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Nonlinear effects on the distribution of crest-to-trough wave heights
Ocean Engng, Vol. 10, No 2, pp.97-106, 1983. Printed in Great Britain.
0029-8018/83/020097-10$03.00/0 Pergamon Press Ltd.
N O N L I N E A R EFFECTS ON T H E D I S T R I B U T I O N OF CREST-TO-TROUGH WAVE HEIGHTS M. A z I z TAYFUN Civil Engineering Department, College of Engineering & Petroleum, Kuwait University, P.O. Box 5969, Kuwait Abstract--The statistical distribution of the crest-to-trough heights of narrowband nonlinear sea waves is derived in a semi-closed form. A quantitative comparison of the resulting density and exceedanee probability distributions with those of the linear theory is given. It is shown that the nonlinearity of waves, even with steepnesses typical of extreme sea states, has an insignificant influence on the distribution of crest-to-trough heights. 1.
INTRODUCTION
FOR A LOW sea state in which waves are essentially linear and characterized with a narrow-band energy spectrum, the statistical distributions of both the crest-to-trough heights and the heights of wave crests above the mean sea level are well known. In particular, the distribution of wave crests is given theoretically by the Rayleigh law (Cartwright and Longuet-Higgins, 1956), and that of crest-to-trough heights is in general non-Rayleigh (Tayfun, 1981 and 1983). Unfortunately, ocean engineering applications quite often involve high sea states in which waves exhibit nonlinear characteristics. For the crest heights of narrow-band nonlinear waves, it has been shown (Tayfun, 1980 and 1982) that the inclusion of nonlinear effects significantly modifies the conventional probabilistic description depicted by the Rayleigh law. The general character of this modification is a spreading of probability mass towards the high crest tail in a m a n n e r consistent with the vertical asymmetry of nonlinear waves with narrower higher crests and wider shallower troughs. However, the corresponding effects of such nonlinearity on the distribution of crest-to-trough heights have not hitherto been investigated. Following Tayfun (1981 and 1983), the present study focuses on the possible effects of nonlinearity on the distribution of the crest-to-trough heights. Retaining the assumption that the sea surface is characterized with a narrow-band spectrum, the analysis proceeds in the next section with an analytic approximation to nonlinear sea waves. This approximation leads to the derivation of the distribution of nonlinear crest-to-trough heights in a semi-closed form and enables us to generate explicit results for comparison with the corresponding linear predictions. The study concludes with a brief summary of results in Section 3. 2.
DISTRIBUTION OF C R E S T - T O - T R O U G H H E I G H T S
2.1. Narrow-band approximation We consider a nonlinear field characterized to first-order by the spectrum S(to) as a function of angular frequency to. The ith spectral m o m e n t is given by 97
98
M. Aziz
TAYFUN
~e
P'i =
/
tois(a~) d~o .
(1)
° O
Of particular interest are ~Lothe first-order variance of the surface displacement, and tb = I~1/~o the mean frequency. The spectrum is narrow banded if (Longuet-Higgins, 1975):
1)2 =
~1'2
1.1,o(~2
1 << 1 .
(2)
Under this condition, it has been shown (Tayfun, 1980) that the surface displacement at a given point in deep water can be expressed as a function of time t in the form
Y(t) = A(t) cos(&-rd~) + ~
1
kAZ(t) cos 2(&t+d~) + o(v~.2/k)
(3)
where k ~o2/g with g being defined as the gravitational acceleration, d~(t) a random phase process with the uniform density 1/2~r in [0, 2"tr], A(t) the amplitude or envelope process, and ~ = k(2~o) ~ represents a measure of steepness or nonlinearity. The typical range of e in the J O N S W A P data is 0.044-0.177 (Huang et al., 1981), the larger values representing high seas. Equation (3) depicts the surface profile as an amplitude - modulated Stokian wave with a mean frequency & and phase d~. The leading term represents the first-order or linear profile whose maxima (crests) and minimia (troughs) lie on the envelopes, A(t) and - A ( t ) , respectively. The scaled amplitudes ~ = A/,4, where ,4 = (~rl~o/2) '/'-, are governed by the probability law corresponding to the invariant Rayleigh form: =
f~(O = ~-~i exp T
~2 , (~ _ O) .
(4)
The second term of Equation (3) concisely represents the physical effect of nonlinear corrections which introduce a vertical asymmetry to the linear profile by causing the crests to become narrower and higher, and the troughs longer and shallower. The crests and troughs associated with Y lie on the upper and lower envelopes defined, respectively, to o(v(2/k) by
A,(t) = A + --~-kA 2 ,
(5a)
At(t) = - A + ~ - k A 2 .
(5b)
Distribution of crest-to-trough wave heights
99
Hence, in contrast with the linear case, the envelopes Au and At are not symmetric with respect to the mean sea level. Both are displaced upwards by an equal amount (1/2)kA 2 in a manner consistent with the vertical asymmetry of the nonlinear profile Y. 2.2. Linear heights A crest-to-trough height is defined as the differential elevation between a wave crest at time, say, t and the proceeding trough at time t + ('r/2), where -r approximately corresponds to the local zero up-crossing wave period. Therefore, the linear crest-to-trough heights are given by
H,= A(t) + A(t +--~-) .
(6)
Normalizing this with respect to the mean height/')t = (2"n'0.o)'/:, we obtain HI =
¢I = H--~- ~
1
(7)
(¢1 + ¢2)
where
A(t) ¢1=
A
A(t + -r/2) '
¢:=
.4
(8)
The joint probability density of 61 and 62, given that r is fixed, is (Middleton, 1960): ( ~) f¢,¢_, 6t62;
7ox;L
~r2 6162 1o [ "trr~6z__ 4 1-r 2 [2(1 - r2)J
=
~ ,~_+ 4 1 - r 2 j (6i ->0) (9)
where Io designates the zero-order modified Bessel function, and r(+)
h
(+) (+t
O
= (92 + h2) '~ ,
= -p.o I
f
(lOa)
S(to) sin(to - &) ~
•
o
= 0.0 1
f o
S(to) cos(to-&) ~
•
dco
dto .
(lOb)
(10c)
Utilizing these in Equation (7), Tayfun (1981 and 1983) has demonstrated that the probability density of ~1 can be expressed as:
tO0
M. AzIz TAYFUN
f~, (~) = 2I; ~'f.('r) f~,~: (2~i-~2, "r
(11)
42; ~--)d42 d r .
O
where (i - 0, and f, represents the probability density of zero up-crossing periods such as that given by Longuet-Higgins (1975). As v 2 ---, 0, S(to) and f, tend to behave as pseudo-delta functions centered at to = ~b, and -r = ~, respectively. On this basis, Equation (11) reduces to
f;i (~I) = 2
(12)
f~,'~2 ( 2~I-42' 42;-~-) d42 • o
Similarly, Equations (10 a, b, and c) become
r
(~_)- p(@) = 1 - ~ - r 1r e v
2,andx.
(~__)= 0
.
(13)
The validities of the preceding set of approximations relative to the corresponding exact expressions have been verified in terms of a parametric spectrum with a variable band width. In particular, it is noteworthy that Equation (12) is remarkably accurate for v: -< 0.04, showing an error of less than 2% relative to Equation (11) for values of ~t in the interval [0.25, 3.50], which corresponds to at least 98% of the total probability mass (Tayfun, 1983). 2.3. Nonlinear heights The crest-to-trough height in the nonlinear case is defined by
Hn = A.(t) - At(t+ -~-) (14)
= At + A2 + ~ - k ( A ~ - A~) ,
where At = A(t) and A2 = A(t + "r/2). Normalizing Hu with respect to the mean height /-?H = /-ii = (2rrl~o) '/2 yields, to o(ve), Hn _ ~11- HH
1 _~ 2 (41 + 42) +
e(42-
42),
(15)
where 41 and 42 are given by Equation (8). The leading term of ~11 is identical with ~l appropriate to linear waves. Therefore, the remaining quadratic term represents the second-order nonlinear correction to o(e).
Distribution of crest-to-trough wave heights
101
The joint probability density of ~n and 62, for r fixed, follows from the standard relation
(
+)
(
.6,
where
61 = h(~ii, 62) --"
eN,/,rr + [eX/'tr
62
eX/'rr
'
s = 0 ( 6 , , 6 2 ) = a__Lh o(~,x,~2) at,, "
(17) (18)
Therefore,
f~,, ~2 ~II,62,-~-
= ~f~,~2
h(~II,62),62;2 -
(19)
"
An integration of the preceding expression with respect to 62 gives the conditional density of 4ii:
f~l, ~II;
----"
t~i I f~,~2 b.~II,62),62;~--
d62.
(20)
,~2~0
Multiplying the preceding expression with f, and integrating out ~" yield the marginal density of ~xl in the form:
f~H (~II) = /I.~,_> 0 d~i--~f,(r) f~,~2 h([n,62),~2;
d62 dr .
(21)
Now, we appeal to the narrow-band approximation (v 2 < < 1) to suggest that f, is a pseudo-delta function centered at r=÷. This then simplifies Equation (21) to
I
Oh
[h(~ix,62),62; + ]
d62
(22)
.~2->0
Finally, we note that the nonnegativity conditions on ~1=h(~ii,62) and 62 impose a split-structure o n ~ii and enable us to rewrite Equation (22) as:
102
M. AZlZTAYFUN Oh
. h
÷
f~" (~II)~'~ ; -'4- f ~Ilf~'~"[ (~II'~2)'~2;-2"-] d~2 o c~+
(23)
,
where
2 [1 + (1-2~/~r ~ ~n)~], if 0 < ~u < 1 a± - ~X/~r - 2~/w 2 1 ~ / ~ r ' if ~n > 2eX/rr
2.4.
(24)
Comparison
A comparison between the distributions of ~ and ~II follows from the numerical intergration of Equations (12) and (23) for fixed values of the parameters v 2 and e. However, so far as the corresponding first-order statistics are concerned, these can be derived in closed-form directly from the definitions of ~i and ~n as given by Equations (7) and (15) respectively, and by utilizing (Middleton, 1960): .
{~n {~ =
m+n
~
r 1 +
l" 1 +
2F1
m
2 '
n
2 ' 1, r 2
,
(25)
where 2Fl represents the hypergeometric function. In particular, noting that ~ = ~u = 1, it can be shown that
Var(t~i) =
2
--
1 +
~-1 2F1(-1/2, -1/2; 1; r2),
Var(~ll) = Var(~i) + ~1 ~2 (l-r:)
(26a)
(26b)
One can now utilize Equation (13) to obtain
A = Var(~n) - Var(~0 =
~_ ~2v2(2-- I/2T¢2p2) ~ y"IT ~2112 -
(27)
Setting ~ - 0.25 and v 2 - 0.04 as a possible extreme case within the validity of the results generated in this study, Equation (27) shows that A is less than 0.4% Of Var (~0. It is therefore suggested that the first-order probability structures of ~ and i~n differ very little from one another. In order to demonstrate this more explicitly, the densities f~, and f;,,, and the corresponding exceedance probability distributions defined by
Distribution of crest-to-trough wave heights
103
09
I/
!
0.6
t
\
Io
\
11
03
I •
i
//
• ,0.25
7 FIG. 1.
I
I t
2
Comparison of densities offe, f~j and f~,, for e = 0.25 and v 2 = 0.041.
F~,(~,) = 1 -
f~, (u) du
(28a)
O
F~,, (~n) -- 1 -
f~,, (u) du
(28b)
o
were e v a l u a t e d numerically for v 2 = 0.041 and v 2 = 0.016, which c o r r e s p o n d to two of several cases c o n s i d e r e d p r e v i o u s l y ( T a y f u n , 1983). In the case o f t n , the p a r a m e t e r e varied f r o m 0.05 to 0.25 in i n c r e m e n t s of 0.05. T h e c o m p a r i s o n s of results in each case c o n f i r m e d definitely that the first-order p r o b a b i l i t y structure of 4,, is nearly the s a m e as that of ~. F o r illustrative p u r p o s e s , the c o m p a r i s o n s b e t w e e n f~, and f~,, for v 2 = 0.041 and 0.016, b o t h c o r r e s p o n d i n g to • = 0.25 as the worst possible case, are given in Figs 1 and 2. respectively. It is worthwhile to n o t e that, for e = 0.25, and v 2 = 0.041. m a x [fg,, f;,I is less than _+3% off~, for gx - values in the interval [0.225, 2.25]. T h e a r e a u n d e r f~., o v e r this interval r e p r e s e n t s a p p r o x i m a t e l y 98% of the total p r o b a b i l i t y mass. A similar analysis of Fig. 2 with e = 0.25 and v 2 = 0.016 indicates that max~eg,, - f~,] in this case is less than 1% off~, o v e r the s a m e interval. T h e insignificance of these differences is also
104
M.
Azlz
TAYEUN
09
OG
7, O
\
r~ 03 I
• 0.25
0
~
I I
,0.016
L 2
tl..._ _ 3
FIG. 2. Comparison of densitiesre, ft, and re, for e = 0.25 and ~2 = 0.016.
reflected in the corresponding exceedance probability predictions illustrated in Figs 3 and 4, respectively. Finally, we note that the density f~ and exceedance probability distribution F~ appropriate to the Rayleigh law are also included in all the figures here for completeness. The detailed comparisons of Var (~), f~, and F~ with Var (~i), f;,, and F;, are given elsewhere (Tayfun, 1981 and 1983) and need not be pursued any further in the present study. 3.
DISCUSSION AND CONCLUSIONS
In the deterministic Stokes wave theory, it is well known that the second-order nonlinear effects impose a vertical asymmetry on the first-order wave profile, causing the crests of waves to become higher narrower, and the troughs longer shallower. However, the vertical asymmetry is uniform over a typical cycle such that the extremes of the wave crest and trough are displaced upwards equally. As a result, the first-order crest-to-trough height remains unaffected by the nonlinear asymmetry. An extension of the preceding argument to irregular sea waves can not immediately lead to the same conclusion since the wave amplitude is modulated and varies in time.
Distribution of crest-to-trough wave heights
105
o
io-2
~2 ~O.016
:0.25
0
L
I
I
2
~' ~;i'~ FIc. 3.
Comparison of exceedance probability distributions F~, F~ and F ~ for c = 0.25 and v 2 = 0.041.
Therefore, in contrast with the deterministic case, the vertical asymmetry is nonuniform over a typical wave cycle so that the extremal crest and trough points in general have unequal upward displacements. The effect of this differential displacement on the first-order crest-to-trough height is represented by (l/2)k ( A 2 - A 2) = k A a v g ~ A , where Aavg = (A1 + A2)/2, and ~A = At - A 2 may be regarded as average and differential amplitudes over half a local cycle, respectively. Evidently, 8A is unrestricted in sign, and kAa~g is a measure of local wave steepness or nonlinearity. Moreover, it can be shown that 8A is o(v) so that ~A ~ 0 as v 2 ~ O. Therefore, under narrow-band conditions, it is reasonable to suggest that the net effect of nonlinearity on the first-order crest-to-trough heights would be negligible. In fact, the results presented in this study verify the validity of this heuristic argument and show definitely that the nonlinearity of narrow-band waves, even with steepnesses typical of extreme sea states, has for all practical purposes an insignificant influence on the distribution of crest-to-trough heights. study was supported by the Research Council of Graduate Studies. Kuwait University, under grant EV017. Thanks are due to Daisy Mathew for the preparation of the manuscript.
Acknowledgements--This
106
M. Aztz TAYFUN ioo
\ x"
e,
~5
g o
c o
10-2
tad
e" =0.2,5
YZ=O.041
1 I
FIG. 4.
I 2
i 3
Comparison of exceedance probability distributions F~, F+r and F+tt for ~ = 0.25 and v 2 = 0.016.
REFERENCES
D.E. and LONGUET-HIGGINS, M.S. 1956. The statistical distribution of the maxima of a random function. Proc. R. Soc. 237A, 213--232. HUANG, N.E., LONG, S.R. and BLWEN, L.F. 1981. On the importance of the significant slope in empirical wind-wave studies. J. Phys. Oceanography 11,569-573. LONGUET-HtGGINS, M.S. 1975. On the joint distribution of the periods and amplitudes of sea waves. J. geophys. Res. ~ 1 8 ) , 2688-2694. MIDOLETON, D. 1960 An Introduction to Statistical Communication Theory, pp. 396-436. McGraw-Hill, New York. TAYFUN, M,A. 1980. Narrow-band nonlinear sea waves. J. geophys. Res. 85(C3), 1548-1552. TAYFON, M.A. 1981. Distribution of crest-to-trough wave heights. J. Wat.Ways Port Coastal Ocean Div, Am. Soc. civ. Engrs. 10"/(WW3), 149-158. TAYFUN, M,A. 1982. Distribution of nonlinear crest heights. Submitted to J. Wat. Ways Port Coastal Ocean Div. Am. Soc. cir. Engrs. TAYFUN, M.A. 1983. Effects of spectrum band width on the distribution of wave heights and periods. Ocean Engng. 10, 107-118. CARTWRIGHT,
0029-8018/83/020097-10$03.00/0 Pergamon Press Ltd.
N O N L I N E A R EFFECTS ON T H E D I S T R I B U T I O N OF CREST-TO-TROUGH WAVE HEIGHTS M. A z I z TAYFUN Civil Engineering Department, College of Engineering & Petroleum, Kuwait University, P.O. Box 5969, Kuwait Abstract--The statistical distribution of the crest-to-trough heights of narrowband nonlinear sea waves is derived in a semi-closed form. A quantitative comparison of the resulting density and exceedanee probability distributions with those of the linear theory is given. It is shown that the nonlinearity of waves, even with steepnesses typical of extreme sea states, has an insignificant influence on the distribution of crest-to-trough heights. 1.
INTRODUCTION
FOR A LOW sea state in which waves are essentially linear and characterized with a narrow-band energy spectrum, the statistical distributions of both the crest-to-trough heights and the heights of wave crests above the mean sea level are well known. In particular, the distribution of wave crests is given theoretically by the Rayleigh law (Cartwright and Longuet-Higgins, 1956), and that of crest-to-trough heights is in general non-Rayleigh (Tayfun, 1981 and 1983). Unfortunately, ocean engineering applications quite often involve high sea states in which waves exhibit nonlinear characteristics. For the crest heights of narrow-band nonlinear waves, it has been shown (Tayfun, 1980 and 1982) that the inclusion of nonlinear effects significantly modifies the conventional probabilistic description depicted by the Rayleigh law. The general character of this modification is a spreading of probability mass towards the high crest tail in a m a n n e r consistent with the vertical asymmetry of nonlinear waves with narrower higher crests and wider shallower troughs. However, the corresponding effects of such nonlinearity on the distribution of crest-to-trough heights have not hitherto been investigated. Following Tayfun (1981 and 1983), the present study focuses on the possible effects of nonlinearity on the distribution of the crest-to-trough heights. Retaining the assumption that the sea surface is characterized with a narrow-band spectrum, the analysis proceeds in the next section with an analytic approximation to nonlinear sea waves. This approximation leads to the derivation of the distribution of nonlinear crest-to-trough heights in a semi-closed form and enables us to generate explicit results for comparison with the corresponding linear predictions. The study concludes with a brief summary of results in Section 3. 2.
DISTRIBUTION OF C R E S T - T O - T R O U G H H E I G H T S
2.1. Narrow-band approximation We consider a nonlinear field characterized to first-order by the spectrum S(to) as a function of angular frequency to. The ith spectral m o m e n t is given by 97
98
M. Aziz
TAYFUN
~e
P'i =
/
tois(a~) d~o .
(1)
° O
Of particular interest are ~Lothe first-order variance of the surface displacement, and tb = I~1/~o the mean frequency. The spectrum is narrow banded if (Longuet-Higgins, 1975):
1)2 =
~1'2
1.1,o(~2
1 << 1 .
(2)
Under this condition, it has been shown (Tayfun, 1980) that the surface displacement at a given point in deep water can be expressed as a function of time t in the form
Y(t) = A(t) cos(&-rd~) + ~
1
kAZ(t) cos 2(&t+d~) + o(v~.2/k)
(3)
where k ~o2/g with g being defined as the gravitational acceleration, d~(t) a random phase process with the uniform density 1/2~r in [0, 2"tr], A(t) the amplitude or envelope process, and ~ = k(2~o) ~ represents a measure of steepness or nonlinearity. The typical range of e in the J O N S W A P data is 0.044-0.177 (Huang et al., 1981), the larger values representing high seas. Equation (3) depicts the surface profile as an amplitude - modulated Stokian wave with a mean frequency & and phase d~. The leading term represents the first-order or linear profile whose maxima (crests) and minimia (troughs) lie on the envelopes, A(t) and - A ( t ) , respectively. The scaled amplitudes ~ = A/,4, where ,4 = (~rl~o/2) '/'-, are governed by the probability law corresponding to the invariant Rayleigh form: =
f~(O = ~-~i exp T
~2 , (~ _ O) .
(4)
The second term of Equation (3) concisely represents the physical effect of nonlinear corrections which introduce a vertical asymmetry to the linear profile by causing the crests to become narrower and higher, and the troughs longer and shallower. The crests and troughs associated with Y lie on the upper and lower envelopes defined, respectively, to o(v(2/k) by
A,(t) = A + --~-kA 2 ,
(5a)
At(t) = - A + ~ - k A 2 .
(5b)
Distribution of crest-to-trough wave heights
99
Hence, in contrast with the linear case, the envelopes Au and At are not symmetric with respect to the mean sea level. Both are displaced upwards by an equal amount (1/2)kA 2 in a manner consistent with the vertical asymmetry of the nonlinear profile Y. 2.2. Linear heights A crest-to-trough height is defined as the differential elevation between a wave crest at time, say, t and the proceeding trough at time t + ('r/2), where -r approximately corresponds to the local zero up-crossing wave period. Therefore, the linear crest-to-trough heights are given by
H,= A(t) + A(t +--~-) .
(6)
Normalizing this with respect to the mean height/')t = (2"n'0.o)'/:, we obtain HI =
¢I = H--~- ~
1
(7)
(¢1 + ¢2)
where
A(t) ¢1=
A
A(t + -r/2) '
¢:=
.4
(8)
The joint probability density of 61 and 62, given that r is fixed, is (Middleton, 1960): ( ~) f¢,¢_, 6t62;
7ox;L
~r2 6162 1o [ "trr~6z__ 4 1-r 2 [2(1 - r2)J
=
~ ,~_+ 4 1 - r 2 j (6i ->0) (9)
where Io designates the zero-order modified Bessel function, and r(+)
h
(+) (+t
O
= (92 + h2) '~ ,
= -p.o I
f
(lOa)
S(to) sin(to - &) ~
•
o
= 0.0 1
f o
S(to) cos(to-&) ~
•
dco
dto .
(lOb)
(10c)
Utilizing these in Equation (7), Tayfun (1981 and 1983) has demonstrated that the probability density of ~1 can be expressed as:
tO0
M. AzIz TAYFUN
f~, (~) = 2I; ~'f.('r) f~,~: (2~i-~2, "r
(11)
42; ~--)d42 d r .
O
where (i - 0, and f, represents the probability density of zero up-crossing periods such as that given by Longuet-Higgins (1975). As v 2 ---, 0, S(to) and f, tend to behave as pseudo-delta functions centered at to = ~b, and -r = ~, respectively. On this basis, Equation (11) reduces to
f;i (~I) = 2
(12)
f~,'~2 ( 2~I-42' 42;-~-) d42 • o
Similarly, Equations (10 a, b, and c) become
r
(~_)- p(@) = 1 - ~ - r 1r e v
2,andx.
(~__)= 0
.
(13)
The validities of the preceding set of approximations relative to the corresponding exact expressions have been verified in terms of a parametric spectrum with a variable band width. In particular, it is noteworthy that Equation (12) is remarkably accurate for v: -< 0.04, showing an error of less than 2% relative to Equation (11) for values of ~t in the interval [0.25, 3.50], which corresponds to at least 98% of the total probability mass (Tayfun, 1983). 2.3. Nonlinear heights The crest-to-trough height in the nonlinear case is defined by
Hn = A.(t) - At(t+ -~-) (14)
= At + A2 + ~ - k ( A ~ - A~) ,
where At = A(t) and A2 = A(t + "r/2). Normalizing Hu with respect to the mean height /-?H = /-ii = (2rrl~o) '/2 yields, to o(ve), Hn _ ~11- HH
1 _~ 2 (41 + 42) +
e(42-
42),
(15)
where 41 and 42 are given by Equation (8). The leading term of ~11 is identical with ~l appropriate to linear waves. Therefore, the remaining quadratic term represents the second-order nonlinear correction to o(e).
Distribution of crest-to-trough wave heights
101
The joint probability density of ~n and 62, for r fixed, follows from the standard relation
(
+)
(
.6,
where
61 = h(~ii, 62) --"
eN,/,rr + [eX/'tr
62
eX/'rr
'
s = 0 ( 6 , , 6 2 ) = a__Lh o(~,x,~2) at,, "
(17) (18)
Therefore,
f~,, ~2 ~II,62,-~-
= ~f~,~2
h(~II,62),62;2 -
(19)
"
An integration of the preceding expression with respect to 62 gives the conditional density of 4ii:
f~l, ~II;
----"
t~i I f~,~2 b.~II,62),62;~--
d62.
(20)
,~2~0
Multiplying the preceding expression with f, and integrating out ~" yield the marginal density of ~xl in the form:
f~H (~II) = /I.~,_> 0 d~i--~f,(r) f~,~2 h([n,62),~2;
d62 dr .
(21)
Now, we appeal to the narrow-band approximation (v 2 < < 1) to suggest that f, is a pseudo-delta function centered at r=÷. This then simplifies Equation (21) to
I
Oh
[h(~ix,62),62; + ]
d62
(22)
.~2->0
Finally, we note that the nonnegativity conditions on ~1=h(~ii,62) and 62 impose a split-structure o n ~ii and enable us to rewrite Equation (22) as:
102
M. AZlZTAYFUN Oh
. h
÷
f~" (~II)~'~ ; -'4- f ~Ilf~'~"[ (~II'~2)'~2;-2"-] d~2 o c~+
(23)
,
where
2 [1 + (1-2~/~r ~ ~n)~], if 0 < ~u < 1 a± - ~X/~r - 2~/w 2 1 ~ / ~ r ' if ~n > 2eX/rr
2.4.
(24)
Comparison
A comparison between the distributions of ~ and ~II follows from the numerical intergration of Equations (12) and (23) for fixed values of the parameters v 2 and e. However, so far as the corresponding first-order statistics are concerned, these can be derived in closed-form directly from the definitions of ~i and ~n as given by Equations (7) and (15) respectively, and by utilizing (Middleton, 1960): .
{~n {~ =
m+n
~
r 1 +
l" 1 +
2F1
m
2 '
n
2 ' 1, r 2
,
(25)
where 2Fl represents the hypergeometric function. In particular, noting that ~ = ~u = 1, it can be shown that
Var(t~i) =
2
--
1 +
~-1 2F1(-1/2, -1/2; 1; r2),
Var(~ll) = Var(~i) + ~1 ~2 (l-r:)
(26a)
(26b)
One can now utilize Equation (13) to obtain
A = Var(~n) - Var(~0 =
~_ ~2v2(2-- I/2T¢2p2) ~ y"IT ~2112 -
(27)
Setting ~ - 0.25 and v 2 - 0.04 as a possible extreme case within the validity of the results generated in this study, Equation (27) shows that A is less than 0.4% Of Var (~0. It is therefore suggested that the first-order probability structures of ~ and i~n differ very little from one another. In order to demonstrate this more explicitly, the densities f~, and f;,,, and the corresponding exceedance probability distributions defined by
Distribution of crest-to-trough wave heights
103
09
I/
!
0.6
t
\
Io
\
11
03
I •
i
//
• ,0.25
7 FIG. 1.
I
I t
2
Comparison of densities offe, f~j and f~,, for e = 0.25 and v 2 = 0.041.
F~,(~,) = 1 -
f~, (u) du
(28a)
O
F~,, (~n) -- 1 -
f~,, (u) du
(28b)
o
were e v a l u a t e d numerically for v 2 = 0.041 and v 2 = 0.016, which c o r r e s p o n d to two of several cases c o n s i d e r e d p r e v i o u s l y ( T a y f u n , 1983). In the case o f t n , the p a r a m e t e r e varied f r o m 0.05 to 0.25 in i n c r e m e n t s of 0.05. T h e c o m p a r i s o n s of results in each case c o n f i r m e d definitely that the first-order p r o b a b i l i t y structure of 4,, is nearly the s a m e as that of ~. F o r illustrative p u r p o s e s , the c o m p a r i s o n s b e t w e e n f~, and f~,, for v 2 = 0.041 and 0.016, b o t h c o r r e s p o n d i n g to • = 0.25 as the worst possible case, are given in Figs 1 and 2. respectively. It is worthwhile to n o t e that, for e = 0.25, and v 2 = 0.041. m a x [fg,, f;,I is less than _+3% off~, for gx - values in the interval [0.225, 2.25]. T h e a r e a u n d e r f~., o v e r this interval r e p r e s e n t s a p p r o x i m a t e l y 98% of the total p r o b a b i l i t y mass. A similar analysis of Fig. 2 with e = 0.25 and v 2 = 0.016 indicates that max~eg,, - f~,] in this case is less than 1% off~, o v e r the s a m e interval. T h e insignificance of these differences is also
104
M.
Azlz
TAYEUN
09
OG
7, O
\
r~ 03 I
• 0.25
0
~
I I
,0.016
L 2
tl..._ _ 3
FIG. 2. Comparison of densitiesre, ft, and re, for e = 0.25 and ~2 = 0.016.
reflected in the corresponding exceedance probability predictions illustrated in Figs 3 and 4, respectively. Finally, we note that the density f~ and exceedance probability distribution F~ appropriate to the Rayleigh law are also included in all the figures here for completeness. The detailed comparisons of Var (~), f~, and F~ with Var (~i), f;,, and F;, are given elsewhere (Tayfun, 1981 and 1983) and need not be pursued any further in the present study. 3.
DISCUSSION AND CONCLUSIONS
In the deterministic Stokes wave theory, it is well known that the second-order nonlinear effects impose a vertical asymmetry on the first-order wave profile, causing the crests of waves to become higher narrower, and the troughs longer shallower. However, the vertical asymmetry is uniform over a typical cycle such that the extremes of the wave crest and trough are displaced upwards equally. As a result, the first-order crest-to-trough height remains unaffected by the nonlinear asymmetry. An extension of the preceding argument to irregular sea waves can not immediately lead to the same conclusion since the wave amplitude is modulated and varies in time.
Distribution of crest-to-trough wave heights
105
o
io-2
~2 ~O.016
:0.25
0
L
I
I
2
~' ~;i'~ FIc. 3.
Comparison of exceedance probability distributions F~, F~ and F ~ for c = 0.25 and v 2 = 0.041.
Therefore, in contrast with the deterministic case, the vertical asymmetry is nonuniform over a typical wave cycle so that the extremal crest and trough points in general have unequal upward displacements. The effect of this differential displacement on the first-order crest-to-trough height is represented by (l/2)k ( A 2 - A 2) = k A a v g ~ A , where Aavg = (A1 + A2)/2, and ~A = At - A 2 may be regarded as average and differential amplitudes over half a local cycle, respectively. Evidently, 8A is unrestricted in sign, and kAa~g is a measure of local wave steepness or nonlinearity. Moreover, it can be shown that 8A is o(v) so that ~A ~ 0 as v 2 ~ O. Therefore, under narrow-band conditions, it is reasonable to suggest that the net effect of nonlinearity on the first-order crest-to-trough heights would be negligible. In fact, the results presented in this study verify the validity of this heuristic argument and show definitely that the nonlinearity of narrow-band waves, even with steepnesses typical of extreme sea states, has for all practical purposes an insignificant influence on the distribution of crest-to-trough heights. study was supported by the Research Council of Graduate Studies. Kuwait University, under grant EV017. Thanks are due to Daisy Mathew for the preparation of the manuscript.
Acknowledgements--This
106
M. Aztz TAYFUN ioo
\ x"
e,
~5
g o
c o
10-2
tad
e" =0.2,5
YZ=O.041
1 I
FIG. 4.
I 2
i 3
Comparison of exceedance probability distributions F~, F+r and F+tt for ~ = 0.25 and v 2 = 0.016.
REFERENCES
D.E. and LONGUET-HIGGINS, M.S. 1956. The statistical distribution of the maxima of a random function. Proc. R. Soc. 237A, 213--232. HUANG, N.E., LONG, S.R. and BLWEN, L.F. 1981. On the importance of the significant slope in empirical wind-wave studies. J. Phys. Oceanography 11,569-573. LONGUET-HtGGINS, M.S. 1975. On the joint distribution of the periods and amplitudes of sea waves. J. geophys. Res. ~ 1 8 ) , 2688-2694. MIDOLETON, D. 1960 An Introduction to Statistical Communication Theory, pp. 396-436. McGraw-Hill, New York. TAYFUN, M,A. 1980. Narrow-band nonlinear sea waves. J. geophys. Res. 85(C3), 1548-1552. TAYFON, M.A. 1981. Distribution of crest-to-trough wave heights. J. Wat.Ways Port Coastal Ocean Div, Am. Soc. civ. Engrs. 10"/(WW3), 149-158. TAYFUN, M,A. 1982. Distribution of nonlinear crest heights. Submitted to J. Wat. Ways Port Coastal Ocean Div. Am. Soc. cir. Engrs. TAYFUN, M.A. 1983. Effects of spectrum band width on the distribution of wave heights and periods. Ocean Engng. 10, 107-118. CARTWRIGHT,