Prandtl-Meyer function

Question

I'm trying to solve for Mach number based on the given Prandtl-Meyer function, however I keep getting odd error.

Here is my code:

g = 1.4

v2 = 54

Solve[v2 == (Sqrt[(g + 1)/(g - 1)]*
        ArcTan[Sqrt[(g - 1) (M2^2 - 1)/(g + 1)]] - 
       ArcTan[Sqrt[M2^2 - 1]])/Pi*180, {M2}];

This is the website that has the function I'm trying to solve.

Any help would be greatly appreciated.

UPDATED:

g = 1.4

v2 = 54.76

M2 := FindRoot[
  v2 == (Sqrt[(g + 1)/(g - 1)]*
        ArcTan[Sqrt[(g - 1) (x^2 - 1)/(g + 1)]] - 
       ArcTan[Sqrt[x^2 - 1]])/Pi*180, {x, 3}]
M2

Answer 1

Here's a short routine for computing the Mach number from the Prandtl-Meyer equation:

prandtlMeyerMachNumber[γ_?InexactNumberQ, ν_?InexactNumberQ] := 
    Module[{prec = Precision[{γ, ν}], β, η, λ, m},
           λ = Sqrt[SetPrecision[(γ - 1)/(γ + 1), ∞]];
           η = ν + π (1 - 1/λ)/2;
           m = ((3 + γ) η^2/12 + 2/(1 - γ))/η;
           Sqrt[β^2 + 1] /. First @ FindRoot[ν == ArcTan[λ β]/λ - ArcTan[β],
                                             {β, Sqrt[m^2 - 1]}, WorkingPrecision -> prec]]

As noted in the comments, a transcendental equation such as the Prandtl-Meyer equation will often not admit an easy symbolic solution, and one thus has to resort to numerical techniques, as embodied in FindRoot[]. For this case, I derived the starting approximation that is subsequently polished by FindRoot[] using Padé approximation.

As an example:

N[prandtlMeyerMachNumber[7/5, 54 °], 20]
   3.2298336339590326897

Note the use of the degree sign, as the function expects radian arguments.

For the case of $\gamma=7/5$ (which seems to be the common case), the routine will fail at exactly $\nu=\dfrac{\pi(\sqrt{6}-1)}{2}$ (the limiting value of $\nu$), but should otherwise be fine.