Oblique Shocks in Supersonic Flow

When a supersonic stream encounters a wedge or a compression corner, it cannot turn the flow abruptly without generating discontinuities. Instead, nature forms an oblique shock that emanates from the corner at an angle . Across this thin region, the flow direction changes by the wedge angle $\theta$, the Mach number drops, and pressure, temperature, and density jump to higher values. Unlike a normal shock, an oblique shock preserves some component of velocity along the shock front, enabling the flow to remain supersonic downstream if the deflection is modest. Understanding the geometry and strength of these shocks is crucial in designing supersonic aircraft, inlets, and nozzles where controlling compression and total pressure loss is paramount.

The Theta-Beta-Mach Relation

The key relationship governing oblique shocks is the transcendental theta-beta-Mach equation. For a perfect gas, it links upstream Mach number $M_1$, wedge angle $\theta$, shock angle $\beta$, and heat capacity ratio $\gamma$:

This implicit expression admits two solutions for $\beta$: a weak shock with a smaller angle and a strong shock with a larger angle that often yields subsonic downstream flow. The weak solution typically occurs in external aerodynamics because it requires less pressure rise and therefore less entropy generation. Solving the equation for $\beta$ requires numerical methods; our calculator uses a simple bisection search between the Mach angle and ninety degrees to locate the physically relevant root for the chosen $\theta$.

Property Changes Across the Shock

Once the shock angle is known, the flow can be decomposed into components normal and tangential to the shock. Only the normal component experiences the abrupt compression; the tangential component remains unchanged. The normal Mach number upstream is $M_{\mathrm{n1}}=M_1\sin \beta$. Applying the normal shock relations yields the downstream normal Mach number:

The downstream Mach number is then $M_2=M_{\mathrm{n2}}/\sin (\beta -\theta )$. Static pressure, temperature, and density follow from the normal shock relations using $M_{\mathrm{n1}}$. For example, the pressure ratio across the shock is

Downstream velocity remains supersonic as long as the deflection angle is below the maximum that still allows a real solution for $\beta$. Exceeding this detachment angle forces the shock to stand off the surface and behave more like a normal shock, dramatically raising drag.

Engineering Applications

Designers of supersonic aircraft intakes rely on controlled sequences of oblique shocks to compress air efficiently before it enters the engine. Each shock increases pressure and temperature while decreasing Mach number. A series of weak shocks results in smaller total-pressure losses than a single normal shock performing the same overall compression. Similarly, supersonic wind tunnels employ adjustable wedges or cones to generate predictable shocks for aerodynamic testing. In rocket nozzles, internal oblique shocks may form if the nozzle is overexpanded, causing flow separation and performance loss. Understanding the theta-beta-M relation helps engineers anticipate these behaviors and shape surfaces to manage shocks deliberately.

Sample Shock Calculations

The following table illustrates how varying the deflection angle affects downstream conditions for a flow at $M_1=3$ in air ($\gamma =1.4$). Only the weak-shock solution is considered.

θ (deg)	β (deg)	M₂	p₂/p₁	T₂/T₁
5	22.5	2.60	1.22	1.05
10	27.8	2.15	1.55	1.13
15	34.6	1.72	2.18	1.28
20	44.3	1.29	3.44	1.56

Using the Calculator

Provide the upstream Mach number, the flow deflection angle, and the heat capacity ratio. The script searches for a shock angle satisfying the theta-beta-M equation, then computes downstream Mach number and property ratios. Results help estimate how much compression and loss occur for a given wedge or corner. Because all calculations run locally in your browser with no external dependencies, you can experiment freely by adjusting parameters or exploring different gases. Try increasing the deflection angle until no solution exists to observe the onset of shock detachment. Such interactive exploration deepens intuition for compressible-flow phenomena encountered in high-speed aerodynamics.