In 2D steady, irrotational, supersonic flow, the method of characteristics defines two families of characteristic Mach lines ($C_+$ and $C_-$). The Riemann invariants (compatibility relations) along these $C_\pm$ characteristic lines in terms of flow deflection angle $\theta$ and Prandtl-Meyer function $\nu(M)$ are:

(A) Along $C_+$: $\theta - \nu = \text{constant}$, and along $C_-$: $\theta + \nu = \text{constant}$

(B) Along $C_+$: $\theta + \nu = \text{constant}$, and along $C_-$: $\theta - \nu = \text{constant}$

(C) Along $C_+$: $\theta \cdot \nu = \text{constant}$, and along $C_-$: $\theta / \nu = \text{constant}$

(D) Along both $C_+$ and $C_-$: $\theta + \nu = \text{constant}$

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### 1. Strategy & Theoretical Principles In 2D steady, homentropic, irrotational supersonic flow, the governing hyperbolic potential equations can be transformed along their characteristic curves into ordinary differential relations called **compatibility equations**. ### 2. Analytical Formulation The two families of characteristic lines run at the local Mach angle $\mu = \arcsin(1/M)$ to the streamlines: - Left-running characteristics ($C_+$): inclination angle is $(\theta + \mu)$ - Right-running characteristics ($C_-$): inclination angle is $(\theta - \mu)$ The compatibility equations integrated along these lines yield the Riemann invariants: $$\text{Along } C_+: \quad \theta - \nu(M) = K_+ = \text{constant}$$ $$\text{Along } C_-: \quad \theta + \nu(M) = K_- = \text{constant}$$ where $\nu(M)$ is the Prandtl-Meyer angle function. ### 3. Step-by-Step Verification At any point where a $C_+$ characteristic and a $C_-$ characteristic intersect, the local flow direction $\theta$ and Prandtl-Meyer function $\nu$ are determined by the simultaneous solution: $$\theta = \frac{K_- + K_+}{2}$$ $$\nu = \frac{K_- - K_+}{2}$$ ### 4. Distractor Forensics - **Option A:** Correct standard formulation found in Shapiro and Anderson. - **Option B:** Inverted signs between $C_+$ and $C_-$. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Remember along left-running $C_+$ waves, $\theta - \nu = \text{const}$, and along right-running $C_-$ waves, $\theta + \nu = \text{const}$. - **Standard Reference:** Anderson, J.D., *Modern Compressible Flow*, Chapter 11 (The Method of Characteristics).

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