How to derive the reflected Mach number relationship?

[theudster]

TL;DR Summary

Deriving reflected mach number relationship

Hi,

when learnig about reflected waves, I keep coming up with this equation;

to calculate the reflected mach number (Mr).
I can't seem to find the derivation for this and would appreciate your help

Thank you

 

[boneh3ad]

This is honestly one of the trickier topics in elementary gas dynamics. There are a few general rules you should keep in mind:

1. Always change your frame of reference to one with a stationary shock, where possible.
2. For situations like this, the important jump condition is the velocity jump, $u_2/u_1$.

If you keep those in mind, you can solve for the ratio for the incident shock and then for the reflected shock and relate the two. The terms under the radical come from normal shock relations. It's not really something that makes a lot of sense to reproduce here in full, though. Do you have a relevant textbook handy?

 

[theudster]

This is honestly one of the trickier topics in elementary gas dynamics. There are a few general rules you should keep in mind:

1. Always change your frame of reference to one with a stationary shock, where possible.
2. For situations like this, the important jump condition is the velocity jump, $u_2/u_1$.

If you keep those in mind, you can solve for the ratio for the incident shock and then for the reflected shock and relate the two. The terms under the radical come from normal shock relations. It's not really something that makes a lot of sense to reproduce here in full, though. Do you have a relevant textbook handy?

I have Compressible FLow by Anderson, but he doesn't provide the derivation

 

[boneh3ad]

Anderson also says "the derivation is left as an exercise for the reader." If you follow along with that chapter and follow the two rules I mentioned, you ought to be able to derive this on your own. It's a good exercise.