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Wavecharacteristicdistributionsfor
Gaussianwaves—Wave-length,
amplitudeandsteepness
ArticleinOceanEngineering·January1982
DOI:10.1016/0029-8018(82)90034-8
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GeorgLindgren
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Ocean Engng, Vol. 9, No 5, pp.411--432, 1982.
0029-8018/82/005 0411-22 $03.00/0
Pergamon Press Ltd:
Printed in Great Britain.
WAVE CHARACTERISTIC DISTRIBUTIONS FOR
GAUSSIAN WAVES -- WAVE-LENGTH, AMPLITUDE AND
STEEPNESS
GEORG LINDGREN and IGOR RYCHLIK
Department of Mathematical Statistics, Lund Institute of Technology, Box 725, S-220 07 Lund, Sweden
Abstract--In a stationary stochastic process the wave-length and amplitude are defined as the
difference in time and height between a crest and the followingtrough. The distributions of these
quantities are of great practical importance, but no closed form expressions are known at present.
In previous papers we have presented an approximation which gives correct upper and lower
bounds, regardless of the covariance structure under Gaussian assumptions. In this paper the
suggested approximations are compared to two simpler approximations, one due to Cavani6 et al.
(1976) based on a cosine process and a new one, derived by replacing the model process by its
regression curve.
1.
INTRODUCTION
A QUESTION that arises in oceanography, fatigue analysis, and other sciences, is that of
the joint statistical distribution of wave-length and amplitude of r a n d o m waves or
random loads. By this we mean the distribution of the difference in time and height
between the crests (local maxima) and the following troughs (local minima). O t h e r terms
used for the same quantities are period and wave-height.
Due to its great importance many attempts have been made to find the exact form of
the distribution of wave-length and amplitude but no theoretically founded closed form
expression has been found, not even for the univariate marginal distributions. However,
several m o r e or less well-founded suggestions do exist. Rice and Beer (1965) proposed a
procedure related to the one in this paper, but without the necessary computational
aspects. Sj6str6m (1961) and Kowalewski (1963) both suggested a distribution based on
the assumption that the amplitude and the mean value of the crest and trough heights are
independent. More recently, Cavani6 et al. (1976) p r o p o s e d to a p p r o x i m a t e the
wave-form by a suitably chosen cosine function, and thereby obtained a simple closed
expression which seems to fit well to their empirical data. Yet a n o t h e r form has been
suggested by Longuet-Higgins (1975), but this distribution deviates considerably from
observed data.
In previous papers, Lindgren (1972) and Lindgren and HolmstrOm (1978), we have
presented a new approximation to the theoretical distribution, which gives correct
solutions regardless of the covariance or spectral structure under Gaussian assumptions,
in the sense that one can compute numerical bounds for the a p p r o x i m a t i o n error. This
solution is based on a model process for the behaviour of the process after maxima, and it
differs from that of Cavani6 et al. (1976) in that the results d e p e n d on the full covariance
function, and not only on a few spectral moments. A further difference is that Cavani6 et
al. adapted their approximation to positive maxima only, which will exclude some of the
shorter waves f r o m giving a c o n t r i b u t i o n to the distribution. A third simple
411
412
GEORG LINDGREN and IGOR RYCHLIK
approximation is obtained in Section 4 by simplifying the model process to a regression
curve on the height and curvature of the maximum.
The approximations involved in the present approach are quite accurate for many
processes of practical interest, including low frequency noise, and most narrow and
moderate band processes whose spectra have a distinct cut-off point, but they work also
for processes with bimodal spectra as will be shown in Section 5 of the present paper.
Since they require a considerable amount of time-consuming numerical integrations it is
desirable to have the simple approximations available in cases where they are accurate.
The aim of the present paper is to compare the approximations for some different
spectra in order to illustrate their merits and limitations.
T h e r e are other quantities of interest which are simple functions of the wave-length
and amplitude, such as the wave-steepness and the run-up height of waves on a sloping
shore; see e.g. Battjes (1971). The distributions of such quantities can be easily
calculated from the joint distribution, and examples of this are given in Section 5.
2.
BASIC D E F I N I T I O N S
In the entire sequel, we let X(t), t --- 0, be a stationary Gaussian stochastic process with
mean zero and covariance function
r(t) = Cov(X(s), X(s + t)).
Its spectrum, supposed to be absolutely continuous, has the spectral density
f(h) = ~
1(
cos (M) • r(t)dt,
and we denote the spectral m o m e n t s by ho, h2, )~4. . . . .
h2k = [ h2kf(h)d h,
noting that ho = r(0) = Var(X(t)), h2 = - r " ( 0 ) = Var(X'(t)), h4 = rlV(0) = Var(X"(t)).
Let (0 -<) t~ < t2 < ... be the times for the crests, U~, U2 . . . . their heights, and let (T~.
H~), (T2, H2) ..... be the corresponding wave-lengths and amplitudes; see Fig. 1.
HT
0
t
FIG. 1. Definition of crests, crest-heights, wave-lengths and amplitudes.
Distribution of wave-length
413
If the process is ergodic, the empirical distribution function of wave-length and
amplitude
F(t,x; "r) =
#{crests in (O,r) such that Tk<-t,Hk<--x}
#{crests in (O,r)}
converges as r ~ oo. The limit, which we shall denote by Fr.H(t,x), is the (ergodic)
distribution function of wave-length and amplitude, and it can be estimated by observing
one single sample path over a long interval of time. Our purpose is to approximate the
density function fr.r~(t,x) of this distribution.
We will also need the ergodic distribution of the crest heights Uk,
O(u)
=
lira #{crests in (0,-r) such that
~-,=
#{crests in (O,x)}
Uk<-u}
Its density function
-u212~,
e
[ hi'V/1 -- ~2./
q~ ~ e V ~ o } '
(2.1)
depends on the spectral width parameter
(
e=
x2/1,2
1-
hoh4]
'
and was given already by Rice (1945). (In g(u) the functions qb and • are the standard
normal density and distribution functions.)
The average number of crests per unit interval of time, 0 <- t _< 1, is
1
2~
implying an average wave-length, i.e. a mean horizontal distance between a local
maximum and the following minimum of
Tm = ~ kX/X-~,/M.
(2.2)
The mean zero crossing distance is similarly
Tz
=
~rX/X,¢X2 .
Furthermore, it follows from the theory of exceedance measures (cf. Cram6r and
Leadbetter (1967), Section 10.8), that the average value of the total increase of X(t) over
a time interval with length To is
414
GEORG LINDGREN a n d IGOR RYCHLIK
E ( I r;' ( X ' ( t ) ) + d t ) =
To(h2/2"rr) ',2 ,
(with x + = max (0, x)), and since there are on the average (Td2"tr)(h4/h2) I/e maxima in
that interval, the (ergodic) mean height difference between a maximum and the
following minimum (or a minimum and the following maximum) is
Hm
(2"rrh22/h4)1/2 = (27rho) 1/2 (1-~2) 1/2
=
(2.3)
The first few spectral moments thus determine both the mean wave-length and the
amplitude but, as we shall see, not the entire distribution.
From (2.1) it follows that the proportion of positive maxima among all maxima is given
by
,
c~= ~ - ( 1 + ( 1 -
.2,1,2,__ ~ 1(
-
1 +
.
In Section 5 we will, besides wave-length T, and amplitude Hk, study the ergodic
distribution of the wave-steepness
(2.4)
St = ~Hk/(2gT 2) •
3.
AN APPROXIMATION OF WAVE-LENGTH AND AMPLITUDE DISTRIBUTION
This section contains a summary of the basic ideas behind the wave-length and
amplitude approximation, as well as some of its theoretical properties. Full details have
been previously given in Lindgren (1972) and Lindgren and Holmstr6m (1978). For
simplicity of notation we normalize the process and assume ho = V ( X ( t ) ) = 1 in the
sequel.
Both the wave-lengths Tk and the amplitudes Hk depend strongly upon the crest
heights Uk. In fact, it is this influence that makes it possible to successfully approximate
the conditional distribution of wave-length T and amplitude H given the crest height U
by means of simple moment bounds. (Here T, H and U are just random variables with
the distributions specified in Section 2.)
Define
A(t)
= (x4r(/) + x 2 , " ( t ) ) / 0 , 4 -
B(t)
=
C(s,t)
= r(s - t) - {h2h4r(s)r(t) +
(hzr(t) + r"(t))/(h4
--
~),
h 2~,
21
+ h~r(s)e'(t) + (M - h~)r'(s)r'(t) + X ~ ' ( s ) r ( t ) +
Distribution of wave-length
415
+ X2r"(s)r"(t)}/{X2(X4 -- X22)},
and
q.(z)
= k . z e x p ( - ( z - k2u)2/2(h4 - h2)), z > 0,
where k. is a normalizing constant. Let g. be a random variable with the density qu(z),
and A(t) a non-stationary Gaussian process with mean zero and covariance function
C(s, t), independent of g~. Then we can define a certain stochastic process
Xu(t) = uA(t) - g~B(t) + A(t),
(3.1)
which we shall use as our main tool.
The process X~(t) was introduced in Lindgren (1970) as a "model process" for the
behaviour of the X-process near local maxima with height u. In fact, the distribution of
X~(t) is equal to the conditional distribution of X ( t , + t), given that the crest height
U, = u at the crest tk. In particular, if T. is the time for the first local minimum of the
process X . ( t ) , and X . ( T u ) is the height of the minimum, then
P{T <- t, H <- x [ U = u} = P { 0 < T. <- t, X . ( T ~ ) >- u - x }
.
By integrating this with the weight function g(u) defined by (2.1) we obtain the
unconditional distribution
Fr, n(t,x) = P{T <-- t,H <- x} =
~o
l"
= I
g(u)P{O < T~ <- t, X~(T,,) >- u-x} du.
J
--0o
To obtain correct approximations to the distributions of T. and X,,(T,,) we start with
T.. If we write
/W~ = #{upcrossing zeros of X;, (s), 0 < s -< t},
it is obvious that
1
E(1Wt) - ~ - E(1V~r(N~t - 1)) <- P (T,, - t} = P{N~/ - 1}
= 1
-
P{N~/=
O} -<
E(N~'t)
and by calculating the first two moments of ~ we can obtain lower and upper bounds
for the probability P{/W~ = 0}, and of the density fr,,(t),
416
GEORG LINDGREN and IGOR RYCHLIK
f~(t) -< fT,,(t) ~ f~'(t).
Here
f~(t) : fx,: (o (0)E(X" (t) ÷ IX" (t) = 0)
t
f~(t) = f'((t) - f f,v,p).~l,)(O , 0).
"r=O
• E(X~j ( t ) + X " ( r ) + }X;(t) = X , ; ( r ) = O)dr,
where the conditional expectations can be calculated numerically, using the explicit
definition (3,1) of X,(t); for more details see Lindgren (1972).
We can take either of f~'(t) and f~(t) as an approximation offr,,(t), thereby obtaining
an upper or a lower bound for the density. In general f~(t) is a more accurate
approximation, but since it is quite timeconsuming one prefers f~(t) e x c e p t for very
wide banded processes.
We shall use the notation f(T,,)(t) tO denote any of the approximations f'((t) and f~(t)
to the conditional wave-length density fr,,(t).
The next step is to approximate the conditional density of the amplitude, given the
wave-length 7", = t (and crest height U = u). The motivation for this procedure - - and
this is where the present method differs from that of Rice and Beer (1965) - - is the
following observation.
For many processes of practical interest, including all low frequency noise, and most
narrow and medium band processes, the distribution of wave-length, conditioned on the
crest height u, has a very distinct cut-off point t~ such that
l - I
(f~(t))+dt and
o-,-<r:,sup (f~(t) - f~ (t))
are both small. This means, first of all, that the probability is almost one that 7",, -< ~'~,
and secondly, that E(/W~° (N~t~- 1)) ~ 0, so that the probability of more than one
minimum in the interval (0, gd0)is almost zero. Thus, with a probability near one, there is
one and only one minimum of X,(t) for 0 < t <- t~; the first minimum falls before t~
and the second somewhere after t~.
From the observation above follows that if Xu(t) is conditioned to have a minimum for
t = ~ ~ t~, it is almost certain that it is the first minimum so that the amplitude is H,, = u
-
x~(~).
In fact, the conditional distribution of X,('r) given a minimum at • can be expressed in
a form similar to (3.1),
uA.~
-
~ .IB I ~ + ~ .2B 2 ~ + A .
,
Distribution of wave-length
417
where (~t, 42) has a certain bivariate distribution (see Lemma 2.1 in Lindgren (1972)),
and A is a normal random variable with mean zero, independent of (~t,, ~).
The conditional density of H, given that T, = ~(-< tg) is therefore approximately
equal to that of
u(1
-
1
2
A,) + ~,,B1,
- ~uB2,
- A,.
If we denote this density by f(H,,IT,,=r)(X), w e obtain as our final approximation of the
conditional distribution of wave-length and amplitude given the crest height U = u,
f(r,,,H,,)(t, x) = ftr,)(t)f(n,,iT=o(X ) ,
(3.2)
and the error in the approximation is small for t -< tg. Exact expressions for the errors
are given in Theorem 2.4 in Lindgren (1972).
By integrating (3.2) over u, weighting with g(u), we get the joint unconditional density
approximation as
r
f~.m,]~(t,x)
= [ g(u)f(T,,.H,,)(t, x)du
.I
--
(3.3)
oo
a~d the marginal amplitude distribution as
/" l 0
f~m)(x) = J, fT,, (t, x)dt,
where to is chosen according to the ranges of accuracy for the approximations (3.2) for
the different u-values. The superscript (m) indicates the use of the model process (3.1).
4.
APPROXIMATIONS BASED ON SIMPLIFIED MODELS
The distributions of the variables T and H studied in the previous section are equal to
the distributions of wave-length and amplitude in the model process Xu(t) obtained from
(3.1) by letting U be a random variable with density (2.1). This model process describes
all statistical properties of the process X(~ + t), conditioned on the presence of a local
maximum at "r. Formula (3.3) gives an approximation of the exact density of T and H in
this model.
Another type of approximation can be obtained by replacing the exact model process
Xu(t) by a simpler process which describes some, but not all, of the properties of X('r +
t). By calculating the exact distributions of wave-length and amplitude in the simplified
model one obtains an approximation of the corresponding distributions in the original
process. This procedure will yield the exact wave-length and amplitude distributions in
an approximating process, while (3.3) produces an approximate distribution in an exact
model process.
GEORG LINDGREN and IGOR RYCHLIK
418
We shall here present two different simplified model processes leading to explicit
formulas for the densityfT.n; one suggested by Cavani6 et aL (1976), and one new, based
on the exact model process (3.1).
By the same arguments as those leading to (2.1) one has the following joint density for
the height and the second derivative of the process at its local maxima,
t
1
flu, z) = c]zlexp - :2{ZZ~2 j (z 2 + 2a2uz + h4u 2)
/,
(4.1)
(assuming h0 = V(X(t)) = 1), where c is a normalizing constant. By restricting the
arguments in (4.1) to u > 0, z < 0 and renormalizing one gets the height and curvature
distribution at the positive maxima. Cavani6 et al. (1976) approximate the model process
X u ( t ) by the process
X~(t) = Ucos(t • ( - Z/ U) I/2),
(4.2)
where U > 0, Z < 0 are random variables with the density (4.1) over {(u,z); u > 0, z <
0}. This approximation is independent of the covariance function and it f u r t h e r m o r e
excludes waves with negative maximum height.
A n o t h e r possibility is to take the deterministic part of the model process Xu(t),
eliminating the normal residual A(t). This gives a model process approximation
Xa(t)= Z.(
U A(t)_
B(t)),
(4.3)
("d" for deterministic), where U,Z < 0 again are random variables with density (4.1),
now over {(u,z); z < 0}. (Note that the variable 4, appearing in (3.1) has the same
distribution as Z conditioned on U = u.)
In both approximations (4.2) and (4.3) the wave-length T is a function only of U/Z. In
fact, in most cases of interest, T is a one-to-one function of U/Z. This is trivial for X'(t),
and to see that it holds also for Xa(t), suppose there are ct :~ c2 and t. > 0 such that
clA'(ta) -- B'(to) = c2A'(to) - B'(t.) = O.
Then A'(to) = B'(to) = 0, which implies that r'(t.) = r'"(t.) = O, (since h4 - h~, > 0 by
assumption). This can certainly happen, even if it must be regarded as a rather special
case.
Thus, we assume here that T is a one-to-one function of U/Z both in X~(t) and in xd(t).
We further define the amplitude H as X~(O) - X ~ ( ~ and Xa(O) - x d ( T ) , respectively.
T h e n there exist functions p and q such that
U
Z
= p(T), Z = Hq(T).
Obviously, q(t) < 0, while p(t) can take any real value in (4.3), being negative in (4.2).
Distribution of wave-length
419
Both p(t) and q(t) are defined for t ¢ D, the set of possible times for the first minimum,
where D depends on the choice of approximation.
Since U, Z have the joint density (4.1) we obtain, by a simple transformation of
variables,
,/1.2
f~,n ,t,x)= c x 2 t q 3 , t ) p ' , t ) l e x p { - 2 ~ x 2 q 2 ( t ) ~
£2
[(~-2p(t) + 1)2 + 1 ~ 2 ] } ,
(4.4)
where Tm is the mean wave-length rV~2/X4, and i = c or d.
In particular, we obtain for the Cavani6 approximation
pC(t) = -t2/~r 2, qC(t) = -~r2/(2t2),
giving the wave-length and amplitude
T = rr(-U/Z) t/2, H = 2U,
with joint density
f~)t-t (t,x) = C(~__)2 (~m) -5 exp{_~e2 (_~__)2(~m)-4 [ ( ( ~ m ) 2
1)2+ 1 - ~] 2~e e2 ]1,
(4.5)
where the normalizing constant is
C = 4(2"trX2/h4) -3/2 h2(l + X 2 / V ~ 4 ) - l e -1.
In fact, Cavani6 et al. study what we would call the "full" wave-length 2T and the
"half" amplitude H/2, but they then normalize by dividing by the expected values. This
means that (4.5) does not agree explicitly with their formula, but that is only a matter of
choosing suitable scales. In the examples in Section 5 we have chosen ho = h2 = 1 and the
distributions are then centered around T,, = "rr/V'~4, Hm = ~ 4 .
For the regression approximation Xa(t) = Z(UA(t)/Z - B(t)), the time T of the first
minimum satisfies
U-A'(t) - B'(t) = O.
Z
420
GEORG LINDGREN and IGOR RYCHLIK
Since we have excluded the possibility A'(t) = B'(t) = 0 for t > 0, we have U/Z =
B'(T)/A'(T), and thus
B'(t)
A'(t)
Pal(t) -- A ' ( t ) ' qa(t) = B'(t) (1 - A-(t))-+A'(t)B(t) "
for t ~ D, where D is the set of possible times for the minimum, i.e. D = Ux{tx}, where t,.
= inf{t>0; x A ' ( t ) - B ' ( t ) = 0}. If pa(t) is increasing in D, then D is an interval (0, to)
where to = inf{t>O; A'(t) = 0}.
The density f~)i-z(t,x) of wave-length and amplitude is obtained by introducing pd(t)
and qd(t) into (4.4), the constant c now being 1/(X/~h4~).
5.
NUMERICAL COMPARISONS
In this section we shall use the model process approximation f~-m]t defined by (3.3) of
the wave-length and amplitude density fT.H as a standard, against which the simplified
densities f~)rt and f ~ of Section 4 shall be compared. Several types of spectra will be
used in order to illustrate the particular merits and drawbacks of the approximations. As
will be seen, all approximations work well for narrow band spectra, but in general the
0 01
4
x
t
(a)
FIG. 2(a).
Approximate density f~"~ of wave-length and amplitude for low
frequency white noise, spectral width parameter ~ = 0.667.
Distribution of wave-length
5.
421
0,01
~0.03
0.2
0.3
~
~
~
,
t
(b)
FXG. 2(b).
Approximate density f~)n of wave-length and amplitude for low
frequency white noise, spectral width parameter ~ = 0.667.
0.5
/"
04
/
\
"\
"%
/
05
02'
t
2
3
4
5
6
7
8
I
(c)
FIG. 2(C).
Amplitude densities for low frequency white noise with spectral width
parameter ¢ = 0.667, - - = f~'),
= f~), . . . . . .
f~.
422
GEORGLINDGRENand IGORRYCHLIK
Cavani6-densities contain an excess of long waves, while the regression approximation
f ~ puts too much weight on short waves in general.
5.1.
L o w frequency white noise
Let a standardized low frequency white noise process have spectral density
1
f(h) = 2N/~ for
Ixl<~3
with covariance function
r(t) -
sin tV'3
t~/3
Such a process has h0 = h2 = 1 and h 4 = r l V ( 0 ) = 1.8 with spectral width ~ =
(1 - h22/hoh4) 1/2 = 0.667, and it has a mean zero crossing distance of
T~=-.
and an average wave-length and amplitude
T m = "tr/X~/2 ~- 2.342,
Hm = (2~/h4) 1/2 ~ 1.4868.
Figures 2a, b show level curves for the joint wave-length and amplitude density as
approximated by f~r.'~ and f~)n respectively. The Cavani6-approximation f~i~ for spectral width E = 0.664 is presented in Fig. 5(b) in connection with a broad-banded
J O N S W A P spectrum; since f~)H depends on the spectrum only via e one can directly use
Fig. 5(b) also for this spectral type. As is seen the general form off~)H a g r e e s well
with f~rm~while f~)t4 does not catch the typical ridge for t between 3 and 5.
Figure 2(c) presents the marginal densities of amplitude f~m), f ~ ) , and f~) obtained
by integrating the joint densities.
5.2. J O N S W A P spectrum
In this section we shall compare the Cavani6-approximation f~-.)H and the m o d e l
approximation f~rm]-tfor some spectra which have been used to describe states of the
North Sea.
The J O N S W A P spectrum is a particular spectral form, designed to fit empirical data
from the North Sea; see e.g. Houmb and Overvik (1976). It can be fairly peaked, with
little energy on high and low frequencies, and it has the analytical form
f * ( h ) = agZh -5
exp( _ [~(~)--4),~exP(-(h-hrn)2/2~r2(h)h~k~0.2
)
(5.1)
Distribution of wave-length
423
Here g = 981 and
~(x) = era for X -< x,,
= Ub otherwise,
h,, is the approximate peak frequency, and 13 is chosen to be 1.25. The parameter a is a
pure energy parameter, while ~/is a so called "peak enhancement factor", which governs
the height of the peak relative to a standard spectrum. It is sometimes suggested to take
% ( = 0.07) ~: trb ( = 0.09) to permit different left and right spectral width. In the
examples below the spectrum was furthermore truncated at hma× = 1.25.
The parameters or, ~/, h,, can be chosen so as to impose a desired form on the spectrum,
and attempts have been made to relate these to the mean zero-crossing distance and
amplitude, see Houmb and Overvik (1976).
In the following examples the total energy and time scale have been standardized to
make ho = h2 = 1 giving the same mean zero crossing distance and mean average
wave-length and amplitude as in the previous example.
Since in (5.1) the parameters a and h,, will be determined by the standardization, only
the peak enhancement factor ~/ varies with e.
Three combinations of significant wave-height H~ -- 4X/h0 and zero crossing distance
Tz were chosen and corresponding parameter values taken from the table given by
Houmb and Overvik. The spectra were then normalized to have ho = h2 -- 1. The
original parameters and the resulting h4 and spectral width parameter are given in Table 1.
TABLE 1.
JONSWAP
SPECTRAL PARAMETERS FOR THE THREE EXAMPLES
n s
(a)
(b)
(c)
5-6
6-7
14-15
5-5.5
2.5-3.0
10-10.5
oL
,~
(in (5.1))
0.0323
6.49
0.0051
3.37
0.0031
1.00
~m
0.9362
0.7536
0.3077
~4
(a)-(c)
•
(after normalization)
1.047
0.212
1.141
0.352
1.787
0.664
Figures 3-5 (a,b) show level curves for f ~ ' ~ and f ~
for the three spectra with
successively increasing width. The agreement between Figs 3(a) and 3(b) is striking and
there is also a reasonable similarity between the two approximations in Fig. 4. The extra
probability found in Fig, 4(a) for T ~ 5, H ~ 2, not present in Fig. 4(b), has some
support in empirical data presented by Chakrabarti and Cooley (1977), Fig. 6. For the
wide banded case in Fig. 5 with e = 0.664, the approximations deviate considerably from
each other, the main difference being the emphasis that 5(a) lays on low wave-lengths
and amplitudes. The reason for this is probably that the Cavani6 approximation is based
on positive maxima only, thereby excluding some, but not all, of the shorter waves.
The marginal density of amplitude, f~") and f~) are shown in Fig. 3-5(c) for the
three cases. Again the agreement is excellent for e = 0.211, and good for e = 0.352, while
424
GEORG LINDGREN and IGOR RYCtlLIK
in the wide banded case with e = 0.664 the Cavani6 approximation has a greater content
of longer waves.
The density of the steepness S = vH/(2gT 2) is also obtained by integration, and the
results are shown in Figs 6(a)-(c). Note that f ~ ' ~ is in most cases based on the lower
bound f~ to the conditional wave-length density fr,,. Tberefore the total probability
found is slightly smaller than 1 in most cases. This is clearly seen in Fig. 6(b).
5.3.
Bimodal spectra
The approximation f~r~ is based on exact upper or lower bounds for the wave-length
density, and strict numerical limits can be calculated for the approximation error.
Therefore it can be used to approximate fTm for all kinds of spectra where these bounds
are small.
The Cavani~-approximation f~)H on the other hand, requires a narrow band spectrum
by construction, since it is based on one single approximating cosine function.
As was seen already for the broad banded J O N S W A P spectrum in case (c) in Section
5.2, the Cavani6 approximation has a tendency to reduce the content of short waves. One
other situation where it is not likely to work properly is in the case of a bimodal spectrum,
where two different main frequencies are mixed. The approximation f~r~, on the other
0.01
o
i
t
(a)
FIG. 3(a). Approximate density f~r~]-tof wavelength and amplitude; JONSWAP spectrum, spectral
width parameter e -- 0.212.
Distribution of wave-length
425
:o°o
i
2
3
4
5
t
(b)
FIG. 3(b).
Approximate density f~)H of wave-length and amplitude; JONSWAP
spectrum, spectral width parameter e = 0.212.
0.5'
0.2
Ol
i
2
3
4
5
,
6
7
8
,It
(cl
FIG. 3(c). A m p l i t u d e densities for J O N S W A P s p e c t r u m , spectral w i d t h p a r a m e t e r e = 0 . 2 1 2 : - = f~,,,,
___ = f~;!
426
GEORG LINDGREN and ]IGORRYCHLIK
4
x
3
2
,
~
~
.....
;
~
'
t
(o}
Fm. 4(a).
Approximate density f~r~ of wave-length and amplitude; JONSWAP
spectrum, spectral width parameter ~ = 0.352.
001
4
x
I
2
3
4
5
6
t
(b)
FIG. 4(b).
Approximate density f~'~. of wave-length and amplitude; JONSWAP
spectrum, spectral width parameter e = 0.352.
Distribution of wave-length
427
03
02
0
I
2
3
4
5
6
7
8
X
(c)
FIG. 4(c). Amplitude densities for JONSWAP spectrum; spectral width, parameter • = 0. 352;
-
f~!
f~.? ....
O.Ol
x
3
I
2
3
4
.5
t
(o)
FIG. 5(a).
Approximate density/'~r".~ of wave-length and amplitude; JONSWAP
spectrum, spectral width parameter • -- 0.664.
428
QEORG LtNDGREN a n d IGOR RYCHLIK
0oi
2
t
4
3
5
t
(b)
FIG. 5(b).
Approximate density f~)H of wave-length and amplitude; J O N S W A P
spectrum, spectral width parameter ~ = 0.664.
/
s
~..
\
,
0.2
\
xx
~\
!i
\x
I
\
0.1 ~
0
",
\\
2
3
~
4
X
5
6
7
8
(c)
FIG: 5(C). Amplitude densities for J O N S W A P and bimodal spectra '(see Section 5,3); spectral
width parameter ~ = 0.664,
_ f ~ m ) for J O N S W A P spectrum, - . . . . .
f~'}'~ for bimodal
spectrum,
_ f,~c) for J O N S W A P spectrum.
Distribution of wave-length
429
2O
I0
0
0.02
0 04
006
0,08
0.10
O. 12
s
(o)
FI~. 6(a).
Density of steepness based on fir~t (
) and on
J O N S W A P spectrum, ~ = 0.212.
f~'!H
(- - -);
I
4o}
30
50
""\'
]
20
20
~i
/
\
,of
\
i
002
004
006
008
0 ~0
0
\
0.02
s
(b)
FIGs. 6(b)-(c).
004
006
0 08
0 I0
s
,~ = 0 . 3 5 2
•
(c)
~=
Density of steepness based on f~-~ ( - - ) and on f ~ ! h ( - - - ) ;
J O N S W A P spectrum.
0.667
430
GEORG LINDGREN a n d IGOR RYCHL[K
hand can catch both long and short waves and thus reflect the full frequency content of the
process.
To illustrate this we have considered a bimodal spectrum of the mixed Gaussian type
(see Fig. 7),
1
.):el
1
m1 2)+
1
We have chosen the center frequencies m~ = 0.75, m2 = 1.62, with weights cl = 1 - c2 =
0.8, and % -- 0.0241, (re = 0.0294. This gives a spectral width p a r a m e t e r e = 0.664 and k4
= 1.787, which is identical with the case in Fig. 5 in Section 5.2 and close to the low
frequency noise in Section 5.1. Thus the Cavani6 approximation f ~ ) n is identical with
that in Fig. 5(b), while f ( r ~ has to be recomputed.
Figure 8 shows f~m~ for the bimodal Gaussian spectrum. Since here the process
consists of two superimposed narrow band c o m p o n e n t s one could p e r h a p s expect a
certain ambiguity as to what should be m e a n t by a "wave".
In fact, the wave-length as defined in Section 2, will be mainly d e t e r m i n e d by the fast
component and the distribution of T and H will therefore have its main part located
20
1.5
1.0
\
0.,5
05
-J
FIG.
7.
.
ib
i
15
......
20
Bimodal spectrum of the Gaussian type (--) and JONSWAP spectrum
( - - - ) ; ( = 0.664.
Distribution of wave-length
431
O.Ol
FIG. 8. Approximate density f~r~,~t of wave-length and amplitude for bimodal
Gaussian spectrum; ¢ = 0.664.
correspondingly. However, it will occasionally happen that the process will slip over the
first potential minimum point without developing a real trough and in that case the
minimum will be considerably delayed; see Fig. 9.
As is seen in Fig. 8 the joint density f~r~t will then also be bimodal, but of course the
Cavani6 approximation has no chance of catching this.
6. CONCLUSIONS
The approximation ]~r~ given by (3.3) and used in this paper is "exact" in the sense
that it gives mathematical bounds for the joint density fr, n of wave-length and
amplitude. It can be used for all types of spectra and depends on the full covariance
structure of the process.
FIc. 9. Possible wave-length ambiguity with bimodal spectrum.
432
GEORG LINDGREN and IGOR RYCHLIK
As have been reported by several authors, the Cavani6 approximation f~-!H given by
(4.5) fits well with empirical data for narrow band spectra with • not exceeding ca 0.7,
i.e. for normalized spectra with h4 -< 2. As is shown in Section 5.2 it also agrees well with
the exact bound f}rm~ at least for the narrow band J O N S W A P spectra with e -< 0.35.
For larger • it deviates from f~rm~4,but this can possibly be explained by the difference
in basic definition of a "wave". Considering its great simplicity it seems to be r e m a r k a b l y
accurate a n d safe to use for unimodal narrow banded spectra.
The regression approximation f ~ ) , based on the simplified model process (4.3)
depends on the full covariance structure. It is almost as simple and explicit as ft~),,
requires no more numerical work, and is accurate in all cases when the latter is. It is also
able to catch some details in the distribution for broad banded spectra which are not
shown by f~)/4, but has been observed in empirical data.
For complex spectra none of f ~ or f ~ is sufficiently accurate, and one should then
use the m o r e elaborate approximation f~r'~.~ even though it requires more numerical
work.
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