### 1. Introduction

### 2. Formulation

### 2.1 Equation of Motion of Buoys in a WEC

*I*means incident wave and

*S*is scattering.

*A*is the wave amplitude,

*ξ*is the

_{j}*j*-th mode of a buoy’s motion,

*ω*is the wave frequency, and

*t*is the time. In particular, we constrain all the modes of a buoy’s motion but its heave mode of motion in this study. Further, the body boundary conditions of the velocity potential are as follows: where

*S*is the mean body surface of the

_{B}*k*-th buoy,

*N*is the number of buoys, and

*n*is the generalized directional cosine expressed by the inner product between the surface normal vector of a buoy and the vertical unit vector (Choi, 2011). Hence,

_{j}*a*) and the damping coefficient (

_{kj}*b*) of the

_{kj}*k*-th buoy for the

*j*-th mode of motion can be obtained for the generalized mode by integrating the product of the radiation velocity potential for the

*j*-th mode of motion and the generalized directional cosine of the

*k*-th buoy. Similarly, the unit water wave exciting force (

*X*) is calculated by integrating the product of the diffraction velocity potential and the generalized directional cosine of the

_{k}*k*-th buoy (Newman, 1994). Exactly, where

*ρ*is the water density. In addition, the damping force induced due to the extracted energy of a power take-off (PTO) device is assumed to be linear in this analysis. So, the PTO damping force of each buoy is: where

*b*is the damping coefficient of the PTO device. Subsequently, the equation of motion of the buoys that considers the PTO damping force is: where

_{PTO}*δ*is the Kronecker delta function,

_{kj}*m*is the mass of the

_{kj}*k*-th buoy for the

*j*-th mode of motion, and

*c*is the hydrostatic stiffness of the

_{kj}*k*-th buoy for the

*j*-th mode of motion.

### 2.2 Energy Extraction of the WEC

*P̄*, is written in the following form.

*C*is the wave group velocity and

_{g}*g*is the acceleration due to gravity. The capture width can be non-dimensionalized by multiplying it with the wavenumber (

*k*). Notably, the maximum non-dimensional capture width is one at the natural frequency of the buoys when considering only one mode of motion of the buoys. Hence,

*θ*is the angle of incidence of water waves onto an array of buoys, and

_{I}*C*is a parameter related to the configuration of an array of buoys. In addition, all results of our multi-body WECs study are compared with those of a single buoy WEC, even though we don’t directly report the corresponding array gain. But, the better energy extraction efficiency of multi-body WECs in this comparison implies the corresponding array gain. Finally, we state that only the normal incidence of water waves onto the buoys of multi-body WECs, expressed by

*θ*=0° or 90°, is considered in this analysis.

_{I}### 2.3 Resonance in Multi-Body WECs

#### 2.3.1 Rayleigh resonance

*m*is the scattering mode,

*θ*is the direction of propagation of the

_{m}*m*-the scattering mode, and

*d*is the distance between two adjacent buoys oriented along the y-direction. Equation (12) shows that the scattered water waves head to the periodic buoy, i.e.,|sin

_{y}*θ*|

_{m}*=*1, at the critical wave number and new scattering modes emerge. This special case is expressed by the following relation:

#### 2.3.2 Bragg resonance

#### 2.3.3 Laue resonance

*k⃗*

_{I}and

*k⃗*are the respective wave number vectors of the incident and the

_{m}*m*-th scattered waves, and

*a⃗*

_{1}is the

*x*-directional lattice vector of an array of buoys.

*a⃗*

_{1}

*= d*. Then, Eq. (17) corresponding to this rectangular array of buoys is:

_{x}î*θ*

_{m}*= π*−

*θ*. Finally, substituting Eq. (14) into Eq. (18) results in the following equation for the normal incidence of water waves onto this rectangular array of buoys.

_{I}*kd*or

_{y}*kd*at each (

_{x}*m, n*) pair. Notably, Tokić and Yue (2019) reported a substantial reduction in the wave energy extraction of periodic WECs due to Laue resonance.

### 3. Numerical Investigation of the WECs’ Configuration

### 3.1 Vertical Cylinder

*T*denoting the draft of the vertical cylinder. In addition, the ratio of the radius of the cylinder to the draft influences the maximum response amplitude operator(RAO) of the said heave motion. Particularly, a large radius of the cylinder compared to the draft results in a small maximum RAO near the resonance frequency. On the other hand, the bandwidth of non-dimensional capture width (

*kW*) becomes large when this ratio increases with the same maximum RAO value. This study considers vertical cylinders with a radius (a) of 1.0 m and a draft (

*T)*of 0.67 m in the array shown in Fig. 4. The orientation of this array of cylinders with respect to the incident water waves results in different types of this array. The analysis of each of these different types is carried out as follows.

### 3.2 Single-column Attenuator Array

*kW*) of a single-column attenuator array, i.e.,

*θ*=0°, shown in Fig. 5. The total non-dimensional capture width of buoys is divided by the number of buoys to obtain this array’s non-dimensional average capture width. Three values for the x-directional distance between adjacent cylinders (

_{I}*d*) are considered in this analysis: 3a, 8a, and 14a. The results of this analysis are shown in Fig. 6. In particular, the upper graphs of this figure show that the energy extraction by this array decreases abruptly near the Bragg resonance frequency, and this abrupt decrease is denoted by a red dotted line in these graphs. This effect is especially large when

_{x}*N*is not small enough. In addition, this resonance effect is small in the low-frequency region, as seen from the rightmost one of these graphs. Finally, the non-dimensional average capture width of a single-column attenuator array is small compared to that of a single buoy. This reduction is large as the number of buoy increases.

_{x}### 3.3 Single-Row Terminator Array

*kW*) of a single-row terminator array of vertical cylinders shown in the left side of Fig. 7 by following a procedure similar to the one followed for the corresponding single-column attenuator array. Three values for the y-directional distance between adjacent cylinders (

*d*) are considered in this analysis: 3a, 8a, and 14a. The results of this analysis are shown in Fig. 8. Overall, the single-row terminator array of vertical cylinders works constructively. In particular, the energy extraction efficiency increases as the number of buoys increases, and then it converges for some number of buoys. However, energy extraction abruptly drops down near the Rayleigh resonance region, and this abrupt drop is denoted by a blue dotted line in the graphs of Fig. 8. Subsequently, this energy extraction becomes almost zero due to the emergence of new radiated water waves (Srokosz, 1980). The lower graphs of Fig. 8 show the non-dimensional capture width of each buoy in this array. These graphs indicate that each of these buoys achieves large energy extraction. In particular, the buoy in the middle of this array achieves high efficiency near the Rayleigh resonance frequency but not exactly at that frequency. Relatedly, the direction of constructive waves in this terminator array becomes the same as the array direction, i.e., |sin

_{y}*θ*|

_{m}*=*1, during Rayleigh resonance. So, the wall effect of periodic buoys resulting when the constructive wave direction becomes the same as that of the array causes the water waves to concentrate between buoys due to the large reflection of them.

### 3.4. Multi-row Terminator Array

*N*) of this terminator array and investigate the energy extraction efficiency of the resulting arrays. Moreover, we use various

_{x}*d*or x-directional distance between adjacent rows in the same multi-row terminator array as the x-directional periodicity of the buoys reduces the energy extraction of this array. In particular, we study the effect of having various

_{x}*d*values within the same 4×8 terminator array of vertical cylinders on the energy extraction of this array under different configurations of it in detail.

_{x}#### 3.4.1 Few rows terminator array

*d*= 3a, 8a, and 14a are given in Fig. 9. The green lines in this figure denote

_{y}*kd*and

_{x}*kd*divided by

_{y}*π*, respectively, for different combinations of array distances (

*d*,

_{y}= d_{x}*d*, and

_{y}= 2d_{x}*d*). The points of intersection between the green lines and the other blue horizontal lines, red vertical lines, and magenta curves satisfy the Bragg, Rayleigh, and Laue resonance phenomena, respectively. Subsequently, we increase the number of rows (

_{y}= 0.5d_{x}*N*) for a fixed number of columns (

_{x}*N*) of eight, as shown in Fig. 10, and analyze the resulting arrays. Notably, these

_{y}*N*and

_{x}*N*values are sufficient to confirm the effect of multiple scattering of water waves by the buoys in these arrays.

_{y}*kd*, as shown in the left side graphs of this figure. This reduction is obvious when the maximum

_{y}*kd*is smaller than the first Rayleigh resonance frequency.

_{y}*W*)

_{a}_{max}

*= d*, in the absence of oblique waves. Subsequently, we compute the maximum array gain

*q*for an array of

_{max}*N*rows in the absence of oblique waves (Tokić and Yue, 2019) using the array gain given by Eq. (11) and the maximum capture width of a heaving isolated buoy(

*kW*)

_{max}

*=*1:

*kd*to achieve array gain. Hence, the increment of the number of rows in the multi-row terminator arrays of vertical cylinders results in a decrease of the average capture width of these arrays for small

_{y}*kd*values.

_{y}*N*and

_{x}*N*. The red, blue, and magenta dotted lines in Fig. 11 denote the Bragg, Rayleigh, and Laue resonances, respectively, at each frequency. In addition, the gray line of this figure denotes the simultaneous existence of two resonance phenomena, and the black vertical line in this figure denotes the occurrence of all resonances simultaneously at this frequency. First, the effect of Bragg resonance seems small in all these array configurations. Next, the magnitude of the non-dimensional average capture width oscillates between resonance points generally. In addition, a lot of resonance points occur when

_{y}*d*and

_{x}*d*are sufficiently large. Particularly, several Laue resonance points are generated under this condition. Therefore, these arrays’ non-dimensional average capture width becomes extremely oscillating if the distance between adjacent buoys of these arrays is relatively large, as shown in the upper right graph of Fig. 11. However, a large amplitude occurs only when the number of rows in these arrays is smaller than five. Further, the extraction efficiency of the multi-row terminator array of vertical cylinders decreases when the number of rows in it is larger than four generally. Consequently, a smaller value for the number of rows in these arrays gives better performance than a corresponding single-row terminator array, provided the former has a moderate

_{y}*kd*.

_{y}#### 3.4.2 A 4×8 multi-row array

*d*and

_{x}*d*), that can vary inside the array. He found that the local maximum of this extracted power could be obtained for various

_{y}*d*and constant

_{x}*d*values within the array under regular water waves. Moreover, a regular array achieves a local maximum of extracted power at least when the inclined angle of irregular water waves is small (Tokić and Yue, 2021). Tokić and Yue (2021) also concluded that a perturbed position of each buoy results in less extracted power of this array.

_{y}*d*is constructive. In addition, achieving a local maximum of extracted power in a regular array under irregular water waves is possible only with an unconstrained total length of the array. On the other hand, a constrained overall length of an array together with various

_{y}*d*values within it results in better energy extraction efficiency of the array under some conditions. So, we investigate the extracted power of a 4×8 multi-row array of vertical cylinders with various

_{x}*d*values within it though under regular water waves. In addition, the distance between rows of this array is changed. In particular, we choose one regular 4×8 multi-row array of vertical cylinders and six variants of it with at least two same x-directional distances between the rows, as shown in Fig. 12. In addition, four combinations of distances are chosen for the analysis (

_{x}*d*;

_{x}= 3a, dy = 5a*d*;

_{x}= 3a, d_{y}= 12a*d*;

_{x}= 6a, d_{y}= 5a*and d*). The

_{x}= 6a, d_{y}= 12a*d*in these arrangements is always greater than

_{s}*d*.

_{x}*regular*4×8 multi-row array of vertical cylinders. Other colorful lines denote the results for the corresponding arrays with various

*d*in them. In particular,

_{x}*d*influences the relative magnitude of the non-dimensional average capture width between this regular array and the other corresponding arrays with various

_{x}*d*. In addition, a small

_{x}*d*is constructive only in these arrays with various

_{x}*d*. Notably, various small

_{x}*d*(2.8a <

_{x}*d*< 3.5a) values result in larger energy extraction efficiency of any of these arrays with various

_{x}*d*than that of a corresponding regular array with the same

_{x}*d*. However, the details of the analyses giving this finding are not reported here. The graphs corresponding to

_{x}*d*in Fig. 13 have the colorful lines representing the non-dimensional average capture widths of these arrays with various

_{x}= 3a*d*which are larger than that (represented by the black line in this figure) of the corresponding regular array. On the other hand, the black line in the graphs corresponding to

_{x}*d*represents the non-dimensional average capture width of a regular 4×8 multi-row array of vertical cylinders, which is greater than those (represented by the colorful lines in this figure) of the corresponding arrays with various

_{x}= 6a*d*values. The energy extraction efficiency of all these multi-row terminal arrays of vertical cylinders with various

_{x}*d*is better than that of the corresponding regular array almost in all frequency ranges. So, we expect their similar high performances under irregular water waves. Unfortunately, it is difficult to find which of the 4×8 multi-row arrays of vertical cylinders with various

_{x}*d*is better. However,

_{x}*SSX*and

*XXS*types of these arrays have relatively large and small efficiencies, respectively, in many cases.

### 5. Conclusions

*d*or x-directional distance between its adjacent rows is studied. This study shows that a small

_{x}*d*results in better energy extraction efficiency of some of these arrays with various

_{x}*d*than that of a corresponding regular array with the same

_{x}*d*.

_{x}