What is the difference between the Method of Characteristics being formulated in terms of theta & alpha and theta & nu?

So I’m doing an MSc thesis involving the method of characteristics in rocket nozzles.

My lecture notes have the MoC formulated such that the positive characteristics are defined as:

C+ = Tan ( $theta+nu$)

and

C- = Tan ( $theta-nu$)

Which is flow angle relative to the centerline axis ($theta$) and the Prandtl-Meyer expansion angle ($nu$). I’ve previously found a few papers that use this formulation, and its the one that Anderson principally uses in his Modern Compressible Flow – although he freely mixes the use of the $nu$ and $mu$ equations:

C+ = Tan($theta+mu$) or Tan ($theta+alpha$)

C- = Tan($theta-mu$) or Tan($theta-mu$)

Where both $mu$ and $alpha$ are the mach angle, sin^-1 (1/M).

This latter formulation is the one given in Zucrow’s Gas Dynamics exclusively.

Is it because that the $nu$ formulation is for flows with a free pressure boundary, and $mu$ or $alpha$ for internal flows with physical boundaries? This would make sense since a lot of the papers that the $nu$ formulation crops up in are dealing with the MOC and formulation of aerospike nozzle contours – but it doesnt make sense with my professor giving this notation for internal flows as well.

Or is it simply a difference in describing the characteristic geometry and compatability equations?

I’d appreciate it if anyone could shed some light on this.