About the Oblique Shock Calculator
This oblique shock calculator will help determine the fluid flow properties for an oblique shock wave. A shock wave is most commonly associated with fast military aircraft or spacecraft when they move faster than the speed of sound. The design of air intakes for the aircraft is based on calculating the desired fluid flow properties. The air-breathing engines in military jets need air to enter the engine at subsonic speeds to ensure the jet engines operate correctly.
This requirement is achieved by introducing wedge-shaped objects in the air intakes to compress the airflow before it reaches the combustion chamber. You see some aircraft with adjustable inlet cones that move axially to increase or decrease the capture area, while others have inlet ramps for similar purposes.
The most common aircraft with inlet cones are the Mikoyan Gurevich MiG-21, Lockheed F-104, and Dassault Mirage 2000. Some aircraft like Concorde, Lockheed Martin F-22, and Mikoyan Gurevich MiG-25 use an inlet ramp instead of the cone to control the airflow. At supersonic speeds, the air passing through these diffusers slows down due to the formation of shock waves and enters the engine at slower speeds.
In addition to the desirable effects of the shock wave, there are also some undesirable effects such as sonic booms and high-pressure wavefronts in a blast wave arising from an explosion that can cause the loss of human lives. To this end, the underlying physics demands you understand what a shock wave is and what the oblique shock relations are. Read on to know how this oblique shock calculator can help you determine the fluid flow properties.
What is a shock wave? — Normal and oblique shock waves
A shock wave is an abrupt discontinuity that causes a change in fluid pressure, temperature, and density. It is formed when a wavefront travels at supersonic speed and high pressure gets built up. This high-pressure wave travels faster than the local speed of sound and is heard as a loud crack or whip that is produced by supersonic aircraft, explosions, and lightning strikes.
Studies indicate a shock wave has a thickness of about 200 nm; therefore, we can consider it a line or plane. The properties of the fluid — pressure, temperature, density, and velocity — change abruptly past this line. These shocks are often forced by the use of different geometries in an aircraft to achieve favorable post-shock conditions. The two common types of shock waves are:
Normal shock wave
The shock front is perpendicular (normal) to the local flow direction. The flow is decelerated from supersonic to subsonic, with no change in flow direction.
Oblique shock wave
The shock front is inclined at an angle to the local flow. The flow is turned (deflected) and decelerated, but can remain supersonic downstream (for a weak shock).
When the shock wave is deviating at an angle, it is known as an oblique shock wave, whereas when the shock is normal to the local flow, it is called the normal shock wave. Apart from shock waves, in the supersonic flow regime, we can observe another phenomenon called an expansion wave (the Prandtl–Meyer expansion).
Oblique shock relations
An oblique shock is described by the upstream Mach number M₁, the flow deflection (wedge) angle θ, and the wave angle β. The three are linked by the θ-β-M relation:
tan(θ) = 2 · cot(β) · [M₁²·sin²(β) − 1] / [M₁²·(γ + cos2β) + 2]
For a given M₁ and θ there are generally two valid wave angles — a smaller one (the weak shock, which usually occurs in practice) and a larger one (the strong shock). The normal component of the upstream Mach number is Mn1 = M₁·sin(β), and the property ratios across the shock follow the normal-shock relations:
p₂/p₁ = 1 + (2γ/(γ+1))·(Mn1² − 1) — static pressure ratio
ρ₂/ρ₁ = (γ+1)·Mn1² / ((γ−1)·Mn1² + 2) — density ratio
T₂/T₁ = (p₂/p₁) / (ρ₂/ρ₁) — temperature ratio
M₂ = Mn2 / sin(β − θ) — downstream Mach number
How to calculate oblique shock wave properties
- Choose your unit system — metric (kPa, K) or American/imperial (psi, °R).
- Enter the upstream (supersonic) Mach number M₁ > 1.
- Enter the flow deflection (wedge) angle θ in degrees.
- Enter the specific heat ratio γ (1.4 for air at standard conditions).
- Pick the weak or strong shock solution.
- Optionally enter the upstream static pressure and temperature to obtain the downstream values.
- The calculator returns the wave angle β, downstream Mach M₂, and the pressure, density, temperature and stagnation pressure ratios.
If the deflection angle θ exceeds the maximum value θmax for the given Mach number, an attached oblique shock cannot form and the shock detaches into a curved bow shock standing ahead of the body — the calculator will alert you in that case.
Example: Using the oblique shock angle calculator
Consider air (γ = 1.4) flowing at M₁ = 2.0 over a wedge with a half-angle of θ = 15°. Selecting the weak solution, the calculator returns a wave angle of about β ≈ 45.3°, a downstream Mach number of about M₂ ≈ 1.45, a static pressure ratio of about p₂/p₁ ≈ 2.19, a density ratio of about ρ₂/ρ₁ ≈ 1.70, and a temperature ratio of about T₂/T₁ ≈ 1.29. With an upstream static pressure of 101.325 kPa, the downstream static pressure is about 222 kPa.
FAQs
For a given Mach number and deflection angle below θmax, two shock angles satisfy the θ-β-M relation. The weak shock has the smaller wave angle and usually keeps the downstream flow supersonic; it is the solution observed in most external flows. The strong shock has the larger wave angle and always produces subsonic downstream flow.
When the required deflection angle exceeds the maximum deflection angle θmax for the upstream Mach number, an attached oblique shock is no longer possible. The shock detaches and forms a curved bow shock ahead of the body.
For air at moderate temperatures, γ = 1.4. For monatomic gases such as helium or argon, γ ≈ 1.667; for combustion gases at high temperature, γ can be lower (≈ 1.3). Enter the value appropriate for your gas.
Yes. Mach number, angles, ratios and γ are dimensionless, so they are the same in any system. The optional upstream and downstream pressure and temperature are reported in kPa and K (metric) or psi and °R (American/imperial), depending on your selection.