# Reflection Analysis of Impermeable Slopes under Bimodal Sea Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reflection Characteristics of Smooth Impermeable Slopes

## 3. Material and Methods

#### 3.1. Design of Model Tests

_{1}= 0, X

_{12}= L/10, X

_{13}= L/4, and X

_{14}= L/3, as shown in the detailed experimental set up. The full details of the experiment can be found in Orimoloye et al. [39].

#### 3.2. Reflection Analysis

#### 3.3. Estimation of Reflection Parameters

## 4. Results and Discussions

#### 4.1. Influence of Wall Slope on Reflection Characteristics

#### 4.2. Influence of Water Depth Variations

#### 4.3. Influence of Wave Steepness

#### 4.4. Effects of the Crest Freeboard

## 5. Reflection Coefficients of Steep Slopes Under Bimodal Waves

**a**) cot $\alpha $ = 1.5; (

**b**) cot $\alpha $ = 3.0).

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\alpha $ | The structure slope |

$\xi $ | Breaker index or surf similarity parameter |

${\xi}_{m-1,0}$ | Breaker Index with ${L}_{o}$ based on ${S}_{m-1,0}={H}_{m-1,0}/{L}_{m-1,0}$ |

${\xi}_{p}$ | Breaker Index with ${L}_{p}$ based on ${T}_{p}$ |

${T}_{p}$ | Peak period |

FFT | Fast Fourier Transform |

g | Acceleration due to gravity |

h | Water depth |

H | Wave height |

${H}_{m0}$ | Significant wave height |

IFFT | Inverse Fast Fourier Transform |

H | Significant wave height |

Jonswap | Joint North Sea Wave Project |

${K}_{r}$ | Reflection coefficient |

${L}_{m-1,0}$ | The linear wave length |

${R}_{c}$ | Crest freeboard |

SSER | Sea-swell energy ratio |

${T}_{m-1,0}$ | The spectra wave period |

${T}_{pS1}$ | Peak periods of swell wave at 11 s |

${T}_{pS2}$ | Peak periods of swell wave at 15 s |

${T}_{pS3}$ | Peak periods of swell wave at 20 s |

${T}_{pS4}$ | Peak periods of swell wave at 25 s |

${T}_{pW}$ | Peak period of wind wave |

UK | United Kingdom |

${S}_{m-1,0}$ | Wave steepness derived from ${T}_{m-1,0}$ |

${K}_{r}$ | Reflection coefficient |

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**Figure 1.**An illustration of the prediction of the reflection characteristics of smooth impermeable slopes.

**Figure 2.**(

**a**) An example of the bimodal spectra. (

**b**) Shifting patterns of swell peak periods from 11–25 s [35].

**Figure 3.**(

**a**) Layout of a schematic cross-section of the wave gauges applied for reflection analysis. (

**b**) Photograph of the constructed model.

**Figure 4.**Comparison of the reflection coefficient ${K}_{r}$ and breaker parameters ${\xi}_{m-1,0}$ between unimodal and bimodal seas across three slopes for: (

**a**) cot $\alpha =3.0$, (

**b**) cot $\alpha =1.5$, and (

**c**) cot $\alpha =0.0$.

**Figure 5.**Variations of the reflection coefficient ${K}_{r}$ with breaker index ${\xi}_{m-1,0}$ across three different slopes.

**Figure 6.**Variations of the reflection coefficient ${K}_{r}$ with non-dimensional water depth $h/L$ across the three different slopes investigated.

**Figure 7.**Relationships between wave steepness and the reflection coefficient ${K}_{r}$ across the three different slopes investigated.

**Figure 8.**Influence of dimensionless crest freeboard ${R}_{c}/{H}_{m0}$ with the reflection coefficient ${K}_{r}$ across the three different slopes investigated.

**Figure 9.**A representation of the non-linear multi-regression fit between ${K}_{r}$ and ${\xi}_{m-1,0}$ for the sloping seawall with: (

**a**) cot $\alpha $ = 1.5; (

**b**) cot $\alpha $ = 3.0.

**Figure 10.**Comparison between the measured ${K}_{r}$ and the predicted ${K}_{r}$ of the wave reflection coefficient (

**a**) cot $\alpha $ = 1.5; and (

**b**) cot $\alpha $ = 3.0.

**Figure 11.**Comparison between the measured ${K}_{r}$ and the predicted ${K}_{r}$ of the wave reflection coefficient (

**a**) cot $\alpha $ = 1.5; and (

**b**) cot $\alpha $ = 3.0.

**Table 1.**Bimodal wave conditions with the peak period of wind waves (${T}_{pW}$) and peak periods of swell waves ${T}_{pS1-S4}$ tested in the present study.

Test No | ${\mathit{H}}_{\mathit{m}\mathbf{0}}$ (m) | ${\mathit{T}}_{\mathit{pW}}$ (s) | ${\mathit{T}}_{\mathit{pS}\mathbf{1}}$ (s) | ${\mathit{T}}_{\mathit{pS}\mathbf{2}}$ (s) | ${\mathit{T}}_{\mathit{pS}\mathbf{3}}$ (s) | ${\mathit{T}}_{\mathit{pS}\mathbf{4}}$ (s) | h(m) | $\mathit{cot}\mathit{\alpha}$ = 0.0 | $\mathit{cot}\mathit{\alpha}$ = 1.5 | $\mathit{cot}\mathit{\alpha}$ = 3.0 | No. of Tests |
---|---|---|---|---|---|---|---|---|---|---|---|

T001 | 0.125 | 1.11 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T002 | 0.125 | 1.26 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T003 | 0.125 | 1.42 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T004 | 0.125 | 1.58 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T005 | 0.1 | 1.11 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T006 | 0.125 | 1.26 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T007 | 0.1 | 1.42 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T008 | 0.125 | 1.58 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T009 | 0.1 | 1.11 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

T010 | 0.1 | 1.26 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

T011 | 0.125 | 1.42 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

T012 | 0.125 | 1.58 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

T013 | 0.075 | 1.11 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T014 | 0.075 | 1.26 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T015 | 0.075 | 1.42 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T016 | 0.075 | 1.58 | 1.74 | 2.37 | 3.16 | 3.95 | 0.65 | 0 | 1.5 | 3.0 | 13 |

T017 | 0.1 | 1.11 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T018 | 0.1 | 1.26 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T019 | 0.1 | 1.42 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T020 | 0.1 | 1.58 | 1.74 | 2.37 | 3.16 | 3.95 | 0.7 | 0 | 1.5 | 3.0 | 13 |

T021 | 0.075 | 1.11 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

T022 | 0.075 | 1.26 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

T023 | 0.075 | 1.42 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

T024 | 0.075 | 1.58 | 1.74 | 2.37 | 3.16 | 3.95 | 0.6 | 0 | 1.5 | 3.0 | 13 |

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**MDPI and ACS Style**

Orimoloye, S.; Karunarathna, H.; Reeve, D.E. Reflection Analysis of Impermeable Slopes under Bimodal Sea Conditions. *J. Mar. Sci. Eng.* **2020**, *8*, 133.
https://doi.org/10.3390/jmse8020133

**AMA Style**

Orimoloye S, Karunarathna H, Reeve DE. Reflection Analysis of Impermeable Slopes under Bimodal Sea Conditions. *Journal of Marine Science and Engineering*. 2020; 8(2):133.
https://doi.org/10.3390/jmse8020133

**Chicago/Turabian Style**

Orimoloye, Stephen, Harshinie Karunarathna, and Dominic E. Reeve. 2020. "Reflection Analysis of Impermeable Slopes under Bimodal Sea Conditions" *Journal of Marine Science and Engineering* 8, no. 2: 133.
https://doi.org/10.3390/jmse8020133