Impact Statement
This study advances the understanding of fundamental flow physics in wave energy conversion by analysing the hydrodynamics of a two-body wave energy converter (WEC) with an annular heave plate. Using linear wave theory, we investigate how varying the disk porosity influences power extraction efficiency and mechanical loading. The study provides insight into the interplay between hydrodynamic forces and structural response, particularly the influence of added mass and radiation damping on energy transfer. Our findings reveal that porous heave plates can enhance energy capture while mitigating excessive oscillations, thus improving system stability.
The translational impact of this research lies in its contribution to the design and optimisation of next-generation wave energy technologies. By demonstrating that annular heave plates can broaden operational conditions without significant efficiency loss, this work informs the development of more resilient and efficient WECs. The findings have direct implications for renewable energy generation, offshore infrastructure and sustainable marine technologies, contributing to the expansion of wave energy as a viable power source for remote and coastal communities.
1. Introduction
Wave energy conversion has strong potential as a sustainable and environmentally friendly source of renewable energy (Drew et al., Reference Drew, Plummer and Sahinkaya2009; Falcao, Reference Falcao2010; López et al., Reference López, Andreu, Ceballos, De Alegría and Kortabarria2013; Sheng, Reference Sheng2019; Soares et al., Reference Soares, Bhattacharjee, Tello and Pietra2012; Guo & Ringwood, Reference Guo and Ringwood2021b; Clemente et al., Reference Clemente, Rosa-Santos and Taveira-Pinto2021; Guo et al., Reference Guo, Wang, Jin, Duan, Yang and Zhao2022; Li et al., Reference Li, Martin, Jiang, Chen, Thiagarajan, Parker and Zuo2021). Unlike fossil fuels, wave energy produces no greenhouse gas emissions and can be developed with minimal impact on marine ecosystems, with some systems even providing artificial habitats. Although not yet widely deployed, continued research and development could enable broader adoption and help reduce reliance on non-renewable energy sources. Wave energy can also support various coastal and offshore applications, such as desalination, renewable power for offshore industries and electricity for remote coastal or island communities. It can power floating sensors, underwater vehicles for ocean monitoring and aquaculture operations that require a steady energy supply for feeding, filtration and monitoring. By complementing other renewable sources, wave energy conversion can contribute to a more resilient and sustainable energy portfolio. Despite this strong potential, the widespread adoption of wave energy conversion faces significant challenges. These include high capital and maintenance costs, the need for robust designs to survive harsh marine environments, difficulties in long-term device reliability and the variable nature of wave energy, which can complicate consistent power delivery and grid integration. These drawbacks have historically limited the deployment of market-ready wave energy devices, despite substantial global investment and research efforts.
When deployed in relatively moderate to deep waters, wave energy converters (WECs) at the utility and even powering-the-blue-economy scales can be viewed generally as two-body assemblies. Such systems consist of a surface-piercing (or near-surface) body, which is the main mover excited primarily by the incident waves, and a deeper subsurface (reaction) body that is designed to provide the greatest relative motion for maximum power capture (see figure 1). The most common reaction body used is the heave plate and its associated variants. The heave plate has been chosen, for the most part, because its added mass-to-volume ratio is one of the best for bodies oscillating in an infinite fluid medium, which is a reasonable assumption for reaction bodies deeper than 20 m below the water surface. While some progress has been made to date in optimising the shape of floating bodies to maximise power capture (Esmaeilzadeh & Alam, Reference Esmaeilzadeh and Alam2019; Garcia-Teruel & Forehand, Reference Garcia-Teruel and Forehand2021; Goggins & Finnegan, Reference Goggins and Finnegan2014; Guo & Ringwood, Reference Guo and Ringwood2021 a), efforts to achieve more efficient power extraction by adjusting the hydrodynamic response of subsurface bodies remain limited.
Figure 1. Schematics of our two-body point-absorber WEC consisting of a floating sphere of radius
$R_s$
and a submerged annular disk of inner radius
$R_p$
and outer radius
$R_d$
. The sphere and disk are connected together by a spring of constant
$c_{pto}$
and a dashpot with a damping coefficient of
$b_{pto}$
.
A practical way to influence the hydrodynamics of heave plates is to alter their permeability, e.g. through perforation (Hamilton et al., Reference Hamilton, Cazenave, Forbush, Coe and Bacelli2021; Rosenberg & Mundon, Reference Rosenberg and Mundon2016). However, little is known about the comparative performance of WECs with porous heave plates versus traditional solid reaction plates. To help fill this gap, here, we theoretically analyse the performance of a two-body WEC consisting of a spherical float linked to a heave plate in the form of an annular disk via a power take-off (PTO) system. The annular disk represents perhaps the simplest perforated disk with only a single hole in the middle. We apply the linear wave theory to calculate the WEC’s frequency-domain response to regular plane waves, assuming small-amplitude harmonic oscillations. Specifically, we examine how varying the disk porosity affects the performance of the WEC under incident waves of different frequencies. Our calculations reveal numerous scenarios where using annular disks instead of solid ones can enhance power generation. We also find that permeable plates can broaden the operational conditions for WECs by lowering the amplitude of oscillations without sacrificing power conversion efficiency. Furthermore, our results indicate that annular plates may lower the strain on the PTO system with minimal impact on converted power.
In what follows, we first describe the problem statement and our solution (§ 2). The results of our calculations are presented and discussed next (§ 3). A short summary and concluding remarks are given at the end (§ 4) and supplemental information is provided in Appendices A, B and C.
2. Problem statement and solution
We seek to calculate the power extracted from regular waves by a two-body WEC consisting of (i) a floating spherical buoy, (ii) a submerged annular disk acting as a heave plate and (iii) a PTO system connecting the two bodies by a spring and damper with adjustable coefficients (see figure 1). Suppose (i) the water depth is infinite, (ii) the thickness of the disk is negligible hydrodynamically, (iii) the wave amplitude is sufficiently small, compared with the characteristic dimensions of the buoy and disk, to allow the system to be reasonably approximated as linear, (iv) the spherical buoy is half-submerged when the whole assembly is at rest, (v) the viscous damping of the floating body and the radiation damping of the submerged body can be ignored, (vi) the hydrodynamic coupling between the floating and submerged bodies is very weak, (vii) the motion of both the buoy and disk is restricted to heave in the
$x$
direction and (viii) the PTO system has negligible mass.
Under these typical conditions (Beatty et al., Reference Beatty, Hall, Buckham, Wild and Bocking2015; Falnes, Reference Falnes1999; Liang & Zuo, Reference Liang and Zuo2017; Wu et al., Reference Wu, Wang, Diao, Peng and Zhang2014), the dynamics of the coupled system is governed by
\begin{align} f_s &= ( m_s + a_s) \ddot {x}_s + b_s \dot {x}_s + b_{pto} ( \dot {x}_s - \dot {x}_d ) + c_{pto} (x_s - x_d ) + c_s x_s, \\[-9pt] \nonumber \end{align}
\begin{align} f_d &= (m_d + a_d ) \ddot {x}_d + b_d \dot {x}_d + b_{pto} (\dot {x}_d - \dot {x}_s) + c_{pto} (x_d - x_s ), \\[12pt] \nonumber \end{align}
where
$f$
,
$m$
,
$\ddot {x}$
,
$\dot {x}$
,
$x$
,
$a$
and
$b$
represent the excitation force, mass, acceleration, velocity, displacement, and added mass and damping coefficients, respectively, with variables belonging to the sphere, disk and PTO being distinguished by the subscripts s, d and pto, respectively. Also,
$c_s$
and
$c_{pto}$
denote the hydrostatic spring coefficient of the buoy and the spring constant of the PTO system, respectively.
Our goal is to solve the above coupled linear ordinary differential equations and use the solution to determine the power converted by the WEC system. In the first step, it is mathematically convenient to express the variables of interest as real parts of complex quantities, i.e.
\begin{align} f_s &= \mathrm{Re} \left [F_s \exp (i \, \omega \, t)\right ], \\[-9pt] \nonumber \end{align}
\begin{align} f_d &= \mathrm{Re} \left [F_d \exp (i \, \omega \, t)\right ], \\[-9pt] \nonumber \end{align}
\begin{align} \dot {x}_s &= \mathrm{Re} \left [\dot {X}_s \exp (i \, \omega \, t)\right ], \\[-9pt] \nonumber \end{align}
\begin{align} \dot {x}_d &= \mathrm{Re} \left [\dot {X}_d \exp (i \, \omega \, t)\right ], \\[-9pt] \nonumber \end{align}
\begin{align} \dot {x}_r &= \dot {x}_s - \dot {x}_d = \mathrm{Re} \left [ \dot {X}_r \exp (i \, \omega \, t) \right ], \\[6pt] \nonumber \end{align}
where
$\omega$
and
$t$
represent the angular frequency of the incident wave and time, respectively, and
$i^2 = -1$
. Next, we can rewrite (2.1) and (2.2) in the frequency domain as
\begin{align} F_s &= Z_s \, \dot {X}_s + Z_{pto} \, \dot {X}_r, \\[-9pt] \nonumber \end{align}
\begin{align} F_d &= Z_d \, \dot {X}_d - Z_{pto} \, \dot {X}_r, \\[6pt] \nonumber \end{align}
where
\begin{align} Z_s &= b_s + i \, \omega \left ( m_s + a_s - \frac {c_s}{\omega ^2} \right ), \\[-9pt] \nonumber \end{align}
\begin{align} Z_d &= b_d + i \, \omega ( m_d + a_d), \\[-9pt] \nonumber \end{align}
\begin{align} Z_{pto} &= b_{pto} + \frac {c_{pto}}{i \, \omega }. \\[6pt] \nonumber \end{align}
Taking the difference of (2.8) and (2.9), after some rearrangements, we arrive at the following relation for the relative velocity of the buoy and disk:
\begin{align} \dot {X}_r = \frac {F_0}{Z_i + Z_{pto}}, \end{align}
where
\begin{align} Z_i &= \frac {Z_s \, Z_d}{Z_s + Z_d}, \\[-9pt] \nonumber \end{align}
\begin{align} F_0 &= \left ( \frac {Z_d}{Z_s + Z_d} \right ) F_s - \left ( \frac {Z_s}{Z_s + Z_d} \right ) F_d. \\[6pt] \nonumber \end{align}
With
$\dot {X}_r$
known, the velocity of the sphere and disk can be obtained through, respectively,
\begin{align} \dot {X}_s &= \frac {F_s - Z_{pto} \, \dot {X}_r}{Z_s}, \\[-9pt] \nonumber \end{align}
\begin{align} \dot {X}_d &= \frac {F_d + Z_{pto} \, \dot {X}_r}{Z_d}. \\[6pt] \nonumber \end{align}
The average power converted by the PTO system during one period of oscillations
$T$
is calculated from
\begin{align} \bar {P} &= \frac {1}{T} \int _0^T \left ( b_{pto} \, \dot {x}_r + c_{pto} \, x_r \right ) \dot {x}_r \;\textrm {d}t = \frac {1}{T} \int _0^T b_{pto} \, \dot {x}_r^2 \;\textrm {d}t \notag \\ &= \frac {1}{2} b_{pto} \lvert \dot {X}_r \rvert ^2 = \frac {1}{2} b_{pto} \left | \frac {F_0}{Z_i + Z_{pto}} \right |^2. \\[6pt] \nonumber \end{align}
Multiplying (2.1) and (2.2) by
$\dot {x}_s$
and
$\dot {x}_d$
, respectively, and adding, gives
\begin{align} f_s \dot {x}_s + f_d \dot {x}_d &= ( m_s + a_s ) \ddot {x}_s \dot {x}_s + b_s \dot {x}_s^2 + b_{pto} ( \dot {x}_s - \dot {x}_d ) \dot {x}_s + c_{pto} ( x_s - x_d ) \dot {x}_s + c_s x_s \dot {x}_s \notag \\ & \quad + (m_d + a_d) \ddot {x}_d \dot {x}_d + b_d \dot {x}_d^2 + b_{pto} (\dot {x}_d - \dot {x}_s) \dot {x}_d + c_{pto} ( x_d - x_s) \dot {x}_d. \\[6pt] \nonumber \end{align}
Combining the PTO terms, we have
\begin{align} b_{pto} (\dot {x}_s - \dot {x}_d) \dot {x}_s + b_{pto} (\dot {x}_d - \dot {x}_s) \dot {x}_d = b_{pto} \left ( \dot {x}_s^2 + \dot {x}_d^2 - 2 \dot {x}_s \dot {x}_d \right ) = b_{pto} \, \dot {x}_r^2. \end{align}
Over one oscillation period, the time average of the inertial terms vanishes, i.e.
\begin{align} \int _0^T \ddot {x}_s \dot {x}_s \; \textrm {d}t = \int _0^T \ddot {x}_d \dot {x}_d \; \textrm {d}t = 0. \end{align}
Also, the spring terms are conservative and thus do not contribute to the net power. Therefore, after averaging over a cycle, the mean converted power is
\begin{align} \bar {P} = \underbrace {\frac {1}{T} \int _0^T f_s \, \dot {x}_s \;\textrm {d}t}_{\bar {P}_s^{e}} + \underbrace {\frac {1}{T} \int _0^T f_d \, \dot {x}_d \;\textrm {d}t}_{\bar {P}_d^{e}} - \underbrace {\frac {1}{T} \int _0^T b_s \, \dot {x}_s^2 \;\textrm {d}t}_{\bar {P}_s^{d}} - \underbrace {\frac {1}{T} \int _0^T b_d \, \dot {x}_d^2 \;\textrm {d}t}_{\bar {P}_d^{d}}, \end{align}
where
$\bar {P}_s^{e}$
and
$\bar {P}_d^{e}$
are the input powers due to the excitation forces exerted on the sphere and disk, respectively, and
$\bar {P}_s^{d}$
and
$\bar {P}_d^{d}$
are the power dissipated due to their motions. For a given
$F_0$
and
$Z_i$
, it can be shown (see, e.g. Falnes, Reference Falnes1999) that
$\bar {P}$
is maximum when
$Z_{pto}$
is equal to the complex conjugate of
$Z_i$
(denoted by
$Z_i^*$
), leading to
\begin{align} \bar {P}_{max} = \frac {\lvert F_0 \rvert ^2}{8 \, \mathrm{Re} (Z_i)}, \end{align}
where
$\mathrm{Re} (Z_i)$
denotes the real part of
$Z_i$
. However, if
$Z_{pto}$
is restricted to be real (i.e. the spring constant of the PTO systems is set to zero), then the maximum power is extracted when
$b_{pto}$
is equal to the magnitude of
$Z_i$
(denoted by
$| Z_i |$
), resulting in
\begin{align} \bar {P}_{max}^{res} = \frac {\lvert F_0 \rvert ^2}{4 \left [ \mathrm{Re} (Z_i) + | Z_i | \right ] }. \end{align}
In practice, PTO systems can maintain near-optimal conditions by actively adjusting the resistive damping through various control strategies.
To evaluate the maximum power in either case, we need first to determine
$F_s$
,
$F_d$
,
$c_s$
,
$a_s$
,
$b_s$
,
$a_d$
and
$b_d$
. The excitation forces on the sphere and disk are related to the corresponding radiation damping of these objects through the Haskind relation (see, e.g. Falnes, Reference Falnes2002). For the sphere, this means
\begin{align} F_s = | F_s | = \sqrt { 8 \, Q \left (\frac {\lambda }{2 \, \pi } \right ) b_s}, \end{align}
where
$Q = \rho \, g^2 A^2 / 4 \, \omega$
,
$\lambda = 2 \, \pi \, g / \omega ^2$
and
$A$
are the power flux, wavelength and amplitude of the incident wave, respectively, with
$\rho$
being the density of water and
$g$
being the gravitational acceleration. Also, for the disk, the reciprocity relation between the radiation resistance and the excitation force demands
$F_d$
to vanish since the radiation damping of the disk is negligible. To obtain explicit expressions for the maximum extracted powers, we first substitute the relations for
$F_s$
and
$F_d$
into (2.15). We then insert the result into (2.23) and (2.24) and use the definition of
$Z_i$
from (2.14) to obtain
\begin{align} \bar {P}_{max} &= Q \left ( \frac {\lambda }{2 \, \pi } \right ) \! \left \{ 1 + \left (\frac {b_d}{b_s} \right ) \! \left [\frac { b_s^2 + \omega ^2 \left (m_s + a_s - c_s / \omega ^2 \right )^2 }{ b_d^2 + \omega ^2 (m_d + a_d)^2 } \right ] \right \}^{-1}, \\[-9pt] \nonumber \end{align}
\begin{align} \frac {\bar {P}_{max}^{res}}{2 \, \bar {P}_{max}} &= \left \{ 1 + \sqrt { 1 + \left [ \frac {\mathrm{Im} (Z_i)}{\mathrm{Re} (Z_i)} \right ]^2 } \right \}^{-1} \notag \\ &= \left ( 1 + \sqrt { 1 + \left \{ \frac { \left (m_d + a_d\right ) + \left [ \frac { b_d^2 + \omega ^2 \left (m_d + a_d \right )^2 }{ b_s^2 + \omega ^2 \left ( m_s + a_s - c_s / \omega ^2\right )^2 } \right ] \left (m_s + a_s - c_s / \omega ^2\right ) } { \frac {b_d}{\omega } + \left [ \frac { b_d^2 + \omega ^2 \left ( m_d + a_d \right )^2 }{ b_s^2 + \omega ^2 \left (m_s + a_s - c_s / \omega ^2\right )^2 } \right ] \left (\frac {b_s}{\omega } \right ) } \right \}^2 } \right )^{-1}, \\[6pt] \nonumber \end{align}
where
$\mathrm{Im} (Z_i)$
denotes the imaginary part of
$Z_i$
. Additionally, given that the sphere is half-submerged at rest, it is straightforward to show that the linearised spring coefficient due to changes in the buoyancy of the sphere takes the form of
\begin{align} c_s = \rho \, g \, \pi \, R_s^2, \end{align}
where
$R_s$
represents the radius of the sphere, and that
\begin{align} m_s + m_d = \rho \, \pi \left [ \frac {2 R_s^3}{3} + t_d \left (R_d^2 - R_p^2\right ) \right ], \end{align}
with
$R_d$
and
$R_p$
denoting, respectively, the outer and inner radii of the disk and
$t_d$
representing its thickness (see also figure 1).
We use the expressions derived by Hulme (Reference Hulme1982) using the linear wave theory to compute
$a_s$
and
$b_s$
. We also use the formulation developed by Usman & Masoud (Reference Usman and Masoud2025), which is based on the linearised Navier–Stokes equations combined with perturbation theory and the reciprocal theorem, to obtain
$a_d$
and
$b_d$
. To maintain clarity and a seamless flow of information in the main text, the exact equations used and the behaviour of the calculated coefficients are provided in Appendices A and B.
While the assumptions made here are consistent with the practical parameter range described in the following section, relaxing these conditions – for example, by considering larger wave amplitudes or stronger hydrodynamic coupling – would require accounting for nonlinear and boundary effects. These factors could modify the resonance behaviour, optimal PTO settings and predicted power performance, motivating future studies using detailed numerical simulations or experimental measurements.
3. Results and discussion
We set the properties and dimensions of our WEC to be consistent with the assumptions outlined at the beginning of § 2 and to align closely with those of Reference Model 3 (Neary et al., Reference Neary, Lawson, Previsic, Copping, Hallett, Labonte, Rieks and Murray2014), a benchmark two-body point-absorber WEC designed by the U.S. Department of Energy’s national laboratories to facilitate standardised evaluation of the performance and cost of various marine hydrokinetic technologies. Table 1 lists our chosen parameters. Additionally, we assume that both the float and the reaction plate are made of steel with a density of
$\rho _{steel} = 7800 \, \mathrm{kg}\, \mathrm{m}^{-3}$
. Note that the mass of the buoy is calculated from (2.29), eliminating the need to explicitly specify the thickness of the spherical shell. These selections ensure that our study covers a range of practical scenarios of current interest. To analyse the performance of our WEC, we focus on the case where the PTO system exerts only a resistive force, achieved by setting the spring constant of the PTO to zero. This simplification avoids the complications of negative spring constants, which could arise if no restrictions are placed on
$c_{pto}$
. In this scenario, the spring’s restoring force acts in the same direction as its displacement rather than opposing it, which is generally destabilising. In our model, the mean tension in the PTO lines is maintained by a mechanical mechanism that allows the system to oscillate around its mean position. However, in practice, a spring term is often included to compensate for weight discrepancies.
Table 1. Incident wave and WEC properties
Figures 2, 3, 4 and 5 present the outcomes of our calculations. Specifically, figure 2 shows the maximum average extracted power (
$\bar {P}_{max}^{res}$
), the associated amplitude of the relative displacement between the buoy and the disk (
$|X_r|$
) and the damping coefficient of the PTO (
$b_{pto}$
) versus the wave period (
$T$
) for disk outer radius to sphere radius ratios of
$R_d / R_s = 1 / 2 , 1 , 3 / 2 \, \text{and} \, 2$
(represented by the blue, black, red and green curves, respectively). In this figure, the left, middle and right columns correspond to the ratios of the inner-to-outer radius of the disk
$R_p / R_d = 10^{-2}, 10^{-1} \, \text{and} \, 1 / 2$
, respectively. Figure 3 illustrates these same variables but plotted against
$R_p / R_d$
, with the left, middle and right columns corresponding to
$T = 5, 10 \, \text{and} \, 15 \, \mathrm{s}$
, respectively. Figure 4 normalises these variables by their respective values against the baseline scenario of solid disk with
$R_p = 0$
(distinguished by the subscript 0), and figure 5 further explores the normalised variables across different wave periods. This normalisation reveals the efficiency gains or losses attributed to the annular design.
Figure 2. Plots of (a–c) the maximum average extracted power when
$c_{pto} = 0$
(denoted by
$\bar {P}_{max}^{res}$
), the corresponding (d–f) amplitude of the relative displacement between the buoy and the disk (denoted by
$|X_r|$
) and (g–i) the damping coefficient of the PTO (denoted by
$b_{pto}$
) versus the wave period (denoted by
$T$
) for the WEC parameters listed in table 1 and the disk outer radius to sphere radius ratio of
$R_d / R_s = 1 / 2, 1, 3 / 2 \, \text{and} \, 2$
. The left, middle and right columns are for
$R_p / R_d = 10^{-2}, 10^{-1} \, \text{and} \, 1 / 2$
, respectively.
Figure 3. Plots of (a–c) the maximum average extracted power when
$c_{pto} = 0$
(denoted by
$\bar {P}_{max}^{res}$
), the corresponding (d–f) amplitude of the relative displacement between the buoy and the disk (denoted by
$|X_r|$
) and (g–i) the damping coefficient of the PTO (denoted by
$b_{pto}$
) versus the ratio of the inner-to-outer radius of the disk (i.e.
$R_p / R_d$
) for the WEC parameters listed in table 1 and the disk outer radius to sphere radius ratio of
$R_d / R_s = 1 / 2, 1, 3 / 2 \, \text{and} \, 2$
. The left, middle and right columns are for
$T = 5, 10 \, \text{and} \, 15 \, \mathrm{s}$
, respectively.
Figure 4. Plots of (a–c)
$\bar {P}_{max}^{res}$
, (d–f)
$|X_r|$
and (g–i)
$b_{pto}$
normalised by their corresponding values for a solid disk with
$R_p = 0$
(distinguished by the subscript 0) versus
$T$
for the WEC parameters listed in table 1 and
$R_d / R_s = 1 / 2, 1, 3 / 2 \, \text{and} \, 2$
. The left, middle and right columns are for
$R_p / R_d = 10^{-2}, 10^{-1} \, \text{and} \, 1 / 2$
, respectively.
Figure 5. Plots of (a–c)
$\bar {P}_{max}^{res}$
, (d–f)
$|X_r|$
and (g–i)
$b_{pto}$
normalised by their corresponding values for a solid disk with
$R_p = 0$
(distinguished by the subscript 0) versus
$R_p / R_d$
for the WEC parameters listed in table 1 and
$R_d / R_s = 1 / 2, 1, 3 / 2 \, \text{and} \, 2$
. The left, middle and right columns are for
$T = 5, 10 \, \text{and} \, 15 \, \mathrm{s}$
, respectively.
Given Eqs. (2.26)–(2.29) and the asymptotic behaviours of the added mass and damping coefficients of the sphere and the disk (see Appendices A and B), it can be deduced that
$\bar {P}_{max}^{res}$
approaches zero in the extreme limits of
$T \to 0$
and
$T \to \infty$
(see also (C.1) of Appendix C). Consequently, the extracted power is expected to peak at an intermediate wave period. This expectation is confirmed by our results, although the wave period at which
$\bar {P}_{max}^{res}$
peaks lies outside
$5 \, \mathrm{s} \leqslant T \leqslant 15 \, \mathrm{s}$
for
$R_d / R_s = 2$
when the disk’s pore size is not large enough (see the first row of figure 2). We find that, within the range of parameters considered, decreasing the size of the reaction plate relative to the float (i.e. lowering
$R_d / R_s$
) shifts the
$\bar {P}_{max}^{res}$
versus
$T$
curves to the left. Increasing the disk’s porosity produces the same effect, hence the appearance of the green curve’s peak for
$R_d / R_s = 2$
and
$R_p / R_d = 1 / 2$
in figure 2c. Due to these shifts, at a given
$T$
and
$R_d / R_s = 2$
, the introduction of a hole in the centre of a solid reaction disk can either enhance or reduce power extraction (see the first rows of figures 4 and 5). Notably, the order of magnitude of
$\bar {P}_{max}^{res}$
at the peak points is approximated well by the corresponding
$Q \left (\lambda / 2 \pi \right )$
, which is the prefactor in (2.26) (see also (C.1) of Appendix C).
Regarding the variations of the amplitude of the relative displacement, the following observations can be made for
$R_p / R_d = 10^{-2}$
(see figure 2d). For
$R_d / R_s = 1 / 2$
,
$|X_r|$
increases from the starting point of around 0.3 m at
$T = 5 \, \mathrm{s}$
before reaching a peak of slightly above 1 m at about
$T = 6 \, \mathrm{s}$
, and then gradually decreases and plateaus around 0.7 m when
$T$
approaches 15 s (see the blue curve). A very similar trend is observed for
$R_d / R_s = 1$
(see the black curve), whereas, for
$R_d / R_s = 3 / 2$
, the amplitude starts above 0.5 m and generally increases with increasing
$T$
to just below 0.7 m at the end, while exhibiting minor fluctuations (see the red curve). For the highest ratio of disk-to-sphere radius,
$|X_r|$
rises from approximately 0.3 to 0.6 m during the interval of wave periods examined (see the green curve). The abovementioned patterns are closely followed for
$R_p / R_d = 10^{-1}$
(see figure 2e). The described behaviours are replicated, although with some deviations, for
$R_p / R_d = 1/2$
, as well (see figure 2f). Overall, we see that, in most cases, the amplitude of the relative displacement is lower for disks with larger outer radius. We also see that, depending on the ratio
$R_d / R_s$
and the period of the wave, increasing the porosity of the disk may increase or reduce
$|X_r|$
(see the middle rows figures 4 and 5). Nevertheless, the order of magnitude of
$|X_r|$
remains comparable to the wave amplitude
$A$
(see (C.2) of Appendix C). Lastly, we find that the peaks of
$\bar {P}_{max}^{res}$
correspond to the troughs of
$|X_r|$
, except for
$R_d / R_s = 1 / 2$
, in which case both quantities peak at the same wave period (see the first and second rows of figure 2). This behaviour partly stems from the difference in the last terms inside the curly brackets in (C.1) and (C.2) of Appendix C.
As for the changes in the resistance coefficient of the PTO, the general trend of the curves shown in the last rows of figures 2, 3, 4 and 5 resembles those of the plots illustrated in the first rows of the same figures for the captured power. Notable differences, however, are seen in the range
$5 \, \mathrm{s} \leqslant T \lesssim 7 \, \mathrm{s}$
. Also, while the peaks of
$\bar {P}_{max}^{res}$
and
$b_{pto}$
line up closely for
$R_d / R_s = 1, 3 / 2 \, \text{and} \, 2$
, the peaks of
$\bar {P}_{max}^{res}$
for the smallest disk radius are aligned with local minima for
$b_{pto}$
(compare the curves in the first row of figure 2 with the same curves in the last row of this figure). Last but not least, we find that the maximum values of
$b_{pto}$
roughly scale with
$\rho R_d^3 \omega$
(see (C.3) of Appendix C).
4. Summary
We theoretically analysed the performance of a WEC consisting of a floating sphere and a submerged annular heave plate coupled through a purely resistive PTO system. We examined various scenarios by adjusting the disk’s porosity and the ratio of the disk’s outer radius to the sphere’s radius over a range of realistic wave periods, focusing on the implications for power extraction efficiency and operational conditions. Our analysis revealed a complex interplay between wave period, geometric configurations and disk porosity in optimising WEC performance. Notably, we found that annular disks can significantly improve power capture efficiency compared with solid disks, particularly for certain ratios of disk-to-sphere radius. This enhancement, often accompanied by a reduced amplitude of oscillation for the relative motion between the float and heave plate, may, however, increase the demands on the PTO system, as evidenced by higher normalised damping coefficients. These findings underscore the necessity of balancing power capture with managing mechanical stress across varying sea states to ensure efficient and resilient WEC operation.
In conclusion, the annular disk design offers substantial advantages in power conversion efficiency, highlighting the importance of optimising the reaction plate’s dimensions and porosity. While these benefits must be weighed against the increased mechanical complexity and stress on the system, the optimal parameters identified in this study provide valuable insights for designing more effective WECs. These findings also provide a clear theoretical foundation for future wave tank experiments and prototype field trials to validate the predicted power performance and guide practical design improvements. Future research should focus on the long-term durability and economic feasibility of these designs under real-world conditions. Additionally, it is crucial to experimentally verify the theoretical predictions made in our study to identify any deviations from assumptions and facilitate the refinement and extension of the model for practical applications. Such validation could be carried out through scale-model wave tank experiments or prototype field tests that measure the relative motion between the buoy and the disk, hydrodynamic forces, PTO damping characteristics and energy extraction efficiency under controlled sea states, enabling direct comparison with the frequency-domain predictions developed here. Moreover, our work paves the way for further investigations into more advanced scenarios involving perforated disks with multiple pores. Of particular interest would be determining the optimum pore size and distribution for a fixed total porosity. Understanding the hydrodynamic effects of various pore configurations is essential for developing more efficient and robust porous reaction plates.
Acknowledgements
We thank Dr. Salman Husain for stimulating discussions and for his feedback on the manuscript.
Funding statement
Financial support from the Water Power Technologies Office of the U.S. Department of Energy through a Seedling grant is gratefully acknowledged.
Competing interests
The authors declare no conflict of interest.
Appendix A. Added mass and radiation damping of a half-submerged sphere
Following Hulme (Reference Hulme1982), the added mass
$a_s$
and radiation damping
$b_s$
coefficients of a half-submerged sphere can be expressed as
\begin{align} \frac {a_s}{\rho \, R_s^3} + i \, \frac {b_s}{\rho \, R_s^3 \, \omega } &= a_s^\star + i \, b_s^\star \notag \\&= -2 \pi \beta _0 \left \{ \frac {1}{2} + \varLambda \sum _{m = 0}^\infty \frac {\left (- \varLambda \right )^m}{m!} \left [ \Psi (n) + i \pi - \ln \varLambda \right ] \mathcal{I}( 1 , m ) - \frac {\partial \mathcal{I}}{\partial \sigma } \bigg \rvert _{\substack {\nu = 1\\ \sigma = m}} \right . \notag \\& \quad + \left . \sum _{n = 1}^\infty \beta _n \left [ \frac {\varLambda }{2 n} \mathcal{I}( 1 , 2 n - 1 ) + \mathcal{I}( 1 , 2 n ) \right ] \right \}, \end{align}
where
\begin{align} \varLambda &= \omega ^2 R_s / g, \\[-9pt] \nonumber \end{align}
\begin{align} \Psi (n) &= \left \{ \begin{aligned} &-\gamma \quad &\text{if} \quad n = 0 ,\\ &-\gamma + \sum _{k = 1}^n \frac {1}{k} \quad &\text{if} \quad n \neq 0, \end{aligned} \right . \\[-9pt] \nonumber \end{align}
\begin{align} \mathcal{I}( \nu , \sigma ) &= \frac { 2 \left [ A(\nu , \sigma )^2 \sin \! \left (\pi \, \nu / 2\right ) \cos \! \left (\pi \, \sigma / 2\right ) - \cos \! \left (\pi \, \nu / 2\right ) \sin \! \left (\pi \, \sigma / 2\right ) \right ]} {\pi A(\nu , \sigma ) \left ( \nu - \sigma \right ) \left ( \nu + \sigma +1 \right )}, \\[-9pt] \nonumber \end{align}
\begin{align} A(\nu , \sigma ) &= \frac {\varGamma \! \left ( \nu / 2 + 1 \right ) \varGamma \! \left ( \sigma / 2 + 1 / 2 \right )}{\varGamma \! \left ( \nu / 2 + 1 / 2 \right ) \varGamma \! \left ( \sigma / 2 + 1 \right )}, \\[6pt] \nonumber \end{align}
with
$\gamma$
being the Euler–Mascheroni constant and
$\varGamma$
representing the gamma function. Here, the coefficients
$\beta _n$
are calculated from
\begin{align} &\beta _0 = \frac {1}{2} \left \{ \mathcal{L}(0,\varLambda ) - \varLambda \sum _{n = 1}^\infty \beta _n \, \mathcal{I}( 0 , 2 n - 1 ) \right \}^{-1}, \\[-9pt] \nonumber \end{align}
\begin{align} &\beta _n + \varLambda \sum _{m = 1}^\infty \beta _m \, \frac {4 n + 1}{2 n + 1} \left [ \mathcal{I}( 2 n , 2 m - 1 ) - 2 \mathcal{I}( 2 n , 1 ) \mathcal{I}( 0 , 2 m - 1 )\right ] \notag \\ &= \frac {4 n + 1}{2 n + 1} \left [ \mathcal{L}( 2 n , \varLambda ) - 2 \mathcal{L}( 0 , \varLambda ) \mathcal{I}( 2 n , 1 )\right ] \quad \text{for} \quad n = 1, 2, \cdots , \\[6pt] \nonumber \end{align}
where
\begin{align} \mathcal{L}( \nu , \varLambda ) &= - \mathcal{I}( \nu , 0 ) \left ( 1 + \varLambda \right ) \notag \\& \quad + \varLambda \sum _{m = 1}^\infty \frac {\left (- \varLambda \right )^m}{(m-1)!} \left [ \Psi (n) + i \pi - \ln \varLambda - \frac {1}{m}\right ] \mathcal{I}( \nu , m ) - \frac {\partial \mathcal{I}}{\partial \sigma } \bigg \rvert _{\substack {\nu = \nu \\\sigma = m}}. \\[6pt] \nonumber \end{align}
Equation (A.1) simplifies to
\begin{align} a_s^\star = 1.74 - \frac {\pi }{2} \, \varLambda \, \ln \varLambda + \mathcal{O}( \varLambda ) \qquad \text{and} \qquad b_s^\star = \frac {\pi ^2}{2} \, \varLambda + \mathcal{O}(\varLambda ^2), \end{align}
when
$\varLambda$
approaches zero, and to
\begin{align} a_s^\star = \frac {\pi }{3} - \frac {\pi }{8} \frac {1}{\varLambda } + \unicode{x1D4F8}(\varLambda ^{-1}) \qquad \text{and} \qquad b_s^\star = \frac {9 \pi }{\varLambda ^4} + \unicode{x1D4F8}(\varLambda ^{-4}), \end{align}
when
$\varLambda \to \infty$
. Figure A1 shows the variations of
$a_s^\star$
and
$b_s^\star$
as a function of
$\varLambda$
.
Figure A1. Plots of dimensionless (a) added mass and (b) radiation damping of the sphere (denoted by
$a_s^\star$
and
$b_s^\star$
, respectively) versus the dimensionless angular wavenumber
$\varLambda$
.
Figure B1. Plots of dimensionless (a) added mass and (b) viscous damping coefficients of a deeply submerged solid disk (denoted by
$a_{d_0}^\star$
and
$b_{d_0}^\star$
, respectively) versus the dimensionless frequency (also known as the square root of the oscillatory Reynolds number)
$\eta$
.
Appendix B. Added mass and viscous damping coefficients of an oscillating annular disk
Zhang & Stone (Reference Zhang and Stone1998) showed that the added mass
$a_{d_0}$
and viscous damping
$b_{d_0}$
coefficients of a deeply submerged solid disk oscillating with small amplitude can be obtained from
\begin{align} \frac {a_{d_0}}{\rho \, R_d^3} - i \, \frac {b_{d_0}}{\rho \, R_d^3 \, \omega } = a_{d_0}^\star - i \, b_{d_0}^\star = 8 \, \alpha _0, \end{align}
with
$R_d$
being the radius of the disk and
\begin{align} \sum _{m = 0}^\infty \alpha _m \int _0^\infty k \left ( 1 - \frac {k}{ \sqrt { k^2 + i \, \eta ^2 } } \right ) J_{2m+1/2}(k) \, J_{2n+1/2}(k) \; \textrm {d}k = \delta _{0n} \quad \text{for} \quad n = 0, 1, \cdots , \end{align}
where
$m$
and
$n$
are integers,
$\eta = \sqrt {\rho \, R_d^2 \, \omega / \mu }$
,
$\mu$
is the water viscosity,
$J$
is the Bessel function of the first kind and
$\delta _{ij}$
is the Kronecker delta function. The hydrodynamics coefficients of the disk can be expressed succinctly as
\begin{align} a_{d_0}^\star = \frac { 64 \sqrt {2} }{ 3 \pi \, \eta } + \cdots \qquad \text{and} \qquad b_{d_0}^\star = \frac {16}{\eta ^2} + \frac {64 \sqrt {2}}{ 3 \pi \, \eta } + \cdots , \end{align}
in the asymptotic limit of
$\eta \to 0$
, and, according to Usman & Masoud (Reference Usman and Masoud2025), as
\begin{align} a_{d_0}^\star = \frac {8}{3} + \frac { \mathcal{A}_a \ln \eta + \mathcal{B}_a }{\eta } + \cdots \qquad \text{and} \qquad b_{d_0}^\star = \frac { \mathcal{A}_b \ln \eta + \mathcal{B}_b }{\eta } + \cdots , \end{align}
in the opposite limit of
$\eta \to \infty$
, with
\begin{align} \mathcal{A}_a = 1.700, \quad \mathcal{B}_a = 8.349, \quad \mathcal{A}_b = 1.235, \quad \mathcal{B}_b = 8.155. \end{align}
Figure B1 shows the variations of
$a_d^\star$
and
$b_d^\star$
as a function of
$\eta$
.
Very recently, Usman & Masoud (Reference Usman and Masoud2025) derived the following approximate expression for the added mass and viscous damping coefficients of an annular disk with an inner to outer radii ratio of
$\varepsilon = R_p / R_d$
\begin{align} \frac {a_d}{\rho \, R_d^3} - i \, \frac {b_d}{\rho \, R_d^3 \, \omega } = a_d^\star - i \, b_d^\star = a_{d_0}^\star - i \, b_{d_0}^\star + 8 {p^\star }^2 \beta _0 \, \varepsilon , \end{align}
where
\begin{align} p^\star = \frac {2 i}{\sqrt {\pi }} \sum _{m = 0 }^\infty \frac {\varGamma (m+1)}{\varGamma (m+1/2)} \, \alpha _m, \\[-24pt] \nonumber \end{align}
\begin{align} \sum _{m = 0}^\infty \beta _m \int _0^\infty \frac {J_{2m+1/2}(k) \, J_{2n+1/2}(k)} {\displaystyle k \left ( 1 - \frac {k}{ \sqrt { k^2 + i \, \varepsilon ^2 \eta ^2 } } \right ) } \; \textrm {d}k = \delta _{0n} \quad \text{for} \quad n = 0, 1, \cdots . \end{align}
Equation (B.6) was obtained by applying the reciprocal theorem (Masoud & Stone, Reference Masoud and Stone2019) in conjunction with perturbation expansion in terms of the inner radius of the annular disk. From comparison with direct numerical simulations, it was shown that (B.6) maintains excellent accuracy over the range
$0 \leqslant \varepsilon \lesssim 0.5$
. Figure B2 presents the variations of
$a_d / a_{d_0}$
and
$b_d / b_{d_0}$
as a function of
$\varepsilon$
for
$\eta = 10^3$
.
Figure B2. Plots of normalised (a) added mass and (b) viscous damping coefficients of a deeply submerged annular disk (i.e.
$a_d / a_{d_0}$
and
$b_d / b_{d_0}$
, respectively) versus the ratio of disk’s inner to outer radius (denoted by
$\varepsilon$
) at
$\eta = 10^3$
.
Appendix C. Dimensionless representations of
$\bar {P}_{max}$
,
$|X_r|$
and
$b_{pto}$
The equations below provide non-dimensional forms of the quantities plotted in figures 2–3, 4 and 5, expressed in terms of the dimensionless parameters introduced here and in the previous two appendices. These formulations aid in rationalising the trends discussed in § 3
\begin{align} \bar {P}_{max}^{res, \star } &= \frac {\bar {P}_{max}^{res}}{Q \left ( \lambda / 2 \pi \right )} \nonumber \\ &= 2 \left \{ 1 + \left ( \frac {R_d}{R_s} \right )^{\!\!-3} \! \left ( \frac {b_d^\star }{b_s^\star } \right ) \left [ \frac { {b_s^\star }^2 + \left ( m_s^\star + a_s^\star - \pi / \varLambda \right )^2 }{ {b_d^\star }^2 + \left ( m_d^\star + a_d^\star \right )^2 } \right ] \left [ 1 + \frac {| Z_i |}{\mathrm{Re} (Z_i)} \right ] \right \}^{-1}, \\[-9pt] \nonumber \end{align}
\begin{align} | X_r^\star | &= \left | \frac {X_r}{A} \right | = \left \{ \left ( \frac {\varLambda ^3}{b_s^\star } \right ) \left [ {b_s^\star }^2 + \left ( m_s^\star + a_s^\star - \pi / \varLambda \right )^2 \right ] \left [ 1 + \frac {\mathrm{Re} (Z_i)}{| Z_i |} \right ] \right \}^{- 1 / 2}, \\[-9pt] \nonumber \end{align}
\begin{align} b_{pto}^\star &= \frac {b_{pto}}{\rho \, R_d^3 \, \omega } = \frac {| Z_s^\star | | Z_d^\star |}{| Z_s^\star + \left ( R_d / R_s \right )^3 Z_d^\star |} \nonumber \\[5pt] &=\sqrt { \frac { \left [ {b_s^\star }^2 + \left ( m_s^\star + a_s^\star - \pi / \varLambda \right )^2 \right ] \left [ {b_d^\star }^2 + \left ( m_d^\star + a_d^\star \right )^2 \right ] } { \left [ b_s^\star + \left (R_d / R_s\right )^3 b_d^\star \right ]^2 + \left [ \left ( m_s^\star + a_s^\star - \pi / \varLambda \right ) + \left ( R_d / R_s \right )^3 \left ( m_d^\star + a_d^\star \right )\right ]^2} }, \\[6pt] \nonumber \end{align}
where
\begin{align} Z_s^\star &= \frac {Z_s}{\rho \, R_s^3 \, \omega },\quad Z_d^\star = \frac {Z_d}{\rho \, R_d^3 \, \omega },\quad m_s^\star = \frac {m_s}{\rho \, R_s^3},\quad m_d^\star = \frac {m_d}{\rho \, R_d^3}, \nonumber \\[5pt] \frac {| Z_i |}{\mathrm{Re} (Z_i)} &= \sqrt { 1 + \left \{ \frac { \left ( m_d^\star + a_d^\star \right ) + \left [ \frac { {b_d^\star }^2 + \left ( m_d^\star + a_d^\star \right )^2 }{ {b_s^\star }^2 + \left ( m_s^\star + a_s^\star - \pi / \varLambda \right )^2 } \right ] \left ( m_s^\star + a_s^\star - \pi / \varLambda \right ) \left (R_d / R_s\right )^3 } { b_d^\star + \left [ \frac { {b_d^\star }^2 + \left ( m_d^\star + a_d^\star \right )^2 } { {b_s^\star }^2 + \left ( m_s^\star + a_s^\star - \pi / \varLambda \right )^2 } \right ] \left [ b_s^\star \left (R_d / R_s\right )^3 \right ]} \right \}^2 }. \end{align}
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