2 Theoretical formulation

The Cartesian coordinate system $(x, y, z)$ is employed, where the $xz$ -plane represents the mean free surface and the y-axis points vertically downwards. The study considers a $\boldsymbol {\sqcap }$ -shaped configuration composed of thin porous plates and a thick rigid block in a fluid domain of finite depth H. The breakwater configuration can either be a floating structure or a bottom-fixed inverted structure, with two porous plates positioned at $x=-w_1$ and $x=w_2$ . The cross-sectional representation of this configuration, shown in Figure 1, illustrates how the entire domain is divided into three distinct sub-regions, $R_j~(j=1,2,3)$ , based on the structural geometry.

Figure 1

Problem sketch for (a) floating $\boldsymbol {\sqcap }$ -shaped and (b) bottom-fixed inverted $\boldsymbol {\sqcap }$ -shaped breakwater.

Under the assumptions that the fluid is incompressible and inviscid, a scalar velocity potential can be defined as $\Phi (x,y,z,t) = \mbox {Re}[\phi (x,y)e^{\mathrm {{i}(\nu z-\omega t)}}]$ , where $\mbox {Re}[~]$ represents the real part of the complex expression. The term $\nu ={k}_{0}\sin \theta $ defines the horizontal wavenumber component, where ${k}_{0}$ is the wavenumber of the incident wave, $\theta $ is the angle of incidence and the angular frequency is denoted by $\omega $ . Since the breakwater extends along the z-direction, the solution exhibits periodicity in that direction. Consequently, the interface between different regions can be analysed using Snell’s law [Reference Kirby and Dalrymple20]. Within the entire fluid domain, the velocity potentials must satisfy the Laplace equation. However, due to the periodic variation in the z-direction, the governing equation can be reformulated as the modified Helmholtz equation [Reference Mandal and Chakrabarti25]:

$$ \begin{align*} \bigg(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}-\nu^2\bigg)\phi(x,y)=0 \quad\text{for all } (x, y)\in(-\infty, \infty) \times (0, H), \end{align*} $$

where $\phi (x,y)$ represents the spatial part of the complex velocity potential.

By combining the kinematic and dynamic boundary conditions under the principles of linear water-wave theory, the boundary condition at the mean free surface can be expressed as

$$ \begin{align*} \bigg(g\frac{\partial}{\partial y}+\omega^2\bigg)\phi(x,y)\bigg|_{y = 0}=0, \begin{cases} x\in \mathbb{R}\setminus (-w_1, w_2)&\mbox{for~floating~structure},\\ x\in \mathbb{R}&\mbox{for~bottom-fixed~structure}, \end{cases} \end{align*} $$

where g and $\mathbb {R}$ have their usual meaning.

The seabed and the horizontal segment of the $\boldsymbol {\sqcap }$ -shaped configuration are considered rigid and impermeable, satisfying the zero normal velocity condition, expressed as follows:

$$ \begin{align*} \frac{\partial \phi(x,y)}{\partial y}\bigg|_{y = H}=0, \begin{cases} x\in \mathbb{R}&\mbox{for~floating~structure},\\ x\in \mathbb{R}\setminus (-w_1, w_2)&\mbox{for~bottom-fixed~structure}, \end{cases} \end{align*} $$

and

$$ \begin{align*} \frac{\partial \phi(x,y)}{\partial y}\bigg|_{y = d}=0,~x\in (-w_1, w_2)\quad\mbox{for~both~structures}. \end{align*} $$

The boundary condition for the rigid thick block and the thin porous plates of the $\boldsymbol {\sqcap }$ -shaped structure is defined as

(2.1) $$ \begin{align} \frac{\partial \phi(x,y)}{\partial x}\bigg|_{x = -w_1,w_2}=0, \begin{cases} y\in (0, d)&\mbox{for~floating~structure},\\ y\in (d, H)&\mbox{for~bottom-fixed~structure}, \end{cases} \end{align} $$

and

(2.2a) $$ \begin{align} \begin{aligned} \frac{\partial \phi(x,y)}{\partial x}\bigg|_{x = -w_1}&= -{i}k_0 G_1\{\phi(-w_1+0,y)-\phi(-w_1-0,y)\}~\\ &\quad \mbox{on} \begin{cases} y\in (d, h_1)&\mbox{for~floating~structure},\\ y\in (h_1, d)&\mbox{for~bottom-fixed~structure}; \end{cases} \end{aligned} \end{align} $$

(2.2b) $$ \begin{align} \begin{aligned} \frac{\partial \phi(x,y)}{\partial x}\bigg|_{x = w_2}&= -{i}k_0 G_2\{\phi(w_2+0,y)-\phi(w_2-0,y)\}~\\ &\quad \mbox{on} \begin{cases} y\in (d, h_2)&\mbox{for~floating~structure},\\ y\in (h_2, d)&\mbox{for~bottom-fixed~structure}; \end{cases} \end{aligned} \end{align} $$

where $G_1$ and $G_2$ are the complex porous effect parameters of front and rear plates, respectively, as defined by Yu and Chwang [Reference Yu and Chwang40]. The real components, $\mbox {Re}[G_1]$ and $\mbox {Re}[G_2]$ , represent the resistance offered by the porous material to seepage flow, whereas the imaginary components, $\mbox {Im}[G_1]$ and $\mbox {Im}[G_2]$ , correspond to the inertial influence of the fluid moving through the porous medium. Throughout the fluid domain, the velocity potential remains bounded, except at the submerged sharp edges of the $\boldsymbol {\sqcap }$ -shaped structure. The boundary conditions at these edges are given as follows [Reference Mandal and Chakrabarti25]:

(2.3) $$ \begin{align} |\nabla\phi|=O(r^{-1/2})\quad\mbox{as } \textit{r}\rightarrow 0, \end{align} $$

where r= ${((x+w_1)^2+(y-h_1)^2)}^{1/2}$ for the front plate and r= ${((x-w_2)^2+(y-h_2)^2)}^{1/2}$ for the rear plate.

Given that the incident wave propagates along the x-axis from negative infinity, the far-field condition is expressed as follows:

$$ \begin{align*} \phi(x,y)\sim \begin{cases} \phi^{\mathrm{inc}}(x,y)+R\phi^{\mathrm{inc}}(-x,y) &\mbox{as } x\rightarrow -\infty,\\ T \phi^{\mathrm{inc}}(x,y) &\mbox{as } x\rightarrow \infty, \end{cases} \end{align*} $$

where the incident potential, $\phi ^{\mathrm {inc}}(x,y)$ , is expressed as

$$ \begin{align*} \phi^{\mathrm{inc}}(x,y)= \frac{\cosh {k}_{0}(H-y)}{\cosh {k}_{0}H}e^{{i}\mu(x+w_1)} \end{align*} $$

with $\mu ={k}_{0}\cos \theta $ .

3 Analytical solution

The mathematical representation of the velocity potential $\phi (x,y)$ for the three sub-regions, namely $R_1,R_2,R_3$ , created in the fluid domain by the $\boldsymbol {\sqcap }$ -shaped breakwater designs (both floating and bottom fixed) have been obtained by using Havelock’s expansion formulae [Reference Mandal and Kanoria26].

In the sub-region, $R_1 = \{(x,y)\mid -\infty <x<-w_1, 0<y<H\}$ , the velocity potential comprises contributions from the incident and reflected waves. The mathematical expressions for $\phi (x,y)$ corresponding to both breakwater configurations are given as

(3.1) $$ \begin{align} \phi(x,y)=\{e^{{i}\mu(x+w_1)}+V_1e^{-{i}\mu(x+w_1)}\}\frac{\cosh {k}_0(H-y)}{\cosh {k}_0H}+\sum\limits_{n=1}^{\infty}{A}_{n} e^{{s}_{n}(x+w_1)}\frac{\cos {k}_n(H-y)}{\cos {k}_nH}, \end{align} $$

where $s_{n}=\sqrt {{k}_{n}^{2}+\nu ^{2}}$ , $A_{n}$ values $(n=1,2,\ldots )$ represent the unknown expansion coefficients and the constant $V_{1}$ relates to the reflection coefficient R through the expression [Reference Evans and Porter12, Reference Sarkar, De and Gayen33, Reference Sarkar, Roy and De34], $V_1=\mbox {Re}^{2{i}\mu w_1}$ .

In the sub-region, $R_2=\{(x,y)\mid -w_1<x<w_2, d<y<H\}$ for floating structure, or $R_{2}=\{(x,y)\mid -w_1<x<w_2, 0<y<d\}$ for bottom-fixed structure, the velocity potential $\phi (x,y)$ is expressed as a series of eigenfunctions given by: for the floating structure,

(3.2a) $$ \begin{align} \phi(x,y)&=({B}_{0} \cosh \lbrace\nu (x+w_1)\rbrace+{C}_{0} \sinh \lbrace\nu (x-w_2)\rbrace) \nonumber\\&\quad +\sum_{n=1}^{\infty}({B}_{n} \cosh \lbrace\xi_n (x+w_1)\rbrace+{C}_{n} \sinh \lbrace\xi_n (x-w_2)\rbrace) \cos\bigg\lbrace \frac{n\pi (y-d)}{H-d}\bigg\rbrace; \end{align} $$

and for the bottom-fixed structure,

(3.2b) $$ \begin{align} \begin{aligned} &\phi(x,y)=({B}_{0} \cos \lbrace({\alpha}_{0}^{2}-\nu^2)^{{1}/{2}} (x+w_1)\rbrace+{C}_{0} \sin \lbrace({\alpha}_{0}^{2}-\nu^2)^{{1}/{2}} (x-w_2)\rbrace)\frac{\cosh {\alpha}_0(d-y)}{\cosh {\alpha}_0d} \\ &\quad+\sum_{n=1}^{\infty}({B}_{n} \cosh \lbrace({\alpha}_{n}^{2}+\nu^2)^{{1}/{2}} (x+w_1)\rbrace+{C}_{n} \sinh \lbrace({\alpha}_{n}^{2}+\nu^2)^{{1}/{2}} (x-w_2)\rbrace) \cos {\alpha}_n(d-y),\\ \end{aligned} \end{align} $$

where $\xi _n=\sqrt {({n\pi }{(H-d)})^2+\nu ^2}~(n=1,2,\ldots )$ . Additionally, ${B}_{n}$ terms and ${C}_{n}$ terms $(n=0,1,2,\ldots )$ are unknown constants.

In the sub-region $R_3=\{(x,y)\mid w_2<x<\infty ,0<y<H\}$ , the complex potential function accounts for the effects of transmitted waves. Consequently, the formal expressions for $\phi (x,y)$ in both breakwater models are provided as

(3.3) $$ \begin{align} \begin{aligned} \phi(x,y)=V_{2}e^{{i}\mu(x-w_2)}\frac{\cosh {k}_0(H-y)}{\cosh {k}_0H}+\sum_{n=1}^{\infty}E_{n}e^{-s_{n}(x-w_2)}\frac{\cos {k}_n(H-y)}{\cos {k}_nH}, \end{aligned} \end{align} $$

where $E_{n}~(n=1,2,\ldots )$ denotes the unknown expansion coefficients, while the constant $V_{2}$ is connected to the transmission coefficient T through the relation [Reference Evans and Porter12, Reference Sarkar, De and Gayen33, Reference Sarkar, Roy and De34], $V_2 = T e^{2{i} \mu w_2}$ . The above-mentioned wavenumbers $\pm k_{0}$ and $\pm \alpha _{0}$ (representing progressive waves) satisfy the following dispersion relation:

$$ \begin{align*} \tanh{ph}=\frac{K}{p} \begin{cases} p=k \mbox{~and~} h=H \quad\mbox{for both breakwater in sub-regions } R_1 \mbox{ and } R_3,\\ p=\alpha \mbox{~and~} h=d \quad\mbox{for bottom-fixed breakwater in sub-regions } R_2, \end{cases} \end{align*} $$

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

Table 1 illustrates some numerical examples of roots of the above dispersion relation.

Table 1

Roots of the dispersion relation.

Also, the wavenumbers $\pm {i}k_{r}$ and $\pm {i}\alpha _{r}~(r=1,2,\ldots )$ (representing evanescent modes) are solutions of the transcendental equation

$$ \begin{align*} \tan{\gamma h}=-\frac{K}{\gamma} \begin{cases} \gamma=k \mbox{~and~} h=H \quad\mbox{for both breakwater in sub-regions } R_1 \mbox{ and } R_3,\\ \gamma=\alpha \mbox{~and~} h=d \quad\mbox{for bottom-fixed breakwater in sub-regions } R_2. \end{cases} \end{align*} $$

Table 2

Roots of the transcendental equation.

Similarly, Table 2 illustrates some numerical examples of roots of the above transcendental equation. Let us define the horizontal component of fluid velocity for both breakwater structures as

(3.4a) $$ \begin{align} f_1(y)&=\frac{\partial \phi(x,y)}{\partial x}\bigg|_{x = -w_1\pm 0},\quad y\in(0,H), \end{align} $$

(3.4b) $$ \begin{align} f_2(y)&=\frac{\partial \phi(x,y)}{\partial x}\bigg|_{x = w_2\pm 0},\quad y\in(0,H). \end{align} $$

Based on the conditions in equations (2.1) and (2.2), the following equations can be derived:

for the floating structure,

$$ \begin{align*} f_1(y)&= \begin{cases} 0,~y\in (0,d),\\ -{i}k_0 G_1\{\phi(-w_1+0,y)-\phi(-w_1-0,y)\},\quad y\in (d, h_1), \end{cases} \\ f_2(y)&= \begin{cases} 0,~y\in (0,d),\\ -{i}k_0 G_2\{\phi(w_2+0,y)-\phi(w_2-0,y)\},\quad y\in (d, h_2); \end{cases} \end{align*} $$

and for the bottom-fixed structure,

$$ \begin{align*} f_1(y)&= \begin{cases} 0,~y\in (d,H),\\ -{i}k_0 G_1\{\phi(-w_1+0,y)-\phi(-w_1-0,y)\}, \quad y\in (h_1, d), \end{cases} \\ f_2(y)&= \begin{cases} 0,~y\in (d, H),\\ -{i}k_0 G_2\{\phi(w_2+0,y)-\phi(w_2-0,y)\},\quad y\in (h_2, d). \end{cases} \end{align*} $$

As a result of the edge conditions outlined in equation (2.3), the functions $f_j(y)~(j=1,2)$ display singularities close to the submerged sharp edges. According to Mandal and Chakrabarti [Reference Mandal and Chakrabarti25], the singularity behaviour at the submerged edges of the thin plates in the $\boldsymbol {\sqcap }$ -shaped breakwater follows a form proportional to $|y - h_j|^{-1/2} (j=1,2)$ .

Substituting the expansions of $\phi (x,y)$ from equations (3.1)–(3.3) into equation (3.4) and using Havelock’s inversion formula [Reference Havelock16], all constants associated with $\phi (x,y)$ in different sub-regions can be expressed as integrals involving $f_{j}(y)~(j=1,2)$ . Additionally, the continuity of the velocity potential across the gap below or above the thin porous plates must be preserved. This continuity condition results in the following coupled integral equations:

$$ \begin{align*} \sum_{j=1}^{2} \int_{l} f_j(u)\mathcal{M}_{ij}(y,u)\, du = \mathcal{K}_{i}(y),\quad y\in \Omega_i~(i=1,2), \end{align*} $$

where $l \equiv (d, H)$ and $\Omega _i \equiv (h_i, H)~(i=1,2)$ for the floating $\boldsymbol {\sqcap }$ -shaped structure, while $l \equiv (0, d)$ and $\Omega _i \equiv (0, h_i)~(i=1,2)$ for the bottom-fixed $\boldsymbol {\sqcap }$ -shaped structure. The definitions of the functions $\mathcal {M}_{ij}(y,u)$ and $\mathcal {K}_{i}(y)$ $(i,j=1,2)$ are given as follows.

For the floating structure,

$$ \begin{align*} \mathcal{M}_{11}(y,u)&= -\frac{\coth\lbrace\nu(w_1+w_2)\rbrace}{\nu(H-d)} \nonumber\\[4pt]&\quad -\sum\limits_{n=1}^{\infty}\frac{2\coth \lbrace\xi_n(w_1+w_2)\rbrace\cos \lbrace\frac{n\pi(y-d)}{(H-d)}\rbrace \cos \lbrace\frac{n\pi(u-d)}{(H-d)}\rbrace}{\xi_n(H-d)}\\[4pt]&\quad-\sum\limits_{n=1}^{\infty}\frac{4k_{n}\cos k_{n}(H-y)\cos k_{n}(H-u)}{s_{n}(2k_{n}H+\sin 2k_{n}H)},\nonumber\\[-2pc] \end{align*} $$

$$ \begin{align*} \mathcal{M}_{12}(y,u)&= \frac{\operatorname{\mathrm{csch}}\lbrace\nu(w_1+w_2)\rbrace}{\nu(H-d)}\nonumber\\[4pt]&\quad +\sum\limits_{n=1}^{\infty}\frac{2\operatorname{\mathrm{csch}} \lbrace\xi_n(w_1+w_2)\rbrace\cos \lbrace\frac{n\pi(y-d)}{(H-d)}\rbrace \cos \lbrace\frac{n\pi(u-d)}{(H-d)}\rbrace}{\xi_n(H-d)}; \end{align*} $$

and for the bottom-fixed structure,

$$ \begin{align*} \mathcal{M}_{11}(y,u)&= \frac{4\alpha_{0}\cot\lbrace({\alpha}_{0}^{2}-\nu^2)^{{1}/{2}} (w_1+w_2)\rbrace\cosh \alpha_{0}(d-y)\cosh \alpha_{0}(d-u)}{({\alpha}_{0}^{2}-\nu^2)^{{1}/{2}}(2\alpha_{0}d+\sinh 2\alpha_{0}d)}\nonumber\\[4pt]&\quad -\sum\limits_{n=1}^{\infty}\frac{4k_{n}\cos k_{n}(H-y)\cos k_{n}(H-u)}{s_{n}(2k_{n}H+\sin 2k_{n}H)}\nonumber\\[4pt]&\quad -\sum\limits_{n=1}^{\infty}\frac{4\alpha_{n}\coth\lbrace({\alpha}_{n}^{2}+\nu^2)^{{1}/{2}} (w_1+w_2)\rbrace\cos \alpha_{n}(d-y)\cos \alpha_{n}(d-u)}{({\alpha}_{n}^{2}+\nu^2)^{{1}/{2}}(2\alpha_{n}d+\sin 2\alpha_{n}d)},\nonumber\\[-1.8pc] \end{align*} $$

$$ \begin{align*} \mathcal{M}_{12}(y,u)&= -\frac{4\alpha_{0}\csc\lbrace({\alpha}_{0}^{2}-\nu^2)^{{1}/{2}} (w_1+w_2)\rbrace\cosh \alpha_{0}(d-y)\cosh \alpha_{0}(d-u)}{({\alpha}_{0}^{2}-\nu^2)^{{1}/{2}}(2\alpha_{0}d+\sinh 2\alpha_{0}d)}\nonumber\\[4pt]&\quad +\sum\limits_{n=1}^{\infty}\frac{4\alpha_{n}\operatorname{\mathrm{csch}}\lbrace({\alpha}_{n}^{2}+\nu^2)^{{1}/{2}} (w_1+w_2)\rbrace\cos \alpha_{n}(d-y)\cos \alpha_{n}(d-u)}{({\alpha}_{n}^{2}+\nu^2)^{{1}/{2}}(2\alpha_{n}d+\sin 2\alpha_{n}d)},\nonumber\\[-1.8pc] \end{align*} $$

$$ \begin{align*} \mathcal{M}_{21}(y,u)=\mathcal{M}_{12}(y,u),~\mathcal{M}_{22}(y,u)=\mathcal{M}_{11}(y,u) \quad (\mbox{for~both~structures}) \end{align*} $$

$$ \begin{align*} \mathcal{K}_{1}(y)&=\frac{\cosh {k}_0(H-y)}{\cosh {k}_0H}(1+V_1),\nonumber\\[4pt]\mathcal{K}_{2}(y) &=\frac{\cosh {k}_0(H-y)}{\cosh {k}_0H}(-V_2)\quad (\mbox{for both structures}). \end{align*} $$

Let us now define the following transformation as

(3.6) $$ \begin{align} f_{j}(y)=\sum_{i=1}^{2}\mathcal{F}_{ji}(y)({A}_{i}+{B}_{i}), \quad y\in(0,H)~(j=1,2)\end{align} $$

where

(3.7) $$ \begin{align} {A}_{1}=1,\quad {A}_{2}=0,\quad {B}_{1}=V_1,\quad {B}_{2}=-V_2. \end{align} $$

By substituting equation (3.6) into equation (3.7), we obtain

(3.8) $$ \begin{align} \sum_{j=1}^{2} \int_{l} \bigg\{\sum_{i=1}^{2}\mathcal{F}_{ji}(u)({A}_{i}+{B}_{i})\bigg\}\mathcal{M}_{ij}(y,u)\,du=({A}_{i}+{B}_{i})\mathcal{P}_{i}(y),\quad y\in \Omega_i~(i=1,2), \end{align} $$

where $\mathcal {P}_{1}(y)=\mathcal {P}_{2}(y)={\cosh {k}_0(H-y)}/{\cosh {k}_0H}$ .

Using Havelock’s inversion formula, the unknown constants $V_1$ and $V_2$ are represented in terms of $f_{1}(y)$ and $f_{2}(y)$ , respectively. Then, by employing equation (3.6), the following equations are obtained:

(3.9) $$ \begin{align} (A_j-B_j) W_j Q_j = \sum_{i=1}^{2}S_{ji}(A_i+B_i) \quad (j=1,2), \end{align} $$

where $W_1=W_2={{i}\mu }/{k_0}$ , $Q_1=Q_2= \{{2k_{0}H+\sinh 2k_{0}H}\}/{4\cosh k_0H}$ and $S_{ji}= \int _{l}\mathcal {F}_{ji}(y) \mathcal {P}_j(y)\,dy, \, (i,j=1,2)$ . By reorganizing the terms in equation (3.9), we obtain expression of $B_j$ values $(j=1,2)$ in terms of $A_1$ and $A_2$ .

To address the singularity and enhance convergence in numerical computations, the Galerkin approximation method, as described by Evans and Porter [Reference Evans and Porter12], is employed. As a result, the unknown functions $\mathcal {F}_{jl}(y)$ are approximated using an $(N + 1)$ -term Galerkin expansion, expressed in the following form:

(3.10) $$ \begin{align} \mathcal{F}_{jl}(y) \simeq \sum_{n=0}^{N} {a}^{(n)}_{jl}{q}_{j}^{(n)}(y),\quad j,l=1,2. \end{align} $$

In this framework, ${a}^{(n)}_{jl}$ signifies the unknown expansion coefficients, whereas ${q}_{j}^{(n)}(y)$ denotes the basis functions. These basis functions, ${q}_{j}^{(n)}(y)~(j=1,2; n=0,1,2,\ldots ,N)$ , are determined according to the procedure details in [Reference Evans and Porter12].

For the floating structure:

$$ \begin{align*} & {q}_{j}^{(n)}(y)=\\ &\quad \begin{cases} \dfrac{d}{dy}\bigg[e^{-Ky}\displaystyle\int_{y}^{{h}_{j}}e^{Ku}\dfrac{2(-1)^n\lbrace({h}_{j}-d)^{2}-(u-d)^2\rbrace^{{1}/{2}}}{\pi (2n+1) H (h_j-d)}U_{2n}\bigg(\dfrac{u-d}{{h}_{j}-d}\bigg)\,du\bigg]&\mbox{for}~y\in (d, h_j),\\[.3cm] \dfrac{2(-1)^n}{\pi \lbrace(H-{h}_{j})^{2}-(H-u)^2\rbrace^{{1}/{2}}}T_{2n}\bigg(\dfrac{H-u}{H-{h}_{j}}\bigg)&\mbox{for}~y\in (h_j, H); \end{cases} \end{align*} $$

and for the bottom-fixed structure:

$$ \begin{align*} {q}_{j}^{(n)}(y)= \begin{cases} -\dfrac{d}{dy}\bigg[e^{-Ky}\displaystyle\int_{y}^{{h}_{j}}e^{Ku}\dfrac{2(-1)^n}{\pi \lbrace{h}_{j}^{2}-u^2\rbrace^{{1}/{2}}}T_{2n}\bigg(\dfrac{u}{{h}_{j}}\bigg)\,du\bigg]&\mbox{for}~y\in (0, h_j),\\[.3cm] \dfrac{2(-1)^n \lbrace(d-{h}_{j})^{2}-(d-u)^2\rbrace^{{1}/{2}}}{\pi (2n+1) (d-h_j) d }U_{2n}\bigg(\dfrac{d-u}{d-{h}_{j}}\bigg)&\mbox{for}~y\in (h_j, d); \end{cases} \end{align*} $$

where $T_{2n}(x)$ and ${U}_{2n}(x)$ denote the even degree Chebyshev polynomials of the first and second kinds, respectively.

Starting with the substitution of $\mathcal {F}_{jl}(y)$ from equation (3.10) into equation (3.8), the equation is then multiplied by ${q}_{j}^{(m)}(y)$ and integrated over the interval l. This process yields a set of linear algebraic equations that determine the unknown coefficients ${a}^{(n)}_{jl}$ for both configurations, as shown as

(3.11) $$ \begin{align} \sum_{n=0}^{N}\sum_{j=1}^{2}{a}^{(n)}_{jl} \mathcal{V}^{(pj)}_{mn}=\delta_{pl}\mathcal{W}^{(p)}_{m},\quad p,l=1,2;\quad m=0,1,\ldots,N, \end{align} $$

where $\delta _{pl}$ is the Kronecker delta function and for $p,j=1,2$ and $m,n=0,1,\ldots ,N,$

$$ \begin{align*} \mathcal{V}^{(pj)}_{mn}&=\int_{l} \bigg\{{q}_{p}^{(m)}(y)\bigg\{\int_{l}\mathcal{M}_{pj}(y,u){q}_{j}^{(n)}(u)\,du\bigg\}\, dy\bigg\},\quad \mathcal{W}^{(p)}_{m}= \int_{\Omega_p}{q}_{p}^{(m)}(y)\mathcal{P}_{p}(y)\,dy. \end{align*} $$

Using the expression of $\mathcal {M}_{pj}(y,u)~(p,j=1,2)$ as in equation (3.11), the expressions for $\mathcal {V}^{(pj)}_{mn}$ and $\mathcal {W}^{(p)}_{m}~(p,j=1,2; m,n=0,1,\ldots ,N)$ are as follows.

For the floating structure:

$$ \begin{align*} &{V}^{(pp)}_{mn}=\nonumber\\&-\frac{\coth\lbrace\nu(w_1+w_2)\rbrace}{\nu(H-d)} \bigg[\bigg\lbrace\delta_{m0}+\frac{2{i}(-1)^m(h_p-d)k_0G_p}{\pi(2m+1)H}\int_{0}^{1}e^{K(h_p-d)u}(1-u^2)^{{1}/{2}}U_{2m}(u)\,du\bigg\rbrace\nonumber\\&\times \bigg\lbrace\delta_{n0}+\frac{2{i}(-1)^n(h_p-d)k_0G_p}{\pi(2n+1)H} \int_{0}^{1}e^{K(h_p-d)u}(1-u^2)^{{1}/{2}}U_{2n}(u)\,du\bigg\rbrace\bigg]\nonumber\\& -\sum\limits_{r=1}^{\infty}\frac{4k_r}{s_r(2k_rH+\sin2k_rH)} \bigg[\bigg\lbrace J_{2m}\{k_r(H-h_p)\}-\frac{{i}k_{0}G_p}{k_rH}\cos k_r(H-d)J_{2m+1}\{k_r(h_p-d)\}\bigg\rbrace\nonumber\\&\times\bigg\lbrace J_{2n}\{k_r(H-h_p)\}-\frac{{i}k_{0}G_p}{k_rH}\cos k_r(H-d)J_{2n+1}\{k_r(h_p{\kern-1pt}-{\kern-1pt}d)\}\bigg\rbrace\bigg]-{\kern-1pt}\sum\limits_{r=1}^{\infty}\frac{2\coth\lbrace\xi_r(w_1+w_2)\rbrace}{\xi_r(H-d)}\nonumber\\&\times\bigg[\bigg\{\frac{2(-1)^m}{\pi}\int_{0}^{1} \frac{T_{2m}(u)}{(1-u^2)^{{1}/{2}}}\cos\bigg\lbrace r\pi \bigg(1-\frac{H-h_p}{H-d}u\bigg)\bigg\rbrace du +(-1)^p \frac{2{i}(-1)^m(h_p-d)k_{0}G_p}{\pi(2m+1)H} \nonumber\\&\times\bigg(\cos\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg)\int_{0}^{1}(1-u^2)^{{1}/{2}}U_{2m}(u)\cos\bigg(r\pi u-\tan^{-1} \bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg)\,du\nonumber\\&-\sin^2\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg) \int_{0}^{1}e^{K(h_p-d)u}(1-u^2)^{{1}/{2}}U_{2m}(u)\,du\bigg) \bigg\}\nonumber\\&\times\bigg\{\frac{2(-1)^n}{\pi}\int_{0}^{1} \frac{T_{2n}(u)}{(1-u^2)^{{1}/{2}}}\cos\bigg\lbrace r\pi \bigg(1-\frac{H-h_p}{H-d}u\bigg)\bigg\rbrace du +(-1)^p \frac{2{i}(-1)^n(h_p-d)k_{0}G_p}{\pi(2n+1)H}\nonumber\\&\times\bigg(\cos\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg)\int_{0}^{1}(1-u^2)^{{1}/{2}}U_{2n}(u)\cos\bigg(r\pi u-\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg)\,du\nonumber\\&-\sin^2\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg)\int_{0}^{1}e^{K(h_p-d)u}(1-u^2)^{{1}/{2}}U_{2n}(u)\,du\bigg) \bigg\}\bigg], \end{align*} $$

$$ \begin{align*} &{V}^{(pj)}_{mn}\nonumber\\[-8pt]&=\frac{\operatorname{\mathrm{csch}}\lbrace\nu(w_1+w_2)\rbrace}{\nu(H-d)}\bigg[\bigg\lbrace\delta_{m0}+\frac{2{i}(-1)^m(h_p-d)k_0G_p}{\pi(2m+1)H}\int_{0}^{1}e^{K(h_p-d)u}(1-u^2)^{{1}/{2}}U_{2m}(u)\,du\bigg\rbrace\nonumber\\&\quad\times \bigg\lbrace\delta_{n0}+\frac{2{i}(-1)^n(h_j-d)k_0G_j}{\pi(2n+1)H}\int_{0}^{1}e^{K(h_j-d)u}(1-u^2)^{{1}/{2}}U_{2n}(u)\,du\bigg\rbrace\bigg]\nonumber\\&\quad+ \sum\limits_{r=1}^{\infty}\frac{2\operatorname{\mathrm{csch}}\lbrace\xi_r(w_1+w_2)\rbrace}{\xi_r(H-d)}\nonumber\\&\quad\times\bigg[\bigg\{\frac{2(-1)^m}{\pi} \int_{0}^{1}\frac{T_{2m}(u)}{(1-u^2)^{{1}/{2}}}\cos\bigg\lbrace r\pi \bigg(1-\frac{H-h_p}{H-d}u\bigg)\bigg\rbrace du - \frac{2{i}(-1)^m(h_p-d)k_{0}G_p}{\pi(2m+1)H}\nonumber\\&\quad\times\bigg(\cos\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg) \int_{0}^{1}(1-u^2)^{{1}/{2}}U_{2m}(u)\cos\bigg(r\pi u-\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg)\,du\nonumber\\&\quad- \sin^2\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg) \int_{0}^{1}e^{K(h_p-d)u}(1-u^2)^{{1}/{2}}U_{2m}(u)\,du\bigg)\bigg\}\nonumber\\&\quad\times\bigg\{\frac{2(-1)^n}{\pi}\int_{0}^{1}\frac{T_{2n}(u)}{(1-u^2)^{{1}/{2}}}\cos\bigg\lbrace r\pi \bigg(1-\frac{H-h_j}{H-d}u\bigg)\bigg\rbrace \,du - \frac{2{i}(-1)^n(h_j-d)k_{0}G_j}{\pi(2n+1)H}\nonumber\\&\quad\times\bigg(\cos\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg) \int_{0}^{1}(1-u^2)^{{1}/{2}}U_{2n}(u)\cos\bigg(r\pi u-\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg)\,du\nonumber\\&\quad-\sin^2\bigg(\tan^{-1}\bigg(\frac{K(H-d)}{r\pi}\bigg)\bigg) \int_{0}^{1}e^{K(h_j-d)u}(1-u^2)^{{1}/{2}}U_{2n}(u)\,du\bigg)\bigg\}\bigg], \end{align*} $$

$$ \begin{align*} \mathcal{W}^{(p)}_{m}= (-1)^m\frac{I_{2m}\{k_0(H-h_p)\}}{\cosh k_0H}; \end{align*} $$

and for the bottom-fixed structure:

$$ \begin{align*} &{V}^{(pp)}_{mn}\nonumber\\& = \frac{4(-1)^{m+n}\alpha_{0}\cot\{\sqrt{{\alpha}_{0}^{2}-\nu^2} (w_1+w_2)\}}{\sqrt{{\alpha}_{0}^{2}-\nu^2}(2\alpha_{0}d+\sinh2\alpha_{0}d)}\bigg\{\kern-2pt\cosh\alpha_{0}d I_{2m}(\alpha_0h_p)-\frac{{i}k_0G_p}{\alpha_0d}I_{2m+1}\{\alpha_{0}(d-h_p)\}\bigg\}\nonumber\\& \quad \times \bigg\{\kern-2pt\cosh\alpha_{0}d I_{2n}(\alpha_0h_p)-\frac{{i}k_0G_p}{\alpha_0d}I_{2n+1}\{\alpha_{0}(d-h_p)\}\bigg\}- \sum\limits_{r=1}^{\infty}\frac{4k_r}{s_r(2k_rH+\sin2k_rH)}\nonumber\\& \quad \times \bigg\{\kern-2pt\cos k_{r}H J_{2m}(k_rh_p)-\frac{{i}k_{0}G_p}{k_rd}\cos k_r(H-d)J_{2m+1}\{k_{r}(d-h_p)\}\bigg\}\nonumber\\& \quad\times \bigg\{\kern-2pt\cos k_{r}H J_{2n}(k_rh_p)-\frac{{i}k_{0}G_p}{k_rd}\cos k_r(H-d)J_{2n+1}\{k_{r}(d-h_p)\}\bigg\}\nonumber\\& \quad - \sum\limits_{r=1}^{\infty}\frac{4\alpha_r\coth\{\sqrt{{\alpha}_{r}^{2}+\nu^2}(w_1+w_2)\}}{\sqrt{{\alpha}_{r}^{2}+\nu^2}(2\alpha_rd+\sin2\alpha_rd)}\bigg\{\kern-2.5pt\cos \alpha_{r}d J_{2m}(\alpha_rh_p)-\frac{{i}k_{0}G_p}{\alpha_rd}J_{2m+1}\{\alpha_{r}(d-h_p)\}\bigg\}\nonumber\\& \quad \times \bigg\{\kern-2pt\cos \alpha_{r}d J_{2n}(\alpha_rh_p)-\frac{{i}k_{0}G_p}{\alpha_rd}J_{2n+1}\{\alpha_{r}(d-h_p)\}\bigg\}, \end{align*} $$

$$ \begin{align*} {V}^{(pj)}_{mn}&= -\frac{4(-1)^{m+n}\alpha_{0}\csc\{\sqrt{{\alpha}_{0}^{2}-\nu^2}(w_1+w_2)\}}{\sqrt{{\alpha}_{0}^{2}-\nu^2}(2\alpha_{0}d+\sinh2\alpha_{0}d)} \notag \\&\quad \times \bigg\{\kern-2pt\cosh\alpha_{0}d I_{2m}(\alpha_0h_p)-\frac{{i}k_0G_p}{\alpha_0}I_{2m+1}\{\alpha_{0}(d-h_p)\}\bigg\} \notag \\&\quad \times \bigg\{\kern-2pt\cosh\alpha_{0}d I_{2n}(\alpha_0h_j)-\frac{{i}k_0G_j}{\alpha_0}I_{2n+1}\{\alpha_{0}(d-h_j)\}\bigg\} \notag \\&\quad +\sum\limits_{r=1}^{\infty}\frac{4\alpha_r\operatorname{\mathrm{csch}}\{\sqrt{{\alpha}_{r}^{2}+\nu^2}(w_1+w_2)\}}{\sqrt{{\alpha}_{r}^{2}+\nu^2}(2\alpha_rd+\sin2\alpha_rd)} \notag \\&\quad \times \bigg\{\kern-2pt\cos \alpha_{r}d J_{2m}(\alpha_rh_p)-\frac{{i}k_{0}G_p}{\alpha_rd}J_{2m+1}\{\alpha_{r}(d-h_p)\}\bigg\} \notag\\&\quad \times\bigg\{\kern-2pt\cos \alpha_{r}d J_{2n}(\alpha_rh_j)-\frac{{i}k_{0}G_j}{\alpha_rd}J_{2n+1}\{\alpha_{r}(d-h_j)\}\bigg\} \quad (p\neq j), \end{align*} $$

$$ \begin{align*} \mathcal{W}^{(p)}_{m}= (-1)^m I_{2m}(k_0h_p), \end{align*} $$

where $I_n$ corresponds to the modified Bessel function of order n, and $J_n$ refers to the Bessel function of the first kind of order n.

Solving the system of equations for ${a}^{(n)}_{jl}$ in equation (3.11) enables the determination of the unknown functions $\mathcal {F}_{jl}(y)$ . Thus, it facilitates the calculation of $S_{jl}~(j,l=1,2)$ as shown as

(3.13) $$ \begin{align} S_{jl}=\sum_{n=0}^{N} {a}^{(n)}_{jl} \mathcal{Z}^{(j)}_{n}, \end{align} $$

where, for the floating structure:

$$ \begin{align*} \mathcal{Z}^{(j)}_{n}= (-1)^n\bigg[\frac{I_{2n}\{k_0(H-h_j)\}}{\cosh k_0H}-{i}k_{0}G_j\frac{\cosh k_0(H-d)}{\cosh k_0H}I_{2n+1}\{k_0(h_j-d)\}\bigg], \end{align*} $$

and for the bottom-fixed structure:

$$ \begin{align*} \mathcal{Z}^{(j)}_{n}= (-1)^n\bigg[I_{2n}(k_0h_j)-{i}\frac{G_j}{d}\frac{\cosh k_0(H-d)}{\cosh k_0H}I_{2n+1}\{k_0(d-h_j)\}\bigg]. \end{align*} $$

Here, $S_{jl}$ values $(j,l=1,2)$ are obtained from equation (3.13) and subsequently $B_j$ terms $(j=1,2)$ are determined from equation (3.9). Here, $B_1$ and $B_2$ are connected to $V_1$ and $V_2$ through the relation equation (3.7). Additionally, $V_1$ and $V_2$ are then used to evaluate the reflection and transmission coefficients, denoted by R and T, respectively.

4 Convergence analysis

To examine the convergence of our models using the Galerkin approximation, we present the values of $|R|$ for varying numbers of terms (N) in the Galerkin approximation, as shown in Table 3. For this analysis, we select the parameters for floating $\boldsymbol {\sqcap }$ -shaped breakwater as ${w_1}/{H}$ = 0.5, ${w_2}/{H}$ = 1.0, ${h_1}/{H}$ = 0.65, ${h_2}/{H}$ = 0.45 and for bottom-fixed inverted $\boldsymbol {\sqcap }$ -shaped breakwater as ${w_1}/{H}$ = 0.5, ${w_2}/{H}$ = 1.0, ${h_1}/{H}$ = 0.35, ${h_2}/{H}$ = 0.55. Also, the porous effect parameter is taken as $G_1=G_2$ = 0.75 for both the breakwaters. It is evident from Table 3 that accuracy up to five figures (that is, an error estimate of $10^{-6}$ ) has been achieved with $N=3$ (that is, four terms in the Galerkin approximation), demonstrating the accuracy of the present numerical scheme.

Table 3

The reflection coefficients for (a) floating $\boldsymbol {\sqcap }$ -shaped and (b) bottom-fixed inverted $\boldsymbol {\sqcap }$ -shaped breakwater.

The convergence of the infinite series sum (arising in equation (3.6)) is verified in Table 4 by considering different truncation size r of that series. The parametric values are taken to be the same as in Table 3, with $\theta =75^\circ $ for both models, accordingly. Almost five-figure accuracy has been obtained by taking $r=25$ , which ensures the convergence of the infinite series. Therefore, four terms in the Galerkin approximation and $r=25$ in the truncated infinite series are considered for further parametric studies.

Table 4

The reflection coefficients for (a) floating $\boldsymbol {\sqcap }$ -shaped and (b) bottom-fixed inverted $\boldsymbol {\sqcap }$ -shaped breakwater.

5 Verification of numerical models

To ensure the accuracy of two models, various well-known results are reproduced in this section. Hereafter, in all graphical depictions, we take the complex porous effect parameters of front and rear plates to be equal, that is, ( $G_1$ = $G_2$ =G), unless otherwise stated.

For  $\boldsymbol {\sqcap }$ -shaped floating breakwater.

1. (i) The estimates of the reflection coefficient of the present model are compared with the graphical depiction by Bai [Reference Bai1, Figure 10] (rectangular floating structure). To do this, we first eliminate the porous side plate by taking ${h_1}/{H}={h_2}/{H}=0.5$ so that the $\boldsymbol {\sqcap }$ -shaped floating model is transformed into a rectangular floating structure of draft ${d}/{H}=0.5$ . For such an approximation, the other parameters are taken as ${w_1}/{H}={w_2}/{H}=0.5$ and $G=0$ . As a result, Figure 2(a) follows fairly well the estimated curves described by Bai [Reference Bai1, Figure 10].

   Figure 2

   Verification of $\boldsymbol {\sqcap }$ -shaped floating breakwater with existing results of the known literature.

   

2. (ii) Another verification of the present numerical results for the transmission coefficient has been conducted by comparing them with the results of Gesraha [Reference Gesraha14] ( $\boldsymbol {\sqcap }$ -shaped floating breakwater with rigid thin plates). To do this, we neglect the porosity of thin plates by setting G=0, effectively converting them into rigid plates. As a result, the present structure is transformed into a $\boldsymbol {\sqcap }$ -shaped floating breakwater with rigid thin plates. For such an approximation, we are taking ${h_1}/{H}={h_2}/{H}=0.3999, {d}/{H}=0.1176$ , ${w_1}/{H}={w_2}/{H}=0.3765$ and $\theta =0^\circ $ . As observed in Figure 2(b), our estimates of $|T|$ closely coincide with those of Gesraha [Reference Gesraha14, Figure 13].

For inverted  $\boldsymbol {\sqcap }$ -shaped bottom-fixed breakwater.

1. (i) To recover the result obtained by Ouyang et al. [Reference Ouyang, Chen and Tsai31, Figure 4(a)] (a rectangular bottom obstacle) for the case of normal incidence, we approximate the bottom-fixed inverted $\boldsymbol {\sqcap }$ -shaped breakwater into a bottom obstacle of height ${d}/{H}=0.5$ . For such an approximation, we remove the porous plates by taking ${h_1}/{H}={h_2}/{H}=0.5$ and $\textit {G}$ =0. The solid line shown in Figure 3(a) is plotted by taking the other parameters as ${w_1}/{H}={w_2}/{H}=0.25$ . Figure 3(a) depicts that the present data (solid line) coincide well with those of Ouyang et al. [Reference Ouyang, Chen and Tsai31] (represented by dots) and thereby establishes a very good agreement between our present results and the results of Ouyang et al. [Reference Ouyang, Chen and Tsai31].

   Figure 3

   Verification of inverted $\boldsymbol {\sqcap }$ -shaped bottom-fixed breakwater with previously published literature.

   

2. (ii) For the purpose of comparison with the numerical results of Mei and Black [Reference Mei and Black27, Figure 2] for the case of a submerged obstacle, we choose the parameters as ${h_1}/{H}={h_2}/{H}={d}/{H}=0.5$ , ${w_1}/{H}=0$ , $\textit {G}$ =0 and $\theta =0^\circ $ . So that the bottom fixed inverted $\boldsymbol {\sqcap }$ -shaped breakwater transformed into a submerged obstacle as studied by Mei and Black [Reference Mei and Black27].

   Our computed results (represented by a solid line) are in excellent agreement with the results obtained by Mei and Black [Reference Mei and Black27] (represented by dots). The coincidence of Figure 3(b) gives another validation of the results determined from the present method.

3. (iii) Again, the present result of reflection coefficient is compared with the published results of Das et al. [Reference Das, Dolai and Mandal10]. We approximate a bottom-fixed inverted $\boldsymbol {\sqcap }$ -shaped breakwater into a pair of thin rigid bottom-standing barriers by taking ${d}/{H}=0.999$ , so that the height of the rectangular structure is diminished. Also, we neglect the effect of porosity to convert the barriers into rigid by taking $\textit {G}$ =0. The other parameters are taken as ${h_1}/{H}={h_2}/{H}=0.2$ , ${w_1}/{H}={w_2}/{H}=0.3$ . Figure 3(c) clearly indicates that our results almost coincide with those of Das et al. [Reference Das, Dolai and Mandal10, Figure 6].

4. (iv) Lastly, we verified our model with the results from Lee and Chwang [Reference Lee and Chwang22]. To compare the results of reflection and transmission coefficients with the analytical results obtained by Lee and Chwang [Reference Lee and Chwang22], for such an approximation, we choose ${h_1}/{H}=0.999, {h_2}/{H}=0.25, {d}/{H}=0.999$ , so that the bottom-fixed inverted $\boldsymbol {\sqcap }$ -shaped structure is converted into the case of a bottom-standing porous barrier. The other parameters are taken as ${w_1}/{H}=0, {w_2}/{H}=0.001$ and $\theta =0^\circ $ . The solid line in Figure 3(d) almost coincides with the result of Lee and Chwang [Reference Lee and Chwang22]. Thus, it is evident from Figure 3(d) that present results are in good agreement with the results of Lee and Chwang [Reference Lee and Chwang22].

All the above-mentioned verifications clearly indicate that the present method is accurate for further numerical estimations.