 Normal shock reflection at a rigid wall

For a perfect-gas normal shock wave reflected from a stationary rigid wall, apply the Rankine-Hugoniot conditions for a perfect gas using signed velocity in each shock frame. The once-shocked state is the common left state for both applications. Its two relative speeds are roots of $v^2-(\gamma +1)u_{s}v/2-\gamma p_s/{\rho }_s=0$. Their product and the pressure jump eliminate the velocity, giving the displayed relation. A strong incident shock wave gives $p_1/p_s\to (3\gamma -1)/(\gamma -1)$.