 Solution

The Taylor series of the shock density ratio at $\beta =0$ isFor a reversible adiabatic process, $p/p_0=(\rho /{\rho }_0{)}^{\gamma }$, so the density ratio predicted by the ideal-gas adiabat isEquations (1) and (2) agree through order ${\beta }^2$. Thus a weak shock satisfies the adiabatic pressure-density relation with an error of order $O({\beta }^3)$, which in particular establishes the requested accuracy through $O({\beta }^2)$.