Oblique Shock Wave Relations Calculator

📝 Configuration

Solve For Unknown:

Upstream Mach number ($M_1$):

Deflection Angle ($\theta$) [degrees]:

Shock Wave Angle ($\beta$) [degrees]:

Specific Heat Ratio ($\gamma$):

Quick Presets:

📊 Results & Diagrams

Configure inputs and click Calculate Oblique Shock to view results.

📘 Calculation Methodology

Mathematical Model & Theory

Oblique shock waves occur when supersonic flow encounters a boundary deflection forcing the flow in on itself. The relations are governed by the **$\theta$-$\beta$-$M$ equation**:

$$\tan\theta = 2\cot\beta \left[ \frac{M_1^2\sin^2\beta - 1}{M_1^2(\gamma + \cos(2\beta)) + 2} \right]$$

Where:

$\theta$: Wedge deflection angle
$\beta$: Shock wave angle
$M_1$: Upstream Mach number
$\gamma$: Specific heat ratio

Downstream properties are determined by resolving the upstream Mach number normal to the shock wave, $M_{n1} = M_1 \sin\beta$, applying the **Normal Shock Equations** to get $M_{n2}$, and calculating $M_2 = M_{n2} / \sin(\beta - \theta)$.

Academic References

Anderson, John D.: Modern Compressible Flow: With Historical Perspective, McGraw-Hill.
Ames Research Staff: Equations, Tables, and Charts for Compressible Flow, NACA Report 1135.

Worked Engineering Example

Problem Statement:
Supersonic air flow ($\gamma = 1.4$) at $M_1 = 2.0$ passes over a wedge deflection of $\theta = 10^\circ$. Calculate the weak shock wave angle $\beta$ and downstream Mach number $M_2$.

Step-by-step Solution:
1. Evaluate the $\theta$-$\beta$-$M$ relation for $\beta$ in range $[\mu, 90^\circ]$ where $\mu = \arcsin(1/2) = 30.0^\circ$.
2. Solving the implicit relation numerically (weak branch root) yields:
$$\beta = 39.31^\circ$$ 3. Compute the upstream normal Mach component:
$$M_{n1} = 2.0 \times \sin(39.31^\circ) = 1.267$$ 4. Apply normal shock relations for $M_{n2}$:
$$M_{n2} = \sqrt{\frac{2 + 0.4 \times (1.267)^2}{2.8 \times (1.267)^2 - 0.4}} = 0.803$$ 5. Calculate downstream Mach number $M_2$:
$$M_2 = \frac{0.803}{\sin(39.31^\circ - 10.0^\circ)} = 1.64$$ Final Results:
• Shock angle: 39.31° (Weak)
• Downstream Mach: 1.64