Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00084992/00001
## Material Information- Title:
- Non-Gaussian properties of waves in finite water depth
- Series Title:
- UFLCOEL-96011
- Creator:
- Robillard, David J., 1966-
University of Florida -- Coastal and Oceanographic Engineering Dept - Publication Date:
- 1996
- Language:
- English
- Physical Description:
- x, 68 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Coastal and Oceanographic Engineering thesis, M.E ( lcsh )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (M.E.)--University of Florida, 1996.
- Bibliography:
- Includes bibliographical references (leaves 66-67).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by David J. Robillard.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 35128646 ( OCLC )
## UFDC Membership |

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UFL/COEL-96/O1 1 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 JI 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 I. 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. ii TABLE OF CONTENTS ACKNOWLEDGMENTS ....................................................................lii LIST OF FIGURES ........................................................................... v LIST OF TABLES .......................................................................... viii ABSTRACT .................................................................................i 1 INTRODUCTION .......................................................................1 1. 1 Purpose of Study....................................................................... 2 2 LITERATURE REVIEW....................................................... ;............ 5 3 BACKGROUND/STUJDY 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 ar 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 ao, Jis/a and as./ay...................................... 52 5.2.3 Relationship Between acy and a/d............................................. 54 6 SUMMARY AND CONCLUSIONS ..................................................... 56 iii APPENDIX COMPARISON OF ORIGINAL AND SMOOTHED "a" VALUES ...............58 LIST OF REFERENCES...................................................................... 66 BIOGRAPICAL SKETCH.................................................................. 68 iv LIST OF FIGURES Figure pae 1. 1. Wave Profile Examples: a) Gage 615, Severe Sea Condition, and b) Gage 625, Mild 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 63 5 10/23 @ 2 15 5 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 V 4.4. Gage 655 10/23 @2155 Histogram Data ......................................... 27 4.5. Gage 665-l10/23 @2155 Histogram Data ......................................... 27 4.6. Gage 675 10/23 @ 215 5 Histogram Data ......................................... 28 4.7. Gage 625 10/23 @ 2155 Histogram Data ......................................... 28 4.8. Gage 615 -10/25 @1315 Hstogram Data ......................................... 29 4.9. Gage 635 10/25 @ 1315 Histogram Data ......................................... 29 4.10. Gage 645 -10/25 @1315 HsogramnData ......................................... 30 4.11. Gage 655 -10/25 @13 15Histogram 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 aT Data Plotted as a Function of Water Depth ................................... 43 5.3. Spatial Comparison of acy at Mild and Severe Sea Conditions .................... 45 5.4. Spatial Comparison of p./ay at Mild and Severe Sea Conditions .................. 46 Vi 5.5. Spatial Comparison of ay./d at Mild and Severe Sea Conditions.................. 47 5.6. aa as a Function of Water Depth.................................................... 49 5.7. 1./a as a Function of Water Depth.................................................. 50 5.8. as./a as a Function of Water Depth.................................................. 51 5.9. Plot of 11/a as a Function of aa for Gages 615 and 625........................... 53 5.10. Plot of as./a as a Function of acy for Gages 615 and 625........................... 53 5.11. Plot of aa as a Function of ay/d for Mild and Severe Sea Conditions............. 55 A- 1. Plot of Gage 615 Origino~ and Smoothed "a" Values .............................. 60 A-2. Plot of Gage 635 Origini1 and Smoothed "a" Values.............................. 61 A-3. Plot of Gage 645 Original and Smoothed "a" Values.............................. 62 k.4. Plot of Gage 655 Original and Smoothed "a" Values.............................. 63 A.5. Plot of Gage 665 Original and Smoothed "a" Values ............................ 64 k6. Plot of Gage 675 Original and Smoothed "a" Values ............................ 65 A.7. Plot of Gage 625 Original and Smoothed "a" Values ............................ 66 vii LIST OF TABLES Table pge 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 Viii Abstract of Thiesis 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 FD=iT 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. ix Presently, wave data records are required to define the probability density fimetion used in this study. Trend analyses were conducted on the parameters of the probability density finction to determine if simnplifications can be made to its form and to see if it is possible to define the probability density fimction in terms of the local conditions, sea severity and water depth, without the need for wave data records. x CHAPTER 1 UINTRODUCTION As wind generated waves transition from deep water to shallow water, a significant change is evident in the wave profile. As an example, Figure I-1I 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 shall low intermediate water depth condition. Figure I1-I (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. Iu A. k~~A~. AAAlALI4 A A4 ! a) Gage 615, Severe Sea Condition b) Gage 625, Mild Sea Condition Figure 1.1 Wave Profile Examples 1 2 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 nonlinear wave-wave interactions augmented prinmaily 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 Aim (1994a) appears to be applicable throughout the coastal region from deep to shallow water and is defined in closed form Equation (1). A)= I exp 1 2(1 -rap.,- exp(-rax ))2 rx (1) where, r L2for: x :0 3.O0for: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 Stud 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 Aim 1994a) and to analyze the transition point where wave 3 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, pi. and cr. 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 fiunction applicability and 4 analysis. This chapter has two distinct sections. The first section reviews the finite water depth probability density function (Ochi and Aim 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*/a and as./a, to sea severity, (ay), 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 GramCharlier 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 (3ram-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 S 6 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 of non-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 Aim (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 Aim 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. 7 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 LandOcean 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 offlshore to 12,000 meters offshore, Figure 3. 1. .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 staff. 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 8 9 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 seycre 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 F.RF 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. 10 54 a' ~ ,~ FIELD RESIAMCH FACILITY 3605461 I ?4S'S.2*W 'f Ova I * z I ICt) S -~ 0 7 of GAGE 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 Ficility 11 A wide range of wave profiles was observed during the experiment. Figures 3.23.8 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 nonlinearity 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). 12 Figure 3-2 Gage 615 10/23 @ 2155 (d=1.74m) Figure 3-3 Gage 635 10/23 @ 2155 (d=2.83m) Figure 3-4 Gage 645 10/23 @ 2155 (d=4.31m) Figure 3-5 Gage 655 10/23 @ 2155 (d=6.Om) 13 Figure 3-6 Gage 665 10/23 @ 2155 (d=6.08m) Figure 3-7 Gage 675 10/23 @ 2155 (d=6.82m) Figure 3-8 Gage 625 10/23 @ 2155 (d-9.66m) 14 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) Figure 3-12 Gage 655 10/25 @ 1315 (d=6.44m) 15 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) 16 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 O~x Z t =HgT cosh(k(d + z)) skx_21r f) 4,rcosh(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 > nr, 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 < n/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 Crossin2 Period, T IDeep Water Waveleneth, Xe 10/23@ 2155 5.17 sec. 41.715 m 10/25 @j 1315 7.43 sec. 86.157mr 17 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 offshore 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. T7he first condition, representing the mildest sea condition, was taken as 2155, 23 October 1980. Thle 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, -- vcos where Oo and 0 are the deep water cos 9 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 surf zone locations: Conservation of Energy: H 2 = H C CE2 18 Breaker Criterion: H b = Kdb, where Kis defined by Weggel as follows: Kc = b(m) a(m) gT2 with a(m) = 43.8(10 -e" b(m) = 1.56(10 + el9s As the beach slope, m approaches zero, this equation reduces to IC = 0.78. The Group Velocity, C. = (ti)1(1i+ 2 d will be determined based on the dispersio reain 2 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 ~H2 height, H,,,,, will be used to represent the irregular wave field. H,,, =m N F 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 H4,,. 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 19 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 camp arison, 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 ShoreMild- 10/23 @2155 1.51 m j70 m Severe- 10/25 @1315 3.32 m 11 ilm 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 Abn (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 j-1 20 The parameters f3, and Xj are determined by finding the eigenfimction and eigenvalues of the integral equation given by f K(w o)w2 )V/ j (C2)dw)2 = AJ/JWI where K(coi o,2 ) =H(o), o2) IS(aI)S(ow2 ) yj()= orthogonal eigenlbnction f { fyf(ai)yf(k)do = 1 forj =k otherwise, Wi(co) = 0 H c, o2 )=second order frequency response function S(io) = output spectral density finction As can be seen from the form of this solution, the probability density fiction of y(t), can not be found without knowledge of the second order frequency response function, H co, c- ) and there is no way to evaluate this response function for irregular seas. 0chi and Alm 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(co, ca-,). The probability density function thus obtained for describing nongaussian random processes is f~x) e x p y~ap1. -exp(-rax))2 aXl 2f a. [ 2(rac.) j where, y = 128for:x : 0 3.Oofor: x < 0 The parameters, a, p. and aT2. are determined from the cumulants evaluated from the wave data. These cumulants are related to the f3j and Xj values of the Kac-Siegert solution by the following relationships 2 N ki = 1:A j= 1 2 N 2 N k2=I /2j + Z 2 J=1 J-1 2 N 2NI k3 = 6 /3 2JA j + 8 2 3J etc. J-1 J-1 21 In the present study, k1 represents the mean water level and is therefore defined by, k,=0. k2 and k3 are equal to the data sample moments, E[x2] and E[X3 ] respectively. Based on this relationship, the parameters, a, A~. and a2. can be determined through the iterative solution of three equations along with the analysis of a data record for the nonGaussian random process being analyzed. The three equations are a c2 +a g.2 +p. =E[x] =0 a 2. 2a 2a4. = E[x2] 2a aY4.(3 8a 2a2.) =E[x3] 3.3.2 Probabilt Density Function Application to Wave Data In order to determine values for the three parameters, a, 11. and 02., which are used to define the degree of non-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 13 15 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) = I Z (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) =IZYX 3. n , 22 Define the standard deviation of the data record as o* = JE(x2) Evaluate the three parameters, a, pi. and aT2. through iterative solution of the following three equations: a a 2 + a A. + p. =E[x] = 0 ar2. 2a2a4. = E[x2] 2a e-.(3 8a2Ca2.) = 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, pi. and 2 ar 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, g* and Y2 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, A* a were determined iteratively. Each of the parameters contain discrete dimensions. Based on dimensional analysis, these 23 dimensions are as follows: It is known that k2, representing the data sets' variance has dimensions of [LI2]. It is also known that k3, representing the data sets' third moment has dimensions of [L2]. It follows from the knowledge of these dimensions that the parameters, a, p* cy 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* ,aC. must be made nondimensional. The non-dimensional quantities that will be used in this study are, aay, pi/ay, a.Iac and ai/d, a 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 Probabiit Densit Function Applicabilt 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 Aim is valid for representing waves in finite whter 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 nonGaussian 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 fimction 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 nonGaussian random processes such as wave displacements in shallow and intermediate 24 25 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 1.4 1.2 0.4 P fp #i 0 4i 4q 9 9 9 0 0 Displace Ment (in) * 4 Table 4.1 Gage 615 10/23 @ 2155 615 a (1/rn) F"(rW a. (Wn ar (in) d (in) ai/d WAVE PROPERTIES 10/23 R,2 2155 .572645 .04948 .275683 .267219 1 .74 .153574 0 0 S. iWWHitOraMData --PDF Gasn Figure 4.1 Histogram Data 26 Gage 635. -IQ= @ 2155 / 1.80 1.40 120 1.00 0.80 0.60 0.40 0.20 0.00 0 Cl IMlMsagram Data PW........ Gaussian Figure 4.2 Histogram Data Table 4.2 Gage 635 10123 @ 2155 635 a (1/rn) CF. (in) ar (in) d (mn) cy/d WAVE PROPERTIES 10/23 a, 2155 .481441 -.0547 .318871 .30989 2.83 .109502 Gage 645 10123 @ 2155 -0.2 0.2 0.6 1 1.4 Table 4.3 Gage 645 10/23 @ 2155 645 a (1/rn) FL. (in) aF. (in) aF (in) d (mn) ar/d WAVE PROPERTIES 10/23 Q) 2155 .200988 .03084 .343446 .340708 4.31 .079051 Figure 4.3 Histogram Data Dsplmaaoet (in) 16 U c0.8 S L * 0.60.4 0.2 1 0- .1 -0.6 r-wnnstogran Data QeF.....Gussian, 27 Gage 66-10123 @ 2166 I -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 DIspbecamant (in) I EM st0U-am Data FRF ..... ta Table 4.4 Gage 655 10/23 @ 2155 655 a (1/rn) P. (in) cr (Mn) d (mn) cy/d WAVE PROPERTIES 10/23 (),2155 104437 .01771 .340056 .339137 6.0 056523 Figure 4.4 Histogram Data Gage 68-10123 @ 2158 0 S0.8 r- 0.4 0.2 0 .. L -0.25 0.25 0.75 Dispacment(m) C== stogram Data PIT.........Gaussian Figure 4.5 Histogram Data Table 4.5 Gage 665 10/23 @ 2155 665 a (1/rn) P. (in) ar. (in) ar (in) d (in) ca/d WAVE PROPERTIES 10/23 Q~ 2155 .126266 .01835 .365155 .364236 6 .08 .059907 1.2 C08 . S0.6 A0.4 0.2 0* bm%- 1.25 28 F -==wt& Datass l Figure 4.6 Histogram Data Gage 626 -10123@ 2166 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0* .15 .1 -0.5 0 0.5 Dispkacement (in) 1 1.5 Table 4.6 Gage 675 10/23 @ 2155 675 a (1rn) P. (M) Cs. (in) ar (in) d (mn) ca/d WAVE PROPERTIES 10/23 Q) 2155 .094349 .01014 .375221 .374 948 6.82 .054978 Table 4.7 Gage 625 10/23 @ 2155 625 a (1/in) fL. (Mn) a. (in) a (in) d (in) ar/d WAVE PROPERTIES 10/23 a, 2155 .069914 .00819 .394571 .394401 9.66 .040828 Figure 4.7 Histogram Data Gage 676 10/23 @ 2168 1.2 I - U c 0.8 U. A0.6 0.4 02 0- O3splacem nt (M) C 0 U. 0 Iw zMsogam ta CF.....ssian7 I 29 I Gage 615 -10/25 @ 1315 Disphes mosl (m) CM=Histogram Data -POF...Gaussian Figure 4.8 Histogram Data Table 4.8 Gage 615 10/25 @ 1315 615 a (1/rn) 1L. (in) cr. (in) a (in) d (mn) ca/d WAVE PROPERTIES 10125 (a) 1315 .3 72027 .09863 .498230 .47981 2.23 .215161 Gage G36 -10125 @ 1315 I -1.8 -1.2 -0.6 0 0.6 1.2 Displacement(m) 1.8 2.4 Table 4.9 Gage 635 10/25 @ 1315 635 a (1/rn) g. (in) CF. (in) aF (in) d (in) ca/d WAVE PROPERTIES 10/25 Q, 1315 .3 18011 .10497 .591069 .5733 97 3 .32 .17271 1.2 0.8 0 IL 0.6 + 0.4 0.2. 0- 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 r C 0 IL 01 W= stogram ta -PM ... Gaussianf Figure 4.9 EHistogramn Data 30 Gage 64" 101260 1316 0.8 0.7 C 0.6 C0. .5 C0.4 0.2 0.1 0 9 9 0 0 4 CN DIsplaCement 4m) MZ=Histogram Data F.........W7ia Figure 4.10 Histogram Data Gage G66 -10126 @1316 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0- "~9 CO.. NN 0 Displacement (m) V Table 4.10 Gage 645-10/25 @ 1315 645 a (1/m) it. (i) ai. (in) ar (in) d (mn) ca/d WAVE PROPERTIES 10125 Q 1315 .256391 .10245 .628243 .612186 4.66 .13137 Table 4.11 Gage 655 -10/25 @ 1315 655 a (1/mn) I'. (Mn) ar. (Mn) ay (in) d (Wn ai/d WAVE PROPERTIES 10/25 a, 1315 .269927 .12356 .658813 .636821 6.44 .098885 0 ZE U.i C 9. MM I-1stogram eta PCF..... ia Figure 4.11 Histogram Data 31 Gage 665 -10125 @ 1315 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0- -M IFstogram Data D .... wasia Figure 4.12 Histogram Da ta Gage 076 10126 @ 1316 -0.4 0.4 1.2 2 2.8 3.6 Displacement (i) Table 4.12 Gage 665 -10/25 @ 1315 WAVE PROPERTIES 665 10/25 Q-1315 a (1/rn) P. (in) ar (in) d (mn) ai/d .269743 - .12978 . 67434 .650698 6.52 .09980 Table 4.13 Gage 675 -10/25 @ 1315 675 a (1/rn) Rt. (Mn) ar. (in) ar (Mn) d (mn) a/d WAVE PROPERTIES 10/25 A 1315 .220143 .12062 .727669 .708477 7.22 .098127 Figure 4.13 Histogram Data 0 IL. 0 Dis placement (mn) 0.7 0.6 LL Z0 U r 0.5 0.4j 031 0.2. 0.1 0 j -2 -1.2 - CZ= Htogram Data PF ....GaussianI 32 06 Gage 62- 1025 @1315 Table 4.14 Gage 625 -10/25 @ 1315 WAVE PROPERTIES 0.5 625 10/25 Q,1315 0 .4 a (1/rn) .140009 IL(rn) .10961 C 0.3 a Cr.(rW .872244 0.2 oCrnW .858792 d (m) 10.04 0.1 ar/d .085537 -2.5 .1.5 -0.5 0.5 1.5 2.5 3.5 Displacement (in) E Immstgamcato a ... 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 33 non-Gaussian characteristics are most pronounced at the shallower gage locations, Gages 615 and 635. 4.2 Gaussian/non-Gaussian Boundga 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, 34 wave standard deviation, old, is approximately 0.06. The non-dimensional parameter, ai/d, is used because it considers both sea severity and water depth. Eight random sea conditions were selected having ai/d values between 0.04 and 0.10. These eight data sets are depicted in Figure 4.15 in terms of aay as a function of ay/d. The data points labeled AH 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. b 'U 0.12 0.1 0.08 0.06 0.04 0.02 0 0 .02 0.04 0.06 aid 0.08 0.1 SA Non- Gaussian Profies & Gaussian Profiles Figure 4.15 Sample Data Points to Analyze GaussianlNon-Gaussian Boundary A F __ B jG F E A D H _ _ 35 Table 4.15 Statistical Properties of Data Points A-H of Figure 4.15 Comparison with Normal Distribution aid = .100 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -2 -1.5 -1 -0.5 0 0.5 Displacement (in) POINT ai/d H.4 Water Depth, d A 0.1000 3.08 m 9.51mi -B 0.0800 2.74 m 9.67 m C 0.0710 2.47 m 9.64 m D 0.0616 2.47 m 10.03mr E 0.0600 2.43 m 10. 13 mn F 0.0520 1.78 m 8.90 m G 0.0398 1.57 mn 9.88 m H 0.0590 4.77 m 20.20 m LL 0 I * S S 1 1.5 2 1 -Prob. Density Fcn.......--NormalI Distribution Figure 4.16 Point A of Figure 4.15 36 Comparison with Normal Distribution ald =.080 0.9 0.8, rj 0.7 LL0.6 ~0.5 CD 0.4 ~0.3 10 0.2a.0.1 0 -2 -1.5 ".1 -0.5 0 0.5 1 1.5 2 Dis placemne nt (in) rob. Density Fcn.------- Normaul Distribution Figure 4.17 Point B of Figure 4.15 Comparison with Normal Distribution ald = .071 -1 -0.5 0 0.5 1 1.5 2 Displacement (mn) Figure 4.18 Point C of Figure 4.15 U LL U) a' 0 0 L. 0.6 0.5 0.4 0.3 0.2 0.1 0 -2 'I -1.5 37 Comparison with Normal Distribution arid = .0616 0.7. 0.6 U.0.5 *~0.4 0.3 .a0.20 L C. 0.1 0'1 2 - -15 -1 -0.5 0 0.5 Displacement (mn) I Pob. Density Fcn .......I'bml DitribuftinI Figure 4.19 Point D of Figure 4.15 Comparison with Normal Distribution ai/d =.060 0.7 d0.6 I-0.5 S0.4 0.3 .~0.20 . 0-- 1 1.5 -2 -1.5 -1 -0.5 0 0.5 Displacement(in) 1 1.5 2 Prob. Dens ity Fcn.... lNbrrrl Distribution Figure 4.20 Point E of Figure 4.15 -2 -1.5 38 Comparison with Normal Distribution ai/d =.062 0.9 0.8 Cu 0.7 L k0. 7~ 0.5 *0.4 ~0.3 S2. 0.2~- 0.1 0~ -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 is placement (in) rob. Density Fcn.......--Normral Distribution Figure 4.21 Point F of Figure 4.15 Comparison with Normal Distribution caid = .0398 1.2 U.0.8 E- 0.6 0.4 0 a.-0.2 0 2 -1.5 -1 -0.5 0 0.5 Displacement (mn) 1 1.5 2 rob. Density Fcn.......--Normal Distribution Figure 4.22 Point G of Figure 4.15 I I 1-i 39 Comparison with Normal Distribution aid =.059 0.35 d0.3 u. 0.250.2 0.15 .d0.1 0 I 0. 0.05 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 Dsplacement (in) Prob. Density Fcn........-Normal [Dstribution Figure 4.23 Point H of Figure 4.15 Figures 4.16 4.23 confirm that the region between aid = 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 aid =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 ald =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, X3. X3 represents the skewness of the wave profile which they determined to be the dominant parameter affecting the non-Gaussian characteristics of 40 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 cyld = 0.0543. It is encouraging that there shows very close agreement between both findings. Additional analysis was contributed by Ochi and Aim (1994b) whose data points are shown in Figure 4.24 as white circles. Ochi and Aim (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. 10 15 Water Depth (mn) *Ochi & Wang (1984) resent Study (1996) Figure 4.24 GaussianlNon-Gaussian Boundary Defined 7 6 0 0 I I I I. c H t~.. 0 I I + AA B A C A E* -.Ieoo Gal i Wave hoffle A -T A Region .60-4 0 G 0 1 5 20 25 Non-Ga ussiani Wave Profile CHAPTER 5 TREND ANALYSIS OF PARAMETERS 5.1 Relationshi between ar and Water Dept As was discussed earlier, the distribution of the wave profile is highly dependent on sea severity and water dept 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 10/2 5 @ 13 15 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 ca 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. 42 43 Plot of a vs; Distance Offshore 0.9: 0 0.8 1 0.7 ____ ___ -2 0.6 3j 0.5 C .0.4.) 0.3 0.2.- -7 0.1 -8 0 .9 0 50 100 150 200 250 300 350 400 450 500 Distance Offshorle (in) -410/23 a215 5 -0-10/25 0 1315 -WN-Mean Watter Depth Praffe Figure 5.1 Spatial Comparison of ay 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. IIFbionhp blen a and D Q8 an 05 '0 Q4 -6 Q3__ ;5 02 __01 ___0 __ 0 1 2 3 4 5 6 7 8 9 10 Dt*h~m) 4 GKP615 0 Gig9 E ACXJM- Xc3P6Z 0GWU5 0oC9ES-GXV Figure 5.2 All a Data Plotted as a Function of Water Depth 44 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, defined by di 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 Analysiss The non-Gaussian probability density function parameters, a, A. 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 nondimensional 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 d istribution. When the wave profile represents exactly the normal distribution, "a"~ equals zero and a. equals ar. It can be seen 0 100 0 -2 CU CU -10 -12 Plot of Bottom Profile and "acr" Comparison Distance Offshore (in) 2W0 3W040 5w0 600 -- I r t f 4U~ 0 I I I IC I - -II I 3.2 U, 0.1 0.05 n -x-Gage Positions a -10/23 @2155 ae10/25 @13151I Figure 5.3 Spatial Comparison of na at Mild and Severe Sea Conditions I., ......................................................................... 4 M Plot of Bottom Profile and "p1icy" Comparison Distance Offshore (mn) 200 300 A -fr- I 1-4] _____ 1 .1 LI 4 -4- 4 -4 4 I~n __ __ __ _I_ I __I_ _II______-0.3 Figure 5.4 Spatial Comparison of tt.Iar at Mild and Severe Sea Conditions 100 0 0 - 2 F 400 A' 500 I -4 -6 -8 0.% 0 or -10 -12 5 - I-x-Depth Profile -,&--10/23 @ 2155 --a-10/25 @ 1315 Plot of Bottom Profile and "cr. icr" Comparison Distance Offshore (in) 300 400 500 Figure 5.5 Spatial Comparison of a./a at Mild and Severe Sea Conditions 0 100 0: -2 CL 0) 0 -8 -10 -12 .05 x~ .04 .0X3 1.02 1.01 0 I x-Bottom Profile Ai10/23 @ 2155 -o-1I0/25 @ 1315 1 48 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 wh en 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 nondimensional parameter, m./s, re almost identical results to those found for as and st/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 1 r r r r r 3 4 5 Water Depth (mn) 6 7 10/230@2155 ....10/25 @ 1315 ---0/2402135 - --10/24 @1315 Figure 5.6 no as a Function of Water Depth 0.2 0.18 0.16 0.14 0.12 0.08 0.06 0.04 0.02 0 I - -- 1.t 1 ____ ____ I ____ ____ ____ ____ --- -- -- - -- --- 0 1 2 8 9 10 'I t -t -I- I -, Water Depth (mn) 5 6 7 8 7 1 1' I t -10/23 @2155 ....10/25 @ 1315 -10/24 A-2135 - --10/24 @1315 Figure 5.7 tL.Ia as a Function of Water Depth 0 1 0 2 3 9 4 I *1 / -0.05 1- -0.1 -1 -0.15 -0.2 -0.25 10 C - I 1.04 1.035 1.03 1.025 1.02 1.015 1.01 1.005 1 4 5 6 Water Depth (in) - -- -10/24 @2135 --10/24 @1315 Figure 5.8 a./a as a Function of Water Depth b --0 b S b S a .5 / / / -S. -.5 .5..- ---5 I I 0 1 2 3 7 8 9 10 52 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 10125 @ 1315 au 0.15 0.18 A.a-0.18 -0.22 to -0.25 a.I/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 Relationshi Between the three non-dimensional parameters. aca. R./ca and g./c 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 ji.Ia and ada/c compared with acy 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 au to ji./ar and adoc. The functional relationship can be determined only after considering all 53 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. Trend of acy vs IpJa 0 0.05 acy 0.1 0.15 0.2 0. -0.05-0.1 -0.2-Iw -0.25*Gage 615So Gage 625] Figure 5.9 Plot of pi./d as a function of acy for Gages 615 and 625 Trend of a vs ci./ca 1.04. 1.0351.03 1.0251.02-V V 1.015 a.Iar 1.01 1.005-0 0.05 0.1 0.15 0.2 2a *Gage 615 0 Gage 625 Figure 5. 10 Plot of cy./cr as a function of acy for Gages 615 and 625 54 5.2.3 Relationshi Between aar and ai/d Based on the apparent correlation between the parameters of the non-Gaussian probability density fimction, aar will be used to represent the general trends of the other two parameters when plotted as a fimction of ai/d. This relationship is provided in Figure 5. 11 to demonstrate general trends between the wave aay, and the local environment represented in terms of sea severity/water depth in non-dimensional form The lower values of ay/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. Tis plot also reinforces the analysis done in Chapter 4 which determined the boundary values of the Gaussian distribution. At ai/d = 0.06, the aay values appear to begin converging towards the Gaussian limit located at the origin of Figure 5. 11. The higher values of ar/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 0.1 0.15 0.2 ai/d 10/23 @2155 ....10/25 A1315 Figure 5.11 Plot of aa as a Function of ai/d 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 Ln vIn 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 Alm (1994b) defined as 27 1 -rap. -exp(-rax))2~rx where, y L28for: x !0 3.Oofor: 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 56 57 Gaussian. This relationship is defined as IL= 0.24 or -= 0.06. Both forms of this d d relationship are identical since the significant wave height, His is defined as four times the standard deviation, ay, 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. Tis 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. Thlis 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 relationsh ip 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 flucuations 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 flucuations can be smoothed without any noticable loss of accuracy. It is highly unlikely that the probability density fimction 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. 58 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 4 4 10/24/80 0:00 10/24/80 12:0O 10/25/80 0:00 10/25/80 12:00 '0 10/26/80 0:00 K 7 Ctual,."a".- .-..-Smootheda Figure A. 1. PlIot of Gauge 615 Originali and Simoollied ( "a" valIups 'U 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 -Actual "a" ....Smoothed "a*, Figure A. 2. Nlot of Gage 635 Original and Smoothedl "a" Values %I 'U I I M' 0) VVAf\AN4iPIV\f Gage 645 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2-K 0.1 0 10/23/80 12:00 10/24/80 0:00 10124/80 12:00 10/25/80 0:00 10125180 12:00 At-u-al a mote a 10/26/80 0:00 Figurei. A. 3. I'lot of' Gage 04-5 OrigililiI md Smioodlwd "a" Values .0- a' I-J Gage 655 0.9 0.6 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 10/25/50 12:00 -Actual "a" -Smoothed ac 10/26/80 0:00 Figure A.4. Plot of Gage 655 Origiual and~ Snioo)tled "a" Values Eu 4 a' Gage 665 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 01 10/23/80 12:00 10/24/80 0:00 10124/80 12:00 10/25180 0:00 10/25/80 12:00 10/26/80 0:00 11Actual "a" .... Smoothed "a;"_Figure A. 5. Plot of'Gage 665 Original and Smioothed "a" Values cc 4 4 0' L~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 10/25/80 0:00 10/25/80 12:00 S moothed "a' 10/26/80 0:00 Figure A.6. Nlot of'Gage 675 Original anld Smoothled "a" VaIles EU 0' wk 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 I Actual "a" ....Smoothed "a8 I igu re A. 7. Plot of ( ge 02 5 Or)iginail andic s iool lied "ai" Valuies (U 10/24/80 0:00 0 12:00 I ....... ... 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. k'(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, Vol.8, pp.383-397. Langley, R S. (1987), "A Statistical Analysis of Non-Linear Random Waves," Ocean Engineering, Vol. 14, No. 5, pp. 389-407. Longuct-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. Oclii, 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 Alin, K (1 994a), "Probability Distribution Applicable to Non-Gaussian Random Process," J. Probabilistic Eng. Mech., Vol. 9, pp.255-264. Ochi, MXK and Alm, K (I1994b), 'Non-Gaussian Probability Distribution of Coastal Waves," Proc. 24th Coastal Engineering Conf., Vol. 1, -pp. 482-496. Tayflin, M.A. (1980), "Narrow-Band Non-Linear SeaWaves," 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.C 10, pp. 5925-5938. 66 Weggel, J.R. (1972), "Maximum Breaker Height," JL Waterways, Harbors Coastal Eng. Div., ASCE, Vol 98, No.WW4, pp. 1245-1267. 67 BIOGRAPICAL 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 199 1, 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. 68 |