Isentropic expansion waves

Isentropic expansion waves are created when a supersonic flow is redirected along a curved surface. These waves are studied to obtain a relation between deflection angle and Mach number. Each wave in this case is a Mach wave, so it is at an angle formula_1 , where M is the Mach number immediately before the wave. Expansion waves are divergent because as the flow expands the value of Mach number increases, thereby decreasing the Mach angle.
In an isentropic wave, the speed changes from formula_2 to formula_3, with deflection formula_4. We have oriented the coordinate system orthogonal to the wave. We write the basic (continuity, momentum and 1st, 2nd law of thermodynamics) equations for this infinitesimal control volume.
Assumptions:
Relation between θ,M and v.
The continuity equation is,
formula_5 ...(1.1)
First term is zero by assumption(1).
 Now, formula_6
 or formula_7... (1.2)
Now we consider the momentum equation for normal and tangential to shock.
For formula_8- component, 
 formula_9...(1.3)
Second term of L.H.S and first term of R.H.S are zero due to assumption (2) and (1) respectively.
Then,
 formula_10
Or using continuity,
 formula_11
Expanding and simplifying [Using the facts that, to the first order, in the limit as formula_12, formula_13 and formula_14], we obtain
 formula_15
 But, formula_16,
 so, formula_17,
 and
 formula_18 ...(1.4)
Derivation of Prandtl-Meyer supersonic expansion function.
We skip the analysis of the formula_19-component of the momentum and move on to the first law of thermodynamics, which is
 formula_20 ...(1.5)
First term of L.H.S,next three terms of L.H.S and first term of R.H.S are zero due to assumption (3),(4) and (1) respectively.
where,
 formula_21
For our control volume we obtain
 formula_22
This may be simplified as
 formula_23
Expanding and simplifying in the limit to first order, we get
 formula_24
If we confine to ideal gases, formula_25 , so
 formula_26 ...(1.6)
Above equation relates the differential changes in velocity and temperature. We can derive a relation between formula_27 and formula_2 using formula_29. Differentiating (and dividing the left hand side by formula_2 and the right by formula_31 ),
 formula_32
Using equation 1.6
 formula_33
 formula_34
Hence,
 formula_35 ...(1.7)
Combining (1.4) and (1.7)
 formula_36 ...(1.8)
We generally apply the above equation to negative formula_37, let formula_38. We can integrate this between the initial and final Mach numbers of given flow, but it will be more convenient to integrate from a reference state, the critical speed (formula_39) to Mach number formula_27, with formula_41 arbitrarily set to zero at formula_39,
 formula_43
Leading to Prandtl-Meyer supersonic expansion function,
 formula_44
References.
 