Weak Cloaking in Water Waves via Spatial-Transformation Metamaterials

Abstract

When water waves encounter marine structures, significant scattering is produced, which affects the stability and safety of marine equipment. The cloaking of marine structures in water waves, achieved by water-wave transformation metamaterials, provides a new approach to solving this problem. However, in the anisotropic water depth required for perfect cloaking, the radial water depth tends to infinity, and the circumferential water depth tends to zero at the edge of the structure, leading to a singularity problem. This results in extreme parameters of the metamaterial near the structure, making fabrication difficult. To address the above issue, a comb-type metamaterial for weak scattering of water waves is designed in this paper based on space-transformation metamaterials, by which optimized control over the scattering characteristics of water waves is achieved. First, a general nonlinear spatial transformation is designed for both weak scattering and perfect cloaking. Then, the anisotropic water depths and equivalent gravitational acceleration parameters for the two cases are calculated and analyzed. Subsequently, the Helmholtz equation is solved using the finite element method, and the distributions of wave fields under anisotropic water depths and metamaterials for weak scattering and perfect cloaking are compared and analyzed. The results show that for weak scattering, the radial and circumferential water depths tend to constant values at the edge of the structure, thus resolving the singularity problem in perfect cloaking, while the control performance over water waves is comparable. Compared with perfect-cloaking metamaterials, the designed weak-scattering metamaterial significantly reduces the water depth parameters near the structure while maintaining water-wave control performance, making it easier to fabricate.

1. Introduction

Water waves, which are disturbance phenomena that occur at free fluid surfaces, possess characteristics such as refraction, reflection, and resonance. Active control of water waves based on these characteristics has attracted extensive research interest from scholars worldwide. Technologies such as cloaking [1,2], concentrators [3,4], rotators [5], shifters [6], waveguides [7], and wavelength modulators [8] have been proposed successively. Among them, cloaking technology was first proposed in the electromagnetic field [9]. Typically, a cloaked structure does not produce scattered waves in the far field.

It is worth noting that significant scattering occurs when water waves encounter marine structures. This scattering changes the water-wave flow field around the structure and subsequently affects the stability and safety of marine equipment. Therefore, how to effectively reduce or control water-wave scattering caused by structures has become a key problem in marine engineering. Water-wave cloaking technology provides a new solution to this problem. Based on wave characteristics, methods such as the scattering cancelation method (SCM) and the space transformation method combined with metamaterials (STM and MM) have been proposed to design cloaking structures in water waves. Cloaking structures based on SCM use wave resonance effects to reduce scattered waves in the far field. Porter [2] first used SCM to reduce scattered water waves around a vertical cylinder by optimizing the bottom topography around the cylinder. Newman [10,11] introduced an array of small cylinders to cloak a fixed truncated cylinder in water. Iida [12] cloaked a bottom-mounted cylinder by using a floating elastic composite plate. The bending stiffness and mass of the plate were designed to minimize the energy of scattered waves.

Compared with SCM, the STM and MM provide a more flexible way to manipulate waves. This method uses the form invariance of the governing equation under coordinate transformation. By designing equivalent medium parameters with specific spatial distributions, waves are guided to propagate along predefined trajectories. The STM & MM, initially conceptualized for electromagnetic waves [13,14], have since been successfully extended to diverse physical domains, encompassing acoustic waves [15], elastic waves [16], seismic waves [17], thermal waves [18], and beyond. In the field of water waves, Farhat et al. [1] first applied the space transformation method to the cloaking problem for liquid surface waves. Wave manipulation was achieved by designing anisotropic viscosity. Chen et al. [5] extended the concept of transformation media to linear liquid surface waves and designed a rotating water-wave metamaterial based on anisotropic water depth. Berraquero et al. [6] designed a low-scattering water-wave shifter. Dupont et al. [19] realized an invisibility carpet in a water-wave flume. Zareei and Alam [20] proposed a nonlinear transformation method that only requires the adjustment of water depth. Iida and Kashiwagi [21] constructed an anisotropic channel network surrounding a cylindrical target and achieved good cloaking performance. Zhang et al. [22] investigated the influence of the no-flux inner boundary condition of anisotropic media on the cloaking performance of structures. The singularity problem of medium parameters near the inner boundary was also analyzed. Zhang et al. [23] applied the spatial-transformation method to design a straight reflector for parabolic water-wave focusing, and found that the straight reflector retains the focusing capability of the parabolic reflector, with the simplified design yielding greater wave amplification in the focal region.

However, although water-wave cloaking based on spatial-transformation metamaterials has achieved significant theoretical success, its physical implementation faces a fundamental challenge: parameter singularity. To achieve perfect cloaking, a typical coordinate transformation maps a physical point to a finite-sized region. This transformation inevitably leads to extreme distributions of equivalent medium parameters at the inner boundary of the cloak, i.e., at the edge of the cloaked object. Specifically, for a perfect water-wave cloaking metamaterial, the water depth in the radial direction tends to infinity, while the water depth in the azimuthal direction tends to zero. Such extreme and discontinuous anisotropic water depth parameters cannot be realized physically. This limits the transition of water-wave cloaking technology from theory to engineering applications. To address this issue, this paper innovatively proposes a weak-cloaking spatial relationship to avoid the infinite water depth in perfect cloaking. A comb-type metamaterial was subsequently designed. While good wave field control performance is maintained, the vertical size of the metamaterial is significantly reduced.

2. Theory

2.1. Water-Wave Governing Equation in Shallow Water

Consider a fluid that is irrotational, inviscid, and incompressible. By adopting the boundary conditions, the following linear governing equations in shallow water can be derived from the continuity equation and the kinematic equation [23],

$${\displaystyle \frac{\partial \zeta }{\partial t}}+\nabla \cdot \left(h_{0}u\right)=0$$

(1)

$${\displaystyle \frac{\partial u}{\partial t}}+g_0\nabla \zeta =0$$

(2)

where ζ (x₁, x₂; t) is the wave elevation; g₀ is the initial gravity acceleration; h₀ is the initial water depth; u (u₁, u₂) is the horizontal fluid velocity.

Combining Equation (1) and Equation (2) into one term of ζ, the governing equation of shallow-water waves can be obtained,

$${\displaystyle \frac{{\partial }^2\zeta }{\partial t^2}}-\nabla \cdot \left(g_0{h}_0\nabla \zeta \right)=0$$

(3)

Assuming harmonic waves, ζ = η(x₁, x₂)e^−iωt, where η denotes the spatial wave elevation, and ω is the angular frequency. η = 0 means the water surface coincides with the calm water level, η > 0 means the water surface is above the calm water level, and η < 0 means the water surface is below the calm water level. Then the following equation can be obtained,

$$\nabla \cdot (\nabla \eta )+{\displaystyle \frac{{\omega }^2}{{g_{0}h}_0}}\eta =0$$

(4)

Considering $k_0=\omega /\sqrt{g_0{h}_0}$, the well-known Helmholtz equation can be obtained,

$$\nabla \cdot (\nabla \eta )+k_0^2\eta =0$$

(5)

2.2. Space Transformation Method

The Helmholtz equation is form-invariant under coordinate transformations [13,14]. Consider an orthogonal coordinate transformation ϒ* = f(ϒ), where the Jacobian matrix is defined as J = ∂$x_1^{*}$/∂x_j. Here, ϒ (x₁, x₂) and ϒ^* ($x_1^{*}, x_2^{*}$) represent the original space and the transformed space, respectively. The transformed coordinate system allows us to construct new wave fields. Thus, determining the appropriate coordinate transformation relationship is a key step for achieving different wave manipulation functions using the STM.

Via the coordinate transformation, the Helmholtz equation can be rewritten as

$${\nabla }^{*}\cdot ({\displaystyle \frac{\mathit{J}{\mathit{J}}^{\mathrm{T}}}{\left|\mathit{J}\right|}}{h}_0{\nabla }^{*}\eta )+{\displaystyle \frac{{\omega }^2}{\left|\mathit{J}\right|g_0}}\eta =0$$

(6)

The Jacobian matrix can be expressed as,

$$\mathit{J}=\left[\begin{array}{cc}{\displaystyle \frac{\partial x_1^{*}}{\partial x_1}}& {\displaystyle \frac{\partial x_1^{*}}{\partial x_2}}\\ {\displaystyle \frac{\partial x_2^{*}}{\partial x_1}}& {\displaystyle \frac{\partial x_2^{*}}{\partial x_2}}\end{array}\right]$$

(7)

The space transformation terms are absorbed into the medium parameters h₀ and g₀. Then, by defining τ_h = JJ ^T/|J| and τ_g = |J|, Equation (6) is rewritten as follows,

$${\nabla }^{*}\cdot ({\tau }_h{h}_0{\nabla }^{*}\eta )+{\displaystyle \frac{{\omega }^2}{{\tau }_g{g}_0}}\eta =0$$

(8)

Here, τ_h and τ_g are spatial scale factors, i.e., anisotropic media. It should be noted that τ_h is a tensor matrix.

$${\tau }_h={\displaystyle \frac{\mathit{J}{\mathit{J}}^{\mathrm{T}}}{\left|\mathit{J}\right|}}=\left[\begin{array}{cc}{\tau }_h^r& {\tau }_h^{r\theta }\\ {\tau }_h^{\theta r}& {\tau }_h^{\theta }\end{array}\right]$$

(9)

Using $k_0=\omega /\sqrt{g_0{h}_0}$, Equation (8) can be rewritten as

$${\nabla }^{*}\cdot ({\tau }_h{\nabla }^{*}\eta )+{\displaystyle \frac{k_0^2}{{\tau }_g}}\eta =0$$

(10)

Equation (5) and Equation (10) are compared, and their forms are shown to be the same. Therefore, the wave distribution in the transformed space ϒ^* can be achieved by redistributing the medium parameters in the original space ϒ with the anisotropic media (τ_h, τ_g).

2.3. Metamaterial Design Method

It can be observed from Equation (8) that τ_g is spatially dependent, which is difficult to realize in practical engineering. A simplified method is proposed to avoid altering the gravitational acceleration. This method makes τ_g a constant and absorbs it into τ_h [20]. The anisotropic Helmholtz Equation (10) in cylindrical coordinates can be expressed as

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\tau }_h^r{\displaystyle \frac{\partial \eta }{\partial r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\left({\tau }_h^{\theta }{\displaystyle \frac{\partial \eta }{\partial \theta }}\right)+{\displaystyle \frac{k_0^2}{{\tau }_g}}\eta =0$$

(11)

It can be found that if the gravitational acceleration scale factor τ_g is constant, it can be absorbed into the radial water depth ${\tau }_h^r$ and the tangential water depth ${\tau }_h^{\theta }$.

$${\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\tau }_g{\tau }_h^r{\displaystyle \frac{\partial \eta }{\partial r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\left({\tau }_g{\tau }_h^{\theta }{\displaystyle \frac{\partial \eta }{\partial \theta }}\right)+k_0^2\eta =0$$

(12)

Then, the anisotropic Helmholtz equation can be rewritten as

$${\nabla }^{*}\cdot ({\tau }_h^{*}{\nabla }^{*}\eta )+{\displaystyle \frac{k_0^2\eta }{{\tau }_g^{*}}}=0$$

(13)

where ${\tau }_h^{*}={\tau }_g{\tau }_h$, ${\tau }_g^{*}$ = 1.0. It should be noted that Equation (13) still satisfies the transformation equation.

Then, the water waves can be controlled just using ${\tau }_h^{*}$. However, such an anisotropic water depth ${\tau }_h^{*}$ does not exist in nature; the metamaterial technology can be adopted to equivalently substitute ${\tau }_h^{*}$. The effective medium method is employed to design a metamaterial. Since ${\tau }_h^{r^{*}}$ and ${\tau }_h^{{\theta }^{*}}$ are discretized into K segments in this study. Each homogeneous anisotropic segment can be mimicked by alternating layers of isotropic layers a and b. The thickness of each layer of the metamaterial should be much smaller than the wavelength to approximate the anisotropic medium.

$${\tau }_h^{r*}(i)={\displaystyle \frac{2{\tau }_a(i){\tau }_b(i)}{{\tau }_a(i)+{\tau }_b(i)}},{\tau }_{\theta }^{r*}(i)={\displaystyle \frac{{\tau }_a(i)+{\tau }_b(i)}{2}}$$

(14)

where i = 1, 2, …, K, and τ_a (i) and τ_b (i) represent the scale factors of water depths for layers a and b, respectively.

3. Results and Discussions

3.1. Transform Space Design

It is well known that when water waves encounter a structure, scattering occurs, and the intensity of scattering is closely related to the characteristic size of the structure. Generally, a smaller structure gives rise to weaker scattered waves, whereas a larger structure produces stronger scattered waves. If the scattered wave field induced by a large structure is exactly identical to that generated by a small structure, the large structure is said to achieve weak scattering. In other words, weak scattering means that the large structure behaves equivalently to a small structure in terms of its scattering characteristics. Furthermore, if the scattered wave field of the large structure is the same as that in the absence of any structure, i.e., the structure size tends to zero, the phenomenon is referred to as perfect cloaking. It is thus evident that perfect cloaking can be regarded as a limiting case of weak scattering.

Based on the above discussion, the manipulation of water waves requires the construction of a transformation space. This study focuses on the weak scattering effect, and therefore a general spatial transformation for weak scattering is first established. Consider the original physical space ϒ, as shown in Figure 1a, in which a circular scatter of radius α is placed. An arbitrary point in this space is described by cylindrical coordinates (r, θ). In the original space, the wave path is a straight line, as indicated by the red line in Figure 1a. To achieve the weak scattering effect, the annular region α < r < γ in the original space ϒ is mapped to the annular region β < r^∗ < γ in the transformed space ϒ^*, where α < β < γ. This mapping is equivalent to enlarging the scatter of radius α in the original space to a scatter of radius β, thereby yielding a transformed space ϒ^* (r^*, θ^*), as shown in Figure 1b. It can be observed that the wave path is bent due to the space deformation. Under this transformation, the scattered wave field induced by the scatter of radius β is exactly identical to that generated by the scatter of radius α. Meanwhile, in order to keep the spatial scale factor τ_g constant, a nonlinear coordinate transformation is adopted, inspired by the space transformation of perfect cloaking in [20]. It should be noted that perfect cloaking is a special case in Equation (15) with α = 0.

$$\left\{\begin{array}{l}{r}^{*}=\sqrt{{\displaystyle \frac{{\gamma }^2-{\beta }^2}{{\gamma }^2-{\alpha }^2}}{r}^2+{\displaystyle \frac{{\beta }^2-{\alpha }^2}{{\gamma }^2-{\alpha }^2}}{\gamma }^2}\\ {\theta }^{*}=\theta \end{array}\right.$$

(15)

Figure 1. Design of weak scattering according to the STM.

3.2. Anisotropic Media for Weak Scattering

Based on the coordinate transformation relationship given by Equation (15), the parameters of the anisotropic medium can be calculated, and thus the same wave field can be realized in the original space rather than in the transformed space, as shown in Figure 1c. According to the definitions τ_h = JJ^T/|J| and τ_g = |J|, the spatial scale factors τ_h and τ_g, i.e., the anisotropic medium, are obtained as presented in Figure 1c.

$$\left\{\begin{array}{l}{\tau }_h={\displaystyle \frac{\mathit{J}{\mathit{J}}^{\mathrm{T}}}{\left|\mathit{J}\right|}}=\left[\begin{array}{cc}{\tau }_h^r& {\tau }_h^{r\theta }\\ {\tau }_h^{\theta r}& {\tau }_h^{\theta }\end{array}\right]=\left[\begin{array}{cc}{\displaystyle \frac{{r^{*}}^2-\frac{{\beta }^2-{\alpha }^2}{{\gamma }^2-{\alpha }^2}{\gamma }^2}{{r^{*}}^2}}& 0\\ 0& {\displaystyle \frac{{r^{*}}^2}{{r^{*}}^2-\frac{{\beta }^2-{\alpha }^2}{{\gamma }^2-{\alpha }^2}{\gamma }^2}}\end{array}\right]\\ {\tau }_g=\left|\mathit{J}\right|={\displaystyle \frac{{\gamma }^2-{\beta }^2}{{\gamma }^2-{\alpha }^2}}\end{array}\right.$$

(16)

Here, τ_h is a tensor matrix, in which ${\tau }_h^r$ ≠ ${\tau }_h^{\theta }$, indicating that the water depth is anisotropic. Note that τ_h = 1.0 and τ_g= 1.0 for r^* > γ. If the scale factors τ_h and τ_g are known, then according to Equation (10), the same wave distribution in the transformed space, as shown in Figure 1b, can be realized in the original space, as shown in Figure 1c. Thus, the original coordinate system r should be used to replace r^* in Equation (16) [24].

From Equation (16), it can be seen that the parameters of the anisotropic medium are determined jointly by α, β, and γ, subject to the condition α < β < γ. In this study, β = 2.0 m and γ = 4.0 m are adopted. It should be noted that the case of α = 0 m corresponds to perfect cloaking, which can be regarded as a limiting case of weak scattering. For comparison, both the weak scattering case with α = 0.5 m and the perfect cloaking case with α = 0 are considered in this study to systematically evaluate the control performance of the anisotropic medium for weak scattering.

The anisotropic medium parameters for the weak scattering case (α = 0.5) and the perfect cloaking case (α = 0) are first compared. From Equation (16), τ_g is found to be a constant, with values of 0.571 for weak scattering and 0.556 for perfect cloaking. The two values are close to each other, with only a small difference. Further comparison is made regarding the distributions of the components of the spatial scale factor τ_h. Figure 2 presents the variations in the radial component ${\tau }_h^r$ and the azimuthal component ${\tau }_h^{\theta }$ along the radial direction r. It can be observed that the variation trends of ${\tau }_h^r$ and ${\tau }_h^{\theta }$ for the weak scattering case are the same as those for the perfect cloaking case, and the two components exhibit opposite variation tendencies with respect to r. Specifically, the radial component ${\tau }_h^r$ in both cases is a monotonically increasing function of r, which rises rapidly with increasing r and gradually converges as r approaches the outer boundary γ. This variation trend exhibits a nonlinear growth characteristic, with a steep increase followed by a gradual flattening. The azimuthal component ${\tau }_h^{\theta }$ in both cases is a monotonically decreasing function of r, which drops sharply from a large value and then gradually converges as r approaches the outer boundary γ. The rate of decrease is also fast at first and then becomes slow. From the perspective of anisotropy, near the inner boundary of both cases, ${\tau }_h^r$ is very small, and ${\tau }_h^{\theta }$ is very large, indicating strong anisotropy of the medium, which enables the bending of wave rays around the structure. As r increases, the difference between ${\tau }_h^r$ and ${\tau }_h^{\theta }$ gradually diminishes, and the anisotropic characteristics weaken continuously. In the far-field region, ${\tau }_h^r$ and ${\tau }_h^{\theta }$ tend to become equal, and the medium approaches isotropy, which ensures the continuity of wave propagation outside the anisotropic medium.

Figure 2. (a) Comparison of the variations in the radial component ${\tau }_h^r$ along the radial direction r; (b) comparison of the variations in the azimuthal component ${\tau }_h^{\theta }$ along the radial direction r.

Although the variation trends of ${\tau }_h^r$ and ${\tau }_h^{\theta }$ along the radial direction are the same for the two cases, the values of ${\tau }_h^r$ for the weak scattering case are slightly larger than those for the perfect cloaking case, whereas the values of ${\tau }_h^{\theta }$ are slightly smaller. As the radial distance r increases, the differences between the two cases gradually decrease, and both tend to similar values at the outer boundary γ. It is worth noting that at the inner boundary β, ${\tau }_h^r$ = 0.0476 for the weak scattering case, while ${\tau }_h^r$ = 0 for the perfect cloaking case. This means that a non-zero radial scale factor is introduced at the inner boundary in the weak scattering case. In addition, in the region close to the inner boundary β,${\tau }_h^{\theta }$ tends to infinity for the perfect cloaking case, whereas it takes a value of only 21.0000 for the weak scattering case. This indicates that the perfect cloaking case exhibits a significant parameter singularity near the inner boundary, in the form of ${\tau }_h^{\theta }$ tending to infinity and ${\tau }_h^r$ being zero. In contrast, the weak scattering case effectively suppresses this singularity, with both ${\tau }_h^r$ and ${\tau }_h^{\theta }$ remaining finite, and thus the parameter singularity problem in the anisotropic medium is resolved.

To evaluate the wave field control performance of the designed anisotropic medium for weak scattering, corresponding numerical simulations were carried out in this study. The finite element method was employed to solve Equation (10), and the numerical implementation was conducted using the COMSOL Multiphysics 6.0 software platform. The wave field distributions of the weak scattering case were compared with those of the perfect cloaking case and the original scattering case without the anisotropic medium, and the model setup is shown in Figure 3. A square computational domain with a side length of 10 m was established for all three cases. The left boundary of the domain was set as the incident-wave inlet, and the original scatter was placed at the center of the domain and treated as a solid-wall boundary. The incident wave was expressed as η = A₀e^ikx, with amplitude A₀ = 0.05 m, wavelength λ = 3 m, and wavenumber k₀ = 2π/λ. To suppress wave reflection at the boundaries of the computational domain and reduce boundary interference, perfectly matched layers (PMLs) were applied along all outer boundaries. Unstructured grids were adopted to discrete computational domain, where at least 20 grids per wavelength were adopted in the simulation of all cases in this study. Seven layers of grids in the PML were adopted. In addition, to quantitatively evaluate the wave field control performance of the designed anisotropic medium, a circular monitoring line with radius r₀ = 8 m was set in the computational domain.

Figure 3. Numerical model setting.

The spatial distributions of the normalized instantaneous total wave elevation Re[η]/A₀ under different cases were calculated, as shown in Figure 4. Here, “Re[ ]” denotes the real part of the physical quantity inside the brackets. Figure 4a presents the case of scattering by the original scatter with a radius of 2 m. It can be observed that obvious wave reflection and diffraction occur in front of the original scatter, and the wavefront is significantly distorted at the upstream side of the original scatter. Behind the original scatter, a long wave shadow zone is formed, and the wave elevation exhibits a highly non-uniform spatial distribution. This indicates that the larger structure causes a strong disturbance to the incident wave field. Figure 4b shows the case of scattering by a small scatter with a radius of 0.5 m. As the scatter size decreases, the degree of wavefront distortion is significantly reduced, and almost no shadow zone is formed behind the scatter. The wave elevation distribution is relatively uniform overall. Figure 4c presents the case of weak scattering for the original scatter with a radius of 2 m. After manipulation by the anisotropic medium for weak scattering, although the physical radius of the structure remains 2 m, the wave elevation distribution is in good agreement with that of the small scatter case shown in Figure 4b. This demonstrates that the designed anisotropic medium for weak scattering can achieve the control target of making a large structure equivalent to a small one in terms of wave field behavior. Figure 4d shows the case of perfect cloaking for the original scatter with a radius of 2 m. In this case, the wave elevation distribution is almost the same as the background wave field without any structure, indicating that perfect cloaking of the structure is achieved.

Figure 4. Spatial distributions of the normalized instantaneous total wave elevation Re[η]/A₀ under different cases: (a) original scatter, (b) small scatter, (c) original scatter weak scattering, and (d) original scatter perfect cloaking.

The spatial distributions of the normalized total wave amplitude |η/A₀| under different cases were then calculated, as shown in Figure 5. It can be observed from Figure 5a that the wave amplitude is significantly enhanced near the upstream side of the original scatter, forming obvious standing-wave interference fringes. Behind the original scatter, a region of wave amplitude attenuation appears, indicating a significant wave shadowing effect. In contrast, as shown in Figure 5b, the reflected wave amplitude in front of the small scatter is considerably reduced, and the shadow zone behind the scatter is comparable to the background amplitude level. This indicates that the smaller structure causes a weaker disturbance to the wave field, and the scattered wave amplitude level is generally lower than that of the original scatter case. The amplitude distribution in Figure 5c is in good agreement with that in Figure 5b, with the positions of the enhancement and attenuation zones, the amplitude levels, and the overall variation trends all matching well. This validates that the anisotropic medium for weak scattering can effectively suppress the scattering of the original scatter, making its amplitude response equivalent to that of the small scatter. In Figure 5d, the normalized total wave amplitude tends to 1.0 throughout the entire domain, indicating that the incident wave field is almost undisturbed by the structure, which verifies that perfect cloaking completely suppresses the scattered waves.

Figure 5. Spatial distributions of the normalized total wave amplitude |η/A₀| under different cases: (a) original scatter, (b) small scatter, (c) original scatter weak scattering, and (d) original scatter perfect cloaking.

To visually analyze the spatial distributions of scattered waves under different cases, the normalized instantaneous scattered wave elevation Re[η_s]/A₀ and the scattered wave amplitude |η_s/A₀| were calculated for the four cases, as shown in Figure 6 and Figure 7, respectively. According to the principle of superposition, the scattered-wave elevation can be calculated using η_s = η − η₀, where η₀ is the wave elevation of the incident waves. For all cases, the scattered waves propagate outward in concentric circular patterns centered at the structure, reflecting the radiation characteristics of scattered waves from the near field to the far field. It can be observed from Figure 6a and Figure 7a that the scattered waves for the original scatter case are the strongest, forming directional scattering lobes extending downstream, and the scattering disturbance can spread to the far field. In contrast, the scattered waves for the small scatter case are much weaker. As shown in Figure 6c and Figure 7c, after manipulation by the anisotropic medium for weak scattering, the far-field scattered waves of the original scatter are the same as those of the small scatter, and the scattered waves in the near field of the scatter exhibit a symmetric distribution. In addition, it can be seen from Figure 6d and Figure 7d that after manipulation by the anisotropic medium for perfect cloaking, the far-field scattered waves are nearly zero, and only scattered waves exist in the near field of the scatter, which are also symmetrically distributed.

Figure 6. Spatial distributions of the normalized instantaneous scattered wave elevation Re[η_s]/A₀ under different cases: (a) original scatter, (b) small scatter, (c) original scatter with weak scattering, and (d) original scatter with perfect cloaking.

Figure 7. Spatial distributions of the normalized instantaneous scattered wave amplitude |η_s/A₀| under different cases: (a) original scatter, (b) small scatter, (c) original scatter with weak scattering, and (d) original scatter with perfect cloaking.

To determine the scattering efficiency, we used the scattering attenuation coefficient ε. A nondimensional parameter ε of the scattering energy attenuation ratio is defined as shown in Equation (17),

$$\epsilon ={\displaystyle \frac{{\Sigma }_j}{{\Sigma }_{\mathrm{original} \mathrm{scatter}}}}$$

(17)

where Σ means the total scattering energy of the scatter. Σ_{original scatter} means the scattering cross-section of the original scatter. j can be original scatter ε_os, small scatter ε_ss, perfect cloaking with ideal medium ε_pi, perfect cloaking with simplified medium ε_ps, perfect cloaking with metamaterial ε_pm, weak scattering with ideal medium ε_wi, weak scattering with simplified medium ε_ws, weak scattering with metamaterial ε_wm. It should be noted that ε = 0 means perfect cloaking.

The scattering energy Σ can be calculated as follows,

$$\Sigma ={\displaystyle \frac{1}{2\pi }}{{\displaystyle {\int }_0^{2\pi }\left|{\eta }_s(r,\theta )\right|}}^{2}d\theta ,r\to \infty$$

(18)

Here, |η_s(r,θ)| is the scattered wave amplitude at point (r,θ). In this study, the scattered wave amplitude at r = 8.0 m is considered.

Then, the scattering energy attenuation ratios of the original scatter ε_os, small scatter ε_ss, weak-cloaking with ideal medium ε_wi, and perfect-cloaking with ideal medium ε_pi were calculated, respectively. The values of ε were marked in each subgraph in Figure 7. It can be found that ε_wi = ε_ss = 0.16, which means the scattering of the original scatter can be reduced to 16% of that of the small scatter using the weak-cloaking ideal medium. The scattering of the perfect cloaking shown in Figure 7d is zero.

To further analyze the influence of the anisotropic medium on far-field scattered waves, the normalized instantaneous scattered wave amplitude |η_s/A₀| at the far field (r₀ = 8 m) was calculated under different cases, as shown in Figure 8. The 0 degree in the radar chart coincides with the positive direction of the x-axis. In the figure, the black dashed line A represents the original scatter case, the red long-dashed line B represents the small scatter scattering case, the blue dotted-dashed line C represents the original scatter weak scattering case, and the green solid line D represents the original scatter perfect cloaking case. It can be observed that the far-field scattered wave amplitude of case A is the largest overall, with a distinct lobe-shaped distribution in the incident direction, indicating that the original scatter has the strongest scattering effect and causes significant disturbance in the far field. The far-field scattered amplitude of case B is considerably lower than that of case A, indicating that the scattering intensity of the small scatter is much weaker. The far-field scattered amplitude of case C is almost identical to that of case B, demonstrating that the anisotropic medium for weak scattering successfully tailors the scattering field of the original scatter to that of the small scatter. The far-field scattered amplitude of case D is near zero, indicating that perfect cloaking of the scatter is achieved.

Figure 8. Distributions of the normalized instantaneous scattered wave amplitude |η_s/A₀| at the far field (r₀ = 8 m) under different cases: A represents the original scatter case, B represents the small scatter case, C represents the original scatter weak scattering case, and D represents the original scatter perfect cloaking case.

3.3. Metamaterials for Weak Scattering

To realize the equivalent anisotropic medium, a corresponding water-wave metamaterial was designed. According to the subsection on the metamaterial design method, the design of the metamaterial requires the anisotropic water depth ${\tau }_h^{*}$. Therefore, based on Equation (12), the gravitational acceleration scale factor τ_g is absorbed into the water depth scale factor τ_h, yielding ${\tau }_h^{*}$ and ${\tau }_g^{*}$, as given in Equation (19).

$$\left\{\begin{array}{l}{\tau }_h^{*}={\tau }_g{\tau }_h=\left[\begin{array}{cc}{\tau }_h^{r*}& {\sigma }_h^{r\theta *}\\ {\tau }_h^{\theta r*}& {\tau }_h^{\theta *}\end{array}\right]={\displaystyle \frac{{\gamma }^2-{\beta }^2}{{\gamma }^2-{\alpha }^2}}\left[\begin{array}{cc}{\displaystyle \frac{r^2-\frac{{\beta }^2-{\alpha }^2}{{\gamma }^2-{\alpha }^2}{\gamma }^2}{r^2}}& 0\\ 0& {\displaystyle \frac{r^2}{r^2-\frac{{\beta }^2-{\alpha }^2}{{\gamma }^2-{\alpha }^2}{\gamma }^2}}\end{array}\right]\\ {\tau }_g^{*}=1.0\end{array}\right.$$

(19)

In this study, the number of metamaterial layers is selected as K = 25. For the sake of convenient comparison, the parameters α, β, and γ of the metamaterial are chosen to be the same as those of the anisotropic medium, with β = 2.0 m and γ = 4.0 m. The cases of α = 0.5 m and α = 0 are also considered to evaluate the control performance of the weak-scattering metamaterial. Then, according to Equations (14) and (19), the water depth ratios τ_a (i) and τ_b (i) for each layer of a and b, as well as the width of each layer w(i) = (γ − β)/2K, can be obtained for both the weak-scattering metamaterial and the perfect-cloaking metamaterial, as shown in Figure 9. For clarity in observing the metamaterial parameters, only the layouts in half of the cross-section are presented for the two metamaterials. The actual structure is a cylindrical body generated by revolving around the r = 0 axis.

Figure 9. Metamaterial parameters under different cases: MM_perfect cloaking denotes the perfect-cloaking metamaterial, MM_weak scattering denotes the weak-scattering metamaterial.

It can be observed from Figure 9 that, as the distance to the inner cylindrical wall decreases, the water depth ratios τ_a (i) required for both the weak-scattering metamaterial and the perfect-cloaking metamaterial increase, while the water depth ratios τ_b (i) decrease. Far from the inner cylindrical wall, the parameters for the two metamaterials tend to become similar. However, it is worth noting that the innermost layer of the perfect-cloaking metamaterial has a water depth ratio τ_a (1) as high as 38.61, while that of the weak-scattering metamaterial is only 17.98, representing a reduction of approximately 53.4%. For the second layer, τ_a (2) is reduced by 26.9%. This indicates that the water depth parameters of the perfect-cloaking metamaterial are extremely large near the inner cylindrical wall, which corresponds to the inherent singularity of perfect cloaking in theory. The large variation in water depth between adjacent layers makes it difficult to realize layered structures with such steep depth gradients in engineering practice. In contrast, at the same radial position, the water depth of the weak-scattering metamaterial is lower than that of the perfect-cloaking metamaterial, with a reduction of more than half for the innermost layer. The gradient of water depth variation is much milder, which promotes the feasibility of the cloaking protection technology.

To evaluate the wave field control performance of the designed weak-scattering metamaterial, corresponding numerical simulations were also carried out. The numerical method and incident wave conditions are the same as those described in the subsection on anisotropic media for weak scattering, and the model setup is shown in Figure 10. Figure 10 shows that the metamaterial contains K layers, while each layer contains two kinds of sublayers, including a-type and b-type. a-type layer is below the seabed surface, while the b-type layer is above the seabed surface. In addition, to quantitatively evaluate the wave field control performance of the designed metamaterial, a circular monitoring line with radius r₀ = 8 m was set in the computational domain.

Figure 10. Numerical model setup for the metamaterial case.

The total wave field distributions under the weak-scattering metamaterial and perfect-cloaking metamaterial are calculated. To compare the performance of the two metamaterials, the total wave field distributions corresponding to the simplified anisotropic parameters ${\tau }_h^{*}$ and ${\tau }_g^{*}$ are further computed, and the results are presented in Figure 11. It can be seen from Figure 11a,b that the total wave field of the original scatter approaches that of the small scatter when the simplified weak-scattering anisotropic parameters are adopted, and the weak-scattering metamaterial can also achieve this equivalent modulation effect. It can be further observed from Figure 11c,d that the simplified perfect cloaking anisotropic parameters can also realize cloaking of the original scatter, while a slight deviation remains relative to the ideal cloaking result. Such deviation stems from the simplification of the gravitational acceleration scale factor in the anisotropic medium. The perfect-cloaking metamaterial can also yield the cloaking effect, yet its cloaking performance is slightly degraded compared with the scheme using simplified anisotropic parameters. This is because the simplified anisotropic parameters are continuous values, whereas the metamaterial performs an equivalent approximation via layered discrete structures. When the number of metamaterial layers is sufficiently large, its performance will gradually converge to the theoretical result corresponding to the simplified anisotropic parameters.

Figure 11. Comparison of total wave field distributions of the original scatter subjected to simplified anisotropic parameters and metamaterials: (a) weak scattering by simplified anisotropic parameters; (b) Weak scattering by weak-scattering metamaterial; (c) perfect cloaking by simplified anisotropic parameters; (d) perfect cloaking by perfect-cloaking metamaterial.

The differences reflected in the above full-wave field contours are only qualitative. To quantitatively evaluate the discrepancies between the simplified anisotropic parameter scheme and the metamaterial scheme, as well as between the weak scattering and perfect cloaking cases, the scattered wave amplitude distributions and scattering attenuation ratios under various cases are calculated, as shown in Figure 12. It can be observed from Figure 12a,b that the far-field scattered waves are effectively suppressed for both the simplified weak-scattering anisotropic parameter case and the weak-scattering metamaterial case. Their scattering attenuation ratios, ε_ws = 0.22 and ε_wm = 0.25, are close to each other, demonstrating that the layered discrete metamaterial can well reproduce the weak scattering modulation effect of the simplified continuous parameters. By comparing Figure 12c,d, the case adopting simplified perfect cloaking anisotropic parameters yields a scattering attenuation ratio of ε_ps = 0.04, with an overall low scattered wave amplitude and favorable cloaking performance. In contrast, the perfect-cloaking metamaterial case gives ε_pm = 0.21, and its scattering level is higher than that of the simplified parameter scheme. This is because, for the same number of layers, perfect cloaking theory suffers from an inherent boundary singularity. Near the inner boundary, ${\tau }_h^{\theta }$ tends to infinity while ${\tau }_h^r$ approaches zero, and such extreme parameters cannot be properly approximated by the discrete layered structure of the metamaterial. By contrast, the inner boundary parameters of the weak scattering scheme are finite values, which are more amenable to equivalent realization using layered metamaterials.

Figure 12. Distributions of dimensionless scattered wave amplitude |η_s/A₀| and scattering attenuation ratio for various cases: (a) simplified weak-scattering anisotropic parameters; (b) weak-scattering metamaterial; (c) simplified perfect cloaking anisotropic parameters; (d) perfect-cloaking metamaterial.

Comparing the scattered wave amplitude distributions of the weak-scattering metamaterial in Figure 12b and the perfect-cloaking metamaterial in Figure 12d, similar spatial scattering patterns are observed, and effective attenuation of far-field scattered waves is achieved in both cases. Meanwhile, their scattering attenuation ratios show little difference. This indicates that under the constraint of finite layers, the weak-scattering metamaterial can achieve comparable scattering suppression performance to the perfect-cloaking metamaterial while reducing the metamaterial thickness and moderating the gradient of water depth-related parameters.

To quantitatively evaluate the wave field control performance of the weak-scattering metamaterial and the perfect-cloaking metamaterial, the normalized instantaneous scattered wave amplitude |η_s/A₀| at the far field (r₀ = 8 m) for the two metamaterials is presented in Figure 13. It can be observed that for the weak-scattering metamaterial, the far-field |η_s/A₀| values are all below 0.3, exhibiting a distinct multi-lobe pattern with scattering peaks occurring at azimuthal angles of approximately 0°, 120°, and 240°. For the perfect-cloaking metamaterial, the far-field |η_s/A₀| reaches its maximum value in the incident direction, approximately 0.35. The |η_s/A₀| values for both metamaterials are below 0.4, indicating comparable low scattering levels. This suggests that the suppression effect on far-field scattered waves is similar for the weak-scattering metamaterial and the perfect-cloaking metamaterial. Therefore, the weak-scattering metamaterial can reduce the vertical size of the metamaterial and mitigate the gradient of water depth variation without significantly compromising wave field control performance, thus offering greater feasibility in practical applications.

Figure 13. Distributions of the normalized instantaneous scattered wave amplitude |η_s/A₀| at the far field (r₀ = 8 m) for the two metamaterials: MM_weak denotes the weak-scattering metamaterial, and MM_cloaking denotes the perfect-cloaking metamaterial.

4. Conclusions

Cloaking in water waves using spatial-transformation metamaterials has been proved to be a potential method to protect offshore structures. To avoid the inherent parameter singularity problem in perfect cloaking, a weak-cloaking spatial relationship is proposed. A comb-type metamaterial for weak scattering of water waves is designed based on the spatial-transformation metamaterial theory, by which optimized control over the scattering characteristics of water waves is achieved. The main findings are summarized as follows,

First, a general nonlinear spatial transformation is established for both weak scattering and perfect cloaking within a unified theoretical framework. By mapping the annular region α < r < γ in the original space to β < r^∗ < γ in the transformed space, weak scattering is achieved, where an original scatter of radius β behaves equivalently to a small scatter of radius α in terms of scattering characteristics. Perfect cloaking is recovered as a special limiting case when α = 0. The corresponding anisotropic medium parameters, including the spatial scale factors τ_h and τ_g, are derived from the coordinate transformation.

Subsequently, the anisotropic media for weak scattering and perfect cloaking are analyzed and compared. In both cases, τ_g is found to be a constant, with only a small difference between them. The radial component ${\tau }_h^r$ and the azimuthal component ${\tau }_h^{\theta }$ of τ_h exhibit the same variation trends along the radial direction, with ${\tau }_h^r$ increasing monotonically and ${\tau }_h^{\theta }$ decreasing monotonically. However, notable differences are observed at the inner boundary r = β. For perfect cloaking, ${\tau }_h^r$ = 0 and ${\tau }_h^{\theta }$ tends to infinity, indicating a severe parameter singularity. In contrast, for weak scattering, both parameters remain finite. This demonstrates that the weak scattering design effectively resolves the singularity problem inherent in perfect cloaking. The wave field distributions under the anisotropic media are further evaluated by solving the Helmholtz equation via the finite element method. The results show that the perfect-cloaking anisotropic medium successfully achieves cloaking of the scatter in water waves, while the weak-scattering anisotropic medium effectively makes the total wave field of the original scatter equivalent to that of the small scatter.

Finally, a comb-type weak-scattering metamaterial is designed using the uniform effective medium method to realize the equivalent anisotropic medium. The water depth ratios τ_a(i) and τ_b(i) for each layer of the two metamaterials are calculated and compared. The innermost water depth ratio of the weak-scattering metamaterial is reduced by more than 50% compared with that of the perfect-cloaking metamaterial. Numerical results for the metamaterial cases demonstrate that the scattering suppression performance of the weak-scattering metamaterial is similar to that of the perfect-cloaking metamaterial. The weak-scattering metamaterial can reduce the vertical size of the metamaterial and mitigate the gradient of water depth variation without significantly compromising wave field control performance, thus promoting the feasibility of the cloaking protection technology.

In summary, a metamaterial that enables weak scattering of water waves around a structure is designed based on the spatial-transformation metamaterial method. The proposed design effectively suppresses scattering while resolving the parameter singularity problem inherent in perfect cloaking. The significant reduction in the required water depth ratios and depth gradients near the structure promotes the feasibility of spatial cloaking metamaterials. In the future, other parameters such as cloaking area and layer thickness can be further studied to reduce the manufacturing difficulty of metamaterials. In addition, experiments will be conducted to validate the proposed weak-cloaking metamaterial.