Mach Angle Calculator

What is Mach angle?

The Mach angle μ is the half-angle of the cone formed by pressure waves around a supersonic object. When an aircraft, rocket, or bullet flies faster than the local speed of sound, the sound waves it emits pile up into a conical shock front. The angle between the direction of motion and this cone is the Mach angle.

For a flow with Mach number M (speed divided by speed of sound), the Mach angle is given by a simple relation:

Mach angle formula: μ = arcsin(1 / M) (for M > 1)

This angle is usually reported in degrees. As the Mach number increases, the Mach angle becomes smaller and the cone becomes narrower.

Mach angle formula and derivation

Consider an object moving at speed V through a medium where the speed of sound is a. The Mach number is defined as:

As the object moves, it emits sound waves that propagate outward at speed a. In a time interval t, the object travels a distance Vt, while a sound pulse travels a distance at. Connecting the wavefronts forms a right triangle, with the Mach angle μ between the direction of motion and the wavefront.

From this geometry, the sine of the Mach angle is:

Rearranging gives the standard Mach angle relation:

μ = arcsin(1 / M), valid only for M > 1.

Key implications:

• At M = 1, 1 / M = 1, so μ = 90° and the wavefront is effectively a plane perpendicular to the motion rather than a cone.
• For M > 1, the arcsine is defined and the result is an angle between 0° and 90°.
• As M → ∞, 1 / M → 0, so μ → 0° and the cone collapses toward the flight path.

How to use the Mach Angle Calculator

1. Enter Mach number M > 1. This is the ratio of the object speed to the local speed of sound. The calculator is meaningful only for supersonic values.
2. (Optional) Enter distance from the source in meters. This is a straight-line distance behind the object along its path. If provided, the calculator estimates how wide the shock cone is at that distance.
3. Click the calculate button. The tool computes the Mach angle in degrees, and if a distance is entered, it also computes the cone radius at that location.

The internal calculations are:

• Mach angle (degrees): μ = arcsin(1 / M), converted from radians to degrees.
• Cone radius (meters) at distance L from the source: r = L × tan(μ).

Output units:

• Mach angle: degrees
• Cone radius: meters

Worked example

Suppose a jet is flying at M = 2.0 and you want to know the Mach angle and the approximate radius of the shock cone 10 meters behind the aircraft.

1. Compute the Mach angle:
   • μ = arcsin(1 / 2) = arcsin(0.5) = 30°.
2. Compute cone radius at 10 m:
   • r = 10 × tan(30°) ≈ 10 × 0.577 ≈ 5.77 m.

The shock cone opens at 30° from the flight path, and 10 meters behind the nose its radius is about 5.8 meters.

Another example is a bullet traveling at M = 3.0:

• μ = arcsin(1 / 3) ≈ 19.5°, a much narrower cone.
• At L = 2 m behind the bullet, r ≈ 2 × tan(19.5°) ≈ 0.71 m.

Interpreting the results

The Mach angle tells you how widely the disturbance from a supersonic object spreads:

• Larger Mach angle (e.g., 40°–60°): occurs at lower supersonic speeds (M just above 1). The cone is wide, so the affected region extends far to the sides.
• Smaller Mach angle (e.g., 10°–20°): occurs at high Mach numbers (hypersonic flow). The cone is narrow and closely hugs the flight path.

The optional cone radius helps you visualize how far from the path the shock front reaches at a given distance behind the object. This is useful for conceptualizing sonic boom footprints, sensor placement, or shock interaction with nearby structures.

Mach angle vs. other aerodynamic quantities

The table below compares the Mach angle with related concepts often used in high-speed aerodynamics.

Connection to sonic booms and applications

As a supersonic aircraft flies overhead, its Mach cone sweeps across the ground. Observers inside this cone experience a rapid pressure rise, perceived as a sonic boom. The Mach angle helps indicate how far to the side of the flight path the boom can be heard, while altitude and trajectory determine when the cone intersects the ground.

Typical applications of Mach angle calculations include:

• Aircraft and missile design: understanding how shock waves interact with the airframe and nearby structures.
• Wind tunnel testing: interpreting schlieren images and planning sensor locations to capture shock features.
• Range safety and instrumentation: positioning microphones and pressure sensors to record sonic booms or shock signatures.
• Ballistics and rocketry: visualizing the conical shock generated by bullets and launch vehicles.

Assumptions and limitations

This calculator uses a simplified geometric model. Important assumptions are:

• Supersonic flow only (M > 1): the Mach angle formula is not defined for subsonic or exactly sonic flow in the same way. If M ≤ 1, no Mach cone exists.
• Constant speed of sound: the local speed of sound is treated as fixed, so changes with temperature, humidity, and altitude are not modeled.
• Far-field, weak wave approximation: the relation μ = arcsin(1 / M) describes Mach waves and the overall cone geometry, not detailed near-field shock structures around complex shapes.
• Straight-line motion: the distance-based cone radius assumes the object travels in a straight path at constant Mach number.
• No loudness prediction: the cone radius is a geometric visualization only. It does not estimate sonic boom intensity or detailed ground footprint.

For rigorous design work, engineers often pair this simple relation with more advanced tools such as oblique shock calculators, computational fluid dynamics, or dedicated sonic boom prediction codes.

Frequently asked questions

Is Mach angle the same as shock wave angle?

Not exactly. The Mach angle describes the cone formed by infinitesimally weak disturbances in a uniform supersonic flow. The shock wave angle usually refers to the angle of a finite-strength oblique shock attached to a body or wedge, which depends on Mach number and flow deflection. At small deflection angles and weak shocks, the two angles can be similar, but they are not generally identical.

What happens to Mach angle as Mach number increases?

As Mach number increases, the Mach angle decreases. For example, at M = 1.2 the Mach angle is about 56°, at M = 2 it is 30°, and at M = 5 it is about 11.5°. This means high-speed vehicles confine their disturbances to a narrow region around the flight path.

Can you have a Mach angle below Mach 1?

No. The formula μ = arcsin(1 / M) requires 1 / M ≤ 1, which implies M ≥ 1. For M < 1, the expression would require the arcsine of a value greater than 1, which is not defined in real numbers. Physically, subsonic objects do not create a Mach cone.

Does altitude change the Mach angle?

The Mach angle depends on Mach number, not directly on altitude. However, altitude affects the speed of sound, so a given true airspeed corresponds to different Mach numbers at different altitudes. If the Mach number changes, the Mach angle changes accordingly.