Supersonic Shock Wave Calculator
When a body travels faster than the local speed of sound (Mach number greater than 1), it generates shock waves. This calculator computes normal shock wave properties using the Rankine-Hugoniot relations for a calorically perfect gas (gamma = 1.4 for air), including the pressure ratio, temperature ratio, density ratio, and downstream Mach number. It also computes the Mach cone half-angle. These relations are fundamental to supersonic aerodynamics and are derived from conservation of mass, momentum, and energy across the shock.
Normal shock relations (Rankine-Hugoniot)
Mach angle = arcsin(1 / M1) degrees
M2^2 = ((gamma-1)*M1^2 + 2) / (2*gamma*M1^2 - (gamma-1))
P2/P1 = (2*gamma*M1^2 - (gamma-1)) / (gamma+1)
T2/T1 = P2/P1 * (2 + (gamma-1)*M1^2) / ((gamma+1)*M1^2)
rho2/rho1 = (gamma+1)*M1^2 / ((gamma-1)*M1^2 + 2)
These are the classical Rankine-Hugoniot relations for a stationary normal shock in a calorically perfect gas. They are derived by applying conservation of mass, momentum, and energy across the shock, with the perfect gas equation of state. Source: NACA Report 1135 (1953) "Equations, Tables, and Charts for Compressible Flow."
Applications of shock wave analysis
- Supersonic inlet design: oblique shocks are used to decelerate air gradually before a normal terminating shock, reducing total pressure loss compared to a single normal shock.
- Sonic boom prediction: the Mach cone angle determines the lateral extent and timing of sonic boom arrival on the ground.
- Hypersonic heating: temperature ratio across the shock is used to estimate aerodynamic heating rates on leading edges and nose tips.
- Wind tunnel design: normal shock tables are used to set test-section Mach numbers and interpret pitot probe readings in supersonic tunnels.
Supersonic shock wave calculator: frequently asked questions
What is a Mach cone?
When an object moves faster than the speed of sound, it creates a conical pressure wave called a Mach cone. The half-angle of this cone (the Mach angle) is given by arcsin(1/M), where M is the Mach number. At Mach 1, the angle is 90 degrees. At Mach 2, it is 30 degrees. The sonic boom heard on the ground is the intersection of this Mach cone with the surface.
What is the difference between a normal shock and an oblique shock?
A normal shock wave is perpendicular to the flow direction. It produces the maximum possible pressure rise for a given Mach number and always produces subsonic flow downstream. An oblique shock is angled to the flow; it produces a weaker pressure rise and can leave the flow supersonic downstream. Oblique shocks occur on supersonic aircraft leading edges and inlet ramps.
How is the pressure ratio across a normal shock calculated?
The normal shock pressure ratio is: P2/P1 = (2*gamma*M1^2 - (gamma-1)) / (gamma+1), where gamma is the ratio of specific heats (1.4 for air), M1 is the upstream Mach number, and P1 and P2 are static pressures upstream and downstream of the shock. This relation comes from the Rankine-Hugoniot conditions for a normal shock in a perfect gas.
What happens to temperature across a shock wave?
Temperature increases across a shock wave. The normal shock temperature ratio is: T2/T1 = (2*gamma*M1^2 - (gamma-1)) * ((gamma-1)*M1^2 + 2) / ((gamma+1)^2 * M1^2). The kinetic energy of the supersonic flow is converted to thermal energy in the shock, which is why hypersonic vehicles experience severe aerodynamic heating at high Mach numbers.
What is the Mach number downstream of a normal shock?
The downstream Mach number M2 is always subsonic (less than 1) after a normal shock. The formula is: M2^2 = ((gamma-1)*M1^2 + 2) / (2*gamma*M1^2 - (gamma-1)). At Mach 2, the downstream Mach number is approximately 0.577. At very high Mach numbers, M2 approaches sqrt((gamma-1)/(2*gamma)) = 0.378 for air.
Official sources
- NACA: NACA Report 1135: Equations, Tables, and Charts for Compressible Flow (1953).
- NASA Glenn Research Center: Normal Shock Wave Relations.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.