New Physical Insights and Design Formulas on …...New Physical Insights and Design Formulas on Wave...

New Physical Insights and Design Formulas on WaveOvertopping at Sloping and Vertical Structures

Jentsje van der Meer, M.ASCE1; and Tom Bruce2

Abstract: Mean wave overtopping discharge is a key design parameter for many coastal structures, typically designed to limit overtoppingdischarge to below a chosen admissible value. Dutch, German, and British design guidance from the 1990s was updated using results of researchprojects supported by the European Commission, and subsequently unified with the publication of the European Manual for the Assessment ofWave Overtopping, or EurOtop [EurOtop. (2007). European Manual for the Assessment of Wave Overtopping, T. Pullen, et al., eds.], nowused all over the world. This paper explores five technical issues that were not well covered in the unified manual. (1) For sloping structures,overtopping at low and zero freeboard conditions: new analysis brings together the conventional exponential formulas with the few reliabledatasets including very low and zero freeboard. In doing so, early Dutch work from the 1970s was revisited. Weibull-type formulas areproposed, describing wave overtopping at slopes for the whole range Rc=Hm0 $ 0. (2) For vertical walls, the manual distinguishes overtoppingresponses depending upon whether wave breaking occurs, with nonbreaking and breaking conditions described by exponential and power lawformulas, respectively. Here, the governing equations are manipulated in such a way as to reunify the methods, enabling direct and intuitivecomparison. (3) For overtopping at vertical walls under nonimpulsive conditions, early Italian/Dutch and British formulas of the 1990s divergesignificantly for higher freeboards. This paper explores why these two reliable studies arrived at two such different predictors. By reanalysis ofthe original datasets augmented by further data drawn from the international database on wave overtopping, a physical distinction based on thenature of the foreshore is proposed and tested. (4) The prediction method for overtopping at composite vertical structures is reworked to enablethe influence of themound to be apparent and to alignwith plain vertical wall formulas for smaller mounds.A new scheme is proposed that treatscomposite structures via a small adjustment to the new vertical wall approach proposed earlier. (5) There is vast literature on overtoppingresponse atmildly sloping structures and substantial literature on verticalwalls, but in the intervening range (approximately 1V : 1:5H to 5V : 1H), thereis a paucity of reported tests. Recently published Belgian data have enabled the development of a continuous prediction scheme spanningmild slopes,steep slopes, andvertical structureswithout foreshore.DOI:10.1061/(ASCE)WW.1943-5460.0000221.© 2014AmericanSociety ofCivil Engineers.

Author keywords: Coastal structures; Wave overtopping; Sloping structures; Vertical structures; Overtopping manual; EurOtop.

Introduction

Mean wave overtopping discharge (usually q, in m3=s per meterwidth) is a key design parameter for many coastal structures that aredesigned to limit q below a selected admissible discharge. It is notsurprising, therefore, that wave overtopping has been the study of somany academic research studies and the topic of engineering designguidance manuals in many nations. The 1990s saw publication ofguidance manuals in (at least) Netherlands [Technical AdvisoryCommittee for Flood Defence in Netherlands (TAW) 2002], theUnited Kingdom [Environment Agency (EA) and Besley 1999], andGermany (EAK 2002). The study of overtopping continued, withmajor projects such as the European Commission Crest Level Assess-ment of Coastal Structures by Full-Scale Monitoring, Neural NetworkPrediction and Hazard Analysis on Permissible Wave Overtopping(CLASH) project (De Rouck et al. 2009), which delivered important

outputs such as a detailed evaluation of scale, model, and laboratoryeffects, and the overtopping database with 10,000 overtopping tests(Van der Meer et al. 2009), which in turn formed the basis of theartificial neural network prediction method. The European Manualfor the Assessment of Wave Overtopping (Pullen et al. 2007) usedresearch outputs such as those from CLASH to update and unify theearlier Dutch, German, and British manuals into a single volume.

Making the CLASH database freely available was intended toenable other researchers to perform further analysis on any parts of thisinformation in future. Goda (2009) has not only done this, but he alsostates that hecame tounifiedwaveovertoppingprediction formulas forseawalls with smooth impermeable surfaces. Goda (2009) did not usethe artificial neural network predictionmethod for comparison, despitethis being themain predictionmethod in theEuropeanManual for theAssessment of Wave Overtopping (EurOtop) manual (EurOtop 2007).The paper does, however, give two useful considerations (also notedduring the writing of the EurOtop manual) as areas that would benefitfrom further research: the influence of foreshores at vertical walls(under both nonimpulsive and impulsive wave attack) and over-topping at structures with geometries lying between vertical walls andsteep slopes (i.e., very steep slopes; his unified formulas cover verticalwalls through to very gently sloping structures of 1:7).

In the five years since publication of EurOtop, the present paper’sauthors continued exploration of wave overtopping phenomena andidentified five areas of interest in need of further research, which aredeveloped in this paper and listed as follows:1. For sloping structures, overtopping at low and zero freeboard

conditions have often been overlooked in physicalmodel studies

1Principal, Van der Meer Consulting BV, P.O. Box 11, 8490 AA,Akkrum, Netherlands; Professor, UNESCO-IHE, 2601 DA Delft, Nether-lands. E-mail: [email protected]

2Senior Lecturer, School of Engineering, Univ. of Edinburgh, EdinburghEH9 3JL, U.K. (corresponding author). E-mail: [email protected]

Note. This manuscript was submitted on August 13, 2012; approved onJune 28, 2013; published online on July 2, 2013. Discussion period openuntil September 14, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Waterway, Port,Coastal, andOceanEngineering, © ASCE, ISSN 0733-950X/04014025(18)/$25.00.

© ASCE 04014025-1 J. Waterway, Port, Coastal, Ocean Eng.

J. Waterway, Port, Coastal, Ocean Eng.

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http://dx.doi.org/10.1061/(ASCE)WW.1943-5460.0000221

mailto:[email protected]

mailto:[email protected]

(perhaps because of the challenges ofmeasurement of very largedischarges), but they represent important situations, e.g., inanalysis of performance of partially constructed breakwatersand of low-freeboard, lower-cost defenses. It is clear thatfamiliar, exponential-type formulas work poorly in theseregions. Analysis has therefore been performed to bring togetherthe conventional exponential formulas with the few reliabledatasets including very low and zero freeboard. In doing so, theauthors revisited early Dutch work from the 1970s, whichoffered a continuous prediction extending to zero freeboard.

2. For vertical or very steep (battered)walls,U.K.guidance from the1990s (EAandBesley1999), extendedbyEurOtop, identifies theneed to distinguish quite different overtopping responsesdepend-ing on whether wave breaking occurs. Nonimpulsive conditionsare described by a familiar exponential formula, whereas impul-sive (breakingor impacting)waveovertopping is better describedby a power law formula (Pullen et al. 2007). An analogousapproach described overtopping at composite vertical break-waters. The downside of this two-formula approach is that itis not at all easy to compareona single plot, in anyvisual/intuitiveway, the overtopping behavior of a single structure becauseconditions move between impulsive and nonimpulsive condi-tions (different nondimensionalization schemes are used for bothdischarge and freeboard axes). Here, the governing equations aremanipulated in such a way as to facilitate this reunification.

3. For overtopping at vertical walls under nonimpulsive con-ditions, the early works of Franco et al. (1994) andAllsop et al.(1995) remain reliable references. For other than lower free-boards, however, their principal prediction formulas divergesignificantly. EurOtop did not address this issue directly,preferring tomake afit to a larger set of data, but the underlyingquestion remains: why did two reliable studies arrive at twosuch different predictors? Here, by reanalysis of the originaldatasets augmented by further data from the CLASH database,a physical distinction is proposed and tested.

4. Composite vertical structures (breakwaters with a vertical mainwall sitting atop a substantial rubble mound whose presencemay influence the hydrodynamics) are treated by EurOtopaccording to a process analogous to, but distinct from, theanalysis of simple vertical walls. It was argued under Point 2that the inability to visualize both impulsive and nonimpulsiveovertopping responses on a single graph curtailed opportunityfor physical understanding of the response of a particularstructure over the full range of its operating conditions. Sim-ilarly, the different formulations for the analysis of structureswhose mound has little or no influence (according to the plainvertical approach) and for those whose mound’s influencedominates (composite structures approach) represents a discon-tinuity in understanding, interpretation, and analysis.Work hereattempts to bridge this divide in a physically rational way.

5. There is a vast amount of reliable literature on overtoppingresponse at mildly sloping structures (to 1:1.5), substantialliterature on vertical walls, and some on off-vertical batteredwalls, e.g., 10:1 (i.e., 10V :1H) or 5:1 steep slopes. In the in-tervening range (approximately 1:1.5 to 5:1), there is a paucityof reported tests. Although this reflects fewer structuresdesigned with very steep slopes, this represents a gap in theavailability of robust and well-supported guidance.

The artificial neural network prediction method will not be al-tered and remains the governing prediction method, especially formore complicated structures. For structures with simpler geome-tries, however, the standard formulas remain widely used in initialanalysis. It is important to realize that even for simple geometries,some parametric regions are currently less well modeled by these

formulas alone, e.g., overtopping at lowest freeboards. Improvingthe formulas in these regions not only assists in analysis of simplestructures but also provides a firmer basis for cross-comparison ofneural network or numerical model predictions.

Sloping Structures with Low and Zero Freeboards

Basis of the EurOtop Manual

It is long established, based on the work of Owen (1980), that waveovertopping discharge, q, on many kinds of coastal structuresgenerally decreases exponentially as the crest freeboard,Rc, increases,with a form

qffiffiffiffiffiffiffiffiffiffiffigH3

m0

q ¼ a exp

�2b

Rc

Hm0

�(1)

whereHm0 5 spectral significant wave height; and a and b are fittingcoefficients. This form of equation has become popular because itgives a straight line on a log-linear graph, and it has only twocoefficients for fitting to the data.

For sloping structures like dikes or levees, EurOtop (Pullen et al.2007) gives the following design formulas:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig� H3

m0

q ¼ 0:067ffiffiffiffiffiffiffiffiffiffitana

p gb � jm21,0

� exp

24:75 Rc

jm21,0 � Hm0 � gb � gf � gb � gv

!

(2)

with a maximum of


m0

q ¼ 0:2� exp

22:6 Rc

Hm0 � gf � gb

!(3)

where a 5 slope angle; jm21,0 5 breaker parameter based on thespectral period Tm21,0; jm21,0 5 tana=ð2pHm0Þ=ðgT2

m21,0Þ0:5; andgx 5 influence factor [see EurOtop (Pullen et al. 2007) for moreinformation]. Eq. (2) generally describes gentle slopes with plungingor breakingwaves. In contrast,Eq. (3)—themaximumovertopping—describes surging or nonbreaking waves on fairly steep slopes. Thereliability of Eq. (2) is described by a SD (s) in the exponentsð4:75Þ5 0:5. Similarly, the reliability of Eq. (3) is described bysð2:6Þ5 0:35.

Most data considered for Eqs. (2) and (3) [Figs. 5.9 and 5.10 inEurOtop (Pullen et al. 2007) and with similar figures in this paper]have relative freeboards Rc=Hm0 . 0:5. The exponential type equa-tions fit the data nicely, except for the data points at zero freeboard,where the equations would significantly overpredict. Overtopping atlow and zero freeboards is the first subject to be described.

It is observed that Goda’s (2009) unified formulas do not describeovertopping at gentle slopes optimally. It is long established that waveperiod and slope angle have large influence on wave overtopping atgentle slopes, where waves are of the plunging (breaking) type. Thisinfluence is not present for steep slopes and surging or nonbreakingwaves and for vertical walls. As there is no influence of wave period inthe unified formulas, these formulas are not valid for gentle slopes, andapplication shouldbe limited to slopes steeper thanabout1:2.Likemostformulas on overtopping, the formulas are of the exponential type. Thismeans that very low or zero freeboard situations will be overpredicted.



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Update on Reliability of Formulas

The EurOtopmanual describes the reliability of the formula by takingone of the coefficients as a stochastic parameter and giving a SD(assuming a normal distribution). Then deterministic and probabilisticapproaches are given. Actually, the deterministic design or safetyassessment approach in the EurOtop manual should be termeda semiprobabilistic approach because a partial safety factor of 1 SD isused. This paper presents the following enhanced approaches:• Deterministic approach: Use the formula as given with the mean

value of the stochastic parameter(s). This should be done topredict or compare with test data. This is not the same as the fordeterministic design approach of EurOtop (Pullen et al. 2007).

• Semiprobabilistic approach: This is an easy approach for designor safety assessment; this is the previous deterministic approach,but now with the inclusion of the uncertainty of the prediction.The stochastic parameters become m1s.

• Probabilistic approach: Consider the stochastic parameter(s) withtheir given SD and assuming a normal distribution.

• The 5% exceedance lines, or 90%-confidence band, can becalculated by using m6 1:64s for the stochastic parameter(s).In this paper, the formulas are given as the mean prediction

(deterministic approach). The formulas and 5% exceedance curvesare shown graphically. Key coefficients are taken as stochasticvariables, and uncertainty is then described by giving the SD s.

Wave Overtopping according to Battjes (1974) andDutch Guidelines

Battjes (1974) derived an expression for the overtopping volume inperiodic waves on smooth gentle slopes and applied this expression toindividual waves in a random wave train. A bivariate Rayleigh dis-tribution was assumed for the wave height and wavelength. Thisresulted in an expression for the mean overtopping discharge, whichwas still a function of the correlation parameter of the bivariateRayleigh distribution k (Battjes 1974, Appendix A). With k5 0,a lower boundwas found, andwith k5 1, an upper boundwas found.Curved lines on a log-linear graph were the result as in Fig. 1

(explained in the next paragraph). The overtopping parts of Battjes(1974) were not subsequently used a lot in Netherlands, the mainreason being that crest height design of dikes was still based on the2%-run-up level and not on wave overtopping. The apparent com-plexity of the formulasmay have also been a factor in the overtoppingparts of Battjes’ work not seeing wider adoption and exploitation.

The TAW (1985) guidance, however, gave the curves of Battjes(1974) in a graphic form and proposed to use the upper boundary,because one large-scale test in the Delta flume of Delft Hydraulics(now Deltares) on a 1:3 slope was close to this boundary. This curveis given in Fig. 1, together with the mentioned test. The x-axiswas given by Rc cota=ðHmL0Þ and the y-axis by qTmðcotaÞ0:5=ð0:1HmL0Þ, where Hm is mean wave height; L0 is mean wavelength5 g=2p T2

m, with Tm being the mean wave period. A little later in theTAW (1989) guidelines, the significant wave height was introducedby H1=3 5 1:6Hm and the significant wave period T1=3 5 1:15Tm,which led to the following parameters along the axes in Fig. 1:

X ¼ Rccota

��H1=3=1:6

�g�T1=3=1:15

�2=2p

0:5

Y ¼ qT1=3ðcotaÞ0:51:6� 1023 � 2p

��0:1� 1:15H1=3g

�T1=3=1:15

�2

The curve for wave overtopping was then approximated by

logðYÞ520:214X220:787X þ 0:103 (4)

The main difference between the usual exponential function forwave overtopping [Eq. (2)] and Fig. 1 is that Battjes’ (1974) curve isnot a straight line on a log-linear graph. An exponential fit for largerrelative freeboards (a straight line) would be close to the curve inFig. 1, but such a fit would deviate for low freeboards. Theparameters at the horizontal and vertical axes can be rewritten byassumingH1=3 5Hm0 and T1=3 5 Tm21,0. The numeric values can becalculated also, giving

X ¼ 1:45Rc�

Hm0jm21,0�and

Y ¼ 46:1q�

gH3m0

�0:5 Hm0

�Lm21,0 tana

��0:5The x- and y-axes are now exactly the same as the EurOtop (Pullenet al. 2007) formula for breaking waves on a gentle slope, except forthe constant multipliers 1.45 and 46.1 [cf. Eq. (2)]. It also means thatthe curve in Fig. 1 can be directly compared with the data and theprediction curve in EurOtop (Pullen et al. 2007) (Fig. 2). The result atfirst sight is startling, because the curve not only matches the data forpositive freeboard and the EurOtop prediction line [Eq. (2)], but italso fits neatly the zero-freeboard data of Smid et al. (2001) (CLASHDataset 102).

One should, however, realize that fitting was done on a reliableand large-scale test in the Delta flume. From that point of view, it isnot surprising that the curve would match part of the overtoppingdata in Fig. 2. However, it is nevertheless pleasing that it also fitsthe zero freeboard data so well. It can be asserted therefore that thetheory developed by Battjes (1974) gives the correct shape of thecurve to describe wave overtopping at gentle slopes for all appli-cations described as Rc=Hm0 $ 0. It also provides an analytical basisfor the EurOtop choices for the parameter groups on horizontal andvertical axes, which were mainly based on analysis of empirical dataonly and not strongly on analytical reasoning.

Fig. 1. Replot of overtopping curve, developed by Battjes (1974) anddescribed in the TAW guidelines (TAW 1989)



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Further Analysis on Sloping Structures withLow Freeboards

Data for zero freeboard are also available for steep slopes andnonbreakingwaves [Eq. (3)]. Fig. 3 gives the replot of Fig. 5.10 fromEurOtop (Pullen et al. 2007) for this type of structure, where Smidet al.’s (2001) data give the points for zero freeboard. Schüttrumpfand Oumeraci (2005) provide Dataset 102 in the CLASH database.

Dataset 108 also gives data with zero freeboard and for a slope of1:1.5. In CLASH, this dataset was assigned a reliability factorðRFÞ5 4, meaning that the data are deemed unreliable (seeSteendam et al. 2004 for a full explanation of RF). The reason wasthat during screening of the dataset in the CLASHwork, the differentmeasures of wave period did not seem to be consistent. The waveperiod is not, however, part of the analysis of overtopping at steepslopes. For this reason, the data of Dataset 108 for zero freeboard

Fig. 2.Replot of Fig. 5.9 fromEurOtop (Pullen et al. 2007) for breakingwaves on gentle slopes, together with the Battjes (1974) approximation as usedin the TAW guidelines (TAW 1989)

Fig. 3. Replot of Fig. 5.10 from EurOtop (Pullen et al. 2007) with Dataset 108 added for zero freeboard and with a Weibull-type of fit for the wholerange [Eq. (6)]



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have been restored to Fig. 3, although the observation that the data ofthis dataset 108 for positive freeboards are lower than expectedcontinues to flag a concern over the reliability of the whole dataset.

Research in CLASH resulted in a lot of new data and in predictionformulas [Eqs. (2) and (3)] for slopes for breaking waves and non-breaking waves. Both formulas overpredict overtopping for very lowand zero freeboard (Figs. 2 and 3). A polynomial fit as in Eq. (4)describes the data but is not easy to use for comparison betweendifferentformulas.Anewfit for lowfreeboardsonly,with anextra setof formulas,would solve theproblem. It ismore elegant andmore physically rational,however, to propose a curved line in an easy way. As the exponentialfunction is a special case of the Weibull distribution, it is possible to goback to a Weibull-type function and use a fitted shape factor. Sucha function still looks very much like Eq. (1) and is described by


m0

q ¼ a exp

�2

�b

Rc

Hm0

�c(5)

Eq. (5) needs fitting of the correct shape factor, c, and then a refit ofcoefficient a and exponent b. Analysis gave a shape factor of c5 1:3for a good fit for both breaking and nonbreaking waves. This is notnecessarily the best fit, but there is an advantage in having the samevalue for both equations. Fig. 3 shows the final curve for non-breaking waves, covering the full range of relative freeboards[Eq. (7)]. The fit for breaking waves [Eq. (6)] is almost on top of thepolynomial fit in Fig. 2 and is not shown for that reason.

Overtopping on sloping structures with zero and positive free-board can then be described by


m0


p gb � jm21,0 � exp

242 2:7 Rc

jm21,0 � Hm0 � gb � gf � gb � gv

!1:335 (6)

with a maximum of


m0

q ¼ 0:09� exp

242 1:5 Rc

Hm0 � gf � gb

!1:335 (7)

The reliability of Eq. (6) is given by sð0:023Þ5 0:003 and sð2:7Þ5 0:20 and of Eq. (7) by sð0:09Þ5 0:013 and sð1:5Þ5 0:15.These formulas give almost the same wave overtopping as the orig-inal formulas [Eqs. (2) and (3)] but represent nature better forRc=Hm0 , 0:521:0. In general, there is no need to replace Eqs. (2)and (3) by Eqs. (5) and (6), as they give similar predictions. Only forlow and zero freeboards will Eqs. (6) and (7) be better. However, thenew equations give better insight in wave overtopping over the fullrange of zero and positive freeboards.

Vertical Walls and Composite Vertical Structures

Background and Motivations

For vertical breakwaters or seawalls, in the absence of wavebreaking, the influence of the wave period seems very small ornonexistent, and the easy formulation of Eq. (1) with simple valuesfor a and b has become a trusted design formula. Early work byFranco et al. (1994) for relatively deep water gave a5 0:2 andb5 4:3, whereas Allsop et al. (1995) gave a5 0:05 and b5 2:78 inconditions of shallower water. The original datasets for both ref-erences have been retrieved to be able to explore and discuss reasonsfor the differences. They are replotted in Figs. 4 and 5, together withthe prediction lines from the formulas. Both formulas fit theiroriginal data quite well, but what is relatively deep and shallowwater? This will be explored as one of the topics in this section of thepaper.

There has long been evidence that the overtopping process atvertical and steepwalls cannot be described for all conditions simplyby exponential form equations such as Eq. (1). Goda’s design charts

(Goda 2000) show quite pronounced peaks for some shallower(relative) water depths. Also, Goda (2009) finds that local waterdepth on a foreshore is important. Physical model studies in the1990s of overtopping at vertical walls, under conditions where wavebreaking on the structure was present, gave rise to new empirical fitsgiving a power law decrease in overtopping discharge with free-board, rather than an exponential one. The original data of Allsopet al. (1995), which were identified as being breaking, are replottedas Fig. 6.

The impulsive overtopping equations [EurOtop (Pullen et al.2007); Allsop et al. 1995] have a power law form

q

h2pffiffiffiffiffiffiffigh3

p ¼ a

�h p

Rc

Hm0

�2b

(8)

Fig. 4. Replot of Fig. 8 from Franco et al. (1994) for relatively deepwater, with the Allsop et al. (1995) formula for comparison



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The coefficient a and exponent b change depending on structure andwave conditions considered. The exponent b takes values of 3.1 forimpulsive overtopping at plain vertical walls; 2.7 for broken wavesat plain vertical walls; and 2.9 for composite vertical structures.These exponents are simply the result of fitting to data; the dif-ferences have no basis in any analytical framework or in physicalreasoning. The fact that the exponents are all different makes itdifficult to carry out a direct, e.g., graphical, comparison between thedifferent but closely related structures and their associated over-topping responses.

Whether anexponential [Eq. (1)] or a power law should be used isdetermined by some discriminating parameter hp

hp[ 1:3 hHm0

2phgT2

m21,0

(9)

The hp parameter is used as a measure of impulsiveness, witha transition from nonimpulsive to impulsive overtopping conditionsat the wall taking place over the range 0:2# hp # 0:3.

There is a strong evidence that these two apparently distinctmethods work well. The fact that discharge and freeboard are non-dimensionalized in quite different ways for impulsive and non-impulsive conditions has until now prevented simple comparison ofthe formulas. Further, these differences hamper improved physicalrationalization of the transition from one regime to the other. This

part of the paper presents a reformulation of the standard equationsfor impulsive overtopping at vertical walls as described in EurOtop(Pullen et al. 2007) to integrate them into a more unified, physicallyrational framework of prediction tools spanning a greater breadth ofstructure types and wave conditions.

Relationship between Nonimpulsive andImpulsive Overtopping

The exponents b in Eq. (8) are all very close to 3; none deviate farfrom the original 2.92 of Allsop et al. (1995), repeated in the pre-EurOtop U.K. guidelines (EA and Besley 1999). Allowing a WhatIf? approach and fixing b5 3 enables the equations to be manip-ulated algebraically, noting that Eq. (9) can be written in a morephysically illuminating way

hp� 1:3 hHm0

hLm21,0

(10)

withLm21,0 being the (fictitious) wavelength based onTm21,0. Eq. (8)can now be rewritten as

qffiffiffiffiffiffiffigh3

p ¼ aHm0

h

Lm21,0h

�Rc

Hm0

�23

(11)

0qffiffiffiffiffiffiffiffiffiffiffigH3

m0

q ¼ a

ffiffiffiffiffiffiffiffiffiHm0

h

r1

sm21,0

�Rc

Hm0

�23

(12)

where sm21,05 (fictitious)wave steepnesswithwavelength based onTm21,0. Now the left side of Eq. (12) is the conventional di-mensionless discharge, as per Eq. (1), but the right side showsa power function of dimensionless freeboard Rc=Hm0 instead of anexponential one. Eq. (12) ismuch clearer than the formulawith hp onboth sides [Eq. (8)]. The form of Eq. (12) is such that it can be seenthat the overtopping under impulsive conditions depends on acombination of breaker indexHm0=h and the wave steepness sm21,0.Both breaker index and wave steepness have a physical meaning asfollows:• Hm0=h, ∼0:4: Waves are not depth limited, but may be influ-

enced by a gentle foreshore;• Hm0=h, ∼0:5: Depth-limited conditions on gentle sloping fore-

shores (1:50 or gentler);• Hm0=h, ∼0:6: Depth-limited waves on steep foreshores;• sm21,0 5 0:06: Wind waves of near-maximum wave steepness

(not reduced by depth-limited wave breaking); and• sm21,0 , ∼0:02: Low steepness, swell, or possibly caused by

wave breaking on a foreshore (which reduces thewave height butnot the period).Now it is possible to plot predictions for impulsive conditions di-

rectly alongside those for nonimpulsive conditions on the familiardimensionless discharge versus dimensionless freeboard axes. How-ever, before doing so, the coefficient a in Eq. (12) and the equationitself will be reexamined using existing data of the CLASH database.

Reanalysis of Vertical Walls with CLASH Data

As stated earlier in this paper, the differences between the formulasof Franco et al. (1994) and Allsop et al. (1995) have not beenexplained. Moreover, Goda (2009) assumes that a foreshore orforeshore depth may have influence on wave overtopping at verticalstructures. To get more physical insight into the influence onovertopping of deep and shallow water and explain the difference

Fig. 5. Replot of Fig. 23 from Allsop et al. (1995) for shallow water(impulsive waves), with the Franco et al. (1994) formula for comparison

Fig. 6. Replot of Fig. 24 from Allsop et al. (1995) and the power lawformula for impacting/impulsive wave attack; also shown is the datafrom De Waal (1994), referred to via its dataset number (224) in theCLASH database (Table 1)



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between the two overtopping formulas, part of the CLASH databasehas been analyzed in depth.

The CLASH database contains about 10,000 tests on over-topping (Van der Meer et al. 2009). A part of this database relates totests with a vertical or battered wall, which can be found by filteringon cotad 5 0 or on very small values of cotad (i.e., battered walls).This cotad is the first slope in a cross-sectional profile above a toe orberm. The cross section of each test setup can easily be checked inthe database and will show whether the dataset is a plain verticalwall, with or without a foreshore, with or without a berm, andwith orwithout a wave return wall or shifted parapet.

In total, 15 different datasets were used from CLASH (shown inTable 1 as vertical seawall, harborwall, or caisson). In addition, threefurther datasets were used (also shown in Table 1): two from theoriginal dataset of Franco et al. (1994) (Fig. 4) and the basic datasetof vertical walls at the end of a 1:50 foreshore of Allsop et al. (1995)(Figs. 5 and 6). The table gives the dataset number and a reference ifthe dataset is in the public domain. To describe the setup of thedataset, information is given on the presence of a foreshore slope, thetype of structure investigated, and whether a berm or toe structurewas present. In all cases where there was no foreshore, the waterdepth was fairly large. All datasets with a foreshore had a straightforeshore with a given fixed slope. Of the 18 vertical wall data sets,12 had a horizontal foreshore (5no foreshore) and six had a slopingforeshore. Two datasets with a slope instead of a vertical wall wereincluded because they contained rare data with zero freeboard.

The type of structures represented in each test set reveals, to someextent, the objective of the tests. A verticalwallmay be found at the endof a foreshore and then represent a seawall, often with more or less

depth-limited waves. A vertical wall with no foreshore is often a floodwall in a harbor. Waves are relatively small with respect to the waterdepth at storm flood situations in the harbor, and the wall may havea quay area as a kind of berm relatively far below thewater level. Othersituations concern a breakwater like a caisson. Caissons are founded ona berm, but this berm is often deep below water and too small for theoverall structure to be termed composite. They also may have someshape of parapetwall, shifted, or returnwall. Another practical situationcould be a gate of a lock at flood situations, and this would be con-sidered as a vertical wall without foreshore or berm. Some datasets hadbatteredwalls (close to verticalwalls but slightly inclined). They alwayswere seawalls at the endof a foreshore slope.Byanalyzing eachdataset,it was kept in mind what type of structure had been investigated.

All 18 datasets were then plotted individually, with four predictioncurves for comparison: Franco et al. (1994), Allsop et al. (1995),a steep smooth slope [Eq. (3)], and one specific curve for impulsivewaves [Eq. (12), with a5 0:000192 with h=Hm0 5 0:9 and sm21,0

5 0:03]. Examples are shown in Figs. 7 and 8. Fig. 7 shows Data-set 802 of Goda et al. (1975) and clearly shows the increased over-topping for seawalls at the end of a foreshore slope, as all data pointsare along or above the curve for impulsive wave attack. There arehardly any points around the Allsop or Franco curves. Fig. 8 showsCLASH Dataset 914 of Cornett et al. (1999), with tests on a verticalwall with deep water without a foreshore and with a small and deepberm. The overtopping is now significantly less than in Fig. 7 and isgrouped well around the line of Franco et al. (1994).

Individual analysis of all datasets led to one clear conclusion:there is a distinct difference between vertical structures with andwithout a sloping foreshore. The results with a sloping foreshore

Table 1. Description of Datasets Used for Further Analysis on Vertical Walls

CLASH dataset Reference Foreshore slope Type of structure Berm

006 Confidentiala 1:20 Battered 10:1 No028 Herbert (1993) 1:10; 1:30; 1:100 Vertical seawall No043 Pullen et al. (2004) 1:30 Composite Yes044 Pullen et al. (2004) 1:30 Composite Yes102 Schüttrumpf and Oumeraci (2005) No Slope No106 Oumeraci et al. (2001) No Vertical seawall No107 Smid et al. (2001) No Vertical seawall No108 Smid et al. (2001) No Slope 1:1.5 No113 Oumeraci et al. (2001) No Harbor wall Yes224 De Waal (1994) 1:50 Vertical seawall No225 De Waal (1994) 1:20 Vertical seawall No228 Confidentiala No Caisson Yes229 Confidentiala No Caisson Yes315 Confidentiala No Caisson Yes351 Confidentiala No Caisson Yes380 Confidentiala No Caisson Yes402 Confidentiala No Vertical seawall No502 Bruce et al. (2001) 1:10; 1:50 Vertical seawall No503 Bruce et al. (2001) 1:10 Battered 10:1 No504 Bruce et al. (2001) 1:10 Battered 5:1 No505 Bruce et al. (2001) 1:10; 1:50 Composite Yes507 Pearson et al. (2002) 1:13 Battered 10:1 No802 Goda et al. (1975) 1:10; 1:30 Vertical seawall No914 Cornett et al. (1999) No Vertical seawall YesAllsop et al. (1995) Allsop et al. (1995) 1:50 Vertical seawall NoCEPYC Confidentiala No Caisson YesENEL CRIS Confidentiala No Caisson Yes

Note: Datasets fromEnte Nazionale per l’energia Elettrica (ENEL CRIS) and from Centro de Estudios de Puertos y Costas (CEPYC) were received subsequentto CLASH and were not given a CLASH dataset number.aMany commercially-confidential overtopping datasetswere provided to theCLASHdatabase on the basis that theywere not identifiedwidely. These aremarkedas confidential in the table.



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always gave larger overtopping. Within the group of datasetswithout a foreshore slope, there was no notable difference betweenthe caisson-type of structures and plain vertical walls. On the basis ofthis conclusion, the datasets were split into two groups, and eachgroup was then analyzed separately.

Vertical Structures without ForeshoreFig. 9 shows all results for tests without a sloping foreshore. Tests ofDataset 113 with Rc 5 0 have been shifted artificially a little to theright to distinguish them fromDataset 107withRc 5 0. For the lowerfreeboards/larger overtopping rates, the scatter is small. The scatterbecomes larger for Rc=Hm0 . 1. It turns out that Franco et al. (1994)describes these smaller overtopping discharges very well, as inFig. 4. However, in Fig. 4, there were no data for lower freeboards.Fig. 9 shows that Franco et al. (1994) will overpredict overtoppingfor lower freeboards. The other line forAllsop et al. (1995), however,covers this area well, as shown in the graph. This means that bothformulas are valid for vertical structures without a sloping foreshore,but each has their own range of application.

Structures in Fig. 9 can be described as caissons, vertical floodwalls in harbors, and gates of locks in flood situations. They mayhave a berm-type structure relatively deep below water, which doesnot affect overtopping. The description of wave overtopping is thengiven by


m0

q ¼ 0:05 exp

�22:78 Rc

Hm0

�(13)

for Rc=Hm0 , 0:91 (Allsop et al. 1995) and


m0

q ¼ 0:2 exp

�24:3 Rc

Hm0

�(14)

for Rc=Hm0 . 0:91 (Franco et al. 1994).The reliability of Eq. (13) is given by sð2:78Þ5 0:17 and that of

Eq. (14) by sð4:3Þ5 0:6.

Vertical Seawalls on Sloping ForeshoreAll available datasets with a foreshore had a straight, single-gradientforeshore slope, where the wave height was always taken at the lo-cation of the vertical wall. First, h2=ðHm0Lm21,0Þ5 0:23 was used asa discriminator between deflecting or nonimpulsive and impulsivewave conditions. This is approximately equivalent to hp 5 0:3, be-cause of the different wave period measure used. Data withh2=ðHm0Lm21,0Þ. 0:23 were plotted in a graph like Fig. 10, and forh2=ðHm0Lm21,0Þ, 0:23, datawere given in a graph similar to Fig. 11.The nondimensionalization of q used for the y-axis of Fig. 11 isa generalized form of that arising from the manipulation of theEurOtop (Pullen et al. 2007) formulas discussed leading to Eq. (12).An optimum was sought for the best value of h2=ðHm0Lm21,0Þ asdiscriminator and the best parameter group on the vertical axis inFig. 11 by changing the exponents a andb in the expression inEq. (15)

0qffiffiffiffiffiffiffiffiffiffiffigH3

m0

q ¼ const��Hm0

h

�a�sm21,0

�b� Rc

Hm0

�23

(15)

In Eq. (12), the values are a5 0:5 and b521:0. This gives quitea large influence of the wave period on wave overtopping, wherethere is little or no such influence observed on steep slopes and atvertical walls in deep water. It was mainly for this reason that anoptimum was sought for b, where it was expected that the optimumwould be for b. 21.

Analysis confirmed that h2=ðHm0Lm21,0Þ5 0:23 was indeed theoptimumvalue to discriminate between nonimpulsive and impulsive

Fig. 7.Overtopping results of CLASHDataset 802: a sloping foreshorewith a seawall (data from Goda et al. 1975)

Fig. 8. Overtopping results of CLASH Dataset 914: a vertical wall atdeep water (data from Cornett et al. 1999)

Fig. 9. Vertical structures on relatively deep water, with no slopingforeshore



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waves, validating the earlier value of hp 5 0:3. By analyzingexponents a and b in Eq. (15), the conclusionwas drawn that a5 0:5and b520:5 were good values, showing the least scatter and a littleless influence of the wave steepness. On the vertical axis, this givesthe parameter group ½q=ðgH3

m0Þ0:5�=½Hm0=ðhsm2 1,0Þ0:5�.The results for nonimpulsive waves on vertical seawalls at

the end of a sloping foreshore are shown in Fig. 10, with h2=ðHm0Lm21,0Þ. 0:23. The graph shows that Allsop et al. (1995)describes the wave overtopping for these kinds of structures and forgiven wave conditions very well.

The remaining data for an impulsive wave attack are given inFig. 11. Dataset 107 for deep water and zero freeboard was alsogiven for comparison because no data were available for zerofreeboard. There is quite some scatter below the average trend of thedata and almost all of that data belongs to Dataset 28. However, the

other data give a trend of a straight line starting from zero freeboardto rather large relative freeboards (Rc=Hm0 up to ∼1:5 or 2) and thenbecomes amore horizontal trend for very large freeboards. Actually,such a more or less horizontal line goes on even beyond relativefreeboards of Rc=Hm0 5 3e5. Fig. 12 shows a picture of SamphireHoe during a storm where a wave impacted on the vertical wall andjumped high into the air. Under such conditions, even very largerelative freeboards will get overtopping, giving an overtoppingresponse more or less independent of the freeboard, which is in linewith the horizontal trend for highest freeboards in Fig. 11.

In this region of an almost horizontal trend for larger freeboards,a power curve like Eq. (12) will fit quite well as shown in Fig. 11.From that point of view, there is no reason to abandon these kind offormulas. However, it is clear, by definition, that a power functioncannot give the trend for small or zero freeboards because it will notcross the vertical axis, but rather, it uses the vertical axis as anasymptote. This is also clearly shown in Fig. 11 with the dashed line.It is for this reason that it was decided to keep the power function forlarger freeboards and to introduce the common exponential functionfor zero and low freeboards. The formulas are described by


m0

q ¼ 0:011

�Hm0

hsm21,0

�0:5exp

�22:2 Rc

Hm0

�for

Rc=Hm0, 1:35

(16)


m0

q ¼ 0:0014

�Hm0

hsm21,0

�0:5�Rc

Hm0

�23

for Rc=Hm0. 1:35

(17)

The reliability of Eq. (16) is given by sð0:011Þ5 0:0045 and that ofEq. (17) by sð0:014Þ5 0:0006.

Wave overtopping at vertical walls is thus nowgiven byEqs. (13)and (14) (relatively deep water and no sloping foreshore); Eq. (13)

Fig. 10. All data of vertical seawalls on sloping foreshore withh2=ðHm0Lm21,0Þ. 0:23, confirmingAllsop et al. (1995) for deflecting ornonimpulsive waves

Fig. 11.All data of seawalls on sloping foreshore for impulsivewaves [h2=ðHm0Lm21,0Þ, 0:23] andwith optimumvalues of a5 0:5 andb520:5: theupper left dotted line represents the extension of the power law [Eq. (17)] to freeboards below the appropriate range, whereas the dotted line in theextreme lower right shows the extension of the exponential [Eq. (16)] to freeboards higher than the appropriate range



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(sloping foreshore and nonimpulsive waves); and Eqs. (16) and (17)(sloping foreshore with impulsive waves).

All equations use the dimensionless dischargeq=ðgH3m0Þ0:5 on the

left side of the equation, which enables a graph with all predictionequations plotted. A family of curves for impulsive conditions needsto be selected, with various combinations of relative depth and wavesteepness. To do this, three indicative values of the breaker index(Hm0=h5 0:3, 0:5, and 0:9) and two of the wave steepness (sm21,0

5 0:01 and 0:06) are used. Fig. 13 shows the straight lines fornonimpulsive conditions [Eqs. (13) and (14)], as well as the straight/curved lines for impulsive conditions [Eqs. (16) and (17)]. Thecombinations of breaker index and steepness show that the lowestlines (steep waves at deep water) coincide more or less with thenonimpulsive lines, which is what should be expected. Heavybreaking on a very steep foreshore gives the highest lines.

Composite Vertical Structures

For impulsive conditions at composite vertical structures, EurOtop(Pullen et al. 2007) gives

q

d2pffiffiffiffiffiffiffigd3

p ¼ 4:1� 1024�d p

Rc

Hm0

�22:9

(18)

where

dp[ 1:3 dHm0

2phgT2

m21,0

(19)

where d 5 water depth above the berm. In the same way that the hpparameter [Eq. (10)] is used in the EurOtop (Pullen et al. 2007)

method as a discriminator between impulsive and nonimpulsiveconditions at a plain vertical wall, dp discriminates between two setsof formulas for composite vertical structures. In almost the samewayas for Eq. (10), Eq. (19) for dp can be written in a form that offerssome sense of its physical origin

dp� 1:3 dHm0

hLm21,0

(20)

As for composite vertical walls, it is not straightforwardly possibleto analyze in a generic way the differences between impulsive andnonimpulsive forms and it is even harder to get a sense of thephysical transition.

Setting the power index in Eq. (18) to 3 and then following thealgebraic approach used for plain vertical walls


m0

q ¼ b

�dh

�0:5�Hm0

h

�0:51

sm21,0

�Rc

Hm0

�23

(21)

with b5 0:00041. The vertical wall reanalysis of the precedingsection found that the influence of steepness was better representedby sm21,0

0:5. The similarity of the physical situation suggests thatthis adjustment should also be included for the composite structures,giving a tentative prediction equation


m0

q ¼ b

�dh

�0:5�Hm0

hsm21,0

�0:5�Rc

Hm0

�23

(22)

It is immediately clear that this equation offers some physical insight—the apparently separate formulations for plain and compositevertical structures have been reduced to a single set, with the dif-ference between Eqs. (17) and (22) being the constant multiplier anda simple factor of ðd=hÞ0:5, which becomes unity for plain verticalwalls with zero berm height (h5 d).

Before an enhanced prediction scheme can be proposed, a num-ber of further issues require exploration based on the CLASH da-tabase data. Composite structures were identified by vertical upperslopes (cotau 5 0) and by the presence of a toe or mound, i.e., wherethe water depth at the toe or berm is less than that offshore.1. The constant multiplier (b) in Eq. (22) is not the same as the

multiplier for plain vertical walls [Eq. (17)]. Thus, for thesituation of no berm, Eqs. (17) and (22) give different results.Can these two equations be brought together rationally?

2. Does the value of the discriminating parameter dp 5 0:3 fortransition between impulsive and nonimpulsive regimes re-main optimal when applied to the wider CLASH dataset?

3. For plain vertical walls, from the earlier analysis, a number ofdifferent physical situations have been identified andmodeled.The presence or absence of a foreshore was shown to beimportant, as was whether the situation was lower or higherfreeboard, and in the case of situations with foreshore, whetherthe overtopping could be impulsive. It was also establishedthat the wave steepness influence was too great, and it wassuitably reduced. Do these same influences exist for compositestructures?

For Point 1, by comparing the new Eq. (12) (for plain vertical)and Eq. (22) (composite), it is apparent that the two predictorscoincide at a value of d=h � 0:6. This suggests that the mound’sinfluence should cease for conditions where d. 0:6 h, which seemsphysically sensible.

For Point 2, the discriminator was examined in isolation ofadjustments to the prediction equation. EurOtop (Pullen et al. 2007)gives the discriminator dp , 0:3 for impulsive conditions. This

Fig. 12. Impulsive wave overtopping at Samphire Hoe, UnitedKingdom [Courtesy of White Cliffs Countryside Partnership (www.samphirehoe.co.uk)]

Fig. 13. Comparison of overtopping at vertical structures for non-impulsive/deflecting waves and for impulsive waves



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http://www.samphirehoe.co.uk

http://www.samphirehoe.co.uk

criterion, when applied to all CLASH database data for compositestructures, separates these into the plots in Figs. 14 and 15, forconditions predicted to be impulsive or nonimpulsive. The dataidentified as impulsive are well matched with the predictions. Fordata predicted to be in the nonimpulsive regime, it is clear that thereis a group of data at higher freeboards that is significantly under-predicted. The underpredicted data belong to Dataset 505, for whichtherewas a 1:10 foreshore present. Before considering this influence,however, the dp 5 0:3 crossover was tested. Resetting the switchupward to a value of 0.85 improved the success of the predictor inidentifying apparently impulsive conditions and removing theoverpredictions for higher freeboards (Figs. 16 and 17).

Moving from dp 5 0:3 to 0:85 as the critical value, the perfor-mance of the scheme improves. Themean error changes from3.15 to0.87, and the geometric error (measuring the SD of the scatter aboutthe mean of the logarithm of the data) changes from 0.47 to 0.38,indicating an average success in the range 3 =42:4, improved from3 =43:0.

The data that are significantly overpredicted, lyingwell below thelines in Figs. 16 and 17 include many data from Datasets 228 and914, neither of which had foreshores.

For Point 3, the adoption of the adjusted forms of the new verticalwall procedures was then explored. In addition to the advantage ofthe consistency of this approach, such a switch would also bring thephysically sensible behavior at lowest freeboards to the analysis ofovertopping of composite walls. Fixing the conclusions of Points1 and 2, the new vertical wall prediction scheme was then applied,adjusted to composite structures by application of a correction fac-tor of 1:33 ðd=hÞ0:5 for all d=h, 0:6. The multiplier of 1.3 allowscomposite and vertical formulas to coincide at d=h5 0:6


m0

q ¼ 1:3

�dh

�0:5

� 0:011

�Hm0

hsm21,0

�0:5exp

�22:2 Rc

Hm0

�

for Rc=Hm0, 1:35

(23)

Fig. 14. CLASH composite vertical wall data identified as impulsive overtopping according to the EurOtop scheme (Pullen et al. 2007)

Fig. 15.CLASH composite vertical wall data identified as nonimpulsive overtopping according to the EurOtop scheme (Pullen et al. 2007); some dataappear to be wrongly identified as nonimpulsive



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m0

q ¼ 1:3

�dh

�0:5� 0:0014

�Hm0

hsm21,0

�0:5�Rc

Hm0

�23

for Rc=Hm0 $ 1:35

(24)

The composite wall data, excluding those with zero freeboard, areplotted with Eqs. (23) and (24) in Fig. 18. The geometric error is0.39. As previously noted, the exponent of d=hwas set at 0.5, whichis an influence of ðd=hÞ0:5, on the basis of the algebraic manipulationof the EurOtop formulation, after the Allsop et al. (1995) equation.Exploring alternative exponents demonstrated that the 0.5 value isoptimal.

For plainverticalwalls, Fig. 13 offered some physical insight intothe influence of relative depth and wave steepness; the influence ofthe berm is indicated in Fig. 19. From this, it can be seen that underconditions established as impulsive, the berm’s influence is to reduceovertopping discharges. The scale of the influence is not that great,however; it is of the same order of magnitude as the influence ofwave steepness and relative depth on impulsive overtopping at plainvertical walls (Fig. 13).

The scheme for composite structures is thus now alignedwith theimproved vertical scheme, giving physically rational behavior atlowest freeboards (which was not the case for the previous, powerlaw–only scheme). In summary, therefore, overtopping at compositestructures may be considered according to the right side of thedecision chart (Fig. 20). In cases where the mound is small(d=h. 0:6), the structure is treated as vertical. For d=h$ 0:6, in theabsence of a foreshore and possible breaking, the structure is againtreated as plain vertical. In the case of possible breaking, however,the overtopping is arrived at according to the method for plain walls,but with a factor of 1:33 ðd=hÞ0:5 included.

Proportion of Waves Overtopping at Vertical Walls

As for mean discharge, existing guidance offers different formulas forthe statistical distribution of the overtopping volumes associated withindividual wave events (and implicitly, therefore, different formulasfor the estimation of the maximum individual event overtoppingvolume, Vmax). The proportion of overtopping waves forms the basisfor the distribution of individual overtopping volumes. For non-impulsive wave overtopping, the equation for the proportion ofovertopping waves is a Rayleigh distribution (Pullen et al. 2007)

Fig. 16. CLASH composite vertical wall data identified as impulsive overtopping according to adjusted dp discriminator switchover (at dp 5 0:85)

Fig. 17.CLASH composite vertical wall data identified as nonimpulsive overtopping according to adjusted dp discriminator switchover (at dp 5 0:85)



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Now

Nw¼ exp

"2

�1

0:91Rc

Hm0

�2#(25)

For impulsive overtopping waves, EurOtop (Pullen et al. 2007)gives

Now

Nw5 0:031 1

hp

Hm0

Rc(26)

with a minimum described by Eq. (25).The hp can be written as h2=ðHm0Lm21,0Þ [Eq. (10)]. To bring

together Eqs. (25) and (26) so that some physical insight can begained, a relationshipwithRc=Hm0 3 h2=ðHm0Lm21,0Þwas found forimpulsive waves, which was then refitted to a Weibull function

Now

Nw¼ exp

"2

�17:6 h2

Hm0Lm21,0

Rc

Hm0

�0:58#(27)

with a minimum described by Eq. (25).

A Weibull function includes a Rayleigh distribution [b5 2 as inEq. (25)] and an exponential distribution for b5 1. Eq. (27) has aneven lower b value than an exponential distribution, which meansthat it is a very steep distribution. Both formulas [Eqs. (25) and (27)]are shown in Fig. 21, with Rc=Hm0 as the horizontal axis and withlines for various values of h2=ðHm0Lm21,0Þ. The way in which in-dividual maximum volumes for impulsive conditions lift off fromthe nonimpulsive line for higher Rc=Hm0 can be identified clearly, ascan the fact that small values of h2=ðHm0Lm21,0Þ give more over-topping waves.

Very Steep Slopes

Fig. 4.1 in EurOtop (Pullen et al. 2007) gives an overall view ofovertopping on various types of structures. Fig. 4.2 in EurOtop(Pullen et al. 2007) shows that smooth steep sloping structureswith nonbreaking wave conditions give the largest wave over-topping, and this should decrease for very steep (battered) andvertical walls. What happens if slopes become steeper than∼1: 1:5? The two boundaries are known: Fig. 3 for steep smooth

Fig. 18. Comparison of all CLASH database tests for composite structures with foreshore with new scheme for composite structures, based on newvertical wall approach, adjusted dp cutoff; applied for all d=hs , 0:6

Fig. 19.Comparison of overtopping at composite and plain vertical structures for nonimpulsive/deflectingwaves and for impulsivewaves (cf. Fig. 13)



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Fig. 20. Decision chart showing new schemes, vertical to left and composite to right: these schemes use identical physical rationalizations, with thecomposite scheme incorporating a berm influence factor of ðd=hÞ0:5 for all d=h, 0:6, plus an adjusted multiplier ensuring that the schemes meet atd=h5 0:6

Fig. 21. Proportion of waves overtopping: nonimpulsive and impulsive conditions, showing effect of impulsiveness parameter, h2=ðHm0Lm21,0Þ[Eq. (10)]: solid line [h2=ðHm0Lm21,0Þ. 0:23] represents nonimpulsive conditions; lower h2=ðHm0Lm21,0Þ conditions are increasingly stronglyimpulsive



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slopes and Fig. 9 for vertical walls at relatively deep water. Thisquestion could be answered most easily if both figures could bebased on similar equations. This is achieved by also fittinga Weibull-type function through the data in Fig. 10. That datatogether with the new fit [Eq. (21)] and the fit for steep smoothslopes [Eq. (7)] are given in Fig. 22. The vertical wall data (deepwater and nonimpulsive with foreshore with Rc=Hm0 , 1) witha new fit give


m0

q ¼ 0:047� exp

242 2:35 Rc

Hm0 � gf � gb

!1:335 (28)

The reliability of Eq. (28) is given by sð0:047Þ5 0:007 andsð2:35Þ5 0:2. Eq. (28) is not the very best fit—that would be an

equation with a smaller exponent than 1.3. However, there is anadvantage in using 1.3, because the equation is then similar toEq. (7), facilitating a comparison between, and joining of, themethods. The resulting curve (Fig. 22) is still a good fit, consideringthe scatter.

Eq. (7), for steep slopes and nonbreaking waves, and Eq. (28),for vertical walls, have the same shape and only differ in co-efficient and exponent. The connecting parameter is the slopeangle cota.Without any data, onewould probably choose a linearinfluence to combine Eqs. (7) and (28) to one general formula.Recently, however, very interesting data from Victor (2012)became available (Victor et al. 2012). In total, 366 tests wereperformed on steep and very steep smooth slopes with relativelylow freeboards (Fig. 23). Tested slope angles were cota5 0:36,0:58, 0:84, 1:0, 1:19, 1:43, 1:73, 2:14, and 2:75. The range of rel-ative freeboards was 0:11 ,Rc=Hm0 , 1:7. Some of the tests onslope angles of cota5 2:14 and 2:75 belonged to the breakingwave region [Eq. (6)]; most were, however, nonbreaking. Thesedata are given in Fig. 23, together with Eqs. (7) and (28) and on thesame scale as Fig. 3. The range of slope angles covers the wholearea between the two curves in Fig. 23, although vertical wallswere not tested.

Eq. (5) was fitted to the data in the nonbreaking region of eachindividual slope angle, using c5 1:3 and fitting a and b. Thesevalues of a and b were then plotted versus slope angle cota inFig. 24. A rough trend would indeed be a linear expression, butthe real trend is a little more curved. The most gentle slopes ofcota5 2:14 and 2:75 were perfectly matched by Eq. (7)(Fig. 25), where a slope angle with cota5 1:73 showed the firstdeviation from this equation. One could say that wave over-topping starts to decrease if cota, 2, although very slowly.Lines were fitted through the data points with a and b and thefollowing equations were found, which should be used in com-bination with Eq. (5):

a ¼(0:092 0:01ð22 cotaÞ2:1 for cota# 2

0:09 for cota. 2(29)

Fig. 22. Vertical wall data (deep water and nonimpulsive waves witha sloping foreshore and Rc=Hm0 , 1) with new fit and curve for slopesunder nonbreaking wave conditions

Fig. 23. Data of Victor (2012) with very steep slopes from cota5 0:36 to 2:75 and fairly low relative freeboards (nonbreaking data only)



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b ¼�1:5þ 0:42ð22 cotaÞ1:5 with a maximum of 2:35 for cota# 21:5 for cota. 2

(30)

c ¼ 1:3 (31)

Fig. 25 gives the data for slope angles of cota5 0:36, 0:58,0:84, 1:19, 2:14, and 2:75 with Eqs. (5) and (29)–(31). The curvesgive a good trend of the data and also show that the data of the mostgentle slopes of cota5 2:14 and 2:75 match the original well[Eq. (7)] for steep slopes. Thus, it is now possible to describe waveovertopping, for nonbreaking waves, for steep slopes up to verticalwalls with only Eq. (5) with a, b, and c in Eqs. (29)–(31).

Conclusions

The theoretical analysis of Battjes (1974) for gentle smooth slopeshas been revisited. Battjes’ method—a curved line on a log-lineargraph—is shown to describe the whole range of overtoppingresponses extending down to zero freeboard,which is something that

is not possible with the conventional exponential-type overtoppingformulas. Weibull-type formulas are proposed, describing waveovertopping at slopes for the whole range of Rc=Hm0 $ 0 [Eqs. (6)and (7)].

To improve the accuracy of prediction methods for verticalstructures, it is demonstrated that those in relatively deep waterwithout a sloping foreshore should be distinguished from seawallsat the end of a sloping foreshore. For no foreshore, the Franco et al.(1994) formula [Eq. (14)] is valid for larger freeboards and theAllsop et al. (1995) formula [Eq. (13)] is valid from zero free-board until crossing with the Franco formula. The discriminatorh2=ðHm0Lm21,0Þ identifies the onset of impulsive (breaking or im-pacting wave) overtopping at a seawall on a sloping foreshore intononimpulsive conditions [h2=ðHm0Lm21,0Þ. 0:23]. Under theseconditions, Allsop et al. (1995) give the prediction formulas. Forsmaller values of h2=ðHm0Lm21,0Þ, breaking waves give larger waveovertopping and may give significant discharge even for largefreeboards.

Overtopping at vertical structures in relatively deep water canalso be described by oneWeibull-type formula [Eq. (28)], similar tothe formulas for slopes [Eqs. (6) and (7)].

Recent work by Victor (2012) provided data in the previouslypoorly covered region of steep slopes. Analysis of the Victor datashows that there can be a continuousmethod between (smooth) steepslopes and vertical walls governed simply by the slope angle cot a.Overtopping at straight slopes and vertical structures (in deep water,no sloping foreshore) can be described by the following set offormulas:


m0


p gb � jm21,0

� exp

242 2:7 Rc

jm21,0 � Hm0 � gb � gf � gb � gv

!1:335Fig. 24.Coefficient a and exponent b in Eq. (5) (with c5 1:3),fitted forslope angles with cota5 0:36e2:75 (data from Victor 2012)

Fig. 25. Very steep slopes with cota5 0:36, 0:84, 1:19, 2:14, and 2:75 and Eqs. (5), (29), and (30) (data from Victor 2012)



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with a maximum of


m0

q ¼ a� exp

"2

b

Rc

Hm0 � gf � gb

!c #where

a5

(0:0920:01ð22cotaÞ2:1 for cota# 2

0:09 for cota. 2

b ¼�1:5þ 0:42ð22 cotaÞ1:5 with a maximum of 2:35 for cota# 21:5 for cota. 2

and c5 1:3.Overtopping at composite structures can be analyzed according

to a close analog of this new scheme for plain vertical structures.Adjustments are applied for berms higher than d=h, 0:6. Theadjustment is simply a factor of 1:3ðd=hÞ0:5. A decision chartsummary of the proposed unified schema for plain vertical andcomposite structures is presented (Fig. 20).

Acknowledgments

The authors thank EurOtop coauthors William Allsop, AndreasKortenhaus, Tim Pullen, and Holger Schüttrumpf, and LeopoldoFranco, for useful discussions and efforts to retrieve original data.With great pleasure, the authors received and used the data of GhentUniversity on very steep slopes and low freeboards (Peter Troch andLander Victor).

Notation

The following symbols are used in this paper:a, b 5 coefficients or exponents in formulas;

c 5 shape factor in the Weibull distribution;g 5 acceleration from gravity (5 9:81m=s2);

Hm 5 mean wave height;Hm0 5 estimate of significant wave height from spectral

analysis 5 4ffiffiffiffiffiffim0

p;

Hs 5 significant wave height defined as highest one-third of wave heights, Hs 5H1=3;

H1=3 5 average of highest third of wave heights;h 5 water depth at (in front of) toe of structure;hp 5 discriminator between nonimpulsive and

impulsive wave overtopping [Eq. (10)];Lm21,0 5 deep water wavelength based on Tm21,0;

Lm21,0 5 gT2m21,0=2p;

L0 5 deepwaterwavelength basedonTm;L0 5 gT2m=2p;

Now 5 number of overtopping waves;Nw 5 number of incident waves;q 5 mean overtopping discharge per meter structure

width;Rc 5 crest freeboard of structure;

sm21,0 5 wave steepness with Lm21,0, based on Tm21,0;sm21,0 5Hm0=Lm21,0 5 2pHmo=ðgT2

m21,0Þ;Tm 5 average wave period from time-domain analysis;

Tm21,0 5 spectral period defined by m21=m0;T1=3 5 average of the periods of the highest third of wave

heights;X 5 parameter group at x-axis;

Y 5 parameter group at y-axis;a 5 angle between overall structure slope and

horizontal;ad 5 angle between structure slope downward berm and

horizontal;gx 5 influence factor in overtopping formula;k 5 correlation parameter of the bivariate Rayleigh

distribution;mðxÞ 5 mean of measured parameter x with normal

distribution;jm21,0 5 breaker parameter based on sm21,0;

jm21,0 5 tana=s1=2m21,0; andsðxÞ 5 SD of measured parameter x with normal

distribution.

References

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http://dx.doi.org/10.1016/j.coastaleng.2008.09.007






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