US. Sas Gag. Coa Ch Tec. Zp. CERC- 90-12. Jul 1990 . TECHNICAL REPORT CERC-90-12 US Army Corps LABORATORY STUDY ON MACRO-FEATURES Or Engineels OF WAVE BREAKING OVER BARS AND ARTIFICIAL REEFS Barred Profiles by Ernest R. Smith, Nicholas C. Kraus Coastal Engineering Research Center DEPARTMENT OF THE ARMY Waterways Experiment Station, Corps of Engineers 3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199 — re a allie DOCUMENT ™ July 1990 Final Report Approved For Public Release; Distribution Unlimited Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 Under Nearshore Waves and Currents Work Unit 31672 wl, ae Eu Coe Gh Teck. Rp. CERC- 90-12 Jul 1990 : TECHNICAL REPORT CERC-90-12 US Army Corps LABORATORY STUDY ON MACRO-FEATURES ob enaineet= OF WAVE BREAKING OVER BARS AND ARTIFICIAL REEFS Barred Profiles by Ernest R. Smith, Nicholas C. Kraus Coastal Engineering Research Center DEPARTMENT OF THE ARMY Waterways Experiment Station, Corps of Engineers 3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199 DOCUMENT LIBRARY Woods Hole Oceanographic insitiution July 1990 Final Report Approved For Public Release; Distribution Unlimited Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 Under Nearshore Waves and Currents Work Unit 31672 Destroy this report when no longer needed. Do not return it to the originator. The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. The contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. REPORT DOCUMENTATION PAGE 1a. REPORT SECURITY CLASSIFICATION Form Approved OMB No. 0704-0188 Exp. Date: Jun 30, 1986 1b. RESTRICTIVE MARKINGS 3. DISTRIBUTION / AVAILABILITY OF REPORT D 2 ate 2a. SECURITY CLASSIFICATION AUTHORITY 2b. DECLASSIFICATION / DOWNGRADING SCHEDULE 4. PERFORMING ORGANIZATION REPORT NUMBER(S) Approved for public release; distribution unlimited. 5. MONITORING ORGANIZATION REPORT NUMBER(S) Technical Report CERC-90-12 6a. NAME OF PERFORMING ORGANIZATION USAEWES, Coastal Engineering Research Center 6c. ADDRESS (City, State, and ZIP Code) 6b. OFFICE SYMBOL (If applicable) CEWES - CW-P 7a. NAME OF MONITORING ORGANIZATION 7b. ADDRESS (City, State, and ZIP Code) 3909 Halls Ferry Road Vicksburg, MS 39180-6199 8a. NAME OF FUNDING / SPONSORING ORGANIZATION US Army Corps of Engineers 8c. ADDRESS (City, State, and ZIP Code) 8b. OFFICE SYMBOL (If applicable) 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER 10. SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK ELEMENT NO. NO. NO. WORK UNIT ACCESSION NO 31672 Washington, DC 20314-1000 11. TITLE (Include Security Classification) Laboratory Study on Macro-Features of Wave Breaking over Bars and Artificial Reefs 12. PERSONAL AUTHOR(S) Smith, Ernest R.; Kraus, Nicholas C. 13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Year, Month, Day) |15. PAGE COUNT eenieeees rrom_Sep 87 toSep 89 July 1990 232 16. SUPPLEMENTARY NOTATION Available from the National Technical Information Service, 5285 Port Royal Road, Springfield, VA __22161. Us COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number) Breaking waves Wave breaker type Wave plunge distance | i Mi elaboratonysotudy Wave breaker vortex Wave reflection (| | Wave breaker index Wave height deca Wave runup 19. ABSTRACT (Continue on reverse if necessary and identify by block number) A laboratory experiment was conducted in a wave tank to examine macroscale features of wave breaking over bars and reefs. Submerged triangular-shaped obstacles representing bars and reefs were installed on a 1/30 concrete slope to cause wave breaking. Seaward and shoreward slopes of the obstacles were varied, as was the deepwater wave steepness H,/L, , which resulted in 108 monochromatic wave tests and 12 irregular wave tests. Empirical expressions were determined for the wave properties investigated, which included breaker type, height, and depth; plunge, splash, and penetration distance; breaker vortex area; wave decay; wave reflection; and wave runup. Additionally, data acquired from other studies involving plane slopes were reanalyzed to determine breaker indices and plunge distance. Differences were found between wave properties on plane slopes and barred profiles, Plunging and collapsing breakers were predominate for regular waves breaking over bars and reefs, whereas spilling breakers occurred on the plane slope. The strength of return flow (Continued) 21. ABSTRACT SECURITY CLASSIFICATION Unclassified 22b. TELEPHONE (Include Area Code) | 22c. OFFICE SYMBOL DD FORM 1473, 84 MAR 83 APR edition may be used until exhausted SECURITY CLASSIFICATION OF THIS PAGE All other editions are obsolete. WANK CU ANIL 0 0301 0 20. DISTRIBUTION / AVAILABILITY OF ABSTRACT CKXUNCLASSIFIED/UNLIMITED =) SAME AS RPT. 22a. NAME OF RESPONSIBLE INDIVIDUAL Optic users Unclassified Unclassified SECURITY CLASSIFICATION OF THIS PAGE 19. ABSTRACT (Continued) . altered the breaking wave form. Return flow was strongest if the bars were terraced or if the seaward slope f, was steep, and the deepwater wave steepness H,/L, was small. The position of the break point was greatly influenced by return flow, which exerted control over the breaker depth. Breaker height was found to increase in the presence of strong return flow. Plunge distance normalized by breaking wave height was found to be shorter over irregular slopes than plane slopes. The results from the study can be used in numerical models simulating beach shoreline change and in studies of wave height decay. Unclassified SECURITY CLASSIFICATION OF THIS PAGE PREFACE The investigation described in this report was authorized as a part of the Civil Works Research and Development Program by Headquarters, US Army Corps of Engineers (HQUSACE). Work was performed under Work Unit 31672, "Nearshore Waves and Currents," at the Coastal Engineering Research Center (CERC), US Army Engineer Waterways Experiment Station (WES). Messrs. John H. Lockhart, Jr., and John G. Housley were HQUSACE Technical Monitors. Dr. C. Linwood Vincent was CERC Program Manager. The study was conducted from September 1987 through September 1989 by Mr. Ernest R. Smith, Hydraulic Engineer, Wave Dynamics Division (WDD), Wave Processes Branch (WPB), CERC, and Dr. Nicholas C. Kraus, Senior Scientist, Research Division (RD), CERC. This report is substantially the same as the thesis submitted to Mississippi State University by Mr. Smith in partial fulfillment of the requirements for an M.S. degree in civil engineering. Dr. Kraus was the thesis advisor. This study was done under the general supervision of Dr. James R. Houston and Mr. Charles C. Calhoun, Jr., Chief and Assistant Chief, CERC, respectively, and under the direct supervision of Mr. C. Eugene Chatham, Jr., Chief, WDD; and Mr. Douglas G. Outlaw, Chief, WPB. The model tests were conducted by Messrs. Smith, George M. Kaminsky, John Evans, WPB, and Messrs. David Daily and Lonnie L. Friar, Instrumentation Services Division. COL Larry B. Fulton, EN, was Commander and Director of WES during report publication. Dr. Robert W. Whalin was Technical Director. CONTENTS PREFACE . LIST OF TABLES LIST OF FIGURES CONVERSION FACTORS, NON-SI TO SI siamo UNITS OF MEASUREMENTS PART I: INTRODUCTION . Background Purpose Content . PART II: REVIEW OF RELATED STUDIES Breaker Type Wave Reflection . Breaker Index . Plunge Distance Breaker Vortex Wave Height Decay . Wave Runup : Analysis of Peaviious ipaea : Summary . PART III: EXPERIMENT ARRANGEMENT . Facility and Equipment Design Wave Conditions Bar Design Test Procedure PART IV: RESULTS Breaker Type Wave Reflection . Breaker Index . Plunge Distance Splash Distance Penetration Distance Breaker Vortex Wave Height Decay . Wave Runup Wave Steepness Sealing Irregular Waves Measurement Errors PART V: SUMMARY AND RECOMMENDATONS Monochromatic Tests Irregular Wave Tests 6 Recommendations for Future Sesndey REFERENCES APPENDIX A: BREAKER DATA . APPENDIX B: APPENDIX C: APPENDIX. D: TRACINGS OF WAVE FORMS WAVE HEIGHT DATA . NOTATION . Pa ° COMNDAUMUEWNHPH LIST OF TABLES Page Summary of Coefficients and Exponents for Breaker Depth Index... . «: | Jy Spee) cag a 25) Summary of Coefficients and Exponents for Breaker Height Index .. . sel dp ue es a, Gale nice ANE 28 Data Summary for Breaker Tadlese aad) Plunge Distance on Plane Slopes . . . So omc ret coNe. 6 44 Maximum Wave Height Observed in the 18- ia. ‘Tank ipsa cy AE ep Bun iS 63 Range of Wave Conditions . sailie, » Hz, , and H,,, as a Ameen of iiseenee., B, = 10 deg Hnax » H, , and H,,, as a function of distance, B, = 15 deg ‘ POE ee oe eee Predicted H,,,/(H; ye of Goda (1975) (A as a function of measured H,,,/(H;), -. - Predicted (H/H,), of Goda (1975) (A = 0.12) as a function of measured (H/H,), Predicted H,,,/(H;), of Goda (1975) (A as a function of measured (H,,,/H,). H.ms/h as a function of distance, m = 1/30 0.18) 0.09) He as a function of distance, f, = 5 deg Hems as a function of distance, 8B, = 10 deg Hoe as a function of distance, Pe = 15 deg R/(H,), aS a function of € .. segs R/(H,); as a function of €, matin m = 1/30 as the predominant angle R/H,ns aS a function of f, Range between maximum and minimum sree oe Yb a function of H,/L, as Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case Case 2130 2210 2320 2440 4110 4220 4340 4430 6120 6230 6340 6410 8140 8220 8310 8430 10130 at incipient breaking 10210 at incipient breaking 10320 at incipient breaking at at at at at at at at at at at at at at at at incipient incipient incipient incipient incipient incipient incipient incipient incipient incipient incipient incipient incipient incipient incipient incipient breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . breaking . B20 Case 10410 at incipient breaking CONVERSION FACTORS, NON-SI TO SI (METRIC) UNITS OF. MEASUREMENTS Non-SI units of measurement used in this report can be converted to SI (metric) units as follows: Multiply By degrees (angle) 0.01745329 feet 0.3048 inches 2.54 square feet 0.0929 To Obtain radians metres centimetres square metres LABORATORY STUDY ON MACRO-FEATURES OF WAVE BREAKING OVER BARS AND ARTIFICIAL REEFS PART I: INTRODUCTION Background 1. Wave breaking is the most dynamic phenomenon in the nearshore zone. A wave approaching a beach enters shallow water, which causes it to become steeper and shorter. The wave eventually becomes unstable, and the crest either spills down or plunges over its front face, releasing its kinetic and potential energy. The crest of a spilling breaker rolls down the front face of the wave and creates intense turbulence at the surface. A plunging breaker crashes into the water surface shoreward of it, displacing a large volume of water that also plunges shoreward. The vortex of entrained air created by the overturning crest penetrates below the water surface. Plunging and, to a lesser extent, spilling waves create turbulence and currents that mobilize and transport beach sediment. Longshore currents, driven by waves breaking at an oblique angle to shore, transport sediment along the coast. Cross-shore cur- rents generated by breaking waves move sediment across shore to form bars. The point of wave breaking and plunging varies according to the height and length of the incident waves, as well as the beach slope and water level (tidal stage). Thus, bars are constantly changing shape and location, moving seaward in storms and shoreward under calmer conditions. Coastal structures such as breakwaters and jetties are constantly subjected to breaking wave forces. Forces exerted by breaking waves can crack concrete dolosse and displace armor units weighing several tons. 2. Placement of clean dredged material offshore in linear berm config- urations to both protect and nourish the beach has become an area of recent interest (McLellan 1990). Depending on depth and orientation, the presence of offshore berms may alter the incident wave characteristics and influence the breaking wave properties. Although offshore berms have desirable qualities in concept, research in this field has been fairly limited. 3. Breaking waves also provide a recreational environment for surfers. Recently, a surfing reef (shoal) was designed to be placed at Patagonia, Cali- fornia, and the project is in the final stages (Pratte 1989). Walker (1974a) stated the bottom configurations parameters that most influenced favorable surfing waves were the seaward slope, width, and orientation of the shoal to the incident waves. 4. It is clear that knowledge of breaking wave properties is essential in design and placement of coastal structures, as well as for prediction of shoreline movement and beach profile change. Breaking waves have been exam- ined in numerous laboratory studies, but all major experiment programs were conducted on plane slopes. Relationships for breaking wave properties have been developed from such studies, but the presence of a bar or a subsurface structure, such as an offshore berm or an artificial reef, is expected to change the character of breaking. This study departs from previous investiga- tions in that it concentrates on wave breaking over barred profiles with a fixed bottom. 5. Breaking wave characteristics may be studied on a microscale, in- volving basic fluid dynamic properties such as turbulence, bottom shear stresses, and individual water particle velocities and accelerations; or on a macroscale, involving standard engineering parameters such as wave height and period, water depth, and plunge distance. This study concentrates on macro- features of wave breaking over barred profiles in order to reveal functional dependencies and properties of direct coastal engineering interest. Purpose 6. The purpose of this study is to examine the macro-features of wave breaking over bars and artificial reefs. Principal wave properties considered are: a. Breaker type. b. Wave height and water depth at breaking. ce. Plunge and splash distance. d. Vortex area. e. Wave decay. £. Reflection. g. Wave runup. Because this study differs significantly from previous studies of breaking waves by examining the effects of bars on wave breaking, empirical relation- ships developed from measurements will have immediate input to other 10 activities, such as numerical models for simulating beach profile change and studies of breaker wave height decay. Measurements of breaking wave proper- ties over bars and offshore berms and reefs on a microscale are also essen- tial, but are left to future work. 7. The experiment was conducted in a wave tank containing a 1 on 30 concrete slope. Bars and reefs of fixed geometry and constructed of wood were placed on the slope to investigate wave breaking under natural conditions with bars and engineered conditions with artificial reefs. Wave breaking was exam- ined over a range of bar and reef geometries, wave periods, and wave heights for a fixed water level. In nature, breaking waves change the geometry and size of the bar, and these changes feed back to alter the breaking wave char- acteristics. Bars subjected to steady, monochromatic waves and fixed water levels develop an equilibrium profile (Kraus and Larson 1988; Larson, Kraus, and Sunamura 1989; Larson and Kraus 1989). However, in reality bars are subjected to irregular waves and fluctuating water levels, so an equilibrium profile is only approached and never reached. Consequently, bars and waves are continuously interacting with each other. The present study focuses on the control that bars and artificial reefs exert on breaking waves. 8. This study pertained only to macro-features of wave breaking; micro- features of bottom velocity, turbulence, and bottom shear stresses were not addressed. Also, the macro-features of wave celerity, return flow velocity, and the important influence of wind on wave breaking, as discussed by Douglass and Weggel (1989), were not obtained in this study. Although return flow was not measured, qualitative observations and implications of the return flow are discussed in Part IV. Data are available from the experiments to examine wave celerity, but because of the considerable data analysis involved, it is left for future work. Content 9. Part II contains a review on essential properties of breaking waves, including breaker type, breaker indices, reflection, plunge distance, wave decay, and runup. Part III describes the design and execution of the experi- ment, and Part IV presents major results. Conclusions and recommendations for future study are given in Part V. Wave data from previous studies used in the present study and data generated in the present study are listed in 11 Appendix A. Sketches of wave forms from this study are presented in Appendix B, and wave decay data are presented in Appendix C. Notation is listed in Appendix D. 12 PART II: REVIEW OF RELATED STUDIES 10. This chapter reviews previous studies on breaking waves to identify principal variables and trends for comparison with results of the present study. Main topics reviewed are breaker type, breaker index, plunge distance, breaker vortex, wave decay, and runup. Previous studies concentrated on plane sloping beaches; therefore, a background of these topics is necessary to determine the effect, if any, of bars and artificial reefs on the various wave breaking properties. The literature on breaking wave properties is vast, and only the more pertinent studies are included. Original analysis of the previous data is made at the end of the chapter. Breaker Type 11. "Breaker type" refers to the form of a depth-limited wave at break- ing and has an influence on other breaking wave properties. Although there are several classifications of breaker type, it is generally accepted that waves break by spilling (Figure 1), plunging (Figure 2), collapsing (Fig- ure 3), or surging. These photographs were taken during the present study, and the case number is explained in Part IV. Galvin (1968) defined the fol- lowing terminology; spilling breakers occur if the wave crest becomes unstable and flows down the front face of the wave producing a foamy water surface; plunging breakers occur if the crest curls over the front face and falls into the base of the wave, resulting in a high splash; collapsing breakers occur if the crest remains unbroken while the lower part of the front face steepens and then falls, producing an irregular turbulent water surface; surging breakers occur if the crest remains unbroken and the front face of the wave advances up the beach with minor breaking. Breaker type is controlled by the bottom slope and deepwater wave steepness. The deepwater steepness is defined as the ratio of deepwater wave height H, and deepwater wavelength L, , which can be calculated by linear wave theory as: L, = (1) 13 CASE en T= 1.0 SEC H=0.43 FT Hy/bLa = 9.09 Figure 1. Spilling wave CASE 10210 T= 2.5 SEC H = 0.30 FT B,= 5.0 B,=20.0 Figure 2. Plunging wave 14 CASE 10230 Ts 2.5 SEC H=0.30 FT - Figure 3. Collapsing wave where g is acceleration due to gravity, and T is the wave period. Review 12. Patrick and Wiegel (1954) classified breaker type as spilling, plunging, and surging. Patrick and Wiegel observed in the field that breaker type depended primarily upon beach slope m and deepwater wave steepness. They determined spilling breakers occur for large values of H,/L, on flat slopes, surging breakers occur for small values of H,/L, om very steep slopes, and plunging breakers occur between the two extremes. 13. Galvin (1968) introduced collapsing breaker type as an intermediate form of breaker between plunging and surging waves. Galvin defined two param- eters, the “offshore parameter" and the "inshore parameter," to classify breaker type, expressed in terms of the beach slope and wave steepness. Transition values were found to be H, surging-collapsing if < 0.09 L,m? a, (2) plunging if 0.09 < < 4.80 L,m? 15 spilling if > 4.80 L,m? for the offshore parameter, and Hp surging-collapsing if —= < OO mL, Hp plunging if 0.019 < — < 0.427 (3) mL, Hp spilling if —=} | > = —O,427 mL, for the inshore parameter, in which H, is the breaking wave height. 14. Battjes (1975) used the Iribarren number (Iribarren and Nogales 1947), commonly called the surf similarity parameter, to describe breaker type. The surf similarity parameter € is defined as 6 ea (4) where H is wave height. The offshore parameter of Galvin (1968) can be written as €¢52 by specifying H, as the wave height. Battjes converted the transition values of Galvin to values of €, , resulting in surging or collapsing if By 2 Sod plunging if 0.5 < €, < 3.3 (5) spilling if S < 0.9 Battjes defined an inshore parameter €, = m(H,/L,) 1/2? and, by reanalyzing the Galvin data, the following transition values were determined: surging or collapsing if &, > 2.0 plunging if 0.4 < & < 2.0 (6) spilling if &, < 0.4 16 Summary 15. Breaker type is a function of the bottom slope and deepwater wave steepness. Spilling waves occur if the beach slope is flat and wave steepness is large. Plunging breakers are expected for either small wave steepness on mild slopes or large wave steepness on steep slopes. Collapsing and surging breakers occur if wave heights are low in relation to wavelength and slopes are steep. 16. Dimensionless parameters based on beach slope and deepwater wave steepness were developed by Galvin (1968) and Battjes (1975) to quantify breaker type. These parameters are also used to predict other wave breaking properties, such as breaker indices, plunge distance, wave reflection, and runup, as will be discussed. Wave Reflection 17. Lo Jer Lo Lo Lo Jer K, = (8) H, Hy 1 < L, Lo Jer 17 where (H,/L,)~, is the critical wave steepness to obtain complete reflection and is defined as: 1/2 H, 2B) sin? Bp alt ane cian (9) ies T 1 in which f is the slope angle in degrees. 19. Battjes (1975) reexpressed Miche’s (1951) equation to obtain K, as a function of the surf similarity parameter. The resulting equation was: im, = Oe (10) Battjes compared Equation 10 with data collected by Moraes (1971) and found good agreement with the data for breaking ee and €, < 2.5 20. Ahrens (1987) conducted experiments to determine the stability, transmission, reflection, and energy dissipation characteristics of reef breakwaters. Ahrens defined the reef reflection parameter P as —— Gis) where z = height of reef L, and h, = wavelength and depth at toe of reef, respectively cross-sectional area of reef e i Ahrens performed a regression analysis to relate wave reflection to the reef reflection parameter and obtained ae oe eer tee yay Pere (12) (LO & 8, QAO) r The coefficient of determination for Equation 12 was 0.8. Summary 21. Equation 8 (Miche 1951) and Equation 10 (Battjes 1975) include beach slope and deepwater wave steepness to estimate the reflection coefficient. Equation 12 (Ahrens 1987) was developed for reef breakwaters, which differ from the bars and reefs modeled in the present study in that reef 18 breakwaters are larger and permeable, have flatter crests, and are not always submerged. However, the reef reflection parameter relates area and height of the structure to reflection and was considered appropriate to examine for describing reflection in the present study. Breaker Index 22. A schematic of a wave at incipient breaking is illustrated in Fig- ure 4. Breaking is shown to take place over an idealized bar, where lh, is SCALE SCALE C) 2 0 5 ud LJ INCHES CENTIMETERS Figure 4. Definition sketch of wave at incipient breaking the water depth at breaking, 6, is the seaward bar angle, and £3 is the shoreward bar angle. The break point can be defined in several ways. Singamsetti and Wind (1980) listed possible definitions as: a. The point where the wave cannot further adapt to the changing bottom configuration and starts to disintegrate. b. The point where the horizontal component of the water particle velocity at the crest becomes greater than the wave celerity. c. The point where the wave height is maximum. d. The point where part of the wave front becomes vertical. e. The point where the radiation stresses start to decrease. £. The point where the water particle acceleration at the crest tends to separate the particles from the water surface. g. The point where the pressure at the free surface given by the Bernoulli equation is incompatible with the atmospheric pressure. 19 In the present study, after considerable inspection of videotapes of breaking waves, the break point was defined as the point where any portion of the forward face of the wave became vertical. Regardless of which method is used, defining the break-point location is rather subjective because a wave does not always transform from a shoaling wave to a broken wave in an abrupt manner. Judgment is always involved; therefore, it is important to state how the break point was determined in reporting of results. Unfortunately, not all authors specify this. 23. It is also important to state the datum at which the depth at breaking was measured. Note from Figure 4 that the water depth at breaking h, is measured from the still-water level (SWL) under the crest of the wave. This definition was used in all tests in the present study, although h, has also been measured from the mean water level (MWL) in some studies. The equipment available for data analysis in the present study made the SWL more convenient to use as a datum. Review of monochromatic wave studies 24. Many studies have been performed to develop relationships to denote the wave height at breaking. The term "breaker index" is used to describe nondimensional breaker height. The two common indices are of the form es (13) in which y, will be called the breaker depth index, and Og ae (14) in which Q, will be called the breaker height index. 25. Although the present study involves only periodic waves, most earlier studies on breaker indices were conducted with solitary waves. Two widely quoted studies are noted here. McCowan (1891) theoretically determined the value of y, for a solitary wave on a horizontal bottom as 0.78. Munk (1949) derived the following expression for breaker height index from solitary wave theory: 20 @, > O28 & (15) The relationships of McCowan (1891) and Munk (1949) have been widely accepted in the past because of the hypothesis that periodic waves near the break point behave as solitary waves. The following paragraphs present selected relation- ships developed for breaker indices using periodic waves. 26. The first major laboratory experiment on breaking waves was con- ducted by Iversen (1952). Iversen generated periodic waves of steepness in the range of 0.0025 < H,/L, < 0.0901 on four different uniform beach slopes (1/10, 1/20, 1/30, and 1/50) and developed curves for the breaker height index versus deepwater wave steepness. The curves show { decreasing with in- creasing H,/L, . Iversen also noted breaker height was higher for the steeper slopes. This experiment provided breaker data over a broad range of slopes and wave steepnesses, and the data set is often referenced in breaking wave studies. 27. Ippen and Kulin (1955) conducted experiments with solitary and oscillatory waves and concluded that oscillatory waves do not behave as solitary waves near the break point because of backwash from preceding waves. They also noted y, increased as wave period increased, which implies the breaker depth index increases as H,/L, decreases, since L, is directly related to wave period. 28. Galvin (1969) performed laboratory experiments with periodic waves on three different uniform slopes (1/5, 1/10, 1/20) and found that if his data were combined with the Iversen (1952) data, —— © 0,92 m > 0.07 (16) — = 1.40 - 6.85m m < 0.07 (17) Yb 29. Collins and Weir (1969) derived the following expression from linear theory and experimental data of Suquet (1950), Iversen (1952), and Hamada (1963): 21 Y = 0.72 + 5.6m (18) Equation 18 described the experimental data well for slopes 1/20 and milder. 30. Goda (1970) developed curves from existing laboratory data for beach slopes ranging from 1/10 to 1/50. Goda noted that bottom slope affects not only breaker height, but also breaker depth. 31. Weggel (1972) developed a relationship for % based on the form of Munk’s (1949) relationship for as -1/3 Q, = F(m) — | + G(m) (GLE) Weggel determined the functions F(m) and G(m) empirically from the Iversen (1952) data, where F(m) = D,[1 + m - G(m)] (20) Cay) = [DG m) = DEG 7S = O.1e8se-25)]| (21) and D, = (01 + O.Sm)H (22) Dy = (OL > Ole e2)~* (23) where D, and D, are empirical functions for breaker height (Weggel 1972). Weggel commented that use of Equation 19 became questionable for m > 1/10 and H,/L, > 0.06 32. Weggel (1972) also developed an equation for the breaker depth index from previous laboratory data (Iversen 1952, Reid and Bretschneider 1953, Galvin 1969, Weggel and Maxwell 1970, and Jen and Lin 1971) collected on slopes of 1/5, 1/10, 1/15, 1/20, and 1/50. The resulting relationship is expressed as Hy, qi = WG) = Gy) == (24) L je} 22 for a(m) = 6.97(1 - e 3%) (25) 56 b(2) = ———_— (26) (1 + QWs) where a(m) and b(m) are empirical coefficients. Equations 24 and 25 are presented in nondimensional form, but the original equation of Weggel (1972) was given in US customary units. 33. The study by Jen and Lin (1971) was conducted to determine impact pressures on a cylindrical tube placed in the tank. Breaking wave properties obtained from these tests are expected to differ from results from other tests used in the Weggel (1972) analysis. Data obtained from Reid and Bretschneider (1953) were used as transition values from limiting wave steepness to depth- limited breaking. A portion of the data reported by Reid and Bretschneider were from Danel (1952), a study on limiting clapotis. 34. Weggel (1972) determined a(m) and b(m) by assigning minimum and maximum limits to y, . The theoretical value of a solitary wave, yy, = 0.78, from McCowan (1891), was used as the minimum breaker depth index, occurring for m=0 . As m approached infinity, such as at a vertical wall, y was assumed to be double the minimum value, 1.56, by adding the incident and per- fectly reflected wave heights. Breaking wave height appears on both sides of Equation 24, so an iterative approach must be used to determine y, . Equa- tion 24 can easily be solved with the use of a computer, but otherwise the form is inconvenient for hand calculations. 35. Komar and Gaughan (1973) derived a semiempirical relationship for Q, from linear wave theory where -1/5 G, = 0,56 | = (27) The coefficient 0.56 was determined empirically from laboratory data (Iversen 1952, Galvin 1969, and Komar and Simmons (as reported by Komar and Gaughan 1973)) and from field data (Munk 1949). 23 36. Singamsetti and Wind (1980) collected laboratory data and combined the results with data from Galvin (1969) and Iversen (1952). Singamsetti and Wind found -0.254 H, %, = 0.575m?-031 | — (28) ih ‘oO valid for 0.02 < H,/L, < 0.065 and 1/40 = cam =| + Gy (33) Equation 58, based on data collected from previous studies, was developed in the present study and is described at the end of this chapter. Table 1 Summary of Coefficients and Exponents for Breaker Depth Index Equation Source C, ny Cy Nz Number McCowan (1891) 0 0 0.78 0 -- z 0 0 1.09 0 16 Galvin (1969) 0 0 =n) * 0 7 Collins and 5.6 1 0.72 0) 18 Weir (1969) Weggel (1972)** -a(m) 0 b(m) 1 24 Singamsetti and 0.568 0.107 0 -0.237 29 Wind (1980) Sunamura (1981) 1.1 0.167 0 -0.083 30 Present Study -a(m) 0 b(m) 1 58 * f£(m) = function of beach slope. ** H, assumed equal to H, 40. Figure 5 (a-d) shows a graphical comparison of the different breaker depth equations as a function of deepwater wave steepness for slopes of 1/10, 1/20, 1/40, and 1/80. To include Equation 24 (Weggel 1972) in Fig- ure 5 (a-d), H, was set equal to H, . Equation 29 was plotted within the limits specified by Singamsetti and Wind (1980). Equations 16, 17 (Galvin 1969), and 18 (Collins and Weir 1969) are functions of beach slope only and give a constant value of y, regardless of deepwater wave steepness. 25 Galvin =k —= Collins and Weir —_ ae 0.47 Weggel 0.24 Present Study - McCowan 0.0 ! eee i = sR 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo a. m= 1/80 Hb/hb A 1.0 Ol 6) S2ase=see ere ee Sunamura 0.67 —*- Galvin =e i i 0.44 Collins and Weir —>— Weggel 0.2 r —)— Present Study - McCowan 0.0 ! ! | ! 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo b. m= 1/40 Figure 5. Breaker depth index predictions (Continued) 26 0.4 Singamsetti and Wind Sunamura Galvin Collins and Weir Weggel Present Study ~ McCowan ! 1 ! 0.02 0.08 0.10 1.4 0.8 0.65 0.4 st —S- Singamsetti and Wind A Sunamura - Galvin —— Collins and Weir —— Weggel —©— Present Study ~ McCowan 1s 1 0.0 0.00 0.02 0.04 0.06 Ho/Lo d. m=1/10 Figure 5. (Concluded) Bi 0.08 0.10 Relationships given by Weggel (1972), Singamsetti and Wind (1980), Sunamura (1981), and the present study show the dependence of y, mot only on beach slope, but also on wave steepness. Figure 5 (a-d) illustrates the variability between equations, even though the extensive Iversen (1952) data set was included in the development of all relationships, with the exception of McCowan. The variability between equations may result from different defini- tions of breaking, experimental variability, and different datums at which depth was measured (MWL or SWL). All of the relationships were developed in a logical manner; however, Singamsetti and Wind (1980), Sunamura (1981), and the present study include both beach slope and deepwater wave steepness in the developed equations. Therefore, Equations 29, 30, and 58 are recommended. Use of Equations 29 and 58 should be restricted to values of m and H,/l, suggested by Singamsetti and Wind (1980) and the present study, respectively. Summary of breaker height 41. Bottom slope and deepwater wave steepness are also used to deter- mine the breaker height index. The basic equation for breaker height can also be expressed in the form of Equation 33. A list of the different coefficients and exponents for breaker height equations are given in Table 2. The equation for determined in the present study can be found at the end of this | chapter. Figure 6 (a-d) gives a graphical comparison of the different breaker Table 2 Summary of Coefficients and Exponents for Breaker Height Index Equation Source C; Ny Cy Ny Number Munk (1949) 0.30 0 0 -0.33 15 Weggel (1972) F(m) 0 G(m) -0.33 19 Komar and 0.56 0 0 -0.20 27 Gaughan (1973) Singamsetti and 0.575 0.031 0 -0.254 28 Wind (1980) Sunamura (1982) iL. 0.2 0 -0.25 By2. Present Study C(m)* 0 0 n(m) 61 * C(m) = empirical coefficient in breaker height equation. 28 3.5 Source 3.02 -4- Sunamura —* Komar and Gaughan —-=- Weggel ae >< Munk 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo a. m= 1/80 Ob 4.0 Source 3.5 =o Singamsetti and Wind -4- Sunamura 3.0/7 = Komar and Gaughan —-=- Weggel 2.5 —><— Munk —- Present Study 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo b m = 1/40 Figure 6. Breaker height index predictions (Continued) 29 4.0 Source 3.5 -S- Singamsetti and Wind -4- Sunamura 3.0 | —*- Komar and Gaughan | —E- Weggel 225 | —><— Munk \ —S- Present Study 0.00 0.02 0.04 0.06 0.08 0.10 (2p) o iH, , in which H, is determined from monochromatic breaking wave criteria. Irregular wave breaking and decay are difficult to separate because there is no distinct breaker line, and broken as well as unbroken waves are present through the surf zone. 43. Goda (1975) developed a numerical model for irregular wave defor- mation based on various laboratory data. Goda used a modified Rayleigh dis- tribution in which the portion of the distribution that represents broken waves (H > H,) is tapered, rather than cut, which gives a range of breaking wave heights. Breaker height is expressed as: Hy L, Ss Ce ee) (34) Hy Hy where ha x = 15) —— 1 Ktean® m) (35) Lo in which the coefficient A ranges from a maximum of 0.18 to a minimum of 0.12, the coefficient K=15 , and s = 4/3 . Goda found that A = 0.1/7 Sue best fit index curves for monochromatic waves previously presented by Goda (1970) and chose the higher A-value to take into account the variability of breaker heights. The lower value was chosen as two-thirds of the higher value. Goda (1975) stated, "The choice was arbitrary, but it has yielded good agreement with laboratory and field data." 44. The relation of Goda (1975) was used by Seelig (1979) to develop curves to estimate nearshore significant wave height over a range of beach slopes and deepwater wave steepnesses. Seelig (1980) used Goda’s relationship to present curves to predict the location and magnitude of maximum wave height in the surf zone. 45. Thornton, Wu, and Guza (1985) introduced the term "mean breaker line" for random waves, defined as the location where an averaged wave height is maximum. Waves at the mean breaker ime eves both broken and unbroken. The averaged wave height used was root-mean-square (rms) wave height H,,, , maxi- mum wave height H,,, , and the average of the highest one-third oF the recorded wave heights H,,3 , also written as H, , the significant wave height. Thornton, Wu, and Guza used field data acquired at Torrey Pines and Santa Barbara, California, and laboratory data of Goda (1975); Seelig, Ahrens, and Grosskopf (1983); and Thompson and Vincent (1984) to compare the analy- tical model of Goda as calculated by Seelig with design curves based on monochromatic waves given in the Shore Protection Manual (SPM) (1977). The model of Goda reasonably estimated H,,; and H,,, for large wave steepness laboratory data, which was expected since the model was based on the same laboratory data. The Goda model overpredicted H,,;3; for small wave steep- nesses, but reasonably predicted H,,, . The depth at breaking was predicted well for all data. The design curves in the SPM were found to be conserva- tive, especially for small wave steepnesses. 46. Sawaragi, Deguchi, and Park (1989) conducted an experiment with an artificial reef placed in a small wave tank. The reef had a 1/30 slope that extended above SWL and a 1/2 offshore slope. Sawaragi, Deguchi, and Park developed an expression for the A-value of Equation 34 (Goda 1975) for wave height over the reef, but the study concerned energy dissipation rather than breaking wave properties. Summary 47. Estimation of breaker height and location for irregular waves is difficult, since breaking occurs at different locations with different 32 heights. The model of Goda (1975) gives a range of breaking wave heights that has compared well with measurements in the field (Thornton, Wu, and Guza 1985). Although there is no distinct breaker location, the "mean breaker line" gives an indication of where waves break. Plunge Distance 48. The crest of a wave at breaking travels some distance shoreward and strikes the water surface (Figure 7), displacing a volume of water which also travels shoreward. Galvin (1969) defined plunge distance as distance from the break point to the crest touchdown point, and splash distance as distance from the crest touchdown point to the splash touchdown point. Sunamura (1987) adds that the plunge point is the "point where breaking waves completely PLUNGE POINT BREAK POINT x, i) SCALE SCALE t) 2 t) 5 f 1 aU INCHES CENTIMETERS Figure 7. Definition sketch of plunge distance and vortex area disintegrate as their crest enters the underlying water." Plunge distance is usually associated with plunging waves, but the crests of spilling and col- lapsing waves also plunge shoreward into the water column and, therefore, have a plunge distance associated with them. Review 49. Galvin (1969) conducted laboratory experiments on three planar slopes (1/5, 1/10, 1/20) and found that plunge distance X, normalized by breaking wave height was given by 33) XK Sue Ole. 25m (36) Hy Galvin noted that X,/H, values reached as high as 4.5 and the average was 3.0. Splash distance X, was approximately the same as the plunge distance. 50. Weisher and Byrne (1979) filmed breaking waves from a pier at Virginia Beach, Virginia, and measured the plunge distance. Average X,/H, values were 5.9, and values ranged from 1 to 10. 51. Singamsetti and Wind (1980) collected plunge distance data in the laboratory and found X,/H, ~ 3 on a 1/5 slope with a range from 4 to 8 ona 1/40 slope. Singamsetti and Wind found that Equation 36 underestimated their data by approximately 50 percent. 52. Visser (1982) conducted laboratory experiments in a basin to mea- sure longshore current. His measurements include plunge distance. Waves were generated at an angle to a 1/20 slope, and Visser found X,/H, ~ 6 53. Larson and Kraus (1989) tested Equation 36 to predict plunge dis- tance, which was a quantity required as input to a numerical model of beach profile change. Equation 36 was found to underestimate X, for steep bar face slopes. Based on results of Singamsetti and Wind (1980), Larson and Kraus used a constant value of X,/H, = 3 in their numerical model. Summary 54. Plunge distance data are limited, and only one relationship has been given to estimate X, . However, Equation 36 (Galvin 1969) gives values of X,/H, of 4 or less, and observations have shown X,/Hp as high as 10 in the field and 8 in the laboratory. These large differences indicate a need for a more reliable relationship to predict plunge distance. Breaker Vortex 55. The cavity of entrapped air created by the overturning crest of the wave at the plunge point is called the breaker vortex. The cross-sectional area of the vortex A, is shown in Figure 7. Air and water mix as the vortex penetrates the water column. On a movable bed bottom, sediment is suspended, which either is transported by currents or settles to the bottom as the vortex loses angular momentum, decelerates, and vanishes. 34 56. Few studies appear to have been conducted on breaker vortices, although the investigations discussed below indicate that vortices contribute to bar formation, sediment transport, and wave height decay. Two studies are discussed in the following paragraphs. The first study involved wave height decay, and the second study concerned breakpoint bar formation. Review 57. Sawaragi and Iwata (1974) conducted experiments in a wave tank on a composite slope consisting of a 1/18 foreslope leading to a horizontal sec- tion. They examined wave deformation shoreward of the break point and esti- mated that 15 to 30 percent of the energy of the plunging waves was converted to kinematic energy of the subsurface vortex. 58. Miller (1976) investigated breaker vortices in a series of tests with a tilting wave tank containing a sand-filled bottom. He observed that vortices created by plunging waves might extend from the surface to the bottom, whereas vortices generated by spilling waves were smaller and confined to the region near the surface. The results indicated that bars were formed in the presence of large vortices generated by plunging waves, but bars tended to be eliminated when subjected to spilling waves. Miller commented that the simplified set of wave tank results did not justify immediate extrapolation to the field, but suggested that this was a promising area for further study. Summary 59. Sawaragi and Iwata (1974) and Miller (1976) discuss implications of breaker vortices on wave dissipation and sediment movement. Since vortex area cannot be measured easily in the field, the present study attempts to relate vortex area to properties that can be measured easily, such as wave height, period, and local bottom shape. Wave Height Decay 60. A broken wave dissipates energy as it progresses shoreward. The wave may remain turbulent to the beach or become stable and reform. A reformed wave will also shoal and eventually break. Wave height decay is needed to calculate radiation stresses, which drive currents, entrain sedi- ment, and create setup in the surf zone. The broken wave height is also needed to design structures to be located in the surf zone. Several models have been developed to predict wave height decay, and some of these are 35) described in the following paragraphs to illustrate the various types of models. A more thorough literature review can be found in Smith and Kraus (1988). Review 61. Wave height decay can be determined from the dissipation of energy in the surf zone. The energy flux equation is written as oF = -€ (37) Ox in which F is energy flux, x is horizontal distance, and « is the energy dissipation rate defined as F = EC, (38) where E is wave energy and C, is group speed of the waves defined as pH? | = = (39) 8 in which p is water density. 62. Le Méhauté (1963); Divoky, Le Méhauté, and Lin (1970); Battjes and Janssen (1979); Stive (1984); and Svendsen (1984, 1985) assumed the energy dissipation rate of a broken wave was similar to the dissipation rate of a hydraulic jump. The dissipation rate of the hydraulic jump is (40) where a is an empirical coefficient and h is the still-water depth. Battjes and Janssen (1979) and Svendsen (1984, 1985) found a@ equal to unity, indicating the dissipation rate of broken waves and hydraulic jumps is the same. Stive determined a to be a function of beach slope and wave steepness. 36 63. Svendsen (1984, 1985) determined F from crude approximations of actual flow in surf zone waves. Svendsen used a nondimensional form of the energy flux B where jo ———— (41) pgcH and C is wave celerity, equal to L/T , where L is wavelength. Svendsen assumed an idealized flow within the wave and the so-called surface roller, and he approximated B by T 2 1 n L A Th B = —]| |—]|odt + — — (42) T H2 DpH a ole in which n is the free-surface water elevation, and A, is the cross- sectional area of the surface roller. The surface roller is defined as the recirculating part of the flow resting on the front face of the broken wave. Svendsen found a relationship for A, based on experimental data of Duncan (1981) where Ae OM9He (43) This reduces Equation 42 to 1 ” h 3 SS S|] |—— || Ge ©.45 —— (44) T Hg L 64. Dally (1980) and Dally, Dean, and Dalrymple (1985a, 1985b) derived an equation to calculate the decay of broken waves, in which e was assumed to be proportional to the difference between local energy flux F anda stable energy flux F, 37 @=— CP = 18,) (45) in which et (46) where K is a dimensionless decay coefficient, and I is an empirical stable wave factor equal to the ratio of the stable wave height to water depth. 65. Ebersole (1987) compared the models of Dally (1980); Dally, Dean, and Dalrymple (1985a, 1985b); and Svendsen (1984) to field data collected at Duck, North Carolina. Both models predicted wave height well, especially in the inner surf zone. 66. Mizuguchi (1981) developed a model that allowed for wave deforma- tion on complex beach profiles, as does models of Dally (1980); Svendsen (1984, 1985); and Dally, Dean, and Dalrymple (1985a, 1985b). The approach in modeling the surf zone energy dissipation, which Mizuguchi states is "physi-. cally obscure," is to replace molecular viscosity with turbulent eddy viscos- ity in solving for internal energy dissipation due to viscosity. Model pre- dictions compared well with laboratory data collected on a horizontal beach, a 1/10 plane slope, and a step-type beach. 67. A simpler and more traditional method of estimating wave height in the surf zone is by the expression suggested by Longuet-Higgins and Stewart (1964). Bowen, Inman, and Simmons (1969) supported the relationship with laboratory data conducted on a 1/12 slope. Equation 47 implies wave height decay is linear; however, laboratory data by Horikawa and Kuo (1967), Street and Camfield (1967), and Van Dorn (1977) indicate decay is steeper than predicted by Equation 47 on gentler bottom slopes. 68. Noting the concave form of the broken wave profile and motivated by analytical studies to use a simple but more accurate prediction of the broken 38 wave height than a constant value of y, , Smith and Kraus (1988) developed a power law equation for wave height decay as h H = yh, |— (48) hy where n was empirically determined through regression analysis from previously acquired data of Kuo (1965), Horikawa and Kuo (1967), Saeki and Sasaki (1973), Sasaki and Saeki (1974), Van Dorn (1977), Mizuguchi (1981), Maruyama et al. (1983), and Stive (1985). Multiple regression of the data gave: 0.0437, 0.0096 m = 0657/45) HK SF 0.082 (49) m m The value of y, was calculated from Singamsetti and Wind (1980) (Equation 29). 69. Sallenger and Howd (1989) conducted field experiments in the vicinity of a longshore bar at Duck, North Carolina, in 1982 and 1985. They determined that wave energy became saturated at H,,,/h = 0.32 in the inner surf zone independent of H, . However, all wave data were collected seaward of an inner bar, and measured values of H,,,/h included broken and unbroken waves. 70. Irregular wave decay models have been developed by Battjes and Janssen (1979), Dally (1980), and Thornton and Guza (1983) applying monochro- matic decay models to a distribution of wave heights. / Summary 71. Several expressions and models have been developed for wave decay. This abbreviated and selective review of the considerable wave decay litera- ture was made to illustrate the different methods used to predict wave decay. The methods may be complex and may attempt to physically explain energy dissipation in the surf zone, or they may be as simple as applying a constant multiplier to the water depth to estimate wave height. 59 Wave Runup 72. Wave runup is defined as the elevation above SWL reached by an incident wave (Figure 8) and is a subject of great coastal engineering inter- est. For example, beach face change is directly controlled by the position of the water level (Larson and Kraus 1989), and the design height of coastal structures is determined by the amount of runup on the structure. The follow- ing paragraphs summarize selected equations developed from wave runup studies. Figure 8. Definition sketch of wave runup Review 73. Miche (1951) obtained an equation for runup R normalized by deepwater wave height on a uniform slope: — = |— (50) H, 2p where fB is the beach slope in radians. Miche determined Equation 50 theoretically for linear standing (nonbreaking) waves on a slope. 74. Saville (1956) conducted laboratory experiments and measured runup of monochromatic waves on smooth-faced structures (structure slopes of 1/1.5, 1/2.25, 1/3, 1/4, and 1/6) on a 1/10 plane slope. He also measured runup on 1/30 and 1/10 plane slopes. Saville combined the results with previously acquired data (Saville 1955) taken under conditions of 1/3 and 1/1.5 smooth- faced structures, 1/1.5 step faced wall, 1/1.5 riprap-faced wall, vertical wall, and recurved wall. The latter tests were also fronted by a 1/10 plane 40 slope. Saville developed often-used curves for R/H, as a function of wave steepness. 75. Hunt (1959) empirically determined an equation for runup from laboratory data as R H, — = 2.3m (51) where R and H, are in feet. Battjes (1975) nondimensionalized the equation of Hunt to give runup as a function of the surf similarity parameter. The resulting equation was R 5 11,02, toe On | ——— (56) Bs (Ls) R Cie , —— = d(tan B) SS (57) (Hs) Cae in which (L,), is the significant deepwater wavelength, calculated using the significant period T, in Equation 1, and the parameters a through e varied depending on whether R , Rx , OF R, was predicted. Summary 79. The parameters used to determine runup on smooth plane slopes are beach slope and wave steepness. Factors that influence runup in nature also include the roughness and porosity of the slope, and corrections to the above equations must be made to account for the effect of these quantities. Analysis of Previous Data 80. In the present study, selected data sets were reanalyzed to deter- mine relationships for breaking wave height and plunge distance as a function of beach slope and deepwater wave steepness on plane sloping beaches. The following criteria were established to select data for this analysis: a. The study was conducted on a fixed plane slope with monochromatic waves. 42 b. Deepwater wave steepness was given, or wave height was measured in the horizontal section of the tank so that H,/L, values could be calculated by linear wave theory. c. No structures were present in the wave tank that would alter wave breaking or introduce reflections not associated with planar beaches. The data sets selected are summarized in Table 3, and the data used in the analysis are listed in Appendix A. Breaker depth index 81. A breaker depth index was developed following the general (but not the particular) method of Weggel (1972) for 11 data sets for beach slopes covering 1/80 to 1/10. Breaker depth index was plotted versus deepwater wave steepness for each individual slope (Figure 9 (a-f)), and lines that visually best represented the data were drawn (dashed line). Calculated values, shown by solid lines, are explained below. The data were considerably scattered, and regression analysis could not be used. The data scatter apparently retlects inconsistencies of breakpoint location, breaker height, and/or water depth datum. Data collected on the 1/80 slope (Figure 9a) were scattered with no apparent trend; therefore, the average value of y, was chosen for that slope. 82. The line slope a(m) and zero intercept b(m) of the best-fit lines from each slope were plotted versus beach slope (Figures 10 and 11). Equations were fit to a(m) and b(m) by using the method of least squares and combined to yield the following relationship for breaker depth index: Hy Yp = b(m) - a(m) (58) Ly in which aim) 2 5,00GL « ess) (59) and Ll b(m) = (60) G@ # oes) for 0.0007 < H,/L, < 0.0921 and 1/80 Visual Fit at i* wk om —— Calculated ie to if 0.7 4 os eae : cy © = © 0.5 fe ! ! 1 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo d. m= 1/30 Figure 9. (Sheet 2 of 3) 46 Hb/hb 8 1.65 : 1.47 2 oxox + Source Horikawa and Kuo Iversen Singamsetti and Wind Galvin Ilwagaki et al. Visser - Visual Fit Calculated Galvin cath an eens ‘ eB x Visser =a8 * # oO . © Maruyama, et al. 1.0 ~ re -- Visual Fit S —* Calculated - ae 0.8 ao 3 0.6 1 1 ! 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo f. m= 1/10 Figure 9. (Sheet 3 of 3) 47 a(m) 6 1 ! 1 = 0.00 0.02 0.04 0.06 0.08 m Figure 10. Function a(m) 0.10 0.6 0.00 Figure 11. Function b(m) 48 0.10 Hb/hb 2 1.1 1.0 0.9 0.8 0.7 0.6 ! 1 =) i 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo Figure 12. Calculated values of y, for plane slopes as a function of H,/L, (a-£) for each beach slope, represented by the solid line. Breaker height index 83. Breaker height index was plotted as a function of H,/L, for each beach slope (Figure 13 (a-f)). A power law regression of Q, as a function of H,/L, was made for each slope and is represented by the dashed line in Figure 13 (a-f). The coefficient C(m) and exponent n(m) values obtained from each regression were plotted versus beach slope in Figures 14 and 15, respectively. The coefficients and exponents for the 1/80 slope and 1/40 slope do not follow the trend of the other data points. The majority of data collected on the 1/40 slope are from one source (Singamsetti and Wind 1980), and all data collected on the 1/80 slope are from Horikawa and Kuo (1967). Singamsetti and Wind may have defined the breaking wave height and/or break point differently than authors of other data sources. The 1/80 slope experiment had a concrete bottom and was performed in a different tank from the other experiments of Horikawa and Kuo (1967), which had a steel plate bottom.* Since C(m) and n(m) for slopes of 1/80 and 1/40 did not follow * Personal Communication, 1986, Kiyoshi Horikawa, Professor, Department of Civil Engineering, University of Tokyo, Tokyo, Japan. 49 1.6 Point excluded in analysis © Source * Horikawa and Kuo 1.4 Regression 1.2 Z pcetONGs Aten cage ok pte 1.0 : ce ae oe saat SoD eta : 0.8 ! i ! ! 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo a. m= 1/80 QO, 1.9 E Source ao * Saeki and Sasaki 1.7 + Iversen Regression a) — Calculated 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo b. m= 1/50 Figure 13. Breaker height index as a function of H,/L, (Sheet 1 of 3) 50 WS) = = . Source ve r2= 0.91 * Singamsetti and Wind 1.31 . + Stive Sy ---- Regression y —— Calculated 1.457 0.95 | 0.7 =i ! fe ! 4 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo c. m= 1/40 O, 3.0 [ Source 25 r2= 0.74 * Horikawa & Kuo é + Iversen + Iwagaki, et al 20+ O QO Walker ; GL Maruyama, et al. Ae 6 KK Present Study 1.54 fe] ---- Regression i e —— Calculated 1.0+ os ! 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo d. m= 1/30 Figure 13. (Sheet 2 of 3) 51 | Source * Horikawa and Kuo Iversen Singamsetti and Wind Galvin lwagaki, et al. ox oOox*ct+ Visser --- Regression Calculated 1.0 | © 0.5 aces sehen lites i ! 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo e m = 1/20 oO; 3.0 Source 2.6 * Iversen r?= 0.77 + lwagaki, et al. * Singamsetti and Wind ae O Galvin Fe X Mizuguchi 1.8 EB © Visser ee . - Regression 1.4 =~ —— Calculated 1.0 oO * © 0.6 —- = = : 0.0 0.02 0.04 0.06 0.08 0.10 Ho/Lo en = 1/110) Figure 13. (Sheet 3 of 3) 52 C(m) 1.0 0.9 Points excluded in analysis 0.00 0.02 0.04 0.06 0.08 0.10 Figure 14. Function C(m) n(m) 0.30 Points excluded in analysis 0.00 0.00 0.02 0.04 0.06 0.08 0.10 m Figure 15. Function n(m) 53 the trend of the other coefficients, linear regressions were made on C(m) and n(m) for values obtained only on 1/50, 1/30, 1/20, and 1/10 slopes. The resulting equation for , is: -n(m) (eo) OQ, = C(m) (61) Lo in which C(m) = 0.34 + 2.47m (62) and n(m) = 0.30 - 0.88m (63) for 1/50 (65) H, Tes 86. Plunge distance is also expressed as a function of the local surf similarity parameter by ieee Hy = 3, Ag (66) shown in Figure 18. Equation 66 was also obtained visually. 87. Plunge distance normalized by deepwater wave height was compared with €, (Figure 19). The best visual fit equation is pie H = hy Daal (67) fe} 88. Figures 17-19 all show scatter in the data, but plunge distance normalized by H, shows considerably more scatter. The high X,/H,—values indicated on Figure 19 were all from the Galvin (1969) data set and from tests in which he observed multiple wave forms, or solitons, present in the tank. The subject of solitons will be discussed further in Part III. However, the effect of solitons is not apparent if X, is normalized by breaking wave height. Therefore, based on the data set, plunge distance normalized by breaking wave height should be used to determine plunge distance, and data are least scattered if X,/H, is plotted as a function of the offshore surf simi- larity parameter. 56 1 Source 10;° * Singamsett and Wind 4 Visser ° * Galvin x Present Study * “0s K ws *K yx mK *K 0 ! zi | 0.0 0.5 1.0 1.5 2.0 2.5 &, Figure 18. X,/H, as a function of €, Xp/Ho 16 14/ 6 Source ° Singamsetti and Wind 12; - © fe) 4 Visser * Galvin xX Present Study O- Galvin Soliton Cases g ©) es (@) JL 1 ! (@) 1 2 3 4 Figure 19. X,/H, as a function of €, 57 Summary 89. The literature and equations cited in this review provide a neces- sary base for comparing wave breaking properties on plane slopes (previous studies) to barred profiles (present study). More than one relationship is usually given for each property, but most equations illustrate that beach slope and deepwater wave steepness describe breaking wave characteristics. 90. Equations were developed for breaker depth index, breaker height index, and plunge distance as a function of beach slope and deepwater wave steepness. Data used to determine the empirical equations were based on data acquired on plane slopes from previous studies and the present study. The data were scattered and equations were determined from visual fit of the data for breaker depth index and plunge distance. The data were better behaved for breaker height index than for breaker depth index and plunge distance, and a power law regression was used to determine this index. 58 PART III: EXPERIMENT ARRANGEMENT Facility and Equipment 91. Tests were conducted in a 150-ft*-long, 1.5-ft-wide, and 3-ft-high glass-walled tank (Figure 20). The tank contained a 1/30 smooth concrete- capped slope that began 69 ft from the generator board. The section between the slope and the generator was horizontal. Waves were generated by an electronically controlled hydraulic system that drove a piston-type wave board. Displacement of the wave board was controlled by a command signal transmitted to the board by a synthesized function generator for monochromatic waves and by a microcomputer for irregular waves. The microcomputer can also produce monochromatic wave signals for an operator-specified length of time. Although the synthesized function generator can only send monochromatic wave signals, it is independent of the microcomputer and operates until it is manually turned off. The synthesized function generator was more convenient to use if data collection was not necessary, such as when visual observations were made. WAVE GAGES PIS ies TYPE WAVE GENERATOR DISTORTED SCALE, 1H = SV TANK WIDTH = 0.46 m Figure 20. Sketch of tank used in study 92. The water surface elevation was measured with eight double-wire resistance-type gages connected to the microcomputer by cables. The gages were calibrated each day prior to testing. Gage 1 was located 30 ft from the wave generator. Gages 2 and 3 were placed "seaward" of the bar and positioned to measure wave reflection using the method described by Goda and Suzuki (1977). Gage 4 was placed near the point of incipient wave breaking. Gages 5 through 8 were distributed through the surf zone and used to measure wave * A table of factors for converting non-SI units of measurement to SI (metric) units is presented on page 8. 59 heights shoreward of breaking. Locations of Gages 2 through 8 were dependent upon bar location and varied from test to test. The distance between Gages 2 and 3 was dependent upon wave period and water depth, and this distance also varied from test to test. A photograph of Gages 3 through 6 is shown in Figure 21. Wave data were analyzed using the Time Series Analysis (TSA) program*. The TSA program was developed at the Coastal Engineering Research Center (CERC) to provide several analyses of the wave record at each gage. The program was used in the present study to perform downcrossing analysis to obtain average wave height H , significant wave height H, , maximum wave height Hx , average wave period T , significant period T, , and average free-surface water elevation n . The program was also used to make a reflec- tion analysis as described by Goda and Suzuki (1977) and execute a single- channel frequency analysis to obtain peak period T, at each gage. Figure 21. Gages 3-6 93. Two Sony 3/4—in. video cameras mounted on tripods were arranged to focus on the vicinity of wave breaking in the tests: one recorded waves at the break point, and the other recorded the plunge and splash distances. A * C. E. Long, 1985, "Time Series Analysis," Unpublished Computer Program, US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS. 60 grid spaced 2 by 2 in. was taped on the front glass wall of the tank (nearest to cameras) for a reference in analysis of the videotapes. White paper was taped to the back glass wall of the tank to enhance visibility of the recorded waves. A stopwatch was placed in the viewing area of each camera to synchro- nize review of the two videos. A photograph of the video equipment is shown in Figure 22. Figure 22. Video equipment used in study 94. A 35-mm camera provided general documentation for each test. Two still photographs were taken of the bar in the tank, and approximately ten photographs were taken of waves transforming over the structure. 95. Bars were constructed of 3/4-in. marine plywood. A schematic of a typical bar is shown in Figure 23. The seaward and shoreward faces of the structure were connected with strap hinges, and the two ends were anchored at the correct length with predrilled 1/4-in. steel bars. Observations during preliminary tests showed that sections of longer bars flexed under the waves. To minimize this condition, the longer structures were supported by legs attached underneath to prevent flexing or "breathing" of the seaward and shoreward faces due to wave action. Styrofoam was placed in the opening at the crest and taped in position to maintain a flush surface. Styrofoam also was used to seal the sides of the bar against the walls of the tank. Prior to 61 ereeree 552504 > Ke HM Steel bar and Weights Bl ~=—« Plywood [_] Styrofoam O Hinge Figure 23. Definition sketch of typical bar installation, steel plates were attached underneath the structure to prevent it from floating or moving when subjected to waves. Design Wave Conditions 96. The deepwater wave steepness in part controls the breaking wave characteristics (Part II). Therefore, it was desirable to generate waves with a broad range of deepwater wave steepnesses. To determine the wave generating characteristics of the tank, maximum wave heights were produced and measured in the horizontal section over a range of periods from 0.7 to 7.0 sec ata fixed water depth. A water depth of 1.25 ft provided the best location in the tank for video taping of the wave breaking process and was maintained for all tests. This water depth also allowed generation of relatively high waves for improved accuracy with videotape analysis and minimized the effect of surface tension. Maximum deepwater wave steepness H,/L, for each wave period was determined by calculating H, and L, from linear wave theory. For waves normally incident over the bottom contours with no losses or input of energy, the expression for deepwater wave height is 62 Ik, dk. 1b. BH. = | == = (68) Diy Ty in which n is 1 2kh a= =|) — (69) 7) sinh(2kh) and H’ and L!’ are the wave height and wavelength, respectively, in the horizontal section of the tank (h = 1.25 ft), and k = 2x/L is the wave number. The wavelength at a depth h can be calculated by linear wave theory as; L = L, tanh(kh) (70) The observed maximum wave heights and resulting deepwater wave steepnesses are tabulated in Table 4. Maximum H,/L, is plotted versus period in Figure 24. A range of wave conditions based on deepwater wave steepness was selected from Figure 24 and is listed in Table 5. Table 4 Maximum Wave Height Observed in the 18-In. Tank h T Hnax L, H, fate sec fate fae fae Hels 62S OFW/ 0.21 Boal 0.21 0.0844 i D) 1.0 0.46 5) 5 2 0.49 0.0965 Loo 2.0 0.58 20.50 0.59 0.0286 Leo 30 0.92 46.12 0.80 0.0174 625) 4.0 0.67 82.00 0.51 0.0063 1.25 5), (0) 0.42 WAG), 2 0.29 0.0023 La2o 6.0 0.29 184.49 0.18 0.0010 Lo2D 150 O25 Bobo ALL OPES 0.0006 63 Ho/Lo (Si WANIK (la = 128 iG) alt 64 0.08 Kesar 0.06 0.04 0.02 0.00 be =] fe) 1 2 3 4 5 6 7 T (sec) Figure 24. Maximum H,/L, as a function of T Table 5 Range of Wave Conditions Wave i Thos 151 1) E Condition sec H,/L, ft ft ft fate il 1.00 0.0965 5512 0.49 4.76 0.46 2 1.00 0.090 5) Le 0.46 4.76 0.43 3 0.70 0.080 Bo Db 0.20 2.50 0.20 4 1.00 0.070 So L2 0.36 4.76 0.33 5 Lo 25 0.060 8.01 0.48 6.64 0.44 6 150 0.050 LiLo SS} 0.58 8.43 0.53 7] 15,0 0.040 lit 6 53) 0.46 8.43 0.43 8 le 7d 0.030 15.69 0.47 IO. 17 0.45 9 2.00 0.020 20.50 0.41 11.87 0.41 10 3.00 0.010 46.12 0.46 18.49 0.53 11 4.00 0.0063 82.00 0.52 24.97 0.67 12 5.00 0.0023 128.12 0.29 31.40 0.43 13 6.00 0.0010 184.49 0.18 37.80 0.29 14 7.00 0.0006 Beil, Mal 0.15 44.18 0.26 eS 250 0.0088 32703) 0.28 LS) 5211 0.30 97. It was also desirable to generate waves that did not separate into multiple wave forms, or solitons, as discussed by Galvin (1970). Galvin defines solitons as multiple waves that separate from individual waves pro- duced by wave generators in shallow water, and formation of these nonlinear waves is not restricted to any particular kind of equipment. Galvin suggests solitons develop because the wave generator motion does not match the water particle motion at the wave board. The wave conditions listed in Table 5 were plotted on Figure 25 (Galvin 1970) to predict the wave form that would be 1.0 BOUNDARIES BASED ON 137 OBSERVATIONS © DATA NOT FITTING WITHIN BOUNDARY B-F @ WAVE CONDITION WAVES 0.5 BROKEN WAVES ro) eae 0.2 e; REGULAR WAV eS er ies BREAKING LIMIT @ *e 2 o1 2 15 ®6 2 SOLITONS 0.05 C e, 12 @ 2 a. OSes © \ @ \ ) \ CONFUSED “4 ¢ 0.02 MANY SOLITONS WAVETEORMS NUNS 2 Fe Re Ne ah " 0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 H/h Figure 25. Predicted wave forms produced by a periodic wave generator, after Galvin (1970) generated in the tank. Wave Condition 1 falls on the broken waves-regular wave border. Wave Conditions 11-13 indicate two and three solitons may form. Condition 14 lies in the confused wave region, and Condition 15 falls on the border of regular waves and two solitons. Condition 15 was examined visually, and it was determined that no solitons formed. Regular waves were expected for Conditions 2-9. In an effort to generate only stable waves, and also to limit the total number of tests (about 100 tests), five stable wave conditions were selected with H,/L,-values ranging from 0.0088 to 0.09. (The maximum wave steepness attainable is given by the Michell (1893) criterion H,/L, = 0.142). The low steepness wave, Condition 15, was included to cover a range in wave steepness of one order of magnitude. Condition 15 had the lowest wave steepness with the highest wave height that could be generated at 1.25-ft SWL without producing solitons. Bar Design 98. Selection of representative bar geometry was based on results of Larson and Kraus (1989). Larson and Kraus examined morphologic features produced in large wave tanks by regular waves (Saville 1957, Kajima et al. 1983, Kraus and Larson 1988). Saville conducted tests in a 635-ft-long, 15-ft-wide and 20-ft-deep wave tank, and Kajima et al. conducted tests in a 205-m-long, 3.4-m-wide, and 6-m-deep wave tank. Mean sand grain sizes were in the range of 0.20 to 0.47 mm in these experiments. Waves were of a field scale with heights reaching 1.8 m, and periods were as long as 16 sec. 99. Larson and Kraus (1989) found that the equilibrium bars formed in these regular wave studies generally exhibited three predominant angles: a lower seaward angle #, , an upper seaward angle f, , and a shoreward angle B3; . The area covered by $, was fairly small and was a secondary angle. The average seaward angle #, of the bars ranged between 8 to 12 deg, with local slopes as great as 20 deg. The average shoreward bar angle for beach profiles near equilibrium was 28 deg, and ranged from 20 to 35 deg. Average secondary seaward bar angle f, ranged between 4 to 8 deg. 100. Bar geometry generated in a small tank was also examined in the present study. Fowler and Smith (1986) conducted movable-bed tests in the same tank used in this study. The purpose was to compare scaling relation- ships developed by Noda (1972), Lepetit and Leroy (1977), Vellinga (1982), 66 Hughes (1983), and Hallermeier* for movable beds by modeling large-scale tests of Saville (1957). Sand sizes modeled in the laboratory cannot usually be scaled the same as distances and lengths because grain sizes with mean diameters less than 0.08 mm become cohesive and behave as clay and silt. The scaling relationships give distorted scales; that is, the horizontal and vertical scales are not the same, to compensate for larger sand sizes. The distorted scales gave initial slopes as steep as 1:3.9. Sand, coal, and glass beads were used as model sediments in the Fowler and Smith study, but only tests conducted with sand that resulted in erosional bar formations were analyzed to determine bar angles. The small-scale cases analyzed in the present study consisted of sand sizes of 0.22 and 0.40 mm, and wave heights ranging from 5.0 to 9.5 cm. 101. The three predominant angles defined by Larson and Kraus (1989) were also found to be present in the smaller tank tests. The average angle for each group was calculated in the same manner as Larson and Kraus to deter- mine f, , #2, and #3; . Average #, was 20 deg, ranging between 4 and 37 deg. Average #3 was 8 deg and ranged between 0 and 14 deg. A O-deg angle indicates a terrace bar system (Figure 26). Average fz was 6.5 deg. The steep seaward angles produced in the small-scale study were a result of the steep initial slopes. TERRACE Figure 26. Sketch of terraced bar * R. G. Hallermeier, 1984, "Unified Modeling Guidance Based on Sedimenta- tion," Unpublished paper, US Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS. 67 102. Bars observed in the field typically have seaward angles less than 10 deg, as discussed by Larson and Kraus (1989). At least four reasons can be given to explain the gentleness of bar slope in the field: a. Waves are irregular in height, period, and direction. b. The water level varies with the tide. ce. Steady wave conditions do not exist. d. Mechanisms other than short-period incident waves, such as undertow and infragravity waves, may contribute to move sand. 103. After examining bar angles from the aforementioned movable-bed studies performed with monochromatic waves, a range of bar angles was selected. The bars used in the experiment consisted of two angles, 8, and B3; , which implies $B, equals #, . The secondary angle was not included since it covered only a small area of the bar, and it was felt that one seaward angle was sufficient for this initial study. The use of two bar angles resulted in bars that were either triangular (Figure 27) or, in the case of $3; = 0 deg , terraced. Seaward bar angles ranged from 5 to 40 deg, and shoreward bar angles ranged from 0 to 40 deg. The greater seaward angles are unrealistic for sand bars in the field; these were included because it was beneficial for engineering purposes to observe breaking waves on shapes that approximate those of submerged breakwaters or reefs, sills of perched beaches, and similar structures. Figure 27. Shape of typical bar used in study 104. The location and size of the bars were required to produce break- ing waves. Sunamura (1987) found a relationship between the depth of the bar 68 erest h, and the breaking wave height. Larson and Kraus (1989) also ob- served that h, was related to breaking wave height and insensitive to wave period and beach and sand parameters. The depth at the bar crest could be determined from a simple equation: h, = cH, (71) in which the value of the empirical coefficient c was found to be 0.59 by Sunamura and 0.66 by Larson and Kraus. Equation 71 gives one point on the bar, and another point is required to specify the horizontal location and size of the bar. 105. Keulegan (1945) determined the ratio of h, , the depth of the bar trough from SWL, to h, was 1.65 for bars in the field, and 1.69 for bars produced in the laboratory. Sallenger, Holman, and Birkemeier (1985) observed a bar during a storm and found h,/h, = 1.24 . Sunamura (1987) gave values of h,/h, ranging from 1.16 to 1.93 based on field measurements on various coasts around the world. From results of large-wave tank tests (Saville 1957, Kajima et al. 1983), Larson and Kraus (1989) found an average value of h,/h, = 1.74 and that h,/h, was weakly dependent on wave steepness as =92: 3) ire (72) 106. Since the values of h,/h, observed by Keulegan (1945), Sallenger, Holman, and Birkemeier (1985), Sunamura (1987), and Larson and Kraus (1989) differ, it was felt that use of a fixed value would be inappro- priate. Equation 73 was selected to determine h,/h, , and values of this ratio ranged between 1.61 to 2.00 for the design wave conditions. 107. The depth of the bar crest was calculated using Equation 71, with c = 0.66 . The coefficient value of Larson and Kraus (1989) was selected, because it was obtained from the same data set used to determine Equation 72. Breaker height was estimated by Equation 30 (Sunamura 1981) for input into Equation 71. A program was written to calculate wave height through shoaling using linear wave theory at water depth increments of 0.1 ft. The wave height to water depth ratio was calculated at each depth increment and compared with 69 Yp - The design breaking wave height was selected as the wave height at which H/h equaled yy, 108. Once estimates of h, and h, were determined by the above- described procedure, the horizontal location of the bar was known, and the size of the bar could be determined. The depth at the bar trough served as a point of origin from which the shoreward face length was calculated from the given value of #3 and the vertical distance between h, and h, . After the shoreward bar face was calculated, the horizontal location of h, was known, and the seaward bar face length could be determined for the given seaward angle. Test Procedure 109. The bar was placed in the tank at the calculated crest depth. Since h, was based on an estimate of breaking wave height, waves were generated, and the actual breaking wave height was measured. The location of the bar was then adjusted to the correct h, if necessary. Adjustments to correct h, were typically less than 0.05 ft vertically, and the maximum deviation from the calculated value was 0.07 ft. 110. Water surface elevations were measured with the gages sampling at 10 Hz, from which statistical and spectrally defined wave parameters could be calculated. Wave breaking was recorded on videotape. Runup was also measured for each test by visually observing the maximum horizontal distance of the leading edge of the water past the shoreline for 12 or more successive waves, and then converting the distance to vertical elevation above SWL. Qualitative observations were made of the surf zone. Included in the experiment logbook were the depth of bubble penetration at the surf zone gages and the wave form (reformed wave or bore). If the broken wave reformed, the horizontal location of reformation was recorded. Additional notes consisted of the horizontal locations of the gages and water depths at the gages; horizontal location of the bar and water depth at the trough, crest, and toe of the bar; and the water temperature during the test. 111. Once a test was completed, the seaward face of the bar was replaced to give a new &, . Wave height, wave period, and shoreward bar angle were held constant, and the newly constructed bar was placed in the tank for the next test. The preceding method was repeated until tests were made 70 with all seaward bar angles for a given shoreward bar angle. A different shoreward bar face was attached to the bar, and waves were generated over the range of seaward angles. Wave height and period were changed to yield a different H,/L, value after tests were made for all bar configurations, and the procedure was repeated. Pilot test 112. A pilot test was performed as a trial of the methodology and vali- dation of the criterion on bar depth given by Larson and Kraus (1989) prior to actual testing. In this pilot test, the wave conditions and equilibrium bar formed in a large wave tank test conducted by Saville (1957) was reproduced at the smaller scale of the present study. 113. Stive (1985) conducted tests in a large and small tank for similar wave steepnesses to make a scale comparison. Stive found that scale effects were not significant for wave height in the range of 0.15 to 1.5m. A large erosional bar was formed in Saville (1957) test CE400 (Kraus and Larson 1988) ; therefore, this case was selected to model. The deepwater wave steepness was scaled using the Froude model law (Stevens et al. 1942) in the small-scale experiment; = (73) in which the subscripts pr and m denote prototype and model, respectively. Combining Equations 1 and 73 results in: (Ho)m = (Ho) pr (74) loss A wave period was selected as input to Equation 74 that would result in a wave height which could be generated in the tank (h = 1.25 ft) and which would also give a height greater than 0.15 m. The model and prototype test results for Case CE400 are listed in Table 6. Although H,/L, was slightly higher in the small-scale tests because of differences between the design and measured waves, the bar caused wave breaking and validated the criteria of Larson and 71 Table 6 Prototype and Model Conditions of Case CE400 (Pilot Test) T Ly H’ Ho B3 By Study Heyes sec fate fate ft deg deg Saville 0.035 6 160.6 D5 a 5.63 eS 8.4 (1957) Present 0.040 1.46 10.9 0.41 0.44 ILL. 8) 8.4 Study Kraus (1989); therefore, this method of determining bar size and location was used in all tests. Monochromatic wave tests 114. Base tests consisted of various wave conditions and bar configura- tions for which three parameters were varied: H,/L, , &, , and &, . The tests involved five deepwater wave steepnesses, six seaward bar angles, and four shoreward bar angles. Design variables are listed in Table 7. Table 7 List of Design™ Parameters for Base Tests Parameter 1 Parameter 2 Parameter 3 i H B3 By Helen sec IEE deg deg 0.09 1.00 0.43 0 5 0.07 1.00 0.33 20 10 0.05 150 053 30 ILS) 0.03 Lo JS 0.45 40 20 0.0088 2, KO) 0.30 30 40 Nominal values for design purposes that were only approximated. 115. Six tests were also conducted for a design value of deepwater wave steepness (H,/L, = 0.05) for different combinations of wave height and period. Bar configurations consisting of three seaward bar angles (5, 10, and 15 deg) and one shoreward bar angle (20 deg) were subjected to two wave conditions UL @ia25ysec) Ohsvymterands 00sec 0n26Et) inyaddi tion to the base tests with H,/L, = 0.05 . The purpose of this variation was to compare wave breaking properties over a range of heights and periods for a given value Of SHe/iS These variations of the base tests are summarized in Table 8. Table 8 isto Design” Parameters for Base Test Variations Parameter 1 Parameter 2 Parameter 3 T H’ B3 By H,/Lo sec ft deg deg 0.05 Le2S 0.37 20 5) 1.00 0.26 10 15 * Nominal values for design purposes which were only approximated. 116. Tests were made with base case waves without a bar for comparison of breaking wave properties with barred profiles. These tests provided a foundation for observing the influence of the bar on breaking wave properties. The plane slope tests were also included in analysis of previous data sets to obtain relationships for breaker indices and plunge distance. 117. Wave data were collected for 2 min for all monochromatic wave tests, but only 15 successive waves were analyzed. Analysis of the wave data was begun after waves reflected off the concrete slope and was ended before reflected waves from the wave board had returned to the bar. This procedure eliminated contamination of the data from waves reflected off the wave board and simulated the natural reflection of waves from the beach and bar. Break- ing waves were also recorded on videotape for 2 min and quantities determined visually for 10 consecutive waves using the same procedure. Irregular wave tests 118. Three irregular wave conditions were generated for three bar con- figurations each, as well as for the control case of no bar. A JONSWAP com- puter signal was generated for peak periods T, of 1.0, 1.5, and 1.75 sec with significant wave heights H, of 0.37, 0.47, and 0.45 ft, respectively. Figure 28 shows the spectral shape generated for T, = 1.75 sec , and the input parameters required to generate the signal; the wave height at the wave board, the water depth at the wave board, the spectral peak frequency 73 11.0 10.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 Spectral Denefty (in'/Hz) Spectral Density (In’/Hz) 2.0 1.0 H 0.0 0.0 0.5 1.0 1.5 0.0 0.5 1.0 Bee) Frequency (Hz) Frequency (Hz) H = 0.458 ft h = 1.25 ft i) = OSV ike G = 3.3 a; = 0.07 Oy = 0.09 Figure 28. Spectral shape of irregular wave signal used to control wave board and predicted shape of water surface spectrum. (0, and oy are low- and high-frequency spec- tral width parameters, respectively) = ye the spectral width parameter o , and the spectral peak p>? enhancement parameter CG. 119. Peak deepwater wavelength (L,), and significant deepwater wave height (H,), were calculated using linear wave theory with T, and H, measured in the horizontal section of the tank as input. Wave data were collected and analyzed for 500 waves for the irregular wave tests. Table 9 lists design parameters for the irregular wave tests. Table 9 Irregular Wave Test Design” Parameters Parameter 1 Parameter 2 Parameter 3 15) H, B3 By (H;/L) sec ft deg deg 0.078 1.00 O37 20 5 0.044 1.50 0.47 10 0.03 I, 7d 0.45 15 “Nominal values for design purposes which were only approximated. ™ Hs, = significant wave height in horizontal section of task. PART IV: RESULTS 120. Principal results of the study are presented in this chapter. Breaker heights and depths, plunge and splash distances, and vortex areas were determined from videotape. Measurements were made for 10 successive waves and averaged for use in analysis. The video playback device was a Sony 5650 pro- fessional editing machine, consisting of two playback machines and one 13-in. and one 19-in. monitor. Playback speed could be varied down to freeze frame. Wave height decay and wave reflection were obtained by analysis of resistance- type wave gage records. The horizontal uprush of the wave was determined visually and converted to a vertical distance above SWL to acquire wave runup. Breaker type was also observed and recorded during testing. 121. Totally, 120 tests were performed, including 96 base tests with monochromatic waves, 11 variations of the base monochromatic wave tests, and 12 tests with irregular waves. The first monochromatic test modeled the wave conditions over a solid model of the bar formed during a movable-bed test conducted by Saville (1957) in a large wave tank. This pilot test was per- formed to validate the bar depth criterion of Larson and Kraus (1989) to be used to design the bars. 122. Visual observations during the series of tests showed that the return flow over the shoreward slope of the bars influenced the breaking wave characteristics. A strong return flow was present if the cross-sectional area of the surf zone was small, such as for tests with terraced bars. The return flow also appeared to promote formation of a secondary wave in the trough of the incident wave if 6, was large compared with H,/L, . As water flowed seaward over the bar, the water surface profile conformed to the shape of the bar, much like critical water flow over a weir. For steeper bars, the water surface profile over the bar was also steep, and the incident wave tended to collapse or "trip" over the bar, rather than shoal and break by the depth- limiting mechanism. The seaward bar angle necessary to cause wave tripping decreased as H,/L, decreased; therefore, collapsing waves occurred for gentler bar slopes with smaller wave steepnesses. Performance of tests over the entire range of seaward bar angles with the smaller deepwater wave steep- nesses was unnecessary because of the abnormal wave tripping effect. 123. The influence of shoreward bar angle on wave breaking was not found, which was attributable to the return flow. To quantify the effect of 75 shoreward bar angle on wave breaking, return flow measurements will be neces- sary in future studies. The data points shown in the figures of this chapter represent results from individual tests with different $3—angles , unless otherwise noted. 124. A list of the base tests, including the control cases with no bar, is given in Table 10. The first three digits of the case number specify the deepwater wave steepness, shoreward bar angle, and seaward bar angle, respectively. A fourth digit was used to differentiate tests that had iden- tical wave and bar conditions. Results of the data analysis for the monochromatic wave tests can be found in Tables Al and A2. Table 11 summarizes the irregular wave tests. The letter "R" preceding the four-digit case number in Table 11 indicates that irregular, or random, waves were generated. Breaker Type 125. Three breaker types were observed during the tests: spilling, plunging, and collapsing. Breaker type transition values observed in the experiment are shown as a function of offshore surf similarity parameter in the upper portion of Figure 29. The offshore surf similarity parameter was calculated by substituting tanf, for the slope m in Equation 4 (Battjes 1975) and using linear wave theory to calculate H, and L, from wave heights and periods measured at the gage in the horizontal section of the tank. Transition values between breaker types given by Battjes for plane slopes are shown in the lower portion of Figure 29. Both plunging and collapsing breakers in the present study for barred profiles occurred for lower €,-values than predicted by the plane-slope values. Transition values for barred profiles in this study were: surging or collapsing if 65 > 12 plktingsnins se Oo < & < lek (75) spilling if BS Oo& The lower transition values show that some waves that would break by spilling on a plane slope will plunge if a bar is present, and some waves that would plunge on a plane slope collapse on a barred profile. 76 Table 10 Summary of Monochromatic Base Tests T Lo Ho L H’ By Bs" Case sec H,/Lo fate ft fate ft deg deg 2000** AO 2 0.092 5) 29 0.49 4.88 0.45 -- -- 2110 02 0.095 55333} 0.50 by, il 0.47 5) 5 (0) 0) 2120 1.02 0.095 5) 3335} 0.51 4.91 0.47 52) 0 2130 1.02 0.094 5.32 0.50 4.91 0.47 1S) 56) 0 2140 LOU 0.091 5), 27/ 0.48 4.87 0.45 IS), 5 0 2150 1.02 0.096 529 0.51 4.88 0.47 30m 0 2160 1.02 0.097 5), Sl OR) 4.90 0.48 40* 0 2211 1.02 0.089 55533} 0.47 4.91 0.44 5.8 20 2212 1.02 0.089 5), 38} 0.48 Oil 0.44 5 20 2220 1.02 0.091 oS 0.49 dil 0.45 11.8 20 2230 1.02 0.092 531 0.49 4.90 0.45 LO 20 2240 1.02 0.093 So Sil 0.49 4.90 0.46 ORO 20 2250 1.02 0.088 5), 30) 0.47 4.89 0.43 30” 20 2260 1.02 0.088 553i 0.47 4.90 0.44 40" 20 2310 1.02 0.091 5), 32 0.48 4.91 0.45 558 30 2320 PO 0.088 5532 0.47 Oil 0.43 9.5 30 2330 1.02 0.093 5) Sul 0.49 4.90 0.46 A 7/ 30 2340 lO? 0.093 ool 0.49 4.90 0.46 20.6 30 2350 1.02 0.091 5g Sul 0.48 4.90 0.45 SO" 30 2360 OZ 0.090 5, Sil 0.48 4.90 0.44 40* 30 2410 O02 0.091 5.33 0.48 4.91 0.45 528) 40 2420 OZ 0.090 5.30 0.47 4.89 0.44 9.8 40 2430 1.02 0.088 oe. 0.47 4.91 0.44 14.8 40 2440 1.02 0.091 55 31l 0.48 4.90 0.45 19), 40 2450 1.02 0.092 5), ol 0.49 4.90 0.45 30° 40 2460 1.02 0.093 532 0.50 4.91 0.46 40* 40 4000** 1.02 0.066 5), Sl 0.35 4.90 0.32 -- -- 4110 1.02 0.069 5.28 0.36 4.87 0.34 4.8 0) 4120 1. @2 0.070 5) BE 0.37 4.87 0.34 9.8 0 4130 AO? 0.070 5.28 O37 4.87 0.34 14, I 0 4140 iL, @il 0.070 5.26 0.37 4.86 0.34 19.0 0 4150 OZ 0.072 5), Xe) 0.38 4.87 OFS) 30)” 0 4160 1.02 0.073 5), 29) 0.39 4.88 0.36 40* 0 4211 1.02 0.082 5) 30) 0.43 4.89 0.40 Dye 20 4212 1.02 0.069 5), 30 0.37 4.89 0.34 Be 20 4220 1.02 0.075 3) 6 30) 0.40 4.89 0.37 Wi A 20 4230 1.02 0.075 54 30) 0.40 4.89 0.37 14.3 20 4240 1.02 0.076 eq 30) 0.40 4.89 0.38 12), @ 20 4250 O2 0.077 5) 5 30) 0.41 4.89 0.38 30" 20 4260 OZ 0.070 5.28 0.37 4.87 0.34 40* 20 (Continued) Nominal values used. San Nom bance (Sheet 1 of 3) Lv Case sec Heyflen 4310 nO. 0.069 4320 1.02 0.070 4330 1.02 0.069 4340 1.02 0.068 4350 iL © 0.069 4360 1.02 0.068 4410 1.02 0.069 4420 1.02 0.069 4430 IL OP 0.069 4440 OZ 0.069 4450 OZ. 0.069 4460 OZ 0.070 6000** 1.49 0.046 6111 1.49 0.046 6112 50 0.047 6120 1.49 0.048 6130 1.50 0.052 6140 1.50 0.048 6210 1.49 0.048 6220 1.49 0.043 6230 1.49 0.042 6240 iL, 50) 0.043 6250 50 0.041 6310 1.47 0.048 6320 1.49 0.043 6330 1.50 0.043 6340 IL, 5X0) 0.042 6410 1.49 0.046 6420 1.49 0.043 6430 1.50 0.043 6440 1.50 0.043 8000** Lo V& 0.030 8110 Lh Ve 0.031 8120 iL, 74 0.031 8130 174 0.032 8140 ib, 7h 0.032 8210 V4 0.032 8220 Le yh 0.034 8230 Lg Ys 0.034 8240 IL, eh 0.034 * Nominal values used. NOM Dace Table 10 (Continued) Lal « AnuOnunnn Anuounmn wn SQA] GOES © OEeEOeSo GOEOEO2 GeEeEEO0Q GBQOQSCOeO 2 O2OOQOOeOo OOOO oO©S 12a) (o) (Continued) cocoecmcmMme MOM MOCO COMO MMO MO OKO MOM MOm Co FEF KKK HE FHKHKSKHKE GOOQe SGOSEO2 © GCOES OGOEOOO9 GEOG Oo GOGOSO © OOOO EGSO OOOEGeES Nor $ rPwwoun * Nore oFfFOoMm Pee . . (So D- oF 6 OU ia) Wrenn DOFWB CONDO GANNN DANE UY N ° NPR SOFOU OUuDOsF 4 oo eae 1 ' ! Pee WrON ofoun DONwW CORD fo NPR (Sheet 2 of 3) Table 10 (Concluded) T Ly H, iby H By B3" Case sec He/ilee ft ft ft ft deg deg 8310 iL, 7a 0.032 15.52 0.50 10.10 0.48 5,0 30 8320 1.74 0.034 iS) 52 0.52 10.10 0.50 11.0 30 8330 Loh 0.034 15), 52 O53 10.10 0.50 I, 5) 30 8340 1.74 0.034 15.52 0.53 10.10 0.50 DP 5 Al 30 8410 ics 0.033 15.933} 0.51 110), LiL 0.48 5.6 40 8420 LoVe 0.034 IS. 52 0.52 10.10 0.50 IL, 2 40 8430 1.74 0.034 15.53 0.52 10.11 0.50 16.5 40 8440 1.74 0.034 15), 50) 0), 53} 10.10 0.50 20.6 40 10000** 2.49 0.009 S165) 0.28 10S), ial 0.30 -- -- 10110 2.49 0.010 iL 65 0.32 15.11 0.34 5), 4: 0 10120 2.48 0.008 ce oy 0.26 15.10 0.27 9.2 0 10130 2.49 0.008 Slo 72 0.25 115), 1133 0.27 12.9 0 10210 2.49 0.009 S65 0.28 15.11 0.30 5.6 20 10220 2.49 0.009 31.67 0.28 15.12 0.30 11.9 20 10230 2.49 0.008 Sil, OS 0.26 15.11 0.27 17.4 20 10310 2.49 0.008 Vile 72 0.26 15.13 0.28 5.0 30 10320 2.49 0.008 Bil 7/3) 0.26 15.14 0.28 233 30 10330 2.49 0.008 Sik. 72 0.25 ALS}, 113) 0.27 LS) 6 7/ 30 10410 2.49 0.008 Silo /S 0.25 15.14 0.27 52 40 10420 2.49 0.008 Silo 72 0.25 ALS), 18) 0.26 9.8 40 10430 2.49 0.008 31.70 0.25 IU) 5 IL 0.27 15.7 40 Nominal values used. No bar. we (Sheet 3 of 3) 7g Table 11 Summary of Irregular Wave Tests By" Tt (L,) Gaede (H), Case deg sec fat fit fae Ghee Cae Che/ Uae R2000** a OW, 5.88 0.30 0.41 0.051 0.064 R2210 5 1.07 5.88 0.29 0.40 0.050 0.063 R2220 10 1.07 5.81 0.30 0.41 0.051 0.064 R2230 ILS) 1.07 5.81 0.30 0.41 0.051 0.065 R6000** -- Looe 11.78 0.38 0.51 0.032 0.040 R6210 5 1.56 12.42 0.38 0.50 0.030 0.037 R6220 10 1.56 12. BY) 0.38 0.50 0.031 0.038 R6230 15 Lod 12.68 0.38 0.51 0.030 0.038 R8000** =< 1.74 15.50 0.35 0.47 0.022 0.029 R8210 5 Lev ILS) oil 0.34 0.47 0.022 0.029 R8220 10 1.74 15.50 0.34 0.47 0.022 0.029 R8230 BS te Y2 15.64 0.34 0.47 0.022 0.028 Nominal values used. aK No bar. Barred Profiles Spill. | Plunging Collapsing Plane Slopes | Spill. Plunging Collapsing | ! ! i i | | 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 35 Figure 29. Breaker type as a function of €, 80 126. Breaker type was also subjectively classified as "well-behaved" or "confused." The well-behaved breakers were waves that broke by depth-limited conditions (spilling and plunging breakers), and the confused breakers were believed to be waves that broke as a result of strong return flow (collapsing breakers). A relative length parameter was introduced to quantify well- behaved and confused breakers. The relative length parameter was defined as the ratio of the length of the seaward bar face s, , i.e., the horizontal distance from the bar crest to the seaward toe of the bar, to the wavelength at breaking lL, . The wavelength at breaking was calculated by linear shallow-water wave theory as L, = T(gh,)7/? 7e)) Figure 30 shows well-behaved and confused breakers as a function of the rela- tive length parameter. Well-behaved breaking occurred for s,/l, = 0.75, which means well-behaved or depth-limited breaking occurs if s, is at least 75 percent as long as the wavelength at breaking. If s, < 0./5lL, , the bar probably had little effect on wave deformation leading up to breaking and, instead of gradually shoaling, waves broke by tripping over the bar. 127. Under natural conditions on a sandy beach, the bar and incident Breaker Type O Well-Behaved * Confused si/Lb Figure 30. Breaker type as a function of the relative length parameter 81 waves interact to provide an equilibrium profile. Bars are gentler under these conditions than those that cause confused breaking. Confused breaking may occur for engineered conditions, such as in areas where dredged material is disposed. However, if waves possess sufficient energy, they will initiate sediment movement, material will be transported, and the interaction between the bar and waves will form a more gently sloping bar shape. Wave Reflection 128. Reflection coefficients K, obtained from wave heights at Gages 2 and 3 and the predicted values of Equation 8 (Miche 1951) and Equation 10 (Battjes 1975) were plotted as a function of €, , shown in Figure 31. The data are scattered, and one data point is anomalously high (K, = 0.48). The reason this point is high is not known, and it was disregarded in all analysis of reflection. Although there is scatter, a clear trend is present for K, to be constant (0.15) for all values of €&, . Predictions of the Miche equa- tion result in a steep curve that approaches a perfectly reflected wave for bar and wave conditions if ¢, = 1 . Equation 10 underpredicts K, at low m= 1/30 Battjes Miche 3.0 3.5 Figure 31. Comparison of measured K, with predicted values of Battjes (1975) and Miche (1951) as a function of €, 82 values of €, (including the plane slope data) and overpredicts K, at r higher €,-values . The Battjes equation predicts increasing reflection as the surf similarity parameter increases, whereas the data show little depen- dence on €, . Neither the equation of Miche nor that of Battjes predicts K, well for barred profiles over the range of €,—-values . These equations were developed for plane slopes and, as Figure 31 illustrates, are not valid if the bottom topography is irregular. 129. Reflection coefficients from the experiment were compared with Equation 11 (Ahrens 1987), shown in Figure 32, but no strong correlation was found with the reef reflection parameter P . Equation 11 predicts K, better than Equations 8 (Miche 1951) and 10 (Battjes 1975), but does not exhibit a constant trend. The Ahrens equation does have the desirable property of remaining bounded, as opposed to Equations 8 and 10. Kr 0.6 — 6. 0.5} 0.4/7 0.3 7 0.27 OFii ls 0.0 ; ; fe) 2 4 6 8 10 P Figure 32. Measured wave reflection and predicted values of Ahrens (1987) as a function of P 130. An empirical equation was developed to predict kK, . Reflection coefficients from tests with and without bars were plotted as a function of B, (Figure 33), which shows a weakly increasing dependence of K, on f, The visually fit curve represents: K, = 0.132 + 0.119tanf, (77) 83 0.4 . ; + 20 O O 30 e) 0.3 r © a8 ale a X Plane Sipe —— Predi O 5 10 15 20 25 30 35 40 45 6 (deg) Figure 33. Wave reflection as a function of ff, for 1.9 deg < B, < 40 deg. The results indicate that K, for barred profiles is effectively independent of wave steepness. Breaker Index 131. The study required analysis of several hundred waves of widely varying forms. Therefore, it was advantageous to use the most consistent and easily applied definition of incipient breaking. Because velocity, accelera- tion, pressure, and radiation stress were not directly measured in this study, the break point could not be defined by these variables (Singamsetti and Wind 1980). Possible definitions compatible with the capability of the video sys- tem and placement of wave gages were (a) the point where the wave cannot adapt to the bottom configuration and begins to disintegrate, (b) the point where the wave height is maximum, (c) and the point where part of the wave front becomes vertical. Several wave gages closely spaced in the vicinity of the break point would be required to accurately define the break point based on maximum wave height. Consequently, it was felt the limited number of wave gages available for the study would be better used to record wave height in 84 the surf zone and wave reflection. Use of video equipment allowed each wave to be examined during frame-by-frame playback as waves shoaled up to the break point. The point where the wave front became vertical was selected for use as the break-point definition because it was a unique point that could be readily observed on the video monitor and measured by reference to the grid placed on the side glass. 132. The horizontal location of the break point was a critical param- eter because the water depth over the bar became increasingly shallow with small horizontal increments, which made h, and thus breaker depth index sensitive to location. Plunge distance was also measured from the break point; therefore, a change in break-point location results in a different plunge distance. Although the remote sensing system was of high quality, determination of the exact location where the wave front became vertical was subject to judgment. The importance of precisely locating the break point required iteration of frame advancements and retreats during videotape anal- ysis, which became tedious, but once the break point was defined, the breaker height and depth could be obtained with consistency. For some waves the break point occurred between videotape frames. In these cases, the height and depth at breaking were measured from the previous frame. Breaker height and depth were measured from the video monitor using the grid placed on the tank wall as a reference. The 2-in. grid spacing could be scaled from the monitor to 1/4-in. increments. 133. The breaker depth was measured under the crest of the wave from SWL, which was well defined on the grid. The SWL is the water level that would exist if waves ceased but tide and storm surge remained (in the case of the field), whereas MWL is the time-averaged water surface elevation including the presence of waves. Since tide and storm surge did not exist in the tank, SWL was simply the quiescent water level in the tank. The MWL can easily be obtained from the wave gages; however, breaker depth was not always located at a gage; thus, it was convenient to use SWL as the datum for analyzing the videotapes. Water depth was also measured under the horizontal face of the wave from SWL for comparison with depth under the wave crest. For spilling and plunging waves, the break point was located near the crest, resulting in little difference in y, . However for collapsing waves, the depth at the break point could be up to one-half the depth under the crest, which doubled the value of yy, 85 134. Breaker heights and depths were determined from videotape for 80 of the 96 total base tests. Measurements of breaker height and depth were made for all tests in which f, < 20 deg. The values of height and depth represent an average of 10 consecutive waves from each test. Breaker depth index 135. Breaker depth index showed considerable scatter if plotted as a function of €, (Figure 34). Values of y, tend to increase for €, < 0.9 , whereas for higher €,-values the breaker depth index appears to decrease more gently with wide scatter in the range 0.9 < €, < 1.6 , shown by vertical lines. Wave breaking for €, > 0.9 was not only influenced by depth, but by a dependent variable involved in the breaking process, the re- turn flow velocity. Breaking waves at higher €,—values were typically of the form shown in Figure 35. A secondary wave shoreward of the main wave crest is created by the return flow, which causes the break point to develop Hb/hb 5 1.35 tet 0.97 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 34. y, as a function of €, 86 By a) -2 ) 2 4 CASE 6240 CENTIMETERS Figure 35. Collapsing wave at incipient breaking on the front face of the wave crest; i.e., breaking begins before the incident wave has reached the depth-limited break condition. The wave crest was located in deeper water at incipient breaking for these conditions, which increased h, and lowered values of y, . The wide scatter of data indicates the strength of the return flow is a major factor influencing breaker depth at higher €,—-values 136. A breaker depth analysis was made for increasing values and decreasing values of y, in Figure 34, excluding the wide scatter data in the transition region (0.9 < €, < 1.6). A linear regression was performed on the increasing values of y, for spilling and plunging waves occurring over bars in which #, < 10 deg, which is approximately the maximum seaward bar angle observed in nature. A relationship for collapsing waves with €, = 1.6 was made by a best-fit line drawn through y,—-values . The resulting relation- ships for breaker depth index was: 2 Oa = 0.988, Hoe 0.95 = 6, = 0.90 (78) ay 2 eAS = O,02G, toe 1.6 36,2 3.5 The coefficient of determination for the regression analysis was 0.85. Equa- tion 78 is shown in Figure 36a and b and also in Figure 34. The decreasing 87 relationship of Equation 78 was extended through the transition region, repre- sented by the dashed line in Figure 34, for illustrative purposes. Although the data are widely scattered in this region, the relationship follows the trend of the higher y,-values and intersects with the increasing relation- ship of Equation 78 near the lower limit of the transition region. 137. An exponential relation was also developed to predict y for plunging and spilling waves that had €,-values greater than 0.90. The visual fit line shown in Figure 3/7 is expressed as: a 2S USGL Ss @Soebe) (79) Equation 79 was obtained for 0.29 < €, < 1.07 and £, < 10 deg. 138. Figure 38 gives a comparison of Y for 5— and 10—deg bar slopes computed using Equation 79 to the equation developed for plane slopes, Equa- tion 58, as a function of deepwater wave steepness. The prediction for planar slopes exceeds the limits of beach slope given for Equation 58. The values shown by the plane slope equation are values for m=1/10 . Equation 58 gives higher values of y, than Equation 79 for $8, = 5 deg for 0.02 < H,/L, < 0.09 , but underpredicts y, for $, = 10 deg. Also shown in Figure 38 are the expected values of y, on a 1/30 slope using Equation 58, which predicts y, Slightly lower for high wave steepness for 6, = 5 deg. Equation 58 significantly underpredicts y, for low wave steepnesses for $6, = 5 deg and for all wave steepnesses for {f, = 10 deg. Breaker height index 139. Values of as a function of deepwater wave steepness are shown in Figure 39 (a-d) grouped by seaward bar angle. The data show increasing breaker height with decreasing wave steepness, which is also typical of wave breaking on a plane slope. Therefore, it is reasonable to express the data collected on barred profiles in the same form as the relationships developed for plane-slope data: Ty = Gh) ||) —= (80) 88 © ® © © © © © Points excluded in analysis “0.2 0.4 0.6 0.8 1.0 1.2 E. Spilling and plunging waves eh) 0.8 © Points excluded in analysis 0.6 : 4 b. Collapsing waves Figure 36. Expression of y, as a function of €, 89 1.3 1.1 0.9 0.7 0.2 0.4 0.6 0.8 1.0 1.2 = Figure 37. Exponential expression of y as a function of €, ——_ Bi=5 deg oe —— Bi= 10 deg 0.7/4 —— Plane Slope 1/10 —=- Plane Slope 1/30 , 0.6 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo Figure 38. Comparison of breaker depth relationships developed for barred profiles and plane slopes 90 2.0 oa Regression 1.87 — Calculated 0.00 0.02 0.04 0.06 0.08 0.10 2.0 Regression 1.85 — Calculated 1.47 0.6 : ! | 7 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo b. B, = 10 deg Figure 39. 9, as a function of H,/L, (Continued) gil re os Regression 1.8 — Calculated 1.4 a ----- Regression — Calculated 1.2 0.8 - 0.6 1 | i 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo d. B, = 20 deg Figure 39. (Concluded) 92 An equation in the form of Equation 80 was developed for each seaward bar angle by performing a regression analysis on {% as a function of H,/L, , represented by the dashed line in Figure 39 (a-d). Equations for the empiri- cal parameters C(f8,) and n(f,) were obtained by plotting these two param- eters as a function of seaward bar angle, Figures 40 and 41, and fitting an equation to the data by regression analysis. The resulting polynomial func- tions were C(B,) = 0.28 + 2.17tanB, - 6.00tan2B, (81) and n(B,) = 0.36 - 1.59tang, + 4.85tan26, (82) The coefficients of determination for C(f,) and n(f,) were 0.93 and 0.99, respectively. The calculated values of 0, using Equation 80 are shown as the solid line in Figure 39 (a-d). The variation of breaker height index calculated from the regression analysis of each f,-angle is presented in Figure 42 as a function of wave steepness. Figure 43 shows a comparison of 2, by seaward angle using Equation 80. 140. Figure 42 shows that 9, decreases for values of f, > 10 deg for high wave steepnesses, whereas for low wave steepnesses {, becomes larger. Since the steeper bars are shorter, shoaling of the incident wave prior to breaking occurs mainly on the 1/30 plane slope rather than the bar, and is approximately the same as the values predicted for the 1/30 slope using Equation 61. Figure 43 shows {-values for the 15-deg bar are higher than those for the 10-deg bar at higher wave steepnesses, which differs from Figure 42. This is simply due to the variability in the regression of the coefficients C(6,) and n(f,) 141. The higher values of { observed at low wave steepnesses may be explained by the presence of a strong return flow visually observed in these cases. A heuristic explanation can be given to describe the influence of return velocity on wave height. The treatment is based on the depth inte- grated energy equation of Phillips (1977), which for the situation of the tank becomes: 93 + a (= 0,08 a | ss 0.85 0.2 ! 1 ! fo) 5 10 15 20 6 (deg) Figure 40. Function C(;) nF, 0.50 asl aA 0.35 + 0.30+ 0.257 0.20 ! | | fe) 5 10 15 20 6 (deg) Figure 41. Function n(A,) 94 1 i | 1 0.5 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo Figure 42. Regression curves of { as a function of H,/L, OQ, 1.9 ish 1.7 ate ; =< 1H) ase 1.5} 3 == 2X0) ogni ie TNE NM Ree arhe ay bk A BANU Sli MNP plane slope 1.1 0.9- 07+ 0.5 : 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo Figure 43. Calculated values of 9, as a function of H,/L, 95 d dU —= (ECU sb G.) ) = 8 —— SO (83) dx dx in which U is the horizontal flow speed taken positive in the shoreward direction and S,, is the shoreward component of the radiation stress. The return flow U is created to balance the mass of water thrown shoreward in the crest of the breaking wave. Equation 83 will be evaluated at the end of the tank near the wave board (called deep water) and at the break point. To simplify the discussion the term S,, dU/dx will be omitted, although it is non-negligible. Also, frictional losses in the tank, expected to be small, are neglected. 142. At the wave board, U must equal zero since the current cannot penetrate through the board. In the open ocean, this is equivalent to U being zero in deep water. With this consideration and the stated assumptions, Equation 83 gives [EGU G2) = [ECZIC (84) in which the prime denotes conditions where return flow is present and the minus sign in front of the magnitude U indicates flow in the seaward direc- tion. If no return flow or only a very small flow is present, Equation 84 is simply: [EC, ], 7 [EC,], (85) Equating the different breaking conditions (with and without a return flow) having identical deepwater wave conditions yields [E(-U + C,)]_ = [EC], (86) Substitution of _the equation for wave energy density (Equation 39) into Equa- tion 86 and elimination of constants on both sides of the resultant equation yields the following expression: (Hy)?(Cgp - Up) = HECg (87) 96 where ae oe | = breaking wave height in presence of return flow Cz, = group speed of waves at breaking in presence of return flow U, = magnitude of the return flow in the vicinity of the break point Assuming Cop = Cy, , Equation 8/7 can be expressed as: (*) begin a , Under the stated assumptions, Equation 88 demonstrates that a return flow pro- duces higher breaking wave heights than if the flow is absent. Figure 44a and b shows as a function of deepwater wave steepness for seaward angles of 5 and 20 deg. The highest {-values result from conditions in which the cross-sectional water area shoreward of the bar was small (f3 = 0 deg) or the shoreward angle was very steep (83; = 40 deg). Return flow increases for B3 = 0 deg because of continuity; i.e., velocity increases if the water area shoreward of the bar decreases. For situations with steep shoreward slopes, the vertical component of return flow velocity is directed opposite (up) to that expected for a plane slope (down). For cases in which the weir flow was observed, steep 6,-angles and low wave steepness, the return flow speed in- creased over the crest of the bar. Equation 88 shows that under these circum- stances H, will increase, which may explain the increase in {,—values for 20-deg angles at low values of wave steepness in Figures 42 and 43. 143. The preceding derivation shows that a stronger return flow increases the breaker height, but observations indicate % is still well behaved in the presence of strong return flow, unlike the breaker depth index. This suggests breaker height is a more stable parameter than breaker depth for situations with steep bar faces. The derivation can be refined by including the wave-current interaction term S,, dU/dx and actual vaiues of (C,), and (Cy)p , leading to an equation that must be solved by iteration. 97 1.8 7 1.6 1.4 N23 1.0 0.8 0.00 0.02 0.04 0.06 0.08 0.10 1.8 D Q fo) (ome) 1.6 30 40 1.4 p * > 1.2 1.0 0.8 0.6 ; 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo b. £8, = 20 deg Figure 44. 0, as a function of H,/L, Plunge Distance 144. After measurements of breaker height, breaker depth, and breaking location were made for monochromatic wave tests, the videotape was advanced to the plunge point, and plunge distance was measured. Plunge distances were averaged for the same waves that breaker height and depth were measured. Occasionally, the plunge point was obstructed from the video camera by tank wall supports. In these cases, the plunge distance was averaged for the number of unobstructed observations that could be made. Figure 45 shows plunge distance normalized by breaking wave height as a function of €, . The solid line drawn through the data by visual fit represents the following equation: Xp Hy = 0.63€5!-°° + 1.81 (89) for 0.29 < €, < 3.46 Xp/Hb ) —— Barred Profiles Plane Slopes 1.0 | ! ! ! 1 il 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 45. X,/H, as a function of €, 99 145. Plunge distance was also plotted as a function of the local surf similarity parameter (Figure 46). The visual-fit solid line through the data is expressed as Xp Hp = 0, 65ere89 eb iLO (90) —— Barred Profiles Plane Slopes 3.0 eee 2.0 WPCA a tanwbes trig eae : 1.0 IL ! ! | aL 0.0 0.5 1.0 2.0 2.5 3.0 1.5 Ep Figure 46. X,/H, as a function of &, for 0.31 < & < 2.74 , where €, is an inshore surf similarity parameter. Both Figures 45 and 46 show comparable scatter of the data, and neither Equation 89 nor 90 appears to be a better predictor of plunge distance. 146. The equations developed in Part II for plunge distance on plane slopes is shown by the dashed line in Figures 45 and 46. Plunge distance for barred profiles are shorter than those for plane slopes with common values of €, . This can be explained by the transformation of breaker type caused by the barred profiles. Spilling and plunging waves would be expected for the range of €-values shown in Figures 45 and 46, whereas spilling, plunging, and collapsing waves occurred for the barred profiles. The data obtained for barred profiles indicate that X,/H, decreases steeply for smaller surf 100 similarity values and approaches a constant value for larger surf similarity parameter values, whereas plunge distance on plane slopes shows a gradual decrease with increasing surf similarity parameter. This could also be an effect of the return flow, since return flow was observed to be stronger at larger €,-values 147. Figure 47 shows no apparent trend for plunge distance normalized by deepwater wave height plotted as a function of €, , which means that plunge distance is insensitive to deepwater wave height. A representative value visually determined from Figure 47 is xX, = 3 H, Xp/Ho ) 5.0 - 4.07, 3.0 - a a F 2.0 596 5 6 .° 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Eo Figure 47. X,/H, as a function of €, Splash Distance 148. The procedure used to measure plunge distance was also applied to measure splash distance. Splash distance normalized by H, as a function of €, is shown in Figure 48. The data are extremely scattered and show no trend. The splash distance is evidently not dependent on breaking wave height. 149. The ratio of plunge distance to splash distance ranged from 0.6 to 2.7, but average plunge distance was approximately equal to the splash distance (X,/X, = 0.95), which agrees with the result of Galvin (1968) for 101 3.5} 7s - a 3.0} ; 3 be: ee 2.0 se . 1.5 ; ; 0.0 0.5 1.0 15 2.0 Figure 48. X,/H, as a function of 6, plane-sloped beaches. Values of X,/X, tended to be greater for smaller €,-values as shown in Figure 49. Plunge distance was also longer for smaller values of €, , which may explain this trend. The dashed line in Figure 49 represents the empirical relation of Galvin (1968), and the solid line is given by the following visually fit equation = X, = 0.14€31:59 + 0.70 (91) ton (OF295= 55 = 3.46 =) igure 50 shows) X>/X_ as a) function of 6,195 with the empirical equation of Galvin shown by the dashed line. visually fit to the data and is identical to Equation 91 with &, LOG Son, fon) 0.31 = 65 =< 2°74) Equation) 91) may ibe rewritten as): 102 The solid line was substituted Galvin prediction 0.5 ! 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Eo Figure 49. X,/X, as a function of €, Galvin Prediction 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Ep Figure 50. X,/X, as a function of & 103 , = = OME & 0.70 (92) X s This implies that either surf similarity parameter may be used. Equations 91 and 92 give good indications of the trend; however, variability in the data is large. From observations of the video recordings, splash distance appeared to be a function of not only breaker type, but also of the angle at which the plunging wave entered the water column. Penetration Distance 150. A wave crest continues to travel shoreward upon entering the water column until its forward momentum ceases. The penetration distance X, was defined as the total travel distance of the wave crest from the break point. Values of X, were estimated from videotape by observing the maximum forward travel of bubbles caused by the turbulence of the wave crest plunging into the water column. Figure 51 shows penetration distance normalized by breaking Xt/Hb 8.0 7.0/7 6.0 5.0 F 0.0 0.5 1.0 1.5 £ 2.0 2.5 3.0 3.5 fe) Imiasbets Hil, Kes, BS El rcwiatSley OE Ee 104 wave height as a function of the surf similarity parameter. Data obtained from terraced bar cases were not included in analysis because the shoreward face of the bar hindered forward travel of the wave crest. The line shown in Figure 51 is given by Xe —= + 0.9562! & 3,08 (93) Hy with limits of 0.31 < €, < 3.46 . The expression for X,/H, as a function of the local surf similarity parameter shown in Figure 52 is Xt/Hb 8 4r o® 3 c 3 ! Nl l Ls 0.0 0.5 1.0 15 2.0 2.5 3.0 Figure 52. X,/H, as a function of &, Xt = = 100652 + 3585 (94) Hp for 0.33 < € < 2.74 . Both Equations 93 and 94 were determined by visual fit. Penetration distance normalized by H, shows much scatter if plotted versus €, (Figure 53). The hypothesis made that X, is primarily dependent on local wave height also applies to penetration distance. Xt/Ho bo Figure 53. X,/H, as a function of €, Breaker Vortex 151. The breaker vortex is formed when air becomes entrapped by the overturning crest at the plunge point. As the vortex penetrates the water column, the angular velocity of the vortex decreases and the cross-sectional area A, of the vortex becomes greater, until the energy is dissipated. Because the vortex area increases as the vortex moves into the water column, measurements of A, were made at a distinct location. The plunge point is unique to all breaking waves and is also the location of vortex formation; therefore, A, was measured at this point. The shape of the vortex was elliptical in most cases (Figure 7); the major and minor axes of the vortex were measured, and the equation for an ellipse was used to estimate vortex area. 152. Vortex area was estimated for 34 tests with seaward angles less than 10 deg. Vortex area normalized by deepwater wave height was plotted as a function of €, in Figure 54, which shows A,/H2 is small for spilling breakers and larger for plunging and collapsing breakers. Although only three cases in the collapsing range were available for analysis, the data show the trend represented by the visually fit equation 106 Av/Ho 0.20 0.157 * 0.10; 0.05> 0.00 — I eas ! 0.0 0.5 1.0 1.5 2.0 2.5 Eo Figure 54. Exponential relation for A,/H2 as a function of €, —— = 0. 16@, = e99:83tc) (95) HG for 0.29 < €, = 2.06 . A linear regression was also made for spilling and plunging waves only, shown in Figure 55, which gave: = > W005 + 00805, (96) He The coefficient of determination was 0.69, and the equation is limited to 0.29 < €, < 1.07 . Vortex area was also plotted versus &, , Figure 56. The line drawn through the data represents 107 Figure 55. Av/Ho- fe) Affi Linear relation for 1.0 1.2 as a function of €, 0.415 0.09 - 0.037 0.01 Figure 56. 0.8 Ep Linear relation for A,/H2 108 1.0 1.2 as a function of €) — = 0.105€, (97) He for 0.31 < € < 1.06 . Equations 96 and 97 give good estimates of vortex area for spilling and plunging waves. For larger values of the surf similar- ity parameter, Equation 95 can be used to estimate A, . Wave Height Decay 153. Wave height measurements were made for all tests, and the data should be of value for surf zone wave model development. Figure 5/(a—e) shows wave height at each gage normalized by H, as a function of distance for five cases. Comparison of measured wave height with predictions of wave decay models is left for future work. Because beach profiles were irregular, numerical models (e.g., Svendsen (1984, 1985); Dally (1980); and Dally, Dean, and Dalrymple (1985a, 1985b)) will need to be applied. Average wave heights for 12 to 15 waves at each gage for all tests are included in Appendix C. Wave Runup 154. Wave runup measurements were made for all tests. Figure 58 shows R/H, plotted as a function of seaward bar angle #6, , and, for the plane- slope data, beach slope angle (m = 1/30, i.e., $8, = 1.91 deg). Except for the plane-slope points, each data point represents the average R/H,—value for the four shoreward bar angles associated with the seaward bar angle for the particular wave steepness. Basically, the seaward bar angle has no in- fluence on runup. Runup also shows no dependence on €&, , as shown in Fig- ure 59. Runup was plotted versus the surf similarity parameter using the 1/30 slope as the primary angle. The data show increasing R/H, with increasing values of €, . The line drawn in Figure 60 represents the average value of (R/H,)/€ , which was 0.76, whereas Equation 52 (Battjes 1975) predicts this ratio to be 1.0. Figure 61 shows runup for the plane slope cases versus the surf similarity parameter. All measured values are overpredicted by Equa- tion 52, shown as the solid line. The cause of the difference in runup deter- mined in this experiment and by Battjes is not known; additional measurements are clearly warranted. 109 > Wave Height and Elevation (ft) -1.5 ! ! 1 ) 5 10 15 20 25 30 3 ZO Distance from Shoreline (ft) a. Gase 2000 Wave Height and Elevation (ft) 1.0 0.5+ KK xX * * Se ih SE ios SWL 0.0 -0.5+ aoe -1.5 fi i it ! i = it ) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) b. Gase 4130 Figure 57. Wave height as a function of horizontal distance (Sheet 1 of 3) 110 > Wave Height and Elevation (ft) -1.5 je i it | | i 1 fe) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) c. Case 6220 > Wave Height and Elevation (ft) -1.5 | L Nl 1 aah Nl 1 0 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) d. Case 8340 mies SY, (Suaee 2 Oe 3) 111 > Wave Height and Elevation (ft) -1.5 | i = | i i JL fe) 5 10 15 20 25 30 35 Distance from Shoreline (ft) e. Case 10410 Figure 57. (Sheet 3 of 3) 112 40 0.25 Ho/Lo -—O- iy 0.20 0.090 —+- 0.070 —*- 0.050 0.157 -—- 0.030 —>— 0.0085 0.10F 25 0.05}+ oak 1 —l iL | Se | | ! 10) 5 10 15 20 25 30 35 40 45 P (deg) Figure 58. R/H, as a function of 6, R/Ho 2 B oO deg DAG *K 5 1.9 F | i a 8 * 10 = Oo 0.20} + 1 KK par ee Ox » x x 20 ye Oo o 30 0.157 ‘ i XA 40 a *K x A TAS la dir Subse 6 E Ky a x InN x © B O1O|p | age * re g o6 nN i A + 0.05 [Ese S| Eee ee || =e = ae 1 i 0.0 0.5 1.0 1.5 a 2.5 3.0 3.5 4.0 oO Figure 59. R/H, as a function of €, 113 — O7/0& 0.0 -- It ! | | 0.10 0.15 0.20 0.25 0.30 0.35 0.40 bo Figure 60. R/H, as a function of €, using m as the predominant angle R/Ho 0.40 0.35 0.30 0.25, 0.20} *K 0.15; * L * 0.10 K * 0.05 4 0.10 0.15 0.20 0.25 0.30 0.35 0.40 Eo Figure 61. R/H, for plane-slope cases as a function of €, 114 155. Holman and Sallenger (1985) analyzed an extensive field data set of runup on a barred beach. Although there was wide scatter in the data, they concluded that runup appeared to depend on €, . However, the choice of slope with which to calculate €, was unclear. The foreslope appeared to be appro- priate for data taken at high tide and midtide, whereas the bar slope appeared to "have at least some influence" on setup at low tide. (Runup is defined as the combination of a superelevated MWL, called setup, and a time dependent quantity called swash.) It is not clear if the bar was a major cause of wave breaking in their low-tide measurements. The present tests with regular waves indicate that the bar has a very weak influence, if any, on runup if waves break on the bar. In agreement with Holman and Sallenger, the foreshore, or wide-area slope, appears to be the best quantity to use in correlating R/H, with €, ; however, only one slope (1/30) was used in the present study, so this conclusion can only be tentative. Wave Steepness Scaling 156. A subtest was performed to determine if the magnitude of wave height and period had an effect on wave breaking properties. Cases 6210, 6220, and 6230 each had a nominal wave steepness H,/L, = 0.05 , but this ratio was obtained with different wave periods and heights, listed in Table 12. Breaker depth and height indices, runup, and reflection coeffi- cients were determined from these tests and are also listed in Table 12. Figures 62(a-c)-64(a-c) show the wave forms at incipient breaking for the base test (a) and the two variations (b,c). The wave forms for the base test and scaled variations show the incipient breaking wave is similar, but not iden- tical, and has the same breaker type for corresponding seaward bar angles. Breaker depth values for the variation cases follow the trend of the base case; however, breaker height indices and reflection coefficients show great variation and no apparent trend. Although runup values show deviations between the variation cases and base cases, the values are nearly constant. Because runup was shown to depend on wave steepness and primary slope (m), not bar slope, this result is not unexpected. 157. The subtest was inconclusive and not successful in proving or dis- proving if wave period and wave height individually exert influence on wave 80°T cL tT 920 °0 88°0 OTT 720 0 96°0 €8°0 9200 86 °0 OTT 7700 86°0 Oc T ¢70°0 S60 S80 S70°0 Te 1 121 L£S0°0 YI'T LOT 4S0'0 78°0 €Z£°0 €S0°0 Ig qh ay a 72°0 L£Z°0 Gc '0 €€°S 020 (cam) Sc '0 €€°S 0€ 0 42°0 Sc'0 €€°S SjSo, osegq peters jo Aaeuuns cL PTdeL £70 °0 870 °0 870 0 SSO 0 ¢S0'0 cS0 0 c70°0 £70 0 870 0 fo} 1/°H CECI 6609 c1c9 TEC9 1229 T1é9 O€C9 0¢2¢9 O1c9 aseg 116 -2 0 2 4 CASE kas 6210 INCHES -5 () 5 10 [is a CENTIMETERS a. Case 6210 CENTIMETERS b. Case 6211 x NY Lif — SCALE SCALE 2 0 2 1N 5 0 5 CM CASE V6212 c. Case 6212 Figure 62. Wave at incipient breaking for H,/L, = 0.05 , Bie omde rn psa 20nder SCALE CASE “2 Q 2 a v6221 foc | INCHES =5 CY) 5 10 os CENTIMETERS a. Case 6220 {|} -5 (i) 5 10 CASE (2.8 a | 6220 CENTIMETERS b. Case 6221 SCALE -2 0 2 4 a _ CASE INCHES WeonD 1 a ° w cS) CENTIMETERS ec. Case 6222 Figure 63. Wave at incipient breaking for H,/L, = 0.05 , B, = 10 deg , Bz = 20 deg i} of Ke) (eee aay (ey CASE 6230 a. Case 6230 SCALE CASE oo v6231 INCHES -5 ° 5 10 (ot 8 8 CENTIMETERS b. Case 6231 SCALE -2 0) 2 4 (ooo | INCHES CASE V6232 =9 0) 5 10 i s_ es CENTIMETERS ec. Case 6232 Figure 64. Wave at incipient breaking for B, = 15 deg, B3 = 20 deg LILY) fl, = 0.05 , breaking properties for the same wave steepness. Reasons the subtest were indeterminate are: a. Deviations of the phenomenon were within the range of devia- tion for the experiment. b. The range of the variables that determined deepwater wave steepness, H, and L, , was only a factor of two. c. The independent variables, such as f#, and H,/L, , changed slightly between the base case and variation cases because of limitations in equipment. d. Accuracy of the data decreased as the wave height decreased because of limitations in measurements. A complicating factor is the wide variability in reflection, which is not known. The validity of steepness scaling should be pursued in future work, and it is recommended that the various wave conditions be generated on a plane slope prior to installing bars to establish a control condition by eliminating bar variables that are difficult to specify with precision. Irregular Waves 158. Irregular waves were generated, recorded, and analyzed for 500 . waves, contrary to the monochromatic wave tests that were analyzed for 15 waves. Because wave height and period varied, a longer record was required for the irregular wave tests to obtain a statistically strong confidence interval for calculating the wave spectrum. Reflection and re-reflection between the beach and the wave board could not be avoided in the irregular wave tests. 159. Maximum, significant, and rms wave height were calculated from the time series of the irregular wave trains and are plotted versus distance from the still-water shoreline in Figures 65(a—c)-68(a-—c). Figures 65(a—c) -68(a—c) show that the difference between H,., , H, , and H,,, decreases as the waves enter shallow water, which indicates the distribution of wave heights becomes narrower, i.e., the waves become more "regular." This behavior was also shown in other irregular wave laboratory and field studies, such as Thompson and Vincent (1984) and Ebersole and Hughes (1987). 160. Wave height decay for the plane-beach tests is extremely gentle, making it difficult to define a mean breaker line as discussed by Thornton, Wu, and Guza (1985). The barred tests depict a decrease in wave height immed- iately shoreward of the bar, but the wave height over the bar (Gage 4) was not 120 > Wave Height and Elevation (ft) > Wave Height and Elevation (ft) 1 1 1 aes Il 1 1 10) 5 10 15 20 25 30 35 Distance from Shoreline (ft) a. Case R2000 40 ! ! 1 It —— | 1 ! 0 5 10 15 20 25 30 35 Distance from Shoreline (ft) b. Case R6000 Figure 65. H,,, , H, , and H,,, as a function of distance, m = 1/30 (Continued) 40 > Wave Height and Elevation (ft) re) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) ec. Case R8000 Figure 65. (Concluded) 122 1. 0.5 KO xO *XO > Wave Height and Elevation (ft) XS -1.5 i | | 1 | fl =] 0 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) a. Case R2210 > Wave Height and Elevation (ft) 1. Oo aa) ¢ On x W 0.0 oe -0.5-+ = Hrms | x Hs © Hmax -1.5 L Nl i Nl L Ll i f@) 5 10 15 20 25 30 SYS) 40 Distance from Shoreline (ft) b. Case R6210 Figure 66r) Hes). He. ands asea function of distance, f, = 5 deg (Continued) 123 > Wave Height and Elevation (ft) -1.5 Saeanlis n L fL_ Nl N Nl ) 5 10 15 20 25 30 35 Distance from Shoreline (ft) c. Case R8210 Figure 66. (Concluded) 124 40 > Wave Height and Elevation (ft) -1.5 | ! 1 es i I 0 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) a. Case R2220 Wave Height and Elevation (ft) 1.0 © © 20 0.5- } A xx x< © Q x Y 30 6 SWU 0.0 _osl salt 0 Hrms xX Hs © Hmax -1.5 | Jl iS | | | (0) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) b. Case R6220 inteabres (7, EL 5 ke 5 Gixel Iho, GIS el sebioveielein of distance, f, = 10 deg (Continued) 125 Wave Height and Elevation (ft) | © Oo» 0.57 X Xx xO x > ax Oo fe) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) c. Case R8220 Figure 67. (Concluded) Wave Height and Elevation (ft) eat IL. © 0.5 0 © J © > X XK A Xx Som falas 0.0 oi sea ° Hrms -1.0} xX Hs © Hmax -1.5 1 i i ET EE | J O 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) a. Case R2230 Wave Height and Elevation (ft) 1.0 © © ©O ; Ch , © ae ‘i M ae g X ° x swu =k | L ! L 10) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) b. Case R6230 Figure 68. Ha, , H; , and H,,, as a function of distance, f, = 15 deg (Continued) 127 > Wave Height and Elevation (ft) re) 5 10 15 20 25 30 Distance from Shoreline (ft) c. Case R8230 Figure 68. (Concluded) 35 40 always the maximum in the shoaling transformation. Because a steep decay in wave height occurred between Gages 4 and 5 for tests involving a bar, the height at Gage 4 was used as the breaking wave height in analysis of both the barred and the plane-sloped cases. 161. Statistical wave heights derived from data from Gage 4 were com- pared with Equation 34, the breaker height expression for irregular waves (Goda 1975). Goda assumed the breaker height would vary and gave a range to the coefficient A to predict this variation. In the present study, it seemed reasonable to compare the higher value of A with the maximum wave height measured at Gage 4. Figure 69 shows the predicted wave height given by Equation 34, in which the coefficient A was the maximum specified by Goda plotted versus H,,, normalized by (H,;), . The line of perfect prediction is also plotted in Figure 69. The Goda equation predicts the lower measured values well, but for tests with steep—faced bars, resulting in higher €,-values , the equation gives an overprediction. 162. Because Equation 34 (Goda 1975) predicted H,,, well using the upper limit of A , the lower limit of A was used to estimate a statisti- cally averaged wave height. Goda arbitrarily selected the lower limit to be two-thirds of the upper limit; therefore, significant wave height was used since it is the average of the highest one-third wave heights. Predicted values of (0), using A = 0.12 in Equation 34 were plotted against the significant wave height normalized by (H,), at Gage 4 in Figure 70. The predicted values of (%&), agree well with the measurements for B; < 10 deg , but are too high for f, = 15 deg. 163. With the reasonable success in associating A-values with H,,, and H a procedure was developed to determine the A-value required to s > predict H,,./(H;), . Because the Rayleigh distribution gives H, = OCH sae; the minimum value of the empirical coefficient recommended by Goda (1975) was reduced by /2, which yielded A = 0.085 . Predictions with this value of A gave reasonable results for all seaward angles, but it underestimated the measured values for f$, < 5 deg. Therefore, the coefficient was rounded to A= 0.09 . The predicted versus measured values using A = 0.09 are shown in Figure 71. 164. Equation 34 predicts the measured wave heights well for the plane- slope cases, and for the barred slope cases for $, < 10 deg. The equation overpredicted wave heights in the tests with the steeper bar angle 129 , Predicted Hmax/(Hs)o (A = 0.18) | B 2.2 z (dea) © 1.9 (m = 1/30) + 8 * 10 a 6 0.8 (LL 1 i fe sift \ l l 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 Measured Hmax/(Hs)o Figure 69. Predicted Hyax/(H;), of Goda (1975) (A = 0.18) as a function of measured H,,;/(Hs)o0 , predicted Hs/(Hs)o (A = 0.12) 8 a (deg) 1.47 a O 1.9 (m = 1/30) = * 10 Vr ¥ O 15 ¥ | ! 0.6 0.8 1.0 1.2 1.4 Measured Hs/(Hs)o Figure 70. Predicted (H/H;), of Goda (1975) (A = 0.12) as a function of measured (H/H;). 130 » Predicted Hrms/(Hs)o (A = 0.09) s Bi (deg) Oo O- 1.9 (m = 1/30) 1.0 a & * 10 Bl! 415 0.8 + 0.6 0.4 0.4 0.6 0.8 1.0 1.2 Measured Hrms/(Hs)o Figure 71. Predicted H,,,/(H,;), of Goda (1975) (A = 0.09) as a function of measured H,,./(H;), (B; = 15 deg). Other procedures are clearly required to predict heights over engineering structures having steep faces. 165. Figures 72(a—c) to 75(a—c) show H,,./h as a function of distance from the shoreline, where h is the depth evaluated from SWL. The tests con- ducted with bars show a significant increase of H,,,/h over the bar. The increase results from the water depth becoming shallow and waves becoming higher by shoaling and becoming nonlinear in shape over the bar. Wave height to water depth decreases directly shoreward of the bar because water depth is deeper and a majority of the waves broke on the bar. The ratio continues to increase as water depth decreases in the surf zone for all tests, including tests on the plane slope. The plots indicate that wave height does not decay consistently with the decrease in water depth; therefore, H,,,/h is not constant through the surf zone for either barred profiles or plane-sloping beaches. 166. On the basis of their field measurements, Sallenger and Howd (1989) found H,,,/h to be constant, independent of offshore wave conditions, and stated that the wave distribution was energy saturated through the inner surf zone. They concluded that offshore migration of nearshore bars is, therefore, not necessarily associated with the break-point processes; however, 131 Hrms/h 5 1. 0 5 10 15 20 25 30 Distance from Shoreline (ft) a. Case R2000 Hrms/h 1.5 1.07 0.5 - RK -1.5 L | L L ! ! (0) 5 10 15 20 25 30 Distance from Shoreline (ft) b. Case R6000 Figure 72. H,,,/h as a function of distance, m = 1/30 (Continued) 372 40 Hrms/h 1.5 0 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) c. Case R8000 Figure 72. (Concluded) 133 | *K -1.5 ! a 1 1 it 1 {i fe) 5 10 15 20 25 30 35 Distance from Shoreline (ft) a. Case R2210 1.5 10) |= * * * 0.5 * KK 0.0 -1.5 1 1 1 1 1 | _ fe) 5 10 15 20 25 30 35 Distance from Shoreline (ft) b. Case R6210 Figure 73. Heme/h as a function of distance, #, = 5 deg (Continued) 134 Hrms/h 1.5 1.05 * -1.5+ 1 1 ! | | l nt (9) 5 10 1) 20 25 30 35 40 Distance from Shoreline (ft) c. Case R8210 Figure 73. (Concluded) 13'5 Hrms/h 5 -1.5 \ N SS n ! eben Nat} ) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) a. Case R2220 panini 1. we 1.0} < as * | Seager heads 0.0 -0.5} -1.0 Bas L i l ! il L | 0 pa ais) 15 20 25 30 35 40 Distance from Shoreline (ft) b. Case R6220 Figure 74. H,,,/h as a function of distance, #, = 10 deg (Continued) 136 Hrms/h 5 1.07 * 0.5 * * = 1.5 EE 1 =| i 1 it ! re) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) c. Case R8220 Figure 74. (Concluded) 137 1.5 1.0- * * [ % 0.5 * x eK 0.0 -0.5+ mee -1.5 Nl site ites fis Nl I \ re) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) a. Case R2230 Hrms/h 5 * 1.0 * * 2 u * 0.5 ‘) 0.0 e| Sido = 1.5 i | —Jt i | |= IL ) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) b. Case R6230 Figure 75. Hy;/h as a function of distance, f, = 15 deg (Continued) 138 Hrms/h 5 1. -1.5 1 1 | 1 al 1 fe) 5 10 15 20 25 30 35 40 Distance from Shoreline (ft) c. Case R8230 Figure 75. (Concluded) the results of the present study bring into question their conclusion that Hims/h is constant through the surf zone on a barred profile. Sallenger and Howd measured wave heights seaward of an inner bar, and Figures /2(a—c) to 75(a-c) show H,,,/h-values are uniform seaward of the bar. Because Sallenger and Howd apparently did not measure wave height shoreward of the inner bar, they did not find a varying H,,,/h . 167. Runup measurements for irregular waves were taken for 13 to 34 waves near the start of the test, prior to estimated re-reflection from the wave board; therefore, the irregular wave runup data are not statistically reliable. Average wave runup normalized by (H,), is plotted as a function of €, using the seaward angle #, in Figure 76. The predicted values by Equation 55 (Ahrens 1981) are shown as the solid line. Equation 55 estimates runup well for the plane-slope cases, but underpredicts the measurements for tests with barred profiles. Runup normalized by (H,), is independent of €, for the barred profiles. Runup was plotted as a function of €, with the 1/30 slope used as the primary angle (Figure 77). Equation 55 gives good results if the 1/30 slope is used in the surf similarity parameter. 139 R/(Hs)o 0.5 0.6 0.8 1.0 1.2 1.4 1.6 Figure 76. R/(H,), as a function of €0 R/(Hs)o 0.25 B 0.207 0.05 0.00 : 0.25 Figure 77. R/(H,), as a function of €, using m= 1/30 as the predominant angle 140 168. Runup normalized by H,,, for irregular waves was plotted versus seaward bar angle in Figure 78. Runup values for monochromatic waves were included in the plot to illustrate the differences between wave types. Runup of irregular waves was higher than that of monochromatic waves of equivalent deepwater wave steepness, which is attributed to some waves overtaking others and causing a higher uprush, and the back wash of large waves obliterating the incoming smaller waves, which results in measurements of only higher runup events. R/Ho and R/(Hrms)o 0.00 i i | | i it 1 (@) 5 10 15 20 25 30 35 40 45 « Monochromatic waves ** Irregular waves p (deg) Figure 78. R/H,,, as a function of #6, Measurement Errors 169. All tests were conducted with utmost care. However, some error was introduced because of equipment limitations. 170. A description of the data collection system used can be found in Turner and Durham (1984). Wave rods were calibrated before the first test of the day using the microcomputer that collected the wave data. Calibration began when a signal was sent from the microcomputer to the motors attached to the wave rods. The signal caused the motors to move the wave rods a predeter- mined vertical distance, and a voltage reading was taken by the rod at the 141 depth. The procedure was repeated until readings had been made for 11 posi- tions over the calibration range of the rods. The voltage readings were con- verted to vertical distances from the horizontal datum, SWL, and compared with the known physical position of the rod at each reading. The voltage readings of the rods were accepted if the maximum deviation for each of the 11 posi- tions was less than 0.01 ft. The maximum deviation was usually less than 0.01 ft, and deviations greater than this were typically caused by residue on the rods. In these cases, the rod was cleaned, and the procedure was repeated for the individual rods that exceeded the tolerance. 171. The SWL was measured by a point gage, graduated to 0.001 ft, lo- cated in the horizontal section of the tank. The water depth was checked before conducting each test. Measurements of water depth at gages and at the bar were made with a rule that was accurate to 0.05 ft. The same rule was used for measuring seaward and shoreward faces during construction of the bars. All measurements during video analysis were made by scaling the 2-in. grid as it appeared on the monitor to 1/4 in., which gave an error of +1/8 in. 172. Monochromatic waves were analyzed and averaged for 15 waves from the wave record, and measurements from videotapes were averaged for 10 waves. Although the waves were monochromatic (constant period) and regular (constant height), it was important to average the measured quantities to reduce fluc- tuations caused by reflections and currents present in the tank. It was also critical to limit the number of waves averaged to those that were not re- flected from the wave board. 173. To illustrate the importance of averaging values, breaker depth index was plotted as a function of H,/L, for five tests, as shown in Fig- ure 7/9. The average value of y, is marked by the symbol, and the range be- tween maximum and minimum values of y, computed from the 10 measured values of each test is shown by the vertical line at each wave steepness. It is seen that differences of 15 percent in individual values occurred despite care to eliminate wave reflection and seiching. The considerable scatter in the data, often much more than 15 percent, shown in many figures of this report is at- tributable in part to the difficulty in defining the given quantity, such as breaking of collapsing waves, and not to direct measurement. The process of averaging 10 values as done in the present study helped to reduce scatter due to individual wave variation and thus presented a more realistic picture of the variability in the developed relationships. 142 1.07 —- ! 0.6 0.00 0.02 0.04 0.06 0.08 0.10 Ho/Lo Figure 79. Range between maximum and minimum values of y as a function of H,/L, 143 PART V: SUMMARY AND CONCLUSIONS 174. The purpose of this study was to examine the macroscale features of wave breaking over bars and artificial reefs in a wave tank. All major experiments on wave breaking prior to this one were conducted on plane beaches, and relationships for breaker indices and plunge distances developed from these studies were expected to be invalid for complex profiles. The present experiments demonstrated that waves break differently (their forms are different) on a barred profile as compared with on a plane slope, and wave breaking properties such as breaker index and plunge distance also differ. Equations were developed incorporating seaward bar angle and deepwater wave steepness for describing breaker indices, plunge distance, splash distance, penetration distance, and vortex area. New relationships were also developed for breaker index and plunge distance based on a large data set developed from previous plane-slope experiments. Monochromatic Tests 175. Breaker type transition values for spilling and plunging waves over barred profiles were lower than the commonly accepted values given by Battjes (1975) that apply to plane slopes. A strong return flow was observed if seaward bar angles were steep or wave steepness was low. The return flow influenced breaker type by causing the wave to "trip" over the bar, rather than to break as a result of becoming unstable due to limiting depth. The return flow was also strong if the shoreward bar angle was horizontal (ter- raced bottom). Depth-limited or well-behaved breaking occurred for conditions in which the seaward length of the bar was at least 75 percent as long as the wavelength at breaking. Complex breaking occurred for conditions in which the bar faces were steep, and the seaward bar length was shorter than the breaking wavelength. 176. Breaker depth index was greatly influenced by return flow, based on visual observations. Breaker depth data were scattered for collapsing waves and for plunging waves with large values of the surf similarity param- eter. A relationship was developed for breaker depth index for plunging and spilling waves having seaward bar angles less than 10 deg, which restricted the analysis to depth-limited breaking (small return flow). The seaward bar 144 angle $, created under natural conditions is typically gentler than 10 deg. The relationship for barred profiles was compared with the equation developed in this study for plane slopes. The plane-slope equation gave smaller values of breaker depth index than the barred profile equation for f$, = 5 deg for low wave steepness and all wave steepnesses for 10-deg seaward angles. The plane-slope equation gave larger breaker depth values than the barred profile equation for f£, = 5 deg and 0.02 < H,/L, < 0.09 177. Breaker height index was a stabler parameter for conditions in- volving a strong return flow. Breaker height as a function of deepwater wave steepness gave the same trend for bars as data collected on plane slopes. An equation for the experimental data was expressed in the same form as the analogous plane-slope equation. Breaker height decreased for 6, > 10 deg at high wave steepnesses, but if wave steepness was low, breaker heights in- creased as $, increased. For high-steepness waves and steep bars, wave shoaling was mostly caused by the plane slope because the bars were short and had little effect on the waves. Return flow was not strong for these condi- tions. If wave steepness was low and bars steep, a strong return flow was present. A heuristic derivation was made to illustrate breaking wave height increases as return velocity increases. 178. Plunge distance normalized by breaking wave height was relatively constant for larger values of the surf similarity parameter, but increased at smaller values of the surf similarity parameter. Plunge distance was shorter by a factor of approximately 2 if bars were present as compared with the analogous plane-slope case, because breaker type differs with bars present. Plunge distance normalized by deepwater wave height showed no dependence on the surf similarity parameter, which indicates the distance traveled by the plunging wave depends on the local (breaking) wave height. 179. The average of plunge distance divided by splash distance was approximately unity, which agrees with findings of Galvin (1969) for plane slopes, but the plunge-to-splash distance ratio increased for smaller valued surf similarity parameters. Plunge distance was also greater for smaller surf similarity values. Splash distance normalized by deepwater and breaking wave height was essentially constant if expressed as a function of surf similarity, but the data were widely scattered. 180. For barred profiles, the total distance traveled by the wave crest from the break point was called the penetration distance. Penetration 145 distance was found to increase for small values of the surf similarity param- eter. Penetration distance normalized by deepwater wave height showed no dependency on the surf similarity parameter and appears to be dependent on local wave height only. 181. Vortex area was measured at the plunge point for 34 tests with the seaward bar face angle f, < 10 deg. The plunge point was chosen because of its uniqueness to every breaking wave, and it is the point of vortex forma- tion. Vortex area was found to increase with values of the surf similarity parameters and approach an asymptotic value for waves in the collapsing region. 182. Wave reflection was measured seaward of the bar for all tests and was found to increase for steeper bars, which was expected, although reflec- tion did not notably increase with bar slope. Reflection coefficients from the experiment were compared with relationships of Miche (1951) and Battjes (1975) developed for plane slopes. The equation of Miche overpredicted re- flection for most of the experimental data. The Battjes equation predicts increasing reflection with the surf similarity parameter; however, measured values plotted as near-constant and were independent of the surf similarity parameter. Reflection coefficients were also correlated with the reef reflec- tion parameter of Ahrens (1987) developed for reef-type breakwaters. The Ahrens equation gave better agreement, but the equation did not fully repro- duce the trend. A parameter to quantify wave reflection from bars and reefs was not identified, and additional study is required. 183. Wave runup was measured for all tests. Runup was normalized by deepwater wave height and plotted as a function of seaward bar angle, but seaward angle showed little effect on runup. Wave runup was also constant, but scattered, if plotted versus the surf similarity parameter; however, runup was found to increase with the surf similarity parameter if the beach slope of 1/30 was used to compute €, instead of seaward bar angle. The average runup normalized by H, was 0.76€, , whereas Battjes (1975) found that R/H, = 1.0€, . Lower measured values of runup could result from additional energy dissipation caused by bars; however, runup was also lower than predicted by Battjes for the present plane-slope tests. A complete analysis of wave height decay and setup is required to reach a relationship for runup for barred profiles. Wave data were collected through the surf zone and are presented in Appendix C. 146 Irregular Wave Tests 184. Maximum breaker height from irregular wave tests was compared with the predictive expression of Goda (1975) using A= 0.18 . Maximum wave height was predicted well for $, < 10 deg . Significant wave height was also compared with the Goda expression with A = 0.12 , which also gave good results for $, < 10 deg . The minimum coefficient specified by Goda (A = 0.12) was reduced by /2 to predict values for rms wave height based on a Rayleigh distribution, which gave A =0.09 . The equation reasonably pre- dicted wave heights for $, < 10 deg and overpredicted wave heights for B, = 15 deg . 185. Wave height-to-water depth ratio was plotted as a function of hor- izontal distance from the shoreline for irregular waves. The ratio was not constant through the inner surf zone, but increased uniformly into shallow water for plane-sloping beaches and was double peaked for single barred profiles. 186. Irregular wave runup was normalized by (H,), , plotted as a func- tion of the surf similarity parameter, and compared with the expression of Ahrens (1981). The Ahrens expression predicted runup for the plane-slope tests well, but overpredicted runup with bars in the tank. The predicted values of Ahrens showed good agreement with the measured values if the fore- shore beach slope (1/30) was used to compute the surf similarity parameter rather than seaward bar angle. Runup thus appears to depend mainly on the foreshore slope and not on the presence of a bar. Recommendations for Future Study 187. This study verified the strategy of using solid but adjustable bars for examining wave breaking on realistic nearshore bottom shapes. It is recommended to continue the line of study to more exact replication of bar forms and sizes, although gently sloping bars will require considerable resources, equipment, and effort to build and emplace. 188. It was clearly demonstrated that bars alter the characteristics of breaking waves. This is due to the steeper seaward bar slope, volume of water shoreward of the bar, and the speed of the return flow. 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"Wave-Induced Geomorphic Response of Eroding Beaches - with Special Reference to Seaward Migrating Bars," Proceedings of Coastal Sediments '87, American Society of Civil Engineers, pp 788-801. Sunamura, T., and Horikawa, K. 1975. "Two-Dimensional Beach Transformation Due to Waves," Proceedings of the 14th Coastal Engineering Conference, Ameri- can Society of Civil Engineers, pp 920-938. Suquet, F. 1950. "Experimental Study on the Breaking of Waves," La Houille Blanche, No. 3. Svendsen, I. A. 1984. "Wave Heights and Set-Up in a Surf Zone," Coastal Engineering, Vol 8, pp 303-329. 1985. "Wave Attenuation and Set-Up on a Beach," Proceedings of the 19th Coastal Engineering Conference, American Society of Civil Engineers, pp 54-69. Thompson, E. F., and Vincent, C. L. 1984. "Shallow Water Wave Height Param- eters," Journal of Waterway, Port, Coastal and Ocean Engineering, American Society of Civil Engineers, Vol 110, No. 2, pp 293-299. Thornton, E. B., and Guza, R. T. 1983. "Transformation of Wave Height Dis- tribution," Journal of Geophysical Research, Vol 88, No. C10, pp 5925-5938. Thornton, E. B., Wu, C. S., and Guza, R. T. 1985. "Breaking Wave Design Criteria," Proceedings of the 19th Coastal Engineering Conference, American Society of Civil Engineers, pp 31-41. Turner, K. A., and Durham, D. L. 1984. "Documentation of Wave-Height and Tidal Analysis Programs for Automated Data Acquisition and Control Systems," Miscellaneous Paper HL-84-2, US Army Engineer Waterways Experiment Station, Vicksburg, MS. Van Dorn, W. G. 1977. "Set-Up and Run-Up in Shoaling Breakers," Proceedings of the 15th Coastal Engineering Conference, American Society of Civil Engi- neers, pp 738-751. 154 Van Oorschot, J. H., and d'Angremond, K. 1969. "The Effect of Wave Energy Spectra on Wave Runup," Proceedings of the 11th Coastal Engineering Con- ference, American Society of Civil Engineers, pp 888-900. Vellinga, P. 1982. 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"Field Study of Breaking Wave Charac- teristics," Proceedings of the 16th Coastal Engineering Conference, American Society of Civil Engineers, pp 487-506. 155 APPENDIX A: BREAKER DATA 1. This appendix consists of data generated in the present study and data obtained from previous laboratory experiments conducted with periodic waves. 2. Results from the present study (Tables Al and A2) represent average values of 10 to 15 consecutive waves. Analysis of the data was begun after waves reflected off the concrete slope (m = 1/30) and was ended before reflec- tions from the wave board had returned to the bar. 3. Data from previous studies (Tables A3-Al13) were selected from exper- iments in which the slope was planar and fixed, deepwater steepness was given or could be calculated from the data, and no structures were present in the wave tank that would alter wave breaking. 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A8 S090 COCCOOO COOCOCOCOO OOOO oO OOOO O Oooo o .050 .050 .020 .020 .020 .020 .020 .020 .020 PNN NNFOP NPRPRPFP OORPRR BEB BPHP PEPPY. Table A3 (Concluded) PR PR Dw) OFA ODWUN WEUUNH OFUDLYO Ww oO AQ e090 C0000 COCCOCOO CCOOOoO oOo oOO aaaao . 264 . 188 . 264 . 168 .234 .137 S090 C9090 00 COCO O CoCOOCoOoO OOOO O aaa MCo 99090 COCOCOOG COC COCOoO AOaeaOOoOO aoOaoaaOaoOoO aaaMdoe So S OOOO Se OOCeS © ooo oocoocoo°o ooo Run SIOooee OOeoee OEGeQee GOOO0O0Of G2eEOOSo GeEOeQe OeEeOeGOSO OOEe© sec PRPRPRP RP PRP RP RPP PRR RP RP PRP RRP PRR RPP PRP RP RPP PRP RP RPP PRP PRP Pe COMO MOCO CO DODOANAAAARAA ADEHL HF FHKE FHKF FKEENNM NNNNNM NHONMWNMNPD Table A4 Results of Horikawa and Kuo (1967) Ib H, Hy hy cm cm H,/L, cm cm 224.6 8.8 0.039 8.6 W265 224.6 9.1 0.041 Mei U2 5 224.6 12,0 @,@53 N22 13.8 224.6 14.8 0.066 14.2 16.3 224.6 11.7. 0.052 11.6 16.3 224.6 13.0 0.058 U3} Ub 16.3 224.6 13635 @,O59 13.2 oS 224.6 135) ORO6O 13.3} 20.0 224.6 14.6 0.065 14.7 Loo2 224.6 FIA 2 ORO6S 13.9 Zoli os) 224.6 ito @,053 18.2 Ale 3} 224.6 16.4 0.073 16.3 26.3 305.8 8.0 0.026 7.8 12.5 305.8 9.0 0.029 9.3 13.8 305.8 90.7 O,@32 9.4 15.0 305.8 11.6 0.038 11.6 15.0 305.8 No O.O37 11.5 16.3 305.8 W269 0,042 1263 18.8 305.8 13.5 0.044 13.0 20.0 305.8 14.1 0.046 13.5 Dibod 305.8 14.8 0.048 13.8 Zod 305.8 15.3. 0.050 16.4 a5) 305.8 16.7 0.055 N7.8 25.0 399.4 7.9 O.O20 79 ities 399.4 9.3 OW .O25 10.2 12.5 399.4 10.5 0.026 10.3 15.0 399.4 I OR028 11.7 16.3 399.4 12.4 0.031 3.7 16.3 399.4 Bol @,033 14.3 16.3 399.4 14.2 0.036 W362 17.5 399.4 15.9 0.040 14.7 Dies 399.4 16.8 0.042 15.0 22 od 505.4 Uo& @.OlS 8.8 11.3 505.4 S55) OOS 9.2 M225) 505.4 9.0 0.018 9.4 15.0 505.4 9,6 @,Ol9 10.0 13.8 505.4 1@.5 @),@zZ2iL 10.8 15.0 505.4 LORS ONO 22 11.6 16.3 505.4 ti.3 O,022 11.8 16.3 505.4 ito @,023 IS 18.8 (Continued) (Sheet 1 of 3) A10 QAQvreo GooQgee OeqQoeoeem OoEeQeoe OQeeQF GOO OF BOO OoO OOo e Se sec RRR ke ke PR rR re rm Fr Bh ND Bh BM PH NH PO NO BR RO RP PS NM BM PN PD PO Bh NM KN PO POS PRR rr PREEEP PREFER EF RENNDN YNNUNNN YNNNOO COCCO COOCO wanna Table A4 (Continued) Lg H, Hy hy cm cm H,/Lo cm cm 505.4 Sl ORO26 12.4 18.8 505.4 13E95 OR02%7 12.6 20.0 505.4 14 oe LO RO29 SS Dios 505.4 H5)55) OSWsKo) 13.8 Ako 3} 505.4 154 0/030 15.0 Dodd) 624.0 8.5 0.014 8.3 13.1 624.0 8.9 0.014 Bod 1378 624.0 9.8 0.016 10.4 15.0 624.0 IO .7 @ Walz dbs 2 15.0 624.0 1S PO FOLS 11.8 ESO 624.0 1S OOS ZO 16.8 624.0 N2oO Oc@il) iiLs 9 16.5 624.0 W256 O.O20 M269) 20.0 624.0 1 SOMO 2s 13.6 20.0 624.0 14.4 0.023 14.6 2S 624.0 15.2 0.024 15.4 ais 3) 624.0 16.3 0.026 Soo) 25.0 755.0 103 OO O17, 12.8 Woe) YD5) 00 T2006 10.0 14.5 755.0 10.4 0.014 PISS ILS) 5 2 755.0 10.8 0.014 iL 2 Doo 755.0 9.5 0.013 7.6 11.8 755.0 8.6 0.011 11.3 1B) 755.0 Soo) Oo @iLt 10.8 13.5 755.0 6.8 0.009 8.7 11.8 755.0 6.3 0.008 52 10.8 755.0 960 OW W07/ 8.4 9.8 Pra!) C7 ORO006 6.0 6.8 305.8 16.1 0.053 GE 20.2 305.8 14.6 0.048 1S).5) 18.4 305.8 14.0 0.046 15.1 18.3 305.8 13.3 0.043 6 U Ise 72 305.8 11.7. 0.038 Wak 18.2 305.8 6.9 | 0,023 6.3 7.5 305.8 8.2 0.027 UD 6.0 305.8 9.2 0.030 9.4 6.0 305.8 10.5 0.034 ORS, 8.0 305.8 2 AOR SV OROS9 ial 5 ab 8.0 305.8 1b SOROS Ii 14.0 305.8 OKO OR033 10.4 58 (Continued) (Sheet 2 of 3) All SOO OOOO O GOEeEeEeQ OOOeC©S NN NYNNNNNY NNNNKP PREP PB WH NNNNNHM NNHNNF RRERE Table A4& (Concluded) SK) 305. 305. 755. YDS VS. 755. 755. 755. YDS YD 755. US « 825. No COOOCOO OG OOOO CO C&O Oo Co oo Al2 NR WWF OMN WOWDHDHO WENN Nh Oo OOCOOOo OOOO O OOOO ®e Hy hy, cm cm 6.9 7.5 13.9 14.0 iL}, IL 16.0 13.2 16.0 15.7 20.5 16.6 20.5 6.5 eS, 7.8 10.8 8.3 9.3 10.5 9.3 123 10.5 11.3 23) 11.6 12.8 Sy, 113} 2S) 12.33 13.3 12.8 5}. 3} 15.6 16.5 (Sheet 3 of 3) Table A5 Results of Galvin (1969) T Lo Hy Run m sec ft ft H,/Lo 1 OROS 2.0 20 FON ORES 0.0089 ZOOS 4.0 8250 Oil} 0.0016 3 @,05 5.0 WAS cL Oo 0.0009 4 O05 4.0 82.0 0.23 0.0028 SOROS 5.0 W2S5iL O.ike/ 0.0013 6 0.05 6.0 184.5 0.13 0.0007 7 0.05 6.0 184.5 -- = 8 0.10 1.0 Doth OsnY 0.0378 9 0.10 2.0 20.5 == Se 10 0.10 5.0 128.1 -- == 11 0.10 6.0 184.5 0.15 0.0008 12 0.10 1.0 Jo O23 0.0448 13 0.10 2.0 20.5 0.09 0.0045 14 0.10 2.0 AVS OWo27/ 0.0133 15 0.10 5.0 HASoIk — Oo) 0.0018 16 0.10 2.0 20.3 @adul 0.0052 17. =0.10 2.0 20.5 0.32 0.0155 18 0.10 4.0 82.0 0.23 0.0028 19 0.20 1.0 Jel Oily) 0.0378 20 0.20 1.0 Sol O23 0.0448 21 0.20 1.0 5.1 0.26 0.0503 22 0.20 2.0 20.5 0.11 0.0052 Table A6 Results of Saeki and Sasaki (1973) oT LS H, Run m. sec cm cm H,/L, iL 0.02 od 263.6 10.3 0.039 2 0.02 25) 975.0 D583 0.005 [ae aro IQ) HOO Googe QAoQqQeq Oeocqe cm 10.6 Ww WwW Qe OQOQOO COEeee OQOeC2 OOO CoS& OE OoOoPrPe PeeoeoP PoOOoOWh WWF iP © Run © ON DH F WwW HS FE NNN NY FP PP BP PP BP PP Pp OM Pooeouvwvpoo aus 6 SS FE Table A7 Results of Iwagaki et al. (1974) T L, H, Hy, hy m sec cm cm H,/Lo cm cm 0.10 1.0 156.1 al 0.058 9.7 11.1 0.10 iL.) 156.1 6.6 0.042 6.8 755 0.10 1.0 156.1 4.4 0.028 5.1 6.1 0.10 1.5 35) 3} 8.1 0.023 10.1 12.0 0.10 1.5 351.3 6.7 0.019 9.9) 9.8 0.10 D) 351.3 4.6 0.013 6.8 6.7 0.05 1.0 156.1 11.4 0.073 10.9 15.8 0.05 1.0 USO «JL 8.0 0.051 8.4 10.6 0.05 1.0 156.1 4.8 0.031 07 6.8 0.05 eS) 351043 iL 2 0.032 12.8 14.8 0.05 135) 351063) sd 0.019 8.3 ORS 0.05 M5 351.3 36d) 0.010 6.2 6.0 0.05 20) 624.5 6.9 0.011 Dak 12.0 0.05 2.0 624.5 5.0 0.008 8.0 Doll 0.05 2.0 624.5 Sod 0.005 o8} 6.3 0.03 1.0 156.1 8.0 0.051 8.1 11.8 0.03 1.0 156.1 6.1 0.039 6.6 Dod 0.03 1.0 156.1 4.1 0.026 4.4 Oo 7/ 0.03 oD S53 8.8 0.025 10.9 12.8 0.03 eS) SIL, 3) 5.6 0.016 Uo) 9.9) 0.03 2.0 624.5 6.9 0.011 9.6 12.3 0.03 2.0 624.5 5) 0.008 8.3 99 0.03 2.0 624.5 Sod 0.005 59) 6.9 Al4 Run o ON DN FF WwW HS FP PPP PP eB UF wWwNHeE O Se 2 2 © 20 2 oSo 2 2 2 © © CGC C6 © .033 .033 .033 .033 Results of Walker (1974b) Pat SS OSS) SS eS SL mS oS eS ee Table A8& iS ¢ ©) Ce) dS 8) &) FO. SS Cr EC WH ws) A15 So 2 eC Leo CEC eC Oo EC CGC ©& © CO CO OC © .145 o LULZ elt )2. . 200 .247 Ooo ff G2 2 2 © © 2&2 © © © eC © .007 .004 .005 .010 .017 So Oo Oo Oo 2 oOo 28 2 2©c& S&F © oO © oC © Yr & 2 2 8S 2 282 2 2 2 2 © © © © Table AQ Results of Singamsetti_ and Wind (1980) Te ibs, H, H, hy, Xp Run m sec m m H,/L, m m m A5-28 0.20 158 3ooy/ 0.105 0.028 0.117 0.131 0.42 A5-39 0.20 Loo) Bdo/6 0.149 0.040 0.193 0.160 0.61 A5-40 0.20 1,23 2.5/7 0.102 0.040 0.156 0.124 0.46 A5-27 0.20 1.23 2.57 0.071 0.027 0.097 0.103 0.34 A5-18 0.20 1.55 3.76 O.067 0.018 0.095 0.078 0.32 A5-47 0.20 1.04 1.68 0.079 0.047 0.106 0.082 0.32 A5-48 0.20 1.23 2.57 0.125 0.048 0.160 0.134 0.57 A5-43 0.20 1.04 1.67 0.072 0.043 0.091 0.079 0.39 A5-60 0.20 1.04 1.68 0.101 0.060 0.117 0.099 0.36 A5-32 0.20 12 461 SORTA6N 5 0032 0.184 0.195 0.64 1 Novo Mn OAC) LjJ2 &,62 0.097 0.021 @, 125 0.117 0.41 A5-54 0.20 1.283 2.97 0.138 0.054 0.162 0.139 0.57 B5-41 0.20 L283 2.57 0.105 0.041 0.121 0.104 0.46 B5-29 0.20 1283 2.5¢ 0.076 0.029 0.089 0.083 0.34 Bxsl7 O20 1.55 3.7/6 O.006 0.017 0.093 0.098 0.32 B5-49 0.20 1.04 1.68 0.084 0.050 0.087 0.082 0.32 B5-50 0.20 1,28 2,57 0.129 0.050 0.150 0.134 0.57 B5-42 0.20 1.04 1.68 0.071 0.042 0.077 0.099 0.39. B5-60 0.20 1.04 1.68 0.102 0.060 0.118 0.099 0.36 B5-31 0.20 1672 463 0.142 0.031 0.184 0.195 0.64 B5-21 0.20 WoI2 &o.6i OO 0.021 0.124 0.177 0.41 A10-29 0.10 eS) Sad 0.108 0.029 @), 137/ 0.129 0.70 A10-39 0.10 1.55 3.79 0.146 0.039 0.169 0.200 1.05 A10-37 0.10 128° 2.56 0.095 0.037 0.118 0.129 0.68 A10-26 0.10 L283 2.57 0.068 0.026 0.086 0.108 0.53 A10-20 0.10 aoon eS. 0.074 0.020 0.111 0.103 0.55 A10-45 0.10 LoO4 1.67 0.075 0.045 0.091 0.113 0.60 A10-47 =0.10 128 2.55 OW.1480 0.047 0.135 0.146 0.83 A10-42 0.10 WoO IL, Od 0.071 0.042 0.073 0.097 0.53 A10-62 0.10 14 i1,67 ©1038 0.062 0.106 0.129 0.65 A10-28 0.10 i572 4,62 0.132 0.029 0.169 0.186 0.93 A10-19 0.10 e672 4,63 0.089 0.019 0.141 0.117 0.62 A10-53 0.10 T6238 2odd 0.134 0.053 0.150 0.184 0.80 B10-28 0.10 Leos 88.75 0.109 0.029 0.141 0.130 0.60 B10-41 0.10 ee esi S) 0.151 0.040 0.170 0.188 0.88 B10-40 0.10 1,28 2.56 0.103 0.040 0.118 0.131 0.55 B10-29 0.10 1,28 2,55 0.075 0.029 0.101 0.090 0.50 B10-20 0.10 155 3.7/5 0.077 0.020 0.119 0.108 0.55 B10-48 0.10 iLO 1,66 0.080 0.048 0.086 0.090 0.58 (Continued) (Sheet 1 of 3) Al6 Table A9 (Continued) T Ie H, Hy hy X Run m sec m m H,/Lo m m m B10-50 0.10 1.23 2.55 Ootwe 0.050 0.143 0.173 0.75 B10-42 0.10 L.@3) 1,66 © ,070 0.042 0.078 0.078 0.43 B10-62 0.10 1.03 1.66 0.102 0.062 0.112 0.135 0.67 B10-30 0.10 PTS ai 5 on OS; 0.030 0.175 0.185 0.90 B10-19 0.10 Lol 4455 0,086 0.019 0.140 0.124 0.65 B10-55 0.10 L283 2.58)» OW 0.055 0.156 0.188 0.90 A20-30 0.05 lea) ° Sod) 0.114 0.030 0.140 0.170 0.53 A20-41 0.05 LoD Sodd 0.156 0.042 0.174 0.202 0.75 A20-39 0.05 1.28 2.54 0.099 0.039 0.115 0.127 0.59 A20-32 0.05 WAS 2682) 0.081 0.032 0.097 0.102 0.60 A20-19 0.05 1.55 3.74 0.072 0.019 0.106 0.103 0.50 A20-52 0.05 1.04 1.68 0.088 0.052 0.088 0.108 0.66 A20-47 0.05 Wes | Asse) 0.121 0.047 0.135 0.174 0.89 A20-42 0.05 1.04 1.68 0.070 0.042 0.079 0.093 0.45 A20-59 0.05 1.04 1.68 0.099 0.059 0.101 0.130 0.51 A20-29 0.05 1.73 4.65 0.135 0.029 0.176 0.202 0.89 A20-20 0.05 1.73 4.69 0.091 0.019 0.133 0.125 0.75 A20-62 0.05 1,23 255) 0.158 0.062 0.163 0.203 0.77 B20-31 0.05 eo® Bole) 0.118 0.031 0.142 0.160 0.82 B20-42 0.05 No95 3075 OoUse 0.042 0.181 0.213 0.78 B20-41 0.05 1.28 2.54 0.103 0.041 0.119 0.135 0.54 B20-33 0.05 Wek Aoos 0.084 0.033 0.101 0.106 0.58 B20-21 0.05 leo So I/O 0.078 0.021 0.106 0.104 0.61 B20-53 0.05 1.04 1.68 0.089 0.053 0.092 0.100 0.59 B20-48 0.05 543 Zoo 0.123 0.048 0.133 0.165 0.67 B20-44 0.05 1.04 1.69 0.074 0.044 0.077 0.083 0.47 B20-61 0.05 1.04 1.68 0.102 0.061 0.101 0.135 0.67 B20-29 0.05 1.73 4.67 0.136 0.029 0.171 0.181 0.81 B20-19 0.05 Weds 45) 0.091 0.019 0.132 0.128 0.68 B20-63 0.05 1.28 2.55 0.160 0.063 0.165 0.193 0.58 A40O-29 0.025 1.55 3.75 0.110 0.029 0.136 0.145 0.65 A40-39 0.025 1.55 3.74 0.146 0.039 0.170 0.203 oo) A40-40 0.025 1.28 2.54 0.102 0.040 0.119 0.140 1.20 A40-28 0.025 1.28 2.54 0.072 0.028 0.093 ORs 0.80 A4O-21 0.025 1.55 3.76 0.080 0.021 0.112 0.118 0.57 A40-51 0.025 1.04 1.68 0.086 0.051 0.096 0.117 =\= A40-42 0.025 1.04 1.67 0.070 0.042 0.079 0.093 0.60 A40-59 = 0.025 1.28 2.54 0.151 0.059 0.159 0.220 -- A40-48 0.025 1.28 2.55 0.122 0.048 0.137 dei 0.60 (Continued) (Sheet 2 of 3) Al7 oo00 ooooooooo0ocoeo0co PPP Pp PRP PRP RPRPRPRP PPP RP Table A9 (Concluded) BePrPPR PrRrPrPNNFPRFWNHDND WW ooo°o oooo0oo 00 00 0000 .095 H,/L, .062 O71 .080 .020 ooo°o 030 .040 041 .029 .022 .050 042 .057 .047 .061 .073 .079 .021 DQOGOOOOOOOOOSO © ooo°o SDOQOQgqoeegeaqeo© hy % m m 0.153 -- 0.169 0.60 0.205 -- O15 ©.75 0.145 0.65 O21 1.25 O19 1.20 0.101 0.70 O13 0.57 0.127 -- 0.093 0.60 0.195 -- 0.166 0.60 0.143 -- 0.164 0.60 0.195 -- O.155 0,57 A18 (Sheet 3 of 3) Run Run Run QoQ OC 2e © Table Al10 Results of Mizuguchi (1981) Ty 1b H, m sec cm cm H,/Lo 0.10 Lo? 224.6 10.0 0.045 Table All Results of Maruyama et al. (1983) wv Ie H, m sec m m H,/L, 0.034 35 dL 14.99 Lo SY 0.091 Table Al12 Results of Visser (1982) ale Ih H, m sec cm cm H,/Lo 10 2.01 630.1 9.8 0.016 10 1.00 156 50 10.2 0.065 10 1.00 156.0 9.6 0.062 05 OZ 162.3 8.5 ORO S2 05 1 83S) 53333663 To® 0.014 05 O70 76.4 6.0 0.079 05 1.02 G2. 3 8.5 0.052 Table Al13 Results of Stive (1985) T Ib H, m sec m m H,/Lo 0.025 eS 5.05 0.16 0.032 0.025 550 39.00 Lo Qil 0.031 Al9 Hp cm 10.0 Hp m 1, 2g) Hp cm 10.5 10.0 Qo7 iL 10.8 5.8 SAO) Hp m NOnOROF Mwowonrun Hh APPENDIX B: TRACINGS OF WAVE FORMS The waveforms in this appendix were traced from a video monitor during analysis of breaker data. The waveforms shown represent a typical wave at incipient breaking for selected classes of cases. Bl v CENTIMETERS Figure Bl. Case 2130 at incipient breaking SCALE -2 ) 2 4 a INCHES = 0 68 CASE sa 2210 CENTIMETERS Figure B2. Case 2210 at incipient breaking B2 -2 0 2 4 CASE os = SS 2320 CENTIMETERS Figure B3. Case 2320 at incipient breaking SCALE -2 ty) 2 4 CASE (Lo. =~ 2440 INCHES -5 i) 5 10 os SS CENTIMETERS Figure B4. Case 2440 at incipient breaking B3 Lk 4110 INCHES -5 0) 5 10 (oo os CENTIMETERS Figure B5. Case 4110 at incipient breaking SCALE PVEPPIOE MHEES a CASE =< = | 4220 INCHES -5 0 5 10 ees CENTIMETERS Figure B6. Case 4220 at incipient breaking B4 -2 () 2 4 CASE ss 4340 INCHES -5 () 5 10 a CENTIMETERS Figure B7. Case 4340 at incipient breaking SCALE =2 0 2 4 INCHES —5 vee i) 10 CASE a 4430 CENTIMETERS Figure B8. Case 4430 at incipient breaking B5 |} CENTIMETERS Figure B9. Case 6120 at incipient breaking al] — (I) CASE 6230 Figure B10. Case 6230 at incipient breaking B6 -2 0 2 4 CASE is 6340 INCHES -5 i) 5 10 kas CENTIMETERS Figure Bll. SCALE =f 0 2 4 = § INCHES “5 0 5 10 (oti: 86 | CENTIMETERS Figure B12. Case 6340 at incipient breaking CASE 6410 Case 6410 at incipient breaking B7 WH CENTIMETERS Figure B13. INCHES CENTIMETERS Figure Bl4. Case 8140 at incipient breaking Case 8220 at incipient breaking B8 |] -2 () 2 4 CASE (oo 8310 INCHES -5 C) 5 10 CENTIMETERS Figure B15. Case 8310 at incipient breaking -2 0 2 4 CASE (o' — | 8430 CENTIMETERS Figure B16. Case 8430 at incipient breaking B9 CENTIMETERS Figure B17. CASE 10130 Case 10130 at incipient breaking SCALE i CASE me @ 2 4 10210 INCHES = (0) 5 10 (oi §6€=~—S ~ CENTIMETERS Figure B18. Case 10210 at incipient breaking B1O CENTIMETERS Figure B19. CENTIMETERS Figure B20. CASE 10320 Case 10320 at incipient breaking Case 10410 at incipient breaking Bll am Ke) APPENDIX C: WAVE HEIGHT DATA MONOCHROMATIC TESTS KEY: T = wave period (sec) H, = breaking wave height (ft) X, = distance from wave board to break point (ft) h, = still-water depth at break point (ft) H,/L, = deepwater wave steepness hice = depth at bar toe (ft) Xtoe = distance from wave board to bar toe (ft) h, = depth at bar crest (ft) X,. = distance from wave board to bar crest (ft) h, = depth at bar trough (ft) X, = distance from wave board to bar trough (ft) 8, = seaward bar angle (deg) 83; = shoreward bar angle (deg) = beach slope = wave height at gage (ft) = still-water depth at gage (ft) Case 2110: T= Mag = O.882 4 = GilabyAs In, = O.508 3%, 3 Oy ise ly = O,302 24 3 OGL883 4 = Se fh SO. H 0.47 0.45 0.44 0.47 0.14 0.12 0.13 0.12 30. 79 81. 83 88 90. 93. 97. mean-water level relative to still-water level (ft) m H X = distance from wave board to gage (ft) h ] 1.02; H, = 0.43; X, = 83.66; hy = 0.60; H./L, = 0.095; xX h n 00 1.25 -0.016 .29 0.93 0.006 00 0.85 -0.003 WD 0.61 -0.041 .04 0.30 -0.025 08 0.30 0.012 00 0.30 0.043 00 0.29 0.054 Cl Case 2120: T= 1.02; H, - 0.45; X, = 86.08; h, = 0.48; H,/L, = 0.095; es = On/ SP Say > OS.008 My SONSOe 4S BEDE h, = 0.30; X, = 96.83; 6, = 10; B3 = 0. H X h n 0.47 30.00 25) -0.010 0.46 82.75 0.81 -0.004 0.43 84.50 0.75 -0.018 0.43 86.25 0.44 -0.054 0.15 89.00 0.30 0.029 @) 113) 91.50 0.30 0.019 0.14 94.00 0.30 0.043 0.12 99.00 0.28 0.060 Case 2130: T = 1.02; H, = 0.37; X = 86.64; hy = 0.43; H,/L, = 0.094; Ihe = O.698 x5 = BS-/98 In, S OSS x, = 87 138 Ie, = W308 % = YO.038 fh = Ws fo = O- H xX h ” 0.47 30.00 iL, 25) -0.024 0.47 83.29 0.78 -0.005 0.44 85.00 0.73 -0.025 0.46 85.96 0.57 -0.049 0.15 89.50 0.29 0.031 0.12 91.50 0.29 0.014 0.14 94.00 0.29 0.040 @), a 99.00 0.28 0.050 Case 2140: T = 1.01; H, = 0.39; X, = 86.73; hy = 0.45; H,/L, = 0.091; ee ORGS Eee 1 O6rli7 hers On Sikh xem more h, = 0.27; X, = 96.83; B, = 20; fs = 0. H Xx h n 0.44 30.00 25 -0.024 0.43 S35 72) 0.72 -0.022 0.42 85.50 0.70 -0.027 0.42 86.50 0.50 -0.031 0.10 89.50 O29) 0.008 Oe 91.50 0.29 0.029 0.14 94.00 0.29 0.041 (0), lil 99.00 0.28 0.044 c2 Case 2150: Heoe = 0.66; Xi,, = 86.50; hy = 0.31; X, = 87.13; T= 1.00; H, 0.00; X = i = O.288 ot, = 96.838 xy = SO3 Sy = On 0 0 0. Case 2160: Nice = 0.66; Xi, = 86.67; h, = 0.31; KX, = 87.13; 0 0 0. 0 0 H -47 -41 42 -42 .12 14 .13 pdkal xX 30.00 83. 85. 86. 89. 91. 94. 99. Ti 75 50 58 50 50 00 00 1.00; H, 1 0 0. So Oo Oo L& © h 25 -0. .76 -0 70 -0 52S) -0 529) 0 528) 0 .28 0 .28 0 = 0.00; X n 003 .025 .025 .030 .010 .031 041 .048 = 0.00; h, h, = 0.28; X, = 96.83; B, = 40; B; = 0. 0 0 0. Case 2211: poe = 0.78; Xiog = 83.08; hy = 0.29; X, = 88.25; 0 0 0. 0 0 H .48 -42 43 .44 o dal 12 .15 0) 30 83 85. 86 89. 91. 94. 99. Xx .00 iS 46 75 50 50 00 00 1 0 0. Qo Oo C&C O& © h .25 .77 71 .50 .30 .30 .29 .28 T = 1.00; H, = 0.00; X U] .002 .012 .015 .020 .014 .040 O51 .055 = 0.00; h, h, = 0.60; X, = 89.08; B, = 5; B3 = 20. H 0 0 0. So Co LC O& © 44 -42 42 .43 . 10 .20 . 16 .17 30 81 82. 85. 88. 90 93. 97. xX .00 .00 75 35 UD 5 US) YD 25 | .012 .007 .003 .015 .013 .032 .029 .032 C3 0.00; hy, = 0.00; H,/L, = 0.090; 0.00; H,/L, = 0.090; 0.00; H,/L, = 0.090; Case 2212: T = 1.02; H, = 0.39; X&, — 82.35; h, = 0.60; H,/L, = 0.089; ese = 0-87; Xtoe = 80.63; h, = 0.30; X, = 85.83; h, = 0.67; X, = 86.58; 6, = 5; ps = 20. H X h oY] 0.44 30.00 25 -0.012 0.41 78.25 0.96 0.007 0.41 79.92 0.88 0.018 0.45 83.38 0.79 -0.007 0.14 87.50 0.65 0.028 0.20 89.50 0.59 0.025 0.17 92.50 0.48 0.029 0.17 97.50 0.33 0.032 Case 2220: T= 1.02; H, = 0.41; X, = 85.91; h, = 0.42; H,/L, = 0.091; Dice = 0.73; Xtce — 84.88; h, = 0.26; X, = 86.83; h, = 0.64; X, = 87.58; 6, = 10; f; = 20. H x h n 0.45 30.00 1.25 -0.003 0.46 82.75 0.81 -0.003 0.45 84.50 0.74 -0.007 0.41 86.08 0.40 -0.019 0.18 88.29 0.62 0.011 0.16 90.25 0.57 0.017 0.17 93.25 0.46 0.017 0.18 97.30 OFS5 0.020 Case 2230: T = 1.02; H, = 0.39; X, = 87.27; h, = 0.43; H,/L, = 0.094; hice = 0.67; Xtoe = 86.50; h, = 0.30; X, = 87.75; h, = 0.62; X, = 88.46; 6, = 15; Bz = 20. H xX h n 0.45 30.00 25) 0.000 0.45 83.75 0.77 0.001 0.46 85.50 OPW! 0.001 0.44 86.92 0.47 -0.016 0.23 S9R25 0.59 0.016 0.21 S25 0.52 0.017 0.19 94.25 0.44 0.026 0.18 98.25 0.30 0.028 C4 Case 2240: T = 1.03; H, Rgoe = 0.67; Xo, = 86.83; h, = 0.30; KX, = 87.75; hy, = 0.62; X, = 88.46; B, X 30.00 83.75 85. 87. H25 91. 94. 98. Case 2250: So 2 Co © © © OS © Case 2260: Sa Oo Co CO LC FGF SCO C© a & 2 2 2 2 © © H H -46 46 .49 -46 .24 21 . 20 . 20 .43 .40 44 47 . 26 .21 .18 .18 H 44 .38 5 oe) a2 .25 .20 .19 .20 89 50 00 25 25 25 = 0.38; X, = 87.32; h, = 0.46; H,/L, = 0.091; = 20; B; = 20. h n 1.25 0.002 0.77 0.010 0.71 -0.009 0.52 0.000 0.59 0.016 ORD2 0.019 0.44 0.026 0.30 0.026 T = 1.00; H, = 0.00; X, hyge = 0N66" |X, 47 — 987521): who = 0230-1 Xiv—n87L75% h, = 0.63; X, = 88.46; B, xX 30. 84. 86. 87. 89. 91. 94. 98. oS 2&2 CO CO OC oO OC PP Ds = 0.00; h, = 0.00; H,/L, = 0.090; 30; B3 = 20. U] 25 -0.005 75 -0.016 68 -0.015 50 -0.021 60 -0.001 52 0.006 44 0.016 29 0.024 T = 1.00; H, = 0.00; X, ly = WoO7/8 B55, = SO2928 Io =] O29 0 SS B7/55s¢ h, = 0.63; X, = 88.08; fy, X 30. 84. 86. 87. 88. 90. .67 OT 98, 00 25 00 04 63 67 67 oO COC CEC OC OC OC OO ff fey = 0.00; h, = 0.00; H,/L, - 0.090; 40; B3 = 20. U] ae) -0.010 76 -0.022 68 -0.027 41 -0.030 61 0.004 54 0.007 45 0.019 32 0.025 C5 Case QGiO2 Ww = W023 kh = 30s , = GAM In, — O.589 EL/it, 2 O09 Myan = Oo889 45, = Ose Ing 2 Oodle 24, — By .25e h, = 0.64; X, = 87.75; 6B, = 5; Bs = 30. H Xx h 7) 0.45 30.00 1.25 -0.003 0.42 We7d 0.90 0.013 0.41 81.50 0.83 0.015 0.44 84.46 0.55 -0.007 0.14 88.67 0.61 0.026 0.20 90.67 0.55 0.025 0.17 93.67 0.45 0.029 0.20 97 57 0.32 0.034 Case 2320: T = 1.02; H, = 0.43; X& = 86.17; hy, = 0.47; H,/L, = 0.088; ian = Oo728 Mon |= 85.258 Ing = 0.298 3G = B77 h, = 0.64; X, = 87.67; B, = 10; B3 = 30. H xX h n 0.43 30.00 625 -0.013 0.47 83.25 0.80 0.002 0.47 85.00 0.73 0.002 0.40 86.38 0.43 -0.033 0.23 88.67 0.61 0.014 0.19 90.67 0.55 0.009 0.19 93.67 0.45 0.017 0.21 97.67 ORS 2 0.019 Case 2330: T = 1.02; H, = 0.38; %&, = 86.91; h, — 0.42; H,/L, = 0.093; yee = 0668) Xe, =) 86.17 he = ON29-0xe = 87 438): h, = 0.64; X, = 87.83; 6, = 15; fy = 30. H X h n 0.46 30.00 2S) 0.018 0.44 83.75 0.77 0.007 0.47 85.50 Oat 0.006 0.47 87.00 OFS -0.017 0.25 88.67 0.61 0.015 0.19 90.67 0.54 0.017 0.19 93.67 0.45 0.029 0.19 97.67 ORS 2 0.029 C6 Case 2340: T = 1.02; H, = 0.38; X, — 86.73; h, = 0.47; H,/L, = 0.093; ice = 0-673 Xtoe = 86.38; h, = 0.29;1X, = 87.21; h, = 0.64; X, = 87.75; B, = 20; By = 30. H xX h n 0.46 30.00 125 0.013 0.42 84.25 0.76 0.002 0.39 86.00 0.68 -0.010 0.50 87.00 0.38 -0.016 0.26 88.67 0.61 0.014 0.22 90.67 0.54 0.016 0.21 93.67 0.45 0.028 0.21 97.67 0.32 0.028 Case 2350: T= 1.00; H, = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.090; hpse = ONG) Xecq = 865675 he = OF 29-0 Xen 8)//2 25); h, = 0.64; X, = 87.75; B, = 30; B3 = 30. H xX h n 0.45 30.00 25 0.001 0.41 84.25 0.76 -0.009 0.36 86.00 0.68 -0.020 0.52 87.00 0.40 -0.025 0.24 88.67 0.61 0.013 0.22 90.67 0.54 0.014 0.19 93.67 0.45 0.026 0.20 96.83 0.34 0.029 Case 2360: T= 1.00; H, = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.090; hos = O68 Xe. = 87.00; ho = 08297 0xe) =88 7225); h, = 0.64; X, = 87.75; B, = 40; B; = 30. H X h ” 0.44 30.00 ~- 1.25 -0.018 0.41 84.25 0.76 -0.022 0.35 86.00 0.68 -0.033 0.54 87.00 0.44 -0.028 0.23 88.63 0.61 0.001 0.21 90.67 0.54 0.007 0), UE) 93.67 0.45 0.016 0.20 97.67 0.32 0.024 C7 Case 2410: h, = 0.63; X, = 87.67; By H 0. 0. 0. Sa 2 2&2 2 © Case 2420: h, = 0.62; X, = 88.08: f, H 0. 0. 0. Sa 2 2 2 © Case 2430: 45 4l 40 -45 o dls} .20 . 16 o dl) 44 48 45 .40 . 16 . 20 . 20 3 Ce) T = 1.02; H, = 0.39; X, = 84.08; h, = 0.58; H,/L, = 0.091; hice = 0.81; Xtoe = 82.33; h, = 0.30; X, = 87.33; Xx 30. 80. 82. 84. 88. 90. 93. SE 00 75) 00 75 67 67 67 67 = 5; B3 = 40. h n 1,25) -0.001 0.88 0.012 0.83 0.011 0.55 -0.007 0.61 0.029 OFS) 0.029 0.45 0.030 OFS 2 0.035 T = 1.02; H, = 0.39; X, = 87.34; hy = 0.37; H,/L, = 0.090; ign = O.098 Wn = 8D./98 ly = O. 308 2 = 87.758 xX 30. 83. 85. 86 89. 91. 94. 98. T= 1.02; H, 00 50 25 al 50 50 50 50 = 10; B; = 40. h n 1225 -0.009 0.78 -0.003 On -0.013 0.47 -0.024 0.59 0.011 0.50 0.019 0.43 0.023 0.30 0.024 = 0.39; X, = 87.50; h, = 0.39; H,/L, = 0.088; Mon = O.668 Yo, = 86.673 In, = 0.203 h, = 0.63; X, = 88.17; 6, = 15; B3 = 40. H 0.44 0.45 47 .43 oe pail oll) .18 X 30.00 83 85. 86. 89. il. 94. 98%: oD 50 92 50 50 50 50 u/] -0. -0. -0. -0. 0. X, = 87.83; 013 012 009 020 008 0.015 0. 0.024 020 C8 Case 2440: Case 2450: Sa 2 Oo 2 © CEC OS © Case 2460: SO oO Oo C2 S&S OC © H 0.45 5 3Y) .42 .47 .22 .21 520 mls T= 1.02; H, Xx 30. 84. 86. 87. 89. 91. 94. 98. 00 25 00 21 50 50 50 50 Ike 0. 0. 0 0 0. 0 0 0.39; X, ees ON 665) Xtcg = 86.96) )ho,— OR SOGy Xou— 87183); h, = 0.62; X, = 88.17; 6; = 20; 65 = 40. h 25 mS 68 .50 coy) 50 .43 . 30 0 -0 -0 -0 0. 0 0. 0 = 87.49; h, = 0.42; H,/L, = 0.091; | .016 .009 .019 .015 006 .005 019 .024 T = 1.00; H, = 0.00; X& = 0.00; h, = 0.00; H,/L, = 0.090; ee OMG 7 LEXeee 6 O58) he — SONS 0; Keu =o 7/1, h, = 0.64; X, = 87.46; B, = 30; B, = 40. H 45 37 Ol 52 .19 21 .20 .19 xX 30. 83r 85. 86. 89. he 94. 98. T S 1.00; Hy 00 75 50 75 50 50 50 50 n .025 -009 -015 011 005 003 021, 027 = 0.00; hy = 0.00; H,/L, = 0.090; hice = 0.67; Xtce = 86.67; hy = 0.30; X, = 87.08; h, = 0.66; X, = 87.42; B, = 40; Bs EQ Oo C2 2 Eo EC OC ©& H -46 .37 .40 55S) .24 aa, 19 .19 xX 30. 83. 85. 86. 88. 90 O3F 97. 00 U5) 50 vg) YD 5 U5) 75 75 h 1.25 0 0.77 -0 0.70 -0 0.49 -0 0.59 0 0.50 0 0.43 0 0.30 0 = 0.00; X, h 1.25) 0 0.77 -0 0.7L -0 0.46 -0 0.61 0 0.54 0 0.45 0 0.32 0 40. Case 4110: T = 1.02; H, = 0.30; X, = 89.35; h, = 0.40; H,/L, = 0.069; isn 2 OoOO§ Mays = SI oNOP Inj O.B33 3 = Dil BSe ls, = O.228 XS YO 33 fh = Be fp = O, H xX h n 0.34 30.00 iL 2S -0.010 0.30 85.50 0), 7il -0.009 0.30 87.00 0.65 -0.004 0.33 89.42 0.39 -0.013 0.11 92.50 O22. 0.021 0.10 94.50 0.22 0.036 0.08 97.00 0) Zl 0.042 0.07 100.00 0.23 0.041 Case 4120: T = 1.02; H, = 0.32; X, = 90.93; h, = 0.29; H,/L, = 0.070; men = O58 KA. = O7o1Os Th, So O.252 Ko O98. h, = 0.22; xk, = 99.13: 6, = 10: 6, = 0. H xX h n 0.34 30.00 2S) -0.004 ORS 87.46 0.64 -0.003 0.28 89.00 0.60 -0.008 0.30 90.54 0.36 -0.013 0.19 9250 0), 2il 0.021 0.11 94.50 0.21 0.036 0.10 97.00 0.22 0.044 0.08 100.00 0.24 0.042 Case 40303 G) = 102+ mF NOMS) 6x4) O1 OO hy =" OMSiIl- Ha Ee =O NO70F Iman = OoBUe S.5 O OO.889 Ih, 6 On229 %, & O18 Im = Oo2is SS O9.193 215s Gs 0, H x h 7) 0.34 30.00 125) -0.020 0.30 88.00 0.63 -0.029 0.30 89.50 0.60 -0.013 0.30 90.83 0.22 -0.018 0), 12) 92.50 0.22 0.016 0.12 94.50 0.22 0.035 0.10 97.00 0.21 0.043 0.08 100.00 0.24 0.046 Case 4140: T= 1.01; H, = 0.30; X%, = 90.68; h, = 0.46; H,/L, = 0.070; Hise = 0.56; Xtge = 90.63; h, = 0.22; X, = 91.33; h, = 0.21; X, = 99.13; B, = 20; B, = 0. H xX h n 0.34 30.00 1.25 -0.006 0.30 88.00 0.62 -0.007 0532 89.50 0.58 -0.004 0.31 90.92 0.37 -0.012 0.10 93.00 0.22 0.022 0.12 95.00 OR22 0.044 0.09 97.00 0.20 0.051 0.07 100.00 0.24 0.047 Case 4150: T = 1.00; H, = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.070; Riss = 0.53; Xtoe = 90.63; hy = 0.23; X, = 91.33; h, = 0.21; X, = 99.13; 6, = 30; B, = O. H X h n 0.35 30.00 2) -0.003 0.30 88.50 0.61 -0.008 0.31 90.00 0.57 -0.004 0.34 91.04 0.37 -0.012 0.12 93.00 0.22 0.021 0.12 95.00 OF22 0.043 0.09 97.00 0.22 0.050 0.07 100.00 0.23 0.045 Gass FGDs 24 NOS T. > W.003 % 5 0.008 me = O00. Ht, = O07: May = Ood93 Meg = QL WOS Ing = Oo.225 3G = Sle do) i, 2 OoMile Ke, = OOQ.1Ss f= M02 fy = O, H xX h n 0.36 30.00 125 -0.002 0.30 88.50 0.67 -0.009 0.31 90.00 0.57 -0.005 0.35 91.08 0.36 -0.020 0.13 93.00 OR22 0.025 0.11 95.00 0.22 0.046 0.09 97.00 0.22 0.051 0.09 100.00 0.24 0.047 Case 4211: 0 0 0. Case 4212: Sa eo 2 2 & H .40 5 3// 35 . 36 3 ILS) 18 . 16 ails) T = 1.02; H, = 0.36; X hice = 0.72; Xtcoe = 85.29; h, = 0.29: X, = 89.38; h, = 0.58; X, = 89.92; B, 30. 3} 4 85. 87. 91. OBE 96 100. T= 1.00; H, Xx 00 25 00 79 00 00 .00 00 = 87.72; h, = 0.43; H,/L, = 0.082; Nias = Oo VIS Meo, = 8.538 In, = O28 XX, = BO AG8 h, = 0.58; X, = 90.00; 6, 0 0 0. Case 4220: 0 0 0. 0 0 H .34 033} 31 03S) g dks} ieY/ o Als) . 16 30. 83. 85. 87. 90. 92. 5). 9). T= O2 Fae = 5; Bz = 20. h n 1,25) 0.022 O79 0.003 0.73 -0.006 0.42 -0.014 0) 53} 0.008 0.46 0.022 0.38 0.018 OF23) 0.012 = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.070; = 52 fy = 20. h ” 25 0.002 0.79 0.001 O73 0.001 0.41 -0.005 0.56 0.003 0.47 0.011 0.40 0.014 0.26 0.015 = 0.33; X, = 90.35; h, = 0.31; H,/L, = 0.075; Hoe = 0.60; Xtgg = 89.25; h, = 0.24; X, = 90.75; LY O02) OE) SP AO ina 0 0 0. SO C2 Oo Oo © H ool 5oe) 34 .38 5 LY) 18 o JLfs} o lb7/ 30 87 88. 90. 92. 94. DY « 100. xX .00 .00 50 29 00 00 00 00 .25 .65 .61 oe -49 44 .34 .24 -0. -0. oO 2&2 2 © © " 008 007 .017 .015 .012 .019 .019 .019 C12 Case 4230: h, = 0.51; X, = 91.58; 6, = 15; B3 Case 4240: SO 2 eC CEC 2S OC Oo CO H so 632 .36 .42 US) .18 .17 . 16 it =3 OD ith, Xx 30. 87. 89. 90. 927 94. 97 100. T = iL OAs Hy 00 50 00 71 50 50 50 50 So CC CO © EC OC OC ff h 0.32; X, = 90.59; h, = 0.35; H,/L, = 0.075; hice = 0.57; Xtoe = 90.08; h, = 0.24; X, = 91.00; 20. .003 .012 .015 .015 .008 .017 .019 .021 = 90.33; h, = 0.54; H,/L, = 0.076; Hee =8085670 X50, = 90533 hoe" ON24) eXeu—1 911400); h, = 0.51; X, = 91.58; 6, = 20; B; So eo 2 2 2 S&S OC © Case 4250: Ee & 2 eo C2 EC eC OS H . 38 BS2 ol .42 .16 .18 5 AL7/ .15 H . 38 .29 al 45 .18 .20 .16 .14 xX 30. 87. .00 90. 2% 94. D0 « 100. 89 T = 1500; H, xX 30. .50 87 88. 90. Oe 94. 97. 100. 00 50 63 50 50 50 50 00 33 50 50 50 50 50 SE of © © 2 © CO fH h 20. .000 .012 .010 .013 .006 .017 .022 | q29 -0 .64 -0 . 60 -0 ao -0 37) 0 43 0 32 0 .22 0 0.35; X, n 52S 0 .64 -0 .60 -0 .35 -0 47 0 43 0 ao 0 .22 0 Se oOo © Eo 2 © CEC h .025 0.00; X, = 0.00; hy = 0.00; H,/L, = 0.070; Tee OMS 7X 90833) B ONO 4) ween —9 O85): h, = 0.52; X, = 91.33; 6, = 30; B; = 20. .25 .64 62 oo) .47 43 3 se " .007 .O11 .004 .013 .014 .019 .025 .028 C13 Case 4260: T = 1.00; H, = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.070; lien = Oo55§ Mian = YFO.07/8 Inn = O.2%3 X, a Vil, Ws H X h n 0.34 30.00 125 -0.001 0.26 87.96 0.63 -0.013 0.29 89.46 0.59 -0.007 0.38 90.75 0.38 -0.018 0.18 92.46 0.47 0.004 0.20 94.00 0.44 0.009 0.19 96.00 0.38 0.015 0.14 100.00 0.23 0.025 Case 4310: T= 1.02; H, = 0.31; & = Geile th = 0.44; H,/L, = 0.069; hice = 0.68; Xice = 86.08; h, = 0.28; XK, = 90.00; h, = 0.56; X, = 90.38; B, = 5; B3 = 30. H xX h u] 0.34 30.00 1.29 -0.005 0.30 84.50 0.75 -0.004 0.33) 86.00 0.68 0.002 0.34 88.25 0.42 -0.009 @), 15) 91.25 @). Sal 0.005 0.17 93533 0.46 0.011 0.16 96.33 0.37 0.010 0.17 99,83 0.25 0.013 Case 4320: T= 1.02; H, = 0.32; X = 90.78; h, = 0.32; H,/L, = 0.070; Mia = Wor3S May, = SOG33 In, = O29 x oS Vil ss h, = 0.50; X = 91.71; B, = 10; Bs = 30. H Xx h n 0.34 30.00 12S -0.005 0.30 88.04 0.62 -0.011 033 89.50 0.58 -0.003 0.37 90.88 OFS -0.010 0.15 92.67 0.47 0.004 0.18 94.67 0.43 0.015 0), L7/ 96.67 0.35 0.014 0.18 100.00 0.24 OROI97, C14 Case 4330: T= 1.02; H, = 0.29; X%, = 91.53; hy = 0.35; H,/L, = 0.069; iy = OoG39 Sa, 2 OOS hy = O,O86 56 S 92.005 h, = 0.48; X, = 92.75; B, = 15; 6 = 30. H Xx h ” 0.34 30.00 eZ) -0.004 0.29 88.96 0.60 -0.014 0.27 90.50 0). 55 -0.010 0.38 91.54 0.31 -0.011 0.12 93.46 0.45 0.002 0.17 95.50 0.39 0.010 0.16 97.50 0.32 0.015 0.15 100.50 0.22 0.018 Case 4340: T = 1.02; H, = 0.30; X, = 91.45; h, = 0.44; H,/L, = 0.068; ice = 0.51; Xtoe = 91.33; hy = 0.23; KX, = 92.04; h, = 0.48; X, = 92.42: 6, = 20; f, = 30. H x h n 0.34 30.00 25 -0.001 0.28 89.00 0.60 -0.012 0.29 90.50 0.55 -0.005 0.38 91.67 0.32 -0.008 0.14 93.50 0.45 -0.001 0.17 95.50 0.39 0.012 0.16 97.50 0.32 0.014 0.15 100.50 0.22 0.020 Case 4350: T= 1.00; H, = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.070; lean @ Oo592 wy, 2 Vile in, Go OR88 de, = ILL SAE h, = 0.50; KX, = 91.92; B, = 30; B3 = 30. H x h n 0.34 30.00 1.25 -0.001 0.26 88.50 ~ O56 -0.014 0.29 90.00 0.57 -0.013 0.39 S33 0.30 -0.020 0.17 93.00 0.46 0.004 0.18 95.00 0.41 0.012 0.16 97.00 0.33 0.015 (0) a5) 100.00 0.23 0.020 C15 Case 4360: T= 1.00; H, = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.070; Roe = 0.51; Gray = 91.42; h, = 0.23; X, = GIL ,7/2 Ime = O.492 % = 92.083 fh = AO8 i, = 30. H xX h n ORSs3 30.00 IL 25) -0.007 0.28 88.75 0.61 -0.013 0.28 90R25 0.56 -0.014 0.39 91.46 ORS /, -0.023 0.18 93.00 0.46 0.004 0.19 95.00 0.41 0.010 (0), sL7/ 97.00 ©3333 0.017 0.14 100.00 0.24 0.023 Case 4410: T = 1.02; H, = 0.30; X, = BO.508 Ih, = 0.378 FL, — O.059s Ina = OnGO3 Wen = GW OAR Ty = O,WAR = C0) S158 Ih, = On929 % = Qioi/s fh = D8 fsa = 4O, H Xx h n 0.34 30.00 2S -0.002 0.32 5)o 33) @. il -0.007 0.33 86.83 0.65 -0.001 0.36 89.00 0.41 -0.011 (0), 172 92.50 0.47 0.011 (0) , 1L7/ 94.75 0.42 0.016 0,115 97.00 33) 0.016 0), IL7/ 100.00 0.23 0.018 Case 4420: T = 1.02; H, = 0.33; X, = 90.35; h, = 0.31; H,/L, = 0.075; nee = O57 exe net = 1.9 OF il ney =| Om ale Xu Oalazae h, = 0.50; X, = 91.96; B, = 10; B3 = 40. H xX h n 0.34 30.00 1.25 -0.005 O51 88.25 0.62 -0.010 0.35 89.75 0.58 -0.001 0.37 91.04 0.31 -0.010 0.16 93.00 0.46 0.004 0), 17 95.00 0.40 0.015 0), 7) 97.00 0.33 0.017 0.17 100.00 0.23 0.018 C16 Case 4430: 0 0 Case 4440: 0 0 0. aS © © © © Case 4450: oo 2 So © CGC COC © H .34 .29 H .34 oy) 29 .40 .18 .20 .19 .17 H .34 og) .29 41 .18 .18 oALy/ .15 Tf = 1,02; H, = 0530: X%, = 91.11; hy, = 0.40; HAL, = 0.069. hice = 0.54; Xtgg = 90.79; h, = 0.23; X, = 91.75; h, = 0.49; X, = 92.00; p, 30. 88. 90. Oi. IS) 95). 7 100. Xx 00 50 00 29 00 00 00 00 a Oo EC 2 2 © © = 15; B3 = 40. h n 2) -0.005 .61 -0.009 .56 -0.008 ool -0.014 -46 0.003 41 0.011 32. 0.016 28) 0.018 T= sl 02 Hes ORS On xXe=) 9ON83 ha =O} Sil sHe/le = 908069): ice = 0:53; Xtoe = 90.83; hy = 0.22: X, = 911.54: h, = 0.50; X, = 91.79; B, 30 88. 90. Sik 93 95. DY « 100. Xx .00 50 00 13 .00 00 00 00 1 0 0. So Oo Oo OC © = 20; B; = 40. h n 25 -0.002 .60 -0.010 56 -0.012 e338 -0.010 46 0.002 41 0.012 .32 0.013 23 0.017 T= 1.00; Hy = 0.00; Xp Men = Wo523 we 2 OL. 298 In, = O.239 Xx, S Sil 7/58 h, = 0.49; X, = 92.00; B, 30 88 90. Sil. 23} 95. oe 100. xX .00 75 25 42 00 00 00 00 = 0.00; h, = 0.00; H,/L, = 0.070; = 30; Bs = 40. h n 525) -0.009 .60 -0.014 56 -0.013 223 -0.017 46 0.003 41 0.012 yd, 0.015 223, 0.022 C17 Case 4460: T = 1.00; H, = 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.070; Mey O52 Kn, O OSE thy = 0.223 X, S 91.67: h, - 0.50; X, = 91.96; B, = 40; B; = 40. So © 2 2 28 2 €& © Case 6111: H .34 ay) .29 -41 o LY/ adl) 6 ALY/ .14 Xx 30. 88. 90. Sib 93. 25) .00 100. D7 T= 1.49; H, 00 75 2S) 46 00 00 00 So 2 2 2 © © © h 22) .60 oD 0.51; X, = 81.60; h, - 0.60; H,/L, = 0.046; liye COS Ty S Pole ty Go 0,202 3, S BBail7s h, = 0.32; X, = 95.63; B, = 5; B3 = Sa Oo oOo Oo 28 2 SO ©& Case 6112: H .48 .48 -47 oe o kd o ALB) oll2 .14 X 30. YD 77. 81. 86. 89. 23.0 98. 1; = SO} Hp 00 00 95 83 50 50 50 50 So O&O CEC CO FC OC PP PF h 62S) 5) 0.00; X, = 0.00; h, = 0.00; H,/L, = 0.050; hice = 1.06; Xoo = 74.83; h, = 0.42; X, = 81.83; h, = 0.46; X, = 92.30; B, = 5; Bs = Se © C&C ©o& eC CO EC CSC H .50 47 oy) 53 IG .14 a dLil . LO Xx 30. HOF 74. U9. 84. 88. WB. 98. 00 92 50 33) 00 00 00 00 So 2 CEC CO 2 fF FP P h > Be) .20 5 (O7/ OS) .42 .42 .44 n oul n .002 .013 .013 .026 .002 Oe .018 .022 u " -0. -0. -0. -0. 0. 0 0. 0 0. OF Case 6120: H OF! OFS 0. Case 6130: So EC OO C& © 67 .61 521 5 LS) lal odbal T = 1.49: Hy = 0.63; X, = 84.17; ‘hy =) 0.47; Hi/l, = 0.048: He = O88 ue = 8248 eh. = OMS0 xe BolTe7- iy = OoS8e % 2 OS.639 fh 30. 79 82. 84. 86. 89. I~ 98. T= Xx 00 .50 00 00 50 50 50 50 1.50; Hy 0. " .019 .017 = 84.33; h, = 0.51; H,/L, = 0.052; IMsan = Oo/98 Uon5 =] BI. HA3 It, = O.298 86 & BD. 17/8 Ie = O933 M a VH65e (sh H Case 6140: So oo Eo 2G Lf 2 © soul 496 .63 .98 25 5 Ly odbil . 10 30. 81. 83. 84. 86. 89. W380 98 2b i SO} Hy xX 00 00 00 30 50 50 50 75,0 0. .082 .083 = 84.27; h, - 0.63; H,/L, — 0.048; Tice = 0.76; Xtoe = 83.96; h, = 0.30; X, = 85.17: hy = 0.32; X, = 95.63; B, = 20; B3 = H 5 yl .58 .63 5oe) .22 .18 wL2 LO a Eo Co So EC fo EC © 30. 81. 83. 84. 86 89. I3\ 98 xX 00 20 20 38 .50 50 50 39.0 = OS by = h 28) -0 0.91 -0 0.81 0.50 -0. 0.29 0.31 0.32 ORS = 0.62; X, = 155 Ba = h L525 -0. 0.85 -0. O79 0.50 -0. 0.30 0.31 0.32 0.30 = O.7ale 3% h 1.25 -0. 0.85 -0. 0.79 0.57 -0. 0.30 0.31 0.32 0.31 0. 0. 0 0. 0 0. uy] Case 6210: Te NoAS)e Vets 0.46; X, = 80.19; h, = 0.63; H,/L, = 0.048; Men C 1OOs Xn = T8G99 be = O.383 xX, So 88,083 h, = 0.76; X, = 84.00; p, Case 6220: eo eo C2 Fo 2&2 © © & H oul .43 . 66 43 o Ddl .28 oe) 22d 30. 73. UY o 82. 85. 89. 94. 98. = IL A2)2 Hy = ©.57/2 X, xX 00 50 50 46 00 00 00 67 il iL i 0) 0. 0 0 0 h 02) . 10 .O1 -44 78 .60 .44 .28 B3 = 20. " = 84.48: h, = 0.45; H,/L, = 0.043; Ina = On@il® WG.5 = S208 ly = O.938 2G > G5. 2s h, = 0.68; X, = 86.08; 6, = 10; 63 = 20. Case 6230: S oO Oo Co 8 2 OC OS H -46 55 .62 5 aks) 22 528) 2D ollY 30. 80. 82. 84. 87 90. 94. 98. Gp = Xx 00 46 46 46 .00 00 00 67 1.49; H, al 0 0. Se 2 eo 2 © Troe = 0-75; Xtoe = 84.42; h, = 0.33; X, = 85.88; Im 6 0,672 % = BG.758 1h Se So 2 2 ©& © 2S ©& H .44 5 ons) 0 oNe) 2) alt 24 523 2 AL7/ Sil. 81. 84. Sr 88. Die 94. 98. Xx 00 25) 00 08 00 00 50 50 h n 25 -0.005 od -0.010 81 0.034 45 -0.020 65 0.003 ny 0.016 45 0.023 29 0.030 = OR63 eX = 85515 uh, = OF52; eHo/ls 1 ONO42%: = 15; B; = 20. h n 25 -0.008 85 -0.006 .76 0.048 Sil -0.016 .63 -0.001 54 0.018 44 0.025 5 28) 0.032 So O22 C&C CGC CO OC OES PF C20 Case 6240: Mey = OsI8 May S BIeIDS Iya WsSiks XG = Geosas T= 1-50; 0H, =10260-0X, = 83. 75:0h, =) 0.77; HL/l, = 0.043; Im, = O70 %%, = 5.675 r= 203 a = 20. Xx 30. 80. 83. 84. 86. .50 93. .50 Case 6250: Ian = OoIDS 255 = VACD8 Io, 2 O.522 x = WEo8I/s Soe 2 2 © Se © H -46 soe) -61 58 gall ss) .24 odky/ 89 98 00 50 00 Zl 50 50 So © CO Oo Oo 2&2 OC PP h 525) .87 79 .50 5 (972 5 oY) .45 529 -0. -0. 0. -0. @ S 1,509 Eh 2 OC0e x6, hy. = 0.69; X, = 95.75; By oO Oo eC OC oO Oo OC CO Case 6310: H .44 .60 oy) .99 ool .30 .22 olf xX S07 80. .00 83 84. 86. 89. 3 98. dh es WAS Meh 3 OS 265 = SLE Ia Mie = L028 X55 = JOcI/ae In = OSES YG S GA0058 00 50 38 50 50 50 50 SO EC CO CO CO SC OC Pf h, = 0.78; X, = 83.42; B, 0 0 0 0 0. 0 0 0 H 49 47 .48 o aul 21 ol) ofal/ o dal X 30.00 73.46 -46 80. 85. 89. 23}. YY 76 9/2. 96 50 50 50 U] 005 009 036 016 .001 .018 .026 .037 = 0.00; h, = 0.00; H,/L, = 0.050; = 30; fs = 20. h U7 58) -0.006 . 86 -0.009 o UY) 0.036 .58 -0.015 oye) 0.003 .58 0.022 45 0.031 5 28) 0.043 ="5)) 83 h 30. n 0.63; H,/L, = 0.048; Gase 6320: T= 1.49; H, = 0.60; X = 84.43; h, = 0.45; H,/L, = 0.043; tiny = OoIOR Mog S B2c898 Ing = W323 2X o (35), Lg) s H X h n 0.46 30.00 25) -0.009 0.56 80.75 0.85 -0.010 OFA! 82.75 0.79 0.016 0.58 84.38 0.45 -0.018 0.18 87.25 0.65 0.007 0.26 89.75 0.58 0.017 0.25 9350 0.46 0.022 0.18 97.50 O32 0.029 Case 6330: T= 1.50; H, = 0.57; X, = 84.13; h, = 0.58; H,/L, = 0.043; ey = On 123 Kien = OS.678 ty 01328 3X, = BS. 103 Ih, = O70 &, = 85.673 fh = W538 fh = 30. H X h ” 0.45 30.00 1528) -0.007 0.59 80.75 0.86 -0.010 OFA: 83.25 0.79 0.016 0.59 84.67 OR -0.020 OFZ: 87.25 0.66 0.000 0.27 SORTS 0.58 0.012 0.23 9350 0.46 0.020 (0), Y) 97.50 0.32 0.031 Gace 6340s WS 1,502 M = O.572 3 = GAOSe im > O.753 EL, = O.0s ay On JOS Say, > BAOGS Ih, > O.328 04 > C0 h, = 0.70; X, = 85.67; B, = 20; B3; = 30. H X h ” 0.45 30.00 RZ -0.007 0.60 81.00 0.85 -0.008 ORS 83.50 0.78 0.014 0.62 84.67 0.44 -0.038 0.24 Sii2> 0.64 -0.004 0.28 89.75 0.58 0.012 0.24 93550 0.46 0.022 (0) 5 aL) D7 30) 0533} 0.032 C22 Case 6410: T= 1.49; H, = 0.51; X, = 80.72; h, = 0.56; H,/L, = 0.046; yon = LoS M5 = 7O.675 Ie = O93 24 = B2507/3 h, = 0.79; X, = 83.08; 6, = 5; B3 = 40. H Xx h 7) 0.49 30.00 1.26 -0.006 0.48 73.50 1.12 -0.006 0.69 76.50 1.15 0.018 0.54 81.33 0.52 -0.011 0.25 84.50 0.75 0.002 0.28 87.50 0.64 0.012 0.28 92.50 0.47 0.017 O27 97.50 0.32 0.018 Case 6420: T = 1.49; H, = 0.54; X, = 84.46; h, = 0.45; H,/L, = 0.043; en = OR Wy = SBSoOWS Inv, = OSS 2G SS Wd. he, = OR705 X: = 85.715 By = 10s B3)= 40°: H xX h n 0.46 30.00 1.26 -0.007 0.56 80.25 0.88 -0.013 0.62 82.75 0.80 0.036 0.60 84.46 0.44 -0.021 0.21 86.75 0.65 -0.001 0.25 88.75 0.61 0.014 0.25 93.00 0.46 0.025 0.19 98.00 0.31 0.033 Case 6430: T= 1.50; H, = 0.59; X, = 84.46; h, = 0.53; H,/L, = 0.043; Teg & OnV7e Seay CSESBS Ing = O5929 3% — CB G25: h, = 0.70; X, = 85.65; f; = 15; B3 = 40. H xX h ” 0.46 30.00 25 -0.007 0.61 80.92 0.85 -0.009 0.71 83.50 0.78 0.026 OFS 84.50 OFoi -0.018 0.23 86.75 0.65 -0.001 0.26 89.75 0.61 0.011 0.24 93.00 0.46 0.023 0.13 98.00 0.31 0.033 C23 Case 6440: T= 1.50; H, = 0.00; X%, = 0.00; h, = 0.00; H,/L, = 0.050; yee =) O76 Xe et 8407 hey = 0833) Xe = oieli7i: h, = 0.70; X, = 85.58; 6, = 20; B = 40. H xX h n 0.46 30.00 2 -0.007 0.62 80.92 0.85 -0.008 0.64 83.50 0.78 0.032 @.55) 84.58 @, Sul -0.022 0.29 86.75 0.65 -0.001 0.28 88.75 0.61 0.008 0.24 93.00 0.46 0.022 (0), JLaL 98.00 0). Sal 0.033 Case 8110: T = 1.74; H, = 0.55; X, = 82.78; h, = 0.53; H,/L, = 0.031; Nice = 0.90; Xtoe = 79.83; hy = 0.31; X, = 85.38; h, — 0.34; X, = 94.90; 6, = 5; Bs = 0. H Xx h ” 0.46 30.00 126 -0.014 0.47 Yo 0) Lo W3) -0.015 0.51 Y) SO) O92 -0.015 OF 635} 5 JL5) OF -0.026 0.18 87/35 @.31L 0.039 0.16 91.00 0.36 0.051 0.12 95.00 0.36 0.053 ORE. 100.00 OR25 0.055 Case 8120: T=1.74; H, = 0.57; X, = 84.46; h, = 0.45; H,/L, = 0.031; len = OF98 Mon = 82.958 Iq = O.308 % = 85.558 ee OMSS ax ena OF ao el Olam Ol H xX h n 0.47 30.00 1.26 -0.015 0.50 V9s50 @.9il -0.014 ORD3 82.50 0.81 -0.016 0.54 83.88 0) 55 -0.021 0.29 87.00 OFS 0.034 (0), E7/ 91.00 35) 0.061 0.12 95.00 0.35 0.064 @), 113} 100.00 0),23) 0.063 Case 8130: T= 1.74; H, = 0.66; X, = 84.45; h, = 0.56; H,/L, = 0.032; Ntce = 0.76; Xeoe = 83.90; hy = 0.32; X, = 85.38; hy = 0.34; X = 94.90; 6, = 15; fs = 0. H xX h n 0.48 30.00 126 -0.014 0.53 80.75 0.86 -0.014 0.75 83.25 0.79 -0.007 0.60 84.45 0.55 -0.018 ORS 87.25 @)., Sal 0.021 0.20 91.00 0), 35) 0.058 0.12 95.00 0) 35 0.064 @ 13 100.00 OR23 0.063 Case 8140: T = 1.74; H, = 0.66; X, = 84.42; h, = 0.66; H,/L, = 0.032; hess = O75) Xeee, = 845305 hy — 0.325) x4 — 85.35- h, = 0.33; X, = 94.90; 6, = 20; B; = O. H xX h Uy) 0.48 30.00 1216 -0.012 0.52 80.70 0.86 -0.010 OFS 83.70 Oo 77 -0.017 0.57 84.38 0.52 -0.024 O)s aby 87). 2e) 0.31 0.019 0.20 91.00 0.35 0.057 (0), hal 95.00 0.34 0.063 (0) 5 a3} 100.00 O23 0.065 Case 8210: T= 1.74; H, = 0.48; X, = 82.83; h, = 0.56; H,/L, = 0.032; Ma, — O99 My, 2 9.852 in, = O.5G8 3 S BS als) H Xx h n 0.48 30.00 25) -0.004 0.46 76.40 One -0.008 0.66 79.40 0.92 0.009 OR 2 83.70 0.49 -0.013 0.19 86.70 0.66 0.008 0.28 90.50 0.56 0.021 0.26 94.50 0.44 0.022 0.17 100.00 0.24 0.029 C25 Case 8220: T= 1.74; H, = 0.59; X, = 85.18; h, = 0.41; H,/L, = 0.034; Ness) = Ones Xecq = SS O0R he = 034) Xo = 85/160); hy = 0867; X, = 86.315 6) = 10; 63 = 20°. H Xx h n 0.50 30.00 iL, 25) -0.004 O57 80.42 0.87 -0.003 0.70 83.42 0.78 0.017 0.62 84.75 0.49 -0.015 OF25 87.67 0.63 -0.001 O27 90.50 0.56 0.016 Ona 94.50 0.43 0.022 (0), 3} 100.00 0.24 0.035 Gase 8290s Wo il 7he it, = O.543 2 So B53 I, = O.S08 LL/k, = O.W84s3 lim = O78 Mo, = GS.4O8 In, = O.508 % = GD.053 hy, = 0.67; X& = 86.35; B, = 15; B3 = 20. H Xx h ” 0.50 30.00 12 -0.004 0.54 80.88 0.82 -0.006 0.74 84.00 0.75 0.030 0.58 85.08 0.50 -0.017 0.31 87.50 0.63 0.004 0.30 90.50 O55 0.018 0) .22 94.50 0.43 0.023 0.12 100.00 0.24 0.039 Case 8240: T = 1.74; H, = 0.58; X, = 84.71; h, = 0.61; H,/L, = 0.034; len = Oo7hS Minn = Gh. 718 In, = O.528 %, = S505 iy, = O.672 %. = BO.993 fh = 203 fa = 20- H xX h n @), Sil 30.00 1,25) -0.006 0.60 80.75 0.86 -0.008 0.70 S)3}5 U/S) 0.76 0.022 0.60 85.00 0.56 -0.011 0.29 87.50 0.63 0.005 0.30 90.50 0.54 0.014 0.23 94.50 0.43 0.027 0.14 100.00 0.23 0.040 C26 Case 8310: T= 1.74; H, = 0.50; X, = 82.80; h, = 0.58; H,/L, = 0.032; hp = 0889) Xone 179-92 he = Of Sere Xen —= 85/05; hy, = 0.71; X = 85.52; 6, = 5; Bz = 30. H xX h n 0.48 30.00 125 -0.007 0.46 76.25 O02 0.000 0.58 19-7 0.90 -0.004 0.54 83.60 0.55 -0.011 0.23 87.00 0.65 0.007 0.28 91.00 0.52 0.017 0.26 95.00 0.41 0.019 0.16 100.00 0.23 0.031 Case 8320: T = 1.74; H, = 0.60; X, = 84.41; h, = 0.53; H,/L, = 0.032; idea = Qo IB8 res = GS} 5508 Io, = Oo353 KGa 85.44; Ins = O0993 % 3 BHs903 fa = WOR fy = 30, H xX h n 0.50 30.00 25) -0.005 0.58 80.25 0.88 -0.001 0.72 33}, 25) 0.79 0.014 0.61 84.45 0.54 -0.012 O29 87.50 0.64 0.004 0.27 90.50 ORS» 0.009 OF23 94.50 0.43 0.022 (0) abs) 100.00 0.24 0.035 Case 8330: T= 1.74; H, = 0.56; X, — 85.26; h, = 0.48; H,/L, = 0.034; ves = Oodle Sens Oo GALGOs ny S O,Ailg = 25), 755 iy = On678 3 = HE.253 ao Se Yo SO, H Xx h n 0.50 30.00 1.25 -0.006 0.58 81.00 0.85 -0.003 0.79 84.00 0.75 0.014 0.60 85.08 0.61 -0.011 0.31 87.50 0.63 -0.006 0.30 90.50 ORD5 0.010 0), Zak 94.50 0.43 0.021 0.12 100.00 0.24 0.035 C27 Case 8340: tt = IL, Ae Hy, h, = 0.67; X, — 86.15; B, = 20; B, = 30. So 2 oOo 2 o 2 2 © Case 8410: ins = O.B8S Xe, = SOs298 In, = Ol 368 26 = 8.3532 H 5.0 .98 74 oH . 30 ol 2. sl 30. 80. 83. 85. 87. 90. 94. 100. xX 00 90 95 10 50 50 50 00 ao © © @2@ © OS [fp h a2) .85 ./76 oo) .63 oOo) 43 .24 -0. =(0), U] 005 007 AOS .009 0.001 0.007 0.023 0.039 mo Iphe th = O58 x, ly, = Oo7/038 % = Bao 8 /shi Case 8420: IMs = OoI8S 255 = 85.508 In, = OSs 2G = Boas SoS eo Co CO OC S&S © OS H -48 47 .56 Sy) 520) 5S) a) o Al) 30 76 80. 83. 87. sib. I5)o 100. xX .00 .50 00 45 00 00 00 00 il 1 0. Sa © ©& © © =) 8327 sah = 5; B3 = 40. h ” 25) -0.005 .O1 -0.007 88 -0.003 soo) -0.014 .66 0.009 3 0.020 41 0.023 soe 0.031 0.56; X& = 84.80; hy = 0.69; H,/L, = 0.034; Hece = 0-74: Xtoe = 84.80; h, = 0.34; KX, = 85.67; 0.54; H,/L, = 0.033; woo Wo vhs ih = O.54e oo = BASDSS In, = O.50¢ EAL, = W.Wshe h, = 0.69: X% = 85.77: By 0 0 OF SO Eo OC OC © H 5 210) Oy, 82 5 Oat 527) 30 .24 .14 30 80 83. 84. 87. it. 5) « 100. Xx .00 .00 00 13 00 00 00 00 SoS Oo Oo 2 C&C © © FP = 10; B, = 40. h n 925 -0.006 88 -0.005 79 0.000 58 -0.009 65 -0.014 5 0.013 41 0.023 28 0.038 C28 Case 8430: T= 1.74; H, = 0.54; X%, = 84.85; h, = 0.49; H./L, = 0.034; Phe = OoiG8 San, & OA.08 Tho 0.383 26 > 85.258 h, = 0.70; X, = 85.60; 6, = 15; 3 = 40. H x h ” 0.50 30.00 625) -0.005 0.56 80.75 0.86 -0.005 0.80 83.25 0.79 0.009 0.62 84.44 0.59 -0.008 0.29 87.00 0.66 -0.008 0.29 91.00 0.54 0.008 0.22 95.00 0.41 0.021 0.13 100.00 0.24 0.037 Case 8440: T=1.74; H, = 0.54; X% = 84.84; h, = 0.65; H,/L, = 0.034; ees = OR 4 wx 6 8447/5) he es OnSy saXce—mOo100); hy, = 0.69; X, = 85.95; B, = 20; B3 = 40. H xX h 7) 0.50 30.00 125 -0.003 0.56 81.25 0.85 0.001 0.82 83.75 0.77 0.008 0.61 84.85 0.64 -0.009 0.30 87.00 0.65 -0.009 0.29 91.00 0.53 0.009 0.22 95.00 0.41 0.021 0), 1 100.00 0.24 0.037 Case 10110: T.= 2:49; H, = 0.44; X, = 89.83; h, = 0.37; H,/L, = 0-010; Itsy = Ons 2G5, = 87.958 lo, = W258 2 = Vil 25s h, = 0.26; X, = 97.80; 6, = 5; 63 = O. H xX h n 0.34 30.00 1.26 -0.007 0.33 83.50 0.78 -0.007 0.35 87.50 0.63 -0.006 0.44 89.90 0.38 -0.016 0.17 92.50 0.24 0.020 0.14 94.50 0.24 0.037 0.11 97.00 0.25 0.048 0.13 100.00 0.23 0.050 C29 Case 10120: T = 2.48; H, = 0.44; X, = 90.26; h, = 0.42; H,/L, = 0.008; eey =) OeSSS 25, EO LOSF in, = W258 oe O25: he = On26) X=! 97/480) By — NO (85 0). H Xx h U] 0.27 30.00 1.26 -0.010 OFS 85.50 0.71 -0.008 0.34 89.50 0.59 -0.010 OF39 90.50 0.40 -0.013 0.20 20 0.24 0.017 OES 94.50 0.24 0.042 0.12 97.00 0.25 0.044 0.08 100.00 0.23 0.051 Case 10130: T = 2.49; H, = 0.40; X, = 90.29; h, = 0.50; H,/L, = 0.008; ice = 0.56; Xtoe = 90.40; h, = 0.24; X, = 91.25; in = O69 % 3 9760s = 158 ye OW. H xX h n O27 30.00 1.26 -0.009 32 85.50 0.71 -0.008 OFS 2 89.50 0.58 -0.012 0), 39 90.75 0.39 -0.016 0.15 92.50 0.24 0.011 0.14 94.50 023 0.038 0.12 97.00 0.25 0.042 @)., dil 100.00 0.24 0.049 Case 10210: T = 2.49; H, = 0.39; X, = 89.65; h, = 0.38; H,/L, = 0.009; lien = O.G45 x5, = 875653 In, = Oo208 *, = YlW5e hy, = 0.50; KX, = 91.54; 6, = 5; B3 = 20. H Xx h n 0.30 30.00 1.26 -0.005 0.31 83.50 0.78 -0.004 Ons 2 87.50 0.64 -0.008 0.40 89.48 0.41 -0.010 (0), L7/ DiS) 0.50 0.000 0.17 94.00 0.45 0.012 0.19 97.00 0.34 0.015 0.14 100.00 0.24 0.022 C30 Case 10220: T = 2.49; H, = 0.37; & = 89.62; hy = 0.39; H,/L, = 0.009; Hoe = 0.60; Xtgg = 89.05; hy = 0.26; X, = 90.35; h, = 0.53; X, = 90.80; B, = 10; B; = 20. H X h n 0.30 30.00 1.26 -0.004 0.33 84.50 0.74 -0.005 0.37 88.50 0.61 -0.006 0.42 89.60 0.40 -0.011 0.18 92.00 0.49 0.000 0.13 94.00 0.44 0.009 0.18 97.00 0), 333) 0.016 0.10 100.00 0.23 0.026 Case 10230: T = 2.49; H, = 0.41; X, = 89.73; h, = 0.56; H,/L, = 0.008; Mas = O088 Xe, = SO.733 I, = O.27/8 x, = YO.a58 hy, = 0.53; X, = 91.00; 8B, = 15; B3 = 20. H x h n 0.27 30.00 1.26 -0.005 0.33 84.75 0.74 -0.005 0.43 88.75 0.61 -0.003 0.39 89.98 0.43 -0.009 0.21 92.00 0.49 0.000 0.20 94.00 0.44 0.005 0.16 97.00 0.34 0.015 0.10 100.00 0.23 0.025 Case 10310: T = 2.49; H, = 0.41; X, = 90.55; h, = 0.36; H,/L, = 0.008; Rice = 0.63; Xtoe = 88.17; hy = 0.27; X, = 91.55; h, = 0.49; X, = 91.85; 6, = 5; B; = 30. H x h 7) 0.28 30.00 1.26 -0.005 0.30 84-00 0.76 -0.004 0.39 88.00 0.63 -0.006 0.41 90.15 O39 -0.012 0.18 93.00 0.46 0.002 0.20 95.00 0.41 0.008 (0), UY) 97.00 0.34 0.015 0.16 100.00 0.23 0.021 c31 Case 10320: Case 10330: Case 10410: h, = 0.49; X, = 92.10: gp, = 5; QO Oo Oo C2 Se Oo Oo © oO Oo 2 2 282 Oo Co © Soo Oo 2 L2& 2 © H H H . 28 2 36 .42 W239 o dL d/ g Al 7/ . 10 .27 2 .34 4l 9 Bal B22. o dl d/ 09 sl) 5 SIL .34 42 4 ils) 19 .20 oll T= 2.49; Hp Baan 2 On598 Xi = 89.558 Th, 2 O.278 34 = 90.85: Me On5M8 Sees Cilio By X SOF 85. 89. 90. .00 .00 .00 .00 a X 30. 85. 89. 90. 92. .00 .00 .00 T= xX 30. 84. 88. 90. 93) 35) .00 7) 100. 00 00 00 25 2.49; H, les = O.D98 on = YO.SO3 In, = O.278 » = Vil,O53 h, = 0.51; X, = 91.40; B, = 15; B3 = 30. 00 25 25 45 00 Oo Co CO Of Oo OC © |fP So Ee Co O&O Oo Oo PP h h . 26 52 . 60 0.36; X, = 90.18; h, = 0.38; H,/L, = 0.008; 10; B3 = 30. u] 26 -0.009 of3 -0.006 . 60 -0.008 .38 -0.015 .49 -0.002 -41 0.005 53S} 0.014 723 0.021 0.40; X, = 90.30; hy = 0.54; H./L, = 0.008; -0.008 .007 .007 -0.011 0.001 0.003 0.013 0.021 2.49; H, = 0.42; X%, = 90.66; hy = 0.38; H,/L, = 0.008; os = On.Gls 54.4, = BG.60s Ty o O27 % = OSs 00 50 50 40 10 00 00 So Oo Eo © OC OC © FP h . 26 6D .61 oe) -46 -41 .34 .24 .007 .006 .008 .012 .002 .007 .O15 .020 C32 Gase 10420: T = 2.49: H, = 0.37; X, = 90.64: h, = 0.40; H,/L, = 0.008: Atoe = 0.573 Gog = BO. 138 he = 0.26; X, = 91.40; hy = On509 5, = O.G8s f, O NOs 5 > AO, H xX h n 0.26 30.00 126 -0.004 0), Sil 85.50 @). vail -0.006 0.37 89.50 0.59 -0.007 0.42 90.70 0.38 -0.010 0.20 92.50 0.47 -0.001 0.19 94.50 0.44 0.005 0.19 97.00 0.34 0.012 0.10 100.00 0.24 0.022 Case 10430: T = 2.49; H, = 0.38; X, = 90.63; h, = 0.52; H,/L, = 0.008; teas 2 OpS58 Nyy 3 CO, ly = OMe 5%, & CHL De h, = 0.50; X, = 91.65; B, = 15; f; = 40. H X h ” 0) 27 30.00 1.26 -0.006 0.32 85.50 @), “il -0.007 0.31 89.50 0.58 -0.007 0.43 90.90 0.39 -0.008 0.17 92.50 0.47 0.000 0.19 94.50 0.44 0.003 0.18 97.00 0.34 0.013 0.11 100.00 0.24 0.023 PLANE SLOPE MONOCHROMATIC TESTS Case 2000: T = 1.02; H, = 0.40; X, = 86.53; h, = 0.67; H,/L, = 0.092; m = 0.033 H X h n 0.45 30.00 1.26 0.000 0.42 83.00 0.80 -0.004 0.43 84.50 ORS -0.004 0.44 86.65 0.66 -0.006 0.27 90.00 OR -0.008 0.19 93.00 0.46 0.006 0.15 96.00 0.38 0.021 OFS 100.00 0.24 0.027 C33 Case 4000: T = 1.02; H, = 0.27; X, = 93.70; h, = 0.43; H,/L, = 0.066; m = 0.033 H X h ” 0.32 30.00 1.26 -0.002 0.30 91.00 0.53 -0.002 0.30 92,25 0.48 -0.007 0.30 93.96 0.44 -0.004 @),2al 95,05 0.41 -0.004 0), 19) 96.50 0.36 -0.003 0.14 98.25 0.30 0.010 @ 112 100.00 0.23 0.016 Case 6000: T = 1.49; H, = 0.51; X%, = 85.32; hy = 0.71; H,/L, = 0.046; m = 0.033 H xX h n 0.48 30.00 26 -0.003 0.54 81.50 0.84 -0.009 0.54 84.00 0.76 -0.011 0.49 85,95 0.68 -0.015 0.32 89.00 0.60 -0.011 0.28 91.00 0.52 -0.007 19 95.00 0.41 0.021 ORS 100.00 0.24 0.039 Case 8000: T= 1.74; H, = 0.54; KX, = 85.09; hy = 0.71; H,/L, = 0.030; m = 0.033 H xX h ” 0.45 30.00 126 -0.008 0.50 80.75 0.86 -0.008 0.56 84.00 0.76 -0.007 0.49 85.90 0.69 -0.007 (0), Sil 89.00 0.60 -0.005 0.32 91.00 @) 53 0.004 0,19) 95.00 0.41 0.021 0.15 100.00 0.24 0.042 C34 Case 10000: T = 2.49; H, = 0.42; X, = 91.68; h, = 0.49; H,/L, = 0.009; m = 0.033 H xX h ” 0.30 30.00 126 -0.003 0.34 87.70 0.63 -0.006 0.44 91.70 0.50 -0.006 0), 35 93.95) 0.45 -0.004 O22 94.95 0.41 -0.003 0.16 96.50 0.35 0.003 0), L7/ 98.25 0.30 0.014 0.08 100.00 0.24 0.023 IRREGULAR TESTS KEY: T = peak wave period (sec) hy oe = depth at bar toe (ft) Xioe = distance from wave board to bar toe (ft) h, = depth at bar crest (ft) X, = distance from wave board to bar crest (ft) h, = depth at bar trough (ft) X, = distance from wave board to bar trough (ft) B, = seaward bar angle (deg) B3 = shoreward bar angle (deg) m = beach slope Hmax = maximum wave height at gage (ft) Hs = significant wave height at gage (ft) Hmean = mean wave height at gage (ft) Hrms = rms wave height at gage (ft) X = distance from wave board to gage (ft) h = still-water depth at gage (ft) n = mean-water level relative to still-water level (ft) 35 Case R2210: h, = 0.56; X, = 90.45; p, Hmax .49 .43 -45 .38 21 .24 523 726 a 2 eo 2 2&2 2 Oo ©& Case R2220: Hmax .50 .43 -41 .49 .24 oO) 24 25) a Oo Oo 2 8 C2 Oe © Case R2230: ly, = O.5923 Xe Hmax 503 -42 -41 .43 5 Oe) 9225) 2D tS ko OO Mas = OoH25 55 = Wc Ios SS Woks 26, = GL) ots}58 = 7 fy = Ade Hs Hmean Hrms Xx 0.37 O25 O27 30.00 0.31 0), Zit 0.23 84.50 0.31 0), Zit OF23 85.00 0.30 OR 28 88.00 0.15 Feat 0.12 91.00 0.18 ORS 0.14 93.00 17 0), 13} 0.14 96.00 0.18 0.14 0.15 100.00 SS 00 lttion = OoGils 5, = SSoEOR Ing = Oe238 0G = YWWsS0s ltr, = O58 de 3 SILOS fe = MOS fe > 2X0). Hs Hmean Hrms X O37 0.25 0.27 30.00 O32 0.22 0.24 87.75 0.31 0.21 0), 23) 88.25 OFS 2 0), 22 0.24 89.65 0.18 0.13 0.14 Ql, 63) 0.18 0.14 0.15 93.00 0.19 (0), 5) 0.15 96.00 0.20 OFPES: 0.16 100.00 T = 1.00 Ite = O.538 265, = BOGOR In, = O.858 x, = VOLO8 = 91.30; 6, = 10; 63 = 20. Hs Hmean Hrms Xx OFS8 O25) O27, 30.00 O33 OR22 0.24 88.10 0.30 0.20 O22 88.60 0.33 0.23 OF25 90.10 0.18 0.13 0.14 Vl, OS ORS 0.14 @}, 5) 93.00 0.20 Opps 0.16 96.00 0.20 0.16 (), L7/ 100.00 SoS Oo OC EG CO eC OC OS ol C36 fap oOo EF LC OO C6 Oo OC Ff So o& © OC © Oo OC PP a Co CEC CO © © © Pf 5 2e) sUd 74 .40 54 -42 38 .24 o Do) . 63 . 63 38 .50 -47 . 38 .24 62s) .63 .62 .38 .50 47 238 .24 So eo oOo oe © SG © © .004 .000 .002 .004 .005 .007 .008 .O11 .003 .002 .001 .004 .004 .006 .009 .010 .003 .002 .001 .003 .002 .005 .008 .009 Case R6210: hoe = 0.88; X.. = 80.10; h, = 0.32; KX, = T = 1.50 hy, = 0.66; X, = 86.50; B, = 5; 63 = Hmax 0. 0. .60 Bo .34 .34 5 2 ool Se & © 2S& © © Case R6220: 70 62 a Oo 2 2 & © © 2b SS 1.50) Hs 0. .45 -45 41 523} .26 . 26 .23 46 Hmean 0. 0. .31 .30 .16 cdl’) .20 . 16 Se Oo Co eC SO CS 45 31 Hrms 0. 0. .34 .32 oll .20 5 Cal .18 So 2 2 2&2 2 © 35 34 hkoe = 01.78; Xtcg = 83.45; h, = 0.32; h, = 0.67; X, Hmax Case R6230: Sa © 2 2&2 © Oo S& © .73 OY) .62 .58 .37 40 3) .34 So C&C eC C2 eC C&O Ce S&S tS 1.50 86.41; B, = 10; B; Hs 47 245 44 48 atoll 729 .27 .21 Hmean 533) .32 . 30 .35 5 ALG) So © oOo OCG Oo Co OC © 22 o Zak .16 Hrms eS & oo eC 2 ee 2&2 © .35 34 33 38 .21 oe) B22, .17 on 2 Oo 792 Meo, = BSS ly, = O.S2e h, = 0.70; X, = 85.62; B, = 15; B3 = Hs Hmax eo C2 COCO Oo Oo Oo © .72 5 Xs) oo) .67 . 36 .43 .34 o28) oO oS CO Oo C2 Ceo OC © .48 -42 .44 oe oll .30 .27 .20 Hmean soe) . 30 520) 5 SU . 20 ode) 9 dD 5 ILS) Sa 2 C2 C2 OC oO EC © Hrms So © © © © © © © .36 aS oO) 40 o call .24 .23 .16 20. 100. C37 .85 oO Oo 2 © © © © Ff So Co Oo 2 CEC oC oC Pf Se © © © © © © F .25 .81 0 UY -47 .65 mod) 41 N23 .009 .005 .006 .009 .010 .014 .014 .016 .008 .006 .007 .O11 .007 .013 .014 .020 .006 .007 .007 .O15 .006 .010 .012 .019 Case R8210: eon = Ostlig en = BOCG iy SS Od22 1G = Hmax . 66 .63 .65 .63 . 36 ofl oD .38 So 2 eo 2 2 2 2 © Case R8220: a r/S) 5; B3 Hrms 0.33 0) 3S} OFS3 OFS3) 0. 0 0 0 18 al) o dul oil 7/ Neon = O25 Gen = G8.898 I, = O2328 h, = 0.66: KX, = 86.57: B, Hmax . 66 5 XS} 65 .62 5 OY) .44 .34 35) Ooo ©oe © © © © Case R8230: Hs Hmean 0.45 OF29 0.44 0.30 0.45 0.30 0.44 0.30 0.24 0.17 0.28 0.20 0.27 0.20 0.23 0.16 fo ed Hs Hmean 0.44 OF29 0.42 0.28 0.45 0.30 0.47 0.33 0.27 0.19 0.30 0.22 0.27 0.21 0.22 0.15 Ss 17d 10; 63 Hrms So Goo © Oo €E 5 31 55) . 36 lines = O.708 Se 2 OS.S79 Im 2 0.308 h, = 0.64; X, Hmax . 66 .60 .58 64 -41 -41 533) .34 So & G2 Ee o& 2G © © = /5aD9 [oa = Hs Hmean 0.44 0.29 0.42 0.29 0.45 0.30 0.48 ORS3 0.28 0.20 0.30 OF23) 0.27 O21: git OFS ISS 85 Hrms S © 62 eo © 6 © © 6 bk 5 Sul 538) oe) Zl 24 B22. .16 20. 100. C38 So Oo eo 2 2 © OS Sp Soe © © © © © FU aD FN wD Sy TS wt Sp eo Sy 24 er SOE age oe Ss So fF Own FF yn NN Pf FF WHS DW A Ww .006 .006 .005 .010 .008 .012 .012 .014 PLANE SLOPE IRREGULAR TESTS Case R2000: T = 1.00 m = 0.033 Hmax Hs Hmean Hrms x h n 0.48 0.38 OR25 0.28 30.00 1.25 -0.001 0.43 0.32 0.21 0.23 84.50 0.75 0.000 0.43 0.31 0.21 0.23 85.00 0.73 -0.003 0.41 0.31 0.21 0.23 88.00 0.63 -0.003 0.34 0.26 0.18 0.20 91.00 0.53 -0.003 0.37 0.28 0.20 0.21 93.00 0.46 -0.002 0.30 0.24 0.18 0.19 96.00 0.33 0.000 0.24 0.18 0.14 0.15 100.00 0.24 0.008 Case R6000: T = 1.50 m = 0.033 Hmax Hs Hmean Hrms xX h ” 0.69 0.43 0F33 0.35 30.00 2) -0.004 0.59 0.45 0.32 0.34 79.10 0.94 -0.002 0.60 0.45 0.31 0.34 79.90 0.90 -0.004 0.56 0.45 0.31 0.34 84.50 0.75 -0.006 0.48 0.37 0.27 0.29 87.25 0.65 -0.006 0.49 0.37 0.27 0.29 91.00 0.53 -0.004 0.35 0.28 0.22 0.23 95.00 0.41 0.004 0.36 0.20 0.15 0.16 100.00 0.23 0.019 Case R8000: T=1.75 m = 0.033 Hmax Hs Hmean Hrms xX h 7) 0.68 0.45 0.30 0.33 30.00 oe -0.004 0.62 0.45 0.30 0.33 79.70 0.90 -0.005 0.64 0.45 0.30 0.33 80.70 0.86 -0.007 0.59 0.45 0.30 0.33 85.00 0.73 -0.007 0.49 0.37 0.26 0.28 88.00 0.63 -0.006 0.49 0.38 0.27 0.29 91.00 0.53 -0.005 0.37 0.30 O22 0.23 95.00 0.41 0.002 0.35 0.22 0.14 0.16 100.00 0.23 0.016 C39 a(m) G(m) APPENDIX D: NOTATION Empirical coefficient in breaker depth equation, Weggel (1972)* and present study. Coefficient in breaker height equation (Goda 1975) Cross-sectional area of reef Cross-sectional area of surface roller Cross-sectional area of vortex Empirical coefficient in breaker depth equation, Weggel (1972) and present study. Nondimensional energy flux Coefficient in equation for depth over bar crest Wave celerity Coefficient in generic breaker index equations Coefficient in generic breaker index equations Group speed of waves Group speed of waves at breaking in the presence of return flow Empirical coefficient in breaker height equation Empirical exponent in breaker height equation for barred profiles Empirical function for breaker height (Weggel 1972) Empirical function for breaker height (Weggel 1972) Wave energy Spectral peak frequency Wave energy flux Function of beach slope Empirical function in breaker height (Weggel 1972) Stable wave energy flux Acceleration due to gravity Spectral peak enhancement parameter Empirical function in breaker height (Weggel 1972) Still-water depth Water depth in horizontal section of wave tank Water depth at breaking Water depth at crest of bar * See References at the end of the main text. D1 he Water depth at reef toe h, Water depth at trough of bar H Wave height H’ Wave height measured in horizontal section of wave tank Hi/3 Average of the highest one-third wave heights in a wave record Hy Breaking wave height Hy, Breaking wave height in the presence of return flow H, Incident wave height Hea Maximum wave height Ho Deepwater wave height (Gn Wave height in model Caer Wave height in prototype. Ge)e Significant deepwater wave height He Reflected wave height Has Root-mean-square wave height He Significant wave height H. Significant wave height measured in horizontal section of wave tank aL y/lly, Deepwater wave steepness (HELA) ee Critical wave steepness for reflection (Miche 1951) (HEALS) Model deepwater wave steepness (Clee ears Prototype deepwater wave steepness k Wave number K Coefficient in breaker height equation (Goda 1975) K Dimensionless decay coefficient (Dally, Dean, and Dalrymple 1985a, 1985b) Ke Reflection coefficient L Wavelength IL? Wavelength calculated in horizontal section of wave tank Ly Wavelength at break point Ihr Deepwater wavelength Cie Significant deepwater wavelength (Ly) Peak deepwater wavelength I Wavelength at reef toe m Beach slope n Ratio of group velocity to individual wave velocity D2 n(m) Empirical exponent wave decay equation (Smith and Kraus 1988) Exponent in generic breaker index equations Exponent in generic breaker index equations Empirical exponent in breaker height equation for plane slopes Empirical exponent in breaker height equation for barred profiles Reef reflection parameter Runup Average runup Two-percent runup Significant runup Exponent in breaker height equation (Goda 1975) Horizontal distance from the bar crest to the seaward toe of the bar Radiation stress Wave period Wave period in model Peak wave period Wave period in prototype Significant wave period Horizontal velocity component Horizontal velocity component at break point Horizontal distance Coefficient in breaker height equation (Goda 1975) Plunge distance Splash distance Penetration distance Reef height Coefficient in hydraulic jump dissipation rate equation Beach slope angle Lower (primary) seaward bar angle Upper (secondary) seaward bar angle Shoreward bar angle Empirical stable wave factor equal to ratio of stable wave height to water depth Ratio of wave height to water depth at breaking Energy dissipation rate D3 o7, OF Free-surface water elevation Surf similarity parameter Offshore surf similarity parameter Inshore surf similarity parameter Water density Spectral width parameter Low-frequency spectral width parameter High-frequency spectral width parameter Ratio of breaking wave height to deepwater wave height D4 in mai : nies | 7 : en ve > eng Lae of Cae Ty ic ne i ; : i } . | " . | ; at 7 Ca ed Ny ' a: ; ae an waa | cine my . ‘ i, ae ‘ 4 i i i Dea ct) diy lol! 4