1 Introduction
The interaction of surface gravity waves with submerged breakwaters has been a subject of significant research and interest among ocean engineers and coastal scientists. Submerged breakwaters, designed to mitigate the impact of waves on coastal regions, play a crucial role in wave energy dissipation and sediment redistribution. Their effectiveness hinges on several factors including their geometrical configuration, structural integrity, material composition and placement relative to the shoreline. Traditional rigid breakwaters often fail under substantial wave loads, leading to collapse and ineffective protection. To address these challenges, hydroelastic analysis has gained considerable attention due to its critical role in the design of various engineering structures across multiple fields, including coastal, marine, offshore, aerospace and civil engineering. The study of the interaction between fluid and structures, in such a way that the hydrodynamic force of the fluid and the elastic force of the solid body are considered, is referred to as hydroelasticity of marine structures [Reference Heller and Abramson13, Reference Wu47]. In recent years, there has been significant growth in the study of hydrodynamic performance of flexible/elastic porous structures. The flexible structures provide an additional feature of wave attenuation through structural deformation compared with a rigid structure. Further, flexible structures are light in weight, cost-effective, reusable and environmentally friendly compared with their rigid counterparts.
Over the past few decades, there has been substantial research interest in the problem of water wave scattering by fixed, rigid obstacles of various geometrical shapes. Dean [Reference Dean9] was among the first to tackle the issue, solving the problem of time-harmonic waves normally incident on a vertical barrier extending downward from a point below the free surface using complex variable methods. Shortly after, Ursell [Reference Ursell45] derived the solution for the complementary problem of a vertical barrier extending upwards through the free surface from a point below the free surface. Subsequently, several researchers developed more complex explicit solutions for vertical barrier problems in both infinitely deep and finite-depth water. These researchers included Porter [Reference Porter31], Smith [Reference Smith41], Williams [Reference Williams46], Losada et al. [Reference Losada, Losada and Roldan22], Porter and Evans [Reference Porter and Evans32], Chakrabarti et al. [Reference Chakrabarti, Banerjea, Mandal and Sahoo4] and others who focused on single barrier scattering problems. Significant progress in the study of wave interaction with two, three or more thin identical or nonidentical vertical barriers was made by Levine and Rodemich [Reference Levine and Rodemich19], Jarvis [Reference Jarvis16], Evans and Morris [Reference Evans and Morris10], Newman [Reference Newman29], Morris [Reference Morris28], McIver [Reference McIver26], Isaacson et al. [Reference Isaacson, Baldwin, Premasiri and Yang14], De et al. [Reference De, Mandal and Chakrabarti8], Roy et al. [Reference Roy, De and Mandal33] and others.
The interaction of water waves with vertical elastic bodies has not been extensively studied within the framework of linear theory, resulting in limited mathematical techniques available in the literature. Meylan [Reference Meylan27] investigated the scattering of water waves by a surface-piercing thin vertical elastic plate using Green’s function method. The time-dependent version of this problem was solved by Peter and Meylan [Reference Peter and Meylan30] using a generalized eigenfunction expansion. Chakraborty and Mandal [Reference Chakraborty and Mandal5] employed the method described by Meylan [Reference Meylan27] to study the scattering of water waves by a thin vertical elastic plate submerged in both deep and finite-depth water. Additionally, Chakraborty et al. [Reference Chakraborty, Mondal and Gayen6] explored the behaviour of elastic plates with both clamped and free ends, floating in deep water or water of uniform finite depth, using the hypersingular integral equation method.
To enhance the environmental friendliness of vertical breakwaters, rigid vertical structures can be replaced with porous barriers placed at various positions in the ocean. Sollitt and Cross [Reference Sollitt and Cross43] initially studied the effect of barrier permeability on incident surface waves and modified Darcy’s law, which many researchers later used in the field of porous breakwaters. Macaskill [Reference Macaskill23] solved the problem of normally incident wave scattering involving a permeable thin barrier in water of infinite depth by reducing it to integro-differential equations using Green’s integral theorem. Chwang [Reference Chwang7] developed a porous wavemaker theory to investigate the propagation of surface waves by the horizontal oscillation of a vertical porous plate. Yu [Reference Yu48] applied an approximate method to solve the problem of diffraction of surface water waves by a semi-infinite porous breakwater using a boundary condition based on Sollitt and Cross [Reference Sollitt and Cross43]. Isaacson et al. [Reference Isaacson, Premasiri and Yang15] used a matched eigenfunction expansion method to develop numerical solutions for the interaction of normally incident waves with a surface-piercing porous barrier and verified their results experimentally. Karmakar and Soares [Reference Karmakar and Soares17] investigated wave scattering by multiple bottom-standing flexible porous barriers using both the direct method and the wide-spacing approximation. Gayen and Mondal [Reference Gayen and Mondal11] explored the scattering of small-amplitude surface water waves by two symmetric vertical thin porous plates using the collocation method. Li et al. [Reference Li, Liu and Li20] developed an accurate solution to water wave scattering by porous barriers of different structures, including surface-piercing and bottom-standing barriers, using a multi-term Galerkin approximation. Manam and Sivanesan [Reference Manam and Sivanesan24] used an analytical approach to solve deepwater wave motion over surface-piercing or bottom-standing porous barriers. Behera and Ng [Reference Behera and Ng2] employed eigenfunction expansions to investigate oblique wave scattering by multiple bottom-standing flexible porous barriers near a rigid wall. Sasmal et al. [Reference Sasmal, Paul and De38] and Sasmal and De [Reference Sasmal and De34, Reference Sasmal and De35, Reference Sasmal and De37] studied the scattering of water waves by various configurations of multiple porous barriers extending to infinite depth in a series of works.
In many of the studies documented above which examine various barrier configurations, the barriers are generally considered to be either impermeable or permeable. Furthermore, hydroelasticity theories for structures like floating elastic plates of arbitrary shapes and disks have been widely studied and reviewed by researchers in recent years. For instance, Sturova [Reference Sturova44] investigated the linear hydroelastic problem of periodic surface pressures acting on plates of arbitrary shapes. Hassan et al. [Reference Hassan, Meylan and Peter12] studied the problem of water wave interaction with a submerged elastic plate of negligible thickness by the eigenfunction-matching method. Zheng et al. [Reference Zheng, Meylan, Fan, Greaves and Iglesias50] developed a semi-analytical model based on linear potential flow theory and an eigenfunction expansion method to study wave scattering by a floating porous elastic plate with an arbitrary shape. The interaction of water waves with submerged porous elastic disks was also investigated by Zheng et al. [Reference Zheng, Meylan, Greaves and Iglesias51]. Liu et al. [Reference Liu, Li and Fang21] developed new analytical solutions for oblique wave scattering by porous rubble-mound breakwaters and seawalls based on contour integrals without finding complex roots. Zheng et al. [Reference Zheng, Meylan, Zhu, Greaves and Iglesias52] further developed a theoretical model based on linear potential flow theory and an eigenfunction matching method to investigate wave scattering by multiple circular floating porous elastic plates. Smith et al. [Reference Smith, Peter, Abrahams and Meylan42] investigated the problem of water-wave scattering by a semi-infinite submerged thin elastic plate, whether porous or non-porous, using the Wiener–Hopf technique. Ashok et al. [Reference Ashok, Gunasundari and Manam1] presented explicit solutions to the problem of scattering of normally incident waves by a submerged flexible porous thin plate. Surface gravity wave interaction with an articulated flexible submerged plate in the presence of lateral compressive force was investigated by Boral et al. [Reference Boral, Sahoo and Meylan3]. The problem of water wave scattering by a vertical porous elastic plate completely submerged in water of infinite depth in the presence of surface tension at the free surface was investigated by Singh and Gayen [Reference Singh and Gayen39]. Singh and Kaligatla [Reference Singh and Kaligatla40] studied the combined effect of refraction-diffraction due to bottom topography on water wave scattering by a vertical flexible-porous structure by employing the Galerkin approximation method. Recently, Sasmal and De [Reference Sasmal and De36] investigated the scattering of water waves with multiple flexible porous barriers by employing the Galerkin approximation method.
This paper investigates the feasibility of integrating hydro elasticity into the analysis of surface gravity wave interactions with multiple flexible, porous plates of arbitrary lengths in infinitely deep water. The potential function is simplified into linear integral equations using the Havelock inversion formula, followed by a matching process. These equations are solved using a single-term Galerkin approximation. Numerical estimates for the reflection coefficient, energy dissipation and wave force coefficients are provided for various parameter values. The method’s convergence and accuracy are validated by comparing our results with existing literature. The findings indicate that the flexural rigidity of the porous plates significantly affects wave reflection, energy dissipation and wave forces. Moreover, the wave load decreases substantially as the number of plates increases.
2 Mathematical model
Here, we consider N number of porous elastic plates. Cartesian coordinate axes are chosen with the x-axis directed along the mean water surface, taking the y-axis vertically downwards. It is assumed that N number of unequal poroelastic plates are placed vertically, as shown in Figure 1. The jth vertical poroelastic plate is situated at
$x=a_{j}$
,
$y\in L_{j}\equiv (h_{2j-1},h_{2j}),~j=1,2,3,\ldots ,N$
. For the convenience of study, the fluid region is divided into
$N+1$
subregions, namely,
$$ \begin{align*} &\kern35pt R_{1}\equiv(-\infty< x<a_{1}, 0<y<\infty), \\&R_{j}\equiv(a_{j-1}< x<a_{j}, 0<y<\infty),\quad j=2,3,\ldots, N, \\&\kern35pt R_{N+1}\equiv(a_{N}< x<\infty, 0<y<\infty). \end{align*} $$
We use
$N+1$
velocity potentials
$\phi _{j}, ~j=1,2,3,\ldots ,N+1$
for the
$N+1$
subregions and
$\overline {L}_{j}\equiv (0,\infty )-L_{j},~j=1,2,3,\ldots ,N$
to denote the fluid area of gaps of plates. It is assumed that the fluid is inviscid, incompressible, and the motion irrotational and simple harmonic in time with angular frequency
$\omega $
. Assuming linear theory, the time-harmonic motions of the fluid are described by the velocity potential
$\Phi _{j}(x,y,z,t)=Re\{\phi _{j}(x,y) e^{i\nu z-i\omega t}\}$
, where
$Re$
denotes the real part and
$\nu $
is defined by
$\nu =K\sin \theta $
, where
$K={\omega ^{2}}/{g}$
, g being the acceleration due to gravity, and
$\theta $
is the angle of the incident wave to the x-axis. The spatial velocity potential
$\phi _{j}(x,y)$
for
$j=1,2,3,\ldots ,N+1$
in each sub-region satisfies the governing Helmholtz equations given by
$$ \begin{align} (\nabla^{2}-\nu^{2})\phi_{j}=0, \quad -\infty<x<\infty, \quad 0<y<\infty, \end{align} $$
where
$\nabla ^{2}$
is the horizontal Laplacian operator and the subscript corresponds to the total wave field in each subregion.
Schematic diagram of the physical domain demonstrating oblique scattering of surface gravity waves by an array of vertical flexible plates of arbitrary lengths.
The linearized kinematic and dynamic boundary conditions on the free surface can be merged into
$$ \begin{align} \bigg( \frac{\partial }{\partial y}+ K\bigg) \phi_{j}=0 \quad \text{on } y=0. \end{align} $$
The no-flux boundary condition on the seabed is
$$ \begin{align} |\nabla \phi_{j}|\rightarrow 0\quad \text{as } y\rightarrow \infty, \end{align} $$
where
$\nabla =(\partial /\partial x,\partial /\partial y)$
.
The plate is deflected horizontally with the displacement
$\eta _{j}(y,t)=Re[v_{j}(y)e^{(-1)^{j}i\omega t}]$
,
$j=1,2,3\ldots , N$
, where
$v_{j}(y),~j=1,2,3,\ldots , N $
is the complex quantity whose absolute value is assumed to be small. The conditions on the porous plates are given by (see the paper by Ashok et al. [Reference Ashok, Gunasundari and Manam1])
$$ \begin{align} \frac{\partial \phi_{j}}{\partial x}=\frac{\partial \phi_{j+1}}{\partial x}=-iKG_{j}(\phi_{j+1}-\phi_{j})-i\omega v_{j}(y),~~ x=x_{j},~~y\in L_{j},~ j=1,2,3,\ldots,N, \end{align} $$
where
$G_{j}=G_{j}^{r}+iG_{j}^{i},~j=1,2,3,\ldots ,N$
are the dimensionless porous effect parameters of poroelastic plates. The complex porous effect parameters can be written as [Reference Yu and Chwang49]
$$ \begin{align*}G_{j}=\dfrac{\varepsilon_{j}(f_{j}+iS_{j})}{K\alpha_{j}(f_{j}^{2}+S_{j}^{2})},\end{align*} $$
in which
$\varepsilon _{j}$
are the porosity constant,
$f_{j}$
are the resistance force coefficient,
$S_{j}$
are the inertial force coefficient and
$\alpha _{j}$
are the thickness of the porous medium. The real part
$G_{j}^{r}$
represents the resistance effect of the porous material against the flow, while the imaginary part
$G_{j}^{i}$
represents the inertia effect of the fluid inside the porous material.
The displacement amplitude
$v_{j}(y), ~j=1,2,3,\ldots ,N $
satisfies (see [Reference Ashok, Gunasundari and Manam1])
$$ \begin{align} \frac{\partial^{4} v_{j}}{\partial y^{4}}-\lambda^{2}v_{j}=-\frac{i\omega \rho}{EI}(\phi_{j+1}-\phi_{j}),\quad y\in L_{j},\ j=1,2,3,\ldots,N, \end{align} $$
where
$\lambda =\omega \sqrt {m_{b}/EI}$
,
$EI$
is uniform flexural rigidity of porous barrier,
$m_{b}$
is uniform mass per unit length and
$\rho $
is the density of water.
In the vicinity of sharp submerged edges, the velocity potential behaves as [Reference Karp and Karal18, Reference Mandal and Chakrabarti25]
$$ \begin{align} |\nabla \phi_{j}|=O(r^{-1/2})\quad \text{as } r\rightarrow 0, \end{align} $$
where
$r=\sqrt {(x-a_{j})^{2}+(y-h)^{2}}$
, where
$h=h_{2j-1}, h_{2j},~j=1,2,3,\ldots ,N$
.
The conditions at infinity are given by
$$ \begin{align} \begin{array}{ll}\phi_{1}(x,y)=T \phi^{inc}(x,y)\quad \text{as }x\rightarrow -\infty , \\ \phi_{N+1}(x,y)=\phi^{inc}(x,y)+R \phi^{inc}(-x,y) \quad \text{as }x\rightarrow \infty , \end{array} \end{align} $$
where R and T are the unknown complex reflection and transmission coefficients to be determined, and
$\phi ^{inc}(x,y)=e^{-Ky-i\mu (x-a_{N})}$
,
$\mu =K\cos \theta $
.
At the common boundary of the region, velocity potential
$\phi _{j}$
and gradient of velocity potential
$\partial \phi _{j}/\partial x$
are continuous across the gap. Let us consider
$$ \begin{align} f_{j}(y)=\frac{\partial \phi_{j}(a_{j},y)}{\partial x}=\frac{\partial \phi_{j+1}(a_{j},y)}{\partial x}, \quad \xi_{j}(y)=\phi_{j+1}(a_{j},y)-\phi_{j}(a_{j},y) \end{align} $$
$\text {for}~j=1,2,3,\ldots ,N.$
Eliminating
$v_{j}(y), ~j=1,2,3,\ldots ,N $
, from (2.4) and (2.5) along with (2.8),
$$ \begin{align} \bigg( \frac{\partial^{4}}{\partial y^{4}}-\lambda^{2}\bigg)\frac{\partial \phi_{j}}{\partial x} &=\bigg( \frac{\partial^{4}}{\partial y^{4}}-\lambda^{2}\bigg)\frac{\partial \phi_{j+1}}{\partial x} \notag \\ &=-iKG_{j}\frac{\partial^{4}\xi_{j}(y)}{\partial y^{4}}+ \lambda^{2}\bigg( iKG_{j}-\frac{\rho}{m_{b}}\bigg)\,\xi_{j}(y) ,~~y\in L_{j},~ j=1,2,3,\ldots,N. \end{align} $$
3 Solution procedure
Applying the method of separation of variables, along with (2.1)–(2.7), Havelock’s expansion of velocity potentials
$\phi _{j}(x,y),~j=1,2,3,\ldots ,N$
for each of the subregions is given by
$$ \begin{align} &\phi_{1}(x,y)=T\phi^{inc}(x,y)+ \int_{0}^{\infty}A(k)e^{\zeta(x-a_{1})} (k\cos ky-K\sin ky)\,dk\quad \text{in } R_{1}, \end{align} $$
$$ \begin{align} &\phi_{j}(x,y) =e^{-Ky}(c_{j} e^{i\mu x}+d_{j} e^{-i\mu x}) \nonumber \\ &\qquad +\int_{0}^{\infty}\{M_{j}(k) e^{-\zeta x}+N_{j}(k) e^{\zeta x}\}(k\cos ky-K\sin ky)\,dk\quad \text{in } R_{j},~j=2,3,\ldots,N, \end{align} $$
$$ \begin{align} \phi_{N+1}(x,y)&=\phi^{inc}(x,y)+R \phi^{inc}(-x,y) \nonumber\\ &\quad+\int_{0}^{\infty}B(k) e^{-\zeta(x-a_{N})}(k\cos ky-K\sin ky)\,dk \quad \text{in } R_{N+1} , \end{align} $$
where
$\zeta =\sqrt {k^{2}+\nu ^{2}}$
, and
$c_{j},~d_{j}$
are arbitrary constants, and
$A(k),~B(k),~M_{j}(k),~N_{j}(k), j=2,3,\ldots ,N$
are unknown functions and, in the mathematical analysis below in which they appear, are convergent.
Now, using the expression of velocity potentials
$\phi _{j}$
from (3.1)–(3.3) in (2.8) and applying Havelock’s inversion formula,
$$ \begin{align} &-Te^{-i\mu (a_{1}-a_{N})}=c_{2} e^{i \mu a_{1}}-d_{2} e^{-i\mu a_{1}},\quad -1+Re^{2i\mu a_{N}}=c_{N} e^{i \mu a_{N}}-d_{N} e^{-i\mu a_{N}}, \end{align} $$
$$ \begin{align} & c_{j}e^{i\mu a_{j}}-d_{j}e^{-i\mu a_{j}} =c_{j+1}e^{i\mu a_{j}}-d_{j+1}e^{-i\mu a_{j}} ,\quad j=2,3,\ldots ,N-1, \end{align} $$
$$ \begin{align} & A(k)=N_{2}(k) e^{\zeta a_{1}}-M_{2}(k) e^{-\zeta a_{1}}, \quad B(k)=N_{N}(k) e^{\zeta a_{N}}-M_{N}(k) e^{-\zeta a_{N}}, \end{align} $$
$$ \begin{align} &N_{j}(k) e^{\zeta a_{j}}-M_{j}(k) e^{-\zeta a_{j}}=N_{j+1}(k) e^{\zeta a_{j}}-M_{j+1}(k) e^{-\zeta a_{j}},\quad j=2,3,\ldots ,N-1, \end{align} $$
$$ \begin{align} &-Te^{-i\mu (a_{1}-a_{N})}+c_{2} e^{i \mu a_{1}}+d_{2} e^{-i\mu a_{1}}= 2K\int_{L_{1}}e^{-Ky}\xi_{1}(y)\,dy, \end{align} $$
$$ \begin{align} &(c_{j+1}-c_{j}) e^{i \mu a_{j}}+(d_{j+1}-d_{j}) e^{-i\mu a_{j}}= 2K\int_{L_{j}}e^{-Ky}\xi_{j}(y)\,dy,\quad j=2,3,\ldots,N-1, \end{align} $$
$$ \begin{align} &1+Re^{2i\mu a_{N}}-c_{N} e^{i \mu a_{N}}-d_{N} e^{-i\mu a_{N}}= 2K\int_{L_{N}}e^{-Ky}\xi_{N}(y)\, dy, \end{align} $$
$$ \begin{align} &M_{2}(k) e^{-\zeta a_{1}}+N_{2}(k) e^{\zeta a_{1}}-A(k)=\frac{2}{\pi(k^{2}+K^{2})}\int_{L_{1}}\xi_{1}(y)(k\cos ky-K\sin ky)\, dy, \end{align} $$
$$ \begin{align} &(M_{j+1}(k)-M_{j}(k)) e^{-\zeta a_{j}}+(N_{j+1}(k)-N_{j}(k)) e^{\zeta a_{j}} \notag \\ &\quad = \frac{2}{\pi(k^{2}+K^{2})}\int_{L_{j}}\xi_{j}(y)(k\cos ky-K\sin ky)\,dy,~j=2,3 \ldots,N-1, \end{align} $$
$$ \begin{align} &B(k)-M_{N}(k) e^{-\zeta a_{N}}-N_{N}(k) e^{\zeta a_{N}}=\frac{2}{\pi(k^{2}+K^{2})}\int_{L_{N}}\xi_{N}(y)(k\cos ky-K\sin ky)\,dy. \end{align} $$
Equations (3.8)–(3.10) can be reconstructed using the expressions from (3.4) and (3.5) as
$$ \begin{align} &2K\int_{L_{1}}\xi_{1}(y)e^{-Ky}\,dy= (i\cot(\mu(a_{1}-a_{2}))-1)Te^{-i\mu (a_{1}-a_{N})} \notag \\ &\quad +i\csc(\mu(a_{1}-a_{2}))(c_{2}e^{i\mu a_{2}}-d_{2}e^{-i\mu a_{2}}), \end{align} $$
$$ \begin{align} &2K\int_{L_{2}}\xi_{2}(y)e^{-Ky}\,dy\nonumber\\ &\quad = -i\csc(\mu(a_{1}-a_{2}))Te^{-i\mu (a_{1}-a_{N})}+i\sin(\mu(a_{3}-a_{1}))\csc(\mu (a_{3}-a_{2})) \notag \\ &\qquad \times\csc(\mu(a_{2}-a_{1})) (c_{2}e^{i\mu a_{2}}-d_{2}e^{-i\mu a_{2}})-i\csc(\mu(a_{3}-a_{2}))(c_{3}e^{i\mu a_{3}}-d_{3}e^{-i\mu a_{3}}), \end{align} $$
$$ \begin{align} &2K\int_{L_{j}}\xi_{j}(y)e^{-Ky}\,dy= -i\csc(\mu (a_{j}-a_{j-1}))(c_{j-1}e^{i\mu a_{j-1}}-d_{j-1}e^{-i\mu a_{j-1}}) \notag \\ &\quad+i\sin(\mu(a_{j+1}-a_{j-1}))\csc(\mu (a_{j}-a_{j-1}))\csc(\mu(a_{j+1}-a_{j}))(c_{j}e^{i\mu a_{j}}-d_{j}e^{-i\mu a_{j}}) \notag \\ &\quad -i\csc(\mu(a_{j+1}-a_{j}))(c_{j+1}e^{i\mu a_{j+1}}-d_{j+1}e^{-i\mu a_{j+1}}),~j=3,4,\ldots,N-2, \end{align} $$
$$ \begin{align} &2K\int_{L_{N-1}}\xi_{N-1}(y)e^{-Ky}\,dy= -i\csc(\mu (a_{N-1}-a_{N-2}))(c_{N-2}e^{i\mu a_{N-2}}-d_{N-2}e^{-i\mu a_{N-2}}) \notag \\ &\quad +i\sin(\mu(a_{N}-a_{N-2}))\csc(\mu (a_{N-1}-a_{N-2}))\csc(\mu(a_{N}-a_{N-1})) \notag \\ &\quad \times(c_{N-1}e^{i\mu a_{N-1}}-d_{N-1}e^{-i\mu a_{N-1}})-i\csc(\mu(a_{N}-a_{N-1}))(1-Re^{2i\mu a_{N}}), \end{align} $$
$$ \begin{align} & 2K\int_{L_{N}}\xi_{N}(y)e^{-Ky}\,dy =-i\csc(\mu(a_{N}-a_{N-1}))(c_{N-1}e^{i\mu a_{N-1}}-d_{N-1}e^{-i\mu a_{N-1}}) \nonumber \\ &\quad +1-i\cot(\mu(a_{N}-a_{N-1}))+(1+i\cot(\mu(a_{N}-a_{N-1})))Re^{2i\mu a_{N}}. \end{align} $$
Substituting the expressions of
$A(k),~B(k),~M_{j}(k),~N_{j}(k),~j=2,3,\ldots ,N$
from (3.11)–(3.13) after using the relation of (3.6) and (3.7) in (2.9),
$$ \begin{align} \sum_{m=1}^{N}\int_{L_{m}}\xi_{m}(t)\mathcal{H}_{nm}(y,t)dt+\mathcal{E}\xi_{n}(y)=S_{n}(y), \quad n=1,2,3,\ldots,N, \end{align} $$
where
$$ \begin{align*} & \kern50pt \mathcal{E}={\lambda^{2}( iKG_{n}-{\rho}/{m_{b}})}/({K^{4}-\lambda^{2}});\\& S_{1}(y)=-i\mu Te^{i\mu (a_{N}-a_{1})-Ky};\quad S_{N}(y)=-i\mu e^{-Ky} (1-Re^{2i\mu a_{N}});\\& \kern9pt S_{j}(y)=i\mu e^{-Ky}(c_{j}e^{i\mu a_{j}}-d_{j}e^{-i\mu a_{j}}),\quad j=2,3,\ldots,N-1, \end{align*} $$
and
$$ \begin{align} \left\{\begin{array}{llll}\displaystyle \mathcal{H}_{nn}(y,t)=-\frac{2iK^{6}G_{n}}{(K^{4}-\lambda^{2})}e^{-K(y+t)}+\frac{1}{\pi(K^{4}-\lambda^{2})} \\[.2cm] \quad \times \displaystyle\int_{0}^{\infty}\bigg( \frac{\zeta(k^{4}-\lambda^{2}) }{k^{2}+K^{2}}-\frac{2iKG_{n}k^{4}}{k^{2}+K^{2}}\bigg) (k\cos ky-K\sin ky)(k\cos kt-K\sin kt)\,dk,\\[.2cm] \displaystyle \mathcal{H}_{mn}(y,t)=\mathcal{H}_{nm}(y,t)= \displaystyle\frac{1}{\pi(K^{4}-\lambda^{2})} \\[.2cm] \quad \times\displaystyle\int_{0}^{\infty}\frac{\zeta(k^{4}-\lambda^{2})}{k^{2}+K^{2}} e^{\zeta(a_{m}-a_{n})}(k\cos ky-K\sin ky)(k\cos kt-K\sin kt)\,dk;\\ m,n=1,2,\ldots,N. \end{array}\right.\\[-24pt]\nonumber \end{align} $$
Let us consider potential difference function
$\xi _{m}(t)$
as
$$ \begin{align} \xi_{m}(t)&= -i\mu Te^{i\mu (a_{N}-a_{1})}X_{m1}(t) -i\mu (1-Re^{2i\mu a_{N}})X_{mN}(t) \notag \\[-4pt] &\quad +i\mu\sum_{j=2}^{N-1} (c_{j}e^{i\mu a_{j}}-d_{j}e^{-i\mu a_{j}})X_{mj}(t), m=1,2,3,\ldots,N. \end{align} $$
After using the expressions of
$g_{m}(t)$
from (3.21), the integral equations in (3.19) can be reduced as
$$ \begin{align} \sum_{m=1}^{N}\int_{L_{m}}X_{mj}(t)\mathcal{H}_{nm}(y,t)\,dt+\mathcal{E}X_{nj}(y)=\delta_{nj}e^{-Ky}, y\in L_{n}, n,\quad j=1,2,3,\ldots,N,\end{align} $$
where
$\delta _{nj}$
is the Kronecker delta symbol.
3.1 Solutions of integral equations using Galerkin’s approximation technique
The integral equations given by (3.22) have been solved using the single-term Galerkin technique. The unknown functions
$X_{ij}(y)$
are chosen in the following form:
$$ \begin{align} X_{mj}(t)=b_{mj}~\mathcal{F}_{m}(t), \quad m,j=1, 2, 3,\ldots,N, \end{align} $$
where
$\mathcal {F}_{j}(t)$
are suitably chosen basis functions and
$b_{mj}$
, unknown constants, are to be found.
To deduce the unknown constants
$b_{mj}$
appearing in (3.23), we substitute the expressions of
$X_{mj}(t)\, (m, j = 1, 2, 3,\ldots ,N)$
from (3.23) into the integral equations in (3.22), then multiply the mth integral equation in (3.22) by
$\mathcal {F}_{m}(t)$
and integrate over
$L_{m}$
. This yields the following linear systems of equations in
$b_{mj}\, (m, j = 1, 2, 3,\ldots ,N)$
:
$$ \begin{align} \sum_{m=1}^{N}b_{mj}\mathcal{C}_{nm}=\delta_{nj}\mathcal{D}_{n},~~ n,j=1 , 2, 3,\ldots,N, \end{align} $$
where
$$ \begin{align} \left\{\begin{array}{lll}\displaystyle \mathcal{C}_{nn}=\int_{L_{n}}\mathcal{F}_{n}(y)\bigg\lbrace \int_{L_{n}}\mathcal{F}_{n}(t)\mathcal{H}_{nn}(y,t)\,dt\bigg\rbrace \,dy+\mathcal{E}\int_{L_{n}}[\mathcal{F}_{n}(y)]^{2}\,dy,\\ \displaystyle \mathcal{C}_{nm}=\mathcal{C}_{mn}=\int_{L_{n}}\mathcal{F}_{n}(y)\bigg\lbrace \int_{L_{m}}\mathcal{F}_{m}(t)\mathcal{H}_{nm}(y,t)\,dt\bigg\rbrace \,dy,\quad n\neq m,\\ \displaystyle \mathcal{D}_{n}=\int_{L_{n}}\mathcal{F}_{n}(y)e^{-ky} \,dy, \quad n,m=1, 2, 3,\ldots,N. \end{array}\right. \end{align} $$
We choose the basis function
$\mathcal {F}_{m}(y), m=1, 2, 3,\ldots ,N$
as follows (see [Reference Singh and Kaligatla40]):
$$ \begin{align} \mathcal{F}_{m}(y)= e^{-Ky}\bigg[\int_{h_{2m-1}}^{y}\frac{(d_{m}^{2}-z^{2})e^{Kz}}{\sqrt{(z^{2}-h_{2m-1}^{2})(h_{2m}^{2}-z^{2})}}\,dz \bigg],\quad h_{2m-1}<y<h_{2m}, \end{align} $$
where
$$ \begin{align*}d_{m}^{2}=\dfrac{\int_{h_{2m-1}}^{h_{2m}}\frac{z^{2}e^{Kz}}{\sqrt{(z^{2}-h_{2m-1}^{2})(h_{2m}^{2}-z^{2})}}\,dz}{\int_{h_{2m-1}}^{h_{2m}}\frac{e^{Kz}}{\sqrt{(z^{2}-h_{2m-1}^{2})(h_{2m}^{2}-z^{2})}}\,dz}, \quad m=1, 2, 3,\ldots,N.\end{align*} $$
Substituting the expressions of
$\mathcal {F}_{m}(y)$
from (3.26) and
$\mathcal {H}_{mn}(y,t)$
from (3.20) into (3.25),
$$ \begin{align*} \mathcal{C}_{nn}&=\frac{1}{\pi( K^{4}-\lambda^{2}) }\int_{0}^{\infty}\bigg( \frac{\zeta(k^{4}-\lambda^{2}) }{k^{2}+K^{2}}-\frac{2iKG_{n}k^{4}}{k^{2}+K^{2}}\bigg)\bigg[\int_{h_{2n-1}}^{h_{2n}}\frac{(d_{n}^{2}-z^{2})\sin kz}{\sqrt{(z^{2}-h_{2n-1}^{2})(h_{2n}^{2}-z^{2})}}\,dz \bigg]^{2}\,dk \notag \\&\quad -\frac{i K^{4}G_{n}}{2(K^{4}-\lambda^{2})}\bigg[\int_{h_{2n-1}}^{h_{2n}}\frac{(d_{n}^{2}-z^{2})e^{-Kz}}{\sqrt{(z^{2}-h_{2n-1}^{2})(h_{2n}^{2}-z^{2})}}\,dz \bigg]^{2}+\frac{\lambda^{2}( iKG_{n}-\frac{\rho}{m_{b}})}{K^{4}-\lambda^{2}}\notag \\&\quad \times\int_{h_{2n-1}}^{h_{2n}}e^{-2Kt}\bigg[\int_{h_{2n-1}}^{t}\frac{(d_{n}^{2}-z^{2})e^{Kz}}{\sqrt{(z^{2}-h_{2n-1}^{2})(h_{2n}^{2}-z^{2})}}\,dz \bigg] ^{2}\,dt,\quad n=1,2,3,\ldots,N, \end{align*} $$
$$ \begin{align*} \mathcal{C}_{mn}=\mathcal{C}_{nm}&=\frac{1}{\pi( K^{4}-\lambda^{2})}\int_{0}^{\infty}\frac{\zeta ( k^{4}-\lambda^{2}) e^{\zeta (a_{m}-a_{n})}}{k^{2}+K^{2}}\bigg[\int_{h_{2m-1}}^{h_{2m}}\frac{(d_{m}^{2}-z^{2})\sin kz}{\sqrt{(z^{2}-h_{2m-1}^{2})(h_{2m}^{2}-z^{2})}}\,dz \bigg]\notag \\ &\quad \times\bigg[\int_{h_{2n-1}}^{h_{2n}}\frac{(d_{n}^{2}-z^{2})\sin kz}{\sqrt{(z^{2}-h_{2n-1}^{2})(h_{2n}^{2}-z^{2})}}\,dz \bigg]\,dk, \quad m,n=1,2,3,\ldots N, \end{align*} $$
$$ \begin{align*} \mathcal{D}_{n}=\frac{1}{2K}\int_{h_{2n-1}}^{h_{2n}}\frac{(d_{n}^{2}-z^{2})e^{Kz}}{\sqrt{(z^{2}-h_{2n-1}^{2})(h_{2n}^{2}-z^{2})}}\,dz, \quad n=1,2,3,\ldots, N. \end{align*} $$
3.2 Determination of reflection and transmission coefficients
To determine the coefficients R and T, we substitute the expression from (3.21) into (3.14)–(3.18), then these equations can be written in the matrix notation as
$$ \begin{align} (P+Q)V=W, \end{align} $$
where
$$ \begin{align*}Q=\left[{\begin{array}{cccccccc} i+\cot(\mu(a_{1}-a_{2})) & -\csc(\mu(a_{1}-a_{2}))&-&-&-&-&-& 0\\ -\csc(\mu(a_{1}-a_{2}))& \sin(\mu(a_{3}-a_{1}))\csc(\mu (a_{3}-a_{2}))\csc(\mu(a_{2}-a_{1})) & -&-&-&-&-& 0 \\-&-&-&-&-&-&-&- \\-&-&-&-&-&-&-&- \\-&-&-&-&-&-&-&- \\ 0& - & -&-&-&-&-& \csc(\mu(a_{N}-a_{N-1}))\\0& 0 & -&-&-&-&-& i-\cot(\mu(a_{N}-a_{N-1})) \end{array}}\right],\end{align*} $$
$$ \begin{align*}V=\left[{\begin{array}{cccc} -Te^{i\mu (a_{N}-a_{1})} \\ (c_{2}e^{i\mu a_{2}}-d_{2}e^{-i\mu a_{2}})\\-\\-\\-\\-\\ (c_{N-1}e^{i\mu a_{N-1}}-d_{N-1}e^{-i\mu a_{N-1}}) \\ Re^{2i\mu a_{N}} \end{array}} \right], \qquad W=\left[{\begin{array}{cccc} 2K\mu P_{1N} \\ 2K\mu P_{2N}\\-\\-\\-\\-\\ 2K\mu P_{(N-1)N}+\csc(\mu(a_{N}-a_{N-1}))\\ 2K\mu P_{NN}-i-\cot(\mu(a_{N}-a_{N-11})) \end{array}} \right],\end{align*} $$
$$ \begin{align*} P=2K\mu \left[{\begin{array}{ccccccc} P_{11} & P_{12}& - & - & - & - & P_{1N}\\ P_{21} & P_{22}& - & - & - & - & P_{2N} \\ - & -& - & - & - & - & - \\ - & -& - & - & - & - & - \\ P_{N1} & P_{N2}& - & - & - & - & P_{NN} \end{array}} \right],\end{align*} $$
with
$P_{mj}=\int _{L_{m}}X_{mj}(t)e^{-Kt}\,dt,~m,j=1,2,3,\ldots ,N$
. Once the unknown constants
$b_{mj}, m,j=1, 2, 3,\ldots ,N$
are obtained after solving the system of equations (3.24), then
$X_{mj}, m,j=1, 2, 3,\ldots ,N$
from relations of (3.23) can be obtained. Hence, R and T can be determined by solving (3.27).
3.3 Dynamic wave force
The dynamic pressure
$P(x,y)$
can be obtained according to the Bernoulli equation (see [Reference Li, Liu and Li20, Reference Singh and Gayen39]),
$$ \begin{align*} P(x,y)=-i \rho \omega\sum_{j=1}^{N}\xi_{j}(y). \end{align*} $$
The magnitude of horizontal wave force acting on plates can be obtained by integrating the dynamic pressure along the porous plates as follows [Reference Li, Liu and Li20, Reference Singh and Gayen39]:
$$ \begin{align*} C_{f}=-i\rho \omega \sum_{j=1}^{N}\int_{L_{j}}\xi_{j}(y)\,dy. \end{align*} $$
The nondimensional form of the horizontal force coefficient on the vertical porous plates is given by
$$ \begin{align*} K_{f}=\frac{K~ |C_{f}|}{\rho g}. \end{align*} $$
3.4 Energy identity relation
The energy identity plays an important role in the theoretical study of water wave scattering by plates. Some of the incident wave energy can be dissipated by the porous plates. The absolute values of the reflection and transmission coefficients are connected by the relation (see [Reference Singh and Gayen39])
$$ \begin{align} |R|^{2}+|T|^{2}=1-J. \end{align} $$
Here, J signifies the amount of dissipated energy due to the permeability of plates and its expression in terms of the potential differences across the plates is found to be (see [Reference Singh and Gayen39])
$$ \begin{align*} J=\frac{2K^{2}}{\mu}\sum_{j=1}^{N}G_{j}^{r}\int_{L_{j}}|\xi_{j}(y)|^{2}\,dy, \end{align*} $$
where
$G_{j}^{r}$
is the real part of the porous effect parameters. It is noticed from the expression of J that the integrand in the right-hand side of (3.28) is always positive for a nonzero
$G_{j}^{r}$
so that
$|R|^{2}+|T|^{2}<1$
for the case of permeable barriers, whereas for impermeable plates, the energy identity relation satisfy
$|R|^{2}+|T|^{2}=1$
, which provides the convergence of result in the present study.
For different values of
$KH_{2}$
, the above relations (3.28) and
$|R|^{2}+|T|^{2}=1$
can be verified as in the last column of Tables 1 and 2 which certainly gives a partial check on the validation of the problem studied here for
$N=4$
. Table 1 represents the set of values
$|R|$
,
$|T|$
,
$|R|^{2}+|T|^{2}$
and
$1-J$
as a function of the wave number
$KH_{2}$
for the case of porous plates. Table 2 shows
$|R|$
and
$|T|$
, and that
$|R|^{2}+|T|^{2}$
satisfies the energy identity for impermeable plates. These two tables imply the convergence of the numerical results obtained in the present manuscript. Table 3 represents the amount of dissipated energy
$|J|$
for the different number of plates. It is observed that
$|J|$
increases monotonically as
$KH_2$
increases, regardless of the number of plates included in the array. Additionally, Table 3 shows that arrays containing more plates lead to larger values of
$|J|$
for the whole computed range of wave conditions. This suggests that the use of multiple plates can be more effective in dissipating wave energy.
Values of
$|R|$
,
$|T|$
,
$|R|^{2}+|T|^{2}$
and
$1-J$
against
$KH_{2}$
with
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, G_{1}=G_{2}=G_{3}=G_{4}=1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =45^\circ $
.
Values of
$|R|$
,
$|T|$
and
$|R|^{2}+|T|^{2}$
against
$KH_{2}$
with
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, G_{1}=G_{2}=G_{3}=G_{4}=0, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =45^\circ $
.
Values of
$|J|$
against
$KH_{2}$
with
$E=0.4, M=0.01, \theta =45^\circ $
and
$G_{N}=1 (N=1,2,\ldots ,6).$
4 Results and discussion
In this section, the characteristics of arbitrary poroelastic plates are demonstrated in attenuating wave forces and dissipating wave energy through numerous results. The numerical results are calculated assuming
$N=4$
for four poroelastic plates. The numerical results are plotted for the reflection coefficient, the transmission coefficient, the total energy and the deflection of the structure. Consider the dimensionless parameters
$M={m_{b}}/{\rho h_{2}}$
,
$E = {EI}/{\rho g h_{2}^{4}}$
for the elastic porous plates. All results are presented versus the nondimensional frequency, which is defined as
$KH_{2}$
. The parametric values that are kept constant throughout the numerical computations are given by fluid density
$\rho =1025.0$
kg/m
$^{3}$
and
$g=9.81$
m/s
$^{2}$
. The nondimensional parameters are given as
$$ \begin{align*}x_{j} =\frac{a_{j}}{h_{2}},~j = 1, 2, 3, 4; \quad H_{j} = \frac{h_{j}}{h_{2}},~j = 1, 2, 8;\quad d_{j}=x_{j}-x_{j-1},~j=2,3,4.\end{align*} $$
4.1 Model validation with existing results
In this section, we compared the present results with known limiting cases when certain parameters are assumed to be absent. It helps validate theoretical models and computational methods. When specific parameters are neglected or set to zero, it induces significant alterations in the behaviour of structures, particularly evident in plate structures.
We verified the results for permeable plates obtained in this study by comparing them with those obtained by Gayen and Mondal [Reference Gayen and Mondal11], who employed the hypersingular integral equation method for analysing two equal plates. Gayen and Mondal [Reference Gayen and Mondal11] plotted
$|R|$
against dimensionless wave number
$K(d+b) (\equiv Kh_{2}~~\text {in this study})$
for three different values of
$\mu =(d-b)/{(d+b)} (\equiv H_{1}=H_{3}~~\text {in this study})$
while keeping the length of separation between plates, porosity and incident wave angle fixed in their figure 4. In this paper, we plotted
$|R|$
against
$KH_{2}$
for specific values:
$x_{1}=-0.3, x_{2}=x_{3}=x_{4}=0.3, H_{4}=1$
,
$G_{1}=G_{2}=1, G_{3}=G_{4}=0,$
and
$H_{1}=H_{3}=0.01,0.05,0.25$
and
$\theta =0^{o}, M=0.001, E=0.999$
in Figure 2. It is observed that the corresponding curves from Gayen and Mondal [Reference Gayen and Mondal11] for two equal plates nearly coincide with those depicted in Figure 2. This comparison provides additional validation for our results and suggests agreement between the methodologies employed in both studies.
$|R|$
against
$KH_{2}$
for
$x_{1}=-0.3, x_{2}=x_{3}=x_{4}=0.3, E=0.999, M=0.001, G_{1}=G_{2}=1, G_{3}=G_{4}=0, \theta =0^{o}, H_{4}=1$
.
Finally, we assessed our solutions for a single elastic barrier by comparing them with numerical results derived from Green’s function, as developed by Meylan [Reference Meylan27]. For this comparison, we normalized the parameters and constructed Figure 3 for normal incident waves with a wavelength denoted by
$\varsigma $
, where
$\varsigma ={2 \pi }/{K}$
, as was done by Meylan [Reference Meylan27]. In Figure 3, we plotted the curves of
$|R|$
against the depth of the plate
$l ({h_{2}}/{\varsigma })$
for various values of
$E_{1}\equiv ({E}/{\varsigma })$
in the case of normal incident wave. The curves of Figure 3 almost coincide with those presented by Meylan [Reference Meylan27]. This agreement further validates the accuracy and reliability of our method.
$|R|$
against depth of the plate
$l(={h_{2}}/{\varsigma })$
for
${a_{1}}/{\varsigma }={a_{2}}/{\varsigma }={a_{3}}/{\varsigma }={a_{4}}/{\varsigma }=0,{h_{1}}/{\varsigma }=0.001, M=0.01, G_{1}=G_{2}=G_{3}=G_{4}=0, \theta =0^{o}, (E_{1}\equiv ({E}/{\varsigma }))$
.
4.2 Effects of various parameters on results
This subsection presents computed results for various parameter values of flexible porous plates. The accompanying graphs provide a visual representation of these findings.
Reflection coefficient
$|R|$
is plotted against wave number
$KH_{2}$
for the angle of incidence
$\theta =0^\circ , 30^\circ , 45^\circ , 60^\circ $
in Figures 4 and 5. The peak value of the reflection coefficient decreases as the angle of incidence increases. This trend is attributed to the mutual interaction between incident and reflected waves within the plates. The maximum and minimum values of
$|R|$
shift from left to right as the angle of incidence increases. This shift suggests changes in the behaviour of reflected waves with varying angles of incidence. Comparing figures of equal and nonequal porosity, there is a significant increase in the maximum and minimum values of
$|R|$
for equal porosity. This indicates that porosity influences the reflection properties of the plates, with equal porosity leading to higher reflection coefficients. The reflection coefficients exhibit an oscillatory nature for larger wave numbers. Moreover, higher oscillations are observed for smaller values of
$\theta $
, indicating a dependence on the angle of incidence. From all these figures, we can conclude that the reflection coefficient tends to be small when the porosity is large.
$|R|$
against
$KH_{2}$
for
$x_{1}{\kern-1.2pt}={\kern-1.2pt}0.5, x_{2}{\kern-1.2pt}={\kern-1.2pt}1.5, x_{3}{\kern-1.2pt}={\kern-1.2pt}2.5, x_{4}{\kern-1.2pt}={\kern-1.2pt}3.5, E{\kern-1.2pt}={\kern-1.2pt}0.4, M{\kern-1.2pt}={\kern-1.2pt}0.01, G_{1}{\kern-1.2pt}={\kern-1.2pt}G_{2}{\kern-1.2pt}={\kern-1.2pt}G_{3}{\kern-1.2pt}={\kern-1.2pt}G_{4}{\kern-1.2pt}={\kern-1.2pt}1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1$
.
$|R|$
against
$KH_{2}$
for
$x_{1}{\kern-1pt}={\kern-1pt}0.5, x_{2}{\kern-1pt}={\kern-1pt}1.5, x_{3}{\kern-1pt}={\kern-1pt}2.5, x_{4}{\kern-1pt}={\kern-1pt}3.5, E{\kern-1pt}={\kern-1pt}0.4, M=0.01, G_{1}=1, G_{2}=1.5, G_{3}=1.75, G_{4}=2, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1$
.
$|R|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1 \theta =0^\circ $
.
$|R|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =45^\circ $
.
$|R|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, M=0.01, G_{1}=G_{2}=G_{3}= G_{4}=1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|R|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, M=0.01, G_{1}=G_{2}=G_{3}= G_{4}=1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =45^\circ $
.
Figures 6 and 7 depict the reflection coefficient
$|R|$
versus the nondimensional wave number
$KH_2$
for different porosity values. The figures illustrate how the reflection coefficient
$|R|$
changes for four identical flexible plates subjected to normal and oblique incident waves. The minimum and maximum values of
$|R|$
occur at specific wave numbers. However, as the porosity increases, the reflection coefficient
$|R|$
decreases. This decrease is attributed to the increase in wave energy dissipation associated with higher porosity. The porous effect parameters influence the behaviour of the reflection coefficient. Similar to previous observations, the reflection coefficients exhibit an oscillatory behaviour for larger wave numbers. This oscillation pattern is characteristic of the interaction between waves and the porous plates.
Figures 8–11 analyse the reflection coefficient
$|R|$
versus the nondimensional wave number
$KH_2$
for different values of flexural rigidity E. Figures 8 and 9 focus on the impact of flexural rigidity on
$|R|$
involving equal plates with identical porosity, corresponding to normal and oblique incident waves, respectively. Figures 10 and 11 examine the influence of flexural rigidity on wave reflection under conditions of nonidentical porosity, covering both normal and oblique incident waves. The trend observed across all figures is that the values of
$|R|$
increase as the flexural rigidity E increases. This suggests that higher flexural rigidity enhances the reflection of waves when flexible porous plates are present. However, there is a notable difference in the peak values of
$|R|$
between curves with equal and nonidentical porosity. In the case of nonidentical porosity, the peak values of
$|R|$
are reduced compared with those with equal porosity. This indicates that variations in porosity between the plates influence the reflection characteristics, likely due to differences in how the waves interact with the plates.
$|R|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, M=0.01, G_{1}=1, G_{2}=1.5, G_{3}=2, G_{4}=2.5, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|R|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, M=0.01, G_{1}=1, G_{2}=1.5, G_{3}=2, G_{4}=2.5, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =45^\circ $
.
Figures 12 and 13 provide an analysis of the reflection coefficients
$|R|$
as a function of the nondimensional wave number
$KH_2$
for various values of plate separation length. As the separation length between the plates decreases, the values of
$|R|$
shift towards the right. This indicates a correlation between plate separation length and the position of the peak reflection coefficient. The peak value of
$|R|$
is observed to decrease in the case of an oblique incident wave compared with a normal incident wave. Across all figures, the oscillation of
$|R|$
increases as the plate’s separation length increases. This phenomenon can be attributed to constructive and/or destructive interference between incident and reflected waves within the flexible porous plates.
$|R|$
against
$KH_{2}$
for
$E=0.4, M=0.01, G_{1}=G_{2}=G_{3}=G_{4}=1, H_{1}=H_{3}=H_{5}= H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|R|$
against
$KH_{2}$
for
$ E=0.4, M=0.01, G_{1}=G_{2}=G_{3}=G_{4}=1, H_{1}=H_{3}=H_{5}= H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =45^\circ $
.
The reflection coefficient
$|R|$
is plotted against
$KH_{2}$
for varying submergence depths of the plates in Figures 14 and 15, corresponding to normal and oblique incident waves, respectively. It is noted that as the submergence depth of the plates decreases, the values of
$|R|$
shift towards the right. Additionally, the peak value of
$|R|$
is lower for oblique incident waves compared with normal incident waves. In the case of oblique incident waves, the overall reflection coefficient is generally lower across all submergence depths, highlighting the effect of wave direction on the interaction with submerged plates.
$|R|$
against
$KH_{2}$
for
$E=0.4, M=0.01, G_{1}=G_{2}=G_{3}=G_{4}=1, H_{1}=H_{3}=H_{5}= H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|R|$
against
$KH_{2}$
for
$E=0.4, M=0.01, G_{1}=G_{2}=G_{3}=G_{4}=1, H_{1}=H_{3}=H_{5}= H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|K_{f}|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|K_{f}|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =60^\circ $
.
Figures 16 and 17 depict the dimensionless amplitudes of wave forces acting on plates plotted against the nondimensional wave number
$KH_2$
for different values of the porous effect parameter. These figures provide insight into how the amplitude of wave forces varies with changes in both the wave number and the porous effect parameter. The observations from Figures 16 and 17 suggest that as the porosity increases, the coefficient of wave force decreases. Additionally, due to inertial effects, the values of
$|K_{f}|$
for complex porosity are lower than those for real porosity. This implies that the presence of pores within the material affects the amplitude of wave forces differently depending on whether the porosity is real or complex. The decrease in wave force coefficient with increasing porosity likely reflects a reduction in the resistance offered by the porous material to the incoming waves. However, Figures 18 and 19 illustrate the effect of wave forces on flexible porous barriers across four different values of flexural rigidity. It is observed from Figures 18 and 19 that the coefficient of wave force increases as the flexural rigidity of the porous plate increases. This suggests that flexural rigidity plays a significant role in enhancing the wave forces experienced by plates. A higher flexural rigidity likely results in increased resistance to bending or deformation when subjected to wave forces, leading to a higher coefficient of wave force. It is also observed that the maximum values of the wave forces
$|K_{f}|$
are lower for obliquely incident waves compared with normal incident waves.
$|K_{f}|$
against
$KH_{2}$
for
$x_{1}{\kern-1pt}={\kern-1pt}0.5, x_{2}{\kern-1pt}={\kern-1pt}1.5, x_{3}{\kern-1pt}={\kern-1pt}2.5, x_{4}{\kern-1pt}={\kern-1pt}3.5, M=0.01, G_{1}=G_{2}=G_{3}= G_{4}=1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|K_{f}|$
against
$KH_{2}$
for
$x_{1}{\kern-1pt}={\kern-1pt}0.5, x_{2}{\kern-1pt}={\kern-1pt}1.5, x_{3}{\kern-1pt}={\kern-1pt}2.5, x_{4}{\kern-1pt}={\kern-1pt}3.5, M{\kern-1pt}={\kern-1pt}0.01, G_{1}=G_{2}=G_{3}= G_{4}=1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =60^\circ $
.
$|J|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|J|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, E=0.4, M=0.01, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =60^\circ $
.
Figures 20 and 21 illustrate the amount of dissipated energy
$|J|$
plotted against wave number
$KH_2$
for various porosity values. These figures likely provide insights into how the dissipation of energy varies with changes in the wave number and porosity of the material. The observations from Figures 20 and 21 suggest that energy dissipation decreases as porosity increases. The addition of an imaginary part to the porous effect parameter results in a decrease in the value of dissipated energy. This suggests that incorporating certain imaginary components, possibly related to inertial effects, leads to a reduction in energy dissipation. However, dissipated energy
$|J|$
for various flexural rigidity values is plotted against the wave number in Figures 22 and 23. Through Figures 22 and 23, it can be noticed that with increasing flexural rigidity of the barriers, the amount of dissipated energy
$|J|$
increases. The higher flexural rigidity likely allows the plates to absorb and dissipate more energy, resulting in higher values of
$|J|$
. Comparing Figures 20–23, it is observed that the maximum values of the dissipated energy
$|J|$
are lower for obliquely incident waves compared with normal incident waves.
$|J|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, M=0.01, G_{1}=G_{2}=G_{3}= G_{4}=1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =0^\circ $
.
$|J|$
against
$KH_{2}$
for
$x_{1}=0.5, x_{2}=1.5, x_{3}=2.5, x_{4}=3.5, M=0.01, G_{1}=G_{2}=G_{3}= G_{4}=1, H_{1}=H_{3}=H_{5}=H_{7}=0.05, H_{4}=H_{6}=H_{8}=1, \theta =60^\circ $
.
5 Conclusion
This paper presents a mathematical model and analysis of how surface waves interact with multiple flexible porous plates, using the Galerkin approximation method. This study computes and presents several hydrodynamics quantities, including reflection and transmission coefficients, hydrodynamic force, and energy dissipation. These hydrodynamics quantities are crucial for understanding how waves interact with the flexible porous plates and how they affect each other. The proposed model’s validity is established by comparing its results with existing data from the literature. The computed reflection and transmission coefficients satisfy the energy identity for both permeable and impermeable flexible plates, affirming the model’s accuracy. A decrease in the reflection coefficient with an increase in the angle of incidence is observed. Vertical plates are found to be better reflectors. Additionally, the oscillation of the reflection coefficient increases with increasing plate separation length, indicating interference effects within the flexible porous plates. The flexural rigidity of the plates significantly influences hydrodynamic coefficients. Higher flexural rigidity leads to increased reflection coefficients, especially in the presence of porosity. Wave load is initially higher at the first plate but decreases as waves propagate over subsequent plates, aligning with expectations. However, increasing porous effect parameters leads to a decrease in the reflection coefficient due to enhanced wave energy dissipation. Higher energy dissipation is achieved by minimizing the inertial effect of flexible porous plates, indicating their effectiveness in dissipating wave energy. Compared with rigid plates, multiple flexible porous plates demonstrate a reduction in reflection, dissipated energy and wave force. This finding suggests their suitability for constructing breakwaters and mitigating wave effects on coastal and marine facilities. Overall, the analysis provides valuable insights into the role of multiple flexible porous structures in dissipating wave energy and attenuating wave load, which is crucial for coastal and marine engineering applications.
Acknowledgements
The authors would like to gratefully acknowledge the Editor and Reviewers for their valuable comments and suggestions to improve the quality of the manuscript. This work is partially supported by a SERB, DST, India (grant number TAR/2022/000107).
