Shock Wave

Why shocks form

A small pressure disturbance in a gas travels at the speed of sound, c = sqrt(γ R T / M_w). For air at 20°C, c ≈ 343 m/s. If a piston pushes the gas slower than c, every infinitesimal pressure pulse outruns the piston and the gas ahead has time to smoothly adjust. If the piston pushes faster than c, pulses cannot escape upstream — they pile up at the leading edge until the gradient becomes so steep that viscosity and thermal conduction balance it in a thin layer. That thin layer is the shock.

Inside the shock front, kinetic theory shows the width is just a few mean free paths — for air at STP, about 100 nm. From outside the shock looks like a mathematical discontinuity.

Rankine-Hugoniot jump conditions

In the rest frame of the shock, conservation of mass, momentum and energy across the front gives:

rho_1 v_1                =  rho_2 v_2                    (mass)
P_1 + rho_1 v_1^2        =  P_2 + rho_2 v_2^2            (momentum)
h_1 + v_1^2 / 2           =  h_2 + v_2^2 / 2              (energy)

Three equations, plus the ideal gas equation of state P = ρ R T / M_w and the enthalpy h = c_p T. Solving for the downstream state in terms of upstream Mach M₁ = v₁/c₁ and γ gives the canonical ratios.

Jump ratios across a normal shock in air (γ = 7/5)

M₁	P₂/P₁	ρ₂/ρ₁	T₂/T₁	M₂
1.0	1.00	1.00	1.00	1.000
1.5	2.46	1.86	1.32	0.701
2.0	4.50	2.67	1.69	0.577
3.0	10.33	3.86	2.68	0.475
5.0	29.0	5.00	5.80	0.415
10.0	116.5	5.71	20.4	0.388
∞	→∞	6.00 (limit)	→∞	0.378

Note: the density ratio is bounded by 6 even as M₁ goes to infinity. Temperature and pressure are not bounded. M₂ (downstream Mach) is always < 1 — downstream is subsonic in the shock frame.

Worked example: Mach 2 normal shock

Upstream: air at sea level, P₁ = 101325 Pa, T₁ = 288 K, ρ₁ = 1.225 kg/m³, v₁ = 2 c₁ = 686 m/s. Shock heading into still air.

gamma = 1.4
M1 = 2.0

rho2/rho1 = (gamma + 1) M1^2 / ((gamma - 1) M1^2 + 2)
          = 2.4 * 4 / (0.4 * 4 + 2)
          = 9.6 / 3.6 = 2.667

P2/P1     = 1 + (2 gamma / (gamma + 1)) (M1^2 - 1)
          = 1 + (2.8/2.4) * 3 = 1 + 3.5 = 4.50

T2/T1     = (P2/P1) / (rho2/rho1) = 4.50 / 2.667 = 1.687

M2 = sqrt( ((gamma-1) M1^2 + 2) / (2 gamma M1^2 - (gamma-1)) )
   = sqrt(3.6 / 11.2) = 0.577

Downstream conditions: P₂ = 4.50 × 101325 = 456 kPa. T₂ = 1.687 × 288 = 486 K (213°C). ρ₂ = 2.667 × 1.225 = 3.27 kg/m³. Downstream gas is hot, dense, and moving subsonically.

Shock-wave types — comparison

Type	Driver	Geometry	Example
Normal shock	Body or piston moving along axis	Plane perpendicular to flow	Shock-tube experiments; bow ahead of blunt body
Oblique shock	Wedge or compression turn	Plane at angle β > μ to flow	Wedge airfoil at supersonic speed
Mach cone	Point source moving supersonically	Cone, half-angle μ = arcsin(1/M)	Sonic boom from bullet or aircraft
Bow shock	Blunt body in supersonic flow	Detached, curved upstream	Re-entry capsule, planetary magnetosphere
Blast wave	Sudden energy release	Expanding sphere	Explosion; supernova ejecta
Astrophysical (collisionless)	Magnetic field, not particle collisions	Sharp jump in low-density plasma	Earth's bow shock; SNR shells

JavaScript — normal shock calculator

// Normal shock jump relations for ideal gas with adiabatic index gamma
function normalShock(M1, gamma = 1.4) {
  const M1sq = M1 * M1;
  const p2_p1   = 1 + (2 * gamma / (gamma + 1)) * (M1sq - 1);
  const rho2_rho1 = (gamma + 1) * M1sq / ((gamma - 1) * M1sq + 2);
  const T2_T1   = p2_p1 / rho2_rho1;
  const M2sq    = ((gamma - 1) * M1sq + 2) / (2 * gamma * M1sq - (gamma - 1));
  return {
    P_ratio:   p2_p1,
    rho_ratio: rho2_rho1,
    T_ratio:   T2_T1,
    M2:        Math.sqrt(M2sq)
  };
}

console.log('Mach 2 shock in air:');
console.log(normalShock(2.0));
// { P_ratio: 4.5, rho_ratio: 2.667, T_ratio: 1.687, M2: 0.577 }

// Mach cone half-angle
function machAngle(M) {
  if (M < 1) return null;
  return Math.asin(1 / M) * 180 / Math.PI;
}
console.log('Mach 2 cone:',  machAngle(2.0).toFixed(1) + ' deg');  // 30.0
console.log('Mach 5 cone:',  machAngle(5.0).toFixed(1) + ' deg');  // 11.5
console.log('Mach 35 cone:', machAngle(35).toFixed(2) + ' deg');   // 1.64 -- Apollo re-entry

// Sedov-Taylor blast: R(t) = ksi * (E/rho)^(1/5) * t^(2/5)
function sedovRadius(E_joules, rho_kgm3, t_s, ksi = 1.033) {
  return ksi * Math.pow(E_joules / rho_kgm3, 1/5) * Math.pow(t_s, 2/5);
}
// Trinity test, 21 kt = 8.8e13 J, ambient air rho = 1.225 kg/m^3, 0.1 s after
console.log('Trinity radius @ 0.1s:', sedovRadius(8.8e13, 1.225, 0.1).toFixed(0) + ' m');
// ~108 m -- matches G.I. Taylor's classic estimate from declassified photos

Where shock waves matter

• Aviation. Supersonic and hypersonic vehicle design hinges on shock-wave drag and heating.
• Re-entry. Apollo, Space Shuttle, SpaceX Dragon — survive shock-heated air via heat shields.
• Sonic booms. Constrains civilian supersonic flight (Concorde retired largely due to overland boom restrictions).
• Astrophysics. Supernova remnants, AGN jets, gamma-ray-burst afterglows, all driven by shocks.
• Lithotripsy. Focused acoustic shocks shatter kidney stones in vivo.
• Detonations. Self-sustained shock + reaction wave in explosives; basis of explosive engineering.
• Inertial fusion. Shock-driven compression of a fuel capsule to ignition (NIF, ICF).

Common mistakes

• Treating shocks as zero-thickness everywhere. They have finite width ~ a few mean free paths. In rarefied flows (high altitude, plasmas) this matters.
• Assuming infinite density compression. Density ratio asymptotes to (γ+1)/(γ-1) — max 6 for air, regardless of how strong.
• Forgetting downstream is subsonic. M₂ < 1 in the shock's rest frame. This is why the shock structure exists.
• Conflating shocks with sound waves. Sound is linear, reversible; shocks are nonlinear and irreversible (entropy generation).
• Using ideal-gas Rankine-Hugoniot at extreme conditions. Real gas effects (dissociation, ionization, radiation) matter for hypersonic and astrophysical shocks.
• Missing the difference between a shock and an expansion fan. Compression turns make shocks; expansion turns make smooth fans (Prandtl-Meyer), not jumps.