Solving for shock wave angle using Mach and deflection angle

[twmggc]

I am trying to show how the shock wave angle varies as I hold the deflection angle constant and increase the Mach number. I am trying to solve for Beta (shock wave angle) using the theta-beta-mach relation:

tan(θ)= 2cot(β) * (M²sin2(β)-1) / (M²(1.4+cos(2β))+2)

This seems like it should be a simple problem, but I can't seem to figure it out.
Note, I am using constant specific heats hence the 1.4 in the equation.

Any hints?

 

[Aero51]

If I remember my gas dynamics correctly that equation is implicit with respect to beta.
However, "Elements of Gas Dynamics", by Liepmann and Rosko give an approximate relationship between β and θ when the mach number is large:

$\beta ≈ (\gamma +1)/2 * \theta$

L&R provide a graphical solution to the weak and strong shocks on page 87. This is probably the best method to show the relationship between deflection angle and shock angle. Also, they note that for a specific mach number there is a maximum possible deflection angle, θ.

 

[twmggc]

Thank you for the response, but since I am attempting to model the change in Beta with constant deflection angle the Liepmann and Rosko equation won't help me too much. As far as the graphical solutions go, I have an anderson book with theta-beta-mach relations plotted but they don't go to into detail for higher mach numbers.

 

[boneh3ad]

It's just an implicit equation. Plug it into fsolve or something similar in Matlab. You can't solve it explicitly exactly.

 

[Vadar2012]

Yep, gogo Newton's method. Gotta love Matlab.