u cS Army Coast. Enq. Res-Ctr, CETA CETA 81-17 Irregular Wave Runup on Smooth Slopes by John P. Ahrens COASTAL ENGINEERING TECHNICAL AID NO. 81-17 DECEMBER 1981 WHO! DOCUMENT COLLECTION \ 7 Li ‘ J ———— ~Sterine 8° Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING TC RESEARCH CENTER 330 Kingman Building Us Fort Belvoir, Va. 22060 No, Sl 09w30 this material Reprint or republication of any of Army Coastal shall give appropriate credit to the U.S. Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: Nattonal Teehntcal Information Service ATTN: Operations Diviston 5285 Port Royal Road Springfteld, Virginta 22161 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. HIN iio | | O soa a MN UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS - REPORT NUMBER 3. RECIPIENT'S CATALOG NUMBER CETA 81-17 ” TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Coastal Engineering Technical Aid 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(s) IRREGULAR WAVE RUNUP ON SMOOTH SLOPES AU THOR(s) John P. Ahrens 10. PROGRAM ELE AREA & WORK PERFORMING ORGANIZATION NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CERRE-CS) Kingman Building, Fort Belvoir, Virginia 22060 11. CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) MENT, PROJECT, TASK UNIT NUMBERS D31229 12. REPORT DATE 13. NUMBER OF PAGES 15. SECURITY CLASS. (of this report) UNCLASSIFIED 15a. DECL ASSIFICATION/ DOWNGRADING SCHEDULE Approved for public release; distribution unlimited. } 16. DISTRIBUTION STATEMENT (of this Report) 17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Irregular waves Wave rundown Smooth plane slopes Wave runup 20. ABSTRACT (Continue on reverse side if necesaary and identify by block number) The results of several laboratory studies have been used to develop a method to estimate the wave runup and rundown on plane, smooth slopes caused by irregular wave action. Curves and equations are presented which can be used to compute the 2-percent runup, significant runup, mean runup, and approximate lower limit of rundown. A procedure is suggested for adapting the smooth-slope results to wave runup on rough and porous slopes. Example problems illustrate the use of the material presented. 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Within the method's range of applicability it supersedes Section 7.212, "Irregular Waves," of the Shore Protection Manual (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977); CETA 77-2 "Prediction of Irregular Wave Runup" by John P. Ahrens; and CETA 78-2 "Revised Wave Runup Curves for Smooth Slopes" by Philip N. Stoa. It also supersedes the parts of CETA 79-1 "Wave Runup on Rough Slopes," by Philip N. Stoa, which estimate wave runup on rough and porous slopes by adjusting the runup for similar wave con- ditions on smooth slopes using a rough-slope correction factor. This report was prepared by John P. Ahrens, Oceanographer, under the gen- eral supervision of Dr. R.M. Sorensen, Chief, Coastal Processes and Structures Branch, Research Division. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. TED E. BISHOP Colonel, Corps of Engineers Commander and Director VI APPENDIX A B CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). SYMBOLS AND DEFINITIONS. . . «. «© 2 «© © «© © e © @ © © © © TNERODU GION ey eulethathie: ploustottotie = of edb athie. io fietiviets eo! tell +e sel entiontie IRREGULAR WAVE RUNUP ON PLANE, SMOOTH SLOPES .... «© « « » IRREGULAR WAVE RUNDOWN . . 2. «© «© «© © © © © © © © © © © © © @ APPLICATION OF RESULTS TO ROUGH AND POROUS SLOPES. .... -» EXAMPIEHPROBGEMS ciive cls hey eypeieeienye ehle fe ie: s te «8 eule aieine SUMMARY. . . » «© » PRTRRAEURE GREED Se ctceeet tahoe) et M ister is se! ss 6 ee 8 eee RUNUP SCALE-EFFECT CORRECTION FACTOR, k, FOR SMOOTH SLOPES RUNUP REDUCTION FACTOR, r, FOR VARIOUS TYPES OF ROUGH AND POROUS STRUCTURES. . . 2. «© « © «© © © © © © © © © © ow RUNUP SCALE CORRECTION FACTOR, k, FOR VARIOUS TYPES OF ROUGH AND POROUS STRUCTURES. ... . . «© © «© © © «© «© © «© «© « TABLES 1 Regression coefficients for runup parameters R2/Hg, R,/Hs, eunidy Restle py seteimce Mon cease Nog mrad Wecmem ef uc) URI ect Pu al te). sh ek bin ca? ete oy 2 Values of the runup parameters for example probleml........ 1 Irregular wave plane, smooth 2 Irregular wave plane, smooth 3 Irregular wave plane, smooth 4 Irregular wave plane, smooth 5 Irregular wave plane, smooth 6 Irregular wave FIGURES runup slope runup slope runup slope runup slope runup slope runup parameters versus wave of 1 on 1,'ds/H, > 3. parameters versus wave of 1 on 1.5, ds/Hs > 3 parameters versus wave of don) 2, do/Hs, > 3). parameters versus wave of I on 2.5, ds/Hs > 3 parameters versus wave of 1 on 35 dg/Hg = 3) e parameters versus wave plane, smooth slope 1 on 4, ds/Hg > 3... steepness steepness steepness steepness steepness steepness 7 Rs/Hs versus the surf parameter for 3 < ds/Hg 8 Rdgg/Hg versus the surf parameter 11 - 10 14 CONVERSION FACTORS, UeS- CUSTOMARY TO METRIC (SL) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SL) units as follows: Multiply inches by 2524 To obtain millimeters square inches cubic inches feet square feet cubic feet yards Square yards cubic yards miles square miles knots acres foot-pounds millibars ounces pounds ton, long ton, short degrees (angle) Fahrenheit degrees 2254 66452 16.39 30.48 0.3048 0.0929 0.0283 0.9144 0.836 0.7646 1.6093 259.0 1.852 0.4047 1.3558 1.0197 28235 453.6 0.4536 1.0160 0.9072 0.01745 By x 1073 centimeters square centimeters cubic centimeters centimeters meters Square meters cubic meters meters square meters cubic meters kilometers hectares kilometers per hour hectares newton meters kilograms per square centimeter grams grams kilograms metric tons metric tons radians Celsius degrees or Kelvins! 1To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F -32). To obtain Kelvin (K) readings, use formula: Ko= (5/9) Ce =32)\ +2273. 15). Rdog SYMBOLS AND DEFINITIONS water depth at the toe of the slope or structure on which runup occurs acceleration of gravity, 32.2 feet per second squared significant wave height at the toe of the structure runup correction factor for scale effects deepwater wavelength, Lo = gTp*/2n mean runup significant runup, i.e., average runup of the highest one-third of wave runups 2-percent runup, i.e., elevation above the stillwater level exceeded by 2 percent of the runups 98-percent rundown, i.e., depth below the stillwater level that is just greater than 98 percent of the rundowns rough-slope runup correction factor, ratio of rough~slope runup to smooth-slope runup, all other conditions the same period of peak energy density of the wave spectrum significant wave period, i.e., average period of the highest one-third of waves angle formed between the slope of the structure and the horizontal surf parameter, & = [@iz/ta)'!2 cot 6]7! IRREGULAR WAVE RUNUP ON SMOOTH SLOPES by John P. Ahrens I. INTRODUCTION This report provides guidance on the magnitude and distribution of wave runup and rundown elevations caused by irregular wave conditions similar to those occurring in nature. The results presented are for plane, smooth struc- tures with relatively deep water at the toe of the structure. For these con- ditions this report supersedes earlier guidance in Section 7.212 of the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977) and Ahrens (1977) which indicate that irregular wave runup has a Rayleigh distribution. Within the range of test conditions this report also supersedes Stoa (1978a) and the parts of Stoa (1979) which esti- mate wave runup on rough and porous slopes by adjusting the runup on a smooth slope by a correction factor. The range of test conditions covered in this report is discussed in the next section. II. IRREGULAR WAVE RUNUP ON PLANE, SMOOTH SLOPES Three sources of data were used in establishing the methods presented in this report: van Oorschot and d'Angremond (1968), Kamphuis and Mohamed (1978), and Ahrens (1979) which discussed data recently collected at the Coastal Engi- neering Research Center (CERC). The conditions considered are a structure with a plane, smooth slope fronted by a horizontal bottom offshore. The water depth at the toe of the structure is relatively deep, i.e., 3 < dg/H, < 12, where d, is the water depth and Hg the significant wave height at the toe of the structure. When there is relatively deep water at the toe of the struc-— ture the offshore slope of the bottom has little influence on the wave condi- tions and therefore little influence on the wave runups. This lack of influence indicates that the runup results presented can be applied to situations where there is an offshore slope. Since the water depth also has little influence on wave runup for conditions when dg/H§ > 8 (Stoa, 1978a), where Hj is the deep- water, unrefracted wave height, Stoa's finding suggests that the results of this study should be good for dg/H, > 12. Three runup parameters were chosen to characterize the runup distribution caused by irregular wave conditions, i.e., the mean runup, R, the significant runup, R,g, and the 2-percent runup, Rj. The significant runup is the aver- age runup of the highest one-third of wave runups and the 2-percent runup is the elevation exceeded by 2 percent of the wave runups. Figure 1 shows trend-line curves for R2/Hg, Rg/Hg, and R/Hg for a plane, smooth slope of 1 on 1. These parameters are plotted as a function of the irregular wave steepness parameter, Hs/gTp’, where T is the period of peak energy density of the wave spectrum and g_ the acceleration of gravity. The approximate relationship between Tp and the average period of the significant waves, Ts, is given by Goda (1974) as Tp = 5 (@)5) qs (1) Denotes + 1.0 std. dev. about trend line Figure 1. Irregular wave runup parameters versus wave steepness for a plane, smooth slope of 1 on 1, d,/Hs >.3. Figures 2, 3, 4, 5, and 6, which are similar to Figure 1, show trend lines for slopes of 1 on 1.5, 1 on 2, 1 on 2.5, 1 on 3, and 1 on 4, respectively. The trend lines in Figures 1 to 5 are all of the general form H fu. \? Rx = Cy + Co =. ar C3 a Hg ety 8Tp (2) where Rx represents Ro, Rg, or R, and C,, Co, and C3 are dimensionless re- gression coefficients. In some cases Cy or C3 is zero; if C3 is zero the trend line is straight. Since a calculator or a-computer may be more convenient for calculating the runup parameters than using the figures, Table 1 provides a tabulation of the regression coefficients, along with some statistical parameters which can be used to evaluate how well the curves fit the data. The standard deviation is the standard deviation of the data about the trend-line curves and is shown in Figures 1 to 6 to give an indication of the magnitude of the scatter about the curves. The coefficient of variation is the standard deviation divided by the mean value of Ry/Hg. Using the coefficient of variation to determine the percent scatter indicates that Rg/Hg can usually be estimated within the range of +5 to 10 percent about the trend-line curves; Ro/H, and R/Hg can be esti- mated within the range of +10 to 15 percent about the curves. ee Boe 5 er LTA Runup/Hs ~m on : He (ae : aS 0.0 = = 2x10” 4x10" 6x10 Bx10° H,/gTp Figure 2. Irregular wave runup parameters versus wave steepness for a plane, smooth slope of 1 on 1.5, davis > Se Denotes + 1.0 std. dev. about trend line Runup/Hs Mm nae ae Ez ca Ee pars an aie a ie oS Figure 3. Irregular wave runup parameters versus wave steepness for a plane, smooth slope of 1 on 2, d,/H, > 3. } Denotes + 1.0 std. dev. about trend line 2 ~ a5 Runup/H, | cy : ie Cey oY aa Bc coe eis Figure 4. Irregular wave runup parameters versus wave steepness for a plane, smooth slope of 1 on 2.5, d,g/H, > 3. Denotes + 1.0 std. dev. about trend line Figure 5. Irregular wave runup parameters versus wave steepness for a plane, smooth slope of 1 on 3, dois oS. 10 i Denotes + 1.0 std dev. about trend line 3.0 R 2.0 "Mis: (61 wo a5 a = Rs 2 ceo Hsia ast Fy, 10 $= 0.84 € 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 2 H,/gTp Figure 6. Irregular wave runup parameters versus wave steepness for a plane, smooth slope 1 on 4, dg,/Hg > 3. Table 1. Regression coefficients for runup parameters R,/Hs, Rs/Hs, and R/Hs (see eq. 2)is Regression coefficients Cot 6 Cj Co C3 Std. dev. Coeff. of variation eae ie eee ee Ra He 1.0° 2.32 7.15 x 10! 0 0.343 0.134 2052" 9 1295 102 0 0.487 0.156 Bak lgphl afl yes aly ) 0.421 0.123 3.39 1.29 =x 102 -=1.61 x 10* 0.420 0.118 3.70 0 =1570) = <0" 0.415 0.120 4.0 3.60 -2.22 x 104 0) 0.330 0.117 R./H, 1.0) 1.34 6.61 x 10) 0 0.133 0.085 1.38 3.18 x 102 <=1.97 x 10% 0.195 0.094 1.64 3.57 * 102 =3.09 x 10* 0.136 0.059 1.94 2.79 x 102 -3.21 x 10% 0.184 0.078 2.11 1.87 x 102 -2.67 x 10% 0.190 0.081 4.0 2.52 -7.94 x 10! ) 0.122 0.053 R/Hs 1.0 0.71 1.10 x 10? -8.07 x 103 0.150 0.157 0:75) 197 x 10>) —114—x: 10* 0.143 0.119 2.0 0.93 2.42 x 10% -1.93 x 104 0.142 0.101 1.00 2.78 x 10% —3.13)< 10? 0.141 0.099 3.0 1.19 2.09 x 102 -2.96 x 10% 0.181 0.123 440° 147 725 x TO =1a70sa0F 0.127 0.085 Figure 6, for a slope of 1 on 4, is somewhat different than Figures 1 to 5 for steeper slopes. Plunging waves become the dominant breaker type on the 1 on 4 slope, indicating that wave runup can be predicted using a type of for- mula suggested by Hunt (1959) and used by van Oorschot and d'Angremond (1968). Figure 6 shows trend-line curves, using equation (2), for the less steep wave conditions, i.e., and a Hunt-type formula is used for the steeper wave conditions, i.e., Hg/gTp* > 0.003 where plunging waves dominate. The Hunt-type formulas for Fig- ure 6 are given by the equations R = 1.61 5 (3) Hy R oa aD) 4c = 1G s EVE V ns (5) Hs where the surf parameter, €, is given by Ee 1 ae tan 0 (Hg/L,)/2 cot @ (g/L) !/? Lo is the deepwater wavelength given by and cot 6 is the cotangent of the angle 9 between the structure slope and the horizontal. Figure 7 provides a different perspective and additional insight on the trends to be expected for irregular wave runup. The R,/Hg curves from Figures 1 to 6 have been transferred to Figure 7 and plotted versus the surf parameter, —&, to show the influence of breaker characteristics on runup. When € < 2.0, most of the larger waves in the incident wave train plunge directly on the structure and Rg/Hg decreases with increasing H,/gT 2 and increasing cot 0. This plunging wave region is where a Hunt-type formula (Hunt, 1959) such as equations (3), (4), and (5) is valid. When &€ > 3.5, no waves plunge on the structure indicating a standing wave condition or surging wave region. The influence of Hg/gTp? and cot ® on Rs/Hg is reversed for surging waves as l2 Transition Region Surging Region (Standing Waves Against Structure) Plunging Region (Waves Plunge Directly on Structure) Cot §=2.0 Cot@=1.0 0 10 20 30 4.0 5.0 6.0 7.0 Surf Parameter, € Figure 7. Rg/Hg versus the surf parameter for dg/Hg > 3. compared to plunging waves; i.e., Rg/Hg increases as Hs/gTp* increases and cot 8 increases. The reversal of influence creates a transition region, 2.0 < — < 3.5, where there is little net influence of He/gTp* and cot @ on Rg/Hg,. It is in this transition region that the largest values of Rg/Hg occur, prob- ably because the most nonlinear surging waves occur in this region. Figure 7 identifies these regions and shows the runup trends. Equations (3), (4), and (5) can be used on slopes flatter than 1 on 4 as long as plunging waves pre- dominate, i.e., & < 2.0. All the results in this report were obtained in relatively small-scale laboratory studies and must be corrected for scale effects (Stoa, 1978a). The correction for scale effects of wave runup on smooth slopes can be found in Stoa (1978b) (shown in App. A). Example problem 1 in Section V illustrates the method of applying this correction. The results in Figures 1 to 7 are all presented in terms of the significant wave height at the toe of the structure, Hg, rather than the deepwater, un- refracted wave height, Hj. If it is desired to convert the results of this study to deepwater conditions, Hg should be multiplied by the shoaling coef- ficient, given in Appendix C of the SPM (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977), calculated using dg and Tp to obtain an estimate of the deepwater, unrefracted significant wave height. IIL. IRREGULAR WAVE RUNDOWN Irregular wave rundown is characterized by the 98 percentile rundown, Rdgg, i.e., the rundown depth below the stillwater level which is greater than 98 percent of the wave rundowns. The irregular wave rundown parameter, Rdgg is analogous to the runup parameter, Ro, since only 2 percent of the rundowns are lower than Rdgg. Figure 8 shows the trend of the relative rundown, Rdgg/H, as a function of the surf parameter, &, and the approximate upper [5 Approx. upper limit of dato scatter Approx. lower limit of data scatter Surf Parameter, €= TLS cre Figure 8. Rdgg/H, versus the surf parameter. and lower limits of data scatter about the trend-line curve. The trend-line curve for relative rundown is given by the equation Rdgg —2.46/é& = -2.32e (6) He The absolute value of relative rundown is small for small values of the surf parameter since the plunging waves which dominate these conditions cause con- siderable wave setup. As the surf parameter increases a standing wave develops against the structure and the relative rundown approaches -1.75, although values occasionally as low as -2.25 were observed. Equation (6) provides a simple way to estimate the approximate lower limit of rundown. There is no scale-effect correction factor specifically developed for wave rundown, so it is recommended that the correction factor for wave runup be applied to rundown as illustrated in example problem 2 in Section V. IV. APPLICATION OF RESULTS TO ROUGH AND POROUS SLOPES The results given in this report can be applied to plane, rough- and porous-slope structures, if there is.relatively deep water at the toe of the structure (as discussed previously in Sec. II). To apply these results it is necessary to have a reliable estimate of the rough-slope runup correction fac- tor, r, which is the ratio of wave runup on a rough or porous slope to the 14 runup on a smooth slope, all other conditions being the same (Stoa, 1978a). Normally, r is determined in laboratory experiments using monochromatic wave conditions but it appears that r factors determined in this manner can also be applied to irregular wave conditions (Battjes, 1974). Values of r for various types of rough and porous slopes are given by Stoa (1979) (shown in App. B). Often wave runup on rough slopes must be corrected for scale effects and the correction factors are given in Stoa (1979) (shown in App. C). Example problem 3 illustrates how the results presented in this report can be applied to a rough and porous slope and the method of applying the rough-slope scale- effect correction factor. V. EXAMPLE PROBLEMS Kok KK Kk RK RK KK OK OK OR & & & EXAMPLE PROBLEM 1 * *& *& ¥ ¥ KKK KKK KARE This example illustrates the use of the runup equation, Figures 1 to 6, and the recommended method of interpolation between slopes. GIVEN: A plane, smooth slope of 1 on 2.75 is subjected to irregular wave action. The significant wave height, significant wave period, and water depth at the toe of the structure are 6.0 feet (1.83 meters), 7.0 seconds, and 24.0 feet (7.3 meters), respectively. FIND: R, Rg, and Rp for the given conditions. Would there be substantial wave overtopping if the freeboard of the structure were 20.0 feet (6.10 meters)? SOLUTION: Since there is no figure or set of coefficients for the runup equation (eq. 2) for a slope of 1 on 2.75 it is necessary to compute R, Rg, and Rg for slopes of 1 on 2.50 and 1 on 3.00 and interpolate between them. To start, calculate the period of peak (maximum) energy density, Tp, using equation (1). Tp = 1.05. T,.= 1.05 (7.0) = 7.35 seconds Then compute the steepness parameter, He/elyo H Ea pe great eats 0.00345 Bloor S252 i-25) = Using the above value of steepness in equation (2) with the coefficient given in Table 1 allows the computation of R,/H,. For example, to calcu- late Ro/H, for a 1 on 2.5 slope Z Bq tees 1129200200345) [-16,100(0.00345)*] = 3.64 Ss The above value of Rj/Hg can be confirmed, using Figure 4. Therefore, Rg = 3.64(Hg) = 3.64(6.0) = 21.8 feet (6.64 meters) The other runup parameters Rg and R can be calculated in a similar manner, then used for interpolation to give the values of the runup parameters for the 1 on 2.75 slope as shown in Table 2. 15 Table 2. Values of the runup parameters for example problem 1. cot 0 Ro/Hg Ro Rs /Hs Re R/Hy R (£t) (£e) (ft) 2.50 3.64 21.8 5152 i be a ae 3.00 3.49 21.0 2263 14.6 1.56 9.4 25 -- 21.4! -- 14.9! -- 9.41 ltInterpolated value. The interpolated values in Table 2 should be corrected for scale effects to yield the required answer. The scale correction factor for a slope of 1 on 2.7/5 is 1.125 (see App. A); therefore, Ro = 21.4 (1.125) = 24.1 feet (7.35 meters) Rg = 14.9 (1.125) = 16.8 feet (5.12 meters) R = 9.4 (1.125) = 10.6 feet (3.28 meters) A freebaord of 20.0 feet falls between Ry» and Rg, so the structure crest would not be overtopped frequently, probably by less than 10 percent of the waves. It is, therefore, expected that the volume of overtopping would not be great. It is difficult to determine how high a smooth structure would have to be to prevent all wave overtopping but a reasonable estimate would be Rmax ~ Ro + Hs where Rmax is the elevation of the maximum runup. kK KK KK KK KK KOK OK KX EXAMPLE PROBLEM 2 * * ¥ & KX KX KK KK KK KARE This example illustrates how to calculate the approximate lower limit of rundown. GIVEN: A plane, smooth 1 on 2.50 slope is subjected to irregular wave action. The significant wave height, significant wave period, and water depth at the toe of the structure are 7.0 feet (2.13 meters), 8.0 seconds, and 30.0 feet (9.14 meters), respectively. FIND: Rdgg for the above conditions; this is the approximate lower limit of wave rundown. SOLUTION: The period of peak energy density is Tp = 1.05(T,) = 1.05 x 8.0 = 8.40 seconds and the surf parameter is 1 i E= 1/2 = vf = 2.87 (Hg/Lo) cot 6 {7.0/ 132.2 x (8.4)71/20} (2.5) 16 Using this value cf §& in equation (6) gives the relative rundown, i.e., RaSé -2.46/E a = -2.32e = -0.99 Ss which can be confirmed in Figure 8. Then Rdgg = 2(7.0)(-0:99) = 6-9 feet. (—2.10 meters) and using Appendix A to correct this rundown for scale effects gives Rdgg (corrected) = -6.9(1.128) = -7.8 feet (-2.38 meters) The same scale correction factor used for runup is used for rundown. kok Kk & kK kK kK Ok OK OK OK OK ® & EXAMPLE PROBLEM 3 * *¥ *¥ ¥ KK KK KKK KK KK This example illustrates how the results of tests with irregular waves on smooth slopes can be applied to situations where the structure is rough and porous. GIVEN: A rubble-mound breakwater is to be built with a slope on the seaward face of 1 on 2 which will be overtopped by wave action only occasionally under the design conditions. The design conditions include a significant wave height, significant wave period, and water depth at the toe of the structure of 15.0 feet (4.57 meters), 12.0 seconds, and 45.0 feet (13.72 meters), respectively. The core of the breakwater will be slightly above the design water level, i.e., a high core breakwater. FIND: The height at which the breakwater will only occasionally be overtopped during the design conditions. SOLUTION: The period of peak energy density is Tp = 1.05(Ts) = 1.05 (12.0) = 12.6 seconds and the steepness parameter is s N50 7 = aa Bie 6052. 202,16) Using equation (2) with the coefficients in Table 1 for a plane, smooth slope of 1 on 2 and RjH/g gives ee Ss =.3.2083 + 71.879 (0.00293) = 3.42 (this value can be checked in Fig. 3) and Ro = 3.42(15.0) = 51.3 feet (15.64 meters) 17 The runup reduction factor, r, for rubble-mound breakwaters with high cores is 0.52 (see App. B) and the scale-effect correction factor is 1.06 (see App. C) so Rg for the breakwater is Ro (breakwater) = 51.3(0.52) 1.06 = 28.3 feet (8.63 meters) Rg and R are found in a similar manner to be Rg (breakwater) = 20.0 feet (6.10 meters) 12.2 feet (3.72 meters) R (breakwater) These calculations indicate that if the freeboard were 28.3 feet only 2 per- cent of the waves with a Hg = 15 feet and Tg = 12 seconds spectrum would overtop the structure while a freeboard of 12.2 feet would allow about half the waves to overtop. A freeboard equal to Rg, i.e., 20 feet, will satisfy the condition of only occasional wave overtopping since about 13 percent of the waves would be expected to overtop the breakwater. KR KKK KKK KK KK KKK KKK KK KKK KKK KKK KK KKK KK KKK VI. SUMMARY Equations and curves are presented for computing three runup parameters and one rundown parameter for plane, smooth slopes exposed to irregular wave condi- ‘tions where dg/Hg > 3. These parameters are R2, the elevation exceeded by only 2 percent of the runups; Rs, the average runup of the highest one-third of the wave runups; R, the mean runup of all the runups; and Rdgg, the depth below the stillwater level which is just greater than 98 percent of the rundown. Example problem 1 illustrates the use of equation (2) in computing the rundowns, parameters, and the method of interpolation for runup on slopes not specifically covered in this report. Example problem 2 illustrates the method of computing rundown. Example 3 illustrates how the study results for smooth slopes can be applied to rough and porous slopes, in this case to com- pute the desired freeboard for a rubble-mound breakwater. LITERATURE CITED AHRENS, J.P., "Prediction of Irregular Wave Runup," CETA 77-2, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., July WOT e AHRENS, J.P., "Irregular Wave Runup," Proceedings of the Conference on Coastal Structures '79, American Society of Civil Engineers, Vol. II, 1979, pp. 998-1019. BATTJES, J.A., "Wave Runup and Overtopping,'' Technical Advisory Committee on Protection Against Inundation, Rijkswaterstaat, The Hague, Netherlands, 1974. GODA, U., "Estimation of Wave Statistics from Spectral Information," Pro- ceedings of the Symposium on Ocean Wave Measurement and Analysts, Vol. I, 1974, pp. 320-337. HUNT, I.A., "Design of Seawalls and Breakwaters," Journal of the Waterways and Harbors Divtston, Vol. 85, No. WW3, Sept. 1959, pp. 123-152. KAMPHUIS, J.W., and MOHAMED, N., "Runup on Irregular Waves on Plane, Smooth Slope," Journal of the Waterway, Port, Coastal, and Ocean Division, Vol. 104, No. WW2, May 1978. STOA, P.N., "Reanalysis of Wave Runup on Structures and Beaches," TP 78-2, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Mar. 1978a. STOA, P.N., "Revised Wave Runup Curves for Smooth Slopes," CETA 78-2, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., July 1978b. STOA, P.N., "Wave Runup on Rough Slopes," CETA 79-1, U.S. Army, Corps of Engi- neers, Coastal Engineering Research Center, Fort Belvoir, Va., July 1979. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Proteetton Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. VAN OORSCHOT, J.H., and D'ANGREMOND, K., "The Effect of Wave Energy Spectra on Wave Runup," Proceedings of the 11th Conference on Coastal Engineering, American Society of Civil Engineers, 1968, pp. 888-900. Vahey ay j fell ee am De it re i 7 Vv, eo be: na. eB Hey ii mn ion Pe init aa Se hy As 5 es) tos A, USN EA oy A fen Na tateen eta ire On . eae, ve) for ry Lory re dois ig’ agit A : 7 i 7 ME at ow 1 Wibiyt mf + at Taig a Rate ‘ vot ! 5 i i ” ’ : Wa put Ri rie o Pi Fi i Dh eA Ny at) ee! ( I ew; i. ane, ( 9 409) adojs BINJINIGS t Cie Derek EEN per ee aa a TeEEEESE HH ereee HATE ESESSSSESS ey saa SPSS fa eee PSEEEES Fraear rr ee TEES eee | (88Z6T °803S) SHdOIS HLOOWS YOd “HX ‘YOLOVA NOILOGWYOD LOMAAI-ATVOS dANny "dl Ss Sereres= all ES es V XICGNdddV v ¢ ra n= BOL 9:0 yO _£0 Al) me) re TTT ae EEE i | ara Eel 201 He ET izee aoe FF 5 57H Hae te eee, 81° 2 | APPENDIX B RUNUP REDUCTION FACTOR, r, FOR VARIOUS TYPES OF ROUGH AND POROUS STRUCTURES (Stoa, 1979) I. VALUE OF r FOR QUARRYSTONE RUBBLE-MOUND STRUCTURE (HIGH CORE) te =sO}D2 Quarrystone armor layer tee Stones thick, ) random placement Fa ®&Oe Ola CERES, a“ a ot ae N Saeko a ey ae Ze h d 075< —2=11 Core "e s $ Underlayers 22 II. VALUES OF r FOR CONCRETE ARMOR UNITS 1. Embankment. a. Gobi Blocks. r°~"0-93 for’ H'/k 2 or —H/K_ 476 (oh r v ! ' (use Hj when qo/HS > 3 and H_ when d/h; <5) 34¢ in (010m) fal roo 7 ro ! ke Lf! Fetevation of Gobi Block Plan View of Gobi Block b. Stepped Slopes. Values of r for stepped slopes. Type of step Slope (cot 6) ri Vertical risers 15S} 0.75 2.0 0.75 3.0 0.70 Rounded edges 3.0 0.86 lj < Hi/k, < 12 where k, is the height of the riser. 23 7A Embankment and Rubble Mound. Values of r for concrete armor units. Armor unit and Length dimension, placement method k, Armor-layer thickness (No. of units) Tetrapod Random | he Uniform Quadripod Random | he Uniform hs Tribar Random Uniform Modified cube Random Uniform Uniform Uniform 1.3 to 24 III. 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