The classic water wave problem consists of finding the irrotational evolution of an ideal fluid with a free interface, subject to gravity. Because of the free interface, the classic water wave equations are nonlinear, and are therefore difficult to analyze theoretically and numerically. Recently, a reformulation of water waves was given in terms of a new nonlocal equation that connects the free interface and the velocity potential evaluated on the free interface. This integral equation, together with the well-known Bernoulli equation expressed in terms of the velocity potential evaluated on the free interface, serve to replace the standard water wave equations. Because the nonlocal equation depends on a free spectral parameter, it is referred to in this thesis as the nonlocal spectral (NSP) equation. One notable advantage of this reformulation is that the vertical coordinate is eliminated, thus reducing the dimensionality of the problem and fixing the domain in which the equations are posed.

In this thesis, the classic water wave problem is investigated using the above NSP formulation. A generalized NSP formulation is also developed for two fluids separated by a free interface and bounded by either a rigid lid or free surface.

In the first part of this thesis, the connection between the NSP formulation and the classic water wave equations is analyzed. Of particular interest, from the nonlocal equation the series expansion for the Dirichlet-Neumann operator associated with the classic water wave equations is obtained. The derivation demonstrates that the NSP formulation captures the kinematic conditions governing classic water waves (that is, the nonlocal equation is equivalent to the classic water wave equations without the Bernoulli equation).

With the NSP formulation on a solid theoretical foundation, an NSP formulation is developed for two ideal fluids separated by a free interface and bounded above by either a free or surface or rigid lid. As in classic water waves, the equivalence of the NSP and classic two-fluid formulations is demonstrated. From the two-fluid NSP formulation, asymptotic reductions of stratified fluids are obtained, including coupled nonlinear Schrodinger (NLS) equations and a (2+1)-dimensional generalization of the intermediate long wave (ILW) equation. Numerical investigation of the (2+1)-dimensional ILW equation indicates that it is close to being integrable. From the two-fluid NSP formulation, nontrivial conservation laws for the two-fluid problem are also derived.