Supersonic Flow: Mach Cones, Shock Waves and Rankine-Hugoniot

The cone that a fast object drags behind it

A body moving through air emits pressure disturbances that spread outward as spherical sound waves. Below the speed of sound those waves outrun the body and warn the air ahead that something is coming. Push past Mach 1 and the body outruns its own waves: each disturbance still spreads as a sphere, but the source has moved on before the wave gets far, so successive spheres pile up along a cone trailing behind the body -- the Mach cone.

The half-angle of that cone, the Mach angle mu, follows directly from geometry: in the time a sound wave has traveled one radius, the body has traveled M radii, so sin(mu) = 1/M. At Mach 1 the cone is a flat wall right at the body; by Mach 3 it has narrowed to about 19 degrees; at very high Mach number it collapses to a thin sliver hugging the direction of travel.

mu = arcsin(1 / M)        Mach angle (half-angle of the Mach cone)
M = 1   ->  mu = 90°   (wave sits flush against the body)
M = 2   ->  mu = 30°
M = 5   ->  mu = 11.5°

Oblique shocks versus the normal bow shock

A sharp, pointed body flying supersonically compresses the air into a thin oblique shock attached at (or near) its nose, angled somewhere between the Mach angle and straight across the flow depending on the body's own wedge angle. A blunt body -- a re-entry capsule, a rounded nose cone -- cannot support an attached oblique shock at all; instead a detached, curved bow shock stands off ahead of it, normal (perpendicular) to the flow right at its centerline and curving back toward the Mach angle far from the body.

What actually happens inside a shock: Rankine-Hugoniot

A shock is not a smooth compression -- to the scale of the flow it is a discontinuity only a few molecular mean-free-paths thick, across which pressure, density, temperature and velocity jump almost instantaneously. Conservation of mass, momentum and energy across that jump gives the Rankine-Hugoniot relations, and for a normal shock in air they reduce to closed-form ratios in terms of the upstream Mach number M1 and the ratio of specific heats gamma (about 1.4 for air):

p2/p1 = 1 + (2g/(g+1))(M1^2 - 1)         pressure jump
rho2/rho1 = ((g+1)M1^2) / ((g-1)M1^2+2)  density jump
M2^2 = (1+((g-1)/2)M1^2) / (gM1^2-(g-1)/2)   downstream Mach
(g = ratio of specific heats, about 1.4 for air; M1 > 1 always gives p2>p1, rho2>rho1, M2<1 for a normal shock)

Regimes: from transonic to hypersonic

Mach number does not just scale one effect, it changes which physics dominates. Below about Mach 0.8 the flow is subsonic and compressibility barely matters. Between roughly 0.8 and 1.2 the flow is transonic -- some regions locally exceed Mach 1 even though the vehicle has not, and shocks flicker in and out. From about Mach 1.2 to 5 the flow is cleanly supersonic: attached oblique shocks, predictable Mach cones, the Rankine-Hugoniot relations apply directly. Above Mach 5 the flow becomes hypersonic, where the shock layer sits so close to the body that it interacts with the boundary layer, and the temperature behind the shock climbs high enough to dissociate air molecules -- a regime that needs real-gas chemistry, not just the ideal-gas relations above.

Frequently asked questions

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