Coast Eng. Res. Ctr CETA CETA 77-5 Wave Setup on a Sloping Beach by John R. Lesnik COASTAL ENGINEERING TECHNICAL AID NO. 77-5 SEPTEMBER 1977 U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER TC Kingman Building 330 Fort Belvoir, Va. 22060 Os Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal 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: 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 Department of the Army position unless so designated by other authorized documents. eee AL EEE SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE 1. REPORT NUMBER 2. GOVT ACCESSION NO.) 3. RECIPIENT'S CATALOG NUMBER GETA 77-5 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Coastal Engineering , Technical Aid WAVE SETUP ON A SLOP ING BEACH 6. PERFORMING ORG. REPORT NUMBER AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s John R. Lesnik 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 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Army September 1977 Coastal Engineering Research Center 13. NUMBER OF PAGES Kingman Building, Fort Belvoir, Virginia 22060 18 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of thie report) UNCLASSIFIED 15a. DECL ASSIFICATION/ 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) Coastal engineering Wave setup Coastal structures Waves Sloping beaches ABSTRACT (Continue an reverse side if necesaary and identify by block number) This report combines the material previously presented in Sections 2.62 and 3.85 of the Shore Protection Manual. Computation of wave setup on beaches as steep as 1 on 10 (m=0.01) can be easily determined by graphical means when incident wave conditions are defined. Practical applications are discussed and two example problems are provided. D oe EDFTION OF 1 NOV 65 IS OBSOLETE Di wauwsl4es UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) At yi "a Lae Tees 7A es aT PREFACE This report describes a method of estimating wave setup for beaches of varying slope. The technical guidelines presented are a combination of procedures discussed in the Shore Protection Manual (SPM), Sections 2.62 and 3.85 (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1975). The methods described in Section 3.85 are best applied to beaches with slopes flatter than 1 on 100 (m= 0.01). This report, by applying methods of Section 2.62, presents a method for estimating wave setup for slopes as steep as 1 on 10 (m = Q.10). The work was carried out under the coastal construction program of the U.S. Army Coastal Engineering Research Center (CERC). The report was prepared by John R. Lesnik, dydraulic Engineer, under the general supervision of R.A. Jachowski, Chief, Coastal Design Criteria Branch, who initially conceived the idea for this technical aid. The author acknowledges Dr. D.L. Harris, whose constructive comments enhanced the utility and clarity of this report. 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 CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (STI). SYMBOLS AND DEFINITIONS. INTRODUCTION . EQUATIONS. SAMPLE DESIGN PROBLEMS . LITERATURE CITED . FIGURES Definition sketch of wave setup. Breaker height index, H,/H} versus deepwater wave steepness, eI fat. BAe ee Ae Lay Sees O S.y/Hp versus Bi ieilicn Definition sketch for example problem 1. Page 14 18 ARS LS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (ST) 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 1.8532 kilometers per hour acres 0.4047 hectares foot-pounds 1, S558 newton meters millibars 10197 x 10"? 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) 0.1745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! 1T> obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formulas C =) (5/9) (-32): To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15. SYMBOLS AND DEFINITIONS dimensionless parameter dimensionless parameter water depth depth of water at breaking wave gravitational acceleration wave height at breaking (breaker height) deepwater significant wave height deepwater wave height equivalent to observed shallow-water wave unaffected by refraction or friction. significant wave height H,)33 average height of highest one-third of waves for specified time period wavelength deepwater wavelength beach slope wave setdown at breaking zone net wave setup at shore wave setup between breaker zone and shore wave period WAVE SETUP ON A SLOPING BEACH by John R. Lesntk I. INTRODUCTION Design of coastal structures requires consideration of abnormally high water levels produced by storms. An important component of the storm surge can be the rise in water level produced by wave action. The wave train approaching the coast and breaking offshore causes the water to pile up on the beach. Depending upon the wave characteristics (height and period) and beach slope, this accumulation of water will continue until the slope of the water surface in the onshore-offshore direction results in a head which balances the forces tending to drive the water onto the beach. This rise in water level is commonly called wave setup. Two conditions that could produce wave setup will be examined in this report. The simplest case is illustrated in Figure 1(a), where the dash- line represents the normal stillwater level (SWL); i.e., the water level that would exist if no wave action were present. The solid line represents the average water level when wave shoaling and breaking occur. A series of waves is shown at an instant in time, illustrating the actual wave breaking and the resultant runup. As the waves approach the shore, the average water level decreases to a minimum at the breaking point, d ,. The differ- ence in elevation between the mean water level (MWL) and the normal SWL at this point is called the wave setdown, Sp. Beyond this point, dp, the MWL rises until it intersects the shoreline. The total rise between these points is the wave setup between the breaking zone and the shore, denoted AS. The net wave setup,, S,,, is the difference between AS and Sp and is the rise in the water surface at the shore above the normal SWL. In this case, the wave runup, R, is equal to the greatest height above normal (SWL) which is reached by the uprush of the waves breaking on the shore. For this type of problem, the runup, R, includes the setup component and a separate computation for S,, is not needed. The reason for this is that laboratory measurements of wave runup are taken in refer- ence to the SWL and already include the wave setup component. Figure 1(b) illustrates a more complex situation involving wave setup on a beach fronted by a wide shelf. At some distance offshore the shelf abruptly drops off to deep water. As waves approach the beach, the larger waves in the spectrum begin to break at the seaward edge of the shelf and a setup is produced. The increase in water level produced by this setup allows larger waves to travel toward shore until they break on the beach. Runup calculations performed at that point would include setup effects from the breaking of these waves. Calculation of the precise value of the wave setup for all conditions is a formidable problem for which a satisfactory solution is not yet 7 Normal SWL MWL On a beach Normal SWL b. With a berm Figure 1. Definition sketch of wave setup. available. The problem can be greatly simplified through an idealization which leads to a satisfactory estimate of the upper limit of this effect for many practical problems. Fortunately, the upper limit of the wave setup is of greatest importance in most design problems. When waves, coming from deep water, are dissipated on the beach with- out refraction, the kinetic energy of the waves is converted to the potential energy of wave setup, and the kinetic energy of longshore cur- rents and turbulence. The wave setup component is maximized by neglecting the longshore currents and turbulence. This situation exists in many laboratory wave tanks and on beaches where the bottom contours are approx- imately parallel to the beach and the waves approach along a line normal to the shore. At most locations, it is also possible for the extreme waves to approach along a line normal to the shore. Where this is not true, a conservative upper limit can generally be obtained by multiplying the value obtained by the procedure given in Section II by the cosine of the angle between the wave crest outside the breaker zone and the shoreline. Where bottom contours are not approximately parallel to the shore, the estimates (Sec. II) will tend to be too large for regions of diverg- ing wave ravs and too small for regions of converging wave rays. A more complex analysis involving refraction analysis and a solution of the radiation stress equations is expected to provide essentially the same answer as the procedures given in Section II where bottom contours are nearly parallel to the shore and the waves approach along a line nearly normal to the shore. When the waves undergo significant refraction over parallel bottom contours, the more detailed calculations are expected to yield lower values. Additional development is needed to provide satis- factory procedures for computing wave setup in regions with complex bathymetry. This report provides the designer with a simplified method of estimat- ing wave setup on a sloping beach. Section 3.85 of the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1975) provides a method for estimating wave setup assuming d, = 1.28 Hp}. This assumption best applies to relatively flat beaches (m < 0.01) with breaker steepness (H,/gT*) values less than 0.01. A method for relating d, to H, for sloping beaches is given in elo SM (See, A.OA)o IY spilt these relationships to the method for estimating wave setup, a family of curves is developed that defines the net wave setup for the breaker height, Hp, and the period, T, for any breaker steepness or beach slope. The computation of wave setup can be an important part of a thorough design effort requiring water level estimation. For major engineering structures such as nuclear powerplants, it is quite important to consider ‘all possible causes of water level rise. Wave runup computations alone will usually be sufficient, but in cases similar to that shown in Figure 1(0), where large waves break offshore, an initial adjustment to the SWL 3g will be necessary to consider the waye setup caused by these breaking waves. In studies of coastal flooding by hurricanes, the engineer could choose to include the effects of wave setup in the water level estimate. This report could be used to compute that setup for both cases (Fig. 1) where runup values are not desired. Additional methods for estimating wave setup are given in James (1974) and Goda (1975). Application of these methods is not discussed in this report. II. EQUATIONS All equations in this memorandum have been previously presented in the SPM. Equation 3-48 of the SPM shows that the net wave setup on a shoreline is the algebraic sum of the wave setup and wave setdown, or 8, 20S > Si, (1) where S,, is the net setup, AS is the wave setup, and Sp is the wave setdown; Sp is defined as a negative value. Equation 3-46 of the SPM defines the setdown, Spe as gee (EB a Sh Lieto pe (2) 64m dy 3/2 where g = gravitational acceleration, Ht = equivalent unrefracted deepwater wave height, T = wave period, dp, = depth of water at breaking wave. Note that H! lo = HoKp, and where Kp SPulia He =H ‘O° Equations 2-91, 2-92, and 2-93 of the SPM define d, in terms of the breaker height, Hp, period, T, and beach slope, nm. H. Ores esta (3) where a and b are approximately: as 48.75 @ 5 oF (4) MS ares (5) Values of dp from equation (3) can then be used in equation (2) when defining the wave setdown. Equation (2) uses the equivalent unrefracted deepwater wave height, Ht, vather than the breaker height, Hp. Figure 2 gives values of Hp/H4A in terms of m and Egil Longuet-Higgins and Stewart (1963) have shown from an analysis of Saville's (1961) data that, AS = 0.15 dp (approximately) . (6) Combining equations (1) to (6) gives ge (aye Sy = 0.15 dp = ()) 64m dp 3/2 where H dye (8) BERET Ty AG67S Gl = ert) Bp 1 + e719-5m gt? Figure 3 plots equation (7) in terms of S,,/H, versus H,/gT? for slopes of m = 0.02, 0.033, 0.05, and 0.10, and is limited to values of 0.0006 < H,/gT? < 0.027. : - Wave setup is a phenomenon involving the action of a train of many waves over a sufficient period of time to establish an equilibrium water level condition. The exact amount of time for equilibrium to be estab- lished is unknown but a duration of 1 hour is considered as an appropriate minimum value. The very high waves in the spectrum are too infrequent to make a significant contribution in establishing wave setup. For this reason, the significant wave height, H,, represents the condition most suitable for design purposes. The designer is cautioned not to confuse the wave setup with wave runup. If an estimate of the highest point reached by water on the shore (SL61 “19}U9D yoTeosay SuTLeouTsuUq TeIseoD ‘Ste9uTSug Fo sdzoy ‘Away *S°p) le fA *‘ssoudesejs oAemM Ioqemdoep snsi0A OH / Fy ‘Xopur JYUSTOY Teyeoerig °*7 oan3Ty 216 oH ¢€00 200 100 8000 900°0 vO000 £000 2000 100:0 80000 9000°0 rn eee tee oy eco see Ele ee eles ! piel eto) 2 | 3 east ak clade sl esaleapcletiee ee | = | Ly | | ; es soesss aa =| sy qt fest | =4 4270 age et 8 ee ! (ee oH : ie Peale (ote ! 4 i | ' es He ot06 ! ie cdl cet he tie fem vO IY . ' i ty ' . 1 a | ae : a 9) 2.00 cic : 90 ! 6 } te heal ; Nes ee Pease wee an ee oem (:0 pias Fc ee ac Ae mi 12 *713/%q snsaon 4/5 100 8000 9000 eat #333 SERES SS CES SESSSR= ST y Fry ae 5252581) 65 \aSs Ses Ss SSSSe es! vOOO £000 2000 je Jor) cre este 6 "¢ omn3sTy 1000 8000':0 9000°0 v0000 900 800 af 220 pe'O 13 is desired, the runup produced by a larger design wave can he estimated after considering the water level produced by wave setup (using H,) and other effects (e.g., astronomical tide, wind setup). The selection of a design wave for runup considerations is left to the designer based upon the requirements of the project. Y The setup estimates using the methods described in this report are based upon the assumption that the waves approach normal to the coast. A wave that approaches the coast at an angle has components normal and parallel to the coast. The normal component produces wave setup, the parallel component produces a longshore current. It is reasonable to assume that the setup is a function of the cosine of the angle between the wave crest at breaking and the shore. Reducing the estimated wave setup in this manner is left to the judgment of the designer. III. SAMPLE DESIGN PROBLEMS The following examples show the use of the techniques presented in the solution of typical design situations. Refer to the SPM for other information related to the total design problem (e.g., wave theory, refraction analysis, tides, storm surges, wave breaking). * ke kK kK K kK kK Kk kK K K KF * * EXAMPLE PROBLEM 1] * * * * * * * * * ® & K kK * GIVEN: A wave gage is located in 22 feet of water at MLW (see Fig. 4). Analysis of the gage record for a period during a storm yields a sig- nificant wave height, H, = 20 feet and period, T, = 12 seconds. Assume the direction of wave approach is normal to a straight coast with parallel contours (i.e., refraction coefficient = 1.0). FIND: The maximum water level at the beach where runup calculations can be made considering an initial SWL at MLW. SOLUTION: From the given conditions in Figure 4, the significant wave will break offshore of the shelf and induce a setup. First, define the unrefracted deepwater wave height, H’, and the breaker height, Hp. Using the methods given in SPM (App. C, Table C-1), the following wave height values were obtained for = SCE ake 0.02984 o 58.1202) H ae a Ao) bis ny) S 77 #OCE by referring to Figure 2 with m = 0.05, and 14 Figure 4. Definition sketch for example problem 1. Q —— = 0.005830 ee H D —= 1.31 \ ach At this point, the problem can be completed by either an algebraic solution of equations (7) and (8) or by using Figure 3 with ni 23.27 feet 0.005019 then . S = 0.100 Hp or Si) = 2.58 feet Si) = 2.6 feet Therefore, the new water level at the beach will be +2.6 feet MLW, which will result in a depth of 3.6 feet at the toe of the beach Slope. The computation of the maximum runup height on the beach would involve the determination of the maximum breaking wave and runup for a range of wave periods. The highest runup elevation computed would be used for design purposes. zk Kk kK kK kK K kK kK kK K * * * EXAMPLE PROBLEM 2 * * * * ®* *®* * * * & & & KK GIVEN: A mathematical model simulation indicates that a particular section of coastline will experience a storm surge of +15 feet for a particular hurricane. The backshore area is protected by a contin- uous line of sand dunes whose lowest elevation is at about +20 feet. An estimate of the deepwater significant wave height and period yields Ho = 30 feet and T, = 12 seconds. The beach slope is a constant m= O05, FIND: Whether continuous flooding of the backshore can be expected when wave setup is considered. SOLUTION: In this case, assume that nL, = Hie. Then, Hp, can be found from Figure 2 with — = 0.00647 gT and m= 0.05 8 H thus, _ = il, lo 1s or Hp = 34.80 feet . From Figure 3, with Hp = 34.80 feet H —_ _ 9.007505 gT? and m= 0,05 8 Sy thus, — = 0.124 Hp or Sy) = 4.3 feet Therefore, the MWL will be at elevation +19.3 feet which is 0.7 feet below the top of the dunes. Extensive flooding should be expected. If desired, Section 7.22 of the SPM could be used to estimate the quantity of flow over the dune. LITERATURE CITED GODA,Y., “Irregular Wave Deformation in the Surf Zone," Coastal Engineering Cp, Herxeip, Oils Ass L754 yas LSA, JAMES, I.D., "Non-Linear Waves in the Nearshore Region: Shoaling and Setup,'' Estuartne and Coastal Marine Setence, 1974, pp. 207-234. LONGUET-HIGGINS, M.S., and STEWART, R.W., "A Note on Wave Setup," Journal of Marine Research, Vol. 21, No. 1, 1963, pp. 4-10. SAVILLE, T., Jr., "Experimental Determination of Wave Setup,'' Proceedings, Second Techntcal Conference on Hurricanes, National Hurricane Research Project, Report No. 50, 1961, pp. 242-252. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protection Manual, 2d ed., Vols. 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BILecn* £0¢2OL “C-/L VIA0 “pte TeoTuydeq ButiseutTsue TeqseoD “Zejue) yoreesey BuTTsauTsugq TeIseoD *S*n :SeTtes “II “eTITL “I ‘adoTs yoeeg *y ‘saaey *¢ ‘dnqes aaey *7 *SutieauT3ua TeIseoOD “| *paptaoid e1e swetTqor1d oeTduexe om} pue pessnostp ere suotjeottdde Teo -T70P1g “peuTyep e1e sUOTITpUOD BAEA JUepTOUT usYyM sure TeoTYydezs Aq PpeuTutejzep ATTSea 9q ued (1 0°0=") OL UO {| se deeaqs se sayoeaq uo dnjes aaem Jo uot eAnduoyj “TeNueW uoTIJIeI0Tg e10YS ey, FO CGg*E pUe 7g°zZ suoT}09§ UT pajueserd ATsnotAead TeTrejzeW ay sautTquod jaiode1 sTuyL "gt ‘d :Aydez3o0TTqQTq (S-LL VLIO ! 19}ueD yoreesey ButTsseuT3uq TeISeOD “S'N - pre TeoTUYyDe, ZJutresuTZue Teqseop) “TIT : *d gy "(16 ‘99TATeg UOTJeEWIOJUT TeOTUYSe] TeuoTIeN worz STqeTTeAe : "ea ‘pTetgsutidg § requep yoIeasey But~iseutSugq Teqseog *s*n : “eA ‘2}oATeg J1Oq — *yTusey *y uyor Aq / Yoreq Butdots e uo dnjzes ane *y uyor ‘yTuseT SUA YO ea1ecn* £0ZOL *G-/Z VIGO “pre TeoOTUYyIe, BuTiseuTSue Teq,seoD *laqueD Yyoieesey Buy~teeut~Zuq TeyseoD *s*n :satzjes “II ‘“eTAFL “I ‘adoTs yoreg “hp ‘seAemM *¢ ‘dnjes aaey *Z “SuTisauT3ua Teqseop *| *pepfAoid aie swetqoid afduwexe omq pue pessnostp aae suotjeottdde Teo -TJ0R8Ig ‘“peutyep e1e SUOTITpUoD aARN JUepTOUT uayA sueow TeoTYyders Aq pouTureajep ATTsee aq ued (1Q°O=W) O, Wo | se daaqs se sayoreq uo dnjes aaen Jo uoT_eAIndwoy “enue uoTOe701g aIOYS ay Jo cg*E puke 79°Z suoT}09S UT pejuesead ATsnotAeid TeTrajeW |ayR SauTquod jaoder sTYyL "eg, *d :AydergotTqtTg (S-LL VLUO £ Tequeg yoreesoy BZutTrseuTsuq TeIseoD “S'n = pre TeoTuYyoe, VufrveuT8ue Teqseop) “TTT : ‘d QI “LL6| ‘20TAIaS UOTIeWIOFJUT TeoTUYoa], TeuoTIeN woasz aTqeTTeae : ‘ea SppTetysutaidg { raquep yorteasey ButiseuTSugq Teqseoj) *S*m : “eA ‘ITOATeg JA0q — "yTuseT *y uyor Aq / yoeeq ButdotTs e uo dnjes aaey “yY uyor ‘yTUseT halt , i" iV ah uy 4 dy | ne ; SY . - oh Wahi “eae ay ; ‘ Bet" > fh vay Ae