Normal-shock relations
For a perfect gas with ratio of specific heats γ, a stationary normal shock at upstream Mach M1 has
M2² = [1 + ((γ−1)/2) M1²] / [γ M1² − (γ−1)/2]
P2/P1 = 1 + [2γ/(γ+1)] (M1² − 1)
ρ2/ρ1 = [(γ+1) M1²] / [(γ−1) M1² + 2]
T2/T1 = (P2/P1) / (ρ2/ρ1)
Stagnation temperature is continuous across the shock (T02 = T01). Stagnation pressure falls: the flow is adiabatic but not isentropic. At M1 = 2, γ = 1.4, the NACA 1135 values are M2 = 0.57735, P2/P1 = 4.5000, T2/T1 = 1.6875, ρ2/ρ1 = 2.6667, and P02/P01 = 0.72087.
Rayleigh pitot
A pitot tube in a supersonic stream measures the stagnation pressure behind a normal shock, P02. The Rayleigh-pitot formula gives P02/P1 directly. Combined with a static-pressure measurement, it is the classical way to obtain supersonic Mach number.
Sources
NACA Report 1135, normal-shock tables; NASA Glenn Beginner’s Guide; Anderson, Modern Compressible Flow. Upstream isentropic stagnation quantities are the same identities used on the isentropic flow calculator.
Frequently asked questions
Can a normal shock occur at Mach 0.9?
No. A normal shock requires M1 ≥ 1. At M1 = 1 the shock has zero strength: all ratios are 1 and M2 = 1.
Why does total pressure drop across the shock?
Entropy rises in the irreversible compression. Total temperature is unchanged because the shock is adiabatic (no heat transfer in the model), but total pressure, which is defined isentropically, falls.
What is the Rayleigh-pitot pressure used for?
It is the stagnation pressure a pitot tube reads in a supersonic flow, after the bow shock in front of the tube. Together with static pressure it yields Mach number; this page reports P02/P1 for that purpose.