# Representative Transmission Coefficient for Evaluating the Wave Attenuation Performance of 3D Floating Breakwaters in Regular and Irregular Waves

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{t}) which is defined as the ratio between the transmitted wave height (H

_{t}) and the incident wave height (H

_{I}). Numerically or analytically, where the incident wave height is already given, the transmission coefficient can be obtained by determining the transmitted wave height at a surface point behind the breakwater (e.g., see [2]). Experimentally, the incident wave height and transmitted wave height can be determined from the measured wave elevations. These approaches for determining the transmission coefficients for floating breakwaters are similar to those discussed for the traditional bottom-founded breakwaters [3,4,5].

## 2. Problem Definition, Methodology and Verification

_{s}and significant wave height H

_{I,s}. The incident wave angle with the x-axis is denoted by θ. The problem at hand is to determine the transmitted wave field and use this information to quantify the performance of the 3D floating breakwater under regular and irregular waves.

_{i}is the wave frequency of the i

^{th}spectral component, ω

_{s}= 2π/T

_{s}, n

_{ω}is the number of discrete spectral components for accurate representation of continuous wave spectrum. The resolution frequency is Δω = (ω

^{U}− ω

^{L})/n

_{ω}, where ω

^{L}and ω

^{U}are respectively the lower and upper bounds of wave frequencies where the spectral density value vanishes. The i

^{th}spectral component has the wave height ${H}_{\mathrm{I}}\left({\omega}_{i}\right)=2\sqrt{2S\left({\omega}_{i}\right)\Delta \omega}$.

_{I}, λ and ω, respectively. In comparing the wave attenuation performances of the floating breakwater under irregular waves and regular waves, we shall use a representative regular wave where its wave period and height are, respectively, equal to the significant wave height H

_{I,s}and the significant wave period T

_{s}[4,17,18,19].

_{t}is used:

_{t}is the transmitted wave height at the point considered.

_{t,s}is given by:

_{t}(ω

_{i}) is the transmitted wave height at the considered point for the component wave frequency ω

_{i}. The squared term under the square root in Equation (4) is usually referred to as a transfer function between the spectrum of the transmitted waves S

_{t}(ω) and the incident wave spectrum S(ω). We have the following relation:

**M**is the global mass matrix,

**M**

_{a}is the matrix of added mass,

**C**

_{d}is the matrix of hydrodynamic damping,

**K**is the global stiffness matrix,

**K**

_{rf}is the global matrix of the restoring force resulting from the combination of the buoyancy force and the gravitation force acting on the breakwater,

**u**is the nodal vector of the complex amplitudes of the displacements,

**F**

_{exc}is the vector of the complex amplitude of the excitation wave force. The vectors of breakwater displacements and excitation wave forces acting on the breakwater at the time t are, respectively, given by:

**K**,

**K**

_{rf}and

**M**are obtained using the finite element method [25]. The boundary condition due to the presence of the mooring system that the breakwater only moves up and down is imposed in the numerical model by modifying the stiffness matrix

**K**using the penalty method [26]. The added mass and hydrodynamic damping matrices

**M**

_{a}and

**C**

_{d}are obtained by applying the boundary element method procedure for the linear hydrodynamic problem where the fluid motion can be expressed in terms of the velocity potential φ

_{re}. In the frequency domain, the velocity potential can be written in the following form:

^{2}), ∂/∂n indicates the differential along the unit formal vector pointing from the structure to the fluid, u

_{j}(where j = {1, 2, 3}) are the complex amplitudes of the displacements along the x-, y- and z- directions, n

_{j}indicates the unit normal vector, |

**x**| is given by $\left|x\right|=\sqrt{{x}^{2}+{y}^{2}}$, k is the wave number and can be obtained by solving the dispersion relation k tanh(kH) = ω

^{2}/g, φ

_{in}is the complex amplitude of the incident velocity potential and is given by:

_{d}of the hydrodynamic water pressure can be calculated from the spatial velocity potential using the following equation [28]:

_{w}(= 1025 kg/m

^{3}) is the mass density of water.

_{3}|/A) of the breakwater are given in Figure 3. The results reported by Diamantoulaki et al. [6] are indicated by ‘3D Ref.’, while the results from the present study are indicated by ‘3D Present’. It can be seen that the present results are in good agreement with the results obtained by Diamantoulaki et al. [6].

## 3. Results and Discussions

_{I,s}= H

_{I}= 1 m).

#### 3.1. Transmitted Wave Field in Regular and Irregular Waves

_{t}between the transmission coefficient predicted for the representative regular wave and that for irregular waves. ΔK

_{t}= K

_{t,regular}—K

_{t,irregular}where K

_{t,regular}and K

_{t,irregular}are, respectively, the transmission coefficients for representative regular waves and irregular waves. Figure 11 shows that the difference ΔK

_{t}may be smaller than −0.4, and larger than 0.1. This finding for box-type heave-only breakwaters indicates that whilst the representative regular wave analysis is still adopted for predicting the wave attenuation performance in some design guidelines [4,17,19], the predicted performance may be significantly different from the actual performance of the breakwater in realistic irregular waves. More studies should be conducted in the future for different types of floating breakwaters and wave spectra to investigate their wave attenuation performance using the representative regular wave and irregular wave analyses.

#### 3.2. Evaluation of Wave Attenuation Performance

#### 3.2.1. From Experimental Studies

#### 3.2.2. From Numerical Studies

_{t,90%}for the case in Figure 12a is about 0.79, which is larger than the mean transmission coefficient by about 20% but smaller than the maximum transmission coefficient by about 10%. Note that the approach to determine the representative transmission coefficient based on the area percentage may also be applicable for experimental studies if the profile of wave elevations within the area of interest can be captured (such as by using stereo-videogrammetry [37]).

^{*}is also considered. This breakwater is similar to FB#1L, but its considered area of interest is equal to that of FB#1. The target area percentage is set to 90%. Figure 13 shows the representative transmission coefficients K

_{t,90%}for various wave frequencies. It can be seen that for regular waves, FB#1 is generally more effective than FB#2 in attenuating incident waves when ω > 0.78 rad /s, but it is less effective for longer waves. This is expected as the width of the breakwater has to be larger for attenuating wave forces for longer waves. For ω > 0.78 rad /s, the difference in the wave attenuation performance between the two breakwaters is up to about 39%. For irregular waves with ω

_{s}> 0.78 rad /s, the difference is much smaller (less than 10%).

^{*}). The significant difference in the wave attenuation performance between these two breakwaters of different lengths implies that it is necessary to perform a 3D hydrodynamic analysis of finite breakwaters so that the breakwater end effects on the wave attenuation performance is fully accounted for. The use of a 2D analysis for a floating breakwater with finite length may result in a non-conservative prediction (i.e., overprediction) of the wave attenuation performance.

_{t,90%}for the heave-only and motionless breakwaters FB#1 and FB#2. It can be seen that when the breakwaters are fixed, FB#2, with a larger width, is more effective than FB#1 in attenuating incident waves. Their associated representative transmission coefficients increase as the wave frequency decreases and are generally significantly lower than the representative transmission coefficients for the heave-only FB#1 and FB#2. The difference in the representative transmission coefficient between the motionless breakwaters and the heave-only breakwaters indicates the necessity of considering breakwater motions in the hydrodynamic analysis. Figure 14 also shows that the representative transmission coefficients for the motionless breakwaters (i.e., diffraction problems) are always smaller than unity, and the large transmission coefficient phenomenon (K

_{t,90%}> 1) only occurs for the heave-only breakwaters (i.e., combined diffraction-radiation problems). Such a large transmission coefficient phenomenon may result from the interaction between the diffracted waves and the radiated waves for wave frequencies close to the heave resonant frequency of the breakwater (as discussed in Section 3.1).

_{t,1/3}). Figure 15 shows the representative transmission coefficients K

_{t,1/3}, K

_{t,85%}, K

_{t,90%}, K

_{t,95%}for FB#1, FB#2, and FB#1L. Here, K

_{t,85%}, K

_{t,95%}are the representative transmission coefficients corresponding to the area percentage of 85% and 95%, respectively. It can be seen that, for the cases considered, K

_{t,1/3}is close to K

_{t,90%}and K

_{t,85%}, and it is smaller than K

_{t,95%}by up to about 10% for irregular waves. Note that the largest transmission coefficient (K

_{t,100%}) is considered when determining K

_{t,1/3}. This consideration is similar to the consideration of the largest wave height in the wave spectrum when determining the significant wave height. The consideration of the largest wave height is necessary as there is a possibility that the largest wave height occurs at any point within a prescribed surface area. However, for transmission coefficients, the largest transmission coefficient may occur at only certain areas such as near the lee side of the floating breakwater or the rear end of the area of interest, as seen in Figure 9b,c. This may make the consideration of the largest transmission coefficient less necessary when compared to the consideration of the largest wave height. Thus, the definition of the representative transmission coefficient based on the area percentage (such as K

_{t,90%}) may be adopted, without considering the largest transmission coefficient as in the definition of K

_{t,1/3}.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Transmission coefficients along y = 0 and 2 m ≤ x ≤ 40 m, θ = 45°: (

**a**) B/λ = 0.3, (

**b**) B/λ = 0.6, (

**c**) B/λ = 1.1.

**Figure 4.**Transmission coefficients for mesh 1 (10 m × 10 m), mesh 2 (5 m × 5 m), mesh 3 (3.33 m × 3.33 m), mesh 4 (2 m × 2 m): (

**a**) ω = 1.57 rad/s, (

**b**) ω = 1.05 rad/s.

**Figure 5.**Contours of normalized wave elevation amplitude for FB#1 under regular waves, θ = 0°, L/B = 10: (

**a**) B/λ = 0.2, (

**b**) B/λ = 0.27, (

**c**) B/λ = 0.36, (

**d**) B/λ = 0.63.

**Figure 6.**Contours of normalized wave elevation amplitude for FB#1L under regular waves, θ = 0°, L/B = 25: (

**a**) B/λ = 0.15, (

**b**) B/λ = 0.2, (

**c**) B/λ = 0.36.

**Figure 7.**Contours of normalized wave elevation amplitude for FB#1 without motion (diffraction problem) under regular waves, θ = 0°, L/B = 10: (

**a**) B/λ = 0.2, (

**b**) B/λ = 0.27.

**Figure 9.**Contours of normalized significant wave height for FB#1 under irregular waves, θ = 0°: (

**a**) ω

_{s}= 0.785 rad/s, (

**b**) ω

_{s}= 0.911 rad/s, (

**c**) ω

_{s}= 1.047 rad/s. Semi-elliptical area bounded by lee side of breakwater and red dashed semi-ellipse is assumed to be the area of interest (in Section 3.2.2).

**Figure 10.**Contours of normalized significant height for FB#1L under irregular waves, θ = 0°: (

**a**) ω

_{s}= 0.785 rad/s, (

**b**) ω

_{s}= 1.047 rad/s.

**Figure 11.**Difference (ΔK

_{t}) in transmission coefficient of FB#1 under representative regular wave and irregular waves: (

**a**) ω

_{s}= 0.911 rad/s, θ = 0°, (

**b**) ω

_{s}= 1.047 rad/s, θ = 0°, (

**c**) ω

_{s}= 0.911 rad/s, θ = 30°, (

**d**) ω

_{s}= 1.047 rad/s, θ = 30°

^{.}

**Figure 12.**Transmission coefficient versus area percentage (black solid line) for FB#1, θ = 0°, irregular wave: (

**a**) ω

_{s}= 0.911 rad/s, (

**b**) ω

_{s}= 1.047 rad/s. Square blue marker indicates mean transmission coefficient and corresponding area percentage. Red star marker indicates 90% area percentage and corresponding transmission coefficient.

**Figure 13.**Representative transmission coefficients K

_{t,90%}for FB#1, FB#1L and FB#2 in regular and irregular waves.

**Figure 14.**Representative transmission coefficients K

_{t,90%}in regular waves for heave-only and motionless FB#1, FB#2.

**Figure 15.**Representative transmission coefficients K

_{t,95%}, K

_{t,90%}, K

_{t,85%}, K

_{t,1/3}in regular and irregular waves: (

**a**) FB#1, (

**b**) FB#2, (

**c**) FB#1L.

Parameter | FB#1 | FB#2 | FB#1L |
---|---|---|---|

L | 200 | 200 | 500 |

B | 20 | 30 | 20 |

d | 10 | 6.7 | 10 |

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**MDPI and ACS Style**

Nguyen, H.P.; Park, J.C.; Han, M.; Wang, C.M.; Abdussamie, N.; Penesis, I.; Howe, D.
Representative Transmission Coefficient for Evaluating the Wave Attenuation Performance of 3D Floating Breakwaters in Regular and Irregular Waves. *J. Mar. Sci. Eng.* **2021**, *9*, 388.
https://doi.org/10.3390/jmse9040388

**AMA Style**

Nguyen HP, Park JC, Han M, Wang CM, Abdussamie N, Penesis I, Howe D.
Representative Transmission Coefficient for Evaluating the Wave Attenuation Performance of 3D Floating Breakwaters in Regular and Irregular Waves. *Journal of Marine Science and Engineering*. 2021; 9(4):388.
https://doi.org/10.3390/jmse9040388

**Chicago/Turabian Style**

Nguyen, Huu Phu, Jeong Cheol Park, Mengmeng Han, Chien Ming Wang, Nagi Abdussamie, Irene Penesis, and Damon Howe.
2021. "Representative Transmission Coefficient for Evaluating the Wave Attenuation Performance of 3D Floating Breakwaters in Regular and Irregular Waves" *Journal of Marine Science and Engineering* 9, no. 4: 388.
https://doi.org/10.3390/jmse9040388