We study the evolution of small-amplitude water waves when the fluid m
otion is three dimensional. An isotropic pseudodifferential equation t
hat governs the evolution of the free surface of a fluid with arbitrar
y, uniform depth is derived. It is shown to reduce to the Benney-Luke
equation, the Korteweg-de Vries (KdV) equation, the Kadomtsev-Petviash
vili (KP) equation, and to the nonlinear shallow water theory in the a
ppropriate limits. We compute, numerically, doubly periodic solutions
to this equation. In the weakly two-dimensional long wave limit, the c
omputed patterns and nonlinear dispersion relations agree well with th
ose of the doubly periodic theta function solutions to the KP equation
. These solutions correspond to traveling hexagonal wave patterns, and
they have been compared with experimental measurements by Hammack, Sc
heffner, and Segur. In the fully two-dimensional long wave case, the s
olutions deviate considerably from those of KP, indicating the limitat
ion of that equation. In the finite depth case, both resonant and nonr
esonant traveling wave patterns are obtained.