Determining Wave Transmission over Rubble-Mound Breakwaters: Assessment of Existing Formulae through Benchmark Testing

Abstract

Low-crested and submerged breakwaters are frequently employed as coastal defence structures. Their efficiency is governed by wave energy dissipation, and the wave transmission coefficient can evaluate this parameter. The current study conducts experimental investigations on both low-crested and submerged breakwaters exposed to different wave conditions to compare their performance with that of emerged breakwaters. The current study provides a comprehensive review of existing formulae and highlights the impact of design variables. To evaluate the reliability of each existing formula, four “reference” configurations are used. Having these structures at the same overall volume, the results also provide a useful tool for engineers involved in the lowering operation of existing breakwaters. Nature and magnitude of governing parameters are investigated, and some points of criticism are outlined. The comparison results show that few of the existing equations give reliable estimates of the transmission coefficient for all the models tested in this study. Higher values of root mean square error are related to the emerged breakwater rather than the submerged ones. To obtain information about the transmitted wave energy, spectral analysis is applied as well. Different behaviours of the transmitted spectrum, n terms of shape and peak frequency, are highlighted. The results improve the overall knowledge on formulae that are in the literature, in order to make the user more aware.

1. Introduction

Breakwaters are structures located nearshore to protect shorelines and harbours that are exposed to waves, tides, and wind [1,2,3]. A comprehensive guideline for the design, construction, and maintenance of coastal defence structures was first presented in the Shore Protection Manual [4] and British Standards on Maritime Structures [5], then, recently, in the Coastal Engineering Manual [6]. Conventional emerged rubble-mound breakwaters have been the most common shore protection or sea defence structures since the beginning of the last century [7]. Although these structures are reliable regarding wave energy dissipation and shoreline protection, they are vulnerable in extreme storms and expensive to maintain and rebuild. In addition, emerged breakwaters caused several undesired phenomena such as tombolo and salient formation, degradation of water quality, downdrift beach erosion, and visual impact [8,9]. Much attention has been given in recent years to the environmentally friendly designs of breakwaters. Low-crested and submerged breakwaters are proposed as alternative solutions due to their low environmental and visual impact, low overall costs and construction time, and increased biodiversity [10,11,12,13]. The environmental aspect of coastal defence structures was highlighted during the THESEUS project [11]. The main aim of the project was to develop an integrated approach for planning sustainable coastal defence structures to mitigate coastal erosion and flooding risks. Extensive research on low-crested breakwaters was conducted within the European DELOS project [7]. The primary goal of that project was to promote the development of low-crested structures that were both environmentally friendly and cost effective. Submerged breakwaters allow exchange of water masses between offshore and onshore regions, reducing pollution on beaches [7,14,15]. More recently, other novel submerged breakwaters have been investigated [16]. The level of coastal protection provided by submerged breakwaters depends on various factors, including the breakwater’s width and freeboard, the distance between the breakwater and the shore, the hydrodynamic characteristics of the breakwater, the wave climate, and the angle of the wave attack [17,18]. Various experimental studies and numerical models have been developed to predict various physical phenomena associated with submerged and low-crested breakwaters. Diffraction and overtopping are two phenomena that can influence wave disturbance at the leeward of the breakwaters. Vicinanza et al. [19] were the first to consider the effects of diffraction and overtopping in predicting transmitted wave height. Previous laboratory experiments, numerical modelling, and field measurements have been conducted to study the hydrodynamic and morphodynamic impacts of these breakwaters in coastal areas, and these studies are still under development. The functional design knowledge of low-crested and submerged breakwaters, including their impacts on wave transformation and reflection, wave overtopping and current fields around the structures, sediment transport, and shoreline response, is still developing.

This study aims to provide a comprehensive review of wave transmission coefficient formulae for low-crested and submerged breakwaters. In addition, an attempt is made to use an up-to-date experimental database in a wave flume to evaluate the validity of the existing wave transmission prediction formulae. The paper includes an evaluation of performance through a benchmark program, illustrated by four reference configurations. These reference configurations consist of a physical model of an emerged breakwater and three low-crested structures with different submergence levels but the same overall volume.

1.1. Governing Parameters of Wave Transmission

Table 1. Wave characteristics and structural parameters involved in the wave transmission phenomenon.

Symbol	Unit	Definitions
Hi	(m)	Incident significant wave height, preferably Hm0i, at the toe of the breakwater
Ht	(m)	Transmitted significant wave height, preferably Hm0t
Tp	(s)	Peak wave period
Lp	(m)	Deep-water wavelength
mo	(m2)	Zeroth-order moment of the wave spectrum
Tm−1,0	(s)	First negative moment of the energy spectrum
Tm0,2	(s)	Zero-crossing period or mean zero-crossing period
hs	(m)	Height of the breakwater
d	(m)	Water depth at the toe of the breakwater
Rc	(m)	Crest freeboard of the breakwater
B	(m)	Crest width of the breakwater
m	(-)	Seaward slope of the breakwater
Sop	(-)	$\mathrm{Wave} \mathrm{steepness} (S_{op}=\frac{2\pi H_i}{g{T}_p^2}=\frac{H_i}{L_p}$)
ξop	(-)	$\mathrm{Surf} \mathrm{similarity} \mathrm{parameter} ({\xi }_{op}=\frac{m}{\sqrt{S_{op}}}$)
Dn50	(m)	Nominal diameter of the armour rock

lists the main wave characteristics and structural parameters that describe wave transmission phenomena. Figure 1 illustrates a definition sketch for a rubble-mound breakwater.

The roughness of the armour layer, the permeability of the breakwater, and the angle of the incident wave height are other parameters that influence the hydraulic performance of the breakwaters [20].

1.2. Existing Wave Transmission Formulae

A common way to study wave transmission is to perform physical model experiments. In such experiments, a scaled model of the entire breakwater profile is constructed and exposed to incident waves in a wave flume or wave basin. Finally, the wave transmission characteristics are evaluated based on analysing the measured wave conditions in front of and behind the breakwater. In this regard, many experimental studies have been performed to investigate the wave transmission characteristics of submerged and low-crested breakwaters. Several authors have proposed a series of wave transmission formulae, some of which are reported in this section along with their respective descriptions. Van der Meer [21] proposed a simplified method to predict the wave transmission coefficient for emerged and submerged rubble-mound breakwaters. This formula linearly relates the relative crest freeboard ($R_c/H_i$) to the wave transmission coefficient ($K_t$), without taking crest width into account. The most available datasets from various investigations were analysed by Van der Meer and Daemen [22]. In this study, they used the nominal diameter of the armour layer (D_n50) to define dimensionless parameters that affect wave transmission, namely relative freeboard ($R_c/D_{n50}$), relative wave height ($H_i/D_{n50}$), and relative crest width ($B/D_{n50}$). This formula considers the crest width and is valid for emerged and submerged rubble-mound breakwaters with minimum and maximum transmission coefficients of 0.075 and 0.75, respectively. D’Angremond et al. [23] reanalysed the existing dataset, except for data with extremely high wave steepness (i.e., $s_{op}\ge 0.6$), breaking waves (i.e., $H_i/d\ge 0.54$), highly submerged ($R_c/H_i<-2.5$), and highly emerged ($R_c/H_i>2.5$) breakwaters. They applied the same approach as adopted by Van der Meer and Daemen [22] to propose a new wave transmission formula for low-crested rubble-mound breakwaters. Seabrook and Hall [24] carried out a large number of laboratory tests to develop a formula for submerged rubble-mound breakwaters, taking into account a wide range of wave conditions and submerged breakwater parameters such as crest widths, submergence depths, and structure slopes. Their results showed that the transmission coefficient is most sensitive to the depth of submergence ($R_c$), the incident wave height at the toe of the structure (viz., the spectral wave height, ($H_i$), defined as four times the square root of the zeroth-order moment ($m_0$) of the wave spectrum), and the crest width ($B$). To a lesser degree, the transmission coefficient is influenced by the period of the incident wave, the breakwater armour dimensions ($D_{n50}$), and the breakwater slopes ($m$).

Regarding the wave period, it was established that the first negative moment of the energy spectrum ($T_{m-1.0}$) can provide a better description than a zero-crossing period or a mean zero-crossing period ($T_{m0,2}$) [25]. This is due to the fact that the value of m₂ is extremely sensitive to measurement errors or variations in integration analysis [26]. Moreover, $T_{m-1.0}$ gives more weight to the longer periods in the wave spectra [27,28,29], providing a conservative approach in the design formulae. However, often, conversion values are used to relate $T_{m-1.0}$ to simpler wave period parameters. For instance, the EurOtop manual uses a conversion factor $T_p$ = 1.1 $T_{m-1.0}$, where $T_p$ is the peak period of the spectrum. A large-scale model test of wave transmission on low-crested and submerged breakwaters was carried out by Calabrese et al. [30]. They developed a transmission coefficient formula for rubble-mound breakwaters, exposed to breaking waves, by adopting the formula of Van der Meer and Daemen [22] and substituting B with $D_{n50}$ to define the dimensionless parameters. The European DELOS project compiled a large database on wave transmission [9]. This new database includes the data by Van der Meer and Daemen [22] and D’Angremond et al. [23] on rock and tetrapod structures, Calabrese et al. [30] on large-scale tests in shallow foreshores, Seabrook and Hall [24] on submerged structures with very wide crests, Hirose et al. [31] on aqua-reef blocks with very wide crests, and Melito and Melby [32] on structures with core-loc. Briganti et al. [33] reanalysed and calibrated the d’Angremond et al. [23] formula using the DELOS project database. The analysis revealed the need for an additional formula that would allow a reliable estimate of the transmission coefficient at wide-crested breakwaters ($B/H_i>10$). For structures with $B/H_i<10$, the d’Angremond et al. [23] formula is still considered accurate. It should be noted that both formulas produce a discontinuity for $B/H_i=10$. Van der Meer et al. [34] argued that the wave transmission coefficient proposed by d’Angremond et al. [23] is based on a limited data set, and the applicability of it could be improved by reanalysing the dataset by d’Angremond et al. [23] and those from the DELOS project [34]. The result led to an improved wave transmission coefficient formula for impermeable structures. A new wave transmission prediction method was presented by Buccino and Calabrese [35]. Their method was based on a highly simplified model of phenomena that govern wave transmission behind breakwaters, including wave breaking, wave overtopping, and energy transfer through the breakwater. The method proposed by Buccino and Calabrese [35] was calibrated and validated using a large dataset to create a general design tool for wave transmission.

Table 2. Summary of currently most relevant wave transmission coefficient formulae for submerged and low-crested rubble-mound breakwaters.

Reference	Formulae	Limitations	Type of Structure	Equation No.
Van der Meer [21]	$K_t=0.80$      for    $-2.00<R_c/H_i<-1.13$ $K_t=0.46-0.3\frac{R_c}{H_i}$ for       $-1.13<R_c/H_i<1.20$ $K_t=0.10$       for    $1.20<R_c/H_i<2.0$	$-2.0<R_c/H_i<-1.13$  $-1.13<R_c/H_i<1.2$  $1.20<R_c/H_i<2$	Emerged Submerged	(1)
Van der Meer and Daemen [22]	$K_t=a\frac{R_c}{D_{n50}}+b$ $a=0.031\frac{H_i}{D_{n50}}-0.24$ $b=-5.42S_{op}+0.0323\frac{H_i}{D_{n50}}-0.0017{\left(\frac{B}{D_{n50}}\right)}^{1.84}+0.51$	$1<H_i/D_{n50}<6$ $0.01<S_{op}<0.05$ $-2<R_c/D_{n50}<2$ $0.075<K_t<0.75$	Emerged Submerged	(2)
D’Angremond et al. [23]	$K_t=-0.4\left(\frac{R_c}{H_i}\right)+0.64{\left(\frac{B}{H_i}\right)}^{-0.31}\left(1-exp\left(-0.5{\xi }_{op}\right)\right)$	$-2.5<R_c/H_i<2.5$ $0.075<K_t<0.80$	Emerged Submerged	(3)
Seabrook and Hall [24]	$$\begin{gathered} K_t=1-\left(exp\left(-0.65\left(\frac{R_c}{H_i}\right)-1.09\left(\frac{H_i}{B}\right)\right)+0.047\left(\frac{B}{L}\frac{R_c}{D_{n50}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-0.067\left(\frac{R_c}{B}\frac{H_i}{D_{n50}}\right)\right) \end{gathered}$$	$0\le \frac{B}{L}\frac{R_c}{D_{n50}}\le 7.08$ $0\le \frac{R_c}{B}\frac{H_i}{D_{n50}}\le 2.14$ $5\le \frac{B}{H_i}\le 74.47$	Submerged	(4)
Calabrese et al. [30]	$K_t=a\frac{R_c}{B}+b$ $a=\left(0.6957\left(\frac{H_i}{d}\right)-0.7021\right)exp\left(0.2568\left(\frac{B}{H_i}\right)\right)$ $b=\left(1-0.562exp\left(-0.0507{\xi }_{op}\right)\right).exp\left(-0.0845\left(\frac{B}{H_i}\right)\right)$	$-0.4\le R_c/B\le 0.3$ $1.06\le R_c/H_i\le 8.13$ $0.31\le H_i/d\le 0.61$ $3\le {\xi }_{op}\le 5.20$	Emerged Submerged	(5)
Briganti et al. [33]	$\mathrm{for} B/H_i>10$: $K_t=-0.35\frac{R_c}{H_i}+0.51{\left(\frac{B}{H_i}\right)}^{-0.65}\left(1-exp\left(-0.41{\xi }_{op}\right)\right)$ for $B/H_i<10$:   $K_t=-0.4\frac{R_c}{H_i}+0.64{\left(\frac{B}{H_i}\right)}^{-0.31}\left(1-exp\left(-0.5{\xi }_{op}\right)\right)$	$-5.0<R_c/H_i<5.0$ $0.37<B/H_i<102.12$ $0.002<S_{op}<0.07$	Emerged Submerged	(6)
Van der Meer et al. [34]	$\mathrm{for} {\xi }_{op}<3$: $K_t=-0.3\left(\frac{R_c}{H_i}\right)+0.75\left(1-exp\left(-0.5{\xi }_{op}\right)\right)$ $\mathrm{for} {\xi }_{op}\ge 3$: $K_t=-0.3\left(\frac{R_c}{H_i}\right)+0.75{\left(\frac{B}{H_i}\right)}^{-0.31}\left(1-exp\left(-0.5{\xi }_{op}\right)\right)$	$-2.5<R_c/H_i<2.5$ $0.37<B/H_i<43.48$ $0.02<S_{op}<0.06$ $0.075<K_t<0.80$	Emerged Submerged	(7)
Buccino & Calabrese [35]	for    $2<R_c/H_i<0.83$: $K_t=\frac{1}{1.18{\left(\frac{H_i}{R_c}\right)}^{0.12}+0.33{\left(\frac{H_i}{R_c}\right)}^{1.5}\frac{B}{\sqrt{H_i{L}_{p0}}}}$ for $R_c/H_i=0$: $K_t={\left[min\left(0.74;0.62{{\xi }_{op}}^{0.17}\right)-0.25min\left(2.2;\frac{B}{\sqrt{H_i{L}_{p0}}}\right)\right]}^2$		Submerged	(8)

provides a summary of the discussed formulae and the types of structures in which they have been calibrated.

2. Experimental Procedure and Setup

2.1. Laboratory Model

The laboratory experiments were conducted in the Hydraulic and Maritime Hydraulic Laboratory at the University of Campania Luigi Vanvitelli’s Engineering Department, using a length scale of 1:25 (Froude scaling) compared to the prototype. A series of 2D tests were performed at the wave flume, which has a width of 0.8 m, a height of 0.6 m, and a length of 13.4 m. Moving away from the wave generator, it was flat for about 3 m before sloping to a 1:22 slope. The facility is equipped with the wave generation section at one end and a dissipative gravel beach for wave absorption at the other (see Figure 2). A piston-type wave maker was applied to generate regular waves with different frequencies and irregular waves. In total, four different structures were tested. It is worth noting that the 2D wave flume was specially assembled within a 3D wave basin, as its partition. The last part of the wave flume, i.e., corresponding to the dissipative gravel beach, was kept laterally open and waves were also run outside the wave flume. In this way, a more natural water circulation at the rear part of the structure was achieved, thereby avoiding unrealistic wave setup behind the breakwater.

Four rubble-mound breakwaters were made of crushed stones with a diameter of 0.063 m ($D_{n50}$) and positioned 7 m from the wave paddles. The cross-sections of the breakwaters were changed in a way to create different configurations while maintaining the same volume of stones. This means that the lower the breakwater height, the greater the width. One was emerged, test (a), one had zero freeboard, test (b), and the other two were submerged, tests (c) and (d). Figure 3 shows the sections of the different configurations that are used in this study. The summary of the models’ details is presented in

Table 3. Wave parameters measured in the laboratory.

Models	Wave Type	Measured Incident		Measured Transmitted
		Hm0-i (m)	Tp0,2-i (s)	Hm0-t (m)	Tp0,2-t (s)	Rc (m)
Test (a)	Regular	0.021–0.256	0.772–1.698	0.003–0.044	0.418–1.748	+0.065
	Irregular	0.068–0.110	0.742–1.314	0.011–0.017	0.664–0.925
Test (b)	Regular	0.017–0.234	0.766–1.750	0.004–0.066	0.489–1.513	0.00
	Irregular	0.068–0.108	0.743–1.305	0.012–0.024	0.642–1.104
Test (c)	Regular	0.019–0.273	0.779–1.768	0.017–0.083	0.713–1.519	−0.065
	Irregular	0.072–0.120	0.763–1.315	0.037–0.043	0.711–0.951
Test (d)	Regular	0.016–0.277	0.766–1.752	0.016–0.089	0.597–1.472	−0.035
	Irregular	0.073–0.120	0.762–1.318	0.032–0.039	0.646–1.002

Note: See parameter definitions in Notation.

.

2.2. Wave Characteristics and Measurements

To separate incident and reflected waves, three wave gauges were installed near the model. The incident and reflected spectra were determined using the approach of Mansard and Funke [36], and the positioning of the wave gauges was based on suggestions by Klopman and Van der Meer [37]. Another two wave gauges were placed at the rear side of the models for measuring the transmitted wave heights. Wave gauges were meticulously calibrated prior to each experiment. The software AwaSys (Aalborg University, Aalborg, Denmark, 2010), allows the generation of regular and irregular waves. Irregular waves were generated using the three JONSWAP spectrum parameters, $H_{m0}$, $T_{p0,2}$, and the so-called peak enhancement factor γ (γ = 3.3 in all tests). Table 3 summarises the test parameters for this study.

2.3. Dimensional Analysis

Finding reliable solutions to engineering problems are rarely achieved only through analytical methods. Laboratory tests are usually required to determine how one factor is dependent on the others. The first step in physical modelling is to identify the variables that influence the physical phenomenon. Dimensional analysis is a technique that can be used not only to find the correct form of a relationship between two variables, but also to suggest how experimental campaigns should be designed. The variables that govern the flow, wave, and structural features affecting the magnitude of transmission coefficients are as follows:

$$f_1\left(\rho , g, \nu ,B,d,H_i,H_t,h_s,L_p, m, R_c,D_{n50}\right)$$

(9)

where $\rho$, $g$, and $\nu$ are mass density of water, gravitational acceleration, and kinematic viscosity of water, respectively, and the other variables are defined in Table 1. Applying dimensional analysis and using the Buckingham π-theorem, the functional relationships can be expressed in dimensionless form as:

$$f_2\left(K_t=\frac{H_t}{H_i},{\xi }_{op}=\frac{m}{\sqrt{S_{op}}},\frac{R_c}{H_i},\frac{B}{H_i},S_{op}=\frac{H_i}{L},\frac{R_c}{D_{n50}},\frac{H_i}{D_{n50}},\frac{B}{D_{n50}},\frac{H_i}{d}\right)$$

(10)

where $K_t$ and $C_r$ are the transmission and reflection coefficients, respectively, $S_{op}$ is wave steepness, and ${\xi }_{op}$ is the surf similarity parameter, which can be defined as:

$$K_t=\frac{H_t}{H_i}=f_3\left({\xi }_{op},\frac{R_c}{H_i},\frac{B}{H_i},S_{op},\frac{R_c}{D_{n50}},\frac{H_i}{D_{n50}},\frac{B}{D_{n50}},\frac{H_i}{d}\right)$$

(11)

These dimensionless parameters are used to show their effect on the transmission coefficient.

3. Results

The transmission coefficient is used to assess the hydraulic performances of breakwaters. Wave transmission is quantified by the wave transmission coefficient, $K_t=H_t/H_i$, where $H_t$ and $H_i$ are the transmitted and incident wave heights, respectively. The wave energy that is transmitted to the rear side of the breakwater is proportional to this value.

3.1. Wave Transmission Performance

3.1.1. Effect of Relative Crest Freeboard ($R_c/H_i$) on Wave Transmission Coefficient ($K_t$)

The relationship between the relative crest freeboard and wave transmission coefficient is presented in Figure 4. As shown in this figure, the transmission coefficient increases as freeboard is reduced from positive to negative. The transmission coefficient changes from low to high values as the freeboard decreases due to an increase in overtopping. The transmission coefficient exhibits relatively high values close to 1 for tests (c) and (d) and large negative values of $R_c/H_i$, as shown in Figure 4. Additionally, for these models, the transmission coefficient remains above 0.2. For an emerged breakwater with a significant relative freeboard, the transmission coefficient remains well above zero.

The relation between the transmission coefficient and relative freeboard, which is nondimensionalised by $D_{n50}$, is shown in Figure 5. This figure shows that the transmission coefficient varies from 0.05 to 0.22 for the emerged breakwater (test (a)), which also has a small crest width. For the zero-freeboard breakwater (test (b)), $K_t$ ranges from 0.16 to 0.33. On the other hand, the value for $K_t$ is between 0.24 and 0.89 and 0.27 and 0.99 for submerged breakwaters, tests (c) and (d), respectively.

3.1.2. Effect of Wave Steepness ($H_i/L_0$) on Wave Transmission Coefficient ($K_{\mathrm{t}}$)

Figure 6 shows the effect of wave steepness on the wave transmission coefficient. At the same wave steepness, the crest freeboard of the breakwaters has a significant impact on the wave transmission coefficient. For the same wave steepness, by decreasing the freeboard of the breakwaters from positive to negative, $K_t$ increases. The results show that test (d) has the highest $K_t$ values regarding the wave steepness.

3.1.3. Effect of Relative Wave Height ($H_i/D_{n50}$) on Wave Transmission Coefficient ($K_{\mathrm{t}}$)

In the literature, the effect of incident wave height was considered as a relative wave height, ($H_i/D_{n50}$), dimensionalised with the nominal diameter of the armour rock [22]. Figure 7 shows the effect of this parameter on the wave transmission coefficient. This figure shows that the transmission coefficient for the emerged breakwater, test (a), which has the lowest value among the others, ranges between 0.05 and 0.22. The transmission coefficient increases from 0.17 to 0.33 for the zero-freeboard breakwater. For the submerged breakwaters, tests (c) and (d), increasing the wave height decreases the transmission coefficient for relative wave heights less than 2, ($H_i/D_{n50}<2.0$), while for the relative wave height greater than 2, ($H_i/D_{n50}>2.0$), as the incident wave height increases, the transmission coefficient remains almost constant. This confirms that higher waves are affected more by the submerged breakwaters while small waves pass the crest unhindered [22], and for the relative wave heights greater than 2, the effect of this phenomenon remains constant.

3.1.4. Effect of Relative Crest Width ($B/L$) on Wave Transmission Coefficient ($K_{\mathrm{t}}$)

The relative crest width, $B/L$, influences the wave transmission over the breakwaters. The effect of relative crest width on $K_t$ is shown in Figure 8. The emerged and zero-freeboard breakwaters (tests (a) and (b)) show the same oscillatory trend for the transmission coefficient by changing the relative crest width. As shown in Figure 7, for tests (a) and (b), by increasing the values of $B/L$, $K_t$ first increases and then decreases. This oscillatory trend was firstly detected by Gómez Pina and Valdés [38], suggesting a strong correlation to the type of wave breaking conditions. For the submerged breakwaters in tests (c) and (d), $K_t$ increases when the values of $B/L$ increase.

3.2. Literature Formulae Performance

A comparison between the measured transmission coefficient and the calculated wave transmission coefficients by existing formulae is presented in Figure 9. Furthermore, root mean squared error, RMSE, was employed as a reliability indicator to assess of the comparison between experimental data and existing formulae [30]. The results of this comparative analysis are shown in Figure 9.

$$RMSE=\sqrt{\frac{{\sum }_N{\left(\frac{K_{tC}-K_{tM}}{K_{tM}}\right)}_i^2}{N}}$$

(12)

where N is the total number of data, and $K_{tC}$ and $K_{tM}$ are the calculated and measured values of the wave transmission coefficient, respectively. The results indicate that for the emerged breakwater, the formulae proposed by Van der Meer and Daemen [22], d’Agremond et al. [23], and Briganti et al. [33] show a relatively reliable agreement. Figure 9 also shows that these formulae have the lowest RMSE values for test (a). The effect of nondimensional crest freeboard, $R_c/D_{n50}$ or $R_c/H_i$, wavelength or steepness and crest width were incorporated into formulae from both of the preceding references. For the zero-freeboard predicted $K_t$, those from the formulae of d’Agremond et al. [23] and Briganti et al. [33] were more reliable than others, as indicated by RMSE 0.210 and 0.266, respectively. The results show that for the submerged breakwaters tested in this study, the formulae presented by d’Agremond et al. [23], Van der Meer et al. [34], and Briganti et al. [33] are sufficiently reliable to predict a suitable transmission coefficient for the tested conditions.

4. Discussion

4.1. Literature Formulae Comparison

The current study designed laboratory tests on wave transmission at emerged, zero-freeboard, and submerged breakwaters. The experimental tests were applied to compare and assess the performance of existing formulae for predicting wave transmission coefficients. Special attention was given to avoid unrealistic wave setup inside the 2D wave flume. The results from the comparison show that some of the equations are not suitable to represent transmission for the breakwaters tested in this study. Wave breaking is an important factor affecting wave transmission. As a result, the crest width plays a significant role in determining the wave transmission coefficient [39]. The effect of this factor is absent from the Van der Meer [5] formula. In this formula, the transmission coefficient only depends on a dimensionless crest height and the transmission coefficient is always equal to 0.46 for zero-freeboard. Hence, the Van der Meer [21] formula failed to give a reliable prediction of wave transmission coefficients for all configurations, as it showed an overestimation of the coefficients (Figure 9a). Van der Meer and Daemen [22] (Figure 9b) by taking into account the effect of crest freeboard and width, wave height, and wave steepness, proposed new formulae for wave transmission coefficient. The comparison results show good agreement for the emerged and zero-freeboard breakwaters, although for the submerged ones the scatters are evident, and for large values of K_t this formula underestimates the prediction. The Seabrook and Hall [24] formula is valid for submerged barriers but overestimate $K_t$ for emerged breakwaters. However, for the submerged breakwaters, there is still evident scatter, and it seems this formula is not applicable over the range of current test data (Figure 9e). A significant overestimation was also found for the formula of Calabrese et al. [30] (Figure 9d). Figure 9f shows that the Van der Meer et al. [34] formula is quite well bounded for the submerged breakwaters, while there is obvious scatter for the emerged and zero-freeboard breakwaters. This formula is proposed in the presence of smooth structures and oblique waves, so this could explain the scatter for the structures mentioned above. In the case of submerged breakwaters, the main parameters that affect the transmission coefficient are the crest freeboard and crest width. These parameters can be effective for smooth breakwaters as well. Another important factor that plays a significant role in the transmission coefficient for emerged breakwaters is the porosity of the structure. This parameter influences the transmission coefficient of emerged and zero-freeboard breakwaters, while its effect is missing in smooth structures. This could be another reason for the observed scatter in the transmission coefficient for the aforementioned breakwaters when predicted by the formula of Van der Meer et al. [34]. The comparison results show that, among the existing transmission formulae, d’Angremond et al. [23] (Figure 9c) and Briganti et al. [33] (Figure 9g) provide the most reliable estimate of the transmission coefficient for all breakwaters tested in this study. Comparison analysis results are shown in Figure 10 based on RMSE. This figure shows that for the emerged breakwater, the worst choice is the Seabrook and Hall [24] formula and the best choice is the Van der Meer and Daemen [22] formula. In the case of zero-freeboard and submerged breakwaters, there are more possible choices to predict wave transmission coefficient; however, the results in Figure 10 show the well-accepted formulae for the tested conditions are d’Angremond et al. [23] and Briganti et al. [33]. It is worth noting that natural uncertainty is expected for a real-case scenario. For instance, wave overtopping-induced transmission could be considerably higher with similar incident spectra under the influence of onshore winds [40,41] or other wave and atmospheric process, i.e., wave setup, tides, storm surge, and sea level rise [42,43,44,45].

4.2. Additional Considerations

A good estimation of the wave transmission coefficient is necessary for the design or assessment of breakwaters [46]. However, that coefficient only provides information about the wave heights behind the breakwaters and does not provide information about the wave period or the total amount of wave energy transmitted over the breakwaters. It is now widely accepted that a proper design cannot be separated from frequency domain analysis, especially when the breakwater serves as a beach erosion controller. When feeding the time series of water surface elevation to the Fourier transform, the result is a complex-valued function representing the Fourier coefficients of that function. These coefficients make it possible to obtain the variance spectrum of such function. For discretely sampled data, the discrete energy spectrum Sp can be represented in terms of Fourier amplitudes,

$${\mathrm{S}}_{\mathrm{n}}=\langle Z_n{Z}_n^{*}\rangle$$

(13)

where Z_n is the complex Fourier series of the function, and being Z_n the complex conjugate of Z_n^*, and < > denotes the expected-value, or average, operator. The discrete Fourier transform Z_n of a wave record ξ_j is:

$$Z_n=\frac{1}{N}\sum _{j=0}^{N-1}{\mathsf{\xi }}_{j}exp\left[-i2\pi jn/N\right]$$

(14)

$${\mathsf{\xi }}_j=\sum _{n=0}^{N-1}{\mathrm{Z}}_{j}exp\left[i2\pi jn/N\right]$$

(15)

For j = 0, 1, …, N − 1; n = 0, 1, …, N − 1.

The single spectrum S_n is therefore:

$$S_n=\frac{1}{N^2}\left[{\left|Z_n\right|}^2+{\left|Z_{N-n}\right|}^2\right];\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}n = 1,2, \dots , (N/2-1)$$

(16)

In comparing the spectral energy distribution of incident and transmitted waves, several authors [46,47] found that the primary peak frequency remained almost unchanged, but a considerable amount of energy (about 40%, as claimed by Van der Meer [46]) was shifted towards high frequencies. Figure 11 shows an example of the incident and transmitted spectra for all configurations tested in this study. As expected, the peak frequency for both spectra remains more or less the same, while the transmitted spectra show different kinds of spreading. For the emerged breakwater, Figure 11(a-II) shows that the transmitted energy decreases significantly (two orders of magnitude less), albeit leading to the formation of two peaks. In that case, waves break on the smooth slope and jump over the crest into the water behind the structure; that jump often creates favourable conditions for further wave generation behind the structure [46]. For the other three configurations, the transmitted spectra were found to spread more uniformly in the range 1.5–3.5 times the primary peak frequency, according to Van der Meer [46]. This amount of energy present at higher frequencies causes the mean period to considerably decrease. Comparing Figure 11(c-II),(d-II), it is possible to note that the second harmonic (with a frequency of about two times the primary peak frequency) decreases with the lowering of the ratio $B/R_c$. This behaviour was already highlighted by Yamashiro et al. [47] in testing two configurations of submerged breakwater with $B/R_c$ equal to 2 and 4, respectively. In this vein, if test (b) (Figure 11b-II) is considered to have a larger value of $B/R_c$, it appears to also explain the reduction in the second peak. These points should be strongly considered when designing low-crested breakwaters because they likely affect the features of sediment transport processes. In fact, despite a lower transmission coefficient, the presence of the bichromatic wave groups behind the emerged breakwater could lead to greater offshore sediment transport than in the case of submerged ones. For instance, Vicinanza et al. [48], in conducting experiments on differences in beach profile evolution for comparable energy flux of the incident short waves during monochromatic, bichromatic, combination, and random waves, showed that the greatest relative vertical variations in the surf zone were obtained in the bichromatic case. Baldock et al. [49] claimed that bichromatic wave groups showed maximum wave height in the swash zone, greatly increasing offshore transport and generating breaker bars that are larger and farther offshore. Hence, it is suggested that the large change in the beach response is a result of the increased significant and maximum wave heights in the wave groups. This aspect could also provide a heuristic explanation for many errors in the design of coastal structures which, relative to beach erosion control, were not as effective as originally believed. Coastal engineers were particularly focused on structural behaviour and hydraulic performance (wave transmission) between the 1970s and 1990s, with no critical assessment of transmitted wave spectra and their effects on sediment transport.

5. Conclusions

This study provides tips on literature formulae performance for estimating wave transmission over rubble-mound breakwaters. A benchmark program was applied, in which the validated hydrodynamics from new laboratory data were compared to wave transmission coefficients calculated by several existing formulae. Experiments were undertaken using a set of emerged and submerged breakwaters with varying submergence. The four reference configurations were built while keeping the overall volume constant, i.e., the lower the breakwater height, the greater the width. Comparison results for the measured and predicted transmission coefficient illustrated that for the zero-freeboard and submerged breakwaters, the formulae proposed by d’Angremond et al. [23] and Briganti et al. [33] give the most reliable estimation, while for the emerged breakwater, Van der Meer and Daemen’s [22] formula gives good agreement with the current data. For all formulae, worst performances in terms of RMSE were found for the emerged breakwater, for which the K_t values were often overestimated. If that could be seen as positive in terms of hydraulic safety, negative effects can be addressed in terms of water recirculation inside the harbour/protected area.

Spectra analysis was used to provide a critical appraisal: the reader is warned that by using the transmission coefficient, detailed knowledge of transmitted hydrodynamics may be neglected. Analysing the experimental data revealed that the transmitted energy spectrum presented two peaks, in all cases—one with the same value as the incident spectrum and the other with a frequency two times greater than the incident peak frequency. It was suggested that the primary peak frequency is independent of the crest freeboard, while the energy spectrum is proportional to it. In particular, a quasi-bichromatic spectrum was found for the emerged configuration. By decreasing the crest freeboard, more energy is transferred to the rear side of the breakwaters, as expected. However, the second peak frequency for the submerged breakwaters seems also affected by the crest width. This can be defined by waves breaking over the submerged breakwaters, and it is confirmed that the crest width plays an important role in the distribution of the transmitted energy.