1. Introduction
Numerous experiments are conducted in ocean engineering basins (OEB) to evaluate and validate the wave-induced motions and loads of ships and offshore structures. Waves are generated under various conditions, often in combination with currents and wind, to assess the motion performance of floating bodies. However, because a basin has finite dimensions, the generated waves cannot leave the basin and must therefore be dissipated by an appropriate wave-absorbing device. Generally, regular waves are generated over multiple wave periods to allow sufficient time for evaluating motion performance, and measurements are taken before reflected waves return to the test area. In irregular wave experiments, which typically require longer durations, tests are performed such that the wave spectrum at the structure location satisfies the target conditions even in the presence of some reflected waves. Because a large number of waves are frequently generated during an experiment, and subsequent tests cannot begin until the remaining waves have sufficiently decayed, minimizing wave reflection has a significant impact on both the efficiency and accuracy of basin experiments.
Most OEBs employ wave-absorption systems installed at the end opposite the wave generator to dissipate incoming waves. Wave-absorption systems can generally be classified into active and passive wave absorbers based on the mechanism used to attenuate wave energy. Active wave absorbers generate counteracting waves that interfere with the incident waves and thereby reduce wave energy through destructive interference. Although this approach can be readily applied in simplified two-dimensional wave flumes or relatively small basins, it is difficult to use as a standalone solution for waves with various propagation directions or strong nonlinear characteristics (Schäffer & Klopman, 2000; De Mello et al., 2013; Herdayanditya et al., 2025). For example, large circular basins such as those at the National Maritime Research Institute (NMRI) and FloWave Ocean Energy Research Facility of the University of Edinburgh employ wave-generating paddles for wave absorption, but passive wave absorbers are also installed (Gyongy et al., 2014). In contrast, the most common approach in three-dimensional basins is the installation of passive wave absorbers. Wave-absorbing plates dissipate wave energy primarily through two mechanisms. The first involves introducing multiple perforations into the plate, thereby inducing energy dissipation through pressure losses across the openings. The second involves reducing the local water depth to induce shoaling and ultimately wave breaking. Wave attenuation through perforations is generally effective over a wide frequency range, whereas inducing wave breaking is particularly effective in the long-wave region, where wave diffraction effects are relatively weak. Additional energy dissipation may occur through vortex generation at the plate edges, whereas phase differences created by the confined space between the plate underside and free surface can produce reflected waves that are out of phase with the incident waves (Lebey & Rivoalen, 2002; Lim, 2014; Urrego-Pabón et al., 2024).
Numerical studies of perforated structures have been extensively conducted based on the potential flow theory (Huang et al., 2011; Han & Wang, 2022). The damping effects associated with perforated plates have generally been modeled using two approaches. The first is the Darcy damping model, which linearly relates the perforation rate to the fluid pressure drop. Cho and Kim (2008) experimentally determined the relationship between perforation rate and fluid pressure and proposed a corresponding damping model. They also investigated the reflection characteristics of inclined wave-absorbing plates using an eigenfunction expansion method and a domain decomposition approach based on the potential flow theory. The second approach employs a drag coefficient based on the principle of energy equivalence. Ko and Cho (2018) incorporated a drag coefficient into a potential-flow-based numerical model and validated the drag coefficient corresponding to different perforation rates through comparison with wave tank experiments.
Wave-absorbing plates are installed in a wide variety of configurations worldwide, and gentle slopes are generally recognized to provide superior performance (Ouellet & Datta, 1986; Laflèche et al., 2023). Key design parameters include the plate geometry, length, and number of plates. The geometry is typically either flat or parabolic. Parabolic configurations are generally known to provide better wave absorption than flat plates over a broad range of wave periods (Neelamani et al., 2004; Hodaei et al., 2016; Izquierdo et al., 2021). Regardless of geometry, increasing the plate length generally improves absorption performance over a wider frequency range because a larger number of perforations can be incorporated. However, longer plates reduce the available test area within the basin. In addition, arranging multiple wave-absorbing plates with overlapping sections near the free surface can improve absorption performance. Nevertheless, because fluid velocity decreases rapidly in such configurations, an appropriate spacing between plates must be considered.
In this study, the reflection coefficients of the wave-absorbing plates installed in the OEB of the Korea Research Institute of Ships and Ocean Engineering (KRISO) were evaluated. Furthermore, the effects of plate geometry, length, and number on wave reflection were numerically investigated to identify methods for improving wave-absorption performance. The following section introduces the theoretical background and numerical methods used for wave absorption and reflection analysis. Section 3 discusses the numerical results, and Section 4 presents the conclusions.
2. Numerical Analysis
2.1 Potential Flow Theory
In this study, a two-dimensional numerical wave tank based on potential flow theory was employed to evaluate wave-absorption performance. A boundary element method (BEM) based on two-dimensional Rankine sources was applied to calculate the velocity potential. The boundary integral equation used in the BEM was derived from Green’s second identity as follows:
Here, P denotes the field point, Q the source point, C the solid angle, G the Green function, and Γ the entire boundary ( Γfree +Γbody +Γbotom + Γwall+Γrad).
The BEM can be applied to both time-domain and frequency-domain analyses (Kim et al., 2025; Sim et al., 2025). For regular waves, a frequency-domain analysis can be performed by introducing a time harmonic assumption. Under a linear assumption and with one side of the domain represented by a rigid wall, the frequency-domain boundary conditions can be expressed as shown in Fig. 1.
In this study, wave attenuation by perforated plates was modeled using two approaches. The model coefficients were adopted from studies by Cho et al. (Cho & Kim, 2008; Ko & Cho, 2018). In addition, the perforation rate was fixed at 0.1, which has been reported as the most effective value in perforated plate experiments (Cho, 2002; Jung & Koo, 2021).
Here, k is the wave number, b = 57.63P – 0.9719,
β = 4 C D 3 π | ∂ ϕ + ∂ n | , C D = 0.6 ( 1 P C c - 1 ) 2
A perforated plate experiences different pressures on its two sides within the perforated region, resulting in distinct velocity potentials on either side of the plate. Consequently, additional treatment is required beyond the conventional BEM formulation. To account for wave attenuation by perforations, this study adopted a multi-domain boundary element method (MBEM). In this approach, the entire fluid domain is divided into multiple subdomains, and boundary conditions are applied independently within each subdomain while enforcing continuity conditions across the interfaces between adjacent domains (Cho & Kim, 2008). For example, when a single perforated plate is present, the fluid domain can be divided into the two regions shown in Fig. 2.
The first region includes the wave-absorbing plate and radiation boundary. The boundary conditions can be discretized as follows:
where Γ(1)=Γfree +Γrad +Γbotom++Γimag++Γbody+
Here,
A i , j = ∫ G ( P i , Q j ) d l ( Q j ) ; B i , j = ∫ ∂ G ( P i , Q j ) ∂ n d l ( Q j )
By substituting the boundary conditions given in Eqs. (2)–(4) and the damping models of Eqs. (5)–(6), the discretized equation can be expressed as
(8)
Next, the integral equation for the second region, which contains the wave-absorbing plate and the wall boundary, can be expressed as
where Γ(2)=Γbody−+Γimag−+Γwall+Γbotom−
Substituting the corresponding boundary conditions yields
(10)
At the artificial interface between the two regions, continuity of pressure and fluid velocity can be imposed through the following boundary conditions (Cho & Kim, 2008):
Combining these conditions with Eq. (8) for the first region results in
(12)
When multiple wave-absorbing plates are present, the fluid domain can be divided into additional subdomains corresponding to the number of plates. The governing equations for each region are then assembled and solved simultaneously. Within the framework of potential flow theory, the reflection coefficient can be derived analytically as follows:
where
N 0 = ∫ - h 0 cosh 2 k ( h + z ) d z = h 2 ( 1 + sinh 2 k h 2 k h )
Here, I is the imaginary unit, w is the wave frequency, g is the gravitational acceleration, and A is the wave amplitude. In addition, N0 denotes the normalization factor. This factor ensures that the reflection coefficient remains dimensionless when the wave-amplitude component is extracted from the velocity potential associated with the progressive wave mode.
2.2 Separation of Reflected Waves
Wave data obtained from basin experiments were analyzed based on linear wave theory to determine the wave reflection coefficient. In a two-dimensional domain, the incident and reflected waves at position y can be expressed as
Assuming that wave elevations are measured at n different locations, the incident and reflected wave components at each measurement position can be expressed as
Using the phase relationships of the measured wave elevations at each location, the complex representation of the wave amplitude can be expressed as
Here,
B T n = | A T | exp ( i σ n )
The discretized form of the governing equations can then be expressed as follows:
(21)
The complex amplitudes of the incident and reflected waves can be obtained using the least-squares method. At least three wave-gauge measurement locations are required to determine the solution uniquely.
3. Results and Discussion
3.1 Validation of the Numerical Analysis for Wave-Absorbing Plates
To validate the potential-flow-based numerical analysis employed in this study, we compared the reflection coefficients of a single wave-absorbing plate at various inclination angles (Θ) with experimental data and previously published numerical results (Fig. 3). The experiments were conducted in a two-dimensional wave tank, and the reflection coefficients were evaluated using both the linear Darcy model and the equivalent nonlinear drag model (Ko & Cho, 2018).
Fig. 4 presents the reflection coefficients as a function of the transverse length-to-wavelength ratio for inclination angles of 5°, 10°, and 15°, respectively. The results obtained using the linear Darcy model are shown on the left, whereas those based on the equivalent nonlinear drag model are shown on the right. The numerical predictions obtained in this study agreed well with both the experimental measurements and the previously reported numerical results, demonstrating the validity of the proposed numerical approach.
Reflection coefficients were subsequently calculated for a variety of wave-absorbing plate configurations. The geometries considered in this study are summarized in Fig. 5. For convenience, each configuration was assigned an ID, which is used throughout the following sections when presenting and discussing the reflection coefficient results.
3.2 Comparison of Reflection Coefficients in the Ocean Engineering Basins
The reflection performance of the wave-absorbing plates currently installed in the OEB of KRISO was first evaluated. In the OEB located in Daejeon, wave-absorbing plates have been installed over a total transverse length of 6 m. At the rear section, vertical wave-absorbing plates are arranged over a distance of 3 m at regular intervals. In front of these, inclined wave-absorbing plates occupy the remaining 3 m. The perforation rate of the vertical plates gradually increases from 0.2 to 0.4 and 0.6, whereas the inclined plates have a perforation rate of 0.1 (Fig. 6). The water depth in the OEB was set to 3.2 m, and wave elevations were measured using wave gauges located near the center of the basin. Waves were generated under two conditions: a fixed wave height (H = 0.1 m) and a fixed wave steepness (H/λ = 0.05). Wave conditions with sufficiently large amplitudes were selected. Reflection coefficients were calculated from the measured wave elevations and compared with the numerical predictions. Fig. 7 presents the reflection coefficients for each condition. The left panel shows the results obtained under the fixed wave-height condition, whereas the right panel corresponds to the fixed wave-steepness condition. Because relatively large wave heights were used, the linear Darcy model predicted lower reflection coefficients than those observed experimentally. In contrast, the equivalent nonlinear drag model produced results that were in closer agreement with the experimental data. For the fixed wave-steepness condition (H/L = 0.05), shown in the right panel, strong wave nonlinearity persisted even in the long-wave region, and the wave height increased with wave period to maintain a constant wave steepness. These characteristics were considered to contribute significantly to the discrepancy between the experimental results and the predictions based on linear potential flow theory. Overall, the reflection coefficient was approximately 0.1 in the short-wave region and increased to approximately 0.2–0.3 in the long-wave region.
Reflection coefficients measured in the same OEB at a water depth of 1.36 m were also examined for comparison (Cho & Kim, 2008). Fig. 8 compares the experimental and numerical results for wave steepness values of H/λ = 0.02 and 0.04. Because these wave conditions exhibited lower steepness than those considered previously, the predictions obtained using the linear Darcy and equivalent nonlinear drag models were very similar, and both agreed closely with the experimental measurements. In addition, the reflection coefficients were generally lower than those observed under higher wave-steepness conditions, remaining below 0.1 for most wave periods.
Subsequently, the reflection characteristics of the deep ocean engineering basin (DOEB) located in Busan were investigated. The wave absorber installed in the DOEB has a transverse length of 10 m and consists of a double-plate configuration with an inter-plate spacing of approximately 0.8 m. In addition, part of the upper section of the front plate is bent at a steeper angle to promote wave shoaling (Fig. 9).
Reflection coefficients were measured for waves propagating toward the absorber installed along the long side of the basin. The experiments were conducted under conditions of a wave height of 0.1 m and water depth of 10 m. In Fig. 10, the left panel compares the numerical and experimental reflection coefficients under the same conditions, whereas the right panel compares the numerical results obtained when the water depth was reduced from 10 m to 3.2 m while maintaining the same wave absorber configuration.
Compared with the wave absorber in the OEB, the DOEB absorber has a greater transverse length and therefore maintains lower reflection coefficients over a broader long-wave range. Reflection coefficients remained below 0.1 for wave periods up to approximately 3 s and increased for longer-period waves. The numerical results indicated that the reflection coefficients predicted by potential flow theory become relatively larger in the long-wave region. This discrepancy is attributed to the inability of potential flow theory to represent wave breaking induced by shoaling at the front section of the absorber. In contrast, when the water depth was reduced from 10 to 3.2 m, lower reflection coefficients were obtained over a range of wave conditions. This behavior is likely due to the increased flow velocity through the perforated plates in shallower water, combined with the sensitivity of the inclined front section to changes in local flow conditions.
3.3 Comparison of Reflection Coefficients for Modified Wave Absorber Configurations in the Ocean Engineering Basin
To improve both basin space utilization and wave-absorption performance, we investigated a new wave absorber configuration. As an initial step, the effect of replacing the existing wave absorber system with alternative configurations was examined by comparing the resulting reflection coefficients. First, the rear vertical wave-absorbing plates were removed, as shown in the left panel of Fig. 11, and the resulting reflection coefficients were compared. As shown in the right panel of Fig. 11, removing the vertical plates led to a slight increase in the reflection coefficient in the long-wave region, whereas a slight change was observed over the remaining wave-period range. This result indicates that vertical wave-absorbing plates contribute only marginally to wave attenuation for waves within the range of periods typically used in basin experiments.
Next, a configuration was considered in which all existing wave-absorbing plates were removed and replaced with a single continuous inclined wave-absorbing plate. The plate was modeled as a single flat perforated plate with a transverse length of 6 m, corresponding to the entire wave absorber region. The reflection characteristics of the single-plate configuration were first examined for various inclination angles. Based on the optimal single-plate configuration, an additional lower plate was introduced, and the influence of the spacing between the two plates was investigated (Fig. 12).
Fig. 13 presents the reflection coefficients of the single wave-absorbing plate as a function of inclination angle. The results were compared with those of the existing OEB wave absorber using the linear Darcy model. For nearly all inclination angles, the 6 m single-plate configuration produced lower reflection coefficients than the existing absorber. This improvement is attributed to the longer absorption region provided by the 6 m plate compared with the 3 m inclined section of the current configuration. The influence of inclination angle indicates that very small angles result in relatively high reflection coefficients, whereas increasing the inclination angle reduces wave reflection. The lowest reflection coefficients predicted by the linear model were obtained for inclination angles of approximately 11°–14°. Notably, this range is very similar to the inclination angles currently used in both the OEB and DOEB.
Based on these findings, a second perforated plate was added beneath the upper plate for the configurations with inclination angles of 12°–14°, which exhibited the best wave-absorption performance. The lower plate arrangement is depicted in the right panel of Fig. 12. The reflection coefficients were then evaluated while the spacing between the upper and lower plates was varied. Fig. 14 compares the reflection coefficients predicted by the linear Darcy and equivalent nonlinear drag models for wave steepness values of 0.01, 0.02, and 0.03 under various inclination angles and plate spacings. First, the reflection coefficients of the single- and double-plate configurations were compared while maintaining a fixed inclination angle. Overall, the double-plate configurations maintained lower reflection coefficients in the long-wave region than the corresponding single-plate configurations. Subsequently, reflection coefficients were compared for inclination angles of 12°, 13°, and 14° while the spacing between the two plates was varied. The results indicated that smaller plate spacings tended to increase reflection coefficients in the long-wave region. However, for the wave-period range most relevant to basin experiments, slightly larger spacings generally resulted in lower reflection coefficients. Among the configurations examined, an inclination angle of 12° combined with an inter-plate spacing of approximately 0.8 m consistently produced the lowest reflection coefficients over a range of wave steepness conditions.
3.4 Comparison of Reflection Coefficients According to Wave Absorber Length and Geometry
Although increasing the length of a wave absorber generally improves wave-absorption performance, it also reduces the usable area of the basin. Therefore, the effect of reducing the absorber length from the existing 6 m configuration was investigated to determine whether basin space utilization could be improved without significantly compromising performance. Fig. 15 presents the reflection coefficients of the double-plate configuration with an inclination angle of 12° and an inter-plate spacing of 0.8 m, which was identified as one of the most effective candidates in the previous analysis. The absorber length was reduced from 6 m to 5, 4.5, and 4 m, and the corresponding reflection coefficients were evaluated. The results are presented for different wave steepness values using both the linear and nonlinear damping models. All the configurations exhibited similar reflection coefficients in the short-wave region. However, as the absorber length decreased, the range of wave periods over which low reflection coefficients were maintained became progressively narrower, resulting in larger differences in the long-wave region. Although waves with periods longer than approximately 2 s are relatively less frequently used in the current OEB, a rapid increase in reflection coefficient was observed for shorter absorber lengths under conditions of higher wave steepness.
As noted by Ouellet and Datta (1986), wave absorbers can be constructed not only as flat plates but also in parabolic configurations mounted on a beach-type structure. Compared with flat plates, parabolic absorbers have a steeper inclination near the front section and a gentler slope toward the rear. They are generally known to provide effective wave absorption over a broader range of wave periods. Accordingly, the absorber geometry was modified from a flat plate to a parabolic profile while maintaining the inclination angle and inter-plate spacing that yielded the best performance for flat-plate configurations of 4–5 m length (Fig. 16). Fig. 17 compares the reflection coefficients predicted by the linear Darcy model for different absorber lengths. Reflection coefficients were evaluated for absorber lengths of 4 and 5 m. Regardless of geometry, the double-plate configurations produced lower reflection coefficients than the corresponding single-plate configurations. Furthermore, the reduction in reflection coefficient resulting from the adoption of a parabolic geometry was more pronounced for the double-plate configurations than for the single-plate configurations.
Note that these results are based on potential flow theory and may represent idealized performance. In practice, parabolic perforated plates are more difficult to manufacture, and achieving a uniform perforation distribution over a curved surface may be challenging. Consequently, detailed flow analyses and reflection coefficient evaluations using computational fluid dynamics (CFD) are considered necessary for the final candidate configurations.
4. Conclusions
A numerical investigation was conducted to evaluate the wave reflection characteristics of wave-absorbing plates used in OEBs. The numerical analyses were based on potential flow theory, and wave attenuation by perforated plates was modeled using both the linear Darcy model and the equivalent linearization of nonlinear drag forces. To address the boundary-condition discontinuities introduced by perforated plates, this study employed an MBEM in which the fluid domain was divided according to the number of wave-absorbing plates. The reflection characteristics of the wave absorbers installed in the OEB and DOEB of KRISO were evaluated and compared with numerical predictions. Both basins were found to maintain reflection coefficients below approximately 0.2 over a broad range of wave frequencies. Because the absorber installed in the DOEB is longer, it maintained lower reflection coefficients over a wider range of wave periods. In addition, various modifications of the wave absorber configuration used in the OEB were investigated, and lower reflection coefficients were obtained when the entire absorber region was replaced with a single continuous wave-absorbing plate.
The influence of absorber geometry on reflection performance was also examined. The optimum inclination angle was found to be approximately 11°–14°, and the installation of an additional lower plate further reduced the reflection coefficient. Longer wave absorbers maintained low reflection coefficients over a broader range of wave periods, whereas for absorber lengths of 4–5 m, reflection coefficients began to increase at wave periods of approximately 2 s. Finally, reflection coefficients were compared for flat and parabolic wave absorbers with the same length and inclination angle. The parabolic configuration exhibited superior wave-absorption performance compared with the flat-plate configuration. Nevertheless, parabolic wave absorbers present manufacturing challenges, and whether the damping coefficients employed in the potential-flow analysis can be directly applied to curved perforated surfaces has not yet been verified. Furthermore, discrepancies between experimental and numerical results were observed in the long-wave region, likely because wave breaking could not be rigorously represented within the potential-flow framework. When the final candidate configurations have been selected, CFD analyses and additional experiments will be conducted to further evaluate their performance.
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