TECHNICAL REPORT CERC-87-17 CHARACTERISTICS OF REEF BREAKWATERS US Army Corps of Engineers by John P. Ahrens Coastal Engineering Research Center DEPARTMENT OF THE ARMY Waterways Experiment Station, Corps of Engineers PO Box 631, Vicksburg, Mississippi 39180-0631 December 1987 Final Report Approved For Public Release; Distribution Unlimited Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 Under Civil Works Research Work Unit 31616 When this report is no longer needed 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. 0030110001157 Unclassified SECURITY CLASSIFICATION OF THIS PAGE REPORT DOCUMENTATION PAGE Ja. REPORT SECURITY CLASSIFICATION Unclassified 2a. SECURITY CLASSIFICATION AUTHORITY 2b. DECLASSIFICATION / DOWNGRADING SCHEDULE 4. PERFORMING ORGANIZATION REPORT NUMBER(S) Technical Report CERC-87-17 6b. OFFICE SYMBOL (If applicable) 6a. NAME OF PERFORMING ORGANIZATION USAEWES, Coastal Engineering Research Center 6c. ADDRESS (City, State, and ZIP Code) PO Box 631 Vicksburg, MS 39180-0631 8b. OFFICE SYMBOL (If applicable) 8a. NAME OF FUNDING/SPONSORING ORGANIZATION US Army Corps of Engineers 8c. ADDRESS (City, State, and ZIP Code) Washington, DC 20314-1000 11. TITLE (Include Security Classification) Characteristics of Reef Breakwaters 12. PERSONAL AUTHOR(S) Ahrens, John P. le Oceanographic Institution Form Approved OMB No. 0704-0188 Exp. Date: Jun 30, 1986 1b. RESTRICTIVE MARKINGS 3. DISTRIBUTION / AVAILABILITY OF REPORT Approved for public release; distribution unlimited. 5. MONITORING ORGANIZATION REPORT NUMBER(S) 7a. NAME OF MONITORING ORGANIZATION 7b. ADDRESS (City, State, and ZIP Code) 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER 10. SOURCE OF FUNDING NUMBERS PROGRAM PROJECT TASK WORK UNIT NO. NO. ACCESSION NO 31616 ELEMENT NO. 13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Year, Month, Day) 15. PAGE COUNT Final report FROM TO December 1987 62 16. SUPPLEMENTARY NOTATION Available from National Technical Information Service, 5285 Port Royal Road, Springfield, COSATI CODES GROUP SUB-GROUP 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number) Breakwaters Water waves (LC) (LC) 19. ABSTRACT (Continue on reverse if necessary and identify by block number) A laboratory study was conducted to determine the stability, wave transmission, wave reflection, and energy dissipation characteristics of reef breakwaters. Reef breakwaters are low-crested structures comprised of a homogeneous pile of stone with individual stone weights in the range of those ordinarily used in the armor and first underlayer of traditional multilayered breakwaters. irregular wave conditions. The study included over two hundred tests, all using Results of the study are discussed and summarized through the use of equations fit to the data. The equations fit the data well, phenomena as they are currently understood, 20. DISTRIBUTION / AVAILABILITY OF ABSTRACT [4] UNCLASSIFIED/UNLIMITED [1 SAME AS RPT 22a. NAME OF RESPONSIBLE INDIVIDUAL DD FORM 1473, 84 MAR OO pric Users 83 APR edition may be used until exhausted All other editions are obsolete are consistent with the physics of the various and approach logical limiting values. (Continued) ABSTRACT SECURITY CLASSIFICATION Unclassified 22b. TELEPHONE (Include Area Code) | 22c. OFFICE SYMBOL SECURITY CLASSIFICATION OF THIS PAGE Unclassified Unclassified SECURITY CLASSIFICATION OF THIS PAGE 19. ABSTRACT (Continued). Important findings include: A reef stability model which can predict the degree of degration of the struc— ture as a function of severity of irregular wave attack. A wave transmission model capable of predicting the amount of wave energy transmitted over and through the structure for both submerged and nonsubmerged reefs. A wave reflection model which makes accurate predictions of energy reflected from the reef for a wide range of wave conditions and structure heights. A model which predicts the amount of incident wave energy dissipated by the reef. SECURITY CLASSIFICATION OF THIS PAGE PREFACE The study reported herein was authorized by the Office, Chief of Engi- neers (OCE), US Army Corps of Engineers, and funded through the Coastal Engi- neering Functional Area of Civil Works Research and Development, under Work Unit 31616. The project was monitored by Messrs. John H. Lockhart, Jr., and John G. Housley, OCE Technical Monitors. The study was conducted at the Coastal Engineering Research Center (CERC) of the US Army Engineer Waterways Experiment Station (WES). Dr. C. Linwood Vincent, CERC, is Program Manager of the Coastal Engineering Functional Area. This report was prepared by Mr. John P. Ahrens, Research Oceanographer, Wave Research Branch (CW-R), Wave Dynamics Division (CW), CERC. Assisting Mr. Ahrens in conducting the study were the following CERC employees: Ms. Karen P. Zirkel and Messrs. Louis Myerele and Martin F. Titus, Engineering Technicians; Messrs. John Heggins, Computer Assistant, and Leland Hennington, Summer Aide, who helped to analyze the data; and Eng. Gisli Viggosson on temp- orary assignment from the Icelandic Harbour Authority, Reykjavik, Iceland. Work was performed under direct supervision of Messrs. D. D. Davidson, CW, and C. Eugene Chatham, Chief, CW; and under general supervision of Dr. James R. Houston and Mr. Charles C. Calhoun, Jr., Chief and Assistant Chief, CERC, respectively. This report was edited by Ms. Shirley A. J. Hanshaw, Information Products Division, Information Technology Laboratory, WES. Commander and Director of WES during publication of this report was COL Dwayne G. Lee, CE. Technical Director was Dr. Robert W. Whalin. PREFACE.... PART I: CONTENTS eoceoeseeeeseceeeee eee ee ee eee eee eee eo oe ee eee eee ee eee ee eo eo ee ee 8 8 8 TONPONOWDIDGIEILON 5 oc GOO GoD dD DDD GND DOOD KD DODD ODDO OOD OOOOHODDNNDNRS Bae) eeaRONLS Good GOOD DODD DDU DOOD OOOO OODNOAD OAD DObDOOGDDDDOOOCDONND BXEOINAISG 6000000000000000000005000000000005000005000000000000000000 PART ITI: LABORATORY SETUP AND TECHNIQUES USED......2...ccccccccccecee Seeloslilayy WOSESs oon c 0c dD OOO DDD DODD ODO ODD DDC ODDDDDDDDDDDODDDDNDNDN?S BrevalousmDamage meses cyerelelelelehelveketelelelshovalclolelelelereleielelenchenelehelekatcl ol Neleke rete sAOEIILE GUEAVESSo5 coo o0d 00D DDD DDO DODDDOODODDODDD ODN OO ONODDONODONNN PART III: STABILITY AND PERFORMANCE RESULTS ........2.ccccccccccecccces Stabislitty to Erreguillar) Wave Attackicysjcr.icjcieleleielelcielslelels/elclclelle)eletelclelelels Wave Wave PART IV: REFERENCES. PHOTO 1 APPENDIX A: APPENDIX B: APPENDIX C: HesINGUSSGtCMs ooo6G0G D000 0U0 DODD DDDNDDDDDO0000000000000000000 Refllectionwand) Energy Diss i patdomleereielelelelelerelelelelelovolcicnelelerensicvons GOMGGUSIONS > ooc0d00d 0060 DDD 0b0D000DDDUDDDDNDOOD0DNDDDD000000 TABULAR SUMMARY OF STABILITY AND PERFORMANCE DATA........ REGRESSION ANALYSIS USED TO DEVELOP FIGURE 29 SHOWING ENERGY DISTRIBUTION IN VICINITY OF REEF.......... INOAUAMEILONN GS 6.60.000000000000000000000000000000050000000000000 Bl Cl CHARACTERISTICS OF REEF BREAKWATERS PART I: INTRODUCTION 1. A reef breakwater is a low-crested rubble-mound breakwater without the traditional multilayer cross section. This type of breakwater is little more than a homogeneous pile of stones with individual stone weights similar to those ordinarily used in the armor and first underlayer of conventional breakwaters. 2. In recent years a number of low-crested breakwaters have been built or considered for use at a variety of locations. Most of these structures are intended to protect a beach or reduce the cost of beach maintenance. Other applications include protecting water intakes for power plants and entrance channels for small-boat harbors and providing an alternative to revetment for stabilizing an eroding shoreline. In situations where only partial attenua-— tion of waves on the leeside of a structure is required, or possibly even advantageous, a low-crested rubble-mound breakwater is a logical selection. Since the cost of a rubble-mound breakwater increases rapidly with the height of the crest, the economic advantage of a low-crested structure over a tradi- tional breakwater that is infrequently overtopped is obvious. Because the reef breakwater represents the ultimate in design simplicity, it .could be the optimum structure for many situations. Unfortunately, the performance of low- crested rubble-mound structures, particularly reef breakwaters, is not well documented or understood. Background 3. A number of papers have noted that armor on the landside slope of a low-crested breakwater is more likely to be displayed by heavy overtopping than armor on the seaward face (Lording and Scott 1971, Raichlen 1972, and Lillevang 1977). Raichlen discusses the characteristics of overtopping over the crest and the inherent complexity of the problem. Walker, Palmer, and Dunham (1975) give a carefully reasoned discussion of the many factors influ- encing stability of heavily overtopped rubble-mound breakwaters. They also show a figure which suggests what armor weight is required for stability on the backside of a low-crested breakwater. Unfortunately, the data scatter shown in the figure undermines confidence in the suggested armor weights. 4. In Australia, the breakwater at Rosslyn Bay was damaged severely during Cyclone David in 1976 (Bremner et al. 1980). The crest height of the structure was reduced as much as 4 m but still functioned effectively as a submerged breakwater for over 2 years until it was repaired. Based on the surprisingly good performance of the damaged Rosslyn Bay breakwater and the findings from model tests, a low-crested design was chosen for the breakwater at Townsville Harbor, Australia. This breakwater is unusual because it was built entirely of stone in the 3- to 5-ton* range (Bremner et al. 1980). Reef breakwaters, as described in this paper, are very similar to the Townsville breakwater except a wider gradation of stone was used in the model breakwater tests discussed herein. 5. Seelig (1979) conducted an extensive series of model tests to deter- mine wave transmission and reflection characteristics of low-crested break- waters, including submerged structures. From these tests Seelig concluded that the component of transmission resulting from wave overtopping was very strongly dependent on the relative freeboard (i.e., freeboard divided by inci- dent significant wave height). Recent work by Allsop (1983) with multi- layered, low-crested breakwaters shows that wave transmission is strongly dependent on a dimensionless freeboard parameter which includes the zero- crossing period of irregular wave conditions. Allsop did not find substantial wave period dependency in his evaluation of breakwater stability. He indi- cates, however, that since wave transmission (which largely results from over— topping) is dependent on period, then possible stability of the backside slope would also be a function of wave period. Scope 6. A study currently being conducted at the US Army Engineer Waterways Experiment Station's Coastal Engineering Research Center is intended to docu- ment the performance of low-crested breakwaters. This paper discusses labora- tory model tests of reef breakwaters and provides information on their stabil- ity to wave attack, wave transmission and reflection characteristics, and wave energy dissipation. a * Metric ton. PART II: LABORATORY SETUP AND TECHNIQUES USED 7. To date, 205 two-dimensional laboratory tests of reef breakwaters have been completed. These tests were conducted in a 61-cm-wide channel within CERC's 1.2- by 4.6- by 42.7-m tank (Figure 1). All tests were SCALE 0 1 2 3 4 5 (m) o DENOTES WAVE GAGE LOCATION WALL OF WAVE TANK BEACH "PONDING RELIEF T ei npe a == REEF PONDING RELIEF : BREAKWATER S3GRAVEL WAVE ABSORBER BEACH TRAINING WALLS GRAVEL WAVE ABSORBER BEACH WALL OF WAVE TANK PLAN VIEW Figure 1. Plan view of wave tank and test setup conducted with irregular waves. The spectra used had wave periods of peak energy density a ranging from about 1.45 to 3.60 sec, and water depth at the structure d. ranged from 25 to 30 cm. Signals to control the wave blade were stored on magnetic tape and transferred to the wave generator through a computer data acquisition system (DAS). For this study four files were stored on the tape which could produce a spectrum with a distinct period of peak energy density. Table 1 gives the nominal period of peak energy density for each file. 8. If there were no attenuation of the signal to the wave generator, the files used were intended to produce a saturated spectrum at all frequences above the frequency of peak energy density for the water depth at the wave blade. For frequencies lower than those of the peak, the energy density de- creased rapidly. This procedure produced a spectrum of the Kitaigorodskii type as described by Vincent (1981). The amplitude of the signal to the wave generator was attenuated by a 10-turn potentiometer in a voltage divider * For convenience, symbols and unusual abbreviations are listed and defined in the Notation (Appendix C). Table 1 Period of Peak Energy Density for Each Tape File Tape ee cubeaas File i 1 15 AS 2 oO) 3 2.86 4 3.60 ———————————————————— network which allowed control of the wave heights generated. In addition, the waves were generated in a water depth 25 cm greater than at the breakwater and shoaled to the water depth at the structure over a 1-V on 15-H slope (see Figure 2). This setup ensures that severe conditions can be developed at the SCALE 1 0 1 2M EME Sl 0 (ie an TRAINING WARE WAVE GAGES|\ =< maya WAVENCAGES ae lis 19 METERS TO WAVE GENERATOR Haas ‘aunt Y) FROM END OF TRAINING WALLS 4. 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 DISTANCE ALONG CHANNEL (M) Figure 2. Cross section of test channel structure site. Incident zero-moment wave heights His ranged from about 1 to 18 cm. 9. Three parallel wire-resistance wave gages were used in front of the breakwater to resolve the incident and reflected wave spectra using the method of Goda and Suzuki (1976), and two wave gages were placed behind the structure to measure the transmitted wave height. The location of gages is shown in Figure 2. During data collection gages were sampled at a rate of 16 times per second for 256 sec by the same DAS which controlled the wave generator motion. 10. Two types of model tests were conducted during this study: stabil- ity and previous damage tests. Each type followed a prescribed sequence. Stability Tests 11. For a stability test the following test sequence was used: a. Rebuild the breakwater from the previously damaged condition. b. Survey the breakwater to document its initial condition. c. Calibrate the wave gages. d. Select the tape file and signal attenuation setting. e. Start the wave generator and run waves. Es Collect wave data (several or more times). g. Stop the wave generator. h. Survey the breakwater to document its final condition. The duration of wave action was from 1-1/2 hr for a test using the File l spectrum to 3-1/2 hr for a File 4 spectrum. Generally, the technicians observing the tests thought that most of the stone movement occurred during the first 10 or 15 min of wave generation, so the final survey is regarded as an equilibrium profile for the structure. In rebuilding the breakwater the technicians rarely touched the stone but merely pushed it around by foot until the shape conformed to the desired initial profile. This procedure was a con- scious effort to avoid overly careful placement of the stone. Outlines of the desired initial profile were fixed to the walls of the testing channel, and a moveable template was used to ensure that the initial profile was reasonably close to the desired profile. Initial configuration of the breakwater for a stability test was a narrow, trapezoidal shape with seaward and landward slopes of 1V on 1.5H (Figure 3). Crest widths were three typical stone dimen- sions wide, using the cube root of the volume of the median weight stone W 50 as the typical dimension d Figure 3 also shows a typical profile after 500s moderately severe wave attack during a stability test. Wave transmission and reflection also were measured during a stability test. 30 = — —SWLSEASIDE Wve LANDSIDE «ivy hvaaie == INITIAL eR PROFILE Sz ao 10 ACCRETION S AREA = 0 10 20 30 40 50 60 70 80 90 100 110 120 DISTANCE ALONG CHANNEL, CM Figure 3. Cross-sectional view of initial and typical damaged reef profiles (swl denotes still-water level) Previous Damage Tests 12. Previous damage tests were conducted to answer the question of how the breakwater would perform for moderate wave conditions after it had been damaged by very severe wave conditions. For previous damage tests there was very little readjustment of the damage profile from test to test; conse- quently, the breakwater was not rebuilt at the end of a test. No stability information was obtained from these tests, and the duration of wave action was only half an hour; however, wave transmission and reflection were measured. Previous damage tests were performed in the following sequence: a. Survey breakwater for last test which becomes initial survey for current test. b. Calibrate wave gages. c. Select wave file and signal attenuation setting. d. Start generator and run waves for half an hour. ie Collect wave data (two or three times). f. Stop wave generator. g- Survey breakwater as noted above in Step l. 13. All 205 of the completed tests of this study can be divided logi- cally into 10 subsets or test series. Because of the test plan, stability test series have odd numbers, and previous damage test series have even num-— bers. Table 2 lists the basic information about each subset. 14. Two different sizes of stone were used during this study. For sub- sets 1 through 6 an angular quartzite with a median weight of 17 g was used, Table 2 Basic Data for Each Subset Area of eer Breakwater N Water Height Median . pee me Depth "as built" Stone Weight ross Section Subset of 4 ot G < ; No. Tests s 2 {oul Cc > cm 50 5 fs t > cm 1 27 25 25 iL7/ 1,170 ; : fe NA* 17 1,170 3 29 25 30 7/ 1,560 4 Le LS NA 7 1,560 ; < 22 35 ly 2,190 : ie 22 NS 17 2,190 t ae me 32 i 1,900 8 26 BS) NA 71 1,900 9 13 30 Sy? all 1,900 x 2 20 NA 71 1,900 * NA denotes not applicable to previous damage test series. and for subsets 7 through 10 a blocky to angular diorite with a median weight of 71 g was used. Photo 1 depicts the stone, and Table 3 summarizes informa-— tion about it. Table 3 Stone and Gradation Characteristics SNR@ Hansa eter lai cnn inn QUaT.071¢ Cnn nD Torte 27% weight (g) 7.0 14.0 Median weight, Weg (g) 7/5 (0) ales) 98Z weight (g) 28.0 139.0 Density (elie) 2.63 2.83 Porosity (Z) 45 44 Profile Surveys 15. Initial and final profiles of the reef were obtained by survey. The survey rods had feet attached with ball-and-socket connectors. For the small stone used for subsets 1 through 6, the foot of the survey rod had a diameter of 2.54 cm; and for the somewhat larger stone used in subsets 7 through 10, the foot of the survey rod had a diameter of 3.81 cm. Three pro- files were used to establish an average profile for the reef. One profile line was exactly in the center of the wave channel, and the other two profile lines were 15 cm on either side of center. The survey interval along the channel was 3.05 cm. 10 PART III: STABILITY AND PERFORMANCE RESULTS 16. The report herein consolidates findings from all of the data sub- sets identified in Table 2 into general conclusions about the stability and performance characteristics of reef breakwaters. Specific characteristics in- clude the stability of reef breakwaters to irregular wave attack, wave trans— mission over and through the breakwater, wave reflection from the breakwater, and dissipation of wave energy. A mathematical model is developed for each characteristic which provides a simple method to summarize findings from this study and a convenient way to furnish results to potential users. These math- ematical models are intended to work together with the stability model fur- nishing the equilibrium crest height to both transmission and reflection models which together are used to estimate the amount of energy dissipated by the reef. Stability to Irregular Wave Attack 17. The stability of reef breakwaters will be quantified by damage or lack of damage during a test, the most important aspect of which is the reduc- tion in crest height caused by wave attack. This aspect of stability is important because the performance of a reef breakwater will be judged largely on its wave transmission characteristics. Wave transmission is very sensitive to crest height relative to water level. Crest height reduction factor 18. One of the most effective methods to evaluate damage to a reef breakwater is to use the ratio of the crest height at the completion of a test to the height at the beginning of the test before waves have been run. This ratio, h o/h? , will be referred to as the crest height reduction factor. For comparing damage within a subset, h o/h? is effective because it inherently accounts for the random variation of one to two centimeters in the constructed crest height from test to test within a subset. Another advantage of the crest height reduction factor is that all stability subsets have the same natural limiting values of 1.0 and 0.0. Stability number and spectral stability number comparison 19. Experience with the stability of traditional rubble-mound break- waters to monochromatic waves suggests that one of the most important til variables to explain damage would be one similar to the stability number used by Hudson and Davidson (1975). The following definition is used for the stability number for tests with irregular waves: Sate 173 (1) where we is the density of stone and we is the density of water. Since these tests were conducted in fresh water, WEES 1.0 afen® . As far as the stability tests of reef breakwaters are concerned, it was apparent that tests with a higher period of peak energy density did more damage than similar tests with a smaller period of peak energy density. This finding is consistent with the results of a study conducted by Gravesen, Jensen, and Sorensen (1980) on the stability of high-crested, rubble-mound breakwaters exposed to irregular wave attack. According to the stability analysis of Gravesen, the spectral stability number is defined (x, 7 ye Ne a mo (2) where os is the Airy wave length calculated using oe and the water depth at the toe of the reef d. 20. Figures 4 through 8 show comparisons of the effectiveness of the stability number and the spectral stability number in accounting for damage to reef breakwaters. In Figures 4, 5, 6, 7, and 8 the crest height reduction factor is plotted versus the traditional stability number and the spectral stability number for stability subsets 1, 3, 5, 7, and 9, respectively. The figures show that there is less scatter in the damage trends when they are plotted versus the spectral stability number. They also show that there is little or no damage for spectral stability numbers less than about six but that damage increases rapidly for spectral stability numbers above eight. In the following analysis the spectral stability number will be used to define 12 RELATIVE CREST REDUCTION, hc/he’ RELATIVE CREST REDUCTION, he/hc’ STABILITY NUMBER, Ns a. Crest height reduction factor versus the stability number 0 2 4 6 8 10 12 14 16 18 SPECTRAL STABILITY NUMBER, Ns* b. Crest height reduction factor versus the spectral stability number Figure 4. Stability comparisons for subset 1 13 S (Co) S fon) ° N RELATIVE CREST REDUCTION, hc/hc’ S (o>) 0.5 STABILITY NUMBER, Ns a. Crest height reduction factor versus the stability number ’ nae ty Uj Cj a C 10) 2 4 6 8 10 12 14 16 1 SPECTRAL STABILITY NUMBER, Ns* 2 wo S fon) ° N RELATIVE CREST REDUCTION, hc/hc’ ° for) 0.5 € b. Crest height reduction factor versus the spectral stability number Figure 5. Stability comparisons for subset 3 14 RELATIVE CREST REDUCTION, hc/hc’ RELATIVE CREST REDUCTION, hc/hc’ STABILITY NUMBER, Ns a. Crest height reduction factor versus the stability number 1.0 0.9 0.8 0.7 0.6 0.5 0 2 4 6 8 10 12 14 16 18 SPECTRAL STABILITY NUMBER, Ns* b. Crest height reduction factor versus the spectral stability number Figure 6. Stability comparisons for subset 5 15 RELATIVE CREST REDUCTION, hc/hc’' RELATIVE CREST REDUCTION, he/hc’ STABILITY NUMBER, Ns a. Crest height reduction factor versus the stability number (0) 2 4 6 8 10 12 14 16 18 SPECTRAL STABILITY NUMBER, Ns« b. Crest height reduction factor versus the spectral stability number Figure 7. Stability comparisons for subset 7 16 RELATIVE CREST REDUCTION, he/hc’ RELATIVE CREST REDUCTION, he/hc’ STABILITY NUMBER, Ns a. Crest height reduction factor versus the stability number 1.0 0.9 0.8 0.7 0 2 4 6 8 10 12 14 16 18 SPECTRAL STABILITY NUMBER, Ns* 0.6 0.5 b. Crest height reduction factor versus the spectral stability number Figure 8. Stability comparisons for subset 9 i the relative severity of wave attack on reef breakwaters. Secondary stability factors 21. Data analysis and observation of the laboratory tests indicate that several factors other than the spectral stability number have a quantifiable influence on the stability of reef breakwaters. Figure 9 will help identify what will be referred to as secondary stability factors or variables. In Figure 9 the damage trends for all five stability subsets are shown using sub- jectively drawn curves. Figure 9 shows the relative crest height Bede (see Figure 3) as a function of the spectral stability number. For intercomparing damage trends between subsets, the variable ho/d, is better than ia 5 When various subsets are plotted using ho/ht » the data trends tend to fall on top of each other, especially for Ne <8. Using ho/d, to show damage trends spreads the data out so that subsets can be distinguished and provides better orientation by showing the swl. 22. Relative exposure to wave action. One secondary stability factor is the relative exposure of the structure to wave action. Submerged break- waters are much less exposed to wave attack than breakwaters with crests above the water level. Water overlying a submerged crest greatly dampens wave impact forces and attenuates the lift and drag forces on the stone. This fac-— tor is illustrated in Figure 9 where structures with the greater initial rela- tive height hi/d. have their height reduced more rapidly with increasing N& than structures with lower initial relative height. In Table 4, which can be used with Figure 9 to evaluate the influence of secondary stability factors, the average value of initial relative crest height hi/d. is given by subset along with two other secondary stability factors, the bulk number and the "as built" effective reef slope C' , which are discussed below. Subsets 1 and 5, which represent tests using the same stone size and water depth, illustrate the influence of hi/d, on stability. Figure 9 shows that the wide dif- ference in initial relative height of these structures is maintained until Ne is about 6.0; however, when noticeable stone movement starts at about Ne = 6 , the difference in relative heights for the reefs of the two subsets tends to decrease with increasing value of Né . For the most severe condi- tions at about NE = 17 , the difference in relative height between the two subsets is not very large. Based on analysis of all the data, it is concluded that the greater the initial height of the reef the more vulnerable it is to wave attack. 18 RELATIVE CREST HEIGHT, h./d, SUBSET 5 SUBSET 7 SUBSET 3 SUBSET 9 SUBSET Ste 1 APPROXIMATE THRESHOLD OF STONE MOVEMENT SUBSET SPECTRAL STABILITY NUMBER, N,°* Figure 9. Damage trends of the relative crest height as a function of the spectral stability number for the stability subsets 1, 3, 5, 7, and 9 Table 4 Average Values of Secondary Stability Variables by Subset Subset No. * kk 1 Woy ST Sy (68) Relative Crest Height Was Built” held, 0.99 1.18 1.41 277, 1.06 BL bulk number, defined by Equation 3. Eftective Reef 2 Reef Slope Size " . Ww as Built B* n C'x** 3h7/ 1.90 450 1.80 631 1.76 222 1.88 Oe: 1.88 C' effective reef slope, "as built," defined by Equation 4. 19 23. Influence of reef bulk. Subsets 1 and 5 can be used also to illustrate the influence of size or bulk of the reef on stability. Even though the difference in relative height for the two subsets narrows with increasing Ne ,» the crest heights of the reefs of subset 5 always are higher than those of subset 1. In fact, Figure 9 shows that the relative position of the trends for subsets 1, 3, and 5 are maintained such that the larger struc— ture always has a greater crest height than the smaller structure for a given value of Né - In order to intercompare the stability of all subsets, a general measure of breakwater size is needed which will be consistent with the data trends shown in Figure 9. Within this context, the variable which best characterizes the size of the reef breakwater is called the bulk number B n and is defined as Bh ot Sea Ce ee (3) ( 50 50 w r where rida area of breakwater cross section, ene Wale unit weight of stone Pyieme deo = dimension of stone, cm 24. Bulk number can be described as the equivalent number of median stones per median stone width in the breakwater cross section. Equivalent is used because BO does not include the influence of porosity which is about 45 percent for the two stone gradations used in this study. The value of the bulk number lies in its ability to explain the rather straightforward behavior of the relative location of the damage trends for subsets 1, 3, and 5 in Fig- ure 9. It also explains the rather anomalous behavior, such as that of the trend for subset 9 crossing the trend for subset 1. At first it seems sur- prising that the reefs of subset 9 degrade faster than those of subset 1, con- sidering that the reefs of subset 9 have the greater cross-sectional area (see Table 2). However, when the bulk number is used to measure the size of the reef rather than the cross-sectional area, the relative behavior of the damage trends for subsets 1 and 9 seems more plausible. Subsets 1 and 9 have bulk numbers of 337 and 222, respectively, indicating that the reefs of subset 1 20 have more stone in the cross section than the reefs of subset 9. All the data appear to indicate that when the relative severity of wave attack is based on the spectral stability number the stability of the reef correlates better with the number of stones in the cross section than with the absolute size of the cross section. Other factors being equal, a reef with a large bulk number is more stable than a reef with a small bulk number because there are more stones to dissipate wave energy and to shelter other stones from wave forces. 25. Effective slope of the reef. The remaining secondary stability factor is a combination of the first two. This factor, referred to as the effective slope of the reef, is obtained by dividing the cross-sectional area by the square of the crest height. Two effective slope variables will be dis- cussed in this report: (a) the effective slope of the structure "as built," defined as CONS aEs (4) (5) These variables are considered a cotangent function since dividing A. by h. one time produces a variable which can be regarded as a horizontal length, and dividing this length by ho creates a cotangent-like variable. For low- crested, or submerged reefs, these variables provide a simple way to charac- terize an average slope or shape for what is sometimes a rather complex shape (e.g., see Figure 3). Table 4 shows that the average values of the effective structure slope "as built" are in a relatively narrow range. Since the land- ward and seaward faces of the reef were built to a slope of 1V on 1.5H (cot 9 = 1.5), the difference between the values of C"' in Table 4 and 1.5 result from the crest width of the trapezoid which increases the effective slope, as illustrated in Equation 6. The "as built" cross section of the reef is a nar- row trapezoid with a crest width three stone diameters wide. For this study 21 the cross-sectional area of the reef is given approximately by /s) W Lhe , (%50 A. = (2) cot @ + 3h! ( 32) (6) where cot @ is the cotangent of the angle 6 between the "as built" sea- ward and landward breakwater slopes and the horizontal. If the severity of wave attack exceeds a value of the spectral stability number of about six, the reef deforms. A convenient method to quantify the deformation is to use effec-— tive response slope for reef breakwaters defined by Equation 5. In Figure 10 the response slope C is plotted as a function of N& . This figure is simi- lar to Figure 14.17 presented by Wiegel (1964) showing the relationships among the grain size, beach slope, and severity of the exposure of a beach to wave action. 26. Because of the narrow range of the effective "as built" reef slope C' (Table 4), it was not possible to quantify the influence of this variable on stability. It is assumed that the flatter the initial slope of the reef the more stable it will be. Future laboratory tests may expand the range of this variable so that the influence of the initial slope can be determined definitively. 27. Figure 10 suggests that a logical form for a reef breakwater sta- bility equation would be > = oH (Ga) (7) Fl QO Nie where C is a dimensionless coefficient. Regression analysis was used to 1 determine the value of Cy for tests where Né > 6.0 3; the value obtained was C, = 0.0945. With this value of Cy ,» Equation 7 explains about 99 percent of the variance in C for the 109 stability tests with Ne > 6.0 . Equation 7 approaches logical limits with C+eo,as N*¥ + o& s and Ge 160 5 as Ne <> (0) 22 EQUATION 7 me 4 OF STONE MOVEMENT BREAKWATER RESPONSE SLOPE, IE,, COTANGENT SPECTRAL STABILITY NUMBER Ns« Figure 10. Reef breakwater response slope versus the spectral stability number for stability subsets 1, 3, 5, 7, and 9 since the natural angle of repose for gravel is about 45 deg, giving C = 1.0 for a triangular reef cross section with side slopes of 1V on 1H. Equation 7 can be compared to the observed data in Figure 10. It is surprising that the response slope of the reef, stone size and density, and severity of wave attack can all be linked with a relation as simple as that in Equation 7. It is difficult to add secondary stability variables to an equation like Equa- tion 7 and improve the ability to predict the response slope over Equation 7 very much. At the same time it is clear from Figure 9 that secondary stabil- ity factors have some influence on reef stability. After trial and error the following equation was developed which includes one secondary stability vari- able and does a better job of predicting the response slope of the reef: d h' = exp {N* [0.0676 + 0.0222 (3) (8) Ss where the relative "as built" crest height of the reef hi/d, was added to an equation like Equation 7 to improve the predictive ability. Equation 8 23 explains 99.5 percent of the variance in C for the 109 tests with Ne >6. 28. It was found that when using Equation 8 to predict the relative crest height ho/d, for values of Né near or below six, illogically high values could result. Higher values are to be expected since Equation 6 was developed for tests where NE > 6 and there was enough rock movement to form an equilibrium reef profile and not for wave conditions where the "as built" reef slope was too stable to be deformed. Since it would be useful to have a stability model which predicts reasonable response crest heights over the entire range of test conditions, another stability equation was developed to predict crest heights for values of Né < 10. This range provides a con- venient overlap with the range of Equation 8 and allows an equation to be developed which will be simple enough to serve as a rule-of-thumb relation for zero to relatively low damage situations. This equation is given by = eR [ -0.00005 aay? 9] (9) er oe Ke) I Equation 9 provides a simple relation which follows the trend of the data well, albeit somewhat conservatively in the range Né < 10 as can be seen in Figure 11. The small levels of damage predicted by Equation 9 for N% < 6 represent settlement and consolidation of the reef under wave action and not conspicuous stone movement. 29. Equations 8 and 9 are used together to compute the response crest height of the reef over a wide range of wave severity. This approach will be referred to as the stability model. The procedure is to use Equation 8 for iN > 10 and Equation 9 for N* < 6. If we let the solution for ho /he in s Equation 9 be denoted (n,/az) and the solution for h/ht in Equation 8 be g denoted (h,/h2) » then the following equation u @ ts "2 h (‘ es :) h. 10-6 / \nt Vos G/T Cl) can be used in the transition region 6 < Ne < 10 to compute the response crest height ho - To judge the effectiveness of this procedure, 24 RELATIVE CREST REDUCTION, hc/hc’ | O OBSERVED DATA SPECTRAL STABILITY NUMBER Ns- Figure 11. Crest height reduction factor versus spectral stability number for stability subsets 1, 3, 5, 6, and 9 Figures 12, 13, 14, 15, and 16 were prepared to compare observed data for sub- sets 1, 3, 5, 7, and 9, respectively, with synthetic data trends generated by the stability model. Figures 12 through 16 show ho/d. versus Ne with synthetic trends for each subset generated using A. and d. from Table 2 and hi/d. from Table 4. Values of ho/d. were generated at integer values of Né for a range of Ne about the same as observed within each subset. Synthetic damage trends comprise the type of information that could be gen- erated by a user of the stability model. In general, synthetic trends follow observed data trends very well. Discrepancies between predicted and observed values appear to occur because the stability model does not include the bulk number. 25 SYNTHETIC DAMAGE TREND LEGEND < EF < ra} fa) Ww > x w LZ) ao fo) ia) cy So 1 1 1 0.9 sP/24LHOISH LS3Y9 SAILV13y SPECTRAL STABILITY NUMBER, Ns« Comparison of data and the synthetic damage trends generated by the stability model for subset 1 Figure 12. a 2 Ww oO Ww a sp/24’LHOISH LS349 SAILV13Y SPECTRAL STABILITY NUMBER, Ns« Comparison of data and the synthetic damage trends generated by the stability model for subset 3 Figure 13. 26 RELATIVE CREST HEIGHT, he/ds 0.9 ie) 2 4 6 10 14 16 18 0.8 07 0.6 0.5 SPECTRAL STABILITY NUMBER, Nse Figure 14. Comparison of data and the synthetic damage trends RELATIVE CREST HEIGHT, he/ds generated by the stability model for subset 5 a “/ Ea eg ee l| 2 ie aa Bieel 38 HH} Fel il les Gia 0 2 4 6 8 10 12 14 16 18 SPECTRAL STABILITY NUMBER, Nse Figure 15. Comparison of data and the synthetic damage trends generated by the stability model for subset 7 27 SYNTHETIC DAMAGE TREND RELATIVE CREST HEIGHT, hc/ds t ie il aS as Fe ee ae ee SPECTRAL STABILITY NUMBER, Ns* Figure 16. Comparison of data and the synthetic damage trends generated by the stability model for subset 9 Wave Transmission 30. For the tests mentioned above the wave transmission coefficient Ke is defined as (11) where He is the zero-moment transmitted wave height, and H. is the zero- moment wave height at the transmitted gage locations with no breakwater in the test channel. Although this is not the most commonly used definition of K. 5 it has some advantages over the traditional definition which is given by the ratio of transmitted to incident wave height. Equation 11 can be stated as the ratio of the transmitted wave height to the wave height which would be observed at the same location without the breakwater in the channel. This definition eliminates wave energy losses occurring between the incident and transmitted gages in the absence of a breakwater in the testing channel. 28 These losses were observed to be considerable for the most severe wave condi- tions during calibration of the channel. In effect, K. measures attenuation of wave energy because of the presence of the breakwater and eliminates addi- tional energy losses caused by natural wave breaking processes occurring be- tween the incident and transmitted wave gages. Using the above definition of K, will allow evaluation of wave energy dissipating characteristics of reef breakwaters in the next section. Because of the definition used, Ke should be somewhat conservative, i.e., higher than the more traditional definition of the transmission coefficient. 31. Wave transmission has proved to be a very difficult characteristic of reef breakwaters to predict partly because this study includes both sub- merged and nonsubmerged rubble-mound structures. Seelig (1980) found that the relative freeboard parameter F/Fio was the most important variable in ex- plaining wave transmission of submerged and overtopped breakwaters, where freeboard F is equal to crest height minus water depth, i.e., F=H -d_. However, a confusing trend will be obtained using this variable when there is a transition in the dominant mode of transmission from that due to wave runup and overtopping to that due to transmission through the structure. Figure 17 identifies the dominant mode of transmission as a function of the relative freeboard and shows a schematized data trend. The difficulty in parameteriz— ing the wave transmission process can be appreciated partly by considering the influence of the wave height. When a reef breakwater is submerged, the pri- mary mode of transmission results from wave propagation over the crest and, generally, the smaller the wave the greater the K. - When the crest is just above the water level, the dominant mode of transmission results from wave runup and overtopping, and the larger the wave the larger the KL ernthne relative freeboard is greater than about one, the dominant mode of transmis— sion is through the structure; and the smaller the wave the greater the K 9 A number of other factors tend to further confuse the above generalities. 32. The easiest way to discuss development of a general wave transmis— sion model for reef breakwaters is to first consider relatively high struc-— tures where relative freeboard F/Hoo is greater than one. When the dominant mode is wave transmission through the reef, Ky is a function largely of one variable which is the product of wave steepness and bulk number. Figure 18 shows a plot of K versus the reef transmission variable (., a5) / Mao A.) for the 37 tests where F/B > 1.0. This one variable caused the wave 29 1 ! ©) H Z Wu ar io 6 0.8 ! B ~y fe Feo] a) t = Ze v ix ©oa 3 ;co>Z iL 06 wiki TRANSMISSION O THROUGH = ' REEF {e) : a eo B a, TRANSMISSION Sons OVER CREST 4 g Pa ee Re = < (7p) jag J 5, ' 0.2 \ 1 t ! 0 0) -3 9) -1 () 1 2 3 RELATIVE FREEBOARD, F/H,_,, Figure 17. Conceptual sketch showing the dominant modes of wave trans— mission for a reef as a function of the relative freeboard 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 TRANSMISSION COEFF, K, 0.25 0.2 0.1 Figure LEGEND O OBSERVED DATA REEF TRANSMISSION VARIABLE, 7 Qo 18. Wave transmission coefficient as a function of the reef transmission variable to illustrate the ability of Equation 12 to predict transmission of relatively high reefs (FE/H > 1.0) 30 transmission data to coalesce into one well-defined trend. A prediction equa- tion was fit to the data shown in Figure 18, and the following relation was obtained: Lo t HOA 0.592 1G) {| eee i 2 Taso for F— > 1.0 mo Equation 12 explains about 97 percent of the variance in K. for the range considered. It is apparent from the composition of Equation 12 why the rela- tive freeboard Hi/e 3 was not a good variable for explaining wave transmis— sion through relatively high breakwaters. 33. For conditions where transmission is not dominated by wave energy propagating through the reef, relative freeboard F/Ho is the most influen- tial variable. Part of the value of the variable is in being able to account for the changing influence of wave height as the dominant mode of transmission shifts between wave propagation over the crest to wave runup and overtopping. For submerged reefs the relative freeboard correctly indicates the interesting property of being able to dissipate energy of large waves more effectively than that of small waves. For reefs being overtopped, the relative freeboard correctly indicates that larger waves have higher transmission coefficients. In spite of these assets, wave transmission for low and submerged reefs is far too complicated to be formulated adequately in terms of ELE alone partly because wave energy is still propagating through low and submerged reefs even though transmission may be dominated by either overtopping or propagation over the crest. In addition, energy going over the reef is quite dependent on crest width and bulk of the structure which introduces the influence of other variables. Considering the multitude of confusing influences and the complex- ity of the phenomenon, the following regression relation was fit to the 167 tests with relative freeboards less than one: 31 we gi Co 3/2 _ ID ae F a ORE a eee exp at = Cy D si P 50 p for F 7 < 1.0 mo where C, = 1.188 C., = 0.261 C, = 0.529 C, = 0.00551 Equation 13 explains about 92 percent of the variance in Ke for the 167 tests where EAS < 1.0. Equation 13 is the result of a considerable amount of trial and error effort to find an equation which fits the data well, makes physical sense based on current understanding of the phenomenon, approaches the correct limiting values, and is reasonably simple. The regression analy-— sis for Equation 13 is shown in Appendix B. 34. If Equations 12 and 13 are used, the transmission coefficient can be predicted over the entire range of conditions tested in this study. Pre- dicted values of Ky were made using Equation 12 for BiB >? 1.0 and Equa- tion 13 for F/H < 1.0. This prediction method will be referred to as the wave transmission model. Figures 19, 20, 21, 22, and 23 show predicted and observed values of Ke as a function of F/G for subsets 1 and 2, 3 ‘and s4); 5 and 6, 7 and 8, and 9 and 10, respectively. Figures 19 through 23 indicate that the wave transmission model does a good job of predicting individual test results and produces trends very similar to those of the observed data. 35. In addition to investigating the attenuation of wave energy passing over and through the reef, it is also possible to determine the relative shift in wave energy caused by the structure. The shift in wave energy is measured by the ratio of the period of peak energy density of the transmitted wave to the period of peak energy density of the incident wave. Figure 24 shows the shift in peak period as a function of relative freeboard. What is surprising 32 TRANSMISSION COEFF, K RELATIVE FREEBOARD, F/Hmo Figure 19. Comparison of data and predicted values of the wave trans- mission coefficient using the transmission model for subsets | and 2 TRANSMISSION COEFF, kK, SS = O OBSERVED K, + PREDICTED K, RELATIVE FREEBOARD, F/Hmo Figure 20. Comparison of data and predicted values of the wave trans-— mission coefficient using the transmission model for subsets 3 and 4 33 LEGEND © OBSERVED k, + PREDICTED K, So °o So = ~ foe} wo oO o+ +lo Oo = 40 ft a | Bese v Lc 0.6 & w ies fe) e acl Zz 05 ac 7) ) B 0.4 ie < Cc FE 0.3 0.2 0.1 = 2 a + “ 1 3 5 RELATIVE FREEBOARD, F/Hmo Figure 21. Comparison of data and predicted values of the wave trans-— mission coefficient using the transmission model for subsets 5 and 6 TRANSMISSION COEFF, K, (CAL) LEGEND O OBSERVED K, + PREDICTED K, RELATIVE FREEBOARD, F/H,,, Figure 22. Comparison of data and predicted values of the wave trans-— mission coefficient using the transmission model for subsets 7 and 8 34 TRANSMISSION COEFF K, Figure 23. 5 ue DO OBSERVED K, + PREDICTED K, RELATIVE FREEBOARD, F/H,,, Comparison of data and predicted values of the wave trans- mission coefficient using the transmission model for subsets 9 and 10 UW (TRANS) /T,, (INCIDENT) Figure 24. 0 OBSERVED DATA of | ape RELATIVE FREEBOARD, F/H,,, Ratio of the transmitted period of peak energy density to the incident period of peak energy density as a function of the rela- tive freeboard, all subsets 35 about this analysis is that the reef does not produce much shift in the peak period of the spectrum. In fact, in only a few tests was the shift as much as 10 percent. Wave Reflection and Energy Dissipation 36. The method developed by Goda and Suzuki (1976) to resolve the wave spectrum into incident and reflected components is the method used in this study to calculate the reflection coefficient. According to Goda and Suzuki, the reflection coefficient is defined as where ED and EL are the reflected and incident wave energy of the spec-— trum, respectively. 37. One variable, the reef reflection parameter, was found to be con- spicuously better than others for predicting wave reflection and is formulated as no 1b c = Be t mice h Cc This parameter can be thought of as approximately the ratio of wave length to horizontal distance between the toe of the reef and the swl on the reef. Since, for many tests, the reefs are deformed and/or submerged, the quantity (a./n2 d. is sometimes only indicative of this horizontal distance. When K. is plotted versus the reef reflection parameter, a very strong data trend re- sults (Figure 25). Such a strong trend seems surprising considering the com-— plex nature of irregular wave reflection and the wide range of conditions represented in Figure 25. A regression equation was fit to the data shown in Figure 25 to provide a convenient rule-of-thumb method to estimate reflection from a reef and to provide insight relating to wave reflection from coastal structures in general. The equation is given by 36 Se (14) 9 2 Boks Okt C) Toa t where C, = 8.284 and C, = -0.951 are coefficients. Equation 14 explains 1 2 about 80 percent of the variance in kK. for the 204 tests considered, follows the trend of the data well, and approaches the correct limiting values. WAVE REFLECTION COEFF, Kr REEF WAVE REFLECTION PARAMETER, slipeS, Figure 25. Wave reflection coefficient versus the reef reflection parameter illustrating the ability of Equation 14 to predict reflection, all subsets 38. While the analysis was being conducted to develop Equation 14, it was clear a relation could be developed which could explain considerably more of the variance in K. if more dependent variables were used. Better esti- mates of reflection from reefs would be valuable since wave reflection causes navigation problems, increases potential for toe scour, and can cause erosion at nearby shorelines by increasing the severity of wave conditions. Im addi- tion, knowledge of wave reflection provides a way to estimate the amount of wave energy dissipated by the reef. The ability of low and submerged rubble structures to dissipate wave energy has long been appreciated, but only in 37 recent years has it been possible to quantify this property. Quantification of energy dissipation by a reef is the property that justified consideration of rubble-mound construction since both wave reflection and transmission are usually undesirable. The basic conservation of energy relation for rubble structures can be written as follows: Kt + K + dissipation = 1.0 (15) where dissipation in Equation 15 refers to the fraction of the incident wave energy dissipated by the structure. 39. The following regression equation will provide an accurate estimate of wave reflection from a reef breakwater: ae 9 Ae Wy K. = exp (2) ate al + C3 TA ate “s(R) (16) P ae he Ss where Cc, = -6.774 C, = -0.293 C, = -0.0860 C, = +0.0833 Equation 16 explains about 99 percent of the variance in K. for the 204 tests considered. The dependent variables and the signs of their coefficients are consistent with current understanding of wave reflection. All the depen- dent variables in Equation 16 affect reflection in a monotonic manner such that, other factors being equal, K. increases as aM, decreases, ho/d. increases, A, [he decreases, and E/Boe increases. However, some care should be exercised in using Equation 16; for example, reflection will in- crease with increasing crest height only until the crest height approaches the limit of wave runup which for a reef would be F/B Sia) Sy isinceyallSterms in Equation 16 are negative for submerged reefs, the equation approaches the correct limiting value of K. = 0 for decreasing structure height. On the other hand, Equation 16 was fit to a data set where reflection was strongly correlated to height of the reef which suggests that the equation might not be satisfactory for reefs with crest heights above the limit of runup. This 38 problem is demonstrated in Figure 26a where the difference between predicted kK. and observed K.. are plotted versus relative freeboard F/H oo . Fig= ure 26a shows that Equation 16 predicts K. usually to within +0.05 with lit- tle systematic error except for high relative freeboards, i.e., Ee > 26D o Because of the possibility of systematic error for high relative freeboards, it is recommended that if the relative freeboard exceeds 2.5, a value of 2.5 be used in Equation 16. When this procedure is applied to the data of this study, it removes the systematic error as shown in Figure 26b. 40. It is intended that the prediction equation for K.. » Equation 16, be used with the wave transmission model (discussed in paragraph 34) in the energy conservation relation given by Equation 15 to compute energy dissipated by the reef. This approach was used to prepare Figure 27 which shows a scat- ter plot of predicted energy dissipation versus "observed" energy dissipation caused by the reef. Figure 27 shows that the procedure outlined above can make good predictions of energy dissipation and the rather surprising fact that, for some conditions, the reef can dissipate up to 90 percent of incident wave energy. Generally, greatest energy dissipation was observed for short-— period waves on reefs which were high enough not to be overtopped. The lowest observed energy dissipation of about 30 percent occurred for the few reefs with a relative crest height less than 0.7, i.e., h/d. < 0.7 . For sub- merged reefs, energy dissipation increases with increasing steepness AO ea and with increasing relative reef width Ao - Reefs with their crest near the swl will dissipate between about 35 to 70 percent of incident wave energy, and dissipation is strongly dependent on relative reef width as shown in Figure 28. For reefs with moderate to heavy overtopping, i.e., 0< F/H Lo < 1.0 , energy dissipation is strongly dependent on the relative reef width but not on wave steepness. 41. Since wave energy dissipation characteristics of reef breakwaters are so important, a special analysis was conducted to illustrate the influence of the most important variables in a simple way that would still be consistent with the data. This analysis used the most effective two variables in pre- dicting Ky and the two most effective variables for predicting K. with the provision that one of the variables be common to both Ke and K so that the predicted values could be plotted on a common axis. Fortunately, the re- lative crest height ho/d, provides a good common variable. Good predictions are obtained for transmission using the variables h/d, and Bo and for 39 -"y "Yous Jy RELATIVE FREEBOARD, F/H,,, F/H mo No upper bound for a. “ea 4-7 Yous I> RELATIVE base a F/Hing As upper bound, b. ion dicting the reflect Equation 16 ing in pre tous icien Error coeff Figure 26. 40 LINE OF PERFECT PREDICTION PREDICTED ENERGY DISSIPATION REEF OBSERVED ENERGY DISSIPATION REEF Figure 27. Scatter plot of the predicted energy dissipation by a reef using the dissipation model versus the observed energy dissipation, all subsets ENERGY DISSIPATION AT, RELATIVE REEF WIDTH, sp Figure 28. Energy dissipation by reefs with crest near the swl as a function of the relative reef width 41 wave reflection when h/d. and relative depth do are used. Regression analysis was used to develop the curves for K. and K. shown in Figure 29. The equations used to compute the curves in Figure 26 explain about 82 percent and 98 percent of the variance in K. and K.. » respectively. Appendix B gives the equations used in Figure 29 and other information related to the regression analysis. The curves shown in Figure 29 fit the general trends of the data quite well. However, the real value of Figure 29 is that it is a compilation of information about wave transmission, wave reflection, and wave energy dissipation of reef breakwaters. Figure 29 is an improvement over Fig- ure 8 in Ahrens (1984) because Figure 29 is based on an analytic model; whereas Figure 8 is based on subjective curve fitting to the observed data. 100 : 0 x 65 uo ty 2 d,/Lp = 0.05 2 J Ww 80 20 ws REFLECTED > 5 ENERGY = ue 30a Oa Zu o6 60 400 cow om I) ou Ww ENERGY 50 o DISSIPATED 40 LIMITS OF OBSERVED DATA 20 TRANSMITTED ENERGY PERCENT WAVE ENERGY TRANSMITTED PAST STRUCTURE 0.4 0.6 0.8 1.0 1.2 1.4 RELATIVE CREST HEIGHT, h,/d, Figure 29. Distribution of wave energy in the vicinity of a reef breakwater 42 42. PART IV: CONCLUSIONS This report summarizes the results from 205 laboratory tests of reef breakwaters conducted using irregular waves. Findings from this study can be categorized as follows: (a) the stability of the structure to wave attack, (b) wave transmission over and through the structure, (c) wave reflec-— tion from the structure, and (d) energy dissipation by the structure. These findings are largely summarized through the use of equations fit to the data which can be used to predict various breakwater characteristics with sur- prisingly high accuracy. 43. The important conclusions from this study are: a. |Q. A stability number was defined by Equation 2 and named the spectral stability number which was found to be the single most important variable influencing the stability of reef breakwaters. There is very little stone movement or damage for spectral sta- bility numbers less than six, but stone movement and damage can be clearly seen for values greater than eight. For values of the spectral stability number above six, the in- fluence of other variables on stability can be identified. Other factors being equal, the stability of the reef increases the lower the relative crest height h/d. 3 as its size de- fined by Equation 3 increases; and as the slope of the struc-— ture, as defined either by Equation 4 or 5, gets flatter. Wave transmission over and through a reef is a very complex process. Part of the complexity relates to the confusing in- fluence of some variables; e.g., for breakwaters with positive freeboards transmission over the reef is directly proportional to wave height, while energy transmitting through the reef is inversely proportional to the wave height. For conditions where transmission is dominated by wave energy propagating through the reef, a simple relation, Equation 12, was found to predict the transmission coefficient very well. When the domi- nant modes of transmission resulted from wave overtopping or wave propagation over the crest of a submerged reef, a rather complex relation, Equation 13, was required to make reasonable estimates of transmission coefficients. Wave reflection is easier to predict than either stability or wave transmission. A simple relation using only one variable, Equation 14, was able to explain about 80 percent of the vari- ance in the reflection coefficients. A more complex relation, Equation 16, was developed which explained about 99 percent of the variance in the reflection coefficient. Other factors being equal, reflection coefficients increase with increasing wave length and increasingly steeper reef slopes. Reflection 43 [Fh coefficients also increase with increasing relative reef height h/d. and increasing relative freeboard F/E Lo until the crest height reaches the upper limit of wave runup. Wave energy dissipation characteristics of a reef are difficult to summarize briefly because of the complexity of the phenome- non. One surprising finding was that for short-period waves an > 0.12 which do not overtop the crest the reef will dis-— sipate 80 to 90 percent of incident wave energy. For reefs with the lowest relative crest height tested 0.63 < h/d. < 0.70 , the structure would dissipate about 30 percent of incident wave energy. Reefs with their crests near the still- water level will dissipate between 30 to 70 percent of incident wave energy depending on the relative reef width Ah The model developed in this study was found to make good esti- mates of energy dissipation. 44 REFERENCES Ahrens, J. P. 1984 (Sep). "Reef Type Breakwaters,"’ Proceedings of the 19th Coastal Engineering Conference, Houston, Tex., pp 2648-2662. Allsop, N. W. H. 1983 (Mar). "Low-Crest Breakwater, Studies in Random Waves," Proceedings of Coastal Structures 83, Arlington, Va., pp 94-107. Bremner, W. D., Foster, N., Miller, C. W., and Wallace, B. C. 1980. ''The Design Concept of Dual Breakwaters and its Application to Townsville, Australia," Proceedings of the 17th Coastal Engineering Conference, Sydney, Australia, Vol 2, pp 1898-1908. Goda, T., and Suzuki, Y. 1976. "Estimation of Incident and Reflected Waves in Random Wave Experiments," Proceedings of the 15th Coastal Engineering Con- ference, Honolulu, Hawaii, pp 828-845. Gravesen, H., Jensen, 0. J., and Sorensen, T. 1980. "Stability of Rubble Mound Breakwaters II," Danish Hydraulic Institute, Copenhagen, Denmark. Hudson, R. Y., and Davidson, D. D. 1975. "Reliability of Rubble-Mound Break- water Stability Models," Proceedings of the ASCE Symposium on Modeling Tech- niques, San Francisco, Calif., pp 1603-1622. Lillevang, 0. J. 1977 (Mar). "A Breakwater Subject to Heavy Overtopping: Concept, Design, Construction, and Experience," Proceedings of ASCE Specialty Conference, Ports '77, Long Beach, Calif., pp 1-33. Lording, P. T. and Scott, J. R. 1971 (May). "Armor Stability of Overtopped Breakwater," Journal of Waterways, Harbors and Coastal Engineering, American Society of Civil Engineers, Vol WW2, Paper 8138, pp 341-354. Raichlen, F. 1972 (May). "Armor Stability of Overtopped Breakwater," Journal of Waterways, Harbors, and Coastal Engineering, American Society of Civil Engineers, Vol WW2, Discussion of paper 8138, pp 273-279. Seelig, W. N. 1979 (Mar). “Effect of Breakwaters on Waves: Laboratory Tests of Wave Transmission by Overtopping," Proceedings of Coastal Structures '79, Alexandria, Va., Vol 2, pp 941-961. Seelig, W. N. 1980 (Jun). "Two-Dimensional Tests of Wave Transmission and Reflection Characteristics of Laboratory Breakwaters," CERC Technical Report No. 80-1, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Vincent, C. L. 1981 (Nov). "A Method for Estimating Depth-Limited Wave Energy, Coastal Engineering Technical Aid 81-6, US Army Engineer Waterways Experiment Station, Vicksburg, Miss. Walker, J. R., Palmer, R. Q., and Dunham, J. W. 1975 (Jun). "Breakwater Back Slope Stability," Proceedings of Civil Engineering in the Oceans/III, Newark, Del., Vol 2, pp 879-898. Wiegel, R. L. 1964. Oceanographical Engineering, Prentice-Hall, Inc., Englewood Cliffs, N. J. Wiegel, R. L. 1982 (Mar). "Breakwater Damage by Severe Storm Waves and Tsunami Waves," Prepared for Pacific Gas and Electric Co., Berkeley, Calif. 45 DIORIT UART ZIT! onnnaimennananannnnennnen Photo 1. Representative samples of the stone used in this study (As a scale, labels in figure are 12.2 by 2.3 cm.) APPENDIX A: TABULAR SUMMARY OF STABILITY AND PERFORMANCE DATA i yi Median Density Area Water = AVE. AVE AVE. Structure Damaged Area File Weight Of Of Depth INC. INC. ‘Trans. Cai. Height Structure Of Subset Test Test Aang WS? Stone Bw, At 3 Hac Tp Hac AVE. Heo = =oas Built Height Damage Hd, NG. Tyo2 Gain qr. wt. ce*2 Ch. cr Sec con Kr ca. he ch. he cm. Ad ce*2 Pay i i 1,100 172.630 1170 25 11.010 1.450 4.450 0,242 10.250 24.900 21.920 49.520 H 2 i 1.080 17-2. 6350 1170 25 «10.140 1.460 5.480 +23 9.500 24.720 23.010 42.920 1 3 { 1.060 PP Zl 1170 25 «8.000 1.4300 ABO) 0.7) 7,566 © 24.110 23.509 17.740 1 4 i 1.086 172.630 1170 250 9.750 1450) 30270) 0.210 © SABO «25.390 © 24. 4eO)©— «10.590 1 S 1 1.020 1? 2.630 1176 25° «2.870 1.440 1680) 0.208 «= 2. 7E0 24.280 © 23.950 = 4. 0 1 é 1 2.100 1 22650 1170 25 «13.430 252350 mes 0.352 11.626 24.410 20.700 77.290 1 7 1 2.08¢ 17 2.630 1170 25 11.500 2.230 97.219 «= 0.217 10.750 924.840) 21.462 §~©— 75.990 i 8 ! 2. 0a0 172.630 1170 29 9.070 2.250 9.800 0.379 8.880 25.480 23.77) 34.750 1 9 1 2,049 17 22630 1170 256.095) 2.270 3.469 «O40 6.075 «25.090 «24.600 © «13.390 t 12 fz 02 17-2. 630 1170 23 2.910 92.280 = 1.58) | OATS = 2.910 © «24.990 «24.540 a a0 { {j 1 2.100 172.630 1170 25° (3.150 802.230 897.890 = 0.327) 11.540 925.050 «19.990 91.790 1 13 1 3.100 LU 2s630) 1179 25° 15.780 3.000 9.360 O.311 11.970 25.720 16.980 213.030 1 14 135080 17 2.630 1170 25 14.350 © 3.000 »= 8.500 = 0.296 )9= 14.680 ©6294. 780 917.590 «168.530 1 15 t 3.060 17.2. 630 1170 25° (11.380 92.780 7.200 = 0.299 910.260 = 24.440 = «19.840 100.410 1 16 i 3.046 Ten OS 1170 29° 7.810 25760) 5040) 08337 Pas) 255270) 22.560) 392580 1 17 12020 7 ~~ 2.630 1170 29 3.890 92.750 = 2.560) 0.425 3.820) 24.660 «24.440 = 2.040 H 12 { 3.100 {725630 1176 25° 15.720 2.950 9.176 0.303 11.960 24.690 17.100 70.010 t 20 1 4,020 7 2.630 117¢ 25 9.460 63.5530 54000461 5.310 | 24.140 © 23.800 = 5. 950 i 2! t 4.040 17) 2.636 1176 25° «(10.076 3.520 6.845 9.354 9.220 24.780 18.870 111.8460 1 22 t 4.060 17 2.630 1170 20 «14.250 «= 3,600 =. B90 1022) 11.690 «25.120 © 16.490 181.720 { 3 { 4.07¢ i 2.630 1176 25° «16.100 3.640 99.470 0.351) 12.330 «24.840 | 15. BED 212.380 i 25 {1 1,100 172.630 1170 20) (L456 1.450 © O30) 0240) 10.620 24.250 = 21.920 35.770 i 26 1 1.080 17. 2..630 117¢ 25° «10.082 «= 1.466 5.7300 0.225 9.450 | 24.570 «9 21. BBD 40. 040 1 27 1 2.060 17 2.630 1170 25 8.830) 02.248 0,650 = 0.483 B.470) | 25.09% = 21.820 = 43.290 i 2 {1 3.060 17 2.630 1170 Zo ff,900 9 2.B00 7.48 512) 10.370 24.990 19.290 102.460 { 29 1 4.040 17.2. 63¢ 1176 25° «10.380 3.590) 07.230) 0.558 9.440 24. BIC |= «18.260 131.180 1 30 t 4. 06¢ 17? 2.638 1176 25 «14.980 3.630 99.080 0.330 «11.980 24.810 16.700 204.110 3 a 1 1.100 17-2630 1565 25° 11.360 891.470 = 9.180) = 0.234 «= 10.540 929.170» 24.720 119.380 3 32 1 1.08¢ 17 2.630 1560 20 «9.460 = 1.450 © 4.040) = 0.237 8.900 930.480) 26.430 105.729 3 33 1 1.069 17. 2..630 1560 25° «-7.820 1.440 = 2.420 0.299 7.430 = 29.600 28.040 © 44.870 3 34 i: 1.046 17-2630 1560 25 «5.000 861.440 1.286 6319) 5.270 «29.630 «29.380 = 5.450 3 35 1 1.020 17-2. 63¢ 156¢ 25 «2.820 «61.440 © 0.720) 0.338 = 2.170 929.630 §=29.500 839 7, 340 3 36 {1 3.100 172.636 1566 25° «15.630 862.980 «= 8.220) 0.203 11.950 = 29.810 919.260 299.890 3 37 1 3.080 17 2.630 1569 25° «13.760 3.000 7.980 0.288 11.480 29.170 19.780 303.420 3 38 { 3.060 17 -2..636 1560 25° «10.980 2.810 6.186 0.319 10.000 29.440 22.340 155.890 3 39 { 3.040 17 2.630 1560 25° «7.496 «= 2.820 93.760 )= 0.430 = 7.220 «29.260 §=25. 880 = 67.730 Zz 40 1 3.02¢ 17 -2,.630 156¢ 25 «3.680 = 2.796) 1.240 = 0.5B4 3.620 §= 29.840 9 28.520 §=—-23, 230 3 41 i 2.100 17 2.630 1560 25 «13.386 © 2.230 7.180) = 511 11.610 © 29.290» 21.730 175.400 3 42 1 2.086 t 2.630 1560 250 11.170 2.270 6.100 0.331 10.540 29.380 24.020 122.260 3 43 1 2.060 17 2.630 1560 25 6300 © 2.250 4140 ALL = 8.230029, 810 © 26.000 = 77.760 Z 44 1 2.046 17 2.630 1560 235.729 2.290) 2.070 390.5102. 710 ©= 29.290 © 27.980 34.000 z 5 1 2.020 17 2.630 1560 25 2.890 2.280 0.860 0.532 2.890 29.440 29.290 9.490 z 46 { 4,020 172.630 1560 25 deol@ 95.560 = 2.160 «= 0.597) 3.350 9=6 29.630 §= 28.070» 29. 40 3 47 { 4.040 17 2.630 1560 25° «10.610 3.580 7.030 0.339 9.610 29.500 20.850 244.610 3 48 1 4.049 17 925630, 1560 25° «10.176 893.520 6.500 0.544 9.290 29.699 21.400 208.010 3 49 { 4.066 17 2.630 1569 25 «14.610 3.570 A330 0.528 11.840 © 30.080 =: 18.350 341,420 3 5¢ 1 4.076 i 2.636 1560 25 «419.820 3.600 9.140 0.332 12.250 29.380 17.570 345, 9B0 Z 51 Det OO W/ ABERS 1560 25 «2.610 3.520 0.890 9.615 2.570 28.999 28.800 5,390 3 52 1 2.100 17 2.639 1566 25° «13.230 «= 2.250 397.200 30.320 11.570 ©29.690 22.070 177.540 3 54 { 3,106 17 2.630 1560) 25° «15.996 «©=2.900 98.290) 0,305) 11.950 «928.740 = 19.450 : 258.270 3 56 1 4.070 17 2. 630 1560 25 «415.840 3.520 8.800 0.317 12.260 29.230 18.010 332.960 Note: Area of BW = cross-sectional area of breakwater; Inc. Ho = incident we H We, WW = alnyeavelenie WwW 8 AeeH, IG = transmitted H 8 Geil, mo P P mo mo I calibrated H ; mo mo A3 Median Density Area Water AVE. AVE. AVE. Structure Dagaged Area File Weight Of Of Depth INC. INC. Trans. Cal. Height Structure Of Subset ‘Test Test And w50 Stone BW, At ds Hao Tp Hao AVE. Hao as Built Height Damage NO. NO. Type Bain qr. wt. ca*2 ca. ca 5ec. coe Kr ca. he ca. he ca. Ad ca*2 3 47 1 1.080 172.630 1560 25° «10.420 1.430 4.610 = 0.242, 9.740 )= 29.140 924.110 119.660 3 68 1 1.100 17 2.630 1560 25° «(11.060 1.460 4.950 0.248 10.290 29.020 25.910 92.530 3 49 1 2.060 17 2.630 1560 25 «8.430 = «2.250 4.110 ©0410 B10 29.440 «26. 610». 430 3 70 1 3.060 172.630 1560 25 «10.890 2.910 6.200 0.317 9.940 29.230 22.010 66.390 3 11 1 4.010 172.630 1560 2 2.590 3.590 0.920 0.600 2.550 28,860 28.100 15.140 5 72 1 1.100 17 2.630 2190 S 10.860 1.460 2.520 0.279 10.120 34.870 29,440 219.440 5 1B 1 1.080 17 2.630 2190 25 «9.380 «1.450 1.680) = 0.272 BBO) 34. SHC | 32,190 113.160 5 74 teent070 17 2,630 2190 25 «7.910 «1.420 1.150 = 0.298 )~= 7.510 35.050 33.890 138.700 5 75 1 1.060 {72s G50 2190 25 «7.520 1.460 0.990 0.289 7.150 34,780 34.560 83.430 5 76 1 1.040 17 2.630 2190 25 «5.460 «1.410 0.720 0.285 5.230 34.560 34.550 23.880 5 71 1 4,070 17 2.630 2190 25 15.720 3,580 7.690 0.322 12.220 34.780 20.120 644.190 5 78 eet 0700 172.63 2196 25 «8.820 1.410 1.560 0.285 8.320 35.450 33.560 163.970 5 79 tz 17 2.630 2190 25 «2.750 1.440 = 0.570 0.354 = 2.650 935.270 35.260 «= 3.900 5 80 fh AL SKN) 172.63 2190 25° «12.960 2.270 5.770 0.303 11.480 36.060 24.200 393.260 5 al { 2,080 GW AER) 2190 25 10.890 2.280 4.320 0.335 10.340 35.170 26.610 345.970 5 82 1 2.060 17 2.630 2190 25 9.640 = 2.300 3.460) 30.384 = 9.370» 35.050 = 28. 680» 235.700 5 Ki 1 2.040 17 2.630 2190 25 «6.790 «2.300 1.440) 0.489.790 «35.270 33.590 116.690 5 84 G2 4020 172 e6S 2190 25 «4.030 2,300 0.840 »= 0.538 = 4.030 «34.410 «34.400 «© 4.090 5 85 1 3.100 17 2.630 2190 25 «(15.340 = 3.000» 7.230 0.312 11.910 34.990 21.610 514.590 5 86 1 3.080 17 2.630 2190 25° «14.080 2.960 6.900 0.311 11.590 35.910 22.160 538.370 5 a7 1 3.070 7 2.630 2190 25 «12.750 2,860 6.320 0.314 11.040 35.540 23.040 429.580 5 88 1 3.060 een6s0 2190 25° «11.160 2.840 2.680 0.352 10.400 35.230 24.870 324.140 5 89 t 3.040 17 2..630 2190 25 «7.580 2.8650 2.680 0.477 7.310 35.140 29.720 184.970 5 50 1 3.020 17 2.630 2190 25 3.780 2.780 0.990 O.5B1 3.720 35.360 35.350 6.970 5 W 1 4,040 172,630 2190 25° «14.290 «3.560 = 6. B90 0.364 «11.710 35.170 21.030 905.190 5 92 1 4,040 17e2nGs0 2190 25 «10.130 3.570 5.350 0.383 9.260 34.590 23.560 387.080 5 93 1 4.020 172.630 2190 25, 55550) 93.960), 7203209 (0N524) VSeRO SANB10 27H ONON 2eliso0 5 94 1 4,010 172.636 2190 25 2.580 3.570 0.760 0.642 2.540 25.870 35.860 5.020 5 5 hath) 172.630 2190 25° 4.280 «= 1.440 0.650241 4110 §= 35.570 35.420 «©: 8. 30 5 96 teetR05 17 2.630 2190 25 «7.020 «1.440 «= 0.9000. 28B = 6.690 35.540 935.330 34. 860 5 7 1 1.080 172.630 2190 25 «9.990 «1.330 «1.980 0.290» 9.370 35.170 31.760 187.200 5 98 eet100 17 2.630 2190 25° «11.350 1.450 3.060 »3= 0.297 10.540 34.960 © 29.540 231.050 5 99 124050 17 2.630 2190 25 «5.480 862.290 «1.030 900.509 «= 5.470» 35.480 «34.440 «©. 560 5 100 1 2,060 17 2R630 2190 25 220 2.290 = 2.400» 0,450 = 8.110 = 34.630 ©—-30.270 +170. 380 5 101 1 2.080 17 2.630 2190 25° «14.030 = 2.280 4.410 = 0.363 10.440 © 35.300 927.340 «293.850 5 102 1 2.050 17 2.630 2190 25 «6.910 2.290 = 1.340 = 0.490). 870» 35.910 31.330 131.360 5 103 lh ALN) 17 2.630 2190 25° «13.020 2.280 4.070 0.318 11.500 35.080 24.050 329.250 5 104 1 3.010 172.630 2190 25 1.819 2.780 0.620 0.590 1.790 35.810 35.690 5.670 5 105 esH030 17 2.630 2190 25 «95.680 2,810 1.590 0.521 5.550 35.750 31.760 146.600 5 106 1 3.050 17 2.630 2190 25 9.310 92.850 4.110 0.413 =. 770) 36.090 © 27.160 271.370 5 107 1 3.080 17 2.630 2190 25 «13.870 2.860 6.500 0.357 11.520 35.630 22.190 502.700 5 108 1 3.100 17 2.630 2190 25 «15.610 2.910 7.340 0.335 11.950 34.930 21.280 531.870 5 109 1 4,010 17 2.630 2190 25 «2.560 3.560 0.780 0.624 2.5270 35.910 35.900 2.690 5 110 li GOR) 17 2.630 2199 25 «8.060 «3.560 9= 3.170 §©= 0.526 «= 7.620)» 35.540 926.970 247.400 5 11 1 4,060 17 -2..630 2190 25° «14.460 3.540 7.330 0.335 11.780 35.360 20.360 588.170 5 112 1 4.070 17 2.630 2190 25° «15.990 3.580 7.880 0.327 12.300 36.030 19.780 696.820 7 124 1 1,100 itvme2s850 1900 25° «14.440 = 1.450 «3.910 0.354 «10.619 «31.460 © 31.210 © 42. 460 7 125 1 1,080 TAN ees 830 1900 25° 10.020 1.450 2.940 0.335 9.400 31.700 31.550 26.570 7 12 1 1.060 Tt 2.830 1900 25 «8.030 1.440 © 2.000 3S 0.352 7.620) 31.360 © 31.350 ©2070 7 127 1 1.040 125830) 1900 25 «69.600 91.430 1.330 0.378 5.360 31.700 31.410 1.770 # 128 11020 Ti —-:2.830 1900 25 «2.600 «= 1.430 = 0.820 = 0.430 = 2.500 31.670 31.660 «= 3.00 7 129 1 2.100 71-2830 1900 25 «13.050 2.226 4.950 0.455 11.500 31.670 31.460) 57.440 7 130 1 2.080 71 2.830 1900 25° «14.110 2.300 4.060 0.471 10.500 32.340 31.970 18.950 7 121 1 2.060 Ta24830 1900 25 8.630 2.280 2.980 0.508 8.490 31.910 31.670 24.900 7 132 1 2.040 71 -2..830 1900 25) 5.580) 25280) 1.720) 05387 SUS7Ol 325060 Sie730) 7660 7 133 fee 24020 71-2830 1900 | EM) OR) NS OLS I) LG) SG 1.390 A4 Median Density Area Water AVE. AVE. AVE. Structure Damaged Area File Weight Of Of Depth _—INC. INC. ‘Trans. Cal. Height Structure Of Subset Test Test And W50 Stone BH, At ds Heo Tp Hao AVE. Hao as Built Height Damage NO. NO. Type Gain qr. wt. ca*2 ca. ca Sec. coe Kr ca. he ca. he ce. Ad ca*2 7 134 1 3.100 Ti 2.830 1900 25 15.660 3.080 6.770 0.426, 11.960 31.640 29.720 95.740 7 35 { 3.080 71-2850 1900 25 14.020 2.880 6.12 0.409 11.580 32.160 29.750 106.840 7 136 1 3.040 712,830 1900 25 11.170 2.790 4.890 0.449 10.130 32.520 30.540 45.240 7 13 1 3.040 Ti 2.830 1900 ae Holl 2,82 2.850 0.502 7.160 31.670 31.540 7.900 7 13 se02 71-2. 830 1900 25 3.550 2.780 1.350 0.556 3.500 31.300 31.120 2-970 7 139 TROT, Th Pek 1900 25 15.860 3.580 8.910 0.409 12.260 31.390 24.320 258.560 7 140 1 4.040 Ti 2.830 1990 mm Ga) | GY 6.42 0.466 11.480 32.250 29.140 100.410 7 141 1 4.040 Ti 2.830 1990 25 10.380 3.550 4.490 0.511 9.440 31.390 30.210 50.450 7 142 1 4.02 71 «2.830 «1900 Seo COMMAS, 1.83 0.586 4.970 32.220 31.210 18.210 7 143 { 4.010 712.830 1900 75 2.350 3.690 0.980 0.596 2.320 31.670 31.660 7.010 7 144 1 1,030 71 2.830 1900 25 3.980 «1.420 1.070 0,382 3.820 31.970 31.040 ‘1.110 7 145 1 1.05 71 2.830 1900 25 6.740 1.390 «1.590 © 0,356 © 430 «31.850 31.790 «= 3. 180 7 146 1 1.080 71 2.830 1900 75 9.980 1.450 2.970 0.380 9.360 31.820 31.810 9.290 7 {47 { 1,100 Th Pah) 1900 25 14200 e450) 9 3.62 0.379 10.590 32.000 31.730 25.230 7 148 1 2.030 71 2,830 1900 25 4.070 2.290 1.310 0,554 4.070 31.820 31.640 4.460 7 149 1 2.050 7 2.830 1900 25 7.070 24290 2.340 0.52 7.030 31.610 31.460 9.330 7 150 1 2.090 71 2.830 1900 25 14.320 2.260 4.250 0.482 10.630 31.490 31.360 25.040 7 151 te 25100 Ti —.2. 830 1990 25 13.119 2.230 5.100 0.461 {1.530 31.610 30.510 40.880 7 152 A ani TAMeReS 1900 25 1.650 2.780 0.800 0.593 1.630 31.700 31.690 9.730 7 153 1 35030 7 2.830 1900 25 5.660 2.790 2.020 0.554 5.540 31.820 31.030 20.200 7 154 1 3.050 71-2. B30 1900 25 9.750 2.800 4.230 0.481 9.110 31.300 31.000 27.870 7 155 1 3.080 7 2.830 1909 75 14.240 2.810 6.040 0.423 11.650 31.240 29.630 96.300 7 156 1 3.190 { 2.830 1900 25 15.420 2.880 6.960 0.418 11.930 32.130 29.590 106.650 7 157 { 4.010 Ti 2.83 1900 oy nal 3.580 1.000 0.588 2.320 32.800 32.770 1.300 7 158 1 4.039 7 2.830 1909 25 «7.810 3.600 2.910 0.564 7.410 32.740 32.460 15.960 7 159 { 4.060 1 2.830 1900 De HELEBKN) at 6.840 0.452 11.800 32.220 26.970 146.420 7 140 1 4,570 71 2.830 1900 75 16.040 3.580 7.680 9.430 12.310 31.940 26.880 142.420 7 {1 1 4,060 71-2. B30 1900 75 14.420 3.540 4.690 C.471 11.760 31.860 28.250 129.910 9 186 1 4,040 71-2, 830 1900 30 10.540 3.560 6.700 0.422 9.870 32.000 29.810 47.660 9 189 1 1.040 Ti 2.830 1900 30 5.760 1.430 3.140 0.256 5.520 31.820 31.790 8.950 9 190 1 1.080 71 2.830 1900 2) 10.940 1.500 4.030 0.301 10.320 31.550 31.940 7.620 9 19! 1 1,100 71-2830 1990 30 12.630 1.500 6.950 0.295 11.800 31.730 31.240 7.840 3 192 1 2,040 TANNe AOS 1900 5 5.800 2.200 3.210 0.443 5.790 31.580 31.520 1.370 3 193 { 2.080 1 2.830 1900 30 12,020 2.190 7.130 0.388 11.620 31.670 31.060 17.280 9 194 { 2,100 7 2,830 1900 30 «14.460 2.22 8.370 0.257 13.280 31.580 29.660 42.990 9 195 1 3.040 Ti 2.83 1900 30 8.200 2.990 5.090 0.43 7.960 32.000 31.760 7.250 9 196 1 43.080 71-2, B30 1900 30 16.090 3.080 9.590 0.388 13.610 31.610 26.610 198.260 9 197 1 43.100 7 2.83 1900 30 18.170 3.060 10.330 0.344 14.250 32.060 25.510 191.290 9 198 1 4.020 Ti 2,830 1900 3 5.22 3.37 2.900 O,497 5.110 32.130 32.060 3.160 9 199 { 4,05 7 2.93 1900 30 13.380 3.310 8.380 0.405 11.970 32.000 28.640 99.310 9 200 1 4.070 71-2830 1900 30 17.600 3.280 10.470 ~ 0.362 14.230 31.610 25.210 198.630 2 12 2 2.040 { 2.630 1170 PSE SECT OMMNEZ S24 OMENS ENt OMRON, 5.960 19.990 19.991 1.770 2 { 2 2.040 { 2.630 {170 75 5.870 2.230 4.610 0.271 5.860 17.100 14.860 1.080 2 24 2 2.040 17 2.630 1170 25 «45.950 2.230 4.690 0.243 5.930 15.880 15.910 0.450 4 Fi 2 2.080 17 2.630 1560 2 5.510 2.260 3.930 0.215 5.500 19.450 19.390 2.420 4 5 2 2,020 17 2,630 1560 mI 7.720 2.240 2.120 0.180 2.720 18.010 17.980 1.020 4 5 2 2.040 iW © Dole) 1560 25 «5. 45 2,22 4.210 0,210 5.440 17.980 17.830 0.740 4 59 2 2.060 17 2.630 1560 Oe) Sat) aks 6.03 0.228 6.230 17.830 17.800 9.960 4 60 2 2.080 17 2.630 1560 75 11.180 2.230 7.330 0.261 10.540 17.800 17.860 -0.650 4 bi 2 2.400 17 2.630 1560 Dol 27 OMe 2S 8.040 0.272 11.580 17.860 18.010 9.190 4 42 2 1.020 17-2630 1560 25 3.170 1.440 2.120 0.125 3.050 18.010 17.890 1.490 4 b3 2 1,940 17 2.630 1540 5 5.560 1.440 73.980 0.150 5,320 17.890 17.740 0.840 A b4 2 1,040 17 2.630 1540 25 7.990 1.440 5.22 0.178 7.580 17.740 17.480 0.190 4 65 2 1,080 17 2,630 1560 75 9,920 1.440 6.110 0.213 9.310 17.620 17,560 9.460 4 tb 2 1.100 17 2.630 1540 25 11.190 1.460 6.680 0.229 10.400 17,560 17.710 1.110 £ 113 D Stofifl {7 2.630 2190 25 «2.840 8©=— 1.43 7.010 0.151 2.730 19.780 19.810 -0.840 A5 Median Density Area Water AVE. AVE. AVE. Structure Damaged Area File Weight Of Of Depth INC. INC. ‘Trans. Cal. Height Structure Of Subset Test Test And w50 Stone BW, At ds Hao Tp Hao AVE. Hao as Built Height Damage NO. NO. Type Gain qr. wt. ca*2 ca. ca SeCe coe Kr ca. heca. heca., Ad ca*2 6 114 2 1.070 WP Botest 2190 2 9.040 1.430 4.870 0.214 8.530 19.810 19.600 4.370 6 HS 2 1.040 172.630 2190 29 «9.090 0 1.400) )= 3.500) = 0.159 = 3.350 919.600 «19.630 ~=— 0.370 6 116 2 1.060 17 2,630 2190 2 8.120 1.440 4.510 0.201 7.700 19.630 19.540 2.140 6 117 2 = 1.080 172.630 2190 25 49.980 1.440 5.240 0.23 9.360 19.540 19.630 NA f) 118 Zell 00 17 2.630 2190 25° «11.470 = 1.450 = 5.680) 0.245) 10.640)» 19.630 «19.960 ©4740 6 119 2 2.020 17 2.630 2190 2 2.490 2.220 1.840 9.185 2.490 19.960 19.780 0.370 6 120 2 2.040 17 2,630 2190 25 = «9.180 29 3.660 0.180 5.170 19.780 19.910 1.300 6 121 2 2.040 7 2.63 2190 25. «7.960 = 2.220 4.990 = 0.206 ~=s 7.870» 19.810 19.811 1.490 ) 12 2 = 2.080 17 -2..630 2190 25 «10.660 «2.250 = 6. 100 30.233 10, 180 919.810 19.960 = 5.440 6 123 22 s100 t 2.630 2190 25 «12,880 2.23 6.6790 0.260 11.450 19.960 19.750 3.160 8 162 2 RONG 71 2.830 1900 25. «1.090 1.430 0.570 0.51 =: 1.050 © 28.250 28.380 «=: 1.5 BO 8 163 Ze ln020 7 2.83 1900 Zo zeas 1.43 0.960 0.284 2.540 28.380 28.190 9.960 8 164 2 1.030 T2180 1900 2 3.990 1.440) 1.4200 6.247) 3.830) 28.190 28.250 NA 8 165 2 1.040 71-2830 1900 25° esB80 440 7 910 9 0.238) S250) 9285250! 9285220) 970.960 g 146 2 1.060 71-2. B30 1900 2 7.800 1.450 2.779 0.249 7.410 28.220 28.190 1.760 Q 1g 2 -1..080 Us Zack) $900 29 «9.780 «= 1.460 = 3.740) 0.272 9.190 928.190 = 28.350 «= 1.110 Q 148 2 flo st) 71 2.830 1900 25 «11.03 1.450 4.360 0.299 10.270 26.350 28.160 1.670 a 169 2 2.010 2850 1900 25 «1.060 = 2.280 00.700) 0.883 1.160 928.160 )©= 28. let © 1.760 8 17 A eal 7 2.83 1900 25 92.550 2.270 1.190 )= 0.454 = 2.550 928.160 928.250 «=: 0.650 8 171 2 2.030 T1 = -2.B30 1900 25 «5.940 492.260 01.760 = 0.43 3.940 28.250 28.190 0.460 g 172 2 2.040 Hh Bock 1900 25° «-9.440) = 2.300 2.50) 0.436 5.430 28.190 28.220 1.670 8 73 2 2.060 TAS 1900 25 «8.730 = 2.260 = 3.980 = 0. 408 = 8.580 © 28.220 28.221 «=: 1. 110 8 17 2 2.080 Th AoGKH) 1900 25° 11.260 2.280 «= «5.150 90.395 19.600 28.220 28.290 «0.190 q 175 2 2.100 T2583 1900 25 «13.510 2.240 6.520 0.391 11.590 28.290 27.650 4.040 8 176 2 3.010 71 -2..B30 1900 25 «1.620 2.780 = 0.920 0.493 1.600) 27.650 = 27.610 = 0.680 8 177 C02 TAY anh 1900 25 «3,550 2.800 »=:1.430 = 0.463 = 3.500 927.610 §9=27.580 9-0. 650 9 178 2ess080 71 2.830 1900 25. «-§.400 = 2.800 2.630 0.440 «= 5.480 027.580 9= 27.610 = 1.110 Q 179 2 3.040 {2.830 1900 250 7.590 = 2,830 3.810 0.400) 7.310 © 27.610 927.650 © 5. 160 Q 18 2 3.060 7 2.830 1900 25 «11.340 = 2.840 5.460 = 0.393 10.240 §=— 27.650 927.651 = 2.040 8 184 2 3.080 71-2830 1900 25 «14.160 2.800 6.680 0.391 11.620 27.650 28.010 0.370 9 182 2 = =3.100 71-2. B30 1900 25 13.520 1.800 5.950 0.350 11.160 28.010 27.550 2.600 8 183 2 4.010 7 2.830 1900 2a eeco Soo80 N30) OFS1S) 9925220) 27.1550) 2BNO10 0 S20) a 194 2 4.020 71-2. B30 1900 25 5.010 3.590 2.510 0.493 4.880 28.010 27.580 2.420 Q 185 2 4,030 TL 2.830 1990 25) 7500) VeSe560) 9S. 940) 10474) Se71S0j 272980) mez CBO mmmeetel 8 186 2 4,040 TAG ZABS 1500 a5) ath 3.540 5.120 0.457 9.120) 27.680 27.580) 3.810 g 187 2 4,050 Dy 28 1900 25 (12.260 3.540 6.040 0.445 10.680 27.580 27.490 2.400 10 201 2 2.020 Ti 2.830 1900 SH) Aoaltl) — Boat Qvas0) 105295) 25580) e2onzl0) an2d 1.860 10 202 2 2,040 eee OS 1900 $0 52570) 25220 45310) 108290) 95560) 925.2110) 2551 0nee2250) 19 203 2 2.040 Fl 25830 1900 SON 80750) M2h220) | wGNSSON Ch 2Bb) mG noaO) cS. GOmmezo 20 MemOnOaa 10 204 2 2.080 Boy 1200 50) ze 200) mone 9.000 0.299 11.800 25.120 24.960 2.970 10 205 2 2.100 mM Boks 1990 SOP A440) 9 2N220) E920) TONS06) tee260) 24960) 8 2oN020memec50 A6 APPENDIX B: REGRESSION ANALYSIS USED TO DEVELOP FIGURE 29 SHOWING ENERGY DISTRIBUTION IN VICINITY OF REEF Ae i aR At ea) ae ee @ 5, i | , ; aa ey ; i iN al f VP Etk. a ; ji a0 me! i; ee ot ae . ts Ue se he : acs y I) } i) ie i hie ei 5 : i i oa 1 some 3 hen ‘ i 7 re N 1, as tah ie he) 5 Mi ‘ ' i 5 ‘ il : ~ 3 i ; i sn { pees ral : Ne x . nas Ch ert can ant 1. For the energy dissipation figure (Figure 29) the following equation was used to predict the wave transmission coefficient: where C, = 0.02945 C. = So 329) C, = 0.585 R? = 0.859 F = 611 2. The wave reflection curves shown in Figure 29 were calculated using the following equation: where Cc, = 0.2899 Cc, = -0.7628 Cc, = -7.3125 R? = 0.984 F = 4,175 B3 4 : # Ne Seis yi) APPENDIX C: NOTATION Area of damage (ena) Cross-sectional area of breakwater (a) Bulk number, defined by Equation 3 Response slope of reef to wave action, defined by Equation 5 Effective "as built" reef slope, defined by Equation 4 Dimensionless coefficient Water depth at toe of breakwater (cm) (Weg/w_) 3 , typical dimension of the median stone (cm) In - a » freeboard of structure which for reef can be either positive or negative (cm) Crest height of breakwater after wave attack (cm) Crest height of breakwater "as built" (cm) Zero-moment wave height at transmitted gage locations with no breakwater in channel (cm) Zero-moment transmitted wave height (cm) Incident zero-moment wave height (cm) Reflection coefficient of breakwater as defined and calculated by method of Goda and Suzuki (1976) H/H » wave transmission coefficient Airy wave length calculated using a and d. (cm) Stability number, defined by Equation 1 Spectral stability number, defined by Equation 2 Wave period of peak energy density of spectrum (sec) Density of stone (aflame) Density of water, tests conducted in fresh water, Wal 1.0 (elem) Median stone weight (subscript indicates percent of total weight of gradation contributed by stones of lesser weight) (g) C3 phy a SRL ee i hi YT ie Dine i * i i ' ? i ti “i i | Ds pts ; i ; i eh i i i ‘ ;