1. **Problem statement:** Helium with specific heat ratio $\gamma = 1.67$ flows at Mach number $M_1 = 1.80$ and encounters a normal shock. We need to find the Mach number $M_2$ and pressure ratio $\frac{P_2}{P_1}$ across the shock. 2. **Formulas and important rules:** - For a normal shock, the downstream Mach number $M_2$ is given by: $$M_2 = \sqrt{\frac{1 + \frac{\gamma - 1}{2} M_1^2}{\gamma M_1^2 - \frac{\gamma - 1}{2}}}$$ - The pressure ratio across the shock is: $$\frac{P_2}{P_1} = 1 + \frac{2 \gamma}{\gamma + 1} (M_1^2 - 1)$$ 3. **Calculate $M_2$:** - Substitute $\gamma = 1.67$ and $M_1 = 1.80$: $$M_2 = \sqrt{\frac{1 + \frac{1.67 - 1}{2} \times 1.80^2}{1.67 \times 1.80^2 - \frac{1.67 - 1}{2}}} = \sqrt{\frac{1 + 0.335 \times 3.24}{1.67 \times 3.24 - 0.335}}$$ - Calculate numerator inside the root: $$1 + 0.335 \times 3.24 = 1 + 1.0854 = 2.0854$$ - Calculate denominator inside the root: $$1.67 \times 3.24 - 0.335 = 5.4108 - 0.335 = 5.0758$$ - Thus: $$M_2 = \sqrt{\frac{2.0854}{5.0758}} = \sqrt{0.4109} = 0.641$$ 4. **Calculate pressure ratio $\frac{P_2}{P_1}$:** - Use formula: $$\frac{P_2}{P_1} = 1 + \frac{2 \times 1.67}{1.67 + 1} (1.80^2 - 1)$$ - Calculate denominator: $$1.67 + 1 = 2.67$$ - Calculate numerator: $$2 \times 1.67 = 3.34$$ - Calculate $M_1^2 - 1$: $$1.80^2 - 1 = 3.24 - 1 = 2.24$$ - Substitute: $$\frac{P_2}{P_1} = 1 + \frac{3.34}{2.67} \times 2.24 = 1 + 1.251 \times 2.24 = 1 + 2.802 = 3.802$$ 5. **Final answers:** - Downstream Mach number after the shock: $M_2 \approx 0.641$ - Pressure ratio across the shock: $\frac{P_2}{P_1} \approx 3.80$ **Solved By : Ibrahim Haval** **Solved for : Dr.Zaineb**