# Bragg Scattering of Surface Gravity Waves Due to Multiple Bottom Undulations and a Semi-Infinite Floating Flexible Structure

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Numerical Solution Based on Boundary Element Method

## 4. Energy Identity

## 5. Analytic Long Wave Solution

## 6. Results and Discussions

#### 6.1. Semi-Infinite Plate in the Absence of Bottom Undulation

#### 6.2. Trenches/Breakwaters in the Absence of Semi-Infinite Plate

#### 6.3. Plate-Trench Combination

## 7. Time Dependent Displacement

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 3.**Semi-infinite plate for different values of (

**a**) rigidity $D/\rho g$ with ${h}_{1}=5$ m; (

**b**) water depth ${h}_{1}$ with $D/\rho g={10}^{5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{4}$ in the absence of trenches/breakwaters.

**Figure 4.**Reflection coefficient ${K}_{r}$ versus width $w/{L}_{1}$ for different number N of rectangular (

**a**) trenches; (

**b**) breakwaters in the absence of semi-infinite ice sheet with gap $d/{L}_{1}=0.6$, ${L}_{1}=40\pi $ m.

**Figure 5.**${K}_{r}$ versus gap $2d/{L}_{1}$ for different number N of rectangular (

**a**) trenches with width $w/{L}_{1}=0.6$; (

**b**) breakwaters with width $w/{L}_{1}=0.35$ in the absence of the semi-infinite plate with ${L}_{1}=40\pi $ m.

**Figure 6.**${K}_{r}$ versus wave number ${k}_{1}{h}_{1}$ in case of single rectangular (

**a**) trench; (

**b**) breakwater in the absence of semi-infinite plate with different widths $w/{L}_{1}$, ${L}_{1}=40\pi $ m.

**Figure 7.**${K}_{r}$ versus ${k}_{1}{h}_{1}$ in case of double rectangular (

**a**) trench width $w/{L}_{1}=0.6$; (

**b**) breakwater with width $w/{L}_{1}=0.35$ in the absence of the semi-infinite plate for different values of gaps $2d/{L}_{1}$.

**Figure 8.**Reflection coefficient ${K}_{r}$ versus wave number ${k}_{1}{h}_{1}$ for different number N of rectangular trenches with (

**a**) width $w/{L}_{1}=0.6$, gap $d/{L}_{1}=0.86$, and breakwaters with (

**b**) width $w/{L}_{1}=0.35$, gap $d/{L}_{1}=0.60$ and (

**c**) width $w/{L}_{1}=0.6$, gap $d/{L}_{1}=0.86$ in the absence of the semi-infinite plate.

**Figure 9.**${K}_{r}$ versus ${k}_{1}{h}_{1}$ in case of different depth ${h}_{2}/{h}_{1}$ of double rectangular (

**a**) trenches with $w/{L}_{1}=0.6$, $d/{L}_{1}=0.86$; (

**b**) breakwaters with $w/{L}_{1}=0.35$, $d/{L}_{1}=0.6$ in the absence of a semi-infinite plate.

**Figure 10.**Reflection coefficient ${K}_{r}$ versus wave number ${k}_{1}{h}_{1}$ in case of single (

**a**) trench and (

**b**) breakwater in the presence of the semi-infinite plate for different values of $w/{L}_{1}$ with ${L}_{g}/{L}_{1}=0.25$ and $D/\rho g={10}^{5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{4}$.

**Figure 11.**${K}_{r}$ versus ${k}_{1}{h}_{1}$ for different number N of (

**a**,

**b**) trenches with $w/{L}_{1}=0.6$, $d/{L}_{1}=0.86$; (

**c**,

**d**) breakwaters with $w/{L}_{1}=0.35$, $d/{L}_{1}=0.6$, in the presence of a semi-infinite plate with ${L}_{g}/{L}_{1}=0.25$, $D/\rho g={10}^{5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{4}$.

**Figure 12.**${K}_{r}$ versus ${k}_{1}{h}_{1}$ in case of different depth ${h}_{2}/{h}_{1}$ of double (

**a**) rectangular trenches with $w/{L}_{1}=0.6$, $d/{L}_{1}=0.86$; (

**b**) breakwaters with $w/{L}_{1}=0.35$, $d/{L}_{1}=0.6$ in the presence of a semi-infinite plate.

**Figure 13.**${K}_{r}$ versus ${k}_{1}{h}_{1}$ for different values of rigidity $D/\rho g$ in the case of (

**a**) trenches with $w/{L}_{1}=0.6$, $d/{L}_{1}=0.86$; (

**b**) breakwaters with $w/{L}_{1}=0.35$, $d/{L}_{1}=0.6$ in the presence of a semi-infinite plate with ${L}_{g}/{L}_{1}=0.25$ and $N=3$.

**Figure 14.**Reflection coefficient ${K}_{r}$ versus gap $2d/{L}_{1}$ for different number of (

**a**) trenches with $w/{L}_{1}=0.33$; (

**b**) breakwaters with $w/{L}_{1}=0.89$ in the presence of a semi-infinite plate with $D/\rho g={10}^{5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{4}$, ${L}_{g}/{L}_{1}=0.25$.

**Figure 15.**Reflection coefficient ${K}_{r}$ versus gap ${L}_{g}/{L}_{1}$ in case of different number N of trenches with (

**a**) $w/{L}_{1}=0.33$, $d/{L}_{1}=0.61$, and breakwaters with (

**b**) $w/{L}_{1}=0.89$, $d/{L}_{1}=1.37$ in the presence of the semi-infinite plate with $D/\rho g={10}^{5}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{4}$.

**Figure 16.**Variation of the free surface displacement ${\zeta}_{c}(x,t)$ (m) versus x (m) and time t (s) in the case of a semi infinite plate in the case of flat seabed.

**Figure 17.**Variation of the free surface displacement ${\zeta}_{c}(x,t)$ (m) versus x (m) and time t (s) in the case of a semi-infinite plate in the presence of a single trench.

**Figure 18.**Variation of the free surface displacement ${\zeta}_{c}(x,t)$ (m) versus x (m) and time t (s) in the case of a semi-infinite plate in the presence of double trenches.

**Figure 19.**Variation of the free surface displacement ${\zeta}_{c}(x,t)$ (m) versus x (m) and time t (s) in the case of a semi-infinite plate in the presence of single breakwater.

**Figure 20.**Variation of the free surface displacement ${\zeta}_{c}(x,t)$ (m) versus x (m) and time t (s) in the case of a semi-infinite plate in the presence of double breakwaters.

**Table 1.**Energy identity as in Equation (31) for waves scattering by semi-infinite floating flexible plate in finite water depth case.

${\mathit{k}}_{1}{\mathit{h}}_{1}$ | $\mathit{D}/\mathit{\rho}\mathit{g}$ in ${\mathbf{m}}^{4}$ | BEM-Based Solution | ||
---|---|---|---|---|

${\mathit{K}}_{\mathit{r}}^{\mathit{2}}$ | $\mathit{\gamma}{\mathit{K}}_{\mathit{t}}^{\mathit{2}}$ | ${\mathit{K}}_{\mathit{r}}^{\mathit{2}}+\mathit{\gamma}{\mathit{K}}_{\mathit{t}}^{\mathit{2}}$ | ||

0.1 | ${10}^{5}$ | 0.00011 | 0.99988 | 0.99999 |

${10}^{6}$ | 0.00088 | 0.99912 | 1.00000 | |

${10}^{7}$ | 0.00830 | 0.99169 | 0.99999 | |

${10}^{8}$ | 0.02147 | 0.97852 | 0.99999 | |

0.2 | ${10}^{5}$ | 0.00146 | 0.99853 | 0.99999 |

${10}^{6}$ | 0.00989 | 0.99010 | 0.99999 | |

${10}^{7}$ | 0.02345 | 0.97654 | 0.99999 | |

${10}^{8}$ | 0.05330 | 0.94669 | 0.99999 | |

0.3 | ${10}^{5}$ | 0.00615 | 0.99384 | 0.99999 |

${10}^{6}$ | 0.01784 | 0.98216 | 1.00000 | |

${10}^{7}$ | 0.04025 | 0.95974 | 0.99999 | |

${10}^{8}$ | 0.09604 | 0.90395 | 0.99999 | |

0.4 | ${10}^{5}$ | 0.01108 | 0.98891 | 0.99999 |

${10}^{6}$ | 0.02560 | 0.97439 | 0.99999 | |

${10}^{7}$ | 0.06206 | 0.93793 | 1.00000 | |

${10}^{8}$ | 0.14006 | 0.85993 | 0.99999 |

**Table 2.**Relative error corresponding to Figure 3a.

${\mathit{k}}_{1}{\mathit{h}}_{1}$ | $\mathit{D}/\mathit{\rho}\mathit{g}$ | Analytic Solution | BEM-Based Solution | Relative Error |
---|---|---|---|---|

0.1 | ${10}^{5}$ | 0.01056 | 0.01102 | 0.04356 |

${10}^{6}$ | 0.02972 | 0.02851 | 0.04071 | |

${10}^{7}$ | 0.09113 | 0.09254 | 0.01547 | |

${10}^{8}$ | 0.14655 | 0.14962 | 0.02094 | |

0.2 | ${10}^{5}$ | 0.03826 | 0.03751 | 0.01960 |

${10}^{6}$ | 0.09944 | 0.09786 | 0.01588 | |

${10}^{7}$ | 0.15316 | 0.15759 | 0.02892 | |

${10}^{8}$ | 0.23088 | 0.23147 | 0.00255 | |

0.3 | ${10}^{5}$ | 0.07847 | 0.07954 | 0.01363 |

${10}^{6}$ | 0.13355 | 0.13627 | 0.02036 | |

${10}^{7}$ | 0.20064 | 0.20283 | 0.01091 | |

${10}^{8}$ | 0.30990 | 0.31006 | 0.00051 | |

0.4 | ${10}^{5}$ | 0.10529 | 0.10725 | 0.01861 |

${10}^{6}$ | 0.16001 | 0.16230 | 0.01431 | |

${10}^{7}$ | 0.24912 | 0.25013 | 0.00405 | |

${10}^{8}$ | 0.37425 | 0.37870 | 0.01189 | |

0.5 | ${10}^{5}$ | 0.12367 | 0.12427 | 0.00485 |

${10}^{6}$ | 0.18771 | 0.18927 | 0.00831 | |

${10}^{7}$ | 0.29405 | 0.29514 | 0.00370 | |

${10}^{8}$ | 0.42597 | 0.42647 | 0.00117 |

**Table 3.**Values of $w/{L}_{1}$, $2d/{L}_{1}$ for common min/ max of ${K}_{r}$ in case N number of trenches/breakwaters.

min$\left\{{\mathit{K}}_{\mathit{r}}\right\}$ | max$\left\{{\mathit{K}}_{\mathit{r}}\right\}$ | ||||
---|---|---|---|---|---|

Trenches | $w/{L}_{1}$ | 0.6 | 1.2 | 1.8 | 0.33 |

Trenches | $2d/{L}_{1}$ | 0.72 | 1.72 | 2.73 | 1.22 |

Breakwaters | $w/{L}_{1}$ | 0.35 | 0.70 | 1.06 | 0.89 |

Breakwaters | $2d/{L}_{1}$ | 0.20 | 1.20 | 2.20 | 1.71 |

**Table 4.**Values of ${k}_{1}{h}_{1}$ for common minimum of reflection coefficient for different values of $w/{L}_{1}$.

min$\left\{{\mathit{K}}_{\mathit{r}}\right\}$ | ||||
---|---|---|---|---|

Single trench | ${k}_{1}{h}_{1}$ | 0.25 | 0.51 | 0.76 |

Single breakwater | ${k}_{1}{h}_{1}$ | 0.25 | 0.50 | 0.75 |

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**MDPI and ACS Style**

Kar, P.; Koley, S.; Trivedi, K.; Sahoo, T. Bragg Scattering of Surface Gravity Waves Due to Multiple Bottom Undulations and a Semi-Infinite Floating Flexible Structure. *Water* **2021**, *13*, 2349.
https://doi.org/10.3390/w13172349

**AMA Style**

Kar P, Koley S, Trivedi K, Sahoo T. Bragg Scattering of Surface Gravity Waves Due to Multiple Bottom Undulations and a Semi-Infinite Floating Flexible Structure. *Water*. 2021; 13(17):2349.
https://doi.org/10.3390/w13172349

**Chicago/Turabian Style**

Kar, Prakash, Santanu Koley, Kshma Trivedi, and Trilochan Sahoo. 2021. "Bragg Scattering of Surface Gravity Waves Due to Multiple Bottom Undulations and a Semi-Infinite Floating Flexible Structure" *Water* 13, no. 17: 2349.
https://doi.org/10.3390/w13172349