# Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Problem Definition

#### 2.2. Dispersion Relations and Eigenfunctions

#### 2.3. Subaerial Porous Breakwaters

#### 2.4. Eigenfunction Matching Method

#### 2.5. Wave Force on the Partially Reflecting Wall

## 3. Results

#### 3.1. A Rectangular Porous Structure near a Partially Reflecting Vertical Wall

#### 3.2. Wave Force on the Vertical Wall

#### 3.3. Multiple Porous Structures near a Totally Reflecting Vertical Wall

#### 3.4. Trapezoidal Porous Breakwaters near a Porous Seawall

## 4. Discussion

#### 4.1. Constructive Bragg Scattering by the Partially Reflecting Wall

#### 4.2. Destructive Bragg Scattering by the Partially Reflecting Wall

#### 4.3. Oblique Incidence

#### 4.4. Periodic Porous Breakwaters

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${\alpha}_{1,m,n}$ | coefficient of vertical eigenfunction |

${\alpha}_{2,m,n}$ | coefficient of vertical eigenfunction |

${\beta}_{1,m,n}$ | coefficient of vertical eigenfunction |

${\beta}_{2,m,n}$ | coefficient of vertical eigenfunction |

$\gamma $ | incidence angle |

$\epsilon $ | porosity of porous media |

$\lambda $ | wavelength of incident wave |

$\sigma $ | angular frequency of incident wave |

$\rho $ | density of water |

${\varphi}_{m}$ | velocity potential on the $m-th$ shelf |

${\eta}_{m}$ | surface elevation on the $m-th$ shelf |

$\delta $ | parameter for porous layer |

${\theta}_{R}$ | phase angle |

${\zeta}_{m,n}\left(z\right)$ | vertical eigenfunction |

${\xi}_{m,n}^{\left(1\right)}\left(x\right)$ | the first horizontal eigenfunction |

${\xi}_{m,n}^{\left(2\right)}\left(x\right)$ | the second horizontal eigenfunction |

${\zeta}_{m,l}^{\mathrm{larger}}$ | vertical eigenfunction for the larger total depth |

${\zeta}_{m,l}^{\mathrm{smaller}}$ | vertical eigenfunction for the smaller total depth |

$\nabla $ | three-dimensional gradient operator |

${\nabla}^{2}$ | three-dimensional Laplace operator |

$\Delta $ | operator for porous pressure |

$\overline{a}$ | amplitude of incident wave |

$a$ | amplitude of the half-cosine shaped breakwater in Section 4 |

$b$ | width of the rectangular porous breakwater in Section 3.1 and Section 3.2 or separation distance between half-cosine breakwaters in Section 4 |

${d}_{m}$ | water depth on the $m-th$ shelf |

$f$ | friction coefficient of porous media |

$g$ | acceleration of gravity |

${h}_{m}$ | total depth on the $m-th$ shelf |

$i$ | unit of complex numbers |

${k}_{y}$ | transverse wavenumber |

${\widehat{k}}_{m,n}$ | lateral wavenumber of the $n-th$ evanescent mode on the $m-th$ shelf |

${k}_{m,n}$ | absolute wavenumber of the $n-th$ evanescent mode on the $m-th$ shelf |

${\widehat{k}}_{m,0}$ | lateral wavenumber of the propagating mode on the $m-th$ shelf |

${k}_{m,0}$ | absolute wavenumber of the propagating mode on the $m-th$ shelf |

$n$ | index of modes |

$m$ | index for shelves and steps |

${p}_{m}$ | pressure on the $m-th$ shelf |

$p$ | index for constructive Bragg scattering |

$q$ | index for destructive Bragg scattering |

$t$ | time |

$s$ | inertial coefficient of porous media |

${u}_{m}$ | fluid velocity or discharge velocity on the $m-th$ shelf |

${x}_{m}$ | $x$ coordinate of the $m-th$ step |

${\overline{x}}_{m}$ | reference location of the |

$\left(x,y,z\right)$ | three-dimensional Cartesian coordinates |

${A}_{m,n}$ | EMM unknown coefficients |

${B}_{m,n}$ | EMM unknown coefficients |

$D$ | Separation distance between the last porous breakwater and the vertical wall in Section 3.1, Section 3.2 and Section 4. |

${G}_{1}$ or ${G}_{2}$ | variable for depth eigenfunction |

${K}_{w}$ | partially reflecting factor of the vertical wall |

${K}_{R}$ | reflection coefficient |

${K}_{F}$ | dimensionless horizontal wave force on the vertical wall |

$M$ | number of shelves plus one |

$N$ | number of evanescent modes |

$T$ | wave period of incident wave |

$2\pi /K$ | wavelength of the periodic bottom in Section 4 |

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**Figure 1.**EMM definitions for the problem of water wave scattering by porous structures near a partially reflecting vertical wall over uneven bottoms.

**Figure 3.**Problem definition of water wave scattering by a rectangular porous breakwater near a partially reflecting vertical wall over a uniform bottom.

**Figure 4.**Reflection coefficient varying against dimensionless wavenumber for normally incident water wave scattering by a rectangular porous breakwater near a partially reflecting vertical wall.

**Figure 5.**Reflection coefficient varying against dimensionless wavenumber for obliquely incident water wave scattering by a rectangular porous breakwater near a partially reflecting vertical wall.

**Figure 6.**Dimensionless wave force varying against dimensionless wavenumber for normally incident water wave scattering by a rectangular porous breakwater near a partially reflecting vertical wall.

**Figure 7.**Dimensionless wave force varying against dimensionless wavenumber for normally incident water wave scattering by a rectangular porous breakwater near a partially reflecting vertical wall with different partially reflecting factors.

**Figure 8.**Problem definition of water wave scattering by multiple rectangular porous breakwaters near a vertical wall over a uniform bottom.

**Figure 9.**Reflection coefficient varying against dimensionless wavenumber for obliquely incident water wave scattering by three porous breakwaters near a totally reflecting vertical wall.

**Figure 10.**Reflection coefficient varying against dimensionless wavenumber for obliquely incident water wave scattering by six porous breakwaters near a totally reflecting vertical wall.

**Figure 11.**Reflection coefficient varying against dimensionless wavenumber for obliquely incident water wave scattering by a single trapezoidal porous breakwater near a porous seawall.

**Figure 12.**Reflection coefficient varying against dimensionless wavenumber for obliquely incident water wave scattering by two trapezoidal porous breakwaters near a porous seawall.

**Figure 13.**Problem definition of Bragg scattering by (

**up**) porous; (

**middle**) partially porous; (

**down**) impermeable half-cosine breakwaters near a partially reflecting vertical wall over a uniform bottom.

**Figure 14.**Reflection and transmission coefficients varying against 2k

_{1,0}/K for Bragg scattering by four periodic half-cosine shaped impermeable breakwaters without vertical wall.

**Figure 15.**Reflection coefficient and dimensionless wave force varying against 2k

_{1,0}/K with D = 0.4 m for normal-wave Bragg scattering by four periodic half-cosine shaped impermeable breakwaters with partially reflecting vertical wall.

**Figure 16.**Reflection coefficient and dimensionless wave force varying against 2k

_{1,0}/K with D = 1.2 m for normal-wave Bragg scattering by four periodic half-cosine shaped impermeable breakwaters near a partially reflecting vertical wall.

**Figure 17.**Reflection coefficient and dimensionless wave force varying against 2k

_{1,0}/K with D = 0.8 m for normal-wave Bragg scattering by four periodic half-cosine shaped impermeable breakwaters near a partially reflecting vertical wall.

**Figure 18.**Reflection coefficient and dimensionless wave force varying against 2k

_{1,0}cosγ/K with D = 0.8 m for oblique-wave Bragg scattering by four periodic half-cosine shaped impermeable breakwaters near a vertical wall with different partially reflecting factors.

**Figure 19.**Reflection coefficient and dimensionless wave force varying against 2k

_{1,0}cosγ/K with D = 0.8 m for oblique-wave Bragg scattering by four periodic half-cosine shaped impermeable breakwaters near a partially reflecting vertical wall with different incidence angles.

**Figure 20.**Reflection coefficient and dimensionless wave force varying against 2k

_{1,0}/K with D = 0.4 m for normal-wave Bragg scattering by four periodic half-cosine shaped porous or impermeable breakwaters near a partially reflecting vertical wall.

$\mathit{N}$ | ${\mathit{k}}_{1,0}{\mathit{h}}_{1}=\mathit{\pi}/10$ | ${\mathit{k}}_{1,0}{\mathit{h}}_{1}=\mathit{\pi}/3$ | ${\mathit{k}}_{1,0}{\mathit{h}}_{1}=\mathit{\pi}$ |
---|---|---|---|

0 | 0.4167 | 0.4419 | 0.4387 |

1 | 0.3960 | 0.4070 | 0.4499 |

2 | 0.3962 | 0.4110 | 0.4555 |

3 | 0.3954 | 0.4085 | 0.4550 |

5 | 0.3953 | 0.4087 | 0.4560 |

10 | 0.3952 | 0.4088 | 0.4566 |

15 | 0.3952 | 0.4087 | 0.4566 |

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**MDPI and ACS Style**

Chang, J.-Y.; Tsai, C.-C.
Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms. *J. Mar. Sci. Eng.* **2022**, *10*, 409.
https://doi.org/10.3390/jmse10030409

**AMA Style**

Chang J-Y, Tsai C-C.
Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms. *Journal of Marine Science and Engineering*. 2022; 10(3):409.
https://doi.org/10.3390/jmse10030409

**Chicago/Turabian Style**

Chang, Jen-Yi, and Chia-Cheng Tsai.
2022. "Wave Forces on a Partially Reflecting Wall by Oblique Bragg Scattering with Porous Breakwaters over Uneven Bottoms" *Journal of Marine Science and Engineering* 10, no. 3: 409.
https://doi.org/10.3390/jmse10030409