Establishment of Numerical Wave Flume Based on the Second-Order Wave-Maker Theory

With growing computational power, the first-order wave-maker theory has become well established and is widely used for numerical wave flumes. However, existing numerical models based on the first-order wave-maker theory lose accuracy as nonlinear effects become prominent. Because spurious harmonic waves and primary waves have different propagation velocities, waves simulated by using the first-order wave-maker theory have an unstable wave profile. In this paper, a numerical wave flume with a piston-type wave-maker based on the second-order wave-maker theory has been established. Dynamic mesh technique was developed. The boundary treatment for irregular wave simulation was specially dealt with. Comparisons of the free-surface elevations using the first-order and second-order wave-maker theory prove that second-order wave-maker theory can generate stable wave profiles in both the spatial and time domains. Harmonic analysis and spectral analysis were used to prove the superiority of the second-order wave-maker theory from other two aspects. To simulate irregular waves, the numerical flume was improved to solve the problem of the water depth variation due to low-frequency motion of the wave board. In summary, the new numerical flume using the second-order wave-maker theory can guarantee the accuracy of waves by adding an extra motion of the wave board. The boundary treatment method can provide a reference for the improvement of nonlinear numerical flume.


Introduction
A wave flume is an important piece of equipment for carrying out research on water waves and water-wave action on maritime structures. It is also a source of irreplaceable data for engineers solving practical problems in ocean engineering. Most laboratory wave-makers are either pistontype or flap-type. Piston-type wave-makers have been widely used to generate waves in laboratory flumes or basins. First-order wave-maker theory has been well established and verified by experiments. The basic assumption of the first-order wave-maker theory is the small-amplitude assumption. If the motion of the wave-maker is sinusoidal, the generated small-amplitude waves will be decomposed into a primary wave and a spurious superharmonic wave (only the sum of frequencies appears in a regular wave), which will affect the stability of the wave profile. In early 1847, Stokes found the superharmonic wave by considering a regular wave in terms of a perturbation series using the wave steepness as the small ordering parameter. Fontanet (1961) was the first to provide a solution for the problem of the generated nonlinear wave. He found the spurious superharmonic wave by a sinusoidal moving piston-type wave-maker in the Lagrangian coordinates and suggested that it could be reduced by using a wave-board control signal with a superharmonic component. Madsen (1971) derived a solution for the nonlinear wave-maker problem by considering a sinusoidal moving piston-type wave-maker in the Eulerian coordinates and established the theory of reducing spurious superharmonic waves in regular waves in shallow water. Swan (2009a, 2009b) provided a new theory for generating regular waves by a flap-type wave-maker using the second-order wave-maker theory, and their theory has been verified by experiments.
For irregular waves, the second-order wave-maker theory has both sum and difference frequencies in the interaction terms. Schäffer (1996) derived the second-order wavemaker theory including sum-frequency and difference-frequency components without the narrowband assumption.
The theory was applied to both piston-type and flap-type wave-makers and has been verified by experiments. Schäffer and Steenberg (2003) extended the second-order wave-maker theory to multidirectional waves.
On the other hand, many researchers have developed numerical wave flumes for simulating the piston-type wavemaker to reproduce the physical wave basin accurately. Commercial CFD code was used to generate waves in a numerical wave tank using a piston-type wave-maker by Wood et al. (2003), Liang et al. (2010), and Prasad et al. (2017). Sriram et al. (2006) made 2-D waves using a pistontype wave-maker and the finite-element approach. A wave flume with a moving boundary that could absorb reflected waves was developed by Lara et al. (2011). Finnegan and Goggins (2012) and Anbarsooz et al. (2013) simulated viscous wave fields by a flap-type wave-maker and verified the accuracy of the results against the flap-type wave-maker theory. Water-wave generation in the SPH method using a moving wave board, which can generate and absorb waves, was carried out by Farahani et al. (2014), Wen et al. (2016), and Altomare et al. (2017). Higuera et al. (2015) extended the IHFOAM numerical model to include moving-boundary wave generation and simulated a three-dimensional focusing wave. A 3D numerical wave flume with a pistontype wave-maker has been used to simulate the coupled fluid-structure interaction (Agamloh et al., 2008;Prasad et al., 2010Prasad et al., , 2014Zullah et al., 2010). However, existing numerical models based on the first-order wave-maker theory will lose accuracy as nonlinear effects become prominent.
In the present study, a numerical wave flume with a piston-type wave-maker has been developed. First-and second-order wave-maker theories are introduced in the numerical flume. The numerical flume is verified by laboratory experiments. The unstable wave profiles generated with first-order wave-maker theory are illustrated, and the reasons for generating unstable wave profiles are analyzed. Furthermore, regular and irregular waves are simulated with the first-and second-order wave-maker theories. For regular waves, the harmonic analysis and spectral analysis are used to compare the waves simulated with the first-and secondorder wave-maker theories. For irregular waves, the numerical flume model is improved to solve the problem of variation in the water depth due to the low-frequency motion of the wave board when the second-order wave-maker theory is used. The low-pass filter and spectral analysis are used to show the superiority of the second-order wave-maker theory. The boundary treatment of the numerical flume based on the second-order wave-maker theory is different from that based on the first-order wave-maker theory. This paper can provide a reference for the boundary setup of nonlinear numerical wave flume.

Numerical model
The numerical model solves the incompressible Navier-Stokes equations for a two-phase flow of water and air by incorporating a VOF scheme for tracking the free surface (see Berberović et al. (2009)). The governing equations were solved by a finite-volume method, discretized on a structured mesh within the framework of the FLUENT commercial CFD software from ANSYS Inc. Second-order wave-maker theory was introduced to simulate nonlinear wave propagation.

Governing equations
The flow motion of an incompressible, viscous fluid can be described by the Navier-Stokes equations. As the wave propagation in a flat bottom without considering boundarylayer can be treated as a turbulence-free problem, the simulations are performed with laminar model.
The continuity equation in the Cartesian coordinate system is: and the momentum equation is given by: where t is time, ρ is the density, and μ is the viscosity; p is the pressure; i, j = 1, 2 (x, y); f 1 = 0, f 2 = -g, g is the gravitational acceleration; x 1 and x 2 are the horizontal and vertical coordinates respectively; u 1 is the horizontal flow velocity; u 2 is the vertical flow velocity; and S i is the source term for the momentum equation. In this study, waves were simulated using a piston-type wave-maker, S i = 0.
To determine the position of the free-surface air-water boundary, the volume of fluid method is used. This method adds another governing equation, given by: where α q = 1 indicates that the cell is filled with the q-th fluid and α q = 0 indicates that there is no fluid in the cell. 0 < α q <1 denotes that the cell contains the air-water interface.
2.2 Second-order wave-maker theory In this section, the second-order wave-maker theory is briefly introduced. For detailed information, the reader can refer to Schäffer (1996). Fig. 1 shows a typical arrangement of a wave flume, together with the arrangement of the wave gauges used to record the wave surface in the following wave simulations. It is assumed that the fluid is incompressible and viscous and that the flow is irrotational. Let η = η(x, t), X = X (y, t), g, h, and t denote the surface elevation, wave-board position, acceleration of gravity, still-water depth, and time, respectively. In classical wave theory, the velocity potential function satisfies the following Laplace equation and the corresponding boundary conditions: Laplace equation: bottom boundary condition: kinematic free-surface boundary condition: dynamic free-surface boundary condition: wave-board boundary condition: where X 0 (y, t) is the wave-board position.
With the wave steepness ε = H/L as the small ordering parameter, the surface elevation, potential, and wave-board position correct to the second order are given by: The second-order potential function and surface elevation can be decomposed into three parts, where ϕ (21) describes the bound waves due to the interaction between the first-order wave components, ϕ (22) denotes the free waves due to the wave-board leaving its mean position and due to ϕ (21) mismatching the boundary condition at the wave board, and ϕ (23) gives the free waves generated by the second-order wave-board motion. If the control signal for the wave-board motion is based on first-order wave-maker theory, the resulting second-order waves are given by ϕ (2) =ϕ (21) +ϕ (22) , and spurious free waves from ϕ (22) will be generated. Let subscript 0 denote the progressive part of a wave field. The purpose of the second-order wave-maker theory is to determine so as to produce free waves that restrain these spurious free waves by requiring: or equivalently, Then, the first-and second-order solutions can be summarized as follows. For a detailed derivation, the reader can refer to Schäffer (1996). The solution to the first-order problem is as follows: where c.c. is the complex conjugate of the previous item. The first-order transfer function can be written as: The relationship between the complex amplitude A of the progressive part of the first-order wave and the complex amplitude X a of the first-order wave-board position can be expressed as: The free-surface elevation of the first-order progressive waves is: When A=a-ib, the first-order wave board motion function is: The solution to the second-order problem is given as: where jnlm means that the previous item exchanges l and j as well as m and n, "+" and "-" denote the sum frequency and the difference frequency contributions, respectively. is the second-order transfer function, and is the solution of the generalized linear dispersion equation: The symbol -:* is defined by where * denotes complex conjugation: For the definitions of the parameters in the equations, one can refer to Schäffer (1996). The free-surface elevation of the second-order progressive waves and second-order wave board motion are: For application convenience, the wave board motion should be introduced to the software using User-Defined functions in the time domain. So, the second-order motion function in the time domain is given in the appendix.

Numerical algorithm method
A 2D numerical model was subdivided into cells, and the governing equations were discretized using the finitevolume method. The governing equations were solved using a 2D pressure-based solver in a second-order, implicit, unsteady-state formulation and iterated to achieve a converged solution. Pressure-velocity coupling used the PISO algorithm. The discretization schemes for the pressure, momentum, and volume fractions were implemented using the PRESTO! scheme, second-order upwind scheme, and Georeconstruct scheme, respectively. The simulations were defined as having converged when all residuals were smaller than 10 -5 . To ensure the stability of the model and improve the accuracy, the maximum Courant number was set to be 0.25 according to the trial tests for wave simulation. The initial time step was set to be 0.001 s and the time step was modified automatically according to the Courant number according to Eq. (34).
where, δt is the maximum time step, δx is the cell size in the direction of the velocity and |U| is the magnitude of the velocity at that location.

Dynamic mesh model
Dynamic mesh model should be used for the moving wave board. In the mesh zone, the dynamic layering method can be used to add or remove layers of cells adjacent to a moving boundary, based on the width of the layer adjacent to the moving boundary. The dynamic layering method sketch is shown in Fig. 2. The ideal layer width on each moving boundary could be specified in the dynamic mesh model. The layer of cells adjacent to the moving boundary (layer j) is split or merged with the layer of cells next to it (layer i) based on the width (L h ) of the cells in layer j.
If the cells in layer j are expanded, the cell widths are allowed to increase until where L hmin is the minimum cell width of cell layer j, L hideal is the ideal cell width, and α s is the layer split factor. α s = 0.4 in the numerical model. If the cells in layer j are being compressed, they can be compressed until where α c is the layer collapse factor. α c = 0.2 in the numerical model. If the moving boundary is an internal zone as the following simulation for irregular waves, the cells on two sides of the internal zone are considered for dynamic layering. Sliding interface must be used to separate the deforming cells and adjacent non-deforming cells, which is shown in Fig. 3.

Absorption zone in the numerical model
The end of the numerical flume is no slip wall boundary condition. To avoid the wave reflection from the end wall and ensure the stability of wave simulation for a long time, an additional momentum source S i is added to Eq. (2) by the user-defined function for the absorption zone shown in Fig. 1 at the end of the numerical tank to eliminate the wave reflection. It is given by: where ρ is the density, and γ is the damping coefficient, taking 0.58 in this paper. x b is the starting point of absorption zone. L b is the length of absorption zone, which can take two times the wave length; u i (i=1, 2) are the horizontal and vertical flow velocity, respectively.

Validation for the wave propagation
To verify the ability of the numerical model to simulate the wave propagation, a grid convergence study was conducted. A two-dimensional numerical domain was considered, as shown in Fig. 1. The key factors affecting the wave propagation accuracy were analyzed, considering both the aspect ratio (AR) of cells and the number of points per wave height (p.p.w.h). The results of the second-order Stokes wave theory was used as a reference solution. A regular wave with the wave period T = 2.5 s and wave height H = 0.15 m at a water depth of h = 0.7 m was propagated in the numerical model for 15 wave periods. In each wave period, a uniform distribution of 15 points was used to calculate the diffusive error using where η xi and denote respectively the numerically computed free-surface elevation and the reference free-surface elevation at Point x i after the correction of the phase error.
⟨·⟩ is the arithmetic mean operator. Fig. 4 shows the variation of the diffusive error with time normalized by the wave period with different aspect ratios (AR). The computational cells have the same number of p.p.w.h, but different aspect ratios are used to compare the diffusive error. Computational cells with different aspect ratios have different diffusive errors. The diffusive error significantly increases with the increasing aspect ratio. The diffusive error is the smallest when AR=1. So, the cell aspect ratio AR=1 was used in the numerical simulations. The grid cells have a resolution of mainly Δx in the horizontal direction and Δy in the vertical direction (Δx=Δy), which are shown in Table 1. The free surface elevation obtained using three mesh schemes are shown in Fig. 5. It can be seen that the model is convergent using the resolution of mainly H/15. The total cell numbers and the simulated time in which the case has been running can be found in Table 1 as well. By comparing the results shown above, in the following numerical simulations, the computational cells for 15 points per wave height and AR = 1 are used. An overview of the vertical mesh can be seen in Fig. 6.

Validation against experiments
Further validation of the numerical solver was conducted by comparing the numerically simulated and laboratory experimental regular waves by Schäffer (1996). The experimental flume was 20 m long, 1.0 m deep and 0.6 m wide.    Their physical flume was equipped with a piston-type wavemaker at one side and an efficient absorber zone at the other side. For validation, a regular wave with the period T = 3.0 s and H = 0.13 m in a flume of depth h = 0.7 m was used. The waves were generated with the first-order and second-order wave-board control signals. Fig. 7 shows comparisons of the experimental and numerically calculated free-surface elevations in the flume using the first-order and second-order wave-board control signals. For the free-surface elevation, good agreement between experimental and numerically simulated results is obvious. By comparing the results shown in Fig. 7, the freesurface elevations along the wave flume with the second-order wave board-control signals are much more stable than those with the first-order wave board-control signals. This phenomenon will be further investigated in detail.

Range of applicability
In general wave experiments, the relative depth (h/L) and wave steepness (H/L) are the main parameters because they define the wave properties. The second-order wave amplitude must be a factor in the second-order wave-maker theory. When secondary peaks appear in the trough of a regular wave, the theory is invalid. The extreme case of the superharmonic amplitude corresponds to one-fourth of the primary wave height. A parameter S has been introduced to measure the degree of nonlinearity (Schäffer, 1996): For irregular waves, H is replaced by the significant wave height. S represents the ratio of the superharmonic amplitude to the primary wave height. According to Eqs. (21) and (32), the limit of S is equal to one.
First, the waves generated with the first-order wavemaker theory can be decomposed into three parts: where , , and denote the free-surface elevations of the first-order progressive waves, second-order progressive waves (bound superharmonics), and spurious free superharmonics, respectively. To obtain an intuitive understanding of the relationship between the dimensionless wave height of a bound superharmonic (H b /H) and the dimensionless wave height of a spurious free superharmonic (H f /H), Fig. 8

shows the variations of H b /H and H f /H with changes in the relative depth kh and the wave steepness H/L. When kh is a constant, H f /H and H b /H increase as H/L increases.
As kh increases, H b /H rapidly decreases and then tends to a constant value that is proportional to H/L. On the other hand, H f /H rapidly decreases and then slowly increases as kh increases. With the H/L and kh keeping unchanged, H b /H is always larger than H f /H. In fact, the spurious free superharmonic waves should be eliminated in the process of the wave propagation. However, when it is small, its effect on the wave propagation can be neglected. So when the dimensionless wave height of a spurious free superharmonic is roughly over 0.05 (i.e., the wave steepness H/L>0.05 and re-  lative water depth kh<1.5), the second-order wave-maker theory is recommended for numerical wave simulation.

Regular wave simulation
With the validated numerical model, various wave cases were simulated using the first-order and second-order wavemaker theory. Table 2 gives the parameters for the regular wave simulation with the wave period T, wave height H, relative depth kh, wave steepness H/L, and nonlinearity parameter S. The nonlinear parameter S varies from 0.410 to 0.999. Fig. 9 gives the typical second-order wave-board displacement (X) and the corresponding first-order wave-board displacement (X (1) ) and second-order frequency sums for the wave-board displacement (X (2)+ ). It can be seen that for regular waves, the first-order (X (1) ) and second-order (X) waveboard displacements are very similar to each other. Unlike the first-order wave-board displacement (X (1) ), the second-order wave-board displacement (X) has an additional term, which is the second-order frequency sum of the wave-board displacements (X (2)+ ). The motion of the second-order wave board is still back and forth in the vicinity of the initial position of the wave board due to the small magnitude of the second-order frequency sum of the wave-board displacements (X (2)+ ).
To highlight the variation in the wave profiles along the flume, Fig. 10 shows wave profiles at different positions after the correction of the phase error for Case R1 as an example. The waves generated by using the first-order wavemaker theory have spatially unstable wave profiles. However, the waves generated by using the second-order wave-maker theory can maintain stable profiles along the wave flume. Fig. 10 shows that in the case of a strongly nonlinear wave, the second-order wave-maker theory can effectively restrain the propagation of spurious free superharmonic waves and generate stable wave profiles in both  To perform further verification of the effectiveness of the second-order wave-maker theory, waves at different positions were selected for the harmonic analysis. Fig. 11 shows a comparison of the wave heights of superharmonic waves separated from their generated waves using the firstorder wave-maker theory (black squares) and second-order wave-maker theory (white circles). The curve and straight lines indicate the theoretical values obtained with first-order and second-order wave-maker theory for the superharmonics. Fig. 11 shows that the superharmonic wave heights separated from the generated waves with the second-order wave-maker theory are distributed in the vicinity of the theoretical values predicted by the second-order wave-maker theory and close to a constant. However, although the separated wave height from the waves generated by the first-order wave-maker theory is also distributed in the vicinity of the theoretical value predicted by the first-order wavemaker theory, the superharmonic wave height changes periodically with the propagation distance. This is the reason for the spatial instability of the wave profile, as shown above. Therefore, it can be concluded that the numerical flume based on the second-order wave-maker theory can provide a better wave profile, especially when nonlinear effects become prominent.
To analyze the advantages of the numerical flume based on the second-order wave-maker theory from the energy aspect, the spectral analysis of the free-surface elevation time series at different positions was conducted using the FFTH transformation with the computed wave time series. Fig. 12 gives the non-dimensional energy spectra for Case R2 as an example, in which is the mean wave height. A logarithmic coordinate system was used for the vertical coordinate axes for clear comparison. The spectral energy at the  Fig. 11. Comparison of the wave heights of superharmonic waves separated from the generated waves using the first-order and second-order wavemaker theory along the wave flume. fundamental frequencies for the first-order and second-order wave-maker theory is in good agreement with the theoretical solution. However, the secondary peak energy for the first-order wave-maker theory does not match well with the theoretical solution. In particular, the first-order wavemaker theory loses accuracy as nonlinear effects become prominent. By contrast, the secondary peak energy for the second-order wave-maker theory is in good agreement with the theoretical solution, and the secondary peak energy maintains a stable value. From the energy aspect, it can be concluded that the numerical flume based on the second-order wave-maker theory shows excellent performance in generating nonlinear waves.

Irregular wave simulation
For the numerical simulation of irregular waves, a wave with the peak period T p = 2.5 s and significant wave height H s = 0.12 m was selected. The water depth h = 0.7 m, and the nonlinearity parameter S was 0.680. Fig. 13 gives the typical time series of the wave-board displacement. This time series is totally different from the wave-board displacement for regular waves as shown in Fig. 9. Compared with the first-order wave-board displacement (X (1) ), the secondorder wave-board displacement (X) has two additional terms: the second-order frequency sum displacements (X (2)+ ) and the second-order frequency difference of the waveboard displacements (X (2)-). The second-order difference frequencies of the wave-board displacement (X (2)-) have large magnitude and low frequency. This means that the wave board does not move back and forth in the vicinity of the initial position. Instead, the motion of the second-order wave board substantially deviates from the initial waveboard position due to the large magnitude and low fre-quency of the second-order frequency difference of the wave-board displacement (X (2)-).
When the control signal for the wave-board motion is based on the first-order wave-maker theory, the wave-maker in the typical numerical flume (see Fig. 1) is at the left boundary. The wave board moves back and forth in the vicinity of the initial position. For regular waves, the typical numerical flume still works when the control signal for the wave-board motion is based on the second-order wavemaker theory due to no low-frequency movement, as shown in Fig. 9. However, if a typical numerical flume model is used to simulate irregular waves by the second-order wavemaker theory, a major problem arises. Fig. 14 shows the water depth changing in the wave flume over time with the second-order wave-board displacement, as shown in Fig. 13 using the wave flume in Fig. 1. It is clear that any substantial deviation in the motion of the second-order wave board from its initial position will lead to the water depth changes over time.
To overcome the problem of changes in the water depth over time due to the deviations from the initial wave-board position during the irregular wave simulation, an improved numerical flume model was proposed as shown in Fig. 15. The wave board is placed at the internal domain. Two absorption zones are arranged on the left and right sides of the flume. At the bottom of the wave board, a small space is set up and allowed to let the water flow freely. Sliding interface technique is used to deal with the gap at the bottom of the wave board. When the wave board moves back and forth, the water depth can maintain a constant value, ensuring the numerical simulation accuracy. In order to determine the size of the gap, the error is obtained by following Eq (41), in which and indicate the low-pass filtered series for the time series calculated with the second-order wavemaker theory and the theoretical solution. The relative size of the gap is measured by L space /d, in which L space is the size of the gap. Fig. 16 shows the error between the theoretical value and calculated value in different relative gaps. The error decreases firstly and then increases as the relative gap increases. When the gap takes 5%-6% of the water depth, the error is the minimum.
Clearly, correcting the difference frequency component is very important for irregular wave simulation based on the second-order wave-maker theory (see Fig. 13). To show clearly the effect of correcting the difference frequency component, the time series of the low-pass filtered series for the time series calculated with the first-order and second-order wave-maker theory and the theoretical solution at x=10 m were compared with the results shown in Fig. 17. The figure shows that the low-pass filtered series obtained using the second-order wave-maker theory is close to that of the theoretical solution. This demonstration shows that the advantage of the second-order wave-maker theory is to restrain the extra spurious free harmonic waves.
Furthermore, Fig. 18 gives the analyzed non-dimensional energy spectra corresponding to Fig. 17. It indicates that the peak frequencies of the energy spectra for the first-order and second-order wave-maker theory are in good agreement with the theoretical solution. However, the low-fre-quency part of the energy spectrum for the first-order wavemaker theory does not match well with the theoretical solution. Because the extra spurious free harmonic waves are reduced by the correction applied by the second-order wavemaker theory, the numerical flume based on the second-order wave-maker theory exhibits excellent performance for simulating the bound wave energy.

Conclusions
A numerical model with a moving boundary using the first-order and second-order wave-maker theory has been developed and presented in this paper. In the numerical flume model, the dynamic mesh technique was developed. The boundary treatments for irregular wave simulation were specially dealt with. It provides a reference for nonlinear numerical flume. The applicability range of the second-order numerical model is discussed. When the dimensionless   Fig. 17. Comparison of the low-pass filtered free-surface elevation time series using the first-order wave-maker theory, second-order wave-maker theory, and the theoretical solution (x = 10 m). The peak period T p = 2.5 s, and significant wave height H s = 0.12 m.
wave height of a spurious free superharmonic is roughly over 0.05 (i.e., the wave steepness H/L>0.05 and relative water depth kh<1.5), the second-order wave-maker theory should be recommended for numerical wave simulation. The numerical model by the second-order wave-maker theory can effectively restrain the propagation of spurious free harmonic waves. It can create a stable wave profile in both the spatial and time domains.
For regular waves, the superharmonic wave height of the second-order wave-maker theory is close to constant, with the spatial variation. The spatial uniformity of the second-order wave-maker theory is better than that of the first-order wave-maker theory when nonlinear waves are generated. From the energy standpoint, the secondary peak energy of the energy spectrum for the second-order wavemaker theory has a stable value. It can be concluded that the numerical flume based on the second-order wave-maker theory can maintain the stability of high-frequency energy prediction.
For irregular waves, an improved numerical flume has been developed for simulating nonlinear irregular waves using the second-order wave-maker theory. The problem of the water depth variation over time in the original numerical flume with the second-order wave-maker theory has been solved. Sliding interface technique is used to deal with the gap at the bottom of the wave board. Spurious free harmonic waves in the irregular waves are suppressed by adding an extra motion to the wave board. Irregular waves making use of the second-order wave-maker theory can obviously improve the accuracy of low-frequency energy.
Future work will be focused on expanding the secondorder wave-maker theory into three-dimensional numerical simulations to reproduce three-dimensional waves using a multi-segment piston wave-maker.

Appendix:
X = X (1) 0 + X (2)± 0 X (2)± 0 A n = a n − ib n A −: * m = a m ∓ ib m Second-order wave board motion is . Comparing with first-order wave board motion, just one more item is added. If and , the real expression of Eqs. (32) and (33) can be written as.