Oblique Shock Calculator
Enter the upstream Mach number, deflection angle (wedge half-angle), and specific heat ratio. The calculator solves the theta-beta-Mach relation for the weak-shock wave angle, then applies the Rankine-Hugoniot oblique shock relations to give you downstream Mach number, pressure ratio, density ratio, temperature ratio, and stagnation pressure ratio. Results update instantly as you type. Works for air (gamma = 1.4) or any ideal gas.
What is an oblique shock wave?
When supersonic flow encounters a solid surface angled to the flow, such as a wedge or ramp, it must turn to follow the surface. Rather than turning gradually, supersonic flow adjusts almost instantaneously through a thin, angled wave called an oblique shock. Unlike a normal shock (which stands perpendicular to the flow and always drops the flow to subsonic), an oblique shock stands at an angle beta to the upstream flow and, for most practical angles, leaves the downstream flow still supersonic. The downstream Mach number is lower, and pressure, density, and temperature are all higher behind the shock. Because oblique shocks are less severe than a normal shock at the same upstream Mach, they are used deliberately in the inlet designs of supersonic aircraft and missiles to compress air more efficiently.
The theta-beta-M relation and how this calculator solves it
Three quantities are linked by the theta-beta-Mach (TBM) relation: the upstream Mach number M1, the flow-deflection angle theta (the wedge half-angle), and the shock wave angle beta. Given M1 and theta, the TBM relation is a transcendental equation with no closed-form solution, so the calculator solves it numerically using a bisection search on the weak-shock branch. A weak shock produces a smaller wave angle and leaves supersonic flow downstream; a strong shock gives a larger beta and typically subsonic downstream flow. In practice, weak shocks occur naturally unless the downstream pressure forces a strong solution. Once beta is found, the calculator applies the Rankine-Hugoniot relations, which use only the component of M1 normal to the shock (called Mn1 = M1 sin(beta)) in the same way as normal-shock relations.
What the outputs mean for engineering and aerodynamics
The pressure ratio p2/p1 directly sets the compression achieved by the shock, which determines the pressure entering a supersonic engine core. The stagnation pressure ratio p02/p01 is the key efficiency indicator: values close to 1 mean the shock is nearly isentropic and very little useful work is lost; values significantly below 1 mean substantial irreversible entropy generation. For an inlet with three oblique shocks each having p02/p01 = 0.98, the overall total pressure recovery is 0.98 cubed = 0.94, far better than a single normal shock at the same upstream Mach would give. The temperature ratio T2/T1 matters for materials: the kinetic heating behind the shock determines the thermal load on the wedge surface. The downstream Mach M2 sets up the conditions for any subsequent shock or expansion in the flow path.
Shock detachment and the maximum deflection angle
For every upstream Mach number, there is a maximum deflection angle theta_max beyond which no attached oblique shock solution exists. If theta exceeds theta_max, the shock detaches from the leading edge and forms a curved bow shock that stands off in front of the body. At the stagnation point of a detached bow shock the wave is locally normal to the flow; it curves around the body becoming progressively more oblique away from the centerline. The calculator shows theta_max for your M1 and gamma. Supersonic aircraft and missile nose cones must keep their half-angles below theta_max for the design Mach number, or accept the high drag of a detached bow shock. As Mach number increases, theta_max also increases, so faster vehicles can use blunter shapes without detachment.
Oblique shock results for air (gamma = 1.4) at selected Mach numbers
| M1 | Beta (deg) | M2 | p2/p1 | T2/T1 | p02/p01 |
|---|---|---|---|---|---|
| 1.5 | 56.68 | 1.114 | 1.666 | 1.162 | 0.9866 |
| 2 | 39.31 | 1.641 | 1.707 | 1.17 | 0.9846 |
| 2.5 | 31.85 | 2.086 | 1.864 | 1.203 | 0.9759 |
| 3 | 27.38 | 2.505 | 2.054 | 1.242 | 0.9631 |
| 4 | 22.23 | 3.286 | 2.506 | 1.329 | 0.9254 |
| 5 | 19.38 | 3.999 | 3.044 | 1.429 | 0.8725 |
Weak-shock solutions for a deflection angle of theta = 10 deg. Beta is the wave angle, M2 is the downstream Mach number.
Frequently asked questions
What is the difference between an oblique shock and a normal shock?
A normal shock stands perpendicular to the flow and always decelerates the flow to subsonic speeds, with large losses in total pressure. An oblique shock stands at an angle beta (greater than the Mach angle) and, for weak solutions, leaves the flow supersonic with smaller entropy generation and less total pressure loss. Engineers deliberately use oblique shocks in supersonic inlets because multiple weak oblique shocks can compress the flow much more efficiently than a single normal shock at the same upstream Mach.
What is the Mach angle and how is it related to beta?
The Mach angle mu = arcsin(1/M1) is the half-angle of the Mach cone: the weakest possible wave attached to a body moving supersonically, producing zero pressure change. The oblique shock wave angle beta always lies between mu and 90 deg. As the deflection angle theta approaches zero, beta approaches mu (a vanishingly weak shock). As theta approaches theta_max, beta approaches the strong-shock value. A beta of 90 deg corresponds to a normal shock.
What specific heat ratio should I use?
For air at standard conditions and moderate temperatures (below about 800 K), use gamma = 1.4. At very high temperatures, vibrational modes of O2 and N2 become active and gamma drops toward 1.3. For monatomic noble gases (helium, argon), gamma = 5/3 = 1.667. For diatomic gases at very high temperature or combustion products, use 1.2 to 1.3. The default of 1.4 is appropriate for most aerodynamics problems with air.
What does the stagnation pressure ratio tell me?
Stagnation (total) pressure p0 represents the pressure the flow would have if brought to rest isentropically. Across a shock, entropy increases and p0 always decreases. The ratio p02/p01 quantifies how much total pressure the shock retains: 1.0 would be a perfectly isentropic process, which no real shock achieves. For oblique shock inlet design, maximizing p02/p01 means you preserve more of the flow energy for combustion or thrust. A ratio of 0.95 means 5% of the inlet total pressure is lost irreversibly.
Why does downstream Mach M2 decrease from M1 across the shock?
A shock wave is a compression process: pressure and temperature rise, while flow speed (relative to the pre-shock frame) drops. Because the speed of sound rises with temperature (c = sqrt(gamma R T)), and the flow speed also drops, both effects work together to reduce the Mach number. For a weak oblique shock the Mach drop is modest and M2 remains supersonic; for a stronger shock the drop is larger and M2 can fall below 1.