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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Governing Parameters of Wave Transmission

#### 1.2. Existing Wave Transmission Formulae

_{n}

_{50}) 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 ($m$).

_{2}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 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

#### 2.2. Wave Characteristics and Measurements

#### 2.3. Dimensional Analysis

## 3. Results

#### 3.1. Wave Transmission Performance

#### 3.1.1. Effect of Relative Crest Freeboard (${R}_{c}/{H}_{i}$) on Wave Transmission Coefficient (${K}_{t}$)

#### 3.1.2. Effect of Wave Steepness (${H}_{i}/{L}_{0}$) on Wave Transmission Coefficient (${K}_{\mathrm{t}}$)

#### 3.1.3. Effect of Relative Wave Height (${H}_{i}/{D}_{n50}$) on Wave Transmission Coefficient (${K}_{\mathrm{t}}$)

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

#### 3.2. Literature Formulae Performance

## 4. Discussion

#### 4.1. Literature Formulae Comparison

_{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

_{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:

_{n}is therefore:

## 5. Conclusions

_{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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic view of the wave flume and experimental arrangement for measuring incident, transmitted, and reflected wave fields in the wave flume.

**Figure 3.**Rubble-mound breakwater configurations tested in the present study. Test (

**a**): emerged breakwater $({R}_{c}=0.065\mathrm{m}$); Test (

**b**): zero-freeboard breakwater (${R}_{c}=0$); Test (

**c**): submerged breakwater (${R}_{c}=-0.065\mathrm{m}$); and Test (

**d**): submerged breakwater (${R}_{c}=0.035\mathrm{m}$).

**Figure 11.**Examples of the incident and transmitted wave spectra for breakwaters: emerged (

**a**); zero-freeboard (

**b**); and submerged (

**c**) and (

**d**).

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

Symbol | Unit | Definitions |
---|---|---|

H_{i} | (m) | Incident significant wave height, preferably H_{m0i}, at the toe of the breakwater |

H_{t} | (m) | Transmitted significant wave height, preferably H_{m0t} |

T_{p} | (s) | Peak wave period |

L_{p} | (m) | Deep-water wavelength |

m_{o} | (m^{2}) | Zeroth-order moment of the wave spectrum |

T_{m−1,0} | (s) | First negative moment of the energy spectrum |

T_{m0,2} | (s) | Zero-crossing period or mean zero-crossing period |

h_{s} | (m) | Height of the breakwater |

d | (m) | Water depth at the toe of the breakwater |

R_{c} | (m) | Crest freeboard of the breakwater |

B | (m) | Crest width of the breakwater |

m | (-) | Seaward slope of the breakwater |

S_{op} | (-) | $\mathrm{Wave}\mathrm{steepness}({S}_{op}=\frac{2\pi {H}_{i}}{g{T}_{p}^{2}}=\frac{{H}_{i}}{{L}_{p}}$) |

ξ_{op} | (-) | $\mathrm{Surf}\mathrm{similarity}\mathrm{parameter}({\xi}_{op}=\frac{m}{\sqrt{{S}_{op}}}$) |

D_{n}_{50} | (m) | Nominal diameter of the armour rock |

**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] | ${K}_{t}=a\frac{{R}_{c}}{{D}_{n50}}+b$ $a=0.031\frac{{H}_{i}}{{D}_{n50}}-0.24$ $b=-5.42{S}_{op}+0.0323\frac{{H}_{i}}{{D}_{n50}}-0.0017{\left(\frac{B}{{D}_{n50}}\right)}^{1.84}+0.51$ | $1<{H}_{i}/{D}_{n50}<6$ $0.01<{S}_{op}<0.05$ $-2<{R}_{c}/{D}_{n50}<2$ $0.075<{K}_{t}<0.75$ | 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] | ${K}_{t}=1-\left(exp\left(-0.65\left(\frac{{R}_{c}}{{H}_{i}}\right)-1.09\left(\frac{{H}_{i}}{B}\right)\right)+0.047\left(\frac{B}{L}\frac{{R}_{c}}{{D}_{n50}}\right)\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-0.067\left(\frac{{R}_{c}}{B}\frac{{H}_{i}}{{D}_{n50}}\right)\right)$ | $0\le \frac{B}{L}\frac{{R}_{c}}{{D}_{n50}}\le 7.08$ $0\le \frac{{R}_{c}}{B}\frac{{H}_{i}}{{D}_{n50}}\le 2.14$ $5\le \frac{B}{{H}_{i}}\le 74.47$ | 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) |

Models | Wave Type | Measured Incident | Measured Transmitted | |||
---|---|---|---|---|---|---|

H_{m0}_{-}_{i} (m) | T_{p0,2-i} (s) | H_{m0-t} (m) | T_{p0,2-t} (s) | R_{c} (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 |

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**MDPI and ACS Style**

Hassanpour, N.; Vicinanza, D.; Contestabile, P.
Determining Wave Transmission over Rubble-Mound Breakwaters: Assessment of Existing Formulae through Benchmark Testing. *Water* **2023**, *15*, 1111.
https://doi.org/10.3390/w15061111

**AMA Style**

Hassanpour N, Vicinanza D, Contestabile P.
Determining Wave Transmission over Rubble-Mound Breakwaters: Assessment of Existing Formulae through Benchmark Testing. *Water*. 2023; 15(6):1111.
https://doi.org/10.3390/w15061111

**Chicago/Turabian Style**

Hassanpour, Nasrin, Diego Vicinanza, and Pasquale Contestabile.
2023. "Determining Wave Transmission over Rubble-Mound Breakwaters: Assessment of Existing Formulae through Benchmark Testing" *Water* 15, no. 6: 1111.
https://doi.org/10.3390/w15061111