When a convergent-divergent supersonic nozzle operates such that the ambient back pressure $p_a$ is higher than the nozzle design exit pressure $p_e$ (i.e., $p_a > p_e$), causing oblique shock waves to form at the nozzle exit plane, the nozzle is operating in which regime?

(A) Over-expanded

(B) Under-expanded

(C) Correctly expanded

(D) Choked subsonic

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Pressure Comparison:** Design exit pressure $p_e < p_a$ (ambient pressure). 2. **Stage 2: Wave Pattern:** Because the gas must re-compress to match ambient atmospheric pressure, compression oblique shock waves attach to the nozzle lip. 3. **Stage 3: Classification:** This is the definition of an over-expanded nozzle. 4. **Stage 4: Selection:** Option A is correct.

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