Let
where the perturbations 
, 
, 
, 
are
suitably small (e.g., 
, 
, 
where 
is a
characteristic group speed of the disturbances).
Consider DP/Dt:
if terms of second order like 
etc. are neglected.
If 
is approximated by 
in (4.6)
then this equation becomes
to first order in the perturbations. Similarly (4.5) can be approximated by
to first order in the perturbations. Using (4.9) this becomes
or
where
which represents a perturbation of potential specific volume.
This follows from the definition
which implies
It also follows that if 
is treated as a constant
that
Equation (4.10) then takes the form
where
which is an approximation of the vertical displacement of
a surface of constant 
from its equilibrium
level in the basic state.
Finally the linearized version of the equation of motion
(4.4), taking into account that 
is
The perturbation in terms of is
simply 
, or using (4.11)
or using (4.13) and (4.15)
In summary (4.16), (4.9) and (4.14) yield the linear homogeneous set:
We can define a matrix operator for the self-adjoint Eqs. (4.18) to (4.20):
