Linear theory

Recall that for isentropic flow over a corner (Prandtl-Meyer Expansion), we obtained:

, 

Now we can argue that for small angles of compression, only weak oblique shocks will be formed, and the entropy increase is minimal. In fact, it can be shown that , so that  is small for small . Therefore, the relations for isentropic flow should hold for weak compressions as well.

Using the expression for dp, we can write, for small changes,

or, 
 
 

Sign of c_p
Note that  by definition should be positive when , i.e. the pressure is higher than the freestream static pressure (compression). This is easier to keep in mind than any sign convention for .
 
 

Lift Coefficient
For small,  where  is the equation to the airfoil surface.

where  is pressure on the lower surface and  is pressure on the upper surface.

Note that  where  is the distance along the chord and is the slope with respect to .

Given the equations to the upper and lower surfaces  and  we can find .

Example 1

Flat plate at angle of attack a .
Upper surface: 

Lower surface: 

(How do we figure out whether it should be +a or -a ? Think about the sign of c_p. The upper surface is an expansion surface; Þ negative c_p while the lower surface is a compression surface Þ positive c_p)

This is a very useful result. In fact, this is valid for relatively large values of a as well. Note that  for airfoils in incompressible flow. Do not attempt to use the above formula for airfoils in subsonic flow either.
 
 

Example 2

Airfoil with thickness at angle of attack.
Upper surface is described by 
Lower surface is described by 

At angle of attack, upper surface slope,
At angle of attack lower surface slope, 

At  =0 and 1,  (leading edge and trailing edge)
So,

Astonishing result! Neither thickness now camber produced any lift in supersonic flow.