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The following tutorial introduces the topic of Oblique Shock Wave Relations, a phenomenon predominantly studied in the field of aerodynamics, specifically gas dynamics. Oblique shock waves occur when a shock wave propagates in a medium at an angle to the flow direction, rather than perpendicular to it. This tutorial will explore the calculations and formulas associated with heat capacity ratio, Mach number of upstream, and oblique shock angle.

Pressure Ratio of Oblique Shock Wave = |

Density Ratio of Oblique Shock Wave = |

Temperature Ratio of Oblique Shock Wave = |

Stagnation Pressure Ratio of Oblique Shock Wave = |

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The following equation relates the upstream Mach number (M_{1}), downstream Mach number (M_{2}), and the wave angle (β) with respect to the flow direction:

tan(β) = 2cot(θ)[(M_{1}^{2}sin^{2}(β) - 1)/(M_{1}^{2}(γ + cos(2β))) + 2]

Where:

- M
_{1}: Mach number of upstream - M
_{2}: Mach number of downstream - β: Oblique shock angle
- γ: Heat capacity ratio

The study of oblique shock waves and their corresponding equations is a fundamental aspect of gas dynamics, a field of physics developed by many individuals over the years. The concept of oblique shock wave relations was further refined with the development of supersonic and hypersonic aerodynamics.

Understanding oblique shock wave relations is essential in the field of aeronautics and astronautics. Engineers use these principles while designing supersonic aircraft and spacecraft to handle and minimize the effects of shock waves, enhancing the safety and performance of these vehicles.

While many have contributed to the field, Theodore von Kármán and Ludwig Prandtl stand out for their significant contributions to aerodynamics, particularly in the study of supersonic flows and shock waves. Their work in the early to mid-20th century set the foundation for modern aerodynamics.

- Oblique shock wave relations are crucial for spacecraft reentry, where the vehicle experiences high-speed flow at an oblique angle.
- The study of shock waves has allowed for the development of supersonic travel, changing the way we understand transport and travel times.
- The analysis of shock waves also has environmental implications as it helps understand and mitigate sonic booms, a significant noise pollution source in areas near air travel routes.

Understanding oblique shock wave relations is fundamental to modern aerodynamics and many aspects of physics. The principles behind these relations continue to influence advancements in high-speed travel, spacecraft design, and our overall understanding of fluid dynamics.

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