Citation
Wave transmission and current patterns associated with narrow-crested submerged breakwaters

Material Information

Title:
Wave transmission and current patterns associated with narrow-crested submerged breakwaters
Series Title:
UFLCOEL-94022
Creator:
Browder, Albert E
University of Florida -- Coastal and Oceanographic Engineering Dept
Place of Publication:
Gainesville Fla
Publisher:
Coastal & Oceanographic Engineering Dept., University of Florida
Publication Date:
Language:
English
Physical Description:
xii, 118 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Breakwaters -- Mathematical models ( lcsh )
Shore protection -- Mathematical models ( lcsh )
Artificial reefs -- Florida -- Palm Beach ( lcsh )
Dissertations, Academic -- Coastal and Oceanographic Engineering -- UF ( lcsh )
Coastal and Oceanographic Engineering thesis, M.S ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (M.S. in Engineering)--University of Florida, 1994.
Bibliography:
Includes bibliographical references (leaves 116-117).
Statement of Responsibility:
by Albert E. Browder.

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University of Florida
Holding Location:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
33143197 ( oclc )

Full Text
UFL/COEL-94/022

WAVE TRANSMISSION AND CURRENT PATTERNS ASSOCIATED WITH NARROW-CRESTED SUBMERGED BREAKWATERS
by
Albert E. Browder
Thesis

1994







ACKNOWLEDGEMENTS
I want to express my sincere thanks to my committee chairman, Dr. Robert J. Thieke, for his support and guidance, especially down the stretch. Considerable thanks go to 1)r. Robert G. Dean, not only for serving as a committee member, but for providing me with numerous opportunities to 'get my feet wet,' for he knows I do not want to just stand around. I also thank Dr. Dean for introducing me to the world of coastal politics. My thanks also go to Dr. Ashish J. Mehta for serving on my thesis committee.
Vernon Sparkman, Jim Joiner, Sidney Schofield, Chuck Broward, Danny Brown, and Sonya Brooks deserve my greatest thanks for their support in the Coastal and Oceanographic Engineering Laboratory, making my lab study an unforgettable experience. I want to thank Vik Adams and George Chappell for including me in their field excursions all over the state.
Thanks go to those people at the Town of Palm Beach, the Department of Environmental Protection, and American Coastal Engineering, Inc. for their help and support of this project.
Many people in the Coastal Engineering office deserve a huge hand for making this work so much fun. Many thanks go to Mike Dombrowski, my P.E.P. Reef partner in crime, for all the field trips and headaches. Kenny, Mike, Paul, Eric, Chris and Monica, Mark, Sue, Darwin, Tom, Santiago, Jie, and Wally made the days seem much shorter. Thanks go to Keeley for taking responsibility. The real reason I can graduate is the support from Becky, Sandra, Lucy, Helen, Cynthia, and Laura, because if you don't know the secretaries, you don't know anyone. Market Street didn't hurt either. Thanks, Pam.
Finally, I want to thank my parents, Larry and Susan Browder, for everything, but most importantly for the opportunity to take opportunity.




TABLE OF CONTENTS

ACKNOWLEDGEMENTS.........................

. . . . . . . 11

LIST OF FIGURES................................................. v
KEY TO SYMBOLS............................................... viii
ABSTRACT...................................................... xi
CHAPTER 1 INTRODUCTION........................................ 1
1.1 Objectives and Rationale...................................... 2
1.2 Report Organization......................................... 4
1.3 Literature Review.......................................... 5
1.3.1 Wave Transmission Studies............................. 5
1.3.2 Structure Induced Current Patterns........................ 13
1.4 Comments............................................... 14

CHAPTER 2 DATA COLLECTION ....
2.1 Description of the Midtown Palm

Beach Monitorin~

Program........................

2. 1.1 Site Description...........
2.1.2 Wave Data Analysis........
2.1.3 Background Wave Climate and 2.1.4 Current Measurements......
2.1.5 Volumetric Changes........
2.1.6 Unit Settlement...........
2.1.7 Scour Rod Measurements ..
2.1.8 Present Status............
2.2 Methodology of Laboratory Study ...
2.2.1 Experimental Equipment ...
2.2.2 Test Plan...............
2.2.3 Wave Height Measurements ..
2.2.4 Current Measurements...... 2.3 Comments.....................

Verificati

... . . . 15
... . . . 15
... . . . 16
... . . . 18
.n. . .. . . 22
... . . . 22
... . . . 22
... . . . 24
... . . . 25
... . . . 25
... . . . 26
... . . . 26
... . . . 28
. . . . . . . .. 3 1
... . . . 32
... . . . 33

CHAPTER 3 WAVE TRANSMISSION...............
3.1 Field Measurements.....................
3. 1. 1 Transmission Coefficient Determination 3.1.2 Spectral Analysis................
3.1.3 Individual Wave Tracking..........




3.2 Laboratory Measurements ................................. 47
3.2.1 Cross-Shore Wave Height Decay . . . . . . . . . . . 47
3.2.2 Laboratory Transmission Coefficients .................... 49
3.3 Analytical M odel ....................................... 52
3.3.1 Totally Submerged Barriers .......................... 52
3.3.2 Partially Submerged Barriers ......................... 54
3.4 C om m ents ........... .... ..... .. ..... .. .... ...... .. 59
CHAPTER 4 STRUCTURE INDUCED CURRENTS ....................... 60
4.1 Motivation from Field Results ............................... 60
4.2 Laboratory Investigations .................................. 65
4.2.1 The 'Pumping Current .............................. 65
4.2.2 Other Current Effects .............................. 69
4.3 Analytical Approach ..................................... 71
4.3.1 Explanation of the Problem .......................... 71
4.3.2 Mass Conservation Approach ......................... 74
4.3.3 Momentum Approach ................................... 76
4.4 Com m ents ........................................... 81
CHAPTER 5 CONCLUSIONS ..................................... 83
5.1 Midtown Palm Beach Installation .............................. 83
5.2 Vero Beach Laboratory Study ............................... 85
5.3 Analytical M odels ...................................... 87
5.4 Design Considerations .................................... 89
APPENDIX A SIGNIFICANT WAVE HEIGHT AND PERIOD DATA FROM
SEADATA WAVE GAGES ............................. 92
APPENDIX B WAVE TRANSMISSION THEORY DERIVATION .............. 94
B. I Totally Submerged Barriers ................................ 94
B.2 Partially Submerged Barriers ............................... 98
APPENDIX C LABORATORY DROGUE TRAJECTORIESNELOCITIES ........ 101
REFEREN CES .............................................. 116
BIOGRAPHICAL SKETCH ...................................... 118




LIST OF FIGURES
1.1 Transmission Coefficient vs. Non-Dimensional Wavelength for Four Analytical
Approaches. Freeboard Ratio (f/h) = -0.25 .......................... 8
1.2 Transmission Coefficient vs. Relative Freeboard for Four Experimental
Studies (H/L = 0.02 to 0.04) ................................. 11
2.1 Location Map Relative to Port of Palm Beach Entrance ................. 17
2.2 Location of P.E.P. Reef ...................................... 17
2.3 Cross-Sectional View of P.E.P. Reef (Courtesy American Coastal
Engineering, Inc.) .......................................... 18
2.4 Survey Profile Plan ......................................... 23
2.5 Wave Basin Schematic ....................................... 27
2.6 Reef Unit Arrangements for Initial Tests ........................... 29
2.7 Reef Arrangements for Additional Model Testing ..................... 31
2.8 Cross-Shore Wave Height Profile Lines ........................... 32
3.1 Transmission Coefficient History, Midtown Palm Beach Installation.
1 January 1994 to 24 October 1994, CDN Gages ..................... 37
3.2 Wave Height Transmission Coefficients, Midtown Palm Beach Installation.
Seadata Gages, 21 September 1994 to 24 October 1994 ................. 38
3.3 Significant Wave Heights, Midtown Palm Beach Installation. Seadata Gages,
21 September 1994 to 24 October 1994 ........................... 40
3.4 Spectral Density of Offshore, Nearshore, and Shoaled Offshore Records.
CDN 26 November 1993, 6:00 PM ............................... 42
3.5 Spectral Density of Offshore, Nearshore, and Shoaled Offshore Records.
Seadata 5 October 1994, 4:00 AM .............................. 42
3.6 Spectral Density of Offshore, Nearshore, and Shoaled Offshore Records.
Seadata 17 October 1994, 8:00 AM .............................. 44
v




3.7 Surface Elevation Records for Offshore and Nearshore CDN Gages.
18 December 1993, 6:00 PM ................................... 45
3.8 Cumulative Frequency Distribution of Wave Heights for the Offshore,
Nearshore, and Shoaled Offshore Records. 18 December 1993, 6:00 PM ...... 46
3.9 Cross-Shore Wave Height Profiles for Initial Laboratory Tests. Freeboard
Ratio, f/h = 0.0 ........................................... 48
3.10 Wave Height Transmission Coefficients for Initial Laboratory Tests versus
Freeboard Ratio, f/h = 0.0 .................................... 49
3.11 Transmission Coefficients Versus Relative Freeboard, f/Hi ................ 51
3.12 Definition Sketch for Transmission Over a Submerged Barrier ............. 53
3.13 Analytical Approaches for Transmission Coefficient Compared to Laboratory
and Literature. Laboratory Scale Values, Hi = 0.012 ft ................. 57
4.1 Isolines of Elevation Change (ft), August 1993 to December 1993. Contours
in 1.0 foot intervals ......................................... 61
4.2 Volumetric Changes for Latest Survey Period and Last Year (yds3) ........... 62
4.3 Bottom Drogue Trajectories, f/h = 0.0 ............................ 66
4.4 Schematic of Mass Transport Profile Under Wave Passing Over a Barrier ...... 72 4.5 Schematic of Wave Basin and Pumping Currents ...................... 73
4.6 Average Exit Velocity vs. Length of Structure. Continuous Structure,
f/h = 0.0. Figure Applicable for Model Conditions .................... 75
4.7 Schematic of Lateral Inflow Model to Determine Water Surface Profile ....... 77
4.8 Ponding Level Calculated for Laboratory Scale, 100% Flow Divergence
H = 0.04 cm (0.012 ft), T = 2 s ................................ 79
4.9 Ponding Levels for P.E.P. Reef Prototype Scale. H = 3.28 ft, T = 6 s.
Velocities Shown Are Exit Plane Values (x = 2,000ft.) ................. 80
A. 1 Offshore Seadata Gage Significant Wave Height, Julian Date, 1994 ........... 93
A.2 Offshore Seadata Gage Modal Period, Julian Date, 1994 .................. 93
A.3 Nearshore Seadata Gage Significant Wave Height, Julian Date, 1994 .......... 93
A.4 Nearshore Seadata Gage Modal Period, Julian Date, 1994 ................. 93
vi




Definition Sketch of Bernoulli Principle Theory .............
Definition Sketch of Weir Flow Theory ..................
Control and Type A Case, f/h = 0.0. Wavemaker is to the right. ft/s ..........................................
B and C Cases, f/h = 0.0. Wavemaker is to the right. Units in ft/s Type D Case, f/h = 0.0. Wavemaker is to the right. Units in ft/s Control and Type A Case, f/h = -0.2. Wavemaker is to the right. ft/s . . . . . . . . . . . . . . . . . . . . .
B and C Cases, f/h = -0.2. Wavemaker is to the right. Units in ft/s Type D Case, f/h = -0.2. Wavemaker is to the right. Units in ft/s Control and Type A Case, f/h = -0.4. Wavemaker is to the right. ft/s ............................................

Units in
Units in Units in. . Units in

B and C Cases, f/h = -0.4. Wavemaker is to the right. Units in ft/s .......
Type D Case, f/h = -0.4. Wavemaker is to the right. Units in ft/s ........
Type E Case, f/h = 0.0. Offset Distances = 2w and 4w. Units in ft/s ...... Type E Case, f/h = 0.0. Offset Distance = 6w. Units in ft/s ............
Type F Case, f/h = 0.0. Offset Distances = 2w and 4w. Units in ft/s ...... Type F Case, f/h = 0.0. Offset Distance = 6w. Units in ft/s ............
Type G Case, f/h = 0.0. Offset Distance = 4w. Units in ft/s ............

. 95 98 102 103
104

C.8
C.9 C.10 C.11 C.12 C.13 C.14




KEY TO SYMBOLS A channel cross sectional area
As'w linearization constants At structure cross sectional area
AV area of opening in barrier
B channel width
C wave celerity
Cf Chezy coefficient
Cg wave group celerity
C1,2,3,4 coefficients, Ahrens (1987) d5o median stone diameter, Ahrens (1987) ds total water depth, Ahrens (1987)
E wave energy per unit surface area
E, specific energy, weir flow
f freeboard
f/h freeboard ratio, freeboard over water depth f/Hi relative freeboard, freeboard over incident wave height Jr" wave energy flux g acceleration due to gravity
h total water depth
hC critical depth over weir




hc height of structure, Ahrens (1987)
Hi,r,t wave height (incident, reflected, or transmitted) H, significant wave height
H,,,o offshore significant wave height, Ahrens (1987) i subscript denoting incident (offshore condition)
k wave number, 21r/L
Kr,t wave height coefficient, reflected or transmitted L wavelength
LP, wavelength, Ahrens (1987) M momentum function
N number of waves in record
P wetted perimeter
q flowrate
qv flowrate through barrier openings qw flowrate over weir
r subscript denoting reflected wave
s spreading parameter
Sf channel friction slope
S, channel bottom slope
t subscript denoting transmitted conditions
T wave period
U velocity
w structure cross-shore width
x distance along structure




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 WAVE TRANSMISSION AND CURRENT PATTERNS
ASSOCIATED WITH NARROW-CRESTED SUBMERGED BREAKWATERS By
Albert E. Browder
December 1994
Chairperson: Dr. Robert J. Thieke
Major Department: Coastal and Oceanographic Engineering
This study presents design guidance as to the performance of narrow crested submerged offshore breakwaters. Performance information is based on a field monitoring program currently underway by the Coastal and Oceanographic Engineering Department of the University of Florida (UFCOE) of the Palm Beach, FL, Prefabricated Erosion Prevention (P.E.P.) Reef Installation and a laboratory investigation of the P.EP. Reef recently carried out by UFCOE. Analytical models developed for submerged barriers and a review of pertinent literature on the subject provide additional insight.
Performance of a submerged barrier focuses on two major aspects. Wave attenuation is the primary objective of a breakwater. Generally, submerged barriers are expected to provide partial reduction of storm waves in order to provide some measure of shore protection. The second consideration involves the alteration of the nearshore current pattern in the vicinity of the reef. Any structure placed in the nearshore zone will change the existing current patterns, which in turn will alter the sediment transport in that region.




Design guidance for submerged barriers is provided in terms of platform arrangement of barriers and relative height of the structure. The relative height of the structure becomes the primary variable in any submerged breakwater design since it directly affects the wave height attenuation. The degree of wave height attenuation is also found to be related to the alteration of the current system in the vicinity of the structure.
To provide a significant amount of wave height attenuation (at least 10%), it is recommended that the height of the barrier be at least 80% of the mean water depth, including any settlement of the barrier and any storm surge associated with the design wave height. Alternatively, it is recommended that the amount of clearance over the structure crest should be no more than 1.3 times the design incident wave height. The determination of the crest elevation then dictates the degree of change in the associated current patterns. To. offset any adverse current generation in the lee of the structure, a problem documented herein through laboratory results, it is recommended that a submerged shore-parallel breakwater be segmented and offset along its length in order to relieve any adverse currents that may be created.
It is recognized that the performance of submerged breakwaters is extremely site specific. Results from one installation should not be applied directly to another. This study is intended to provide general design criteria and platform options.




CHAPTER 1
INTRODUCTION
In recent years there has been a desire to find a long lasting solution to beach erosion. "Soft" solutions such as beach nourishment and beach dewatering have been shown to be effective in certain applications. In many cases, however, these solutions either prove to be ineffective or costly to maintain. Beach nourishment, for example, often requires additional placement of material on the beach to maintain the success of the project. In light of these difficulties, attention has turned to permanent "hard" structures placed in the coastal zone to prevent beach erosion. One such structure is a submerged breakwater, which is the focus of this study. A submerged breakwater is a rubble-mound or solid barrier whose crest is at or below the Mean Water Level (MYvrL) and is usually placed parallel to shore. These breakwaters are used to provide partial protection against incident wave attack, primarily against large storm waves.
Submerged breakwaters are an attractive alternative to large emergent structures. The primary benefit is the financial savings of submerged barriers. Since the crest elevation is considerably lower than that of an emergent barrier, the amount of material needed for construction is obviously considerably less. Another benefit is the aesthetic value of a submerged structure which does not obstruct an otherwise attractive ocean view. An additional benefit is that the barrier often serves as an artificial reef, attracting its own biological community. The immediate drawback to a submerged breakwater is the lower level of protection it affords against wave attack. Another possible problem is the alteration of the natural current and sediment transport systems. These aspects require attention when considering such a structure for use along a given stretch of beach.




2
Most of the research concerning submerged breakwaters focuses on wave transmission over such barriers. Laboratory studies are plentiful, dating back to the 1930s in some cases. This material covers many wave conditions and barrier configurations, both experimentally and analytically. Very little data exist from field measurements as most actual installations have either not been monitored or have not been monitored for wave climate. Very little information exists about the behavior of the current patterns in the vicinity of such barriers. Some numerical analysis has been conducted, and a handful of laboratory experiments discuss this behavior, but field data are nearly non-existent due to the intensive measurements required to adequately address the problem.
The State of Florida passed legislation in 1989 to allow experimental projects to be installed to try to solve the beach erosion problem. Since that time, two submerged breakwater projects have been installed, one in Palm Beach fronting the DuPont property, and the; second in Midtown Palm Beach. The first installation was not extensively monitored and has since suffered wave attack which has scattered the units in the project and rendered it ineffective. Tbe second installation is currently being monitored for its performance in the modification of both the sediment transport patterns and the wave climate.
A third experimental project is currently under consideration in Vero Beach, Florida. This project builds on the experiences of the previous two installations and has prompted additional engineering study into the behavior of submerged barriers. 'Me focus of the additional work lies in physical and numerical modelling. For projects currently under consideration and for future installations, a basis for design is needed. This study attempts to fill that need.
1. 1 Obiectives and Rationale
The purpose of this paper is to present results of recent investigations into the performance of submerged breakwaters and to provide basic design information to those planning




3
to install such a structure. As in most coastal engineering endeavors, results can be very site specific and many times predicting the outcome of a project can be difficult. It is necessary to have a basic idea of how a submerged barrier will perform when installed, particularly in terms of wave attenuation and potential side effects.
This paper presents information on wave height attenuation for submerged breakwaters, specifically narrow crested barriers. These are structures whose crest width is much smaller than the wavelengths of the waves approaching the barrier. Field measurements, laboratory studies, analytical developments, and pertinent literature are presented in an attempt to define the wave transmission behavior of a given submerged barrier arrangement. Design guidance is given in general terms to establish the necessary height of a barrier to achieve a given level of wave attenuation.
Guidance is also offered to the changes in the current and sediment transport systems that may result from installing a submerged barrier. This focuses on relieving or preventing currents that may induce erosion in the vicinity of the barrier. This erosion runs counter to the objectives of installing a submerged breakwater and it is obviously desirable to minimize or prevent such a problem. Recommendations for platform design based upon physical modelling and field observations are given.
The rationale for this report stems from the three year monitoring program of the Nfidtown Palm Beach Prefabricated Erosion Prevention (P.E.P.) Reef. This program, conducted by the University of Florida Coastal and Oceanographic Engineering Department and sponsored by the Florida Department of Environmental Protection and the Town of Palm Beach, FL, involves monitoring of the wave climate, sediment erosion/accretion patterns, scour behavior, sediment size analysis, and structure settlement of a 4,000 foot long pre-cast concrete submerged breakwater. In order to determine the performance of this structure, information regarding the




5
1.3 Literature Review
The dominant information related to submerged breakwaters concerns the wave transmission characteristics of such structures. Of the many sources found regarding this topic, most involve rubble mound structures, since these are practical to build and have been constructed in several areas around the world. The first portion of this review attempts to define the history of wave transmission studies and discuss some of the more relevant works that involve submerged breakwaters. Where possible, an attempt is made to focus on the portions of these works that involve narrow crested impermeable structures.
A considerably smaller body of knowledge exists regarding the current patterns around offshore submerged breakwaters. The material found on this topic is presented herein, and will be developed and compared to both laboratory and analytical data in Chapter 4.
1.3.1 Wave Transmission Studies
The use of artificial submerged breakwaters is not new to the coastal engineering field. Hall (1939) mentions the installation of a 'low pre-cast concrete "artificial reef" parallel to shore in shallow water' intended to increase sand accretion in the Hollywood, Florida, area. While no details of that installation were given, laboratory experiments are described for triangular, trapezoidal, and thin walled (narrow crested) breakwaters. The study measured the wave transmission of monochromatic waves of varying heights in varying water depths over the structures. Hall (1939) concluded that a submerged barrier parallel to shore will reduce the rate of littoral drift in its lee. From this he concluded that the barrier would cause accretion on the protected shoreline and provide 'a means of protecting a beach without disfiguring it.' The report also noted that a vertical wall is the most effective shape in attenuating. wave height and that for storm wave protection, the structure height should be at least 0.8 times the average water depth.




6
Dean (1945) presented an analysis of the reflection of waves by a submerged plane barrier. Using linear wave theory, reflection and transmission coefficients for deep water waves were derived. This development did not consider any loss of energy through turbulence or wave breaking over the barrier. Johnson et al. (1951) presented an energy flux approach to determining transmission coefficients, K,, defined as the ratio of transmitted to incident wave height. By calculating the average energy flux above a submerged barrier crest (equation (1.1))
- i T o cosh 2k (h+z
f pafog 2 h h+z) dzdt (1.1)
T J_,P cosh (kh) sinh (kh)
and redistributing that portion of the energy flux over the entire water column on the lee side of the barrier, a transmission coefficient is obtained (equation (1.2)). This equation is applied for a given value of barrier submergence.
K 1- sinh(2k(h-f) ) + 2k(h-f) (1.2)
sinh(2kh) + 2kh
At this point it is useful to define the term freeboard ratio. This non-dimensional value, f/h, is used to describe the distance from the mean water level (MWL) to the barrier crest divided by the total water depth at the barrier toe. It is defined herein as a negative quantity in order to distinguish submerged structures from emergent structures. Several other non-dimensional values are also used to plot transmission coefficient variation. The freeboard, -f, is often divided by the incident wave height, Hi, to create a relative freeboard value. This ratio is also negative, again to reflect the fact that it describes a submerged barrier. This notation means that, for the same water depth, a structure with a lower freeboard ratio (such as -0.6) lies further below the surface than a structure with a larger freeboard ratio (such as -0.2).
Ogilvie (1960) provided an analytical treatment of wave transmission of shallow water waves over thin walled barriers. His results were presented as a function of relative wave length




7
(water depth divided by wavelength). Mei and Black (1969) presented coefficients for a full range of relative wave lengths from solutions of the Laplace equation for wave propagation over submerged obstacles. Mei and Black provided coefficients for both thin walled barriers as well as barriers of finite width. An interesting finding of this work involves the oscillating transmission coefficient that results from varying the crest width. The relation between the wavelength of the attenuated waves and the crest width affects the degree of attenuation experienced. Massel (1983) verified this phenomenon, stating that the finite barrier width causes higher harmonics to be generated in the space above the barrier and transmitted forward as free waves. Harmonic generation also provides a means of distinguishing between barriers with finite crest widths and those considered to be narrow crested. VVhile this study focusses on narrow crested structures where harmonic generation does not occur, this phenomenon is noteworthy in the design considerations of submerged breakwaters.
Figure 1. 1 presents a comparison of the analytical developments discussed above for a freeboard ratio of -0.25. The plot presents transmission coefficient versus relative wave length. Again, the transmission coefficient is defined as the ratio of the transmitted wave height to the incident wave height. The plot shows the range of applicability of the various methods. Ogilvie (1960) presented results for shallow water waves, therefore these results pertain to values less than kh = 0.314. Similarly, Dean (1945) developed equations for deep water waves (kh > 3.14). The results indicate the effect of wavelength on wave transmission. The work of Mei and Black (1969) provides a value for all kh, and shows that the maximum wave height reduction for this freeboard ratio value is only approximately 8%.
As the length of the incident wave increases, the barrier effect diminishes as only a small portion of the wave experiences the barrier at one time. An extreme example of this would the astronomical tides. These waves obviously experience no effect of the barrier. As the wave length decreases relative to the water depth, the wave motion does not extend deep enough to




. . . . .. . .

/

1.0 0.9 0.8
0.7 0.6 0.5

/
. . . . . / .
/"
/ /,
/
I
......K ..im

Dean(1945)
Johnson et al. (1951) ...... Ogilvie (1960)
Mei & Black (1969)

0 1
Long Waves

5 6
Short Waves

Figure 1.1 Transmission Coefficient vs. Non-Dimensional Wavelength
Analytical Approaches. Freeboard Ratio (f/h) = -0.25.

for Four

experience the effects of the barrier and the waves pass over the barrier unaffected. Therefore, only an intermediate range of wavelengths experiences an appreciable amount of wave height attenuation. Fortunately, this range of wavelengths applies to typical wind wave climates approaching most coastlines. An interesting note from Figure 1.1 is that although the energy flux approach of Johnson et al. (1951) is given for all kh, laboratory experiments indicate that realistic results are only obtained for deep water cases. This is observed for the full range of f/h values.
Goda (1969) and Goda et al. (1967) conducted extensive laboratory measurements of submerged breakwaters. He presented transmission coefficients as a function of the relative freeboard and developed an empirical expression for Kt as well as best fit curves through the individual data points. The tests included various crest widths, from 0.9 cm (0.03 ft, considered

I.
/ / i'




9
to be narrow crested) to 40 cm (1.31 ft) in width, and various incident wave heights, from 3 to 34 centimeters (0.10 to 1.12 ft). The tests also included a variation in water depth, creating a range of freeboard ratios from -0.2 to 0.0 for the submerged barrier tests. Equation (1.3) presents the empirical expression for K, from these experiments. The values of the coefficients ce and ( are proposed as 2.0 and 0.4 for the thin-walled barriers in the re-analysis (Goda et al., 1969), respectively.
- 0..5 [1-sin- (- + (1.3)
H, 2at H,
Dattatri et al. (1978) tested a range of shapes and depths of submergence to determine transmission coefficients. This report concluded that the relative depth of submergence is the most important parameter in the performance of submerged breakwaters. Quantitative data from the thin-walled barrier tests were not presented for analysis.
As stated previously, many submerged breakwaters are rubble mound structures composed of large stones. A large laboratory test was conducted by Seelig (1980) to study the wave transmission characteristics of various breakwater designs. Most of the designs involved rubble mound structures and all involved barriers of finite crest width. Ahrens (1987) conducted over 200 laboratory tests of various submerged rubble mound configurations to determine damage, stability, and wave attenuation characteristics of these barriers. From the wave transmission data, the following expression for K, was developed, where C1 = 1.188, C2 = 0.261, C3 = 0.529, and C4 = 0.00551, At = cross sectional area, LP = incident wavelength, d5o = median stone diameter, and f/H,,o = freeboard ratio < 1.0.
1.0
Kh: Lc cl At c2 A32 (1.4)
1.0+( ( Hm +C4(




10
While equation (1.4) was developed for rubble mound barriers, the influence of the stone size is small. This is seen since the coefficient relating to median stone diameter, C4, is two orders of magnitude smaller than C3- It will be shown in the results section that equation (1.4) yields results that compare well with solid, non-rubble barriers and that the stone size effect is small.
Van Der Meer and d'Angremond (1992) reviewed several works on rubble mound submerged breakwaters, including Ahrens (1987) and several experiments conducted at Delft Hydraulics in the Netherlands. They also cite the relative freeboard of the barrier as the most important design parameter of such structures. An interesting and counter-intuitive result from their literature survey is the suggestion that the transmission coefficient is constant at 0.8 in the lower range of relative freeboards; tested (-2.0 < flHi < -1.0). As the relative freeboard approaches -oo, it is expected that the transmission coefficient would approach 1.0 as the barrier height shrinks to 0.0. Thus it is expected that there would be an asymptotic behavior of Kt as the relative freeboard decreases.
Cornett et al. (1993) investigated the performance of reef-type breakwaters, both as standalone shore protection structures and in tandem with larger emergent breakwaters. This work investigated not only the transmission of a single representative wave, but also the transmission of the full wave spectrum over a structure. This provides a more detailed look into the energy transmission and dissipation over submerged barriers. Other methods of investigating wave transmission characteristics are given. The surface elevation records of the offshore and onshore sides of the reef area are plotted to show the reduction in wave height of individual waves. From these types of records a cumulative distribution of wave heights can be computed. Comparison of the offshore and onshore records is provided in this work to show how the higher waves in a record experience more wave attenuation. This would indicate that submerged breakwaters




11
would have a greater effect on storm waves than on mild waves. However, this improvement in wave attenuation during storms is often countered by an increased water level due to the associated storm surge.
Figure 1.2 presents the results of several of the laboratory studies discussed above. The transmission coefficient is plotted versus relative freeboard, consistent with most of the laboratory works surveyed above. In each case the wave steepness and water depth to wavelength ratios are
1 .0 _, 1.. ... ).. ..j I,, I I I I I I I

-4 -3 -2

0 1 2

f/Hi
Transmission Coefficient vs. Relative Freeboard for Four Experimental Studies (H/L = 0.02 to 0.04).

approximately equal. Goda et al. (1967, 1969) presented a curve plotted through experimental points as well as an equation taken from the entire set of data from the experiments. Both representations are shown, and they show some variation, differing by 30 percent atf/Hi = 0.0. Equation (1.3) does include data from finite crest width structures, which serves to lower the transmission coefficient in comparison with the narrow-crest only cases.

0.8 0.6 0.4 0.2

0.0 ...
-5

. . .
. . . . . . . . . ... . . . . ... . . .\ _.. .. . . . .
'. \
Goda(1967)exp
...... Goda (1969) eq. (1.3) --.......
Ahrens (1987) eq. (1.4)
-Van Der Meer
& d' Angremond (1992)
Cornett et al. (1993)

submerged
.. .

I I I I i i

Figure 1.2

I

.........
- - - - -




13
1.3.2 Current Patterns Associated with Submerged Breakwaters
The addition of a structure in the coastal zone obviously changes the current patterns in that area. Knowledge of the changing patterns is important in terms of safety and shoreline response. In some instances structures added to a coastal system can create rip currents which are hazardous to swimmers. On another level, the purpose of adding a submerged breakwater is to protect the area behind it and to promote the accretion of sand. The current patterns in the vicinity of the barrier have a strong effect on the deposition of sand on the protected beach.
Longuet-Higgins (1967) described the rise in Mean Water Level (MWL) behind a submerged breakwater or sand bar. He presented a simple calculation to determine the vertical change in MWL based on the Bernoulli equation and requiring knowledge of the incident and reflected wave amplitudes. The rise in MWL behind a breakwater has been labeled 'ponding,' and while Longuet-Higgins provides a means of predicting the ponding level, he does not discuss the consequences of this rise.
Seelig and Walton (1980) presented a method of estimating the flow through offshore breakwater gaps. They reported that wave overtopping of breakwaters creates an offshore flow through breakwater gaps and around the ends of such structures. The goal was to provide design information for breakwaters in order to avoid high velocities through these gaps that could cause erosion. Using the ponding level as a gradient to drive a flow, they presented a simple continuity-energy calculation to predict the average flow through a given gap size. Seelig and Walton expressed the need to limit offshore flows to below 0.5 ft/s to avoid eroding material from the supposedly protected area behind a breakwater.
Lin (1986, 1988) investigated the performance of the P.E.P. Reef during the initial development of the unit. His investigations included numerical simulations and a limited set of field data from the DuPont site P.E.P. Reef I installation. Results from the field studies shed




14
little light on the change in velocity of the longshore current behind the Reef. Lin concludes that the Reef converts some wave energy into current energy and thus creates a strong current along the top of the Reef. He also concludes that the Reef does not alter the longshore current and that the Reef is very effective in stopping offshore sediment loss during summer storms. In addition, it is reported that the shoreline behind the Reef is very stable and insensitive to changes in offshore wave conditions.
1.4 Comments
While information regarding the performance of submerged breakwaters is available to design engineers, these data are not entirely consistent and cannot replace project specific wave climate and sediment transport information. As stated previously, the need for comprehensive monitoring of coastal engineering projects, especially experimental ones, is imperative. Often political and environmental interests cloud the issue as to the actual performance of coastal installations. It is important to evaluate data from such monitoring objectively and not apply data from one site directly to another with an expectation of a complete predictive capability. This report is intended to provide a basis for the performance of submerged narrow crested breakwaters and should be treated as such.




CHAPTER 2
DATA COLLECTION
Data for this report were obtained from two sources. The first source is the three year monitoring program of the Midtown Palm Beach Prefabricated Erosion Prevention (P.E.P.) Reef. This program is being conducted by the Coastal & Oceanographic Engineering Department of the University of Florida (UFCOE) and is currently in its second year of operation. The monitoring program is quite extensive, including hydrographic surveys, continuous wave climate analysis, scour analysis, and unit settlement surveys. Data from the surveys are available from July, 1992, which is just prior to installation, and data from the wave climate analysis are available from October, 1993, to the present.
The second source of information for this report is a laboratory study conducted by UFCOE at the UFCOE Laboratory in Gainesville, FL, during the summer of 1994. The study was sponsored by the Indian River County Board of Commissioners to provide physical model guidance for a permit request to install a 4,000 foot long P.E.P. Reef in Vero Beach, FL.
This chapter details the methodology of both the monitoring program and the laboratory study. This information is provided in order to present both the extents and the limitations of the field measurements and the physical modelling.
2.1 Description of the Midtown Palm Beach Monitoring Program
In 1989, the State of Florida passed legislation to permit experimental projects to attempt to solve Florida's beach erosion problems. One such experimental project is the P.E.P. Reef,




16
a pre-cast concrete structure placed parallel to shore in approximately ten feet of water depth at the Midtown Section of the Town of Palm Beach, FL. The Reef is composed of interlocking units, each twelve feet long. 330 units were placed approximately 250 feet from the shoreline, forming a 4,000 foot long barrier broken only by a 220 foot gap to allow an AT&T cable crossing, and a twelve foot gap to accommodate an emergency storm water outfall. Placement of the units began in the summer of 1992. 57 units had been installed when Hurricane Andrew struck South Florida in August, 1992. One additional unit was placed immediately after the hurricane when it was noticed that a significant amount of settlement had occurred for the first 57 units. Installation was suspended until the following summer. The remaining 273 -units were placed during the period of May to August, 1993.
As part of the project, a program to monitor the performance of the Reef was implemented. This program is sponsored by the Florida Department of Environmental Protection (D.E. P.) and the Town of Palm Beach, FL. The program began in the summer of 1993 with the installation of scour rods and wave gages and a complete hydrographic survey conducted immediately after installation was complete. Survey data taken in July, 1992, serve as a baseline for the monitoring program surveys.
2. 1.1 Site Description
The Midtown Palm Beach P.E.P. Reef Installation is located off the Town Of Palm Beach, FL, approximately 4.5 miles south of the Port of Palm Beach Entrance as shown in Figure 2. 1. Figure 2.2 indicates the location of the Reef relative to DEP monument R-95. The Reef lies in roughly 10 feet of water and is approximately 250 feet from the shoreline. The tidal range in this area averages 2.80 feet with a spring tide range of 3.03 feet. The Reef units themselves are six feet high, twelve feet long (longshore direction), and fifteen feet wide. A cross sectional view of the unit is shown in Figure 2.3.




I' 8'Woi

15'
note: unit sections 12 feet long

Seaward -

Figure 2.3 Cross-Sectional View of P.E.P. Reef (Courtesy American Coastal
Engineering, Inc.).
2.1.2 Wave Data Analysis
One of the expectations of a submerged breakwater is that it will provide protection against wave attack by reflecting or dissipating wave energy. To monitor the effectiveness of the Reef in attenuating wave heights, two wave gages were installed, one on either side of the Reef. These gages, as indicated in Figure 2.2 lie fifty feet on either side of the Reef along the same perpendicular to the shoreline. The gages lie in different water depths; the offshore gage lies in 13.5 feet of water while the nearshore gage lies in roughly 6.5 feet of water. The gages have been in concurrent operation since the middle of October, 1993.
The gages measure both pressure and two-directional currents in order to produce wave height and direction and current magnitude and direction data. The gages are cylindrical




19
packages with the pressure sensors and electromagnetic current meters located at the top of the package. These packages are secured to pipes jetted into the seafloor. The offshore gage is attached to a jetted-in tripod for protection. The nearshore gage is attached to a single jetted pipe near the seafloor to prevent the sensors from lying too close to the water surface. The packages are installed and their orientations measured to obtain the correct directional information.
The data collected from the sensors is transferred to temporary memory storage inside the package. Data is stored there until it is downloaded by telephone to the UFCOE Laboratory for analysis. The packages are hard wired by buried cable to shore where a shore station/modem package connects to a telephone line for modem communications. The cable connection also supplies power to the wave gages. The packages do contain batteries and a hard disk drive for backup use should the cable be damaged or the power supply lost from shore. These batteries can power the package for several weeks, long enough to allow a field crew time to visit the site and make necessary repairs.
Both wave gages collect data on an hourly basis. The collection scheme for the gages includes sampling the pressure and two directional current components at a frequency of one hertz for 17.1 minutes (1,024 seconds). Every hour the average pressure and current are recorded. Every sixth hour the entire 1,024 second record is stored. The hourly averages are used to determine tidal records and current variations with the tidal cycle, if any. The full pressure and current records are used to calculate wave heights, directions, and spectral information.
Using information from the gages, performance information can be obtained regarding the wave attenuation characteristics of the reef. When the package is 'called' via modem, the internal clock in the package is set to the clock time of the laboratory computer. This ensures the data from both gages is truly concurrent. Wave records from both gages can be compared, both in the time and frequency domains, to calculate transmission coefficients.




20
Data from the gages are analyzed in a format consistent with the Florida Coastal Data Network, operated for many years by UFCOE. This format provides significant wave height, modal period, modal wave direction, current magnitude, and current direction data for each sixhour record. The 1,024 second record is divided into 128 second blocks that overlap by fifty percent. A Fast Fourier Transform (FFT) is performed on each block, and the results of each block are averaged to create an energy spectrum. The significant wave height is computed from the energy spectrum as shown in equation (2.1).
H,= 44 f-E(a)dcd (2.1)
The modal period is taken as the period associated with the peak of the energy spectrum. The modal wave direction is determined by a directional spectra model which uses a symmetric cosine-power function (Cartwright 1963). Equation (2.2) presents the directional spectrum, E(r,O), in terms of the one dimensional spectrum, E(o), and the Gamma function.
E(a,4) = E(o) 22s-1 r 2(s+l) Cos-2s (' (2.2)
71 r(2S+l) 2
s is the spreading parameter, calculated from the co-spectra of the pressure/current data. Coastal and Oceanographic Engineering (1993) provides a detailed explanation of the directional spectrum analysis. The peak of the function in equation (2.2) is taken as the predominant direction of wave travel.
To compare the two records to determine transmission coefficients in a consistent manner, the offshore record must be modified to include the shoaling effect which naturally results from the gages lying in different water depths. To accomplish the comparison, the offshore significant wave height of each record is shoaled using linear wave theory and the modal period to the




21
corresponding depth at the nearshore gage. Linear wave theory computes shoaled wave heights according to equation (2.3):
ahoaled c Cgffsh6a (2.3)
Cg0ffshoze shoaled
Where C. is the group speed of the wave. Once the shoaled incident wave height is determined, the transmission coefficient, Kt, can be calculated as shown in equation (2.4):
Kt H Hs8.S" (2.4)
Soffshore shoaled
This provides a fair evaluation of the wave attenuation provided by the Reef by comparing the wave height at the nearshore gage to the wave height that would have occurred at the same point in the absence of the Reef.
Additional analyses are performed on the wave records to determine K4 values for individual waves. By performing an FFT on the full 1,024 second record, adapting the Fourier coefficients to the surface conditions, and then performing an inverse FFT on the new coefficients, the surface elevation record is found. This record can be examined for individual waves, or it can be processed by a zero-upcrossing routine to obtain the distribution of wave heights. The actual elevation record can be used to directly compare wave heights and to verify the clock times of the wave gages. The distribution of wave heights from each record can be compared to determine the dependency of transmission coefficient on wave height. This is accomplished by first shoaling the offshore record appropriately (as previously described), then calculating the cumulative distribution of wave heights for both records. Differences in the distributions indicate which waves are affected more by the presence of the Reef.




22
2.1.3 Background Wave Climate and Verification
Additional wave gages were installed in September, 1994, to provide further information about the wave climate near the Reef. These gages were first installed next to the FCDN gages to provide a check of the wave data already collected. The gages were then moved to a nearby location south of the Reef to provide background wave climate information, specifically the amount of natural wave height reduction occurring in that vicinity as a result of wave breaking.
2.1.4 Current Measurements
The electromagnetic current sensor on each wave gage provides the current magnitude and direction information for each hourly record. This provides a general description of the overall currents in the vicinity of the Reef. It does not, however, provide significant information about the spatial variation of current patterns around the Reef. The two gages provide point measurements roughly 3 feet from the seafloor at the gage locations.
2.1.5 Volumetric Chane
A large portion of the monitoring program focuses on the hydrographic surveying. The surveying provides erosion/accretion data as well as changes in the position of the Mean High Water (MWL) level. Figure 2.4 depicts the profiles along which the surveys are conducted every three months. 75 profile lines are surveyed by land surveying methods, wading/swimming surveys with a rod and level, and boat fathometer measurements. These three techniques are overlapped to provide repeatability in the data. The quarterly surveys extend offshore to 1,200 feet. An annual survey extends offshore to 3,500 feet. The profile lines shown in Figure 2.4 that extend from DEP monuments are surveyed to 6,500 feet offshore. The surveying is conducted by an outside firm, Sea Systems, Inc., of Pompano Beach, FL, and the data from their surveys are compiled and sent to UFCOE on computer disk and in printed form.




. ,- --..,- N 85E

- Most Northerly ULines Except
-S 850E for Those at DNR Monuments

---- E 'E NOTES:
. ON 80*E All Profile ULines at 900 Azimuth Except, as Noted,
8-75' ....-" for 12 of the 15 Lines From DNR Monuments
-BREAKWATER Total of 75 Lines. (DEP Monuments Only Surveyed
.-...N 800E Annually, Others Quarterly).
4.150- N 802E
.. --N750E
"r *6- 75 '
S1-150 Denotes 6Spaces at 75 ft
Between Profile Lines
-------- j-o-- N 900E
.-N 800E
- ...Most Southerly Line Except
,..--" for Those at DNR Monuments
N

0 5000 ft

Figure 2.4 Survey Profile Plan.




24
The data are analyzed to produce volumetric change information for each intersurvey period as well as a cumulative volumetric change based on the July, 1992, baseline survey. The volumetric changes are calculated using a trapezoidal rule method. This method calculates the volumetric change along a profile line in terms of volume per unit distance. The volumetric change is then computed by multiplying by the distance halfway to the next survey lines. For instance, if the survey lines were 200 feet apart, the volume per unit distance value would be multiplied by 200 feet (100 feet north + 100 feet south).
The volumetric change along a given profile line is divided into eight cells each roughly sixty feet in length. These cells are used to create seafloor elevation change contour plots. These cells vary in length depending on the exact distance of the Reef from the shoreline. Four cells make up the distance from the Reef to the shoreline, and four more cells extend roughly the same distance offshore from the Reef. This provides approximately 500 points to create reasonably smooth contour plots.
Additional information from the surveys includes the change in Mean High Water shoreline position. This information is determined from the individual profile lines and the M1UW mark on each. Tidal information taken from Lake Worth Pier, a few miles south of the Reef, places the MHW mark at + 1. 87 feet relative to the National Geodetic Vertical Datum (NGVD). Observation of the individual two dimensional profile changes provides additional insight into the sedimentation patterns in the area.
2.1.6 Unit Settlement
During the surveys, the top elevation of each individual unit is measured. These data have been collected since installation of the Reef began, and they document the settlement of the units. Following Hurricane Andrew, it was noticed that the units settled a great deal, and the




25
settlement became an important issue in the performance of the installation. Settlement of the units is an important consideration since it strongly affects the amount of wave transmission over the Reef. Too much settlement can render a submerged breakwater ineffective.
2.1.7 Scour Rod Measurements
Twenty-eight scour rods were installed around the Reef to determine if the Reef causes problematic scour around its periphery. The scour rods are two inch diameter copper pipes eight feet long jetted into the sand approximately six feet. 'Me pipes have large disks that move freely up and down the length of the pipes. These disks are set on the sand surface and follow the surface during periods of erosion due to scour. The disks are then measured relative to the pipe cap and the scour depth determined. The disks often sink to a given level then are covered over with sand during recovery periods. In these cases the scour depths are measured and the disks are reset to the current sand level.
Twenty-two pipes are located near the edges of the Reef. Eighteen of these are at the ends of the Reef or at the AT&T gap to measure scour. Four pipes are located at the centerline of the structure, and six pipes are located well north and south of the Reef for control purposes.
2.1.8 Present Status
The monitoring program is entering its second year. Update reports for the performance are presented every six months (Dean et al., 1994b; Browder et al., 1994). These include the compilation of the wave data for the period, the analysis of the survey data for the surveys conducted during that time, and updated conclusions as to the performance of the Reef. The wave data obtained since October are verified and augmented by additional wave gages installed in September, 1994, to verify the data and provide background transmission data.




26
2.2 Methodology of Laboratory Study
A physical model study of the hydrodynamic performance of the P.E.P. Reef was conducted in the summer of 1994 at the UFCOE Laboratory in Gainesville, FL. As mentioned previously, the study was sponsored by the Board of County Commissioners of Indian River County, FL. The purpose of the study was to provide design guidance to the county to assist in the permitting of an experimental P.E.P. Reef in Vero Beach, FL. This laboratory study provides design information regarding the relative height of the barrier and the planform arrangement of Reef units relative to the shoreline. The study was conducted on a 1:16 scale, which results in time and velocity scales based on Froude modelling of 1:4.
2.2.1 Experimental Eguipmn
The tests were conducted in the three-dimensional wave basin at the UFCOE Laboratory. A schematic of the basin is shown in Figure 2.5. This basin consists of a multi-paddle wave maker, a sloping offshore section leading up to a concrete horizontal bed fronting an area where a fixed or movable bed beach may be installed. For this study, concrete block walls were installed to create a 47 foot long beach. Gravel was used to make an immovable bed beach of 1:8 slope. This slope was chosen to recreate the beach-face slope at the site. The horizontal bed area was painted with a one foot square grid to facilitate velocity measurements.
Forty-eight individual 1: 16 scale model Reef units were fabricated from dimensions (see Figure 2.3) provided by American Coastal Engineering, Inc. These models were concrete units cast in fiberglass molds constructed at the UFCOE Laboratory. The model units are 4.5 inches high by 9.0 inches (longshore length) by 11.25 inches (cross-shore width). Blocks inserted during fabrication provided the vents in the units. The units were placed in various planform arrangements roughly 15.6 feet (250 feet prototype) from the still water shoreline in the model.




/ P.E.P. Reef Model

1:8 Gravel Gridded
Beach Test Area

Paddle
Wavemaker

Figure 2.5 Wave Basin Schematic

Regular waves with 2 second periods and 4 cm wave heights (0.012 ft) were generated in the basin by the multi-paddle wavemaker. The paddle wavemaker, made of nine inch wide paddles, is capable of generating waves at various angles and heights by setting the individual phases and amplitudes of each paddle. For this test, however, all 63 paddles were set to the same phase and amplitude to create a near-uniform wave height across the basin approaching normal to the gravel beach. The wave height was chosen to provide a compromise between the average wave height conditions at the site and the need to limit basin effects in the model study. Large wave heights put too much energy into the model and make the basin effects dominant over the desired experimental effects. The two second period in the model translates to an eight second period in prototype, which is typical of swell periods at Vero Beach, FL.




28
Wave heights were measured with a standard capacitance type wire wave gage. A wire partially immersed in the water functions as the capacitor in a tuned circuit. The amount of water covering the wire changes the capacitance in the circuit and changes the output frequency from the wave gage. A signal conditioner receives the signal from the wave gage and translates that signal from a frequency to a voltage signal, which is then sent to a strip char-t recorder. The estimated error in the wave height measurements from such an arrangement is estimated to be approximately 10% of the measured wave heights.
Currents were measured in the experiments using drogues. These drogues consisted of table tennis balls injected with water to make them slightly negatively buoyant and sealed. This resulted in a bottom drogue that would experience only minor friction against the concrete bed. The moving drogues were videotaped on the horizontal concrete bed, which was painted with a one-foot square grid to provide distance measurements. The videotape was post processed to obtain current patterns and magnitudes for each test.
2.2.2 Test Plan
The test plan consisted of twenty initial tests using four platform arrangements and a control run at four different depths. The control run for each depth consisted of the same measurement pattern performed in the basin with no Reef units present. This was done to establish background wave height profiles and current patterns. The four different water depths were chosen to be 12, 15, 20, and 30 cm. (0.037, 0.046, 0.061, and 0.091 ft). These depths provide freeboard ratios, f1h, of 0.0, -0.2, -0.4, and -0.6, respectively. The platform arrangements are shown in Figure 2.6. Each arrangement is labelled. with a letter, and future references to a particular arrangement will use these letters. The Type A case consists of 45 Reef




30
units placed in a continuous line 33.75 feet long. This represents a prototype length of 540 feet, and is the longest continuous arrangement tested in the 47 foot long basin. The Type B case has three 11 unit segments separated by 10 unit wide gaps. This represents 132 foot long prototype segments separated by 120 foot long gaps. The Type C case uses 45 units staggered in five unit segments. The stagger distance used in the initial tests was 1.875 feet, equivalent to two crossshore unit widths. The 45 units are equivalent to 540 feet of actual Reef length, and each segment would be 60 feet long. The Type D case derives from the Type C case only in this instance the offshore segments have been removed to create 60 foot long (prototype) gaps along the total 540 feet of Reef units.
Following the initial tests, the results were analyzed and additional tests were conducted to further explore the most promising arrangement seen in the initial tests. Figure 2.7 depicts the three additional arrangements tested in the basin. These cases were derived from the Type C case seen previously, and were only tested at f1h = 0.0 (crest at SWQ. The type E case consists of 45 units divided into five nine-unit segments. These segments represent 108 foot long segments in prototype. The offset distance was varied from twice the cross-shore unit width, 2w, to six times the cross-shore width, 6w (30 to 90 feet in prototype). 'Me Type F case shortened the offshore segments to seven units, creating a small gap along the shoreline between segments. The offshore segments represented 84 foot long sections in prototype. Again the cross-shore offset distance was varied as before. The Type G case was conducted based on the results of the Type F case and consisted of segments and gaps that were twice as long as the Type F case, 216 foot prototype onshore segments, and 132 foot offshore segments separated by two unit length longshore gaps.




E) 9 Unit Segments

-FT II 2wto 6wF) 9 & 7 Unit Segments

MHHHHWHHHHH

I I

I I I I

2w to 6w-

G) 18 & 11 Unit Segments

4-4I

mI I I LE

If I ~ II]

4

'l I I1I

Figure 2.7

Reef Arrangements for Additional Model Testing.

2.2.3 Wave Height Measurements
For each test (27 total) wave heights were measured at 70 different locations. These measurements fell along five profile lines defined by the grid on the horizontal bed. Figure 2.8 indicates the profile lines relative to the basin centerline. Wave heights were recorded at each intersection point on the grid to measure the wave height decay along each profile line. The four

I I t J

7=

--T-T

... . . . . . . . I

I - -




12 ft. -6 ft. 0 ft. 6 ft. 12 ft.
Figure 2.8 Cross-Shore Wave Height Profile Lines.
2.2.4 Current Measurements
For each test (including control tests), dye and drogues were used to determine the current patterns both qualitatively and quantitatively. In each test, approximately 30 drogues were tracked on the floor, moving over the painted grid on the bottom of the horizontal bed. The

32
points on the landward and seaward extents of each measurement line on the grid were averaged together, respectively, for all five lines to determine overall incident and transmitted wave heights for each experiment. This provided twenty measurements in the average for both the incident (seaward) and transmitted (landward) wave heights. The measurements on the offshore side were extended past the grid in cases where the offshore segments fell too close to the edge of the grid. Some of the cases contained profile lines that fell along breakwater gaps. These data were included to indicate sections of shoreline that would be unprotected from wave attack. Results from the wave height measurements were compared in terms of wave height decay profiles and transmission coefficients. These coefficients are presented as functions of both the freeboard ratio, f1h, and the relative freeboard, flHi.




33
drogues were tracked with videotape. The videotape was then analyzed to plot the trajectory of each drogue and calculate average velocities of the drogues.
The drogues were placed in both longshore patterns and cross-shore patterns to obtain adequate coverage of the area around the Reef. Of particular interest was the area between the end of the Reef and the sidewalls of the basin. A higher concentration of drogues was placed in and near the gaps to determine the magnitude and direction of flow in those areas. Average bottom velocities were determined and compared for each test to study the effects of the Reef. These values were compared to the control cases where no Reef was placed in the basin.
2.3 Comments
Field monitoring and laboratory studies have been conducted to gather information on the performance of the P.E.P. Reef submerged breakwater system. This system is classified as a narrow crested submerged breakwater (justification of this is discussed later). It is useful to discuss some of the limitations of the monitoring and of physical modelling.
The field monitoring provides the best information on submerged breakwaters, i.e. actual prototype data. Some limitations of the program, however, must be recognized. Surveys are conducted on a quarterly basis, and are subject to the seasonal variations and timing of episodic events. A survey may fall during a relative calm time in which the protected beach has experienced accretion of sediment, or it may fall after a particularly damaging storm, where the beach may have lost a considerable amount of material. Thus it is important to look at the performance of a project over a sufficiently long time so as to distinguish the overall trend from episodic events. In addition, current measurements from the monitoring program are taken at only two points along a 4,000 foot long structure. Therefore no current pattern information can be gained from the current meters. It would be very useful to have field measurements of the




34
current patterns, and aerial dye studies have been conducted, but obtaining quantitative current data would be far too measurement intensive and costly to be effectively realized.
As in any physical model study, limitations and basin effects must be discussed. In this model, only shore normal waves were tested. This is obviously not the case in nature, but these waves were used to provide basic performance information. Attempts were made to minimize basin effects, primarily by reducing the wave height used so as not to 'drown out' the effects of the Reef itself. Control tests were conducted to document the basin effects, and these effects were considered in the conclusions.
Finally, it is again stressed that coastal engineering projects are extremely site specific, and nothing can replace actual field data from the site of interest. The field and laboratory work presented herein are given as a basis for initiating a submerged breakwater project, and should not be blindly applied to a project at another site.




CHAPTER 3
WAVE TRANSMISSION
The first objective of any breakwater is to do just that, 'break' the water, i.e., reduce the height of the incoming waves to protect the area behind the structure. The transmission of waves past a submerged breakwater is the focus of this chapter. The amount of wave height attenuation provided by a submerged barrier is the primary variable used to describe the success or failure of such structures.
Performance of a breakwater for wave height reduction is provided here in terms of a transmission coefficient, Kt, which describes the ratio of the transmitted wave height to the incident wave height. In cases of sloping bottoms, the incident wave height is corrected for shoaling effects before comparison, as described in Chapter 2. Transmission measurements have been taken for the Midtown Palm Beach P.E.P. Reef installation and the Vero Beach P.E.P. Reef laboratory study and are discussed in this chapter. In addition, a simple analytical approach to determining Kt is presented and compared to both the field and laboratory results as well as previously discussed literature. From this work, performance criteria and design recommendations can be made in terms of freeboard ratio, f~h, and relative freeboard, f/Hi.
3.1 Field Measurements
Wave data have been collected continuously from the Midtown Palm Beach P. E. P. Reef installation since mnid-October, 1993. As discussed in Chapter 2, the data are analyzed for significant wave height, modal period, direction, and energy spectrum. Comparison of values




36
from both gages produces transmission coefficients based on significant wave heights. Compilation of the wave data can be found in Dean et al. (1994b) and Browder et al. (1994).
The focus of the field data concerning transmission coefficients will be the one month period from 21 September 1994 to 24 October 1994. During this time two additional Seadata wave gages were installed at the site to verify the data from the existing CDN gages and provide background wave climate information. These two gages were installed adjacent to the CDN gages for two weeks during this period to provide concurrent data, then they were moved to a location 500 feet south of the Reef (at the same cross-shore distances) where the background measurements were taken for the remaining two weeks. The data from the Seadata instruments (significant wave heights and modal periods) are compiled in Appendix A.
3.1.1 Transmission Coefficient Determination
Wave height reduction is customarily measured in terms of a non-dimensional transmission coefficient, K,. Chapter 2 outlined the procedure for determining the transmission coefficients from significant wave heights, which actually provides an indicator of the amount of energy transmitted over the structure. Figure 3.1 shows a plot of the transmission coefficient versus time for the Midtown Palm Beach P.E.P. Reef installation. The plot indicates a fairly consistent range of Kt from January, 1994, to roughly the end of June, 1994, between 65% to 85%. Starting at the end of June, the values began to drift, reaching an average value of 1.0 by October, 1994. This behavior suggests that one of the CDN gages has lost its calibration. Both gages have been in place continuously since October, 1993, which is a long time for an instrument in the ocean to survive. It is believed that the rubber portions of the pressure sensor have degraded over time, resulting in small changes in the measured wave heights. These small changes have a large effect on K4, especially for smaller wave heights.




1.2 I
* 40
0.84 4"
0. .........~
0 50 100 150 20 50
~~ula Date 19944.~4 ~
Fiur Trnmiso Cofiin Hitoy Mitw amBac ntlain
1 Jnury 99 to24Octbe 194,CDNGaes
An attmp wa mad to corec th coficet thog th iernemaue tec
gage.~ ~ ~ ~ ~~4 Sic th9prahue ooti ,i ieroe h ai ftetd agscnb
Fiue 3.1 cretthtransmission coefficients istrey, MiTwn Palmo Bfiea rns easun.
however, was very nearly one over the range of suspected poor values. This would not account for the larger drift seen in the values. The apparent problems are high frequency dynamic problems (roughly 0.1 hertz) while the calibration method using the tide ranges is a low frequency (0.00005 hertz) static calibration. The use of the static calibration does not correct the dynamic problems seen in the CDN gages over the last three months. As a result, the comparison of the Seadata to CDN transmission coefficients will be studied between times of similar wave conditions with the CDN data taken from the previous winter season.
Figure 3.2 shows the transmission coefficients calculated from the Seadata gages during the month long test period. The first two weeks of the period indicate the time when the gages were installed within the confines of the P. E. P. Reef. Here the average values of K, range from




1 .2 I .
.0V ..... ...
0.6
0.4
0.2 a Seadata Reef
V Seadata Control
0 0I I I r
260 265 270 275 280 285 290 295 300
date
Figure 3.2 Wave Height Transmission Coefficients, Midtown Palm Beach Installation.
Seadata Gages, 21 September 1994 to 24 October 1994.
roughly 0.70 to 0.90, similar to the results seen in the CDN gages during the first six months of 1994. This provided a level of confidence in the quality of the data from both the Seadata packages and the CDN data from the first half of the year. The second half of the test period indicated in Figure 3.2 shows the background (control) transmission coefficients measured south of the Reef. Here the average values are slightly higher, ranging from 0.79 to 0.97. The lower values seen in the graph from day 282 to day 288 have been excluded from the averages, since these data are obviously anomalous and are believed to be caused by biological growth on the nearshore Seadata gage. The sudden jump in Kt at day 288 indicates that the growth or blockage on the pressure port was removed either by a large wave event or another biologic entity. It is noted here that this data set is quite small and that additional measurements will be conducted in the future once the mechanical difficulties in the CDN gages are resolved. While it is unfortunate that the equipment problems occurred during the Seadata test period, such is the nature of ocean field measurements, and to have two gages operate successfully together for nearly nine months is indeed fortunate.




39
The preliminary results of the Seadata test period indicate that between 5 and 20% of the wave height reduction seen in the Palm Beach data previously published can be attributed to the natural reduction of wave heights caused by energy losses between the gages. In that regard, for the cases where larger reduction in wave heights are reported, more of the reduction is likely due to wave breaking. For example, in the instances where the CDN gages reported a 35% reduction in wave height, it is likely that as much as 15% of this reduction is attributable to causes other than the Reef. This is due to the fact that the larger reduction of wave heights occurs during larger wave events such as storms. It is also during this time that more natural energy losses would occur between the two gages. As a result, the Reef itself is most likely contributing roughly 10 to 20% of the wave height reduction previously reported to be 15 to 35%. This degree of wave height reduction is more comparable to previously published works (see Chapter 1) for a structure whose crest elevation is relatively low in the water column. The Midtown Palm Beach installation has an average freeboard ratio of -0.56 relative to the NGVD water depth. At normal high tide the ratio decreases to an average of -0. 62, and at low tide the ratio increases to
-0.51. Most data for these ratios predict the transmission coefficient to be 90% or more.
To try to compare coefficients more closely, storm data are examined from both gages. Figure 3.3 shows the significant wave height history for the one month Seadata test pe-riod. Of particular interest is the six day period beginning 15 October 1994 (Julian Date 288). During this time a large storm passed through the area, generating significant wave heights of up to 4.92 feet. During this time the average significant wave height was recorded to be 3.12 feet. The corresponding average transmission coefficient was 0.91. This storm was compared to an eight day period beginning 15 February 1994 (Julian Date 46). The February storm produced similar significant wave heights (up to 4.60 feet) and posted an average significant wave height of 3.12 ft as well (the data were searched specifically to find a storm with the same average wave height).




5 SIe I i i Offshore
4 Seadata Offshore 4 + Seadata Onshore
3- 3
2+
0''S
01 ..
260 265 270 275 280 285 290 295 300
Julian Date
Figure 3.3 Significant Wave Heights, Midtown Palm Beach Installation. Seadata Gages,
21 September 1994 to 24 October 1994.
The average Kt calculated during the February storm was 0.77. Considering the similarity in the two storm events, it is presumed the Reef was responsible for a 14% reduction in the wave heights approaching the beach, while other natural causes were responsible for the other 9% reduction in wave height measured during that time.
While the data set described above is quite short, it does address the question of background wave height reduction in the vicinity of the Midtown Palm Beach P.E.P. Reef. This comparison raises a good point that will be discussed further in Chapter 4. A submerged breakwater or any hard structure does not act alone to defend upland development, rather, it works in combination with the natural beach to cause waves to dissipate their energy, either by breaking or reflection, before they impact the beachface and upland areas.
3.1.2 Snectral Analysis
Other features of the transmission of waves over a submerged breakwater have been investigated during this study. One is the spectral transformation of the irregular wave climate




41
passing over the structure. The CDN and Seadata gages both provide time series of pressure that can be translated into spectral density information. Chapter 2 outlined this process, where the pressure records are passed through a Fast Fourier Transform routine and their spectral components transferred to the surface level. This results in a spectral density plot that describes which frequencies contain the most energy and how much energy is transmitted past the structure. Plots such as these also provide information about the generation of higher harmonic components of the incident waves. These harmonics would appear as peaks in the spectral density plot at multiples of the incident frequency. As discussed in Chapter 1, the generation of higher harmonics is characteristic of broad-crested structures, where the high frequency components are generated in the space above the crest and transmitted as free waves (Massel, 1983). This criterion is used to further classify the structure as narrow or broad crested.
Figure 3.4 shows the spectral density for a wave record recorded 26 November 1993. The figure indicates a very narrow frequency wave climate consisting of swell of ten second period. The significant wave height measured from the spectrum for this record was 4.26 feet at the offshore gage. The transmission coefficient for this record was calculated at 0.72. A field crew visit to the site during this particular storm visually verified the period and wave heights coming from the consistent offshore swell. The storm conditions here persisted for several days, causing noticeable erosion and two to three foot scarps to be cut in the beach in some locations. No other frequency components are noticed in the figure, indicating the absence of either local wind generated waves (typically 0.25 to 0.33 hertz components) or higher harmonic generation.
Figure 3.5 depicts the wave spectral density recorded 5 October 1994. This record was obtained from the Seadata test period data when the gages were in place within the confines of the P.E.P. Reef. The shape of the spectrum looks quite similar to Figure 3.4, indicating long period swell conditions with an offshore significant wave height of 2.96 feet and a transmission




0
0.

0

0.3
f(Hz)

Figure 3.4 Spectral Density of Offshore,
CDN 26 November 1993, 6:00
20 II
15
CI\
4 10 I
I i
SI I
5
0
0.0 0.1 0.2

Nearshore, PM.

0.4

0.5

and Shoaled Offshore Records.

0.3 0.4

f(Hz)
Figure 3.5 Spectral Density of Offshore, Nearshore, and Shoaled
Seadata 5 October 1994, 4:00 AM.

0.5

Offshore Records.

- offshore ....... nearshore
shoaled offshore
i i ' "'' 7




20 I
15 offshore
....... nearshore
shoaled offshore
C\2
10
0.0 0.1 0.2 0.3 0.4 0.5
f(Hz)
Figure 3.6 Spectral Density of Offshore, Nearshore, and Shoaled Offshore Records.
Seadata 17 October 1994, 8:00 AM.
3.1.3 Individual Wave Tracking
In addition to examining significant wave heights and average values for transmission coefficients, individual waves can be 'tracked' to see how they are attenuated as they pass over a submerged breakwater. The wave records from the Midtown Palm Beach installation can be analyzed to provide the water surface elevation records as a function of time. By performing a Fast Fourier Transform (FFT) on the pressure record, the Fourier coefficients in the frequency domain are obtained. The record is a pressure signal, thus the values must be transferred to surface elevation values through the respective pressure response function at each frequency component. The result is a frequency domain description of the surface. To obtain the true surface displacement, an inverse FFT is performed to obtain the time series. If both the offshore and the nearshore gage are treated this way, individual wave records are obtained that travel over




45
the structure from one gage to the other. In this way each wave height can be examined and a transmission coefficient for a single wave can be calculated.
Figure 3.7 shows a portion of the surface record from 18 December 1993. Here the two records are matched by moving the offshore record ahead in time to account for wave travel time between the gages. In the plot the first large peak indicates how the height for that wave has been noticeably reduced. Thirteen such high waves were measured in this record, shoaled appropriately to the nearshore gage depth, and their transmission coefficients calculated. The average Kt for these waves was 0.76 while the value for the entire record was 0.78. Another feature of the plot is the area around the 516 second mark where the surface fluctuations are quite small. This matching area indicates that the gages are sampling the wave climate at the same time, which is obviously valuable for concurrent comparisons between the gages.
2
-2 offshore + 8 s
.....nearshore
480 496 512 528 544
t(s)
Figure 3.7 Surface Elevation Records for Offshore and Nearshore CDN Gages.
18 December 1993, 6:00 PM.




46
As mentioned previously, submerged breakwaters have a more pronounced effect on waves of larger heights. This can be demonstrated from surface elevation records such as in Figure 3.7. If a zero-upcrossing routine is used to identify the individual waves in a record and their periods, then a distribution of wave heights can be calculated. Figure 3.8 indicates the cumulative frequency distribution of wave heights for the record taken 18 December 1993. The plot shows the nearshore, offshore, and shoaled offshore records. The shoaled record represents what the distribution would look like in the absence of the breakwater. In this example the distributions do not differ noticeably until the wave height exceeds two feet. At this point the breakwater begins to affect the waves more, reducing their heights by a larger percentage. For example, at a cumulative frequency of 0.5 the corresponding Kt value is 0.93. At a cumulative frequency of 0.9, Kt = 0.81. Thus the higher waves in the record are affected more by the presence of the breakwater than the smaller waves are.
1.0 -
/* /
0.8 .~I
. / /
:!
0.6/
0.4.
.- offshore
. . ....... nearshore
02/ shoaled offshore
0.2 ry
0.0
0 1 2 3 4 5 6
H (ft)
Figure 3.8 Cumulative Frequency Distribution of Wave Heights for the Offshore,
Nearshore, and Shoaled Offshore Records. 18 December 1993, 6:00 PM.




0.20 0.16
,4 0.12
0.08
0.04
A nn

0 1 l 2 3 4 5 6 7 8 9 10 11 12 13 Offshore Distance (ft./grid)
Figure 3.9 Cross-Shore Wave Height Profiles for Initial Laboratory Tests. Freeboard
Ratio,f/h = 0.0.
basin. These cases have a continuous longshore stretch of units, with no longshore gaps where waves will pass unaffected. The cases that have gaps along the shoreline (B and D) indicate slightly higher wave heights at the beach toe. This is because the gaps allow waves to approach the beach unattenuated. These profile lines serve to raise the average landward side wave height at each cross-shore distance for these cases, resulting in higher transmission coefficients overall. At lower freeboard ratios, the wave attenuation decreases in similar fashion. At a freeboard ratio of -0.6, the greatest water depth tested, none of the arrangements indicated significant wave attenuation. This behavior was consistent with all aspects of the laboratory tests. At higher freeboard ratios (0.0 and -0.2) the structure induced effects (wave transmission, currents, etc.) were readily apparent. As the freeboard increased, the structure effects diminished 'until they were indistinguishable from the control cases.

SI I I I I I
I q
. . .... .. . .. .. . . . . .. ..
Control
....... 45 Continu pus (A)
3 Segment (3)
---- 45 Staggered (C)
5 Unit Graps (D) reef I line*
i I i I i i i I A ,, ,




3.2.2 Laboratory Transmission Coefficients
Taking the landward and seaward data points described above and averaging them across the basin, a transmission coefficient was calculated for each test. In the laboratory test, no shoaling corrections were made to the offshore values since the tests were conducted on a horizontal bottom where no shoaling should occur. Figure 3.10 depicts the K, values determined for the initial twenty tests. The values are plotted as a function of freeboard ratio, f~h. For each arrangement, the transmission coefficient increases as the freeboard ratio decreases (right to left). At a freeboard ratio of -0.6, all four cases produced coefficients of roughly 1.0, indicating no reduction of wave height at all. Recall that the idtown Palm Beach installation has an average freeboard ratio of -0.56, indicating that it should produce little wave attenuation according to these results. This apparent contradiction was one reason for conducting the extra wave gage testing described in Section 3. 1.

-0.6 -0.4 -0.2
f/h
Wave Height Transmission Coefficients for Freeboard Ratio, f/h.

Initial Laboratory Tests versus

. I I I I I I I

A 45 Continuous (A)A o- 3 Segments (B) o 45 Staggered (C)........

-Ahrens (1987) eq. (1.-4)

Figure 3.10

I




50
The arrangement that produced the most wave height attenuation was the continuous 45 unit case, Type A. This arrangement produced a 25% reduction in wave height at a freeboard ratio of 0.0. The next best case was the Type C, 45 unit staggered arrangement, which performed nearly as well as the Type A case at all depths. The Type B and Type D cases provided considerably less wave height reduction, again due to the fact that these cases have gaps along their lengths that allow waves to pass to the beach unattenuated. At greater depths all the arrangements perform similarly, producing transmission coefficients within 10% of each other. For such a structure to be effective in reducing wave heights, it is generally accepted that a transmission coefficient of 90% or less is required. Otherwise no noticeable protection for the beach will be realized. Figure 3. 10 also includes the empirical equation developed by Ahrens (1987). Equation (1.4) is plotted here using the laboratory scale values. Good agreement is seen with laboratory data, particularly for the continuous structure case. The percent difference between Ahrens' work and the Type A case is less than ten percent at f1h = 0. 0, decreasing to less than 1% at a -0.6 freeboard ratio. Ahrens' work was conducted with rubble mound structures, which are porous. It will be shown later that the porosity of a rubble mound structure and the holes in the P.E.P. Reef model units behave quite similarly in affecting the transmission of wave heights.
Figure 3. 10 describes the influence of water depth and freeboard on Kt. Intuitively, as the clearance over a submerged breakwater increases (decreasing freeboard ratio), the incident waves are less affected. Figure 3. 10 does not, however, describe the relationship between the wave height and the freeboard. Both relationships are important in the performance of a submerged breakwater and demand design consideration, but the two are obviously related by the freeboard. The freeboard is the single most important variable in the design of such structures, as it directly affects the wave transmission and the generation of structure induced currents.




Figure 3.11 Transmission Coefficients Versus Relative Freeboard, flHj.

. . . . . . . . . . . . . . . . . . . . . . . . . .. i . . . . . . . . . .
. . . . . .. . - -- . . . . . . . . . . . . . . .
A
. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . ..
A 45 Continuous (A)
Ahrens (1987) eq. (1.4)

51
The effect of wave height and freeboard on the transmission coefficient are indicated in Figure 3. 11. This is the same data of the previous figure plotted versus the relative freeboard, flHi. Here only the Type A case is compared against Ahrens' (1987) work. The other cases compare similarly to this plot as in the previous plot. With a fixed incident wave height in the laboratory, the water depth is varied to create a range of freeboards over the structure. As the freeboard increases in magnitude, the relative freeboard becomes more and more negative and the waves pass over the barrier more easily. The data indicate that below a relative freeboard of roughly -1.3, the wave heights are reduced by less than 10%, indicating the structure is fairly ineffective. The results of both Figure 3. 10 and Figure 3. 11 indicate that in order to significantly reduce the incident wave height, the barrier must occupy at least 80% of the mean water column, a result which mirrors the conclusion drawn by Hall in 1939.

5

1.0 0.9 0.8 0.7 0.6

-4 -3 -2
f /Hi

-1 0




52
3.3 Analytical Model
In the study of submerged breakwaters for design purposes, it is useful to have some simple means of predicting the transmission associated with a certain design. Several previously published works have attempted to predict values for K, with varying degrees of success. Many methods present transmission coefficients in terms of the wave number, k, and the water depth, with no inclusion of wave height. Other methods have resulted in empirical expressions from laboratory data, which produce good results for the range of conditions tested. Presented herein is a simple analytical approach to predicting the transmission coefficient for narrow crested submerged breakwaters. The approach includes the influence of freeboard, total water depth, and wave height. This approach, like most others, has its limitations and drawbacks, which are discussed herein. The development includes barriers that are completely submerged and partially submerged, including any openings in the barrier, up to the point where the time varying water surface on the seaward side does not reach the crest elevation of the barrier. The basis for the analyses is discussed in this chapter, while the entire derivations are found in Appendix B.
3.3.1 Totally Submerged Barriers
The first portion of the solution applies to structures whose crest elevation never exceeds the trough elevation of the waves on the seaward side. A schematic of the problem is shown in Figure 3.12. The offshore water surface of the barrier is composed of the incident and reflected wave patterns. The two offshore components, as well as the transmitted wave, are all assumed to be in phase. The water surface profiles of each component are described by the following:
11 Hi, Z' t Cos (a t) (3.1)
2




53
where t1 = water surface profile, H = wave height, a = wave angular frequency, and i,r, and t denote incident, reflected, and transmitted, respectively. Each term is spatially averaged so that the kx term that would occur in the cosine argument equals zero.
Hi Hr Ht
MWL
f flow (q)
h
~~barrier -~Seaward
Figure 3.12 Definition Sketch for Transmission Over a Submerged Barrier
If the difference in elevation of the water surfaces on either side of the barrier is taken as the driving head for a flow over the barrier, equation (3.2) defines the velocity, u. This velocity is linearized with the coefficient, A.

u = /2g(T+1'11-Tl t) = A. (TIi+T11-T1 c)

(3.2)

The linearization constant can be determined in many ways. Here it is found by equating the maximum velocities of the two velocity descriptions. Equation (3.3) shows the result of the linearization constant:

2

(3.3)




54
where Krt represents the reflection and transmission coefficients, respectively. The linearized velocity expression is then integrated over the freeboard above the barrier to obtain the flowrate. Also included in the flow is the contribution of the flow through any holes in the barrier. The P.E.P. Reef units have three large holes across the length of each unit. The flow through these holes is included in the same fashion as the flow over the barrier. The area of the holes, A, per unit structure length, replaces the area over the barrier in that calculation. The total flow is then equated to the flow on the transmitted side, assuming shallow water conditions. Since the transmission coefficient is the variable of interest, the reflection coefficient is removed from the equation by invoking the conservation of mass over the barrier:
Ui+Ur = Uc -. 1-K, = Kt (3.4)
Equating the two flowrates and solving for K, results in the implicit equation, (3.5).
1
Kt 1+ Hi 2 (3.5)
4 (Av-f) HI
3.3.2 Partially Submerged Barriers
When the water reaches an elevation where the incident/reflected wave trough is at or below the structure crest, the barrier becomes only partially submerged, and the above approach is not valid. In this section partially submerged barriers will be addressed. The approach taken here will be that of intermittent critical flow over a sharp-crested weir. The flow associated with any holes in the barrier will be incorporated in this development as well.
Consider the specific energy in the critical flow of water over a sharp-crested weir as shown in equation (3.6). The derivative of the specific energy, set equal to zero, determines the value at which the flow becomes critical, that is, no information downstream of the barrier is




dEC q- 2 0 (3.6)
transmitted upstream to affect the flow. Here, hC is the critical depth of water over the crest from the crest elevation to the water surface. The energy in the flow is generated by the head of water created as the wave form approaches the barrier. This head is the difference in the surface elevation and the structure crest. It is recognized that during certain portions of the wave cycle there will be no flow over the barrier, only through the holes, if any. The portion of the flow over the barrier is then written as equation (3.7) when 71i + nrf> 0, q," = 0 otherwise.
23
3 2 =A(1+1-)(3.7)
The linearization constant is determined here by matching the maximum flow of the two, resulting in equation (3.8).
2 ~- 2
AW= [(1+K 2cs~ (3.8)
3 2 Hi
The flow over the barrier can be represented by a Fourier series with components at an infinite number of frequencies. Here the primary frequency component is found and assumed to be much larger than any secondary harmonic components. The limits of integration for the calculation of the first Fourier coefficient are determined from the geometry of the problem. These limits are taken to be the times during the wave cycle when the water surface elevation on the incident side is above the crest elevation of the barrier. The first Fourier component of the flow then becomes equation (3.9). Again, the details of this development are contained in Appendix B.
qw= -AW Hi1 I+K) Cos,_( H.2f (3.9)
IC 2 2 i (1 +Kr)




56
This intermittent portion of the flow is then added to the flow through any holes in the barrier. The flow through any holes is calculated in the same manner as the flow over a totally submerged barrier, via a Bernoulli-type argument. The linearization constant for the hole flow, As, is of the same form as equation (3.3). The area of the holes is included in the development as the height of the holes per unit width of barridr. For the case of the P.E.P. Reef, the three holes represent 46% of the longshore length of the barrier. In addition, a flow contraction coefficient for the holes of 0.6 is included. Therefore, the effective area of the holes, A is the height of the holes multiplied by the longshore percentage and the contraction coefficient. The flow through the holes is then given by equation (3.10).
q,= AsAv(fli+1r-*t) (3.10)
Combining the two flows, equating the sum to the flow on the transmitted side, and solving for K, results in another implicit equation for the transmission coefficient, shown in equation (3.11).
Ke ={ i(Z~ 2 -_2f] 1
K t 2T 3 H1
t 2 2-Kt) COS ( ) -2f
2f ),h (3 3.(2)
2 Hi (2-Kc)
+ 2 2 A (1 -K)
hHi (1-Kt)
The second term on the right hand side indicates the contribution from the flow through any holes in the barrier.
The two equations for Kt are solved iteratively over an appropriate range of relative freeboards for the laboratory scale values discussed in section 3.2. Figure 3.13 plots the results of the two analytical expressions as well as the laboratory results and the published works of Goda (1969) and Ahrens (1987). The ranges of relative freeboards are extended slightly past their realistic ranges in order to show how the two methods overlap. As the plot indicates, the




1.0
A
0.8 7 ..
0 .4 .. .. .. .. .
A 45 Continuous (A) V
- Ahrens (1987) eq. (1.4)' Goda(1969)exp
0.2 Goda (1969) eq. (1.3) .......-----equation (3.5)
0.0 I
-5 -4 -3 -2 -1 0 1
f/Hi
Figure 3.13 Analytical Approaches for Transmission Coefficient compared to Laboratory
and Literature. Laboratory Scale Values, Hi = 0.012 ft.
two methods do not smoothly intersect. This discontinuity is a result of the fact that the two mechanisms of the flow actually occur together beginning at some point near f/H, = -0.5. At this point the flow over the barrier begins to be intermittent, but still strongly resembles the submerged barrier approach of equation (3.5). As the barrier becomes less and less submerged the flow begins to more closely resemble the weir flow approach. The actual mechanics of how this transition occurs are not fully understood at this time. The two equations are plotted here to indicate the behavior of each. The weir flow approach is terminated in the figure at. the point where the incident/reflected water surface drops to the top of the holes in the laboratory study. At that point the structure begins to more resemble an emergent structure, which is not the focus of this study. Note that beyond this point, flow over the top of the barrier is now accomplished by run-up and overtopping, neither of which resemble weir flow.




58
The analytical expressions show good agreement with the present laboratory study and the experimental work of Ahrens (1987) at lower relative freeboards. As the barrier crest approaches the Mean Water Level (MW.L) the agreement worsens. At a relative freeboard of
-0.35 (where the two methods intersect) the predicted K, differs from the measured value in the laboratory by 41%. In addition, the holes in the model units contribute about 38% of the wave transmission. The calculated value of Kt at that relative freeboard is 0.48. Without the flow through the holes, the calculated value would be roughly 0.30.
The differences between the analytical data and the laboratory data can be seen in the assumptions used in the development. The flow over the barrier is modelled as a steady flow, when in fact the flow of a wave over an obstacle is quite unsteady. This means that acceleration effects are neglected. It is recognized that the flow of a wave over such a barrier would have considerable acceleration components, particularly as the barrier occupies more of the water column. These acceleration components would serve to increase the flow over the structure, thus increasing the transmission coefficient. This is seen in the plot where the discrepancy widens as the barrier crest approaches the MWL. Also the approach has been linearized to facilitate calculation of the transmission coefficient. Although there are many ways to linearize the equations, all techniques are still approximations to the exact solution.
The agreement of the laboratory work with that of Ahrens (1987) is explained by the fact that Ahrens' work was conducted using porous rubble mound structures. The holes in tthe P.E.P. Reef model units represent porosity, which increases the level of wave transmission. The results of Goda (1969) were obtained in the laboratory with impermeable structures with no holes in them. Consequently the transmission coefficients are lower. The analytical approaches presented herein would be expected to predict higher transmission, but without the acceleration effects the predictions are too optimistic. These values could be taken as a lower limit for design purposes,




59
but the results from experimental and field work would likely provide a better estimate for the wave height reduction at a given relative freeboard.
3.4 Comments
Wave height attenuation over submerged narrow-crested breakwaters has been investigated in several manners in the field, in the laboratory, and analytically. The interest in studying this phenomenon is obviously to determine the degree of protection against storm wave attack provided by a given barrier at a given depth of submergence. It is apparent that the freeboard, f, is the most important variable in the design of a submerged breakwater. 'Ilie greater the freeboard, the less attenuation afforded by the barrier.
From a wave height reduction standpoint, the freeboard should be kept as small as possible in order to reduce the wave heights significantly. However, the freeboard affects much more than just the wave attenuation. Chapter 4 discusses the effects of structure induced currents in the nearshore zone, and it will be shown the amount of freeboard can contribute to undesired current effects. It is desirable to have at least a 10% reduction in the heights of large waves for a submerged barrier to be judged effective, and to achieve this the freeboard must be limited to a certain value. But both issues must be addressed simultaneously and a compromise reached. Considering only wave height attenuation, however, it is noted that the barrier should occupy at least 80% of the mean water column in order to be effective. Alternatively, the ratio of freeboard to incident wave height should be -1.3 or greater in order to have a substantial effect on wave height reduction.




CHAPTER 4
STRUCTURE INDUCED CURRENTS
The second focus of this paper is the effect of a submerged barrier on the nearshore current patterns. Obviously the presence of any structure in the surfzone will alter the existing current patterns, diverting the flow around the barrier. Of primary concern is the possible generation of currents strong enough to transport sediment away from the project area. This concern is the focus of this chapter.
4.1 Motivation from Field Results
The motivation for this concern stems from the initial performance of the Midtown Palm Beach P.E.P. Reef installation. During the first three months after the installation of the full Reef, the area directly in the lee of the Reef lost a substantial amount of material, 35,000 cubic yards (yds3), while the 2,000 foot long stretches of beach to the north and south of the Reef gained 3,600 yds3 and 13,200 yds3 of material, respectively (Dean et al., 1994b). The movement of such substantial amounts of sand clearly suggests that the current patterns in the area had been modified from the natural condition.
Figure 4.1 shows the changes in seafloor elevation in the vicinity of the Reef during the four month period from August to December, 1993. The solid contours indicate areas of accretion and the dashed lines indicate erosion. The plot clearly shows the losses experienced in the lee of the Reef during this period. In the lee the seafloor dropped an average of 1.0 feet, while the region landward and south of the Reef gained 0.75 feet in elevation overall. The figure




250
200 7H
150 LI
100
I 50 15
0
, -50
W 100 ,J.,:.i -.:.. ..' '
-150 /. ."'-200
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-- North Distance from monument 92F (ft)
Figure 4.1 Isolines of Elevation Change (ft), August 1993 to December 1993. Contours
in 1.0 foot intervals.
suggests that material has been removed from the lee of the Reef and deposited to the north and south of the Reef, predominantly to the south, which is the prevailing direction of longshore sediment transport in the area.
Dean et al. (1994b) presented a hypothesis for the sedimentation patterns observed above. The mechanism proposed derives from the fact that waves passing over the Reef transport mass (water and possibly suspended sediment) shoreward over the Reef. Wave mass transport is a well understood and documented phenomenon. This transport of mass is normally balanced by an offshore return flow in the bottom portion of the water column. Dean et al. proposed that this seaward return flow was reduced by the presence of the barrier parallel to shore. This interruption causes at least a portion of the return flow to be diverted alongshore, where it is superimposed upon any existing longshore current. This diverted flow impedes the ]longshore




62
flow in the updrift direction and augments the longshore flow in the downdrift direction. The addition of this diverted flow may have increased the level of suspended sediment transport, carrying water and sediment to the ends of the Reef where the restriction to the natural flow was removed and the currents were reduced, thus causing the deposition of sediment, particularly in the direction of the natural longshore current.
Further monitoring results from the Midtown Palm Beach Installation are consistent with this conclusion. Figures 4.2a and 4.2b show the volume changes in six regions around the Reef (Browder, 1994). Figure 4.2a shows the volume changes in the vicinity of the Reef for the most recent survey period (as of this writing), March 1994 to July 1994. This period indicates that the only region to lose material is the area in the lee of the Reef, while the remaining five regions have posted modest gains. These depositional patterns are indicative of the current patterns described above. The mechanism discussed is referred to herein as a 'pumping current.'
March 1994 to July 1994 August 1993 to July 1994
94A 94A 7
+2,000 +5,500 2000' -5,500 +1,800 2000'
9 5 E - --- ------- .......9 5 E ........ ....
R- Reef IReef
t t
-4,400 +5,600 4000' -37,000 +3,800 4000'
0 0
99B 99B
+4,300 +2,300 2000' +17,100 -5,100 2000'
101A --_---- 101A
240'+/--- 240' +/-' 240' +/--- 240'+/-(a) (b)

Volumetric Changes for Latest Survey Period and Last Year (yds3).

Figure 4.2




63
Figure 4.2b indicates the cumulative changes in the vicinity of the Reef for the two year period from August, 1993, to July, 1994. The August, 1993, hydrographic survey was taken immediately after the placement of the entire 330 P.E.P. Reef units and is considered by the monitoring program to be the post-installation baseline survey for the performance of the project. The plot indicates substantial losses in the lee of the Reef, resulting in a drop of the seafloor by 1.0 feet during this time. The region landward and south of the Reef has accumulated 17,100 yds3 of material during this time. This behavior would support the mechanism of the pumping current superimposed on the southerly directed longshore current.
The net longshore sediment transport in the Palm Beach area is to the south, driven mainly by the northeaster storms experienced during the winter months. It is during these times that the longshore current is strong and southerly directed, and it is also during these times that the most sediment is suspended in the water column. It is recognized here that much of the volume changes observed to the north of the Reef are influenced by other structures in the nearshore zone, and the losses seen in that region in Figure 4.2b may not be directly attributable to the presence of the Reef. The magnitudes of the changes seen leeward of and south and landward of the Reef, however, indicate that these are attributable, to some degree, to the Reef.
Definitive field measurements of the current patterns are not available due to the measurement intensive nature of the problem. The wave gages in place at the site provide only point measurements of the currents on either side of the Reef. These gages primarily provide an indication of the longshore current. Current data have been compiled for both gages since October, 1993. The data suggest that the longshore current is actually slightly higher during the summer months, ranging between 0.3 to 0.6 ft/s and directed most often to the north. During the winter the magnitude of the current is typically decreased to approximately 0.3 ft/s and directed frequently to the south due to northeast storm conditions.




64
It appears that there exists a natural northerly current at the P.E.P. Reef location in Midtown Palm Beach. This current is associated with the Gulf Stream, which flows closest to the Florida Coast along the Palm Beach County shoreline. During the summer months when the mild wave climate does not generate a substantial longshore current, this current predominates. However, during the winter northeast storms, the longshore currents generated by wave action overcome the northerly currents. The result is a net southerly directed longshore current of lesser magnitude. Again it is noted that the times of highest suspended sediment concentration are during the strong winter storms, which results in the net southerly longshore transport.
Limited dye and drogue studies conducted at the Palm Beach site have not documented the presence of an increased current in the lee of the Reef. It is suggested that the: pumping currents hypothesized herein exist as secondary flows, flows whose magnitudes are substantially smaller than the longshore current. Secondary currents have been documented, particularly in flows around bends, such as river flows. There the result of the secondary flow is to accumulate material on the inside of the river bend, which results in meanders and ultimately in the formation of ox-bow lakes when the bend in the river becomes too severe. The presence of a secondary flow in the lee of a submerged barrier manifests itself in an increase in the magnitude of the velocity of the longshore current in the direction of flow, and a decrease in the magnitude of the longshore flow where the flow is directed updrift. The increased velocity of the current can act to increase the amount of sediment transport over that of the natural longshore current.
Again, no direct field measurements support the existence of the pumping mechanism. Depositional features observed during the monitoring program of the Midtown Palm Beach installation suggest that such a mechanism exists, although perhaps as a secondary, or minor, current, prompting research into the problem.




65
4.2 Laboratory Investigations
One of the objectives of the laboratory study described in Chapter 2 was to verify the existence of the pumping current proposed from the field work. The laboratory study was conducted primarily to document the flow patterns around various planform arrangements for the Vero Beach, FL, P. E. P. Reef design, including any pumping currents that could lead to erosion problems. This section details the results of the laboratory experiments and the verification of the presence of a pumping current. It is noted that the pumping current, while of considerable interest, was not the only focus of the lab study regarding currents. The study was conducted to document the entire current field around the model structure, including the possible occurrence of rip currents.
As described in Chapter 2, bottom-moving drogues were used on a gridded floor in the wave basin to trace their trajectories and calculate average velocities along their paths. Drogues were placed both in the 'nearfield region,' six inches or less from the Reef model units, and further away in the 'farfield region,' extending all the way to the beach toe in the wave basin.
4.2.1 The 'Pumpviniz Current'
The laboratory study demonstrated the presence of an alongshore flow directed to both ends of each Reef arrangement tested. Since the waves used in the study were all normally incident on the beach, this current is appropriately considered to be a pumping current. Both dye streaks and drogues were used to document this phenomenon. Figure 4.3 shows the trajectories of the bottom drogues for a test in which a continuous Reef (Type A case) 33.75 feet long was placed in a water depth of 12 cm (0.037 ft), creating a freeboard ratio of 0.0 (crest at Still Water Level, SWL). The drogues placed in the lee of the Reef were all carried to the ends and offshore. The laboratory study used sidewalls that directed the flow offshore; in the field the




67
Reef effects as the water depth increased. At a freeboard ratio of -0.6, the deepest case tested, the velocities were indistinguishable from the control cases.
Changes in the planform arrangements affected the magnitudes of the currents around the ends of the Reef. In the Type B Case where three separate segments were used, at f/h = 0. 0 pumping patterns were observed around each segment. The velocities around each segment were slightly reduced, averaging approximately 0.4 ft/s prototype at each exit plane, with slightly higher velocities noticed at the ends of the entire test where the flow area was restricted near the side-walls.
The Type C case provided little improvement over the continuous Reef case, with exit plane bottom velocities measuring up to 0.52 ft/s at f/h = 0.0. The Type D case offered some reduction of the pumping currents, indicating velocities of up to 0.40 ft/s. The Type B and Type D cases provided gaps parallel to the shoreline which provided some offshore relief of the mass transport accumulating in the lee of the structure. The difficulty with this type of arrangement is that it leaves stretches of the shoreline unprotected from incident wave attack, which can lead to an irregular shoreline.
The last three cases tested, Types E, F, and G, were performed as compromise arrangements between the need for wave attenuation and the desire to limit the magnitude of the pumping currents induced. The Type E case further investigated the offset case, Type C, to determine if a different cross-shore separation distance would promote offshore flow between the segments, thus reducing the pumping currents. All three types tested after the initial tests were tested at a freeboard ratio of 0.0 in order to clearly demonstrate the Reef effects. Increasing the separation distance to approximately 60 feet in prototype appeared to slightly reduce the induced pumping currents, to roughly 0.4 ft/s prototype at the exit plane. Increasing the separation distance to 90 feet appeared to provide no additional benefit.




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The Type F case was investigated to further promote offshore flow. This was done by shortening the lengths of the offshore segments. As a result small gaps in the Reef were created parallel to shore, but the lengths (12 ft prototype) were not long enough to leave the shoreline exposed to unattenuated waves due to a mechanism termed herein as bridging (to be discussed). The results of this test reduced the pumping currents slightly in comparison to the continuous Reef case, from roughly 0.52 ft/s to just under 0.4 ft/s in the most extreme cases.
The Type G case was an extension of the Type F case. The Type F case presented some improvements in design including protection of the entire shoreline in the lee of the Reef from wave attack. The Type G case was conducted to determine if longer individual segments would increase the currents over that of the Type F case. It will be shown later that the magnitude of the pumping current is approximately proportional to the length of the barrier segment. Lengthening of the segments reduces the cost and time required for installation. Longer segments of units would require fewer end tie-downs and less time to install. This test produced currents of the same magnitude as the previous case, reaching a maximum of just under 0.4 ft/s prototype.
Table 4.1 summarizes the exit plane velocities measured for the seven arrangements at a freeboard ratio of f~h = 0.0. It was observed in all tests that reducing the freeboard ratio (from 0.0 to -0.2 to -0.4, etc.) produced similar patterns of lesser magnitudes. This trend continues until the effect of the structure becomes indistinguishable above the control behavior of the basin. Table 4.1 indicates that in all cases the current patterns flowing from the middle of the lee of the structure to the ends are of similar magnitude, and that certain arrangements provide small reductions in the pumping currents. The cases where significant gaps are left in the planform (Types B and D) allow for some offshore flow along the line of the structure, but again they leave stretches of the shoreline unprotected from wave attack. In these cases, pumping currents of slightly smaller magnitude were observed as well.




69
Table 4.1 Exit Plane Velocities for Seven Test Arrangements. f/h = 0.0, Types are
Illustrated in Figures 2.6 and 2.7.
Case IArrangement [Prototype Exit Plane Velocity, t/
A 45 Continuous Units 0.52
B Three 11 Unit Segments 0.40
C 45 Staggered Segments 0.52
D Five 5 Unit Segments 0.40
E 9 Unit Segments 0.40
F 9 & 7 Unit Segments 0.40
G 18 & I1 Unit Segments 0.40

In all cases where the wave heights were reduced by the structure to a noticeable degree (transmission coefficient, Kt of 0.9 or lower) the pumping current was observed. In these cases, the barrier appears to block a sufficient percentage of the return flow and divert it alongshore, creating the pumping currents. No one test arrangement appeared to be a clear solution to the problem but each represented some degree of compromise between the wave attenuation and the generation of structure induced longshore currents, which are inextricably linked.
4.2.2 Other Current Effects
The laboratory study also indicated other current patterns and effects of interest. In particular are two items: 1) The presence of a 'bridging effect' between barrier segments, and 2) The lack of a strong near-bottom return flow in the lee of the barrier. These two aspects are important in both the actual and perceived performance of a submerged breakwater.
The bridging effect mentioned previously appears to be a combined wave diffraction effect from both segments on either side of a gap as the waves pass through the gap. The result




70
of this bridging mechanism is to provide some degree of wave attenuation in gaps between segments and to create flow channels between segments. In cases where the barrier segments were sufficiently close together, no significant offshore flow between segments was observed and no reduction of wave attenuation was measured. The presence of the bridging mechanism prompted the investigation into lengthening the cross-shore separation distances enough to just overcome the bridging and provide some offshore relief of the pumping currents along the line of the barrier. This bridging mechanism may prove to be beneficial in reducing the likelihood of rip-current generation by such a structure, which is a concern along any recreational beach. However, this bridging mechanism has been observed in the laboratory only, and does not negate the need for concern about rip-currents in the field.
The second interesting observation regarding the current patterns was the lack of a nearbottom return flow in any of the laboratory tests. In both the field and the laboratory, upwelling of currents is observed over the line of the structure, particularly during the passing of wave troughs. This upwelling has been attributed to the diversion of the seaward return flow up along the landward face of the structure. The upward flow is then credited with blocking any seaward transport of sediment from the lee of the structure. In two dimensional laboratory studies of submerged barriers, the longshore flow component is obviously not present, and the return flow must be forced back offshore over the barrier. The three dimensional tests conducted as part of the Vero Beach study do indicate this upwelling, however, the belief that the upwelling would prevent a substantial offshore loss of sediment is not supported by the lab findings. In very few instances was a bottom flow observed to be moving offshore in the lee of the Reef. This would suggest that the upwelling patterns seen in the field are localized effects generated by the transfer of momentum of waves passing over the barrier. The momentum transfer to the water column generates an eddy just landward of the barrier, and it is this eddy that appears as the upwelling




71
of water and sediment. This indicates that sediment suspended over the barrier comes primarily from the nearfield region close to the barrier. Further away from the structure in the lee of the structure, most bottom flows (those that would carry the most sediment) were directed onshore or alongshore with the pumping current. Neither of these would carry sediment into the upwelling region.
4.3 Analytical Approach
Having verified the existence of a pumping mechanism in the laboratory, it is desirable to attempt to develop some analytical basis for both the existence and the prediction of such currents. This section explains the basis for the generation of these currents and presents simple analytical approaches for predicting the magnitude of such flows.
4.3.1 Explanation of the Problem
The pumping current discussed previously has been observed in a laboratory setting and is proposed as the mechanism responsible for some part of the deposition patterns measured at the idtown Palm Beach P.E.P. Reef Installation. A physical explanation of the generation of these currents is as follows. As waves propagate toward a beach, they transport a net mass of fluid in the direction of wave propagation. In an Eulerian perspective, this net mass transport exists in the upper portions of the wave water column between the trough and crest elevations of the wave. Obviously, this net mass transport in the upper portions of the water column must be balanced somewhere by an equal mass transport directed offshore, otherwise the water level on the beach would rise continually which is an unrealistic situation. In the absence of strong three-dimensional effects, this offshore directed flow is termed the return flow and is usually




73
water surface elevation; the development was a two dimensional representation only. In three dimensions, however, the effect of a ponding level in the lee of a finite-length structure is to drive flows in the direction of the longshore water surface elevation gradient. This ponding level, or superelevation of the water level, would be highest at the longshore centerline of the structure and approximately zero at the ends of structure, and is consistent with the pumping currents discussed above.
A simple representation of the three dimensional problem is shown in Figure 4.5. The figure shows a schematic of the wave basin used in the experiments previously described. The arrows in the figure indicate the transport over the structure along its length, and the relief of the 'ponded water' around the ends of the structure. In the laboratory, sidewalls on the basin forced all the relief flow to eventually be directed offshore. In the prototype, this pumping current can either be directed offshore or continue alongshore after reaching the ends of the structure (with a subsequent reduction in magnitude, analogous to an expanding jet discharge).
P.E.P. Reef Model
/.dMM

Figure 4.5 Schematic of Wave Basin and Pumping Currents.

1:8 Gravel
Beach

Paddle
Wavemaker




0.15 1
Type A Case, f/h =0.0
0 1 0 -- -- . .
7. Diverted
4_1 .01~1007.
0. 5 ........................ ......... ........
.... ..*..207.
0.00. . . .
0 4 8 12 16 20
Distance from Structure Centerline (ft) Figure 4.6 Average Exit Velocity vs. Length of Structure. Continuous Structure, f/h
0.0. Figure Applicable for Model Conditions.
smooth concrete bottom, while the remaining area is over the rough gravel beach. ][f the total flow were accordingly confined within 86% of the exit plane area, the average velocity in this case would be 0.09 ft/s, which is still lower than the measured velocity. Another possible explanation of this difference is that this method assumes the flow to be distributed uniformly over the entire exit plane, while the actual velocity distribution is most likely non-uniform, and may well have a local maximum shifted toward the structure.
Seelig and Walton (1980) presented a simple method for the calculation of the average velocity through the exit plane of a submerged barrier based on the ponding level. This method simply used a Bernoulli equation statement relating the difference in head between the lee of the breakwater and the seaward side to the velocity generated. The length of the structure was not




76
considered in the development. Using this method for the laboratory situation above, Seelig and Walton predict an average exit velocity of 0.16 ft/s, twice that predicted by the simple mass conservation method. Seelig and Walton also emphasize that the value calculated is an average velocity and that local velocities around the structure could be higher.
4.3.3 Momentum Conservation Approach
Using a momentum balance approach, the velocity profile along the line of the structure and the water surface elevation, or ponding level, can be determined. Longuet- Higgins (1967) presented a momentum approach to the determination of the ponding level that relied on knowledge of the transmitted and reflected wave heights. This two dimensional model represents a limit of the increase in water surface elevation since it does not include any three dimensional effects, such as the driving of currents due to the elevation change. Using the laboratory scales discussed above, and assuming conservation of energy, shown in equation (4.2)
1. = Kr2 + K:2 (4.2)
Longuet-Higgins' approach results in a ponding level of nearly 40% of the incident wave height (4 cm (0.012 ft)). The value of Kt was taken from the laboratory measurements for the calculation. If this ponding level is used in a simple channel flow manner to compute an average velocity at the exit plane of the structure, the resulting velocity is over 2.0 ft/s, clearly an unrealistic value for the model. This suggests that the maximum ponding level predicted by Longuet-Higgins is never attained for the finite length structure of the laboratory experiments.
To approach the problem in a simple three-dimensional manner, the structure is modelled as an open channel loaded with a uniform lateral inflow. The lateral inflow is taken to be some percentage of the wave mass transport over the structure, which is assumed to be invariant along the line of the breakwater. Solutions for such flows exist in open-channel flow texts, such as




77
Henderson (1966). Figure 4.7 shows a simple schematic of the approach to the problem; the perspective in the schematic is an elevation view looking seaward from the shoreline. The lateral inflow is taken to be the wave volumetric transport and is uniform over the length of the structure. The approach uses a local control volume (the dashed box in Figure 4.7) to consider the local change in momentum, M, across the control volume.
Wave Transport Lateral Inflow
iF + + + + + + i
MWL
flo Q :Q + d
S'f:
IO 0
Figure 4.7 Schematic of Lateral Inflow Model to Determine Water Surface Profile.
If the balance of momentum is written and the bed resistance and slope are considered, the following equation (4.3) results:
yAM + yAbAz = --r0,PAx (4.3)
where the first term on the left hand side denotes the local change in momentum, the second term includes the bottom slope (0 in this case), and the right hand term includes the bottom friction effects. If this equation is written in differential form and the derivative taken with respect to x,




the following, equation (4.4), results:
-2q 2
dh S -gA 2x (4
dK~~ q2B(4)
gA 3
where dh/dx = local water surface gradient, So bottom slope (taken to be 0 in this analysis), Sf= friction slope, q = volumetric flow rate, A = cross-sectional area, and B == width of channel (distance from structure to shoreline). The friction slope, Se., is taken to be
Sf U u2P (4.5)
Cf2A
where u = average velocity at cross section, P = wetted perimeter of channel, and C = the Chezy coefficient (calculated as the hydraulic radius divided by the Manning coefficient to the 1/6 power).
Using this method, the profile of the water surface can be calculated, similar to backwater curve calculations, starting from a known elevation and moving the control volume stepwise 'up the channel' to the structure centerline. In this case the known elevation is taken to be the elevation of the water surface at the end of the structure, where the elevation must return to the ambient level. Figure 4.8 shows the alongshore profile of the water surface elevation in the lee of the structure tests in the laboratory. The profile indicates a minuscule centerline water surface elevation of only 0.0002 ft, but this elevation difference taken over a length of 16.9 feet produces an average exit velocity of 0.07 ft/s (for 100% divergence of the return flow). If this elevation head is used in a simple Manning's equation manner as a check (with a linear water surface slope from centerline to end), the average velocity calculated at the exit plane is 0.08 ft/s, which is in line with previous calculations and the measured results for bottom velocities.




ft X 104
S0.5 Water Surface Elevation --~0.01 .
0 4 8 12 16 20
Distance from centerline (ft)
Figure 4.8 Water Surface Elevation Profile Calculated for Laboratory Scale, 100% Flow
Divergence. H = 0.04 cmn (0.012 ft), T = 2 s.
The proportions of the graph present the appearance of a substantial discontinuity in the water surface at the end of the structure, however this difficulty is attributed to the lack of an exit loss in the development. Similar to a jet discharge, the flow at the end of the structure will have an exit loss that will extend the superelevation of the water surface beyond the end of the structure and will remove the discontinuity. The water surface slope at the centerline of the structure is zero, reflecting the stagnation point along the breakwater where the flow must be diverted to one direction or the other, since in the absence of other effects the flow must be symmetric.
Figure 4.9 shows the water surface elevation profiles calculated for the P.E.P. Reef prototype. Using a wave height of 3.28 ft. (1 .0 m) and 6 second period, the elevation profile along 2,000 feet of Reef is determined for various percentages of diverted return flow. The centerline elevations range from 0.30 ft. for the 100% diverted case to 0.01 ft. when only 20% of the return flow is directed alongshore. Again the profiles behave parabolically, decreasing to zero at the control depth, taken as 9 feet for this example outside the lee of the Reef. For




80
0 .4 . . . .. .. .. . . . . ..
% Diverted
- 20Z
0 ... ...Vi....... 407t 40.
---- 607
----- 807
------------------ t.lf/
0 .1 .1. ............... . '
. ........................................ "......
......0.95 ft .......................... .-0.48 .. . .. ....../.. .
0 .0 ..
0 250 500 750 1000 1250 1500 1750 2000
Distance from Reef Centerline (ft) Figure 4.9 Ponding Levels For P.E.P. Reef Prototype Scale. H = 3.28 ft, T = 6 s.
Velocities Shown Are Exit Plane Values (x = 2,000 ft.).
comparison, the method by Longuet-Higgins (1967) predicts a maximum setup of 0.27 inches. This would indicate that the Midtown Palm Beach Installation is sufficiently long to reach the 'potential setup' predicted by the two-dimensional momentum approach. The difficulty in both approaches presented herein is that the resulting exit plane velocities seem too high to be realistic. The exit velocities calculated for each percent divergence case are noted in Figure 4.9. The lowest velocity (for 20% divergence) is 0.48 ft/s. In the Longuet-Higgins approach, the set-up predicted by a simple Manning's equation calculation would result in an exit velocity of roughly 0.6 ft/s, which is on the order of magnitude of the longshore current measured in the area. Such a high current would definitely be noticeable in the field. Such a high current has not been observed by the UFCOE monitoring team nor by lifeguards patrolling the area. It is noteworthy to recall that with the low freeboard ratio of the Midtown Palm Beach Installation (roughly -0.6), the percentage of the flow that is diverted alongshore is most likely closer to 20 to 40% and certainly not 80 or 100%.




81
Several possibilities exist for the discrepancies in the velocity calculations, One assumption made in the model is that of linear wave mass transport. Obviously waves in the prototype are irregular and non-linear. The linear assumption can lead to an overestimation of the mass transport of as much as 15 % depending on the non-linear representation used to describe the mass transport and the nature of the waves. Another possibility is the relief of the return current in the upper levels of the water column. The mass transport profile shown in Figure 4.4 is a simplistic view of the profile. It is possible that the return current is relieved in part at the surface, creating a situation in which the incoming waves are propagating against an adverse current. This phenomenon was occasionally observed in the laboratory via the use of dye streaks. However, this behavior was not observed on a consistent basis, nor was the effect widespread in the laboratory basin. The result of this possibility would be an even smaller percent divergence of the return flow, leading to smaller, more realistic exit velocities at the ends of the structure.
4.4 Comments
The current patterns generated by a submerged breakwater in the nearshore zone have been investigated, both experimentally and analytically. The purpose of the investigation was to verify the patterns and provide some means of predicting the magnitude of the currents generated in the vicinity of such a structure. The existence of a 'pumping current,' hypothesized from field data, has been verified in the laboratory. Currents that flow from the centerline of a submerged breakwater to its ends are most likely a contributing factor to the erosion patterns seen at the &dtown Palm Beach Installation. Analytical approaches to predicting the magnitudes of these currents yield reasonable results in comparison to laboratory data, however, these methods overestimate the structure-induced longshore currents in the prototype. The analytical models and




82
previously published literature do indicate that there is a fundamental hydrodynamic basis for such currents. A fundamental question arises as to how to accurately model the amount of volumetric flow diverted by the barrier in the prototype, where measurements of the velocities along the line of the structure are unavailable.
From a practical standpoint, the designer of such a structure needs to be aware of the potential problems associated the generation of the pumping currents discussed herein. The loss of a substantial amount of sediment in the lee of a submerged breakwater significantly compromises its main objective, which is the protection of the beach and upland development behind it. If the beach in the lee of a structure has very little or no sand, such as the idtown Palm Beach Installation, the smaller waves that pass over the structure, while having been reduced in size and energy, may still produce a similar amount of damage to the shoreline. The structure and the sandy beach act as a system to provide defense of upland developments.
The results of the field monitoring and the laboratory suggest that segmenting and offsetting a submerged breakwater installation may provide some relief from the structure induced longshore currents. It is, however, not possible to separate wave attenuation over such structures and the generation of these currents. Each design must anticipate the wave climate and the desired amount of wave attenuation, and plan appropriately to mitigate the currents created. Further recommendations can be found in Chapter 5.




CHAPTER 5
CONCLUSIONS
Submerged breakwaters have become topics of considerable interest in coastal engineering in recent years. Reactions to beach nourishment regarding environmental concerns and the expense of periodic renourishment have turned the attention of coastal engineers to more permanent solutions to the beach erosion problem. Interest in the state of Florida has been high in light of two experimental field projects currently in place in the state and a third project under consideration. While data exist regarding the performance of these structures in some aspects, comprehensive performance data and field results are sparse. The purpose of this study was to provide general design guidance for the installation of a submerged breakwater, particularly a narrow-crested breakwater, such as the Midtown Palm Beach Prefabricated Erosion Prevention (P.E.P.) Reef.
5.1 Midtown Palm Beach Installation
Field data from a comprehensive monitoring program of the Midtown Palm Beach P.E.P. Reef installation have been collected and compiled for one year of the three year monitoring program. Wave data collected during that time indicate a 15 to 35% reduction in significant wave height between wave gages located offshore and onshore of the P.E.P. Reef. Additional wave gages installed at the site verify the amount of wave height reduction calculated by the permanent wave gages and provide an indication of the background reduction provided by wave breaking. This reduction would occur irrespective of the presence of the Reef. Control




86
published by Ahrens (1987) which compared well to these data. The data indicate that a submerged barrier must occupy at least 80% of the mean water depth in order to effect a transmission coefficient of less than 90%. Any transmission higher than that would be ineffective in providing substantial protection to the shoreline. The arrangements with full coverage of the shoreline (no large longshore gaps) provided the best wave height reduction along the entire shoreline. Wide gaps along the shore would allow unattenuated waves to pass through the gaps and strike the beach with more energy than the stretches protected by the structure. The result would be an irregular shoreline, with more erosion in the unprotected areas.
It was found in the laboratory that the wave height reduction by a submerged breakwater was directly linked to the generation of pumping currents in the lee of the structure. The blockage of return flow and the transfer of momentum into the water column in the lee result in an elevation of the water level which drives a flow from the center of the structure to the ends of the structure. The linking of these two mechanisms prompted the desire to find a compromise solution. Different planform arrangements were tested to determine an arrangement that would reduce the incident wave heights and minimize the longshore pumping currents.
The laboratory studies were unable to provide an "ideal" solution to the problem in which the wave heights were attenuated sufficiently and no significant adverse currents were generated. Some compromise solutions were recommended, but no solution reduced the wave heights and simultaneously removed the pumping currents. Recommended values of relative freeboard were determined to provide at least a ten percent reduction in incident wave height, and planform suggestions were offered to offset the longshore flow generated by the structure. To reduce wave heights sufficiently, a relative freeboard ratio, f/H, of -1.3 or greater is recommended. This value must take into account any settlement of the structure that may increase the freeboard. To offset the pumping currents generated by the structure, the structure should be constructed in segments so as to provide some means of offshore return flow along the line of the structure.




87
By offsetting the segments of an installation, gaps are created perpendicular to shore which allow for offshore return flow while still protecting the entire stretch of shoreline. The concern in placing gaps along the structure is the generation of potentially dangerous currents. VVhile no rip currents were observed in the laboratory, this does not mean a rip current could not occur along the segments of a submerged barrier. However, the premise of segmenting and offsetting a submerged breakwater is to reduce any structure induced increase in the magnitude of the existing current. Another effect that might prevent rip current generation is the presence of a bridging mechanism between segments. This mechanism was observed in the laboratory and is believed to be responsible for preventing stron offshore flows between segments.
The goal then is to provide gradual relief of the structure induced currents along the entire length of the structure so that no significant amount of sediment is transported away from the area and no strong currents are generated by the structure anywhere along the project. An arrangement of offset segments was found to be the most promising compromise soludon tested. The offshore segments were set back roughly 45 to 60 feet (prototype) from the line of the onshore segments. In addition the segments were kept as short as possible to reduce the magnitude of the pumping currents generated initially. Segment lengths of no longer than 216 feet (prototype) were tested in the laboratory. It is recommended that the lengths be kept as short as possible, although the economic feasibility of the actual lengths must also be assessed.
5.3 Analytical Models
Analytical models were developed to describe and predict wave transmission over submerged narrow-crested breakwaters and the pumping currents generated by the wave height reduction. Both models use simple hydrodynamic approaches and both are subject to the assumptions made in their development. The goal of the models was both to predict the




88
performance of a submerged breakwater and to provide a fundamental explanation for the interaction of such structures with incident waves.
The wave transmission model described herein uses both a Bernoulli energy balance approach and a weir flow approach to predict the transmission coefficient for waves passing over barriers that are completely submerged as well as partially submerged. The two approaches are patched together near the mean water level, although their combined effects are not considered. The transmission over the structure resembles the Bernoulli approach when the barrier is completely submerged. As the seaward water surface drops below the crest of the barrier, the flow over the barrier becomes intermittent and resembles a weir flow for par-t of the wave cycle. As the structure becomes more exposed, the flow begins to behave more like a wei r flow. In addition, the flow through the holes of the P.E.P. Reef was found to contribute substantially to the transmitted wave height.
The model predicts the transmission coefficient fairly well for small values of relative freeboard, when the barrier is completely submerged. As the barrier occupies more and more of the water column, the prediction worsens. It is recognized that this model is a steady flow model of an unsteady flow situation. No acceleration effects are considered. These acceleration effects would cause convergence of the flow over the barrier, resulting in more transmission and higher wave heights on the leeward side. This problem becomes more important as the barrier occupies more of the water column, as shown in the results (Figure 3.13). Figure: 3.13 also indicates that the laboratory results from Ahrens (1987) agree well with the P.E.P. Reef model tests. This would indicate that the holes in the P.E.P. Reef design simulate the effects of rubble mound structures, providing porosity and increased transmission.
A model was also developed to predict the increased water surface elevation and pumping currents seen in the laboratory and proposed in the field. The model depicts the problem as an




89
open channel flow loaded with a lateral inflow. By solving the local momentum balance along the channel the water surface profile can be determined. The longshore velocity profile of the pumping currents is determined through the conservation of mass of the wave mass transport over the structure. The model predicted the pumping velocity in the laboratory setting fairly well when the structure blocked a substantial amount of the water column. As the barrier becomes more and more submerged, the question arises as to how much of the seaward return flow is diverted alongshore when the structure does not block most of the water column. This seems to be of particular interest in the field case, where the pumping current velocities predicted by this method are unrealistically high. It is noted that an increased current in the lee of the P.E.P. Reef has not been observed in the field.
It is proposed that the pumping current exists in the field as a secondary current, one whose magnitude is much smaller than the longshore current. The current is not observable in the field, but augments the longshore current and contributes to the sediment transport, carrying sediment from the area in the lee of the Reef to the ends. This is supported by the survey data of the volume changes seen in the idtown Palm Beach installation. It has yet to be established how much of the seaward return flow is diverted into this pumping current.
5.4 Design Considerations
The monitoring of a field installation as well as laboratory and analytical investigations have provided general design data for use in future submerged breakwater installations. These design data provide general information to be augmented by site specific information and numerical models. It is important to recognize the site specific nature of coastal engineering projects and attempt to consider the site conditions as much as possible.
Hall (1939) concluded that in order for a submerged breakwater to have a substantial effect on the wave height transmission the barrier must occupy at least 80% of the mean water




APPENDIX A
SIGNIFICANT WAVE HEIGHT AND PERIOD DATA FROM SEADATA GAGES
Consistent with the Florida Coastal Data Network format for wave data, units are given in meters (1 meter = 3.28 feet). The time span for the Seadata test period was 21 September to 24 October 1994 (Julian days 264 to 297).




1.5
1.0 0.5
0.0
260 Figure A.1

0

260 Figure A.2

1.5
1.0
0.5 0.0
2
Figure A.3

0-

60

260 260

265 270 275 280 285 290 295 300
date
Offshore Seadata Gage Significant Wave Height, Julian Date, 1994.

265 270 275 280 285 290 295 300
date
Offshore Seadata Gage Modal Period, Julian Date, 1994.

A Seadata Reef V Seadata Control
22
-V

265 270 275 280 285 290 295 300
date
Nearshore Seadata Gage Significant Wave Height, Julian Date, 1994.

265 270 275 280 285 290 295 300
date

Nearshore Gage Modal Period, Julian Date, 1994.

SSeadata Concurrent Seadata Control
, i i i i i i , , i i i ,0, ,

7 A Seadata Concurrent
y Seadata Control
, i i , i i i I i , I i I i b b ,

Figure A.4




Figure B.1

Definition Sketch of Bernoulli Principle Theory.

If the difference in elevation of the water surfaces on either side of the barrier is taken as the driving head for a flow over the barrier, equation (B.2) defines the resulting velocity, u. This velocity is linearized with the coefficient, A.

u = /2g(lj+1,-ntl) = As(11i+r-11t)

(B.2)

The linearization constant can be determined in many ways. Here it is found by equating the maximum velocities of the two velocity descriptions. The maximum velocity occurs when the cosine portions of the terms of equation (B.2) equal unity. Each term is then divided by the incident wave height, Hi, producing a relationship in terms of the reflection and transmission coefficient. Equation (B.3) gives the value of the linearization constant:

(B.3)

As g2
-2 (I+K -

Hi Hr Ht
7 MWL --f _. flow (q)
h
Seaward[




96
where Krt represents the reflection and transmission coefficient, respectively. The linearized velocity expression is then integrated over the freeboard above the barrier to obtain a volumetric flow rate, shown in equation (B.4).
f 2 udz (B. 4)
The resulting flow rate is then:
q = As[-f(1+ d-1]) + (Ili+11r) 2 12 (B.5)
2 2
If the water surface on the transmitted side is composed of an average and a fluctuating term, the flowrate then is represented by a steady term and oscillating terms at both the incident frequency and the second harmonic of that frequency:
H. 1t
Aq =A(-f = [ (l+K,-Kt) cos ((;t) 2rl]
Hi (B.6)
+ As H2 [ (l+K,) 2Cos(at)-(2 -L- + KtCOS(at))2]
8 H1
It is then assumed that the contribution from the mean elevation terms and the second harmonic terms is neglible. Therefore only the first term in brackets from equation (B.6) is retained. This flow rate is equated to the discharge on the transmitted side, assuming shallow water wave conditions, as indicated in equation (B.7).
q = As(-f ) (1+K -Kt) cos(at) = KV1ghCOS (at) (B.7)
2 2
At this point it is necessary to invoke conservation of mass in order to relate the transmission coefficient to the reflection coefficient and remove Kr from equation (B.7). To do




97
so, the flow in and out of a narrow control volume surrounding the barrier and water surface is considered. Equating the flow on both sides produces equation (B.8).
u+u t = H )cos(at) = -jcos(at)
2 2 2 (B.8)
1-K, = Kt
Substituting the result of equation (B.8) into equation (B.7) and solving for K, yields the implicit expression for the transmission coefficient (note f < 0.0):
1
K t -- I H 1L ( t)( .9
4f Hi1
The flow rate found in equation (B.6) describes the flow over the structure. To include the effect of the flow through any openings, an additional term is added to that equation. The flow through the holes is calculated exactly as the flow over the submerged barrier since in this development only completely submerged openings are considered. This is done to include the three openings in the P.E.P. Reef modules, which are nine inches high in prototype and cover 46% of the longshore length of each unit. This 46% is included in the area calculation along with a contraction coefficient (roughly 0.6). The effective area of the holes, A., replaces the freeboard area over the barrier in that calculation. Both the hole area and the freeboard area have units of length (per unit width basis). The total flow is then equated to the flow on the transmitted side, assuming shallow water conditions. The full expression for Kt is then given by equation (B.10).
1
K; = + Hi 1 (B. 10)
4 (Av-f) 2 Hi-)




98
B.2 Partially Submerged Barriers
When the water reaches an elevation where the incident/reflected wave trough is at or below the structure crest, the barrier becomes only partially submerged, and the approach used above is not as valid. In this section partially submerged barriers will be addressed. The approach taken here will be that of intermittent critical flow over a sharp-crested weir. The flow associated with any holes in the barrier will be incorporated in this development as well. A definition sketch of the approach is shown in Figure B.2
Consider the specific energy in the critical flow of water over a sharp-crested weir as shown in equation (B.11). The derivative of the specific energy, set equal to zero, determines
dE, = 1- = 0 (B.11)
dhc gh C3
the value at which the flow becomes critical, that is, no information downstream of the barrier

Definition Sketch of Weir Flow Theory.

7tt
"3i+ T r / weir ___W
A [ 'flow (q)
h
award barrier
IFIII/11111111111//11/11111 SeawardIIIII

Figure B.2




99
is transmitted upstream to affect the flow. Here, hC is the critical depth of water over the crest from the crest elevation to the water surface. Solving for q in the previous equation produces equation (B.12):
q = (gh3) 2(B. 12) The energy in the flow is generated by the head of water created as the wave form approaches the barrier, given in equation (B. 13).
E= 'ij'1r,- = 3 C(B. 13)
2
This head is the difference in the offshore side surface elevation and the structure crest. It is recognized that during certain portions of the wave cycle there will be no flow over the barrier, only through any openings. Thie portion of the flow over the barrier is then equation (B.14):
q = 2 3B14
2 (B 14)1i+~rf
3
The linearization constant, A,, is determined here by matching the maximum flow of the two, resulting in equation (B.15).
r- [(+r c2 (t -2- 2 (B. 15)
W 3 2 H1
Previously, the transmitted oscillating component was determined by resolving '0, into steady and oscillating components, then using only the fundamental frequency of the resulting flow rate equation (B.6). In this case, the flow over the barrier can be represented by a Fourier series with components at an infinite number of frequencies. Here the primary frequency component is found and assumed to be much larger than any secondary harmonic components.




100
The limits of integration for the calculation of the first Fourier coefficient are determined from the geometry of the problem. These limits are taken to be the times during the wave cycle when the water surface elevation on the incident side is equal to the crest elevation of the barrier, a conditon defined in equation (B.16) when ni +nr = f.
at. = COS-1 [ 2f (B. 16)
Hi (I +Kr)
The first Fourier component of the flow then becomes equation (B.17).
q = 2 f q,,cos ((t) dt (B.17)
Performing the integration over the time when the water surface is above the structure crest provides the intermittent weir flow over the barrier that contributes to the flow on the transmitted side. The result of the integration is shown in equation (B.18).
qw = -- [1 (1+K,) Cos--( 2f (B.18)
ICW2 2 Hi (1I+Kr)
This intermittent portion of the flow is then added to the flow through any holes in the barrier. The flow through any holes is calculated in the same manner as the flow over a totally submerged barrier, via a Bernoulli argument. The linearization constant for the hole flow, A,, is of the same form as equation (B.2). The addition to the flow due to the openings is given in equation (B.19).
q, = A8Av(1)i+T_1-,1) (B. 19)
Summing the two flows, equating the sum to the flow on the transmitted side, and solving for Kt results in another implicit equation for Kt, shown in equation (B.20). The second term on the right hand side indicates the contribution from the flow through any holes in the barrier.
Kt 2 { Hi ( 2 -K)- (TC)T 2 f 2
-2- -) [ (2-Ke) cos a ____ f 2 TIt 2h3 Hi
2f (B.20)
2 Hi (2-Kt)
+ 2 (2 A (1-Kt)
+ 2 hH (1 -K)




I I I
II
* I i I !I
1I i
* I I
* ~ .3
____________ I I I
I I
S I I I I I
I~It!ii I!
zimv I
___ I i t
Lfi :! hI I S I ______ 11i I
I i i I-.:~z~zjj :
i

Figure C.1 Control and Type A Case,f/h = 0.0. Wavemaker is to the right. Units in ft/s.




.I
rO
IAt
I I
i k i h ig i in fts
, ,,l I L -.'r
i At t U In f
Figure C.2 B and C Cases, flh =0.0. Wavemaker is to the right. Units in ft/s.