an inviscid fluid. A periodic wave train with fundamental frequency moves obliquely,

with representing angular direction measured with respect to the positive x axis. The

harmonics have phases where Small disturbances in the form

of sidebands with frequencies are introduced and, over time, energy is transferred

from the primary wave to the sidebands. Using methods introduced by Benjamin and Feir

for two-dimensional motion, we show that the magnification will become unbounded if the

sideband frequencies meet the condition

where a denotes the fundamental amplitude of the perturbed wave train. The limiting

cases and yield the two dimensional Benjamin-Feir instability. Indeed, if

we write then this condition for instability may be expressed as