Attached weak oblique shocks over the same wedge are sketched for three different upstream Mach numbers \(M_1, M_2, M_3 > 1\). Which ordering is TRUE?
Show Hint
For the weak attached oblique shock at fixed wedge angle: bigger \(M\) \(\Rightarrow\) shock rotates toward the surface (smaller \(\beta\)); as \(M\to 1^+\), \(\beta\to 90^\circ\).
Step 1: Dependence of shock angle on Mach number for fixed wedge angle (weak branch). For a given wedge deflection \(\theta\) and weak attached shock, the shock angle \(\beta\) satisfies the \(\theta\)–\(\beta\)–\(M\) relation. As the upstream Mach number \(M\) increases, the weak-shock angle \(\beta\) decreases, approaching the Mach angle \(\mu=\sin^{-1}(1/M)\). Conversely, as \(M\) approaches \(1^+\), \(\beta\) increases toward \(90^\circ\) (normal shock limit).
Step 2: Read the sketches qualitatively. From left to right, the drawings show shocks that are (i) at a moderate angle to the wedge, (ii) tightly hugging the wedge (small \(\beta\)), and (iii) very open/near-normal (large \(\beta\)). Thus the corresponding Mach numbers satisfy \[ \text{smallest }\beta \Rightarrow \text{largest }M, \qquad \text{largest }\beta \Rightarrow \text{smallest }M. \] Therefore, \[ M_2\; \text{(middle)}\; >\; M_1\; \text{(left)}\; > \; M_3\; \text{(right)}. \] \[ \boxed{M_3 < M_1 < M_2} \]
