Supersonic Shock Waves

What a shock wave actually is

Sound is just a small pressure perturbation propagating at the local speed of sound a = √(γRT). When a body moves slower than sound, the air ahead "knows" the body is coming — pressure waves run ahead and the body slips smoothly into the gap.

When the body moves faster than sound, the air ahead can't be warned. Pressure waves emitted by the body can't outrun it; they pile up at the leading edge into a thin transition region — only a few mean-free-paths thick, around 10⁻⁷ m at sea level — across which pressure, density, temperature, and velocity all change abruptly. That transition region is the shock wave.

Three conservation equations describe a shock — mass, momentum, energy (Rankine–Hugoniot). With an equation of state they give post-shock conditions in closed form for any upstream Mach. Flow always decelerates; pressure, density, and temperature rise; entropy increases.

Three shock geometries

NORMAL SHOCK                OBLIQUE SHOCK              DETACHED BOW SHOCK
(in front of                (attached to wedge)        (around blunt body)
 a Pitot probe)
                                       ╲ shock
       │                                ╲
   M>1 │ M<1                  M>1        ╲      M>1            ╲ ╱
   ────│────→                ──────────→ ╲────→             M>1 ╲ ╱   subsonic
       │                                  ╲                      ╲ region
       │ ↑ T,P,ρ all                       ╲ θ                  ──┤   behind
       │   jump up                          ╲                     │   ────
       │                                ─────╲────                ╱
                                              ╲                  ╱
   flow stops                              wedge              blunt body
   normal to shock                         angle θ            (cylinder)
                                          < θ_max

Normal, oblique, and detached compared

Type	Geometry	Post-shock flow	Total-P loss at M=2	Where you see it
Normal shock	Perpendicular to flow	Always subsonic	~28%	Pitot probes, terminal inlet shock
Weak oblique shock	Shallow β	Still supersonic	~5–15%	Wedge leading edges
Strong oblique shock	Steep β	Subsonic	~25–28%	Backpressure-imposed, rare
Detached bow shock	Curved, stands off	Subsonic at stagnation	~28% at centreline	Capsule reentry, blunt noses
Prandtl-Meyer expansion	Continuous fan	Faster, lower P, T, ρ	0% (isentropic)	Outward corners, nozzle exits
Mach wave	Infinitely weak	Same as upstream	0%	Vanishing-turn-angle limit

The two oblique-shock solutions for a given turn angle (weak and strong) come from the θ-β-M relation having two β roots. Nature picks the weak solution unless backpressure forces the strong one — uncommon outside laboratory test sections.

The θ-β-M relation in numbers

For γ = 1.4 air, the maximum turn angle θ_max above which no attached oblique shock exists rises with Mach number:

Mach number	Mach angle μ	θ_max (max turn angle)	β at θ_max
1.5	41.8°	12.1°	66.6°
2.0	30.0°	22.9°	64.7°
3.0	19.5°	34.1°	65.2°
5.0	11.5°	41.1°	67.1°
10.0	5.7°	44.4°	69.8°

At Mach 2 a 20° wedge produces an attached oblique shock (20° < 22.9°); a 30° wedge detaches into a curved bow shock. This is why supersonic inlets use 8–12° ramps stacked in series rather than a single steep one.

Worked example: oblique shock on a Mach 2 wedge

A 10° wedge in air at M = 2.0. From θ-β-M, the weak shock angle β ≈ 39.3°. The Mach component normal to the shock:

M_n1 = M·sin β = 2.0 · sin 39.3° = 1.27

Apply normal-shock relations on M_n1 = 1.27: P₂/P₁ ≈ 1.71, T₂/T₁ ≈ 1.17, M_n2 ≈ 0.80. Then M₂ = M_n2/sin(β−θ) = 0.80/sin 29.3° ≈ 1.64.

The wedge has decelerated the flow from Mach 2.0 to Mach 1.64, raised the pressure 71%, and dropped the total pressure by under 7%. Compare a normal shock at Mach 2.0: flow decelerates to Mach 0.58, pressure jumps 4.5×, total pressure drops 28%. The oblique shock is dramatically gentler — the entire reason supersonic intakes use stacked oblique compression.

Real-world cases

• Concorde, Mach 2 cruise. The variable-geometry intake ramp produced two oblique shocks plus a terminal weak normal shock just inside the lip, recovering 92% of free-stream total pressure. Without that scheme the engines would not have made the SFC needed for trans-Atlantic range.
• F-22 Raptor leading edges. At Mach 1.5+ supercruise, the wing's sharp leading edge attaches an oblique shock. Body-conformal antennas, embedded sensors, and weapons-bay doors all sit downstream of this shock; design pressures use post-shock conditions.
• SR-71 inlet "unstart". The Blackbird's spike-controlled inlet stacked oblique shocks; if the terminal normal shock was pushed forward of the throat, total-pressure recovery collapsed and asymmetric thrust spun the aircraft hard. Pilots learned to recover by retracting the spike forward.
• Apollo capsule reentry. Blunt-body reentry at Mach 25 produces a detached bow shock standing off the heat shield by ~10% of its diameter. The standoff kept hot post-shock gas away from the surface while the strong shock dissipated most kinetic energy in the airstream rather than the spacecraft.
• Sonic boom over a city. A Concorde 18,000 m up at Mach 2 produces an N-wave overpressure of about 100 Pa at the ground. Supersonic overland flight was banned in the US in 1973 because the boom carpet from coast-to-coast operations was untenable. NASA's X-59 aims at an 80 dB "thump" rather than a boom by reshaping the aircraft.
• Scramjet combustor. A scramjet inlet uses external and internal compression shocks to slow free-stream M=8 air to M=2 in the combustor, where supersonic combustion completes in milliseconds. Shocks both make scramjet propulsion possible and constrain its operating envelope to a narrow Mach band.

Variants

• Weak vs strong oblique shock. Two solutions for the same θ. Weak is the natural choice; strong appears only when backpressure forces it (e.g., inside a CD duct with high exit pressure).
• Detached vs attached oblique. Below θ_max, attached oblique. Above θ_max — or for blunt bodies — the shock detaches into a curved bow with a normal-shock element at the centreline.
• Conical (Taylor-Maccoll) shocks. A sharp cone at supersonic flow generates a conical shock with continuous post-shock region governed by ODEs. Cone half-angles to 30°+ remain attached at Mach 4 — sharper geometries are tolerated than 2D wedges.
• Prandtl-Meyer expansion fan. Outward corners produce isentropic expansion fans, the supersonic mirror of shocks. Used on rear surfaces of supersonic airfoils.
• Mach reflection. When two oblique shocks intersect at high turn angles they merge into a Mach stem connected to the originals by a triple point. Important inside intakes and explosive-driven shock tubes.
• Hypersonic shock layer. Above M ≈ 5, the shock sits close to the body and post-shock temperatures dissociate molecules. Real-gas effects make calorically-perfect-gas tables wrong by 10–50%.

Common failure modes

• Shock-induced boundary-layer separation. When a shock impinges on a boundary layer with a large adverse pressure jump, the boundary layer can separate. On compressor blades and inlets this manifests as λ-shock structures, vibration, and total-pressure loss far worse than the inviscid prediction.
• Inlet unstart. If the terminal normal shock in a supersonic inlet is expelled forward of the throat, the inlet "unstarts" — total pressure collapses, mass flow plummets, engine thrust crashes. Recovery requires reducing back-pressure or changing geometry.
• Shock-shock interaction heating. Where two shocks intersect, peak heating can locally spike 10× the bow-shock baseline. The 1967 X-15 incident saw a strut heat to 1100°C from such an intersection at Mach 6.7.
• Wave drag and Mach buffet. At transonic speeds (M ≈ 0.8–0.95) shocks form on the upper wing surface, causing local separation and unsteady buffet. Civil aircraft have a Mach buffet boundary above which sustained flight isn't safe.
• Sonic boom carpet. Overpressure from a supersonic aircraft sweeps a 50–80 km wide ground track. Acceptable boom levels constrain routes and altitudes; this is why Concorde was restricted to over-ocean cruise.
• Water hammer. The same Rankine-Hugoniot physics applies in liquids. A valve closure that drops local pressure below vapour can produce a collapsing-cavity pressure pulse that bursts plastic pipework.