How to Use Oblique Shock Calculator
The Oblique Shock Calculator solves the implicit θ–β–M relation numerically to find weak and strong wave angles for a given deflection, or conversely computes the flow deflection from a measured wave angle.
- Enter upstream Mach number M₁ — must be supersonic (M₁ > 1) for an oblique shock to exist.
- Enter specific-heat ratio γ — default 1.4 for air; adjust for other gases (CO₂ ≈ 1.3, He ≈ 1.67).
- Choose input type — “deflection θ” if the wedge or ramp geometry is known; “wave angle β” if the shock is visible in schlieren or shadowgraph imagery.
- Read β weak and β strong when θ is the input, or read θ when β is the input.
- Check the detachment warning — if θ exceeds θ_max for the given M₁, the shock detaches and a bow shock forms; the calculator flags this condition.
- Review downstream Mach M₂ and the static pressure ratio p₂/p₁ in the result panel.
Formula & Theory — Oblique Shock Calculator
The Oblique Shock Calculator solves the θ–β–M relation, derived by applying Rankine–Hugoniot conditions across the oblique shock:
tan θ = 2 cot β · [ M₁² sin²β − 1 ] / [ M₁²(γ + cos 2β) + 2 ]
M₂² sin(β − θ) = (M₁² sin²β + 2/(γ−1)) / (2γM₁² sin²β/(γ−1) − 1)
p₂/p₁ = 1 + 2γ(M₁² sin²β − 1)/(γ + 1)| Symbol | Meaning | SI Unit |
|---|---|---|
| M₁, M₂ | Upstream / downstream Mach numbers | — |
| β | Wave angle from upstream flow direction | ° |
| θ | Flow deflection (wedge or ramp half-angle) | ° |
| γ | Specific-heat ratio | — |
| p₂/p₁ | Static pressure ratio across shock | — |
For any M₁ there is a maximum deflection θ_max; below it two solutions exist — the weak shock (smaller β, M₂ typically supersonic) and the strong shock (larger β, M₂ subsonic). In practice the weak solution is realised unless back pressure forces a strong shock.
Use Cases for Oblique Shock Calculator
- Supersonic inlet ramp design — size multi-ramp deflection angles to compress airflow progressively to near-sonic without producing a detached bow shock.
- Wind-tunnel schlieren analysis — convert measured shock angles from optical images into flow deflections and pressure ratios for test-section calibration.
- Diamond-foil and double-wedge airfoils — compute leading-edge shock properties and trailing-edge expansion fans for wave-drag and lift prediction.
- Hypersonic vehicle forebody — calculate the shock-layer temperature and pressure ratio at the stagnation region for thermal protection system sizing.
- Shock-tube and detonation studies — assess whether an oblique detonation front angle is consistent with Chapman–Jouguet oblique shock conditions.
- Compressible-flow coursework — solve θ–β–M homework problems numerically and verify against graphical Mach-angle charts in gas dynamics textbooks.