Mach Cone and Oblique Shock

What is modelled

A source moving at speed U through air with sound speed a emits waves that travel at a. When U exceeds a the wave fronts cannot get ahead, and their common envelope is a cone with half angle mu. A wedge in a supersonic stream cannot send a warning upstream either, so the flow turns abruptly through a thin oblique shock attached to the tip.

Equations

• Speed of sound a = √(γ R T), Mach angle sin μ = 1 / M.
• Theta-beta-M relation tan θ = 2 cot β (M² sin²β − 1) / (M² (γ + cos 2β) + 2); the weak solution is shown.
• Normal Mach number Mn1 = M sin β, p2/p1 = 1 + 2γ(Mn1² − 1)/(γ + 1), ρ2/ρ1 = (γ + 1) Mn1² / ((γ − 1) Mn1² + 2).
• Downstream M2 = Mn2 / sin(β − θ).

Checks run in the page

• Sound speed against the handbook form and the Mach angle found as the tangent to a wave front.
• Mass, normal momentum and energy conserved across the shock.
• Turning angle recovered from the velocity triangle.
• Positive entropy rise and the largest attached turning angle.

Honest simplifications

Air is a calorically perfect gas with gamma = 1.4 and the shock is a thin discontinuity with no boundary layer or shock-boundary-layer interaction. At hypersonic speeds real air dissociates and the perfect-gas numbers become unreliable. Beyond the maximum turning angle the real shock detaches into a curved bow shock, which this page flags but does not draw.

• Raise the Mach number: the cone narrows and the shock lies closer to the wedge surface.
• Choose Detached bow shock: the turning angle exceeds the maximum an attached shock can supply.
• Lower the temperature to see the speed of sound and the source speed change at fixed Mach number.