Irrotational Deep Hydroelastic Waves

• The paper demonstrates a detailed derivation of the linear dispersion relation that captures the interplay between gravity, tension, and bending stiffness.
• It employs potential flow analysis, spectral discretization, and bifurcation theory to analyze nonlinear traveling wave solutions and well-posedness in low regularity regimes.
• Experimental validations confirm theoretical predictions by matching phase velocity minima and wave pattern formations across varying elastic and tension parameters.

Irrotational deep hydroelastic waves are dispersive wave phenomena that arise at the interface between a deep, incompressible, and inviscid fluid and an overlying thin elastic sheet, usually of negligible inertia. These systems are characterized by the interplay between fluid hydrodynamics (governed by irrotational flow and potential theory) and the out-of-plane elastic response (controlled by bending and in-plane stretching of the sheet). The canonical setting assumes infinite fluid depth ("deep water") and is amenable to linear and nonlinear analyses. This topic underlies a range of physical scenarios, from the wake patterns of floating elastic sheets to the existence and dynamics of periodic nonlinear traveling waves and the mathematical theory of quasilinear dispersive PDEs with nontrivial geometric free boundaries.

1. Governing Equations and Physical Setup

The configuration consists of an incompressible, inviscid, irrotational fluid occupying the half-space beneath a deformable elastic sheet (or interface) parametrized by displacement $\zeta(x,y,t)$. The fluid is described by a velocity potential $\phi(x,y,z,t)$, satisfying the Laplace equation $\nabla^2\phi=0$ for $z<\zeta(x,y,t)$ and subject to a no-flux condition as $z\to -\infty$ (deep water) (Ono-dit-Biot et al., 2018, Wan et al., 26 Dec 2025). The deformation of the elastic sheet obeys, in the linearized regime, the Föppl–von Kármán constitutive relations where the sheet exerts both bending (proportional to $B\nabla_{x,y}^4\zeta$) and stretching or surface tension forces (proportional to $-T\nabla_{x,y}^2\zeta$) on the underlying fluid. The dynamic boundary condition at the fluid–sheet interface couples the fluid pressure (via a linearized Bernoulli equation) to these elastic restoring forces.

For two-fluid setups (e.g., with densities $\rho_1$ below and $\rho_2$ above the interface), the elastic boundary is described parametrically, and the pressure jump across the sheet includes contributions from both bending stiffness and possible inertia per unit length $m_e$ (Akers et al., 2017). The essential assumption throughout is irrotationality in the fluid domain (existence of a potential).

2. Linear Dispersion and Limiting Regimes

Assuming small-amplitude oscillations ($|\zeta|\ll\lambda$), the combined fluid-elastic system admits plane wave solutions. The closed-form linear dispersion relation for irrotational deep hydroelastic waves on a thin elastic sheet is (Ono-dit-Biot et al., 2018): $$\omega^2(k) = gk + \frac{T}{\rho}k^3 + \frac{B}{\rho}k^5$$ where $g$ is gravity, $\rho$ the fluid density, $T$ the in-plane tension, $B=Eh^3/[12(1-\nu^2)]$ the bending modulus, and $k$ the wavenumber.

Distinct asymptotic regimes emerge:

• Pure gravity waves ($B=T=0$): $\omega^2=gk$, recovering classical deep-water gravity wave motion.
• Tension-dominated (capillary-analog) waves ($B=0$): $\omega^2=gk + (T/\rho)k^3$; for short wavelengths, tension dominates and $c\sim k^{1/2}$.
• Bending-dominated (flexural) waves ($T=0$): $\omega^2=gk + (B/\rho)k^5$; at short wavelengths, $\omega^2\approx (B/\rho)k^5$ and $c\sim k^{3/2}$.

The minimum of the phase velocity curve $c(k)=\omega/k$ sets a threshold speed $v^*$ below which no steady waves (wake) can form, analogous to the classic Kelvin threshold (Ono-dit-Biot et al., 2018).

3. Nonlinear and Interfacial Wave Formulations

Beyond the linear regime, a broad class of hydroelastic waves involves periodic, traveling, or even multi-valued interface solutions. When considering two immiscible fluids separated by an elastic sheet, the interface motion is described by a pair of potential flows $\phi_1$, $\phi_2$, with kinematic and dynamic boundary conditions on $y=\eta(x,t)$. The elastic pressure includes both bending stiffness and tension, and, if present, the inertia per unit length modifies only the higher-order nonlinear terms (Akers et al., 2017).

The traveling wave problem is naturally cast in an arclength-parametrized formulation:

• The interface $z(\alpha)=x(\alpha)+iy(\alpha)$, with tangent angle $\theta(\alpha)$ and curvature $\kappa=\theta_\alpha/|z_\alpha|$,
• The system reduces to a nonlocal, pseudo-differential set of equations involving the Birkhoff–Rott integral for vortex sheet strength and a Bernoulli-elastic balance,
• The global bifurcation approach (via a “identity plus compact” mapping in functional-analytic framework) ensures the existence of continua of periodic traveling wave solutions for each integer wavenumber.

4. Bifurcation Structure and Branch Classification

The global bifurcation analysis, as in Akers–Ambrose–Sulon (Akers et al., 2017), demonstrates that for given physical parameters (bending stiffness $S$, tension $L^s$, density difference $A$, mass $M_e$), families of periodic traveling wave solutions emerge. For each wavenumber $k$, the linearization about the flat state yields critical speeds $c_{\pm}(k)$ where nontrivial branches emanate: $c_\pm(k) = A \pm \sqrt{A^2 + 2S k^4 + \cdots}$ The corresponding solution branches $C_\pm(k)$ can:

• Grow unboundedly (amplitude diverges),
• Reconnect to the trivial (flat) state at a different wave speed,
• Terminate via self-intersection or loss of regularity (overturning sheets).

The mass $M_e$ does not influence the linearized bifurcation points $c_\pm(k)$, but crucially affects the global structure of solution branches; for $M_e=0$, unbounded amplitude or self-intersecting branches are possible, while for sufficiently large $M_e$, reconnection to another trivial solution or limitless growth may occur (Akers et al., 2017).

Numerical continuation of these branches employs Fourier spectral discretizations and Broyden iteration, with diagnostic quantities such as total displacement and critical scenarios tabulated (see table below).

Point	Speed $c$	Displacement $h$	Scenario
1	1.231	0.12	Small-amplitude
2	1.120	0.85	Moderate wave
3	1.057	2.37	Near self-intersection

For the representative parameters $k=2$, $S=0.5$, $A=0.8$, $M_e=0.1$, the branch leads to self-intersection at point 3 (Akers et al., 2017).

5. Analytical Well-posedness in Low Regularity Regimes

Recent advances address well-posedness and regularity of the Cauchy problem for deep hydroelastic waves, notably in the presence of low regularity initial data (Wan et al., 26 Dec 2025). The physical system is formulated as a free-boundary Euler–elastic PDE, which admits a reduction (following Zakharov–Craig–Sulem) to a coupled dispersive system for the free boundary displacement $\eta$ and velocity potential $\psi$ restricted to the interface: $$\begin{aligned} &\eta_t = G(\eta)\psi \ &\psi_t + \tfrac{1}{2}\psi_x^2 - \tfrac{1}{2}\frac{(\eta_x\psi_x + G\psi)^2}{1+\eta_x^2} + \sigma E(\eta) = 0 \end{aligned}$$ where $G(\eta)$ is the Dirichlet–Neumann operator and $E(\eta)$ the geometric elastic pressure.

By employing a paradifferential (Bony) calculus and constructing a cubic modified energy tailored to the quasilinear structure and non-resonant cubic contributions, local well-posedness is established for initial data in

$$\mathcal{H}^s = H^{s+3/2}(\mathbb{R}_x)\times H^s(\mathbb{R}_x),\text{ for } s > 3/4$$

This regularity threshold is sharp relative to prior work and relies critically on a paradifferential diagonalization and quartic corrections to the energy, capitalizing on the non-resonant character of the $|\xi|^{5/2}$ dispersion (Wan et al., 26 Dec 2025).

6. Experimental Validation and Parameter Measurements

Direct experimental investigations, as in Ono-dit-Biot et al., examine hydroelastic wakes by towing a localized perturbation at controlled velocities along floating thin elastic sheets (Ono-dit-Biot et al., 2018). The bending modulus $B$ and tension $T$ are independently measured, enabling precise comparison between observed wave patterns (e.g., wavelength selection, phase velocity minima) and the theoretical dispersion relation. The observed agreement across a range of $B$ and $T$ values confirms the robustness of the theoretical framework and validates the prediction of a minimal velocity threshold for steady wake generation. This threshold distinguishes regimes where only dynamically forced, non-steady waves are possible (Ono-dit-Biot et al., 2018).

7. Assumptions, Limitations, and Context

The established results and experimental agreements are predicated on the following assumptions:

• Deep-water regime ($h_\text{fluid}\gg\lambda$),
• Linearity (small amplitude, $|\zeta|\ll\lambda$) except where nonlinear or global bifurcation analyses are explicitly conducted,
• Invincid, incompressible, irrotational fluid conditions,
• Homogeneous, isotropic, and thin elastic sheets with negligible inertia (except as a parameter in nonlinear extensions),
• No mean flow other than that induced by the deforming interface,
• Neglect of viscosity in both fluid and sheet,

The inclusion of more complex rheologies, finite-thickness effects, or viscosity alters the system fundamentally and is outside the scope of the cited results. The current theory unifies and generalizes several classical dispersive wave systems (gravity-capillary, pure flexural waves), positioning hydroelastic waves as a widely applicable model in geophysics, engineering, and mathematical analysis of dispersive free-boundary problems (Ono-dit-Biot et al., 2018, Akers et al., 2017, Wan et al., 26 Dec 2025).