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A general theory of three-dimensional wave groups Part I: The formal derivation
Ocean Engng, Vol. 24, No. 3, pp. 265-280, 1997 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All fights reserved 0029-8018/97 $1%00 + 0.00
Pergamon S0029--8018(96)00013.3
A GENERAL THEORY OF THREE-DIMENSIONAL WAVE GROUPS PART I: THE FORMAL DERIVATION Paolo Boccotti Department of Fluid Mechanics and Offshore Engineering, University of Reggio, Calabria, Italy.
(Received 21 November 1995; accepted in final form 31 January 1996) A b s l ~ e t - - I f we know that a wave of given exceptionally large crest-to-~ough height occurs at a fixed point x_oat an instant to in a random wind-generated sea state, we can predict what happens with a very high probability before and after to in an area surrounding xo. The expressions of the surface displacement and velocity potential in this area are obtained in closed form. They are exact to the first order in a Stokes expansion and hold for nearly arbitrary bandwidth and solid boundary. It will be shown in Part II that these expressions represent either the evolution of a single threedimensional wave group or the collision of two wave groups, according to the configuration of the solid boundary. The theory was developed in a series of papers starting on 1981. This paper presents the whole theory in a compact form thanks to a radical simplification of the mathematical proof. Copyright © 1996 Elsevier Science Ltd
NOMENCLATURE
A,B,C,D,
determinants of covariance matrices
M
Ao, Bo, Co, cofactors of covariance matrices D O, Mij expected number per unit time EX H crest-to-trough wave height jth moment of the frequency spectrum about the origin wave length probability density function probability, probability of exceedance time time lag 1" abscissa of the absolute m i n i m u m of the autocovariance ~ T ) dummy variables U, W x_ =(x,y) point in the horizontal plane x~ =(xo,Yo) point where a wave of given very large height occurs x_ =(x,r) relative location of a point with respect to x_o vertical coordinate Z mj
L P P t T
or=H/o" '1 o~ T
4,
surface displacement quotient of the crest elevation and the crest-to-trough wave height variance of the random surface displacement at x__o variance of the random surface displacement at x_o + _X time lag between the crest and trough of a wave velocity potential covariance of the surface displacement and the velocity potential 265
266
Paolo Boccotti one-dimensional autocovariance of the surface displacement space-time autocovariance of the surface displacement 1. INTRODUCTION
A steady wind wave field can be thought of as the sum of a very large number of small periodic waves with frequencies and directions such as to form some characteristic twodimensional spectra, and with uniformly distributed random phase angles (Longuet-Higgins, 1963; Phillips, 1967). According to this theory, both the surface displacement and the velocity potential, to the first order in a Stokes expansion, represent stationary Gaussian processes of time at any fixed point. This paper deals with the conditional probability of the surface displacement if it is known that a zero up-crossing wave with an exceptionally large crest-to-trough height H occurs at a fixed point x~, at a given time instant. The mathematical analysis is based on the general techniques set up by Rice (Rice, 1944; Rice, 1945; Rice, 1958). The assumption is made that H/tr---~oo, which is not necessarily inconsistent with Stokes' assumption that H/d---+O and H/L---+O (or being the root mean square surface displacement of the random wave field as a whole; d the water depth; L the wave length). Under this assumption, the probability approaches 1 that the surface displacement surrounding x o approaches a well-defined deterministic configuration in space and time. Really, for a wave with a realistically large height H, say H = 8+9o-, the residual random noise will be not negligible. Moreover, in nature, this wave may be spilling or almost breaking near the crest and this will produce a localized profile distortion. But the overall configuration in space and time of the three-dimensional wave group including this wave should be close to the deterministic configuration obtained in the limit H&r ~ oo, which will be confirmed in Part II by the results of a field experiment. This theory was introduced bit by bit by the writer in a series of papers (Boccotti, 1981; Boccotti, 1982; Boccotti, 1989). Here, a radical simplification is achieved thanks to the new proof given in section 2 and section 3. Specifically, it is proved that a special condition becomes necessary and sufficient for the occurrence of a wave of given height H, as H/o.--,oo. This condition is 1 1 ~l(to) = ~ H, rl(to + T*) = - ~ H,
(1)
where ,/(t) is the surface displacement at a fixed point x_o ; to is an arbitrary instant; T" is equal to the abscissa of the absolute minimum of the autocovariance function $, which is assumed to be also the first minimum on the positive domain of this function. This will be referred to hereafter as 'condition (1)'. The proof that condition (1) becomes necessary and sufficient for the occurrence of a wave of given height H, as H/o,--.oo, permits to substitute the condition 'given that a zero up-crossing wave of crest-to-trough height H occurs at a fixed point x~' by the much more simple condition 'given (1) at x~'. Hence, the theory can be proved in only a few steps. In the above cited papers (Boccotti, 1981; Boccotti, 1982; Boccotti, 1989) this theory was carried out together with the closed form solution for the wave height probability, under general bandwidth assumptions (in the limit HIo.--.o*). Here, the expression of the wave height probability is reobtalned as a corollary of the formal proof that condition (1) is necessary. This expression proves to fit very well Forristall's data of simulated heights
General theory of three-dimensional wave groups: I
267
(Forristall, 1984) even for H/o->4, and hence we get the first evidence that the hypothesis of H/o'--*~ can be succesful. 2.
CONDITION (1) IS SUFFICIENT FOR THE OCCURRENCE OF A WAVE OF GIVEN VERY LARGE HEIGHT H
Let us consider the surface displacement r/(t) at a fixed point x~ under the assumption that */(t) is a stationary Gaussian random process of time (theory of the wind waves to the first order in a Stokes expansion). Let us examine the probability density function (pdf) of this surface displacement at any fixed time instant, given condition (1). For any fixed time lag T we have
p *~(to + 13 = ul*/(to) = ~ H, *~(to + T ~) = - ~ H = A33
27rA
A23-A13
• exp - ~ -
u
A33
2
'
where Aij and A are, respectively, the i, j cofactor and the determinant of the covariance matrix of the random variables */(to), */(to + T~), */(to + 13: A 0- = i, j cofactor, l ~ T") A = determinant of \ ~ ( T ) qJ(T- T')
(3)
D
gJ(O)
Equation (2) implies that the expected value of *~(to + 7) given condition (1) is A23-A13 H ~ T ) - ~ T - T ~) H A33 2 - ~b(0)-~b(T~) 2-"
~l(to + T) -
(4)
Moreover, Equation (2) implies that the standard deviation of *~(to + 13 given condition
(1) is
J
A A33 -
J
[ qJ(0) 1 -
q?(T) + tka(T-T*)-20(13~T-T~)[~T~)/~O)]] ~b2(0)_qj2(T, )
,
(5)
which is smaller than or = [~(0)] 1'2,
(6)
since I~T')/~0)I is smaller than unity. Therefore, the quotient of the standard deviation and the expected value of *~(to + 13 given condition (1), generally, approaches 0 as H/o ~-* ~, and hence *~(to + 13 is asymptotically equal to ~(to + 13 as H/o ~-,o~. This result prescinds from the definition of 7"; that is, 7" can be any given time lag. The particularity of 7" will be used in the following analysis of the expected surface displacement h as a function of the time lag T. Since ~ 0 ) and ~ T ' ) = ~ ( - T ~) are, respectively, the absolute maximum and the absolute 1
minimum of ~13, ~ has its absolute maximum (equal to ~ H) at T -- 0 and its absolute
268
Paolo Boccotti
1 minimum (equal to - ~ H) at T = T*, and hence at T = 0 there is a wave crest and at T
= T' there is a wave trough. Since 7" is the abscissa of the first minimum of ~ T ) after T = 0, the derivative ~(T) is negative on (0, T'): ~(T)<0 on (0,T'),
(7)
which implies (since qJ is an even function)
~ T ) > 0 on
(-T',0),
(8)
or, alternatively,
~(T - 1") > 0 on (0,T').
(9)
The inequalities (7) and (9) imply that ~/is strictly decreasing for T in [0,T']. Therefore the wave crest at T = 0 and the wave trough at T = 7"* belong to the same zero upcrossing wave, and this wave has a crest-to-trough height equal to H - see Fig. 1. Hence the conclusion: as H/cr---}~, condition (1) becomes sufficient for the occurrence of a wave of given height H. 3.
CONDITION (1) IS NECESSARY FOR THE OCCURRENCE O F A WAVE OF GIVEN VERY LARGE HEIGHT H
A general necessary condition for the occurrence of a wave of given height H is that the surface displacement is equal to ~ r / ( 0 < ~ < 1 ) at an instant to (wave crest) and that it is equal to (~ - 1 ) H at a later instant to + T (wave trough):
rl(to) = (;H, ~q(to + "r) = ( ~ - l ) n ,
0<~<1.
(10)
the derivative is always negative on this interval absolute maximum
~H
,
/
T*
Fig. 1. Generalfeaturesof function (4). The actual surfacedisplacementgiven the condition(1) is asymptotically equal to this function as H/o,--~oo.
General theory of three-dimensional wave groups: I
269
To fix the ideas, Fig. 2 shows a few waves with a fixed height H and different values of ~ and ~-. Let us consider the probability that the surface displacement falls between ~ and + d~/l at an instant to, and that it falls between (~ - 1 ) H and (~ - 1 ) H + d~72 at a later instant to + "r;, to, H, "r, and ~ being arbitrarily fixed values, and dr/l, d7/2 being fixed intervals of small amplitude. It is given by
1 exp 2 I r [ ~ ( 0 ) - ~(1-)1 u2
P(H,'r,~ =
1
[ H \ z] ]d'thd'tl2,
~f('r,O~)
(lla)
(
(llb)
where I(r,
9 - 2
+ 2 q,(0) +
Now, let us examine this probability as a function of "r and ~, as I-IIo'--*~. The first term on the RHS of Equation (1 lb) is independent of ~ and its absolute minimum occurs at ~= 7~ because the absolute minimum of q4~') on (0,oo) has been assumed to occur at ~-= ~ . The second term on the RHS of Equation ( l l b ) is zero when ~ = 1/2 and is greater than zero when ~:~ 1/2. Therefore, the absolute minimum of f(~',0 for I" in (0,oo) and ~ in 1 ( - ~, + m) occurs at ~-= T', ~ = ~ , which implies
P(H,~',OIP H , T ~,
---,0 as -----,~
(12)
or
1 for every fixed pair (z, ~) such that ~'¢7 ~ and/or ~4= ~ . This suggests that a wave with a given crest-to-trough height H necessarily has ~- = 7~ 1 and ~ = ~ , as HIo'---~. In other words,
as I-I/o'---,~, condition (1) becomes not only sufficient but also necessary for the occurrence of a wave of given height H.
I,
T2
i i
I
7"1
%
,i
I
Fig. 2. Waves with a fixed height H and different values of ~ (the quotient of the crest elevation and the wave height) and ~"(the time lag between crest and trough).
270
Paolo Boccotti
A formal proof of this property is given in the Appendix through an analysis of the expected number per unit time of local wave maxima of fixed elevation ~H which are followed by a local wave minimum of elevation (~ - 1 ) H after a fixed time lag r. The above mentioned closed solution for the probability of the wave heights will be a corollary of this formal proof. 4.
THE EXPECTED CONFIGURATION OF THE WATER SURFACE IN SPACE AND TIME IF A WAVE OF GIVEN VERY LARGE HEIGHT OCCURS AT A FIXED LOCATION
If a wave with a given very large crest-to-trough height H occurs at a fixed point ~ , what is the expected configuration in space and time of the water surface surrounding X_o? From section 2 and section 3, to say 'a wave with a crest-to-trough height H occurs at 1 a point xj , as H/o'--*~, is equivalent to say 'the surface displacement at x~ is ~ H at some 1 instant to and is - ~ H at the instant to + T " . Then, in order to answer the question put above, we examine the pdf of the surface displacement at any fixed point x~ + X and instant to + T, given the condition 1 1 "O(x~,to) = ~ H, "O(x~,to + T') = - ~ H,
(13)
(this is the same as condition (1), but we write ag(X_o,to)instead of ~l(to) because now we deal with waves in the space-time). The pdf under examination can be written in the form
[
,]
1
p rlX(~o + X, to + T) = ulrl(x~,to) = ~ H,'oX(yo,to + T*) = - ~ H
B•33
=~/2~'B
exp -
~
u
~
2-
(14)
'
in which B 0 and B are, respectively, the i, j cofactor and the determinant of the covariance matrix of r/(x~, to), ~j(Xo, to + T*),~(x~ + X, to + T) •
o~ B o = cofactor, B = determinant of
q~(_0,T~)
q~(_X,T)
~(0,T") oa ~(X,T-T*)|. l ~(X_,T) ~ ( X , T - T') -O~x /
(15)
In this matrix, • is the two-dimensional autocovariance defined by ~(X,T) = ,
(16)
and oa and O'Zxare the variances of the surface displacement at the points x__oand X_o+ X, respectively. Equation (14) implies that the expected value of ~/(X__o+ X, to + T) given condition (13) is
General theory of three-dimensional wave groups: I
~ ( ~ + X_,to + T) -
271
B23-B13 H qt(X_,T)-qt(~_,T-T') H B 3 ~ 2 - qt(O,0)-qt(_0_0,2W) 2-"
(17)
Moreover, Equation (14) implies that the standard deviation of ~ x(8_o + X_,to + T) given condition (13) is JO2x - o ~ qt2(-X'T) + *2(X_,T-T')-2*(X_,T)*(X_,T-T')[*(O_,T~)/*(Q,O)] =
_
q,2(0_,0)_q~2(0,T,
)
,
(18)
which is smaller than O2x since IW(9_,T')/W(_0_,0)I<1. Let us express the expected value (17) in terms of the dimensionless autocovariance fft(X,T) = ~(_X,T)/o-o'x,
(19)
whose value can range between 1 and - 1 in consequence only of the relative phase of (X_o,t) and (x~ + X,t + T). We have 1 fft(X_,T)-fft(~_,T- T') H ~l(X_o+ X, to + T) = ~ 1-ffr(O_,T') ~rx_o.--,
(20)
which makes it evident that the quotient of the standard deviation and the expected value of ~(xo+X_,to + T) given condition (13), generally, approaches 0 as H/o,--,oo (since the standard deviation is smaller than orx), and hence rlx(_~o+ X_,to + T) is asymptotically equal to ~l(x__o+ X, to + T) as HIo,--.oo. Part II of this paper will show that ~/, as a function of _Xand T, represents either the evolution of a single three-dimensional wave group or the collision of two wave groups, according to the shape of the solid boundary. 5.
THE EXPECTED DISTRIBUTION OF VELOCITY POTENTIAL IF A WAVE OF GIVEN VERY LARGE HEIGHT OCCURS AT A FIXED LOCATION
Associated with the configuration (17) is a distribution of velocity potential in the water, which to the lowest order in a Stokes expansion is given by
qb(x_o+ X_,z,to + T) = ~- [
qt(0,0)_W(Q,T. )
j,
(21)
where X ,z and T are the independent variables and ~ is the cross-covariance of the surface displacement and the velocity potential of the random wave field:
dp(X,z,T) = <~7(x_o,t)~b(xo+ X,z,t + T) > .
(22)
The fact that the velocity potential associated with the surface displacement (17) is given by Equation (21) can be proved under the assumption that ~ and qb are solutions to the linear problem. Particularly, here we shall prove the equality
272
Paolo Boccotti
From Equation (17) and Equation (21) of ~1 and ~b, and definitions (16) and (22) of the covariances, this equality takes the form H <~/(x__o,t)7](x~+ X,t + T) > - <~/X(~o,t)~/(x_o+ X,t + T-T") > 2
- 1Hb
-
g 2 bT
(24)
• ( < r l ( x ~ ' t ) ~ b ( x - ° + X - ' z ' t +>T )~> -~< r+l ( -x ~~' t )rc ~' ( x- - °)+~X - ' z ' t + T - T * ) >
z=o'
where the common factor
- cancels since it does not depend on T. With the temporal means in the explicit form, equality (24) becomes At ~ ~ At Jo r/(x_o,t)[7/(X__o+ X,t + T)-rl(x_o + X,t + T - T " ) ] d t
1
=
lim
At
(x__o,t)
(
+ X_,z,t + T)
(25)
At---* - ¢hx(~ + X,z,t + T - T " ) ] z = o dt, and is proved since rl(x o + X,t + T) -
g ~1. dp(X_o+ X,z,t + T) z =o,
rl(x~ + X,t + T - T " ) -
g ~TCl~_o + X,z,t + T-T")z=O,
(26) (27)
(these equations being a direct consequence of the assumption on ~ and ¢h)The proof that Equation (17) and Equation (21) are solutions to the linear problem can be completed in a similar way. 6. THE THEORY HOLDS FOR NEARLY ARBITRARY SOLID BOUNDARY AND BANDWIDTH We have assumed that the surface displacement represents a stationary Gaussian process of time at any fixed location, but we have not assumed that the random process be homogeneous in space. Specifically, the variance o-2_xof the random surface displacement at + X has been assumed to be generally different from the variance oa at x~, which happens if some solid obstacle causes wave reflection and/or diffraction, or the sea bottom causes wave refraction• As a confirmation that the theory holds also in these cases, we can verify that expression (Equation (21)) of (b automatically fulfils any given solid boundary condition. To this end, let us assume that a point (X',Z') belongs to a solid surface and that n is the unit normal vector to the surface at this point• We wish to show that
273
General theory of three-dimensional wave groups: I
(28)
n . V ~ = 0 at X = x ' - X o , z = z', that is, from E q u a t i o n (21) o f ~,
{
- < n ( ~ , t ) ~ ( x ~
n.V
Z < ~ r / ~ +
+ X,z,t + T - T ~) > 7") >
H} = 0,
(29)
atX=
x " -x~, Z : z •
Since _X and z are the only i n d e p e n d e n t space variables, this can be rewritten in the form - < r / ( x ~ , t ) n . V ~ b ( ~ + X,z,t + T - T •) > = O,
(30) at X_ = x_"--x~, Z = Z, which shows the equality to h o l d under the a s s u m p t i o n that the velocity potential o f the r a n d o m w a v e s satisfies the given solid b o u n d a r y condition at (x',z'). Finally, we w o u l d e m p h a s i z e that the theory is general also as to the w a v e spectrum. Indeed, the o n l y restriction is that ~ T ) (the F o u r i e r transform o f the frequency spectrum) m u s t have an absolute m i n i m u m , w h i c h m u s t be the first m i n i m u m after T = 0. In the course o f several field m e a s u r e m e n t s (which will be quoted in Part II), the writer f o u n d this a s s u m p t i o n to be not satisfied only in a few special cases in w h i c h the spectrum was very b r o a d b e c a u s e o f the o v e r l a p o f w i n d w a v e s and swells o f nearly the s a m e height. REFERENCES Boccotti, P. 1981. On the highest waves in a stationary Gaussian process. Atti Acc. Ligure 38, 271-302. Boccotti, P. (1982) On the highest sea waves. Report Istituto Idranlica Universith di Genova, 1-160. Boccotti, P. 1989. On mechanics of irregular gravity waves. Atti Acc. Naz. Lincei, Memorie, VIII 19, 111-170. Cartwright, D.E. and Longuet-Higgins, M.S. 1956. The statistical distributions of the maxima of a random process. Proc. Roy. Soc A-237, 212-232. Forristall, G.Z. 1984. The distribution of measured and simulated height as a function of spectra shape. J. Geoph. Res 89, 10547-10552. Longuet-Higgins, M.S. 1963. The effect of non-linearities on statistical distributions in the theory of sea waves. J. Fluid Mech 17, 459-480. Longuet-Higgins, M.S. 1980. On the distribution of the heights of sea waves: some effects of nonlinearity and finite bandwidth. J. Geoph. Res 85, 1519-1523. Phillips, O.M. 1967. The theory of wind generated waves. Adv. in Hydroscience 4, 119-149. Rice, S.O. 1944. Mathematical analysis of random noise. Bell Syst. Tech. J 23, 282-332. Rice, S.O. 1945. Mathematical analysis of random noise. Bell Syst. Tech. J 24, 46-156. Rice, S.O. 1958. Distribution of the duration of fades in radio transmission: Gaussian noise model. Bell Syst. Tech. J 37, 581-635. APPENDIX F o r m a l p r o o f that condition (1) is necessary, a n d solution f o r the probability o f the w a v e heights under general bandwidth assumptions Definition and equation o f the expected number per unit time EX(a,'c,~) The general assumptions about the existence of the derivatives of the autocovariance ~ T ) are made as in the well-known paper of Rice (Rice, 1944; Rice, 1945). The compact symbols *b and ~br are used in place of ~/(t) and ~T). Without loss of generality, both the zeroth and the second moment o f the frequency spectrum about the origin are assumed unit:
274
Paolo Boccotti mo = 0 -2 = 1, m2 = 1,
(A1)
which implies
q,o =
(A2a)
1, 0)o = - 1,
Iq,~l
(A2b)
where the dot denotes the derivative. Finally, the following symbol is defined:
ot = H/o'.
(A3)
The expected number per unit time of local wave maxima whose elevation falls between ~ct and
(~ + d~)a, which are followed by a local wave minimum whose elevation falls between (~ -1)c~ and (~ - 1)a - dot, after a time lag whose amplitude falls between ~" and T + d r (a, ~', and ~ being arbitrarily fixed values, and da, dT, and d~ being fixed intervals of small amplitude) is denoted by
EX(ot,'r,O da dr d~. The equation of EX(et,'r,~) proceeds from the general solution introduced by Rice (Rice, 1944; Rice, 1945), and can be written in the form
EX(ot,7,~)=otp[rlo=,Ot, r/o = 0, ~/. = 0, T/T= ('--1)C~] f ~ f ] [ U l W
(A4)
"P[~o = u, ~ = wlr/o = ~a, ~1o = 0, i/. = 0, "q. = (~-- 1)a] dw du. The expression of the joint pdf in this equation is
p[no = ~a, ho = 0, h, = o, n, = ( ~ - 1)~1 - (2~r)2,~ exp
-
~ f(r,~)a
,
(A5)
where ~)2-7,-- = 4(Mn + M14)(~2-~) + 2Mn jr( M '
(A6)
and M o .and M are, respectively, the i, j cofactor and the determinant of the covariance matrix of r/o, "t/o, 77. ~/. : .,
Mij = i, j cofactor, M = determinant of ~
0
1
- qs.
- ~.
1
. -qJ.
0
,
'
(A7)
1
Note that M44 is equal to M1~, which has been used in Equation (A6).
Evaluation of the double integral in the equation of EX(a,'r,~, in the limit ov--~ Let us consider, preliminarily, the conditional pdf of the second derivative ~o, given the condition r/o = ~a, 4/o = 0, 41,--0,~, = ( ~ - 1)ct.
(A8)
It can he written in the form P[~o = ulr/o = ~ot, 7/o = 0 , h , = 0, r / , = (~:- 1)a] = • exp - ~ - ~
u+
Css
a
,
/C55
X/27rC
(A9)
General theory of three-dimensional wave groups: I
275
in which C o and C are, respectively, the i, j cofactor and the determinant of the covariance matrix
of'Oo, 41o, 41. n . Oo:
("
o
o
i
C o = i, j cofactor . . . . C = determinant of ~b. - ~ .
~ -~,~ 1
0
q,~
~7
-1 o
1 o
0 1
"@.. ~7
"~7
~.
m4
(AIO)
Equation (A9) implies that the expected value of qJo given condition (A8) is = K,('r,C")a,
(A11)
where K,(%O -
-C,5~-G5(~-1)
(A12)
Css
and that the standard deviation is (A13) Let us consider now the random variable ~o/~ given condition (A8), as O/-----~
•
From Equation (A11) the expected value of this random variable is K~(I",O, and from Equation (A13) the standard deviation approaches 0, so that we can assume p[(~jda) = u['qo = ~a,41o = 0,417 = 0,n¢ = ( ~ - 1)a] = 6[u-K,(~',~)],
(A14)
where 6(u-fi) denotes the delta function. In a similar way we arrive at
p(~Ja) = w['qo = ~a,41o = 0,41, = 0,~1, = ( ~ - 1)a] = 6[w-K2(~',0],
(A15)
in which K2(~',O =
- O l s ~ - D 4 s ( ~ - 1) , 055
(A16)
and D 0 is the i, j cofactor of the covariance matrix of 90, 41o, 41, 9¢, 07 :
Di: = i, j cofactor of
-q,7
1
o
o
-'tp~.
0
-1
m4
ii -(O~. o q,7 q,7 0 1 -q,7 1
(A17)
From Equation (A14) and Equation (A15) we have p[(qjo/a) = u,('OJot) = wlr/o = ~ot,41o = 0,41. = 007. = ( ~ - 1)a]
= 6[u-K,(%0, w-K2(r,g¢')],
(A18)
where ~(u-fi, w-fv) denotes the delta function of two variables. This equation, which is valid as a---*~, enables us to straightforwardly evaluate the integral on the RHS of Equation (A4). The result is
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Paolo Boccotti
(o fSlulwp[~o=U,~ =wlrlo=~a,ilo=O, il =O, rb=(~_l)~]dwd u ~]KI(~',~IK2(~',~o/2
if
(AI9)
KI(~',,)<0 and K2(~',~ > 0,
otherwise. The values of K~ and K2 depend on the choice of r and ~ and that is because we write K~(r.~)
1
and K2(r,~). If r = T' and ~ = ~ , Ki and Kz take on the following special values x , r( ,~ )
-ll+~r*'.
= - ~ i - - , , 7 *,
K ~({)_211+t~T*l_0r. r, ,
(A20)
and hence
The substitution of Equation (A5) and Equation (A19) into Equation (A4) and the use of Equation (A21) yield
EX(c~,r,¢) - (2rr)2 M~~
exp -
f(r,C")a2 IK,(l-,C3K2(r,~[c~3 if r = r , ~ = Z '
1 ^ EX(a,r,~) <_ 1 ~----- e x p [ - ~ f(r, Oa2]lK,(r,~K2(r, Ola 3 otherwise, (2~-)~~/M('r) I_
(A22a)
(A22b)
where we write M(r) instead of M, in order to specify that the determinant of the covariance matrix (AT) depends on r.
Proof that the absolute maximum of EX(a,'r,¢3 as a---,~ occurs at r = T*, ~ =
1
We seek the pair(s) (r,¢") at which ~'(r,0-defined by Equation (A6)-has its absolute minimum. First, let us think of )~(r,~ as a function of ~ for any fixed r. Its absolute minimum occurs at
1
= ~ and is fmin(r) =
r,
--
M
(A23)
Then, let us examine this function of r. The numerator and the denominator on the extreme right of Equation (A23) are given by MI1--MI
4 = 1 - - 1 ~ r - - i ~ r + i~.r dl-
~2rlPr--Or~,
(A24)
and are related to each other by the following equation: M= (M,,-M,4)(1-0,)-~[(1
+ 0,)(1 + ~ , ) - qr~].
(A26)
After a few attempts it was possible to prove that the function of r within square brackets in this equation is greater than zero for every r:/:0. First, it was found that [(1 + 0~)(1 + ~ ) - ~/r~]- MEZ + M=3 , 1-0,
(A27)
General theory of three-dimensional wave groups: I
277
then it was proved that the RHS in this equation is greater than zero. This can be verified by noting that the joint pdf p[~o = 0,~o = w, 4/, = uw, 7, = 01 - (2zr)2/-~ exp - ~-~ (M33u2 + 2M23u + M22)w2 (A28) must be bounded for all u as w-.oo, which implies M22M33-M~23 > 0,
(A29)
Mz2 + M23 > 0,
(A30)
and hence
since M22 is equal to M33 and is greater than zero. Inequality (A30) completes the proof that the term within square brackets in Equation (A26) is greater than zero. Since the term within square brackets in Equation (A26) is greater than zero, we have
M<_(Mll --Mi4)(1 - ~O,),
(A31 )
^ fmin(T) ~-~
(A32)
and hence 1
1-qJ~"
Moreover, we have ~min(T~) =
(A33)
1
1-~r" '
because ~br" = 0. Since the absolute minimum of ~b, for ~"in (0,oo) occurs at 1- = T', Equation (A32) and Equation (A33) imply that also the absolute minimum of)~min('/")occurs at "/"= 7"."lHence the absolute minimum of i(T,O for T in (0,oo) and ~ in ( - oo,+ oo) occurs at ~- = 7* and ~ = 2" This result and Equation (A22a) and Equation (A22b) imply that
1
for every fixed pair (~',0 such that ~-:#T* and/or ~:# ~. Specifically, the quotient on the LHS of Equation (A34) is of smaller (or equal) order than e x p { - ~ [ i ( % 0 - i ( T " , ~ ) ] o t 2} as a---.oo, in which the difference within square brackets is greater than zero.
Definition of the expected numbers per unit time EX(a), EXs,w.(a,r,O, and proof of the necessary condition Let 1. EX(oOda = expected number per unit time of waves whose crest-to-trough height H falls in a fixed small interval (a, a + d~x); 2. EX,.w.(a,z,~) d a d r d~ = expected number per unit time of local wave maxima whose elevation falls between ~a and (~ + dOa, which are followed by a local wave minimum whose elevation
278
Paolo Boccotti
falls between (~ - l ) a and (~ - 1 ) a - da after a time lag whose amplitude falls between z and r + d,c, with the local wave maximum and the local wave minimum that must be, respectively, crest and trough o f the same zero up-crossing wave. (This condition will be referred to as the same wave condition, and hence the subscript s.w.).
Then the equation relating EX(a) to EXs.w.(a,'r,~) is
EX(~)=f ~ flEXs.w.(a,~,~d~d~.
(135)
1 A corollary of section 2 is that a local wave maximum of elevation ~ a at an instant to and a 1 local wave minimum of elevation - ~ a at to + T" certainly satisfy the same wave condition as a,---,
~, and hence the following property of E X .... follows 1
1
1. EX~.w.(a,T*, ~ ) is asymptotically equal to EX(a,T*, ~ ) as a---~oo.
Two more properties of E X .... are 2. EX~.w.(a,z,~) is always smaller than or equal to EX(a,'r,~) since they have the same definition apart from the same wave condition; 3. when r is greater than a few times the mean wave period, EXs.~.(a,z,~) can be assumed 0 because the probability that the same wave condition is satisfied approaches 0. These properties of EXs.w. together with Equation (A34), imply that, as a---*~, 1. the whole contribution to integral (A.35), apart from a negligible share, proceeds from a small 1 neighborhood of point ~-= T', ~ = ~ ; 2. the expression of E X .... in this small neighborhood can be obtained from Equation (A22a) of EX: 1
1
(27r)2/M(T, ) •
.'exp[ X:( _l-T,
(A36a)
1
where 11+~:
M(V') = ( l - ~ * ) ( ~ - ~ ' ) ,
(A36b)
1-~:' ~ =
~]
l
~= T*, ~= 12
(In examining Equation (A36a) the reader should bear in mind that
General theory of three-dimensional wave groups: I
279
is equal to zero.) Since/~, and/~e are nonzero, Equation (A36a) shows that, as oe-,% EX.... a,7 ~ + 8r, ~ + 8
has
the same order as EXs.~,(m~, ~ ) o n l y if &r and 8~ are of order ol-', and hence the neighborhood 1
of point T = T*, ~ = ~ making the whole contribution (apart from a negligible share) to integral (A35) has a radius of order a 1. Therefore, a wave of given height H necessarily has the following two characteristics as o~--~: 1. crest-trough lag equal to 7"* (apart from a random difference 6r of order o~-1), 1
2. quotient of crest elevation/and crest-to-trough height H equal to ~ (apart from a random difference 8~ of order a J), which completes the proof of the necessary condition. Corollary: a closed solution for the probability of the crest-to-trough heights under general bandwidth assumptions As a corollary, the expression of EX(a) as cr--.oo is deduced by integration of EX~.w.(a,r,~) over l
the neighborhood of point r = T ~, ~ = ~. Letting u = a 6r, w = ct ~:,
(A37)
then from Equation (A36a) we obtain 1
1 .... exp[-~'f(T~'2)]f_~f_.
1
(2rr)2/M(T')
~,,
1
exp[- ~ (K'~ u2 + I~e wi)]dw du, and hence 2lK'(iw' ~)K2(T~' ~)1
e x p [ - 1 3~(1W,~)c~Z],
(A39)
where the constants are specified by Equation (A36b). The probability p(a)d~ that a wave has a crest-to-trough height in a fixed small interval (a, c~ + da) is given by p(oOda = EX(a) dot/EX+ ,
(A40)
where EX+ is the expected number per unit time of zero up-crossing waves, whose expression was deduced by Rice (Rice, 1944; Rice, 1945). Under assumption (A1), we have EX+ = 1/27r,
(A41)
so that O~
p ( a ) - ~/2~r~(U~Or. ) 2(1--Or.)
Ot2
4(1
Finally, the probability of exceedance associated to this pdf is
(A42)
280
Paolo Boccotti PIERSON and MOSK0~ITZ spectrum
1.O0
~Pr* =-0.65 , ~;r*=0,40
0.95 c01~ Second rectangular spectrum v
~Pr. =-0.55 , "~r. =0.88
0.90
I
Np
I
First rectangular spectrum
0.85
~T,=-0.41
0.80
1
..........................................
10-s
l0 -4
10-s
10-2
10-1
,
~T*=0.65
i
1
Fig. 3. Quotient Q(P) defined by Equation (A44). The continuous lines represent Equation (A45) which is exact as P---~0.The data were obtained by Forristall (1984) from numerical simulation of stationary Gaussian processes. O~2
~]2~Jr'(1-~r*) These are the forms assumed by p(ct) and P(a) as ~-.-,oo. We see that the probability of a nondimensional wave height (HI~) exceeding a fixed level a depends on the autocovariance minimum
(qJr* = ~(T')lmo ) and on the relevant second derivative (t~r" = ~(T')lm2). If the spectrum is very narrow, the autocovariance approaches a cosine, so that ~ r ' - " - 1 and tpr'---'l and Equation (A42)-Equation (A43) become equal to the well-known expressions of Cartwright and Longuet-Higgins (1956). If the bandwidth is finite, ~r" is greater than - 1 and hence the probability of exceedance gets smaller than in the case of the very narrow spectrum, which is consistent with the conclusion of Longuet-Higgins (1980) on the effects of finite bandwidth. From a numerical simulation of Gaussian random processes, Forristall (1984) obtained a few data of the quotient
Q(P) =
ot(e)gi
....
p,c,r,~/a(e)ve~y. . . . . .
sp .......
(A44)
where a(P) is the wave height H that has a probability P to be exceeded. Now, from Equation (A43) we know that the exact form of Q(P) as c~---,oo(or, alternatively, as P---.0) is
Q(P)
[(1
[
-
$r') In(K/P)] 1 2-~-n(17P)
/j
,
(A45)
where K-
1+ (~-
(A46)
12(br-(1- Or*) " Fig. 3 shows that the agreement between this equation and Forristall's data is very good. Particularly, we can see that the convergence of the data points to Equation (A45) occurs starting on P = 0.20, in case of a typical spectrum of the wind waves.
Pergamon S0029--8018(96)00013.3
A GENERAL THEORY OF THREE-DIMENSIONAL WAVE GROUPS PART I: THE FORMAL DERIVATION Paolo Boccotti Department of Fluid Mechanics and Offshore Engineering, University of Reggio, Calabria, Italy.
(Received 21 November 1995; accepted in final form 31 January 1996) A b s l ~ e t - - I f we know that a wave of given exceptionally large crest-to-~ough height occurs at a fixed point x_oat an instant to in a random wind-generated sea state, we can predict what happens with a very high probability before and after to in an area surrounding xo. The expressions of the surface displacement and velocity potential in this area are obtained in closed form. They are exact to the first order in a Stokes expansion and hold for nearly arbitrary bandwidth and solid boundary. It will be shown in Part II that these expressions represent either the evolution of a single threedimensional wave group or the collision of two wave groups, according to the configuration of the solid boundary. The theory was developed in a series of papers starting on 1981. This paper presents the whole theory in a compact form thanks to a radical simplification of the mathematical proof. Copyright © 1996 Elsevier Science Ltd
NOMENCLATURE
A,B,C,D,
determinants of covariance matrices
M
Ao, Bo, Co, cofactors of covariance matrices D O, Mij expected number per unit time EX H crest-to-trough wave height jth moment of the frequency spectrum about the origin wave length probability density function probability, probability of exceedance time time lag 1" abscissa of the absolute m i n i m u m of the autocovariance ~ T ) dummy variables U, W x_ =(x,y) point in the horizontal plane x~ =(xo,Yo) point where a wave of given very large height occurs x_ =(x,r) relative location of a point with respect to x_o vertical coordinate Z mj
L P P t T
or=H/o" '1 o~ T
4,
surface displacement quotient of the crest elevation and the crest-to-trough wave height variance of the random surface displacement at x__o variance of the random surface displacement at x_o + _X time lag between the crest and trough of a wave velocity potential covariance of the surface displacement and the velocity potential 265
266
Paolo Boccotti one-dimensional autocovariance of the surface displacement space-time autocovariance of the surface displacement 1. INTRODUCTION
A steady wind wave field can be thought of as the sum of a very large number of small periodic waves with frequencies and directions such as to form some characteristic twodimensional spectra, and with uniformly distributed random phase angles (Longuet-Higgins, 1963; Phillips, 1967). According to this theory, both the surface displacement and the velocity potential, to the first order in a Stokes expansion, represent stationary Gaussian processes of time at any fixed point. This paper deals with the conditional probability of the surface displacement if it is known that a zero up-crossing wave with an exceptionally large crest-to-trough height H occurs at a fixed point x~, at a given time instant. The mathematical analysis is based on the general techniques set up by Rice (Rice, 1944; Rice, 1945; Rice, 1958). The assumption is made that H/tr---~oo, which is not necessarily inconsistent with Stokes' assumption that H/d---+O and H/L---+O (or being the root mean square surface displacement of the random wave field as a whole; d the water depth; L the wave length). Under this assumption, the probability approaches 1 that the surface displacement surrounding x o approaches a well-defined deterministic configuration in space and time. Really, for a wave with a realistically large height H, say H = 8+9o-, the residual random noise will be not negligible. Moreover, in nature, this wave may be spilling or almost breaking near the crest and this will produce a localized profile distortion. But the overall configuration in space and time of the three-dimensional wave group including this wave should be close to the deterministic configuration obtained in the limit H&r ~ oo, which will be confirmed in Part II by the results of a field experiment. This theory was introduced bit by bit by the writer in a series of papers (Boccotti, 1981; Boccotti, 1982; Boccotti, 1989). Here, a radical simplification is achieved thanks to the new proof given in section 2 and section 3. Specifically, it is proved that a special condition becomes necessary and sufficient for the occurrence of a wave of given height H, as H/o.--,oo. This condition is 1 1 ~l(to) = ~ H, rl(to + T*) = - ~ H,
(1)
where ,/(t) is the surface displacement at a fixed point x_o ; to is an arbitrary instant; T" is equal to the abscissa of the absolute minimum of the autocovariance function $, which is assumed to be also the first minimum on the positive domain of this function. This will be referred to hereafter as 'condition (1)'. The proof that condition (1) becomes necessary and sufficient for the occurrence of a wave of given height H, as H/o,--.oo, permits to substitute the condition 'given that a zero up-crossing wave of crest-to-trough height H occurs at a fixed point x~' by the much more simple condition 'given (1) at x~'. Hence, the theory can be proved in only a few steps. In the above cited papers (Boccotti, 1981; Boccotti, 1982; Boccotti, 1989) this theory was carried out together with the closed form solution for the wave height probability, under general bandwidth assumptions (in the limit HIo.--.o*). Here, the expression of the wave height probability is reobtalned as a corollary of the formal proof that condition (1) is necessary. This expression proves to fit very well Forristall's data of simulated heights
General theory of three-dimensional wave groups: I
267
(Forristall, 1984) even for H/o->4, and hence we get the first evidence that the hypothesis of H/o'--*~ can be succesful. 2.
CONDITION (1) IS SUFFICIENT FOR THE OCCURRENCE OF A WAVE OF GIVEN VERY LARGE HEIGHT H
Let us consider the surface displacement r/(t) at a fixed point x~ under the assumption that */(t) is a stationary Gaussian random process of time (theory of the wind waves to the first order in a Stokes expansion). Let us examine the probability density function (pdf) of this surface displacement at any fixed time instant, given condition (1). For any fixed time lag T we have
p *~(to + 13 = ul*/(to) = ~ H, *~(to + T ~) = - ~ H = A33
27rA
A23-A13
• exp - ~ -
u
A33
2
'
where Aij and A are, respectively, the i, j cofactor and the determinant of the covariance matrix of the random variables */(to), */(to + T~), */(to + 13: A 0- = i, j cofactor, l ~ T") A = determinant of \ ~ ( T ) qJ(T- T')
(3)
D
gJ(O)
Equation (2) implies that the expected value of *~(to + 7) given condition (1) is A23-A13 H ~ T ) - ~ T - T ~) H A33 2 - ~b(0)-~b(T~) 2-"
~l(to + T) -
(4)
Moreover, Equation (2) implies that the standard deviation of *~(to + 13 given condition
(1) is
J
A A33 -
J
[ qJ(0) 1 -
q?(T) + tka(T-T*)-20(13~T-T~)[~T~)/~O)]] ~b2(0)_qj2(T, )
,
(5)
which is smaller than or = [~(0)] 1'2,
(6)
since I~T')/~0)I is smaller than unity. Therefore, the quotient of the standard deviation and the expected value of *~(to + 13 given condition (1), generally, approaches 0 as H/o ~-* ~, and hence *~(to + 13 is asymptotically equal to ~(to + 13 as H/o ~-,o~. This result prescinds from the definition of 7"; that is, 7" can be any given time lag. The particularity of 7" will be used in the following analysis of the expected surface displacement h as a function of the time lag T. Since ~ 0 ) and ~ T ' ) = ~ ( - T ~) are, respectively, the absolute maximum and the absolute 1
minimum of ~13, ~ has its absolute maximum (equal to ~ H) at T -- 0 and its absolute
268
Paolo Boccotti
1 minimum (equal to - ~ H) at T = T*, and hence at T = 0 there is a wave crest and at T
= T' there is a wave trough. Since 7" is the abscissa of the first minimum of ~ T ) after T = 0, the derivative ~(T) is negative on (0, T'): ~(T)<0 on (0,T'),
(7)
which implies (since qJ is an even function)
~ T ) > 0 on
(-T',0),
(8)
or, alternatively,
~(T - 1") > 0 on (0,T').
(9)
The inequalities (7) and (9) imply that ~/is strictly decreasing for T in [0,T']. Therefore the wave crest at T = 0 and the wave trough at T = 7"* belong to the same zero upcrossing wave, and this wave has a crest-to-trough height equal to H - see Fig. 1. Hence the conclusion: as H/cr---}~, condition (1) becomes sufficient for the occurrence of a wave of given height H. 3.
CONDITION (1) IS NECESSARY FOR THE OCCURRENCE O F A WAVE OF GIVEN VERY LARGE HEIGHT H
A general necessary condition for the occurrence of a wave of given height H is that the surface displacement is equal to ~ r / ( 0 < ~ < 1 ) at an instant to (wave crest) and that it is equal to (~ - 1 ) H at a later instant to + T (wave trough):
rl(to) = (;H, ~q(to + "r) = ( ~ - l ) n ,
0<~<1.
(10)
the derivative is always negative on this interval absolute maximum
~H
,
/
T*
Fig. 1. Generalfeaturesof function (4). The actual surfacedisplacementgiven the condition(1) is asymptotically equal to this function as H/o,--~oo.
General theory of three-dimensional wave groups: I
269
To fix the ideas, Fig. 2 shows a few waves with a fixed height H and different values of ~ and ~-. Let us consider the probability that the surface displacement falls between ~ and + d~/l at an instant to, and that it falls between (~ - 1 ) H and (~ - 1 ) H + d~72 at a later instant to + "r;, to, H, "r, and ~ being arbitrarily fixed values, and dr/l, d7/2 being fixed intervals of small amplitude. It is given by
1 exp 2 I r [ ~ ( 0 ) - ~(1-)1 u2
P(H,'r,~ =
1
[ H \ z] ]d'thd'tl2,
~f('r,O~)
(lla)
(
(llb)
where I(r,
9 - 2
+ 2 q,(0) +
Now, let us examine this probability as a function of "r and ~, as I-IIo'--*~. The first term on the RHS of Equation (1 lb) is independent of ~ and its absolute minimum occurs at ~= 7~ because the absolute minimum of q4~') on (0,oo) has been assumed to occur at ~-= ~ . The second term on the RHS of Equation ( l l b ) is zero when ~ = 1/2 and is greater than zero when ~:~ 1/2. Therefore, the absolute minimum of f(~',0 for I" in (0,oo) and ~ in 1 ( - ~, + m) occurs at ~-= T', ~ = ~ , which implies
P(H,~',OIP H , T ~,
---,0 as -----,~
(12)
or
1 for every fixed pair (z, ~) such that ~'¢7 ~ and/or ~4= ~ . This suggests that a wave with a given crest-to-trough height H necessarily has ~- = 7~ 1 and ~ = ~ , as HIo'---~. In other words,
as I-I/o'---,~, condition (1) becomes not only sufficient but also necessary for the occurrence of a wave of given height H.
I,
T2
i i
I
7"1
%
,i
I
Fig. 2. Waves with a fixed height H and different values of ~ (the quotient of the crest elevation and the wave height) and ~"(the time lag between crest and trough).
270
Paolo Boccotti
A formal proof of this property is given in the Appendix through an analysis of the expected number per unit time of local wave maxima of fixed elevation ~H which are followed by a local wave minimum of elevation (~ - 1 ) H after a fixed time lag r. The above mentioned closed solution for the probability of the wave heights will be a corollary of this formal proof. 4.
THE EXPECTED CONFIGURATION OF THE WATER SURFACE IN SPACE AND TIME IF A WAVE OF GIVEN VERY LARGE HEIGHT OCCURS AT A FIXED LOCATION
If a wave with a given very large crest-to-trough height H occurs at a fixed point ~ , what is the expected configuration in space and time of the water surface surrounding X_o? From section 2 and section 3, to say 'a wave with a crest-to-trough height H occurs at 1 a point xj , as H/o'--*~, is equivalent to say 'the surface displacement at x~ is ~ H at some 1 instant to and is - ~ H at the instant to + T " . Then, in order to answer the question put above, we examine the pdf of the surface displacement at any fixed point x~ + X and instant to + T, given the condition 1 1 "O(x~,to) = ~ H, "O(x~,to + T') = - ~ H,
(13)
(this is the same as condition (1), but we write ag(X_o,to)instead of ~l(to) because now we deal with waves in the space-time). The pdf under examination can be written in the form
[
,]
1
p rlX(~o + X, to + T) = ulrl(x~,to) = ~ H,'oX(yo,to + T*) = - ~ H
B•33
=~/2~'B
exp -
~
u
~
2-
(14)
'
in which B 0 and B are, respectively, the i, j cofactor and the determinant of the covariance matrix of r/(x~, to), ~j(Xo, to + T*),~(x~ + X, to + T) •
o~ B o = cofactor, B = determinant of
q~(_0,T~)
q~(_X,T)
~(0,T") oa ~(X,T-T*)|. l ~(X_,T) ~ ( X , T - T') -O~x /
(15)
In this matrix, • is the two-dimensional autocovariance defined by ~(X,T) =
(16)
and oa and O'Zxare the variances of the surface displacement at the points x__oand X_o+ X, respectively. Equation (14) implies that the expected value of ~/(X__o+ X, to + T) given condition (13) is
General theory of three-dimensional wave groups: I
~ ( ~ + X_,to + T) -
271
B23-B13 H qt(X_,T)-qt(~_,T-T') H B 3 ~ 2 - qt(O,0)-qt(_0_0,2W) 2-"
(17)
Moreover, Equation (14) implies that the standard deviation of ~ x(8_o + X_,to + T) given condition (13) is JO2x - o ~ qt2(-X'T) + *2(X_,T-T')-2*(X_,T)*(X_,T-T')[*(O_,T~)/*(Q,O)] =
_
q,2(0_,0)_q~2(0,T,
)
,
(18)
which is smaller than O2x since IW(9_,T')/W(_0_,0)I<1. Let us express the expected value (17) in terms of the dimensionless autocovariance fft(X,T) = ~(_X,T)/o-o'x,
(19)
whose value can range between 1 and - 1 in consequence only of the relative phase of (X_o,t) and (x~ + X,t + T). We have 1 fft(X_,T)-fft(~_,T- T') H ~l(X_o+ X, to + T) = ~ 1-ffr(O_,T') ~rx_o.--,
(20)
which makes it evident that the quotient of the standard deviation and the expected value of ~(xo+X_,to + T) given condition (13), generally, approaches 0 as H/o,--,oo (since the standard deviation is smaller than orx), and hence rlx(_~o+ X_,to + T) is asymptotically equal to ~l(x__o+ X, to + T) as HIo,--.oo. Part II of this paper will show that ~/, as a function of _Xand T, represents either the evolution of a single three-dimensional wave group or the collision of two wave groups, according to the shape of the solid boundary. 5.
THE EXPECTED DISTRIBUTION OF VELOCITY POTENTIAL IF A WAVE OF GIVEN VERY LARGE HEIGHT OCCURS AT A FIXED LOCATION
Associated with the configuration (17) is a distribution of velocity potential in the water, which to the lowest order in a Stokes expansion is given by
qb(x_o+ X_,z,to + T) = ~- [
qt(0,0)_W(Q,T. )
j,
(21)
where X ,z and T are the independent variables and ~ is the cross-covariance of the surface displacement and the velocity potential of the random wave field:
dp(X,z,T) = <~7(x_o,t)~b(xo+ X,z,t + T) > .
(22)
The fact that the velocity potential associated with the surface displacement (17) is given by Equation (21) can be proved under the assumption that ~ and qb are solutions to the linear problem. Particularly, here we shall prove the equality
272
Paolo Boccotti
From Equation (17) and Equation (21) of ~1 and ~b, and definitions (16) and (22) of the covariances, this equality takes the form H <~/(x__o,t)7](x~+ X,t + T) > - <~/X(~o,t)~/(x_o+ X,t + T-T") > 2
-
g 2 bT
(24)
• ( < r l ( x ~ ' t ) ~ b ( x - ° + X - ' z ' t +>T )~> -~< r+l ( -x ~~' t )rc ~' ( x- - °)+~X - ' z ' t + T - T * ) >
z=o'
where the common factor
1
=
lim
At
(x__o,t)
(
+ X_,z,t + T)
(25)
At---* - ¢hx(~ + X,z,t + T - T " ) ] z = o dt, and is proved since rl(x o + X,t + T) -
g ~1. dp(X_o+ X,z,t + T) z =o,
rl(x~ + X,t + T - T " ) -
g ~TCl~_o + X,z,t + T-T")z=O,
(26) (27)
(these equations being a direct consequence of the assumption on ~ and ¢h)The proof that Equation (17) and Equation (21) are solutions to the linear problem can be completed in a similar way. 6. THE THEORY HOLDS FOR NEARLY ARBITRARY SOLID BOUNDARY AND BANDWIDTH We have assumed that the surface displacement represents a stationary Gaussian process of time at any fixed location, but we have not assumed that the random process be homogeneous in space. Specifically, the variance o-2_xof the random surface displacement at + X has been assumed to be generally different from the variance oa at x~, which happens if some solid obstacle causes wave reflection and/or diffraction, or the sea bottom causes wave refraction• As a confirmation that the theory holds also in these cases, we can verify that expression (Equation (21)) of (b automatically fulfils any given solid boundary condition. To this end, let us assume that a point (X',Z') belongs to a solid surface and that n is the unit normal vector to the surface at this point• We wish to show that
273
General theory of three-dimensional wave groups: I
(28)
n . V ~ = 0 at X = x ' - X o , z = z', that is, from E q u a t i o n (21) o f ~,
{
n.V
+ X,z,t + T - T ~) > 7") >
H} = 0,
(29)
atX=
x " -x~, Z : z •
Since _X and z are the only i n d e p e n d e n t space variables, this can be rewritten in the form
(30) at X_ = x_"--x~, Z = Z, which shows the equality to h o l d under the a s s u m p t i o n that the velocity potential o f the r a n d o m w a v e s satisfies the given solid b o u n d a r y condition at (x',z'). Finally, we w o u l d e m p h a s i z e that the theory is general also as to the w a v e spectrum. Indeed, the o n l y restriction is that ~ T ) (the F o u r i e r transform o f the frequency spectrum) m u s t have an absolute m i n i m u m , w h i c h m u s t be the first m i n i m u m after T = 0. In the course o f several field m e a s u r e m e n t s (which will be quoted in Part II), the writer f o u n d this a s s u m p t i o n to be not satisfied only in a few special cases in w h i c h the spectrum was very b r o a d b e c a u s e o f the o v e r l a p o f w i n d w a v e s and swells o f nearly the s a m e height. REFERENCES Boccotti, P. 1981. On the highest waves in a stationary Gaussian process. Atti Acc. Ligure 38, 271-302. Boccotti, P. (1982) On the highest sea waves. Report Istituto Idranlica Universith di Genova, 1-160. Boccotti, P. 1989. On mechanics of irregular gravity waves. Atti Acc. Naz. Lincei, Memorie, VIII 19, 111-170. Cartwright, D.E. and Longuet-Higgins, M.S. 1956. The statistical distributions of the maxima of a random process. Proc. Roy. Soc A-237, 212-232. Forristall, G.Z. 1984. The distribution of measured and simulated height as a function of spectra shape. J. Geoph. Res 89, 10547-10552. Longuet-Higgins, M.S. 1963. The effect of non-linearities on statistical distributions in the theory of sea waves. J. Fluid Mech 17, 459-480. Longuet-Higgins, M.S. 1980. On the distribution of the heights of sea waves: some effects of nonlinearity and finite bandwidth. J. Geoph. Res 85, 1519-1523. Phillips, O.M. 1967. The theory of wind generated waves. Adv. in Hydroscience 4, 119-149. Rice, S.O. 1944. Mathematical analysis of random noise. Bell Syst. Tech. J 23, 282-332. Rice, S.O. 1945. Mathematical analysis of random noise. Bell Syst. Tech. J 24, 46-156. Rice, S.O. 1958. Distribution of the duration of fades in radio transmission: Gaussian noise model. Bell Syst. Tech. J 37, 581-635. APPENDIX F o r m a l p r o o f that condition (1) is necessary, a n d solution f o r the probability o f the w a v e heights under general bandwidth assumptions Definition and equation o f the expected number per unit time EX(a,'c,~) The general assumptions about the existence of the derivatives of the autocovariance ~ T ) are made as in the well-known paper of Rice (Rice, 1944; Rice, 1945). The compact symbols *b and ~br are used in place of ~/(t) and ~T). Without loss of generality, both the zeroth and the second moment o f the frequency spectrum about the origin are assumed unit:
274
Paolo Boccotti mo = 0 -2 = 1, m2 = 1,
(A1)
which implies
q,o =
(A2a)
1, 0)o = - 1,
Iq,~l
(A2b)
where the dot denotes the derivative. Finally, the following symbol is defined:
ot = H/o'.
(A3)
The expected number per unit time of local wave maxima whose elevation falls between ~ct and
(~ + d~)a, which are followed by a local wave minimum whose elevation falls between (~ -1)c~ and (~ - 1)a - dot, after a time lag whose amplitude falls between ~" and T + d r (a, ~', and ~ being arbitrarily fixed values, and da, dT, and d~ being fixed intervals of small amplitude) is denoted by
EX(ot,'r,O da dr d~. The equation of EX(et,'r,~) proceeds from the general solution introduced by Rice (Rice, 1944; Rice, 1945), and can be written in the form
EX(ot,7,~)=otp[rlo=,Ot, r/o = 0, ~/. = 0, T/T= ('--1)C~] f ~ f ] [ U l W
(A4)
"P[~o = u, ~ = wlr/o = ~a, ~1o = 0, i/. = 0, "q. = (~-- 1)a] dw du. The expression of the joint pdf in this equation is
p[no = ~a, ho = 0, h, = o, n, = ( ~ - 1)~1 - (2~r)2,~ exp
-
~ f(r,~)a
,
(A5)
where ~)2-7,-- = 4(Mn + M14)(~2-~) + 2Mn jr( M '
(A6)
and M o .and M are, respectively, the i, j cofactor and the determinant of the covariance matrix of r/o, "t/o, 77. ~/. : .,
Mij = i, j cofactor, M = determinant of ~
0
1
- qs.
- ~.
1
. -qJ.
0
,
'
(A7)
1
Note that M44 is equal to M1~, which has been used in Equation (A6).
Evaluation of the double integral in the equation of EX(a,'r,~, in the limit ov--~ Let us consider, preliminarily, the conditional pdf of the second derivative ~o, given the condition r/o = ~a, 4/o = 0, 41,--0,~, = ( ~ - 1)ct.
(A8)
It can he written in the form P[~o = ulr/o = ~ot, 7/o = 0 , h , = 0, r / , = (~:- 1)a] = • exp - ~ - ~
u+
Css
a
,
/C55
X/27rC
(A9)
General theory of three-dimensional wave groups: I
275
in which C o and C are, respectively, the i, j cofactor and the determinant of the covariance matrix
of'Oo, 41o, 41. n . Oo:
("
o
o
i
C o = i, j cofactor . . . . C = determinant of ~b. - ~ .
~ -~,~ 1
0
q,~
~7
-1 o
1 o
0 1
"@.. ~7
"~7
~.
m4
(AIO)
Equation (A9) implies that the expected value of qJo given condition (A8) is = K,('r,C")a,
(A11)
where K,(%O -
-C,5~-G5(~-1)
(A12)
Css
and that the standard deviation is (A13) Let us consider now the random variable ~o/~ given condition (A8), as O/-----~
•
From Equation (A11) the expected value of this random variable is K~(I",O, and from Equation (A13) the standard deviation approaches 0, so that we can assume p[(~jda) = u['qo = ~a,41o = 0,417 = 0,n¢ = ( ~ - 1)a] = 6[u-K,(~',~)],
(A14)
where 6(u-fi) denotes the delta function. In a similar way we arrive at
p(~Ja) = w['qo = ~a,41o = 0,41, = 0,~1, = ( ~ - 1)a] = 6[w-K2(~',0],
(A15)
in which K2(~',O =
- O l s ~ - D 4 s ( ~ - 1) , 055
(A16)
and D 0 is the i, j cofactor of the covariance matrix of 90, 41o, 41, 9¢, 07 :
Di: = i, j cofactor of
-q,7
1
o
o
-'tp~.
0
-1
m4
ii -(O~. o q,7 q,7 0 1 -q,7 1
(A17)
From Equation (A14) and Equation (A15) we have p[(qjo/a) = u,('OJot) = wlr/o = ~ot,41o = 0,41. = 007. = ( ~ - 1)a]
= 6[u-K,(%0, w-K2(r,g¢')],
(A18)
where ~(u-fi, w-fv) denotes the delta function of two variables. This equation, which is valid as a---*~, enables us to straightforwardly evaluate the integral on the RHS of Equation (A4). The result is
276
Paolo Boccotti
(o fSlulwp[~o=U,~ =wlrlo=~a,ilo=O, il =O, rb=(~_l)~]dwd u ~]KI(~',~IK2(~',~o/2
if
(AI9)
KI(~',,)<0 and K2(~',~ > 0,
otherwise. The values of K~ and K2 depend on the choice of r and ~ and that is because we write K~(r.~)
1
and K2(r,~). If r = T' and ~ = ~ , Ki and Kz take on the following special values x , r( ,~ )
-ll+~r*'.
= - ~ i - - , , 7 *,
K ~({)_211+t~T*l_0r. r, ,
(A20)
and hence
The substitution of Equation (A5) and Equation (A19) into Equation (A4) and the use of Equation (A21) yield
EX(c~,r,¢) - (2rr)2 M~~
exp -
f(r,C")a2 IK,(l-,C3K2(r,~[c~3 if r = r , ~ = Z '
1 ^ EX(a,r,~) <_ 1 ~----- e x p [ - ~ f(r, Oa2]lK,(r,~K2(r, Ola 3 otherwise, (2~-)~~/M('r) I_
(A22a)
(A22b)
where we write M(r) instead of M, in order to specify that the determinant of the covariance matrix (AT) depends on r.
Proof that the absolute maximum of EX(a,'r,¢3 as a---,~ occurs at r = T*, ~ =
1
We seek the pair(s) (r,¢") at which ~'(r,0-defined by Equation (A6)-has its absolute minimum. First, let us think of )~(r,~ as a function of ~ for any fixed r. Its absolute minimum occurs at
1
= ~ and is fmin(r) =
r,
--
M
(A23)
Then, let us examine this function of r. The numerator and the denominator on the extreme right of Equation (A23) are given by MI1--MI
4 = 1 - - 1 ~ r - - i ~ r + i~.r dl-
~2rlPr--Or~,
(A24)
and are related to each other by the following equation: M= (M,,-M,4)(1-0,)-~[(1
+ 0,)(1 + ~ , ) - qr~].
(A26)
After a few attempts it was possible to prove that the function of r within square brackets in this equation is greater than zero for every r:/:0. First, it was found that [(1 + 0~)(1 + ~ ) - ~/r~]- MEZ + M=3 , 1-0,
(A27)
General theory of three-dimensional wave groups: I
277
then it was proved that the RHS in this equation is greater than zero. This can be verified by noting that the joint pdf p[~o = 0,~o = w, 4/, = uw, 7, = 01 - (2zr)2/-~ exp - ~-~ (M33u2 + 2M23u + M22)w2 (A28) must be bounded for all u as w-.oo, which implies M22M33-M~23 > 0,
(A29)
Mz2 + M23 > 0,
(A30)
and hence
since M22 is equal to M33 and is greater than zero. Inequality (A30) completes the proof that the term within square brackets in Equation (A26) is greater than zero. Since the term within square brackets in Equation (A26) is greater than zero, we have
M<_(Mll --Mi4)(1 - ~O,),
(A31 )
^ fmin(T) ~-~
(A32)
and hence 1
1-qJ~"
Moreover, we have ~min(T~) =
(A33)
1
1-~r" '
because ~br" = 0. Since the absolute minimum of ~b, for ~"in (0,oo) occurs at 1- = T', Equation (A32) and Equation (A33) imply that also the absolute minimum of)~min('/")occurs at "/"= 7"."lHence the absolute minimum of i(T,O for T in (0,oo) and ~ in ( - oo,+ oo) occurs at ~- = 7* and ~ = 2" This result and Equation (A22a) and Equation (A22b) imply that
1
for every fixed pair (~',0 such that ~-:#T* and/or ~:# ~. Specifically, the quotient on the LHS of Equation (A34) is of smaller (or equal) order than e x p { - ~ [ i ( % 0 - i ( T " , ~ ) ] o t 2} as a---.oo, in which the difference within square brackets is greater than zero.
Definition of the expected numbers per unit time EX(a), EXs,w.(a,r,O, and proof of the necessary condition Let 1. EX(oOda = expected number per unit time of waves whose crest-to-trough height H falls in a fixed small interval (a, a + d~x); 2. EX,.w.(a,z,~) d a d r d~ = expected number per unit time of local wave maxima whose elevation falls between ~a and (~ + dOa, which are followed by a local wave minimum whose elevation
278
Paolo Boccotti
falls between (~ - l ) a and (~ - 1 ) a - da after a time lag whose amplitude falls between z and r + d,c, with the local wave maximum and the local wave minimum that must be, respectively, crest and trough o f the same zero up-crossing wave. (This condition will be referred to as the same wave condition, and hence the subscript s.w.).
Then the equation relating EX(a) to EXs.w.(a,'r,~) is
EX(~)=f ~ flEXs.w.(a,~,~d~d~.
(135)
1 A corollary of section 2 is that a local wave maximum of elevation ~ a at an instant to and a 1 local wave minimum of elevation - ~ a at to + T" certainly satisfy the same wave condition as a,---,
~, and hence the following property of E X .... follows 1
1
1. EX~.w.(a,T*, ~ ) is asymptotically equal to EX(a,T*, ~ ) as a---~oo.
Two more properties of E X .... are 2. EX~.w.(a,z,~) is always smaller than or equal to EX(a,'r,~) since they have the same definition apart from the same wave condition; 3. when r is greater than a few times the mean wave period, EXs.~.(a,z,~) can be assumed 0 because the probability that the same wave condition is satisfied approaches 0. These properties of EXs.w. together with Equation (A34), imply that, as a---*~, 1. the whole contribution to integral (A.35), apart from a negligible share, proceeds from a small 1 neighborhood of point ~-= T', ~ = ~ ; 2. the expression of E X .... in this small neighborhood can be obtained from Equation (A22a) of EX: 1
1
(27r)2/M(T, ) •
.'exp[ X:( _l-T,
(A36a)
1
where 11+~:
M(V') = ( l - ~ * ) ( ~ - ~ ' ) ,
(A36b)
1-~:' ~ =
~]
l
~= T*, ~= 12
(In examining Equation (A36a) the reader should bear in mind that
General theory of three-dimensional wave groups: I
279
is equal to zero.) Since/~, and/~e are nonzero, Equation (A36a) shows that, as oe-,% EX.... a,7 ~ + 8r, ~ + 8
has
the same order as EXs.~,(m~, ~ ) o n l y if &r and 8~ are of order ol-', and hence the neighborhood 1
of point T = T*, ~ = ~ making the whole contribution (apart from a negligible share) to integral (A35) has a radius of order a 1. Therefore, a wave of given height H necessarily has the following two characteristics as o~--~: 1. crest-trough lag equal to 7"* (apart from a random difference 6r of order o~-1), 1
2. quotient of crest elevation/and crest-to-trough height H equal to ~ (apart from a random difference 8~ of order a J), which completes the proof of the necessary condition. Corollary: a closed solution for the probability of the crest-to-trough heights under general bandwidth assumptions As a corollary, the expression of EX(a) as cr--.oo is deduced by integration of EX~.w.(a,r,~) over l
the neighborhood of point r = T ~, ~ = ~. Letting u = a 6r, w = ct ~:,
(A37)
then from Equation (A36a) we obtain 1
1 .... exp[-~'f(T~'2)]f_~f_.
1
(2rr)2/M(T')
~,,
1
exp[- ~ (K'~ u2 + I~e wi)]dw du, and hence 2lK'(iw' ~)K2(T~' ~)1
e x p [ - 1 3~(1W,~)c~Z],
(A39)
where the constants are specified by Equation (A36b). The probability p(a)d~ that a wave has a crest-to-trough height in a fixed small interval (a, c~ + da) is given by p(oOda = EX(a) dot/EX+ ,
(A40)
where EX+ is the expected number per unit time of zero up-crossing waves, whose expression was deduced by Rice (Rice, 1944; Rice, 1945). Under assumption (A1), we have EX+ = 1/27r,
(A41)
so that O~
p ( a ) - ~/2~r~(U~Or. ) 2(1--Or.)
Ot2
4(1
Finally, the probability of exceedance associated to this pdf is
(A42)
280
Paolo Boccotti PIERSON and MOSK0~ITZ spectrum
1.O0
~Pr* =-0.65 , ~;r*=0,40
0.95 c01~ Second rectangular spectrum v
~Pr. =-0.55 , "~r. =0.88
0.90
I
Np
I
First rectangular spectrum
0.85
~T,=-0.41
0.80
1
..........................................
10-s
l0 -4
10-s
10-2
10-1
,
~T*=0.65
i
1
Fig. 3. Quotient Q(P) defined by Equation (A44). The continuous lines represent Equation (A45) which is exact as P---~0.The data were obtained by Forristall (1984) from numerical simulation of stationary Gaussian processes. O~2
~]2~Jr'(1-~r*) These are the forms assumed by p(ct) and P(a) as ~-.-,oo. We see that the probability of a nondimensional wave height (HI~) exceeding a fixed level a depends on the autocovariance minimum
(qJr* = ~(T')lmo ) and on the relevant second derivative (t~r" = ~(T')lm2). If the spectrum is very narrow, the autocovariance approaches a cosine, so that ~ r ' - " - 1 and tpr'---'l and Equation (A42)-Equation (A43) become equal to the well-known expressions of Cartwright and Longuet-Higgins (1956). If the bandwidth is finite, ~r" is greater than - 1 and hence the probability of exceedance gets smaller than in the case of the very narrow spectrum, which is consistent with the conclusion of Longuet-Higgins (1980) on the effects of finite bandwidth. From a numerical simulation of Gaussian random processes, Forristall (1984) obtained a few data of the quotient
Q(P) =
ot(e)gi
....
p,c,r,~/a(e)ve~y. . . . . .
sp .......
(A44)
where a(P) is the wave height H that has a probability P to be exceeded. Now, from Equation (A43) we know that the exact form of Q(P) as c~---,oo(or, alternatively, as P---.0) is
Q(P)
[(1
[
-
$r') In(K/P)] 1 2-~-n(17P)
/j
,
(A45)
where K-
1+ (~-
(A46)
12(br-(1- Or*) " Fig. 3 shows that the agreement between this equation and Forristall's data is very good. Particularly, we can see that the convergence of the data points to Equation (A45) occurs starting on P = 0.20, in case of a typical spectrum of the wind waves.