As before, for the perturbed state take 
, 
, and 
where 
, 
, 
now
depend on x, y, z, but subject to the constraints (5.1)
to (5.3).
Then
are the linear approximations where terms like 
are neglected.
Eq. (4.4) can be written as
Using (5.2), (5.4c) and neglecting 
gives
the linear perturbation equation
or using (4.11) this becomes
of which (4.18) is a special case when 
,
bearing in mind (4.13), (4.14) and that 
for .
Using (5.4a,b), the linear approximation of (4.5) is
Since 
is assumed independent of time and that its
spatial variation in the direction of is negligible
by (5.1), then (5.8) can be written as
using definitions (2.31) and (4.11) and noting that
.
Bearing in mind that 
by (2.35),
we note that for , 
must be vertical
(i.e., , depend only on z) and (5.9)
reduces to (4.10) as a special case.
Finally, using (5.3) and (5.4a), the linear perturbation form of (4.6) is
of which (4.19) is a special case for .
It will be convenient in this analysis to define the following quantities
The last of these is equivalent to (4.15). For ,
the vectors 
and 
(which are non-dimensional) both
reduce to the unit vector 
.
In general, for 
, , and
are no longer parallel, however the inclination of and
from the vertical is small for typical basic state conditions and the
magnitudes of and are nearly unity.
With the above definitions, (5.7), (5.9) and (5.10) then take
the final form
4= 5.75truein
which reduce to (4.18) to (4.20) for .