Sonic Boom (Mach Cone)

The intuition — outrunning your own sound

Every moving object pushes the air aside, sending out pressure ripples — sound waves — that spread spherically at the local speed of sound, about 340 m/s in air at 15 °C. As long as the object is slower than that, its sound runs ahead of it: you hear an approaching car or subsonic jet before it arrives.

Now speed the object up to the sound speed and beyond. It can no longer outrun the waves it is making; it catches up to and overtakes its own sound. The ripples it emitted a moment ago, a second ago, ten seconds ago, all bunch together along a single sloped surface trailing behind it. Thousands of weak waves stack edge-to-edge into one abrupt, nearly discontinuous jump in pressure — a shock wave. In three dimensions, that pile-up wraps around the object as a cone. When the cone's surface sweeps over your ears, the near-instantaneous pressure jump hits you as the bang we call a sonic boom.

How the cone forms

Picture the object emitting a pressure pulse every instant. Each pulse expands as a sphere centered on where the object was when it emitted it.

• Subsonic (M < 1). The object stays inside all its own spheres. The spheres bunch up ahead (higher pitch) and spread behind (lower pitch) — that's the Doppler effect, but no shock forms.
• Sonic (M = 1). The object keeps pace with the leading edge of its spheres. They all touch at one point in front of it, forming a flat wall of piled-up sound — the famous "sound barrier."
• Supersonic (M > 1). The object runs ahead of its own spheres. The common tangent to all those spheres is a cone whose apex sits at the object. Everything inside the cone is the "zone of action"; outside, the air hasn't heard the object yet — it's the "zone of silence." That's why a supersonic jet arrives in dead quiet, then booms.

The geometry is exact. In the time the object travels a distance v·t, the earliest sphere has expanded to radius c·t (c = sound speed). The cone's half-angle μ is set by the right triangle linking those two:

sin μ = (c·t) / (v·t) = c / v = 1 / M

This μ is the Mach angle, the single number that describes the whole cone.

The governing physics

The dimensionless Mach number compares object speed to the local sound speed:

M = v / c

The sound speed in an ideal gas depends only on temperature (not pressure), via:

c = sqrt(γ · R · T / Mₘ)     (γ = 1.4 for air, R = 8.314 J/mol·K, Mₘ = 0.0289 kg/mol)

c ≈ 331.3 · sqrt(1 + T_C / 273.15)  m/s     (T_C = temperature in °C)

At sea level, 15 °C: c ≈ 340 m/s. At a typical cruise of 11 km, the air is about −56 °C, so c drops to ≈ 295 m/s — which is why the same true airspeed means a higher Mach number up high.

The cone half-angle (Mach angle):

μ = arcsin(1 / M)

And the full cone (apex) angle is 2μ. For a blunt body the leading shock is a curved "bow shock" standing slightly detached from the nose; for a slender body it collapses onto the sharp Mach cone above.

Across the shock itself the gas state jumps discontinuously. The Rankine–Hugoniot relations set the downstream pressure, density, and temperature from conservation of mass, momentum, and energy. The pressure ratio across a normal shock is:

p₂ / p₁ = 1 + (2γ / (γ+1)) · (M² − 1)

For an oblique shock at the cone surface, only the velocity component normal to the shock matters, so you replace M with M·sin μ, which makes the on-ground overpressure far gentler than this normal-shock value.

Speed regimes

Regime	Mach number	What the air does
Subsonic	M < 0.8	Smooth, compressible-but-shock-free flow; sound runs ahead. No boom.
Transonic	0.8 ≤ M < 1.2	Local pockets of supersonic flow + weak shocks form on the wing; buffet and drag rise sharply ("sound barrier").
Supersonic	1.2 ≤ M < 5	Well-defined Mach cone; persistent boom carpet. Concorde cruised here at M ≈ 2.
Hypersonic	M ≥ 5	Shock layer hugs the body; air dissociates/ionizes; intense heating. Re-entry capsules, scramjets.

Mach angle and cone by the numbers

Mach M	Speed (340 m/s sound)	Mach angle μ = arcsin(1/M)	Full cone 2μ
1.0	340 m/s (1225 km/h)	90° (flat wall)	180°
1.2	408 m/s	56.4°	112.9°
1.5	510 m/s	41.8°	83.6°
2.0	680 m/s (Concorde ≈ M2)	30.0°	60.0°
3.0	1020 m/s (SR-71 ≈ M3.3)	19.5°	38.9°
5.0	1700 m/s	11.5°	23.1°
10.0	3400 m/s	5.7°	11.5°

The trend is the headline: faster flight makes a narrower, more swept-back cone. At hypersonic speeds the shock is almost wrapped onto the body itself.

Worked example — when does the boom arrive?

An aircraft cruises level at altitude h = 15 km, speed M = 2 (so v = 680 m/s with c = 340 m/s at the surface — we'll use a constant c for simplicity). The Mach angle is μ = arcsin(1/2) = 30°.

The shock cone reaches the ground point directly below where the plane was, but trails behind by a horizontal distance set by the cone slope:

x_lag = h / tan(μ) = 15 km / tan(30°) = 15 / 0.577 ≈ 26 km

So by the time the boom hits a listener, the plane is already about 26 km past overhead — roughly 26 km / 680 m/s ≈ 38 seconds downrange. The listener sees the jet pass in silence, and only ~38 s later does the cone edge sweep over them with the bang. That delay is the dead giveaway that the boom is the trailing cone, not a "moment" of breaking the barrier.

Loudness, the N-wave, and damage

What hits the ground is not a single click but an N-wave: the pressure jumps up sharply at the nose shock, ramps smoothly downward through the expansion over the body, dips below ambient, then jumps back up at the tail shock. Plotted against time it traces the letter N. The two near-vertical edges are what the ear hears as the characteristic double bang.

Source	Overpressure Δp	Effect
High-altitude airliner boom	~50 Pa (1 psf)	Distant thunder; startles, no damage
Concorde at cruise	~100 Pa (2 psf)	Sharp double bang heard over ocean
Fighter, low/fast pass	~150–500 Pa (3–10 psf)	Loud crack; can rattle/crack windows
Very low supersonic pass	>~500 Pa (>10 psf)	Plaster cracks, glass breaks; rare and regulated
NASA X-59 design target	~6 Pa (~75 PLdB)	A soft "thump" — quiet enough to revisit overland bans

Overpressure scales roughly with the aircraft's weight and length and inversely with altitude — heavier, shorter, lower planes boom harder. It does not grow without limit with speed; once supersonic, going faster mostly tightens the cone rather than amplifying the bang.

Where it shows up

• Supersonic aircraft. Concorde, military fighters, the SR-71. The boom carpet is why supersonic civil flight over land has been banned in the U.S. since 1973.
• Bullets and projectiles. A supersonic rifle round drags its own tiny Mach cone; the "crack" downrange is its sonic boom, distinct from the muzzle "bang."
• The bullwhip crack. The whip's tip exceeds Mach 1 and sheds a mini shock — the oldest human-made sonic boom.
• Spacecraft re-entry. Returning capsules and the Space Shuttle produced clearly heard double booms over their landing tracks.
• Cherenkov radiation. The optical analog: a charged particle moving faster than light in a medium emits a light cone with exactly the same μ = arcsin(1/(v/c_medium)) geometry — the blue glow in reactor pools.
• Astrophysical bow shocks. Stars and the heliosphere plough supersonically through interstellar gas, forming Mach-cone-like bow shocks.
• Low-boom design. NASA's X-59 reshapes the airframe so the shocks don't coalesce into one big N-wave, softening the boom into a thump.

Common misconceptions and edge cases

• "The boom happens only when the plane breaks the barrier." No — the cone trails the plane continuously the whole time it's supersonic. Everyone under the path hears it as the cone reaches them.
• "You hear it directly below the plane." The boom reaches a listener after the plane is well past, trailing by h/tan μ (≈26 km in the example above).
• "Faster = louder." Beyond Mach 1, more speed mainly narrows the cone. Loudness (overpressure) is driven more by altitude, weight, and length than by raw Mach number.
• "It's a one-time event in the air." It's a moving surface that drags a continuous boom carpet along the ground for the whole supersonic leg of the flight.
• "The vapor cone around a transonic jet is the boom." That misty cone is a Prandtl–Glauert condensation cloud — humid air briefly cooling below the dew point in the low-pressure expansion. It can appear near M ≈ 1, but it is not the shock and not the boom.
• "Sound speed depends on how hard you push the air." In an ideal gas, c depends only on temperature, not on pressure — colder air aloft means a lower c, hence a higher Mach number for the same true airspeed.