Recall the definition of pressure coefficient for incompressible flows

When the flow is compressible, it makes more sense to use  as the normalizing parameter. Thus c_p can be written as,

(show this for homework, and also define limiting values, and the corresponding flow conditions)

Let us derive a linearized form of c_p, to use along with out small-perturbation equations. From the isentropic flow relations,

From the energy equation,
; where 

or,  where 

The second term on the RHS is << 1. Thus,
is of the form   where 

This can be expanded using the Binomial theorem:

as:
smaller terms
where 
 

or, for 
 

Thus, 

This is valid for a 2-D or 3-D flow over flat wings.