Solving for shock wave angle using Mach and deflection angle

AI Thread Summary
The discussion focuses on determining the shock wave angle (Beta) while keeping the deflection angle constant and increasing the Mach number using the theta-beta-Mach relation. The equation provided is implicit in Beta, making it challenging to solve directly. Reference is made to "Elements of Gas Dynamics" for an approximate relationship between Beta and theta at high Mach numbers, but this is deemed less useful for the specific modeling of Beta changes. Graphical solutions and numerical methods, such as using fsolve in Matlab or Newton's method, are suggested as practical approaches to tackle the problem. The conversation emphasizes the complexity of the relationship and the need for computational tools to derive solutions effectively.
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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?
 
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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.
 
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.
 
It's just an implicit equation. Plug it into fsolve or something similar in Matlab. You can't solve it explicitly exactly.
 
Yep, gogo Newton's method. Gotta love Matlab.
 
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