Three-dimensional gravity-capillary standing waves: computation, resonance and instability

TL;DR

This paper develops a numerical framework for analyzing three-dimensional gravity-capillary standing waves, exploring their bifurcations, resonance patterns, and instabilities with high accuracy and efficiency.

Contribution

It introduces a novel spatio-temporal collocation method that avoids numerical stiffness and instability, enabling detailed study of complex 3D standing wave patterns.

Findings

Excellent agreement with existing models and full potential-flow results.

Identification of bifurcations leading to square, hexagonal, and flower-like patterns.

Revealed an instability mechanism related to oblique instability in 3D waves.

Abstract

We present a numerical study of three-dimensional gravity-capillary standing waves by using cubic and quintic truncated Hamiltonian formulations and the Craig-Sulem expansion of the Dirichlet-Neumann operator (DNO). The resulting models are treated as triply periodic boundary-value problems and solved via a spatio-temporal collocation method without executing initial-value calculations. This approach avoids the numerical stiffness associated with surface tension and numerical instabilities arising from time integration. We reduce the number of unknowns significantly by exploiting the spatio-temporal symmetries for three types of symmetric standing waves. Comparisons with existing asymptotic and numerical results illustrate excellent agreement between the models and the full potential-flow formulation. We investigate typical bifurcations and standing waves that feature square, hexagonal, and more complex flower-like patterns under the three-wave resonance. These solutions are generalisations of the classical Wilton ripples. Temporal simulations of the computed three-dimensional standing waves exhibit perfect periodicity and reveal an instability mechanism based on the reported oblique instability in two-dimensional standing waves.