For a weak shock wave with Mach number $M_1 = 1 + \epsilon$ where $\epsilon \ll 1$, the entropy change $\Delta s$ across the shock wave is proportional to:

(A) $\Delta s \propto (M_1^2 - 1)^3 \approx \epsilon^3$

(B) $\Delta s \propto (M_1 - 1)$

(C) $\Delta s \propto (M_1 - 1)^2$

(D) $\Delta s = 0$

Answer Key:

Verified in official key

### 1. Strategy & Theoretical Principles From Taylor expansion of the Rankine-Hugoniot shock relations for weak shock waves: - Across a weak normal or oblique shock, static pressure jump $\Delta p / p_1 \propto (M_1^2 - 1) = O(\epsilon)$. - Velocity and density jumps are also of first order: $O(\epsilon)$. - However, the entropy generation $\Delta s$ across a shock wave vanishes to both first and second orders: $$\Delta s = c_v \frac{2\gamma(\gamma - 1)}{3(\gamma + 1)^2} (M_1^2 - 1)^3 + O((M_1^2 - 1)^4)$$ ### 2. Significance in Gas Dynamics Because $\Delta s$ is of third order in $(M_1 - 1)^3$, very weak shock waves (such as sound waves or Mach waves) can be treated as **isentropic** to second-order accuracy. This forms the foundational justification for linearized supersonic potential flow theory and acoustic wave theory. ### 3. Distractor Forensics - **Option A:** Correct 3rd-order dependence. - **Option B/C:** First and second order terms vanish identically in the entropy expansion. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Pressure jump is 1st order in $\epsilon$, but entropy jump is 3rd order in $\epsilon^3$. - **Standard Reference:** Liepmann, H.W. & Roshko, A., *Elements of Gasdynamics*, Chapter 2.

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