= Solution
Assume $\gamma>1$. The upstream <ideal gas> is at rest in the laboratory frame and has <mass density> $\rho_1=CR^{-\beta}$. In the <shock frame> its incident speed is $V$. The strong-shock limit of the <Rankine-Hugoniot conditions for a perfect gas> gives compression $\chi=(\gamma+1)/(\gamma-1)$ and downstream shock-frame speed $V/\chi$. Transforming back to the laboratory frame gives $u_2=V(1-1/\chi)=2V/(\gamma+1)$.
The normal momentum jump yields
$$
p_2+\rho_2(V-u_2)^2=\rho_1V^2,\qquad
p_2=\frac{2}{\gamma+1}\rho_1V^2,
$$
where the cold upstream <pressure> has been neglected. Thus the boundary values in the normalization of part (b) are
$$
\boxed{U(1)=\frac2{\gamma+1},\qquad
D(1)=\frac{\gamma+1}{\gamma-1},\qquad
P(1)=\frac2{\gamma+1}}.
$$
The <velocity> jump uses the relative speed $V-u_2$, not the laboratory speed $u_2$. This distinction is essential to all three <shock wave> values.
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