A normal shock wave stands in a supersonic air stream with upstream Mach number $M_1 = 2.0$. Taking the ratio of specific heats $\gamma = 1.4$, what is the downstream Mach number $M_2$ behind the shock wave? (Round off to two decimal places)

Answer Key:

0.58

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Normal Shock Relation:** For an ideal gas with ratio of specific heats $\gamma$: * $M_2^2 = \frac{2 + (\gamma - 1) M_1^2}{2\gamma M_1^2 - (\gamma - 1)}$. 2. **Stage 2: Parameter Substitution:** * $\gamma = 1.4$, $\gamma - 1 = 0.4$, $2\gamma = 2.8$. * $M_1 = 2.0 \implies M_1^2 = 4.0$. 3. **Stage 3: Calculation of Numerator & Denominator:** * Numerator: $2 + 0.4 \times 4.0 = 2 + 1.6 = 3.6$. * Denominator: $2.8 \times 4.0 - 0.4 = 11.2 - 0.4 = 10.8$. 4. **Stage 4: Downstream Mach Number:** * $M_2^2 = \frac{3.6}{10.8} = \frac{1}{3} \approx 0.333333$. * $M_2 = \sqrt{\frac{1}{3}} = \frac{1}{1.73205} \approx 0.57735$. * Rounding to two decimal places gives $0.58$. 5. **Stage 5: Official Reference:** John D. Anderson Jr., *Modern Compressible Flow*, Chapter 3.

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