Stability of Fully Nonlinear Stokes Waves on Deep Water: Part 1. Perturbation Theory

We consider a full set of harmonics for the Stokes wave in deep water in the absence of viscosity, and examine the role that higher harmonics play in modifying the classical Benjamin-Feir instability. Using a representation of the wave coefficients due to Wilton, a perturbation analysis shows that the Stokes wave may become unbounded due to interactions between the $N^{th}$ harmonic of the primary wave train and a set of harmonics of a disturbance. If the frequency of the $n^{th}$ harmonic is denoted $\omega _{n} =\omega \left( {1 \pm \delta } \right)$ then instability will occur if $$0<\delta <\frac{\sqrt 2\ k\,n^ns_n }{\left( {n-1} \right)!}$$ subject to the disturbance initially having sufficiently large amplitude. We show that, subject to initial conditions, all lower harmonics will contribute to instability as well, and we identify the frequency of the disturbance corresponding to maximum growth rate.