Experimental and Numerical Investigation On Wave ...

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Experimental and Numerical Investigation On Wave Transmission Past Rubble-Mound Submerged Breakwaters F. CIARDULLI Sogreah Gulf (Artelia Group), Dubai, UAE G. CUOMO HR Wallingford Ltd, Wallingford, UK M. BUCCINO University of Naples “Federico II”, Naples, Italy M. CALABRESE University of Naples “Federico II”, Naples, Italy

Summary In this paper the wave transmission past submerged breakwaters is investigated with the use of a 2D Boussinesq-type model (BOUSS-2D) developed by the U.S. Army Corps of Engineers (USACE) included in the AquaveoTM Surface-water Modelling System (SMS). Due to the nature of the model the submerged barrier has been treated as surface-piercing porous structure implemented in the model as a porous layer boundary. The numerical model has been calibrated and validated against 2D physical model tests (Ciardulli, 2009) performed at the LInC Laboratory of University of Naples “Federico II”. A simple method to derive the porous layer parameters has been proposed providing engineers with guidelines for the use of porous layer to predict transmission coefficient behind submerged structures when using Boussinesq-type 2D (horizontal) numerical model. Numerical predictions have been found to be in good agreement with measurements (R2 statistics around 0.9) as well as with outcomes of the semi-empirical equations by Buccino and Calabrese and d’Angremond et al. (maximum difference 0.08).

Introduction Submerged breakwaters are frequently adopted by engineers as structural solution for shoreline stabilization and beach erosion control in areas where a low visual and environmental impact is desired. The submerged barriers enhance the shoreline stability since they reduce the incoming wave action by forcing waves to break and dissipate the incoming wave energy effectively. Although a number of engineering issues related to the usage of submerged barriers have been addressed in the last few decades (e.g, Dean et al., 1997; Zanuttigh and Martinelli, 2008; Calabrese et al., 2003; Calabrese et al., 2011), the prediction of wave climate leeward the barriers remains indeed the main task to their functional design. A primary worldwide used hydraulic response parameter is the transmission coefficient Ct (transmitted to incident wave profile standard deviation ratio), which is clearly related to the global wave energy dissipation. Significant efforts have been made in the past by many researchers to provide engineers with simple and reliable tools to estimate Ct (d’Angremond et al., 1997; Seabrook and Hall, 1998; Briganti et al., 2003 Buccino and Calabrese, 2007; Van der Meer et al., 2005) as function of the main structural and hydraulic parameters. To meet the design criteria, empirical equations are used for the design of the barrier cross-section. In the practical applications, the use of numerical models is nevertheless often required to achieve a comprehensive assessment of the hydro/morphodynamic response of the protected beach to the construction of these type of structures. Whilst offshore to nearshore wave transformation has been so far well described and reproduced by various numerical models, the interaction of waves with low crested structures is still Coasts, Marine Structures & Breakwaters 2013

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the subjject of scientific efforts in both physical and num merical mode elling. In fa act, when submerged breakwa aters are pla aced in the nearshore n arrea, assessing the trans smitted wave e field in the e shadow zone is a key point of o the design process. In this paper, p wave transmission n behind rubble-mound submerged s b breakwaters measured during d 2D physical model testts (Ciardulli, 2009) perfformed at th he LInC Lab boratory of University U of Naples ompared to predictions by a 2D nu umerical mod del solving the t Boussine esq type “Federicco II” are co equation ns. The latterr is the BOUSS-2D Bousssinesq wave e model deve eloped by the e U.S. Army Corps of Enginee ers (USACE) included in the Aquaveo oTM Surface--water Modelling System (SMS). The model is based o on the solutio on of Bousssinesq-type equations e in their “exten nded” version derived byy Nwogu (1993). The model is able to simulate s partial wave re eflection and d transmissiion through surfacepiercing porous stru uctures such as low-cres sted structure es (LCS) an nd breakwate ers. In this paper p the submerg ged barrier used u in the ph hysical mode el tests have been incorp porated in the e numerical model m as porous layer. A calib bration of the e porous med dium parame eters has bee en performed d providing e engineers mple guidelines to assist in the modelling m o porous la of ayer bounda aries and with w their with sim impleme entation in th he model dom main to accu urately model the transmission coeffic cient in the le ee of the submerg ged structure es. Numericcal results are also com mpared with predictions by semi-empirical equation by Buccino and Calabresse (2007) and a d’Angrem mond et al. (1996). The use of different mode elling approaches is discusse ed, highlighting pros and cons of diffe erent tools avvailable to de esigners.

Experrimental Investig gation Laborato ory experime ents have be een performe ed in the sma all scale cha annel (SSC, Figure 1) of the LInC laborato ory of the Hyydraulic, Geo otechnical an nd Environm mental Engine eering Department (DIGA A) of the University of Naples s Federico II. I The flume e is 23.50m m long, 0.80m m wide and 0.75m deep and is d with a pistton type wavvemaker, capable of gen nerating both regular an nd spectral waves. w A provided rubble sspending bea ach with an average 1:10 slope proffile has been n placed at the t end of th he flume, opposite e to wave-m maker; this in n order to minimize m the effects of re eflection. Th he adopted spending s beach arrangement was found to o induce refle ection coefficcients on ave erage not larrger than 5%.

Figure e 1. The SSC C flume of the LInC Labo oratory. erent structu ure layouts ha ave been tessted (Ciardullli, 2009), butt only two of them are co onsidered Five diffe in this p paper, that iss, homogene eous rubble mound barrriers with a median m diam meter (Dn,50) equal to 0.058m and a mean n porosity, n, n of 0.4. The e structures had a front slope angle e (tanαoff) off 1:2, the eight (hc = 0.25m), 0 but different d crow wn widths and namely B = 0.25m (S Structure 3) and B = same he 0.8m (S Structure 4). The breakw waters were placed in th he flume over a flat botttom and ha ave been subject to regular wave w attackss with varying g wave heig ght and perio od. The still water depth h, h, has ept constant at 0.315m throughout t the tests. Wa ave height has h been inccreased from m 3cm to been ke 12cm att a step of 1ccm: this to ca arefully check k the transitio on from non--breaking to breaking wa ave at the structure e crest. For each e wave height, three wave w period, T, have bee en used, nam mely 1s, 1.5ss and 2s. The free e surface os scillations ha ave been accquired at 15 1 positions along the flume f using resistive probes ssampled at 25Hz (Figurre 2). Each test was pe erformed witth and witho out structure and the transmisssion coefficient Ct was estimated as s the ratio between b the measured wave w profile standard deviation ns with and without w struccture (with/without ). Notice e that the reccorded regula ar wave sign nals have w been an nalysed only within the tim me window where w the re e-reflection effects from the wave ma aker were not significant. Coasts, M Marine Structuress & Breakwaters rs 2013

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Measurement locations

Piston type wave-maker

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Spending beach

Hi , T

HT , T d

1/10

B 1/20

h

1/10

Figure 2. General scheme of measurement layout.

The model set up can be considered as representative of real submerged breakwater at a reference Froude length-scale of 1:16. In the following, the experimental measurements and the numerical simulations are presented at the prototype scale. The parameters of the data set adopted in the present study are summarized in Table 1. Table 1. Parameters of the data set (Prototype Scale) Symbol Value Parameter 0.93 Armour rock size (m) Dn50 Porosity 0.4 n 1:2 Front slope angle tanoff Crown width (m) 4 and 12.8 B Crown width to wave length ratio 0.04 to 0.51 B/L0 Submergence (m) 1.04 d 0.44 to 1.92 Submergence to wave height ratio d/Hi

2D Numerical Investigation Description of the Model The 2D numerical investigation was conducted using the BOUSS-2D wave model developed by the U.S. Army Corps of Engineers (USACE) included in the AquaveoTM Surface-water Modelling System (SMS) package. BOUSS-2D is a two-dimensional finite difference numerical model based on the time domain solution of the non-linear and depth integrated Boussinesq-type equations in their “extended” version derived by Nwogu (1993, 1996). The vertical structure of the flow is first resolved by expanding the kinetic potential, Φ, as a Taylor series about an arbitrary elevation in the water column (zα). Then the depth integrated balance of mass and momentum can be obtained, transforming the original 3D problem into a 2D one. The governing equations can be written in their fully non-linear version as follows (Nwogu, 1996): ∂η ∂t

∙

∂ ∂t

g

0 (1) ∙

w

w

z

∂ ∙ ∂t ∂ ∙ ∂t

∙

∂ ∂t

∂ ∂ 1 ∙ ∙ z ∂t ∂t 2 ∂ ∂ ∂z ∙ ∙ ∙ ∂t ∂t ∂t 0 (2)

in which η is the surface elevation, h is the still water depth and Z 0.465 along the water column. Coasts, Marine Structures & Breakwaters 2013

is the velocity vector at a depth of

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represents the velocity vector at the free surface (function of is the depth integrated flow velocity. and

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and w is its vertical component

The effect of the wave energy dissipation due to wave breaking is modeled by introducing an eddy viscosity term at the right hand side of Eq. 2: 1

∙

(3)

The eddy viscosity term is proportional to the amount of turbulent kinetic energy produced by wave breaking (determined through a one-equation turbulence model) via a turbulence length, , which is a leading calibration parameter. The waves are assumed to start breaking as

exceeds the phase celerity.

Approach Submerged breakwaters reduce transmission of incident wave energy by means of four main mechanisms, namely: •

Wave breaking;

•

Skin friction;

•

Seepage;

•

Wave reflection.

All the above are of dissipative nature except wave reflection, which is conservative. In this paper we have chosen to act only on one of them (wave breaking) during the calibration phase; in order to limit the number of parameters involved. Once the primary damping source has been characterised, the related parameters have been varied in the model, till a zero of a Convergence Index (CI) was been reached (or inferred) for each experiment. CI is defined as the relative scatter between measured and predicted transmission coefficients:

,

. ,

, .

(4)

where , . represents the arithmetic mean of the transmission coefficients measured at the 15 gauge array displayed in Figure 2. Data on Structure 3 are used as training set, whereas Structure 4 is employed for “verification” and “extrapolation”. Note that this choice implicitly assumes that the transmission coefficient could be estimated via a single smooth function of non-dimensional wave/structure quantities, so that the same relationship should apply to breakwaters of different outer geometry. This approach is usually adopted when an empirical formula is fitted to experimental data.

From Wave Breaking to the Equivalent Porous Layer The most elementary assumption is that of implementing the barrier as a submerged impermeable body, that is including the structure geometry within the seabed. In this way, the structure acts on the wave by inducing breaking intensity through the turbulence scale length . However this leads to the appearance of two unphysical phenomena: a) the velocity at the free surface remained below the phase celerity even for relatively steep waves, delaying and reducing wave breaking; b) very steep waves propagated over the structure crest leading to numerical instability. This suggested introducing a porous layer, which in BOUSS-2D must extend throughout the water column, as it is considered as representative of a surface piercing structure. It should be therefore be borne in mind that the use of this feature to model submerged structures leads in principle to an effective discrepancy between the model assumptions and the effective behaviour of the structure, which imposes to consider an idealised structure having dissipation characteristics equivalent to that of the real breakwater. The governing equations in the porous medium are:

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∂ ∂ ∂ ∂

∙

5

0 (5) gg |

∂ ∂ 1 ∂ ∙ ∙ h ∙ 2 ∂ ∂ ∂ 0 (6) ∙

|

z

in which h is the po orosity and Engelun nd (1953) reccommended: 1 1

and

are e the lamina ar and turbu ulent friction factors resp pectively.

(7) 1

(8)

matic viscosity of the wa ater, is the e diameter off the porous material an nd where is the kinem are empirical consstants that ra ange from 78 80 to 1,500, and a 1.8 to 3.6.

and

d for calibrattion, namely , and . In order to reduce It followss that three parameters can be used these de egrees of fre eedom, the motion m has been b assum med to be rou ugh turbulen nt and accord dingly has bee en set equal to zero. Thiss hypothesis, as well as being realisttic for most practical p app plications, is suited d to model th he present te est configura ation as the structures s arre homogene eous with larrge voids 0.4) has and high h Reynolds numbers n within the medium (no core e). The actua al value of porosity p ( then bee en fixed and d used ass unique caliibration para ameter. The structure cro oss-section h has been schemattized as an equivalent rectangle, r th he mean wid dth of which h equalled th he ratio betw ween the cross se ectional area of the barrie er (Structure 3) and its he eight. The optimal friction coefficients have been found f to be in general very v small, well w below th he typical ove. This is not surprisin ng, as the model m does not n reproduc ce the mechanism of values rreported abo overpassing and forrces the wa aves to entirely seep thrrough the po orous mediu um. Unfortun nately, in some ca ases no convvergence hass been achie eved, even th hough β0 van nished (Figurre 3).

3. Compariso on of measurred and simu ulated transm mitted wave signals s for β0 = 0. Left pa anel: Hi = Figure 3 0.48m, T = 4s (Pro obe 6). Right panel: Hi = 1.92m, T = 8 8s (Probe 4).

The Eq quivalent Porosity Index

Previouss results led to the force ed choice off setting 0 and looking for an eq quivalent po orosity ∗ . From a p physical poin nt of view thiss means that the reductio on of wave height h in the lee of the bre eakwater is entirely ruled by wave w reflectio on, as well as a by the inertial forces within w the porrous medium m (Eq. 6). In fact this t is an exxtremely cru ude approach which hass the advanttage of bein ng rather sim mple and pragmattic, but that can c be used d only to sim mulate the tra ansmitted wa ave field, since wave refllection in front of the t structure is likely overestimated. “Virtues and vices” of o the adopte ed solution are a highlighte ed in Figure 4. In the sim mulations the e width of the poro ous layer hass been set eq qual to the mean m width off Structure 3. Coasts, M Marine Structuress & Breakwaters rs 2013

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Figure 4 4. Features of o the selecte ed solution: a) a T = 4s; b)) T = 8s; c) T = 10s. Leftt panels: exa amples of convergence curves. Right panels: simulated d vs. measurred (probe 3)) transmitted signals. A first p point of stren ngth of the proposed app proach is tha at it is easy to calibrate: left panels indicates that CI ssmoothly incrreases from negative to positive, p allo owing to identify easily the e optimal value of ∗ . More im mportantly, th he simulated transmitted wave signa als have bee en found to well w approxim mate the measure ed ones in te erms of crest to trough as symmetry (rig ght panels in n Figure 4). On the o other hand, the t main wea akness is tha at the distribu ution of wave e energy in the frequencyy domain is not p properly reproduced. This is due to the fact tha at the nonlinear respons se of the submerged breakwa aters is basiccally a consequence of the abrupt change c in th he flow prope erties cause ed by the presencce of a steep p-faced subm merged step. Since we are a using a porous layer instead, which w is a smoothe er perturbation source, the free su urface energ gy will stay concentrate ed around the t peak frequenccy. The prop posed appro oach thereforre leads to a linearizatio on of the pro oblem, simila ar to that implicitlyy assumed when w using empirical e forrmulae for th he prediction n of wave tra ansmission. However

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the practical consequences of the strong nonlinearity of the transmitted wave field have been not yet clarified till now. After the optimal values of ∗ have been identified for the narrow crested breakwater, an effort has been addressed to assess if any relationships existed between those values and the main wave/structure variables, such as the incident wave height (Hi), the incident wave period (T), the submergence (d) and the crest width (B). The following expression has been found based on a multiple non-linear regression model: .

. ∗

0.373 ∙

for 0.04

∙ ⁄

0.16 and 0.44

(9) ⁄

1.92

Note ∗ increases with growing the relative submergence d/Hi (assumed positive), leading to a larger transmission rate; oppositely, the equivalent porosity decreases with the crest width to deep water wavelength ratio B/L0, implying a reduction of Ct. This is consistent with the physical principles that rule wave transmission process. It is of interest to underline that although the value of B is constant in the “training data” (Structure 3), we will be assuming the dependence of Ct on the non-dimensional variable B/L0 to hold also with varying crown width, and possibly beyond the original validity bounds reported above. This is made clearer in the following of the paragraph. As shown in Figure 5, Eq. 9 provides reasonable estimates of the optimal ∗ for Structure 3; the value of the R2 statistics has been found to be about 0.87.

Figure 5. Optimal values of

∗

vs. predictions of Eq. 9. Data refer to Structure 3.

According with previous reasoning, Eq. 9 has been now tentatively applied to get estimates of the equivalent porosities also for the Structure 4 (wide crested breakwater); then BOUSS-2D has been used to assess the values of the transmission coefficient to compare with measurements. It is worth noting that in the simulations the width of the equivalent porous layer has not been changed. The dependency of the transmission coefficient Ct on the crest width being accounted for in the equivalent porosity parameter (Eq. 9). Numerical and experimental results compare relatively well in Figure 6, with the exception of some experiments with short period T = 4s, which leads to very steep waves. This suggests that the proposed approach could be successfully employed in practical applications, however empirical it may seem. As a further consequence, the validity bounds of the Eq. 9 might be reasonably extended ⁄ ⁄ to: 0.04 0.51 and 0.44 1.92.


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Figure 6. Predicted and measured transmission coefficients for Structure 4

Application to Random Waves A possible source of uncertainty of the analysis presented above is that experimental data refer to regular waves; it is therefore useful to clarify how previous results may apply to irregular wave trains. For this purpose, the response of ideal submerged breakwaters subjected to random waves has been studied using BOUSS-2D. The structures have been assumed to be installed on a water depth of 5m and have both the seaward and shoreward slope inclined 1:2 to the horizontal. In a first simulation step, the submergence has been fixed at 1.0m (from the still water level) whilst the crest width has been gradually increased from 5m to 16m. Successively, a single crown has been selected (B = 10m) and the crest freeboard has been varied between 1.0m and 2.50m. A JONSWAP spectrum random sea state with a zero moment significant wave height Hm0 = 2.7m and a peak period Tp = 8s has been generated at a depth of 10m and propagated over a 1/50 foreshore, up to the barriers. Here the significant wave height has been found to be 2.2m. Thus the value of n* has been estimated as a function of the incident Hm0 and Tp using Eq. 9; simulations have then been run and Ct numerically predicted. Note that this implies that an equivalence has been established between regular and random seas based on the significant wave height and the peak period; although questionable from a theoretical point of view, several studies proved this approach to be valid for practical purposes (e.g. Adams and Sonu, 1987, Loveless et al., 1997, Buccino and Calabrese 2007). The values of Ct estimated by the described procedure have been plotted vs. B/L0 and d/Hm0 in Figures 7 and 8. In the graphs, the predictions of the empirical formulae by Buccino and Calabrese (2007) and d’Angremond et al. (1996) are also reported. The results are not completely satisfactory; despite the values and the trends of the transmission coefficient being in a reasonable agreement with those of the design equations, BOUSS-2D predictions decay less rapidly as the crest width increase (Figure 7) and increases more gently with growing the relative submergence d/Hm0 (Figure 8). This might be due to the fact that Eq. 1 does not account for the effect of the Iribarren number which is among the most important variables in the empirical models. Further modelling is needed to investigate on this issue.

Conclusion Wave transmission past submerged rubble-mound breakwaters has been numerically investigated with a well validated Boussinesq-type numerical model available for commercial use (USACE-SMS BOUSS-2D). The numerical model has been calibrated and validated against experimental data obtained from laboratory tests carried out in the LinC laboratory of University of Naples “Federico”. A simple method to predict the transmission coefficient in the sheltered area by incorporating the submerged barriers in the numerical model as porous layers has been proposed.


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1.0 Bouss 2D

Ct

Buccino & Calabrese d'Angremond et al.

0.8

0.6

0.4

0.2

0.0 0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

B/L0

Figure 7.Predicted values of Ct vs. the relative crown width 1.0

Ct 0.8

0.6

0.4

Bouss 2D

0.2

Buccino & Calabrese d'Angremond et al.

0.0 0.40

0.60

0.80

1.00

d/Hm0 1.20

Figure 8.Predicted values of Ct vs. the relative crest freeboard Due to the nature of the model, the submerged breakwater has been treated as an idealized surface piercing structure that extends throughout the entire water column with width equal to the mean width of the real structure.Results of calibration indicated that, when dissipative terms due to the turbulent flow through the porous medium are activated in the momentum balance equation, high dissipations take place leading the model predictions to match experimental data only when using friction coefficients well below the typical values reported in literature. This behavior suggested setting the turbulent friction factor equal to zero 0 and searching for an equivalent porosity ∗ , that is forcing the reduction of the wave height in the lee of the barrier by means of wave reflection and inertial dissipation within the porous medium. Optimum values of ∗ achieved for the narrow crested breakwater seem to be well correlated with the main wave/structure parameters, in agreement with physical principles that rule wave transmission. The transmitted wave signals reproduced by BOUSS-2D have been found to well approximate the measured ones in terms of crest to trough distance. On the other hand, the distribution of wave


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energy in the frequency domain is not well captured by the model since the presence of the porous layer leads to a weak non-linear wave-structure interaction. A predictive curve (Eq. 9) has been proposed to compute the value of ∗ as function of ⁄ and ⁄ ⁄ ⁄ , valid for 0.04 0.51 and 0.44 1.92. The curve provides reasonable estimates with a determination index of about 0.87. The proposed method has then been applied to the wide crest breakwater showing quite good comparisons between numerical predictions and experimental data. Extremely simple and pragmatic, the proposed method is recommended to be used only to simulate the transmitted wave field, since wave reflection in front of the structure is likely overestimated. An application to random wave trains has also been performed. The proposed approach has been tested for a set of typical submerged breakwaters under a selected random sea state. BOUSS-2D predictions are in reasonable agreement with Buccino and Calabrese (2007) and d’Angremond et al. (1996), being the maximum difference among the values of the transmission coefficients about 0.10. However the observed dependence on both the crest width and submergence is not completely satisfactory. Further investigations are to be carried out to explore this matter in more detail and test the model performances for random sea states.

References Adams, C. B., and Sonu, C. J. (1987). “Wave transmission across submerged near-surface breakwaters.” Coastal engineering.B. L. Edge, ed., ASCE, New York, 1729–1738. Briganti R, Van der Meer J.W., Buccino M., Calabrese M. (2003). “Wave transmission behind low crested structures”, Proc. 3rd Coastal StructuresConference, Portland, USA, 580-592 Buccino, M. and Calabrese, M. (2007),”Conceptual Approach for Prediction of Wave Transmission at Low-Crested Breakwaters”, Journal of Waterway, Port, Coastal, Ocean Eng., 133(3), 213–224; Calabrese, M., Vicinanza, D., Buccino, M. (2003). “2D wave set up behind low crested and submerged breakwaters”, Proceedings of the 13th International Conference ISOPE, Honolulu, Hawaii, 831-836; Calabrese, M., Di Pace, P., Buccino, M., Tomasicchio, G.R. and Ciralli, E. (2011). “Nearshore Circulation at a Coastal Defence System in Sicily. Physical and Numerical Experiments”. Journal of Coastal Research, SI 64, 474-478. Ciardulli, F. (2009), “La risposta idraulica di barriere sommerse in campo non lineare”, Ph.D. Thesis (in Italian), University of Naples “Federico II”; d’Angremond, K, van der Meer, J W and de Jong, R J (1997), “Wave transmission at low-crested structures”,Coastal engineering (1996),B.L. Edge, ed.,ASCE,New York,2418-2426; Dean, R.G., Chen, R., Browder, A. E. (1997). “Full scale monitoring study of submerged breakwater, Palm Beach, Florida, USA”. Coastal Engineering, (29):291–315. Engelund, F. (1953). “On the laminar and turbulent flow of ground water through homogeneous sand,” Trans. Danish Academy of Technical Sciences 3(4); Goda, (2010) “Random Seas and Design of Maritime Structures“, 3rd edition, World Scientific; Loveless, J. H., Debski, D., and Leonard, P. (1997). “The design and performance of submerged breakwaters.” Rep. UK MAFF, Contract CSA 2606, Univ. of Bristol, U.K. Nwogu O. (1993), “Alternative form of Boussinesq equations for nearshore wave propagation”, Journal of Waterway, Port, Coastal and Ocean Eng.,ASCE,119(6),618-638; Nwogu O. (1996), Numerical prediction of breaking waves and currents with a Boussinesq model, Paper presented at the 25th International Conference on Coastal Engineering, ICCE 96, Orlando, FL.; Seabrook, S. R. and Hall, K. R. (1998), ”Wave transmission at submerged rubble-mound breakwaters”, Proc. 26TH Int. Conf. on Coast. Engineering, 1998; van der Meer, J.W., Briganti, R., Zanuttigh, B., Wang, B., 2005. Wave transmission and reflection at low crested structures: design formulae, oblique wave attack and spectral change. Coastal Engineering 52 (10–11), 915–929. Zanuttigh, B. and Martinelli, L. (2008). “ Transmission of wave energy at permeable low crested structures”. Coastal Engineering 55 (12), 1135-1147.
