# Linear diffraction and radiation theory of water waves by a truncated vertical circular cylinder with heave plate in deep and shallow drafts

Seung-Yoon Han, Benjamin Bouscasse, Vincent Leroy, David Le Touzé

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Seung-Yoon Han, Benjamin Bouscasse, Vincent Leroy, David Le Touzé. Linear diffraction and radiation theory of water waves by a truncated vertical circular cylinder with heave plate in deep and shallow drafts. International Journal of Naval Architecture and Ocean Engineering, 2024, 16, pp.100580. 10.1016/j.ijnaoe.2023.100580. hal-04616811

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# Linear diffraction and radiation theory of water waves by a truncated vertical circular cylinder with heave plate in deep and shallow drafts

Seung-Yoon Han*, Benjamin Bouscasse, Vincent Leroy, David Le Touzé

Nantes Université, École Centrale Nantes, CNRS, LHEEA, UMR 6598, F-44000 Nantes, France

## ARTICLE INFO

**Keywords:**

Linear diffraction/radiation problems

Potential flow theory

Eigenfunction expansion method

Boundary element method

Model testing

Heave plate

Floating offshore wind turbine

Deep and shallow drafts

## ABSTRACT

A first-order analytical theory on a truncated and surface-piercing vertical circular cylinder with a circular plate mounted at the bottom of the cylinder was generalized to solve the linear wave diffraction and radiation problems based on potential flow and the hypothesis of small wave and motion amplitudes. The domain was decomposed, and the linearized velocity potentials were derived in each subdomain and matched on each subdomain interface employing pressure and normal velocity continuity. The linear hydrodynamic loads obtained with the analytical method were compared with the results of linear boundary element method (BEM) solvers. Dedicated experimental campaigns were performed on fixed models with regular waves (diffraction) and then on models oscillating in surge, heave, and pitch motions without incident waves (radiation). The analytical method describes well the wave excitation loads from the experiments for small wave steepness ($H/λ = 0.02$). The predictions of added mass and damping are shown to be applicable to the surge motion, only for small Keulegan-Carpenter numbers ($KC < 1$). On the other hand, the analytically predicted radiated waves demonstrate satisfactory agreement with experiments for all motions.

## 1. Introduction

In marine hydrodynamics, potential flow theory is well established by assuming that the fluid is incompressible and inviscid, and that the flow is irrotational. In a linear context combining small wave amplitude and motion displacement, the entire wave-body interaction problem can be split into two typical subproblems: diffraction and radiation problems (e.g., see Faltinsen (1990)). These subproblems correspond to a fixed body in incident waves and an oscillating body in calm water. In diffraction problems, the hydrodynamic loads are the result of the interaction between the body and the incident waves, known as wave diffraction loads. The radiation problems describe the hydrodynamic loads caused by the body motions. The loads are typically interpreted as added mass and damping representing the load's components proportional to the acceleration and velocity, respectively.

The diffraction-radiation problems have been extensively studied for the conventional cylindrical structure commonly used in ocean engineering for fixed or floating foundations. A monopile substructure installed at the sea bed is an example represented by a bottom-mounted circular cylinder, and floating substructures such as TLP (Tension Leg Platform), SPAR, and semi-submersible platforms can also be characterized

as truncated circular cylinders. These fundamental shapes have long been studied by hydrodynamicists to understand better the complex mechanisms involving both linear and nonlinear hydrodynamics.

For the bottom-mounted vertical circular cylinder, MacCamy and Fuchs (1954) introduced a mathematical model to solve the diffraction problem in finite depth. Malenica and Molin (1995) developed semi-analytical solutions up to the third order to compute the nonlinear horizontal wave forces. Faltinsen et al. (1995) introduced nonlinear diffraction theory, so-called FNV methods, to evaluate the nonlinear horizontal loads up to the third-order harmonics on the cylinder at infinite water depth. Later, Kristiansen and Faltinsen (2017) generalized the FNV methods to an arbitrary order of harmonics in finite water depth. They focused on the third harmonic load and showed good agreement with their regular wave experiments for small and medium wave steepnesses.

On the other hand, the truncated circular cylinder, which can be considered a floating body, introduces several challenges related to diffraction and radiation. Garrett (1971) presented the analytical model for the diffraction problem of a truncated vertical circular cylinder to evaluate the first-order horizontal and vertical forces and moments. The fluid domain was separated into external and internal domains. In addition

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: seung-yoon.han@ec-nantes.fr (S.-Y. Han).</footnote>
tion, matching conditions were implemented on the interface to ensure the continuity of pressure and normal velocity. This method was named the matched eigenfunction expansions method (MEEM) (e.g. see Linton and McIver (2001)). Later, attempts to solve the non-linear problem were made by Huang and Taylor (1996) and Teng and Kato (2002) up to the second and third order.

Regarding the radiation problem, Yeung (1981) computed the added mass and damping coefficients on the truncated vertical circular cylinder by solving the radiation problem analytically. A particular solution was presented to satisfy the body boundary condition due to the body motions. A similar explanation for the heave motion was given by McIver and Evans (1984), where the occurrence of negative added mass is discussed using kinetic and potential energy.

Bhatta and Rahman (2003) presented the generalized solutions for the radiation and diffraction problems on the truncated cylinder. The computational results of wave excitation, added mass, and damping were provided in Bhatta (2011). Malenica (2012) summarized the generalized solutions by adding various physical aspects regarding vertical cylinders, including hydroelasticity, array of cylinders, and small forward speed. The analytical expressions of the cylinder for submerged bottom-standing and floating conditions can be found in Li and Liu (2019). Additionally, analytical solutions for other shapes, such as a hemisphere (Hulme (1982)) and a sphere (Linton (1991)), are also available in the literature.

Recently, some analytical studies have been done on a circular plate attached to the truncated cylinder. This so-called “heave plate” has been widely used on floating platforms to reduce wave-induced motion responses, particularly heave motion, by increasing the added mass and damping (Haslum and Faltinsen (1999)). The heave plate is now frequently used in floating offshore wind turbines as well. Cong et al. (2019) focused on the diffraction problem to evaluate analytically the first-order wave excitation and the mean drift force. Koh and Cho (2016) solved the heave radiation problem on the truncated cylinder with dual damping plates based on the MEEM. Moreau et al. (2022) presented the analytical expressions of diffraction and radiation to solve the motion responses of the bottomless circular cylinder with a moonpool and bilge boxes by introducing particular solutions on the box regions. The analytical models were successfully adjusted to match their experimental measurements by including additional viscous damping, which is not considered in the potential flow theory.

The viscous effects of heave plates with the vertical cylinder were examined through experiments or numerical analysis by focusing on heaving motions in Thiagarajan et al. (2002); Tao and Thiagarajan (2003); Tao et al. (2007); Lopez-Pavon and Souto-Iglesias (2015). Considerable knowledge has been obtained on hydrodynamic loads, specifically on the added mass and damping governed by flow parameters (e.g., Keulegan-Carpenter number, KC, and Reynolds number, Re) and geometrical parameters (e.g., diameter, thickness, and porosity of the plate). In general, viscous damping of the heave plate is obtained by assuming no wave contribution to the total (or equivalent) damping. This approach is appropriate in the context of classical offshore structures, such as SPAR and TLP, which typically operate at deep submerged depths. For floating wind turbines, however, the wave damping induced by the heave plate is significant because free surface interactions occur due to relatively shallow depths. Therefore, the damping needs to be carefully computed using potential flow theory to obtain the precise contribution of viscous damping.

In engineering applications, boundary element methods (BEMs) based on boundary integral equations with free-surface green functions are practically used to solve the diffraction-radiation problem. The methods involve discretizing the body boundary into a finite number of panels, which allows for handling complex body geometries. Various forms of the integral equation were presented by Lee et al. (1996) and Malenica and Chen (1998) to eliminate the irregular frequency, which is a numerical problem related to the interior boundary value problem.

Recently, He et al. (2021) proposed a combined boundary integral equation method based on two flow models for more efficient computation.

Numerical methods are widely used in hydrodynamic modeling for various applications, including evaluating the performance of wave energy converters (WECs) (Kim and Cho (2021); Cheng et al. (2022a,b)). A comparative study of open-source codes (Nemoh and HAMS) and a commercial code (WAMIT) was conducted by Sheng et al. (2022) to guide the hydrodynamic modeling of WECs. The study focused on fundamental cylindrical structures and concluded that open-source software generally provides accurate results. However, the codes gave different hydrodynamic coefficients for the thin heave plate due to the different numerical schemes employed. Therefore, hydrodynamic modeling of the thin plate remains a challenge.

The objective of this study is to establish an analytical approach for robust hydrodynamic modeling of heave plates. We present generalized first-order analytical solutions for a truncated circular cylinder with a heave plate mounted at the bottom of the cylinder. The solutions cover the linear diffraction and radiation of water wave problems, including surge, heave, and pitch motions. Linearized velocity potentials in the regime of small wave and motion amplitudes are presented on the decomposed fluid domain. The problems were solved using the matched eigenfunction expansion method (MEEM) following a similar approach to that of Koh and Cho (2016), Cong et al. (2019), and Moreau et al. (2022). The first-order hydrodynamic loads were calculated for the cylinder with thick and thin plates and verified using BEM solvers.

Dedicated experiments were carried out in the hydrodynamic and ocean engineering tank at Ecole Centrale Nantes for the circular cylinder with and without the thin plate to identify the region of validity of the analytical method. Two experimental campaigns were conducted to measure the hydrodynamic loads induced by waves and motions: one with a fixed body in incident waves and the other with a forced oscillating body in the absence of incident waves. A comparison was made with small wave steepness ($H/\lambda = 0.02$) and motion amplitudes ($KC < 1$) to minimize the nonlinear effects. Results from the experiments and the analytical solutions are presented and discussed, focusing on the physical quantities of interest: the wave excitation loads, the added mass, damping, and radiated wave elevations.

## 2. Mathematical description

A truncated circular cylinder with a circular plate mounted at the bottom of the cylinder is considered as shown in Fig. 1. The fluid domain, $\Omega$, is filled with an inviscid and incompressible fluid, and the velocity field is assumed irrotational. The right-handed and body-fixed coordinate system is defined on the undisturbed free surface (at $z = 0$) with the z-axis pointing upward. The incident waves propagate in the positive x-direction.

The geometry of the body is represented by three parameters: the radius of the cylinder $R_c (=D_c/2)$, the radius of the plate $R_d (=D_d/2)$, and the thickness $t_d$. Here, $D_c$, and $D_d$ stand for the diameter of the cylinder and the plate, respectively. The body is immersed at the draft $d$, and the distance from the undisturbed free surface to the top of the heave plate is $d_c (=d - t_d)$. The bottom of the sea is assumed to be flat and placed at the water depth, $h$.

### 2.1. Governing equation and boundary conditions

With the assumptions of potential flow and periodicity of the problem with frequency $\omega$, the flow is determined by the complex velocity potential $\Phi(r, \theta, z, t) = \phi(r, \theta, z)e^{-i\omega t}$. The spatial function, $\phi$, satisfies the following governing equation in the fluid domain, $\Omega$,

$$ \nabla^2 \phi(r, \theta, z) = 0 \quad \text{in } \Omega \qquad (1) $$

where $\nabla^2$ is the Laplace operator in the cylindrical coordinate.

To solve the governing equation, the boundary conditions are established on the free surface ($S_F$), the bottom boundary ($S_b$), and the body
<figure><img src="image_5.png" /><figcaption>Fig. 1. Description of the problem for a truncated circular cylinder with a circular plate mounted at the bottom.</figcaption></figure>

boundary ($S_B$). After linearization using Taylor series expansion by assuming small wave amplitude and motion displacement, the boundary conditions can be obtained with respect to the mean position:

$$-\frac{\omega^2}{g} \phi + \frac{\partial \phi}{\partial z} = 0, \quad z = 0 \quad \text{on } S_F \qquad (2)$$

$$\frac{\partial \phi}{\partial z} = 0, \quad z = -h \quad \text{on } S_b \qquad (3)$$

$$\frac{\partial \phi}{\partial n} = \mathbf{V} \cdot \mathbf{n} \quad \text{on } S_B \qquad (4)$$

$$\lim_{k_0 r \to \infty} r^{1/2} \left( \frac{\partial \phi}{\partial r} - i k_0 \phi \right) = 0 \quad \text{on } S_{\infty} \qquad (5)$$

where $\mathbf{V}$ is the velocity vector of the moving body, $\mathbf{n}$ is the normal vector pointing out of the fluid domain, $k_0$ is the wave number, and $g$ stands for the gravitational acceleration.

The velocity potential $\phi$ is decomposed into the sum of the diffraction potential $\phi_D$ and the radiation potential $\phi_R$.

$$\phi(r, \theta, z) = \phi_D(r, \theta, z) + \phi_R(r, \theta, z) \qquad (6)$$

The body boundary condition in Equation (4) becomes two body boundary conditions after using the linear superposition expressed as,

$$\frac{\partial \phi_D}{\partial n} = 0 \quad \text{on } S_B \qquad (7)$$

$$\frac{\partial \phi_R}{\partial n} = \mathbf{V} \cdot \mathbf{n} \quad \text{on } S_B \qquad (8)$$

where the boundary condition in Equation (7) is for a diffraction problem, in which the structure is fixed in the incident waves, and Equation (8) is for a radiation problem, in which the body is oscillating harmonically in the absence of incident waves.

In this paper, the general solutions to the diffraction and radiation problems are developed using domain decomposition methods. The entire fluid domain is divided into three subdomains, which are one outer domain ($\Omega_I$) and two inner domains ($\Omega_{II}$ and $\Omega_{III}$) shown in Fig. 2. Two additional boundary conditions are introduced on the interface between the outer and inner domains (at $r = R_d$) through the continuity of pressure and normal velocity.

## 2.2. Diffraction problem

This section introduces the general solution to the diffraction problem expressed in each of the three domains. The diffraction potential that satisfies the governing equation in Equation (1) can be expressed

<figure><img src="image_6.png" /><figcaption>Fig. 2. Decomposed fluid domain.</figcaption></figure>

as an eigenfunction expansion considering the boundary conditions of each domain.

In the outer domain, $\Omega_I$, the diffraction potential is further decomposed into the incident wave potential and perturbed potential,

$$\phi_D^I(r, \theta, z) = \phi_I^I(r, \theta, z) + \phi_P^I(r, \theta, z) \qquad (9)$$

with the superscript indicating the fluid domain.

The incident wave potential of constant finite depth is given in cylindrical coordinates $(r, \theta, z)$ as,

$$\phi_I^I(r, \theta, z) = -\frac{igA}{\omega} \sum_{m=0}^{\infty} \epsilon_m i^m J_m(k_0 r) \frac{\cosh k_0 (z + h)}{\cosh k_0 h} \cos m\theta \qquad (10)$$

where $A$ is the wave amplitude, $J_m$ is a Bessel function of the first kind with the order $m$, and $\epsilon_m$ is the Neumann symbol defined by $\epsilon_0 = 1$ and $\epsilon_m = 2, m \ge 1$.

The perturbed wave potential satisfies the radiation condition and can be expressed with an eigenfunction expansion:

$$\phi_P^I(r, \theta, z) = -\frac{igA}{\omega} \sum_{m=0}^{\infty} \epsilon_m i^m \left[ \sum_{n=0}^{\infty} A_{mn} U_{mn}(r) f_n(z) \right] \cos m\theta \qquad (11)$$

where $A_{mn}$ are the unknown coefficients that the matching conditions will determine.

The functions $U_{mn}(r)$ are defined by:

$$U_{mn}(r) = \begin{cases} H_m^{(1)}(k_0 r), & n = 0 \\ K_m(k_n r), & n \ge 1 \end{cases} \qquad (12)$$

where $H_m^{(1)}$ is a Hankel function of the first kind and $K_m$ is a modified Bessel function of the second kind.

The basis functions $f_n(z)$ that are orthogonal over the interval $-h < z < 0$ are given as,

$$f_n(z) = \begin{cases} \cosh k_0(z + h) / \cosh k_0 h, & n = 0 \\ \cos k_n(z + h) / \cos k_n h, & n \ge 1 \end{cases} \qquad (13)$$

where $k_n$ are the eigenvalues determined by the dispersion relation $\omega^2/g = k_0 \tanh k_0 h = -k_n \tan k_n h$. The eigenvalues, $k_n$, correspond to the propagating ($n = 0$) and the evanescent wave modes ($n \ge 1$), respectively.

In the domain $\Omega_{II}$, the diffraction potential is different from that of $\Omega_I$, as the top boundary condition is not a free-surface condition but a body boundary condition given by the plate:
$$ \phi_D^{II}(r, \theta, z) = -\frac{igA}{\omega} \sum_{m=0}^{\infty} \epsilon_m i^m \left[ \sum_{n=0}^{\infty} B_{mn} V_{mn}(r) Z_n(z) \right] \cos m\theta \quad (14) $$

where $B_{mn}$ are the unknown coefficients, and the functions $V_{mn}(r)$ are given as,

$$ V_{mn}(r) = \begin{cases} (r/R_d)^m, & n=0 \\ I_m(\alpha_n r)/I_m(\alpha_n R_d), & n \ge 1 \end{cases} \quad (15) $$

where $I_m$ is the modified Bessel function of the first kind. The vertical basis functions, $Z_n(z) = \cos \alpha_n (z + h)$, are orthogonal in the interval from $-h$ to $-d$, where $\alpha_n = n\pi/(h-d)$ are the eigenvalues.

Lastly, the diffraction potential of the remaining domain ($\Omega_{III}$) is written as an eigenfunction expansion,

$$ \phi_D^{III}(r, \theta, z) = -\frac{igA}{\omega} \sum_{m=0}^{\infty} \epsilon_m i^m \left[ \sum_{n=0}^{\infty} (C_{mn} X_{mn}(r) + D_{mn} W_{mn}(r)) F_n(z) \right] \cos m\theta \quad (16) $$

where $C_{mn}$ and $D_{mn}$ are the unknown coefficients, and the functions $X_{mn}(r)$, $W_{mn}(r)$ are given as

$$ \begin{aligned} X_{mn}(r) &= \begin{cases} J_m(\beta_0 r)/J_m(\beta_0 R_c), & n=0 \\ I_m(\beta_n r)/I_m(\beta_n R_c), & n \ge 1 \end{cases} \\ W_{mn}(r) &= \begin{cases} Y_m(\beta_0 r)/Y_m(\beta_0 R_c), & n=0 \\ K_m(\beta_n r)/K_m(\beta_n R_c), & n \ge 1 \end{cases} \end{aligned} \quad (17) $$

where $Y_m$ is a Bessel function of the second kind.

The vertical basis functions, $F_n(z)$, are orthogonal within the interval of $-d_c < z < 0$ in $\Omega_{III}$.

$$ F_n(z) = \begin{cases} \cosh \beta_0 (z + d_c) / \cosh \beta_0 d_c, & n = 0 \\ \cos \beta_n (z + d_c) / \cos \beta_n d_c, & n \ge 1 \end{cases} \quad (18) $$

where the eigenvalues $\beta_n$ are the roots of $\omega^2/g = \beta_0 \tanh \beta_0 d_c = -\beta_n \tan \beta_n d_c$.

## 2.3. Radiation problem

In the radiation problem of the frequency domain, the motions are given as,

$$ \xi(t) = \operatorname{Re} \left\{ \xi_j e^{-i\omega t} \right\} \quad (19) $$

where $\xi_j$ stands for a complex amplitude of motion in the j-th degree of freedom; here, we restrict the degrees of freedom to the surge, heave, and pitch motions ($j = 1, 3, 5$).

The radiation potential, $\phi_R$, can be expressed by linear superposition to handle the body boundary condition in Equation (8) as:

$$ \phi_R(r, \theta, z) = \sum_{j=1,3,5} -i\omega\xi_j \phi_j(r, z) \cos m\theta \quad (20) $$

where $\phi_j$ denotes a unit amplitude velocity potential in the j-th degree of freedom.

After introducing body velocity and $\phi_j$ into the body boundary condition, we obtain

$$ \frac{\partial \phi_j}{\partial n} \cos m\theta = n_j \quad (21) $$

where the normal vector is defined, $n = (n_1, n_2, n_3)$ and $r \times n = (n_4, n_5, n_6)$ and $r$ is the position vector with respect to the reference point, $O$.

The general solution in each of the three domains is almost identical to the solution of the diffraction problem, with the difference that a particular solution $\psi_j(r, z)$ is needed for heave and pitch motions to satisfy the inhomogeneous body boundary conditions due to the presence

of a forcing term. Compared to the previous diffraction problem, the solution will be considered only with $m=0$ for heave (axisymmetric) and $m=1$ for surge and pitch motions, thanks to the cosine functions appearing in the body boundary conditions of the radial direction.

The boundary value problem of the exterior domain ($\Omega_I$) is expressed as,

$$ \begin{aligned} \nabla^2 \phi_j^I &= 0 \quad \text{in } \Omega \\ -\frac{\omega^2}{g} \phi_j^I + \frac{\partial \phi_j^I}{\partial z} &= 0 \quad \text{on } z=0 \\ \frac{\partial \phi_j^I}{\partial z} &= 0 \quad \text{on } z=-h \quad (22) \\ \frac{\partial \phi_j^I}{\partial r} &= Q_j^I = \begin{cases} 1, & j=1 \\ 0, & j=3 \\ z, & j=5 \end{cases}, \quad \text{on } r=R_d \end{aligned} $$

The linear velocity potential for the exterior domain ($\Omega_I$) is very similar to the solution of the diffraction problem in Equation (11),

$$ \phi_j^I(r, z) = \epsilon_m \sum_{n=0}^{\infty} A_{mn} U_{mn}(r) f_n(z) \quad (23) $$

where the radiation condition is satisfied.

The interior domain ($\Omega_{II}$) consists of the seabed ($z=-h$) and the body surface ($z=-d$). The boundary value problem is,

$$ \begin{aligned} \nabla^2 \phi_j^{II} &= 0 \quad \text{in } \Omega \\ \frac{\partial \phi_j^{II}}{\partial z} &= 0 \quad \text{on } z=-h \quad (24) \\ \frac{\partial \phi_j^{II}}{\partial r} &= \begin{cases} 0, & j=1 \\ 1, & j=3 \\ -r, & j=5 \end{cases}, \quad \text{on } z=-d \end{aligned} $$

where the inhomogeneous body boundary conditions need to be handled using particular solutions, especially for the heave and pitch motions.

The velocity potential in domain $\Omega_{II}$ is written as the sum of the homogenous solution and the particular solution $\psi_j^{II}(r, z)$ to satisfy the body boundary condition,

$$ \phi_j^{II}(r, z) = \epsilon_m \sum_{n=0}^{\infty} B_{mn} V_{mn}(r) Z_n(z) + \psi_j^{II}(r, z) \quad (25) $$

where the functions $V_{mn}(r)$ and $Z_n(z)$ are shown in Equation (15).

The particular solutions, $\psi_j^{II}(r, z)$, of the surge, heave, and pitch satisfying the governing equation, the bottom and body boundary conditions at $z=-d$, and $h$ are given,

$$ \psi_j^{II}(r, z) = \begin{cases} 0, & j=1 \\ \frac{-1}{4(h-d)} [r^2 - 2(z+h)^2], & j=3 \\ \frac{1}{8(h-d)} [r^3 - 4r(z+h)^2], & j=5 \end{cases} \quad (26) $$

Similar expressions for the particular solutions can be found in Bhatta (2011); Koh and Cho (2016) for heave and Moreau et al. (2022) for pitch motions.

The boundary value problem in domain $\Omega_{III}$ consists of the governing equation, the free-surface boundary condition, and the body boundary conditions on the cylinder and the heave plate surface.

$$ \begin{aligned} \nabla^2 \phi_j^{III} &= 0 \quad \text{in } \Omega \\ -\frac{\omega^2}{g} \phi_j^{III} + \frac{\partial \phi_j^{III}}{\partial z} &= 0 \quad \text{on } z=0 \end{aligned} $$
$$ \frac{\partial \phi_j^{III}}{\partial z} = \begin{cases} 0, & j = 1 \\ 1, & j = 3 \\ -r, & j = 5 \end{cases}, \quad \text{on } z = -d_c \qquad (27) $$

$$ \frac{\partial \phi_j^{III}}{\partial r} = Q_j^{III} = \begin{cases} 1, & j = 1 \\ 0, & j = 3 \\ z, & j = 5 \end{cases}, \quad \text{on } r = R_c $$

The general solution of the potential can be written as,

$$ \phi_j^{III}(r, z) = \epsilon_m \sum_{n=0}^{\infty} \left[ C_{mn} X_{mn}(r) + D_{mn} W_{mn}(r) \right] F_n(z) + \psi_j^{III}(r, z) \quad (28) $$

where the functions $X_{mn}(r)$, $W_{mn}(r)$ and $F_n(z)$ are already given in the diffraction problem.

The particular solution, $\psi_j^{III}(r, z)$, satisfying the governing equation, the body boundary condition at $z = -d_c$ and the free-surface boundary condition is expressed as,

$$ \psi_j^{III}(r, z) = \begin{cases} 0, & j = 1 \\ z + \frac{g}{\omega^2}, & j = 3 \\ -(z + \frac{g}{\omega^2})r, & j = 5 \end{cases} \qquad (29) $$

where the particular solution is also discussed in Koh and Cho (2016) for heave and Moreau et al. (2022) for pitch motions.

## 2.4. Determination of the unknown coefficients

The general solutions developed with a series of eigenfunctions are matched on the interface based on the hypothesis of normal velocity and pressure continuity (see Garrett (1971)). The unknown coefficients in the diffraction and radiation potentials ($A_{mn}$, $B_{mn}$, $C_{mn}$, $D_{mn}$), are obtained by the linear sets of the equations from the velocity and pressure continuity. The infinite number of series in the eigenfunction expansions is truncated as a finite number. Here, we use different numbers of eigenmodes, $N$, in each domain as $N_1$, $N_2$, and $N_3$ since each subdomain shows a different convergence behavior to $N$, as reported in Moreau et al. (2022).

The first set of equations can be obtained using the continuity of the normal velocity as well as the body boundary condition on $r = R_d$ and $-h < z < 0$.

$$ \int_{-h}^{0} \frac{\partial \phi^I}{\partial r} f_k(z) dz = \int_{-h}^{-d} \frac{\partial \phi^{II}}{\partial r} f_k(z) dz + \int_{-d}^{-d_c} Q_j^I f_k(z) dz + \int_{-d_c}^{0} \frac{\partial \phi^{III}}{\partial r} f_k(z) dz, \quad k = 0, 1, 2, \dots, N_1 \qquad (30) $$

where $Q_j^I$ is the forcing term of the body boundary condition on the heave plate thickness in the radial direction (e.g. $Q_j^I = 0$ in the diffraction and heave radiation problems).

Then we have two more sets from the pressure continuity on $r = R_d$, $-h < z < -d$ and $-d_c < z < 0$.

$$ \int_{-h}^{-d} \phi^I Z_k(z) dz = \int_{-h}^{-d} \phi^{II} Z_k(z) dz, \quad k = 0, 1, 2, \dots, N_2 \qquad (31) $$

$$ \int_{-d_c}^{0} \phi^I F_k(z) dz = \int_{-d_c}^{0} \phi^{III} F_k(z) dz, \quad k = 0, 1, 2, \dots, N_3 \qquad (32) $$

The last set of equations is deduced using the non-penetration condition on the cylinder surface on $r = R_c$ and $-d_c < z < 0$.

$$ \int_{-d_c}^{0} \frac{\partial \phi^{III}}{\partial r} F_k(z) dz = \int_{-d_c}^{0} Q_j^{III} F_k(z) dz, \quad k = 0, 1, 2, \dots, N_3 \qquad (33) $$

where $Q_j^{III} = 0$ in the diffraction and heave radiation problems

Substituting the velocity potentials of the diffraction or radiation problems into the above equations (from Equation (30) to (33)) gives the generalized linear system of equations for each Fourier mode, $m$:

$$ A_m = L_m^{-1} R_m \qquad (34) $$

where $A_m = [A_{m,0}, \dots, A_{m,N_1}, B_{m,0}, \dots, B_{m,N_2}, C_{m,0}, \dots, C_{m,N_3}, D_{m,0}, \dots, D_{m,N_3}]^T$ is the unknown coefficient matrix for the diffraction or radiation problems. A detailed description of $L_m$ and $R_m$ is given in Appendix A.

# 3. Physical quantities of interest

After determining the unknown coefficients of velocity potentials, interesting physical quantities can be obtained, such as flow velocity, pressure, loads, and free-surface elevations in the entire fluid domain, $\Omega$. This section focuses on the hydrodynamic loads and the radiated wave fields to compare with the numerical methods and experimental measurements.

## 3.1. Linear hydrodynamic loads

Linear hydrodynamic loads are calculated by integration of hydrodynamic pressure on the mean wetted body surface $S_B$. The hydrodynamic pressure is obtained from Bernoulli's equation. Firstly, using the diffraction potentials, the hydrodynamic loads considered as wave excitation forces and moments are expressed as,

$$ F_i = i \rho \omega \iint_{S_B} \phi_D n_i dS \qquad (35) $$

where subscript $i$ represents the unit normal vector component of the body.

Secondly, with the radiation potential $\phi_R$, the hydrodynamic loads induced by the body motion are expressed as,

$$ F_{\text{rad},i} = \rho\omega^2 \sum_{j=1,3,5} \iint_{S_B} \xi_j \phi_j \cos(m\theta)n_i dS \qquad (36) $$

where $j = 1, 3$, and $5$ for surge, heave, and pitch motions.

Furthermore, in the radiation problem, the hydrodynamic loads ($F_{\text{rad},i}$) are often interpreted as the added mass and damping by the following equations,

$$ a_{ij} = \text{Re} \left\{ \rho \iint_{S_B} \phi_j \cos(m\theta)n_i dS \right\}, \\ b_{ij} = \text{Im} \left\{ \rho\omega \iint_{S_B} \phi_j \cos(m\theta)n_i dS \right\} \qquad (37) $$

where, $a_{ij}$, $b_{ij}$ are the added mass and damping coefficients which are proportional to the acceleration and velocity of the body. The analytical expressions for wave excitation loads, added mass, and damping can be found in Appendix B.

## 3.2. Horizontal mean drift loads

Horizontal mean drift loads can be calculated with the linear velocity potential using Far-field methods introduced by Maruo (1960) and Newman (1967). Using momentum conservation, the expression for the horizontal mean drift forces is given in Mei et al. (2005):
$$ \overline{F}_x = - \int_0^{2\pi} \rho r \cos \theta \left\{ \int_{-h}^0 \left( \frac{1}{4} \left( \frac{\partial \phi_D}{\partial r} \frac{\partial \phi_D^*}{\partial r} - \frac{\partial \phi_D}{\partial z} \frac{\partial \phi_D^*}{\partial z} \right) \right) dz + \frac{\omega^2}{4g} |\phi_D|_{z=0}^2 \right\} d\theta \quad (38) $$

where $r \to \infty$ and * denotes the complex conjugate. Here we focus on the mean drift forces for the diffraction problem.

With large $r$, the diffraction potential in Equation (9) can be expressed using the asymptotic formula for the Hankel function of the first kind, $H_m^{(1)}$, as,

$$ \phi_D^I \sim \frac{-igA}{\omega} f_0(z) \left[ e^{ik_0 r} \cos \theta + \left( \frac{2}{\pi k_0 r} \right)^{1/2} e^{i(k_0 r - \pi/4)} \times \sum_{m=0}^{\infty} \epsilon_m i^m A_{m0} e^{-\frac{m\pi i}{2}} \cos m\theta \right] \quad (39) $$

where the evanescent eigenmodes vanish.

Substituting the asymptotic expression of the velocity potential in Equation (39) into Equation (38), the horizontal mean drift is obtained using the stationary phase method and the cosine identities given in Malenica and Molin (1995). The analytical expression of mean drift forces given in Cong et al. (2019) is reported as,

$$ \overline{F}_x = - \frac{\rho g A^2}{k_0} \frac{2k_0 h + \sinh 2k_0 h}{2 \sinh 2k_0 h} \operatorname{Re} \left\{ 2 \sum_{m=0}^{\infty} \left( A_m A_{m+1}^* + A_{m+1} A_m^* + \epsilon_m A_m \right) \right\} \quad (40) $$

where $A_m = i^m A_{m0} e^{-\frac{m\pi i}{2}}$.

### 3.3. Wave elevations

The free-surface elevation is obtained from the dynamic free-surface boundary condition.

$$ \eta(r, \theta) = \frac{i\omega}{g} \phi(r, \theta, 0), \quad \text{where } z=0 \quad (41) $$

We can calculate the diffracted or radiated free-surface elevations in the fluid domains using the diffraction or radiation potentials.

### 3.4. Verification

The analytical method is examined with two configurations of a truncated circular cylinder with a heave plate. These configurations are based on the semi-submersible type of floating offshore wind platforms: DeepCWind (Robertson et al. (2014)), which has a thick heave plate, and HiPRWind (Lopez-Pavon and Souto-Iglesias (2015)), which is equipped with a thin plate. The dimensions of the models are given in Table 1. To compute the analytical solutions, the number of eigenmodes is selected for each subdomain as $N_1 = 100$, $N_2 = 100$, and $N_3 = 80$, and the number of Fourier modes is chosen as $M = 15$. The selected mode numbers are applicable for the thick and thin heave plate configurations.

To compare with the analytical method, the linear boundary element method (BEM) solvers named HydroStar (Bureau-Veritas (2020)) and NEMOH (Kurnia and Ducrozet (2023)) were used. Both BEM software solve boundary integral equations using the source method that satisfies the same governing equations and boundary conditions as the analytical method in Section 2.1. Additionally, the direct method in HydroStar, which is mixed with source and dipole methods, is used to handle the thin plate configuration.

#### 3.4.1. Thick heave plate

Fig. 3 shows a panel model of the circular cylinder with the thick plate ($t_d/D_d = 0.25$) of the DeepCwind platform. The panels are evenly distributed on the wetted cylinder and plate surfaces, including the top, side, and bottom surfaces. The wetted surfaces refer to only one

<table><caption>**Table 1**
Dimensions of vertical circular cylinder with circular plate.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Cylinder with thick plate</th><th>Cylinder with thin plate</th></tr></thead><tbody><tr><td>D<sub>c</sub></td><td>m</td><td>12.0</td><td>7.0</td></tr><tr><td>D<sub>d</sub></td><td>m</td><td>24.0</td><td>20.0</td></tr><tr><td>d</td><td>m</td><td>20.0</td><td>15.5</td></tr><tr><td>t<sub>d</sub></td><td>m</td><td>6.0</td><td>0.1</td></tr><tr><td>h</td><td>m</td><td>100.0</td><td>100.0</td></tr><tr><td>D<sub>d</sub>/D<sub>c</sub></td><td>-</td><td>2.0</td><td>2.9</td></tr><tr><td>d/D<sub>c</sub></td><td>-</td><td>1.7</td><td>2.2</td></tr><tr><td>t<sub>d</sub>/D<sub>d</sub></td><td>-</td><td>0.25</td><td>0.01</td></tr><tr><td>Reference model</td><td>-</td><td>DeepCwind</td><td>HiPRWind</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Panel model for the truncated cylinder with the thick heave plate of the DeepCwind platform under the undisturbed free-surface level at $z=0$.</figcaption></figure>

side of the panel, which is defined by the normal vectors pointing out of the body. Small-sized panels are selected to ensure high-resolution source distributions, while a symmetric body is considered to reduce the number of panels. As a result, the total number of panels used in the symmetric body is approximately 8,000.

Computations of the linear diffraction and radiation theory (MEEM) are shown in Fig. 4. Non-dimensional linear hydrodynamic loads are compared with the source methods of HydroStar and Nemoh, including wave excitation loads, added mass, and damping in surge, heave, and pitch. The results of the cylinder with the thick plate indicate good agreement between the analytical and the numerical results.

#### 3.4.2. Thin heave plate

It can be challenging to use the source method in BEM solvers for the thin plate because the source and field points on the plate are in close proximity to each other. Therefore, two different approaches are studied to handle the thin plate of the HiPRWind ($t_d/D_d = 0.01$). The first approach utilizes the source method with the panel model similar to that of the thick plate. This model, presented in Fig. 5a, involves evenly distributed panels on both the cylinder and plate surfaces, resulting in a total of around 7,000 panels on the symmetric body. The second approach assumes that the thin plate has a zero-thickness (see Fig. 5b), meaning that both sides of the panels on the plate are defined as wetted surfaces. This approach uses the mixed panel method (Bureau-Veritas (2020)) by distributing dipoles on the zero-thickness plate and sources on the cylinder surface. Since the dipoles cannot be used for the symmetric body in HydroStar, about 9,000 panels are used for the entire body. In general, the size of the panel is almost identical between the thin plate and zero-thickness models.

In Fig. 6, the linear hydrodynamic loads on the cylinder with the thin plate are presented. The predictions made by Nemoh using the source method and by Hydrostar using both source and dipole methods
<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Non-dimensionalized wave excitation loads, added mass and damping on the truncated circular cylinder with the thick plate (lines: MEEM, △: NEMOH, and ○: HydroSTAR).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Panel models for the truncated cylinder with the thin plate of the HiPRWind platform under the undisturbed free-surface level at $z = 0$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Non-dimensionalized wave excitation loads, added mass and damping on the truncated circular cylinder with the thin plate (lines: MEEM, △: NEMOH, ○: HydroSTAR and ×: HydroSTAR with dipole).</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 7. Top and side views of the experimental set-up in a hydrodynamic and ocean engineering tank at Ecole Centrale Nantes. The body-fixed reference frame is given ($O_b, x_b, y_b, z_b$) with load convention ($F_x^b, F_z^b, M_y^b$) expressed in the body reference frame.</figcaption></figure>

<table><caption>**Table 2**
Dimensions of the experimental model.</caption><thead><tr><th>Parameter</th><th>Unit</th><th>Cylinder alone</th><th>Cylinder with plate</th><th>Description</th></tr></thead><tbody><tr><td>$D_c$</td><td>m</td><td>0.35</td><td>0.35</td><td>Diameter of cylinder</td></tr><tr><td>$D_d$</td><td>m</td><td>0.35</td><td>1.00</td><td>Diameter of plate</td></tr><tr><td>$l_d$</td><td>m</td><td>0.005</td><td>0.005</td><td>Thickness of plate</td></tr><tr><td>$d$</td><td>m</td><td>0.775/0.250</td><td>0.775/0.250</td><td>Draft (deep/shallow)</td></tr><tr><td>$D_d/D_c$</td><td>-</td><td>1.0</td><td>2.9</td><td>Diameter ratio</td></tr><tr><td>$d/D_c$</td><td>-</td><td>2.2/0.7</td><td>2.2/0.7</td><td>Draft ratio (deep/shallow)</td></tr></tbody></table>

are generally close to the results from the analytic method. Nonetheless, in contrast to the thick plate, some deviations can be seen in the vertical wave excitation forces ($F_z$), as well as the heave added mass ($a_{33}$) and damping ($b_{33}$), and pitch added moment of inertia ($a_{55}$). The source methods for both Hydrostar and Nemoh underpredict these hydrodynamic coefficients compared to the analytical method. In comparison, the dipole method of Hydrostar provides a good agreement of the heave damping, but the added mass of heave and pitch is overestimated compared to the analytical method. Further comparison will be made with the experiment in the following Section 5.

## 4. Experiments

Dedicated model tests were carried out in the hydrodynamic and ocean engineering tank at Ecole Centrale Nantes to validate the analytical solutions. The dimensions of the tank are 46.4 m in length, 30.0 m in width, and 5.0 m in water depth. The tank is equipped with a flap-type wave maker (on the left-hand side in Fig. 7) and an absorbing beach (on the right).

The experimental models were designed as a single vertical and surface-piercing circular cylinder with and without the thin plate mounted at the bottom to investigate the effects of the plate on the hydrodynamic loads. The experimental model is a scaled model (1/20) of the HiPRWind 2MW floating offshore wind turbines with a semi-submersible substructure (Lopez-Pavon and Souto-Iglesias (2015)). Two different drafts were considered, deep ($d/D_c = 2.2$, design draft of HiPRWind) and shallow ($d/D_c = 0.7$), to investigate the interactions between the heave plate and the free surface. The detailed dimensions of the model are summarized in Table 2.

The model was installed in the wave tank, piercing the water surface, and held from its top by a hexapod. The hexapod can keep a constant position at different drafts or impose motions such as surge, heave, and pitch. A multi-component load cell was installed between the model and the hexapod to measure the loads induced by the incident waves or the motions. The hydrodynamic loads were obtained by

extracting the inertial, gravitational, and hydrostatic load components from the measured loads. Seven wave gauges were also placed around the model to measure the incident, diffracted, and radiated wave fields. A motion tracking system designed by Qualisys was utilized to confirm body motions.

Fig. 8 shows a photograph of the general arrangement of the experiment. Diffraction/radiation campaigns were performed in this study. During the diffraction test, the hexapod held the model firmly to restrict the movement in the tank where the wave maker generated regular waves. Eight regular wave frequencies were selected with 0.02 of the wave steepness ($H/\lambda$). For the radiation tests, forced oscillation tests in calm water were conducted. The hexapod imposed the harmonically oscillating motions to the model in the pure surge, heave, and pitch. The center of rotation of the pitch motions is located at the origin of the body-fixed reference frame, $O_b$. Several motion frequencies were

<figure><img src="image_12.png" /><figcaption>Fig. 8. The experimental model with the heave plate connected on the hexapod in the tank.</figcaption></figure>
<table><caption>**Table 3**
Wave and motion conditions in model scale for diffraction/radiation tests.</caption><thead><tr><th>Test type</th><th>Wave/motion periods, T [s]</th><th>Wave steepness, H/λ [-]</th><th>Motion amplitude, ξ [m or rad]</th><th>Description</th></tr></thead><tbody><tr><td>Regular waves</td><td>0.9 - 3.0</td><td>0.02</td><td>-</td><td>Diffraction tests</td></tr><tr><td>Regular motions</td><td>0.9 - 3.0</td><td>-</td><td>0.05</td><td>Radiation tests</td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** The non-dimensional first harmonic of the horizontal wave excitation forces and mean drift forces induced by regular waves with 0.02 of wave steepness (H/λ) on the cylinder with (Δ) and without the heave plate (○) at the deep (d/D_c = 2.2, blue colors) and shallow (d/D_c = 0.7, red colors) drafts.</figcaption></figure>

considered in the same range as the wave frequencies. Small constant motion amplitudes were selected in order to minimize nonlinear effects on the problem: 0.05 m for surge and heave motions and 3.0 deg for pitch motion to limit the Keulegan-Carpenter (KC) number to below 1. The corresponding KC numbers for surge, heave, and pitch motions are given by

$$KC = \frac{2\pi\xi}{D_c}, \quad \text{for surge, heave} \qquad (42)$$

$$KC = \frac{2\pi\xi d}{D_c}, \quad \text{for pitch}$$

where ξ is the amplitude of the motion and d is the draft. The KC number for rotational motion is referred to ITTC (2021). The tested wave and motion parameters are summarized in Table 3.

## 5. Results

The results obtained with the analytical solution are compared with the experimental measurements to identify the method's region of validity and provide a global overview of the heave plate effects on the hydrodynamic loads. The same number of modes verified in Section 3.4 are used for comparison. Two geometries of a truncated circular cylinder with and without the heave plate are investigated in deep and shallow drafts (see Table 2 for details). A thin plate is considered in this section.

The results are summarized in the following section as a set of four experimental measurements (EXP) where the circles represent the truncated cylinder and the triangles represent the cylinder with the heave plate, and two different color series describe the drafts: deep draft (blue) and shallow draft (red). The distinct line types illustrate the analytical results (MEEM). The diffraction tests are presented first through the amplitude of the first harmonic and the mean components of the wave excitation load. Surge, heave, and pitch radiation tests are then presented through the first harmonic of the wave elevations and the diagonal terms of the added mass and damping matrices.

### 5.1. Diffraction tests

Figs. 9 and 10 provide the first-harmonic loads and the mean horizontal forces obtained from the diffraction tests. The computational results of the wave excitation loads, which include the contribution of the incident wave and the perturbed waves, are compared with the experimental data. In general, the measured first harmonics of the wave-induced loads are well described by the analytical method, whereas the far-field method only predicts the trends of mean horizontal drift forces. The potential theory is able to capture the effects of the heave plate on the wave excitation loads at both deep and shallow drafts.

On the non-dimensional horizontal forces, the heave plate effect is noticeable within the frequency range of 0.5 < ω²D_c/g < 2.0. The presence of the plate increases the first-harmonic forces by around 5% at the deep draft and 50% at the shallow draft resulting in a sharp peak at ω²D_c/g = 1.2. However, the effect of the heave plate on the horizontal force becomes smaller in the low (ω²D_c/g < 0.2) and high (ω²D_c/g > 2.0) frequency regions. The first harmonic forces at the shallow draft are generally lower than at the deep draft due to the reduction in the wetted body surface on the immersed cylinder.

On the other hand, the heave plate does not noticeably affect drift forces at the deep draft, but it increases significantly the drift forces at the shallow draft. The theoretical and experimental results are comparable, although there is a shift in the peak frequency of the horizontal drift force predicted by the analytical method at the shallow draft.

The effect of the plate is more significant on the vertical forces and moments, as illustrated in Fig. 10. Theoretical analysis describes that, without the plate, the vertical forces decrease as the wave frequency increases. However, the presence of the plate is indicated to cause an increase in the vertical forces, which is consistent with experimental measurements. Nevertheless, some differences can be noticed at low frequencies in the case of the cylinder with the plate. According to the theory, the plate leads to a cancellation of the vertical forces at the deep and shallow drafts. The experiment shows a similar trend, but the theory underestimates the vertical forces in this low-frequency regime.

The wave excitation moments given at the reference point, O_b, present good agreement between the theoretical prediction and the ex-
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** The non-dimensional first harmonic of the vertical wave excitation forces and moments induced by regular waves with 0.02 of wave steepness ($H/λ$) on the cylinder with (△) and without the heave plate (○) at the deep ($d/D_c = 2.2$, blue colors) and shallow ($d/D_c = 0.7$, red colors) drafts.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** The non-dimensional first harmonic amplitude of radiated waves on the cylinder with (△) and without the heave plate (○) at the deep ($d/D_c = 2.2$, blue colors) and shallow ($d/D_c = 0.7$, red colors) drafts.</figcaption></figure>

experiments. The presence of the heave plate increases the moments at the deep and shallow drafts. It can be observed that reducing the draft causes higher moments on the cylinder with the plate while having the opposite effect on the cylinder without the plate.

## 5.2. Radiation tests

Fig. 11 presents radiated waves obtained at $r/D_c = 4.0$ with $\theta = \pi$ which is the position of wave gauge number 3 (WG3) in our experimental setup given in Fig. 7. The radiated waves calculated by using the analytical method agree with the measurements, showing the influence of the heave plate and the draft.

The surge motion generates waves that increase monotonically as the motion frequency increases. At the deep draft, the first-harmonic wave amplitude is slightly higher with the plate than without the plate, but the overall trend remains similar. This illustrates that the presence of the cylinder is more dominant than the plate for generating the waves in the surge oscillation. In contrast, at the shallow draft, the effect of the plate on the radiated waves increases the wave amplitude significantly.

In heave motion, there is a strong interaction between the heave plate and the free surface. Without the plate, the radiated waves generated by the cylinder alone are insignificant compared to the waves in surge motions. But with the heave plate, the radiated waves become visible in both deep and shallow drafts. As expected, the plate and free-surface interactions are increased at the shallow draft. The waves no longer monotonically increase with the motion frequency. Instead, a

peak appears at deep and shallow drafts at the frequencies ($\omega^2 D_c / g$) around 1.2 and 1.4, respectively.

For the pitch motion of the cylinder alone, larger waves are generated at the deep draft than at the shallow draft. Conversely, the radiated waves are increased at shallow drafts in the presence of the plate except for the very low frequency ($\omega^2 D_c / g < 0.75$).

The non-dimensional added mass and damping are presented in Figs. 12-13. In surge motion, the theoretical predictions show good overall agreement with the experimental measurements. Notably, the heave plate provides more added mass ($a_{11}$) and damping ($b_{11}$) in both deep and shallow drafts. However, at the shallow draft, the added mass of the cylinder with the plate is slightly lower than the solitary cylinder in the high-frequency region.

In contrast, the theory is insufficient to predict the added mass and damping for the heave and pitch motions. In heave motions, the experimental results demonstrate a large amount of damping with the heave plates, while the theoretical prediction shows almost zero damping at the deep draft. The theory presents more damping at the shallow draft than at the deep draft but still shows notable discrepancies compared to the experimental results. This indicates that a dominant physical quantity is missing in the hydrodynamic forces, although the theory predicts the radiated waves well compared to the experimental measurements.

According to Garrido-Mendoza et al. (2015), this discrepancy may be attributed to the predominant influence of flow separation on the heave damping coefficient ($b_{33}$) at the deep draft. They presented an explicit equation of the heave damping as a function of the entrophy
<figure><img src="image_16.png" /><figcaption>Fig. 12. Non-dimensional added mass coefficients for surge ($a_{11}$), heave ($a_{33}$), and pitch ($a_{55}$) RAO on the cylinder with ($\triangle$) and without the heave plate (○) at the deep ($d/D_c = 2.2$, blue lines) and shallow ($d/D_c = 0.7$, red lines) drafts.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 13. Non-dimensional damping coefficients for surge ($b_{11}$), heave ($b_{33}$), and pitch ($b_{55}$) RAO on the cylinder with ($\triangle$) and without the heave plate (○) at the deep ($d/D_c = 2.2$, blue lines) and shallow ($d/D_c = 0.7$, red lines) drafts.</figcaption></figure>

term based on the energy dissipation by neglecting the wave contributions. It is worth noting that in the potential flow theory, dissipation is only taken into account through the radiated waves. The heave damping at the deep draft is not significantly affected by frequency, which is consistent with the findings of Lopez-Pavon and Souto-Iglesias (2015). However, at the shallow draft, the experiments indicate that the motion frequency is a dominant parameter of heave damping, and wave damping makes a considerable contribution to the total damping, as suggested by the theory.

The difference in the heave damping may vary depending on the thickness of the plate, as reported in Tian et al. (2017). This is because the thickness of the plate affects the vortex shedding pattern (Tao and Thiagarajan 2003) based on the KC number, and thick plates shed weak vortices (individual vortex shedding).

## 6. Conclusions

This study described a first-order analytical theory to solve the flow around a truncated and surface-piercing vertical circular cylinder with a circular plate mounted at the bottom of the cylinder. The potential flow was considered for the entire fluid domain to solve the linear diffraction and radiation of water wave problems based on the assumption of small wave and motion amplitudes in surge, heave, and pitch motions to linearize the problems. Matched eigenfunction expansion methods were implemented into the decomposed fluid domain to obtain the general solutions of the linear velocity potentials satisfying the boundary value problems. By using the velocity potentials, analytical expressions were introduced for the wave excitation loads, added mass, and damping.

As a verification, two vertical circular cylinders were examined: one with a thick heave plate and the other with a thin heave plate. For the thick plate, the linear hydrodynamic loads predicted by the derived analytical solutions were in good agreement with the results of the linear BEM solvers based on the source method (HydroSTAR and NEMOH). In comparison, the source method underestimated the vertical hydrodynamic components on the thin plate. The dipole method in Hydrostar improved the prediction of heave damping, but the added mass of heave and pitch was overestimated for the thin plate compared to the results from the analytical method.

The region of validity of the analytical method was further studied through two experimental campaigns carried out in the ocean engineering tank at Ecole Centrale Nantes. The analytical method described well the measured first harmonics of wave excitation loads within the limited range of wave steepness ($H/\lambda = 0.02$). The method captured the effects of the heave plate shown experimentally, with a significant increase in the wave excitation loads. Interestingly, the method indicated that the vertical wave forces cancel out at the low-frequency regime with the plate. However, the forces were underestimated compared to the experiment. In this regime, the flow velocity decays slowly in the vertical direction which may cause dominant flow separation. This is a hypothesis to justify why the method underestimates the forces. Conversely, in short waves, where the discrepancy between the computed and measured force is negligible, the flow velocity is minimal at the heave plate draft.

For the mean horizontal drift forces, both the analytical method and the experiment showed that the plate effects are negligible at the deep
draft. However, the effects are noticeable at shallow draft even though there is a shift in the peak of the analytical solution.

In the radiation problem under a low KC regime ($KC < 1$), the analytical solution was acceptable for estimating the radiated wave fields of experimental measurements. However, the prediction of the added mass and damping was only applicable to surge motion. For the heave and pitch motions, the analytical solution presented discrepancies in the added mass and damping due to the effects of the flow separation. The experimental results demonstrated that the heave plate substantially increases the added mass and damping for surge, heave, and pitch motions at the deep and shallow drafts.

Further experimental studies are planned to investigate the effects of nonlinear waves on wave excitation loads and the contributions of vorticity and viscosity to added mass and damping with the heave plate. Parametric studies will be conducted using different plate geometries, such as the plate diameter, porosity, and draft, which are essential to understanding hydrodynamics for the diversity of floating offshore wind platforms. Lastly, a comprehensive hydrodynamic database, including viscous drag, will be established based on the present analytical solution to achieve a reasonable comparison with high-fidelity models.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgement

This research is part of the EU project FLOAWER (FLOAting Wind Energy netwoRk). This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement N° 860879. This work was also carried out within the HP_Flow project in the WEAMEC framework, West Atlantic Marine Energy Community, and with funding from the Pays de la Loire Region (France) and European Regional Development Fund. The work presented in this paper was supported by Centrale Nantes - Bureau Veritas research chair.

## Appendix A. Matrix for determining unknown coefficients

The matrices $L$ and $R$ from the linear system of equations in Equation (34) consist of the submatrices as,

$$L = \begin{bmatrix} \mathbb{L}^{11} & \mathbb{L}^{12} & \mathbb{L}^{13} & \mathbb{L}^{14} \\ \mathbb{L}^{21} & \mathbb{L}^{22} & 0 & 0 \\ \mathbb{L}^{31} & 0 & \mathbb{L}^{33} & \mathbb{L}^{34} \\ 0 & 0 & \mathbb{L}^{43} & \mathbb{L}^{44} \end{bmatrix}, \quad R = \begin{bmatrix} \mathbb{R}^{1} \\ \mathbb{R}^{2} \\ \mathbb{R}^{3} \\ \mathbb{R}^{4} \end{bmatrix} \qquad (A.1)$$

where the first row is established from the normal velocity continuity on $r = R_d$ and $-h \le z \le 0$ in Equation (30),

$$\mathbb{L}_{i,i}^{11} = U'_{mi}(R_d) \int_{-h}^{0} f_i^2(z) dz \qquad (A.2)$$

where $i = 0 \sim N_1$.

$$\mathbb{L}_{i,j}^{12} = -V'_{mj}(R_d) \int_{-h}^{-d} Z_j(z) f_i(z) dz \qquad (A.3)$$

where $i = 0 \sim N_1$, and $j = 0 \sim N_2$.

$$\mathbb{L}_{i,j}^{13} = -X'_{mj}(R_d) \int_{-d_c}^{0} f_j(z) f_i(z) dz \qquad (A.4)$$

where $i = 0 \sim N_1$, and $j = 0 \sim N_3$.

$$\mathbb{L}_{i,j}^{14} = -W'_{mj}(R_d) \int_{-d_c}^{0} f_j(z) f_i(z) dz \qquad (A.5)$$

where $i = 0 \sim N_1$, and $j = 0 \sim N_3$.

The second row is from pressure continuity on $r = R_d$ and $-h \le z \le -d$ in Equation (31),

$$\mathbb{L}_{i,j}^{21} = U_{mj}(R_d) \int_{-h}^{-d} f_j(z) Z_i(z) dz \qquad (A.6)$$

where $i = 0 \sim N_2$, and $j = 0 \sim N_1$.

$$\mathbb{L}_{i,i}^{22} = -V_{mi}(R_d) \int_{-h}^{-d} Z_i^2(z) dz \qquad (A.7)$$

where $i = 0 \sim N_2$.

The third is from pressure continuity on $r = R_d$ and $-d_c \le z \le 0$ (Equation (32)),

$$\mathbb{L}_{i,j}^{31} = U_{mj}(R_d) \int_{-d_c}^{0} f_j(z) F_i(z) dz \qquad (A.8)$$

where $i = 0 \sim N_3$, and $j = 0 \sim N_1$.

$$\mathbb{L}_{i,i}^{33} = -X_{mi}(R_d) \int_{-d_c}^{0} F_i^2(z) dz \qquad (A.9)$$

where $i = 0 \sim N_3$.

$$\mathbb{L}_{i,i}^{34} = -W_{mi}(R_d) \int_{-d_c}^{0} F_i^2(z) dz \qquad (A.10)$$

where $i = 0 \sim N_3$.

Lastly, the no penetration condition on cylinder surface $r = R_c$ and $-d_c \le z \le 0$ (Equation (33)) gives the fourth row as,

$$\mathbb{L}_{i,i}^{43} = X'_{mi}(R_c) \int_{-d_c}^{0} F_i^2(z) dz \qquad (A.11)$$

where $i = 0 \sim N_3$.

$$\mathbb{L}_{i,i}^{44} = W'_{mi}(R_c) \int_{-d_c}^{0} F_i^2(z) dz \qquad (A.12)$$

where $i = 0 \sim N_3$.

On the other hand, the matrix $R$ in diffraction problems is established based on the contribution of the incident wave potential as,

$$\mathbb{R}_i^1 = -J'_m(k_0 R_d) \int_{-h}^{0} f_0(z) f_i(z) dz \qquad (A.13)$$

where $i = 0 \sim N_1$.

$$\mathbb{R}_i^2 = -J_m(k_0 R_d) \int_{-h}^{-d} f_0(z) Z_i(z) dz \qquad (A.14)$$

where $i = 0 \sim N_2$.

$$\mathbb{R}_i^3 = -J_m(k_0 R_d) \int_{-d_c}^{0} f_0(z) F_i(z) dz \qquad (A.15)$$
where $i = 0 \sim N_3$.

$$ \mathbb{R}_i^4 = 0_{(N_3+1)\times 1} \qquad (A.16) $$

In radiation problems, the matrix $\mathbb{R}$ is determined by the particular solutions and forcing terms on the body boundary condition as,

$$ \mathbb{R}_i^1 = \frac{1}{\epsilon_m} \int_{-d}^{-d_c} Q_p^I f_i(z) dz \qquad (A.17) $$

where $i = 0 \sim N_1$.

$$ \mathbb{R}_i^2 = \frac{1}{\epsilon_m} \int_{-h}^{-d} \psi_p^{II} Z_i(z) dz \qquad (A.18) $$

where $i = 0 \sim N_2$.

$$ \mathbb{R}_i^3 = \frac{1}{\epsilon_m} \int_{-d_c}^{-0} \psi_p^{III} F_i(z) dz \qquad (A.19) $$

where $i = 0 \sim N_3$.

$$ \mathbb{R}_i^4 = \frac{1}{\epsilon_m} \int_{-d_c}^{0} Q_p^{III} F_i(z) dz \qquad (A.20) $$

where $i = 0 \sim N_3$. Here, $p = 1, 3$, and $5$ are for surge, heave, and pitch, respectively.

## Appendix B. Analytical expression of the hydrodynamic loads

By inserting the diffraction potentials in Equations (10), (11), (14), and (16) into Equation (35), the analytical expressions of the wave excitation forces and moments are obtained as below.

### Linear horizontal wave force:

$$ F_x = -2\pi i \rho g A \left[ R_d \left\{ J_1(k_0 R_d) \int_{-d}^{-d_c} f_0(z) dz + \sum_{n=0}^{N_1} A_{1n} U_{1n}(R_d) \int_{-d}^{-d_c} f_n(z) dz \right\} + R_c \sum_{n=0}^{N_3} (C_{1n} + D_{1n}) \int_{-d_c}^{0} F_n(z) dz \right] \qquad (B.1) $$

where the integrations of vertical function are given by,

$$ \int_{-d}^{-d_c} f_n(z) dz = \begin{cases} \frac{1}{k_0 \cosh k_0 h} (\sinh k_0(h - d_c) - \sinh k_0(h - d)), & n = 0 \\ \frac{1}{k_n \cos k_n h} (\sin k_n(h - d_c) - \sin k_n(h - d)), & n \ge 1 \end{cases} \qquad (B.2) $$

$$ \int_{-d_c}^{0} F_n(z) dz = \begin{cases} \frac{1}{\beta_0} \tanh \beta_0 d_c, & n = 0 \\ \frac{1}{\beta_n} \tan \beta_n d_c, & n \ge 1 \end{cases} \qquad (B.3) $$

### Linear vertical wave force:

$$ F_z = 2\pi \rho g A \left[ \sum_{n=0}^{N_2} B_{0n} Z_n(-d) \int_{0}^{R_d} r V_{0n}(r) dr - \sum_{n=0}^{N_3} F_n(-d_c) \left\{ C_{0n} \int_{R_c}^{R_d} r X_{0n}(r) dr + D_{0n} \int_{R_c}^{R_d} r W_{0n}(r) dr \right\} \right] \qquad (B.4) $$

where the integrations of radial function are presented as,

$$ \int_{0}^{R_d} r V_{0n}(r) dr = \begin{cases} \frac{1}{2} R_d^2, & n = 0 \\ \frac{1}{a_n} R_d I_1(\alpha_n R_d), & n \ge 1 \end{cases} \qquad (B.5) $$

$$ \int_{R_c}^{R_d} r X_{0n}(r) dr = \begin{cases} \frac{1}{\beta_0} (R_d J_1(\beta_0 R_d) - R_c J_1(\beta_0 R_c)), & n = 0 \\ \frac{1}{\beta_n} (R_d I_1(\beta_n R_d) - R_c I_1(\beta_n R_c)), & n \ge 1 \end{cases} \qquad (B.6) $$

$$ \int_{R_c}^{R_d} r W_{0n}(r) dr = \begin{cases} \frac{1}{\beta_0} (R_d Y_1(\beta_0 R_d) - R_c Y_1(\beta_0 R_c)), & n = 0 \\ \frac{1}{\beta_n} (-R_d K_1(\beta_n R_d) + R_c K_1(\beta_n R_c)), & n \ge 1 \end{cases} \qquad (B.7) $$

### Linear wave moment:

$$ M_y = -2\pi i \rho g A \left[ R_d \left\{ J_1(k_0 R_d) \int_{-d}^{-d_c} z f_0(z) dz + \sum_{n=0}^{N_1} A_{1n} U_{1n}(R_d) \int_{-d}^{-d_c} z f_n(z) dz \right\} + R_c \sum_{n=0}^{N_3} (C_{1n} + D_{1n}) \int_{-d_c}^{0} z F_n(z) dz + \sum_{n=0}^{N_2} B_{1n} Z_n(-d) \int_{0}^{R_d} r^2 V_{1n}(r) dr - \sum_{n=0}^{N_3} F_n(-d_c) \left\{ C_{1n} \int_{R_c}^{R_d} r^2 X_{1n}(r) dr + D_{1n} \int_{R_c}^{R_d} r^2 W_{1n}(r) dr \right\} \right] \qquad (B.8) $$

where the integrations of radial and vertical functions are provided as follows,

$$ \int_{0}^{R_d} r^2 V_{1n}(r) dr = \begin{cases} \frac{1}{4} R_d^3, & n = 0 \\ \frac{1}{a_n} R_d^2 I_2(\alpha_n R_d), & n \ge 1 \end{cases} \qquad (B.9) $$

$$ \int_{R_c}^{R_d} r^2 X_{1n}(r) dr = \begin{cases} \frac{1}{\beta_0} (R_d^2 J_2(\beta_0 R_d) - R_c^2 J_2(\beta_0 R_c)), & n = 0 \\ \frac{1}{\beta_n} (R_d^2 I_2(\beta_n R_d) - R_c^2 I_2(\beta_n R_c)), & n \ge 1 \end{cases} \qquad (B.10) $$
$$ \int_{R_c}^{R_d} r^2 W_{1n}(r) dr = \begin{cases} \frac{1}{\beta_0} (R_d^2 Y_2(\beta_0 R_d) - R_c^2 Y_2(\beta_0 R_c)), & n=0 \\ \frac{1}{\beta_n} (-R_d^2 K_2(\beta_n R_d) + R_c^2 K_2(\beta_n R_c)), & n \ge 1 \end{cases} \quad (B.11) $$

$$ = \begin{cases} \frac{1}{k_0^2 \cosh k_0 h} \left( -k_0 d_c \sinh k_0 (h-d_c) - \cosh k_0 (h-d_c) \right. & , n=0 \\ \left. + k_0 d \sinh k_0 (h-d) + \cosh k_0 (h-d) \right) & , n \ge 1 \end{cases} $$

$$ \int_{-d_c}^{0} z F_n(z) dz = \begin{cases} \frac{-1}{\beta_0^2 \cosh \beta_0 d_c} (\cosh \beta_0 d_c - 1), & n=0 \\ \frac{1}{\beta_n^2 \cos \beta_n d_c} (\cos \beta_n d_c - 1), & n \ge 1 \end{cases} \quad (B.13) $$

In the radiation problem, substituting the unit amplitude velocity potential (Equations (23), (25), and (28)) into Equation (37), gives the analytical expression for surge, heave, and pitch motions as,

$$ \iint_{S_B} \phi_1 \cos \theta n_1 dS = -2\pi \left\{ R_d \sum_{n=0}^{N_1} A_{1n} U_{1n}(R_d) \int_{-d}^{-d_c} f_n(z) dz + R_c \sum_{n=0}^{N_3} (C_{1n} + D_{1n}) \int_{-d_c}^{0} F_n(z) dz \right\} \quad (B.14) $$

$$ \iint_{S_B} \phi_3 n_3 dS = 2\pi \left[ \begin{aligned} & \sum_{n=0}^{N_2} B_{0n} \cos(n\pi) \int_0^{R_d} rV_{0n}(r)dr \\ & - \frac{1}{4(h-d)} \left( \frac{R_d^4}{4} - (h-d)^2 R_d^2 \right) \\ & - \sum_{n=0}^{N_3} F_n(-d_c) \left\{ C_{0n} \int_{R_c}^{R_d} rX_{0n}(r)dr + D_{0n} \int_{R_c}^{R_d} rW_{0n}(r)dr \right\} \\ & + \frac{1}{2} \left( d_c - \frac{g}{\omega^2} \right) (R_d^2 - R_c^2) \end{aligned} \right] \quad (B.15) $$

$$ \iint_{S_B} \phi_5 \cos \theta n_5 dS = -2\pi \left[ \begin{aligned} & R_d \sum_{n=0}^{N_1} A_{1n} U_{1n}(R_d) \int_{-d}^{-d_c} z f_n(z) dz \\ & + \sum_{n=0}^{N_2} B_{1n} \cos(n\pi) \int_0^{R_d} r^2 V_{1n}(r) dr \\ & + R_c \sum_{n=0}^{N_3} (C_{1n} + D_{1n}) \int_{-d_c}^{0} z F_n(z) dz \\ & - \sum_{n=0}^{N_3} F_n(-d_c) \left\{ C_{1n} \int_{R_c}^{R_d} r^2 X_{1n}(r) dr \right. \\ & \left. \qquad + D_{1n} \int_{R_c}^{R_d} r^2 W_{1n}(r) dr \right\} \end{aligned} \right] \quad (B.16) $$

$$ + \frac{1}{16(h-d)} \left( \frac{R_d^6}{6} - R_d^4 (h-d)^2 \right) \\ - \frac{1}{8} \left( d_c - \frac{g}{\omega^2} \right) (R_d^4 - R_c^4) \\ - \frac{R_c^2}{2} \left( \frac{d_c^3}{3} - \frac{1}{2} \frac{g}{\omega^2} d_c^2 \right) $$

## References

Bhatta, D.D., 2011. Computations of hydrodynamic coefficients, displacement-amplitude ratios and forces for a floating cylinder due to wave diffraction and radiation. Int. J. Non-Linear Mech. 46, 1027–1041. https://doi.org/10.1016/j.ijnonlinmec.2011.04.022.

Bhatta, D.D., Rahman, M., 2003. On scattering and radiation problem for a cylinder in water of finite depth. Int. J. Eng. Sci. 41, 931–967. https://doi.org/10.1016/S0020-7225(02)00381-6.

Bureau-Veritas, 2020. Hydrostar for Experts User Manual, Version 8.10.

Cheng, Y., Fu, L., Dai, S., Collu, M., Cui, L., Yuan, Z., Incecik, A., 2022a. Experimental and numerical analysis of a hybrid wec-breakwater system combining an oscillating water column and an oscillating buoy. Renew. Sustain. Energy Rev. 169, 112909. https://doi.org/10.1016/j.rser.2022.112909.

Cheng, Y., Fu, L., Dai, S., Collu, M., Ji, C., Yuan, Z., Incecik, A., 2022b. Experimental and numerical investigation of wec-type floating breakwaters: a single-pontoon oscillating buoy and a dual-pontoon oscillating water column. Coast. Eng. 177, 104188. https://doi.org/10.1016/j.coastaleng.2022.104188.

Cong, P., Liu, Y., Gou, Y., Teng, B., 2019. Theoretical modeling of hydrodynamic characteristics of a compound column-plate structure based on a novel derivation of mean drift force formulation. Proc. Inst. Mech. Eng., Part M, J. Eng. Marit. Environ. 233, 1022–1036. https://doi.org/10.1177/1475090218808019.

Faltinsen, O.M., 1990. Sea Loads on Ships and Offshore Structures.

Faltinsen, O.M., Newman, J.N., Vinje, T., 1995. Nonlinear wave loads on a slender vertical cylinder. J. Fluid Mech. 289, 179–198. https://doi.org/10.1017/S0022112095001297.

Garrett, C.J.R., 1971. Wave forces on a circular dock. J. Fluid Mech. 46, 129–139. https://doi.org/10.1017/S0022112071000430.

Garrido-Mendoza, C.A., Thiagarajan, K.P., Souto-Iglesias, A., Colagrossi, A., Bouscasse, B., 2015. Computation of flow features and hydrodynamic coefficients around heave plates oscillating near a seabed. J. Fluids Struct. 59, 406–431. https://doi.org/10.1016/j.jfluidstructs.2015.10.003.

Haslum, H., Faltinsen, O., 1999. Alternative Shape of Spar Platforms for Use in Hostile Areas. OTC.

He, J., Wu, H., Yang, C.J., Zhu, R.C., Li, W., Noblesse, F., 2021. Diffraction–radiation of regular water waves and irregular frequencies: a straightforward flow-modeling approach and analysis. Eur. J. Mech. B, Fluids 90, 7–14. https://doi.org/10.1016/j.euromechflu.2021.07.011.

Huang, J.B., Taylor, R.E., 1996. Semi-analytical solution for second-order wave diffraction by a truncated circular cylinder in monochromatic waves. J. Fluid Mech. 319, 171. https://doi.org/10.1017/S0022112096007306.

Hulme, A., 1982. The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. J. Fluid Mech. 121, 443. https://doi.org/10.1017/S0022112082001980.

ITTC, 2021. Estimation of roll damping. In: ITTC - Recommended Procedures and Guidelines 7.5-02-07-04.5.

Kim, J., Cho, I.H., 2021. Wave power extraction by multiple wave energy converters arrayed in a water channel resonator. Int. J. Nav. Archit. Ocean Eng. 13, 178–186. https://doi.org/10.1016/j.ijnaoe.2021.02.004.

Koh, H.J., Cho, I.H., 2016. Heave motion response of a circular cylinder with the dual damping plates. Ocean Eng. 125, 95–102. https://doi.org/10.1016/j.OCEANENG.2016.07.037.

Kristiansen, T., Faltinsen, O.M., 2017. Higher harmonic wave loads on a vertical cylinder in finite water depth. J. Fluid Mech. 833, 773–805. https://doi.org/10.1017/jfm.2017.702.

Kurnia, R., Ducrozet, G., 2023. Nemoh: open-source boundary element solver for computation of first- and second-order hydrodynamic loads in the frequency domain. Comput. Phys. Commun. 292, 108885. https://doi.org/10.1016/j.cpc.2023.108885.

Lee, C.H., Newman, J.N., Zhu, X., 1996. An extended boundary integral equation method for the removal of irregular frequency effects. Int. J. Numer. Methods Fluids 23, 637–660. https://doi.org/10.1002/(SICI)1097-0363(19961015)23:7<637::AID-FLD437>3.0.CO;2-3.

Li, A., Liu, Y., 2019. New analytical solutions to water wave diffraction by vertical truncated cylinders. Int. J. Nav. Archit. Ocean Eng. 11, 952–969. https://doi.org/10.1016/j.ijnaoe.2019.04.006.

Linton, C., 1991. Radiation and diffraction of water waves by a submerged sphere in finite depth. Ocean Eng. 18, 61–74. https://doi.org/10.1016/0029-8018(91)90034-N.

Linton, C.M., McIver, P., 2001. Handbook of Mathematical Techniques for Wave/Structure Interactions. CRC Press.
Lopez-Pavon, C., Souto-Iglesias, A., 2015. Hydrodynamic coefficients and pressure loads on heave plates for semi-submersible floating offshore wind turbines: a comparative analysis using large scale models. Renew. Energy 81, 864–881. https://doi.org/10.1016/j.RENENE.2015.04.003.

MacCamy, R., Fuchs, D., 1954. Waves Forces on Piles: a Diffraction Theory.

Malenica, S., 2012. Semi-analytical methods for different problems of diffraction-radiation by vertical circular cylinders. Int. J. Ocean Syst. Eng. 2, 116–138. https://doi.org/10.5574/IJOSE.2012.2.2.116.

Malenica, S., Chen, X., 1998. On the Irregular Frequencies Appearing in Wave Diffraction-Radiation Solutions.

Malenica, S., Molin, B., 1995. Third-harmonic wave diffraction by a vertical cylinder. J. Fluid Mech. 302, 203–229. https://doi.org/10.1017/S0022112095004071.

Maruo, H., 1960. The drift of a body floating on waves. J. Ship Res. 4, 1–11.

McIver, P., Evans, D.V., 1984. The occurrence of negative added mass in free-surface problems involving submerged oscillating bodies. J. Eng. Math. 18, 7–22. https://doi.org/10.1007/BF00042895.

Mei, C.C., Stiassnie, M., Yue, D.K.P., 2005. Theory and Applications of Ocean Surface Waves, vol. 23. World Scientific.

Moreau, M., Kristiansen, T., Ommani, B., Molin, B., 2022. An upright bottomless vertical cylinder with baffles floating in waves. Appl. Ocean Res. 119, 102934. https://doi.org/10.1016/j.APOR.2021.102934.

Newman, J.N., 1967. The drift force and moment on ships in waves. J. Ship Res. 11, 51–60. https://doi.org/10.5957/jsr.1967.11.1.51.

Robertson, A., Jonkman, J., Masciola, M., Song, H., Goupee, A., Coulling, A., Luan, C., 2014. Definition of the Semisubmersible Floating System for Phase II of OC4.

Sheng, W., Tapoglou, E., Ma, X., Taylor, C., Dorrell, R., Parsons, D., Aggidis, G., 2022. Hydrodynamic studies of floating structures: comparison of wave-structure interaction modelling. Ocean Eng. 249, 110878. https://doi.org/10.1016/j.oceaneng.2022.110878.

Tao, L., Molin, B., Scolan, Y.M., Thiagarajan, K., 2007. Spacing effects on hydrodynamics of heave plates on offshore structures. J. Fluids Struct. 23, 1119–1136. https://doi.org/10.1016/j.JFLUIDSTRUCTS.2007.03.004.

Tao, L., Thiagarajan, K., 2003. Low kc flow regimes of oscillating sharp edges I. Vortex shedding observation. Appl. Ocean Res. 25, 21–35. https://doi.org/10.1016/S0141-1187(03)00031-2.

Teng, B., Kato, S., 2002. Third order wave force on axisymmetric bodies. Ocean Eng. 29, 815–843. https://doi.org/10.1016/S0029-8018(01)00047-6.

Thiagarajan, K.P., Datta, I., Ran, A.Z., Tao, L., Halkyard, J.E., 2002. Influence of heave plate geometry on the heave response of classic spars. In: ASMEDC, pp. 621–627.

Tian, X., Tao, L., Li, X., Yang, J., 2017. Hydrodynamic coefficients of oscillating flat plates at $0.15 \le kc \le 3.15$. J. Mar. Sci. Technol. 22, 101–113. https://doi.org/10.1007/s00773-016-0401-2.

Yeung, R.W., 1981. Added mass and damping of a vertical cylinder in finite-depth waters. Appl. Ocean Res. 3, 119–133. https://doi.org/10.1016/0141-1187(81)90101-2.