Prandtl Meyer Expansion Calculator

The formula

tan θ = 2cot β (M₁²sin²β − 1)/(M₁²(γ + cos 2β) + 2)

An oblique shock is a normal shock, tilted

Split the velocity into components normal and tangential to the wave. The tangential component passes through completely unchanged; the normal component behaves exactly as it would across a normal shock. Every oblique-shock relation follows from that one decomposition, which is why the same normal-shock formulas appear with M₁ sin β in place of M₁.

Because only the normal component is shocked, a weak oblique shock usually leaves the flow supersonic and destroys far less stagnation pressure than a normal shock at the same Mach number. That is the whole basis of supersonic inlet design: several weak oblique shocks in series recover much more pressure than one strong normal shock.

Two solutions, and a limit

For every achievable deflection the θ-β-M relation has two roots — a weak shock at a shallow angle and a strong one at a steep angle. Nature almost always picks the weak one; the strong solution requires downstream conditions to force it.

Above a maximum deflection, which rises with Mach number, neither root exists and no attached shock is possible. The shock detaches and stands off the body as a curved bow wave, with a subsonic region behind its nose. That is why supersonic aerofoils are thin and sharp — a blunt leading edge guarantees a detached shock and the drag that comes with it.

Expansions are free, compressions are not

Turning a supersonic flow away from itself produces a Prandtl-Meyer fan, which accelerates and cools the flow through a continuous, perfectly isentropic turn — no stagnation pressure is lost at all. Turning it into itself produces a shock, which always loses some. That asymmetry between compression and expansion is the deepest structural fact in supersonic aerodynamics.