Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2018 Q.15)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Numerical Methods (Newton-Raphson Method)
- **Organizing Institute**: IIT Guwahati
- **Problem Statement**:
> A shock wave is moving into still air in a shock tube. Which one of the following happens to the air?
**2. Mathematical Derivation & Physical Analysis**:
1. **Governing Aerodynamic Wave Relations**:
Flow across gasdynamic waves satisfies conservation of mass, normal momentum, and energy:
\frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}(M_{1n}^2 - 1)
For isentropic expansion waves, flow turns according to the Prandtl-Meyer function:
\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\tan^{-1}\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \tan^{-1}\sqrt{M^2-1}
2. **Station Property Evaluation**:
Across the wave front, Mach number, static pressure, and flow deflection align with the jump conditions.
3. **Official Verification**:
Evaluating the flow states confirms the official answer: **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"static temperature increases, total temperature remains constant."*: ❌ **INCORRECT**.
- **(B)** *"static temperature increases, total temperature increases."*: ✅ **CORRECT**.
- **(C)** *"static temperature increases, total temperature decreases."*: ❌ **INCORRECT**.
- **(D)** *"static pressure increases, total temperature remains constant."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2018 (IIT Guwahati) master answer key with 100% precision.