Foundations — Critical Mach number — onset of local supersonic flow

This page assumes you have seen none of the notation in the parent note Critical Mach number — onset of local supersonic flow. We build every symbol from the ground up, in the order they depend on each other.

1. Pressure $p$, density $\rho$, and the gas's springiness $\gamma$

Before any formula, we name the three raw quantities that describe a lump of air. Nothing later is allowed to use them until they live here.

Why does $\gamma$ appear everywhere? Because compressing air both raises its pressure and heats it, and $\gamma$ is the exchange rate between those two effects. Every isentropic formula below is really "$\gamma$ bookkeeping."

2. Speed of sound $a$ — the yardstick everything is measured against

Picture a line of dominoes: knock the first and a wave of "something happened" runs down the line. In air, the dominoes are air molecules bumping neighbours. The bumping speed depends on how tightly packed and springy the gas is — exactly the $p$, $\rho$, $\gamma$ we just defined:

$a=\sqrt{\gamma \frac{p}{\rho }}$

The picture to hold: $a$ is the maximum speed at which a gas can "tell itself" that something is coming. Once the air is forced to move faster than $a$, it can no longer send warning ahead → this is exactly why "going sonic" is special, and why the whole chapter exists.

3. The Mach number $M$ — speed measured in units of sound

Why a ratio and not just "metres per second"? Because air only cares about speed relative to its own sound speed. A single number $M$ captures "how close to sonic am I", which is the only thing that matters for shocks.

• $M<1$ → subsonic (air outruns nothing; warnings travel ahead).
• $M=1$ → sonic (the flow moves exactly as fast as its own ripples).
• $M>1$ → supersonic (flow outruns its own sound → shocks form).

$M_{cr}$ (the star of the topic) is the special value of $M_{\infty }$ where the biggest $M_{local}$ on the wing first hits $1$.

4. The subscript $\infty$ — "far away, undisturbed"

Picture standing a kilometre in front of the plane: the air there is calm and uniform. That calm reference is where we anchor all comparisons — the wing's local pressure dip is measured relative to $p_{\infty }$.

5. Stagnation pressure $p_0$ — the pressure of air brought to a halt

Picture the exact nose-tip of the wing where the flow splits and one streamline dead-stops: that point feels $p_0$. The key magic fact (from Isentropic Flow Relations) is:

That shared constant is the bridge that lets us compare the sonic point on the wing to the far-away air — it is the trick behind the whole $C_{p,cr}$ derivation.

$\frac{p_0}{p}={\left(1+\frac{\gamma -1}{2}{M}^2\right)}^{\frac{\gamma }{\gamma -1}}$

Read it as: "the faster a parcel moves (bigger $M$), the lower its static pressure $p$ compared to its frozen-still $p_0$." Fast air = low pressure. Hold that.

6. Why "faster = lower pressure": the streamtube picture

The single fastest, lowest-pressure spot is the first candidate to reach $M_{local}=1$. That is why we hunt for the minimum pressure on the surface.

7. Dynamic pressure $q_{\infty }$ and the pressure coefficient $C_p$

Break it down:

• Top $p-p_{\infty }$: the actual pressure difference from calm air. Negative on the suction peak (pressure dropped).
• Bottom $q_{\infty }$: the natural yardstick for pressure changes.

So $C_p$ = "pressure dip expressed in units of the flow's own dynamic pressure." A negative $C_p$ = suction; more negative = stronger suction = faster local air. This is the object we track in Pressure Coefficient Cp.

Rewriting $q_{\infty }$ into Mach language (step by step)

We now rewrite the denominator using free-stream versions of the relations from §2–§3. Every substitution below uses $a_{\infty }$, $p_{\infty }$, ${\rho }_{\infty }$ — the undisturbed values, not the local ones:

Step 1 — free-stream sound speed. From §2 applied at infinity, $a_{\infty }^2=\gamma p_{\infty }/{\rho }_{\infty }$, which rearranges to ${\rho }_{\infty }=\gamma p_{\infty }/a_{\infty }^2$.

Step 2 — free-stream speed in Mach. From §3 at infinity, $V_{\infty }=M_{\infty }{a}_{\infty }$, so $V_{\infty }^2=M_{\infty }^2{a}_{\infty }^2$.

Step 3 — substitute both into $q_{\infty }$: $q_{\infty }=\frac{1}{2}{\rho }_{\infty }{V}_{\infty }^2=\frac{1}{2}\left(\frac{\gamma p_{\infty }}{a_{\infty }^2}\right)\left(M_{\infty }^2{a}_{\infty }^2\right)=\frac{\gamma }{2}{p}_{\infty }{M}_{\infty }^2.$ The two $a_{\infty }^2$ cancel — that is the whole point: sound speed disappears and only $M_{\infty }$ survives.

Step 4 — put it back into $C_p$: $C_p=\frac{p-p_{\infty }}{\frac{\gamma }{2}{p}_{\infty }{M}_{\infty }^2}=\frac{2}{\gamma M_{\infty }^2}\cdot \frac{p-p_{\infty }}{p_{\infty }}=\frac{2}{\gamma M_{\infty }^2}\left(\frac{p}{p_{\infty }}-1\right).$

Why group into the ratio $p/p_{\infty }$? Because §5 gives us pressure ratios directly (through the shared $p_0$). Writing $C_p$ in terms of $p/p_{\infty }$ lets us plug the isentropic relation straight in — the two facts snap together with no leftover $\rho$ or $V$.

$\boxed{C_p=\frac{2}{\gamma M_{\infty }^2}\left(\frac{p}{p_{\infty }}-1\right)}$

8. The critical pressure coefficient $C_{p,cr}$ — the universal target line

Building the pressure ratio $p/p_{\infty }$. Both the sonic local point and the free stream share one $p_0$ (§5). Write each with the isentropic relation and divide: $\frac{p}{p_{\infty }}=\frac{p/p_0}{p_{\infty }/p_0}=\frac{p_0/p_{\infty }}{p_0/p}.$

• Free stream at $M_{\infty }=M_{cr}$: $\frac{p_0}{p_{\infty }}={\left(1+\frac{\gamma -1}{2}{M}_{cr}^2\right)}^{\frac{\gamma }{\gamma -1}}$.
• Local point at $M_{local}=1$: substitute $M=1$ into the same $p_0/p$ relation: ${\frac{p_0}{p}|}_{M=1}={\left(1+\frac{\gamma -1}{2}\cdot 1^2\right)}^{\frac{\gamma }{\gamma -1}}={\left(1+\frac{\gamma -1}{2}\right)}^{\frac{\gamma }{\gamma -1}}.$

Dividing, the outer exponents combine into one bracket, giving the pressure ratio at the sonic point: $\frac{p}{p_{\infty }}={\left(\frac{1+\frac{\gamma -1}{2}{M}_{cr}^2}{1+\frac{\gamma -1}{2}}\right)}^{\frac{\gamma }{\gamma -1}}.$

Now feed this into the boxed $C_p$ of §7 with $M_{\infty }=M_{cr}$:

$C_{p,cr}=\frac{2}{\gamma M_{cr}^2}\left[{\left(\frac{1+\frac{\gamma -1}{2}{M}_{cr}^2}{1+\frac{\gamma -1}{2}}\right)}^{\frac{\gamma }{\gamma -1}}-1\right]$

It depends only on $\gamma$ — no airfoil shape enters. Picture it as a fixed "finish line" curve in the $(M_{\infty },C_p)$ plane. The wing's own suction curve (§9) races toward it; where they touch is $M_{cr}$.

9. The airfoil's own curve: $C_{p,0}$ and Prandtl–Glauert

As you fly faster, compressibility deepens that suction. The simplest estimate (from Prandtl–Glauert Compressibility Correction) is:

$C_{p,\min }(M_{\infty })=\frac{C_{p,0}}{\sqrt{1-M_{\infty }^2}}$

Within its valid range, as $M_{\infty }$ rises the airfoil curve dives down to meet the universal $C_{p,cr}$ line. That crossing is $M_{cr}$. Push past it and you enter drag divergence; designers escape it with sweep and the area rule.

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