# Bragg Resonance of Water Waves by Multiple Permeable Thin Barriers over Periodic Breakwaters

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. The Mathematical Model

#### 2.2. Method of Solution

## 3. Results

#### 3.1. Water Wave Interactions with a Single Permeable Barrier over Flat Bottom Topography

#### 3.2. Water Wave Scattering by Dual Permeable Barriers over Uniform Bottom

#### 3.3. Water Wave Scattering over Periodic Breakwaters

#### 3.4. Water Scattering by Fully Submerged Barriers behind an Undulated Bottom

## 4. Discussion

#### 4.1. Permeable Thin Barriers with Trapezoidal Breakwaters

#### 4.1.1. Bottom-Standing Barriers

#### Influence of Breakwater Amplitudes

#### Influence of Permeable Parameters

#### 4.1.2. Surface-Piercing Barriers

#### Influence of Breakwater Amplitudes

#### Influence of Permeable Parameters

#### 4.1.3. Fully Submerged Permeable Barriers

#### 4.2. Permeable Thin Barriers with Half-Cosine Breakwaters

#### 4.2.1. Bottom-Standing Barriers

#### Influence of Breakwater Amplitudes

#### Influence of Permeable Parameters

#### 4.2.2. Surface-Piercing Barriers

#### Influence of Breakwater Amplitudes

#### Influence of Permeable Parameters

#### 4.2.3. Fully Submerged Permeable Barriers

#### 4.3. Energy Loss

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic of water wave reflection by multiple permeable barriers over periodic breakwaters.

**Figure 2.**(

**a**) Surface-piercing barrier and (

**b**) bottom-standing barrier for separated abrupt shelves.

**Figure 3.**$\left|R\right|$ and $\left|T\right|$ with respect to ${k}_{1,0}{h}_{1}$ for a single permeable (

**a**) surface-piercing and (

**b**) bottom-standing barrier over a flat bottom with different values of $G$.

**Figure 4.**$\left|R\right|$ and $\left|T\right|$ with respect to ${k}_{1,0}{h}_{1}$ for a single permeable (

**a**) surface piercing and (

**b**) bottom-standing barrier over the flat bottom with different values of $N$.

**Figure 5.**Varying $\left|R\right|$ with respect to incidence $\gamma $ for a single permeable surface-piercing and bottom-standing barrier over the flat bottom.

**Figure 6.**$\left|R\right|$ varying incidence $\gamma $ for dual permeable (

**a**) surface-piercing and (

**b**) bottom-standing barriers over the flat bottom.

**Figure 7.**Varying $\left|R\right|$ with respect to ${k}_{1,0}{h}_{1}$ for dual permeable (

**a**) surface-piercing and (

**b**) bottom-standing barriers over the flat bottom with different values of permeable parameters $G$.

**Figure 8.**Two types of periodic breakwaters: (

**a**) trapezoidal breakwater; (

**b**) half-cosine breakwater.

**Figure 9.**Varying $\left|R\right|$ with respect to dimensionless $2{k}_{1,0}/K$ of the (

**a**) trapezoidal shape and (

**b**) half-cosine bottom for different numbers of $J$.

**Figure 10.**Water waves are scattered by undulation bottom and multiple fully submerged permeable barriers.

**Figure 11.**Variations in $\left|R\right|$ with respect to $v/\lambda $ for fully submerged permeable barriers over undulation bottom.

**Figure 12.**Variations in (

**a**) reflection $\left|R\right|$ and (

**b**) transmission $\left|T\right|$ coefficients with respect to $2{k}_{1,0}/K$ for permeable bottom-standing barriers over trapezoidal breakwaters with different values of ${h}_{s}/{h}_{1}$.

**Figure 13.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}/K$ for permeable bottom-standing barriers over trapezoidal breakwaters with different values of $G$.

**Figure 14.**Variations in (

**a**) reflection $\left|R\right|$ and (

**b**) transmission $\left|T\right|$ coefficients with respect to $2{k}_{1,0}/K$ for permeable surface-piercing barriers over trapezoidal breakwaters with different values of ${h}_{s}/{h}_{1}$.

**Figure 15.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}/K$ for permeable surface-piercing barriers over trapezoidal breakwaters with different values of $G$.

**Figure 16.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}/K$ for fully submerged permeable barriers with different values of $G$.

**Figure 17.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}\mathrm{cos}\gamma /K$ for fully submerged permeable barriers with different values of $\gamma $.

**Figure 18.**Variations in (

**a**) reflection $\left|R\right|$ and (

**b**) transmisstion $\left|T\right|$ coefficient with respect to $2{k}_{1,0}/K$ for permeable bottom-standing barriers over half-cosine breakwaters with different values of ${h}_{s}/{h}_{1}$.

**Figure 19.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}/K$ for permeable bottom-standing barriers over half-cosine breakwaters with different values of $G$.

**Figure 20.**Variations in (

**a**) reflection $\left|R\right|$ and (

**b**) transmision $\left|T\right|$ coefficient with respect to $2{k}_{1,0}/K$ in the permeable surface-piercing barriers over half-cosine breakwaters with different values of ${h}_{s}/{h}_{1}$.

**Figure 21.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}/K$ for permeable surface-piercing barriers over half-cosine breakwaters with different values of $G$.

**Figure 22.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}/K$ for fully submerged permeable barriers over half-cosine breakwaters with different values of $G$.

**Figure 23.**Variations in $\left|R\right|$ with respect to $2{k}_{1,0}\mathrm{cos}\gamma /K$ for fully submerged permeable barriers over half-cosine breakwaters with different values of incident angles $\gamma $.

**Figure 24.**Variations in ${E}_{loss}$ with respect to dimensionless parameter $2{k}_{1,0}/K$ over (

**a**) trapezoidal breakwaters and (

**b**) half-cosine breakwaters with different values of $G$.

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**MDPI and ACS Style**

Tran, C.-T.; Lin, C.; Tsai, C.-C. Bragg Resonance of Water Waves by Multiple Permeable Thin Barriers over Periodic Breakwaters. *Water* **2023**, *15*, 495.
https://doi.org/10.3390/w15030495

**AMA Style**

Tran C-T, Lin C, Tsai C-C. Bragg Resonance of Water Waves by Multiple Permeable Thin Barriers over Periodic Breakwaters. *Water*. 2023; 15(3):495.
https://doi.org/10.3390/w15030495

**Chicago/Turabian Style**

Tran, Chang-Thi, Chitsan Lin, and Chia-Cheng Tsai. 2023. "Bragg Resonance of Water Waves by Multiple Permeable Thin Barriers over Periodic Breakwaters" *Water* 15, no. 3: 495.
https://doi.org/10.3390/w15030495