Mach Cone Simulator — Sonic Boom from Supersonic Flight
Visualise the conical shock wave trailed by a supersonic body (M>1) and the sonic boom that strikes the ground beneath it. Vary the Mach number, sound speed, flight altitude and observer offset to compute the Mach angle mu = arcsin(1/M), the cone reach and the boom delay in real time. Compare subsonic, sonic and supersonic wave-front patterns side by side and replay the geometry that left SR-71 and Concorde footprints all over the planet.
Parameters
Mach number M
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Sound speed c
m/s
Flight altitude h
m
Observer ground offset x
m
The aircraft moves left to right at altitude h with the cone trailing behind. The observer stands on the ground at horizontal offset x; the cone reaches them when x equals l = h/tan mu, the cone reach.
Results
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Mach angle mu
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Flight speed V
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Boom arrival delay
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Cone reach distance
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Mach cone and ground boom
The orange aircraft flies right at altitude h. Blue lines show the Mach cone (half-angle mu) trailing behind it; the green dot is the observer at (x, 0); a red cross marks where the cone meets the ground. For M ≤ 1 expanding spherical waves replace the cone.
Mach-angle curve mu(M)
Horizontal axis: M from 1.0 to 5.0. Vertical axis: mu in degrees. The curve is mu = arcsin(1/M); it tends to 90 degrees as M tends to 1 (plane wave) and to 0 degrees as M tends to infinity (sharp cone). The yellow marker shows the current M; sweeping M reveals how the cone sharpens.
For $M<1$ disturbances propagate ahead of the body and no cone forms; $M=1$ is a plane wave; only for $M>1$ does the Mach cone appear. With the defaults $M=2,\ c=343\ \text{m/s},\ h=1000\ \text{m}$ this gives $\mu=30^\circ$, $V=686\ \text{m/s}$, $\ell=1732\ \text{m}$ and $t_{\text{delay}}=2.52\ \text{s}$.
What is the Mach Cone Simulator?
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With the defaults at M=2 the Mach angle is exactly 30 degrees and the cone hits the ground 1732 m behind the aircraft. That is suspiciously round.
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Nice spotting. The relation is sin mu = 1/M, so M=2 gives sin mu = 0.5 and mu = 30 degrees, the same trigonometry you see in optical critical angles. The ground reach is h/tan(30 deg) = 1000/0.5774, which is 1732 m, because tan(30 deg) = 1/sqrt(3) and sqrt(3) is roughly 1.732. Memorising the Mach cone for a few integer Mach numbers (2, 3, 4) lets you do these estimates in your head.
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When I push M down to 0.8 the cone disappears and I see expanding ripples instead. Is that the subsonic regime?
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Yes. In subsonic flight (M<1) the speed of sound c is faster than the body, so each spherical wave overtakes the aircraft and travels ahead of it. Disturbances reach both forward and backward, so they smear out and never pile up on a focused conical sheet. At M=1 the body keeps pace with its own waves and the envelope degenerates into a plane wave (the Prandtl-Glauert singularity). For M>1 the waves fall behind and form a cone, the phenomenon Ernst Mach first photographed with shadow imaging of bullets in the 1880s, which is why we call them Mach number, Mach angle and Mach cone.
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If I lift the altitude to 12000 m at M=2 the cone reach jumps to about 20 km. Is that why people heard Concorde long after it had passed overhead?
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Exactly. Concorde cruised at 18 km altitude and Mach 2.04, so the Mach angle was about 29.4 degrees and the cone reach was 18000/tan(29.4 deg) which is roughly 32 km. That means the boom hit the ground 32 km behind the aircraft, about 56 seconds after it passed overhead. By the time you heard the bang the jet had already disappeared over the horizon. The overpressure was around 100 Pa, which is ten million times the audibility threshold and rattles window panes, which is why supersonic operations were banned over land. NASA's X-59 QueSST is being flight-tested in 2026 to bring that overpressure down below 25 Pa.
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At M=5 the angle drops to 11.5 degrees and the cone looks like a needle. Is that the hypersonic regime?
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It is. Hypersonic flight (M>5) makes the cone almost parallel to the body axis. The SR-71 Blackbird (M=3.3) already had mu about 17.6 degrees, the Space Shuttle on re-entry (M=25) about 2.3 degrees and only a few-kilometre-wide footprint. Beyond that the simple sin mu = 1/M no longer captures the physics, because air starts to dissociate, ionise and radiate. Mars-entry vehicles are simulated with chemically non-equilibrium Navier-Stokes plus the Park model (eleven species and vibrational relaxation) coupled to DSMC at low density. The Mach cone is the textbook starting point, but a wide world of physics is layered on top.
Physical model and key equations
The Mach cone is the envelope of the spherical wave fronts emitted by a point source in supersonic motion, a Huygens-style construction. Ernst Mach (1838 to 1916) and Peter Salcher photographed shadowgraphs of supersonic bullets in Vienna in 1887, providing the first direct visualisation of shock fronts.
If the body was at $(-Vt, h)$ at time $-t$, the spherical wave it emitted then has expanded to radius $ct$ by time $0$, while the body has reached the origin $(0, h)$. The envelope of all such wave fronts is a cone with half-angle $\mu$ given by
where $M = V/c$ is the Mach number. For $M < 1$ the right-hand side exceeds 1 and there is no real solution: the cone does not exist. For $M = 1$ we have $\mu = 90^\circ$ (a plane wave), and for $M > 1$ the cone half-angle decreases monotonically, approaching $0$ as $M \to \infty$.
If the aircraft cruises horizontally at altitude $h$, the cone surface meets the ground at a horizontal distance $\ell$ behind the aircraft equal to
$$\ell = \frac{h}{\tan\mu} = h\sqrt{M^2 - 1}$$
The second form follows from $\sin\mu = 1/M$, $\cos\mu = \sqrt{1 - 1/M^2}$ and $\tan\mu = 1/\sqrt{M^2 - 1}$. For an observer directly below the aircraft path the boom arrives $t_{\text{delay}} = \ell/V = h/(M\,c\,\tan\mu)$ seconds after the aircraft passes overhead. With the defaults $M=2$, $c=343$ m/s, $h=1000$ m the simulator returns $\ell = 1732$ m and $t_{\text{delay}} = 2.52$ s.
The actual pressure signature of a sonic boom is the so-called N-wave: a sharp positive overpressure $+\Delta p$ at the bow shock, a smooth decay to $-\Delta p$ through the expansion behind the body and a second jump back to ambient at the tail shock. Typical values are $\Delta p \approx 50$ Pa for an F-18 at Mach 1.4 and 10 km altitude, and $\Delta p \approx 100$ Pa for Concorde at Mach 2 and 18 km altitude.
Real-world applications
Supersonic aircraft design and certification: The FAA, EASA and JCAB restrict supersonic flight over land via overpressure limits (effectively a ban in the United States). Next-generation supersonic transports such as NASA X-59 and Boom Overture stretch the airframe to more than 30 m and shape the bow, wing and tail shocks so that they merge into a low-amplitude signature, targeting $\Delta p$ below 25 Pa. CFD codes (OVERFLOW, Cart3D) and atmospheric propagation tools (PCBoom, ZEPHYRUS) are coupled to optimise the design. The Mach cone geometry sets the baseline.
Atmospheric re-entry: The Space Shuttle and HTV (Kounotori) entered the atmosphere at about M=25 with an extremely sharp Mach cone and a strong bow shock around the heat shield. CFD analyses combine direct simulation Monte Carlo (DSMC) with chemically non-equilibrium Navier-Stokes to predict surface heat flux peaks of about 1 MW/m². The Mach cone theory provides the boundary conditions for these large-scale solvers.
Internal and external ballistics: Military and hunting rifles fire bullets at M=2 to M=4. As they pass an observer the conical shock produces the familiar crack of a supersonic round, and Mach's original 1880s shadowgraphs visualised exactly this geometry. Modern acoustic shooter detection systems (Boomerang, PILAR) triangulate the firing position from the time delays of the Mach cone arriving at multiple microphones, a tool widely used in urban combat.
Cavitation and underwater propulsion: Supersonic motion in water is essentially impossible, but supercavitating torpedoes (such as the Russian VA-111 Shkval) wrap themselves in a vapour cavity and reach about 200 km/h underwater, equivalent to M about 0.07 in water. The two-phase boundary problem is structurally similar to the Mach cone construction and is studied with homogeneous flow models.
Common misconceptions and pitfalls
The most common misconception is that a sonic boom only happens at the moment a vehicle breaks the sound barrier. In reality the cone trails the aircraft throughout supersonic flight, so the boom footprint follows the entire flight path, typically a strip about 80 km wide. Chuck Yeager's 1947 first supersonic flight gave us the phrase "the sound barrier", but there is no single bang at breakthrough; the boom is heard everywhere along the supersonic track.
Another popular pitfall is the belief that the Mach cone is independent of the airframe shape. It is for an idealised point source, but a real aircraft emits separate shocks from the nose, wing leading edge, canopy, wing trailing edge and tail; on the ground these merge into the N-wave signature. NASA's SonicBAT flight tests in 2017 with an F/A-18B confirmed that ground overpressures at twenty-five microphones agreed with CFD predictions to within five percent. The Mach cone tells you where the principal shock lies, but CFD is essential for the waveform details.
Finally, people often think the Mach angle depends on altitude. The expression sin mu = 1/M is purely a local Mach-number relation; altitude enters only through the speed of sound $c$, which drops from 340 m/s at sea level to 295 m/s at 11 km in the standard atmosphere. So at fixed ground speed $V$ the Mach number $M = V/c$ does change with altitude, and so does mu. The simulator exposes $c$ as an independent slider so you can mimic this effect.
FAQ
A Mach cone is the conical shock-wave envelope trailed by a body moving faster than the speed of sound (M>1). Because the body's speed V exceeds the sound speed c, the spherical waves emitted at every instant fall behind the vehicle and their envelope is a cone. The half-angle mu (Mach angle) is given by sin mu = 1/M. With the tool's defaults M=2.0, c=343 m/s, h=1000 m, you get mu=30 degrees, the cone hits the ground 1732 m behind the aircraft and the boom arrives 2.52 s after the aircraft passes overhead.
The Mach cone is a moving sheet of pressure jump trailed by the vehicle. When a ground observer crosses this sheet they feel a sudden pressure rise followed by a sharp drop, which is heard as the sonic boom. Typical overpressure is 50 to 100 Pa for a fighter at Mach 2 cruising at 12 km altitude, enough to rattle window panes on the ground. This is precisely why Concorde was restricted to over-water routes.
In the subsonic regime (M<1) the body moves slower than the sound waves it emits, so each spherical wave overtakes the vehicle and travels ahead of it. Because waves can reach the front, disturbances spread out smoothly rather than piling up on a focused conical sheet, so no shock cone exists. At M=1 the envelope of the spherical waves degenerates into a plane wave and only for M>1 do the waves fall behind to form the Mach cone. The tool draws expanding spherical waves instead of a cone whenever M is less than or equal to 1.
mu = arcsin(1/M) is monotonically decreasing: 90 degrees at M=1 (a plane wave), 30 degrees at M=2, 19.5 degrees at M=3 and asymptotically 0 degrees at M=infinity (an infinitely sharp cone). The faster the vehicle the narrower the cone, so the sonic-boom footprint becomes a long thin strip behind the aircraft. The SR-71 Blackbird (M=3.3) produced a footprint about 80 km wide, and a re-entering Space Shuttle (M=25) leaves an almost point-like focused boom.