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

Authors

Shahrdad Sajjadi, Embry-Riddle Aeronautical UniversityFollow
David L. Ross, Division of Geophysics and Planetary Physics, CAMAC

Submitting Campus

Daytona Beach

Department

Physical Sciences

Document Type

Article

Publication/Presentation Date

4-2010

Abstract/Description

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 Ꙍ_n = Ꙍ(1 ± ꭉ) then instability will occur if

√2 k nⁿ s_n

0 < ꭉ < (n -1) !

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.

Publication Title

Advances and Applications in Fluid Mechanics

Publisher

Pushpa Publishing

Scholarly Commons Citation

Sajjadi, S., & Ross, D. L. (2010). Stability of Fully Nonlinear Stokes Waves on Deep Water: Part 1. Perturbation Theory. Advances and Applications in Fluid Mechanics, 2(7). Retrieved from https://commons.erau.edu/publication/1015