Oblique Shock Wave Calculator

An oblique shock is a shock wave inclined at an angle beta to the upstream flow direction, formed when supersonic flow is turned into itself by a wedge, cone, or compression corner. Unlike a normal shock, it deflects the flow by an angle theta while also decelerating the component of velocity normal to the shock.

Given M1 and beta, the deflection angle theta has a direct closed-form formula. Given M1 and theta instead, finding beta requires solving a cubic-like equation numerically since two valid shock angles (weak and strong) can produce the same theta. Taking beta as the input avoids that iterative root-finding step entirely.

The Mach angle mu = asin(1/M1) is the angle of the weakest possible disturbance (a Mach wave) that supersonic flow can generate. An oblique shock only exists for beta strictly greater than mu; at beta = mu the shock strength drops to zero and it becomes a Mach wave, and beta below mu is not physically possible for this M1.

At beta = 90 degrees the oblique shock relations reduce exactly to the normal shock relations: the deflection angle theta becomes zero and Mn1 equals M1. A normal shock is simply the limiting case of an oblique shock at the maximum possible shock angle.

For a given M1 and deflection angle theta less than theta_max, two shock angles beta satisfy the theta-beta-M relation: a smaller beta (weak shock, usually leaves M2 supersonic) and a larger beta (strong shock, always leaves M2 subsonic). This calculator takes beta directly as an input, so it always returns whichever solution corresponds to the beta you enter, and the weak branch is what naturally forms on an unconfined wedge.

Yes. Unlike a normal shock, where M2 is always subsonic, an oblique shock only decelerates the velocity component normal to the shock. For most weak-shock solutions, M2 remains supersonic, which is why oblique shocks are used in supersonic inlets to slow the flow gradually while avoiding the large total-pressure loss of a single normal shock.

theta_max is the largest deflection angle a given M1 can achieve with an attached oblique shock. If a wedge half-angle exceeds theta_max for the approaching Mach number, no attached oblique shock solution exists, and the shock detaches into a curved bow shock standing off the wedge.

The stagnation pressure ratio p02/p01 depends only on the Mach number normal to the shock, Mn1 = M1 sin(beta). A smaller beta gives a smaller Mn1 and therefore a weaker shock with less total-pressure loss, which is exactly why multi-shock supersonic inlets use a series of shallow oblique shocks rather than one steep shock.

Yes, though the effect is secondary to M1 and beta. Air uses gamma = 1.4, but combustion products, monatomic gases, and high-temperature real-gas effects have different gamma values, and this calculator lets you override the default to model any calorically perfect gas.

A normal shock is a special case of an oblique shock at beta = 90 degrees, always facing the flow straight on and always leaving M2 subsonic. This calculator handles the general inclined case for any beta above the Mach angle, which is the relevant geometry for wedges, cones, and supersonic inlet ramps rather than a shock standing perpendicular to the flow.