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UFL/COEL-96/011
NON-GAUSSIAN PROPERTIES OF WAVES IN
FINITE WATER DEPTH
by
David J. Robillard
Thesis
1996
NON-GAUSSIAN PROPERTIES OF WAVES IN FINITE WATER DEPTH
By
DAVID J. ROBILLARD
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF ENGINEERING
UNIVERSITY OF FLORIDA
1996
ACKNOWLEDGMENTS
David J. Robillard wishes to express sincere thanks to his wife and children for
their understanding and support. The author would also like to thank the members of the
graduate committee, Dr. Michel K Ochi, Dr. Robert G. Dean and Dr. Ashish Mehta, for
their input and guidance during this study. Special thanks are extended to the committee
chairman, Dr. Ochi, whose instruction and insight enabled me to learn a great deal on the
topic of this study.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ......................................................................................... ii
LIST OF FIGURES................................................................................................... v
LIST OF TABLES ....................................................................................................... viii
ABSTRACT........................................ ..................... .............................................. ix
1 INTRODUCTION ............................................................................................... 1
1.1 Purpose of Study.......................................................................................... 2
2 LITERATURE REVIEW ..........................................................................................5
3 BACKGROUND/STUDY APPROACH........................................................... 8
3.1 Wave Record Data...................................... ............................................... 8
3.1.1 Data Acquisition.................................................................................. 8
3.1.2 Wave Profile Transformations ............................................ ............... 9
3.2 Nearshore Zone Characteristics................................................................ 16
3.2.1 Discussion of Water Depth Regions During ARSLOE......................... 16
3.2.2 Discussion of the Surf Zone During ARSLOE ........................................ 17
3.3 Non-Gaussian Probability Density Function................................ ............ .. 19
3.3.1 Non-Gaussian Probability Density Function Background ..................... 19
3.3.2 Probability Density Function Application to Wave Data......................... 21
3.3.3 Dimensional Analysis of PDF Parameters.......................... ........... .. 22
4 PROBABILITY DENSITY FUNCTION ANALYSIS.............................................. 24
4.1 Probability Density Function Applicability Verification................................... 24
4.2 Gaussian/Non-Gaussian Boundary Definition .............................................. 33
5 TREND ANALYSIS OF PARAMETERS...........................................................42
5.1 Relationship Between a and Water Depth.................................... ............ 42
5.2 Non-Dimensional PDF Parameter Analysis................................. ............ .. 44
5.2.1 Spatial Analysis of Data...................................................................... 44
5.2.2 Relationship Between aa, .,/o and ./ ............................................... 52
5.2.3 Relationship Between ao and a/d........................................ ............ 54
6 SUMMARY AND CONCLUSIONS .............................................. ........ 56
APPENDIX
COMPARISON OF ORIGINAL AND SMOOTHED "a" VALUES.............................. 58
LIST OF REFERENCES....................................... ................................................ 66
BIOGRAPHICAL SKETCH .................................................................................. 68
LIST OF FIGURES
Figure page
1.1. Wave Profile Examples: a) Gage 615, Severe Sea Condition, and b) Gage
625, M ild Sea Condition .................................................. ................................. 1
3.1. Coastal Engineering Research Center Field Research Facility ......................... 10
3.2. Gage 615 10/23 @ 2155 Wave Profile................................. ......................... 12
3.3. Gage 635 10/23 @ 2155 Wave Profile............................................... 12
3.4. Gage 645 10/23 @ 2155 Wave Profile..................... ......................... 12
3.5. Gage 655 10/23 @ 2155 Wave Profile..................... ............................ 12
3.6. Gage 665 10/23 @ 2155 Wave Profile..................... ................................ 13
3.7. Gage 675 10/23 @ 2155 Wave Profile..................... ......................... 13
3.8. Gage 625 10/23 @ 2155 Wave Profile...................................... ............ 13
3.9. Gage 615 10/25 @ 1315 Wave Profile......................................................... 14
3.10. Gage 635 10/25 @ 1315 Wave Profile....................................................... 14
3.11. Gage 645 10/25 @ 1315 Wave Profile...................................... ............ 14
3.12. Gage 655 10/25 @ 1315 Wave Profile..................... ......................... 14
3.13. Gage 665 10/25 @ 1315 Wave Profile..................... ................................ 15
3.14. Gage 675 10/25 @ 1315 Wave Profile..................... ......................... 15
3.15. Gage 625 10/25 @ 1315 Wave Profile.......................................................... ... 15
4.1. Gage 615 10/23 @ 2155 Histogram Data .................................................. 25
4.2. Gage 635 10/23 @ 2155 Histogram Data ........................................................26
4.3. Gage 645 10/23 @ 2155 Histogram Data ........................................... ............. 26
4.4. Gage 655 10/23 @ 2155 Histogram Data .................................. ............. 27
4.5. Gage 665 10/23 @ 2155 Histogram Data .................................. ............. 27
4.6. Gage 675 10/23 @ 2155 Histogram Data .................................. ............. 28
4.7. Gage 625 10/23 @ 2155 Histogram Data .................................. ............. 28
4.8. Gage 615 10/25 @ 1315 Histogram Data ................................................ 29
4.9. Gage 635 10/25 @ 1315 Histogram Data .......................................................29
4.10. Gage 645 10/25 @ 1315 Histogram Data .................................. ............. 30
4.11. Gage 655 10/25 @ 1315 Histogram Data .................................. ............. 30
4.12. Gage 665 10/25 @ 1315 Histogram Data .......................................................31
4.13. Gage 675 10/25 @ 1315 Histogram Data ................................................ 31
4.14. Gage 625 10/25 @ 1315 Histogram Data ........................................................32
4.15. Sample Data Points to Analyze Gaussian/non-Gaussian Boundary.................... 34
4.16. Probability Density Function of POINT A of Figure 4.15 ................................. 35
4.17. Probability Density Function of POINT B of Figure 4.15 .................................36
4.18. Probability Density Function of POINT C of Figure 4.15 .................................36
4.19. Probability Density Function of POINT D of Figure 4.15 ................................. 37
4.20. Probability Density Function of POINT E of Figure 4.15.................................. 37
4.21. Probability Density Function of POINT F of Figure 4.15.................................. 38
4.22. Probability Density Function of POINT G of Figure 4.15................................. 38
4.23. Probability Density Function of POINT H of Figure 4.15................................. 39
4.24. Gaussian/Non-Gaussian Boundary Defined................................................41
5.1. Spatial Comparison of a at Mild and Severe Sea Conditions ............................43
5.2. All a Data Plotted as a Function of Water Depth........................................... 43
5.3. Spatial Comparison ofaa at Mild and Severe Sea Conditions........................... 45
5.4. Spatial Comparison of p./o at Mild and Severe Sea Conditions........................ 46
5.5. Spatial Comparison of a./a at Mild and Severe Sea Conditions........................ 47
5.6. aa as a Function of Water Depth................................................................ 49
5.7. p./a as a Function of Water Depth.................................................................. 50
5.8. c./a as a Function of Water Depth.......................................................... 51
5.9. Plot of p*/o as a Function ofaa for Gages 615 and 625 ................................. 53
5.10. Plot of y/o as a Function ofaa for Gages 615 and 625 ................................... 53
5.11. Plot of aa as a Function of c/d for Mild and Severe Sea Conditions ................. 55
A.1. Plot of Gage 615 Original and Smoothed "a" Values........................................ 60
A.2. Plot of Gage 635 Original and Smoothed "a" Values....................................... 61
A.3. Plot of Gage 645 Original and Smoothed "a" Values....................................... 62
A.4. Plot of Gage 655 Original and Smoothed "a" Values......................................... 63
A.5. Plot of Gage 665 Original and Smoothed "a" Values .................................... 64
A.6. Plot of Gage 675 Original and Smoothed "a" Values .................................... 65
A.7. Plot of Gage 625 Original and Smoothed "a" Values ........................................ 66
LIST OF TABLES
Table page
3.1. Deep Water Wave Statistics for Mild and Severe Sea Conditions........................ 16
3.2. Surf Zone Locations for Mild and Severe Sea Conditions................................... 19
4.1. Wave Profile Statistics for Gage 615 10/23 @ 2155........................................ 25
4.2. Wave Profile Statistics for Gage 635 10/23 @ 2155........................................ 26
4.3. Wave Profile Statistics for Gage 645 10/23 @ 2155........................................ 26
4.4. Wave Profile Statistics for Gage 655 10/23 @ 2155........................................ 27
4.5. Wave Profile Statistics for Gage 665 10/23 @ 2155........................................ 27
4.6. Wave Profile Statistics for Gage 675 10/23 @ 2155........................................ 28
4.7. Wave Profile Statistics for Gage 625 10/23 @ 2155....................................... 28
4.8. Wave Profile Statistics for Gage 615 10/25 @ 1315......................................... 29
4.9. Wave Profile Statistics for Gage 635 10/25 @ 1315..........................................29
4.10. Wave Profile Statistics for Gage 645 10/25 @ 1315......................................... 30
4.11. Wave Profile Statistics for Gage 655 10/25 @ 1315.........................................30
4.12. Wave Profile Statistics for Gage 665 10/25 @ 1315.........................................31
4.13. Wave Profile Statistics for Gage 675 10/25 @ 1315.........................................31
4.14. Wave Profile Statistics for Gage 625 10/25 @ 1315................................ 32
4.15. Statistical Properties of Data Points A-H in Figure 4.15..................................... 35
5.1. General Shallow Water Limits of Parameters ............................................... 52
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Engineering
NON-GAUSSIAN PROPERTIES OF WAVES IN FINITE WATER DEPTH
By
David J. Robillard
May 1996
Chairman: Michel K. Ochi
Major Department: Coastal and Oceanographic Engineering
As waves propagate into shallower waters, their profile changes from a Gaussian
distribution to an increasingly non-Gaussian profile. The non-Gaussian properties of
waves in finite water depths are analyzed through the use of a probability density function
defined in closed form and that can be solved by the application of wave displacement
data. Initially, the applicability of the probability density function is verified using wave
data records obtained from the Coastal Engineering Research Center (CERC) at Duck,
North Carolina. A broad range of sea conditions and water depths are represented to
ensure the probability density function correlates well with actual histogram data over
varied conditions. With the applicability verified, the probability density function is used
to define criteria for the boundary where the wave field can no longer be considered
Gaussian. The ability to determine when a shoaling wave profile can no longer be
considered Gaussian is of paramount importance in order to ensure proper wave theories
are being applied. This boundary between Gaussian and non-Gaussian wave profiles will
be determined in terms of significant wave height and water depth.
Presently, wave data records are required to define the probability density function
used in this study. Trend analyses were conducted on the parameters of the probability
density function to determine if simplifications can be made to its form and to see if it is
possible to define the probability density function in terms of the local conditions, sea
severity and water depth, without the need for wave data records.
CHAPTER 1
INTRODUCTION
As wind generated waves transition from deep water to shallow water, a significant
change is evident in the wave profile. As an example, Figure 1-1 shows portions of wave
records obtained by the Coastal Engineering Research Center, (CERC), during the
ARSLOE Project. Figure 1-1(a) is the wave profile recorded at a distance of 60 meters
offshore in a shallow intermediate water depth condition. Figure 1-1 (b) is the wave
profile taken at the same time but at a distance of 12 km offshore in a deep water
condition. Deep water wave profiles in the sea characteristically reflect a Gaussian (linear)
random process. The height of the wave crests are probabilistically equal to the height of
the wave troughs. As waves move from deep to shallow water, the wave profile
transforms from a Gaussian random process to a non-Gaussian (nonlinear) random
process in which the wave profile shows a definite excess of high crests and shallow
troughs.
u AJ i 4.AJJ ,Ij,,o. AN ..,! A Al A Lv AA 4gA
we j^JJ^J WIy IV Y-b 'Mr V V
a) Gage 615, Severe Sea Condition b) Gage 625, Mild Sea Condition
Figure 1.1 Wave Profile Examples
This remarkable transformation in the wave profile, as an irregular sea transitions
from deep to shallow water, has also been evidenced in laboratory experiments (Doering
and Donelan 1993). The transformation of wave profiles can be attributed to the non-
linear wave-wave interactions augmented primarily by decreasing water depth. The
"degree" of non-linearity a wave field has at a particular site of interest in the coastal
region appears to depend largely on sea severity and water depth.
Although there has been awareness that Gaussian-based solutions do not always
represent accurate results in the coastal region, clarification of wave characteristics from
deep to shallow water has been a subject of considerable interest in ocean and coastal
engineering. There have been few studies which have offered non-Gaussian solutions
describing the probability density function of waves in coastal regions. A commonly
known study was by Longuet-Higgins (1963), which statistically predicts coastal wave
properties using a power series expansion. This probability distribution is based essentially
on the Gram-Charlier series distribution. A probability density function developed by Ochi
and Ahn (1994a) appears to be applicable throughout the coastal region from deep to
shallow water and is defined in closed form Equation (1).
Ax) = -exp 1 ) (1 ya p. exp(-yax))2 yax (1)
2) 7r-. 2(rao-.)
28 for: x 0
where, y = {
3.00for:x <0
This is the probability density function that will be used as the basis to analyze
coastal wave profiles for this study.
1.1 Purpose of the Study
The overall objective of this study involves two components. The first component
will be to validate the applicability of the probability density function developed for waves
in finite water depth (Ochi and Ahn 1994a) and to analyze the transition point where wave
profiles can no longer be considered Gaussian. In order to accomplish the first objective of
this study, a statistical analysis was carried out on wave time histories obtained during a
storm by the Coastal Engineering Research Center (CERC) during the Atlantic Ocean
Remote Sensing Land-Ocean Experiment (ARSLOE) Project in 1980. To interpret the
transformation of waves from a Gaussian to a non-Gaussian random process, wave
records at seven locations were taken where the water depths ranged from two meters to
twenty-one meters. In total, approximately 1000 wave records were analyzed with each
wave record consisting of 4800 water displacement measurements. The individual wave
displacement measurements were recorded at a frequency of 4 Hertz in twenty minute
intervals.
Upon confirmation of the probability density functions' applicability, the histogram
data will be compared closely with the corresponding normal distribution to determine the
conditions under which the wave profile can no longer be considered Gaussian. In
particular, general relationships to sea severity and water depth will be determined. A
criteria will be established which will define the limit to when the Gaussian wave profile
becomes non-Gaussian. This limit will be based on the nonlinearity parameters analyzed
from the wave profile data. The second objective of the study will then be to analyze the
parameters of the probability density function. The three parameters, a, p. and o. will be
statistically analyzed in both dimensional and non-dimensional form. Trends based on the
analysis will be described and relationships linking these parameters of the probability
density function to site conditions will be developed.
This study consists of six chapters. Chapter 2 provides a general review on the
development of probability density functions applicable to coastal regions. Chapter 3
describes the ARSLOE Project of 1980 and addresses the statistical approach taken in this
study. The data used as the basis for this study, were obtained from the ARSLOE Project.
This chapter will also describe the local field conditions at CERC during the time the data
were collected. Chapter 4 addresses the probability density function applicability and
analysis. This chapter has two distinct sections. The first section reviews the finite water
depth probability density function (Ochi and Ahn 1994b) and verifies the applicability of
the probability density function. The second section discusses sea severity and water
depth and the transition between Gaussian and non-Gaussian wave profiles. Chapter 5
relates the non-dimensionalized parameters of the probability density function, aa, 1A./o
and c /o, to sea severity, (a), and water depth in both dimensional and non-dimensional
form. The parameters will also be compared to each other and relationships determined.
This chapter describes trends evident based on statistical analysis. These trends are
qualitative and are intended to give a better understanding of the interaction the wave
profile has with its environment as it propagates in shallow waters. Finally, Chapter 6
provides the summary and conclusion for this study.
CHAPTER 2
LITERATURE REVIEW
As stated in the Introduction, the profile of waves in finite water depths is quite
different from that observed in deep water. From the stochastic view point, waves in finite
water depth are categorized as a non-Gaussian random process for which a precise
description of the process does not exist. This is in contrast to the Gaussian random
process which can be defined by a rigorous mathematical formulation. Although
considerable attention has been given to Gaussian waves in deep water, little information
is available on the non-Gaussian characteristics of shallow water waves.
One of the most commonly known presentations describing non-Gaussian waves is
the Gram-Charlier series probability density function. This density function is given in a
series consisting of an infinite number of terms. Longuet-Higgins (1963) theoretically
showed that the statistical properties of nonlinear waves can be presented by the Gram-
Charlier series distribution. Bitner (1980) found reasonable agreement in a comparison
between the Gram-Charlier series distribution and the histogram constructed from wave
data obtained in a relatively shallow water area.
On the other hand, Ochi and Wang (1984) analyzed more than 500 wave records
obtained by the Coastal Engineering Research Center (CERC) at Duck, North Carolina
during the ARSLOE Project (1980). They found that while the statistical properties of
waves at any water depth can be well represented by the Gram-Charlier series distribution,
there is a drawback to applying this density function in practice. The probability density
function becomes negative for large negative wave displacements and this negative portion
does not vanish with increasing number of terms in the series distribution. Because of this
difficulty, the transition property of wave characteristics from gaussian to non-gaussian
distribution can not be well clarified. They derived, however, a limiting sea severity for a
specified water depth below which wind-generated coastal waves may be considered to be
a Gaussian random process.
Another approach for describing the surface elevation ofnon-Gaussian random
waves is to assume that individual waves can be expressed as a Stokes wave. There
appears to be some reservation to applying this approach for representing the statistical
distribution of the wave profile. This method presupposes a preliminary form of the wave
profile before establishing its probability density. This supposition may impose
unfavorable bias on the outcome.
Tayfun (1980) derived the probability density function based on the Stokes profile
approximation up to the second order. Huang et al (1983) developed the probability
density function considering all terms up to the third order. These probability density
functions are superior to the Gram-Charlier series distribution in that they do not have any
negative portion over the entire range of wave displacement. Comparison of these
probability density functions and wave data show good agreement.
Langley (1987) applied a probability density function representing the response of
a non-linear system developed by Kac and Siegert (1947), to nonlinear random waves.
This probability density function is given in the form of a series and can only be evaluated
numerically. Ochi and Ahn (1994a) modified the Kac Siegert density function so that it is
presented in closed form with parameters determined through spectral analysis. They
applied the probability density function to non-Gaussian coastal waves. In the application
process, they decomposed the spectra into linear and nonlinear components to clarify the
degree of non-linearity involved in shallow water waves (Ochi and Ahn 1994b). The
probability density function does not have any negative portion in its entire domain and
appears to represent the statistical distribution of the wave profile in any sea state during a
storm in deep, intermediate and shallow water.
Studies concerned with the non-Gaussian properties of waves in finite water depth
include Thornton and Guza (1983) who claimed that the results of their analysis of field
data obtained at Torrey Pines Beach, California, show that the Rayleigh probability
distribution can be applied to represent wave heights observed from shallow to deep (10.8
meter) water. This implies that all observed waves can be considered a Gaussian random
process. However, the sea severity in relation to water depth at all locations in the
nearshore zone where their measurements were taken was too mild and below the limit for
waves to be a non-Gaussian random process. This subject will be discussed in detail in
this present study.
In closing this chapter, it is important to note that there still remains a significant
amount of work to do in refining our ability to accurately describe irregular seas in finite
water depths and over a wide range of sea severity.
CHAPTER 3
BACKGROUND / STUDY APPROACH
3.1 Wave Record Data
An experiment jointly hosted by the U.S. Army Coastal Engineering Research
Center (CERC) and the National Ocean Survey (NOS) was conducted from 06 October
1980 to 30 November 1980. An aspect of this Atlantic Ocean Remote Sensing Land-
Ocean Experiment (ARSLOE) involved ocean wave studies implemented near the
CERC's Field Research Facility (FRF) at Duck, North Carolina. In situ wave measuring
devices were deployed in mean water depths ranging from 1.35 meters to 24.4 meters and
were placed in a linear array extending from 60 meters offshore to 12,000 meters offshore,
Figure 3.1.
3.1.1 Data Acquisition
The wave displacement data analyzed in this study came from a total of eight wave
measuring devices. The FRF has a 550 meter research pier oriented perpendicular to the
local shore which was equipped with seven Baylor resistance-type wave staffs. These
Baylor wave staffs function by determining the change in electrical resistance experienced
by changes in water elevations. Round-trip travel time of an electrical current from an
above-water emitter to the water surface is measured. Following along the line of the
pier off-shore was a Waverider Buoy Gauge. This is a surface-following class buoy. It
functions by double-integrating the output of an accelerometer held vertical by a pendulum
to produce the time history of buoy heave motion. The statistics of the buoy heave then
represent the wave displacements. In the Waverider buoy, the pendulum is contained
entirely within the spherical float so the buoy does not have to be stabilized. These
devices recorded both wave displacements and wave periods but only the displacement
data were analyzed for this study. The wave gauges recorded water surface fluctuations
simultaneously at all eight locations at a frequency of 4 Hertz over twenty minute
intervals. This resulted in 4800 individual wave displacements recorded for each twenty
minute segments continuously. Due to the significant amount of data obtained during this
unique experiment, only a sub-set of the data was analyzed. This chosen data sub-set
began at 1200 on 23 October 1980 and ended at 1315 on 25 October 1980. This two day
segment was unique in that a severe storm occurred from 23-25 October which was the
most significant meteorological event recorded during the experiment. Wind speeds on
the order of fifteen meters per second were recorded during the severe storm. This two
day sub-set of data was used as the basis for this study since it included the entire range of
wave profiles experienced at each of the eight wave gauge locations. From this two day
sub-set, a total of nearly 5 million individual wave displacements were analyzed and the
wave profile data covered the entire range of sea conditions present during the complete
56 day ARSLOE Project.
3.1.2 Wave Profile Transformations
The data collected at the FRF during ARSLOE is unique and impressive. It is very
well suited for analyzing changes in the wave field as the waves propagate through
shoaling depths. The bathymetry at the experiment location is relatively simple showing
straight and parallel contours. This provided favorable conditions which maximized
control over the environmental impact on the quality of the data received. The wave
profile data provide a matrix of wave profiles allowing both spatial and temporal analyses
to be performed.
-
S .\
I P
p .
9 ^.* '
i 0
4
54
FIELD IESIARCH FACILITY
10'S4.6"N1 75'4S'S.2"W
S ? ~
.#\
I., I %
(15
m.i l
\\
t '
GAGE
I
12 K 710
456 625
310- 675
200-665
151 655
115-645
90--635
60-615
ZERO-MSL
Figure 3.1 Coastal Engineering Research Center Field Research Facility
a
A wide range of wave profiles was observed during the experiment. Figures 3.2-
3.8 represent a mild sea condition recorded on 10/23 @ 2155 for all finite water depth
gage locations shown in Figure 3.1. Figures 3.9-3.15 represent a severe sea condition
recorded at the height of a storm on 10/25 @ 1315 for the same gage locations. It is
evident from the sample wave data that the wave field transitions as it shoals from deep to
shallow water. The wave data representing the mild sea condition at Gage 625, Figure
3.8, are characterized by an almost symmetrical distribution of positive and negative wave
profile displacements. This is characteristic of Gaussian random processes. As the wave
shoals into shallower water, the wave profile transforms showing progressively higher
peaks and shallower troughs. This process is clearly non-Gaussian and must be analyzed
differently. It is also clear from observing these wave records that the degree of non-
linearity increases as the relative water depth decreases and as sea severity increases. This
implies that analysis of this nonlinearity must include a dependence on water depth as it
relates to the magnitude of the waves' displacement (sea severity).
j a I 0 A A l A'" 1 n a' A / ' 4 " J -A A A
Figure 3-2 Gage 615 10/23 @ 2155 (d=1.74m)
IV v "Y %.vw %d ,.. -wwy wv TV -v v w" - "vv -vl% IV Yv -V ,kr 4 wv lv "
Figure 3-3 Gage 635 10/23 @ 2155 (d=2.83m)
Figure 3-4 Gage 645 10/23 @ 2155 (d=4.31m)
v^ ^M "4v^^ '1 ^\S v VW ,vy\ ^^ v1- ,
Figure 3-5 Gage 655 10/23 @ 2155 (d=6.0m)
Figure 3-6 Gage 665 10/23 @ 2155 (d=6.08m)
Figure 3-7 Gage 675 10/23 @ 2155 (d=6.82m)
L A kA 1 k A A
Figure 3-8 Gage 625 10/23 @ 2155 (d=9.66m)
Figure 3-9 Gage 615 10/25 @ 1315 (d=2.23m)
Figure 3-10 Gage 635 10/25 @ 1315 (d=3.32m)
Figure 3-11 Gage 645 10/25 @ 1315 (d=4.66m)
VV V v vvvvV vvJ vv -' ^ LUv AjvVN VUN V6-V-
Figure 3-12 Gage 655 10/25 @ 1315 (d=6.44m)
Figure 3-13 Gage 665 10/25 @ 1315 (d=6.52m)
Figure 3-14 Gage 675 10/25 @ 1315 (d-7.22m)
Figure 3-15 Gage 625 10/25 @ 1315 (d=10.04m)
A k L L A .A 6-a 4 pA q A A
3.2 Nearshore Zone Characteristics
All wave fields propagating in finite water depths are influenced by the attributes
of the local sea environment. These attributes include water depth and beach slope. An
awareness of the characteristics of these attributes is important to understand the type of
influence they have on the wave field.
3.2.1 Discussion of Water Depth Regions During ARSLOE
Based on linear theory, there are three water depth regions which are defined using
the velocity potential for a progressive wave. The velocity potential is defined as
HgTcosh(k(d +z)) 2n 2* t.
(x,z,t)= sin(kx ).
4r cosh(kd) T
The distinct water depth regions are determined by the asymptotes of the hyperbolic
function. Deep water wave data obtained at gage location 710 and shown in Table 3.1
allow these regions to be defined for the mild sea condition and the severe sea condition.
In general, the deep water wave condition is defined as kd > n, where k is the wave
number and h is the water depth. This condition occurs at d = 21m for the mild sea
condition on 10/23 @ 2155 and d = 43m for the severe sea condition on 10/25 @ 1315.
The shallow water condition is defined as kd < x/10. This condition occurs at d = 0.6m for
the mild sea condition and d = 1.3m for the severe sea condition on 25 October. The
intermediate water condition occurs between these bounds. All wave profile data
obtained from ARSLOE are located in the intermediate water depth region.
Table 3.1 Deep Water Wave Statistics for Mild and Severe Sea Conditions
Date/Time Zero Crossing Period, T Deep Water Wavelength, Xo
10/23 @ 2155 5.17 sec. 41.715 m
10/25 @ 1315 7.43 sec. 86.157 m
3.2.2 Discussion of the Surf Zone During ARSLOE
An integral aspect of any study which covers the nearshore region must analyze
and consider the surf zone. The surf zone is a transient entity which varies in width off-
shore depending on the local sea conditions. These local conditions include the beach
profile, sea severity and direction of wave field. In order to define the surf zone in this
study, wave field data were analyzed which represented the range of sea conditions
experienced over the two day period. The first condition, representing the mildest sea
condition, was taken as 2155, 23 October 1980. The second condition, representing the
most severe sea condition was taken as 1315, 25 October 1980.
Linear wave theory will be used to determine the effects of shoaling as the wave field
enters shallow water. Refraction will not be considered since this process tends to reduce
the shoaling wave heights and thus move the surf zone shoreward. This is shown by the
definition of the refraction coefficient, K, = cos- where Oo and 0 are the deep water
V cos0
wave angle and the finite water depth wave angle respectively. This ratio will always be
less than one causing a reduction in the resulting wave height. In order to ensure that
conservative results are obtained, refraction will be neglected in this study. The beach
profile at the ARSLOE project site shows slightly varying beach slopes so a wave
breaking criteria which considers the beach slope will be used. Weggel (1972)
reinterpreted laboratory results and showed a dependency of breaker height on beach
slope. The following equations were used to define the surfzone locations:
CCEgl
Conservation of Energy: H 2 = H C g
CE2
Breaker Criterion: H, = Kdb, where K is defined by Weggel as follows:
K = b(m) a(m) H with a(m) = 43.8(10 e-")
gT
b(m) = 156(L0 + e-9m)-
As the beach slope, m approaches zero, this equation reduces to K = 0.78.
The Group Velocity, C, = (I 1+)( 2kd will be determined based on the
2 \ 2
dispersion relation, = gk tanh(kd), where L = wave length, T = wave period
(assumed constant), g = gravitational acceleration, k = the wave number and d = local
water depth.
In order to standardize the location of the surf zone, the root-mean-square wave
height, Hn,, will be used to represent the irregular wave field. H,,,,, = -V H
where Hi is defined as the wave height of the i' wave of the irregular wave field. The surf
zone which results from the use of HI,. will represent the approximate "center of mass"
location where the most significant set of individual waves in the wave field begin to
break. The surf zone calculations were based on a root-mean-square wave height that was
determined from the deep water wave displacement data. This single wave height was
used to represent the irregular sea and shoaled landward. Since the real sea will have a
significant number of waves with heights larger and smaller than the root-mean-square, the
actual breaking zone is an obscure entity. This technique for determining the surf zone
locations is adequate to make general comparisons when analyzing the spatial changes in
the nonlinearity of the irregular sea in finite water depths. The surf zone locations during
the mildest and most severe sea conditions over the two day period are shown in Table
3.2.
In Table 3.2 it is shown that during the mild sea condition, the breakpoint, indicating
the beginning of the surf zone, is located 70 meters from the shoreline in 1.51 meters of
water. This breakpoint is the location where a wave having a height equivalent to the
root-mean-square wave height of the wave field will break. During the severe sea
condition, this breakpoint occurs 110 meters from the shoreline in 3.32 meters of water.
Realistically, waves would be observed breaking considerably seaward of the breakpoints
determined in the respective sea conditions due to the spectrum of wave heights present in
an irregular sea. The root-mean-square wave height breakpoint is used in order to
standardize the position of the surf zone for comparison purposes.
Table 3.2 Surf Zone Locations for Mild and Severe Sea Conditions
Sea Condition Depth where a wave with Start of Surf Zone
height H,, Breaks -Dist. from Shore-
Mild 10/23 @ 2155 1.51 m 70 m
Severe 10/25 @ 1315 3.32 m 110 m
3.3 Non-Gaussian Probability Density Function
The probabilistic approach used in this study will make use of the probability
density function applicable to non-gaussian random processes developed by Ochi and Ahn
(1994a). This probability density function appears to represent non-Gaussian waves in
shallow and intermediate water depths as well as Gaussian waves in deep water.
3.3.1 Non-Gaussian Probability Density Function Background
The probability density function has a background of the Kac-Siegert solution
developed for the response of non-linear systems. The density function, y(t), is given as
the sum of the standardized normal variates and its squares as follows
N
y(t)= Z (pZ, + A ZJ)
j=l
The parameters Pj and j are determined by finding the eigenfunction and eigenvalues
of the integral equation given by
f K(col,co2)y (c.2)dco2 = yj,(i,)
where K(co 2 )= H(oo, o2 ) S(acI)S(wo2)
j(o) = orthogonal eigenfunction = { fy (co)yl (k)dco = 1 forj = k
otherwise, Wj(co) = 0
H(mc 02 ) = second order frequency response function
S(co) = output spectral density function
As can be seen from the form of this solution, the probability density function ofy(t),
can not be found without knowledge of the second order frequency response function,
H(co c2 ) and there is no way to evaluate this response function for irregular seas. Ochi
and Ahn showed however, that Kac-Siegerts' solution can be obtained from the time
history of the wave record without any knowledge of the second order frequency response
function, H(o, oh ). The probability density function thus obtained for describing non-
gaussian random processes is
f(x) = exp (1 ya. exp(-ax))2 Tax
2;cra. [ 2(yao-.)
28for:x 2 0
where, y =
3.00for: x <0
The parameters, a, p. and o2. are determined from the cumulants evaluated from the wave
data. These cumulants are related to the 3j and Xj values of the Kac-Siegert solution by
the following relationships
2N
ki = Z-'A 2
j=
2N 2N
k2 = p 2 + Z 2
J=1 J-1
2N 2 N
k3=61/ p21J + 8 2 l3J etc.
ji- j=-
In the present study, ki represents the mean water level and is therefore defined by,
kl = 0. k2 and k3 are equal to the data sample moments, E[x2] and E[x3] respectively.
Based on this relationship, the parameters, a, p. and a2. can be determined through the
iterative solution of three equations along with the analysis of a data record for the non-
Gaussian random process being analyzed. The three equations are
a oa2 + a .2 + p. = E[x] = 0
2. 2a a = E[x2]
2a 4.(3 8a202.)= E[x3]
3.3.2 Probability Density Function Application to Wave Data
In order to determine values for the three parameters, a, 1,. and o2., which are
used to define the degree ofnon-linearity a shoaling wave field has at a given location and
sea state, data records must be analyzed. With the applicability of the probability density
function verified based on the wave data collected during ARSLOE, the approximately
1000 data records from 1200 on 23 October to 1315 on 25 October will be analyzed to
determine values for the three parameters defining the probability density function. The
procedure used during this study to perform this task is as follows:
Compute the standard deviation of each twenty minute wave record consisting of
4800 discrete wave displacement data points E(x2) = x2
n ,
(E[x] does not need to be computed since it is defined as zero and the data are
already standardized).
Compute the third moment of the same twenty minute wave record
E(x'3)= xY .
n ,
Define the standard deviation of the data record as o" = E(x) .
Evaluate the three parameters, a, p. and o2., through iterative solution of the
following three equations:
a a2. + a A.2 + p.= E[x] = 0
02. 2a2a. = E[x2]
2a a4.(3 8a22*) = E[x3].
It is important to note that the standard deviation of the data record is used in the
analysis as an indicator of the sea severity and not the significant wave height. This is
because the definition of significant wave height is not applicable to non-Gaussian waves.
All available data sets covering the seven finite water depth gage locations
between 1200, 23 October 1980 and 1400, 25 October 1980 were analyzed and a, p. and
a2* were determined.
Based on a plot of the individual parameters in the space domain, it was
determined that a smoothing technique should be used to refine the "a" parameter values.
This smoothing was done in order to remove the sporadic tertiary effects while ensuring
the smoothed "a" value does not lose any authenticity. An exponential smoothing
technique was used which resulted in the lowest mean absolute deviation (MAD) between
the raw "a" parameter value and the smoothed "a" parameter. The remaining two
parameters, p. and o2* did not show the same susceptibility to tertiary effects so no
smoothing was done. Comparisons between the actual and smoothed "a" parameters are
shown in Appendix A for all Gage positions.
3.3.3 Dimensional Analysis of PDF Parameters
For each of the approximately one-hundred-fifty data sets per gage position, the
probability density function parameters, a, t o* were determined iteratively. Each of
the parameters contain discrete dimensions. Based on dimensional analysis, these
dimensions are as follows: It is known that k2, representing the data sets' variance has
dimensions of [L2]. It is also known that k3, representing the data sets' third moment has
dimensions of[L3]. It follows from the knowledge of these dimensions that the
parameters, a, p* a. must have dimensions [1/L], [L] and [L] respectively.
In order to minimize any dependence on external conditions and to ensure the
solutions are universally applicable, these parameters, a, .,, a. must be made non-
dimensional. The non-dimensional quantities that will be used in this study are, aa, p./o,
c./J and o/d. o is defined as the standard deviation of that wave displacement data and d
is defined as the water depth.
CHAPTER 4
PROBABILITY DENSITY FUNCTION ANALYSIS
4.1 Probability Density Function Applicability Verification
As stated in the introduction, one of the objectives of this study is to determine
whether or not the probability, density function developed by Ochi and Ahn is valid for
representing waves in finite water depths. To verify the applicability of this probability
density function, wave displacement data sets obtained from the ARSLOE experiment
were used in comparing the histogram of the data record to the probability density
function. Each data set contained 4800 wave displacement readings taken over a twenty
minute period. To ensure a thorough verification of the probability density function, the
data records used for the analysis were from all seven "finite" water depth gage locations
and analysis was done over a wide range of sea state conditions. One analysis was done
on data from 23 October 1980 at 2155. This represented a mild sea state. The other
analysis was done on data from 25 October 1980 at 1315. This data set was at the peak
of the most severe storm experienced during the entire ARSLOE study.
Figures 4.1-4.14 show comparisons between the histogram data and the non-
Gaussian probability density function. Included also in the figures is the Gaussian
(normal) probability density function for comparison. Figures 4.1-4.7 pertain to the mild
sea condition while Figures 4.8-4.14 represent the severe sea condition. As can be seen
from these figures, the probability density function agrees very well with the histogram
generated from the data record in all cases. This implies that the probability density
function derived by Ochi and Ahn can be used with confidence when analyzing non-
Gaussian random processes such as wave displacements in shallow and intermediate
waters. It can also be seen that the probability density function approximates the normal
distribution when the wave field is in the deeper intermediate water condition under mild
sea conditions.
Gage 615 -10123 @ 2155
Di9 9 0t
OlDsplacm Mnt (m)
4 -
Table 4.1 Gage 615 10/23 @ 2155
615
a (1/m)
p.(m)
a. (m)
a (m)
d (m)
cr/d
WAVE PROPERTIES
10/23 2155
.572645
-.04948
.275683
.267219
1.74
.153574
IStHistogramData ----- PDF *** Gaussian
Figure 4.1 Histogram Data
Gage 635- 10IZ 21z' Table 4.2 Gage 635 10/23 @ 2155
1.60
WAVE PROPERTIES
1.40
1.20 ', 635 10/23 a 2155
1.00
a (1/m) .481441
0. (m) -.0547
S0.40 a .(m) .318871
020
a (m) .30989
0.00
S a "9 d (m) 2.83
Displacment (m)
c/d .109502
SIg l Histogram Dta PP ....... Gaussian
Figure 4.2 Histogram Data
Gage 645 10123 @ 2155
Table 4.3 Gage 645 10/23 @ 2155
1.2
WAVE PROPERTIES
1
6 645 10/23 ( 2155
S0.8 a (1/m) .200988
S06 (m) -.03084
C F. (m) .343446
S0.4
IaC ((m) .340708
0.2 .
d (m) 4.31
o /d .079051
-1 -0.6 -0.2 02 0.6 1 1.4
Displacement (m)
I gnm ~lt stogramData ---PF **** Gaussan
Figure 4.3 Histogram Data
Table 4.4 Gage 655 10/23 @ 2155
I
-1 -0.8 -0.6 -0.4 -02 0 0.2 0.4 0.6 0.8 1 1.2 1.4
[Dspcaement (m)
I HMstogram I~at FF ....... Gauss n
655
a (1/m)
*(m)
P. (m)
a (m)
d (m)
cy/d
WAVE PROPERTIES
10/23 (, 2155
.104437
-.01771
.340056
.339137
6.0
.056523
Table 4.5 Gage 665 10/23 @ 2155
WAVE PROPERTIES
665 10/23 ( 2155
a (l/m) .126266
p.(m) -.01835
C. (m) .365155
a (m) .364236
d(m) 6.08
a/d .059907
Figure 4.5 Histogram Data
0.8
0.6
2 0.4
A.
0.2
0
0age 665 10/23 2l5
Gage g76 -10/23 @ 215 Table 4.6 Gage 675 10/23 @ 2155
12
WAVE PROPERTIES
1 675 10/23 1 2155
U
0.8 / a (1/m) .094349
0. p. *(m) -.01014
S a.(m) .375221
0.4
k (m) .374948
0.2
d (m) 6.82
0
99 C o 0 0 a/d .054978
Olsplacm ent (m)
a stgram Data PF -....... -Guss
Figure 4.6 Histogram Data
Gage 626 -10W23 @ 2166
Table 4.7 Gage 625 10/23 @ 2155
1
0.9 WAVE PROPERTIES
0.8
- 625 10/23 (a 2155
0
t 0.7
u 0.6 a (l/m) .069914
S0.5 f. (m) -.00819
- 0.4
(. (m) .394571
S0.3
02 a (m) .394401
0.1 d(m) 9.66
-1.5 -1 -o.s 0 o.s 1 1.5 a/d .040828
Displacement (m)
I m IsamlHstogram Dta PCF *.... Gaussian
Figure 4.7 Histogram Data
Table 4.8 Gage 615 10/25 @ 1315
WAVE PROPERTIES
615 10/25 s 1315
a (1/m) .372027
p. (m) -.09863
(m) .498230
a (m) .47981
d (m) 2.23
a/d .215161
Figure 4.8 Histogram Data
Gage 636 10126 @ 1316
-1.8 -1.2 -0.6 0 0.6 12
Displacement(m)
1.8 2.4
Table 4.9 Gage 635 10/25 @ 1315
635
a (1/m)
g. (m)
a. (m)
a (m)
d (m)
o/d
WAVE PROPERTIES
10/25 1315
.318011
-.10497
.591069
.573397
3.32
.17271
I M Hstogram Data PF ...**** Gaussian
Figure 4.9 Histogram Data
Oage 64 -10126 @ 1316 Table 4.10 Gage 645-10/25 @ 1315
0.8
WAVE PROPERTIES
0.7
0.- 645 10/25 a 1315
0
0.5 a (1/m) .256391
0.4 j p.t (m) -. 10245
Sa. (m) .628243
0.2
Sa (m) .612186
0.1
Sd (m) 4.66
0- dm
7 99; q 0 0 ar/d .13137
DIsplacement (m)
mm= Histogram Data PF .... Gaussan
Figure 4.10 Histogram Data
Gage 566 10128 @ 1316
Table 4.11 Gage 655 -10/25 @ 1315
0.8
0.7 WAVE PROPERTIES
S0.6 655 10/25 ( 1315
S0.5 a (l/m) .269927
0.4 't. (m) -.12356
J 0.3 a".(m) .658813
oa a (m) .636821
0.1
d(m) 6.44
0
0 N o c /d ..098885
Figure 4.11 Histogram Data
Table 4.12 Gage 665 -10/25 @ 1315
WAVE PROPERTIES
665 10/25 Q 1315
a (1/m)
p. (m)
a. (m)
a (m)
d (m)
a/d
.269743
-.12978
.67434
.650698
6.52
.09980
Figure 4.12 Histogram Data
Gage 76 10126 @ 1316
0.4 +
-2 -1.2
-0.4 0.4 12 2 2.8 3.6
Displacement (m)
Table 4.13 Gage 675 -10/25 @ 1315
675
a (1/m)
a (in)
t. (m)
d (m)
a/d
WAVE PROPERTIES
10/25 1315
.220143
-.12062
.727669
.708477
7.22
.098127
Figure 4.13 Histogram Data
-m HIstogram Data PF ....... Gaussian
Gage 62 -10/26 @ 1316 Table 4.14 Gage 625 -10/25 @ 1315
0.6
WAVE PROPERTIES
0.5 625 10/25 (, 1315
S.4 a (1/m) .140009
sc. (t) -.10961
S0.3
S0o. r. (m) .872244
0.2 o(m) .858792
d (m) 10.04
0.1
c/d .085537
-2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
EiiB Hstogram Data PF .... Gaussian
Figure 4.14 Histogram Data
Based on a comparison with the water depth regions defined in Section 3.2.1, it
can be seen that the histogram data and the probability density function closely resemble
the normal distribution even after the wave field enters intermediate water for milder sea
conditions. This is shown in Figures 4.6 and 4.7. This implies that the characteristics of
the wave profile are dependent on the sea condition as well as the water depth.
The histogram data for both the mild and severe sea conditions in Figures 4.1-4.14
show the distribution of the wave profiles progressively becomes more non-Gaussian as
the waves propagate toward shallower depths. The non-Gaussian trend becomes more
extreme as the sea condition becomes more severe. It is significant to note also that the
non-Gaussian characteristics are most pronounced at the shallower gage locations, Gages
615 and 635.
4.2 Gaussian/non-Gaussian Boundary Definition
The histogram data shown in Figures 4.1 4.14 represent the same wave profiles
depicted in Figures 3.2 3.15. The transformation from Gaussian to non-Gaussian
profiles begin with subtle changes in the amplitude of the negative displacements and the
width of the troughs. This initial transformation can be seen be comparing Figures 4.4 and
4.6. Figure 4.6 shows a nearly normal distribution. Figure 4.4 shows a slight elevation of
the apex and a narrowing of the negative tail. It is also significant to note that the positive
part of the distribution remains relatively unchanged. This indicates that the initial changes
affect the trough region of the profile more drastically than the positive region. Figure 4.4
demonstrates that the trough region becomes shallower and wider. As the profile
continues to shoal, changes begin to occur in the positive displacement region of the
distribution. The peaks become narrower and higher. This can be seen graphically by
comparing Figures 4.2 and 4.4. The positive region of the wave profile is not as full when
compared with the normal distribution and the positive tail extends farther than the normal
curve.
As described above, significant transformations occur in the wave profile as the
wave train propagates into shallower water. There is a point in this transformation when
the wave profile deviates significantly enough from the normal distribution that the waves
can no longer be considered a Gaussian random process. Identification of this boundary is
important to ensure assumptions made regarding the waves remain valid. Analysis of the
histogram data shows that the non-Gaussian characteristics of coastal waves depends on
water depth and sea severity. Based initially on analysis of the histogram data shown in
Figures 4.5 and 4.6, it appears this transition boundary occurs when the non-dimensional,
wave standard deviation, o/d, is approximately 0.06. The non-dimensional parameter,
a/d, is used because it considers both sea severity and water depth. Eight random sea
conditions were selected having c/d values between 0.04 and 0.10. These eight data sets
are depicted in Figure 4.15 in terms of ac as a function of /d. The data points labeled A-
H in Figure 4.23, each correspond to a distribution shown in Figures 4.15-4.23. Data
points A-G were intentionally grouped around the 10 meter water depth. This minimized
any impact water depth and/or beach slope may have on the analysis results. The
statistical data for these points are provided in Table 4.15.
0.12
0.1
0.08
0.06
0.04
0.02
0
0
.02
0.04
0.06
o/d
0.08
A Non-Gaussian Profiles A Gaussian Profiles
Figure 4.15 Sample Data Points to Analyze Gaussian/Non-Gaussian Boundary
A
B
G F^
H^
Table 4.15 Statistical Properties of Data Points A-H of Figure 4.15
POINT a/d H. Water Depth, d
A 0.1000 3.08m 9.51m
_B 0.0800 2.74 m 9.67 m
C 0.0710 2.47m 9.64m
D 0.0616 2.47m 10.03 m
E 0.0600 2.43 m 10.13 m
F 0.0520 1.78 m 8.90 m
G 0.0398 1.57 m 9.88 m
H 0.0590 4.77 m 20.20 m
Comparison with Normal Distribution
old = .100
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Displacement (m)
-- Prob. Density Fcn. - --- Normal Distribution
Figure 4.16 Point A of Figure 4.15
Figure 4.17 Point B of Figure 4.15
Comparison with Normal Distribution
l/d = .071
0.6-
0.5
0.4
2 0.3
0.2
C. 0.1
0--
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Displacement (m)
S Prob. Density Fcn. ----.... Normal Distribution
Figure 4.18 Point C of Figure 4.15
Figure 4.19 Point D of Figure 4.15
Figure 4.20 Point E of Figure 4.15
Comparison with Normal Distribution
o/d = .0616
0.7
0.6
0.5 "
S0.4
S0.3
0. 2 0.2
o.
0. 0.1
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Displacement (m)
Rob. Density Fcn. ...... Normal Distribution
Comparison with Normal Distribution
aod = .062
0.9
0.8
u 0.7
S0.6
7' 0.5
5 0.4
S0.3
2 0.2
S0.1
0
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Displacement (m)
Prob. Density Fcn. -..--... Normal Distribution
Figure 4.21 Point F of Figure 4.15
Comparison with Normal Distribution
old = .0398
1.2
.0.8
c 0.6
0.4
. 0.2
0 ---- ,~ -----,-------, --1 ---
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Displacement (m)
SRob. Density Fcn. -- ---- Normal Distribution
Figure 4.22 Point G of Figure 4.15
i
Comparison with Normal Distribution
aid = .069
0.35
0.3
u 0.25
S 0.2
0.15
.d 0.1 -
L. 0.05
-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5
Displacement (m)
Prob. Density Fcn. -- ---- Normal Dstribution
Figure 4.23 Point H of Figure 4.15
Figures 4.16 4.23 confirm that the region between a/d = 0.06 and a/d = 0.08,
shown in Figures 4.20 and 4.17 respectively, appears to be when the wave profile
transforms from Gaussian to non-Gaussian. A closer inspection of this region is shown in
Figures 4.18 4.20. The distribution shows a considerable increase in deviation from the
normal between a/d = 0.06 and a/d = 0.0616 as shown in Figures 4.20 and 4.19
respectively.
Due to the subjective nature of this boundary, a general boundary can be defined
based on probabilistic analysis. This boundary, defining the conditions under which the
wave profile can no longer be considered Gaussian, occurs when o/d =0.06.
A similar study was conducted by Ochi and Wang (1984); however, they defined
the boundary condition between Gaussian and non-Gaussian wave profiles in terms of the
wave record parameter, .3. X3 represents the skewness of the wave profile which they
determined to be the dominant parameter affecting the non-Gaussian characteristics of
coastal waves. Ochi and Wang concluded that coastal waves in seas for which X3 is less
than 0.2 can be considered as a Gaussian random process. For comparison purposes, this
boundary condition is plotted along with the findings of this present study in Figure 4.24.
In terms compatible with the approach of this present study, their boundary condition can
be defined as a /d = 0.0543. It is encouraging that there shows very close agreement
between both findings. Additional analysis was contributed by Ochi and Ahn (1994b)
whose data points are shown in Figure 4.24 as white circles. Ochi and Ahn (1994b)
defined the transition criterion between Gaussian and non-Gaussian as occurring when a <
0.01 where "a" is one of the parameters in the non-Gaussian probability density function.
The white circle data points define the limiting cases based on his criterion. This third
method for determining the boundary between Gaussian and non-Gaussian wave profiles
agrees very well with the other two further verifying the validity of this finding.
I
6
?5
<4
2
1
0
I~ I
Water Depth (m)
- - - Ochi & Wang (1984) --- Present Study (1996)
Figure 4.24 Gaussian/Non-Gaussian Boundary Defined
I-.
A
Non-Gaussiai Wave Profile
B A
--Gaussia i WaY Phofile
o F" A Region
o
H
~e~0'
CHAPTER 5
TREND ANALYSIS OF PARAMETERS
5.1 Relationship between a and Water Depth
As was discussed earlier, the distribution of the wave profile is highly dependent
on sea severity and water depth with sea severity being defined in terms of the wave
profiles' standard deviation, a.' Since these two terms will be used considerably in the
following trend analyses, their relationship with each other will be examined.
Figure 5.1 shows examples of simultaneous measurements of a plotted as a
function of distance offshore for two cases. It is evident from Figure 5.1, that the
standard deviation of the wave profile is influenced by water depth. As water depth
decreases, the relative magnitude of the standard deviation decreases. It is also seen that
for more severe sea conditions, represented by 10/25 @ 1315 data in Figure 5.1, the
overall change in a as the wave field propagates into shallower waters is greater as
compared with milder sea conditions (10/23 @ 2155). Based on theoretical
considerations of a, it can be assumed that a approaches zero as distance offshore
approaches zero. This trend is observed in Figure 5.1 since both cases shown appear to
converge to the origin. From the information on the surf zone locations for these two
cases computed in Chapter 3, it appears that a begins to decrease rapidly towards zero
once the break point is reached.
The information shown in Figure 5.1 based on two cases is shown in Figure 5.2 for
all data sets analyzed in this study. Instead of a being plotted as a function of distance
offshore, Figure 5.2 plots a as a function of water depth. This will take into consideration
any local changes in mean sea level.
Figure 5.1 Spatial Comparison of a at Mild and Severe Sea Conditions
The trends evident in Figure 5.2 reinforce the analysis done for the two cases in
Figure 5.1. The upper bounds of the discrete data represent the severe sea conditions
while the lower bounds represents the mild sea conditions.
Relationip Hn a and D
2 3 4 5
Depth(m)
6 7 8 9 10
*Ce~3615 g96 A g8a xg6es aGBogB oCas .C65]
Figure 5.2 All a Data Plotted as a Function of Water Depth
It can also be inferred from Figure 5.1 and Figure 5.2 that as the sea state becomes
milder, the waves' standard deviation approaches a constant value and hence the sea
severity approaches a constant before decreasing towards the origin.
In the absence of elaborate wave data, the value for a can be readily determined
with a knowledge of the local and breaking wave heights and the relative water depth,
d
defined by where g = gravitational constant and T = wave period, using a method
gT
such as stream function theory (Dean 1974). To ensure the use of accurate wave height
values in applying the stream function theory, nonlinear shoaling methods should be used
in finite water depths to determine the breaking and local wave heights.
5.2 Non-Gaussian PDF Parameter Analysis
The non-Gaussian probability density function parameters, a, p. and a., will be
analyzed in non-dimensional form spatially as a function of distance offshore, against each
other in non-dimensional form, and with aa as a function of a/d. The first analysis of the
non-Gaussian probability density function parameters will involve comparing the non-
dimensional parameters against the distance the gage was located offshore. This will
provide a spatial orientation of the parameters which will disregard any local fluctuations
in water depth due to tidal cycle of storm surge. The results are shown in Figures 5.3-5.5.
5.2.1 Spatial Analysis of Parameters
All three parameters shown in Figures 5.3-5.5 appear to gradually increase in
absolute magnitude until approximately 150 meters to 175 meters offshore. The mild
condition represented by 10/23 @ 2155 consistently yields lower values when compared
with the more severe sea condition on 10/25 @ 1315. As the sea condition becomes more
mild, the wave profile resembles the normal distribution. When the wave profile
represents exactly the normal distribution, "a" equals zero and ar equals a. It can be seen
Plot of Bottom Profile and "acr" Comparison
Distance Offshore (m)
0 100 200 300 400 500 600
0 0.3
.2 ~- -----------^ .- ..- --- ------.--- ---------- ------- -------------------------.0.2
-2 -------------0.25
E -4 -0.2
a)
6 0.15 b
I -8 0.1
-10 .05
-12 n
-x-Gage Positions ---10/23 @2155 -e-10/25@ 1315
Figure 5.3 Spatial Comparison of aa at Mild and Severe Sea Conditions
Plot of Bottom Profile and "p.lo" Comparison
Distance Offshore (m)
Figure 5.4 Spatial Comparison of t./Co at Mild and Severe Sea Conditions
.15 ^
:L
-x-Depth Profile -a-10/23 @ 2155 ---10/25 @ 1315
Plot of Bottom Profile and "r. la" Comparison
Distance Offshore (m)
Figure 5.5 Spatial Comparison of a./o at Mild and Severe Sea Conditions
.0o
--t1.05
E-4
-10.
a)
(0
-10
-x-Bottom Profile -A-10123 @ 2155 -o-10/25 @ 1315
from Figures 5.3 and 5.5 that as approaches zero under mild conditions and st/s
approaches one. Figure 5.4 shows that m./s approaches zero under these same conditions
which is also consistent with the theoretical limit of the non-Gaussian probability density
function.
There is also a significant perturbation evident in Figures 5.3-5.5 around 150
meters offshore. The mild sea condition shows this abrupt change occurring seaward
when compared with its location under severe sea conditions and the perturbation is more
pronounced in the severe sea condition case. In general, all trends analyzed for the non-
dimensional parameter, m./s, are almost identical results to those found for as and s./s but
in the mirror image. This is due to the fact that the value of m. is negative. The negative
sign is supported by the theory used to develop the probability density functions' three
governing equations.
The next spatial analysis will consist of comparing the non-dimensional parameters
with water depth. This comparison will be sensitive to any influence of local depth
changes occurring from tidal cycles and/or storm surges. Figures 5.6 5.8 show the
results of this analysis.
The trends from this analysis are the same as those determined by Figures 5.3-5.5.
Due to the difference in scale of the abscissa the curves based on water depth appear to
show much smoother transitions. Figures 5.6-5.8 include two additional cases
representing intermediate sea conditions between those represented by 10/23 @ 2155 and
10/25 @ 1315. These two additional cases approximate the trends displayed by the mild
sea condition of 10/23 @ 2155. This is consistent with the data and time of these cases
since they occurred prior to the severe storm so the sea condition was not yet impacted
significantly.
As was discussed earlier, in the deeper water conditions all three parameters tend
to show convergence towards their respective values for a Gaussian wave profile. In
extremely shallow water conditions, the parameters also tend to converge towards a
0.2
0.18
0.16
0.14
0.12
o 0.1
0.08
0.06
0.04
0.02
0
1 2 3 4 5 6 7 8 9
Water Depth (m)
-- 10/23 2155 - 10/25 1315
S- -10/24 2135 - -10/24 1315
Figure 5.6 ao as a Function of Water Depth
0 1
0
-0.05
-0.1
-0.15
-0.2
-0.25
Water Depth (m)
4 5
--- 10/23 @ 2155 - 10/25 1315
-- -10/24 2135 ----10/24 @.1315
Figure 5.7 tL./o as a Function of Water Depth
1.04
1.035
1.03
1.025
1.02
1.015
1.01
1.005
1
0 1 2
3 4 5 6
Water Depth (m)
- 10/23 @ 2155 - 10/25 @ 1315
- -10/24 @ 2135 -10/24 @ 1315
Figure 5.8 a./o as a Function of Water Depth
I]
constant value which is different depending on the sea condition. For the milder sea
conditions, the parameters tend to converge to the values shown in Table 5.1.
Table 5.1 General Shallow Water Limits of Parameters
Non-Dimensional Parameter Mild 10/23 @ 2155 Severe 10/25 @ 1315
ao 0.15 0.18
A,/o -0.18 -0.22 to -0.25
Q./a 1.033 1.04 to 1.045
When the separate plots for each non-dimensional parameter are compared in both
spatial cases, it is apparent that similar trend descriptions fit all three parameters. For this
reason, a closer analysis will be made among the three variables to determine if this
similarity can be quantified. The next trend analysis consists of comparing the three
parameters with each other.
5.2.2 Relationship Between the three non-dimensional parameters, ar., p./o and a./o
As discussed earlier, the three non-dimensional parameters appear to be highly
correlated with each other. A closer look will be taken to see if specific relations can be
developed connecting the three non-dimensional parameters together. Figure 5.9 and
Figure 5.10 show plots of the raw data representing t./o and c6./ compared with aa
respectively at the extreme water depth locations, gage 615 and gage 625.
All data sets from Gages 615 and 625 were plotted since they represent the
shallowest and deepest gage locations. There are very definite relationships which hold
constant between the extreme range of water depths and sea conditions, that link aa to
ji./a and a./o. The functional relationship can be determined only after considering all
the data points from all water depth and sea severity conditions to ensure it remains
consistently valid. A deterministic relationship between these non-dimensional parameters
is a topic of active research.
Gage 615o Gage 625
Figure 5.9 Plot of pj/a as a function ofac for Gages 615 and 625
Gage 615 0 Gage 625
Figure 5.10 Plot of a./c as a function ofacy for Gages 615 and 625
Trend of ac vs pJa
5 aa 0.1 0.15 0.2
0 0.0
0
-0.05
-0.1
~Ja-0.15
-0.2-
-0.25-
Trend of aa vs a./a
1.04
1.035
1.03
1.025
1.02
1.015
a.ra 1.01
1.005
1
10 0*
,- ---------- ,,ow't
-----&Wf^te-
0.05 0.1
ac
d '9 C~ P~_)
0
0.15 0.2
5.2.3 Relationship Between aa and a/d
Based on the apparent correlation between the parameters of the non-Gaussian
probability density function, aa will be used to represent the general trends of the other
two parameters when plotted as a function ofo/d. This relationship is provided in Figure
5.11 to demonstrate general trends between the wave ao, and the local environment
represented in terms of sea severity/water depth in non-dimensional form. The lower
values of /d correspond to the deeper gage locations and the trend is for these values to
approach the values defined by wave profiles having a normal distribution. This plot also
reinforces the analysis done in Chapter 4 which determined the boundary values of the
Gaussian distribution. At c/d = 0.06, the aa values appear to begin converging towards
the Gaussian limit located at the origin of Figure 5.11.
The higher values of o/d represent the shallower water depths. It appears that at
these values aa tends to approach a finite asymptotic limit. This is consistent with the
findings from the spatial analysis figures. In between these two limits of Figure 5.11 there
is a depression in the severe sea condition case. This phenomenon must be analyzed more
thoroughly to ascertain a reasonable explanation for its existence and will be the subject of
a later study.
0.05
0.15
o/ld
-- 10/23 @ 2155 .-.-..10/25 A 1315
Figure 5.11 Plot of aa as a Function of old for Mild and Severe Sea Conditions
0.2
0.18
0.16
0.14
0.12
m 0.1
0.08
0.06
0.04
0.02
0
*1'''"
0.25
CHAPTER 6
SUMMARY AND CONCLUSIONS
It is shown, based on the results of this study, that wind generated waves exhibit
significant changes in their profile forms as they propagate into finite water depths. The
actual nature of this transformation has been a subject that has not been covered in any
significant detail. The unique wave data records collected during the ARSLOE Project of
1980 enabled a detailed study on this topic to be produced.
One objective of this study was to verify the applicability of the non-Gaussian
probability density function developed by Ochi and Ahn (1994b) defined as
f(x) = exp (1 yap. exp(-yax))2 yax
f) ar-. I 2(yact.)2
L28for: x 0
where, =<
3.00for:x < 0
Through statistical analysis of the ARSLOE data, it has been confirmed that this
probability density function can be used with confidence to represent wave profiles in
finite water depths. It correlates very well with histogram data over an extensive range of
sea conditions. The distribution of the wave profile in finite water depths shows a
dependence on water depth and sea severity. In particular, as water depth decreases, the
distribution becomes increasingly non-Gaussian. Also, as sea severity increases the same
trend occurs.
With the applicability of the probability density function verified, an analysis of the
boundary condition where the wave profile can no longer be considered Gaussian was
conducted. A general relationship has been proven to exist between the largest significant
wave height and the local water depth where the wave profile can still be considered
HS a
Gaussian. This relationship is defined as s = 0.24 or = 0.06. Both forms of this
d d
relationship are identical since the significant wave height, Hs is defined as four times the
standard deviation, a, in deep water conditions. This relationship has been verified using
three different approaches with each yielding similar results. In shallower water
conditions, this boundary, defining the transition from Gaussian to non-Gaussian wave
profiles, appears to shows a dependence on beach slope. This is a qualitative observation
from the present study and additional research must be done to quantify this finding. This
dependence on beach slope may be similar to the influence of beach slope on breaking
waves. The linear result defined by this study is akin to the earliest breaking criterion
based on Solitary Wave Theory.
At present, the only way this probability density function can be defined is through
knowledge of statistical properties of the wave field of interest. In particular, the mean,
variance and third moment of the wave profile must be known. This requires that wave
profile data be obtained in order to evaluate the parameters of the probability density
function.
Another objective of this study was to determine relationships linking the
parameters of the probability density function to local field conditions. If the wave profile
statistics can be defined in terms of local environmental conditions such as sea state and
water depth, the probability density function can be defined without wave data. This study
has revealed qualitative trends between the parameters of the probability density function
and local field conditions based on the extensive data complied from ARSLOE. From
these qualitative trends, it is evident that the specific relationships between the parameters
and environmental conditions are complex and may require consideration of additional
parameters besides sea severity, a, and water depth. The pursuit of a functional
relationship between the parameters and the local environment continues to be a topic of
considerable interest and active research.
APPENDIX
COMPARISON OF ORIGINAL AND SMOOTHED "a" VALUES
It can be seen that the nonlinearity parameter, "a" is characterized by a significant
amount of local fluctuations superimposed on the overall trend of the parmeter. Based on
the definition of"a" as a nonlinearity parameter used to describe the probability density
function of a wave profile, these small fluctuations can be smoothed without any noticable
loss of accuracy. It is highly unlikely that the probability density function of a wave profile
changes so sporadically over such short periods of time as indicated by the minor
variations of"a". There was not any one smoothing technique that was used in the
smoothing of"a" at all gage positions. Instead, several smoothing techniques were
evaluated and the technique which produced the lowest mean absolute deviation (MAD)
was the technique used for that particular gage position.
Gage 615
0.9
0.8 -----
0.7
0.6 -
0.5
0.4 -
0.3 ---
0.2
0.1
0
10/23/80 12:00
I. r 1.
1
10/24/80 0:00
10/24/80 12:00
10/25/80 0:00
10/25/80 12:00
10/26/80 0:00
I--- Actual "a" -- - Smoothed "a"
Figure A. I. Plot of Guge 615 Original ami Smoollied "a" Values
;,h I
u er
1-
---- -'-`---- ~~-'--- ~---'
--
i/
"'
Gage 635
0.9
0.8 ----
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10/23/80 12:00
10/24/80 0:00
10/24/80 12:00
10/25/80 0:00
10/25/80 12:00
10/26/80 0:00
S Actual "a" ...... --- .moothed "'a"
Figure A.2. Plot of Gage 635 Original and Smoothed "a" Values
-11_1
I I
~-I
w.P-
--_~.~... _
"--~1 `I
1-~
Gage 645
0.9
0.8
0.7 -
0.6
0.5 -
0.4
0.3
0.2
0.1
0
10/23/80 12:00
10/24/80 0:00 10/24/80 12:00 10/25/80 0:00 10/25/80 12:00
|-- Actual "a" I-Smoothed "a" 1
10/26/80 0:00
Figurei. A. 3. I'lot of'G(agte 04-5 OrigililiI md Smioodlwd "a" Values
5vw
''
i_- ..
~ii~v
Gage 655
0.9
0.8 -
0.7
0.6 --
0.5
0.4 -
0.3
0.2
0.1 -
0
10/23/80 12:00
10/24/80 0:00 10/24/80 12:00 10/25/80 0:00 10/25/80 12:00
Actual "a" -. --. -Smoothed ."a"
10/26/80 0:00
Figure A.4. Plot of Gage 655 Original and Smootled "a" Values
4
Gage 665
0.9
0.8 ---
0.7
0.6 ---
0.5
0.4
0.3
0.2
0.1
10/23/80 -12:00
10/23/80 12:00
10/24/80 0:00
10/24/80 12:00
10/25/80 0:00
10/25/80 12:00
10/26/80 0:00
-I ." Actual "a" -....: Smoothed "a;'"
Figure A.5. Plot ol'Gage 665 Original and Smioothed "a" Values
4 4
. ~~(- -~u\~A 7\ Nlft(le --L'
ii' _
I
Gage 675
0.9
0.8
0.7
0.6
0.5 -
0.4
0.3 -
0.2
0.1
0
10/23/80 12:00
10/24/80 0:00 10/24/80 12:00
---Actual "a"
K- r
10/25/80 0:00 10/25/80 12:00
- ---Smoothed "a"
10/26/80 0:00
Figure A.6. Plot ol'Gage 675 Original and Smoothed "a" Values
__II
U( wk
IKMt
Gage 625
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10/23/8
10/24/80 12:00
1~
10/25/80 0:00
10/25/80 12:00
10/26/80 0:00
S--Actual "a"
. - -Smoothed "a"
I igu re A. 7. Plot of ( ge 02 5 Or)iginail andic s iool lied "ai" Valuies
10/24/80 0:00
0 1
0 12:00
I
ITM"
EL
.. .. .
.j
LIST OF REFERENCES
Bitner, E.M. (1980), "Nonlinear Effect of the Statistical Model of Shallow Water Wind
Waves," Applied Ocean Resources VoL 2,No.2, pp. 63-73.
Dean, R.G. (1974), "Evaluation and Development of Water Wave Theories for
Engineering Application," Vols. 1 and 2, Spec. Rep. 1, U.S. Army, Coastal
Engineering Research Center, Fort Belvoir, VA.
Doering, J. C., Donelan, M. A (1993), "The Joint Distribution of Heights and Periods of
Shoaling Waves," Journal Geophy. Res., Vol 98, No. C7, pp. 12,543-12,555.
Huang, N.E., Long, S.R., Tung, G.G. and Yuan, Y. (1983), "A Non-Gaussian Statistical
Model for Surface Elevation of Nonlinear Random Waves," Journal Geophy. Res.,
Vol.88, No.C12, pp.7597-7606.
Kac, M. and Siegert, A.J.F. (1947), "On the Theory of Noise in Radio Receivers with
Square Law Detectors," J. Applied Physics, VoL8, pp.383-397.
Langley, R S. (1987), "A Statistical Analysis of Non-Linear Random Waves," Ocean
Engineering, Vol.14, No.5, pp.389-407.
Longuet-Higgins, M.S. (1963), "The Effect of Non-Linearities on Statistical Distributions
in the Theory of Sea Waves," J. Fluid Mech., Vol.17, Part 3, pp. 459-480.
Ochi, M.K and Wang, W.C. (1984), "Non-Gaussian Characteristics of Coastal Waves,"
Proc. 19th Coastal Eng. Conf., Vol.1, pp. 836-854.
Ochi, M.K and Ahn, K. (1994a), "Probability Distribution Applicable to Non-Gaussian
Random Process," J. Probabilistic Eng. Mech., Vol. 9, pp.255-264.
Ochi, M.K and Amn, K (1994b), "Non-Gaussian Probability Distribution of Coastal
Waves," Proc. 24th Coastal Engineering Conf., Vol.1, pp. 482-496.
Tayfni, M.A. (1980), "Narrow-Band Non-Linear Sea Waves," J. Geophy. Res. Vol.85,
No.C3, pp. 1548-1552.
Thornton, E.B. and Guza, R.T.(1983), "Transformation of Wave Height Distribution," J.
Geophy. Res., Vol.88, No.C10, pp. 5925-5938.
Weggel, J.R. (1972), "Maximum Breaker Height," J. Waterways, Harbors Coastal Eng.
Div., ASCE, Vol 98, No.WW4, pp. 1245-1267.
BIOGRAPHICAL SKETCH
The author was born on September third, 1966, in Albany, New York. His affinity
toward the ocean was evident in high school Upon graduation from high school, he
attended the U. S. Naval Academy and graduated in 1988 with a B.S. degree in Ocean
Engineering. He began his naval career as a Navy Diver serving on a minesweeper. In
October 1991, he transferred into the U. S. Navy Civil Engineer Corps.
He arrived in Gainesville in January 1995 to begin graduate work in the field of
coastal engineering under Dr. Michel K. Ochi A short one and one-half years later, in
May 1996, he is on his way to Panama City, Florida, to serve as the Engineering Officer at
the Navy Experimental Diving Unit.
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