A considerable simplification and an added flexibility in the analysis
occurs if one neglects the vertical acceleration. This is
justifiable for gravity modes if
, and is certainly justifiable for
quasi-geostrophic (Rossby) modes for
which 
(i.e. subinertial modes). As in
the previous analyses, we ignore compressibility and employ
the traditional approximation for the Coriolis terms. However,
we will now remove the restriction of a rigid lid and thus
allow surface (barotropic) gravity modes.
Finally, we treat the sea bed as level (D = constant).
The hydrostatic approximation (i.e.
( 
)
and the restriction to constant D allow us the completely
separate the z dependency of admissible modes from the
x, y, t dependency.
Stated another way, the equations of motion for the linear,
undamped, free disturbances are converted into a set of
linearly independent equations controlling the x, y, t
dependency of each of the admissible vertical modes.
Thus the equations for which solutions are to be sought are
plus the boundary conditions
Here 
and the subscript 1 is dropped for
simplicity on all other variables.