# Step Approximation on Water Wave Breaking and Dissipation over Variable Bottoms across the Surf Zone

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Problem Definition

#### 2.2. Eigenfunction Matching Method

#### 2.3. Wave Breaking and Dissipation

#### 2.4. Wave Force

## 3. Results

#### 3.1. Rectangular Breakwater Close to a Partially Reflecting Vertical Wall

#### 3.2. Constructive and Destructive Bragg Scattering by Periodic Half-Cosine Shaped Breakwaters

#### 3.3. Oblique Incidence

#### 3.4. Water Wave Breaking and Dissipation by Mild Slopes

## 4. Discussion

#### 4.1. Parametric Analysis on Wave Scattring by Rectangular Breakwater in Front of Partially Reflecting Vertical Wall

#### 4.2. Wave Breaking and Dissipation by Steep Slopes

#### 4.3. Wave Breaking and Dissipation by Composite Slopes

#### 4.4. Wave Breaking and Dissipation by a Barred Beach

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Problem definition of wave scattering by a rectangular breakwater near a partially reflecting wall.

**Figure 3.**Dimensionless wave force varying against dimensionless breakwater width for scattering of incident waves by a rectangular structure placed near a partially reflecting vertical wall [35].

**Figure 4.**Problem definition of the oblique constructive and destructive Bragg scattering by four periodic half-cosine-shaped breakwaters placed near a partially reflecting wall.

**Figure 5.**Variation of wave reflection and dimensionless wave force varying against $2{k}_{1,0}/K$ for the constructive Bragg scattering by four periodic half-cosine-shaped breakwaters placed near a partially reflecting wall [39].

**Figure 6.**Variation of wave reflection and dimensionless wave force varying against $2{k}_{1,0}/K$ for the destructive Bragg scattering by four periodic half-cosine-shaped breakwaters placed close to a partially reflecting vertical wall [39].

**Figure 7.**Reflection coefficient and dimensionless wave force varying against $2{k}_{1,0}\mathrm{cos}\gamma /K$ for the oblique constructive Bragg scattering by four periodic half-cosine shaped breakwaters placed near a partially reflecting vertical wall [39].

**Figure 8.**Wave heights for breaking and dissipation of waves by a slope of $1/10$ for various numbers of (

**a**) evanescent modes and (

**b**) shelves.

**Figure 10.**The change in surface elevations at various wave phases for scattering of waves (

**a**) t = T/4 (

**b**) t = T/2 (

**c**) t = T3/4 (

**d**) t = T by a slope of $1/10$.

**Figure 12.**Effects of the wall partially reflecting factors K

_{w}on the dimensionless wave force and reflection coefficient with normal incidence.

**Figure 13.**Effects of the wall partially reflecting factors K

_{w}on the dimensionless wave force and reflection coefficient with oblique incidence of $\gamma =30\xb0$.

**Figure 14.**Effects of the breakwater heights on the dimensionless wave force and reflection coefficient.

**Figure 18.**Wave heights for wave breaking and dissipation by a barred beach with various numbers of shelves for the (

**up**) spilling and (

**down**) plunging breaking cases.

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**MDPI and ACS Style**

Chang, J.-Y.; Tsai, C.-C.
Step Approximation on Water Wave Breaking and Dissipation over Variable Bottoms across the Surf Zone. *J. Mar. Sci. Eng.* **2023**, *11*, 62.
https://doi.org/10.3390/jmse11010062

**AMA Style**

Chang J-Y, Tsai C-C.
Step Approximation on Water Wave Breaking and Dissipation over Variable Bottoms across the Surf Zone. *Journal of Marine Science and Engineering*. 2023; 11(1):62.
https://doi.org/10.3390/jmse11010062

**Chicago/Turabian Style**

Chang, Jen-Yi, and Chia-Cheng Tsai.
2023. "Step Approximation on Water Wave Breaking and Dissipation over Variable Bottoms across the Surf Zone" *Journal of Marine Science and Engineering* 11, no. 1: 62.
https://doi.org/10.3390/jmse11010062