Perfect gas, adiabatic, 1-D, steady flow is assumed. The shock is physical only when M_1n = M_1·sin(β) \gt 1; otherwise the stat values are shown as `—`.
Bottom = horizontal upstream (cool color, long arrows = high speed) / red line = oblique shock at angle β / downstream flow turned by θ along the wedge surface (warm color)
x = flow-deflection angle θ / y = shock angle β / lower branch = weak solution, upper branch = strong solution / yellow dot = current (θ, β)
An oblique shock is a thin discontinuity inclined at the shock angle β to the upstream flow. Crossing it, the flow is turned by an angle θ. The three quantities are linked by the θ-β-M relation.
θ-β-M relation (gives θ explicitly from M_1 and β):
$$\tan\theta = \frac{2\cot\beta\,(M_1^2\sin^2\beta - 1)}{M_1^2(\gamma + \cos 2\beta) + 2}$$Normal shock-component M_1n and the downstream normal component M_2n (normal-shock relations applied to M_1n):
$$M_{1n} = M_1\sin\beta, \qquad M_{2n}^2 = \frac{1 + \tfrac{\gamma-1}{2}M_{1n}^2}{\gamma M_{1n}^2 - \tfrac{\gamma-1}{2}}$$Downstream Mach and pressure ratio (the same form as a normal shock applied to M_1n):
$$M_2 = \frac{M_{2n}}{\sin(\beta - \theta)}, \qquad \frac{P_2}{P_1} = 1 + \frac{2\gamma}{\gamma+1}\bigl(M_{1n}^2 - 1\bigr)$$A shock is physical only for M_1n \gt 1. For each θ there are two values of β: the smaller is the weak solution, the larger is the strong solution.