## Abstract

In this paper, to determine the efficacy of the porous bed on damping far-field wave energy, the wave dynamics around a circular plate is studied. By combining the appropriate boundary conditions, the unknown potential is attained for the free surface and the plate-covered region. The Bessel series solution is attained further, by employing the matched eigenfunction expansion technique. Wave force excitation on the circular plate, deflection of the plate, and flow distribution is calculated and examined to comprehend the efficacy of the current investigation. Moreover, the motion of the plate is assessed in the time domain. The study reveals a substantial reduction in wave amplitude on the plate’s leeward side because of the energy dissipation by the porous bed. Also the study concludes that for intermediate values of porosity with larger wavenumbers, approximately 50% of wave power is dissipated with respect to incident wave power.

## 1 Introduction

A significant period in the historical evolution of floating breakwater technologies was the design and eventual construction of the floating breakwaters “Bombardon” Headland [1]. Floating elastic constructions are widely utilized as a breakwater for wave protection on gently sloping beaches, ports, and harbors. The important feature of floating structures is that they enable currents to freely travel through them, eliminating challenges like sediment accumulation near the structures. Floating breakwaters are an alternative option compared to regular permanent breakwaters to safeguard an area from wave attacks. Moreover, floating breakwaters are highly effective when their size is half the wavelength and/or if their normal oscillation duration is much larger than the wave period. The latest advances and technological innovations in the usage of very large floating structures (VLFS) for shoreline and inshore implementations were examined by Lamas-Pardo et al. [2]. Since they are lightweight, economical, reusable, and ecologically benign, floating flexible structures are utilized because they may enhance wave reduction through wave–structure interaction. Many experimental, theoretical, and numerical studies of the hydro-elastic behavior of floating plates have been conducted [3–6] and cited therein.

It can be clearly seen that there are lots of significant progress in the hydrodynamic analysis of floating and submerged elastic plates by different researchers [7–9]. Ohkusu [10] employed the boundary integral method to analyze the behavior of an elastic plate floating horizontally. Sturova [7] investigated the obliquely incident waves on an elastic band using Fox and Squire [11] method. Sahoo et al. [12] reported that the highest wave reflection and the minimal wave transmission can be attained in the case of built-in edge condition by considering the semi-infinite elastic plate. The Euler–Bernoulli model for the ice sheet in association with the potential flow has been used by Mohapatra et al. [13] to simulate the ice deflections. Lin and Lu [14] assessed the scattering of waves in a two-layer fluid by a thin floating semi-infinite plate and reported that the varying thickness of the structure affect the wavenumbers and the critical angles. Behera et al. [8] analytically examined the impact of a permeable bed on a floating elastic plate in both single- and two-layered fluids. Later, in the hydrodynamic analysis of pontoon-type VLFS, the modal expansion method is applied to determine the motion vibration [15]. The coalescence of the three wave modes for a specific value of compressive force at the saddle point was reported by considering a semi-infinite floating elastic plate [16]. Most recently, Boral et al. [9] investigated the performance of an elastic plate on a Winkler foundation, which is submerged within the framework of blocking wave dynamics.

Even though the wave action of an elastic plate in a homogeneous fluid has been studied in a two-dimensional setting, a 3D framework is required for implementation in ocean waves to use as an efficient breakwater to safeguard coastal regions from incident wave action [17–22]. Furthermore, circular structures are considered to be optimal for efficient utilization of ocean space and for facilitating future development operations as they are less prone to damage from waves and currents due to their streamlined shape [18,23]. Zilman and Miloh [24] studied the behavior of circular thin plate and found the solution for shallow water in closed form with the aid of Stoker’s shallow water theory. The stress resultants of a circular plate’s response were analyzed using the Mindlin circular plate theory [25]. The behavior of an external load on the plate was investigated by Sturova [26], whereas in the case of finite and infinite depth, the problem of the circular elastic plate was investigated by Andrianov and Hermans [27]. Pham et al. [28] concluded from his study that a submerged annular plate could be fixed to the perimeter of a circular VLFS to minimize the hydroelastic reaction. Due to its use in minimizing wave vibration at sheltered/harbor places during extreme storm/tide occurrences, the wave diffraction by an elastic plate was examined in both single-/two-layer fluids [19]. Furthermore, the motion of the circular plate was studied by Hua et al. [20] with the aid of finite-element method and concluded that as compared to stretching energy, bending energy is higher, which accounts for the major portion of potential energy. To examine wave scattering by a floating porous elastic disc, Meylan et al. [29] developed a theoretical and numerical model. Zheng et al. [30] used the conceptual model to examine a floating elastic plate of any shape. Later, the reduction of hydrodynamic load on the inner plate in the existence of the exterior membrane was studied by Gayathri and Behera [31].

In the aforementioned studies, the bottom of the sea was considered rigid; however, physically, the seabed is slippery, porous, or sloping. Recently, the effects of porous bottom on wave scattering by different marine structures of various configurations were studied by some researchers [8,32,33]. Furthermore, in their study, they used a new interface boundary condition between open water and porous region [8]. The different dissipation phenomena related to the porous media lead to wave dampening and therefore change the wave properties in the case of a permeable seabed [8]. In a single-layer fluid, Martha et al. [34] investigated the wave scattering by a sinusoidal undulation on the bottom using the Fourier transform technique. Belibassakis [35] developed a one-solution mild-slope equation to describe wave propagation over a permeable rippled substrate, based on the prior work by Athanassoulis and Belibassakis [36]. Maiti and Mandal [37] investigated the interaction of a single-layer fluid confined across a permeable bed with a finite elastic plate having symmetric edges. Behera et al. [8] studied the role of a porous bed for wave interaction with a floating elastic plate in both single-/two-layered fluids. Using the Laplace–Fourier transform technique, Kundu and Mandal [38] investigated axisymmetric gravity wave motion in the presence of a permeable bed. The scattering of waves by a surface-piercing barrier over a permeable bed was studied by Shagolshem et al. [32]. Their study also reports that a unique combination of barrier porosity and seabed porosity dissipates a significant amount of wave energy. Recently, Mohanty and Sidharth [33] used the spectral technique to study flexural gravity wave motion in the existence of current and the stationary phase method to get asymptotic solutions.

As stated above, a floating elastic structure can be used as a breakwater, solar panel, wave energy converter, etc., and the permeability of the bottom of the sea has a significant effect on wave damping. Therefore, it is necessary to consider the bottom of the sea as porous, as this remains lacking in this field and comprises the motivation for this study. The objective of this manuscript is to analyze the role of porous bottom on wave scattering by a circular elastic plate in finite water depth by employing the eigenfunction expansion method. The rest of the paper is structured as follows: The comprehensive formulation and solution of the physical problems are illustrated in Secs. 2 and 3, respectively. To comprehend the effectiveness of the suggested system, the influence of different physical parameters (wavenumber, rigidity of the plate, compressive force on the plate, permeability of the seabed, and radius of the plate), hydrodynamic load on the structure, flow distribution, and deflection of the plate is evaluated and examined in Sec. 4. In Sec. 5, the important observations are emphasized (Table 1).

List of symbols | |
---|---|

h | Water depth |

a | Plate radius |

d | Plate thickness |

E | Youngs’ modulus |

G | Seabed porosity |

g | Acceleration due to gravity |

H | Wave amplitude |

k_{n} | Wave number |

N | Compressive force |

ω | Angular frequency |

ϕ | Velocity potential |

ρ | Water density |

ρ_{p} | Plate density |

ν | Poisson ratio |

η_{1}, η_{2} | Free surface elevation |

List of symbols | |
---|---|

h | Water depth |

a | Plate radius |

d | Plate thickness |

E | Youngs’ modulus |

G | Seabed porosity |

g | Acceleration due to gravity |

H | Wave amplitude |

k_{n} | Wave number |

N | Compressive force |

ω | Angular frequency |

ϕ | Velocity potential |

ρ | Water density |

ρ_{p} | Plate density |

ν | Poisson ratio |

η_{1}, η_{2} | Free surface elevation |

## 2 Mathematical Formulation

*r*,

*θ*,

*z*) is defined, in which,

*r*−

*θ*and

*z*− are taken in horizontal and pointing downward negative directions, respectively. The radius of the plate is

*a*, which is floating at

*z*= 0. The sketch of the fluid domain is illustrated in Fig. 1. The fluid is modeled as irrotational, inviscid, and incompressible. The velocity potential of the flow field satisfies the following three-dimensional Laplace equation:

*z*= 0 can be written as

*η*

_{1}is the elevation in the open water region. The kinematic condition on the open water and the plate-covered region is given by

*G*is the porosity parameter of the sea bottom. The boundary condition at the mean free surface by eliminating

*η*

_{1}from Eqs. (2) and (3) is given by Mondal et al. [19]

*g*is the acceleration due to gravity. The radiation condition in the far field is given as

*H*being the incident wave amplitude,

*J*

_{m}is the first kind Bessel function of order

*m*for

*m*= 0, 1, 2, 3, …, and $\epsilon m=1$ when

*m*= 0, $\epsilon m=2im$ when

*m*= 1, 2, 3, …

*E*denotes the Young’s modulus,

*ρ*

_{p}represents the density, and

*I*=

*d*

^{3}/[12(1 −

*ν*

^{2})] with

*d*being the thickness of the plate and

*ν*indicates the Poisson ratio of the flexible circular plate. From (5) and (9), it can be found that

*η*

_{2}can be expressed as

*r*=

*a*for 0 <

*θ*< 2

*π*

## 3 Method of Solution

*ϕ*

_{j}for

*j*= 1, 2 satisfy Eq. (1), and the boundary conditions (Eqs. (2)–(5)) are expressed as

*m*= 0, 1, 2, …,

*J*

_{m}represents the Bessel function, and $Hm(1)$ denotes the Hankel functions, and

*I*

_{m}is the modified first kind of Bessel functions, respectively, of the first kind of order

*m*. The eigenfunctions

*f*

_{n}(

*z*) can be expressed as follows:

*k*

_{n}satisfying the dispersion relation

*k*

_{n}= 2

*π*/

*λ*, with

*λ*being the wavelength. The eigenfunctions

*g*

_{n}(

*z*) can be expressed as

*p*

_{n}for

*n*= 0, 1, 2, … satisfies the dispersion relation [8]

*K*=

*ω*

^{2}/

*g*, $D=EI/\rho g$, and

*M*=

*m*

_{p}/

*ρ*as in Ref. [37] and when the porous-effect parameter

*G*is complex, all the roots of the dispersion relation in Eq. (17) and Eq. (19) are complex in nature. The system of linear equations is attained by employing the matching conditions as in Eq. (7) along with the orthogonality to determine the constants

## 4 Results and Discussion

*h*= 50 m, the density of water

*ρ*= 1025 kg m

^{−3}, Poisson ratio

*ν*= 0.3, Young’s modulus

*E*= 1 GPa, plate density

*ρ*

_{p}= 922.5 kg m

^{−3}, and the dimensionless quantities such as dimensionless porosity of the seabed

*Gh*= 0.25, dimensionless plate thickness

*d*/

*h*= 0.07, dimensionless radius of the plate

*a*/

*h*= 2, compressive force $N=0.6EI\rho g$ are held constant unless it is highlighted in the appropriate figure’s caption. The hydrodynamic vertical load acting on the circular plate can be expressed as

### 4.1 Model Validation.

The verification of the method and code is conducted in Fig. 2 which illustrates the vertical wave force on the plate *V*_{p} against *k*_{0}*h* for a floating plate over a rigid seabed. The results from the current method are compared with those from the earlier study as a special case in the case of the rigid seabed by Selvan et al. [21]. It is clear that the current method is fairly accurate as the results from the present study and the previous studies are in very good agreement.

### 4.2 Wave Force on the Plate.

Figure 3 portrays the hydrodynamic wave load acting on the circular plate *V*_{P} over the range of non-dimensional wavenumber *k*_{0}*h* for varying edge conditions, seabed porosity *Gh*, and the radius of the plate *a*/*h*, respectively. Figure 3(a) reveals that the plate with a clamped edge exhibits a larger wave load than the plate with a free edge. The maxima of the hydrodynamic force shift toward the left because of the phase change caused by a change in the edge condition. The free edge condition is employed in the following analysis to study the impact of different physical quantities. Initially, for increasing values of porosity of the seabed, the vertical wave load on the plate increases for smaller wavenumber as seen in Fig. 3(b). For porous seabed (i.e., *Gh* > 0), the hydrodynamic load follows an oscillatory trend due to the dissipation of wave energy. However, for *Gh* = 0, the hydrodynamic load on the plate is high because of the absence of dissipation of incident wave energy by the impermeable seabed. For the smaller wavenumber, the distribution of wave force acts differently than when the wavenumber is larger (i.e., the wave force increases for higher values of *Gh* in the long wave region, whereas it shows an opposite pattern for the larger wavenumber). In addition, the minima of the wave forces occur because of the variability in the dissipation of wave energy by the porous seabed. The increase in wave force for smaller wavenumbers can be attributed to the increased interaction between the waves and the seabed as the porosity increases. Furthermore, in longwave, the force on the plate becomes more oscillatory as the porosity of the seabed increases. In addition, for shortwaves (*k*_{0}*h* > 1), the wave load becomes stable (i.e., the amplitude of the wave force reaches a steady-state value beyond a certain wavenumber and remains constant thereafter), it could be due to the fact that the seabed becomes more effective in absorbing the wave energy as its porosity increases.

On the other hand, it is evident from Fig. 3(c) that the hydrodynamic load increases for an increase in the radius of the plate. This is due to the larger surface area of the circular plate. For wavenumbers ranging 0 ≤ *k*_{0}*h* ≤ 0.5, the wave force increases with increasing wavenumber, attained a maximum value, and then decreased to a minimum value. Subsequently, for larger values of wavenumber, the wave force increases again. This phenomenon can be attributed to the interaction between the waves, plate, and seabed. As the radius of the plate increased, the surface area of the plate in contact with the waves also increased, resulting in a larger force being exerted by the waves on the plate. The oscillatory pattern observed in the wave force can be attributed to the interference of the incident and reflected waves, resulting in constructive and destructive interference at certain wavenumbers. However, for the smaller radius of the circular elastic plate (i.e., *a*/*h* = 0.5), the wave force is negligible due to the smaller surface area of the plate.

Figure 4 illustrates the hydrodynamic force against the wavenumber for increasing values of dimensionless plate thickness *d*/*h* and compressive force *N*. It is noticeable that as the thickness of the plate increases, the wave load on the plate decreases (Fig. 4(a)). As the plate thickness increases, it becomes stiffer and more resistant to deformation, and it is plausible that the structural dynamics and hydrodynamic behavior of the plate could be significantly altered, thereby affecting the interaction with the surrounding fluid and wave-induced response of the plate. This cause the plate to transmit less waves, which results in a decrease in the wave force on the plate. Also, the porous seabed can dissipate some of the wave energy, which can alter the fluid–structure interaction between the plate and the water. Furthermore, for an increase in the compressive force *N*, the vertical wave force increases as observed in Fig. 4(b). The wave force patterns resemble those seen in Fig. 4(a). The increase in the wave load might be owing to the fact that for increasing values of *N*, the compressive force exceeds the buckling limit of an elastic plate. Eventually, as mentioned in Refs. [21,39], the plate becomes more unstable. As the compressive force increases, the plate becomes stiffer and the deformation decreases, resulting in a more uniform response to the incident waves, and hence a more stable wave force occurs.

### 4.3 Power Dissipation.

*E*, attains maximum value of $10\u221220kg \, m\u22121s\u22123$. Therefore, the dimensionless power dissipation is represented as $E=k0Ediss/PIn$.

The contour plot in Fig. 5 portrays the wave power dissipation by a circular elastic plate floating over a porous seabed as a function of wavenumber *k*_{0}*h* and porosity of the seabed, for varying dimensionless radius *a*/*h*. From the sub-figures, it is evident that as the dimensionless radius increases, the amount of wave power dissipated by the floating plate also decreases. In the long wave region, when Re(*G*) varies, more energy is dissipated; this suggests that the drag force on the plate increases, which results in higher energy dissipation. This is likely due to the fact that smaller wavelengths interact more strongly with the plate, resulting in greater energy transfer from the wave to the plate. In addition, the drag force on a structure increases as the wave frequency increases, which is in accordance with that obtained for the wave force on the plate (see Fig. 3). At larger wavenumbers, the waves have shorter wavelengths and higher frequencies, which makes them more sensitive to changes in the properties of the seabed. In particular, a highly porous seabed can absorb more energy from the waves, leading to higher levels of energy dissipation. This is because the porosity of the seabed allows the waves to penetrate deeper into the bed, which leads to the dissipation of the wave energy. The contour plot clearly shows this effect, with an increase in the area of high energy dissipation values as the porosity of the seabed increases. Thus, the range of power dissipation by the circular elastic plate varies in the order 10^{−20} to 10^{−70}. On comparing with the maximum value of incident wave power (see Eq. (25)), approximately 50% of wave power is dissipated by the plate for a larger range of long wavenumber.

### 4.4 Flow Distribution Around the Circular Plate.

The effect of seabed porosity on the free surface elevation is plotted in Figs. 6(a) and 6(b). For a non-porous bed (*Gh* = 0), the wave amplitude is higher on the sea side of the structure, however, in the existence of porous seabed as shown in Fig. 6(b), the wave amplitude is less on the lee side of the plate. In addition, the deflection of the plate is consistent with the findings of Mondal et al. [19]. The results indicate that the flow around the plate is significantly affected by the type of seabed. It is noticeable that the flow around the plate is less in the case of a porous seabed as compared to a rigid seabed. This can be attributed to the fact that the porous seabed allows some of the incoming wave energy to be dissipated through the porous medium, resulting in reduced flow around the plate. Also, on increasing the *Gh* value, there is a dissipation due to the damping of wave energy around the plate resulting in a decrease in wave elevation on the leeward side of the plate.

In Figs. 7(a) and 7(b), the effect of plate thickness on the amplitude of surface wave elevation is plotted. It is noticeable from Fig. 7(a) that the circular plate having a smaller thickness (*d*/*h* = 0.01), the wave amplitude is more. On the other hand, for a plate with a larger thickness (*d*/*h* = 0.07), the circular plate transmits fewer waves. This is true since, as the plate thickness increases, the structural flexibility decreases, and the plate reflects more waves. From the sub-figures, it is evident that for smaller thicknesses, the flow distribution is quite irregular, with regions of high and low velocity appearing randomly around the plate. However, as the thickness of the plate increases, the flow distribution becomes more uniform, with the velocity of the water around the plate increasing steadily from the center of the plate toward the edges.

### 4.5 Circular Plate Deflection.

Figure 8 exhibits the deflection amplitude of the floating plate over an impermeable (*Gh* = 0) and permeable (*Gh* = 0.25) seabed. The amplitude of deflection is more in the case of a rigid bed as seen in Fig. 8(a). Conversely, the amplitude of the deflection reduces in the case of a permeable bed (*Gh* ≠ 0). From the sub-figures, it is evident that in the case of a rigid seabed, the deflection of the plate is symmetric, with the maximum deflection occurring at the center of the plate. However, in the case of a porous seabed, the maximum deflection occurs at one side of the plate. This can be attributed to the fact that the porous seabed causes a damping effect on the wave motion, leading to a reduction in the amplitude of the wave. The plate deflection is plotted for the different radii of the structure in Figs. 9(a) and 9(b). As the radius of the structure increases, the deflection of the circular flexible plate increases. This fact is noticeable owing to the increase in the surface area of the plate. As the radius of the plate increases, the wave energy is lost, resulting in reduced transmission on the leeward side of the structure. This stems because of the argument that the larger plate undergoes high wave force as observed in Fig. 3(b). Furthermore, as the radius of the plate increases, the wave force distribution around the plate becomes more symmetric. This can be observed in the deflection patterns, which become more circular in shape as the radius of the plate increases.

The plate deflection for different values of compressive force *N* is plotted in Fig. 10. In the absence of compressive force (*N* = 0), the wave amplitude and the plate deflection are more. Whereas, in the presence of compressive force (*N* = 0.6(*EIρg*)^{1/2}), the wave amplitude and the plate deflection reduce. Moreover, with an increase in compressive force beyond *N* = 1.4(*EIρg*)^{1/2}, the values of compressive force become close to the buckling limit and the irregular pattern may account for the initiation of structural instability (as in Refs. [19,40]). It can be concluded from the sub-figures that the presence or absence of compressive forces in wave–structure interaction problems has significant engineering implications. When compressive forces are present, the structural response is dominated by the bending deformation of the structure. This is because the porous seabed resists the plate’s downward motion due to the wave action, resulting in additional drag force and a reduction in the wave force on the plate. Conversely, when compressive forces are absent, the response is dominated by the stretching deformation of the structure, with bending and stretching deformations being associated with different modes. At the same time, in this case, the plate is free to move downward due to wave action, with no additional drag force due to the porous seabed, leading to an increase in the wave force on the plate. Thus, considering the compressive force within the buckling limit of the plate allows us to design floating structures, such as offshore platforms, buoys, and ships.

Figures 11 and 12 illustrate the deflection amplitude of the plate for an increasing dimensionless plate thickness (*d*/*h*) and Young’s modulus (*E*). For an increase in the thickness of the plate, the plate deflection decreases as observed in Fig. 4(a). This is because the increase in plate thickness necessitates a decrease in the flexibility of the structure; furthermore, the plate reflects more waves. The deflection was found to decrease as Young’s modulus *E* increases (Fig. 12). This behavior can be attributed to the fact that the plate becomes more rigid and less flexible with increasing Young’s modulus. Furthermore, for larger Young’s modulus, the plate becomes more resistant to bending and resulting in less deflection.

In Fig. 13, the results are presented for a different edge condition of a circular plate in terms of deflection. The results showed that the deflection amplitude of the plate is largest for the clamped edge case, as illustrated in Fig. 13(a), indicating that the additional support provided by the clamped edges allows for a larger deflection of the plate. This observation can be explained by the fact that the clamped edges restrict the movement of the plate in the radial direction, which results in a greater deformation of the plate in the vertical direction. These findings have practical implications for the design of floating structures, particularly those that require a large deflection amplitude, such as wave energy converters. On the other hand, the deflection amplitude is intermediate for the free edge case, as shown in Fig. 13(b). This stems that the lack of support at the edges leads to a reduced deflection of the plate compared to the clamped edge case. This is consistent with our understanding that the edges of the plate experience less pressure from the incident wave compared to the center of the plate. These findings have practical implications for the design of floating structures that require a moderate deflection amplitude, such as offshore platforms. In contrast, the deflection amplitude is less for the simply supported edge case, as depicted in Fig. 13(c), suggesting that the constraint imposed by the support edges results in a more rigid response of the plate. This behavior is characterized by a symmetric deflection shape, with the maximum deflection occurring at the center of the plate. This observation can be attributed to the fact that the simply supported edges provide a partial constraint on the plate’s movement, which limits the deformation of the plate in both the radial and vertical directions.

### 4.6 Flow Field.

In Fig. 14, instantaneous velocity field in regions 1 and 2 as shown in Fig. 1 is plotted for increasing values of *Gh*. Due to the interaction of waves, it can be noticed that the flow around the plate is more toward the windward side. For the case of a rigid seabed, the velocity field exhibits a symmetric pattern about the vertical axis passing through the center of the plate. The maximum velocity is observed at the crest of the incoming wave and decreases toward the trough. However, in the case of a porous seabed, the presence of porosity affects the velocity distribution around the plate. The maximum velocity is observed on the upstream side of the plate, where the waves enter the porous seabed. As the waves pass through the seabed, the velocity decreases due to the resistance offered by the seabed. Thus, as compared to the non-porous seabed, the wave transmitted on the leeward side of the plate is less in the case of a porous seabed. This is evident from Fig. 14(b), where well-spaced arrows and the values denote the flow and velocity, illustrating less transmission on the leeward side.

Figure 15 illustrates the velocity field around the plate for increasing values of compressive force of the plate. It is evident from the results that as the compressive force acting on the plate increases, the velocity field becomes more intricate. The maximum velocity still appears on the upstream side of the plate, but it is not solely influenced by the wave crest. The plate deformation resulting from the increased compressive force also contributes to higher velocities. Moreover, the flow of waves through the seabed becomes more turbulent and disordered, causing variations in the velocity field. As the compressive force increases, the plate becomes stiffer and the deformation decreases. This suggests that the compressive force plays a significant role in determining the velocity field around the floating plate.

### 4.7 Time Simulation of Solution.

*η*) is computed using the following transformation rule:

*j*= 1, 2, and $\u211c$ denotes the real part. Furthermore, the parameters such as $k\xafc$ and

*s*are the center frequency and spreading function, respectively. The values have been fixed as

*s*= 1 and

*k*

_{c}= 3/

*h*for simulation.

The time simulation of the scattering of waves by a floating circular plate is shown as static in Figs. 16(a)–16(d). The videos that correspond to Figs. 16(a)–16(d) are given in a multimedia file. As the wave interacts with the circular plate (Fig. 16), the wave amplitude reduces on the lee side of the structure. Furthermore, the deflection of the plate increases due to the impermeability of the sea bed. This occurs as a result of the partial reflection by the plate. In the case of a permeable bed for *Gh* = 0.25 (Fig. 17), the wave amplitude is less than the rigid bed because of the dissipation of wave energy. As a consequence, the scattered wave amplitude in both the windward and leeward sides of the plate reduces. Thus, it is evident from Fig. 12 that the plate deflection is less as compared to Fig. 16. Thus, Figs. 16 and 17 show that the circular plate floating over a porous bed dissipates more wave energy than the rigid bed. On the other hand, from Fig. 18, it is noticeable that in the presence of compressive force, the bending deformation of the structure becomes the predominant mode of structural response. This is attributed to the resistance offered by the porous seabed against the plate’s downward motion caused by wave action, leading to the generation of additional drag force and a decrease in the wave force acting on the plate.

## 5 Conclusion

A semi-analytical model is used in this study to examine the wave dynamics around a floating elastic circular plate over a porous bed. Based on the linear potential flow theory, the eigenfunction expansion method is applied in this analysis. The impacts of wave conditions, plate thickness, the radius of the plate, and compressive force on wave scattering are investigated through the wave load on the plate, flow distribution around the plate, and time simulation analysis. The study reveals that the permeability of the seabed plays a significant role in mitigating the wave force on the plate. Moreover, for an increase in the permeability of the seabed and the radius of the plate, the hydrodynamic load on the plate decreases in an oscillatory pattern. In addition, the flow field and the deflection results reveal that the amplitude of the wave on the leeward side of the plate is significantly decreased because of the dissipation of wave energy by the porous seabed. Due to the presence of the porous seabed, energy dissipation occurs, which results in a serene environment on the lee side of the plate. Finally, it can be concluded from the time simulation results that the circular plate floating over a porous bed reduces significantly the wave amplitude on the lee side of the plate as compared to that for a rigid bed. In addition, increasing the values of compressive force results in more wave force on the plate and high amplitude on the lee side of the plate. Hence, it is crucial to consider compressive forces when designing floating structures, such as offshore platforms, buoys, and ships. The study concludes that for intermediate values of porosity with larger wavenumbers, a range of 10^{−20} to 10^{−70} of wave power can be dissipated. The findings of this result, which demonstrate the effect of porosity on compressive forces, can offer valuable insights for designing such structures in porous seabed environments. Thus, the existence of seabed porosity, moderate compressive force, and plate radius minimizes the damages induced by the incident wave energy.

## Acknowledgment

CCT gratefully acknowledges the financial support from the National Science and Technology Council of Taiwan (Grant No. MOST 109-2221-E-019-061-MY3).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.