Spectral Stability of Traveling Water Waves: Analytic Dependence of the Spectrum

The motion of the free surface of an ideal fluid under the effects of gravity and capillarity arises in a number of problems of practical interest (e.g. open-ocean pollutant transport, deep-sea oil platform design, and the generation and propagation of tsunamis), and, consequently, the reliable and accurate numerical simulation of these “water waves” is of central importance. In a pair of recent papers the author, in collaboration with F. Reitich (Proc. Roy. Soc. Lond., A, 461(2057): 1283–1309 (2005); Euro. J. Mech. B/Fluids, 25(4): 406–424 (2006)), has developed a new, efficient, stable and highorder Boundary Perturbation scheme (the method of Transformed Field Expansions) for the robust numerical simulation of traveling solutions of the water wave equations. In this paper we extend this Boundary Perturbation technique to address the equally important topic of dynamic stability of these traveling wave forms. More specifically, we describe, and provide the theoretical justification for, a new numerical algorithm to compute the spectrum of the linearized water-wave problem as a function of a parameter, ε, meant to measure the amplitude of the traveling wave. In order to demonstrate the utility of this new method, we also present a sample calculation for two-dimensional waves in water of infinite depth subject to quite general two-dimensional perturbations.