1   Introduction

In recent decades, research on wave energy extraction has aimed to maximize energy capture while keeping costs manageable. Nonetheless, only a limited number of wave energy converters (WECs) are currently applicable in practical engineering due to their cost-effectiveness (Wang and Dong, 2023; Han and Dong, 2021). A recommendation is to combine WECs with various marine structures (Hallak and Guedes Soares, 2025), such as offshore platforms and breakwaters, while considering environmental factors like angled waves and currents, to develop a multi-functional system that can lower construction and design expenses.

The placement of this present integrated wave energy system (WES) in coastal zones is extremely beneficial for several grounds, which contribute to its appeal. These reasons include: ⅰ) the availability of existing structures like breakwaters and seawalls that can be leveraged for the integrated WES to extract wave energy due to the heaving buoy, ⅱ) the reflected waves from these structures that enhance wave energy extraction due to the porous-effect parameter that dissipates wave energy, and ⅲ) significantly reduced construction and design costs in comparison to deep sea locations. Therefore, to save the fabricating cost of the WECs (Wang et al., 2022b; Zhao et al., 2019; Vicinanza et al., 2019; Mustapa et al., 2017), the present integrated WES can be used in practical engineering.

Emerging studies are exploring a new WES design that integrates buoys or oscillating water columns (OWCs) via either floating or breakwaters that are fixed (Rezanejad and Guedes Soares, 2018). Numerical and theoretical approaches have been employed to analyze these models. The theoretical studies on the hydrodynamic efficiency of hybrid wave energy extraction (WEE) systems (Wang and Dong, 2021, 2023). These systems consisted of a toroidal oscillating body, a coordinated porous cylindrical structure, and a toroidal oscillating buoy. All these components were positioned on a bottom heaving WEC located centrally within a semi-circular aperture in a vertical solid breakwater. Proximity to a rigid rear wall significantly amplifies horizontal forces on a WEC and the wall itself. Significantly, these effects occur at frequencies that match the piston mode gap and the heave resonance frequencies. Using an analytical approach and subjected to oblique waves, the influence of incident wave angle on a floating structure with no motion of rectangular geometry close to a vertical wall with a stepped bottom was investigated in Bhattacharjee and Guedes Soares (2011). Schay et al. (2013) used the commercial software WAMIT to simulate various floater shapes for heaving point absorbers positioned close to a vertical rigid wall, subsequently analyzing the resulting forces and hydrodynamic coefficients.

Using a matching technique, Wang et al. (2022a) analyzed a WEE system comprising a heaving buoy and a porous wall, designed to work cohesively. Hsu and Wu (1997) investigated the effect of a rigid vertical wall and bottom sill on the wave quantities and hydrodynamic characteristics pertaining to a rectangular floating structure (Shen et al., 2005). It was found a positive correlation between sill height and reflection coefficient in different water depths; conversely, the transmission coefficient decreased as sill height increased. The effectiveness of a floating breakwater (FB), involving mooring cables and a vertical wall was investigated through the wave quantity by Elchahal et al. (2008). To broaden the effective frequency bandwidth, Ning et al. (2017) developed an FB featuring a dual-pontoon design and then investigated the impact of WEC power take-off (PTO) damping on wave characteristics, response amplitude operator (RAO), and performance of the resulting integrated WEC configuration. Rezanejad and Guedes Soares (2021) examined the hydrodynamic efficiency of an innovative floating OWC device consisting of two chambers placed in the front and back regions in the upstream region.

Zhang et al. (2020) examined how wave reflection affects the hydrodynamic behavior of a combined FB and WEC and their results demonstrated that the inclusion of the wall significantly changed the system's hydrodynamic properties, energy conversion efficiency, and wave reduction performance. Ji et al. (2021) introduced a novel design for a reversed L-type breakwater combined with a WEC, which includes an FB of rectangular geometry equipped with an anchored vertical curtain-type wall anchored by piles. Guo et al. (2022) investigated the effects of a porous vertical wall on a submerged, flexible, perforated membrane situated above a varying seabed.

Chen et al. (2020) conducted a numerical investigation into the hydrodynamic efficiency of an FB, which was designed as a vertical pile-restrained WEC. Wang et al. (2022b) conducted a numerical study to investigate breaking wave impact, focusing on wave profile during impact, the pressure exerted by breaking waves, and the interplay between waves and breakwaters with perforations. Cheng et al. (2024) presented a comprehensive approach that combines multiple OWCs with an adaptable FB to harness wave energy. Xu et al. (2022) conducted a review of the latest advancements in wave energy marine buoys, while the aspects of mooring design are discussed by Xu et al. (2019).

Prior research conducted by Mohapatra et al. (2024a) and Islam et al.(2018, 2019) examined wave radiation from a floating rectangular box using an analytical approach in twodimensional, oblique, and three-dimensional contexts, without incorporating a vertical wall. Conversely, a threedimensional theoretical model was created to analyze wave interaction with a horizontally floating flexible membrane moored by springs, and this model was compared against experimental test results (Mohapatra et al., 2023). Recently, Mohapatra et al. (2024b) developed a theoretical model of a horizontal floating flexible membrane in a 3D channel, and results are compared with experimental data sets, applications to WEC and breakwater. Amouzadrad et al. (2024) reviewed the hydrodynamics and gap resonance of multibody floating structures based on analytical and numerical approaches.

Existing theoretical studies lack a comprehensive investigation into the oblique wave interaction between a rectangular heaving buoy WEC and in the presence of a vertical flexible perforated membrane (VFPM) wall.

Hence, Section 2 introduces a mathematical model developed under linearized wave theory and incorporates the VFPM to describe oblique wave radiation from a rectangular box. In Section 3, radiation potentials are derived via Fourier and eigenfunction expansion. These and the diffraction potential enable the calculation of wave excitation force and hydrodynamic coefficients. The mathematical expressions for the heaving buoy's motion and power capture, concerning damping PTO, are also given. In Section 4, the accuracy of the present theoretical result for hydrodynamic coefficients is analysed at various oblique angles, and the results, achieved without incorporating a VFPM wall and a limiting case of vertical rigid perforated wall, are validated by comparison to existing BEM results. In addition, the effects of oblique wave angles and structural design on hydrodynamic coefficients, wave forces, power capture, optimal PTO damping, and transmission coefficients are investigated through numerical results, which are then presented and analysed in detail. Finally, in Section 5, significant concluding remarks and the future direction of the current analysis are discussed.

2   Mathematical formulation

The physical problem is represented by a mathematical model in a 3D Cartesian coordinate system $(x, y, z)$, where the $x-z$ lies in the horizontal plane that lines up with the undisturbed water surface, and vertically downward is considered the positive direction for the y-axis. The new WES consists of a heaving floating box and a VFPM wall is placed after the right end of the floating box at a distance $l_{1}$ is connected by the spring moorings with stiffness $q$. The origin O is considered to be the central point of the rectangular box.

Hence, the distance between the origin O and the vertical membrane is $l+l_{1}$. The wave radiation arises from the heaving movement of a rectangular box that is floating on a water surface, with a width of 2l, a draft of d, and extending infinitely in the z-direction. Therefore, the entire fluid domain is categorized into four zones as established by: $x \in(-l, -\infty), x \in(0, h), x \in(-l, l), y \in(d, h), x \in\left(l, < l+l_{1}\right)$, $y \in(0, h)$, and $x \in\left(l+l_{1}, \infty\right), y \in(0, h)$ are referred to as $\Omega_{1}$, $\Omega_{2}, \Omega_{3}$, and $\Omega_{4}$, respectively (as illustrated in Figure 1).

Figure  1  The model of a wave energy extraction system

Download: Full-Size Img

It is assumed that the water is inviscid and incompressible, with the flow being irrotational and exhibiting simple harmonic motion over time, characterized by an angular frequency ω, as well as in the z-direction.

Therefore, a velocity potential exists $\Phi(x, y, z, t)=\operatorname{Re}\{\phi (x, y, z) \exp (-\mathrm{i} \omega t)\}$, where Re denotes the real part of the complex expression and $\phi(x, y, z)$ denotes the spatial velocity potential. The potential $\phi(x, y, z)$ can be divided into three components: potential attributable to the incoming, the diffracted, and the radiated potentials. These components are indicated $\phi_{I}, \phi_{D}, \phi_{R}$ respectively. Hence, the global potential $\phi$ can be written as

A progressive wave, characterized by its angular frequency ω, interacts with a rectangular box and it is positioned at an angle θ relative to the x-axis. The spatial velocity potential due to the incident wave resulting from this wave interaction is given by $\phi_{A}(x, y, z)=\varphi_{A}(x, y) \mathrm{e}^{\mathrm{i} \gamma z}, y=k_{0} \sin \theta$, where $k_{0}$ represents the z-component of the wavenumber linked with the incident wave. Consequently, the potential of the incident wave for linear waves traveling in the negative x-direction is indicated as

with $g$ represents the gravitational constant, $A$ (= one half of the wave height $=H / 2$ denotes the amplitude of the incoming wave, $H$ denotes the wave height, and for the sake of simplicity in numerical computation, $A$ is assumed to be equal to 1 without any loss of generality, and $s=k_{0} \cos \theta$ is the $x$-component of the wave number $k_{0}$ related to the incident wave fulfils the dispersion relation as follows.

Considering that the box's movement is minimal and its oscillations fluctuate sinusoidally in the z-direction, we represent the amplitude of the box's heave motion as $I_{R}$. Consequently, the radiated potential, denoted as $\phi_{R}$, is formulated as follows:

with $\varphi_{R}$ that does not depend on $z$ and associated with the reduced wave equation.

The free surface condition can be read as

where $\varepsilon=\omega^{2} / g$.

The conditions at the solid bottom are indicated by

The boundary condition (BC) resulting from the heaving box is expressed as (see Islam et al., 2019)

The term governing the rigid structural boundary is defined by

The linear kinematic and dynamic conditions on the VFPM can be expressed as

with $P_{j}(x, y, z, t)=-\rho g y-\rho\left(\partial \Phi_{j} / \partial t\right), j=3, 4$ is the hydrodynamic pressure.

Since the VPFM is tethered to the seafloor via spring moorings, the boundary constraints on the VFPM at $y=0, h$ are defined by:

with $m_{m}, \tau_{f}, \sigma$, and $\xi$ refer to the mass density, the membrane tension, the porous-effect parameter, and the membrane displacement of the VFPM, respectively.

It is considered that the far-field condition can take the following form

3   Method of solution

Applying the method of variable separation across four distinct zones $\Omega_{1}, \Omega_{2}, \Omega_{3}$, and $\Omega_{4}$, the radiated potentials $\varphi_{R}$ satisfying (5), (6)‒(11) are expressed

where $f\left(\alpha_{0}, y\right)=\left\{-\cosh \alpha_{0}(h-y)\right\} /\left\{\alpha_{0} \sinh \alpha_{0}(h-d)\right\}$

with $P_{n}(y)$ 's, $Q_{n}(y)$ 's and $R_{n}(y)$ 's are the eigenfunctions.

The eigenvalues $k_{0}$ are defined identically as stated in Equation (3) and $k_{n}=\mathrm{i} k_{n}, s_{n}, \alpha_{n}, \beta_{n}$, and $\mu_{n}$ 's that adhere to these dispersion relations include:

The eigenfunctions $\left(P_{n}(y)\right.$ 's, $Q_{n}(y)$ 's, and $R_{n}(y)$ 's) exhibit orthogonality within their respective intervals (see Mohapatra et al., 2024a).

The continuity of velocity at $x= \pm l$ can be read as

Further, the continuity of pressure at $x= \pm l$ can be read as

and by applying the orthogonal relations, a linear system of equations $4(N+1)$ is derived to determine the unknown constants $A_{1 n}$ 's, $A_{2 n}$ 's, $A_{3 n}$ 's and $B_{2 n}$'s.

Proceeding as in Mohapatra et al. (2024a), the system of linear equations 4(N + 1) solved by developing MATLAB codes by trimming the infinite series into a finite number of terms N in the expansion as in Equations (12)‒(15). Once the unknown coefficients are obtained, then the complete solution will be obtained concerning radiated potentials.

By substituting the velocity potentials $\varphi_{R 3}$ and $\varphi_{R 4}$, as defined in Equations (17) and (18), into the normal velocity continuity condition on the left-hand side of Equation (13b), we arrive at an equation for the unknown variables for $n=0, 1, 2, \cdots$ as

3.1   Wave excitation force

Since the floating box is considered infinite in the z-direction and the potential exhibits periodicity in that direction, it is necessary to deal exclusively with the wave-induced forces on a cross-section that is orthogonal to the z-axis. Therefore, the wave forces can be articulated in relation to the incident and radiated potentials as follows:

In this context, $W_{S}$ represents the wetted surface in the $x y$-plane, while $n$ indicates the inward normal to the floating box within the same plane, defined as $n=n_{y}$, where $n_{y}$ signifies the components of the outward-pointing unit normal to the surface of the box.

The dimensionless wave force of heave can be computed by using the following formula.

The dimensionless heave added mass (C) and heave damping coefficient (D) are expressed as

3.2   Power capture and efficiency of the system

The motion of the rectangular box is described by

with $M=\rho B d, K=\rho g B$, and $D_{\text {PTO }}$ constitute the buoy mass, hydrostatic spring stiffness, and PTO damping, respectively. Moreover, the symbols $C, D, I_{R}$, and $F_{v}$ have been previously defined.

The calculation of the optimal PTO damping D_OPTO for the heaving buoy are presented as (Wang et al., 2022b)

The power harnessed by the rectangular buoy is computed as

Conversely, the VFPM's transmission coefficient, K_t, is determined by

4   Numerical results and analysis

This section analyses validation of the current solution, several crucial parameters on the hydrodynamic coefficients, the efficiency of capturing power, and the VFPM wall's energy transmission characteristics.

MATLAB R2023b (64-bit, win64) with a desktop PC, equipped with an Intel® Core i7-4790 CPU running at 3.60 GHz, 16 GB of RAM with a frequency of 3 601 MHz, 4 physical cores, and 8 logical processors, and an SSD drive was used to perform the study. The evaluated time for each calculation was between 8 and 10 minutes.

4.1   Convergence analysis and validation

The hydrodynamic coefficients values are accurate to 3-decimal places when N is 30 or greater, as shown in Table 1. Therefore, to ensure numerical accuracy, the calculation of the radiation velocity potential formulas is limited to 30 terms.

Hereafter, all computations are carried out using the current solution and incorporating the water density $\rho= 1025 \mathrm{~kg} / \mathrm{m}^{3}$, gravitational constant $g=9.8 \mathrm{~m} / \mathrm{s}, d / h=0.4$, $l / h=0.42$, and incident angle $\theta=30^{\circ}$ unless otherwise stated.

Figure 2 presents a validation of the current solution for the vertical wave force ($F_{v}$) on a heaving box (without a VFPM wall) against BEM results from Zheng et al. (2006), plotted against the non-dimensional wavenumber $k_{0} h$. The current results for the vertical force, in the absence of a VFPM wall, demonstrate strong agreement with the BEM data.

Figure  2  Validation for F_v between the current and that gained via BEM

Download: Full-Size Img

The non-dimensional added mass A (no VFPM wall) from the current solution is compared to the BEM results of Zheng et al. (2006) in Figure 3, as a function of k₀h. The excellent agreement between the analytical solution and the BEM results validates the present approach, and the solution also captures the trend observed in the BEM data.

Figure  3  Validation of C between the present and those obtained from a BEM

Download: Full-Size Img

Figure 4 shows a comparison between the damping coefficient (D) obtained from the current analytical model (without a VFPM wall) and the BEM results of Zheng et al. (2006), plotted against k0_h. The results demonstrate a good agreement in both their trends and numerical values.

Figure  4  Validation of D between the current results and those obtained from a BEM

Download: Full-Size Img

In Figure 5, the present solution of the transmission coefficient $K_{t}$ is validated with a limiting case with incident angle $\theta=0^{\circ}, l / h=0, d / h=0$, and a rigid porous wall; existing in the literature (Wang et al., 2022b) versus $k_{0} h$. It is observed that they aligned well with their pattern and data points.

Figure  5  Comparison of K_t between the present and a limiting case; rigid perforated wall

Download: Full-Size Img

4.2   Influence of oblique angle and structural parameters

Figure 6 presents the influence of obliquely incident angle θ on the F_v against $k_{0} h$. It has been noted that as the values of θ increase, the vertical wave force diminishes. This suggests that the angle of the incoming wave aligns parallel to the floating box, resulting in a diminished effect.

Figure  6  Influence of θ on the F_v versus $k_{0} h$

Download: Full-Size Img

Figure 7 simulates the impact of $d / h$ on the dimensionless vertical wave force ($F_{v}$) for $l / h=0.18$ and oblique angle $\theta=30^{\circ}$ against $k_{0} h$. It can be seen that the values of $F_{v}$ increase with a decrease in d/h. The reason is that the deeper drafts create greater horizontal loads on the floating box while resulting in smaller vertical loads.

Figure  7  Effect of d/h on the F_v versus $k_{0} h$

Download: Full-Size Img

Figure 8 demonstrates the influences of $l / h$ on the $F_{v}$ for $d / h=0.4$ and $\theta=30^{\circ}$ against $k_{0} h$. It has been noted that the vertical wave forces rise as the width of the box increases. Figure 9 demonstrates the impact of $\theta$ on the $A$ for $d / h=$ 0.4 and $h / l=6$ against $k_{0} h$. The added mass rises with $\theta$ at lower wavenumbers, $k_{0} h$ but this trend reverses at larger wavenumbers.

Figure  8  Influence of l/h on the F_v versus $k_{0} h$

Download: Full-Size Img

Figure  9  Effect of θ on the added mass versus wavenumber

Download: Full-Size Img

In Figure 10, the effects of non-dimensional draft $d / h$ on the non-dimensional added mass of the heave, with $l / h=$ 0.18 and oblique angle. versus $k_{0} h$ are plotted. It is observed that the added mass of heave decreases with an increase in non-dimensional draft $d / h$, which is due to the fact that higher in the draft yields larger in the mass of the structure, thus, the heave structure resonates less in heave motions.

Figure  10  Influence of d on C vs. the wavenumber

Download: Full-Size Img

Figure 11 gives the impact of non-dimensional width $l / h$ on the $C$, with $d / h=0.18$ and oblique angle $\theta=30^{\circ}$ versus $k_{0} h$ are plotted. The added mass of heave is observed to decrease as the values of $l / h$ increases for $k_{0} h \geqslant 5$. However, this pattern does not consistently hold for smaller values $k_{0} h$.

Figure  11  Influence of l/h on the added mass against the wavenumber

Download: Full-Size Img

Figure 12 reveals observations similar to those in Figure 6. Nevertheless, at larger $k_{0} h$, the damping coefficients ($D$) exhibit negligible variation across various oblique angles, effectively approaching zero. Figure 13 illustrates the relationship between the non-dimensional draft $d / h$ and the $D$, plotted against $k_{0} h$ for $l / h=0.18$. The plots show that as the structural draft increases, the heave damping coefficients diminish.

Figure  12  Influence of θ on the damping coefficient D versus $k_{0} h$

Download: Full-Size Img

Figure  13  Effect of d/l on D versus $k_{0} h$

Download: Full-Size Img

In Figure 14, the damping coefficients of heave become lower for longer structures. This occurs because, as the structural length increases while the draft and width remain constant, the mass of the structure also rises, resulting in reduced damping.

Figure  14  Effect of l/h on D versus wavenumber

Download: Full-Size Img

4.3   Impact of oblique angle and structure parameters on the Pcapture and PTO damping

Figure 15 illustrates the relationship between oblique wave angle and the $P_{\text {capture }}$ as a function of $k_{0} h$ with $d / h=0.4$ and $l / h=0.18$. The plots show that as $\theta$ increases, the WEC captures more power at lower $k_{0} h$. However, this fashion reverses at higher $k_{0} h$.

Figure  15  Influence of oblique angle θ on the power capture

Download: Full-Size Img

Also, in Figure 15, wave-capturing power is more significant in shallower water compared to deeper water. This recommends that such a wave energy system could be particularly effective in coastal regions for energy extraction.

Figures 16‒17, the influence of the nondimensional draft and the width of the box on the $P_{\text {capture }}$ for oblique wave angle θ = 30° versus $k_{0} h$ are plotted. In both cases, it was seen that the power captured by the WEC becomes greater for the higher draft and wider structure as the wavenumber increases. However, this effect seems to be negligible for smaller wavenumber. This suggests that this type of wave energy system may be more effective for extracting wave energy in coastal areas.

Figure  16  Effect of d/h on the power capture with l/h = 0.18 and θ = 30°

Download: Full-Size Img

Figure  17  Effect of l/h on the power capture with d/h=0.4 and θ = 30°

Download: Full-Size Img

The effect of $\theta$ on the optimal PTO damping coefficient versus $k_{0} h$ is shown in Figure 18. The results indicate that higher $\theta$ lead to increased optimal damping. Nevertheless, at higher $k_{0} h$, this increase diminishes slightly, exhibiting a minor reversal. In Figures 19-20, the impact of structural parameters (draft and width) on the optimal PTO damping with $d / h=0.4$ and $\theta=30^{\circ}$ versus $k_{0} h$ are illustrated. The observations resemble those shown in Figure 16 and Figure 17.

Figure  18  Impact of θ on the optimal PTO damping

Download: Full-Size Img

Figure  19  Effect of d/h on the optimal PTO damping for θ = 30°

Download: Full-Size Img

Figure  20  Effect of l/h on the optimal PTO damping versus $k_{0} h$

Download: Full-Size Img

Figure  21  Influence of θ on the transmission coefficients

Download: Full-Size Img

4.4   Effect of oblique angle and design parameters on transmission coefficients

Figures 21 ‒ 23 present the impact of various oblique angles, non-dimensional structural draft, and width on the transmission coefficients K_t versus $k_{0} h$. It is seen that as the incidence angle, draft, and width of the structure increase the transmission coefficient becomes lower. Further, with larger values of wavenumber, the impact of angle and structural parameters on the K_t is minimal.

Figure  22  Impact of the structural draft on $K_{t}$ vs. $k_{0} h$

Download: Full-Size Img

Figure  23  Impact of structural width on the $K_{t}$

Download: Full-Size Img

In Figure 24, the impact of porous-effect parameter on the K_t is analyzed. Three values are used for the porous effect parameter σ. The second and third of these values included coefficients for both inertial and frictional resistance. It is found that as the values of σ rises the K_t becomes higher. Also, it can be observed that the when both exist (inertial and frictional resistance) in the porous effect parameter that gives a greater performance in wave transmission than that the case of only inertial force exist.

Figure  24  Influence of porous-effect parameter σ on the K_t

Download: Full-Size Img

5   Conclusions

This study introduces a mathematical model that illustrates a WES comprised of a floating rectangular heaving box and a moored, VFPM wall subjected to oblique waves. The eigenfunction expansion method and the orthogonal condition are used to derive the analytical expressions of the radiated potentials in each region. The hydrodynamic coefficients, the motion equation, and the power capture formulas related to the damping PTO are provided. The accuracy of the analytical results for the hydrodynamic coefficients on different oblique angles is demonstrated and the efficiency of the power capture on the influence of oblique wave angles is examined. The analysis leads to the following key conclusions:

1) The paper offers a novel contribution through the integration of a moored VFPM wall with a floating rectangular heaving box, specifically addressing the scenario of oblique wave conditions and analysing the impact of these waves on hydrodynamic coefficients and wave power. In addition, the current results concerning hydrodynamic coefficients and wave quantity were consistent with prior findings.

2) It has been found that as the oblique angle increases, both the vertical wave force added mass and the damping coefficient decrease. The power harnessed by the rectangular buoy is substantial in coastal areas; however, as the incidence angle increases, the ideal PTO damping rises for smaller values of $k_{0} h$.

3) Regarding the transmission coefficients, it is found that the oblique angle and structural parameters play a vital role in the transmission of waves below the rectangular buoy.

4) This analysis could aid in understanding the hydrodynamic performance of WEC concerning their PTO systems. This formulation and analysis could contribute to developing an integrated WES, featuring a combination of an array of heaving buoys and a moored flexible perforated membrane wall under current load.