Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 314 / 2 / d / Solution

The pressure jump relation with $p_2/p_1=1+ϵ$ givesIf ${\beta }_1$ is the angle between the upstream velocity and the shock front, then $M_{x1}=M_1\sin {\beta }_1$. To leading order,the Mach angle. Expansion of the compression ratio givesBecause $u_y$ is continuous and $\tan \beta =u_x/u_y$,Writing $\theta ={\beta }_1-{\beta }_2$ and linearizing the tangent about $\sin {\beta }_1=1/M_1$ gives the weak-oblique-shock deflectionThe normal component decreases while the tangential component is unchanged, so ${\beta }_2<{\beta }_1$: the flow turns toward the shock front.