# Laboratory Experiments on the Influence of the Wave Spectrum Enhancement Factor on a Rubble Mound Breakwater

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

_{s}), peak frequency (f

_{p}) or peak period (T

_{p}) (f

_{p}= 1/T

_{p}), and of form parameters such as the Peak Enhancement Factor (PEF) [12]. Several spectral functions are used to represent the wave signal: Jonswap, Pierson-Moskowitz [13], ISSC, ITTC, JONSWAP, SCOTT, and Liu [14]. Among these, the Jonswap spectrum is one of the most widely used for different sea conditions. The JONSWAP spectrum was established following a series of tests in the North Sea carried out by Hasselmann et al. [15]. This spectral density function of wave energy is currently the most adopted for the design of coastal structures because the shape parameters of this function can be set to match the shape of the wave spectrum with the local wave energy distribution. In the case of the Moroccan Atlantic coast, no study has been conducted thus far to define the local wave spectrum model of offshore waves. Mean spectral shape parameters are commonly considered for the design of different types of breakwaters; nevertheless, the choice of an inappropriate shape parameter may lead to underestimation or overestimation of breakwater stability during physical tests.

_{p}, or period with highest spectral energy density. For another function form of spectrum energy such as a double peak spectrum, Schtittrumpf et al. [21] and Van der Meer [22] showed the pertinence of the spectral wave period T

_{m−1,0}. In fact, T

_{m−1,0}gives more weight to the longer periods in the spectrum than an average period.

## 2. Materials and Methods

#### 2.1. Model Set-up

^{3}placed on the wave-exposed side, and a reduced volume of 1.5 m

^{3}in the upper part of the rear slope. Cubipod® is a novel precast block that can be placed in a single or double layer; and it has an economical advantage in comparison with commonly used blocks such as tetrapods and Antifer blocks. Cubipods® have a proven ability of resistance to a large range of wave conditions in the Mediterranean Sea and the Atlantic Ocean [24]. The breakwater core is made of rubble components ranging in weight from 1 to 1000 kg; a stone layer separates the core from the armour blocks. Figure 3 presents a cross section of the studied breakwater.

#### 2.2. Wave Spectrum

- $r={e}^{[-\frac{{\left(f-{f}_{p}\right)}^{2}}{2{\sigma}^{2}{f}_{p}^{2}}]}$
- α: Phillips constant
- f
_{p}is the frequency which corresponds to the peak value of the spectral density function - σ = (σ
_{a}if f ≤ f_{p}; σ_{b}if f ≥ f_{p})

_{1/3}and T

_{1/3})

- ${\beta}_{J}=\frac{0.0624\times \left(1.094-0.01915\times \mathrm{ln}\gamma \right)}{0.23+0.0336\gamma -0.185{\left(1.9+\gamma \right)}^{-1}}$
- $r={e}^{[-\frac{{\left(f-{f}_{p}\right)}^{2}}{2{\sigma}^{2}{f}_{p}^{2}}]}$
- γ varies from 1 to 7
- H
_{1/3}: Significant wave height - T
_{1/3}: Significant wave period - T
_{p}: Peak period.

_{p}) [15]. During wave propagation, nonlinear interactions between waves involves energy transfer from peak frequencies to low wavelengths and very long wavelengths [26,27].

#### 2.3. Wave Conditions

_{p}), as well as the retained PEF (γ = 1 and γ = 3.3). The duration of the tests covered a period sufficiently long to represent a real sea state. The natural wave conditions retained are summarized in Table 3.

#### 2.4. Measurements of Damage, Pressure on the Crown Wall and Overtopping

- Beginning of damage: Corresponds to the displacements of the armour blocks over a distance greater than or equal to D
_{50}(mean diameter of armour blocks); - Irribaren damage: Holes created in the armour surface cause the exposure of the sub-layer;
- Beginning of destruction: Corresponds to the beginning of damage to the sub-layer;
- Destruction: the sub-layer is exposed to the effect of incident waves.

^{®}was recorded using a high-precision digital video camera. After each test, the moved and flipped units were counted before reconstituting the layer for the next test. In order to measure the instantaneous variation of wave pressure on the crown wall, the latter was equipped with a sensor placed at mid-height (Cf. Figure 4).

- V: Total of the overtopping volume measured at the end of the test
- B: Width of the receptacle
- T: Test duration.

## 3. Results

#### 3.1. Overtopping

- The measured average overtopping rate (q) for a spectrum with γ = 3.3 is 35% to 100% greater than with the spectrum with γ = 1.
- For a wave spectrum generated with γ = 1, the measured average overtopping has almost the same value for the two peak periods T
_{p}= 16 s and T_{p}= 18 s.

#### 3.2. Water Pressure on the Crown Wall

- For a given value of the peak period (T
_{p}), the measured maximum pressure for a spectrum with γ = 3.3 is 20% higher than the spectrum with γ = 1. - For a given value of the PEF, the maximum peak period generated the higher value of maximum wave pressure.

#### 3.3. Armour Block Stability

- For the first tests established for waves with Hs varying from 4 to 5 m, the variation of the PEF parameter has no consequence on armour stability. This is mainly due to the fact that the structural response is below the threshold of the damage beginning level;
- For the rest of the tests where we approached the destabilization limits (Hs varying from 5 to 6 m), we noted that the influence of the PEF becomes more meaningful for higher peak periods.

## 4. Discussion

_{mean}), significant wave height (H

_{S}or H

_{1/3}), and the mean of the 10% highest waves (H

_{1/10}).

_{1/10}heights for the two studied JONSWAP spectra with γ = 3.3 and γ = 1 is approximately 1%. This difference cannot be the source of the measured pressure differences. The variation of maximum wave pressure on the crown wall is therefore due to other factors associated with the spectral energy distribution.

## 5. Conclusions

- The PEF variation has a significant effect on the pressure exerted on vertical structures and on mean overtopping flow;
- The stability of the armour layers is not sensitive to the variation of the PEF for intermediate wave periods (12 to 14 s). This observation is consistent with the results of armour stability tests conducted by Van der Meer and Pilarczyk [36] for narrow and wide wave spectra. However, for longer waves, higher values of the PEF lead to more severe damage levels. The influence of the PEF on armour layer response is therefore highlighted for long period ocean waves.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Photo of the wave flume with the model breakwater in the foreground (

**a**) and components of the breakwater (

**b**) front slope (1), crown wall (2), and back slope (3).

**Figure 4.**Location of the pressure sensor to measure the instantaneous variation of wave pressure impact on the crown wall.

Notation | Unit | Scaling Factor |
---|---|---|

Length, width, wave height | m | N |

Surface | m^{2} | N^{2} |

Volume | m^{3} | N^{3} |

Time | s | N^{1/2} |

Velocity | m/s | N^{1/2} |

Mass | kg | N^{3} |

Density | kg/m^{3} | 1 |

Test Reference | Density (kg/m ^{3}) | Full-Scale Median Weight (kg) | Modelled Median Weight (kg) |
---|---|---|---|

Front armour layer | 2.4 | 7200 | 0.267 |

Rear armour layer | 2.4 | 3600 | 0.133 |

Underlayer in front side | 2.6 | 600 | 0.022 |

Underlayer in rear side | 2.6 | 350 | 0.013 |

Core | 2.6 | 500 | 0.018 |

Test Reference | Return Period | T_{p} (s)
| PEF | H_{s} |
---|---|---|---|---|

DAK21135 | 200 years | 14 | 1 | 6 m |

DAK21136 | 200 years | 14 | 3.3 | 6 m |

DAK21137 | 100 years | 16 | 3.3 | 5.5 m |

DAK21139 | 100 years | 16 | 1 | 5.5 m |

DAK21138 | 100 years | 18 | 3.3 | 5.5 m |

DAK21146 | 100 years | 18 | 1 | 5.5 m |

DAK21154 | 200 years | 16 | 3.3 | 6 m |

DAK21155 | 200 years | 16 | 1 | 6 m |

DAK21156 | 200 years | 18 | 3.3 | 6 m |

DAK21157 | 200 years | 18 | 1 | 6 m |

Test | Return Period | T_{p} [s] | γ | H_{s} | Q (L/s/m) |
---|---|---|---|---|---|

DAK21137 | 100 years | 16 | 3.3 | 5.5 | 26 |

DAK21139 | 100 years | 16 | 1 | 5.5 | 19 |

DAK21138 | 100 years | 18 | 3.3 | 5.5 | 46 |

DAK21146 | 100 years | 18 | 1 | 5.5 | 24 |

DAK21154 | 200 years | 16 | 3.3 | 6 | 62 |

DAK21155 | 200 years | 16 | 1 | 6 | 41 |

DAK21156 | 200 years | 18 | 3.3 | 6 | 88 |

DAK21157 | 200 years | 18 | 1 | 6 | 43 |

Water Level [m/Zh] | H_{s} (In −25 m/Zh) | T_{p} (In −25 m/Zh) | Observations for γ = 3.3 | Observations for γ =1 |
---|---|---|---|---|

+4.21 | 4 m | 12 s | No damage | No damage |

+3.61 | 4.4 m | 14 s | No damage | No damage |

+4.21 | 5 m | 12 s | No damage | No damage |

+4.21 | 5 m | 18 s | No damage | No damage |

+3.61 | 5 m | 18 s | No damage | No damage |

+4.21 | 5.5 m | 14 s | Oscillation of 2 blocks | Oscillation of 1 block |

+4.21 | 5.5 m | 18 s | Oscillation of 2 blocks | Oscillation of 2 blocks |

+ 3.61 | 5.5 m | 18 s | Oscillation of 2 blocks | Oscillation of 2 blocks |

+ 4.21 | 6 m | 18 s | Extraction of 3 blocks | Oscillation of 3 blocks |

+ 3.61 | 6 m | 18 s | Extraction of 3 blocks | Oscillation of 4 blocks |

+ 4.21 | 6.6 m | 18 s | Extraction of 3 blocks | Oscillation of 4 blocks |

γ =1 | γ =3.3 | Rayleigh Distribution | |
---|---|---|---|

H_{1/10}/H_{s} | 1.248 | 1.253 | 1.271 |

H_{mean}/H_{s} | 0.639 | 0.636 | 0.626 |

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**MDPI and ACS Style**

Bakali, H.; Aouiche, I.; Serhir, N.; Zahir, Y.; Ziane, E.h.; Harti, A.; Zerhouni, Z.; Anthony, E.
Laboratory Experiments on the Influence of the Wave Spectrum Enhancement Factor on a Rubble Mound Breakwater. *J. Mar. Sci. Eng.* **2022**, *10*, 2035.
https://doi.org/10.3390/jmse10122035

**AMA Style**

Bakali H, Aouiche I, Serhir N, Zahir Y, Ziane Eh, Harti A, Zerhouni Z, Anthony E.
Laboratory Experiments on the Influence of the Wave Spectrum Enhancement Factor on a Rubble Mound Breakwater. *Journal of Marine Science and Engineering*. 2022; 10(12):2035.
https://doi.org/10.3390/jmse10122035

**Chicago/Turabian Style**

Bakali, Hosny, Ismail Aouiche, Najat Serhir, Youssef Zahir, El hassan Ziane, Abderrazak Harti, Zakariae Zerhouni, and Edward Anthony.
2022. "Laboratory Experiments on the Influence of the Wave Spectrum Enhancement Factor on a Rubble Mound Breakwater" *Journal of Marine Science and Engineering* 10, no. 12: 2035.
https://doi.org/10.3390/jmse10122035