# Surface wave scattering by multiple flexible fishing cage system

Siluvai Antony Selvan
School of Mathematics and Statistics
The University of Melbourne
Parkville VIC 3010, Australia
<EMAIL_ADDRESS>
&R. Gayathri
Department of Mathematics
SRM Institute of Science and Technology
Kattankulathur-603203, India
<EMAIL_ADDRESS>
&Harekrushna Behera*
Department of Mathematics
SRM Institute of Science and Technology
Kattankulathur-603203, India
<EMAIL_ADDRESS>
&M H Meylan
School of Mathematical and Physical Sciences
University of Newcastle
NSW 2308, Australia.
<EMAIL_ADDRESS>

###### Abstract

A study of the wave dynamics around a multiple fishing cage system is carried
out under the assumption of the linear water wave theory and small-amplitude
wave response. The Fourier–Bessel series expansion of the velocity potential
is derived for regions enclosed under the open-water and cage systems and in
the immediate vicinity. Further, the scattering between the cages is accounted
for by employing Graf’s addition theorem. The porous flexible cage system is
modelled using Darcy’s law and the three-dimensional membrane equation. The
edges of the cages are moored along its circumferences to balance its position
in the deep sea. The unknown coefficients in the potentials are obtained by
employing the matched eigenfunction method in conjunction with the least-
squares approximation method. In addition, the far-field scattering
coefficients for the entire system are obtained by expanding the Bessel and
Hankel functions in the plane wave representation form. Numerical results such
as the hydrodynamic forces, scattering coefficients, and power dissipation are
investigated for various cage and wave parameters. The wave loading on the
cage system can be significantly damped by the spatial arrangement, membrane
tension, and porous-effect parameter. Moreover, the far-field results suggest
that the cage system can also be used as a breakwater.

_K_ eywords Multiple flexible cylinders $\cdot$ porous membrane; mooring edge
$\cdot$ eigenfunction expansion method $\cdot$ Graf’s addition theorem $\cdot$
hydrodynamic forces $\cdot$ far-field dissipation

##  1 Introduction

There is a huge demand on seafood owing to the increasing population (Duarte
et al. [1] and Golden et al. [2]). Simultaneously, there is a need to save the
endangered species in the ocean due to global warming and ocean acidification
(McCormick et al. [3]). Both can be made possible using fish farming, which
was widely employed in the local ponds and lakes in the past decades (Helal et
al. [4] and Zhang et al. [5]). In order to implement the same in the ocean,
there are a series of challenges posed by the marine environment. Hence,
suitable modeling needs to be carried out for explaining the wave response of
such a system. The fish cage can be modeled as the cylindrical perforated
membrane-like structure with perforated bottom moored along its circumference
(Chan and Lee [6] and Mandal and Sahoo [7]). Although wave attenuation by a
single flexible porous cage has been well studied, however, there are no
studies in the literature regarding the wave response of the multiple flexible
perforated fishing cage system, which forms the ideal model for the fish farm.
Therefore, the authors are motivated to study the hydrodynamic response of a
system of multiple fishing cages. In the case of such a cage system, the
parameters such as porosity, flexibility and geometry play a major role in
explaining the wave response of a single/multiple cage systems. The
significance of such parameters have been investigated by many researchers in
different models for various applications.

In general, the presence of porosity in any off-shore structures can be
modeled using Darcy’s law (Chwang [8] and Yu and Chwang [9]), which damps the
high amplitude wave by dissipating its energy. Several studies have carried to
study the wave dissipation by the rigid cylindrical porous structures
(Williams et al. [10], Fredriksson et al. [11] and Sarkar and Bora [12]). The
permeability, size and location of the perforation significantly control the
hydrodynamic loads acting on the porous cylinder (Williams et al. [10]). The
theoretical and experimental studies on a porous cylinder were carried out by
Zhao et al. [13], which suggests that there exist additional components
corresponding to the porous effect effects along with the wave-radiating
damping. In the case of a surface piercing cylinder, the variation of draft,
porosity and radii influences the wave run-up and hydrodynamic loading on the
porous cylinder (Sarkar and Bora [14]). Sarkar and Bora [12] investigated the
wave interaction with the surface-piercing bottom-mounted porous cylinder,
where the resonance effect was observed at a particular wavenumber in the
presence of porosity. Later, Behera et al. [15] studied an influence of
truncated concentric porous cylinder on the inner rigid platform, where the
optimum number of porous structures were estimated for damping the wave
amplitude. Recently, the cnoidal wave diffraction from the dual concentric
cylinders in the presence of an arc-shaped outer cylinder was analyzed by Zhai
et al. [16].

In addition to the porosity, the inclusion of flexibility into the off-shore
structures reduce the damage experienced due to the wave loading. Further, it
also proves to be cost-effective and easily portable (Lee and Chen [17],
Williams et al. [18], Meylan et al. [19]). The wave passage through the
vertical flexible porous membrane was analyzed by Chan and Lee [6], where the
tensile force was incorporated in the standard two-dimensional
Euler–Bernoulli’s equation. They observed that the deformation of structure
increases for larger flexibility and the presence of perforation substantially
decreases the deformation. Other than this, the significance of both the
porosity and flexibility have also investigated in the case of horizontal
floating/submerged two-/three- dimensional structures (Behera and Sahoo [20],
Meylan et al. [21], Behera et al. [22], Selvan et al. [23] and Zheng et al.
[24]). In the case of cylindrical structures, Mandal et al. [25] studied the
effect of a flexible porous cylinder on the inner truncated rigid platform,
where the hydrodynamic wave loads were analyzed for various flow and
structural parameters. The dynamics of the flexible porous cylinder mounted on
the truncated rigid bottom was investigated by Su et al. [26]. Recently, the
gravity wave interaction with the cylindrical fish cage having bottom
perforated membrane was analyzed by Mandal and Sahoo [7] and found that the
proper selection of membrane tension along with the cage radius consequently
reduces the deflection of the cylindrical side-wall.

In the previous study of wave scattering from the single fishing cage (Su et
al. [26] and Mandal and Sahoo [7]), the bottom of the fish cage was either
mounted to a rigid bed or clamped. However, in the physical viewpoint, the
fishing cage can be moored along its circumferences, which was not addressed
in the previous studies of Su et al. [26] and Mandal and Sahoo [7]. To the
best of authors’ knowledge, the topic of fish farm constituting of multiple
fishing cages haven’t addressed in any of the previous studies in the
literature. In the present study, the surface wave interaction with the
multiple fishing cage system has investigated. Moreover, the fishing cages are
moored along its cylindrical circumferences using the frictionless linear
spring. In order to consider the scattering between the cages, Graff’s
addition theorem has employed. This theorem forms the basis of modeling the
wave scattering from any multiple floating structures (Park et al. [27, 28],
Zheng et al. [29] and Zheng et al. [30]). Further, the quantities like power
dissipation reflected and transmitted wave amplitudes at the far-field have
analyzed in the present study to check the usage of multiple fishing cage
system as a breakwater.

The present manuscript is organized as follows: Section 2 contains the
mathematical formulation, where the details of the governing equation and the
associated boundary conditions for the physical problem are discussed; Section
3 showcases the method of solution, in which the procedure for solving the
physical problem by employing the method of an eigenfunction in conjunction
with the technique of least-squares approximation are explained in detail.
Further, the brief description about the derivation of far-field amplitude
function and the power dissipation is given; Section 4 contains the result and
discussion, which further subdivided into the dual, triple and multiple
fishing cage systems along with the time-domain simulation. In the case of the
dual and triple fishing cages, the hydrodynamic wave forces, far-field
amplitude functions and power dissipation are investigated. Moreover, for
multiple fishing cage systems, far-field amplitude functions, power
dissipation and flow distributions are analyzed. The technique of time-domain
simulation is given in the last subsection of results and discussion. The
supplementary files are attached to show the surface wave scattering by the
single and multiple fishing cage system; In the last Section (i.e. Section 5),
the detailed summary and significance of the present work are highlighted.

##  2 Mathematical formulation

The dynamics of surface waves around the system of N multiple flexible fishing
cages [Fig. 1(a)] is studied in the global Cartesian coordinates $(x,y,z)$
with origin overlap with the still water having the final depth $H$ and
$z$-axis pointed upward direction. The position of arbitrary $j$th and $k$th
fishing cages are marked as $(x_{j},y_{j})$ and $(x_{k},y_{k})$ in the global
co-ordinate system, respectively [Fig. 1(b)], which form the origins of the
local polar coordinate systems $(r_{j},\theta_{j},z)$ and
$(r_{k},\theta_{k},z)$ with $k=1,2,\dots,\text{N}$ and $j=1,2,\dots,\text{N}$.
The thicknesses of entire cylindrical fishing cages are assumed as $d$, which
is assumed very small on comparing with the wavelength of an incident
monochromatic wave ($\lambda$) as a consequence of small amplitude wave
theory. Further, the radii of an array of floating fishing cages are
considered as $b_{k}$ with $k=1,2,\dots,\text{N}$ [Fig. 1(d)]. The
incompressible and inviscid fluid is assumed to be having an irrotational and
time-harmonic motion of the form $\mathrm{exp}({-\mathrm{i}\omega t})$. The
domain enclosed between the still water and a rigid bottom is divided into an
exterior and N interior regions. The general velocity potential potential is
defined as
$\Phi(r,\theta,z,t)=\Re\\{\phi(r,\theta,z)\,\mathrm{e}^{-\mathrm{i}\omega
t}\\}$ with $\phi(r,\theta,z)$ be the spatial velocity potential of $k$th
region where $\omega$ be the angular frequency and $\Re$ denotes the real
part. This general potential $\phi(r,\theta,z)$ satisfying all the assumption
of fluids obeys the Laplace equation, which can be expressed as

$\displaystyle\frac{\partial^{2}\,\phi}{\partial
r^{2}}+\frac{1}{r}\frac{\partial\,\phi}{\partial
r}+\frac{1}{r^{2}}\frac{\partial^{2}\,\phi}{\partial\,\theta^{2}}+\frac{\partial^{2}\,\phi}{\partial\,z^{2}}=0,$
(1)

Further, the spatial velocity potential in the $k$th region is given as
$\phi_{2}^{k}(r_{k},\theta_{k},z)$ and $\phi_{3}^{k}(r_{k},\theta_{k},z)$ with
$k=1,2,\dots,\text{N}$. In an outer region, it is denoted by
$\phi_{1}(r_{k},\theta_{k},z)$.

Figure 1: Schematic sketch of the defined physical problem; (a) Top view
showing the global Cylindrical coordinate ($O_{g}r_{g}\theta_{g}z$), (b) Top
view showing the $k$th\- and $j$th-cylinders, (c) Lateral view of a flexible
perforated moored fishing cage and (d) Lateral view showing the array of
multiple fishing cages.

The fluid depth enclosing the fishing cage is subdivided into the barrier
region $\varLambda_{B}:=\\{\,z\,|\,-h\leq\,z\leq\,0\\}$ and gap region
$\varLambda_{G}:=\\{\,z\,|\,-H\leq\,z\leq\,-h\\}$. The normal velocity becomes
zero along the rigid bed in both the interior and outer regions, which is
given by

$\displaystyle\frac{\partial\phi_{1}}{\partial z}=0,\quad\text{at}\quad z=-H,$
(2) $\displaystyle\frac{\partial\phi^{k}_{3}}{\partial
z}=0,\quad\text{at}\quad z=-H,$ (3)

where $k=1,2,\dots\text{N}$. The linear form of the free surface boundary
condition at both the open-water and cage covered regions are given as

$\displaystyle\frac{\partial\phi_{1}}{\partial
z}-K\,\phi_{1}=0,\quad\text{at}\quad z=0,$ (4)
$\displaystyle\frac{\partial\phi^{k}_{2}}{\partial
z}-K\,\phi^{k}_{2}=0,\quad\text{at}\quad z=0,$ (5)

where $K=w^{2}/g$ and $g$ is the acceleration due to gravity. Using Darcy’s
law, the proportionality between the hydrodynamic pressure and normal velocity
along the cage system is obtained, which are given as

$\displaystyle\frac{\partial\,\phi^{k}_{2}}{\partial\,z}=\mathrm{i}k_{0}G_{v}\,\\{\phi^{k}_{2}-\phi_{1}\\}+\mathrm{i}\omega\eta_{k},\quad\text{at}\quad
r_{k}=b_{k},\quad z\in\varLambda_{B},$ (6)
$\displaystyle\frac{\partial\,\phi^{k}_{2}}{\partial\,z}=\mathrm{i}k_{0}G_{h}\,\\{\phi^{k}_{2}-\phi^{k}_{3}\\}+\mathsf{i}\omega\zeta_{k},\quad\text{at}\quad
0<r_{k}<a_{k},\quad z=-h,$ (7)

where $G_{v}$ and $G_{h}$ be the complex porous-effect parameters of
cylindrical and circular membrane, respectively. The deflection of $k$th
cylindrical and submerged membranes are denoted as
$\eta_{k}(r_{k},\theta_{k},z)$ and $\zeta_{k}(r_{k},\theta_{k},z)$,
respectively. The linear form of dynamical boundary condition corresponding to
the bottom circular membrane reads as

$\displaystyle\left(Q\,\frac{\partial^{3}\phi^{k}_{3}}{\partial\,z^{3}}-m_{m}\,\frac{\partial\phi^{k}_{3}}{\partial
z}-K\,\phi^{k}_{3}\right)-\mathrm{i}k_{0}G_{h}\left(Q\frac{\partial^{2}\phi^{k}_{3}}{\partial
z^{2}}-m_{m}\right)\bigg{(}\phi^{k}_{2}-\phi^{k}_{3}\bigg{)}+K\phi^{k}_{2}=0,\,$
(8)

where $Q=T_{1}/\rho g$ and $m_{m}=\rho_{m}d/\rho g$, with $T_{1}$ and
$\rho_{m}$ being the tensile force and density of bottom perforated circular
membrane, respectively. The deflection of $k$th cylindrical perforated
membrane $\eta_{k}(r_{k},\theta_{k},z)$ is governed by

$\displaystyle
T_{2}\frac{d^{2}\eta_{k}}{dz^{2}}+m_{s}\omega^{2}\eta_{k}=2\mathrm{i}\omega\rho
b_{k}\int_{0}^{\pi}\bigg{(}\phi_{1}-\phi^{k}_{2}\bigg{)}\,\mathrm{e}^{\mathrm{i}\,(\pi-\theta_{k})}d\theta_{k}\,\,\text{at}\,\,z\in\varLambda_{B}\,\text{and}\,\,r=b_{k},$
(9)

where $k=1,2,\dots\text{N}$. Further, $T_{2}$ and $m_{s}$ are the tensile
force and uniform mass per unit length of the cylindrical perforated membrane,
respectively. The matching of velocity and pressure along the interface of
$k$th fish cage can be expressed as

$\displaystyle\frac{\partial\,\phi_{1}}{\partial
r}=\frac{\partial\,\phi^{k}_{2}}{\partial r}\quad\text{at}\quad
z\in\varLambda_{B}\quad\text{for}\quad r_{k}=b_{k},$ (10)
$\displaystyle\frac{\partial\,\phi_{1}}{\partial
r}=\frac{\partial\,\phi^{k}_{3}}{\partial r}\quad\text{at}\quad
z\in\varLambda_{G}\quad\text{for}\quad r_{k}=b_{k},$ (11)
$\displaystyle\phi_{1}=\phi^{k}_{3},\quad\text{at}\quad
z\in\varLambda_{G}\quad\text{for}\quad r_{k}=b_{k},$ (12)
$\displaystyle\frac{\partial\phi^{k}_{2}}{\partial
z}=\frac{\partial\phi^{k}_{3}}{\partial z},\quad\text{at}\quad
r_{k}=b_{k}\quad\text{and}\quad z=-h,$ (13)

Moreover, the condition for mooring along the circumference at both the ends
of the cylindrical perforated membrane is given by

$\displaystyle
T_{2}\frac{d\eta_{k}}{dz}-2\mu\sin^{2}\gamma_{u}\eta_{k}=0,\,\quad\,\text{at}\quad
z=-h,\,\,z=0\quad\text{and}\quad r_{k}=b_{k},$ (14) $\displaystyle
T_{2}\frac{d\eta_{k}}{dz}-2\mu\sin^{2}\gamma_{d}\eta_{k}=0,\,\quad\,\text{at}\quad
z=-h,\,\,z=0\quad\text{and}\quad r_{k}=b_{k},$ (15)

where $\mu$, $\gamma_{u}$ and $\gamma_{d}$ are the mooring spring constant,
the angle extended by the upper and lower mooring lines with respect to
negative $z$-axis, respectively.

##  3 Method of solution

In the outer region, the velocity potential $\phi_{1}(r_{k},\theta_{k},z)$
satisfying the Laplace equation (1) with the associated bottom boundary
condition (2) is derived as

$\displaystyle\phi_{1}=\sum_{m=-\infty}^{\infty}\left(\mathcal{I}_{k}J_{m}(k_{0}\,r_{k})\mathrm{e}^{\mathrm{i}m(\pi/2-\beta)}g_{0}(z)+\sum_{k=1}^{\text{N}}\sum_{q=0}^{\infty}A^{k}_{mq}\,H^{(1)}_{m}(k_{q}r_{k})\,g_{q}(z)\right)\mathrm{e}^{\mathrm{i}m\theta_{k}}$
(16)

where
$\displaystyle\,\mathcal{I}_{k}=(\mathsf{i}g/\omega)\mathrm{e}^{\mathrm{i}k_{0}(x_{k}\cos\beta+y_{k}\sin\beta)}$
is the phase factor with $\beta$ being the phase angle. Here, $J_{m}$ and
$H^{(1)}_{m}$ indicate the Bessel and first kind Hankel functions of order
$m$. The unknown coefficients of open-water region for $k$th fishing cage is
denoted by $A^{k}_{mq}$. Further, the above Eq. (16) satisfies the far-field
radiation condition

$\displaystyle\lim_{r_{k}\to\infty}\phi_{1}=\sum_{m=-\infty}^{\infty}\left(\mathcal{I}_{k}J_{m}(k_{0}\,r_{k})\mathrm{e}^{\mathrm{i}m(\pi/2-\beta)}g_{0}(z)\right),$
(17)

where the scattered portion of the Eq. (16) becomes zero. By employing Graff’s
addition theorem as in Abramowitz and Stegun [31] and Zheng et al. [30], the
Hankel function in the scattered portion of Eq. (16) can be rewritten as

$\displaystyle\phi_{1}=\sum_{m=-\infty}^{\infty}\bigg{[}\mathcal{I}_{k}J_{m}(k_{0}\,r_{k})\mathrm{e}^{\mathrm{i}m(\pi/2-\beta)}\,\mathrm{e}^{\mathrm{i}m\theta_{k}}\,g_{0}(z)+\sum_{q=0}^{\infty}\bigg{(}A^{k}_{mq}\,H^{(1)}_{m}(k_{q}r_{k})\mathrm{e}^{\mathrm{i}m\theta_{k}}+\sum_{{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}}^{\text{N}}A^{j}_{mq}\,\sum_{n=-\infty}^{\infty}\,(-1)^{n}\,$
$\displaystyle\hskip
199.16928ptH^{(1)}_{m-n}(k_{q}R_{jk})\,\mathrm{e}^{\mathrm{i}(m\alpha_{kj}-n\alpha_{jk})}\,J_{n}(k_{q}r_{k})\mathrm{e}^{\mathrm{i}n\theta_{k}}\bigg{)}g_{q}(z)\bigg{]},$
(18)

where the eigenfunction $g_{q}(z)$ associated with the outer region is given
by

$\displaystyle g_{q}(z)=\frac{\cosh k_{q}(z+H)}{\cosh(k_{q}H)},$ (19)

in which, the eigenvalues $k_{q}$ satisfies the following dispersion relation

$\displaystyle\omega^{2}=gk_{q}\tanh(k_{q}H),$ (20)

which can be solved numerically to obtain the one real root $(k_{0})$ and
infinite number of imaginary roots $(k_{1},k_{2},k_{3},\dots)$. Similarly, the
velocity potentials corresponding to the interior region of $k$th fishing cage
after solving the equation (1) along with Eq. (3) are given by

$\displaystyle\phi^{k}_{2}=\sum_{m=-\infty}^{\infty}\left(\sum_{q=0}^{\infty}B^{k}_{mq}I_{m}(p_{q}r_{k})M_{q}(z)\right)\,\mathrm{e}^{\mathrm{i}m\theta_{k}},$
(21)
$\displaystyle\phi^{k}_{3}=\sum_{m=-\infty}^{\infty}\left(\sum_{q=0}^{\infty}B^{k}_{mq}I_{m}(p_{q}r_{k})N_{q}(z)\right)\,\mathrm{e}^{\mathrm{i}m\theta_{k}},$
(22)

where $I_{m}$ is the modified Bessel function of first kind having the order
$m$, and the unknown coefficients of the interior region of $k$th fishing cage
is denoted as $B^{k}_{mq}$. The associated eigenfunctions are given as

$\displaystyle M_{q}(z)=\frac{\cosh p_{q}(z+H)-E_{q}\sinh
p_{q}(z+H)}{\cosh(p_{q}H)-E_{q}\sinh(p_{q}H)},$ (23) $\displaystyle
N_{q}(z)=\frac{\\{\tanh p_{q}(H-h)-E_{q}\\}\\{\cosh p_{q}(z+H)\\}}{\\{\tanh
p_{q}(H-h)\\}\\{\cosh(p_{q}H)-E_{q}\sinh(p_{q}H)\\}},$ (24) $\displaystyle
E_{q}=\frac{V_{q}p_{q}\tanh^{2}p_{q}(H-h)}{V_{q}p_{q}\tanh
p_{q}(H-h)-\\{1-\tanh^{2}p_{q}(H-h)\\}\\{K-\mathrm{i}k_{0}G_{2}V_{q}\\}},$
(25)

where $V_{q}=T_{1}p_{q}^{2}-m_{s}w^{2}$. Further, the eigenvalues $p_{q}$
satisfies the following dispersion relation in the interior cage region, which
is expressed as

$\displaystyle K-p_{q}\tanh(p_{q}H)=E_{q}(K\tanh(p_{q}H)-p_{q}).$ (26)

Solving the Eq. (26) numerically yields infinitely many complex roots
$(p_{0},p_{1},p_{2},\dots)$ for $G\neq 0$. On the other hand, in the case of
$G=0$, the one real roots $(p_{0})$ and infinitely many imaginary roots
$(p_{0},p_{1},p_{2},\dots)$ are obtained. The deflection of $k$th cylindrical
membrane obtained by solving Eq. (9) can be represented as

$\displaystyle\eta_{k}=D^{k}_{m1}\frac{\cos\lambda
z}{\cos\lambda_{1}H}+D^{k}_{m2}\frac{\sin\lambda
z}{\sin\lambda_{2}H}+\frac{\mathrm{i}\omega\pi\rho
b_{k}}{T_{2}(k_{0}^{2}+\lambda^{2})}\bigg{[}\sum_{q=0}^{\infty}B^{k}_{mq}\,I_{m}(p_{q}b_{k})\,M_{q}(z)-\mathcal{I}_{k}J_{m}(k_{0}b_{k})$
$\displaystyle\hskip
5.69046pte^{\mathrm{i}m(\pi/2-\beta)}\,g_{0}(z)-\sum_{q=0}^{\infty}\bigg{[}A^{k}_{mq}H_{m}^{(1)}(k_{q}b_{k})+\sum_{{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}}^{\text{N}}\sum_{n=-\infty}^{\infty}A^{j}_{nq}H^{(1)}_{m-n}(k_{q}R_{jk})\mathrm{e}^{\mathrm{i}(m\alpha_{kj}-n\alpha_{jk})}J_{m}(k_{q}b_{k})\bigg{]}g_{q}(z)\bigg{]},$
(27)

where $\displaystyle\lambda=\sqrt{\frac{m_{s}}{T_{1}}}\omega$. By employing
the velocity continuity as in Eqs. (10) and (11), the following the linear
system of equation is obtained

$\displaystyle\mathcal{I}_{k}J^{\prime}_{m}(k_{0}b_{k})Y_{lq}\,\mathrm{e}^{\mathrm{i}m(pi/2-\beta)}\delta_{lq}+\sum_{q=0}^{\infty}\bigg{[}A^{k}_{mq}H^{\prime(1)}_{m}(k_{q}b_{k})Y_{lq}\delta_{lq}+\sum_{{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}}^{N}\sum_{n=-\infty}^{\infty}A^{j}_{nq}(-1)^{m}H^{(1)}_{n-m}(k_{q}R_{jk})$
$\displaystyle\hskip
156.49014pt\mathrm{e}^{\mathrm{i}(n\alpha_{kj}-m\alpha_{jk})}J^{\prime}_{m}(k_{q}b_{k})Y_{lq}\delta_{lq}-B^{k}_{mq}I^{\prime}_{m}(p_{q}b_{k})\big{[}M_{ql}+N_{ql}\big{]}\bigg{]}=0,$
(28)

where $m=0,1,2,\dots$, $l=0,1,\dots$ and $q=0,1,2,\dots$. The orthogonal
relations $Y_{l}$, $M_{ql}$ and $N_{ql}$ with respect to open water region are
given by

$\displaystyle Y_{lq}=\int_{-H}^{0}g_{l}(z)g_{q}(z)\,dz,\quad
M_{ql}=\int_{-h}^{0}M_{q}(z)\,g_{l}(z)\,dz,\quad\mbox{and}\quad
N_{ql}=\int_{-H}^{-h}N_{q}(z)\,g_{l}(z)\,dz,$ (29)

with $q=0,1,\dots$ and $l=0,1,\dots$. Using Eqs. (6) and (12), the coupled
relation is obtained for $k=1,2,\dots,\text{N}$, which is given by

$\displaystyle
X^{k}(z)=\sum_{q=0}^{\mathbf{n}_{0}}\bigg{[}A^{k}_{mq}\,R^{k}_{mq}(z)+\sum_{{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}}^{N}\sum_{n=0}^{M}A^{j}_{nq}R^{j}_{nq}(z)\bigg{]}+\sum_{q=0}^{\mathbf{n}_{0}}B^{k}_{mq}S^{k}_{mq}(z)+\sum_{i=1}^{2}D^{k}_{mi}W^{k}_{mi}(z)+L^{k}_{m0}(z),$
(30)

where

$\displaystyle R^{k}_{mq}(z)=\begin{cases}H^{(1)}_{m}(k_{q}b_{k})g_{q}(z)\quad
z\in\varLambda_{G},\\\\[4.30554pt]
\left(\frac{b_{k}\omega^{2}\rho\pi}{T_{2}(k_{0}^{2}+\lambda^{2})}-\mathrm{i}k_{0}G_{v}\right)H^{(1)}_{m}(k_{q}b_{k})g_{q}(z)\quad
z\in\varLambda_{B},\end{cases}$ $\displaystyle
R^{j}_{nq}(z)=\begin{cases}H^{(1)}_{n-m}(k_{q}R_{jk})\mathrm{e}^{\mathrm{i}(n\alpha_{kj}-m\alpha_{jk})}J_{m}(k_{q}b_{k})g_{q}(z)\quad
z\in\varLambda_{G}\\\\[4.30554pt]
\left(\frac{b_{k}\omega^{2}\rho\pi}{T_{2}(k_{q}^{2}+\lambda^{2})}-\mathrm{i}k_{0}G_{v}\right)H^{(1)}_{n-m}(k_{q}R_{jk})\mathrm{e}^{\mathrm{i}(n\alpha_{kj}-m\alpha_{jk})}J_{m}(k_{q}b_{k})g_{q}(z)\quad
z\in\varLambda_{B}\end{cases}$ $\displaystyle
S^{k}_{mq}(z)=\begin{cases}-I_{m}(p_{q}b_{k})N_{q}(z)\quad
z\in\varLambda_{G},\\\\[4.30554pt]
\left(\left(-\frac{b_{k}\omega^{2}\rho\pi}{T_{2}(p_{q}^{2}+\lambda^{2})}-\mathrm{i}k_{0}G_{v}\right)I_{m}(p_{q}b_{k})-I^{\prime}_{m}(p_{q}b_{k})\right)M_{q}(z)\quad
z\in\varLambda_{B}\end{cases}$ $\displaystyle
L^{k}_{m0}(z)=\begin{cases}\mathcal{I}_{k}J_{m}(k_{0}b_{k})\mathrm{e}^{\mathrm{i}m(\pi/2-\beta)}g_{q}(z)\quad
z\in\varLambda_{G},\\\
\left(\frac{b_{k}\omega^{2}\rho\pi}{T_{2}(k_{0}^{2}+\lambda^{2})}-\mathrm{i}k_{0}G_{v}\right)\mathcal{I}_{k}J_{m}(k_{0}b_{k})\mathrm{e}^{\mathrm{i}m(\pi/2-\beta)}g_{q}(z)\quad
z\in\varLambda_{B},\end{cases}$

$\displaystyle W^{k}_{m1}(z)=\begin{cases}0\quad
z\in\varLambda_{G},\\\\[4.30554pt] \frac{\mathrm{i}\omega\cosh(\lambda
z)}{\cosh(\lambda H)}\quad z\in\varLambda_{B},\end{cases}\hskip
113.81102ptW^{k}_{m2}(z)=\begin{cases}0\quad
z\in\varLambda_{G},\\\\[4.30554pt] \frac{\mathrm{i}\omega\sinh(\lambda
z)}{\cosh(\lambda H)}\quad z\in\varLambda_{B}\end{cases}$

The above relation Eq.(30) satisfies the least square approximation method as
given by,

$\displaystyle\int_{-H}^{0}|X^{k}|^{2}dz=\text{minimum},$ (31)
$\displaystyle\int_{-H}^{0}X^{k}\frac{\partial X^{k*}}{\partial
A^{k*}_{mq}}dz=0,$ (32)

where ∗ denotes the complex conjugate. The series expansion for eigenvalues
are truncated for $q=\mathbf{n}_{0}$ roots for both the open water and
submerged membrane covered regions for $k$th fishing cage. Similarly, the
Bessel series are truncated for $m=M$ for $k$th fishing cage. This leads to
$2NM(\mathbf{n}_{0}+1)$ system of equations (Eqs. (28) and (30) along with the
edge conditions (14) and (15)), which can be solved for all values of $m$, $q$
and $k$ simultaneously.

###  3.1 Far-field amplitude functions and power dissipation

As $r_{k}\to\infty$, there exists only the progressive wave mode in the
scattered part of the total potential Eq. (16) corresponding to the exterior
region, which can be expressed in the following asymptotic form

$\displaystyle\phi^{s}_{1}=\sum_{k=1}^{N}\sum_{m=-\infty}^{\infty}A^{k}_{m0}\sqrt{\frac{2}{\pi
k_{0}r_{k}}}e^{-\mathsf{i}m\pi/4}e^{\mathsf{i}k_{0}r_{k}}e^{\mathsf{i}m(\theta_{k}-\pi/2)}g_{0}(z).$
(33)

Further, the above expression Eq. (33) is transformed from the local
cylindrical co-ordinate $(r_{k},\theta_{k},z)$ to the global cylindrical
coordinate $(r_{g},\theta_{g},z)$, which can be rewritten as

$\displaystyle\phi^{s}_{1}=\sum_{k=1}^{N}\sum_{m=-\infty}^{\infty}A^{k}_{m0}\sqrt{\frac{2}{\pi
k_{0}r_{k}}}e^{-\mathsf{i}m\pi/4}e^{\mathsf{i}k_{0}R_{gk}\cos(\alpha_{gk}-\theta_{g})}e^{\mathsf{i}m(\theta_{g}-\pi/2)}g_{0}(z).$
(34)

Similarly, the incident portion of the total potential Eq. (16) is expressed
in the asymptotic form and extended to the global cylindrical coordinate,
which is given as

$\displaystyle\phi^{in}_{1}=\sum_{m=-\infty}^{\infty}\mathcal{I}_{k}e^{-\mathsf{i}k_{0}R_{gk}\cos(\alpha_{gk}-\theta_{g})}e^{\mathsf{i}m(\pi/2-\beta)}g_{0}(z).$
(35)

From the Eqs. (34) and (35), the scattering and incident amplitude functions
are denoted as

$\displaystyle\mathbf{D}(\theta_{g})=\frac{\mathsf{i}\omega}{g\pi}\sum_{k=1}^{N}\sum_{m=-\infty}^{\infty}A^{k}_{m0}e^{-\mathsf{i}k_{0}R_{gk}\cos(\alpha_{gk}-\theta_{g})}e^{\mathsf{i}m(\theta_{g}-\pi/2)}~{}~{}~{}\mbox{and}~{}~{}~{}\mathbf{A}(\theta_{g})=\frac{A}{2\pi}\sum_{m=-\infty}^{\infty}e^{\mathsf{i}m(\theta_{g}-\beta)}.$
(36)

The reflected and transmitted wave amplitude functions are defined as

$\displaystyle
R(\theta_{g})=\mathbf{D}(\beta+\theta_{g}+\pi)~{}~{}~{}\mbox{and}~{}~{}~{}T(\theta_{g})=\mathbf{D}(\beta+\theta_{g})+\mathbf{A}(\beta+\theta_{g})~{}~{}~{}\mbox{provided}~{}~{}-\pi/2<\theta_{g}<\pi/2.$
(37)

Further, the power dissipation is obtained directly from the hydrodynamic
pressure acting on the vertical and submerged porous membrane, which is given
as

$\displaystyle\mathcal{P}_{D}=\frac{k_{0}\rho\omega}{2}\left[(G_{v}+G_{v}^{*})\sum_{k=1}^{N}\iint_{\Omega_{h}}|\phi^{k}_{2}-\phi^{k}_{3}|^{2}dS+(G_{h}+G_{h}^{*})\sum_{k=1}^{N}\iint_{\Omega_{v}}|\phi^{k}_{2}-\phi_{1}|^{2}dS\right],$
(38)

where $\Omega_{v}=\\{(\theta,z):0\leq\theta_{k}\leq
2\pi~{}~{}\mbox{and}~{}~{}-h\leq z\leq 0\\}$ and
$\Omega_{h}=\\{(r,\theta):0\leq r\leq
b_{k}~{}~{}\mbox{and}~{}~{}0\leq\theta_{k}\leq 2\pi\\}$ are the domains of
integration, and ∗ denotes the complex conjugate. The incident power per unit
width of wave front is given as

$\displaystyle\mathcal{P}_{I}=\frac{\rho\omega
gA^{2}}{4k_{0}}\bigg{(}1+\frac{2k_{0}H}{\sinh(2k_{0}H)}\bigg{)},$ (39)

which can be used for scaling the power dissipation $\mathcal{P}_{D}$. Hence,
the non-dimensional power dissipation is denoted as
$E=k_{0}\mathcal{P}_{D}/\mathcal{P}_{I}$.

##  4 Results and Discussion

The MATLAB program have developed to solve the system of equations for
determining the unknown coefficients. For numerical discussion, the following
physical parameters such as water depth $H=30$m, density of water $\rho=1025$
kgm-3, wave amplitude $A=1$m, height of fish cage $h/H=0.5$, density of
submerged membrane $\rho_{m}=100$ kgm-3, non-dimensional tensile force
$T_{1}/\rho gH^{2}=T_{2}/\rho gH^{2}=T$ with $T=0.4$, non-dimensional mass of
cylindrical cage $\alpha=0.01$, the porous-effect parameter of cage system
$G_{h}=G_{v}=G=3+3\mathsf{i}$, the mooring spring constant for mooring lines
connected to both the ends of cage $\mu=10^{12}$N/m, the mooring angles
$\gamma_{u}=60^{\circ}$ and $\gamma_{d}=30^{\circ}$, and the phase angle
$\beta=0^{\circ}$ are fixed, unless it is highlighted in the appropriate
figure’s caption. The hydrodynamic wave load acting on a fishing cage in
vertical direction $\mathcal{S}_{v}$, can be expressed as

$\displaystyle\mathcal{S}_{v}=\mathsf{i}\rho\omega\,\int_{0}^{2\pi}\int_{0}^{b_{k}}\,\big{[}\phi_{3}^{k}(r_{k},\theta_{k},-h)-\phi_{2}^{k}(r_{k},\theta_{k},-h)\big{]}r_{k}dr_{k}\cos(\pi-\theta_{k})d\theta_{k}.$
(40)

Further, the non-dimensional form of vertical wave force can be represented as

$\displaystyle C_{v}=\bigg{|}\frac{\mathcal{S}_{v}}{\rho gb_{k}hH}\bigg{|}.$

###  4.1 Single fishing cage

Fig. 2 portrays the amplitude of wave-field in both the presence and absence
of a fishing cage. In the first window [Fig. 2(a)], the incident wave
propagating with the phase angle $\beta=30^{\circ}$ and unit peak-to-peak
amplitude is illustrated. When it interacts with the single fishing cage as
seen in Figs. 2(b)–(d), the waves get scattered, and the portion of waves is
dissipated due to the presence of porosity in the cage. Further, the amplitude
of scattered wave-field is less than the absolute value of incident amplitude
[i.e. $|A|<2$]. On increasing the non-dimensional wavenumber $k_{0}h$, the
wave energy decreases in the deep water. Thus, in the presence of a fishing
cage, more energy dissipation occurs at the larger values of $k_{0}h$ as
compared to that of the smaller values. It is clear from Fig. 2(a) that the
waves with the smaller wavenumbers are damped more efficiently as compared to
the waves having larger wavenumbers [Figs. 2(b)–(d)].

(a) Absence of cage ($k_{0}h=1.25$) (b) Presence of cage ($k_{0}h=1$)

(c) Presence of cage ($k_{0}h=1.25$) (d) Presence of cage ($k_{0}h=1.5$)

Figure 2: Surface amplitude distributions $|\eta|$ for the (a) absence and
(b)–(d) presence of fishing cage at different non-dimensional wavenumber. The
other parameters are $G=3+3\mathsf{i}$, $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$
and $\mu=10^{6}$N/m.

Figs. 3(a)–(c) illustrate the energy dissipation as a function of real porous-
effect parameters (where the inertial effects [i.e. $Im(G)=0$ are neglected]
for different values of $T$ at various wavenumbers. In general, the energy
dissipation increases initially and attains maximum, then decreases for larger
values of $G$. There is a reducing energy dissipation for larger values of $G$
due to the transparency of the cage to the incoming wave, where the
dissipation does not occur. With an increase in the values of $T$, the
dissipation follows an increasing pattern for smaller $G$, and it follows the
reverse pattern for moderate and larger $G$. When the porous-effect parameter
is less, the increasing tensile force develops more restoring force against
the wave interacting with the cage. Thus, it dissipates more energy. On the
other hand, for moderate and larger $G$, the structure gradually becomes
transparent, and the waves are more likely reflected than dissipated. This is
a reason behind the decreasing energy dissipation for moderate and larger
values of $G$. For the waves with larger wavenumber [Fig. 3(c)], the energy
dissipation increases for smaller and moderate values of $G$, whereas it falls
significantly for larger values of $G$.

(a) $k_{0}h=1$ (b) $k_{0}h=1.25$

(c) $k_{0}h=1.5$

Figure 3: Power dissipation $E$ against the real part of $G$ for different
values of $T$ at various nondimensional wavenumber $k_{0}h$. The other
parameters are $G=3+3\mathsf{i}$, $h/H=0.4$, $\beta=30^{\circ}$ and
$\mu=10^{6}$N/m.

The modulus of scattering coefficient along the different directions from the
single fishing cage is plotted in Fig. 4 for different porous-effect parameter
$G$ at various spring constant $\mu$. It is observed that the scattering
coefficient peaks along the incident wave direction [i.e.,
$\theta_{g}=30^{\circ}$]. There exist two sidebands on either side of the
central band, which indicate that the wave scattering decreases periodically
on either side of incident wave direction. On increasing the mooring spring
constant, the membrane stretches more and the wave scattering decreases as a
consequence of less oscillation by a moored cylindrical membrane. On
increasing the porous-effect parameter, the wave scattering decreases owing to
an increasing wave energy dissipation. However, the deviations are quite
significant for smaller and moderate values of $\mu$. In the case of larger
$\mu$ [Fig. 4(c)], the effect of $G$ becomes negligible and the sideband
dampens, which is due to less scattering and dissipation by membrane having
very high stiffness.

(a) $\mu=10^{5}$N/m (b) $\mu=10^{6}$N/m

(c) $\mu=10^{7}$N/m

Figure 4: Modulus of scattering coefficient $|A(\theta_{g})|$ for different
values of $G$ at various mooring spring constant $\mu$. The other parameters
are $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$ and $k_{0}h=1.25$.

In Figs. 5(a), (b) and (c), the vertical wave force acting on the fishing cage
is plotted as a function of non-dimensional wavenumber $k_{0}h$ for different
values of $G$ at $\mu=10^{5}$N/m, $\mu=10^{6}$N/m and $\mu=10^{7}$N/m,
respectively. The vertical wave force increases and attains maximum, then
decreases for increasing values of $k_{0}h$. The maximum wave force happens at
a certain wavenumber as a result of more wave reflection from the cage, in
which the dissipation decreases. This specific wavenumber shifts based on the
cage parameter. With an increase in the mooring spring constant, the magnitude
of wave force increases initially; then, there is a moderate decrease for
increasing $\mu$. It is interpreted that there exists an optimum value of
mooring spring constant [i.e., $\mu=10^{6}$N/m], where the wave force attains
maximum value resulting in more reflection. Further, there is a sharp rise in
the wave force for increasing $\mu$, which is due to increasing membrane
stiffness.

(a) $\mu=10^{5}$N/m (b) $\mu=10^{6}$N/m

(c) $\mu=10^{7}$N/m

Figure 5: Vertical wave force acting on the fishing cage against the
wavenumber for different values of $G$ at various mooring spring constant
$\mu$. The other parameters are $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$ and
$k_{0}h=1.25$.

###  4.2 Dual fishing cage

(a) $R_{12}=H$ (b) $R_{12}=2H$

Figure 6: Surface amplitude distributions $|\eta|$ of the dual fishing cage
system for different spacing parameter $R_{12}$. The other parameters are
$G=3+3\mathsf{i}$, $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$ and $\mu=10^{6}$N/m.

In Fig. 6, the effect of spacing on the amplitude of surface elevation is
plotted. It is noticed the wave interacts the dual system with phase angle
$\beta=30^{\circ}$ from both the figures . On interacting the first cage, the
wave amplitude inside the cage increases initially, then there is an energy
loss that happens inside the cage due to destructive interference between the
incoming waves and inner scattered waves. The same phenomenon of destructive
interference occurs inside the second cage. However, the constructive
interference occurs between the cages on keeping the both cages close to each
other (i.e. $R_{12}=H$). This results in the high amplitude waves in the
region confined between the two cages as observed in Fig. 6(a). For larger
spacing between the cages (i.e. $R_{12}=2H$), there is a loss in wave energy
resulting in the damping of wave amplitudes between the cages as observed in
Fig. 6(b). Moreover, on comparing both the figures, the wave propagation in
the lee-ward side of the structure decreases while increasing the spacing
between the cages. This, in turn, reduces the damages to the fishing cage
system.

(a) $T=0.1$ (b) $T=0.2$

(c) $T=0.4$

Figure 7: Modulus of scattering coefficient $|A(\theta_{g})|$ for different
values of $G$ at various tensile force $T$. The other parameters are
$h/H=0.4$, $\beta=30^{\circ}$, $k_{0}h=1.25$, $R_{12}=H$ and $\mu=10^{6}$N/m.

The modulus of scattering coefficient along the different directions from the
dual fishing cage system is plotted in Fig. 7 for different porous-effect
parameter $G$ at various tensile force $T$. It is noticed that the central
band peaks at the incident wave direction for $\theta_{g}=30^{\circ}$ in all
figures. For smaller tensile force [Fig. 7(a)], the central band reduces and
scatters less wave as compared to the sidebands, which scatters more waves. As
the value of $T$ increases, the central band’s increases, and the sideband
decreases owing to increasing restoring force developed inside the membrane.
Moreover, there exists an optimum value of $T$ [i.e., $T=0.2$] at which the
maximum waves are scattered, after which it decreases slowly for increasing
$T$. On increasing the value of $G$, the wave scattering from the cage system
increases as a consequence of impedance offered by the cages for membrane
having less tensile force. For $T\geq 0.2$, the waves scatter from the dual
cage system decreases for increasing porous-effect parameter, where the
impedance decreases and wave energy dissipation increases.

(a) $T=0.1$ (b) $T=0.2$

(c) $T=0.4$

Figure 8: Power dissipation $E$ against the real part of $G$ for different
values of $h/H$ at various tensile force $T$. The other parameters are
$G=3+3\mathsf{i}$, $\beta=30^{\circ}$, $k_{0}h=1.25$, $R_{12}=H$ and
$\mu=10^{6}$N/m.

The energy dissipation by the dual fishing cages against the real values of
$G$ is plotted for various $h/H$ at varying $T$ in Fig.8, respectively.
Initially, the power dissipation increases with respect to $G$, then it
attains maximum and befalls gradually for larger values of $G$. It is noticed
that for $T=0.1$, the energy dissipation increases for an increasing $h/H$ as
a result of increasing energy dissipation due to increased surface area of the
cage. In the case of $T\geq 0.2$, the energy dissipation behaves differently
for $G<6$ and $G>6$. In the case of $G<6$, the energy dissipation decreases
for increasing submergence depth $h/H$, where more scattering occurs as
compared to the dissipation. On the other hand, for $G>6$, the energy
dissipation increases for increasing $h/H$ resulting in less wave scattering
as compared to the energy dissipation. Further, the membrane with less tension
dissipates more energy for larger depth ratio owing to the increased energy
dissipation.

###  4.3 Multiple fishing cage

Figure 9: Surface wave interaction with an inline arrangement of the multiple
fishing showing the (a) scattering coefficient for varying
$(k_{0}h,\theta_{g})$ at $\beta=30^{\circ}$, (b) scattering coefficient for
varying $\theta_{g}$ at various $\beta$ with $k_{0}h=1.25$, and (c) amplitude
of surface elevation with $k_{0}h=1.25$ and $\beta=30^{\circ}$. The other
parameters are $h/H=0.4$ and $\mu=10^{6}$N/m.

In Fig. 9, the scattering coefficient and flow distributions around the
multiple fishing cage system having inline arrangement are discussed. In Fig,
9(a), the surface plot depicting the scattering coefficients for varying
values of $(k_{0}h,\theta_{g})$ is investigated. In the inline arrangement,
the incident wave is directed with the phase angle $\beta=30^{\circ}$ towards
the system. It is observed that the wave scattering is less for smaller and
larger wavenumbers along the direction of the incident wavefield (i.e.,
$\theta_{g}=30^{\circ}$), where most of the incoming waves pass through the
system with less scattering. Around $1<k_{0}h<1.5$, the major portions of
waves belonging to this bandwidth range are scattered more, resulting in more
reflection from the system. For $k_{0}h=1.25$, the effect of phase angle on
the scattering coefficient is plotted in Fig. 9(b). It is noticed that the
scattering peaks at an incident wave direction, and it follows the oscillatory
pattern along the other direction due to the periodic arrangement of the cage
system. However, the wave scattering decreases on either side of the central
band due to the inline arrangement, which allows most of the incoming waves
through the cage system. Further, the wave scattering decreases for an
increasing value of $\beta$ due to the less screening of waves by the other
cages for higher phase angle. The corresponding flow distributions around the
inline system is demonstrated in Fig. 9(c), where the phase angle of the
incident wave is considered as $\beta=30^{\circ}$. In general, the wave
amplitude of the surface waves decreases downstream of the cage system due to
the wave dissipation by the porous cage system. Further, the wave amplitudes
around the cage’s circumference increases as a result of scattering between
the consequent cages, and accumulated waves inside the cages are dissipated by
the vertical porous structure. This reduces the transmission on the lee-side
of the cage system.

Figure 10: Surface wave interaction with a square arrangement of the multiple
fishing showing the (a) scattering coefficient for varying
$(k_{0}h,\theta_{g})$ at $\beta=30^{\circ}$, (b) scattering coefficient for
varying $\theta_{g}$ at various $\beta$ with $k_{0}h=1.25$, and (c) amplitude
of surface elevation with $k_{0}h=1.25$ and $\beta=30^{\circ}$. The other
parameters are $h/H=0.4$ and $\mu=10^{6}$N/m.

Fig. 10 exhibits the scattering coefficients and flow distributions around the
multiple fishing cage system arranged in a square formation. Unlike inline
arrangement, there exists a wave scattering for smaller wavenumbers in all
directions. Then, the scattering coefficients befall and attain a maximum
between $1<k_{0}h<1.5$, where more waves are scattered. Further, the wave
scattering considerably increases in this arrangement as compared to the
previous arrangement (Fig. 9(a)) due to wave interaction with the column of
cages. By fixing $k_{0}h=1.25$, the response of phase angle on the scattering
coefficients is illustrated along a different direction. It is interesting to
note that the scattering coefficient increases for a certain optimum angle of
$\beta=20^{\circ}$, where most of the waves are scattered. Moreover, the
oscillatory patterns are reduced as compared to other arrangements owing to
the simultaneous scattering of the incident wave by a column of cylinders. For
$\beta>20^{\circ}$, the wave scattering decreases due to the less screening of
waves by the cages along incident wave direction. The flow distributions
corresponding to the square arrangement is plotted in Fig. 10(c), where
$\beta=30^{\circ}$ is fixed. The constructive interference occurs in between
the cages in the first column and it gradually decreases as the wave passes
through the adjacent column of cages. It is observed that the high amplitude
waves are damped by an array of fishing cage due to the wave energy
dissipation. However, the dissipated wave amplitudes are further trapped
inside the system resulting in less wave propagation on the lee-side of the
cage system. The damages to the cage system are greatly avoided in such type
of arrangement as a consequence of multiple scattering between the cages.

###  4.4 Flow field

(a) $N=1$ (b) $N=2$

(c) $N=9$ (d) $N=9$

Figure 11: Flow field around the (a) single fishing cage, (b) dual fishing
cage, (c) multiple fishing cage having inline arrangement ($N=9$) and (d)
multiple fishing cage having square arrangement ($N=9$). The other parameters
are $G=3+3\mathsf{i}$, $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$, $k_{0}h=1.25$
and $\mu=10^{6}$N/m.

In Fig. 11, the flow field around the fishing cage system is plotted for
different values of $N$. It is noticed that the incident surface wave with
phase angle $\beta=30^{\circ}$ is passing through the cage system in all
figures. For $N=1$ (Fig. 11(a)), there are more flows around the circumference
of the cage facing the wind-ward side due to the wave interaction. Then, it is
passing through the cage, which is due to the wave reflection and dissipation,
resulting in less wave transmission in the corresponding lee-ward side. This
is true from the figure 11(a), where the less transmission on the lee-ward
side is shown by well-spaced arrows denoting the flow. In the case of a dual
fishing cage (Fig. 11(b)), there is an interaction of scattered waves in
between the cages resulting in the large accumulation of flows. However, the
flow field inside the second cage decreases, resulting in less transmission on
the lee-side of the cage system. The flow field through the multiple fishing
cages ($N=9$) arranged in inline configuration [Fig. 11(c)] shows that the
wave passage gradually decreases as it propagates through the cages, which
reduces the wave transmission in the lee-side of cages. Moreover, from Fig.
11(d), the wave transmission through the first column of cages decreases for
square configuration. However, there is scattering occurs between the internal
cages, and more waves are transmitted on the lee-side of the cage system.

###  4.5 Time simulation of solution

Figure 12: Wave scattering by dual fishing cages ($N=1$). The other parameters
are $G=3+3\mathsf{i}$, $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$, $k_{0}h=1.25$
and $\mu=10^{6}$N/m. Multimedia view:

The simulation of a surface elevation ($\eta$) can be done using the following
transformation rule, which can be expressed as

$\displaystyle\eta(x,y,0,t)=\Re\biggl{[}2A\sqrt{\pi
s}\int^{\infty}_{-\infty}\eta(x,y,0)\,\mathrm{exp}\\{-s(\bar{k}-\bar{k}_{c})^{2}-\mathsf{i}\omega
t\\}d\bar{k}\bigg{]},$ (41)

where $\Re$ denotes the real part. Further, the parameters such as $s$ and
$\bar{k}_{c}$ are known as spreading function and centre frequency,
respectively. For the purpose of simulation, the values are fixed as $s=2.5$
and $k_{c}=1/H$.

The time simulation of the wave scattering by the single, double and triple
fishing cage system having the inline arrangement are shown as static figure
in Figs. 12, 12 and 12, respectively. The movies corresponding to the static
figures 12, 12 and 12 are given in the multimedia file. It is noticed that the
amplitude of wave interacting with the single fishing cage as in Fig. 12
decreases as it passes through the cage. This happens due to the partial
reflection and dissipation from the single fishing cage. In the case of dual
fishing cage system as in Fig. 13, there is a multiple scattering occurs
between the cages. This consequently reduces the scattered wave amplitude in
both wind-ward and lee-ward sides of the cage system. For triple cage system
as in Fig. 13, at certain wavenumber, the constructive interference occurs
resulting in the increased wave amplitude in between the second and third cage
in the Fig. 13. In general, the wave amplitude in the triple cage system
decreases significantly when compared to the single and dual cage system due
to the increased rate of energy dissipation and multiple scattering between
the cages. Moreover, the wave amplitudes inside the cages are consequently
less in Fig. 14 when compared to Figs. 13 and 14. This shows that the fluid
circulation becomes smoother inside the cages with an increase in the number
of fish cages.

Figure 13: Wave scattering by dual fishing cages ($N=2$). The other parameters
are $G=3+3\mathsf{i}$, $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$, $k_{0}h=1.25$
and $\mu=10^{6}$N/m. Multimedia view:

Figure 14: Wave scattering by triple fishing cages ($N=3$). The other
parameters are $G=3+3\mathsf{i}$, $T=0.4$, $h/H=0.4$, $\beta=30^{\circ}$,
$k_{0}h=1.25$ and $\mu=10^{6}$N/m. Multimedia view:

##  5 Conclusions

The manuscript deals with the surface wave scattering by the system of
multiple fishing cages by employing the linear water wave theory and the
small-amplitude wave response. The general velocity potential in the form of
the Fourier-Bessel series is obtained while solving the Laplace equation.
Further, the matched eigenfunction expansion and least-square approximation
are implemented for solving the system of the linear equation obtained from
the interface and edge conditions. The numerical results, such as hydrodynamic
wave loads, far-field wave amplitudes, and power dissipation, are investigated
for different wave and cage parameters. Moreover, the effect of spatial
arrangements on the far-field wave amplitudes and flow distributions are
investigated. The following observations are made from the present study:

  * •

In the single fishing cage, the wave amplitude decreases on the lee-side of
cage due to the wave energy energy dissipation by the porous structure. The
energy dissipation increases gradually for the waves having larger wavenumbers
resulting in the less wave amplitude on the lee-side of the cage.

  * •

The wave scattering from the single fishing cage decreases when it connected
with the mooring spring of larger spring constant. This ultimately reduces the
hydrodynamic wave loads acting on the cage. Thus, the cage system connected
with the high stiffness spring reduces the damages caused by hydrodynamic wave
loading acting on the cage.

  * •

For the dual fishing cage, there is a constructive interference on placing the
two cages close enough and it decreases as the spacing between the cages
enlarges. This results in decreasing wave amplitude in the confined region
between the cages. However, the wave amplitude inside the cage decreases as a
consequence of destructive interference between an incoming wave and waves
scattered from an inner region of a cage.

  * •

The wave scattered from the dual fishing cage increases as compared to the
single cage, which increases for moderate values of tensile force. Further,
the membrane with less tension dissipates more energy for larger depth ratio
owing to the increased energy dissipation. Thus, this can also be employed as
the power absorber.

  * •

The deployment of the multiple fishing cage in different spatial arrangements
result in the oscillatory pattern in the scattering coefficients. In the case
of inline deployment, the periodic optima in the scattering coefficients are
more and approach zero for certain cage parameters resulting in the Fabry-
Perot effect at certain directions in the far-field. In the case of a square
arrangements of multiple cages, this effect becomes weaker. However, the
damages to the cage system are greatly avoided in square arrangement as a
consequence of multiple scattering between the cages and it further enables
the smooth water circulation inside the cage owing to the wave amplitude
reduction.

## Acknowledgment

HB gratefully acknowledges the financial support from SERB, Department of
Science and Technology, Government of India through “CRG” project, Award No.
CRG/2018/004521.

## Data Availability

The data that supports the findings of this study are available within the
article, highlighted in the related figure captions and corresponding
discussions.

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