Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2021 / Section B / 4 / c

Let $U_1,U_2$ denote the total flow speeds, keeping the normal components distinct from them. The original diagram measures $\beta$ from the initial wall direction to the shock wave itself. The outgoing velocity is turned by $\theta$, so it makes angle $\beta -\theta$ with the shock.

Resolve each velocity tangent and normal to the oblique shock. Tangential momentum conservation, with nonzero mass flux and no tangential surface force, makes the tangential velocity continuous:The normal flow obeys the strong normal shock wave result from part (b):Divide the second relation by the first to obtainThe approximation is controlled by the upstream normal Mach number $M\sin \beta \gg 1$, not just by $M\gg 1$. This distinction matters on the small-angle weak branch in part (d). The angle convention is also important: formulas using the angle to the shock normal instead would interchange sine and cosine.