Normal Shock Wave Calculator

A normal shock is a thin, nearly discontinuous region perpendicular to the flow direction where a supersonic flow abruptly decelerates to subsonic speed, with a sudden jump in pressure, density, and temperature. It is an irreversible, entropy-generating process, unlike the isentropic flow upstream and downstream of it.

The Rankine-Hugoniot jump conditions only have a physically valid (entropy-increasing) solution when the upstream Mach number M1 is supersonic. If M1 were 1 or less, the formulas would either give no real solution or would require entropy to decrease, which violates the second law of thermodynamics, so no shock can exist.

Yes. For every value of M1 greater than 1, the normal shock relations always give M2 less than 1. A normal shock always decelerates supersonic flow to subsonic flow; this is a mathematical consequence of the Rankine-Hugoniot relations, not just a typical outcome.

The stagnation pressure ratio p02/p01 is always less than 1 across a normal shock because the shock is an irreversible process that generates entropy. Even though static pressure p2 rises sharply, the loss of available energy (exergy) shows up as a drop in total pressure, which directly reduces the thrust or efficiency of any propulsion system relying on that pressure.

A single strong normal shock produces a large total-pressure loss, worsening as M1 increases. A series of weaker oblique shocks, each turning the flow through a smaller angle at a lower effective Mach number, achieves the same overall deceleration with substantially less total-pressure loss, which is why supersonic aircraft inlets (like on the Concorde or MiG-25) are shaped to generate oblique shocks ahead of any final normal shock.

T2/T1 is not computed from an independent formula, it equals (p2/p1) divided by (rho2/rho1), the ratio of the pressure jump to the density jump. Computing it this way, rather than from a separate closed-form expression, is a useful check: if your p2/p1 and rho2/rho1 values are both correct, dividing them must give the correct T2/T1.

As M1 approaches infinity, the density ratio rho2/rho1 approaches a finite limit of (γ+1)/(γ-1), which is 6 for air (γ=1.4), while the pressure ratio p2/p1 and temperature ratio T2/T1 grow without bound. This means a normal shock can compress density by at most a factor of 6 for air, no matter how strong the shock, but can raise pressure and temperature arbitrarily high.

The classic Rankine-Hugoniot relations assume a calorically perfect gas with constant γ. At very high Mach numbers (roughly M1 above 5 to 6), real air undergoes vibrational excitation, dissociation, and ionization behind the shock, which changes the effective γ and makes these constant-γ formulas increasingly approximate; real-gas effects must be included for accurate hypersonic shock analysis.

Entropy always increases across a normal shock, consistent with the second law of thermodynamics, since the shock is an irreversible process. The entropy increase is what distinguishes the shock solution (M1 supersonic to M2 subsonic) from the mathematically possible but physically impossible reverse process (which would require entropy to decrease and is therefore rejected).

Yes, supersonic wind tunnel diffusers commonly use a normal shock, or a series of oblique shocks followed by a weak normal shock, to decelerate the test-section flow back to subsonic speed for the diffuser and downstream ducting. This calculator's pressure, density, and temperature ratios directly describe the jump across such a diffuser shock.