Question

A supersonic aircraft has an air intake ramp that can be rotated about the leading edge \(O\) such that the shock from the leading edge meets the cowl lip as shown. Select all the correct statement(s) as per oblique shock theory when flight Mach number \(M\) increases.

• A. A
• B. B
• C. C
• D. D

Hint:

Always recall: \(\mu = \sin^{-1}(1/M)\) is the Mach angle; oblique shocks always lie between \(\mu\) and \(90^\circ\). As Mach number increases, shock angle decreases.

Answer 1

The Correct Option is C

Step 1: Relation between Mach number, deflection angle, and shock angle.
For an oblique shock, \[ \tan \theta = 2 \cot \beta \cdot \frac{M^2 \sin^2 \beta -1}{M^2(\gamma+ \cos 2\beta)+2}. \] For fixed wedge/ramp angle \(\theta\), increasing Mach number changes \(\beta\).

Step 2: Examine statement (A).
Claim: "It is always possible to find a ramp setting to keep \(\beta\) fixed." But oblique shock angle \(\beta\) decreases with increasing Mach number for fixed \(\theta\). To keep \(\beta\) constant, \(\theta\) must adjust appropriately, but there exists an upper limit: beyond a certain Mach, no solution exists for fixed geometry. So (A) is false.

Step 3: Examine statement (B).
For fixed ramp deflection angle \(\theta\), as Mach number increases, the shock angle \(\beta\) decreases. Wait—check carefully: - At low Mach just above 1, \(\beta\) is large (close to \(90^\circ\)). - As \(M\) increases, \(\beta\) decreases toward Mach angle. Therefore for fixed ramp angle, \(\beta\) actually decreases, not increases. But the option says "increases". Let's verify. Using known trend: - Mach angle: \(\mu = \sin^{-1}(1/M)\). - For very high \(M\), \(\beta \to \mu\), which is small. So as \(M\) increases, \(\beta\) decreases. Therefore (B) is false.

Step 4: Examine statement (C).
At too high Mach, wedge angle might exceed maximum allowable for attached oblique shock. Then shock detaches. So indeed, for large \(M\), it is not always possible to maintain geometry with shock at cowl lip. So (C) is true.

Step 5: Examine statement (D).
Shock angle \(\beta\) always satisfies \[ \beta \;>\; \mu = \sin^{-1}\!\left(\tfrac{1}{M}\right), \] since the Mach wave is the lower bound. Therefore the claim "\(\beta<\sin^{-1}(1/M)\)" is false. Correction: So the true statement(s) are (C) only.

Final Answer:
\[ \boxed{\text{(C) only}} \]