The instability of Wilton ripples

• Numerical solutions to Euler’s equations for periodic gravity–capillary waves.
• A variant of the boundary integral method for traveling wave solutions is introduced.
• Stability of Wilton ripple solutions to Euler’s equations is examined.
• New instabilities are present due to the resonance condition being satisfied.

Wilton ripples are a type of periodic traveling wave solution of the full water wave problem incorporating the effects of surface tension. They are characterized by a resonance phenomenon that alters the order at which the resonant harmonic mode enters in a perturbation expansion. We compute such solutions using non-perturbative numerical methods and investigate their stability by examining the spectrum of the water wave problem linearized about the resonant traveling wave. Instabilities are observed that differ from any previously found in the context of the water wave problem.