Oblique Shock Calculator

An oblique shock wave is a shock inclined at an angle β to the incoming supersonic flow, formed when the flow is turned through an angle θ, for example by a wedge or a compression corner. Only the Mach component normal to the shock, Mₙ₁ = M₁·sin β, is compressed like a normal shock: static pressure, density and temperature all rise, while the tangential component passes through unchanged. Example: M₁ = 2.0 turning through θ = 15° produces a weak shock at β ≈ 45.34° with p₂/p₁ ≈ 2.19, as this oblique shock calculator shows.

The θ–β–M relation is tan θ = 2·cot β·(M₁²·sin²β − 1) / (M₁²·(γ + cos 2β) + 2). It connects upstream Mach number M₁, flow-deflection angle θ and shock angle β. Given M₁ and θ it is an implicit equation for β with two solutions; given M₁ and β it gives θ directly. This calculator solves it numerically in both directions.

For any deflection below θ_max the θ–β–M relation has two shock-angle solutions. The weak shock, with the smaller β, compresses the flow mildly and the downstream flow often stays supersonic. The strong shock, with β near 90°, compresses strongly and leaves subsonic flow behind it. For M₁ = 2.0 and θ = 15°: weak β ≈ 45.34° gives p₂/p₁ ≈ 2.19 and M₂ ≈ 1.45, while strong β ≈ 79.83° gives p₂/p₁ ≈ 4.35 and M₂ ≈ 0.64. The weak solution is the default here because it is the one normally realized in external supersonic flow.

It means no straight attached oblique shock can turn the flow through the requested angle. Every Mach number has a maximum deflection θ_max; asking for more forces the shock to detach and stand ahead of the body as a curved bow shock. For example, at M₁ = 2.0 the maximum is θ_max ≈ 22.97°, so θ = 25° is impossible as an attached shock and this calculator reports it as detached.

First decompose the upstream Mach into its normal component Mₙ₁ = M₁·sin β, then apply the normal-shock relations: p₂/p₁ = 1 + 2γ(Mₙ₁² − 1)/(γ + 1), ρ₂/ρ₁ = (γ + 1)Mₙ₁²/((γ − 1)Mₙ₁² + 2), T₂/T₁ = (p₂/p₁)/(ρ₂/ρ₁), and the total-pressure ratio p₀₂/p₀₁ from the Rayleigh pitot formula. The downstream Mach follows from M₂ = Mₙ₂/sin(β − θ).

Yes. The default γ = 1.4 suits air, but other gases differ: a monatomic gas with γ = 1.667 compresses less across the same shock and allows a larger θ_max, while gases with lower γ compress more. Every mode of this calculator lets you edit γ so the results match your working fluid.

They are opposite phenomena, and this boundary matters. An oblique shock COMPRESSES supersonic flow turning into itself: pressure, density and temperature rise and total pressure falls. A Prandtl–Meyer expansion fan does the reverse: flow turning away from itself around a convex corner EXPANDS, so pressure falls, Mach number rises, and the process is isentropic. Shocks use the θ–β–M relation; expansions use the Prandtl–Meyer function. A companion Prandtl–Meyer expansion calculator is planned for this site.

Yes, after a weak oblique shock. Only the normal component must become subsonic (Mₙ₂ below 1); the full downstream Mach M₂ = Mₙ₂/sin(β − θ) can still exceed 1 because of the untouched tangential component. In the worked example above the weak shock leaves M₂ ≈ 1.45, supersonic, while the strong shock leaves M₂ ≈ 0.64, subsonic.

The maximum flow-deflection angle an attached oblique shock can produce at a given Mach number. It is the peak of the θ(β) curve from the θ–β–M relation; this calculator locates it with a numerical search over β between the Mach angle and 90°. Mode 3 tabulates and plots θ_max against M₁ so you can see the attached-versus-detached boundary at a glance.

The Mach angle μ = asin(1/M₁) is the angle at which infinitesimal disturbances propagate in supersonic flow. It is the lower limit of the shock angle: as θ approaches zero, β approaches μ and the shock degenerates into a Mach wave with all property ratios equal to 1. For M₁ = 2.0, μ = 30°.