller than at the two boundaries

of the validity range),

The first conclusion is that a wave of given height a(a400), with a probability

approaching 1, has a crest elevation approaching ~,Therefore the wave of given

height a(a ....00), with probability approaching 1, approaches deterministic

function

1jI(T)-1jI(T -

r,«

D(T)- I-a

:2

(24)

that is function (17) for

13

=~ and ~ --I.

The second conclusion is

PH(a) 0 {a exp[- a2

J}

and PH(a) 0 {exp[- a2

J}

for a400

4(I-a) 4(I-a) (25)

where marginal probability PH follows from the integral with respect to 6

of the joint probability (22), and probability of exceedence Pil proceeds from

PH'

Result (25) is equivalent to

cexp[- a2 ] for a4 00

4( I -a) (26)

where c is a constant to be found, In order to get constant c we come back to

theorem C and the concerned probability P(~I3,T) that the wave elevation is

equal to

~13

at time instant T provided that at time instant 0 there is a

maximum of elevation

13

400, It is given by eq,(18) where function

f(T,O

for

a fixed ~ has its absolute minimum at T' (this being theorem C), Thus, in a

neighbour of T' the probability takes the form

P(~~;T'+6T) 0

{e

x

p{-[f(T',o+~ï(T',06T2]13

2

}}

(27)

where clearly

f

denotes the second derivative of

f

with respect to T, and

ï(T')

is greater than zero, Eq, (27) shows that the order of probability

P

keeps constant in any neighbour 6T of T', of order

₁₃

_-

1 -see Fig,4-, As a

consequence, given a wave crest of elevation

13

4co at instant 0, and given

that the trough elevation is

~13,

with probability approaching 1, the trough

falls at a time instant

T'+6

where 6 is random and has order

13

-

1,

The exact form of conditional probability PTlcnl3l(3) was achieved in the

second part of the demonstration (1989) where the small random difference 6

was taken into account, On ce the solution for PlIcnl3l(3) was computed, the

SEA WAVES ON DEEP WATER

₅₁

same flowP11C .... P1.C .... PII.C and so on -cf. eqs. (20), (21), (22)- was retraced.

The only difference was that they becams true equalities. Ths final result

in the general form for ma not unity, is

P((P;T)

range of order

p

-

I T

o

Tl<

Fig.4 P(~~:T) denotes the probability that the wave elevation at a fixed

instant T be equal to a fixed value ~~, given that at instant 0 there is a

maximum of elevation~. For ~ ....ooand (-I:5~:Sa), P(~(3:T),as a function of T

looks like in the picture.

H I+b

lor

r=-

...."'

.

wilh c>

ymo J2b(l-a) (28)

where b=1 ;V(T·)/~(O)I and a=1II(T·)/1jI(O) as defined at the beginning.

In the case that the frequency spectrum is infinitely narrow, the autocovariance

approaches a eosine so that a ....-I. b ....1and eq. (28) reduces to the well-known simple equation

PII(If)=exp[- H2 ] (29)

8mo

valid for the infinitely narrow frequency spectrum.

The abscissa in Fig.5 is the probability that a given height will be exceeded

by a wave, and the ordinate is the ratio between that height and the height

having the same probability according to classic eq. (29). In other words:

ordinate 1 means that the wave height is equal to that predicted by eq.(29).

The particular coordinate axes were suggested by Forristall (1984) and the

data are taken from his picture and are relevant to his numerical simulations

of the Gaussian process. The curves represent eq.(28) which is valid for

H

/

.[m.

....

00, that is, for Pil .... O. Thus the curves are expected to fit the data

in the left of the figure, and that is what really happens. In particular we

see that for a typical spectrum of the sea waves ( Pierson & Hoskowitz spectrum

) eq.(28) fits the data for PH<O.2 or equivalently for H/.rm-o> 3.5.

52

PAOLO BOCCOTII

LlNES:eQ.(28)exactIerP-o

POINTs:numerical simulationsby Forristell

3 Piereen& Moskowitz spectrum

I

(\

a= -065 ~b~0.40

095

Rectanguier spectrum :iI.~..1.2

!i s: g

'"

!!_a. o 090 s:

'"

,

.

~

a: o 1: .2'

~

_..

085

.

>

..

_J .--0.55 b-0.69

..

.

Rectengular spectrum :..!!.=1.5

Ol.

b_

..

0--0.41

b-0.65

Ol.

P:Probabilily of exceemtoce

Fig.5 The abscissa is the probability PH that a given crest to trough height be exceeded by a wave, and the ordinate is the ratio between that

height and the height ha ving the same probability to be exceeded according to classic eq.(29). The data points are taken from numerical simulations of

Gaussian processes published by Foristall (1984).

5 The wave groups

5.1 A recent theory

For the condition

H

2 (30)

where 10 is an arbitrary time instant, is sufficient and necessary to have a

wave of given height H in the time domain at a given point xo. Yo' It is

sufficient because of theorems A and B: for theorem A, given condition (30),

the wave approaches a well defined deterministic profile, and for theorem B,

the crest-to-trough height of th at profile is H. It is necessary because of

theorem C and eq(22): eq(22) implies that the crest elevation of a wave of

given very large height H approaches ~, and theorem C implies that the time

interval between the crest and the trough of the wave approaches T·.

For theorem A, given condition (30) with ~ --> "", that is equivalent to say

vmo

"given a wave of very large height H at point xo. Yo", with probability

SEA WA YES ON DEEP WATER

₅₃

approaching 1, the wave elevation at given point xo.Yo. at any instant

'0 •

is equal to determinist ic function

n (x y , .T) = 1jI{T)-1jI{T-T·)!!.

"0 O· O' 0 1jI{O)-1jI{T') 2

T

(31)

plus a random noise of order

(fm.

r..r;;;,

.

From the definition of autocovariance,

determinist ic function (31) can also be rewritten in the form

H

TJo{xo•yo"o· T) =

"2

{E X[TJ.{xo• yo.'o)TJ.{xo• yo.'o· T)]·

- E X[TJ.(xo'

v

«. io· T' )TJ.(xo' yo.'o· T)]}/{EX[TJ~(xo' Yo"o)]·

- EX[TJ.(xo•

v

«.'o)TJ.(x •• Yo.

'

0·

T')]} (32)

where EX {.} represents a mean value with respect to the ensemble of the

stochastic process.

Theorem A can be straightforwardly generalized to the form: 'given condition

(30) with

.

~

..

.

....

=.

the probability approaches 1 that the wave elevation and the

velocity potential at any fixed point xo.X.yo.Y and instant 'o.T. are equal

to

- E X[TJ.(xo•

v

«.

'

0

·

T' )TJ.(xo•

x.

Yo· Y.

'

0

·

T)]}/ {EX[TJ~(xo'

v«

.

'0)]·

- E X[TJ.(xo.

v«.

'o)TJ.(xo• Yo.

'

0

·

T')]}

H

41o{xo·X.Yo·Y.z.lo·T)

="2

{EX[TJ.(xo·Yo·'o)41.(xo·X'Yo·Y.z.lo·T].

(33.0)

- EX[TJ.(xo. Yo''0·T')41.(xo•

x

.

Yo· Y. z.' •• T)]}/{EX[TJ~(x •.Yo''0)]·

- E X[TJ.(xo. Yo.'o)TJ.(x •. Yo.

'0·

T')]} (33.b)

plus some lower order random differences'.

The generalization of theorem A to space-time is possible because, in random

wave field (1), the joint probability of the wave e1evation at different

points and the joint probability of the wave elevation and the ve10city

potential are multivariate Gaussian.

Then, since the wave field is stationary, the averages with respect to the

ensemble of the stochast ic process are equal to the avèrages with respect to

time, and thus eq.(33) can be rewritten in the more practical form

H

TJo(xo·X.Yo·Y.'o·T) = "2[<TJ.(xo,Yo")TJ.(xo+X.Yo·Y.'.T».

(34.0)

54

PAOLO BOCCOTII

- <TJ,(xo'Yo'I +T')t ,(x0+X.Yo+Y.z,I +T) >]I[ <TJ!(xo•Yo'I)>+

- <TJ,(xo.Yo.I)'1,(x o.Y»»I +T') >]

where the angle brackets denote a mean with respect to time t

IJ.'

<1(1)> • lirn- 1(1)dl

'C.... "( 0

(34.b)

(35)

and T'is the abscissa of the minimum of the autocovariance function, which

also can be written as a'time-average

~(T)

-<

'l,(xo.yo.I)TJ,(xo.yo.I+T». (36)

H

In conclusion, given condition (30) with {in;....co. that is equivalent to say

given a wave with height

H(

f.,;

-+co) at point xo. Yo.' the generalized theorem A

say. that the wave elevation i. equal to deterministic component 'ID'which is

of order (~)~ plus a random component of order

(

f.,;

r~.

Where determini.tic compönent TJD approaches zero, .ay for large X and/or Y

and/or T, the wave elevation approaches its random component, that i., it

r.turns to be fully random.

In the same way, the velocity potential i. the .um of deteministic compon.nt

+D,

which is of order

(f.,;)Jmotl

and of a random component of order

(;;rJm

otl

mo. being the variance of velocity potential

+•

.

To see the order of

'D'

.q. (34,b)

should be divid.d and multipli.d by

moJmo •

.

with the consequ.nc.

-

--•• ,(xo +X. Yo+Y.z ,I +T)>- <TJ,(xo.Yo' I +T')+,(x 0+X. Yo+Y.z ,I +T)>]1

I[

~:(Xo' Yo'I)> - < ~,(Xo. s«.1)~,(Xo. Yo'I+

T'» ]

(37)

where 11,is the elevation scal.d to its root mean square, and

+,

i. the velocity

potential scal.d to its root mean square.

Result (34), that has been proved for the wave fields where both the fr.e

surface elevation and the velocity potential are stationary random aaussian

proc •••• s of time at any point, is not confined to the classic sea stata (1)

in an open saa. In particular, if we place a vertical reflacting wall along

axis

y-O,

th. random wave field (1) is modified and takes the form

N

'l,(x,y.l) -

2La

,

cos(k,sin9Ix-w,l+el)cos(k,cos9,y)

"1

(38.a)

SEA WAVES ONDEEPWATER

55

of

_Icosh[k,(h+z)]

4>.(x,y,z,l) - 2gL a,w, sin(k,sina,x-w,I+E,)'

'

.

1

coshk,h

.cos(k,cosa,y) (38.b)

where, under the hypotheses of seet. 1 (N4~,E, distributed purely at random in0, 2nandw,~wI ij i~j), both 11.and

4>

.

represent stationary Gaussian processes of time at any point.

Another example is got if we plaee a semi-infinite vertical refleeting wall

along line y=O. Then the random wave field (1) takes the form H TJ,(r. a, I) =

L

a,[F(r, a;w" a,)cos(w,' + E,) +G(r, a;co,;a,) sin (wi' + E,] I·I (39.a)

of

-Icosh[k,(h +z)] 4>.(r,a, z ,I) - gL a,w, k: h [G(r,a;w" a,)cos(w,1 + E,)+

'.1

cosh , - F(r, a;w" a,)sin(w,1 +E,)] where

F(r,a;w,a) - A(ul)cosql+A(u2)cosq2-B(ul)sinq,-B(u2)sinq2

G(r,a;w,a) - A(u,)sinq, +A(u2)sinq2+ B(u,)cosq, +B(u2)cosq2

(39.b) (40.a) (40.b) (41 ) ( 42.a) q, - krcos(a-a'), q2 - krcos(a+a') (42.b)

and rand crare the polar coordinates with the origin at the wall-end,

e

as

usual is the angle of the wave direction with the y-axis, and ~ is equal to

~-

a.

Also eqs (39) are exact to Stokes' first order and are based on the

solution by Penny & price (1952) for the diffraction of the regular periodie waves. Here too, both 11.and

4>

,

represent stationary Gaussian processes of time under the usual hypotheses, and thus result (34) applies also to this random wave field.

It is possible to verify that, provided random wave elevation 11. andvelocity potential

4>

,

satisfy the wave differential equations to Stokes's first order

as well as a set of boundary conditions, also determinist ic wave e1evation

11D and velocity potential 4>D -eq.(34)- satisfy those equations and boundary conditions.

Here we show this property for one of the differential equations, say the

Bernoul1i equation. Then it will be clear that the same way of reasoning can

be applied to the other differential equations as weIl as to the boundary

conditions. We have to show

l(a

4>

,)

pr ouided '1 --- - ---4 '1

0-, g al zoO

(43)

The second equality, which is to be shown, in explicit form is

56 PAOLO BOCCOTfI

(44) where the averages in eq (34.a) of llD and in eq (34.b) of 'D have been put into the explicit form, and the denominator, which is the same both in the equation of TlD and in the equation of

'Do

and which is independent of T, has been simplif\ed. Here, to show the equality, we have only to pospone the derivative to the integral and note that, for hypothesis, the first equality

(43) (the ODa dealing with Tl. and ,,) holds whatever the point and the time instant. Therefore it holds at point xo+X.yo+Y at time instant t + T.

The calculation of determinist ie waves (34) can be done once the spectrum of t~e sea state is given. Indeed the averages in eq.(34) of llD and ~D are relateei< to spectrum S(w.9). In the case of sea states interacting with walls, the .veragee in eq(34) prove to be related to the usual directional spectrum S(w.9) of tne waves that there would be if the wall was not there. As an example let us write the equation for the first average <.> in eq(34.a) in the çase that the sea state interacts with an infinitely long wall. From eq(38.~ of the random wave elevation, we can write

<ll.(xo'Yo.l)ll,(xo+X.yo+Y.I+T»

-. cDS(k,siu9,x- w,1+E,)cos[(kIsinOlx - wi' +EI)+(k IsinOIX -wIT)]dl (45) and then putting the last eosine in the form cos' cos- sin' sin, posponing the integra! to the sum and recalling that w,~ w1if i ~

t,

we get

<lJ,(x0.yo.l)ll,(xo+X.yo+Y.I+T» =

. N 1

- 4

L

-a~cos(k,cos9,Yo)cos[k,cos9,(yo+Y)] cos(k,sin9,X-

w,T)-"12

- 410-

f:

S(w.9) cos(kcos9yo) cos[kcosO(yo+Y)] cos(ksin9X-wT) dOdw (46)

where the second equality proceeds directly from the definition of spectrum - eq(3) - . The integral can be solved numerically, and a convenient way to do this is to put the last eosine in the form coa-cos+strr-utn so that the integral becomes of the type

t"

COSI,uTJ'

f

l(w.9;yo. X. Y)d9dw'"

i"

SinwTf' f2(w.9;yo. X. Y)d9dw (47)

Jo

-

.

J

o

-

a

.where the integrals with respect to

e

are independent of T, and therefore we can considerably reduce the number of operations for calculating the determinist ie waves at different time instants in a point grid. The calculation proves to be expedite because of the quick convergence of the integrals, so that it can be quite euccessfully done by means of personal computers.

SEA WAVES ON DEEP WATER

57

The theory was introduced in two papers (Boccotti, 1988 and 1989) where

more details can be found.

5.2 How a high wave forms in an open eea

Eq.(34.a) shows that the wave of given very large height H at given point xo.Yo within sea state (1) on a open sea, forms because

Fig.6 Typical mode of formation, within a random sea state, of a wave of given very large height H at point Xo.Yo in the center of the framed area. The waves are in a open sea on deep water and the framed area is 6 wave lengths lIJalong x-axis per 8 ld along y-axis. We see that the wave of the given very large height forms because of the transit of a weIl precise wave group.

of the transit of a wave group -Fig.6-. The group has a development stage

during which both the envelope and the wave front narrow till a minimum. Then

a decay stage follows, with the opposite features. Each single wave having a

celerity greater than the group runs along the envelope from the tail where

it is born to the head where it goes to die (in the pictures a single wave

is followed by an arrow during its evolution). Because of this phenomenon,

single waves experience some really big transformations which have been

recently confirmed by an experiment at sea, whose results will be shown at

the 23rd Conference on Coastal Engineering.

The wave of given very large height H proves to be that at the center of

the group at the apex of its development stage. Thus, theory in poor words

says: if you record a wave with a height H very large with respect to the

mean, the probability is very high that it is the central wave of a well

precise wave group at the apex of its development stage.

The wave group of Fig.6 is on deep water and it has been calculated from a

characteristic spectrum that is the mean JONSWAP frequency spectrum (Hasselmann

58

PAOLO BOCCOTII

& al. 1973) with cosZ"C9-90) as spreading direction function; n is taken equa1

to 5 that is a charateristic va1ue, to judge on data of storm seas (Forrista11

&a1.,1980). Here it should be remarked that, if we assume another wind wave

spectrum, only a few details change but the essential features of the wave group mechanics do not change. In particular if the bandwidth grows, e.g. on passing from the JONSWAP to the Pierson &Moskowitz spectrum, only the enveiope narrows; and if the directional spread grows, e.g. on reducing the value of n, only the width of the wave front reduces.

The angle 80, that the dominant direct ion of the spectrum makes with the

y-axis, has been assumed t-o be zero; we see also that the wave group moves along the y-axis. This means that avery high wave, with a very high probability,

belongs to a group which moves along the dominant direction of the spectrum.

5.3 How a high wave forms at a wall

The question is 'how does a wave with a given very large height H form at a point Xo.Yo in contact with the wall? " and Fig.7 is the answer given by eq. (34.a).

o

2

Fig.7 Typical mode of formation of a wave of given very large height H at the breakwater. The wall is along the upper x-parallel side and point xo. Yo

of the wave of given height H is at the wall center. The framed area is 8 wave lengths Ld along x-axis per 6 Ld along y-axis, the dominant direction of the sea state makes a 20' ang1e with the wa1l-orthogona1; as a consequence we see the wave group that approaches the wa11 from the 1eft and then is ref1ected mirrorwise.

SEA WA YESONDEEP WA TER ₅₉

The wall is along the upper x-parallel side of the framed area and given

point Xo.Yo is at the center of the wall. The waves naw are observed from

offshore. The spectrum is the same as for the foregoing pictures, and its

dominant direct ion makes a 200 angle with the wall-orthogonal. The water is

deep.

The first four pictures taken at regular intervals of 2Td(Td- peak period

of the spectrum) from each other show a wave group approching the wall. At

time instant to (0 in the picture) the wave of the given very large height H

is forming at the wall. It is the result af the reflection of the central

wave af the graup. Then, in the last three pictures, the wave group being

reflected mirrorwise goes back seaward.

It will have been nated that, meanwhile the wave group is appraaching the

wall bath the envelope and wave front narrow, and the reverse is the case

when the group goes back. This reveals that the group reaches the wall at the

apex af its development stage when its central wave obtains its maximum height.

Thus the answer of the theory, in words, is 'a very high wave at the wall,

with a very high probability, takes place because a weIl defined wave group

hits the wall when it is at the apex af its development'.

5,4 How a high wave forms off a wall

~.10

~

----

~

.

~

Fig.8 Typical mode.of formation of a wave of given very large height H at a point xo.Yo two wave lengths far off a breakwater. We see two wave groups

that approach the wall: the first group af ter having being reflected collides the second group that approaches the wall (the particle velocity

before of the collision is shown in Fig.9). The wave of given very large height H forms where the central waves of the two groups overlap ..

60 PAOLO BOCCOITI

Now the question is 'how does a wave with a given very large height H form at a point xo.Yo two wave lengths before the wall?', and Fig.8 is the answer given by eq.(36.a). This time, the dominant direct ion of the spectrum is wall-orthogonal.

At instant '0-lOT., two wave groups are distinguishable: the center of the

first is nearly 3L

d (Ld-~

:

)

off the wall, the center of the second group is nearly 7 La from the wall. The tail of the second group is outside of the frame. The two groups approaching the wall appear more clearly in the next two pictures at instants '0- 8Td and '0- 6Te- The fourth picture, time instant

10-4T.,shows the central wave of the first group reflected by the wall. The

group at that time is still building up so that the wave height at the wall is not so large as in Fig.7.

The fifth picture, time instant '0-2Td, shows a complex scene: at the wall, the re ar wave of the first group is being reflected; the next wave (A) is the central wave of the first group which is regaining the sea; wave B is a standing wave generated by the head wave of the first group advancing seaward and by the head wave of the second group advancing landward; wave C is progressive like wave A but travels in the opposite direction: it is the central wave of the second group, which advances landward; finally wave D is the rear wave of the second group, which likewise travels towards the breakwater . Fig.9 provides a readily understood confirmation of the nature of waves A,B,C and D. Indeed it shows the horizontal component of the particle velocity calculated from the velocity potential -eq.(34.b)-. It will be seen that under wave crest A, the horizontal velocity vy is negative, that is seaward oriented;

under wave crest B, vy is practically zero, which reveals a peculiar characteristic of the standing waves; under wave crests C and D,vy is positive,

that is landward oriented. In conclusion, the scene at time instant '0-2T.

is that of two wave groups approaching collision.

A B

c

o

-h

.L__!_L_I

.

Fig.9 Horizontal partiele velocity below the crests of waves A,B,C,D in Fig.8 at instant 10-2Ta.

SEA WAVES ON DEEP WA TER

₆₁

Fig.l0 Typical mode of formation of a wave of given very large height H at

a point in contact with the inward face of the wall, 1 wave length from the

wall-end (the wall is parallel to x-axis and the wall-end is at the center

of the framed area). The given height H is meant to be very large with

respect to the mean wave height at the given point behind the wall. We see

a wave group whose front-center impacts the wall-end.

Two wave periods later, at time instant 10 (0 in the picture), the collision is at its climax. The two central waves are overlapping and the wave of the given very large height H is forming at given point xo.Yo, that is the point at the center of the wave front below the maximum crest. What is shown at time instant 10 is an offshore standing wave field being generated by two

62 _{PAOLO BOCCOTTI}

opposing wave groups. Indeed, not only the centra 1 waves of the two groups are overlapping, but also the head wave of the first group and the rear wave of the second, and the rear wave of the first group and the head wave of the second are overlapping.

5.5 How a high wave farms in the shadow cone behind a wall

Here the question is 'how does it form a wave with a given very large height H at a point in contact with the inward face of the wall? The coordinates of the point are ro = ld' ao=O. Fig.l0 is the answer given by eq.(34.a) for the usual spectrum and deep water. Here too the wall is not shown, but it is easily recognizable because clearly along its line the water surface is discontinuous. We see a wave group that approaches the wall and its front center targets on the wall-end. Thus one half of the wave front goes beyond the breakwater and penetrates into the protected area, and the other half impacts the wall and is raised because of reflection. In line with the above mentioned question, at time instant 10 (0 in the picture) at given point ro.ao there is the crest of the wave of given very large heigth H. It is the first diffracted wave crest, beyond the wall-end.

The given very large wave height H in the figures of this paper is thought of as the maximum expected wave height during a sea state at given point xo. Yo. For this reason wave height H in Fig .10 is smaller than in the foregoing figures.

Some more pictures can be found in the papers by Boccotti (1988, 1989).

The predictions on ref 1ection and diffraction of the wave groups were

verified at the maritime laboratory of the Reggio Ca1abria University. To

that end, a special ref1ecting breakwater of 12m x 2.1m was built in front

of the city beach, on I.Sm of bottom depth, and 30 wave gauges were p1aced

before the wa11. A first communication on the resu1ts was given to the Journées

Nationales Genie Cotier & Genie Civi1, Nantes, 1992.

SEA WAVES ON DEEP WATER

₆

₃

Appendix A : deduction of eq,(17)

The conditional probability of

n.

at instant T given condition (16) is

p['1,(T) - '11 '1,(O)-I3.'1,(T·) - ~13]-~eXP[J('1)] (48.a)

where

J('1) - -2~ (M11132+M 22~2132+M33'12+2M 12~132+2M _1313'1+2M 23~nl3) +

(48.b)

and M 'I and 11 are respectively the i, j cofactor and the determinant of the

covariance matrix of '1,(0). '1,(T'). '1,(T) which is

a I

1jI(T-T')

and 11'1 and 11are respectively the i,j cofactor and the determinant of the covariance matrix of '1,(O).'1,(T') that we get by e1iminating the 3rd rowand

the 3rd column of the matrix, Function J('1) has a maximum for

M13+M23~ [-a1jl(T - T')+ 1jI(T)] '1m~ - 13~{ 2 + M33 I- a [1jI(T -T')- a1l'(T)]~ + }'13 (49) 1-a2

and, in terms of '1m. J('1) can be rewritten in the form

1.133 2

{('1) - J('1..,)- 21.1('1- '1..,) (50)

The maximum {('1..,) proves to be 0 (that can be verified if cofactors M" and

determinant N1 are put in their explicit form),

Therefore the conditional probability takes the form

p(·I.. ·)- ;:(~~exP[-~;('1-'1m)2J (51)

where the identity 1I- 1.133has been applied in the root mean square,

The conditional probability is Gaussian with mean value '1m and varianee

1.1/1.133' The mean value depends on the given elevation (3, the variance does not depend on (3,

64 PAOLO BOCCOlTI

Appendix B : deduction of eq.(18)

The probability P(~~;T)is the ratio of the two probabilities of the Rice type defined after the statement of theorem c. lts general form is

P(~~;T)=p(Tlo=~.1ÎO=O.TlT=~~) f:P(~o=WITlo=~.tlO=O.TlT=~~)IWldW/

Ip( Tlo= ~.1Îo= 0)

f~

p( ~o =W I Tlo=~.1Îo =0) I WIdW (52) where TIris for Tl,(T), pC.·) is a joint probability density function and p('I') is a conditional probability density function. The probability functions in eq.(52) are Gaussian and depend on the covariance matrix of Tl.(O).tl,(O).D,(O)

and TlR(T). The analytical solution for the denominator of eq.(52) was given by Rice (1944), and , if ~-4ao, the order of the denominator proves to be

{

~

ex

p

(

_

~~

2)}

.

As to the numerator of eq. (52), the following two theorems permit a simplification.

TJ.(O) -~. Tl,(O) - O. TJ,(T) - ~~

with ~-4ao and T and ~ arbitrary, the probability approaches 1 that

(53)

Tbeorem C.l: given the condition

Tl.(O) =

-g

e

T.~)~

with

[I

+1jI(T)~(T)_~2(T)]-[1jI(T)+ ~(T)h

g(T.~)= .

1-1jI2(T)-1jI2(T)

plus a random difference of order ~o.

(54.a)

(54.b)

Theorem C.2 : the function g(T.O -eq.(54.b)- is greater than zero for T~O

and -I:S ~:S I.

For the two theorems, the probability p(~o-WITJo-~'~O-O.TlT-~~) is like in

Fig.ll, that is, it concentrates at the negative value -g(T.~)~.Thus the

integral in the numerator of eq. (52) is equal simply to g(T.~)~, and form (18) of probability P(~~;T)can be proved straightforwardly.

SEA WAYES ONDEEP WATER

65

o

_w

-g(T.O~

Fig.ll The probability density function of second derivative ~R at instant 0, given that Th(O)=~, it.(O)=O and TJ.(T)=~~, lor (3-HO, T>O, -1~~~1.

The same reasoning made for theorem A can be repeated also to show theorem

C.1: the probability of TJ.(O), given condition (53), is Gaussian, its mean value depends on given elevation (3, and its variance does not depend on (3.

To show theorem C.2, the only case to be considered is that 1jI(T)+1jI(T) ..O,

else the theorem is trivially proved. In that case a particular value ~p of

~ exists for which g(T,~) is zero

~p-[I+",(T)~(T)-~2(T)]/[1jI(T)+ ~(T)] (55)

The strategy to prove theorem C.2 is

(i) to show that l~pl>1 whatever the T, which implies that

g(T,~) is greater than zero only or smaller than zero only, in the range

-1~~~1.

(ii) to show that, whatever the T, at least one value of ~ exists in the range

[-1.1] for which g(T.~»O.

As to (i), I~pl>l is equivalent ~~> 1, and this in its turn is equivalent

(56)

whieh is satisfied beeause the first member proves to be equal to the determinant

.

.

of a eovarianee matrix: the matrix of TJ.(O), TJ.(O), TJ.(T), TJ.(T). As to (ii),

a partieular value of ~ in the range (-1,1], for whieh g(T,~»O whatever the

T, is ~ -1jI(T) (here it should be reealled that -1 ~ 1jI(T) ~1 beeause of the

assumption ma - 1) •

Appendix C: the inequality dealin9 with funetion l(T,~)-eq(18)-.

We have to prove that

[1jI(T)-~f [1jI(T')_~]2

_-=..;c...:...--'..__:c.::.__ > 2'

1-1jI2(T)-~2(T) l-1jI(T)

lor T>O and "T' and -1~~~a·1jI(T') (57)

66 PAOLO BOCCOlTI

and since 1-1jIz(T)-1jIZ(T) is greater than zero (it is the determinant of the

covariance matrix of D,(O).D,(O).D,(T» it is sufficient that we prove

(58) Fig.l2 shows the two members of inequality (58) to be proved, as functions of ~ for a fixed T. Since 1jI(Te) is defined as the absolute minimum of the

autocovariance function, we have that 1jI(T) >1II(Te) and consequently the origin

of the first parabola, that is the first member of the inequality, falls at the right of the origin of the second parabola, like in the picture.

1

r

Fig.12 First member (MI) and second member (Mz) of inequality (58), for a

fixed T, as functions of ~.

Therefore the first parabola exceeds the second in the concerned range

-I

:s ~ :s

a, if the first parabola exceeds the second at ~ --I, that is if

[I+ 1jI(T)]2 [I+ 1II(T')]z

> (59)

1-1jI2(T) 1-1jI2(T')

which is satisfied because -I<1jI(T') <1jI(T) < I.

SEA WA YES ON DEEP WATER

67

Main symbols

a ratio of the minimum and the maximum of the autocovariance function ai amplitude of the ith email wave forming a sea state

b ratio (absolute value) of the curvature of the minimum and the curvature of the maximum of the autocovariance function

E frequency spectrum

H crest-to-trough wave height h water depth

k wave number L wave length

Ld dominant wave length

mI jth order moment of the frequency spectrum

m..

variance of the velocity potential at a fixed depth P probability

PH probability of exceedence of the wave height p probability density function

S directional spectrum T time interval

Td dominant wave period

T* abscissa of the absolute minimum of the autocovariance function t time

x-y horizontal orthogonal axes

z vertical upward axis with the origin at the mean water level ~ water surface displacement

e

angle that the wave direct ion makes with y-axis ~ velocity potential

~ autocovariance function vu angular frequency

68

PAOLO BOCCOITI

References

Boccotti P., On the highe st waves in stationary Gaussian processes, Atti

Acc. Ligure, 38, 1981

Boccotti P., Sea waves and quasi-determinism of rare events in random

processes, Atti Acc. Naz. Lincei, Rendiconti, 76, 2, 1984

Boccotti p.,Refraction, reflection and diffraction of irregular gravity

waves, in Excerpta of the Italian contribution to the field of hydraulic

engineering, vol.3, 47-89, Libreria Progetto padova Publ., 1988

Boccotti P.,On mechanics of irregular gravity waves, Atti Acc. Naz. Lincei,

Memorie, 19, 5, 111-170, 1989

Bretschneider C.L., Wave variability and wave spectra for wind-generated

waves, US Army Corps of Engr., BEB, Tech. Memo., 1959

Cartwright D.E. and Longuet-Higgins M.S., The statistical distribution of

the maxima of a random process, Proc. Roy. Soc. London, Ser. A,237, 1956

Forristal1 G.Z. et al., Directional wave spectra in hurricane Carmen and

E10ise, Proc. Conf. on Coastal Engng., ASCE, 1980

Forristall G.Z., The distribution of measured and simulated heights as a

function of spectra shape, J. Geoph. Res., 89, 1984

Hasselmann K. et. al., Measurements of wind wave growth and swell decay

during the North Sea Wave Project (JONSWAP), Deut. Hydrogr. Zeit., A-8, 1973

Liu P., A representation for the frequency spectrum of wind-generated waves,

Ocean Engng., 10, 1983

Longuet-Higgins M.S., The distribution of intervals between zeros of a

stationary random function, Phil. Trans., Roy. Soc. London, Ser. A., 254,

1962

Longuet-Higgins M.S., The effects of non linearities on statistical

distributions in the theory of sea waves, J. Fluid Mech., 17, 1963

Mitsuyasu H. et al., Observation of directional spectrum of ocean waves

using a clover-leaf buoy, J. Phys. Oceanography, 5, 1975

Ochi M. K., Hubble E.B., Six-parameter wave spectra, Proc 15th Conf. on

Coastal Engng., ASCE, 1976

Osborne A.R., The simulation and measurement of random ocean wave statistics,

in Topics in Ocean Physics, North Holland, Amsterdam, 1980

Penny W.G. and price A.T., The diffraction of sea waves by breakwater, Phil.

Trans. Roy. Soc., A-244, 1952

Phillips O.M., The theory of wind generated waves, Adv. in Hydroscience,

4, 119-149, 1967

Pierson W.J. and Moskowitz L., A proposed spectral form for fully developed

wind seas based on the similarity theory of S.A. Kitaigorodskii, J. Geoph.

Res., 69, 1964

Rice S.O., The mathematical analysis of random noise, Bell Syst. Tech. J.,

24, 1944

Rice S.O., Distribution of the duration of fades in radio transmission:

Gaussian noise model, Bell Syst. Tech. J., 37, 1958