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Journal of Hydro-environment Research 1 (2008) 216e225www.elsevier.com/locate/jher

Wave breaking macrofeatures on a submerged rubble mound breakwater

Mario Calabrese a, Mariano Buccino a, Francesco Pasanisi b,*

a University of Napoli Federico II, Department of Hydraulic and Environmental Engineering ‘‘Girolamo Ippolito’’, Via Claudio, 21. 80125 Napoli, Italyb ENEA, Department Environment, Global Changes and Sustainable Development, Portici Research Center, Localita Granatello. 80055, Portici (Napoli), Italy

Received 13 February 2007; revised 10 October 2007; accepted 20 November 2007

Abstract

Results of an extensive experimental study on the interaction between waves and submerged barriers are presented. In particular, the presentpaper deals with identification of breaking conditions and morphological description of breaker types occurring on a submerged rubble moundbreakwater. Barriers with varying geometry and permeability were used for tests. Video data were collected and analyzed and, in case of break-ing, different classes and subclasses of breaker types were described and classified, considering: (a) the features of the incipient mechanism ofwave rupture, (b) the shape of wave profile at breaking and (c) the evolution of breaker travelling above and leeward of the barrier. The breakerindex, i.e. wave height to crest freeboard ratio at incipient breaking was studied as a function of the structure layout and wave parameters; ex-perimental results were found to reasonably agree with Goda and Morinobu (1998) findings and a best-fit expression for scale parameter in Godabreaking formula was derived. A first parameterization of breaker types was then proposed, considering both wave and structure characteristics,described by appropriate non-dimensional numbers.� 2008 Published by Elsevier B.V. on behalf of International Association for Hydraulic Engineering and Research, Asia Pacific Division.

Keywords: Wave-structure interaction; Submerged breakwater; Wave breaking

1. Introduction

Over the last decades, submerged breakwaters have receiveda growing appreciation from engineers, authorities and public,as they appear to be capable of providing coastal protectionand, at the same time, avoiding negative impacts on landscapeand coastal environment. For particular site conditions,submerged barriers can be designed with dual objectives ofproviding coastal protection and enhancing surfing conditions,as the artificial reef at Gold Coast, Australia (Black, 2001).

However, despite recent research, a certain lack of knowl-edge about qualitative and quantitative features of wave-struc-ture interaction can still be noticed, that can dramaticallyaffect reliability on predicting the hydrodynamic conditionsin the sheltered area and shoreline response to such kind of

* Corresponding author. Tel.: þ39 0817723306; fax: þ39 0817723344.

E-mail addresses: calabres@unina.it (M. Calabrese), buccino@unina.it

(M. Buccino), francesco.pasanisi@portici.enea.it (F. Pasanisi).

1570-6443/$ - see front matter � 2008 Published by Elsevier B.V. on behalf of Internatio

doi:10.1016/j.jher.2007.11.003

structures (Dean et al., 1997; Stauble and Tabar, 2003; Rana-singhe and Turner, 2006).

In a 2D framework the interaction between a wave anda submerged barrier basically consists of two main processes:

1. wave energy reduction caused by reflection, frictionand wave breaking over the barrier;

2. wave spectrum change due to non-linear processes in theshallow water area above the breakwater.

A large body of literature exists on the first aspect. Exper-imental studies and curve-fitting aimed at predicting the ratiobetween transmitted and incident wave height (i.e. transmis-sion coefficient) can be found in d’Angremond et al. (1996),Seabrook and Hall (1998), Calabrese et al. (2005), van derMeer et al. (2005) and Buccino and Calabrese (2007).

Regarding the second aspect, propagation of monochro-matic non-breaking waves above a submerged rectangular im-permeable step was studied theoretically by Massel (1983) andGoda et al. (1999) using Stokes’ formulation at 2nd order

nal Association for Hydraulic Engineering and Research, Asia Pacific Division.

217M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

approximation. It was pointed out that above the barrier, due tolimited depth, multiple-frequency harmonic components aregenerated, either as bound waves (i.e. with the same celerityand initial phase as the fundamental harmonic) or free waves(i.e. with their own initial phase and celerity, obeying disper-sion equation). As wave train crosses the step, both types of su-per-harmonic components are propagated in the sheltered zone,affecting the frequency distribution of transmitted wave energy.

Investigations through both mathematical and physical modelstudies on various types of submerged structures, mostly dealingwith non-breaking waves, have been reported by Ohyama andNadaoka (1992, 1994), Ting and Kim (1994), Ohyama et al.(1995), Losada et al. (1997) and Huang and Dong (1999).

Experimental studies dealing with breaking waves over thebarrier generally showed a reduction of the transmitted super-harmonic wave components, even with different features(Grue, 1992; Yamashiro et al., 1999; Brossard and Chagdali,2001). It suggests that wave breaking could induce a saturationin the non-linear processes on top of the structure, although themechanism was not further investigated and some aspects re-main unclear.

In general, not much is reported on wave breaking oversubmerged breakwaters.

Nakamura et al. (1966), Battjes and Beji (1992) and Bleckand Oumeraci (2002) focused essentially on wave energy dis-sipation and on the consequences of breaking rather than onthe process itself. Further, the above authors used smooth im-permeable structures in their tests. Effects of porosity were in-vestigated by Hattori and Sakai (1994) based on tests ontrapezoidal submerged breakwaters made of different mate-rials. Empirical formulae for the prediction of the limitingwave height to submergence ratio were proposed, based onthe so-called ‘‘modified surf-similarity parameter’’, originallyintroduced by Hara et al. (1992) for solitary waves.

Takikawa et al. (1997) investigated wave breaking featuresby means of numerical and physical models, and pointed outthat breaker types at submerged barriers can be quite differentfrom those observed on slopes or over natural beaches. Variousbreaker forms were classified on the basis of video analysis.However, a rectangular smooth impermeable barrier wasused; results on real rubble mound breakwaters could be sig-nificantly different from those observed in the model.

0.50 m

13.00 m

PLAN

CROSS-SECTION

0.75 m

glass wallbarrier

barrier

Fig. 1. Wave flume

Kawasaki and Iwata (2001) performed tests using smoothimpermeable barriers having a horizontal step with an offshoreapproach slope. Breaking limits for different values of off-shore slope and different breaker forms were observed at thebarrier.

To have a better understanding of the interaction betweena wave and a submerged barrier, considering, specifically,the influence of wave breaking, an extensive experimentalstudy was conducted at University of Napoli Federico II. Itcovers different aspects of the wave-structure interaction, in-cluding the amount of transmitted wave energy, harmonic gen-eration on the top of the barrier, free and bound wavestransmission at the trailing side and the characteristics of thewave breaking over the structure.

2. Experimental setup

Experiments were performed in a wave flume (Fig. 1) atUniversity of Napoli Federico II, Department of Hydraulicand Environmental Engineering ‘‘Girolamo Ippolito’’. Theflume was 24.50 m long, having two different sections;the first, closer to the wavemaker, had converging walls, andthe width varied from 0.80 m at the wavemaker to 0.50 m ata distance of 8.50 m; the second section, with glass walls,was 13.00 m long, with a constant width of 0.50 m. The heightof flume was 0.75 m. Waves were generated by a piston-typewavemaker, provided with dynamic wave absorber. Absorbingshingle beaches were provided at both ends of the flume.

Two different geometries of rubble mound breakwater weretested, one was with a seaward face slope of 1:2 (steep struc-ture) and the other with a 1:10 slope (gentle-slope structure),with all other parameters constant (Fig. 2).

In order to control the permeability of breakwater, a homo-geneous section was adopted, with D50¼ 6.50 cm, obtaininga bulk porosity equal to 0.44. Permeability of the structurewas then varied by inserting or removing impermeablewooden plates in the cross-section of the barrier. The resultsreported pertain to both the cases of the permeable breakwater(absence of plate) and the impermeable breakwater (plate highas the entire cross-section).

Regular waves were used for the tests, with target waveheights Hi at the wavemaker ranging from 2.0 cm to 12.0 cm,

wave makerpaddle

0.80

m

8.50 m

steel wall

used for tests.

h s =

25.

0 cm

Lm = 112.5 cm

incident wave

1.51 1

2

Rc = 6.5 cm

h =

31.5

cm

B = 25.0 cm

110

Lm = 312.5 cm

Fig. 2. Layout of the structures used for tests.

218 M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

having three wave periods T, namely 1.0 s, 1.5 s and 2.0 s. Re-sistive twin-wire gauges were used to measure surface eleva-tion in four positions, namely, offshore, at the centre of theseaward slope face, at the centre of the crown and in the leeside of the structure, outside the wave reforming area. Surfaceelevation data were analyzed in both time and frequencydomain.

Video camera recordings were used to investigate the oc-currence of breaking and the related morphological featuresof breakers. In order to reduce subjectivity of the analysis,observations were recorded independently by differentobservers.

3. Classification of breaker forms

3.1. Classification criteria

Defining the criteria for the classification of breakerforms is not easy; in fact, given the qualitative nature ofthe analysis, any classification runs the risk of being arbi-trary. Moreover, breaker types form a continuous spectrum.Hence, any classification is a necessary compromise be-tween an accurate description and the obvious need to avoidproliferating the number of classes loosing physicalmeaning.

It was therefore assumed that a reasonable and as-objec-tive-as-possible method for classification could be based onthe features of incipient mechanism of wave rupture, the shapeof wave profile at breaking and the evolution of breaker trav-elling above and leeward of the barrier.

For all cases of breaking, independent of the breakershapes, structure-induced collapsing of waves was observedto get initiated with a water jet moving from the wave frontwhich is projected in the wave direction striking the onwardwater. According to early literature, the above water wedgewill be referred to as ‘‘plunging jet’’. As soon as plungingjet hits the water surface (plunging point), supplementaryjets arise, which govern the breaker evolution across thebarrier.

Thus, classification criteria proposed in the following werefirst based on:

- the part of the wave front from which plunging jet is pro-jected forward;

- the shape of water surface at breaking point.

Altogether, based on above discriminators, four main cate-gories of breakers were distinguished, namely:

- ‘‘spilling-to-plunging’’ breakers;- ‘‘bore’’ breakers;- ‘‘collapsing-surging’’ breakers;- ‘‘two-steps’’ breakers.

Within the main classes, further subclasses were then rec-ognized, depending on the shape of the free surface as thebreaker travels over the barrier.

3.2. Spilling-to-plunging breakers (StP)

Spilling-to-plunging breakers were observed when thestructure is wide enough compared to the incident wavelength.

In the incipient breaking, the plunging jet starts from thecrest and the wave profile at breaking is skewed in sucha way that onward face is steep and rear one slopes down tothe sea (Fig. 3a, c). These breaker forms are substantially iden-tical to the corresponding ones described by Galvin (1968) onslopes and natural beaches.

As previously observed by several authors (Peregrine,1983; Basco, 1985), both spilling and plunging breakingscan be regarded as part of the same hydraulic process, differ-ing from each other only by the scale of vortices induced bythe wave rupture. As soon as plunging jet hits the water sur-face, a so-called plunger vortex is generated at the plungingpoint, while another vortex (surface roller) arises on the steepfront due to the gradient in the velocity field. Difference be-tween spilling and plunging breaking is essentially related tothe intensity of vortices, that depends on the strength of plung-ing jet. In Fig. 3 the scheme proposed by Basco (1985) isreported.

In ‘‘spilling’’ breakers (S ), momentum of the plunging jetis weak (Fig. 3b) and its impact with water occurs very closeto the wave crest, under conditions of which the surface rolleris much more intense than the plunger vortex.

In the case of ‘‘plunging’’ breakers (P), the strength of thejet is high and the water wedge curls over the shoreward faceof wave profile and falls into the base of the oncoming wave(Fig. 3d); during the process of which, a continuous air cylin-der is entrapped in wave profile and the momentum of thewedge allows it to penetrate the water, so that a secondarywave is possibly generated and pushed forward; in this case

Fig. 3. a-b: Spilling breaker; c-d: plunging breaker. Comparison with hydrodynamic scheme proposed by Basco (1985).

219M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

the intensity of plunger vortices and surface roller at breakingcan be considered comparable.

The so-called ‘‘Spillingeplunging’’ breakers (SP) mayoccur as intermediate cases between spilling and plunging; ob-viously, defining boundaries between spilling and spillingeplunging, as well as between spillingeplunging and plungingis based on the judgment of the observer. However, the ab-sence of air bubbles under the wave trough and the presenceof an air cushion between plunging jet and wave profileseemed to be reasonable criteria for defining previous borders.

3.3. Bore breakers (B)

The terms ‘‘bore’’ (B) or ‘‘hydraulic jump’’(HJ ) are usedherein to indicate, by analogy with river hydraulics, thatwave profile at incipient breaking consists of two parallelnearly horizontal stretches at different levels, connected bya central turbulent zone, coming from the plunging down ofone or more jets detaching from wave crest (Fig. 4a, c).

This form was typically observed at the permeable steep-faced breakwater and closely resembles an ‘‘intermediate’’ hy-draulic jump, the shape of which is halfway between classicalHJ (direct HJ, with a continuous foam cushion) and that of un-dular unbroken one.

Generally, as the breaker travels over the breakwater, itsshape remains almost stationary, gradually vanishing in the

lee of the structure (Fig. 4b). Sometimes, however, after firstrupture the wave profile steepens on the top of the barrier(Fig. 4d) and breaker evolves in a way quite similar to StP,with a descending slope profile in the offshore direction. Thesecases were indicated as ‘‘BoreeStP’’ (BeStP) breakers.

3.4. Collapsing-surging breakers (C)

Collapsing-surging breakers were observed only at the 1:2seaward slope impermeable structure.

These breaker types are characterized by a plunging jet ini-tially moving from the lower part of the wave profile (Fig. 5a,c). In some cases, the leading edge of the wave front appearsbubbled, with unbroken wave profile similar to classical‘‘surging’’ breakers (Fig. 5a) observed on steep slopes (Galvin,1968); in other cases, the wave front collapses in the lower partof profile, analogously to ‘‘collapsing’’ breakers also describedby the same author on steep slopes. A common feature of thesebreaker types is that, opposite to StP, the free surface behindthe breaker zone has a rise in slope towards offshore direction(Fig. 5a, c).

Within this class, two different ‘‘sub-forms’’ were recog-nized, depending on the evolution of wave profile after break-ing. For the lower heights, wave profile on the breakwater topevolves quite similar to a hydraulic jump (Fig. 5b) and was re-ferred to as ‘‘collapsingebore’’ breakers (CeB). Otherwise, as

Fig. 4. a-b: Bore breaker; c-d: boreeStP breaker. Comparison with hydraulic jump.

220 M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

incident wave momentum increases, water surface on thecrown becomes skewed, wave profile assumes a descendingslope in the offshore direction and a supplementary jet is pro-jected forward from the wave crest (Fig. 5d), similar to StPbreakers described above; in which case, the breaker was clas-sified as a ‘‘collapsingeStP’’ (CeStP) breaker.

3.5. Two-steps breakers (2)

These particular breaker types were noticed only over the1:10 seaward slope breakwater exposed to longer waves.

The general morphology is quite similar to StP breakers,with profile showing a steep onward face and descending slopein offshore direction. In these cases, however, prior to

c

COLLAPSING

SWL

(Galvin, 1968)

a

(Galvin, 1968)

SURGING

SWL

Fig. 5. a-b: Collapsing-bore breake

detaching of the plunging jet from the crest, a small jet is rec-ognized to initially move from the middle part of the front, ini-tiating the breaking (Fig. 6a, c).

After this ‘‘first’’ breaking, the process continues witha ‘‘second’’ breaking, i.e. the collapse of wave crest, like inStP. If the momentum of both the lower and upper plungingjets is relatively weak, the first wedge may disappear (‘‘two-steps spilling’’, 2S, Fig. 6b), whilst for more intense breaking,i.e. for larger wave heights, the two wedges join in a uniqueplunging jet that is pushed shoreward (‘‘two-steps spillingeplunging’’, 2SeP, Fig. 6d).

It should be mentioned that an analogous kind of breakerwas possibly observed by Takikawa et al. (1997) on a rectangu-lar impermeable step and by Kawasaki and Iwata (2001) for

b

d

r; c-d: collapsingeStP breaker.

Fig. 6. a-b: Two-steps spilling breaker; c-d: two-steps spillingeplunging breaker.

221M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

a trapezoidal geometry; however, both works do not providesufficient information on breaker morphology to assess ifsome differences exist with the present study.

Table 1

Classification of breaker types (all tests)

Wave Breaker type

4. Discussion of results conditions

T (s) Hi Slope Slope Slope Slope

4.1. General (cm) 1:2 imperm. 1:2 perm. 1:10 imperm. 1:10 perm.

1.0 2.00 NO NO NO NO

1.0 3.00 NO NO NO NO

1.0 4.00 S NO S NO

1.0 5.00 S NO S NO

1.0 6.00 S S S S

1.0 7.00 SeP S S S

1.0 8.00 SeP SeP SeP S

1.0 9.00 SeP SeP SeP SeP

1.0 10.00 P P SeP SeP

1.0 11.00 P P SeP SeP

1.5 2.00 NO NO NO NO

1.5 3.00 NO NO NO NO

1.5 4.00 B NO NO NO

1.5 5.00 CeB S 2S NO

1.5 6.00 CeB BeStP 2S NO

1.5 7.00 CeB BeStP 2S S

1.5 8.00 CeStP BeStP 2SeP S

1.5 9.00 CeStP BeStP 2SeP SeP

1.5 10.00 CeStP BeStP SeP SeP

1.5 11.00 CeStP SeP SeP SeP

2.0 2.00 NO NO NO NO

2.0 3.00 NO NO NO NO

2.0 4.00 NO NO NO NO

2.0 5.00 B NO BeStP NO

2.0 6.00 CeB B BeStP NO

2.0 7.00 CeB B SeP S

2.0 8.00 CeB B SeP S

2.0 9.00 CeB B SeP 2SeP

2.0 10.00 CeB B SeP 2SeP

2.0 11.00 CeB B SeP SeP

NO: non-breaking; S: spilling; SeP: spillingeplunging; P: plunging; B: bore;

BeStP: boreeStP; CeB: collapsingebore; CeStP: collapsingeStP; 2S: two-

steps spilling; 2SeP: two-steps spillingeplunging.

Observed breaker forms for all tests are summarized inTable 1; from first observations of results some general quali-tative considerations are discussed.

First, it can be observed that, for any structure geometry, aswave height grows, keeping period constant, transition fromnon-breaking to breaking conditions takes place at imperme-able barrier for a lower value of wave height than the one ob-served over the permeable structure. This is in agreement withprevious study by Hattori and Sakai (1994), who comparedsmooth impermeable barrier and rubble-mound structuresmade of gravel or blocks; authors noticed the effect of struc-ture porosity in delaying wave rupture, and proposed formulasin which breaking wave height grows as an exponential func-tion of porosity.

From video analysis, a possible interpretation of this cir-cumstance can be hypothesized; in fact, at impermeable bar-rier, a more intense depression of water profile at seawardface of breakwater takes place, compared to permeable break-water (Fig. 7a, c); this produces a steeper wave profile(Fig. 7b, d), making the occurrence of breaking easier. Forthe 1:2 slope, the depression at offshore tip can be of sucha magnitude that seaward face of the breakwater is nearly ex-posed, leading to collapse of the lower part of wave front. Thereason for that can be probably ascribed to wave reflection,which is more intense for impermeable structure. However,in addition, the effect of a return current flowing above andthrough the structure must be considered. This current shouldarise from the need of compensating the water mass conveyedby wave in the sheltered area and, of course, enhances the

Fig. 7. Comparison between plunging breaking sequences on steep-slope structure. a-b: Impermeable breakwater; c-d: permeable breakwater. Incident wave

conditions are the same: H¼ 11.0 cm, T¼ 1.0 s.

222 M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

steepening of wave profile. In the case of an impermeablestructure, the current is forced in the shallow water region abovecrown, and becomes likely more intense than in the case of per-meable breakwater. The stronger the flow, the steeper wave pro-file is expected to be, so that breaking becomes more probable.This assumption appears to be reasonable, though further inves-tigations with velocity measures are needed.

It is noteworthy that in present study, porosity of breakwaterwas kept constant for all tests, given that permeability was variedonly by inserting or removing a sheet inside the cross-section;thus, results seem to show that overall permeability, rather thanporosity, affects wave breaking features above the barrier.

Another general consideration can be made on the effect ofoffshore slope. Experimental data show that for growing waveheights, keeping invariant wave period and structure perme-ability, first breaking takes place at lower values of waveheight for the steeper structure with respect to the milderone. Results are consistent with the findings of Goda andMorinobu (1998) for horizontal shelf with approach slope,on contrary to what happens on natural beaches, for whichbreaker index grows with profile slope. The authors explainedthis result as a consequence of higher order free waves gener-ated at the junction point between slope and flat bottom, due tobathymetric discontinuity. Even if this aspect was not furtherinvestigated, it can be hypothesized that low-period harmonicsin wave train are more likely to initiate wave breaking, due totheir steepness, as well established in literature for monochro-matic waves (Peregrine, 1983; Rapp and Melville, 1990).

To confirm, it can be observed that as wave period grows,transition from non-breaking to breaking waves tends to occurat a higher value of wave height, in agreement with results ofHattori and Sakai (1994).

Table 2

Best-fit values for scale parameter A in Eq. (1)

4.2. Breaker index

1:10 Seaward slope 1:2 Seaward slope

Impermeable structure 0.160 0.155

Permeable structure 0.240 0.203

For each structure geometry and permeability, the transition

from non-breaking to breaking conditions was examined. In

particular, the breaker index, i.e. the lower value of incidentwave height (Hi) to submergence (Rc) ratio for which wavebreaking was observed to occur, was studied as a function ofstructure layout; as a preliminary step of the analysis, the for-mulation of Goda (1974), given below, was used:

Hi

Rc

¼ AL0

Rc

�1� exp

�� 1:5

pRc

L0

��ð1Þ

where, beside known symbols, L0 is the deep water wave-length, function of wave period T:

L0 ¼gT2

2pð2Þ

being g the gravitational acceleration.The scale parameter A was fitted to data. The results are re-

ported in Table 2.In Figs. 8e11, experimental points and Eq. (1) are plotted

for the different layouts of breakwaters used for tests. Despitesome scatter in Fig. 10, results show that Eq. (1) can be effi-ciently used for separating breaking and non breaking zones.As expected from previous discussion, scale parameter de-creases as seaward slope increases, and increases with perme-ability of breakwater.

By introducing notational permeability P (van der Meer,1988), data from Table 2 may be condensed in the followingformula:

A¼ 0:17 expð0:56P� 0:28mÞ ð3Þ

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Rc/L0

Hi/R

c

Non-breakingSpillingSpilling-plungingPlungingBoreCollapsing-boreCollapsing-StPEq. (1) - A=0.155

impermeable breakwater - 1:2 slope

Fig. 8. Breaker index for the 1:2 seaward slope impermeable breakwater.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Rc/L0

Hi/R

c

Non-breakingSpillingSpilling-plungingBore-StP2 Steps spilling2 Steps S-PEq. (1) - A=0.160

impermeable breakwater - 1:10 slope

Fig. 10. Breaker index for the 1:10 seaward slope impermeable breakwater.

223M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

where, according to author, P¼ 0 for impermeable structureand P¼ 0.60 for permeable one; m represents seaward slope(m¼ 0.5 for steep slope, m¼ 0.1 for milder slope).

4.3. Parameterization of breaker types

In the study of wave breaking on natural beaches, a univer-sally accepted criterion for prediction of breaker shapes isavailable, based on the well known surf similarity parameter,originally introduced by Galvin (1968). On contrary, in earlyliterature no general method is reported for breaker forms atsubmerged breakwater.

In the present work, a parametric study was attempted, con-sidering both wave parameters and structure properties. Fromexperimental data analysis two main non-dimensional param-eters are derived, as given below:

R�d ¼mffiffiffiffiffiHi

L0

r Hi

Rc

expð � 1:7PÞ ¼ m

ffiffiffiffiffiffiffiffiffiHiL0

p

Rc

expð � 1:7PÞ ð4Þ

and

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Rc/L0

Hi/R

c

Non-breakingSpillingSpilling-plungingPlungingBoreBore-StPEq. (1) - A=0.203

permeable breakwater - 1:2 slope

Fig. 9. Breaker index for the 1:2 seaward slope permeable breakwater.

xB ¼hs

hs

mþB

�Hi

L0

��0:5

ð5Þ

where Hi is the incident wave height, B is the crown width andhs is the height of the top of the structure over the bottom.

First parameter R�d includes the surf similarity parameter atstructure, the incident wave height and the freeboard above thebarrier; thus, it is proportional to run-down to depth ratio, thatreasonably governs the instability of wave trough and, conse-quently, the occurrence of collapsing-surging breakers. Effectof permeability, that significantly influences run-down, wasempirically included in Eq. (4) by means of an exponentialfunction of P.

The second parameter xB can be considered as a ‘‘fictional’’surf similarity parameter that includes both offshore slope andbarrier width; the first term in the product in Eq. (5) representsthe mean slope between the breakwater toe and the leeward tipof the crown. This parameter should rule the effects of thebarrier width, compared to wave parameters; note that it re-duces either as the crown width enlarges or the front slopediminishes.

In Fig. 12 all experimental data are plotted against the pa-rameters above introduced.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Rc/L0

Hi/R

c

Non-breaking

Spilling

Spilling-plunging

2 Steps S-P

Eq. (1) - A=0.240

permeable breakwater - 1:10 slope

Fig. 11. Breaker index for the 1:10 seaward slope permeable breakwater.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.00 0.50 1.00 2.001.50 2.50 3.00 3.50 4.00

StP (including 2-Steps)BoreBore-StPCollapsing-BoreCollapsing-StP

limit RD* for collapsing occurrence

limit ξB for skewedprofile on barrier top

limit ξB for “pure” StP

B

RD

Fig. 12. Breaker types parameterization.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.00 0.50 1.00 1.50 2.00 2.50

Lm/L0

Hi/R

Ceq

Spilling (including 2-Steps)

Spilling-plunging (including 2-Steps)

Plunging

permeable breakwater

Fig. 14. StP breakers evolution for permeable breakwater.

224 M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

Despite a certain scatter, some interesting reasoning seemsto arise from the plot. First, a limit value of R�d seems to exists(say 3.5, horizontal dashed line), beyond which wave breakingoccurs only in the form of collapsing-surging; in these cases, itcan be argued that run-down at structure is so high to cause thecollapse of the lower part of the front, that is the initial mech-anism of such type of breaker.

On the other hand, analogously to what happens on naturalbeaches, a limit value of xB(say 2.4, vertical dash-dotted line)seems to rule the skewing of wave profile over the breakwater,which, reasonably governs the occurrence of spilling andplunging breakers; according to this reasoning, for xB > 2:4bore and collapsing-bore breakers occur, whilst for xB < 2:4breakers evolve likewise StP, with a descending slope in therear part of the wave profile. Finally, a vertical dotted line ten-tatively marks the border for ‘‘pure’’ StP breakers. The graphsuggests that a wave that would break with a StP form ata steep-faced structure may change breaker type into a bore-type one when crown width is reduced (xB increases) or ina ‘‘collapsingeStP’’ breaker if wave height increases, or crestfreeboard reduces.

However, despite above considerations sound physicallyconsistent, further experiments are needed to confirm the ex-plication power of the two parameters here introduced.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.00 0.50 1.00 1.50 2.00 2.50

Hi/R

Ceq

Lm/L0

Spilling (including 2-Steps)

Spilling-plunging (including 2-Steps)

Plunging

impermeable breakwater

Fig. 13. StP breakers evolution for impermeable breakwater.

Finally it is noteworthy that, when focusing only on StPbreakers, transition from spilling to plunging type seems tobe ruled only by relative submergence, i.e. the ratio betweenincident wave height and structure freeboard. This is illus-trated in Fig. 13 for the impermeable barrier and in Fig. 14for permeable one; results are analogous.

In the graph, Lm is the major base of the trapezoidal break-water section and Rceq is an ‘‘equivalent freeboard’’, obtainedby the following formula:

Rceq ¼ h� As

Lm

ð6Þ

where h is water depth over bottom and As is the cross-sectionarea. The ratio in Eq. (6) can be regarded as an ‘‘equivalentstructure height’’, considering the effects of both crown widthand seaward slope, that, as observed before, significantly af-fect StP breakers evolution.

5. Conclusions

A classification of different breaker types occurring on sub-merged rubble mound breakwaters with varying offshore slopeand permeability was proposed, on the basis of video analysis.Different classes and subclasses of breaker forms were quali-tatively identified, according to discriminating criteria basedon both hydrodynamic and morphological considerations.

Analysis of transition from non-breaking to breaking condi-tions shows that breaker index can be described by a Goda-type formula, in which the scale parameter A is expressedby a best-fit formula as a function of structure permeabilityand seaward slope. In agreement with earlier studies, breakerindex increases as permeability grows, and decreases as off-shore slope reduces.

The dependence of different breaker forms on both struc-ture and wave parameters was then investigated through a para-metric study considering two non-dimensional numbers. Thefirst parameter, proportional to wave run-down to submer-gence ratio at structure, affects the stability of lower part ofwave profile and governs collapsing-surging breakers. Thelatter, which can be regarded as a fictional surf similarity

225M. Calabrese et al. / Journal of Hydro-environment Research 1 (2008) 216e225

parameter, influences the skewness of wave profile over thebarrier, and governs the occurrence of spilling-to-plungingbreakers. It was also observed that, among StP forms, transi-tion from spilling to plunging seems to be ruled only by rela-tive submergence.

Although the results are physically reasonable and are sub-stantially consistent with earlier literature, further investiga-tions are needed in order to extend data set to a wider range ofstructure layouts and incident wave parameters. Moreover, ve-locity measurements in the breaking area are needed in orderto support some of the qualitative observations hereby proposed.

Acknowledgements

Contribution provided by Roberta Gentile, Giuseppe DiCarlo and Alessandra Ferrara to experimental activities anddata analysis is hereby acknowledged.

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