Oblique Shock Calculator

Introduction

When a supersonic stream encounters a wedge or a compression corner, it cannot turn the flow abruptly without generating a discontinuity. Instead, the flow creates an oblique shock that leaves the corner at an angle $\beta$. Across that thin shock layer, the flow turns by the deflection angle $\theta$, the Mach number decreases, and the static pressure, density, and temperature increase. Unlike a normal shock, however, an oblique shock does not destroy all of the streamwise motion. The velocity component parallel to the shock remains unchanged, so modest turns can still leave the downstream flow supersonic.

This page is designed for the common engineering case of an attached oblique shock in a perfect gas. That is the classic situation for wedges, external compression surfaces, and many textbook compressible-flow problems. The calculator helps you find the shock angle, identify whether the weak or strong branch is being used, and estimate how much compression occurs across the shock. If your selected turn angle is too large for the chosen upstream Mach number and gas, the page also warns you that an attached solution does not exist and reports the estimated maximum attached deflection angle.

That combination of geometry and thermodynamics is why the oblique shock problem matters so much. Designers of supersonic inlets, missile forebodies, nozzles, and wind-tunnel hardware all need fast ways to predict how much a compression corner will turn the flow and how severe the losses will be. In practice, the most useful first answer is often not just the shock angle itself, but the whole set of downstream properties that follow from it. This calculator gives those values in one place so you can move quickly from a chosen wedge geometry to a physically interpretable result.

How to Use

Start by entering the upstream Mach number $M_1$. Because oblique shocks are a supersonic phenomenon, this value must be greater than 1. Next enter the deflection angle $\theta$ in degrees. This is the amount by which the wall or wedge turns the flow. Finally, enter the heat capacity ratio $\gamma$. For dry air at ordinary high-speed-aerodynamics conditions, $\gamma =1.4$ is the usual starting point.

The branch selector lets you choose between the weak and strong solutions of the theta-beta-Mach relation. The weak branch is the one most frequently observed in external aerodynamics because it produces a smaller shock angle and usually preserves a larger downstream Mach number. The strong branch is still physically meaningful, but it corresponds to a much steeper shock and often drives the flow subsonic after the turn. If you are comparing alternatives, it is useful to compute both branches and notice how strongly the downstream Mach number and property ratios change.

After you click Compute Shock, the result panel reports the shock angle $\beta$, the upstream normal Mach number, the downstream Mach number, and the pressure, density, and temperature ratios across the shock. If the page tells you that no attached solution exists, the message does not mean the flow suddenly becomes impossible. It means that the simple attached oblique-shock model has reached its limit for that geometry. In that regime the shock detaches from the corner and stands off from the body, so a different physical picture is needed.

A practical way to interpret the result is to read it in three passes. First, look at $\beta$ and ask whether the shock is relatively shallow or close to normal. Second, inspect $M_2$ to see whether the downstream flow remains supersonic. Third, look at the pressure ratio and temperature ratio to judge how aggressive the compression is. Those three checks together usually tell you whether a proposed wedge angle is gentle, efficient, and likely to fit the intended design goal.

Formula

The central relationship 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 equation is implicit in $\beta$, so there is usually no tidy algebraic formula you can solve by hand. Instead, the script numerically searches between the Mach angle and ninety degrees and finds the root corresponding to the requested branch. That matters because the same upstream state and turn angle can admit two attached solutions: the weak shock with a smaller $\beta$, and the strong shock with a larger $\beta$. In ordinary external flows, the weak solution is the one that the flow naturally selects.

Once the shock angle is known, the oblique-shock problem becomes a normal-shock problem applied only to the velocity component perpendicular to the shock. The upstream normal Mach number is $M_{\mathrm{n1}}=M_1\sin \beta$. Applying the normal-shock relations then gives the downstream normal Mach number:

Mn₂² = [1 + ((γ - 1) / 2)Mn₁²] / [γMn₁² - ((γ - 1) / 2)]

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

The picture to keep in mind is simple: the stronger the normal component of the incoming flow, the stronger the compression. A larger shock angle produces a larger normal Mach number, which raises the pressure jump and usually lowers the downstream Mach number more dramatically. That is why the strong branch is much more punishing than the weak branch. It is also why small changes in geometry can create large changes in flow behavior when the turn angle approaches the maximum attached value.

Worked Example

Consider a flow with upstream Mach number $M_1=3$ turning through $\theta =10°$ in air with $\gamma =1.4$. If you choose the weak branch, the calculator finds a shock angle near 27.4°. That angle is noticeably larger than the wedge angle because the shock must sit ahead of the corner and create a finite normal velocity component before the flow can turn.

From that shock angle, the upstream normal Mach number comes out to about 1.38. That is the quantity that actually controls the shock strength. Feeding it through the normal-shock relations gives a downstream Mach number of roughly 2.50, a pressure ratio p₂/p₁ ≈ 2.05, a density ratio ρ₂/ρ₁ ≈ 1.66, and a temperature ratio T₂/T₁ ≈ 1.24. The important physical reading is that the flow has compressed substantially, but it is still moving at a clearly supersonic speed after the turn.

If you repeat the same case on the strong branch, the shock angle shifts much closer to ninety degrees and the compression becomes far harsher. That is a useful comparison because it shows why the weak branch is the standard expectation in external aerodynamic applications. It achieves the required turn while keeping the downstream flow faster and the losses lower. The worked example therefore illustrates the main design instinct behind oblique-shock systems: prefer several gentle compressions over one severe one whenever the geometry allows it.

The pattern is the part worth remembering. As the deflection angle increases, the shock angle rises, the compression becomes stronger, the pressure ratio climbs, and the downstream Mach number falls. That trend is what the calculator makes easy to explore. You can quickly see how close a proposed corner is to the maximum attached deflection, and you can observe how sensitive the downstream state becomes as you move toward that limit.

Limitations and Assumptions

This calculator assumes a perfect gas with constant $\gamma$. That is a very useful approximation for many classroom and early-stage design problems, but it is not exact for high-enthalpy flows, strong real-gas effects, chemical reactions, or situations where the specific heats vary substantially with temperature. If you are analyzing hypersonic or very high-temperature applications, you should treat the result as a first estimate rather than a final answer.

The page also models a two-dimensional attached oblique shock. It does not include detached bow shocks, three-dimensional cone shocks, boundary-layer interaction, shock-induced separation, wall curvature, viscous effects, or unsteady motion. Real hardware can depart from the ideal model quickly, especially near the detachment limit or inside inlets where multiple shocks and boundary layers interact. When the calculator reports that no attached solution exists, that is the model telling you that the simple wedge picture has broken down.

Finally, remember that all of the reported ratios are based on static properties across the shock, and the inputs are dimensionless except for the deflection angle in degrees. The tool is excellent for comparing cases and building intuition, but it should be supplemented with fuller compressible-flow analysis when detailed geometry, losses, or inlet performance margins matter. In other words, this calculator is best used as a fast, transparent first-pass solver for attached-shock behavior.

Engineering Applications

Controlled oblique shocks are central to supersonic inlet design. Rather than allowing a single normal shock to slow the flow abruptly and waste total pressure, engineers often use one or more oblique shocks to compress the air in stages before it reaches the engine face. Each stage increases static pressure while limiting the severity of the losses. The calculator is therefore useful whenever you want to estimate how much one wedge, ramp, or corner contributes to that overall compression system.

External aerodynamics offers another common use case. Fins, forebodies, control surfaces, and test wedges in wind tunnels all generate oblique shocks when the local flow is supersonic. In rocket nozzles or high-speed ducts, internal oblique shocks can also appear when the geometry or back pressure forces the flow to compress. In each of those settings, knowing the likely shock angle and downstream Mach number helps you judge loading, drag, heating trends, and whether the chosen geometry is gentle enough to remain attached.

Sharing Your Shock Results

After you evaluate a case, you can use the Copy Result button to place the main outputs on your clipboard. That is convenient when you want to compare several wedge angles, paste values into notes, or send a quick design check to a teammate. Because everything runs in your browser, the page is also useful as a lightweight study tool: change one variable at a time, compare weak and strong branches, and watch how the maximum attached deflection shifts with Mach number and gas properties.