= Solution
The <Taylor series> of the shock density ratio at $\beta=0$ is
$$
\frac{\rho_1}{\rho_0}
=1+\frac{\beta}{\gamma}
-\frac{\gamma-1}{2\gamma^2}\beta^2
+O(\beta^3). \qquad (1)
$$
For a reversible <adiabatic process>, $p/p_0=(\rho/\rho_0)^\gamma$, so the density ratio predicted by the ideal-gas adiabat is
$$
\left(\frac{p_1}{p_0}\right)^{1/\gamma}
=(1+\beta)^{1/\gamma}
=1+\frac{\beta}{\gamma}
-\frac{\gamma-1}{2\gamma^2}\beta^2
+O(\beta^3). \qquad (2)
$$
Equations (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 $O(\beta^2)$.
Back to article page
