Wave Scattering by Inverse T-Type Compound Breakwater with Ocean Currents: An Analytical and Numerical Study

Abstract

The present work focuses on wave scattering generated by an inverse T-type compound breakwater in the presence of the ocean current. The boundary value problem (BVP) is investigated using two distinct strategies: an exact formulation derived from the eigenfunction expansion method (EEM) and a computational framework developed with the boundary element method (BEM). A comparison of outcomes from both techniques with established studies confirms the consistency and accuracy of the present formulations. Reflection and transmission coefficients, along with the time-domain simulations of the free surface, are evaluated under different wave conditions and structural configurations. In the long-wave region, the reflection coefficient exhibits strong dependence on the wavenumber, with higher values observed as the height and width of the porous section increase. Increasing the friction coefficient within the porous layer considerably reduces wave transmission to the leeside, demonstrating the important role of friction in energy dissipation. Furthermore, greater ocean current velocity leads to an increase in the reflection curve, highlighting the significant effect of hydrodynamic conditions on wave–structure interaction. The time-domain simulations of the free surface are also presented to provide a clear visualization of the wave behavior on the surface, both with and without the presence of an ocean current. The findings shed light on the combined influence of breakwaters and ocean currents, enabling the development of coastal protection measures that enhance resilience, sustainability, and safety from erosion and damage.

1. Introduction

Coastal regions are facing unprecedented challenges as rapid population growth combines with the accelerating impacts of climate change and sea-level rise [1,2]. With projections indicating continued expansion of coastal settlements [3], the demand for effective defense systems has never been more critical. Storm surges, extreme waves, and erosion now pose serious risks to the safety of communities and the resilience of coastal infrastructure. In this context, breakwaters have emerged as one of the most vital engineering solutions. Acting as protective barriers between land and sea, they not only dissipate incoming wave energy but also safeguard harbors, beaches, and vulnerable shorelines from destructive forces. By weakening, redirecting, and dissipating wave forces, breakwaters provide a critical line of defense, reinforcing the stability, sustainability, and long-term protection of coastal zones.

Given their importance in coastal protection, understanding the interaction between waves and breakwaters has become a central topic of research in coastal engineering. Over the past several decades, considerable effort has been devoted to analyzing the performance of various breakwater forms, including both submerged [4,5,6,7,8,9] and surface-piercing structures [10,11,12,13]. These investigations have provided essential insights into transmission, reflection, and energy dissipation, thereby strengthening the scientific foundation for designing effective coastal protection systems. Building on this progress, researchers have increasingly turned their attention to porous breakwaters, which have demonstrated several significant advantages over rigid rectangular barriers. Compared to conventional solid structures, a porous rectangular dock allows partial flow of water through its internal medium, enabling enhanced wave energy dissipation and reducing the intensity of reflected waves. This not only decreases the likelihood of scouring and erosion near the shoreline but also minimizes turbulence and environmental disturbance. In contrast, rigid docks tend to reflect a large portion of incident wave energy, often amplifying hydrodynamic forces and potentially causing negative impacts on adjacent coastal regions. Owing to their ability to mitigate wave loads while maintaining natural water exchange and ecological balance, porous docks have emerged as a more sustainable and adaptable solution for modern coastal defense. The work by Sollitt and Cross [14] is considered pioneering in the study of porous rectangular docks, where they calculated the reflection and transmission coefficients of waves interacting with the dock under various wave and structural parameters. Furthermore, Chwang [15] investigated the porous wavemaker theory under small-amplitude surface waves. By following the model of Sollitt and Cross [14] and Chwang [15], a lot of research has been conducted with the bottom-mounted porous structures, see Refs. [16,17,18], and floating porous structures, see Refs. [19,20,21]. Recently, Hu et al. [22] investigated the new type rectangular floating breakwater with porous baffle via the finite volume method.

Building on the advancements in both rigid and porous breakwater research, attention has increasingly shifted toward identifying geometries that can achieve superior wave attenuation while remaining structurally efficient and economically viable. To address these challenges, researchers have proposed various fixed and floating surface breakwater configurations, such as rectangular, $\mathrm{\Pi }$-shaped, inverse $\mathrm{\Pi }$-shaped, circular, and triangular types. In addition to these engineered structures, natural solutions have also gained interest; for example, mangrove trees have been effectively utilized to reduce wave impact in shoreline regions [23]. Among the engineered configurations, T-shaped and inverse T-shaped breakwaters have shown significant potential in satisfying the requirements of enhanced wave energy dissipation and structural efficiency. The research conducted by Neelamani and Rajendran [24] may be considered as pioneering work in the study of T-shaped breakwaters, marking a significant advancement in coastal engineering. Through experimental investigations, they demonstrated the effectiveness of T-shaped breakwaters in mitigating wave energy. Their findings revealed that these structures could reduce wave energy by approximately $65\%$, making them highly efficient in protecting coastal regions from wave-induced impacts. To theoretically assess the effectiveness of T-shaped breakwaters, Wang et al. [25] and Deng et al. [26] employed the eigenfunction expansion technique alongside linear water wave theory. Their studies revealed that increasing the length of the vertical screen of the T-shaped breakwater significantly enhances its ability to reduce transmitted wave energy, particularly in the long-wave region. This finding underscores the importance of optimizing the vertical dimensions of the breakwater to achieve improved wave attenuation.

While T-shaped breakwaters have demonstrated strong wave attenuation capabilities, recent trends in water wave research indicate a growing interest in inverse T-type breakwaters, which have proven to be more commonly used free-surface structures in practical coastal engineering applications. Owing to their enhanced interaction with wave fields and improved hydrodynamic stability, inverse T-type configurations have attracted extensive attention, leading to a substantial body of research supported by both numerical simulations and laboratory experiments [27,28,29]. Continuing this line of development, Sharma et al. [30] explored wave scattering by bottom-mounted dual inverse T-type breakwaters using Havelock’s expansion and the Galerkin method. Their study revealed that both the breakwater width head and tail significantly influence the performance and efficiency of the bottom-mounted inverse T-type breakwater. They emphasized that these dimensions play a crucial role in determining the effectiveness of the structure in dissipating wave energy.

Despite the extensive investigations discussed above, the aforementioned studies have predominantly focused on wave interaction under still-water conditions, without accounting for the presence of ocean currents. In reality, waves commonly propagate in environments influenced by steady or varying currents, which can significantly modify their direction, speed, and energy distribution. Ocean currents—driven by winds, Earth’s rotation, and density variations—may either amplify or diminish wave motion, leading to noticeable changes in wave height and energy flux. Therefore, breakwaters become essential not only in moderating incident waves but also in mitigating the combined impacts of waves and currents, thereby protecting vulnerable coastlines from erosion and structural damage. The importance of considering current effects in wave studies was first recognized by Taylor [31], who introduced the concept of currents acting as a form of breakwater. This idea was extended by Peregrine [32], who analyzed the joint influence of waves and currents. Subsequently, Jonsson and Wang [33] investigated wave propagation over a sloping seabed in the presence of currents, assuming an irrotational flow. Since then, numerous mathematical models have been formulated to examine the interaction of gravity waves with currents [34,35,36,37]. A theoretical model explaining the interaction of oblique waves with a porous rectangular structure under the influence of ocean current was established more recently by Dora et al. [38], underscoring the ongoing significance of current–wave coupling in present coastal research. The impact of ocean current on wave interaction with porous breakwaters has recently gained attention. Dora et al. [39] presented a BEM-based analysis for a porous breakwater configuration, and Swami and Koley [40] further extended this research to a thick porous breakwater installed over a sloped rigid seabed.

In continuation of the work reported in the literature, which primarily evaluated wave reflection, transmission, and hydrodynamic forces acting on breakwaters to understand how incident waves approach, impact, and transmit energy past the structure, it becomes evident that most existing analyses were restricted to the frequency-domain framework. Such an approach, while valuable, fails to fully capture the transient and highly dynamic nature of real ocean wave–structure interactions. Consequently, recent research has shifted toward time-domain analysis, driven by the need for a more realistic representation of evolving wave behavior. The time-domain technique enables direct visualization of fluid motion and provides deeper insight into unsteady and non-linear phenomena that are difficult to observe using conventional steady-state methods. The paper by Meylan and Sturova [41] can be regarded as the pioneering work that introduced time-dependent simulations of wave interactions with floating elastic plates in two dimensions. Building upon this foundational work, Das et al. [42] conducted time-marching simulations to investigate wave motion with a positive phase velocity. These simulations provided a detailed visualization of the wave propagation. Subsequently, the study presented in [42] was further extended by Das et al. [43], who explored time-marching simulations of wave motion in the presence of a shear current. For many years, time-marching simulations remained confined to two dimensions. This changed in 2019 when Meylan [44] introduced the concept of time-marching simulations in three dimensions. Their work demonstrated the interaction of waves with elastic circular plates, revealing how the waves decay after interacting with the plates. Furthermore, the three-dimensional simulations introduced by Meylan [44] provided a more realistic representation of wave behavior compared to the two-dimensional case. Later, following [44], several studies (see, e.g., Refs. [45,46,47]) investigated various wave–structure interaction problems using the time-marching approach. These works collectively contributed to a deeper understanding of fluid dynamics by offering clear and detailed visualizations of fluid motion in various complex scenarios.

While numerous studies have investigated waves and currents as separate phenomena, comparatively fewer have examined their coupled interaction [48,49,50,51]. This gap in the literature underscores the importance of understanding combined wave–current effects, which play a critical role in designing efficient and reliable coastal protection systems. Furthermore, although existing research has explored the performance of various breakwater configurations, including T-type structures and porous systems, a comprehensive evaluation of their effectiveness under simultaneous wave–current conditions remains limited. Motivated by the advantages reported in earlier studies on T-type geometries, porous breakwaters, and current-induced modifications in wave dynamics, the present work focuses on analyzing wave transmission through inverse T-type compound breakwaters—both rigid and porous—in the presence of uniform ocean currents. Additionally, to achieve a realistic representation of wave interaction with coastal structures, static view of time-domain simulations are conducted to capture the dynamic behavior of wave propagation and structural response under combined hydrodynamic forcing.

The rest of the paper is organized as follows: Section 2 delves into the detailed formulation of the problem, with particular attention to the boundary conditions central to this study. Section 3 discusses the approach for solving the BVP, where both the EEM and BEM are applied to determine the unknown coefficients of the spatial velocity potentials. An analysis of the effects of various factors on transmission and reflection coefficients, along with the time-domain simulations of the free surface, is shown in Section 4. This paper also presents a validation of the current investigation using existing studies. Lastly, Section 5 presents a concise summary of the key findings and conclusions derived from this study.

2. Mathematical Formulation

The problem is setup in a Cartesian coordinate system to analyze the interaction of waves and ocean currents with a fully extended inverse T-type compound breakwater placed over a rigid seabed. The physical setup is illustrated in Figure 1, where the $xy$-plane lies horizontally and the z-axis points vertically upward. The total water depth is h, measured vertically from the free surface of water (at $z=0$) down to the seabed (at $z=-h$). The breakwater structure consists of two components—a porous rectangular upper section and a rigid base below it. The upper porous section has height $h_1$ and width $b_2$. The rigid base extends from the seabed to the bottom of the porous section, giving a total height $-(h-h_1)$. The total width of the rigid base is denoted by $s_2=b_1+b_2+b_3$. An incoming wave propagates from the negative x-direction. Furthermore, we assume that a uniform ocean current with constant velocity U is flowing parallel to the x-axis, directed from the negative to positive x-direction. The ocean current represents an idealized condition, since in real coastal environments, it vary with depth due to seabed friction, turbulence, and wave–bed interaction, which also affect wave height and wave speed. Such simplification is adopted to enable clear evaluation of the fundamental effects of steady currents on wave transmission across the breakwater configuration. However, the model does not capture vertical shear flow or nonlinear depth-dependent current variations.

2.1. Problem Setup Using the EEM

To apply the EEM, we divide the problem domain into five regions, as shown in Figure 1. These regions are defined as follows: region 1 is ${\mathsf{\Omega }}_1=\{-\infty <x\le 0,-h\le z\le 0\}$; region 2 is ${\mathsf{\Omega }}_2=\{0\le x\le b_1,-h_1\le z\le 0\}$; region 3 is ${\mathsf{\Omega }}_3=\{b_1\le x\le s_1,-h_1\le z\le 0\}$; region 4 is ${\mathsf{\Omega }}_4=\{s_1\le x\le s_2,-h_1\le z\le 0\}$; and region 5 is ${\mathsf{\Omega }}_5=\{s_2\le x<\infty ,-h\le z\le 0\}$, where $s_1=b_1+b_2$.

2.2. Problem Setup Using the BEM

To investigate the problem using the BEM, it is essential to enclose the entire computational domain completely. This is accomplished by introducing two auxiliary boundaries, $L_1$ and $L_8$, positioned at $x=-l_0$ and $x=r_0$, respectively, as shown in Figure 2, where $l_0>0$ and $r_0>0$. The fluid domain is divided into three distinct regions, namely $R_1$, $R_2$, and $R_3$. To formulate the boundary integral equation used in the BEM, each region is bounded by a set of boundary segments defined as follows. Region $R_1$ is enclosed by the combined boundary ${\mathcal{L}}_{R_1}=L_1\cup L_2\cup L_3\cup L_4\cup L_{m_1}\cup L_{11}\cup L_{12}$, region $R_2$ is bounded by the combined boundary ${\mathcal{L}}_{R_2}=L_{m_1}\cup L_b\cup L_{m_2}\cup L_f$ and region $R_3$ is bounded by the combined boundary ${\mathcal{L}}_{R_3}=L_5\cup L_6\cup L_7\cup L_8\cup L_9\cup L_{10}\cup L_{m_2}$. This closed boundary setup defines the complete geometric model needed for the BEM implementation.

2.3. Governing Equation and Boundary Conditions

In this subsection, the governing equations and associated boundary conditions used to describe the wave–structure interaction are presented. Assuming the fluid motion to be irrotational, there exists a velocity potential function $\mathrm{\Phi }(x,z,t)$ in the entire fluid domain. The total velocity potential $\mathrm{\Phi }(x,z,t)$ is expressed as the superposition of two potential functions [52,53,54]

$$\begin{array}{c}\hfill \mathrm{\Phi }(x,z,t)={\mathrm{\Phi }}_c\left(x\right)+{\mathrm{\Phi }}_w(x,z,t),\end{array}$$

(1)

where ${\mathrm{\Phi }}_c\left(x\right)=Ux$ represents the steady current potential and ${\mathrm{\Phi }}_w(x,z,t)=\mathbb{R}\left[\varphi (x,z)e^{-\mathrm{i}\omega t)}\right]$ is the unsteady wave potential. Here, $\mathbb{R}$ is the real part of the complex function, x and z denote the spatial coordinates, and t denotes time. Both the total and wave potential satisfy the Laplace equation [54]. In each region, the spatial velocity potential ${\varphi }_j(x,z)$ (for $j=1,2,3,4,5$) adheres to the Laplace equation, represented as:

$$\begin{array}{c}\hfill {\nabla }^2{\varphi }_j=0\mathrm{in}\mathrm{the}\mathrm{whole}\mathrm{fluid}\mathrm{domain}.\end{array}$$

(2)

The seabed is considered impermeable, restricting any vertical flow across the bottom boundary at $z=-h$, and is given by

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_j}{\partial z}}=0\mathrm{for}j=1,2,\dots ,5.\end{array}$$

(3)

The free-surface boundary condition in the linearized form at $z=0$ for regions $j=1,2,4,5$ reads

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_j}{\partial z}}-K{\varphi }_j=0\mathrm{for}j=1,2,4,5,\end{array}$$

(4)

where $K={\omega }_d^2/g$ with ${\omega }_d=(\omega -U{k}_0)$ being ‘Doppler shift in the angular frequency’ as discussed in Ref. [32]. For region 3, which contains the porous structure, the free-surface condition at $z=0$ includes inertial s and frictional f effects and is given as

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_3}{\partial z}}=(s-\mathrm{i}f)K{\varphi }_3\mathrm{for}j=3,\end{array}$$

(5)

where s and f are the inertial and frictional coefficients, respectively.

At $x=0$ and $x=s_2$, the pressure and velocity continuities are given as

$$\begin{array}{cc}\hfill \left.\begin{array}{cc}\hfill {\varphi }_1& ={\varphi }_2\hfill \\ \hfill {\displaystyle \frac{\partial {\varphi }_1}{\partial x}}& ={\displaystyle \frac{\partial {\varphi }_2}{\partial x}}\hfill \end{array}\right\}& \mathrm{at}x=0\mathrm{for}z\in (-h_1,0),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \left.\begin{array}{cc}\hfill {\varphi }_4& ={\varphi }_5\hfill \\ \hfill {\displaystyle \frac{\partial {\varphi }_4}{\partial x}}& ={\displaystyle \frac{\partial {\varphi }_5}{\partial x}}\hfill \end{array}\right\}& \mathrm{at}x=s_2\mathrm{for}z\in (-h_1,0),\hfill \end{array}$$

(7)

and at $x=b_1$ and $x=s_1$, the pressure and velocity continuities are given as [40,55]

$$\begin{array}{cc}\hfill \left.\begin{array}{cc}\hfill {\varphi }_2& =(s-\mathrm{i}f){\varphi }_3\hfill \\ \hfill {\displaystyle \frac{\partial {\varphi }_2}{\partial x}}& ={\epsilon }_p{\displaystyle \frac{\partial {\varphi }_3}{\partial x}}\hfill \end{array}\right\}& \mathrm{at}x=b_1\mathrm{for}z\in (-h_1,0),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill \left.\begin{array}{cc}\hfill {\varphi }_4& =(s-\mathrm{i}f){\varphi }_3\hfill \\ \hfill {\displaystyle \frac{\partial {\varphi }_4}{\partial x}}& ={\epsilon }_p{\displaystyle \frac{\partial {\varphi }_3}{\partial x}}\hfill \end{array}\right\}& \mathrm{at}x=s_1\mathrm{for}z\in (-h_1,0),\hfill \end{array}$$

(9)

where ${\epsilon }_p$ is the porous-effect parameter. Additionally, the structural boundary conditions read

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\varphi }_1}{\partial x}}& =0\mathrm{at}x=0\mathrm{for}z\in (-h,-h_1),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\varphi }_5}{\partial x}}& =0\mathrm{at}x=s_2\mathrm{for}z\in (-h,-h_1).\hfill \end{array}$$

(11)

The radiation conditions when $x\to \pm \infty$, are provided by [55]

$$\begin{array}{cc}\hfill & {\varphi }_1=(e^{-\mathrm{i}{k}_{0}x}+R_0{e}^{\mathrm{i}{k}_{0}x})f_{10}\left(z\right)\mathrm{as}x\to -\infty ,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & {\varphi }_5=T_0{e}^{-\mathrm{i}{k}_{0}x}{f}_{10}\left(z\right)\mathrm{as}x\to \infty ,\hfill \end{array}$$

(13)

where the reflection and transmission coefficients are denoted by $R_0$ and $T_0$, respectively, $f_{10}\left(z\right)$ is defined as [55]

$$\begin{array}{c}\hfill f_{10}\left(z\right)={\displaystyle \frac{\mathrm{i}gA}{\omega -U{k}_0}}{\displaystyle \frac{cosh{k}_0(h+z)}{cosh{k}_{0}h}},\end{array}$$

(14)

where $k_0$ is the wavenumber in regions 1 and 5, which satisfies

$$\begin{array}{c}\hfill k_{0}tanh\left(k_{0}h\right)={\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}.\end{array}$$

(15)

3. Solution Strategy

The detailed formulation and solution procedures for both the EEM and BEM are presented in the following subsection.

3.1. Analytical Solution via EEM

To obtain an analytic solution to the BVP, the velocity potentials ${\varphi }_j$ in each region $(j=1,2,3,\dots ,5)$ are determined. These velocity potentials satisfy the Laplace equation given in Equation (2). The general expressions for ${\varphi }_j$ that satisfy these governing relations are formulated as follows:

$$\begin{array}{cc}\hfill {\varphi }_1& =A{e}^{-\mathrm{i}{k}_{0}x}{f}_{10}\left(z\right)+\sum _{n=0}^{\infty }{R}_n{e}^{\mathrm{i}{k}_{n}x}{f}_{1n}\left(z\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\varphi }_2& =\sum _{n=0}^{\infty }\left[A_n{e}^{-\mathrm{i}{p}_{1n}x}+B_n{e}^{\mathrm{i}{p}_{1n}x}\right]f_{2n}\left(z\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\varphi }_3& =\sum _{n=0}^{\infty }\left[C_n{e}^{-\mathrm{i}{p}_{2n}x}+D_n{e}^{\mathrm{i}{p}_{2n}x}\right]f_{3n}\left(z\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\varphi }_4& =\sum _{n=0}^{\infty }\left[E_n{e}^{-\mathrm{i}{p}_{1n}x}+F_n{e}^{\mathrm{i}{p}_{1n}x}\right]f_{4n}\left(z\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\varphi }_5& =\sum _{n=0}^{\infty }{T}_n{e}^{-\mathrm{i}{k}_{n}x}{f}_{5n}\left(z\right),\hfill \end{array}$$

(20)

where $R_n$, $A_n$, $B_n$, $C_n$, $D_n$, $E_n$, $F_n$, and $T_n$ (for $n=0,1,2,3,\dots$) are unknown coefficients, $k_n$ to be determined from the dispersion relation [52]

$$\begin{array}{c}\hfill k_{n}tanh\left(k_{n}h\right)={\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}},\end{array}$$

(21)

$p_{1n}$ and $p_{2n}$ satisfy the following dispersion relations

$$\begin{array}{cc}\hfill p_{1n}tanh\left(p_{1n}{h}_1\right)& ={\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill p_{2n}tanh\left(p_{2n}{h}_1\right)& =(s-\mathrm{i}f){\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}},\hfill \end{array}$$

(23)

and the vertical eigenfunctions $f_{jn}\left(z\right)$ (for $j=1,2,3,\dots ,5$) are given by

$$\begin{array}{c}\hfill f_{jn}\left(z\right)=\left\{\begin{array}{cccc}\hfill & \left({\displaystyle \frac{\mathrm{i}gA}{\omega -U{k}_0}}\right){\displaystyle \frac{cosh{k}_n(h+z)}{cosh\left(k_{n}h\right)}}\hfill & \hfill & \mathrm{for}j=1,5\hfill \\ \hfill & \left({\displaystyle \frac{\mathrm{i}gA}{\omega -U{k}_0}}\right){\displaystyle \frac{cosh{p}_{1n}(h_1+z)}{cosh\left(p_{1n}{h}_1\right)}}\hfill & \hfill & \mathrm{for}j=2,4\hfill \\ \hfill & \left({\displaystyle \frac{\mathrm{i}gA}{\omega -U{k}_0}}\right){\displaystyle \frac{cosh{p}_{2n}(h_1+z)}{cosh\left(p_{2n}{h}_1\right)}}\hfill & \hfill & \mathrm{for}j=3.\hfill \end{array}\right.\end{array}$$

(24)

We substitute the velocity potentials (16)–(20) into the matching conditions for pressure and velocity continuity (6)–(7) as well as the structural boundary conditions (10)–(11). By applying the orthogonality of the eigenfunctions, this leads to the following system of equations

$$\begin{array}{cc}\hfill & {\displaystyle A{\mathcal{X}}_{1m0}+\sum _{n=0}^{\infty }{R}_n{\mathcal{X}}_{1mn}=\sum _{n=0}^{\infty }\left[A_n+B_n\right]{\mathcal{Y}}_{mn},}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & -\mathrm{i}{k}_{0}A{\mathcal{Z}}_{m0}+\sum _{n=0}^{\infty }\mathrm{i}{k}_n{R}_n{\mathcal{Z}}_{mn}=\sum _{n=0}^{\infty }\mathrm{i}{p}_{1n}\left[-A_n+B_n\right]{\mathcal{Y}}_{1mn},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\left[A_n{e}^{-\mathrm{i}{p}_{1n}{b}_1}+B_n{e}^{\mathrm{i}{p}_{1n}{b}_1}\right]{\mathcal{Y}}_{mn}=(s-\mathrm{i}f)\left[C_n{e}^{-\mathrm{i}{p}_{2n}{b}_1}+D_n{e}^{\mathrm{i}{p}_{2n}{b}_1}\right]{\mathcal{X}}_{2mn},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\left[E_n{e}^{-\mathrm{i}{p}_{1n}{s}_1}+F_n{e}^{\mathrm{i}{p}_{1n}{s}_1}\right]{\mathcal{Y}}_{mn}=(s-\mathrm{i}f)\left[C_n{e}^{-\mathrm{i}{p}_{2n}{s}_1}+D_n{e}^{\mathrm{i}{p}_{2n}{s}_1}\right]{\mathcal{X}}_{2mn},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\left[E_n{e}^{-\mathrm{i}{p}_{1n}{s}_2}+F_n{e}^{\mathrm{i}{p}_{1n}{s}_2}\right]{\mathcal{Y}}_{mn}=\sum _{n=0}^{\infty }{T}_n{e}^{-\mathrm{i}{k}_n{s}_2}{\mathcal{X}}_{1mn},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\mathrm{i}{p}_{1n}\left[-A_n{e}^{-\mathrm{i}{p}_{1n}{b}_1}+B_n{e}^{\mathrm{i}{p}_{1n}{b}_1}\right]{\mathcal{Y}}_{mn}=\sum _{n=0}^{\infty }\mathrm{i}{p}_{2n}\left[-C_n{e}^{-\mathrm{i}{p}_{2n}{b}_1}+D_n{e}^{\mathrm{i}{p}_{2n}{b}_1}\right]{\mathcal{X}}_{2mn},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\mathrm{i}{p}_{1n}\left[-E_n{e}^{-\mathrm{i}{p}_{1n}{s}_1}+F_n{e}^{-\mathrm{i}{p}_{1n}{s}_1}\right]{\mathcal{Y}}_{mn}=\sum _{n=0}^{\infty }\mathrm{i}{p}_{2n}\left[-C_n{e}^{-\mathrm{i}{p}_{2n}{s}_1}+D_n{e}^{\mathrm{i}{p}_{2n}{s}_1}\right]{\mathcal{X}}_{2mn},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \sum _{n=0}^{\infty }\mathrm{i}{p}_{1n}\left[-E_n{e}^{-\mathrm{i}{p}_{1n}{s}_2}+F_n{e}^{\mathrm{i}{p}_{1n}{s}_2}\right]{\mathcal{Y}}_{1mn}=-\sum _{n=0}^{\infty }\mathrm{i}{k}_n{T}_n{e}^{-\mathrm{i}{k}_n{s}_2}{\mathcal{Z}}_{mn},\hfill \end{array}$$

(32)

where

$$\begin{array}{cc}\hfill & {\displaystyle {\mathcal{X}}_{1mn}={\int }_{-h_1}^0{f}_{1n}\left(z\right)f_{2m}\left(z\right)\mathrm{d}z,{\mathcal{X}}_{2mn}={\int }_{-h_1}^0{f}_{3n}\left(z\right)f_{2m}\left(z\right)\mathrm{d}z,{\mathcal{Y}}_{1mn}={\int }_{-h_1}^0{f}_{2n}\left(z\right)f_{1m}\left(z\right)\mathrm{d}z,}\hfill \\ \hfill & {\displaystyle {\mathcal{Y}}_{mn}={\int }_{-h_1}^0{f}_{2n}\left(z\right)f_{2m}\left(z\right)\mathrm{d}z,{\mathcal{Z}}_{mn}={\int }_{-h}^0{f}_{1n}\left(z\right)f_{1m}\left(z\right)\mathrm{d}z}\hfill \end{array}$$

for $n,m=0,1,2,3,\dots$. Note that m is given non-negative integers and for fixed value of m, Equations (25)–(32) form a system of simultaneous linear equations involves an infinite number of unknowns, namely $R_n$, $A_n$, $B_n$, $C_n$, $D_n$, $E_n$, $F_n$, and $T_n$ for $n=0,1,2,3,\dots$. Solving for all these infinite unknowns is impractical. Therefore, the series is truncated by retaining only a finite number of terms, say $n=N$, which makes the system solvable using computational methods. After solving for the unknowns, substituting them back into Equations (25)–(32) gives the explicit expressions for the velocity potentials in each region.

3.2. Computational Approach Using the BEM

The integral equations for the two-dimensional BVP are derived in this subsection. Since the domain is divided into three distinct regions—$R_1$ and $R_3$ representing the fluid domains without a porous structure, and $R_2$ denoting the domain occupied by the porous medium—the BEM is developed as a multidomain approach [56]. Let ${\varphi }_1$, ${\varphi }_2$, and ${\varphi }_3$ be the potentials in regions $R_1$, $R_2$, and $R_3$, respectively. Applying Green’s second identity to the velocity potentials ${\varphi }_l$ for $l=1,2,3$ and the Green’s function $\mathcal{G}$ yields the following boundary integral equations:

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{\varphi }_l(x_0,z_0)={\int }_L\left({\varphi }_l{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}(x,z;x_0,z_0)-\mathcal{G}(x,z;x_0,z_0){\displaystyle \frac{\partial {\varphi }_l}{\partial n}}\right)\mathrm{d}L,if(x,z)\in \mathcal{L},\end{array}$$

(33)

where $\mathcal{L}$ is the combined boundary $\mathcal{L}={\mathcal{L}}_{R_1}\cup {\mathcal{L}}_{R_2}\cup {\mathcal{L}}_{R_3}$ and the Green’s function $\mathcal{G}(x,z;x_0,z_0)$ is the fundamental solution satisfying

$$\begin{array}{c}\hfill {\nabla }^2\mathcal{G}=\delta (x_0-x)\delta (z_0-z),\end{array}$$

(34)

and given by

$$\begin{array}{c}\hfill \mathcal{G}(x,z;x_0,z_0)=-{\displaystyle \frac{1}{2\pi }}ln\left(r\right),\end{array}$$

(35)

where $\mathsf{r}=\sqrt{{(x_0-x)}^2+{(z_0-z)}^2}$. The normal derivative of $\mathcal{G}$ is obtained by

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{G}}{\partial n}}=-{\displaystyle \frac{1}{2\pi r}}{\displaystyle \frac{\partial \mathsf{r}}{\partial n}}.\end{array}$$

(36)

For clarity and to facilitate the numerical implementation using the BEM, the physical boundary conditions previously described are now reformulated and applied over the discretized computational boundaries corresponding to each subdomain. The boundary conditions on rigid boundaries for all corresponding segments are given by

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_j}{\partial n}}=0\mathrm{at}{L}_p\mathrm{for}p=2,3,4,\dots ,7,b.\end{array}$$

(37)

Additionally, Equations (12) and (13) are written in the following form as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial }{\partial n}}({\varphi }_1-{\varphi }_{inc})=\mathrm{i}{k}_0({\varphi }_1-{\varphi }_{inc})\mathrm{at}{L}_1,\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial {\varphi }_3}{\partial n}}=\mathrm{i}{k}_0{\varphi }_3\mathrm{at}{L}_8,\hfill \end{array}$$

(39)

where ${\varphi }_{inc}=e^{\mathrm{i}{k}_{0}x}{f}_{10}$ as detailed in Ref. [55]. The linearized free-surface boundary conditions in regions $R_1$ and $R_3$ are given as

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_1}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\varphi }_1=0\mathrm{at}{L}_{11}\mathrm{and}{L}_{12},\end{array}$$

(40)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_3}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\varphi }_3=0\mathrm{at}{L}_9\mathrm{and}{L}_{10}.\end{array}$$

(41)

The linearized free-surface boundary condition at $L_f$ is given as

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\varphi }_2}{\partial n}}-(s-\mathrm{i}f){\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\varphi }_2=0\mathrm{at}{L}_f.\end{array}$$

(42)

Finally, at the interfaces $L_{m_1}$ and $L_{m_2}$ between the porous and non-porous regions, the matching conditions are given by

$$\begin{array}{cc}\hfill \left.\begin{array}{cc}\hfill {\varphi }_1& =(s-\mathrm{i}f){\varphi }_2\hfill \\ \hfill {\displaystyle \frac{\partial {\varphi }_1}{\partial n}}& ={\epsilon }_p{\displaystyle \frac{\partial {\varphi }_2}{\partial n}}\hfill \end{array}\right\}& \mathrm{at}{L}_{m_1}\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \left.\begin{array}{cc}\hfill {\varphi }_3& =(s-\mathrm{i}f){\varphi }_2\hfill \\ \hfill {\displaystyle \frac{\partial {\varphi }_3}{\partial n}}& ={\epsilon }_p{\displaystyle \frac{\partial {\varphi }_2}{\partial n}}\hfill \end{array}\right\}& \mathrm{at}{L}_{m_2}.\hfill \end{array}$$

(44)

By imposing the above boundary conditions as defined in Equations (37)–(44) into Equation (33), the integral equations in each region are obtained as follows.

• For region 1:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\varphi }_1& +{\int }_{L_1}\left({\varphi }_1{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-\mathcal{G}\left({\displaystyle \frac{\partial {\varphi }^{inc}}{\partial n}}+\mathrm{i}{k}_0{\varphi }_1-\mathrm{i}{k}_0{\varphi }^{inc}\right)\right)\mathrm{d}L+{\int }_{L_2}{\varphi }_1{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}\mathrm{d}L+{\int }_{L_3}{\varphi }_1{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}\mathrm{d}L\hfill \\ \hfill & +{\int }_{L_4}{\varphi }_1{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}\mathrm{d}L+{\int }_{L_{m_1}}\left((s-\mathrm{i}f){\varphi }_2{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\epsilon }_p{\displaystyle \frac{\partial {\varphi }_2}{\partial n}}\mathcal{G}\right)\mathrm{d}L\hfill \\ \hfill & +{\int }_{L_{11}}\left({\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}\mathcal{G}\right){\varphi }_1\mathrm{d}L+{\int }_{L_{12}}\left({\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}\mathcal{G}\right){\varphi }_1\mathrm{d}L=0,\hfill \end{array}$$

(45)

• For region 2:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\varphi }_2& +{\int }_{L_{m_1}}\left({\displaystyle \frac{1}{(s-\mathrm{i}f)}}{\varphi }_1{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\displaystyle \frac{1}{{\epsilon }_p}}{\displaystyle \frac{\partial {\varphi }_1}{\partial n}}\mathcal{G}\right)\mathrm{d}L+{\int }_{L_{m_2}}\left({\displaystyle \frac{1}{(s-\mathrm{i}f)}}{\varphi }_3{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\displaystyle \frac{1}{{\epsilon }_p}}{\displaystyle \frac{\partial {\varphi }_3}{\partial n}}\mathcal{G}\right)\mathrm{d}L\hfill \\ \hfill & +{\int }_{L_b}{\varphi }_2{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}\mathrm{d}L+{\int }_{L_f}\left({\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-(s-\mathrm{i}f){\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}\mathcal{G}\right){\varphi }_2\mathrm{d}L=0,\hfill \end{array}$$

(46)

• For region 3:

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\varphi }_3& +{\int }_{L_5}{\varphi }_3{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}\mathrm{d}{L}_5+{\int }_{L_6}{\varphi }_3{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}\mathrm{d}L+{\int }_{L_7}{\varphi }_3{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}\mathrm{d}L+{\int }_{L_8}\left({\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-\mathrm{i}{k}_0\mathcal{G}\right){\varphi }_3\mathrm{d}L\hfill \\ \hfill & +{\int }_{L_9}\left({\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}\mathcal{G}\right){\varphi }_3\mathrm{d}L+{\int }_{L_{10}}\left({\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}\mathcal{G}\right){\varphi }_3\mathrm{d}L\hfill \\ \hfill & +{\int }_{L_{m_2}}\left((s-\mathrm{i}f){\varphi }_2{\displaystyle \frac{\partial \mathcal{G}}{\partial n}}-{\epsilon }_p{\displaystyle \frac{\partial {\varphi }_2}{\partial n}}\mathcal{G}\right)\mathrm{d}L=0.\hfill \end{array}$$

(47)

The boundary integral Equations (45)–(47) are then discretized by evaluating the fundamental solution $\mathcal{G}$, which is centered at the node of the i-th element—treated as the source element $(x_0,z_0)$. Then, Equations (45)–(47) can be written in a discretized form for the i-th element are given as follows.

• For region 1:

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}{\varphi }_{1i}+\sum _{j=1}^{N_1}{\int }_{L_1}\left({\varphi }_{1j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\mathcal{G}}_{ij}\left({\displaystyle \frac{\partial {\varphi }_j^{inc}}{\partial n}}+\mathrm{i}{k}_0{\varphi }_{1j}-\mathrm{i}{k}_0{\varphi }_j^{inc}\right)\right)\mathrm{d}L+\sum _{j=1}^{N_2}{\int }_{L_2}{\varphi }_{1j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L\hfill \\ & +\sum _{j=1}^{N_3}{\int }_{L_3}{\varphi }_{1j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L+\sum _{j=1}^{N_4}{\int }_{L_4}{\varphi }_{1j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L+\sum _{j=1}^{N_{m_1}}{\int }_{L_{m_1}}\left((s-\mathrm{i}f){\varphi }_{2j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\epsilon }_p{\displaystyle \frac{\partial {\varphi }_{2j}}{\partial n}}{\mathcal{G}}_{ij}\right)\mathrm{d}L\hfill \\ & +\sum _{j=1}^{N_{11}}{\int }_{L_{11}}\left({\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\mathcal{G}}_{ij}\right){\varphi }_{1j}\mathrm{d}L+\sum _{j=1}^{N_{12}}{\int }_{L_{12}}\left({\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\mathcal{G}}_{ij}\right){\varphi }_{1j}\mathrm{d}L=0,\hfill \end{array}$$

(48)

• For region 2:

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}{\varphi }_{2i}+\sum _{j=1}^{N_{m_1}}{\int }_{L_{m_1}}\left({\displaystyle \frac{1}{(s-\mathrm{i}f)}}{\varphi }_{1j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\displaystyle \frac{1}{{\epsilon }_p}}{\displaystyle \frac{\partial {\varphi }_{1j}}{\partial n}}{\mathcal{G}}_{ij}\right)\mathrm{d}L+\sum _{j=1}^{N_b}{\int }_{L_b}{\varphi }_{2j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L\hfill \\ & +\sum _{j=1}^{N_{m_2}}{\int }_{L_{m_2}}\left({\displaystyle \frac{1}{(s-\mathrm{i}f)}}{\varphi }_{3j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\displaystyle \frac{1}{{\epsilon }_p}}{\displaystyle \frac{\partial {\varphi }_{3j}}{\partial n}}{\mathcal{G}}_{ij}\right)\mathrm{d}L+\sum _{j=1}^{N_f}{\int }_{L_f}\left({\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-(s-\mathrm{i}f){\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\mathcal{G}}_{ij}\right)\hfill \\ & \times {\varphi }_{2j}\mathrm{d}L=0,\hfill \end{array}$$

(49)

• For region 3:

$$\begin{array}{cc}& {\displaystyle \frac{1}{2}}{\varphi }_{3i}+\sum _{j=1}^{N_5}{\int }_{L_5}{\varphi }_{3j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L+\sum _{j=1}^{N_6}{\int }_{L_6}{\varphi }_{3j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L+\sum _{j=1}^{N_7}{\int }_{L_7}{\varphi }_{3j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L\hfill \\ & +\sum _{j=1}^{N_8}{\int }_{L_8}\left({\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-\mathrm{i}{k}_0{\mathcal{G}}_{ij}\right){\varphi }_{3j})\mathrm{d}L+\sum _{j=1}^{N_9}{\int }_{L_9}\left({\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\mathcal{G}}_{ij}\right){\varphi }_{3j}\mathrm{d}L\hfill \\ & +\sum _{j=1}^{N_{10}}{\int }_{L_{10}}\left({\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{\mathcal{G}}_{ij}\right){\varphi }_{3j}\mathrm{d}L+\sum _{j=1}^{N_{m_2}}{\int }_{L_{m_2}}\left((s-\mathrm{i}f){\varphi }_{2j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}-{\epsilon }_p{\displaystyle \frac{\partial {\varphi }_{2j}}{\partial n}}{\mathcal{G}}_{ij}\right)\mathrm{d}L=0,\hfill \end{array}$$

(50)

where $N_1,N_2,N_3,\dots ,N_{12},N_{m_1},N_{m_2},N_b$ and $N_f$, respectively, denote the total number of segments on the boundaries $L_1,L_2,L_3,\dots ,L_{12},L_{m_1},L_{m_2},L_b$ and $L_f$. For simplicity, we define

$$\begin{array}{cc}\hfill {\displaystyle {\mathcal{M}}_{ij}^l}& ={\displaystyle \frac{1}{2}}{\delta }_{ij}+{\int }_{L_j}{\displaystyle \frac{\partial {\mathcal{G}}_{ij}}{\partial n}}\mathrm{d}L,\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill G_{ij}^l& ={\int }_{L_j}{\mathcal{G}}_{ij}\mathrm{d}L,\hfill \end{array}$$

(52)

where $l=1,2,3$ corresponds to region 1, region 2, and region 3, respectively. By substituting Equations (51) and (52) into Equations (48)–(50) for each region, the following system of equations are as follows.

• For region 1:

$$\begin{array}{cc}\hfill & \sum _{j=1}^{N_1}\left({\varphi }_{1j}{\mathcal{M}}_{ij}^1-G_{ij}^1\left({\displaystyle \frac{\partial {\varphi }_j^{inc}}{\partial n}}+\mathrm{i}{k}_0{\varphi }_{1j}-\mathrm{i}{k}_0{\varphi }_j^{inc}\right)\right){|}_{L_1}+\sum _{j=1}^{N_2}{\varphi }_1{\mathcal{M}}_{ij}^1{|}_{L_2}+\sum _{j=1}^{N_3}{\varphi }_{1j}{\mathcal{M}}_{ij}^1{|}_{L_3}+\sum _{j=1}^{N_4}{\varphi }_{1j}{\mathcal{M}}_{ij}^1{|}_{L_4}\hfill \\ \hfill & +\sum _{j=1}^{N_{m_1}}\left((s-\mathrm{i}f){\varphi }_{2j}{\mathcal{M}}_{ij}^1-{\epsilon }_p{\displaystyle \frac{\partial {\varphi }_{2j}}{\partial n}}{G}_{ij}^1\right){|}_{L_{m_1}}+\sum _{j=1}^{N_{11}}\left({\mathcal{M}}_{ij}^1-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{G}_{ij}^1\right){\varphi }_{1j}{|}_{L_{11}}\hfill \\ \hfill & +\sum _{j=1}^{N_{12}}\left({\mathcal{M}}_{ij}^1-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{G}_{ij}^1\right){\varphi }_{1j}{|}_{L_{12}}=0,\hfill \end{array}$$

(53)

• For region 2:

$$\begin{array}{cc}\hfill & \sum _{j=1}^{N_{m_1}}\left({\displaystyle \frac{1}{(s-\mathrm{i}f)}}{\varphi }_{1j}{\mathcal{M}}_{ij}^2-{\displaystyle \frac{1}{{\epsilon }_p}}{\displaystyle \frac{\partial {\varphi }_{1j}}{\partial n}}{G}_{ij}^2\right){|}_{L_{m_1}}+\sum _{j=1}^{N_b}{\varphi }_{2j}{\mathcal{M}}_{ij}^2{|}_{L_b}\hfill \\ \hfill & \sum _{j=1}^{N_{m_2}}\left({\displaystyle \frac{1}{(s-\mathrm{i}f)}}{\varphi }_{3j}{\mathcal{M}}_{ij}^2-{\displaystyle \frac{1}{{\epsilon }_p}}{\displaystyle \frac{\partial {\varphi }_{3j}}{\partial n}}{G}_{ij}^2\right){|}_{L_{m_2}}+\sum _{j=1}^{N_f}\left({\mathcal{M}}_{ij}^2-(s-\mathrm{i}f){\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{G}_{ij}^2\right){\varphi }_{2j}{|}_{L_f}=0,\hfill \end{array}$$

(54)

• For region 3:

$$\begin{array}{cc}\hfill & \sum _{j=1}^{N_5}{\varphi }_{3j}{\mathcal{M}}_{ij}^3{|}_{L_5}+\sum _{j=1}^{N_6}{\varphi }_{3j}{\mathcal{M}}_{ij}^3{|}_{L_6}+\sum _{j=1}^{N_7}{\varphi }_{3j}{\mathcal{M}}_{ij}^3{|}_{L_7}+\sum _{j=1}^{N_8}\left({\mathcal{M}}_{ij}^3-\mathrm{i}{k}_0{G}_{ij}^3\right){\varphi }_{3j}{|}_{L_8}\hfill \\ \hfill & +\sum _{j=1}^{N_9}\left({\mathcal{M}}_{ij}^3-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{G}_{ij}^3\right){\varphi }_{3j}{|}_{L_9}+\sum _{j=1}^{N_{10}}\left({\mathcal{M}}_{ij}^3-{\displaystyle \frac{{(\omega -U{k}_0)}^2}{g}}{G}_{ij}^3\right){\varphi }_{3j}{|}_{L_{10}}\hfill \\ \hfill & +\sum _{j=1}^{N_{m_2}}\left((s-\mathrm{i}f){\varphi }_{2j}{\mathcal{M}}_{ij}^3-{\epsilon }_p{\displaystyle \frac{\partial {\varphi }_{2j}}{\partial n}}{G}_{ij}^3\right){|}_{L_{m_2}}=0.\hfill \end{array}$$

(55)

Considering the fundamental solution centered at each node on every segment, the boundary integral equation is evaluated at every node $i=1,2,3,\dots ,N_p$, where $p=1,2,3,\dots ,12,m_1,m_2,f,b$. This leads to total of N equations corresponding to N unknowns, forming an $N\times N$ linear system, where $N=N_1+N_2+N_3+\dots +N_{12}+N_{m_1}+N_{m_2}+N_f+N_b$. Solving this system of equations, we obtain the spatial velocity potential $\varphi$ and its normal derivative ${\displaystyle \frac{\partial \varphi }{\partial n}}$ at each node on the boundary element.

4. Results and Discussion

The reflection (R) and transmission (T) coefficients were evaluated for a wide range of structural configurations, wave conditions, and uniform current velocities to study the influence of different parameters. These coefficients were computed based on the following relations:

$$\begin{array}{c}\hfill R={\displaystyle \frac{R_0}{A}}\mathrm{and}T={\displaystyle \frac{T_0}{A}}.\end{array}$$

(56)

For consistency throughout the simulations, the following baseline parameters are used unless stated otherwise: $h=10$ m, $A=1$, $h_1=0.7h$, ${\epsilon }_p=0.5$, $f=0.5$, $s=1$, and $U/\sqrt{gh}=0.5$. In this study, all computations have been executed utilizing the Matlab^® software (2022b).

4.1. Validation

As noted by Dalrymple et al. [16], the inverse T-shaped structure simplifies to the case of a completely extended porous breakwater where there is no ocean current ($U=0$) and the rigid base height is minimal. The change in the modulus value of the reflection coefficient in relation to the dimensionless wavenumber $k_{0}h$ is seen in Figure 3. The solid black line and blue circles, respectively, indicate the outcomes of the current investigation utilising the EEM and BEM, whilst the star symbol represents the results given by Dalrymple et al. [16]. Excellent agreement among these results validates the accuracy of the present methodology.

4.2. Reflection and Transmission Coefficients

This subsection presents the influence of various structural and wave parameters on the reflection and transmission coefficients by the inverse T-type compound breakwater. Figure 4 displays the (a) reflection and (b) transmission coefficients against $k_{0}h$ for different values of the current velocity. In Figure 4a, the presence of the velocity of ocean current $(U/\sqrt{gh}\ne 0)$ leads to distinct oscillatory behavior in the curves compared to the gradually decreasing trend observed when no ocean current is present $(U/\sqrt{gh}=0)$. For cases with a nonzero velocity of ocean current, both the number and prominence of local maxima in the reflection coefficient increase with the velocity of the ocean current. The narrowing of the bandwidth of the curve with increasing current velocity demonstrates that strong reflection is confined to a limited range of wavenumbers, resulting in sharper and higher peaks. In contrast, the transmission coefficient plot (Figure 4b) shows that, for very small wavenumbers $(k_{0}h\to 0)$, nearly full transmission is observed. As $k_{0}h$ grows, the transmission coefficient gradually decreases. At higher current velocity, the transmission coefficient becomes more pronounced, confirming a greater reduction in energy passing through the structure. Across both Figure 4a,b, the results from the analytical approach via the EEM and numerical approach via the BEM show good agreement. Figure 5 presents the modulus value of the reflection and transmission coefficients for different heights of the porous vertical section of the inverse T-type structure.

Figure 5a shows that the reflection coefficient exhibits noticeable oscillations across the range of $k_{0}h$ for all values of $h_1/h$. As the height of the porous section increases, the peak of the reflection coefficient within the range $0.5\lesssim k_{0}h\lesssim 1$ becomes more prominent, indicating a stronger reflective response in $k_{0}h<1$. Figure 5b shows that the transmission coefficient initially increases slightly in the long-wave region ($0\le k_{0}h\lesssim 0.6$) as $h_1/h$ increases. This trend suggests that for very low-frequency (long) waves, a taller porous section may offer less resistance, allowing more wave energy to pass through. However, this behavior reverses in the mid-frequency ranges. Specifically, in the intervals $0.5\lesssim k_{0}h\lesssim 1.4$ and $1.5\lesssim k_{0}h\lesssim 2.25$, the transmission coefficient consistently decreases as $h_1/h$ increases. This implies that as waves become shorter (i.e., as $k_{0}h$ increases), the porous section becomes increasingly effective at attenuating wave energy. The reduction in transmission is more significant in these intervals, showing that the taller porous section acts as a more efficient energy-absorbing or scattering barrier for intermediate to shorter waves.

The width of the porous section ($b_2/h$) of the inverse T-shaped structure also significantly affects wave behavior, as shown in Figure 6. In Figure 6a, increasing the dimensionless width $b_2/h$ of the porous section leads to noticeably higher reflection coefficients, particularly in the long-wave regime (lower $k_{0}h$ values). As $b_2/h$ increases, the initial peak and subsequent local maxima of the reflection curve rise in the long wave region. In Figure 6b, a wider porous section ($b_2/h$) results in a more substantial and persistent reduction in the transmission coefficient across the entire wave spectrum examined. As $b_2/h$ increases, the transmission curve consistently falls lower, demonstrating that broader structures are more effective at dissipating and reflecting incoming wave energy, and thereby reducing the amount of wave energy transmitted towards the back side of the structure. These results emphasize that optimizing the width $b_2/h$ is crucial for the design of stable and efficient coastal protection systems.

Figure 7 demonstrates the effect of the friction coefficient f, associated with the porous section of the structure, on wave reflection and transmission as functions of the dimensionless wavenumber $k_{0}h$. In Figure 7a, the reflection coefficient exhibits oscillatory behavior for all values of f, consistent with the trends observed earlier in Figure 5a and Figure 6a. As the friction coefficient increases within the porous medium, the peaks of the reflection coefficient become less pronounced, indicating enhanced energy dissipation due to internal flow resistance. This reduces the amount of wave energy reflected by the structure. In Figure 7b, the transmission coefficient consistently decreases with increasing f, showing that more wave energy is either reflected or absorbed within the porous layer, resulting in less energy being transmitted beyond the structure. These results highlight the important role of the porous friction coefficient in governing the dissipation characteristics of the breakwater and its overall effectiveness in attenuating wave energy on the leeside.

Figure 8 illustrates the effect of porosity on the reflection and transmission coefficients. As shown in Figure 8a, the peak of the reflection coefficient decreases as the porosity parameter ${\epsilon }_p$ increases. This occurs because higher porosity allows more wave energy to pass through the structure, reducing the amount of energy reflected back. Consequently, as seen in Figure 8b, the transmission coefficient increases. The decrease in reflection directly contributes to greater transmission since more wave energy is able to traverse the porous medium.

4.3. Time-Domain Simulations of the Fluid Flow

The temporal variation in the free surface generated by a wave packet propagating from $x\to -\infty$ can be obtained through the application of the Fourier transform [41,57]. The evolution of the free-surface elevation, denoted by $\eta (x,t)$, is defined by

$$\begin{array}{c}\hfill {\eta }_j(x,t)=\mathbb{R}\left\{{\int }_0^{\infty }\widehat{f}\left(\omega \right){\eta }_j(x,\omega )e^{\mathrm{i}\omega t}d\omega \right\}\mathrm{for}j=1,2,3,4,5,\end{array}$$

(57)

where ${\eta }_j(x,\omega )$ describes the frequency-dependent surface response in each region. In Equation (57), $\widehat{f}\left(\omega \right)$ denotes the Fourier transform of the incoming wave pulse. For the present analysis, the incident wave packet is modeled as a Gaussian distribution [47], defined as

$$\begin{array}{c}\hfill \widehat{f}\left(\omega \right)=e^{\left(-{\displaystyle \frac{{\omega }^2}{4\sigma }}\right)},\end{array}$$

(58)

with $\sigma$ representing the scaling parameter. The results are conveniently expressed in terms of the non-dimensional quantities $\overline{t}=t\sqrt{g/h}$ and $\overline{x}=x/h$. The temporal evolution of these normalized surface elevations reveals the dynamic response of the free surface, offering detailed insights into the transient characteristics of wave motion across the fluid domain.

Figure 9 and Figure 10 illustrate the time-domain simulations of the free-surface elevation $({\eta }_j/h)$ across all regions under various hydrodynamic conditions, and similar observations have been reported in Ref. [47]. The color distribution indicates the relative height of the free surface, where the red shades represent the maximum elevation (wave crest) and the blue shades indicate the minimum elevation (wave trough). Figure 9 corresponds to the scenario without the influence of ocean current, while Figure 10 represents the case where an ocean current is present. The results clearly demonstrate how the free surface evolves as the Gaussian wave packet propagates from the negative x-direction to the positive x-direction. Over time, the amplitude of the deflection steadily diminishes. This behavior is primarily due to the dissipation of energy, which occurs as a result of the dynamic interaction between the inverse T-shaped structure and the surrounding fluid. In region 1 [Figure 9a], which corresponds to the incident side, the surface elevation exhibits a prominent rise as the incoming Gaussian wave packet approaches the structure. In region 2 [Figure 9b], located just before the porous section, the wave amplitude begins to decrease slightly due to partial energy dissipation and reflection at the boundary. The most notable behavior occurs in region 3 [Figure 9c], where the porous structure is positioned. Here, the wave elevation is comparatively high near the entrance of the porous zone, but as the wave propagates through the medium, its amplitude gradually diminishes owing to the combined effects of energy absorption, internal friction, and damping within the porous material. This demonstrates the efficiency of the porous section in attenuating the wave energy. Moving further downstream, in region 4 [Figure 9d], the transmitted wave amplitude becomes significantly lower and smoother, indicating that most of the high-frequency components have been filtered out by the porous barrier. Finally, in region 5 [Figure 9e], the free surface becomes nearly steady, and only a small elevation is observed compared to that in region 1, indicating that the transmitted wave energy is weak. Overall, the figure highlights the gradual transformation of the wave profile from a strong incident wave to a damped transmitted wave, confirming the dissipative nature of the porous structure in the absence of ocean current velocity.

Figure 10 illustrates the time-dependent response of the free surface in the presence of an ocean current $(U/\sqrt{gh}=0.5)$. Compared to the case without ocean current shown in Figure 9, the propagation pattern of the wave packet becomes asymmetric due to the combined effects of current-induced Doppler shifting in angular frequency. In region 1 [Figure 10a], the incident wave amplitude is slightly reduced, indicating that part of the wave energy is reduced by the ocean current. As the wave advances to region 2 [Figure 10b], the interaction between the ocean current and the structure alters the reflection characteristics, resulting in a shifted phase pattern and a slower decay rate near the interface. In region 3 [Figure 10c], where the porous structure is located, the wave amplitude decreases more rapidly than in the no-current case, revealing that the presence of current enhances energy dissipation within the porous medium through increased turbulence and viscous resistance. Moving into regions 4 [Figure 10d] and 5 [Figure 10e], the transmitted waves exhibit smaller amplitudes and smoother temporal variation, confirming stronger damping and reduced energy transmission. Overall, the comparison between Figure 9 and Figure 10 demonstrates that the ocean current significantly influences the dynamic behavior of the free surface by modifying wave propagation speed, enhancing energy loss, and accelerating the decay of oscillations throughout the flow domain.

5. Conclusions

This study focuses on understanding how water waves interact with an inverse T-type compound breakwater under ocean current. The analysis is carried out within the framework of linearized wave theory, assuming small-amplitude surface oscillations. The corresponding BVP is addressed via the EEM and BEM. A comparison of analytical and numerical results shows good consistency. Using this method, the unknown coefficients associated with the velocity potentials are identified, and the resulting equations are solved computationally. The reflection and transmission coefficients for different structural configurations are then calculated using these velocity potentials. The observations of the present study are as follows:

• It is observed that the presence of an ocean current in wave propagation significantly enhances water dissipation through the compound inverse T-shaped structure, leading to less wave transmission. Moreover, it can be concluded that selecting an optimal porosity for the structure ensures minimal wave load, as water waves are efficiently dissipated through the compound structure when subjected to a following current in wave propagation.
• In general, as the current velocity ($U/\sqrt{gh})$ increases, the amplitude of the reflection curve rises, accompanied by a more noticeable decrease in wave transmission. Furthermore, an increase in the height of the perforated section of the structure results in a higher amplitude of the reflection curve in the long-wave region, while the transmission coefficient exhibits an opposite trend in the same region. Additionally, a higher value of the friction parameter $\left(f\right)$ consistently contributes to a reduction in the transmission coefficient across all wave conditions.
• A higher value of the porosity parameter ${\epsilon }_p$ leads to a marked decrease in amplitude of the reflection coefficient as a function of $k_{0}h$, while simultaneously resulting in a significant increase in the transmission coefficient with respect to these parameters.
• The time-domain simulations of the free surface indicate that, over time, the surface elevation decreases more significantly when an ocean current is present compared to the scenario without a current across all regions.

Overall, optimizing the porosity parameter, friction coefficient, velocity of the ocean current, and structural parameters effectively minimizes wave transmission to the leeside zone. This strategic approach significantly reduces incoming wave energy, ensuring enhanced protection for the leeside area. Furthermore, this study can be extended to explore the dynamics of two-layer fluid systems, wave trapping near rigid walls, and wave interactions over a porous bed, offering broader applications and insights into wave-structure interactions.