beach profile equilibrium and patterns of wave decay

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Journal of Coastal Research 21 3 522-534 West Palm Beach, Florida May 2005

Beach Profile Equilibrium and Patterns of Wave Decay andEnergy Dissipation across the Surf Zone Elucidated in aLarge-Scale Laboratory ExperimentPing Wang' and Nicholas C. Kraus*'Department of GleologyUniversity of South Florida4202 E. Fowler Ave.Tampa, FL 33620, U.S.A.pwang@chuma 1 .cas. usf.edu

*U.S, Army EngineerResearch and DevelopmentCenterCoastal and HydraulicsLaboratory3909 Halls Ferry RoadVicksburg, MS 39180, U.S.A.ABSTRACT!WANG, P. and KRAUS, N.C., 2005. Beach profile equilibrium and patterns of wave decay and energy dissipationacross the surf zone elucidated in a large-scale laboratory experiment. Journal of Coastal Research, 21(3), 522-534West Palm Beach (Florida), ISSN 0749-0208,The widely accepted assumption that the equilibrium beach profile in the surf zone corresponds with uniform wave-energy dissipation per unit volume is directly examined in six cases from tbe large-scale SUPERTANK laboratoryexperiment. Under irregular waves, the pattern of wave-energy dissipation across a large portion of the aurf zonebecame relatively uniform as the beach profile evolved toward equilibrium. Rates of wave-energy dissipation acrossa near-equilibrium profile calculated from wave decay in the surf zone support the prediction derived by DKAN(1977)Substantially different equilibrium beach-profile shapes and wave-energy dissipation rates and patterns were gen-erated for regular waves as compared to irregular waves of similar statistical significant wave height and spectralpeak period. Large deviation of wave-energy dissipation from the equilibrium rate occurred at areas on the beacbprofile witb active net cross-shore sedim ent tran spo rt and subs tantial se dimentation and erosion. The rate of wave-energy dissipation was greater at the main breaker line and in tbe swash zone, as compared to middle of tbe surfzone. Based on analys is of the SUPERTANK data, a simple equation is developed for predicting the height of irregularwaves in the surf zone on an equilibrium profile. The decay in wave height is proportional to the water depth to theone-half power, as opposed to values of unity or greater derived previously for regular waves.ADDITIONAL INDEX WORDS: Beach profile, equilibrium, cross-shore sediment transport, wave breaking, coamorphology, SUPERTANK, physical modeling.

INTRODUCTIONThe concept of an equilibrium shape of the heach profilehas proved fruitful in a variety of applications in coastal sci-ence and engineering, such as beach n ourishmen t design andstudies concerned with morphological evolution in the near-shore. Under this concept, if the incident waves and waterlevel remain constant, the beach profile is expected to evolvetoward a stable shape for which the net cross-shore sedimenttransport rate across the profile approaches zero. Assump-

tions underlying the simplest such situation involving ana-lytical solutions are alongshore uniformity, uniform or nearlyuniform sized sediment, absence of obstructions such as hardbottom on the beach profile, and dominance of wave actionas the driving process shaping the profile. These assum ptionscan be relaxed in numerical solutions, such as that of LARSONand KRAUS (2000) for describing profile evolution in the pres-ence of hard bottom. In the study described here, we inves-tigate the equilibrium profile shape in response to breakingwaves, i.e., the profile shape in the surf zone on sandy heach-

es . INMAN et al. (1993) discussed equilibrium profile shapesinside and outside the surf zone, with the offshore bar actingas a hinge point.In a pioneering study of the beach profile, BRUUN (1954)noted th at the shape of the n earshore heach profile along thecoast of Denmark followed a simple power law of depth withdistance seaward from the shoreline. In a macro-scale ap-proach depending on general characteristics of the incidentwaves an d representative heach material grain size. DEAN(1977) derived classes of beach profile shapes that depend onchoice of the governing mechanism assumed to control thetransport processes. The choice that best corresponded tofield data for Atlantic Ocean and Gulf of Mexico beachesyielded a shape similar to that found by BRUUN (1954) in aderivation for which the wave-energy-fiux dissipation perunit water volume is uniform. The result of DEAN (1977) ha s!ed to a standard equilibrium beach profile shape given hy:

h = (1)DOI:10.2112103-003.1 received 8 April 2003; accepted in revision 29January 2004.

where h is the still-water depth, x is the horizontal distancefrom shoreline ih - 0), and A is a dimensional scale param-eter having units of m"-' and a value determined mainly by


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Beach Profile Equilibrium and Wave-Energy Dissipation 623

sediment grain size (DEAN, 1977; MOOR E, 1982; DEAN, 1991;MOUTZOURIS, 199 1).Many studies have shown that Equation (1) or a similarform represents the t ime- and locally-averaged shape of thebeach profile with ample sediment supply on a coast exposedto waves and uninfluenced hy s t ructures and hard bottom{e.g., MOOR E, 1982; LARSON, 1991; WORK and DEAN, 1991; PRU-SZAK, 1993; LARSON and KRAUS, 1994a; WANG and DAVIS,1998, 1999; LARSON et al., 1999; GONZALEZ et a/., 1999). LAR-SON and KRAUS (1989) derived an analytical form for thebeach profile shape at equilihrium under the realistic wavedecay model of DAI.LY et al . (1985) tha t has a planar shapeat the foreshore joining to the form of Equation (1) further inthe surf zone. LARSON ei al . (1999) d erived the power-functionheach profile (Eq. 1) from a non-Iocal balance of t ime-aver-aged offshore and onshore sediment transport. These andother studies give confidence that Equation (1) hoth descrihesnature and has a consistent theoretical b asis. However, to ourknowledge, no equilibrium mechanism has been directly test-ed by comparisons to laboratory or f ield measurements ofwave dissipation and gi-ain size. Hereafte r, this pa per consid-ers wave-energy dissipation per unit volume in the surf zonean d its relation to the equilihrium profile.

In the D E A N (1977) derivation of Equation (1) hased on uni-form dissipation of wave-energy flux per unit volume acrossthe surf-zone profile, the spilling breaking wave assumptionwas invoked, meaning the height H of broken waves in thesurf zone is proportional to the depth h. The equilihriumwave-energy dissipation D. across the surf zone is then(DEAN, 1977):

(2)where p,,, is the density of water, g is the acceleration of grav-ity, and -y is the breaker index for depth-limited waves, de-fined as wave height over water depth iHlh) for breaking orbroken waves. KAMINSKY and KRAUS (1993) found that thecommonly used value for incipient breaking of depth-limitedindividual breaking waves, / = 0.78, is supported by a largelaboratory database for regular (monochromatic) w aves. Theyalso found limits for -y m the more than 400 individual mea-su rements to lie within 0.60 and 1.59.

Fo r a t rain of waves with different heights (irregularwaves), at a given location in the surf zone, the 7 value forsignificant wave height //,. is also found to be close to 0.78.However, root-mean-square wave height //, is appropriatefor calculation of mean wave energy in an irregular wavefield. At a given location, this quantity has a 7 (rms) valuein the approximate range of 0.4 to 0.6 for irregular waves inthe surf zone {e.g., THORNTON and GuzA, 1982, 1983; W ANG etai , 2002a, 20a2b), hecause the ratio Hlh is formed by thecombination of breaking, broken, and non-breaking waves.Field measurements analyzed by THORNTON and GUZA (1982,19831 gave a value of 7,.^, in the range of 0.42 to 0.44.

Wave-energy-flux dissipation per unit water volume acrossshore D{x) can be calculated from measurements of waveheight and water depth at ac^jacent, closely spaced wavegauges as:

dx(3)

where E is the wave energy per unit area, C^ = ^^gh is thewave group velocity in shallow water, andft,,,,,, s the waterdepth at the mid-point between the wave gauges used hereto approximate the energy dissipation per unit volume.The concept of heach-profile equilibrium and the corre-sponding uniform wave-energy dissipation underlie a numherof practical cross-shore sedim ent-transpo rt and beach-profileevolution models {e.g., KRIEBEL and DEAN, 1985; KRIEBEL,1986; LARSON and KiiAUS, 1989; ZHENG and DEAN, 1997). Indevelopment of these m odels, it is assumed that wave-energydissipation rates (Eq. 3) will deviate substantially from theequilibrium rate (Eq. 2) during high-wave energy storms, be-cause the equilibrium beach profile and associated wave-en-ergy dissipation would tend to represent the average condi-tion of all waves, as demonstrated with high-accuracy fielddata (LARSON and KRAUS, 1994a). Therefore, the net cross-shore sediment transport rate q and heach-profile evolutiondepend on the magnitude of deviation from an equilibriumstate {e.g., KRIEBEL and DEAN, 1985; KRIEBEL, 1986; K R A U San d LARSON, 1988; LARSON and KR AUS, 1989):

q - K(D -D.r (4)where /L and n are empirical coefficients. The original studiesemploying this form took n = 1, but larger values such as n 1.5 have been advocated to increase the time rate of changeat longer elapsed time (ZHENG and DEAN, 1997).

Because Equ ation (2) quantifies th e equilibrium ra te of dis-sipation per unit water volume in both theoretical and ap-pl ied t reatments of beach-profile change, it is of central in-teres t to assess its validity under realistic conditions. Thevalidation should answer two questions: 1) Is the wave-en-ergy dissipation rate uniform across the equilibrium surf-zone profile as assumed, and 2) If (1) is t rue , how well doesEquation (2) predict the m agnitude of the equilibrium r ate ofwave-energy dissipation per unit volume?The SUPERTANK large-scale laboratory data collection

project (KRAUS et al., 1992; KRAUS and SMITH, 1994; SMITHan d KRAUS, 1995) documented beach-profile response to ac-cretionary waves, erosional waves, and different boundaryconditions, together with high-density measurements ofwaves and currents in the surf zone. The SUPERTANK datase t has enabled examination of a wide range of cross-shoreprocesses from, for example, beach profile response to a sea-wall (McDoUGAL et ai, 1996) through detailed investigationsof net cross-shore sand transport (LARSON and KRAUS, 1994b)and surf-zone hydrodynamics (KRIEBEL and SMITH, 1994;SMITH, 1994).

For the present study, profile survey data and wave mea-surements across the surf zone made at SUPERTANK areanalyzed to tes t the assumptions that at or near equilibriumof the heach profile, wave-energy-flux dissipation per uni twater volume is uniform across the surf zone, and with a

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524 Wang and Kraus

Table 1. SUPERTANK data used in this study.

Run IDS T. 10-1-20ST.lO-1-270ST_30-l-9ST.30-1-279S T . 10-2-5ST..10-2-170ST_30-2-9ST_30-2-209ST.GO-1-9ST.GO-1-210ST_10-1-20ST_10-1-559

Durationif waveactionimin)2 0

2 7 09

2 7 95

1709

2099

2102 0

5 5 9

//, b eforebreakingim )0.80.80.40.40.80.80.40.40.80.80.50.5

T, .(s)3.03. 09. 09. 04 .54 .58. 08. 03. 03. 08. 08. 0

-y..202 02 02 03. 33. 33. 33. 3

M o nM o nMonMon

Commentsinitial profileend profileinitial profileend profileinitial profileend profileinitial profileen d profileinitial profileen d profileinitial profileen d profile

value close to tha t given by E quation (2). Six cases from theSUPERTANK dataset were selected that had constant wavesrun for at least 3 hours. The cases cover considerably differ-ent wave conditions. The objectives of this study are to ex-amine (1) patterns of surf-zone-profile evolution under vari-ous wave conditions; (2) patterns of wave decay and wave-energy dissipation associated with the profile change and in-cident waves; (3) hyp othesis th at uniform wave-energy-fluxdissipation per unit volume occurs over an equilihrium pro-file; and (4) the accuracy of Equation (2) if uniform wave-energy dissipation is achieved on an near-equilibrium beachprofile. In addition, a new equation is derived that describesdecay in height of irregular waves on an equilihrium profile.

SUPERTANK EXPERIMENTSSUPERTANK was designed to collect data without scalingdistortions to improve understanding and predictive capabil-ities on cross-sbore sediment transport processes (KRAUS etai , 1992). The ex periments were conducted in the large wavechannel at the O. H. Hinsdale Wave Research Laboratory atOregon State University. The channel is 104 m long, 3.7 mwide, and 4.6 m deep, into which a 76-m long sand beach wasemplaced for th e S UPERTANK exp eriments. The fiap-typewave generator was equipped with a sensor and feedbackmechanism to absorb reflected waves at th e peak energy. Thebeach was composed of approximately 600 m^ of very-wellsorted fine sand with a median diameter of 0.22 mm and anaverage fall speed of 3.3 cm/s. Based on equations developed

by MooiUi: (19821 and DEAN (1991), tbe A-value in Equation(1) corresponding to this grain size is 0.10 m'-'. A larger A-value of 0.17 m"^ was specified in the initial beach construc-tion to create a steeper profile to ensure adequate waterdepth in the offshore area (KRAUKand SMITH, 1994). In designof the SUPERTANK tests, erosion or accretion of the profilewas predicted prior to running the waves by relations givenhy KRAUS ef a/. (1991) to arrang e ins trum ents, video cam eras,and satisfy requirements for participating investigators.Six cases from the SUPERTANK data set were judged suit-able (with sufficient duration) for this study (Table ll. In thefollowing discussion and figures, the run ID, e.g., ST^lO- l -20 , indicates the following information: I) the first five char-

acters, ST.IO, denote the SUPERTANK test case, (which in-cludes a number of wave-run segments typically with 10 to70 min of wave action each); 2) the next two ch aracters, 1indicate a selected block in the ST_10 experiment, which iscomposed of several bursts or time intervals with the sameinput waves, and 3) the characters following the last hyphenindicate the number of minutes of the particular wave action(could be the sum of several wave-run seg men ts, particularlyfor the end profile). Still-water level in the channel was con-stant and the same for these tests.

Narrow-peaked (swell type) irregular waves with x,, (spectral width parameter) ^ 20 (Bouws et al., 1985) were generated during the first two cases, ST_1O-1 and ST..30-1Steep erosional waves with a significant wave height //, of0.8 m and spectra l peak wave period 7* of 3 s, were generatein ST_1O-1. The initial heach profile was constructed basedon Equation (1) with an A-value of 0.17 m"'. The end profileresulted from a total of 270 min of wave action. Mild-steepness accretionary waves with a significant height of 0.4 mand peak period of 9 s were generated in S T_30 -l. The initiaprofile was the final profile of a previous 130-min wave run(//, = 0.5 m, 7;, - 8 s, 7,,,, - 3.3). The end profile of ST _301 w as shaped hy a total of 279 min of wave action.

Broad-peaked (sea type) irregu lar waves with 7 ,, - 3.3were generated during the cases, ST_10-2 and ST_3O-2Steep erosional waves with significant height of 0.8 m andpeak period of 4.5 s were g enerated during ST.-10-2. The initial profile was the fin al profile of a previous 280-min narrow -peak wave run (//. = 0.8 m, T^ = 4.5 s, y,,,. = 20). The endprofile of ST_10-2 was shaped by a total of 170 min of waveaction. Gentle accretive waves, with significant height of 0.4m and peak period of 8 s were generated in ST_30-2. Theinitial profile was the final profile of a previous wave run ofvarying wave heights and periods. The end profile wasshaped hy a total of 209 min of wave action.

Regular (monochromatic! waves were generated during thelast two cases of SUPERTANK, S T..G0-1 and ST_IO-l. Steeperosional waves with height of 0.8 m and period of 3 s weregenerated in ST-GO-1. The significant w ave height and peakwave period are similar to those of ST_1O-1, except the waveswere monochromatic. Different energy levels are carried byregular and irregular waves witb identical significant waveheights. This is because for regular waves, significant waveheight equals root-mean-square (rms) wave height, which isdirectly proportional to wave energy, whereas for irregulawaves, significant wave height equals 1.4 times the rms waveheight for Ray]eigh-distributed waves. The initial profile wasthe final profile of a previous wave run of varying waveheights and periods. The end profile of ST_GO-1 resultedfrom a total of 210 min of wave action. Mild-steepness accretionary waves with height of 0.5 m and period of 8 s weregenerated in ST_IO-1. The significant wave height and peakperiod are similar to those of ST_30_2, except that the waveswere monochromatic. The initial profile was the final profileof a previous wave run of varying wave heights and periodsThe end profile of ST_IO-1 was shaped by 559 min total waveaction.

The beacb profile was surveyed across tbe middle of the3.66-m wide channel at the end of each wave-run segment

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typically 10 to70 min long, A Geodimeter T140 survey sta-tion was positioned outside of the tank to track by an infraredbeam tbe position of the prism mounted on the top of the rod.Reproducibility of tbe elevation measurement was typicallywithin 0.5 cm except at the shoreward face of steep break-point bars, where the variation in reproducibility decreasedto a nominal 2 cm (KRAUS ef al., 1992).Waves were measured at 26 cross-shore locations by resis-tance gauges, spanning from tbe sw asb zone to approximate-ly 70 m seaw ard of tbe shoreline. Tbe wave gauges w ere sam-pled at 16 Hz. Both spectral analysis and time-series zero-upcrossing analysis were applied to calculate wave heightand period (K RALIS and SMITH, 1994). A low-pass filter wasapplied to separate incident-band wave motion from low-fre-quency motions. A bigh-pass time series was tbus obtainedby subtracting the low-pass time series from the preprocesseddata (KRAUS and SMITH, 1994, pp.62-66). Tbe period cutofffor tbe filter was set to twice tbe peak period of tbe incidentwaves. Tbis data filtering bas considerable infiuence on tbecalculation of wave heigbt and period in tbe swash zone,where long-period motion was significant and sometime dom-inant. Tbe wave data analyzed in this paper were processedwith the above data filtering and, therefore, the wave heightreferenced in tbe following analysis does not include contri-butions from low-frequency motions. The root-mean-squarewave height and mean wave period calculated from tbe zero-upcrossing analysis were used tocalculate wave-energy dis-sipation.

The rate of wave-energy-flux dissipation was calculatedfrom tbe measurements at two adjacent wave gauges basedon Equation (3) and is represented (plotted in the followingfigures) at the mid-point between them. Typically, eight wavegauges were located in the swash zone, spaced at 0.9-m in-terval. Two wave gauges were located seaward of tbe swashgauges, spaced at 1.8 m. Sixteen gauges were located tbrougband beyond tbe surf zone, spaced at 3.7-m interval. Given tbeclose spacing between tbewave gauges. Equation (3) isex-pected toprovide reliable estimates of wave-energy dissipa-tion rate per u nit w ater volume for tbe S U PERTANK waveconditions, except possibly in areas of extreme gradients inwave beight, e.g., in tbe vicinity of plunging b reakers.

R E S U L T SIn tbe six cases examined and nearly all tbe SU PERTANKruns, a breakpoint bar, or bar in tbe area of incipient depth-limited wave breaking, developed on the initial monotonicpower-function beach profile as described by Equation (1).Little change occurred in the offshore region outside tbe surfzone in tbe cases examined here. All profiles examined in tbisstudy are displayed in Figure 1. Tbe surf-zone portions of tbeprofiles are not directly comparable among different tests,particularly in tbe vicinity of shoreline wbere tbe beacbmight have been reconfigured for different objectives. How-ever, the offshore area was not reconstructed. The followingdiscussion pertains tbe beacb profile at and landward of thebreak-point bar (the surf zone).

Profit* Summary

ST_ 3 O- l -2 7 9 : n ( JST_ 3 -2 -2 0 9 :o n OST I0-1-55S end

Figure 1. All beach profiles examined in this study; note th at the profilechanged little seaward of the har.

Beach-Profile EvolutionTest ST_1O-1 with steep, narrow-band erosional waves wastbe first run of S U PERTANK . The initial profile was tbemonotonic power-function profile constructed witb A = 0.17m"^. Considerable sboreline recession of approximately 2.5 mand development of a 30-cm high break-point bar occurred(Figure 2A). The erosion in the vicinity of the shorelineroughly equaled tbe accumulation in the bar area, wbilebeacb elevation remained largely uncbanged over a consid-erable area from 15 to 20 m between tbe swasb zone and themain breaker line over the bar (Figure 2A). This morphologicchange indicates tbat the sand eroded from tb e shoreline was

transported offshore across an area in the mid-surf zone anddeposited n ear the main bre aker line to form tbe break-pointbar. Moderate erosion in the trough and slight accumulationdirectly landward of it also occurred.The initial surf-zone profile represents the power-functionprofile witb A = 0.17 m'-' well, as constructed. The construct-ed profile was slightly gentler in tbe inner surf zone andsteeper in tbe outer surf zone, as compared to A = 0.17 m"^profile. After 270 min of wave action, the end surf-zone pro-file bad a mucb gentler overall slope tban tbeinitial slopebecause of erosion in the vicinity of the original sho reline anddevelopment of the break-point bar. An equilibrium profileshape with A - 0.17 m" ' over-predicts the water depth across

most of the surf zone, except at tbe bar trough. The value A- 0.10 m ' \ which corresponds to the SU PEK TANK grainsize of 0.22 mm according to MoORE (1982) and DEAN (1991),predicts tbe profile sbape reasonably well across most of tbesurf zone, except at tbe trough, where water depth isunder-predicted considerably.For the remaind er of the three irreg ular w ave cases (Figure2B and Figure 3), tbe magnitude of tbe beacb-profile changewas mucb smaller as compared to the initial changes (Figure2A), especially during tbe accretionary wave cases witb lowerwaves (Figure 2B and Figure 3B|. The bar became slightlywider and flatter, as compared to the initial profile after tbeaccretionary wave actions. Considerable berm erosion above

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526 Wang and Kraus

ST 10-1

15 20Distanc* (m)

- ST _10-1-Z0: Initial proflla-A=0.17en


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S T GO-1

0*ST GO-1-9

5 10init ial profile

15 20D ls t a n c B ( m )

ST_G0-1-Z10:BtidprolH(i25 30

- A = 0 . 1 0

-ST_IO-1-2O: miUal praHe - ST_IO-1-559: en d proIHe

Figure 4. Initial and end heach profiles for ST_GO-1 and ST JO -l , andthe power-function profiles.

contrast to the smooth profile under irregular waves. Thebeach profile in the surf zone remained reasonably similarunder irregular waves with different spectral charac teristics,as well as statistical wave height and wave period. Constancyof profile shape supports present understanding that the A-value defining the equilibrium heach-profile shape is not sig-nificantly infiuenced by wave height, period, and spectralcharacteristics, leaving sediment grain size as the dominantfactor controlling the beach-profile shape, as found in nu-merous previous studies using field data ie.g., DEAN, 1977;MOORE, 1982; MouTZOimis. 1991).Patterns of Wave Decay and Breaker Index in the SurfZone

In thiB study, the main breaker line for both regular andirregular wave cases was defined to he at the location land-ward of which a significantly steeper rate of wave-height de-cay was measured. This criterion was based on comprehen-sion that substantial wave-energy loss or wave-height de-crease should follow major wave breaking. The rate of wavedecay at the breaker line may also provide information onthe intensity of wave breaking. For example, WANG et al.(2002a) measured a much greater rate of wave-height de-crease following plunging breaking than that following spill-ing breaking in a mid-scale laboratory facility.

Different patterns of wave decay were measured in ST.IO-1 at the beginning and the end of the wave run (Figure 5A).At the breaker-point h ar, a steeper decay in wave height oc-

- sn d " * - - B I mmal - - - A -

- e i In i tia l - . a . . B l end

Figure 5. Cross-shore distributions of wave-height decay and hreake rindex over the initial and end beach profile for ST_1O-1 and ST_3O-1.

curred at th e end of the wave run than at the beginning. Thisevolution in pattern of wave decay follows the evolution ofthe breakpoint bar and change in its seaward slope. SMITHand KRAUS (1991) showed that waves breaking on bars havedifferent characteristics than those breaking on uniformlysloping beaches. In the vicinity of the shoreline, a gentlerdecay was measured at the end, probably influenced by theconcave beach-profile shape there. Wave conditions seawardof the surf zone remained nearly identical during the 270 minwave action.The breaker index (BI) 7^,,,^ H^^Jh for irregular waves inthe surf zone exhibited an increasing trend in value with dis-tance from the breakpoint land ward , from 0.37 at the break er

line to 1.05 in the swash zone at the beginning of the waverun (Figure 5A). At the end, a greater breaker-index value of0.60 was measured at the breaker line, followed by a decreaselandward to 0.45. Thereafter, a steady increasing trend toapproximately 0.88 in the swash zone was measured, mean-ing that virtually all waves were breaking or had broken inthat swash zone.The patterns of wave decay at the beginning and end of thewave action for the other three irregular wave cases werenearly identical (Figure 5B; Figures 6A and B), especially forthe two accretionary cases with relatively small wave heights(Figures 5B and 6B). Similar wave patterns are consistentwith the stable beach profiles. The slight broadening and flat-

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528 W ang and K raus

- inMalIS 20 25D l la n c a ( m )

- e n d ---- Bl Init ia l - BI end

Figure 6. Cross-ahore distrihutions of wave-height decay and breakerindex over the initial and end heach profile for ST.10-2 and ST_30-2.

tening of the bar did not result in any substantial change inwave pattern over the bar, Breaking of the infrequent largerwaves probahly caused the shgbt flattening over the bar,which did not induce significant change in the overall waveheight and wave-height decay over the bar,The b rea ke r index -y ,,,, increased from 0.20 over th e sea-ward slope of the bar to 0.75 in tbe swash zone (Figures 5Band 6B). The decrease of//., and H^,^ over the har was notas steep as that in tbe higher wave cases {Figures 5A and6A), indicating that wave breaking over the har was not assignificant. Steep wave decay, indicating major wave hreak-ing, occurred close to the shoreline.For the erosional wave case ST_10-2, the m ain breake r Hneas indicated hy a steep decay of wave height was located overthe bar crest (Figure 6A}. At the beginning of the wave run,tbe wave decay in the vicinity of the shoreline was slightlygentler than tbat at the end of the run. Tbe sligbt seawardmovement of tbe b ar du ring the 170 min wave action did notresult in any distinguishable cbanges in the patte rns of wavedecay (Figure 6A).

The breaker index 7^,,,, exhibited similar cross-sbore distri-bution pattern as tbat of ST_1O-1, with a peak at the har,followed by a moderate decrease, and then a landward in-creasing trend (Figure 6A). Over the bar, the hreaker index

t -;0,0 0-2 0,4 0.6 O.B 1.0 1.2 1-4

Figure 7. Cross-shore distribution of y,

equaled roughly 0.65, followed hy a decrease to 0.44 over ttrough, and then increased to 1,10 in tbe swash zone.Overall, the breaker index y,.,,,^ sbowed a landward increing trend for all tbe four irregular wave cases. An increasey^,,, value over the hreaker-point bar was produced by tcombination of an increased wave height due to shoaling antbe shallow water or the har. Figure 7 compiles all y^,,^ valufor th e four cases. The horizontal axis represen ts the dimesionless distance to tbe still-water shoreline, with .v being tcross-shore distance to shoreline, and x,, is the width of tsurf zone. This landward increasing trend of the hreaker idex across the surf zone is described reasonably well withpower function as:

0 . 4 1 6 1 -

(where the term with the power -0.36 represents the leassquare power-function fit of the measured 7,,,,, with the corelation coefficient R '^ equals 0,68. Tbe rigbt side of Equati(5) was obtained by substitution witb Equation (1), and threpresentative value of 7^,,,j, fits well in the range of 0.40.44 reported by THORNTON and GuzA (1982, 1983) for fiemeasurements.Combining Equations (5) and (1), a simple equation for prdicting the decay of H^,,^ across the surf zone of an equirium heach profile is obtained as

* * rmo !^VAh(

where 7^j, is tbe value of 7 at the point of incipient deplimited wave breaking. The function at the second and thilines of tbe equation is obtained hy substituting tbe powefunction beach profile (Eq. 1) into the first line. Care shoulbe exercised in determining 7^,,._;, over a harred profile anmight best be defined past the plunge area of incipient breaing waves. For comparison, THORNTON and GUZA (198obtained the r esult H^,,,, ~ /i* '" for irreg ular wave hrea kinon a plane sloping bottom a s /i ^ 0 in the surf zone.


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Beach Profile Equilibrium and Wave-Energy Dissipation 52 9

0.60

0.00

ST 10-1: Stable Profile

Measured Calculated5 10 15

Distance from shoreline (m)20

0.35 ST 30-1: Stable Profiie

5 10 15Distance from shoreline (m)

0,60 ST 10-2: Stable P rofile

5 10 15 20Distance from shoreline (m)

0.00Measured -Calculated

5 10 15 20Distance from ahoraiine (m) 25

Figure 8. Measured and predicted lEq. 6) wave heights across the surf zone, irregular w ave cases.

Measured //, values were compared with the predictedones from Equation (6). The surf-zone width x^ was measuredfrom the still-water shoreline to the breaker line where rel-atively steep wave-energy dissipation occurred. The hreakingdepth /),, is determined based on Eq uation (1) from the m ea-sured X,.. The depth h,, can also be measured from the beachprofile. However, measured h ,, is not directly used here be-cause it is sensitive to the relief of the bar. The best-fit y^^^j,value of 0.42 (Eq. 5) was used in the calculation. More dataare necessary to improve the reliability and accuracy of ageneral y,.^,sj,.

The //, was predicted reasonably well for the two ero-sional cases with higher waves (Figure 8: two left panels).The overall trend of wave decay was predicted well althoughthe w ave height was under-predicted a t m ost locations exceptin the swash zone. A 10% increase in "y^^j, value would im-prove the match with the measured data. The wave heightover the breaker-point b ar was substantially under-predicted,and a discrepancy is expected because Equation (6) pertainsto a monotonic beach profile.

For one of the two accretionary cases, ST_3O-1 with nar-row-peak wave and long period of 9 s, wave-height increaseswere measured at two locations, a short distance landwardof the bar and directly seaward of the swash zone. These arelikely due to wave reformation after breaking on the bar andshoaling before further breaking in the swash zone. Although

the complex variations of wave height were not predicted byEquation (6), the general trend was reproduced reasonablywell (Figure 8: upper right panel). Equation (6) is developedbased on the analysis of the monotonic equilibrium beach pro-file as given by Equation (1) and cannot reproduce compli-cated changes in wave height over a multiple bar profile.Wave-height decay was reproduced well for the other ran-dom wave accretionary case, ST_30-2, except for over-predic-tion at the b arred breaker-line (Figure 8: bottom right panel).Wave decay at the breaker line for the accretionary wave casewith lower wave height was m uch gentler as compared to theerosionai cases. The minor peak consisting of a reduced rateof wave decay followed by an increased rate in the mid-surf

zone was not predicted.The patterns of wave decay differed at the beginning andthe end of the erosionai regular wave case, ST_GO-1 (Figure9A). At the beginning, a steady decrease in wave height vnthapproach to shore was measured, whereas much steeperwave decay was measured over the bar after the 210 minwave action. The large wave-height decay from approximate-ly 1,1 m to 0.4 m over the bar apparently resulted from theuniform breaking of regular waves over a narrow zone, incontrast to wider breaker zones of irregular w aves. The waveheight rem ained almost constant across most of the surf zone,followed by a steep decrease at the shoreline. Some wave-

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530 Wang and Kraus

10 15 20Diatinct (m)Bl initial - -a -- Bl end

Figure 9. Cross-shore distributions of wave-height decay and breakerindex over the initial and end beach profile for ST.GO-1 and STJO-l. Thestraight line represents 7 ^^ = 0.62 (solid line) and 7 - 0.78 (dashed line).

height variations were measured seaward of the surf zone,probably resulting from formation of standing waves.At the beginning of the wave run, the breaker index re-mained relatively uniform across shore, typically rangingfrom 0.53 to 0.87. At the end of the wave action, the value ofthe breaker index was nearly 1.20 at the breaker line, fol-lowed by a sharp drop to 0.24 over the trough and then asteady increase to approximately 1.04 in the swash zone. Thestandard y 0.78 found in regular wave studies (KAMINSIO'an d KRAUS, 1993) yielded over-predictions across most of thesurf zone, except at the breaker line and in the swash zone(Figure 9A). Time-series wave analysis yielded an averagefir,Jii^,ii ratio of 0.80 in the surf zone for botb of the tworegular wave cases. This value is greater than the 0.71 ob-tained from the Rayleigh-distribution assumption. Outsidethe surf zone, the H,,,JH,,^ ratio averaged 0.96, which is closeto unity as expected for regular waves. A sharp decrease inth e H^,JH,^g ratio from nearly unity to approxim ately 0.8 oc-curred in the vicinity of the b reaker line, or the narrow zoneof initial wave break ing. Th e H^,JH^^^. ratio of 0.8 would yielda 7rm6 value of 0.62 given the standard 7^,^, = 0.78. The value^rniB ~ 0.62, also plotted in Figure 9A, yielded an improvedmatch with the m easurements, with an average y ^ , ^ value of0.66.

Substantially different wave-decay patterns, as comparedto the irregular wave cases, were also observed in the accre-

tionary regular wave case, STJO-l. The main breaker linas indicated by the steep wave-height decay moved onshorconsistent with the substantial onshore movement of the ba(Figure 9B). Somewhat regulated variations of the wavheight, perhaps due to formation of partial standing waveoccurred both inside and outside the surf zone. The breakeindex varied from 0.36 to over 1.41 across the surf zone. Thlandward increasing trend was not as steady as in the othcases. A breaker index greater than 1.0 was measured ovethe bar crest, i.e., at the main breaker line, both at the beginning and at the end of the wave run. The standard 7 0.78 matched the measured values (with an average of 0.74better than the y^^^ = 0.62. The reason for this is not clearSubstantially different patterns and trends of wave-heighdecay were produced during regular-wave experiments, acompared to the results with irregular waves of similar statistical offshore significant wave height and peak period. Thlandward increasing trend of -Y,.,,,, during the regular wavcases was not as clear as that during the irregular wave cases . Equation (5) developed for irregular waves, does not dscribe the regular wave situation. A much greater value oy ^ , ^ exceeding 1.0, was measured at the breaker line ovethe bar for the regular wave cases. As found from severprevious regular wave studies (DALLY et ai, 1985; SMITH anKRAUS, 1987; SMITH and KRAUS, 1991), the wave-height-decacurve tends to be concave (curved or rounded downwardThis is in co ntrast to the present irregular wave-decay curvwhich has the sense of convex upward (somewhat roundeupward). The difference indicates that results obtained fromsimpler (in terms of wave generation) regular wave studiemight not be directly applicable to irregular waves, similato tbe situation of beach-profile change as discussed abovWave irregularity should be considered in the modeling owave-height decay and beach-profile evolution in the surzone.

Patterns of Wave-Ener^ Dissipat ion Per Unit VolumeThe key assumption of the DEAN (1977) equilibrium theoris that wave-energy dissipation per unit volume is uniform the beach profile is in equilibrium. In the following, the asumption is examined with the SUPERTANK data.Significantly different patterns of wave-energy dissipatioper unit volume were measured at the beginning and end oST_1O-1 (Figure lOA). At the beginning of the run, a sustantial landward increasing trend was measured over th

initial monotonic power-function profile. The wave-energdissipation rate per unit volume increased from nearly zerjust seaward of the surf zone to over 400 Nm/mVs in thswash zone. After 270 min of wave action, the rate of waveenergy dissipation per unit volume became much more unform, rangin g from 54 to 155 Nm/m'Vs with a n av erage of 10Nm/mVs and a standard deviation of 43 Nm/mVs. The equlibrium energy dissipation rate calculated from Equation ((A = 0.1 m"-'; 7ni>s ^ 0.56) was 65 Nm/mV s, approxim ate39% lower than the average value of the measured dissiption rate. The value 0.56 for y ^ ^ was obtained by dividinthe standard 7 value of 0.78 (for significant wave height) bV2, on the assumption of a Rayleigh distribution in wav


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- e n d *i n l l i a l _ ( li s s i p a 6 o n - - - e n( l _d l ss i p3 t io n Calculalad

end *in iHal_d i8s ip8t i on * < - enddisa ipat inn ca l cu ta led

Figure 10, Wave-energy dissipation rat e per unit volume over the initialand end beach profiles, and the computed equilibrium dissipation rate forST-10-1 and ST.JO-1.

height. At the end of the wave action, a high rate of wave-energy dissipation was measured over the bar crest at themain hreaker line, followed hy a sharp drop over the troughand a gradual increase toward the swash zone.Although th e m easured -y ^ valu es were not uniform acrossthe surf zone (as discussed in the previous section) and onaverage were greater than 0.56, they were not used in theEquation (2) calculation, hecause the equation was developedunder th e assumption of a constant 7 under spilling breakers.Predictions from Equation (2) based on the simple equilibri-um profile theory matched the measured values within 50%for this case.Based on the overall heach-profile shape and the relatively

minor profile changes in a series of following runs, it is rea-sonable to believe that the end profile after the 270 min ofwave action should be close to the equilibrium shape. Largedeviations from both measured and predicted equilibrium en-ergy dissipation occurred near the shoreline and at the barcrest at the beginning of the ST_1O-1 wave run. These devi-ations corresponded to substantial heach-profile change and,therefore, net cross-shore sediment transport. The resultsqualitatively support the transport equations (e.^.. Eq. 4) inthe profile-change models of KRIEBEL an d DEAN (1985) andLARSON and KRAVS (1989).

For the two irregular accretionary wave cases, the observedwave-energy dissipation was nearly the same at the begin-

ning and end of wave action (Figures lOB and llB). Thisresult implies nearly identical wave-decay pa ttern and beach-profile shape, indicating th at th e beach was already in near-equilibrium state at the beginning of tbe run. Due to the rel-atively small wave height, wave breaking over the bar crestmay not be as significant as the high-wave cases, as also in-dicated by the small wave-energy dissipation rate at the harcrest. Major wave breaking occurred in a narrow zone nearthe shoreline, as indicated hy increased dissipation of waveenergy there. Overall, measured rates of wave-energy dissi-pation were moderately greater than the predictions fromEquation (2) near the shoreline. In contrast, across the restof the surf zone, the predicted rate was greater th an the mea-sured rates, which were only slightly above zero. Some waveregeneration was evident, particularly for ST_3O-1 (the nar-row-peaked swell type accretive wave case), as indicated byenergy production (negative value of wave-energy dissipa-tion). The 3.7-m spacing may also limit accuracy of calcula-tion of energy dissipation where grad ients in wave height ar ecomplicated or where waves are generated at tbe plungepoint. On average, the wave-energy dissipation rate for thetwo accretionary cases was 46 Nm/m'Vs for the narrow-peakswell-type w aves and 44 Nm/m-Vs for the broad-p eak sea-typewaves. The predicted value of 65 Nm/mVs (Eq. 2) is 44%greater than tbe average measured value, although the mea-sured v alues showed considerahle variation across shore witha slight increasing trend toward the shoreline.

For the broadband erosional wave case, the patterns ofwave-energy dissipation per unit volume were similar at theheginning and end of the w ave action, ranging from 53 to 173Nm/mVs (Figure llA). Little change in the beach profile oc-curred, indicating that it was probably near equilibrium atthe beginning of the run. A great rate of wave-energy dissi-pation per unit volume was measured over the bar crest atthe main breaker line, followed by a sharp decrease and aslight increase toward the swash zone. A sharp increase oftbe dissipation rate occurred in the swash zone, The wave-energy dissipation rate per unit volume across most of themid-surf zone was relatively uniform and matched well withthe value obtained from Equation (2). The two peaks in dis-sipation at the bar crest and in the swash zone were morethan twice the predicted value from Equation (2).

Different patter ns of wave-energy dissipation rate per u nitvolume were measured during the two regular wave cases,as compared to the irregular wave cases. Substantially dif-ferent patterns were also measured at the heginning and theend of the wave run, consistent witb tbe different beach-pro-file shapes and wave-height decay. At the heginning of tberegular erosive wave case, the wave-energy dissipation rateper unit volume ranged from - 4 3 to 402 Nm/m^/s with noapparent trend of changing, A negative wave-energy dissi-pation rate indicates an increase in wave height. This is ap-parent for the accretionary regular wave case. At tbe end ofthe 210 min wave action, a high dissipation rate of nearly624 Nm/m% was obtained over the bar crest at the mainbreaker line, followed by a sbarp decrease to 24 Nm/m'Vs justlandward of the trough (Figure 12AI. The wave-energy dis-sipation rate remained close to zero across most of the mid-surf zone, and then increased to nearly 292 Nm/mVs in the

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532 Wang and Kraua

DIslanca (tn)i)_ dissipat ion (

-Inltlal_0i5s{pation anddisslpatio nFigure 11. Wave-energy dissipation rate per unit volume over the initialand end beach profiles, and the computed equilibrium dissipation rate forST-10-2 and ST-30-2.

Figu re 12. Wave-energy dissipa tion ra t per unit volume (ivci- tlic imii^and end beach profiles, and the computed equilibrium dissipation rate foST.GO-1 and STJO-l .

swash zone. The pa ttern of wave-energy dissipation r ate perunit volume deviated greatly from uniformity and differedsignificantly from the pred icted value from Equa tion (2). De-viation from uniformity of wave-energy dissipation rate wasmuch greater at the end of the wave run than at the begin-ning, influenced hy the intense wave breaking over a narrowzone over the bar.For the regular wave accretionary case, the energy dissi-pation rate per unit volume ranged from -204 to 637 Nm/mVs at the beginning of the wave action, which is substan-tially different from the -296 to 751 Nm/mVs at the end (Fig-ure 12B). The dissipation patterns were far from being uni-form and also differ substantially from the predicted value of65 Nm/m'Vs.

The modulation of standing waves might have significantinfluence on the pattern of wave-energy dissipation for theregular wave cases, especially for the long-period accretion-ary wave case, as indicated by the large negative dissipationrate in the surf zone. The extremely high peak of wave-en-ergy dissipation at the main breaker line in both of the reg-ular wave cases is apparently related to the concentratedwave breaking over the har.The more gradu al break ing of irregular as compared to reg-ular waves tended to form a smoother beach profile andwave-decay pattern , w hereas the localized hreak ing of regu-

lar waves produced more irregularity in the beach profile anshape of wave decay. Given the steadiness of the monochromatic waves, concentrated wave breaking and energy dissipation should be expected, especially for plunging type breaking. The deviation of H,.,,JH.,,^ from unity in the surf zoneregular waves indicates a certain am ount of wave irregularitis generated in the break ing process. No substan tial diffeences were found hetween the narrow-banded swell-typwaves and hroad-banded sea-type waves in the data examined in this study.CONCLUSIONS

In the six SUPERTANK cases analyzed, a nearshore badeveloped and was maintained over an initially monotonipower-function profile. Except in the direct vicinity of thbreak-p oint ba r. the power-function mode! of DEAN (1977) dscribed the equilihrium beach profile reasonably well. For irregular waves, the assumption that an equilibrium profilshape develops under uniform wave-energy-flux dissipatiois supported hy the SUPERTANK data across a targe portioof the surf zone. Relatively greater rates of wave-energy dissipation were measured at the main breaker line and in thswash zone. The equilihrium rate of wave-energy dissipatiowas predicted mostly within 50*7 hy the simple model deriveby DEAN (1977) with a spilling breaking assumption. Signi


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Beach Profile Equilibrium and Wave-Energy Dissipation 53S

icant net cross-shore sediment transport and heach-profilechange occurred at locations where the rate of wave-energydissipation deviated substantially from theequilihrium rate.

For irregular waves, in thesurf zone the root-mean-squarewave height on an equilibrium beach profile was found to beproportional to the square root of local water depth, exceptover the hreak-point bar. Themore gradual wave-height de-cay contrasts to the common spilling wave hreaker assump-tion forwhich the wave height is proportional to local waterdepth in the surf zone. The simple model, requiring the inputof the equilihrium A value and thebounds of the surf zone,reproduced the measured wave decay reasonably well underdifferent large-scale laboratory wave and beach conditions.

Both beach-profile evolution and wave-energy dissipationpattern differ substantially under regular waves as comparedto those under irregular waves of comparable statistical sig-nificant wave height and peak period. Different beach profileshapes were developed under regular waves, particularly inthe vicinity of the break-point bar. For the regular wave cas-es, uniformity of wave-energy dissipation did not improve be-tween the start and end ofwave actions. These results indi-cate studies with regular waves may not represent all sedi-ment-transport processes as associated with irregular waves.A key difference is that wave-energy dissipation for regularwaves tends to be concentrated in a narrow zone of hreaking.Knowledge of cross-shore sediment transport gained fromregular-wave experiments will carry uncertainties if appliedto irregular waves without appropriate interpretation ortreatment of the regular waves, such as through a MonteCarlo approach.

For the irregular wave cases analyzed, there were relative-ly small differences between breaker index values, wave de-cay, and energy-dissipation rate per unit volume for thehroad and narrow-handed waves. This result, based on lim-ited laboratory data but at large scale, indicates that the in-fluences of spectral characteristics of irregular waves on thebeach-profile evolution are not significant for typical waveswith single-peaked spectra.

ACKNOWLEDGMENTSThe work of P. Wang was jointly funded by the Coastal

Inlets Research Program (CIRP) administered at the U.SArmy Engineer Research and Development Center and bythe Louisiana Sea Grant College Program. The work of N.C.Kraus was conducted under the Inlet Geomorphology andChannel Evolution work unit of the CIRP. We appreciate crit-ical reviews and helpful comments hy Drs. Magnus Larsonand Jane McKee Smith. Permission was granted by Head-quarters, U.S. Army Corps of Engineers to publish this paper.

LITERATURE CITEDWR, E.H.; RO.SENTHAL, G,W., and ViNCENT, C.L., 1985. Similarityof the wind wave spectral in finite water depth, 1: spectral form.Journal of Geophysical Research. 90, 975-986.BRLFUN, P., 1954. Coast erosion and the development of beach pro-files. Technical Memorandum N o. 44 , Beach Erosion Board, U.S.Army Corps of Engineers.DALLY, W.R.; DEAN, R.G., and DALRYMPLE, R.A., 1985. Wave height

variation across beaches of arbitrary profile. Journal ofGeophys-ical Research. 90, 11,917-11,927,DEAN, R.G., 1977. Equilibrium beach profiles: U .S. A tlantic and Gulfcoasts. Ocean Engineering Report No. 12. Department of Civil En-gineering, U niversity of Delaware, Newark, DE.DEAN, R.G., 1991. Equilibrium beacb profiles: characteristics and ap-plications. Journal of Coastal Research, 7, 5384.GONZALEZ, M.; MEDINA. R., and LOSADA, M.S., 1999. Equilibrium

beacb profile model for perched beaches. Coastal Engineering, 36,343-357.INMAN D.L.; ELWANY, M. H. S., and JENKINS, S.A., 1993. Shore riseand bar-berm profiles on ocean beaches. Journal of GeophysicalResearch. 98(C10), 18,181-18.199.KAMINSKY', G. M . and KRAUS, N.C, 1993. Evaluation of depth-limitedwave breaking criteria. Proceedings of 2'"' International Sympo -.sium on Ocean Wave Measurement and Analysis, Waves 93 (ASCNew York), pp. 180-193.KRAUS, N.C: LARSON, M.. and KRIEBEL, D.L., 1991. Evaluation ofbeach erosion and accretion predictors. Proceedings Coastal Sedi-ments VI (ASCE, New York], pp. 572-587.KRAUS, N.C. and SMITH, J.M ., 1994. SUPERTANK laboratory datacollection project. Volume 1: Main text. Technical Report CERC-94-3. U .S. Army Engineer W aterways Experiment Station, Coast-al Engineering Research C enter, Vicksburg, M ississippi,KRAUS, N.C: SMITH, J.M., and SOLLITT, O.K., 1992. SUPERTANKLaboratory Data Collection Project, Proceedings 23"' C oastal En-gineering Conference (ASCE, New York), pp, 2191-2204.KRAUS, N.C and LARSON, M ., 1988. Prediction of initial profile ad-justment of nourished beaches to wave action. Proceedings of 1988National Conference on Beach Preservation Technology, FloridaShore and Beach Preservation Association, Tallahassee, Elorida,pp.125-137.KRIEBEL, D.L., 1986. Verification study of a dune erosion model,Shore andBeach, 54(3l, 13-21.KRIEBEL, D.L. and DEAN. R.G., 1985. Numerical simulation of time-dependent beach anddune erosion. Coastal Engineering, 9, 2 2 1 -245.KRIEBEI,. D.L. and SMITH, J.M., 1994. Wave transformation mea-surements at SUPERTANK. Proceedings Coastal Dynamics '94(ASCE, New York], pp. 223-247.LARSON, M ,, 1991. Equilibrium profile of a beach with varying grainsize. Proceedings Coastal Sediments Vl (ASCE, New York), pp.861-874.LARSON, M . and KRAUS. N.C. 1989. SBEACH: Numerical modelingfor sim ulating storm-induced beach changeReport 1: empiricalfoundation and model development. Technical Report. CERC-89~9. U.S, Army Engineer Waterways Experiment Station. CoastalEngineering Research C enter, Vickshurg, M ississippi.LARSON, M . and KRAUS, N .C . 1994a. Temporal and spatial scales ofbeach profile change. Duck, North Carolina. Marine Geology, 117,75-94.LARSON, M , and KRAUS, N .C . 1994b. Cross-shore sand transpo rt un-der random waves at SUPERTANK examined at mesoscale. Pro-ceedings Coastal Dynamics '94 (ASCE. New Yorkl. pp. 204-219,LARSON, M , and KRAUS, N , C . 2000. Representation of non-erodibleIhard) bottoms in beach profile change modeling. Journal of Coast-al Research. 16(1), 1-14.LARSON, M ,; KRAUS, N.C, and WISE, R.A, , 1999, Equilibrium beachprofiles under breaking and non-breaking waves. Coastal Engi-neering. 36,59-85,McDouGAL, W,G.; KRAUS, N.C, and AJIWIBOWO, H ,. 1996, Tbe effectsof seawalls on thebeach: Part ILnumerical modeling of SUPER-TANK seawall tes ts. Journal of Coastal Research, 12(3), 702-713 .MOORE, B,D., 1982, Beach profile evolution in response to changesin water level and wave height, Ma.sters Tlwsis, Department ofCivil Engineering, U niversity of Delaware. Newark, D elaware.MoLtTZOlJRIS, CI., 1991. Beach profile vs. cross-shore distribution ofsediment grain size. Proceedings of Coastal Sediments '9J {ASCE,New York), pp. 419-431,PRUSZAK, Z., 1993. Theanalysis of beach profile changes usingDean's method andempirical orthogonal functions. Coastal Engi-neering, 19, 245-261.

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53 4 Wang and KrausSMITH, J.M .. 1994. Undertow at SUPERTANK. Proceedings CoastalDynamics '9 4 (ASCE, New York), pp. 220-232.SMITH, E.R. and KRAUS. N.C. 1991. Laboratory study of breakingwaves on bars and artificial reefs. Journal of Waterway. Port,Coastal an d Ocean Engineering, 117(4), 307-32 5.SMITH, J .M . and KHAMS, N.C, 1987. Longshore current model basedon power law wave decay. Proceedings Coastal Hydrodynamics '8 7(ASCE, New York), pp, 155-169.SMITH, J .M . andKRAUS. N.C. (Editors), 1995. SUPERTANK labora-tory data collection project. Volume II: Appendices A-I. TechnicalReport CERC-94-3, U.S. Army Engineer Waterways ExperimentS tation, Coastal Engineering Research Center, Vickaburg, M issis-sippi,THORNTON, E . B . and GUZA, R.T., 1982. Energy saturation and pbasespeeds measured on a natural beacb. Journal ofGeophysical Re -search. 84, 9,499-9,508.THORNTON, E.B. and GuzA, R.T., 1983. T ransformation of waveheight distribution. Journal ofGeophysical Research, 88, 5,295-5,938.

WANG, P. and DAVIS, H .A., 1998. A beacb profile m odel for a barredcoastcase study from Sand Key, West-Central Florida. Journaof Coastal Research. 14, 981-991.WANG. P. and DAVIS, R.A., 1999. D epth of closure and the equilibrum beacb profileA case stu dy from S and Key, West-CentraFlorida. Shore and Beach. 67, 33-42.WANG, P.; S M I T H . E.R. and EBERSOLE, B.A., 2002a. Large-scale laboratory measurements of longshore sediment transport unde

spilling and plunging breakers. Journal ofCoastai Research, 1118-135.WANG, P.; EBERSOLE, B.A.; S M I T H , E.R., and J O H N S O N , B.D. 2002bTemporal and spatial variations of surf-zone currents and supended-sediment concentration. Coa.stat Engineering. 46.175-2WORK, PA. and DEAN. R.G ., 1991 . Effect of varyin g sedim ent size oequilibrium beach profiles. Proceedings ofCoastal Sediments '{ASCE, New York), pp, 891-903.ZHENC;, J. and DEAN. R.G .. 1997. Numerical models and intercomparsion.s of beacb profile evolution. Coastal Engineering, 30(3-169-201.


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