Let 
be the elevation of the sea-air
interface above mean water level at location x, y at
time t.
We will suppose that there exists no mass transfer across the interface (i.e, no net evaporation, for example). The kinematic condition is then a statement that a fluid parcel on the surface stays on the surface at all times:
Expanding this in Cartesian form gives
where u, v, w are understood to be evaluated
at z = 
.
This is a non-linear condition because of the second and
third terms and because w as well and u, v
apply at which is a dependent variable.
If we expand u, v, w about z = 0 by Taylor's
series (3.3) becomes
The linearized version, which may be justifiable for suitably
small , and if u, v, w have magnitudes
linearly dependent on , is simply
