Focusing Monochromatic Water Surface Waves by Manipulating the Phases Using Submerged Blocks

Abstract

Focusing water surface waves is a promising approach for enhancing wave power in clean energy harvesting. This study presents a novel method that simplifies the wave-scattering problems of large-scale three-dimensional (3D) focusing blocks by decomposing them into scattering problems of two-dimensional (2D) phase regulators. The phase lags of transmitted waves over such 2D structures of various heights and thicknesses are investigated using both linear potential flow theory and numerical simulations based on smoothed-particle hydrodynamics (SPH). Due to propagation path differences of a converging wave, our approach compensates for circular phase differences within a maximal collection angle by optimizing the geometries of 2D phase regulators. Based on this concept, we designed three types of submerged structures and tested them in a 3D numerical water tank. All three structures successfully converted monochromatic plane waves into circular waves, which then converged at the designated focal point. This study offers a potential method to enhance the collection efficiency of monochromatic and regular waves for wave energy converters.

1. Introduction

The ability to control and focus water surface waves is crucial for enhancing the efficiency of wave energy harvesting [1,2]. Since wave power is proportional to the square of wave height, concentrating this energy into a smaller area could significantly increase the power that can be harnessed. This approach could lead to more efficient energy collection systems and improve the viability of wave energy in regions with smaller or less consistent waves. By amplifying power in low-energy environments, such structures could transform wave energy into a more competitive and sustainable renewable resource. Traditionally, redirecting and focusing water surface waves to a designated area can be achieved using underwater structures as wave-guiding devices. Mehlum and Stamnes [3,4] experimentally showed that a submerged Fresnel-type lens can transform diverging waves into converging waves. Although the efficiency of the focused energy was not significant, they pioneered the feasibility of focusing water surface waves using underwater structures. Griffith and Porter [5] studied the focusing of water surface waves via an elliptical plateau raised from the seabed. They successfully developed a mathematical model from the potential flow theory that can approximately describe the intensities of the focusing waves observed in the experiments. They also showed that elliptical lenses performed better than bi-convex lenses. Motivated by the light waves that focus on caustics, C. Gerardo Ruiz et al. [6] experimentally investigated the focused water surface waves generated by a parabolic wavemaker. They found that water surface waves follow simple optical rules when the amplitude of the wavemaker is small relative to the wavelength. When the amplitude is large, the analogy between the light and the water surface waves is not applicable. Wang et al. [7,8] investigated the focusing effects of water surface waves via gradient–index lenses. According to the shallow water approximation, the refractive index of water waves can be controlled by water depth. They designed focusing lenses based on the refractive indices that are given by the conformal transformation theory. Their experimental results agreed well with the theory and the numerical simulations. Hu and Chan [9] studied the waves passing through periodic cylinder arrays mounted at the bottom of a wave tank. They showed that the refractive index of the arrays is determined by the filling factor. Based on this finding, they designed a biconvex lens using periodic cylinder arrays and successfully focused surface waves into a region roughly the size of a wavelength. Using the Hankel transformation approach, Zheng et al. [10] investigated the interaction of water waves with an array of N submerged horizontal, thin, circular plates. They found that a staggered arrangement of two elastic circular plates can focus waves with minimal energy dissipation. In a different approach, Mayon et al. [11] used a parabolic wall to concentrate the energy of reflected waves. Their research demonstrated that a wave energy converter positioned at the focus of the parabolic wall performs more efficiently compared to open-sea conditions.

In recent years, a novel technique has emerged in nanophotonics, involving the use of meta-surfaces to manipulate the phase, amplitude, and polarization of light at will by rationally configuring the shape, size, position, and orientation of unit cells [12]. Such properties, which originate from the generalized Snell–Descartes law [13], have brought a lot of breakthroughs in developing ultrathin sheets of meta-surfaces mimicking the properties of conventional bulky lenses [14,15,16,17]. Due to the similar wave nature, phase manipulation from the point view of optical waves should also be applicable to water surface waves. To the authors’ knowledge, neither experiments nor simulations have been conducted to investigate the focusing of water surface waves using the idea of phase manipulations.

In this study, we investigate the phase behaviors affected by two-dimensional (2D) submerged rectangular blocks in the regime of linear potential flow theory. The wave focusing can be achieved by manipulating the phase lag induced by the submerged blocks following the generalized Snell’s law, analogous to optical lenses. The smoothed-particle hydrodynamics (SPH) method is used to numerically validate the proposed physical models and focusing performances.

2. Methods

2.1. Theoretical Model

The geometry of the theoretical model under investigation is depicted in Figure 1a. The water is assumed to be inviscid and incompressible, and the motion is irrotational, so the linear potential flow theory is valid. The water depth is H. The rectangular block with height h, thickness d, and infinitive width is mounted at the bottom of the water. The block is subjected to an incident train of regular, monochromatic, small-amplitude waves with an angular frequency ω and an equilibrium free surface at z = 0. The positive direction of z is upward. The whole fluid domain is divided into three regions: x ≤ 0 for Region I, 0 < x ≤ d for Region II, and x ≥ d for Region III. Neuman [18], Mei, and Black’s [19] pioneering work provides a solution to this problem through the eigenfunction expansions of the velocity potential. The coefficients in the expansions can be solved numerically by matching the boundary conditions between the regions. Consider water surface waves with wavenumber k_I = 2π/λ_I traveling in the x direction. λ_I is the wavelength. The dispersion relation is given by [20]:

$${\omega }^2=g{k}_I\tan h\left(k_{I}H\right),$$

(1)

where g is the magnitude of gravitational acceleration. The phase lag when the wave propagates over the rectangular block is:

$$\mathsf{\phi } =\left(k_{II}-k_I\right)d,$$

(2)

where k_II is the wavenumber in Region II. As k_II is determined by the block height, the phase lag induced by the block can be manipulated by its geometry parameters, that is, height h and thickness d. The phase lags (color contrast) as a function of h and d are expressed using a pseudo-color map in Figure 1b for the case of H = 0.5 m and T = 1.0 s, where T is the period of waves. The corresponding wavenumber and wavelength are k_I = 4.15 m⁻¹ and λ_I = 1.51 m, respectively. Three linear approaches with varying phase lags are taken as examples. For the green dashed line, d is fixed, while h varies; for the red dashed line, h is fixed while d varies; and for the magenta dashed line, both h and d vary following the function h/H = 0.8 d/λ_I.

Now, we manipulate the phase lags by varying the height and thickness of the block such that the plane waves are converted into circular waves and converged on the x-axis at x = f, as shown in Figure 2a, where f is the focal length. The transmitted and incident wavenumbers remain the same as k_III = k_I, since the water depth is the same. Then, the phase difference Δ of the waves arriving at the focal point from the origin and the position (d, y) is given by:

$$\Delta =\left(k_I\sqrt{f^2+y^2}+\mathsf{\phi }\left(y\right)\right)-\left(k_{I}f+\mathsf{\phi }\left(0\right)\right),$$

(3)

where φ(y) is the phase lag induced by the block at position y. Constructive interference of the waves at the focal point is ensured if Δ = 2nπ for n = 0, 1, 2, …, +∞. The focusing effect is observable under these conditions. Thus, to achieve focusing, the rectangular block should at least provide a phase lag of:

$$\mathsf{\phi }\left(0\right)-\mathsf{\phi }\left(y\right)=k_I\left(\sqrt{f^2+y^2}-f\right).$$

(4)

The left side represents the phase due to the block, while the right side accounts for the phase difference caused by the different propagation paths from the lens to the focal point. This condition is commonly used in optics for flat lens design [14].

Using the 2D rectangular blocks as building blocks, 3D blocks whose geometry parameters satisfy Equation (4) serve as a focusing lens for the incidence waves. Figure 2b shows three types of focusing lens by varying the block height (Type I), block thickness (Type II), and both (Type III).

2.2. Numerical Method

The focusing effects of these lenses are simulated in a numerical water tank (NWT), which is modelled using the SPH method [21]. SPH is a meshless Lagrangian approach that provides a great advantage in addressing problems associated with complex wave–structure interactions. The idea of the SPH method is to achieve a coarse grain of a continuum media as a collection of smoothed particles. Each of the particles is assumed to obey the Navier–Stokes equation in a Lagrangian form:

$${\displaystyle \frac{d{\overrightarrow{v}}_i}{dt}}=-{\displaystyle \frac{1}{\rho }}\nabla P+ \overrightarrow{\Gamma } +\overrightarrow{g},$$

(5)

where ${\overrightarrow{v}}_i$ is the velocity of particle $i$. $P$, $\rho$, $\overrightarrow{\Gamma }$, and $\overrightarrow{g}$ are the pressure, fluid density, energy dissipation, and the gravitational field at the position where the particle $i$. is.

According to the SPH, the value of a field $X$ at the position $\overrightarrow{r}$ can be extrapolated by using the properties of the nearby particles through a kernel function $W\left(r_{ij},l\right)$:

$$X\left({\overrightarrow{r}}_i\right)={{\displaystyle \sum }}_{j}X\left({\overrightarrow{r}}_j\right){\displaystyle \frac{m_j}{{\rho }_j}}W\left(r_{ij},l\right),$$

(6)

where $r_{ij}=\left|{\overrightarrow{r}}_j-{\overrightarrow{r}}_i\right|$ is the distance between particles $i$ and $j$. $l$ is the smoothed distance. $m_j$ and ${\rho }_j$ are the mass and the density of particle $j$, respectively. If a system is simulated using $N$ number of particles, $m_j$ is simply set to be the total mass divided by $N$. To ensure the conservation of mass, the kernel function needs to be positive and normalized within the interaction region. For our two-dimensional model, the Wendland [22] form is selected:

$$W\left(\xi ,l\right)=\left\{\begin{array}{c}{\displaystyle \frac{7}{4\pi l^2}}{\left(1-{\displaystyle \frac{\xi }{2}}\right)}^4\left(2\xi +1\right), 0\le \xi \le 2\\ 0, \xi >2\end{array}\right., \xi =r_{ij}/l.$$

(7)

By using the extrapolation in Equation (6), The momentum balance equation and the continuity equation can be rewritten as:

$${\displaystyle \frac{d{\overrightarrow{v}}_i}{dt}}=-{{\displaystyle \sum }}_j{m}_j\left({\displaystyle \frac{P_i+P_j}{{\rho }_i{\rho }_j}}+{\mathsf{\Pi }}_{ij}\right){\nabla }_{i}W\left(r_{ij},l\right)+\overrightarrow{g},$$

(8a)

$${\displaystyle \frac{d{\rho }_i}{dt}}={\displaystyle \sum }_j{m}_j\left({\overrightarrow{v}}_{ij}+{\displaystyle \frac{2{\delta }_{s}l{c}_0}{{\rho }_j{r}_{ij}^2}}\left({\rho }_j-{\rho }_i\right) {\overrightarrow{r}}_{ij}\right)·{\nabla }_{j}W\left(r_{ij},l\right),$$

(8b)

$P_i$ is the pressure of particle $i$, which is determined by fluid density according to the following equation [23]:

$$P={\displaystyle \frac{c_0^2{\rho }_0}{\gamma }}\left[{\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^{\gamma }-1\right],$$

(9)

where ${\rho }_0=1000 \mathrm{kg}/{\mathrm{m}}^3$ and $\gamma =7$. $c_0$ is the speed of sound in the media defined by ${\left.{\displaystyle \frac{\partial P}{\partial \rho }}\right|}_{{\rho }_0}$. ${\mathsf{\Pi }}_{ij}$ is the empirical viscosity proposed by Monaghan [24]:

$${\mathsf{\Pi }}_{ij}=\left\{\begin{array}{c}-{\displaystyle \frac{\alpha c_{ij}l}{{\rho }_{ij}}}{\displaystyle \frac{{\overrightarrow{v}}_{ij}·{\overrightarrow{r}}_{ij}}{r_{ij}^2+0.01l^2}} , {\overrightarrow{v}}_{ij}·{\overrightarrow{r}}_{ij}<0\\ 0, {\overrightarrow{v}}_{ij}·{\overrightarrow{r}}_{ij}>0\end{array}\right.,$$

(10)

where $\alpha$ is an adjustable coefficient. Altomare et al. [25] showed that $\alpha =0.01$ is the best value to prevent instability and spurious oscillations in the numerical scheme. Barreiro et al. [21] also validated that with $\alpha =0.01$. The model is good for reproducing the experimental data. $c_{ij}$ and ${\rho }_{ij}$ are the average sound speed defined by $c_{ij}=0.5\left(c_i+c_j\right)$ and density ${\rho }_{ij}=0.5\left({\rho }_i+{\rho }_j\right)$, respectively. ${\overrightarrow{r}}_{ij}={\overrightarrow{r}}_i-{\overrightarrow{r}}_j$ and ${\overrightarrow{v}}_{ij}={\overrightarrow{v}}_i-{\overrightarrow{v}}_j$ are the relative position vector and velocity of particles $i$ and $j$.

To reduce the density fluctuations, a diffusive term with a free parameter ${\delta }_s$ was introduced to the continuity equation in Equation (8b) by Molteni and Colagrossi [26]. In the present study, ${\delta }_s=0.1$ is used.

In the simulations, boundary particles are labelled as a different set of particles from the fluid. They satisfy the same continuity equation. However, they do not move according to Equation (8a). The positions of boundary particles always remain unchanged or change according to a given function. In the present simulations, the wavemaker piston particles are set in motion according to:

$$e\left(\mathrm{t}\right)={\displaystyle \frac{S_0}{2}}\sin \left({\displaystyle \frac{2\mathsf{\pi }}{T}}t\right)+{\displaystyle \frac{A^2}{32H}}\left[\left({\displaystyle \frac{3\cosh \left(k_{I}H\right)}{{\sinh }^3\left(k_{I}H\right)}} \right){\displaystyle \frac{\sinh \left(k_{I}H\right)\cosh \left(k_{I}H\right)+k_{I}H}{{\sinh }^2\left(k_{I}H\right)}}\right]\sin \left({\displaystyle \frac{4\mathsf{\pi }}{T}}t\right),$$

(11)

where $S_0$ is the stroke of the piston. $A$ is the wave height. The wavenumber $k_I$ is determined by the dispersion relation in Equation (1) with a given wave period $T$ and water depth $H$. When energy dissipation is neglible, this piston motion has been proven to be capable of generating second-order Stoke waves [27]:

$$\eta \left(x,\mathrm{t}\right)={\displaystyle \frac{A}{2}}\cos \left(k_{I}x-{\displaystyle \frac{2\mathsf{\pi }}{T}}t\right)+k_I{\mathrm{A}}^2{\displaystyle \frac{3-{\tanh }^2{k}_{I}H}{16{\tanh }^3{k}_{I}H}}\cos \left(k_{I}x-{\displaystyle \frac{4\mathsf{\pi }}{T}}t\right).$$

(12)

When $k_{I}A\ll 1$, the generated waves approach Cosine waves.

After the initial conditions are set, Equation (8a,b) are ready to be integrated for each particle using the Symplectic method [28].

$${\overrightarrow{r}}_i\left(t+\delta t/2\right)={\overrightarrow{r}}_i\left(t\right)+{\displaystyle \frac{\delta t}{2}} {\overrightarrow{v}}_i$$

(13a)

$${\overrightarrow{v}}_i\left(t+\delta t\right)={\overrightarrow{v}}_i\left(t+\delta t/2\right)+{\displaystyle \frac{\delta t}{2}} {\displaystyle \frac{d{\overrightarrow{v}}_i}{dt}}\left(t+{\displaystyle \frac{\delta t}{2}}\right)$$

(13b)

$${\rho }_i\left(t+\delta t/2\right)={\rho }_i\left(t\right)+{\displaystyle \frac{\delta t}{2}} {\displaystyle \frac{d{\rho }_i}{dt}}$$

(13c)

$${\overrightarrow{r}}_i\left(t+\delta t\right)={\overrightarrow{r}}_i\left(t+\delta t/2\right)+{\displaystyle \frac{\delta t}{2}} {\overrightarrow{v}}_i\left(t+\delta t\right)$$

(13d)

$${\rho }_i\left(t+\delta t\right)={\rho }_i\left(t\right)+\delta t {\displaystyle \frac{d{\rho }_i}{dt}}\left(t+\delta t\right)$$

(13e)

To avoid the effect of the reflected waves on the wavemaker, the active wave absorption system discussed in [29] is imposed in the simulations. The position of the wavemaker is corrected based on the water surface elevation in front of the wavemaker so that consistent regular waves are generated. To avoid the interference of reflected waves downstream, a damping zone is set as a passive wave absorber. The velocities of all the particles in this region are imposed in a quadratic decay:

$${\overrightarrow{v}}_i\left(t+\delta t\right)={\overrightarrow{v}}_i\left(t\right)\left(1-\delta t\beta {\left({\displaystyle \frac{x_i\left(t\right)-x_0}{x_1-x_0}}\right)}^2\right)$$

(14)

where $x_0$ and $x_1$ define the damping zone. $\beta$ is the damping coefficient.

3. Results

To investigate the phase lags of transmitted waves caused by rectangular submerged blocks, a 2D NWT is modeled, as shown in Figure 3. The NWT has a length of 12.5 m and is filled with water to a depth of 0.5 m. A piston-type wavemaker is located on the left side of the tank, and a wave absorber is placed on the right side. A rectangular submerged block with height h and thickness d is mounted 3.0 m from the left. Three wave gauges, G1, G2, and G3, are placed at distances d_G1, d_G2, and d_G3, measured from the left side.

The numerical solver used in this study is an open-source code called DualSPHysics [30]. In DualSPHysics, the integration time step is determined by l/c₀. l is determined by the particle resolution d_p. Therefore, d_p is the crucial parameter that affects the numerical convergence and the total time of computation. Despite many studies having validated the reliability of DualSPHysics [31,32], to ensure the solver also works for our system, several tests and validations are performed.

3.1. Convergence Tests

In this section, we compare the water surface elevations $\eta$ obtained from the SPH simulations with the theoretical wave height given in Equation (12). Theoretically, the wavelength of the propagating wave with $T=1.5 \mathrm{s}$ at water depth $H=0.5 \mathrm{m}$ is ${\lambda }_I=2.826 \mathrm{m}$. The generated wave height is set as $A=0.01{\lambda }_I$. Figure 4 shows damping percentages of the wave height measured at the three wave gauges. The wave height decreases as d_p decreases. The values become almost stable when d_p is smaller than 0.003${\lambda }_I$, indicating the convergence of the numerical method. The time series of water surface elevations measured at the three wave gauges using d_p = 0.002${\lambda }_I$ and d_p = 0.053${\lambda }_I$ are plotted in Figure 5a and Figure 5b, respectively. The spatial and temporal simulations generally agree with Equation (12) without introducing any free parameters. However, if a lower particle resolution d_p = 0.053${\lambda }_I$ is used, there are slight phase differences (<0.027π) between the numerical and theoretical results. These phase lags are totally eliminated if a higher particle resolution d_p = 0.002${\lambda }_I$ is used. To make a compromise between reasonable computational time and numerical accuracy, d_p = 0.002${\lambda }_I$ is used for all the following simulations. For each 2D simulation, approximately 400,000 particles are generated. It takes 30 min to simulate 24 s of waves on a PC with an Intel core i5 CPU and RTX4070 graphic card. For a 3D simulation, about 9,000,000 particles are required. Each simulation takes about 10 h to complete.

3.2. Validation of SPH

To validate the SPH simulations generating reliable surface waves, we compare the wave profile with Equation (12) at different wave steepness values. When the values of wave steepness are 0.003 and 0.010, linear propagating waves are generated. The wave crest and wave trough are symmetrical to the free surface level. When $A/{\lambda }_I=0.029$, the wave crest and trough become asymmetrical and distorted. When the value goes to 0.045, the generated wave has an obvious sharper crest and a flattened trough. This is a typical Stokes wave. The four surface waves generated in the SPH simulations are plotted in Figure 6. Due to energy dissipation, the wave height is attenuating when the wave is propagating along the x direction. Despite this slight attenuation, all the data still closely align with Equation (12) without introducing any free parameters. The average deviations between the simulation data and the theoretical values for the four cases are 0.043, 0.037, 0.058, and 0.038, respectively. The corresponding root-mean-square errors are 3.6%, 3.3%, 3.3%, and 2.6%.

By analyzing the time series of the surface elevation and the wave profile using Fourier transformation, we can determine the wave frequency and wavelength. The relationship between the wave frequency and the wavelength in our simulations satisfies the dispersion relation described in Equation (1). Figure 7 shows a comparison between the SPH results and Equation (1) for the case $A/{\lambda }_I=0.010$. The good agreement between the simulation results and Equation (1) ensures the validity of our analysis of the phase of transmitted waves.

3.3. Numerical Phase Lags

Now we introduce a rectangular block to the NWT, as depicted in Figure 3. We conduct simulations using three different values of thicknesses: $d=0.088{\lambda }_I$, $0.177{\lambda }_I$, and $0.265{\lambda }_I$, at various heights ranging from $0.1 H$ to $0.96 H$. For each height and thickness, simulations are conducted three times with different wave steepness values: $A/{\lambda }_I=0.003, 0.010,$ and $0.029$.

Once the water surface elevations are obtained from the simulations, we calculate the phase of the wave using the coefficients of the Fourier series:

$${\mathrm{\varphi }}_{hd}=〈{\tan }^{-1}{\displaystyle \frac{b_{hd}\left(t\right)}{a_{hd}\left(t\right)}}〉,$$

(15)

where $a_{hd}\left(t\right)={{\displaystyle \int }}_{d_{G2}}^{d_{G2}+{\lambda }_I}{\eta }_{III}\left(x,t\right)\cos k_{I}xdx$, $b_{hd}\left(t\right)={{\displaystyle \int }}_{d_{G2}}^{d_{G2}+{\lambda }_I}{\eta }_{III}\left(x,t\right)\sin k_{I}xdx$. $〈\dots 〉$ represents the time average over a wave period. ${\eta }_{III}\left(x,t\right)$ is the water surface elevation of the transmitted wave. ${\mathsf{\Phi }}_{hd}$ is the phase of the transmitted wave over a block with height $h$ and thickness $d$. The phase lag is the phase difference between the transmitted wave with and without a block:

$$\mathsf{\Phi }={\mathrm{\varphi }}_{00}-{\mathrm{\varphi }}_{hd}.$$

(16)

The transmission coefficient is the ratio of the wave amplitude with and without a block:

$${\mathrm{K}}_{\mathrm{T}}=\sqrt{{\displaystyle \frac{{\mathrm{a}}_{\mathrm{hd}}^2+b_{hd}^2}{{\mathrm{a}}_{00}^2+b_{00}^2}}}.$$

(17)

Figure 8 shows the vertical velocity field near the rectangular block at specific instants with wave steepness $A/{\lambda }_I=0.010$. It can be seen that for a given block thickness $d=0.088{\lambda }_I$ in Figure 8a, waves in the tank with a block are delayed compared to waves in the tank without a block. This delay is quantified by the phase lags defined in Equation (16), increasing monotonically with the block height. The arrows and the dashed lines in the figures serve as guides for observing the phase lags. In Figure 8b, a greater block thickness of $0.265{\lambda }_I$ is used. The delay also increases monotonically with the block height but results in greater phase lags.

For relatively small block height and thickness, the transmitted waves remain regular despite the phase lag. However, when the block height approaches the water depth, the transmitted waves become distorted due to the emergence of high-order harmonic (HOH) modes [33]. These HOH modes can appear also at a lower block height as the block thickness increases, but they are consistently observed only in the transmitted waves. The main mechanism involves the accumulation of wave height during the shoaling process, followed by their release in the deeper region, leading to the decomposition of waves. This finding aligns with the previous theoretical and experimental studies [34,35,36,37]. Although our simulations successfully capture the nonlinear behaviors of these waves, this aspect lies beyond the scope of our linear potential flow theory and will not be further discussed here.

Figure 9 shows the vertical velocity field near the rectangular block with the same parameters as in Figure 8, but with a larger wave steepness of $A/{\lambda }_I=0.029$. In addition to HOH modes, wave breaking is observed on top of the block and vortices form near the block’s corners when the block height approaches the water depth. The relationship between phase lags and the block height becomes non-monotonic at this higher wave steepness.

For a given block thickness $0.088{\lambda }_I$, the phase lags as a function of $h$ are plotted on a logarithm scale in Figure 10. The red circles, blue triangles, and green crosses in the figure represent the simulation results, with wave steepness values of 0.003, 0.010, and 0.029, respectively. Generally, the phase lags approach zero as $h$ approaches zero. When $h$ increases, the phase lags initially increase exponentially and then drop suddenly when $h$ approaches $0.9H$. When the block height is lower than 0.7H, the wave steepness does not significantly affect the phase lags. The phase lags as a function of $h$ are independent of wave steepness. When the block height exceeds 0.7H, the phase lags at a wave steepness of 0.029 become significantly lower compared to the phase lags at the other two values of wave steepness. The phase lags predicted by the present model described in Equation (2) are plotted as a dotted line in Figure 10. It shows good agreement with the simulations up to $h=0.9H$ at wave steepness values of 0.003 and 0.010. These results are also consistent with those obtained by Abul [38] using full potential flow theory, plotted as dashed lines.

Figure 11 and Figure 12 show the phase lags on a logarithm scale as a function of $h$ at block thicknesses $0.177{\lambda }_I$ and $0.265{\lambda }_I$, respectively. The data show good consistency with the theory at small block heights and small wave steepness. However, the deviation of the phase lags under a wave steepness of 0.029 becomes greater, starting from a block height of 0.6H. These deviations are likely due to the emergence of HOH modes. When HOH modes occur, the algorithm might fail to identify the correct phase lags. In summary, our linear model works consistently with the simulations results for $h/H<0.9$ and $d/{\lambda }_I<0.2$ when $A/{\lambda }_I<0.01$. Under these conditions, the nonlinearity is negligible.

3.4. Surface Wave Focusing

Since the phase of linear potential flow has no dependence on the block width in accordance with Equation (2), we can use the 2D rectangular blocks as the unit cells of a phase regulator. By carefully selecting the height and the thickness of each block and placing them along the y-direction of the 3D water tank, we can control the propagation of the transmitted waves by manipulating the phase. As a rule of thumb, we construct the 3D structure capable of focusing water surface waves by equalizing Equations (2) and (4):

$$\left(k_{II}-k_I\right)d=k_I\left(\sqrt{f^2+y^2}-f\right).$$

(18)

For given a focal length f, we solve the values of h and d at different positions y. The finite rectangular elements with the parameters (h, d) and equal widths are deposited at position y to form a composite 3D block. To test our idea, we construct three types of blocks, as shown in Figure 2, by finding the corresponding d, y, and h that satisfy Equation (18), which can converge the transmitted waves at a focal distance f = λ_I. The maximum height for Type I is 0.75 H, with constant thickness of 0.37λ_I, while the maximum thickness for type II is 0.37λ_I, with a constant height of 0.75 H. The height of Type III is determined by the thickness according to equation h/H = 0.75 d/λ_I. Both blocks provide a maximum phase lag of 0.43π. The parameters of the blocks are listed in

Table 1. Parameters of the focusing blocks.

Lenses	h/H	d/λI	L/λI	φ [Radians]
Type I	[0, 0.75]	0.37	1.4	[0, 0.43π]
Type II	0.75	[0, 0.37]	1.4	[0, 0.43π]
Type III	h/H = 0.75 d/λI	[0, 0.37]	1.4	[0, 0.43π]

. We conduct simulations by placing these blocks in a 3D NWT 5.5λ_I long and 1.5λ_I wide. Figure 13 shows the schematic picture of the 3D NWT. The focusing effects are shown in Figure 14, Figure 15 and Figure 16 using the pseudo-color map of water surface elevations. The red color indicates the wave crest, and the blue color indicates the wave trough. At the top of each figure, plane waves in the NWT without a block are shown for comparison. As we expected, all blocks successfully convert the plane waves into circular waves whose center is located at a distance λ_I from the blocks, working like focusing lenses.

4. Discussions

To be an effective wave energy collection device, it is necessary to consider both the phases and the amplitudes of the reflected and transmitted waves. Instead of amplitude, wave intensity, which is proportional to the square of the amplitude, is usually used in numerical or experimental measurements. This study mainly focuses on the phases, from points of views of both linear potential flow theory and numerical simulations. Regarding the intensities, we calculate the relative wave intensity using the time average of the wave elevation square normalized with the initial wave amplitude square. Figure 17a shows the comparison of relative wave intensities along the x-axis, i.e., y = 0 for Type I, Type II, and Type III blocks. Although all the blocks produce transmitted waves with a local maximum intensity at a distance λ_I from the blocks, precisely at the focal point we designed, the maximum intensities differ between the three types. Type III produces the highest intensity compared to the other two. The relative wave intensity at the focal point of Type III can achieve 1.36, while Type II gives the lowest relative wave intensity at 1.1. On the other side of the block, Type II reflects stronger waves than Type I and Type III. Figure 17b shows the comparison of relative wave intensities along the y-axis at the focal point (x/λ_I = 1.25). It shows that the plane wave is concentrated to a Gaussian-like wave packet.

In order to reflect minimal wave energy while converging the wave into a small region as effectively as possible, maximum phase difference as well as transmission efficiency are required. The phase differences induced by different positions y of the composite block are expected to cover as big a range as possible, ideally 2π, so that a large collection angle of such a “focusing lens” can be achieved. However, from the study on 2D cases, we find that high transmission is normally achieved with relatively small phase lags. The transmission coefficients versus the phase lags induced by varying the block height h for three kinds of block thicknesses d are shown in Figure 18. It implies that increasing the phase lag results in more energy being reflected, which counteracts the ideal design of a focusing block. Indeed, numerous theoretical studies indicate that the resonance of surface waves can happen at appropriate geometric structures. When resonance happens, waves can be either mostly transmitted or reflected. This phenomenon is particularly evident in periodic structures [39,40,41,42]. By systematically exploring the parameters database, it is possible to inversely design an optimized structure with the best performance in both phase lag and transmitted efficiency using state-of-the-art methods, such as the genetic algorithm, the adjoint-based method, and artificial neural networks [43].

However, building such a database from 3D cases would be time-consuming and computationally extensive. The present study offers an efficient approach to exploring the parameter space from 2D cases. Such an approach is similar to the so-called “locally periodic approximation” in electromagnetism, which simplifies the scattering problem of large-scale 3D structures by a composition of periodic scattering problems from 2D grating structures [44]. In our case, we use a similar principle; the difference lies in that a large-scale 3D block with irregular shapes in the height and thickness are first discretized to finite rectangular block unit cells, which are then approximated by a composition of 2D elements. The precondition to applying this approximation is that the phase lag shows no dependence on the infinite dimension (width) and the variations in the other two dimensions are not significant along this infinite dimension.

5. Conclusions

In this study, we propose a method to design 3D blocks capable of focusing water surface waves by manipulating the phase of transmitted waves. We begin by obtaining the phase lags of transmitted waves induced by 2D block unit cells with varying heights and thicknesses using linear potential theory and a numerical method based on SPH. The key strategy is to compensate for the circular phase difference in a maximal collection angle by manipulating the geometries of 2D elements. This ensures that in-phase waves converge at a focal point, thereby achieving wave focusing. In summary,

• Success:

• We successfully present a method to design 3D blocks capable of focusing water surface waves by manipulating the phase of transmitted waves. SPH simulations validate the effectiveness of this method and demonstrate that the relative wave intensity can achieve 1.36 at the focal point.

• Applicability:

• This method can be applied to enhance the monochromatic and regular wave power by ensuring in-phase waves converge at a focal point through geometric manipulation of 2D block elements.

• Primary Limitations:

• The model currently addresses only phase manipulation and does not fully optimize the transmission coefficient of wave scattering. Additionally, the method does not account for the complexities of real sea conditions, such as irregular and multidirectional waves, which make optimization more challenging.

• Future Scope of Model Formulations:

• Future work should focus on optimizing both the phase and the transmission coefficient to improve performance in diverse sea conditions. This study lays the groundwork for more efficient wave energy collection methods, with potential for faster, more effective computational optimizations.