| One of the important open questions in the theory of free surface ideal fluid flows is the dynamic stability of traveling wave solutions. In a spectral stability analysis, the first variation of the governing Euler equations is required, which raises both theoretical and numerical issues, With Zakharov's 1968 and Craig & Sulem's 1993 formulation of the Euler equations in mind, we address the question of analyticity properties of first, and higher, variations of the Dirichlet-Neumann operator. This analysis will have consequences not only for theoretical investigations, but also for numerical simulations of spectral stability of traveling enter Waves.;We present first the main theoretical result regarding analyticity of the first variation of the Dirichlet-Neumann Operator. Following this, we present additional results regarding analyticity of all higher variations as well. After a presentation of the details of the simulation, we give the outcome of some computational experiments. In conclusion, we describe several advantages and disadvantages pertaining to the methodology used, and suggest future applications of these results. |
