Math 2.0

Fluid mechanics Definition

Shock wave in gases

Builds on Shock waveEuler equationsRankine–Hugoniot conditionsSpeed of sound

Definition

Consider one-dimensional flow of a polytropic gas with adiabatic exponent γ>1\gamma > 1. The state is given by the density ρ>0\rho > 0, the velocity uu and the pressure p>0p > 0, and the specific enthalpy is h=γp(γ−1)ρh = \dfrac{\gamma p}{(\gamma - 1)\rho}. Let a weak solution of the Euler equations have a jump along the line x=σtx = \sigma t. On one side it has the constant state (ρ1,u1,p1)(\rho_1, u_1, p_1), and on the other the constant state (ρ2,u2,p2)≠(ρ1,u1,p1)(\rho_2, u_2, p_2) \ne (\rho_1, u_1, p_1). Let wi=ui−σw_i = u_i - \sigma be the velocities relative to the jump, for i=1,2i = 1, 2. The jump is a shock wave if the following hold.

  1. The mass flux m=ρ1w1=ρ2w2m = \rho_1 w_1 = \rho_2 w_2 across the jump is not zero.
  2. The momentum balance p1+ρ1w12=p2+ρ2w22p_1 + \rho_1 w_1^2 = p_2 + \rho_2 w_2^2 holds.
  3. The energy balance h1+12w12=h2+12w22h_1 + \tfrac12 w_1^2 = h_2 + \tfrac12 w_2^2 holds.
  4. The gas is compressed when it crosses the jump. So the density is larger on the downstream side, towards which the gas moves relative to the jump.

Remarks

Clauses 1 to 3 are the Rankine–Hugoniot conditions for the Euler equations, written in the frame of the jump. Clause 4 is the admissibility condition. For a polytropic gas it is equivalent to an increase of the entropy across the jump. It is also equivalent to supersonic relative flow upstream and subsonic relative flow downstream, measured with the local speed of sound. A jump with m=0m = 0 is a contact discontinuity: then w1=w2=0w_1 = w_2 = 0 and p1=p2p_1 = p_2, and only the density may jump.

Example: take γ=1.4\gamma = 1.4, σ=0\sigma = 0, side 11 on the left and upstream, with state ρ1=1\rho_1 = 1, p1=1p_1 = 1, u1=21.4u_1 = 2\sqrt{1.4}, so the upstream Mach number is 22. The downstream state ρ2=8/3\rho_2 = 8/3, u2=31.4/4u_2 = 3\sqrt{1.4}/4, p2=4.5p_2 = 4.5 gives m=21.4m = 2\sqrt{1.4} on both sides, both sides of clause 2 equal 6.66.6, and both sides of clause 3 equal 6.36.3. Since ρ2>ρ1\rho_2 > \rho_1, it is a shock wave.

Non-example: the same two states with the roles of upstream and downstream exchanged satisfy clauses 1 to 3, but the gas would expand across the jump. This expansion shock fails clause 4 and does not occur in nature.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Conservation law, concept, no evidence levelConservation lawconceptEuler equations, concept, no evidence levelEuler equationsconceptShock wave, concept, no evidence levelShock waveconceptSpeed of sound, concept, no evidence levelSpeed of soundconceptWeak solution, concept, no evidence levelWeak solutionconceptRankine–Hugoniot conditions, statement, no evidence levelRankine–Hugoniot cond…statementShock wave in gases, definition, no evidence levelShock wave in gasesdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Conservation law concept
  2. Euler equations concept
  3. Shock wave concept
  4. Speed of sound concept
  5. Weak solution concept
  6. Rankine–Hugoniot conditions statement
  7. Shock wave in gases definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Shock_wave
About this page
TagTV2BAWJET9
URIm2://TV2BAWJET9
Kinddefinition
WritingW0 stub
MSC76L05, 76N15, 35L67
DefinesShock wave in gases
UsesShock wave, Euler equations, Rankine–Hugoniot conditions, Speed of sound
Changed3 November 2026

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