  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.
