Calculator inputs
Enter the upstream flow state and choose a branch. The page solves the theta-beta-M relation numerically and reports attached-shock properties.
Example data table
These examples use the same equations as the calculator. They help you compare how beta shifts with Mach number, turning angle, and branch choice.
| M₁ | θ (deg) | γ | Branch | β (deg) | M₂ | p₂/p₁ |
|---|---|---|---|---|---|---|
| 2.00 | 10.00 | 1.40 | Weak | 39.3139 | 1.6405 | 1.7066 |
| 2.50 | 15.00 | 1.40 | Weak | 36.9449 | 1.8735 | 2.4675 |
| 3.00 | 20.00 | 1.40 | Weak | 37.7636 | 1.9941 | 3.7713 |
| 3.50 | 18.00 | 1.40 | Strong | 84.0220 | 0.4946 | 13.9700 |
Formula used
The calculator solves the oblique shock relation between flow deflection angle
θ, shock angle
β, upstream Mach number
M₁, and specific heat ratio
γ.
Main theta-beta-M relation
tan(θ) = 2 cot(β) [M₁² sin²(β) - 1] / [M₁²(γ + cos(2β)) + 2]
Normal Mach components
Mₙ₁ = M₁ sin(β) and
Mₙ₂² = [1 + (γ-1)Mₙ₁²/2] / [γMₙ₁² - (γ-1)/2]
Static property ratios
p₂/p₁ = 1 + 2γ(Mₙ₁² - 1)/(γ + 1),
ρ₂/ρ₁ = (γ + 1)Mₙ₁² / [(γ - 1)Mₙ₁² + 2],
T₂/T₁ = (p₂/p₁)/(ρ₂/ρ₁)
The weak branch uses the smaller beta root. The strong branch uses the larger root. If the requested turning angle exceeds the attached limit, the page warns that no attached oblique shock exists.
How to use this calculator
- Enter the upstream Mach number above one.
- Type the flow deflection angle in degrees.
- Set gamma for your working gas.
- Choose weak or strong shock behavior.
- Optionally enter upstream pressure and temperature.
- Press Calculate beta to solve the relation.
- Review beta, downstream Mach, and thermodynamic ratios.
- Use the chart and exports for reports or checks.
Frequently asked questions
1) What does beta represent in an oblique shock?
Beta is the angle between the incoming flow direction and the shock front. It sets the normal Mach component, which controls compression strength and all downstream property changes.
2) Why can two beta solutions exist?
For many attached cases, the theta-beta-M relation crosses the selected turning angle twice. The smaller root is the weak solution, while the larger root is the strong solution.
3) Which branch is usually physical?
The weak branch is most common in external aerodynamic flows because it usually gives a supersonic downstream state. The strong branch is possible, but it often drives the downstream flow subsonic.
4) What happens when theta exceeds the maximum attached value?
No attached oblique shock can satisfy the input turning angle. In practice, the shock detaches and forms a curved bow shock instead of a straight attached wave.
5) Why does the calculator need gamma?
Gamma links pressure, density, and temperature changes in compressible flow. Different gases and thermal assumptions change the shock strength and therefore shift the solved beta angle.
6) Can this page compute absolute downstream pressure and temperature?
Yes. Enter upstream pressure and temperature in the optional fields. The page multiplies them by the calculated ratios to estimate downstream static pressure and temperature.
7) Why is M₁ required to be greater than one?
Oblique shock waves require supersonic upstream flow. When the upstream Mach number is one or below, the classical attached oblique shock relation used here no longer applies.
8) What does the plot show?
The graph plots the theta-beta curve for the chosen Mach number and gamma. Your solution point is placed on that curve, so you can visually compare weak and strong root locations.