WaterWave: Advances in Fluid Wave Control

• WaterWave is a multidisciplinary field that defines the control and manipulation of water wave phenomena through engineered seabed properties and metamaterials.
• Broadband wall-less waveguides and double-negative metamaterials are leveraged to confine waves, achieve negative refraction, and improve energy focusing.
• Transformation aquatics and advanced computational models enable precise simulations for applications in energy, coastal protection, and microfluidic systems.

WaterWave is a term encompassing both theoretical, computational, and applied advances in the physics and technology of water waves. It designates not only physical wave phenomena in fluids, but also modern implementations for controlling, confining, and manipulating water waves for energy, coastal protection, microfluidics, and computational purposes. Architectures such as broadband wall-less water-wave waveguides, negative-index metamaterials, transformation aquatics, and optimized wave resonators all fall under WaterWave as recent research-driven technologies.

1. Broadband Wall-less Waveguide Principles

WaterWave architectures enable the confinement of shallow water waves without rigid sidewalls by engineering the viscoelastic properties of the seabed. In the canonical model, an incompressible and irrotational fluid overlays a bottom carpet of continuous springs (stiffness per unit area $k^*$) and dashpots (damping per unit area $b^*$) (Zareei et al., 2018). The governing equations are:

• Laplace’s equation in the fluid: $\nabla^2\phi = 0$ for $-h \leq z \leq 0$
• Free-surface kinematic and dynamic BCs at $z=0$:

$\eta_t = \phi_z$, $\phi_t + g \eta = 0$

• Bottom ($z=-h$) kinematic: $\zeta_t = \phi_z$
• Viscoelastic carpet dynamic: $$\phi_t + g\zeta + \frac{b^*}{\rho}\zeta_t + \frac{k^*}{\rho}\zeta = 0$$

The key result from shallow-water expansion ($kh \ll 1$) is renormalization of gravity:

$$\tilde{g}(\omega) = g \left[ 1 + \frac{\rho g}{k^* + i b^* \omega} \right]$$

The dispersion relation becomes $\omega^2 = \tilde{g} h k^2$. By spatially varying $k^*, b^*$, one engineers local refractive index, enabling a graded-index (GRIN) core with strong wave confinement analogous to optical fibers. This mechanism yields broadband operation, as confinement is not frequency-selective.

2. Metamaterial WaterWave Control via Double Negative Media

WaterWave metamaterials can be fabricated as gear-and-split-tube arrays yielding regions with negative effective gravity ($g_e < 0$) and negative effective depth ($u_e < 0$), realizing Veselago-Pendry "double negative" media (Ge et al., 7 Aug 2025). In the deep-water regime, the effective dispersive wave equation is:

$\omega^2 = g_e k_e \tanh(k_e h_e)$

The Coherent Potential Approximation (CPA) links the geometry (split width $A$, radii $R_i$, teeth $N_i$, splits $N_o$) to $g_e$, $u_e$ via explicit algebraic relations:

$$u_e/u_0 = \frac{1 - S D_1}{1 + S D_1},\quad g_e/g_0 = \frac{1 + S D_1}{1 - S D_1}$$

with $S=4/(i \pi a^2 k^2)$ and $D_1$ (mode-dependent scattering coefficient).

These metamaterial slabs produce tunable omnidirectional bandgaps, all-angle negative refraction, and wave isolation. Experimentally, wave fronts bend by measured angles (e.g., $45^\circ$), closely matching numerical predictions. Applications include harbor calming, energy harvesting via negative refraction concentration, and riverbank erosion prevention with precision geometry-to-wave mapping.

3. Transformation Aquatics and Maxwell's Fishpond

Transformation aquatics generalizes optical transformation devices (Maxwell's Fisheye) to water waves by mapping spherical wave propagation onto planar disks with tailored depth profiles (Kinsler et al., 2012). The Maxwell Fishpond uses a depth profile $h(r) = h_0 [1 + (r/r_0)^2]^2$, inducing a radially varying wave speed

$c_w(r) = \sqrt{g h(r)}$

which replicates the geodesic focusing of waves on a sphere; pulses launched from one point reconverge at the antipode. Experiments reveal up to five repeated refocusing events, with refocus times and amplitude persistence matching theoretical predictions (corrections for surface tension, viscosity included).

This paradigm extends to "Eaton" and "Luneburg" analogues and demonstrates the ability to refocus or steer energy in highly controllable fashions.

4. Topological Manipulation and Particle Control

Structured interference of water-wave fields enables generation of topological wave objects—vortices, skyrmions, polarization Möbius strips—that exert mechanical forces and torques on floating particles (analogous to optical tweezers) (Wang et al., 11 Jun 2024). The chief mechanisms are:

• Gradient force: $$\mathbf{F}_\text{grad} = \operatorname{Re}\alpha \nabla W$$, trapping at field maxima
• Radiation-pressure force: $$\mathbf{F}_\text{press} = \omega \operatorname{Im}\alpha \mathbf{P}$$, driving azimuthal motion
• Spin torque: $T_\text{spin} = \omega \operatorname{Im}\alpha S_z$, inducing rotation

Experimental implementations show stable trapping, controlled orbiting, and spinning for particles from subwavelength ($a\sim$ mm) to macroscopic scales, programmable via the topological charge of structured waves.

5. Computational WaterWave Modeling in Complex Environments

Exact simulation of water waves in arbitrary transient environments exploits conformal mapping methods which transform the physical fluid domain to canonical computational strips, avoiding series expansions and minimizing numerical diffusion (Akselsen, 12 Feb 2025). The mapping is double-layered:

• Prescribed outer map: geometry/boundaries fixed analytically
• Dynamic inner map: propagates free surface motion, exact up to machine precision

The method supports static/moving boundaries, overhangs, bathymetric variation, and wavemaker dynamics, with fast spectral (FFT-based) time stepping (typically seven FFTs/step). Validation against high-resolution FEM and HOS benchmarks shows phase velocities and amplitudes accurate to five significant digits, robust under strong nonlinearity and rapid bathymetric change.

6. Applications in Energy, Protection, and Fluidics

WaterWave technologies enable a spectrum of applied utilities:

• Wave energy concentration: GRIN waveguides and double-negative slabs reduce transverse spread and boost local energy density, multiplying harvesting efficiency (up to $4\times$ amplitude retention over 30 wavelengths) (Zareei et al., 2018, Ge et al., 7 Aug 2025).
• Shoreline protection: Waveguide carpets and metamaterial barriers redirect storm energy, lowering runup and erosion by significant factors (e.g., $\sim0.64$ attenuation in example geometries).
• Microfluidics and sorting: Topologically structured fields sort, trap, and transport particles in centimeter-to-micron scale processes, potentially enabling hydrodynamic tweezers and automated sorting by wave topology.
• Wave focusing and imaging: Negative refraction and transformation aquatics enable subwavelength resolution ($\delta \approx 2.1a$ for $\lambda \approx 5.4a$), critical for imaging and sensing operations.

7. Limitations, Optimization, and Outlook

While WaterWave systems provide extended functionality, their performance is constrained by:

• Bandwidth: Wall-less waveguides are limited to $kh \ll 1$ regime; metamaterial bandgaps set operational windows.
• Nonlinear effects: For large amplitude or dense arrangements (e.g., floating disk arrays), non-linear dissipation (overwash, collision, rafting) must be modeled for accuracy (Bennetts et al., 2014).
• Viscosity and finite-depth corrections: Real fluid properties and boundary-layer effects may alter dispersion and damping, requiring calibration in design and optimization (Wang et al., 11 Jun 2024).

Advances in implicit neural representation and wavelet-based analysis further optimize temporal consistency and restoration in computational WaterWave applications, notably in underwater video stream enhancement (Zhu et al., 5 Dec 2025).

WaterWave, as a research term, thus synthesizes control, confinement, manipulation, and simulation of water-waves across scales, leveraging seabed viscoelasticity, metamaterial engineering, transformation maps, and computational innovation to deliver tunable, broadband, and precise wave management for scientific, industrial, and environmental domains.