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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 takes 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.

🙋 Check the speed of sound for given conditions with our speed of sound calculator.

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 F22, 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.

Different type engine intakes for aircraft.
Engine intakes of different aircraft. (Credit: Wikimedia commons)

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 to understand what is a shock wave and what are oblique shock relations. Read on to know how this oblique shock calculator can help you determine the fluid flow properties.

Fluid flow around the wedges and formation of oblique shock waves.
Fluid flow around the wedges and formation of oblique shock waves.

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 changed 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; and
  • Oblique shock wave.

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 expansion wave. Please do check out our Prandtl Meyer expansion calculator to learn about expansion waves.

Velocity triangles at upstream and downstream of oblique shock wave.
Velocity triangles at upstream and downstream of oblique shock wave.

Oblique shock relations

As we know that the properties of flow change abruptly, let's look at the relations that one can use to define the pre (upstream) and post-shock (downstream) conditions. The subscripts 1 and 2 represent upstream and downstream conditions, respectively. Consider an airflow of pressure pp, temperature TT, and density ρ\rho and traveling at a Mach number, MM.

🔎 You can quickly convert any speed to a Mach number with our Mach number calculator.

The downstream Mach number is given by:

M2=2+(γ1)M12sin2β(2γM12sin2βγ1)sin2(βθ)\scriptsize M_2 = \sqrt { \frac {2+(\gamma -1)M_1^2 \sin^2\beta} {(2 \gamma M_1^2 \sin^2\beta - \gamma - 1)\sin^2(\beta - \theta)} }


  • γ\gamma – Specific heat ratio;
  • β\beta – Wave angle; and
  • θ\theta – Turn angle.

The above equation is rewritten to eliminate the turn angle, θ\theta, such that:

 ⁣M2=(γ+1)2M14sin2β4(γM12sin2β+1)(M12sin2β1)[2γM12sin2β(γ1)][2+(γ1)M12sin2β]\!\scriptsize M_2 = \sqrt{ \frac {(\gamma + 1)^2 M_1^4\sin^2\beta - 4(\gamma M_1^2 \sin^2\beta + 1)( M_1^2 \sin^2\beta - 1)}{[2\gamma M_1^2 \sin^2\beta - (\gamma - 1)] [2 + (\gamma -1)M_1^2 \sin^2\beta]}}

You can obtain the Mach numbers for normal shock wave from respective oblique shock Mach numbers such that:

Mx=M1sinβ My=M2sin(βθ)\quad M_x = M_1 \sin \beta \\ \ \\ \quad M_y = M_2 \sin (\beta - \theta)


  • MxM_x – Upstream Mach number for normal shock; and
  • MyM_y – Downstream Mach number for normal shock.

Pressure ratio: The relationship between upstream and downstream pressure is:

p2p1=2γγ+1(M12sin2β1)\quad \frac{p_2}{p_1} = \frac{2\gamma}{\gamma+1} (M_1^2 \sin^2\beta - 1)

Stagnation pressure ratio: The stagnation conditions for upstream and downstream can be calculated using the Mach number and wave angle. Such that:

   p02p01=[(γ+1)M12sin2β2+(γ1)M12sin2β]γγ1×[γ+12γM12sin2β(γ1)]1γ1\ \ \ \scriptsize \frac{p_{02}}{p_{01}} = \left [ \frac{(\gamma +1) M_1^2 \sin^2\beta}{2 + (\gamma - 1) M_1^2 \sin^2\beta} \right ] ^{\frac{\gamma}{\gamma - 1}} \\ \qquad \qquad \times \left [\frac{\gamma+1}{2\gamma M_1^2 \sin^2\beta - (\gamma - 1)} \right ]^{\frac{1}{\gamma-1}}

Temperature ratio: One can obtain the downstream temperature using the relation:

 ⁣T1T2=[2γM12sin2β(γ1)][2+(γ1)M12sin2β](γ+1)2M12sin2β\! \scriptsize \frac{T_1}{T_2} = \frac{[2\gamma M_1^2 \sin^2\beta - (\gamma - 1)] [2 + (\gamma -1)M_1^2 \sin^2\beta]}{(\gamma +1)^2 M_1^2 \sin^2\beta}

Density ratio: The ratio of densities at upstream to downstream is:

ρ1ρ2=(γ+1)M12sin2β2+(γ1)M12sin2β\quad \frac{\rho_1}{\rho_2} = \frac{(\gamma +1) M_1^2 \sin^2\beta}{2 + (\gamma - 1) M_1^2 \sin^2\beta}

We can also write the stagnation pressure, density, and static pressure as:

p02p01=[ρ1ρ2]γγ1/[p2p1]1γ1\quad \frac{p_{02}}{p_{01}} = \left [ \frac{\rho_1}{\rho_2} \right ]^{\frac{\gamma}{\gamma - 1}} / \left[\frac{p_2}{p_1} \right ] ^{\frac{1}{\gamma-1}}

This oblique shock calculator uses the above set of equations to determine the following upstream and downstream parameters:

  • Mach numbers;
  • Static pressure;
  • Stagnation pressure;
  • Temperature; and
  • Density.

🙋 On a related topic, you may also be interested in Omni's isentropic flow calculator.

How to calculate oblique shock wave properties

To find the oblique shock wave properties:

  1. Enter the upstream Mach number, M1M_1.
  2. Fill in the wave angle, β\beta.
  3. The oblique shock calculator return the following properties:
    • Upstream Mach number for the normal shock wave, MxM_x;
    • Turn angle, θ\theta;
    • Ratios for pressure, temperature, density, and stagnation pressure;
    • Downstream Mach number for the oblique shock wave, M2M_2; and
    • Downstream Mach number for the normal shock wave, MyM_y.

Flow properties
If you have the upstream properties, you can use the advanced mode of this calculator to find the values of downstream pressure, temperature, density, and stagnation pressure, or vice versa.

Example: Using the oblique shock angle calculator

Find the pressure, temperature, and density ratio for an oblique shock wave having a wave angle of 2020^\circ and upstream Mach number, 5.

To find the oblique shock wave properties:

  1. Enter the upstream Mach number, M1=5M_1 = 5.
  2. Fill in the wave angle, β=20\beta = 20^\circ.
  3. Using the oblique shock relations:
    • Turn angle, θ=10.664\theta = 10.664^\circ;
    • Pressure ratio, p2/p1=3.245p_2/p_1 = 3.245 ;
    • Density ratio, ρ2/ρ1=2.214\rho_2/\rho_1 = 2.214 ; and
    • Temperature ratio, T2/T1=1.465T_2/T_1 = 1.465 .


What is an oblique shock wave?

The shock wave developed from the supersonic flow inclined to the local fluid flow is known as the oblique shock wave. This phenomenon results in a decrease of stagnation pressure and increases in entropy of the system. It has both desirable and undesirable effects.

What is a normal shock wave?

When the developed shock wave is perpendicular to the direction of the local fluid flow, it is known as the normal shock wave. The shock wave is considered as a very thin line across which the velocity of the flow decreases and so does the downstream Mach number.

How do I calculate pressure ratio for oblique shock wave?

To obtain the pressure ratio:

  1. Multiply the upstream Mach number by the sine of the wave angle.
  2. Find the square of the product in the previous step.
  3. Subtract 1 from the square.
  4. Multiply it by 2 and specific heat ratio.
  5. Divide the product by the sum of specific heat ratio and 1 to obtain the pressure ratio.

How do I calculate density ratio for oblique shock wave?

To obtain the density ratio:

  1. Multiply the upstream Mach number by the sine of the wave angle.
  2. Find the square of the product in the previous step.
  3. Multiply the square with the sum of specific heat ratio and 1 to obtain the numerator.
  4. Subtract 1 from the specific heat ratio.
  5. Multiply the difference with the square obtained in step 2.
  6. Add 2 to the product to obtain the denominator.
  7. Divide the numerator in step 3 with the denominator in step 6 to obtain the density ratio.
Rahul Dhari
Determine the flow properties before and after the oblique shock wave. Enter the upstream Mach number and wave angle to begin.
Mach number (M1) (Oblique shock)
Mach number (Mx) (Normal shock)
Wave angle (β)
Turn angle (θ)
Pressure ratio (p2/p1)
Stagnation pressure ratio (p02/p01)
Density ratio (ρ2/ρ1)
Temperature ratio (T2/T1)
Mach number (M2) (Oblique shock)
Mach number (My) (Normal shock)
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