I am a new learner in compressible aerodynamics. When studying the shock tube, I struggle to derive the relationship between $M_{s}$ and $M_{R}$ after the shock is reflected from the end wall. How does this formula form step by step?
$$ \frac{M_{R}}{M_{R}^{2} - 1} = \frac{M_{s}}{M_{s}^{2} - 1} \sqrt{1 + \frac{2 \left( \gamma_{1} - 1 \right)}{\left( \gamma_{1} + 1 \right)^{2}} \left( M_{s}^{2} - 1 \right) \left( \gamma_{1} + \frac{1}{M_{s}^{2}} \right)} \tag{1} \label{1} $$
Updates: I understand it is about the normal shock relationship. Also, they share the same Region 2. I think I should use the velocity $u_{p}$ or the pressure $P_{2}$ to derive the equation. But I have no idea how it ends there or if my starting point is wrong.
