CETA 78-2 CAb- A058 407) Revised Wave Runup Curves for Smooth Slopes by Philip N. Stoa COASTAL ENGINEERING TECHNICAL AID NO. 78-2 July 1978 eunors DOCUMENT | \ COLLECTION / & v4 Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING +c RESEARCH CENTER 330 Kingman Building Fort Belvoir, Va. 22060 National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 The findings in this report are not to be construed as an official epartment of the Army position unless so designated by other ithorized documents. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM T. REPORT NUMBER 2. GOVT ACCESSION NO, 3. RECIPIENT'S CATALOG NUMBER CETA 78-2 4. TITLE (and Subtitle) 6 5. TYPE OF REPORT & PERIOD COVERED Coastal Engineering Technical Aid REVISED WAVE RUNUP CURVES FOR SMOOTH SLOPES 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Philip N. Stoa 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS Department of the Army Coastal Engineering Research Center (CEREN-CD) F31234 Kingman Building, Fort Belvoir, Virginia 22060 V1. Neaa ee Ement Die Ae AND ADDRESS 12. REPORT DATE ae © t e a July 1978 oastal Engineering esearch Center Bi, 13 Sa Kingman Building, Fort Belvoir, Virginia 22060 242) PLP CLO 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of thie report) UNCLASSIFIED 15a. DECLASSIFICATION/ DOWNGRADING SCHEDULE DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES KEY WORDS (Continue on reverse side if necessary and identify by block number) Breakwaters Runup Coastal engineering Scale effects Coastal structures Wave runup ABSTRACT (Continue am reverse sides if neceseaary and identify by block number) Results of previous tests of monochromatic wave runup on smooth structure slopes were reanalyzed. The runup results for both breaking and nonbreaking waves are presented in a set of curves similar to but revised from those in the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977). The curves are for structure slopes fronted by horizontal and 1 on 10 bottom slopes. The range of values of (continued) DD, een 1473 ~—s Ep Tion oF 1 NOV 65 IS OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) d,/H, was extended to d,/H} = 8; relative depth (d,/H}) is important even for d,/H) > 3 for waves which do not break on the structure slope. A flow chart is given to assist in choosing the proper figure and in inter- preting the results when applied to untested bottom slopes (i.e., bottom slopes flatter than 1 on 10). Also given are example problems and a curve for scale-effect corrections. 2 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report describes a means of determining wave runup on coastal structures having uniformly sloping, smooth surfaces. The report is based principally on small-scale test results and analyses of Saville (1956) and Savage (1959) as reanalyzed by Stoa (1978). The work was conducted under the coastal engineering research program of the U.S. Army Coastal Engineering Research Center (CERC). The technical guidelines presented in this report supersede the design runup curves for smooth slopes given in the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977). The revised runup curves given here include a wider range of relative depth, d,/H3. These results are based on experiments using regular waves. Ahrens (1977a, 1977b) presented methods for estimating runup and overtopping, respectively, from irregular waves based on results of regular wave testing. The report was prepared by Philip N. Stoa, Oceanographer, under the general supervision of Robert A. Jachowski, Chief, Coastal Design Criteria Branch. 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. JOHN H. COUSINS Colonel, Corps of Engineers Commander and Director II CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) SYMBOLS AND DEFINITIONS . INTRODUCTION. RUNUP CURVES. : j 1. Smooth Structure Fromeed by Homimoncall lpoceen : 2. Smooth Structure Fronted by 1 on 10 Bottom Slope and Zero Toe’ Depth (d, = 0). : Smooth Structure Fronted by 1 on 10 Bottom Slope and Toe Depth Greater than Zero (d, > 0) Ww MAXIMUM RUNUP . SMOOTH-SLOPE SCALE-EFFECT CORRECTION. EXAMPLE PROBLEMS. LITERATURE CITED. TABLES Example runup for T = 7 seconds, constant depth, and (Coe ene = LO 2eSiec Example runup for T = 13 seconds, constant depth, and CAS nav Example runup for constant wave steepness, Bele = 0.0101. = 16 feet. Summary of maximum runup for different conditions FIGURES Definition sketch of variables applicable to wave runup . Relative runup for smooth slope on horizontal bottom; d,/H3 = 5 Relative runup for smooth slope on horizontal bottom; d./HZ = 2 Relative runup for smooth slope on horizontal bottom; apts = Be Relative runup for smooth slopes on 1 on 10 bottom; d, = 0; d/a} = 3 Page 31 31 33 33 10 12 13 14 15 10 11 12 13 Relative cl. = OS Relative Lepil & O« Relative Lop iy 2 Ov Relative Oni 2 Os Relative JO, 2. 0). CONTENTS FIGURES--Continued runup for smooth slopes ; d/H> =5. runup for smooth slopes Cys ts runup for smooth slopes 55 Gales 2 O26 runup for smooth slopes Spe de HH esel.0) runup for smooth slopes 53 d,/HS eal) Washi runup for smooth slopes oF d,/H) = So > on on on on on on 1 Flow chart for the evaluation of wave Runup scale-effect correction factor, on on on on on 10 10 10 10 10 bottom; bottom; bottom; bottom; bottom; bottom; Page 16 LY 18 IY) 20 CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 Square meters cubic feet 0.0283 cubic meters yards 0.9144 meters Square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars 1.0197 x 1073 kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.01745 radians Fahrenheit degrees 5/9 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: K = (5/9) (F -32) + 273.15. SYMBOLS AND DEFINITIONS water depth water depth at toe of structure acceleration of gravity (32.2 feet per second squared or 9.81 meters per second squared) wave height the deepwater wave height, neglecting refraction, equivalent to the wave height, H, measured in a given water depth shoaling coefficient, H/H* runup scale-effect correction factor wavelength deepwater wavelength; wavelength in water depth, ad, such that d/h = O05 horizontal length of slope fronting toe of structure runup; the vertical rise of water on structure face resulting from wave action wave period bottom slope; used for the slope fronting a structure and is different from the structure slope structure slope; may be beach slope if runup on the beach face is being investigated tl i hail A OY reek Ophane 4 i Ne a ies i sigsdags i tile ws: rid ni tess lags 4 « Kuba yee —_—-i., a re kil Sy: 4 papel sue Pgh: a aoe be a = oe us "i b Misty Fit fig Maia. goed. cma Gh), or8i! aim & eR : ay : oo ie of ; j RPV es tei & as Sie: Cty wht» i) by, aie aay ‘ re ih a ee ; VG) Mowe ais: oe Homer LE wget: te lee ey bs ’ ie 4 mie Ea 4 Age . nee & ort ‘an Le J) nconheLe Ditiee G of? ee ee Uy i Wien s+ shhenlnidn selenremnrehuhareate desl » Fi tain. Ce tae ws! > Seeres rarity ra mala 7 iormente- “ih >) ae | ‘ ij WWLGLN FOLIO LP TOWERS, aie Nara 8 REVISED WAVE RUNUP CURVES FOR SMOOTH SLOPES by Philip N. Stoa I. INTRODUCTION Wave runup is the vertical distance above stillwater level (SWL) reached by a wave incident to a structure or beach. Prediction of wave runup on coastal structures is necessary to determine an adequate crest elevation to prevent overtopping or to help determine the extent of overtopping. Wave runup curves for structures with either smooth or rough slopes have previously been presented in the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977). Runup data of Saville (1956) and Savage (1959), together with data from other reports, have been reanalyzed (Stoa, 1978). This report presents revised smooth-slope runup curves which vary in certain regions from those presented in the SPM. A scale-effect correction curve is also given for application to smooth-slope runup. Wave runup is primarily a function of characteristics of the structure and incident wave; wave characteristics are also a function of water depth and bottom slope. The variables are shown in Figure 1 and aLemdetinedsas mes a GUNUpI: Te) sanclerot Structure slopes ads." water depth; d,, water depth at toe of structure; 8, angle of bottom slope at the structure toe; and £, horizontal length of the bottom slope seaward of the structure toe. L and H are the wavelength and wave height, respectively, as measured in a water depth, d. The same wave may be described by an equivalent deepwater wave (d/L 2 0.5) for which the dimensions would be Lg, and Hj. Lo is the deepwater wavelength and HS is the equivalent unrefracted deepwater wave height. L, may be determined if the wave period, T, is known (Ly = gT2/2m); this report uses pT? as the principal measure of deepwater wavelength. Hi is used because it avoids the problem of defining the wave height in varying depths over a sloping bottom where the wave may already have broken. The wave height in deep water is related to wave height in a shallower depth by the shoaling coefficient, H/H3i or Kg. The shoaling coefficient and wavelength, L, may be determined from Tables C-1 or C-2 in the SPM when Lo and the required depth are known. The runup curves are given for three different cases: (a) horizontal bottom at the structure toe; (b) 1 on 10 sloping bottom at the structure toe, with a zero toe depth (d, = 0); and (c) 1 on 10 sloping bottom at the structure toe, with toe depths greater than zero (d, > Os Case () has, generally, the potential for the largest waves attacking the struc- ture. A bottom slope of 1 on 10 is relatively steep for ocean coastlines, and its occurrence would be restricted to beach faces with coarse sediments (see Fig. 4-33 in the SPM), backshore areas subject to flooding, or some nearshore areas. However, most bottom slopes would be flatter than 1 on 10. Experimental data for runup on structures fronted by flatter slopes “(8261 ‘v03Ss) dnuna ovem 03 aTqeottdde setqetaea Fo yoIey¥s UOTIIUTFAGQ “T oansTy SS Vihoy. L SYS VES? MS INS VESTS y Sieg Ca a WA LS Zp adojs woyjog aS 7 < Pp \‘a! 6 H Sp met OH Ve a, —_—<— CN Nao = — vin — on a = aS Nib GAD Bulyoesg e\qissog QINJINIYS 10 are very limited; brief qualitative comments regarding runup in such circumstances are given in later sections. The incident wave characteristics seaward of the toe of the bottom slope are partly determined by the corresponding water depth and are important in determination of runup. The methods presented in Sections II,2 and II,3 are designed to account for the incident wave character- istics at the toe of the bottom slope as determined in model experiments. Natural underwater slopes are rarely so well defined; straight-line approximations of irregular slopes should be determined by the designer. Intersections of the straight lines will define the location of a change in slope. II. RUNUP CURVES 1. Smooth Structure Fronted by Horizontal Bottom. Relative runup, R/H} for a smooth structure fronted by a horizontal bottom is given in Figures 2, 3, and 4 for specific values of relative depth, d,/HS. As shown by comparing the figures, relative runup on the flatter slopes is not a function of dg/H§. However, relative runup on the steep slopes is sensitive to depth effects; relative runup for a given wave steepness, H}/gT~, is largest at the lowest dg/H} value. Thus, proper consideration of depth effects must be included in design. Relative depth values of 2 < dg/H$ < 3 may occur for structures on horizontal bottoms, but experimental data are limited. Figure 2 (dg/HZ = 3) is recommended for cases in which d,/H3 < 3. Large d,/Hg values may occur, for example, in reservoirs; runup determinations for d,/HS > 8 should be based on Figure 4 (d,/Hg = 8). 2. Smooth Structure Fronted by 1 on 10 Bottom Slope and Zero Toe Depth (ds = 0). When dg = 0, wave conditions are determined using the depth, d, at the toe of the 1 on 10 bottom slope. Figures 5, 6, and 7 show the results for d/HS (mot dg/HZ) values of 3, 5, and 8 with a 1 on 10 bottom slope. Runup on a structure fronted by a beach slope flatter than 1 on 10° would be expected to be less than indicated in Figures 5, 6, and 7 for comparable wave conditions. However, these figures are recommended for use when a flatter bottom slope is present and d, = 0. 3. Smooth Structure Fronted by 1 on 10 Bottom Slope and Toe Depth Greater than Zero (dg > 0). Design curves for runup on a smooth structure with d, > 0, fronted by a 1 on 10 bottom slope, are given in Figures 8 to 1l. The curves apply to cases where the relative bottom-slope length is Cj = 0.5). “For values Of C/ii< OS) but) tor high da/H values (agi dae 2s 5) .therrunup values from figures for structures on horizontal bottoms (Figs. 2, 3, and 00108 09 Tie $u0}0q [e}UOZTIOY UO asdo[Ts yOOUS TOF dnunzr sATIeTOY Ov OF tielid Heal EH O02 O!l 8 | “(8Z61I “B03S) ¢ (Q 409) adojs ainyonsys 9 eS © 1 ! | | TH/*P "7 9anstj SO GO WO tO: GO l2 ‘wO3IO0q Te UOZTIOY OOO OF Oy Of Of Ol Opn “(8261 ‘e03s) ¢ = TH/*P uo edots yoous 1oZ dnunz sATeLOyY (@409) adojS einjonsys 3 &) vy & A (| "¢ o1nsT4y + fi {e} 13 ae Be SG a Suet H Bi ett a‘ ian di oe a aS iis i aifees | ae iecee tiger sy ee ay dine ; ne Het ater iOPaD se iss HG PERT IE Sbsuy sis Vc bd ea i nie ae Pare cacit “a tee Hira BS asses a eee HEE = free ieee Haas a0 HSE =e ig sacs ssese Saeew anssdanses SSesscscssseir == it NE i ee HTIEEES a sil a a — [st = efeo Bees ‘o? ° OD ° Eat HELE Hil isda, cee fee Hg HEEEEEA EE EEE THE EEEE Har gs i oe HEE HLNETIEEEGE byt LA Waceeaee ie qu gees ait ze | Tn MMI - 0.2 0.1 O 60 8010 30 40 20 Uy gesess O44 Of OG OS OSs | 0.1 Structure Slope (Cot @) Relative runup for smooth slope on horizontal bottom; Figure 4. = 8 (Stoa, 19/78). 1 0) d,/ “(9461 ‘e01s) ¢ = SH/p {0 = °P £‘wo170q QT UO T uo sodots Yoows 10x dnunz sATIeITOY °g 9ANsTY (Q 409) adojg e4nyon4ys 5 00108 09 Ob Of Zz Ol 8 ets seo | 8090 voe0 ZO 10 me) | 20 Sih nun eaoaeae 20 £0 £0 66 : : v0 i t — 90 3 ie 9°0 80 a Sa = : : == SS etises= 80 Oy ! Se. MU ALES ra - aa: 2 ¢ b 9 g 15 “(8261 ‘e03S) ¢ = SH/p {0 = °P $u02120q QT UO [T uo sedoj[s yOous toF dnunz aATIeTOY °9 9aNnSTY oo O8 > OF Orv iOe O02 Ol 9 Vee aS (Q@ $09) adojg ainjoniys | SO VO WO so Zo 1'O ¢ | 16 jpoues OL) Ow CK2- O Ol fs — ©) Veet g | FO) GO wots GO ie 6 | Witt i iT int tt MUHA BER UE sar 2 OR eH et : ZHO © Ops 7 eS £0 b ORR : == El v0 { erjssee ; in pera ne : : 'S) (0) ee ae eis = saiaan el 80 ii A ; r inn | 9 2 pe 3 @ ¢ pee E YF ise a aa SaSeee 174 9 : ; 9 8 =e = = Z = = Seay eae 8 go Hee SOEESES Ol £W0720q OT uo “(8Z61 ‘20I3S) 8 TH/P £0 s. 2 IT uo sodots yoous toF dnunaz sAtTIeTOY (@ 109) adojs aanyonsys *Z gins Ty 17 "(QL61 “20RS) OO = TED {S70 = Wy $woi30q QI UO JT UO sadotTs ujZOOUS OF dnuNA sATIETOY °g oaINSTY (9409) edojs einyonsys come 09 Ov OF O02 Ol © yp && S | GO VFO wOteoQ GO lO 18 :WO3370qG QT UO T UO sodoTs YyOOWUS TOF dnuNI sATIeTOY "(Q1G “HAS) OF S Fi 6e°o = ay "6 INST i (@ 309) adojs asnyoniys 00108 09 Ob O€ OZ Ol 8 9 bp ¢ d 18090 vO€¢O ZO me t Ms +t veteefe t HN naan | i Fs ; Pes! He | rau @) i Bayt T Sor [ = t TT a { os H pean ae i an iisueseee =: ¢ O aI ime = = c=e a =A WS b O Fr bees 7 i USSU p sees rer free Je 9'0 : = = . = z = i se geet eet es ze i SS SS > ft © Bee EEE SSS : > be He Sg5EEs a — 9 = = ——E fe) = Ol '(QIEL “FONS) Got =] Wey S°0 S Wiz $u02120q QI uo [ uo sedotTs uzOOUS TOF dnunI 9ATIeETOY (Q 109) edojS asnyonsys OO!I08 09 OF OF O02 O|ls 9 ip d ImGiOnIIOM=iOEGiOs =ciOm = me) me) Ua Fa I iH i ; | ANAT ime) ffl 7H eli THEE “OL oansty ty 20 (GG BONS) OOS = Hey» 86°) 2 4h; -£wo120q OT UO [ Uo saedo[s yIoous LoF dnunx sATIeTOY (@ 409) adojs asnyonsys “TT onns ty 00108 09 Ov O£ Od ols 9 21S @ | B30) Sor (70) SO) = ¢O) m0) 10 3 ue | i 7 ! y fF tk ! i a0) : sel ceneimiee anit iuvanaie 20 7 =: i : I 3 ot t : te = F ¢ O i" it was sre BEE ¢ O vO Saas Se : 6:0 90 ‘ 90 80 é 4320001 KAS ate 80 | fi = Sat. Sa a oe < = - — | é ; : : St a ca £ 4 — == = bs sa2== i Hs: BE £ b se gessaseeae === pie ies iesegSeeee seagate E : b 9 = = 9 8 == See = ae = SUS =! 0! oe 401 2| 4) should be used as upper bounds of relative runup on structures fronted by a 1 on 10 slope with the same d,/H} value. In the case of £/L < 0.5 with low values of ide /Hoes (cae. mOLOelametc.) ut. Shouldubesexpected that relative runup will be somewhat higher than predicted from the curves (Figs. 8 to 11), and probably not exceeding 15 to 20 percent higher. However, the effect of the length of a 1 on 10 bottom slope diminishes as the structure slope decreases, and effectively ceases to be significant for cot 6 2 4. These comments are incorporated in a flow chart (Fig. 12) for determining which figure to use to find the runup on a structure fronted by a sloping bottom. Because there are insufficient data available for cases where bottom slopes are flatter than 1 on 10, it is recommended that the curves given in this report, applicable to structures fronted by 1 on 10 bottom slopes, be used; in most cases, results are expected to give higher estimates of R (see Fig. 12). For the larger d./H3 values (e.g., d,/H, > 2.5), relative runup on structures fronted by gentle bottom slopes will be equal to or less than that given in Figures 2, 3, and 4 (horizontal bottom) for the appropriate d,/H4 value. Relative runup on structure slopes flatter than 1 on 4 is largely unaffected by changes in bottom slope. Relative runup on steep structures fronted by a gentle bottom slope will be equal to or less than values given in Figures 8 and 9 but may be slightly higher than those given in Figure 10 (d,/Hg = 1.5). ITI. MAXIMUM RUNUP This section discusses the maximum runup from regular waves when a range of conditions is possible. Maximum runup from irregular waves is not discussed, but an approach to estimation of maximum runup from irregular waves is given by Ahrens (1977a). In his method, runup result- ing from a significant wave is determined from design curves such as given here, and then runup for the irregular waves is assumed to follow a Rayleigh distribution. Maximum runup, R, for a range of regular wave conditions, is not necessarily associated with the maximum relative runup, R/H3. For structures sited on horizontal bottoms, and for a gtven wave steepness, He /enes both the maximum relative runup and the maximum dimensional runup occur at the minimum value of d,/H%. For structures sited on a 1 on 10 sloping bottom, maximum dimensional runup, R, may or may not be coincident with the maximum relative runup determined for a range of wave conditions. If depth, d,, and wave steepness are assumed constant, then maximum relative runup occurs when 1.0 < d./HS £ 1.5, but maximum dimensional runup, R, is found when d,/H} is a minimum (in this report, when d, > 0, then (de/H3) min = 0.6). In cases where a bottom slope flatter than 1 on 10 is present, for a given wave steepness, the maximum relative runup will occur for somewhat higher d,/H} values (1.5 < d,/H} < 2.0). However, if wave height, Hj, and wave steepness are held constant, the maximum dimensional runup, R, will be coincident with maximum relative runup as d,/H3 varies (i.e., as 22 “dnunz oAemM FO UOTJENTLASD YI OF WABVYD MOTY “ZT oaANBtT 10njoo J PO48( O74) “POjINPUOD B49™ $40) jOpOW j! pojrOdxO yjnsos oy) SUDBW joNJoy, “¢ 6'g:8614 esn *B4DW14Sa UD AjuO S! UBAIB Bbuds BY) "MO} 00) 0G AoW BeAUND BY) WOJ) POIIIPAId GNUNs YIIYM JOy SUOIJIPUOD BJODIPUI SyIJOISY “2 “SUOIJIPUOD PaISAJUN jUBSOIdas SOUI|YSOQ ‘| :S3L0N Jonyao < PP8(OH4 74) 2'9'¢ “sb1y asn | O1SeH/% | 1014200710) BO & H Jonyoo > Pe4d( O74) 0Nj30 01 0) 8" 107}200'1 0180 & Ol'6'8 “S814 asn yonya0 > PPS(OH7y) fhe —— — — — — joH/* me : WS2>°H/*P >O1L: Ol 614 asn bs 1onj20 < POH8(oH/y) 8'S © °H/%p 40) be S5iy e8N ‘Ol< g 402 4Oy 443042 SIYyy OS “2 > SH/SP ¥] 1)Ows si °H/%p $S9jUN WO4JOQ [OjUOZIZOY D 40) UDY) juasajsip AjyuDdIjIUbiS 2q jOu Aow uoljipuo? siy) ‘(adojs |jows) gf 402 abso) 104 = ! SO < 1/7 puo b >@402 | JOMj2D wy PP48(SH47y) TE EAE) O16‘ ‘sb13 asn 01'6'a ‘s614 asn y < 6102 [6 10 | ! H € > °H/*P >0 I JO 1ON4DD gy POMS (OH 7y) J0Nja0 J P44 OH 7) [DN 49D gy PO49( OH 7y) I 1 110) @ ‘sb)y esn ! b'e'2 ‘sis asn G2S°H/'*p | 2'9's “sb1g asn | I I 10M 420 zy PO44(CH/y) ! . 0<°H/* p'e'2 ‘S614 esn H eG ' ! ONnjoo = Pe2d(0 mw p0sd/0, < Fe © 102] aes OTA = SH/*P GVA | papa nee ae baED ae ed ee 1 < 9109 G2 < °H/*P € < °H/"p v'e'2 ‘sb14 asn 2'9'g ‘sb14 asn 0 = °H/*P ' ' | ! 1 ! ysbua; adojs-w0jjog yibua} adojs- wosjog G0> 1/7 so< Ws 23 d, changes). The maximums (R/HS and R) may occur at any value of d,/Hg (including d,/Hi = 0) depending on the wave steepness being considered. Runup ae eae Sonn occur at Hones eae AES values of d e/g (sO < d,/HS < 1.5) for high values of eon but at low walines OE Gla/is\s or ton values of Riots For a gtven wave pertod and constant depth, d, (with wave steepness varying as d,/H4 varies), maximum dimensional runup is generally not coincident with maximum relative runup; furthermore, the maximum dimen- Sional runup may occur at other than the minimum d,/H3 value. The designer of a structure subject to runup will usually have a range of wave conditions for which maximum runup must be determined. The preceding discussion emphasizes the need to determine the maximum actual runup by finding the runup for each of several wave conditions. Example problem 3 (Sec. V) highlights some of the relationships discussed here and shows the maximum runup values for different sets of initial wave conditions. IV. SMOOTH-SLOPE SCALE-EFFECT CORRECTION The smooth-slope runup curves plotted in Figures 2 to 11 are based on small-scale wave-flume tests. A limited number of large-scale tests (Saville, 1958) indicated scale effects were present in the runup results. Figure 13 presents values of the correction factor, k, as a function of structure slope; the curve is modified from that given in the SPM, and is extended over steeper slopes. Selection of a particular structure slope may be dependent on evalua- tion of runup on different slopes. The trends in runup on different structure slopes are presumed correct as given by the design curves (Figs. 2 to 11). Comparisons of runup for different structure slopes should be based on the design curves, with the scale-effect correction applied only to the final selected runup value. V. EXAMPLE PROBLEMS The following example problem solutions use Tables C-1 or C-2 in the SPM and the applicable curves in this report. BA A 2s nd Ge ee Ey ee ta ee ENP RIPON SLIEYL JL te cS) RS C2 RO RP ns RS te ep te Ne Se GIVEN: An impermeable structure has a smooth slope of 1 on 3 and is subjected to a design wave, H = 8 feet (2.4 meters), measured at a gage located in a depth, d = 30 feet (9.1 meters). Design wave period is T = 8 seconds. The structure is fronted by a 1 on 90 bottom slope extending seaward of the point of wave measurement. Design depth at structure toe is d., = 25 feet (7.6 meters). (Assume no wave refraction between the wave gage and structure.) 24 “(8Z6T ‘e01S) YY {20}Z9eF UOT}IOIIOD YOEeFFZO-aTeOS dnuny “ST sANnsTAJ (9 409)adojs ainyoniys FO FO VO CoO - Fo me) 00'! aeaeee cO | oa (EEES sea le r= b0'| Eee = eae: : 901 eee SSccs a : =| 80'| = ot | eaeeeeee Sees e===— (0) | 3 segsseus goes cae: : H aaa + - 20 — te oI eee = || Eth soessSs AEE : eS 25 FIND: The height above SWL to which the structure must be built to prevent overtopping by the design wave. SOLUTION: The wave height must be converted to a deepwater value. Using the depth where wave height was measured, calculate Gi et ob an tn lo Dieta pe lapeeee Oe L, milan | (G2.A/AMt » So12(@)e f= 0.08255 - Lo To determine the shoaling coefficient, H/H} Table C-1 in the SPM is used, and = 0.9406 . H6 Therefore, Hf eee eee ee @ 0.9406 0.9406 nh 8 feet °(2.6 meters) Calculate, also, H! EOL Ae SE SEE NE Quinto ar $2.,.D((8)7 and, for d, = 25 feet d 25 — = —_ = 2.94 He as) The bottom slope is very gentle (1 on 90). Assuming that the slope approximates a horizontal bottom, the appropriate set of curves for d,/H} = 2.9 is Figure 2 (for dg/H{ = 3). For a 1 on 26 3 slope and The runup, uncorrected for scale effects, is Ke (Boil) (eis) = (2.1) (8.5) R = 17.9 feet (5.5 meters) The scale-correction factor, k, can be determined from Figure 13, and, for cot 6 = 3, the correction factor is k = 1.12. Thus, the corrected runup is R = (1.12) (17.9) = 20.0 feet (6.1 meters) xe Kk Kk kK Kk kk & * * * EXAMPLE PROBLEM 2 * * * * * * & * & * & & *& & GIVEN: An impermeable, smooth, 1 on 2 structure is fronted by a 1 on 10 bottom slope. Toe depth for the structure is Ga 2 IO ese (G meters), but the bottom slope extends seaward to a depth of 50 feet (15.2 meters), beyond which the slope is approximately 1 on 100. The design wave approaches normal to the structure and has a height of H = 9 feet (2.7 meters) and period of T = 9 seconds, measured at a depth of 55 feet (16.8 meters). FIND: The height of wave runup using the appropriate set of curves. SOLUTION: The wave height given is not the deepwater wave height; it is measured, however, above the gentle 1 on 100 bottom slope which approximates a horizontal surface. To determine the shoaling coeffi- cient, K> for the location of measurement, calculate GU Pune! eT2/2n ES 9 55 (5.12) (9)2 = 0.1326 . Cll, From Table C-1 in the SPM, Therefore, H 9 S al? = = ——— = 9.82 feet (3.0 meters) O Kg 0.9162 d US On & 1.0 , He 9.82 and He 9.82 2. = —_-—=___ = ().00377 . ane | (B2.2) (9) ey Because there is a steeply sloping bottom fronting the structure, the value of &/L must be determined: £ = (50 - 10)(10) = 400 feet (122 meters) Next determine the wavelength in water depth of 50 feet (the depth at the toe of the 1 on 10 slope). For days EO as, |, L (32.2/21) (9) 2 and from Table C-1, =0.1585 . |e Therefore, 6 Lk BIS ae Bowie (96.1 meters) ay S-osisss 28 Then, (eS w HS w|o * |oO sf oO thus, L L The appropriate set of design curves is then determined; the flow chart in Figure 12 shows that Figure 9 has the appro- priate curves, and that the results are presumed correct at model scales. From Figure 9, for H}/gT? = 0003538. The runup is R = = (HS) = (3.0) (9.82) R = 29.5 feet (9.0 meters) For cot @ = 2, the scale-correction factor, from Figure 13, is k= 1.136: Thus, the corrected runup is R= (15186) (9.5) 2 55.5 ese (Os meeeies)) See kak Maken xT K eke x EOD Ee PROIBIEEM) Sucks kits (cece oe) ek Rp ene GIVEN: A design geometrically similar to that in example problem 2, where an impermeable, smooth, 1 on 2 structure is fronted by a 1 on 10 bottom slope. Toe depth for the structure is d, = 10 feet, but the bottom slope extends seaward to a depth of 50 feet beyond which 29 the slope is approximately 1 on 100. However, a range of wave periods and deepwater wave heights are known; el LG feet (4.9 meters). FIND: Maximum runup for three different wave conditions: Tyg, = 7 seconds; Tmgy = 13 seconds; and constant wave steepness, Hh /gT? = 0.0101, with Tyg, = 7 seconds. SOLUTION: For any given d,/H} value, the design curves show that relative runup is highest for the longest wave period (or the lowest wave steepness, ED) However, for constant toe depth, d,, and for constant wave steepness, the largest wave height (or lowest d./H3 value) usually results in the largest absolute runup, R. When a sloping bottom is present, and wave period and toe depth (dg) are held constant, the maximum runup may occur at other than the minimum d,/H4, value. Thus, runup for a range of d,/H5 values should be investigated. In the following development, preliminary determinations of runup are not corrected for scale effect. Only the final runup, as determined for selected wave conditions and structure slope, is corrected. (a) The maximum wave height given is Hj} = 16 feet; for this location, the resultant dg/Hg value is a, ig | 6 which is approximately the lowest value used in Figures 8 to 11. The maximum runup may be determined by constructing a table for varying conditions. Because the maximum wave period is less here than in example problem 2, L is also less; thus, @/L > 0.5 and Figures 8 to 11 may be used. Furthermore, Figure 12 indicates that the results in Figures 8 to 11 are approximately correct, to models scales SFOrida — OU teethwl — Seconds asand gT2 = 1,577.8 feet. Table 1 may be constructed with T held constant at 7 seconds because the maximum wave period results in the highest relative runup for each value of d,g/Hi. The maximum runup of 23.5 feet (7.2 meters) in Table 1 does not occur for the largest wave height because the largest waves break seaward of the structure for the given wave period. 30 Table 1. Example runup for T = 7 seconds, constant depth, and (H3)may, = 16 feet. Fige) “de/Hta. One Hi/eT2 = R/H = (ft) (ft) So R056 AGACO. 1) OROROIG. Nusa | 22sho 9 IAO)-¢ FUOROOF 60 00634enNNN2E35.1n25850 10 1S Gee OOS SAL 18.70 11 SHON Wuit553) MAOA002 WIE ANG OMNMESEOE 1d,/H) values selected to correspond with values in figures; d, = 10 feet. *Rnae = 23.5 feet. (b) For the second condition where Tyg, = 13 seconds, the maximum runup would occur for the lowest d,/H4 value. To check 2&/L, for d = 50 feet: re ot, = 0.0577 i) BGS) oS 0.1020 L L = 490.2 feet ; LE LOOM ian) SVONsie Doom Table 2 may be constructed for dg = 10 feet, T = 13 seconds, gT? = 5,441.8 feet, and using Figures 8 to 11. Table 2 shows that, in this case, not only is runup higher for the longer wave period than in Table 1, but the maximum runup occurs at a lower d,/H5 value for the maximum deepwater wave. Table 2. Example runup for T = 13 seconds, constant depth, and (H})may = 16 feet. Fig. aa /i ae H!/gT? RES OR (ft) (ft) 8 “6 1600 Omen Ben ane? 9 iO 10,00 O©.001YO S80 8.0 10 1.5 Gr O,0mne0 6:00" 2560 11 5.0 3.85 O,000Gl2 S.1S 10.8 lds = 10 feet. OR AG West. 3| (c) For the third condition, suppose that wave steepness is expected to be most important, and that the structure is being designed for a constant wave steepness of H}/gT* = 0.0101 and a maximum period of 7 seconds. Table 3 shows the characteristic relationship that the largest runup, R, occurs for the lowest dg/H} value when H!/gT2 and d, are constant; the largest relative runup has lower dimensional runup. However, Table 3 does not indicate the maximum runup to be expected on this structure for the given conditions; Table 1 shows the maximum (uncorrected for scale effects) to be 23.5 feet when a maximum period of 7 sec- onds is given. Thus, care should be exercised in determining runup for a particular structure. The results of the three parts of this problem are summarized in Table 4, and the calcu- lated values are corrected for scale effect based on Figure 13. x kk kk k &k * & * * * * EXAMPLE PROBLEM 4 * * * * * * * * ¥ ¥ H % ¥ ¥ GIVEN: An impermeable structure has a smooth slope of 1 on 1.5 and is subjected to a design wave, Hi = 5 feet (1.5 meters). Design wave period is T = 6 seconds. The design water depth at the toe of the structure is dg = 0.0 foot. The bottom has a 1 on 10 slope from the structure toe to a depth, d = 15 feet (4.6 meters), at which point the bottom slope changes to 1 on 200. FIND: Determine runup on the structure caused by a wave train approaching normally. SOLUTION: The toe depth is zero, and the bottom slope is 1 on 10; assuming that the more seaward 1 on 200 bottom slope approximates a horizontal bottom, Figures 5, 6, and 7 are applicable, subject to the value jof | d/H,- Therefore, Figure 5 is applicable; H! ED AAD NE DAB ae (62274) (G)F The relative runup for a 1 on 1.5 structure slope is determined by interpolation to be OZ Table 3. Example runup for constant wave steepness, H3/gT? = 0, ONON, Fig, MO/ee c/a ae yee = (ft) (8) (2a) 8 0.0101 0.6 LOO 7oO 5s MD i” 9 0.0101 1.0 LOO 5.5 oes toe 10 0.0101 55) ©.0/ 4.5 Lo75 UNa/ Ia 0.0101 Sr0) §o85 SoZ MoS 5.8 1d, = 10 feet. eile = 7 seconds. Seon O = QB. Roe e2Oelabeets Table 4. Summary of maximum runup for different conditions. Table Wave Meseinmenn” Scale-effect Maximum condition correction R k R (Gee) (ft) 1 Constant period; BoD Is LOO AS a T = 7 seconds 2 Constant period; 41.6 Lo LEO 47.3 T = 13 seconds 3 Constant steepness: Boas Il LISS Zoya A/a = OONONs Tmax = 7 seconds lUncorrected for scale effect. 35 A_ 1.23 . H6 Therefore, R = (1.23) (5) R = 6.15 feet (1.87 meters) The scale-correction factor, k, from Figure 13, is KS sila The corrected runup is R = (1.14) (6.15) = 7.0 feet (2.1 meters) 34 LITERATURE CITED AHRENS, J., ''Prediction of Irregular Wave Runup,'' CETA 77-4, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., July 1977a. AHRENS, J., "Prediction of Irregular Wave Overtopping,'' CETA 77-7, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Dec. 1977b. SAVAGE, R. P., "Laboratory Data on Wave Runup on Roughened and Permeable Slopes,"' TM-109, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Mar. 1959. SAVILLE, T., Jr., "Wave Runup on Shore Structures," Journal of the Water- ways and Harbors Diviston, American Society of Civil Engineers, Vol. 82, No. WW2, 1956. SAVILLE, T., Jr., "Large-Scale Model Tests of Wave Runup and Overtopping, Lake Okeechobee Levee Sections," U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., unpublished, 1958. 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. 1978. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protectton 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. 3S) r * a7) e a Vred F ges 7A 7 i D i i Basi § ‘a. 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