Critical Mach number — onset of local supersonic flow: Visual walkthrough

Step 1 — A wing squeezes the air, so the air speeds up

WHAT. Air flowing left-to-right meets a wing. Over the humped top surface the streamlines (the thin lines the air follows) get pinched closer together.

WHY. Think of a river forced through a narrow gorge: the same amount of water must pass every second, so where the channel is narrow the water runs faster. Air over a wing does exactly this — the curvature narrows the "channel," so the air accelerates. Faster air has lower pressure (this is the "suction peak" on top of the wing).

PICTURE. Watch the red patch — that fastest, lowest-pressure point is the one we will track for the whole page.

The whole game: at what $M_{\infty}$ does the red-patch $M_{l oc a l}$ first hit $1$ ? That $M_{\infty}$ is the critical Mach number $M_{cr}$ .

Step 2 — Total ("stagnation") pressure is the same everywhere on a streamline

WHAT. Pick one streamline. Far upstream the air has static pressure $p_{\infty}$ and Mach $M_{\infty}$ . At the red patch it has $p$ and $M_{l oc a l}$ . We claim these are tied together by a single conserved quantity.

WHY. For smooth, no-friction, no-heat flow (isentropic — see Isentropic Flow Relations), energy is conserved along the streamline. If you imagine bringing the air gently to rest, it always returns to the same "stagnation pressure" $p_{0}$ . So $p_{0}$ is a fixed label carried along the whole line.

PICTURE. The dashed reservoir $p_{0}$ sits above both stations; each station's static pressure is a fraction of that same $p_{0}$ .

Step 3 — Cancel $p_{0}$ to get the pressure ratio between the two stations

WHAT. Write $p_{0} / p$ at the red patch and $p_{0} / p_{\infty}$ far upstream, then divide. The unknown reservoir pressure $p_{0}$ cancels.

WHY. We can't measure $p_{0}$ directly, but we don't need to — only the ratio $p / p_{\infty}$ matters for aerodynamics. Cancelling a nuisance constant is the whole trick of this step.

PICTURE. Two "towers" of pressure sharing the same roof $p_{0}$ ; dividing them erases the roof.

Step 4 — Repackage pressure as a coefficient $C_{p}$

WHAT. Aerodynamicists never quote raw pressure; they quote a dimensionless pressure coefficient $C_{p}$ .

WHY. A number like " $p = 41, 000 Pa$ " means nothing without context. $C_{p}$ measures how far below free-stream the pressure has dipped, scaled by the "dynamic pressure" $\frac{1}{2} ρ_{\infty} V_{\infty}^{2}$ (the pressure the moving air could deliver if stopped). Negative $C_{p}$ = suction. See Pressure Coefficient Cp.

PICTURE. A dial from $C_{p} = + 1$ (full stagnation) down through $0$ into negative suction territory.

Step 5 — Force the red patch sonic: build $C_{p, cr}$

WHAT. Set $M_{l oc a l} = 1$ (red patch just reached sound) and rename the free stream $M_{\infty} \to M_{cr}$ (this is the moment we're after). Feed Step 3's ratio into Step 4's formula.

WHY. "Critical" means the first instant local flow is sonic. Substituting $M_{l oc a l} = 1$ is literally writing that condition into algebra.

PICTURE. The red patch turns gold at exactly Mach 1 — the trigger for everything downstream.

Step 6 — The airfoil brings its OWN curve

WHAT. Each wing has a fixed incompressible minimum pressure coefficient $C_{p, 0}$ (a negative number set purely by its shape). As speed rises, compressibility deepens that suction. A simple model is Prandtl–Glauert.

WHY. $C_{p, cr}$ told us what gas dynamics demands. Now we need what the airfoil actually delivers at each $M_{\infty}$ . Only by comparing "demand" vs "supply" can we find where they meet. See Prandtl–Glauert Compressibility Correction.

PICTURE. A shallow incompressible dip getting stretched deeper and deeper as $M_{\infty} \to 1$ .

Step 7 — The intersection IS $M_{cr}$

WHAT. Plot both curves against $M_{\infty}$ on one axis: the universal $C_{p, cr}$ (rising toward zero) and the airfoil $C_{p, m i n}$ (plunging). Where they cross, the wing's suction peak has just met the sonic requirement.

WHY. Left of the crossing the airfoil's suction is shallower than required — no point is sonic yet. Right of it, suction exceeds the requirement — supersonic pocket already born. The crossing is the knife-edge: the first sonic instant. That $M_{\infty}$ is $M_{cr}$ .

PICTURE. Two pastel curves kissing at one dot — the whole derivation in a single frame.

Step 8 — Edge & degenerate cases (never get surprised)

WHAT / WHY / PICTURE for the corners the smooth story hides:

A flat plate at zero lift ( $C_{p, 0} \to 0^{-}$ ): the airfoil curve barely dips; it only meets $C_{p, cr}$ as $M_{cr} \to 1$ . A perfectly thin, unloaded surface has $M_{cr} \approx 1$ — nothing accelerates, nothing goes sonic early.
$M_{\infty} \to 1$ in $C_{p, cr}$ : the bracket $\to 0$ and the prefactor $2/ (γ M_{cr}^{2}) \to 1.43$ , so $C_{p, cr} \to 0^{-}$ . The universal curve touches zero exactly at Mach 1 — sensible: at flight-Mach 1 you need essentially no extra suction to be sonic.
Very cambered / thick wing (large $∣ C_{p, 0} ∣$ ): the plunging curve crosses early, $M_{cr}$ can fall below $0.6$ . This is why transonic jets use sweep and the area rule to claw $M_{cr}$ back up.
Beyond $M_{cr}$ : the local sonic pocket closes with a shock, triggering drag divergence — the topic starts where this page ends.
Prandtl–Glauert caution: near $M_{cr}$ , P–G under-predicts suction; real crossing (Kármán–Tsien) sits slightly lower. Use it for the picture, not the final digit.

The one-picture summary

Everything collapses to: a universal gas-dynamics curve (what physics demands to reach Mach 1) meeting the airfoil's own suction curve (what the shape supplies) — the crossing is $M_{cr}$ .

Recall Feynman retelling — the whole walk in plain words

Air racing over a wing's hump speeds up and its pressure drops (Step 1). Along any streamline, one hidden "total pressure" stays fixed (Step 2), so I can relate the pressure at the fast spot to the pressure far away and the mysterious constant just cancels (Step 3). I repackage that pressure into a tidy dimensionless dip called $C_{p}$ (Step 4). Now I ask: what dip is needed to make that spot exactly sonic? Setting local Mach to 1 gives one clean formula, $C_{p, cr}$ , that only cares about the gas, not the wing (Step 5). But the wing has its own suction that deepens as I fly faster — the Prandtl–Glauert curve (Step 6). I draw both against flight speed; where they cross, the wing's suction has just met the sonic demand — and that flight speed is the critical Mach number (Step 7). Corner cases behave: flat plates stay sonic-free until Mach 1, fat wings go critical early, and just past $M_{cr}$ a shock forms and drag explodes (Step 8).

Recall

One-line meaning of $M_{cr}$ ::: The flight Mach number where the wing's lowest-pressure point first reaches local Mach 1. What makes the $C_{p, cr}$ curve universal ::: It contains only $γ$ and $M_{cr}$ — no airfoil geometry. Why the intersection gives $M_{cr}$ ::: It's where the airfoil's actual suction first equals the suction demanded to be sonic. Effect of a thinner wing ::: Smaller $∣ C_{p, 0} ∣$ shifts the crossing right → higher $M_{cr}$ .