Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 39C (Waves)

The one-dimensional mass flux and momentum flux are respectivelyIn a steady flow, the conservation laws make both the mass flux and the given energy fluxconstant. Wherever $J_m\ne 0$, their ratio is therefore constant:where $W=p/(\gamma -1)$ is the internal energy per unit volume of the ideal gas. This is the required steady-flow energy integral.

Work in the shock frame, and let the positive upstream and downstream speeds be $w_0$ and $w_1$. The Rankine-Hugoniot conditions for a perfect gas give conservation of mass and momentum:PutMass conservation gives $w_1=w_0/r$, and momentum conservation then gives

Applying the energy integral from part (a) on the two sides givesSubstitution of (1) into (2), followed by cancellation of $p_0/{\rho }_0$, yieldsSolving this linear equation for the density ratio gives

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 $O({\beta }^2)$.