Stability of periodic surface gravity water waves

Euler's equations describe the dynamics of gravity waves on the surface of an ideal fluid with arbitrary depth. In this dissertation, we discuss the stability of traveling wave solutions to the full set of nonlinear equations via a non-local formulation of the water wave problem in two-dimensions. We demonstrate that this new non-local formulation is equivalent to the original formulation of Euler's equations using an extension of the results of Ablowitz and Haut. Transforming the non-local formulation into a traveling coordinate frame, we obtain a new equation for the stationary solutions in the traveling reference frame as a single equation for the surface in physical coordinates. Using this new equation, we develop a numerical scheme to determine traveling wave solutions by exploiting the bifurcation structure of the non-trivial periodic solutions. Finally, we numerically determine the spectral stability for the periodic traveling wave solution by extending Fourier-Floquet analysis to apply to the non-local problem. The full spectra for various traveling wave solutions is generated. In addition to recovering past well-known results such as the Benjamin-Feir instability for deep water, the presence of high-frequency instabilities in shallow water is confirmed. Additionally, new instability regions are found that appear unpublished in the literature.