# Experimental Investigation on Bragg Resonant Reflection of Waves by Porous Submerged Breakwaters on a Horizontal Seabed

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Conditions

#### 2.1. Wave Flume and Porous Submerged Breakwaters

**T**

_{1}and

**T**

_{2}. The sampling interval of the wave gauges was 0.02 s, and each sampling time was 81.92 s. In order to ensure the reliability of the experiment, these gauges were calibrated in a small water tank before the tests and each case was repeated three times, after which the average values of the results were taken.

**h**is the water depth,

**B**is the bar width,

**B**

_{0}is the short bottom edge of the trapezoidal bars,

**D**is the bar height, and

**S**is the spacing of the two adjacent bars.

**V**

_{1}. Next, the fillers were put into the container so they were flush with the water surface, and the volume was recorded as

**V**

_{2}. Then, the permeability of the porous submerged breakwaters

**φ**can be expressed as

**φ**=

**V**

_{1}/

**V**

_{2}

**V**

_{1}is the volume of water in the initial state in cubic centimeters (cm

^{3}), and

**V**

_{2}is the total volume of the filler flush with the water surface in cubic centimeters (cm

^{3}).

#### 2.2. Wave Parameters and Experimental Conditions

**h**, wave period

**T**, and wave height

**H**were the main parameters controlling wave making. The wave height was 0.04 m, and the water depth was 0.25 m, 0.30 m, and 0.40 m, respectively. The wave period

**T**varied with the ratio of twice the distance between the adjacent submerged breakwaters to the wavelength is 2

**S**/

**L**, and 2

**S**/

**L**is equal to 0.80, 0.86, 0.91, 0.94, 0.97, 1.00, 1.04, 1.12, and 1.21. The main factors affecting the Bragg resonant reflection are the permeability, height, width, spacing, section shape, and number of submerged breakwaters. In this study, the number of bars

**N**was fixed at 4, and the effects of permeability

**φ**, relative width

**B**/

**S**, relative height

**D**/

**h**, and section shapes on the Bragg resonant reflection were investigated respectively, as shown in Table 3.

## 3. Experimental Results

**S**/

**L**is near an integral multiple [14,15]. Therefore, this phenomenon and the change in reflectivity are discussed using the parameter 2

**S**/

**L**. This part analyzes the correlation between

**φ**,

**B**/

**S**,

**D**/

**h**, section shapes, and the wave reflection coefficient

**K**, which is expressed by the ratio of the amplitude of the reflected wave to that of the incident wave.

_{R}#### 3.1. Influence of Bar Permeability on Bragg Resonant Reflection

**N**= 4,

**B**= 0.2 m,

**D**= 0.1 m,

**S**= 1.0 m,

**h**= 0.25 m, and

**H**= 0.04 m. The submerged breakwater section was rectangular. We let 2

**S**/

**L**vary from 0.8 to 1.21, and

**T**varied from 1.2s to 1.7s. The permeability of the submerged breakwater was 0 (solid bars), 31.26%, 49.81%, 68.85%, and 100% (no bar), respectively.

**S**/

**L**0.8–1.21, and its peak value is near 0.95. This agrees with the results of Liu et al. [34], Mase et al. [36], Heathershaw [41], and Chang et al. [42]. Theoretically, the peak value of Bragg resonance dominant frequency appeared at a 2

**S**/

**L**equal to 1, and this difference could be explained by the downshift of wave frequency for Bragg resonance [22]. Another phenomenon that could be observed is that the Bragg resonant reflection coefficient decreased with the increased permeability. The same conclusion could be obtained from the numerical results of Ni and Teng [29]. When the permeability of submerged breakwaters was 0, 31.26%, 49.81%, and 68.85%, the measured peak reflection coefficient was 0.287, 0.210, 0.171, and 0.152, respectively. When there was no bar on the bottom bed, there was still a small reflection coefficient, which could be due to the errors as a result of the process of separating the measured waves into incident and reflected waves. The calculations and analyses showed that the reflection coefficients of the submerged breakwaters with a permeability of 31.26%, 49.81%, and 68.85% were reduced by 26.83%, 40.42%, and 47.04%, respectively, compared with the submerged breakwaters with a permeability of 0. Furthermore, it was found that a slowing trend occurred when the Bragg resonant reflection coefficient peak decreased with the increased permeability, as shown in Figure 9.

#### 3.2. Influence of the Relative Width on Bragg Resonant Reflection

**B**/

**S**on the Bragg resonant reflection coefficient, which is defined by the ratio of a single bar width

**B**to spacing between two adjacent bars

**S**. In this study, we set

**N**= 4,

**φ**= 68.85%,

**D**= 0.1 m,

**S**= 1.0 m,

**h**= 0.25 m, and

**H**= 0.04 m. The submerged breakwaters were rectangular and

**T**varied from 1.2s to 1.7s. The values of

**B**/

**S**were 0.1, 0.2, and 0.3, respectively. The reflection coefficient

**K**against 2

_{R}**S**/

**L**for 2

**S**/

**L**varying from 0.8 to 1.21 was calculated and is presented in Figure 11. When 2

**S**/

**L**was near 1, the Bragg resonant reflection coefficient reached its maximum and there was a trend of first increasing and then decreasing to near 1. When the relative bar width was 0.1, 0.2, and 0.3, respectively, the measured peak reflection coefficient was 0.119, 0.152, and 0.162. The situation without submerged breakwaters has been described in the previous section. The calculations and analyses show that the peak reflection coefficient with a relative bar width of 0.2 and 0.3 increased by 27.73% and 36.13% compared with the relative bar width of 0.1.

**B**/

**S**= 0.1 to

**B**/

**S**= 0.5, the peak value of the Bragg resonant reflection coefficient increased gradually. However, when the relative width continued to increase from

**B**/

**S**= 0.5 to

**B**/

**S**= 0.9, the peak value of the Bragg resonant reflection coefficient decreased instead, as shown in Figure 12. Furthermore, this coefficient increased with the increased relative width in the range of 0.1–0.3. This is somewhat consistent with the numerical results from Liu et al. [28], and Ni and Teng [29]. Moreover, additional experimental conditions should be implemented to prove this conclusion.

#### 3.3. Influence of the Relative Height on Bragg Resonant Reflection

**D**/

**h**, where the porous submerged breakwater section is rectangular,

**N**= 4,

**φ**= 68.85%,

**B**= 0.3 m,

**S**= 1.0 m, and

**H**= 0.04 m. The relative height

**D**/

**h**had three values of 0.25, 0.33, and 0.40, where the bar height

**D**was fixed at 0.1 m, and the relative bar height varied by changing the water depth

**h**. Moreover, 2

**S**/

**L**varied from 0.80 to 1.21, and

**T**increased from 1.05 s to 1.70 s. Similarly, the overall trend increased first and then decreased, and the maximum reflection coefficient appeared when the incident wavelength was almost twice the spacing between the two adjacent bars. Moreover, according to the experimental results, when the relative height was in the range of 0–0.40, the Bragg resonant reflection was positively correlated with the relative bar height, that is, a higher relative height indicated a greater reflection coefficient. When the relative height was 0.25, 0.33, and 0.40, the peak values of the Bragg resonant reflection coefficient were 0.078, 0.130, and 0.162, respectively. The peak reflection coefficient with a relative height of 0.33 and 0.40 was twice and two-thirds higher than that with a relative bar height of 0.25.

#### 3.4. Influence of Section Shapes of Bars on Bragg Resonant Reflection

**N**= 4,

**B**= 0.2 m,

**D**= 0.1 m,

**S**= 1.0 m,

**φ**= 68.85%,

**h**= 0.25 m, and

**H**= 0.04 m, the section shapes of submerged breakwaters are fixed as a rectangle, triangle, and trapezoid. Moreover, 2

**S**/

**L**varied from 0.8 to 1.21, and

**T**varied from 1.2 s to 1.7 s. The results are plotted in Figure 15, which shows the experimental values of the Bragg resonant reflection coefficient with different section shapes of bars. The variation of the reflection coefficient with 2

**S**/

**L**was the same as in the previous three sections. Moreover, it was shown that in the same conditions, the rectangular submerged breakwaters had the best Bragg resonant reflection coefficient with a value of 0.162. Moreover, the trapezoidal submerged breakwaters were ranked second with a reflection coefficient of 0.133, while the triangular ones had a poor reflection effect with a value of 0.107.

**S**/

**L**was close to 0.8, and the primary frequency had an obvious downshift, which was not obvious in this experiment.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**Sketch of porous submerged breakwaters with different section shapes and equal spacing ((

**a**) rectangle; (

**b**) triangle; (

**c**) trapezoid.

**h**is the water depth,

**B**is the bar width,

**B**

_{0}is the short bottom edge of the trapezoidal bars,

**D**is the bar height, and

**S**is the spacing of the two adjacent bars).

**Figure 8.**Influence of submerged breakwater permeability

**φ**on Bragg resonant reflection, rectangle,

**N**= 4,

**B**= 0.2 m,

**S**= 1.0 m,

**h**= 0.25 m, and

**H**= 0.04 m.

**Figure 10.**Comparison of experimental values of porous bars on the horizontal seabed (Mase et al. [36]).

**Figure 11.**Influence of the relative width

**B**/

**S**on Bragg resonant reflection, rectangle,

**N**= 4,

**φ**= 68.85%,

**S**= 1.0 m,

**h**= 0.25 m, and

**H**= 0.04 m.

**Figure 13.**Influence of the relative height

**D**/

**h**on the Bragg resonant reflection, rectangle,

**N**= 4,

**φ**= 68.85%,

**B**= 0.3 m,

**S**= 1.0 m, and

**H**= 0.04 m.

**Figure 15.**Influence of section shapes of bars on Bragg resonant reflection,

**N**= 4,

**B**= 0.2 m,

**D**= 0.1 m,

**S**= 1.0 m,

**φ**= 68.85%,

**h**= 0.25 m, and

**H**= 0.04 m.

**Figure 16.**Comparison of the Bragg resonant reflection coefficient

**K**with section shapes of bars (Cho et al. [5]).

_{R}Section Shapes | Long (m) | B (m) | D (m) | B_{0} (m) | S (m) |
---|---|---|---|---|---|

Rectangle | 0.5 | 0.1 | 0.1 | - | 1.0 |

Triangle | 0.5 | 0.1 | 0.1 | - | 1.0 |

Trapezoid | 0.5 | 0.1 | 0.1 | 0.06 | 1.0 |

Fillers | V (cm_{1}^{3}) | V (cm_{2}^{3}) | φ (%) |
---|---|---|---|

A | 30,937.48 | 44,934.61 | 68.85 |

B | 29,186.32 | 58,595.26 | 49.81 |

C | 32,370.26 | 103,551.70 | 31.26 |

Case | h (m) | T (s) | H (m) | φ (%) | Section Shapes | B (m) | D (m) | N |
---|---|---|---|---|---|---|---|---|

1 | 0.25 | 1.20–1.70 | 0.04 | 0, 31.26, 49.81, 68.85, 100 | Rectangle | 0.2 | 0.1 | 4 |

2 | 0.25 | 1.20–1.70 | 0.04 | 68.85 | Rectangle | 0.1, 0.2, 0.3 | 0.1 | 4 |

3 | 0.25, 0.30, 0.40 | 1.05–1.70 | 0.04 | 68.85 | Rectangle | 0.3 | 0.1 | 4 |

4 | 0.25 | 1.20–1.70 | 0.04 | 68.85 | Rectangle, triangle, trapezoid | 0.2 | 0.1 | 4 |

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**MDPI and ACS Style**

Xu, W.; Chen, C.; Htet, M.H.; Sarkar, M.S.I.; Tao, A.; Wang, Z.; Fan, J.; Jiang, D. Experimental Investigation on Bragg Resonant Reflection of Waves by Porous Submerged Breakwaters on a Horizontal Seabed. *Water* **2022**, *14*, 2682.
https://doi.org/10.3390/w14172682

**AMA Style**

Xu W, Chen C, Htet MH, Sarkar MSI, Tao A, Wang Z, Fan J, Jiang D. Experimental Investigation on Bragg Resonant Reflection of Waves by Porous Submerged Breakwaters on a Horizontal Seabed. *Water*. 2022; 14(17):2682.
https://doi.org/10.3390/w14172682

**Chicago/Turabian Style**

Xu, Wei, Chun Chen, Min Han Htet, Mohammad Saydul Islam Sarkar, Aifeng Tao, Zhen Wang, Jun Fan, and Degang Jiang. 2022. "Experimental Investigation on Bragg Resonant Reflection of Waves by Porous Submerged Breakwaters on a Horizontal Seabed" *Water* 14, no. 17: 2682.
https://doi.org/10.3390/w14172682