TECHNICAL REPORT CERC-91-7
US Army Corps WAVE HEIGHT DISTRIBUTIONS
of Engineers IN MULTIPLE-PEAKED SEAS

by
Charles E. Long
Coastal Engineering Research Center

DEPARTMENT OF THE ARMY
Waterways Experiment Station, Corps of Engineers
3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199

July 1991
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July 1991 Final report

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Wave Height Distributions in Multiple-Peaked Seas

Civil Works Research
Work Unit 32484

6. AUTHOR(S)

Charles E. Long

8. PERFORMING ORGANIZATION
REPORT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)
USAE Waterways Experiment Station
Coastal Engineering Research Center
3909 Halls Ferry Road
Vicksburg, MS 39180-6199

Technical Report
CERC- 91-7

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US Army Corps of Engineers
Washington, DC 20314-1000

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Available from National Technical Information Service, 5285 Port Royal
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13. ABSTRACT (Maximum 200 words)

Knowledge of distributions of heights of sea waves under realistic
oceanic conditions is critical to the successful design of coastal shore
protection projects. In this report, a simple examination is made of wave
height distributions in sea states characterized by energy spectra with
multiple peaks, i.e., having energy centered at two or more distinct
frequencies. Sea states with broad frequency spectra are also included
since such seas are composed of waves of diverse frequencies. These cases
violate the assumptions of unimodal, narrow spectra that are formally
required for the conventionally used Rayleigh distribution of wave heights
to apply.

Reported here are tests of the Rayleigh and Modified Rayleigh wave
height distribution models. These models are compared with wave heights

(Continued)

15. NUMBER OF PAGES
63
16. PRICE CODE

20. LIMITATION OF ABSTRACT

14. SUBJECT TERMS
Height distributions Wave heights
Multimodal spectra Wind waves

19. SECURITY CLASSIFICATION
OF ABSTRACT

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)
Prescribed by ANSI Std. 239-18
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13. (Concluded).

from idealized, synthetic time series having spectra with variable widths,
modes, and modal separations. The Modified Rayleigh model is the two-
parameter, deepwater asymptotic form of the Beta-Rayleigh distribution.

The first part of this report involves determining some of the limits
implied by the phrase "unimodal, narrow spectrum" in the synthesis of time
series having Rayleigh distributions of wave heights. It is found that for
unimodal, band-limited, white spectra, there are constraints on both the
overall bandwidth and the number of component waves for a close approxima-
tion to a Rayleigh process to occur.

The primary part of this investigation is examination of wave height
distributions in wave signals derived from multimodal spectra. Using cri-
teria established in the unimodal tests, bimodal spectra are constructed
from two unimodal spectra, each of which yields a Rayleigh wave height
distribution by itself. The two primary variables in these bimodal tests
are a measure of modal separation in the frequency domain and the relative
amount of energy in one mode compared with the other.

Spectra with broad ranges of modal separation parameter and with a
ratio of low-frequency modal energy to high-frequency modal energy roughly
greater than 1.0 have wave height distributions that are well-represented
by the Modified Rayleigh model. The Rayleigh model tends to overpredict
wave height averages for these parametric ranges in much the same way as
was found for time series derived from broad-banded spectra and as have
been found frequently in natural observations. For large modal separations
and small high-frequency modal energy relative to low-frequency modal en-
ergy, synthetic data deviate strongly from the Rayleigh model and deviate
significantly from the Modified Rayleigh model. Further study is needed
to determine the relative importance of such conditions. Further research
is also required to determine the parameters of the Modified Rayleigh
model in terms of spectral parameters. Finally, these conclusions need to
be tested with natural data to ensure that the artifice of synthesis has
not biased the results.

PREFACE

The study in this report was authorized by Headquarters, US Army Corps
of Engineers (HQUSACE), Coastal Engineering Area of Civil Works Research and
Development. Work was performed under Civil Works Research Work Unit 32484,
"Directionality of Waves in Shallow Water," Coastal Flooding Program, at the
Coastal Engineering Research Center (CERC) of the US Army Engineer Waterways
Experiment Station (WES). The HQUSACE Technical Monitors were Messrs. John H.
Lockhart, Jr.; John G. Housley; James E. Crews; and Robert H. Campbell.

Dr. C. Linwood Vincent was the CERC Program Manager.

This investigation was conducted by Dr. Charles E. Long at the WES/CERC
Field Research Facility (FRF) in Duck, NC. The purpose of this study was to
help clarify the behavior of wave height statistics for sea states character-
ized by broad or multimodal spectra.

The report was prepared under the direct supervision of Mr. William A.
Birkemeier, Chief, FRF, and under the general supervision of Mr. Thomas W.
Richardson, Chief, Engineering Development Division; Mr. Charles C. Calhoun,
Jr., Assistant Chief, CERC; and Dr. James R. Houston, Chief, CERC. This
report was edited by Ms. Lee T. Byrne, Information Technology Laboratory, WES.

Commander and Director of WES during publication of this report was

COL Larry B. Fulton, EN. Dr. Robert W. Whalin was Technical Director.

CONTENTS

Page
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Rayleigh Model .. eB ae ere ONE SV UNS ORE Nisa L IM GREY Noes ae Esc e a 9
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Additional Parametersimi. ai oo seek, Mere MeN lh aklisalh nel rsilise chen Uiciens. oti) ay RELY RP il

PART “isk: TEST “DATA GENERATION ye) ofa Ura lee res ache ae ie) ane Set tt se rye eae 14
Inverse Fourier Transform=Lechnique::) 25 050 glue eases ee 14
Unimodal Data... bor Blair WePhhop hg, (BRU hak. Ae ale Vea Il tee fe Dee.
Empirical Probability iSeuuneee Sy PRIS RR ER RUTENT Soy GMa. dca lc ag eee ae aS 24
Bimodal Test. DatayGenerattome: gh sis) es til want cna teak cohen ne athe mea 27

PART LV’: UN EMODATE "EES DSH Ses ae el ies ae IE es ails Le Vintec beta oet Ae a OMe ale 29)
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Test Results ... RR SEINE OLE LU TREN Ree OE Aa a Sei DS)
Implications for Spectral Filtering tn pRaReA aU Smee tra am Sees ern ERM, Ie, alten 38

PART V: BIMODAIE ODE SS ce ieee 08 ey a) phim Geer ah) we Raalcis ot tent aan at cr Yaa 41

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PART VI: CONGLUSTON 5) eset i, cee BP OSes Oy eral a ees 3)
REEERENGES© Aa. 0sc sol ten come eas ahi cigs Yee lGsy whey a cal fe aticntac ieee Mcaile OS (cian neh bee igre aaeserM Lagi cee a 56

APPENDIX, Ax NOTATION: 4 ey sip echt vii dats de, hana tent ayllteey alae la hy tees ah eee as OCU A Al

WAVE HEIGHT DISTRIBUTIONS IN MULTIPLE-PEAKED SEAS

PART I: INTRODUCTION

1. Successful design of shore protection projects depends critically on
knowledge of water level variation. Sea surface elevations change because of
a suite of processes that include tides, storm surge, and wind waves. Wind
waves do significant amounts of work on coastal boundaries, in general. Of
particular importance is that water level changes associated with heights of
wind waves contribute to extremes of beach and structural runup and hence to
high beach erosion rates or potential for overtopping and flooding behind
coastal defenses. For engineering design purposes, it is useful to have a
statistical description of the heights of sea waves so that probabilities can
be assigned to particular water level extremes. With these probabilities, a
likelihood of project survival can be estimated. Ideally, probable distribu-
tions of wave heights in a given sea state can be described with a mathemati-
cal model. An important problem in coastal engineering is the determination
of a model that can represent realistic sea states with reasonable fidelity.

2. One of the earliest and most widely used models is the Rayleigh
probability density function (pdf), first applied in ocean work by Longuet-
Higgins (1952). It is a one-parameter model, relying only on specification of
a root-mean-square (RMS) wave height to allow estimation of the probability
that a randomly chosen wave height will fall within a small range about a
specified wave height. As derived by Longuet-Higgins (1952), the Rayleigh pdf
is valid formally only when an ocean surface can be described by the linear
sum of a very large number of randomly phased and directed sinusoidal com-
ponents which conform to a unimodal frequency spectrum wherein all frequencies
are very nearly (but not quite) equal to some central, characteristic frequen-
cy. Unfortunately, not all sea states have unimodal, narrow-banded spectra.
In a rather broad study of naturally occuring sea states, Thompson (1980)
noted that well over half of his observed spectra were distinctly multimodal.
This condition means that one can expect wave height distributions to deviate
to some extent from a Rayleigh distribution. If such a deviation is impor-

tant, alternative pdf models must be sought.

3. A comparison of the Rayleigh pdf model with a large set of observa-
tions is described in the Shore Protection Manual (SPM) (1984). That com-
parison indicates that if both the number of waves and the RMS wave height
from a sea surface elevation record are modified to minimize differences
between model and data, a discrepancy of 10 to 15 percent remains in the low-
probability but high-wave tails of the distributions. These tails are
important for estimating extreme wave conditions, so it is important that they
be modeled correctly. The above result suggests that either the model or the
data (or both) are not representative of real ocean conditions to within the
stated percentages.

4. On the other hand, Longuet-Higgins (1980) cites favorable com-
parisions of the Rayleigh pdf with data reported by Earle (1975) and
Forristall (1978). To achieve a favorable comparison, Longuet-Higgins
renormalized Forristall’s data with a spectrally based characteristic wave
height modified for effects of finite bandwidth. His argument was justified
in that the RMS wave height is the controlling parameter in a Rayleigh
distribution and the relationship between RMS wave height and spectrum-based
wave height can be altered when a narrow-band process is perturbed by low-
level energy at frequencies away from the peak frequency, as occurs often in
nature.

5. Thornton and Guza (1983) extended tests of the Rayleigh pdf to
include highly nonlinear, actively breaking wave conditions in very shallow
water. They used data from a number of sensors in water depths as shallow as
1 m in the breaker zone at Torrey Pines Beach, California. The Rayleigh pdf
provided very good estimates of wave height statistics in comparison with
their observations. There was, however, a very slight overprediction of the
wave population in the high-wave tail of the pdf, somewhat like the result
given in the SPM (1984).

6. None of the above tests specifically address the problem of height
distributions under conditions where an energy spectrum is distinctly multi-
modal. A simple example of such a condition occurs where two narrow-band
processes coexist but at well-separated frequencies. In nature, such a
condition can arise where a local, wind-driven sea is generated in the
presence of low-frequency swell. In general, one would not expect wave
heights from this scenario necessarily to be Rayleigh distributed because the

requirement of a unimodal, narrow-band process has been violated. In light of

Thompson’s (1980) findings, it appears that multimodal processes are common in
nature so that one is obligated to investigate wave height distributions in

them.

Purpose of Stud

7. The intent of the study reported here is to conduct a preliminary
examination of wave height distributions that arise in multimodal processes.
Only bimodal processes are considered here in order to keep the investigation
simple. To maintain maximum control of experiment conditions, time series
from which to analyze extrema are artificially produced linear sums of
sinusoidal components. This procedure makes the study somewhat idealized, but
avoids some of the vagaries associated with field data. Amplitudes and
frequencies of component wave signals are established through a spectral
definition. Phases of component signals are selected at random.

8. Statistical distributions of artificially generated wave heights are
compared with two distribution models. The first is the Rayleigh model dis-
cussed above. The second model is the deepwater asymptotic form of the so-
called Beta-Rayleigh distribution introduced by Hughes and Borgman (1987).
This second model is a two-parameter function that collapses to the Rayleigh
model for a specific ratio of its parameters. In general, it is less restric-
tive than the Rayleigh model (because it has two parameters instead of just
one) and is called the Modified Rayleigh model. It is included in this
analysis because an analysis by Long (in preparation) indicates that it
represents observations better than either the Rayleigh model or the full
Beta-Rayleigh model for waves having a variety of directional distributions in
intermediate and shallow water. The hypothesis posed here is that the
Modified Rayleigh pdf may represent adequately distributions that deviate
significantly from Rayleigh distributions due to either broad-bandedness or

multimodality.

Scope of Investigation

9. Mathematical descriptions of the models used in this study are given
in Part II. Part III describes the way in which test data are generated.
Part IV discusses an analysis of unimodal processes. This analysis is
necessary because it is not clear how many components distributed over what

bandwidths are necessary to approximate reasonably a Rayleigh process. Such

treatment also helps in understanding what happens in spectral filtering of
time series to eliminate unwanted parts of a signal. The analysis discussed
in Part V uses the results of Part IV to generate some simple bimodal proces-
ses where each mode is individually a Rayleigh process. Results can then be
classified purely by modal separation and ratio of modal energies. Comparison
of analyzed synthetic time series with model curves then reveals which
processes are approximately Rayleigh and under which, if any, circumstances

the Modified Rayleigh model excels.

PART II: MODEL DEFINITIONS

Basic Statistical Notation

10. The models used here are all defined” basically as probability
density functions p of specified crest-to-trough wave height H in the form
p(H) , having dimensions of inverse length. When multiplied by an incremental
range of height dH , the resulting expression p(H) dH is dimensionless and
gives the probability that a randomly chosen height fA lies in the range

between H and H+ dH. This relationship is written as

p(H) dH = Prob[H < A < H + dH] (1)

where Prob means the probability that.
11. If Equation 1 is integrated over all increments of height between
zero and some specified height H , one obtains the cumulative distribution

function (cdf) denoted as P(H) , which is the probability that a randomly

chosen height A is less than a specified height H . This expression takes
the form
H
PG) = | p(x) dx
)
= Prob[A < H] @2))

where x is simply the dummy integration variable. In Equation 2, it can be
seen that P(o) = 1 because all wave heights are less than infinitely high.
One can take the complement of the cumulative probability to obtain the
exceedence probability function Q(H) , which is the probability that a
randomly chosen height A is greater than a specified height H . Formally,

Q(H) is found from the probability density function by

(i p(x) dx
H

Prob[H > H] (3)

Q(H)

bid

For convenience, symbols and abbreviations are listed in the Notation
(Appendix A).

If P(H) is known, Q(H) can also be computed from the result of the follow-

I. p(x) dx - If p(x) dx
0 0

Bice) x= PCH)

ing derivation

Q(H)

ll

lbe= PGES) (4)

12. Conventional statistical definitions allow other properties of a
wave height distribution to be found from basic probability density functions.
For instance, the mode, or most probable wave height, is the maximum of
p(H) . The mean, or average wave height is the integral of the product
H p(H) over all H.. The point here is that all statistical properties of
interest for a given process can be found once the probability density
function is known. It is this function, therefore, that is most fundamental

to define.
Two-Wave Model

13. In the analysis described below, synthetic series of points are
computed by summing a number of sinusoidal components. Aside from the trivial
case of a single sine wave (where mean, mode, RMS, and all other measures of
wave height are constant and equal to each other), the simplest sum is of two
waves of equal amplitudes, very slightly different frequencies, and arbitrary
initial phases. These two wave trains will beat together, forming the
familiar grouping pattern where the waves are nearly in phase and then become
out of phase over relatively long periods.

14. Statistical properties of this two-wave process were derived by

Longuet-Higgins (1952). In particular, he found the pdf to be

2
(2H2 a 2) 1/2 Hi< fe Jal

n rms

P2(H) = (5)
0 HS: /20He.

where the subscript 2 indicates this two-wave model and H,,, is the RMS wave

height, determined from a set of observed wave heights through the equation

1 N i 1/2
Hens ~ E » He | (6)
n=1
In Equation 6, N is the number of observed waves and H, is the height of
the nt? wave. It is useful to treat mathematical models in dimensionless
form, where practicable. The only parameter in Equation 5 is H,,, . Hence,

the pdf can be made dimensionless in the form

2
i nen ee Re ra 0/2.
seat :
C7)

0 Oy)

The exceedence probability corresponding to this pdf is found by integrating
Equation 7 in accordance with Equation 3. The result, in dimensionless form,

is

2 - Ty ig! H
1-= sin? = iD)
H v 2 Hees Hee
Ql a = (8)
rms H
0 ees J2

15. The two-wave model is useful for testing synthetic data generation.
When used with two wave trains of arbitrarily different frequencies, synthetic
wave heights should approximate this statistical distribution, becoming
asymptotically identical as the two frequencies become nearer to (but not the
same as) each other. For these conditions to hold, synthetic time series must
be long enough to include sufficient cycles of the beat period to obtain a

meaningful sample of wave heights.

Rayleigh Model

16. As described by Longuet-Higgins (1952), the Rayleigh pdf is defined

by the equation
Duler CH/ Hires ie
pa(H) = gee e (H/Hams)

rms

(9)

where the subscript R identifies the pdf as Rayleigh. Equation 9 can be

written in dimensionless form by multiplying both sides by H,,, to form

H HS (H/HE 2)
ene pe(z—| a H e ( / ms) (10)

rms. rms

17. The cumulative probability function for the Rayleigh distribution

is found in dimensionless form by integrating Equation 10 with respect to

H/H.,, over the limits from,zero, to. H/He = The result is
L 2
Pale ) =e H/Amns) (11)
ee

The Rayleigh exceedence probability function is found by substituting

Equation 11 in Equation 4 to yield

H

rms.

a {e-] sigateil ante (12)

Modified Rayleigh Model

18. The Modified Rayleigh pdf is the deepwater asymptotic form of the
Beta-Rayleigh model introduced by Hughes and Borgman (1987). They defined the
Modified Rayleigh pdf as

Hos soli Hie) e se) sa(H/Miee ye
Hae Pale} - Fea a) oe

where the subscript MR stands for Modified Rayleigh, [I is the gamma

function (see Abramowitz and Stegun 1970), and a is a parameter defined as

2 (14)
fe 2
IBles
In Equation 14, a@ depends on H,,, , which stands for root-mean-quad wave

height and is computed from a set of N_ observed wave heights H, through

n

the expression

11 N i 1/4
Hing = E d Hs | (15)

10

Noterithat whens a. sla) (or Hl /Hot 21/4 from Equation 14), Equation 13
becomes identical to Equation 10 and the Modified Rayleigh pdf becomes a
Rayleigh pdf. It should be noted that Hughes and Borgman (1987) defined Hie
using a square root instead of the fourth root used here on the right side of
Equation 15. It makes more sense to use the fourth root since it gives a
height parameter with dimensions of length instead of length squared. This is
only a modification of notation and not of the basic model. One would simply
use the square root of the H,,, in Equation 14 above.

19. To find the cumulative distribution function of the Modified
Rayleigh model, one must integrate the pdf numerically. For computations used
in this study, integration was done using Simpson’s rule in double precision
with integration steps of 0.001 in H/H,,,

20. Examination of Equations 13 and 14 reveals that the shape of the
dimensionless pdf depends only on one parameter, the ratio Hyyg/Hims
Figure 1 shows a variety of Modified Rayleigh distributions found for select
wallues) of ©H.\./Hms - The bold curve in Figure 1 is the Rayleigh pdf to which

the Modified Rayleigh pdf degenerates when H,,,,/H = 2/4 | For complete-

rms

ness, the two-wave pdf is also shown in Figure l.
Additional Parameters

21. Where spectra are used for synthetic time series generation, some
additional parameters can be defined. These parameters are based solely on
spectral shape. Perhaps the most frequently used spectral scale of wave

height is denoted H,, and is defined by
Ho = 4 m2 (16)

In Equation 16, m, is the spectral zeroth moment. The more general n‘

spectral moment m, is defined by

fo
m, = I) PGs) oGbe (17)
1

where f is frequency; f, and f, are, respectively, low and high
frequency bounds of the spectral definition; and S is signal variance

spectral density.

Nae

S T So ea | T [recon eer Raa | ae | T es Ft Gall Dea Lar an T T SEY PT Le T | Saea esr T T
Rayleigh, Modified Rayleigh, |
! and Two—Wave pdf Models
|
|
ey Pages Ss oilwo=Wavie
Modified Rayleigh |
1 Hrmq/Hrms =
= 1.050
” !
ev 1.100 |
a 1.150
= r 1.189 Rayleigh 1
a 1.250 |
E 1.316
ao

0.5

3.0

0.0

1.5 2.0 2.5
H/Hrms

0.0 0.5

Figure 1. Examples of model curves

22. Ideally, f, =0 and f, =o , but in practice, the two frequen-
cies are bound to less diverse values.

used, a low frequency of f, = 0 is realizable, but it results only in a

constant offset of the signal that would normally be removed in subsequent

For computer-generated discrete time series, the lowest

Where purely synthetic time series are

wave height analysis.
practical frequency f, is that which allows enough cycles that wave heights

extracted from the time series include the effects of these low-frequency
waves.

EZ

23. The constraint on the upper frequency f, is related to the time
step At used to simulate a time series. If a frequency is too high, there
will be too few time steps per wave. The probability will then be small that
the crests and troughs of these waves will be represented in the discrete
sample. For practical purposes, about 20 time steps per wave is a reasonable
sampling. A crest or trough is then always within one-fortieth of a wave
period of some discrete time step. The fractional error in crest or trough
estimation is then the ratio of the cosine of one-fortieth of a wave cycle
(9 deg) to the cosine of zero degrees. This ratio is about 0.988, correspond-
ing to an error of about 1.2 percent in crest elevation estimation or about
2.4 percent overall. A frequency upper limit of 20 time steps then requires
£, = 1/(20 At)

24. Another parameter of use is one that gives a measure of spectral
width. A wave height distribution is theoretically Rayleigh if the cor-
responding frequency spectrum is narrow-banded. A measure of such spectral
width has been given by Cartwright and Longuet-Higgins (1956) as the parameter

e defined by

2

Gr a) ees (18)
Mom,
where m, , m, , and m, are determined through Equation 17. A small e

indicates a narrow spectrum, and an e near 1 indicates a broad spectrum.

25. Further parameters associated with spectral definitions are given
in Part III, wherein test data generation is described. Parameters associated
with averages of particular subsets of wave heights (for example, average of
the highest one-third) are defined in Part IV, where model test criteria are

described.

13

PART III: TEST DATA GENERATION

26. In this study, wave height distributions are extracted from
synthetic time series that have specific spectral definitions in the frequency
domain. A very efficient computational scheme for synthesizing a time series
from spectra is by way of inverse Fourier transforms. In this chapter, this

technique and the nature of the spectra used in this study are described.

Inverse Fourier Transform Technique

27. Following the description by Bendat and Piersol (1971), a time
series of N water surface elevations or an artificial representation thereof
can be represented by the notation x, = x(t,) = x[(n - 1)At] , for n=1, 2,
3.) he, Nie -gl Ne wherewixaiiisiache n™® discrete sample of a continuous
function x(t) and t, is the n™® discrete increment of time At after an
initial time of zero for the first sample (n = 1). A second, independent,
N-point time series can be represented by the notation y, = y(t,)
= yl @y= 1)At] ,, for “m= 1, 25 .3.5.. Ne Each of these .cimeyseries7eangie
subjected to a discrete Fourier transformation (DFT). For time series x, ,

the kt® element of the DFT is denoted by X, , and the whole set is defined by

N
X= ) x, e72mi(n-1) (k-1)/N eae iD Diet), ane (19)

For the second time series, DFT elements are denoted by Y, and are defined

by
N -
Visa) ear es nea ee ky Sieg Diy ee ek (20)
n=1
28. Because each time series represents a piecewise continuous func-

tion, each can also be represented as a Fourier series of points. The first

time series can thus be written

N/2
X= y {ho cos |7# - Ban = =
k=1

+ Bi, so nee DAs eh 5 (21)

14

where A, and B,, are the kt Fourier cosine and sine coefficients, respec-
tively, of time series x, . Similarly, the second time series can be

expressed in the form

l
a F {oy cos | == - 1){n - 2]
k=1

ons sin| 27 - a = 2) Teil asdhy lay (22)

where A, and By are the kt Fourier cosine and sine coefficients, respec-
tively, of time series y, . The Fourier coefficients can be found from the

time series points through the formulae

il N
Aya non » Xn (23)
N =
Au = 2 y a Se acer Te OM ea eee 5 (24)
Bua = 0 >)
N ng =
Bue = 2 es sin| 22k — 2m = 2) ee eae (26)

for time series x, , and the formulae

:
— 2 27
Ay Ty Dace (27)
Dh id 2n(k - 1 =i
By, = 0 )
2% Peali2 Gk swell N
Dri marian i ey es Be

for time series y, .

5:

29. Noting that the real part of Equation 19 is identical to the
summation term in Equation 24 and that the imaginary part of Equation 19 is

identical to minus the summation term in Equation 26, it is clear that

Alf ae Ee ox, (31)
and

We eee ies TIRE rns» (32)
Similarly,

Ae Boe Ma (33)
and

RAD rg aera = 2 a age (34)
for time series y, . These relationships show how the Fourier series

coefficients and Fourier transform components are related.
30. It is also useful to note that the Fourier series of Equation 21
can be written as

7d USA 9 CRE Oa be

x, = ») Gar cos| N NS 2a ea (35)
=1

where the cosine and sine amplitudes A, and B, , respectively, of

Equation 21 are related to the modulus C,, and phase #¢,, of Equation 35 by

Aya = Cy COS bx (36)
and

Bue = Cye SiN $x (37)
or

Ce = Ade + Br (38)
and

16

B
ox = tan at (39)

These relationships allow definition of an individual wave component in terms
of a wave amplitude C,, and initial phase ¢,, . These relationships are
useful when defining a wave field from a spectrum as is done below.

31. The spectrum or variance spectral density S,, of the k‘® frequency
component of time series x, is found by multiplying Equation 32 by its
complex conjugate, dividing the result by 2, and dividing again by the
frequency bandwidth represented by element k . This frequency bandwidth Af

is defined as

AE= (40)

The set of mathematical operations outlined above yields

ak P :
Sun = DAF (Ay. - IBy.) (Age + 1B) (41a)

iL 2 2

= one Anca! Baw) (41b)
1¢2

ae tex

Se (41c)

= “OE x xe (41d)

where the asterisk (*) denotes complex conjugate. In the above derivation,
Equation 38 has been used in the step from Equation 41b to Equation 4lc.
Equation 4lc is essentially the definition of spectral density since the
variance of a sinusiodal wave of amplitude C,, is 4C2, , and the variance
density is the variance divided by the frequency bandwidth represented by the
single sinusiodal wave. Equation 4lc also shows how the k** spectral element
relates to the Fourier series representation of the process. Equation 41d
comes about by using the right side instead of the left side of Equation 32 in
Equation 4la. Equation 41d shows how the spectral density of element k is
related to element k of the DFT.

32. Clearly, if one knows the spectral density S,, , the time step

At , and the number of time steps N ina time series, the resolution

17

frequency bandwidth can be determined from Equation 40, and wave amplitude

C,, can be found by solving Equation 4lc to form

Gicm(Qe AL Senate (42)

or, if Equation 40 is used to define Af ,

eee cae be 43

The frequency of element k is the (k - 1)t* increment of the resolution

frequency bandwidth Af in the form
f= (Ck > 1) AE (44)
or, if Equation 40 is used to define Af ,

Ges 1)

f. NAt

(45)

The value of the independent variable time at the n*® time step of both time

series! x. and sy, is
t=: (ni =) Ac (46)

Finally, the initial phase is a uniform random deviate in the range from 0 to
2x radians. This definition conforms to the assumption of a Gaussian random
process for natural ocean waves. Notation for random phase of the k‘® wave

component is

Pk on, U,,.[0,1] (47)

where U,,({0,1] represents a uniform random deviate in the range O to 1, and,

in this example, one associated with time series x, . A variable of this

type can be obtained routinely from a computer-based random number generator.
33. With the definitions of Equations 43, 45, 46, and 47, the contribu-

th

tion to a time series at the n™ time step from the k™ frequency component is

18

(48a)

C,, cos(2nf,t, + do) = Cy es 4 $a

D iSuuay 2 i :
- Fe cos{“*( ao 2+ 2n,.(0,11} (48b)
Equation 48a is recognized as the k™ contribution of the Fourier series of
xX, given by Equation 35. The Fourier coefficients A,, and B,, of Equa-

tions 36 and 37, respectively, then become

2) Su yal2

Ae = a cos{2aU,,[0,1]} (49)
2 Sue 1/2 ?

By, = Nat ] sin{2aU,,[0,1]} @o)

Using these Fourier coefficients in Equations 32 and solving for X, yields

N [(2 Syq)2/2 DASeyai2
X= 3 ( NAt cos{2aU,,[0,1]} -i NAC sin(2n0,(0,1])] 51)
as an expression for the k™* element of the DFT of time series x, . Equa-

tion 51 can be written in more abbreviated form by combining terms and using

Euler notation, which results in

2's sl 2st
ee cal 97 1 2mU,a [0,1] ep)

NAt

A similar set of steps can be created for time series y, . If the k™
component of the frequency spectrum of y, is S,, and its kt® initial phase

is 2nxU,,[0,1] , the corresponding DFT element Y, is given by

2S ff eae
ve (Fe) pee A at (Ott) (53)

34. By taking advantage of complex notation and certain properties of
DFT’s, an efficient means of computing multiple time series is achieved. If
the DFT for time series y, (Equation 20) is multiplied by i and added to
the DFT for time series x, (Equation 19) and the k™® element of the result

is denoted as Z, , then

19

Zoi ee RUN:

= 3 (x, + iy,) e7tl2nde-2) (n-2) 10] Eee By
n=1 * ie)
N : .N ; N
=o (AQ = 1B) in (Ay LBS) Nol Aer > (54)
where Equations 32 and 34 were used to obtain the last line. The complex
conjugate of element N - k + 2. of Equation 54 has the form
Zy-k+2 a Xy-ict2 zi LYy-1e+2
N
ay y (x, - iy.) etf2edi-kt2-2)(n-1) 181]
n=1
X i[2m(k-1) (n-1)/N] N
= - ] i = n> = =
2, & > tq) Kia hee (55)
where use is made of the identity e!@*N(@m-1)/N] = 1 Again using Equations 32
and 34, Equation 55 can be written
* N : _ N : N
Zy-K42 = > (Ane = UB) = 2 2 (Ay, 7 -EBy,) esis 2 teens > (56)
35. Adding Equations 54 and 56 and multiplying by 1/N yields an
expression for just the Fourier coefficients o£ time series x, ,; i.e;
F 1 *
GV Re) bbe N Z. + Zy-1+2 (97)

Subtracting Equation 56 from Equation 54 and multiplying by -i/N yields an

expression for just the Fourier coefficients of time series y, in the form
F -i *
Aye ~ IBye = yy [2 - Zy-1+2 (58)

Equations 57 and 58 can be inverted to recover Equation 54 representing Z,
for k=1, 2, ..., N/2 and the complex conjugate of Equation 56 representing
Z, for k =N/2 +2, N/2 +3; 30. N\. che vA.) and) Be (for. ecumersernies
xX, are assigned values from user-defined spectral densities S,, anda set

of random phases 272U,,[0,1] following Equations 49 and 50. A similar set of

relations defines the A, and B, for time series y, . With these

20

definitions made, the complex DFT Z, can be inverse transformed to recover a
complex array for which the real part is the time series x, and the imagi-
nary part is the second time series y, .

36. In summary, two time series can be synthesized from two distinct
spectral density definitions using a single algorithm. Complex numbers are
formed from the Fourier coefficients of the time series. For one time series,

the complex Fourier coefficients are formed from

; 2S, )1 =12xU_ 10,1] N
Ay BS = Fa) e Bic Less ee Zs veierae 5 (59)
and for the other time series, the Fourier coefficients are
2)So\e!2, Si xU 0), N
ss pag) ee i2aU,, [0,1] = N
ASE TBee NAt e y k Hea iis pales 9 (60)

Equations 59 and 60 are then combined to define the DFT of the complex sum of
the two time series in the form
(61)

Zi = | =i Bea) oe CAN ii iB.) | St 2S eee,

N12

and

N N
Zy-K+2 9 9 (62)

|e. Siete) ia Ate a iB.) he 25 MSHS ak
with Z, = Zyjo,, = 0 arbitrarily assigned without loss of generality in the
present application for the mean values and Nyquist frequency values, respec-
tively. Finally, the complex array Z (k=1, 2, ..., N) is inverse

Fourier transformed to recover the time series through the sum

) Zhen in cn ehve ys ny ne 2a EN (63)

net eal Vinee
k=1

In Equation 63, the real part is the time series x, and the imaginary part
is the time series y, . Where a collection of such time series (or their
properties) is useful for statistical purposes, the above algorithm can be
applied repeatedly, with the procedure providing two new time series in each
application. Computer techniques known as Fast Fourier Transforms enable sums

like those on the right sides of Equations 19, 20, and 63 to be computed very

21

rapidly. In this way, statistically stable data sets can be synthesized with

reasonable efficiency.

Unimodal Data

37. In using synthetic data as described above, some assurance must be
gained that the procedure used replicates, in fact, the process it intends to
synthesize. In the present case, a narrow-banded process containing a large
number of spectral lines should yield a Rayleigh distribution of wave heights.
This distribution is the result predicted by Longuet-Higgins (1952). Although
ideally a Rayleigh process contains an infinite number of spectral lines, it
is a practical impossibility to synthesize such a process using discrete
computational techniques. However, it should be possible to approximate the
process if a sufficiently large number of lines is used. Thus, an important
question in the current context is how many lines it takes for a spectrum of a
given width to approximate closely a Rayleigh process.

38. A second question is related to the first. That question is how
narrow a spectrum must be before a Rayleigh process is realized. The deriva-
tion given by Longuet-Higgins (1952) simply states that the process must have
a narrow spectrum, but does not give a practical definition of narrowness.

39. Part of the investigation described here is devoted to examining
these two questions. The idea is to find what is necessary to simulate a
Rayleigh process in a unimodal spectrum and then combine several of these
independently Rayleigh processes to investigate wave height distributions from
processes with multimodal spectra. The approach for this part is to generate
time series from band-limited white spectra having variable bandwidths and
different numbers of spectral lines, with each spectral component being
assigned a random initial phase. Due to the random phasing process, a given
sample time series can have a highly variable maximum wave height depending on
how nearly all the components are in phase at some point in the synthetic time
series. To get a better estimate of a characteristic maximum height, as well
as other high-wave statistics, an average of the wave height distributions of
several time series, having the same generating parameters (except for
phasing) and truncated at the same number of waves, is computed. This average
is compared with both the Rayleigh model, Equations 10 and 12 for the pdf and

exceedence curves, respectively, and the Modified Rayleigh model, Equation 13

22

for the pdf and its intgral using Equation 4 for the exceedence curve. The
fundamental parameters of these time series averages are the defining spectral
widths and numbers of components.

40. Mathematically, the spectra are defined by first specifying the
number of time steps N ina time series and a time step dt between time
steps. By Equation 40, the basic frequency increment of the discrete spectrum
is df = 1/Ndt so that the n™ frequency of the spectrum is f, = (n-1)df
It then remains to assign variance densities to each of the N/2 +1 frequen-
cy bands at or below the Nyquist frequency. For band-limited white spectra,
four more variables are needed: a total variance, a reference center frequen-
cy, a bandwidth, and a number of spectral lines to which to assign energy. In
all of the tests discussed here, the total variance is constrained to be

equivalent to 0.25 m?

This yields an H,, = 2.0 m , which is a convenient
number of order one for computational purposes, but is otherwise unimportant
as all results are normalized with the computed RMS upcrossing wave height
ple

41. The variables in the problem are then the center frequency f,
the overall bandwidth Af , and the number N of spectral lines within Af
which are assigned a finite variance. The bandwidth is normalized by the
center frequency to take the form Af/f, as a parameter. The bandwidth is
then found from the product f,°*Af/f, . The low-frequency bound of this band
is the discrete raw frequency band nearest to f, - Af/2 . The high-frequency
bound is the discrete frequency band nearest to f, + Af/2

42. Assignment of the number of bands within the bounding frequencies
is determined by specifying the interval between raw frequency increments to
which finite variance is assigned. If every line is of finite variance, the
number of lines is N = Af/df . If every other line has finite variance (the
intervening lines having zero variance), the number of lines is N = Af/2df
In this case, the effective overall bandwidth of the spectrum is still the
same Af , but it is represented with only half as many lines, as if the
resolution bandwidth is twice as wide. If the variance in every other band of
the second case is twice the variance in each band of the first case, the
total energy represented by the spectrum remains the same. This same
procedure can be followed by assigning finite variance to every third line,

every fourth line, and so on until as few as two lines remain to represent the

spectrum. Two lines is clearly a lower limit for the number of lines with

23

which to represent a spectrum because the pdf is no longer even approximately
Rayleigh, as derived by Longuet-Higgins (1952), as expressed in Equation 7,
and as shown in Figure 1. Figure 2 illustrates the effect of varying the
number of lines with which to define a band-limited white spectrum. It also
shows the equivalent bandwidths and spectral densities of a "continuous"
spectrum having the reduced number of lines and yet retaining the same overall
bandwidth.

43. For the unimodal spectra, both the overall bandwidth Af and the
number of lines N within this band are varied within the limits imposed by
time series of finite length. In all tests, time series lengths are
N = 65,536 points. With a time step of dt = 0.5 sec , this represents a
record of 32,768 sec or 9 hr, 6 min, 8 sec, a record much longer than is
normally obtained in nature. In all cases, the center frequency is kept at
f. = 0.1 Hz , corresponding to 10-sec waves. Hence, each simulation contains
in excess of 3,000 waves, enough with which to compute some reasonably concise
statistics.

44. For each simulation, 20 runs were made to establish a mean charac-
teristic pdf. With 20 samples, other statistics can be computed as well. For
the present tests, a standard deviation for the exceedence curve is also
computed. Since the number of waves is not exactly constant in the full
length of the time series, the run results are scanned to find the case-with
the fewest waves and all other runs are truncated in time at this same number
of waves. In this way, all 20 runs for each case have the same number of
waves. When wave heights are placed in order of ascending height, there are

then 20 samples at exactly the same discrete exceedence probability estimate.

Empirical Probability Estimates

45. A set of N discrete wave heights H, can be used to estimate the
pdf of the governing process. If all H, are normalized by H,,, (defined in
Equation 6), the number J, which fall in the range uAH/H,,, to
(u+1)AH/H,,, can be counted and divided by N_ to compute the fraction (or
estimated probability) of this occurance. When this fraction is divided by
AH , a wave height range arbitrarily chosen by the investigator, and
multiplied by H,,, to make the result dimensionless, the result is an

estimate of the normalized pdf for that data set.

24

S(f) (m?/Hz)
ORO! —2ZOHOR S10..0. 4070

S(f) (m?/Hz)
O20 2aIO205- Z2OnO! SONO: -40h0

S(f) (m?/Hz)

10.05. 20).0- 30:0' :4'050

S(f) (m?/Hz)
10.0° 20:0 30:07 40.0

0.0

0.0

0.0

Defined Spectra

Equivalent Spectra

:
Every Line

al ae

Every 2” Line

Every 4" Line

|

Every 8'" Line

Lt |

Lo

SSeS

Je

(ee loll

0.1
f (Hz)

Figure 2.

O1270),0 0.1
f (Hz)

Examples of spectral line decimation

2D

46. If more than one set of wave heights is available, a better
estimate of the pdf of the underlying process is found from the average number
J, of contributions to each range bin. Since H,,, is unlikely to vary much
from run to run, the effect is simply to increase the sample population of
wave heights. Such an increase gives a better pdf estimate because there are
more degrees of freedom for the population of each range bin.

47. The same set of normalized wave heights H,/H,,,; can be used to
estimate the cumulative or exceedence probability functions. If the H,/H,,,
are ordered from smallest to largest, the probability that any of the wave
heights is less than the height of wave n is the fraction n/ CNL) atinals
computation provides an estimate of the cumulative distribution function. The

data estimate of exceedence probability Q)(H,/H,,,) is one minus the cumula-

tive distribution at height nm, or

H, a
Spel ie (64)

48. A third set of statistics that are of use is the average of the
highest fraction r of an observed set of waves, denoted HS) > One Yofuthe

most commonly cited members of this group is the average of the highest one-

third of the waves in a given sample. This is given the symbol H(‘?/®

following the notation of Longuet-Higgins (1952). Note that the SPM (1984)
and some other references use the symbol H,,; , instead. From the set of

normalized, ordered wave heights H,/H , the computation of the j‘® dis-

rms

crete, monotonically increasing fraction r,; is given by

a3 (65)
and the average of that fraction of the highest waves is
(z;) a aan
Hang & L y N-it1 (66)
eee J i=1 Ieee

The result of Equation 66 can be interpolated if a particular desired fraction
does not fall on one of the discrete fractions r,
49. For the Rayleigh and Modified Rayleigh models being tested here,

the corresponding average heights are found following the derivation given by

26

Longuet-Higgins (1952). The fraction r of waves that are larger than a

given value of H/H,,, is the exceedence probability for that H/H The

rms

average value of waves higher than a given H/H

H/H

is the integral from that

rms

to infinity of the normalized heights weighted by the pdf and divided

rms

by the area over the same limits under the weighting function (i.e., the pdf).

The general forms for r and H‘/H,,, are

esol esell ae be (67)
H“? 1 H H H
sme vere ca eso rele (68)

The terms in square brackets are the dimensionless pdf models under considera-

tion (Equation 10 for the Rayleigh model or Equation 13 for the Modified
Rayleigh model). Where the above integrals have no simple analytic form,
numerical methods are used for evaluation. The simple trapezoid rule with
increments of AH/H,,, = 0.001 and an upper limit of integration of 6.0 (in

place of infinity) has been found to yield quite accurate results.

Bimodal Test Data Generation

50. The above definitions are used to establish criteria for simulating
a Rayleigh process using discrete inverse Fourier transform techniques to
produce time series from unimodal, band-limited, white spectra having given
bandwidths and numbers of component waves. Once such criteria are estab-
lished, a bimodal spectrum can be constructed of two unimodal spectra, each of
which yields a Rayleigh wave height distribution by itself. Wave height
distributions derived from such bimodal spectra can then be compared with test
models to determine the quality of representation.

51. Since each mode in such a bimodal spectrum yields a Rayleigh wave
height distribution by itself, the details of the mode structure (bandwidth
and number of component wave trains) are no longer important. The two primary
variables in bimodal tests are a measure of modal separation in the frequency
domain and the relative amount of energy in one mode compared with the other.
Modal separation is important because the two modes could be very close to

each other and so just yield a slightly wider unimodal spectrum (and

27

corresponding wave height distribution). If the modes are well separated,
low- and high-frequency waves are present at the same time. Under these
conditions, one would expect a rather diverse character to a wave height
distribution.

52. However, if the energy (i.e., variance) in one mode is much less
than in the other, the high-energy mode would be expected to dominate, and the
process should become asymptotically Rayleigh. Hence, the ratio of energy in
one mode relative to that in the other is important. It seems clear that the
greatest deviations would occur when the two modes are of approximately equal
energy so that neither clearly dominates.

53. To characterize modal separation, a dimensionless difference
between the modal center frequencies is used. If f,, is the center fre-
quency of the first, lower frequency mode and f,, is the center frequency

of the other mode, a separation parameter can be defined as

modal separation = poeya senate (69)
a(f£..2 + f.,1)

It can be as small as zero (if the two modal center frequencies are co-

located) or as’ large vas 2 (ii) i << Eo)

54. To characterize relative energy, the ratio of the variance in the
second mode to the variance in the first mode is used. Since four times the
square root of the variance in a spectral mode can be identified as an H,,
for that mode, the square of the ratio H,, 2/Hno,1 can be used to characterize

relative energy, i.e.,

2
relative energy = (F2| (70)
mo,1

Relative energy can vary from zero to infinity, but for the practical cases

considered here, the range is from 0.25 to 4.0.

28

PART IV: UNIMODAL TESTS

Test Conditions

55. A total of 53 cases were examined for the unimodal tests. Table 1
lists the parameters used in the tests. The primary variables were the
bandwidth Af and the number of spectral lines N . In all cases, the center
frequency was f, = 0.1 Hz , the characteristic spectrum-based wave height was
Ho = 2.0 m , and the number of runs for obtaining statistical averages was
20. Note that spectral component wave amplitudes were found by dividing total
signal variance (equal to the square of jH,,) evenly among all N_ compo-
nents. Note also that there is an upper limit to the number of component
waves for a given spectral bandwidth because of the finite raw bandwidth
imposed by the discrete Fourier technique used to create the time series.
Hence, the narrowest case (Af/f, = 0.05) was limited to 164 bands, but the
broadest case (Af/f, = 1.6) had 5,243 bands.

56. Test criteria were the percentage differences of H(‘!/190) | y(t/10)

H?/3) | and H“/1) from the synthetic results with those from the test

models. Computations followed the pattern shown here for the comparison of
synthetic data with the Rayleigh model estimate of the average of the highest

1 percent of waves

H(1/100) _ 441/100)
% Difference = om Cepia x 100% Gi)

Comparison with the Modified Rayleigh model entails replacing H{!/1° with
Hoe” in Equation 71. Replacement of the highest 1-percent averages with the

Hitio) Ga /29)

highest 10-percent averages , and HG? allows comparison

?

among data and models for this statistic, and so on.
Test Results

57. Results of such comparisons are shown in Figures 3 and 4. Figure 3
shows synthetic data results compared with Rayleigh model statistics for the
four wave height averages mentioned above. Percentage differences are shown
as displacements on ordinate axes. Abscissa displacements are the numbers of

component waves N . Symbols denote synthetic data of constant Af/f, . The

29

Table 1

Parameters* of Unimodal Test Data

Group 1: Af/f, = 0.05 Group 2: Af/f, = 0.10 Group 3: Af/f, = 0.20
Case # # of Lines Case # # of Lines Case # # of Lines
38 164 1 328 11 655
39 82 2 164 12 328
40 41 3 82 13 164
41 20 4 41 14 82
42 10 5 20 15 41
43 5 6 11 16 20
44 3 7 10 L7 10

8 5 18 5
9 3 19 3
10 2
Group 4: Af/f, = 0.40 Group 5: Af/f, = 0.80 Group 6: Af/f, = 1.60
Case # # of Lines Case # # of Lines Case # # of Lines
20 1311 29 2621 45 5243
21 655 30 1311 46 2621
22 328 31 655 47 Sau
23 164 32 328 48 655
24 82 33 164 49 328
25 41 34 82 50 164
26 20 35 41 51 82
Ze 10 36 20 52 41
28 5 37 10 53 20

* For all cases: f, = 0.1 Hz , H, = 2.0 m , 20 runs averaged for each case

inset in the upper part of Figure 3 shows the correspondence between Af/f,
and the spectral width parameter e¢ of Cartwright and Longuet-Higgins (1956),
defined in Equation 18. Figure 4 shows the same pattern of comparison but for
the Modified Rayleigh model vice the Rayleigh model.

58. Figure 3 reveals some interesting characteristics of synthetic
data. Perhaps the most obvious is that the smaller the fraction r_ in the

average H‘*? | the more wave components are required to approximate a

HO/1)

Rayleigh distribution. For the average heights the curves are

relatively flat for all numbers of component lines. The same is true for

HO) H{1/19)

For it appears that about 20 component waves are necessary

HO/10)

to differ from a Rayleigh by less than 10 percent. For HST oe Meine

30

% Difference

% Difference

% Difference

% Difference

Synthetic Data vs. Rayleigh Model

{a} as on fe T Tore lonvatiotntsl cat eay LE eae ERT PLERIELE PLIST Sa | La T Tp akypelaackn Weta a Tenia T T To lupalentant si
; H6Y109) ems |
|
i
i 1
|
S a
i 4
| 4
eS di)
~m |
| J
|
fo) eae)
+ SS ae
| 4
fo)
| ©
|
Ie)
TN
| |
fo)
7™
1 Yh hb eS eh N n at |
=
ro)
S
|
2
ee ee eee ee 1 = 2 eee eee ee SS eee n 4 eae fees feee eine |
1 10 100 1000 10000
Number of Lines
Figure 3. Synthetic wave height averages compared with Rayleigh averages

31

% Difference

% Difference

% Difference

% Difference

10

=Al®,

= 5.0

-40

10

—10

Synthetic Data vs. Modified Rayleigh Model

Visgreslccautrciles tect Wail: T T Te alenlextag celal T T Vissaglsnwalnoe | tact oo) pees T T Laan rm PT

HOY) ems

10 100 1000 10000
Number of Lines

Figure 4. Synthetic wave height averages compared with
Modified Rayleigh averages

32

takes about 100 component waves to stay within 10 percent of a Rayleigh
62/100)

59. The reason for this behavior can be seen in the sequence of
Figures 5, 6, 7, and 8, which show the synthetic, Rayleigh, and Modified
Rayleigh probability density and exceedence functions for the cases with f,
= 0.1 Hz , Af/f, = 0.1 , and N = 2, 3, 10, and 328 component waves, respec-
tively. Because the synthetic curves are averages of 20 different runs,
standard deviations could also be computed and plotted. Standard deviations
are shown as dashed lines in the exceedence graphs. Figure 5 represents the
case of two waves. For this case, the synthetic pdf in the upper part of
Figure 5 looks very much like the two-wave model pdf shown in Figure 1. Also,
the standard deviation is very small. This result is as it should be since
the two-wave case has an exact solution (Equation 7) and the synthetic results
reflect this solution very well.

60. When a third wave train is added, the problem becomes more compli-
cated, having no analytic solution. The result for a single run depends
strongly on the initial phases of the three component waves. If all three are
nearly in phase at some point in the time series, the maximum wave heights
will be larger than if the waves are not nearly in phase at any point in the
time series. Since initial phases vary from run to run, the variation in wave
height distribution is suggested by the standard deviation of the set of runs.
This is shown in Figure 6 as the dashed lines on either side of the average
exceedence curve in the lower part of the figure. The mean of the 20 runs is
higher, however, than the mean of the two-wave results of Figure 5. This
trend continues when the number of component waves increases to 10, as shown
in Figure 7. Figure 7 also shows that the smaller waves begin to conform to
the Rayleigh model more than the two-wave or three-wave cases. This is the
result summarized in Figure 3. At the point where there are 10 component
waves in Figure 3, averages over the highest 1/10, 1/3, and all the waves are
very close to the Rayleigh values. However, the H‘/!°) still differs
substantially from the Rayleigh curve, evidently due to the limited number of
wave components.

61. In Figure 8, the synthetic data are composed of 328 wave trains and
appears to conform to the Rayleigh curve over the whole plotted domain.
Presumably, if a longer time series had been generated, there would be a

region of limiting heights at some point on the high-wave tail. This effect

33

2.6

blgere P(H/H ms)

Exceedence Probability

10

10

—3

10

10

Wave Height Distributions
Averages from 20 Unimodal, White Spectra
Af/fc = 0.10000

No= 2 ENO OO naz
€ = 0.056316 Hmo = 2.000 m
Binmsij=: hd 78. OA OONh mn Ainmaq = oS2) £ OL OOO im
as T a TT Taney Ses eo ae ee

3332 waves sampled
Rayleigh
Modified Rayleigh

k
x 3332 waves sampled t
Penan + fl Sib. ‘
Rayleigh k
PSRE CR --- Modified Rayleigh x
'
x
I
|
ce
iil 4 1 1 ory n [eee ee 1 1 | [eet [Aue ena 1 n L 1 (owl Ls feelin
0.0 0.5 1.0 (IS) 2.0 2.5
Ale

Figure 5. Probability and exceedence curves for two wave trains

34

3.0

bias e(ayAcb =e)

Exceedence Probability

—1

Wave Height Distributions
Averages from 20 Unimodal, White Spectra
Af/fc = 0.10000

N = 3 hey=VOwlOOW Hz
€ = 0.066861 Hmo 2.000 m
Hrms = 1.524 + 0.000 m Higmnah = 1760. O83: im

3303 waves sampled
Rayleigh
Modified Rayleigh

2 S ‘
\ \
\
\ h
| (
( a
i) t
' 1h
! \
\ l
! '
2
' 1
= j ‘
' d
| : \
I $ 1
| % \
| 3 I
x 3303 waves sampled aes
Seer ak Se Dy ' s
” Sa Rayleigh Wena
(20) Sone neeeor Modified Rayleigh Lay ee
ieee
' 1
| 1
J \
- x
es 1 1 fenton 1 1 1 ent ll 4 1 1 I a SPUR VE pe I | 1 1 1 n
Na ORO) 0.5 1.0 1.5 2.0 205
Aber

Figure 6. Probability and exceedence curves for three wave trains

35

3.0

wo
=

Exceedence Probability
10°

Wave Height Distributions
Averages from 20 Unimodal, White Spectra
Af/fe = 0.10000

N = 10 fie =O. (OOM kz
€ = 0.056316 Hmo = 2.000 m
inns’ =" MSO! ee OOO GS: inn Hrmq = 1.640 + 0.091 m
fo a a eal ee a eee
t —- 3224 waves sampled
b =——— | Rayleigh |
[ig ae PRETENSE Se ae hag eI enc inl Ne cereals Modified Rayleigh

{
i]
a)
E \
c x 1
E 3224 waves sampled Re \
f ----- +15S.D. :
| ———— Rayleigh x
sieseie fafa siz}e Modified Rayleigh x \
x
|
|
|
x
a a eerie rem AS Shee TU pe :
0.0 0:5 1.0 iH) 2.0 no) 3.0
H/Hens

Figure 7. Probability and exceedence curves for 10 wave trains

36

Exceedence Probability

ila)

10

10%

Wave Height Distributions
Averages from 20 Unimodal, White Spectra
Af/fc = 0.10000
N = 328 Fev Oe 100M kz
€ = 0.057657 Hmo 2 O00; Mm

Hrms = 1.408 + 0.002 m Hrmq = 1.677 + 0.024 m

SSS Se =} SSS SS SSS ES —S

— 3268 waves sampled
Rayleigh
abOoneeCHe Modified Rayleigh

3268 waves sampled
+ 1-S.D: \

Rayleigh \
Modified Rayleigh

Figure 8.

0.5 1.0 1.5 2.0 2.5
H/H

rms

Probability and exceedence curves for 328 wave trains

37

is suggested in Figure 8 by the widely spaced and nearly vertical standard
deviation curves at high H/H,,, and by extrapolation of the results shown in
Figures 5, 6, and 7. However, this has not affected adversely the statistics
chosen for the present tests.

62. A second observation of the results shown in Figure 3 is that
beyond a certain bulk spectral bandwidth, the parameters do not appear to
approach Rayleigh parameters no matter how many component waves are used in
the spectral definition. This is most clear for the widest case,

Af/f£, = 1.60 , but is also slightly evident in the next-to-widest case

Af/f£, = 0.80 . It is notable that even the mean wave height H‘'/!) and the
average of the highest one-third waves H‘“'/*) also show some effect of
spectral broadening. One of the most extreme examples of this is shown in
Figure 9. Though there are over 5,000 component wave trains, the mean curve
deviates everywhere from the Rayleigh curve and the deviation is significant
at least at the level of the standard deviation of the synthetic data.

63. Also shown in Figure 9 are the Modified Rayleigh pdf and exceedence
curves. These curves clearly represent the synthetic data better than the
Rayleigh curves. This is evident in the broader sense in Figure 4, where all
four measures of comparison are much closer than the Rayleigh comparison of
Figure 3 as long as enough component wave trains are present. This result
suggests that the Modified Rayleigh curve is a better model for broad spectra
if the added parameter H,,, can be determined directly from the spectra. It
is not yet evident that this determination can be made.

64. These tests reveal certain properties about unimodal, band-limited,
white spectra and the probable wave height distributions contained in the
corresponding time series. If the spectra contain in excess of 100 conponent
wave trains and do not have a width such that Af/f, is greater than about
0.4, the resulting wave height distributions will conform very nearly to the

Rayleigh model.

Implications for Spectral Filtering

65. A common way to remove unwanted noise or extraneous signal from a
time series is by spectral filtering. In this method, a time series is
Fourier transformed to obtain a spectrum after which energy in specified

frequency bands is either reduced or eliminated according to the filter

38

Exceedence Probability

emis P(H/H ms)

1

=2

Wave Height Distributions
Averages from 20 Unimodal, White Spectra
Af/fe = 1.60000

N =5243 fe OR MNO Om riz
€ = 0.613751 Hmo = 2.000 m
Hrms = 1.308 + 0.004 m Hrmq = 1.504 + 0.007 m
2 a
— — 3561 waves sampled
Rayleigh J

SHR Gees Modified Rayleigh

1.0

(8)35)

(e)
*% J
‘\
° %
Q
Gk 3
N
Non
x 3561 waves sampled NEAR
ee he £180. en
Sanaa, hayleigh eae J
Ryo i Nopete
oO eRe Hara: Modified Rayleigh WARENe 3
) \ q
\ Osh Sk
\ SY
<
\ 50 =
ea Ne OS UR Ta tn OE) ESU, P ae a|CU E
OBO O25 1.0 1kS YANO) 25 3.0
Yale

Figure 9. Probability and exceedence curves from a broad spectrum

39

characteristics. The filtered spectrum is then inverse Fourier transformed to
generate a filtered time series.

66. A clear danger in this method is that the possibility arises to
remove enough of the spectrum that insufficient energy-containing spectral
lines remain to recover enough of the wave height distribution to have
reasonably good statistics. This is especially true where the parent time
series is short so that the raw spectral frequency bands are wide. Figure 3
indicates that, for spectra that are not too broad to begin with, at least
40 lines must remain to obtain H/!® to within about 2 percent and at least

150 lines must remain to obtain H(‘1/10%)

to within about 5 percent.

67. There may be advantages to spectral filtering as well. If there is
truly noise in the time series such that the spectrum becomes broad in its
presence, then filtering can remove the noise and make the spectrum narrower.
If such a filter reduces spectral width from an equivalent Af/f, of order

1.0 to something less than about 0.5, a clear improvement in all statistics is

suggested by Figure 3.

40

PART V: BIMODAL TESTS

Test Conditions

68. A total of 55 cases have been examined for the bimodal tests.
Table 2 lists the parameters used in the tests. The primary variables are
modal separation, as defined by Equation 69, and relative modal energy, as
defined by Equation 70. Each mode has been constrained to a width of 0.01 Hz

by adjusting the parameter Af/f, along with the parameter f, so that their

c
product yields the desired bandwidth. With a time step At = 0.5 sec and the
time series length of 65,536 points, the raw spectral bandwidth is about
0.0000305 Hz. With this raw bandwidth, a modal bandwidth of 0.01 Hz contains
about 328 spectral lines. Each mode is independently a Rayleigh process, as
determined from the results of Part IV.

69. Modal separation has been allowed to vary from zero to about 1.33.
At zero separation, the two modes coincide on the frequency axis. In this
case, the resulting process should be Rayleigh for all values of relative
modal energy because the effective spectrum is unimodal and narrow, and
contains enough spectral lines. It is no longer a white spectrum, however,
because there are now two randomly phased waves at each frequency. Two waves
at the same frequency can interfere constructively or destructively, depending
on their relative phases, so that the resulting wave amplitude can vary from
zero (destructive interference with waves of equal amplitude) to the sum of
the two component wave amplitudes (constructive interference with the waves in
phase). Corresponding wave energy will vary from zero to a number proportion-
al to the square of the sum of the two component wave amplitudes. In the
derivation by Longuet-Higgins (1952), there was no constraint on the distribu-
tion of energy within the narrow band process. Hence, coincident modes in the
test spectra are expected to yield wave height distributions that follow the
Rayleigh model.

70. The largest modal separation has one mode center frequency at
0.05 Hz and the other at 0.25 Hz. These frequencies are near the limits of
the overall band normally associated with wind waves, corresponding to wave
periods of 20 and 4 sec, respectively. This separation is considered to be a
practical limit for wind waves. Such a case might occur where young or short-

fetch waves are riding on longer, swell-like waves.

41

Table 2

Parameters* of Bimodal Test Data

Mode 1 Mode 2 Mode 1 Mode 2
Case: # fe CHz)y AL fos Lo ECHz) AE ees Case # f.(Hz) Af/f.. £, (Hz) Af io
122 0.100 0.100 0.100 0.1000 144 0.100 0.100 0.100 0.1000
123 0.095 0.105 0.105 0.0950 145 0.095 0.105 0.105 0.0950
124 0.090 o)jatatat 0.110 0.0909 146 0.090) 0.111 0.110 0.0909
125 0.085 0.118 0.115 0.0870 147 0.085 0.118 0.115 0.0870
126 0.080 0.125 0.120 0.0833 148 0.080 0.125 0.120 0.0833
127 0.070 0.143 0.130 0.0769 149 0.070 0.143 0.130 0.0769
128 0.060 0.167 0.140 0.0714 150 0.060 0.167 0.140 0.0714
129 0.050 0.200 0.150 0.0667 151 0.050 0.200 0.150 0.0667
130 0.050 0.200 0.160 0.0625 152 0.050 0.200 0.160 0.0625
131 0.500 0.200 0.200 0.0500 153 0.050 0.200 0.200 0.0500
132 0.500 0.200 0.250 0.0400 154 0.050 0.200 0.250 0.0400
Hno,1 = 0.894 m Bno,2 = 1.789 m Bno,1 = 0.667 m Hno,2 = 0.943 m
100 0/:100/), )0'100 0.100 0.1000 133 0.100 0.100 0.100 0.1000
101 O095." 102105, 0.105 0.0950 134 0.095 (0.105 0.105 0.0950
102 0090!) 0/7 22:7! 0.110 0.0909 135 0.090 0.111 0.110 0.0909
103 0.085 0.118 0.115 0.0870 136 0.085 0.118 OTS) 0)-108:7,0
104 0.080 0.125 0.120 0.0833 137 0.080 0.125 0.120 0.0833
105 0.070 0.143 0.130 0.0769 138 0.070 0.143 0.130 0.0769
106 0.060 0.167 0.140 0.0714 139 0.060 0.167 0.140 0.0714
107 0.050 0.200 0.150 0.0667 140 0.050 0.200 0.150 0.0667
108 0.050 0.200 0.160 0.0625 141 0.050 0.200 0.160 0.0625
109 0.500 0.200 0.200 0.0500 142 0.050 0.200 0.200 0.0500
110 0.500 0.200 0.250 0.0400 143 0.050 0.200 0.250 0.0400
Bno,1 = 1.414 m Eno, 2 = 1.414 m Ano, 1 = 0.943 m Hno,2 = 0.667 m
111 0.100 0.100 0.100 0.1000
112 0.1095) 0/105 0.105 0.0950
113 0.090 0.111 0.110 0.0909
114 0.085 0.118 0.125) -10).087.0
115 0.080 0.125 0.120 0.0833
116 0.070 0.143 0.130 0.0769
117 0.060 0.167 0.140 0.0714
118 0.050 0.200 0.150 0.0667
119 0.050 0.200 0.160 0.0625
120 0.500 0.200 0.200 0.0500
121 0.500 0.200 0.250 0.0400

Hno 1 = 1-789 m Hy, > = 0.894 m

* 20 runs averaged for each case.

71. Variation of relative amounts of energy so that one or the other of
the modes dominates suggests what happens to wave height distributions in such
situations as high-energy short waves on low-level swell or the initial stages
of a growing sea in the presence of an active swell. Five levels of relative
energy have been used in the present tests. The ratio varies from 0.25 to 4.0
following a geometric progression. Clearly, if the ratio approaches zero or
infinity, then one or the other of the two modes will become the only energy
containing mode so that the process will revert to unimodal behavior. It is
expected that the most interesting cases will be those with energy ratios of
order unity.

72. Test criteria in this set of tests are the same as those in the

unimodal tests. Wave height averages HSE 100) tO ie! IS aamdamHoaay

42

from synthetic data are compared with Rayleigh and Modified Rayleigh model
predictions of these statistics. Percentage differences are then computed
following the pattern of Equation 71, and comparisons are done among these

percentages.
Test Results

73. Figures 10 and -11 summarize the results of the bimodal tests.
Figure 10 shows percentage differences of synthetic data from the Rayleigh
model. Figure 11 shows synthetic data compared with the Modified Rayleigh
model. In both figures, peak separation (defined by Equation 69) is along the
abscissa. Symbols denote points of constant relative energy ratio (defined by
Equation 70).

74. In Figure 10, the synthetic data clearly approximate the Rayleigh
model for small modal separations, as expected. Synthetic data begin to
deviate from the Rayleigh model when the peak separation parameter reaches
about 0.40, especially for averages on the high-wave tails of the distribu-
Gilonshwier eg, oh oO He cand HOMO H Curves £oxr)all ivallues ef, the
energy ratio behave similarly up to peak separations of about 0.60, and then
diverge radically depending on the energy ratio. At the largest peak separa-
tion, and for a small energy ratio, i.e., where the energy in the high-
frequency mode is small relative to that in the low-frequency mode, the
average of the highest 1 percent of the waves exceeds the Rayleigh prediction
by almost 20 percent. This result suggests a rather severe problem in
structural design if the Rayleigh model is used for estimating extreme waves,
since it will be low by 20 percent under these conditions. For intermediate
and large energy ratios and for large peak separations, the statistics of the
higher wave averages are less, by up to 7 percent, than the Rayleigh model
prediction.

75. The reasons for this severe dichotomy in the extreme average wave
heights at large modal separations are not immediately obvious. Part of the
explanation is how H,,, behaves relative to the absolute average heights in
the various test conditions. If waves in the high-frequency mode are as
energetic or are more energetic than those in the lower frequency mode (energy
ratio => 1), the effect is simply to broaden the spectrum. High-frequency

waves are carried by lower frequency waves of comparable energy, and the

43

% Difference

% Difference

% Difference

Difference

°

7

Synthetic Data vs. Rayleigh Model

S [ aes a lta aa es tl ee ps ate Pen maar |e T T eine) Ioana L aN of, =|
L H6Y109) /Urmns |
[ Energy Ratio |
ONE ae 4.00 |
O 2.00 |
A 1.00
=e 0.50 ;
[ x O25
|
lk |
eae EELS

0,0) 0.1) O20 OS 0014.) 0-5" i Oren minOlz (OLS ll OxO bem ah Oim ape! AND IS nae
Peak Separation

Figure 10. Wave height averages from bimodal spectra compared with
Rayleigh wave height averages

44

% Difference

% Difference

% Difference

% Difference

20

15

10

Synthetic Data vs. Modified Rayleigh Model

HUF sapale sl T T Tel Tereearsecresnect Tee T aVenaalU ch T T Teal
H6°Y199) Hires
Energy Ratio
4.00
2.00
1.00
0.50

ye a ep

x ot bomb

ge ee

OO OA) OL - Ox8) WLP Os ON \ Oe) OS OKe YO ae A Calas
Peak Separation

Figure 11. Wave height averages from bimodal spectra compared with
Modified Rayleigh wave height averages

45

effect on average heights is similar to results from the broad, unimodal cases
of Part IV. In fact, the cases in Figure 10 with energy ratios of 1.0, 2.0,
and 4.0 have properties very much like the two broadest unimodal cases (Af/f£,
= 0.80 and 1.60) in Figure 3. Mean wave heights H“!/)) are higher than

Rayleigh predictions by 2 to 3 percent; the H‘/*)

H(1/10)

are smaller than Rayleigh

by 2 to 3 percent; the H61/100)

are smaller by 3 to 5 percent; and the
are smaller by 3 to 10 percent. The evident conclusion is that from some
bimodal spectra with intermediate and large peak separation, the effect on
wave height averages is the same as for broad, unimodal spectra.

76. For bimodal spectra with low energy ratios (energy in the high-
frequency mode somewhat smaller than that in the low-frequency mode), average
wave height behavior is very different at large peak separations. Here,
again, short waves are being carried on longer waves, but now the short waves
tend to have less amplitude than the long waves. This situation leads to the
condition that at the crests and troughs of the long waves, there are a number
of short waves whose extrema do not cross the line of zero displacement.

There tend to be higher crests because short wave crests add to long wave
crests, and deeper troughs because short wave troughs drop below long wave
troughs, but fewer waves overall because the short waves do not have as many
zero crossings as when they have higher energy. In regions of the time series
near nodes (zero-crossings) of the low-frequency waves, the number of zero
crossings is about the same as for higher levels of high-frequency energy.
Hence, there is a relative reduction in the number of waves of intermediate
height, those which would occur if the high-frequency waves reached zero level
from extrema of the low-frequency waves. There becomes a relative abundance
of both large and small waves, at the expense of the number of waves of inter-
mediate height.

77. This redistribution of wave heights also affects the value of

Hm; » @ parameter used to normalize all distributions and parameters in this

‘mo

study. The net results of this alteration of H,,, (relative to H,, , the
parameter used to govern total spectral energy and to generate the time
series) and the redistribution of wave heights are shown as the normalized
wave height averages in Figure 10. The higher wave height averages are very
much higher than the Rayleigh prediction. For the lower averages, H'‘t!*)

hovers near the Rayleigh prediction, and the average wave height H“/) tends

to be less than the Rayleigh value.

46

78. A remarkable improvement over the results shown in Figure 10 is the

comparison of the same synthetic data with the Modified Rayliegh model as
shown in Figure 11. This improved comparison could be expected for two
reasons. One reason is that the Modified Rayleigh model is a two-parameter
model and is therefore much more highly adaptable than the Rayleigh model.
The second reason is that the effective broadening of the spectra by medium-
to-large modal separations and medium-to-large energy ratios is rather like
the broad unimodal cases for which the Modified Rayleigh model worked excep-
tionally well, as shown in Figure 4.

79. However, bimodal cases with large modal separations and small
energy ratios were not entirely well-fitted by the Modified Rayleigh model.
There is some improvement over the Rayleigh model as can be seen by comparing
Figures 10 and 11 (they are plotted at the same scale), but this improvement
is likely due to the improved adaptability of the Modified Rayleigh model
rather than any inherent ability to represent the underlying process. This
result suggests that for wide modal separations and small energy ratios, there
is another, as yet undefined, distribution which better represents these
processes.

80. Some examples will clarify the above arguments. Figures 12, 13,
14, and 15 show pdf and exceedence curves for time series derived from bimodal
spectra with small, intermediate, and two cases of large modal separation.
Figure 12 represents a case where modal separation is small ( = 0.10) and
energy ratio is also small ( = 0.25). In this case, the synthetic data lie
very close to the Rayleigh curve, and the Modified Rayleigh model is very near
the Rayleigh model asymptotic shape. Hence, under these conditions, the
process is well-modeled by the Rayleigh pdf. Figure 13 represents a case with
the same energy ratio as in Figure 12, but with an intermediate modal
separation ( = 0.60). The effective spectral broadening is apparent, espe-
cially in the exceedence curves. Synthetic data and Modified Rayleigh results
agree very closely, and both differ from the Rayleigh model on the high-wave
tail of the distribution. This behavior is very much like that shown in
Figure 9, representing broad-banded, unimodal spectra.

81. Figures 14 and 15 depict results from cases where the governing
bimodal spectra have the largest separation parameter ( = 1.33), but extremes
of relative energy parameter ( = 4.0 and 0.25, respectively). Figure 14

represents a case that follows the pattern of large relative modal energy.

47

Wave Height Distributions
Averages from 20 Bimodal, Band—Limited, White Spectra
Mode fc(Hz) Af/fc Hro(m) N
1 0.095 0.10500 12789 S27.
Zz OMOSa7 WxKO95O00 pm ONS 94 wo 27,
Hams = 1.407" (O)..002" "Anna 126724) 01014 € = 0. 102978

15

a es
= | SIZ2ZawiaVvesrsamiple div
Rayleigh
ee re oe a Or a ana A em nb ede birice honraee Modified Rayleigh
2
Aa |
E IL 4
ais
SS
<7
a
on
Eo
Io
°
°
% 3
‘o

Exceedence Probability
10

x 3172 waves sampled ‘ ax
Sok oo 1 SyDy, re SA
n Rayleigh MS x aS
' “gr . N
2) Bay aucceanes Modified Rayleigh Se x S
\ : a2
\ \
\ \
\ \
P Y
lb se pe ee os SN aT a ee rie Se ee Me nats Sr Sree
1020) 0.5 1.0 es) 2.0 2:5 3.0
Aah es

Figure 12. Probability and exceedence curves from bimodal spectra with
small modal separation

48

Exceedence Probability

pe P(H/H ms)

0.5

1.5

Wave Height Distributions
Averages from 20 Bimodal, Band—Limited, White Spectra
Mode fc(Hz) Af/fc Hro(m) N
1 OFO7O SHOM4SO0 Cy 789" 9 S28
2 ©.130 0807690 "0.894 1528

Hrms = 1.374 + 0.008 Hrmg = 1.611 + 0.012 e = 0.551182

1.0

[Naa ent ae aaa ae a 1 I a

— 7 2759 waves sampled
—— Rayleigh
er aa Modified Rayleigh |

44

10° 0.0

3

10

2759 waves sampled
eel SDs

Rayleigh

Modified Rayleigh

arene — Hn 1 n esl 1 <a ! n nr 1 n 4. 1 1 1 be

0.0

Figure

13h

0.5 1.0 1.5 2.0 2.5 3.0
Ana
Probability and exceedence curves for bimodal spectra with
intermediate modal separation

49

Wave Height Distributions
Averages from 20 Bimodal, Band—Limited, White Spectra
Mode fc(Hz) At/ fie \ktmo (nm) N
1 OF050 NOV ZOOOO MONS 94N N28
2 0.250 0.04000 Wii SOV RS ZS

Hras = 1.317 + 0.006 Hrmg = 1.530 + 0.017. € = 0.429501
2 ia T T z T i a 7 u T i u i Zi T iF Tamer came econ aman) u u T eS ee
r — *— 7206 waves sampled
—————_ JRayleigh

Pecbine 2s Modified Rayleigh

i
7G
1)
a
le)
[=
[aw N
to) 1
one
c
o
;U
o
o
O
us 7206 waves sampled "
ASS ESD): ‘5
rn —— _ Rayleigh ) in 4
eye . \
(S| BORE e Modified Rayleigh
Ib hea SUE page | gol (ee ee Rae noes
rae O20) 0.5 1.0 1.5, 2.0) 22S 3.0
Vania

Figure 14. Probability and exceedence curves from bimodal spectra with
large modal separation and high energy ratio

50

Wave Height Distributions
Averages from 20 Bimodal, Band—Limited, White Spectra
Mode fc(Hz) Af/fc — Hro(m) N
1 0.050 0.20000 IE PEQe 8 S28
2 0.250 0.04000 0.894 328

nmsa=ol ove OvOIS. mbnmav=) 1476) OnO25 € = 0.855972
eT
i — 7 3778 waves sampled
ll Rayleigh |
aes als Modified Rayleigh
pS
E
ag
SS
S
a
Ew
To
is)
°
©
’

Exceedence Probability
10

3778 waves sampled
rahagias ol dass
n — Rayleigh
Oy Sr ipeceacneue Modified Rayleigh
i RB co res W ar BrcReO ee APS VI Ret yt Pwr Nery EEN [RL a
Te NOLO 0.5 1.0 igs) 2.0 25) 350
BVAmbes

Figure 15. Probability and exceedence curves from bimodal spectra with
large modal separation and small energy ratio

Si

Synthetic wave heights average to an exceedence curve that is more nearly
represented by the Modified Rayleigh model than the Rayleigh model. Dramatic
differences from Figure 14 are seen in Figure 15, which represents a case with
small relative energy. In Figure 15, synthetic data differ dramatically from
both Rayleigh and Modified Rayleigh pdf curves. The exceedence estimate of
the synthetic data clearly differs from the Rayleigh model, as shown in the
lower part of Figure 15. Synthetic data also deviate by more than one
standard deviation from the Modified Rayleigh curve at low and intermediate
wave heights. The Modified Rayleigh model does appear to conform reasonably
well with synthetic data in the high-wave tail of the distribution, but this
may simply be a fortuitous circumstance, given the poor agreement elsewhere in
the distribution.

82. To summarize, it appears that for spectra with broad ranges of
modal separation parameter and with a moderate range of relative modal
energies roughly greater than 1.0, derived wave height distributions tend to
be well-represented by the Modified Rayleigh model. Most distributions in
these ranges of parameters tend to be overpredicted by the Rayleigh model, as
was found for time series derived from broad-banded spectra and as have been
found frequently in natural observations (SPM 1984). For large modal separa-
tions and small relative modal energy, synthetic data deviate strongly from
the Rayleigh model and deviate significantly from the Modified Rayleigh model.
This result suggests the need for a third model to represent wave height
distributions under conditions where governing spectra have wide modal
separations and low relative energies. It also suggests that further study be
performed to determine how frequently such conditions occur in nature, so as

to determine their relative importance.

52

PART VI: CONCLUSION

83. In this report, a simple examination is made of wave height
distributions in synthetic sea states characterized by energy spectra with
multiple peaks, i.e., having energy centered at two or more distinct frequen-
cies. Sea states with broad frequency spectra are included in this study
since such seas are also composed of waves of diverse frequencies. These
cases violate the assumptions of unimodal, narrow spectra that are formally
required for the conventionally used Rayleigh distribution of wave heights to
apply. Because of common usage of this distribution in engineering design, it
is important to determine what errors are incurred by these violations and if
an alternate model can compensate for them.

84. Reported here are tests of the Rayleigh and Modified Rayleigh wave
height distribution models. These models are compared with wave heights from
idealized, synthetic time series having spectra with variable widths, numbers
of modes, and modal separations. The Modified Rayleigh model is the two-
parameter, deepwater asymptotic form of the Beta-Rayleigh distribution
introduced by Hughes and Borgman (1987). Synthetic time series are produced
by inverse Fourier transform techniques and consist of 65,536 points at a
nominal time step of 0.5 sec, so the record simulates in excess of 9 hr of
sampling. Synthetic wave periods are maintained near 10 sec so the samples
contain typically about 3,000 waves, enough to compute some reasonably concise
statistics. Random phases are used in signal generation, so there is the
possibility of some variation between runs with otherwise constant generating
parameters. Hence, 20 runs were done for each case, and the results averaged
to produce a stabler estimate of expected behavior.

85. The first part of this work involves determining some of the limits
implied by the phrase “unimodal, narrow spectrum" in the synthesis of time
series having Rayleigh distributions of wave heights. It is found that for
unimodal, band-limited, white spectra, there are constraints on both the
overall bandwidth and the number of component waves for a close approximation
to a Rayleigh process to occur. Average heights H‘/!) and the average of
the highest one-third waves H‘/!) are within 2 to 3 percent of Rayleigh
estimates as long as bandwidths normalized by center frequencies do not exceed
about Af/f, = 0.4 . For H°/!9 | it appears that about 20 component waves

are necessary to differ from a Rayleigh H‘/! by less than 10 percent. For

53

H°2/190) it takes about 100 component waves to stay within 10 percent of a

Rayleigh H‘1/109) | Beyond a certain bulk spectral bandwidth, the parameters
do not appear to approach Rayleigh parameters, no matter how many component
waves are used in the spectral definition. This is most clear for the widest
case investigated here, Af/f, = 1.60 , but is also slightly evident in the
next-to-widest case Af/f, = 0.80 . It is notable that even the mean wave
height H‘/!) and the average of the highest one-third waves Hot/>) show
some effect of spectral broadening.

86. The Modified Rayleigh pdf and exceedence curves clearly represent
the synthetic data better than the Rayleigh curves. With roughly the same
constraints on numbers of component waves as for the Rayleigh comparison,
averages of synthetic wave height distributions were within 2 percent of
averages from the Modified Rayleigh model for all bandwidths and for averages

64/100)

out to This result suggests that the Modified Rayleigh curve is a

better model for broad spectra if the added parameter H,,, can be determined
directly from the spectra. It is not yet evident that this determination can
be made, and further research is required to characterize H,,, in terms of
spectral shape.

87. The primary part of this investigation is examination of wave
height distributions in wave signals derived from multimodal spectra. To keep
the study simple, only bimodal cases are considered. Using criteria es-
tablished in the unimodal tests, bimodal spectra are constructed from two
unimodal spectra, each of which yields a Rayleigh wave height distribution by
itself. Given this characteristic, the details of the mode structure
(bandwidth and number of component wave trains) are no longer important. The
two primary variables in these bimodal tests are a measure of modal separation
in the frequency domain and the relative amount of energy in one mode compared
with the other.

88. Modal separation is characterized by differences between modal
center frequencies normalized by the mean of the two modal center frequencies.
In the present tests, the smallest modal separation is zero, corresponding to
modes coincident in the frequency domain and representing a directional wave
field where the modes have two different peak directions. The largest modal
separation has one mode center frequency at 0.05 Hz and the other at O25 Hize
corresponding to wave periods of 20 and 4 sec, respectively, and considered to

be a practical limit for wind waves. Five levels of relative energy have been

54

used in the present tests. The relative energy ratio varies from 0.25 to 4.0
following a geometric progression.

89. Results of this part indicate that for spectra with broad ranges of
modal separation parameter and with a ratio of low-frequency modal energy to
high-frequency modal energy roughly greater than 1.0, derived wave height
distributions tend to be well-represented by the Modified Rayleigh model. The
Rayleigh model tends to overpredict all the wave height averages for these
parametric ranges in much the same way as was found for time series derived
from broad-banded spectra and as have been found frequently in natural
observations (SPM 1984). For large modal separations and small high-
frequency modal energy relative to low-frequency modal energy, synthetic data
deviate strongly from the Rayleigh model and deviate significantly from the
Modified Rayleigh model. This result suggests the need for a third model to
represent wave height distributions under conditions where governing spectra
have wide modal separations and low relative energies. It also suggests that
further study be performed to determine how frequently such conditions occur
in nature, so as to determine their relative importance.

90. The two key findings in this study are, first, that spectra that
are not narrow-banded or are bimodal with moderate-to-wide modal separations
tend to have wave height distributions that are overpredicted by the Rayleigh
wave height model, which may account for the common observation that the
Rayleigh distribution is conservative for engineering design. Second, the
Modified Rayleigh wave height model has a much better capability of charac-
terizing distributions from broad-banded or multimodal wave fields. Further
research is required to determine the parameters of the Modified Rayleigh
model in terms of spectral parameters. Finally, these conclusions need to be
tested with natural data to ensure that the artifice of synthesis has not

biased the results.

DD

REFERENCES

Abramowitz, M., and Stegun, I. A. 1970. Handbook of Mathematical Functions
Dover, New York.

Bendat, J. S., and Piersol, A. G. 1971. Random Data: Analysis and Measure-
ment Procedures, Wiley-Interscience, New York.

Cartwright, D. E., and Longuet-Higgins, M. S. 1956. "The Statistical Dis-
tribution of the Maxima of a Random Process," Proceedings of the Royal Society
of London, Series A, Vol 237, pp 212-232.

Earle, M. D. 1975. “Extreme Wave Conditions During Hurricane Camille,"
Journal of Geophysical Research, Vol 80, pp 377-379.

Forristall, G. Z. 1978. "On the Statistical Distribution of Wave Heights in
a Storm," Journal of Geophysical Research, Vol 83, pp 2353-2358.

Hughes, S. A., and Borgman, L. F. 1987. "Beta-Rayleigh Distribution for
Shallow Water Wave Heights," Proceedings of a Conference on Coastal Hydrody-
namics, American Society of Civil Engineers, 28 June-1 July 1987, Newark, DE,
pp 17-31.

Long, C. E. "Use of Theoretical Wave Height Distributions in Directional
Seas," Technical Report in preparation, US Army Engineer Waterways Experiment
Station, Vicksburg, MS.

Longuet-Higgins, M. S. 1952. "On the Statistical Distribution of the Heights
of Sea Waves," Journal of Marine Research, Vol 11, pp 245-266.

1980. "On the Distribution of the Heights of Sea Waves: Some
Effects of Nonlinearity and Finite Bandwidth," Journal of Geophysical Re-
search, Vol 85, pp 1519-1523.

Shore Protection Manual. 1984. 4th Ed., 2 Vols, US Army Engineer Waterways
Experiment Station, Coastal Engineering Research Center, US Government
Printing Office, Washington, DC.

Thompson, E. F. 1980. "Energy Spectra in Shallow U.S. Coastal Waters,"
Technical Paper No. 80-2, US Army Corps of Engineers, Coastal Engineering

Research Center, Fort Belvoir, VA.

Thornton, E. B., and Guza, R. T. 1983. "Transformation of Wave Height
Distribution," Journal of Geophysical Research, Vol 88, pp 5925-5938.

56

APPENDIX A: NOTATION

Fourier cosine coefficient at k*® frequency for time series x
Fourier cosine coefficient at k*® frequency for time series y
Fourier sine coefficient at k™* frequency for time series x
Fourier sine coefficient at k™*® frequency for time series y
Modulus of sinusoidal component k of time series x
Infinitesimal increment of frequency

Infinitesimal increment of wave height

Time step

Infinitesimal increment of x

Frequency

Center frequency of a spectrum

Specific frequencies

Wave height

Spectrum-based characteristic wave height

The n‘® wave height ina set

Average of highest fraction r of all wave heights
Root-mean-quad wave height

Root-mean-square wave height

Average of highest one-third of all wave heights

Alternate notation for H‘1/*»

Randomly chosen wave height

Summing index

Index of an observed wave height; index of an element of
discrete function

Number of wave heights in accumulation bin u

Al

M2

Mm,

Summing index; frequency index

The nt® moment of a frequency spectrum
Zeroth moment of frequency spectrum
Second moment of frequency spectrum
Fourth moment of frequency spectrum

Index of a set of discrete wave heights; index of a sequence of
time steps

Number of observed waves in a record; number of observed water
surface elevations

Probability density function

Two-wave probability density function
Probability density function

Modified Rayleigh probability density function
Rayleigh probability density function
Cumulative probability function

Two-wave cumulative probability function
Modified Rayleigh cumulative probability function
Rayleigh cumulative probability function
Probability that expression in [ ] is true
Exceedence probability

Two-wave exceedence probability

Exceedence probability estimated from data
Modified Rayleigh exceedence probability
Rayleigh exceedence probability

Fraction between zero and one

Element j of function representing fraction of highest observed
waves

Root-mean-square

A2

Pack

Sea surface variance spectral density

Discrete spectral density at kt frequency of time series

Discrete spectral density at kt® frequency of time series

Element n of a sequence of N discrete times

Bin index for histogram of wave heights

Uniform random deviate at kt® frequency of time series
Uniform random deviate at k™® frequency of time series
Dummy integration variable; a time series

Element n of discrete time series x

Discrete Fourier transform element k of time series
A time series

Element n of discrete time series y

Discrete Fourier transform element k of time series

Discrete Fourier transform element k of complex time series

x Ei aly,

Parameter of Modified Rayleigh pdf
Gamma function

Discrete increment of frequency
Discrete increment of wave height
Discrete time step

Frequency spectral width parameter

Phase of sinusoidal component k of time series x

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