Foundations — Wave drag — transonic and supersonic

This page builds every letter, ratio, and picture the parent note (Wave drag — transonic and supersonic) leans on. Read top to bottom: nothing is used before it is drawn.

0. The characters in the play

Before symbols, meet the physical actors with plain words:

• The air — a springy fluid. Squeeze it and it pushes back; that push is pressure.
• A disturbance — any nudge to the air (the nose of a plane passing by). It spreads outward as a pressure ripple.
• The body — the thing moving through the air (wing, wedge, nose cone).

Everything else is a number attached to one of these three.

1. Pressure $p$ — the "push per area"

The picture: imagine a swarm of tiny air molecules hammering a wall. Each hit is tiny; billions per second add up to a steady push. That steady push per square metre is $p$.

Why the topic needs it: wave drag is ultimately a difference in pushes. If the air pushes harder on the front of a body than on the back, the leftover is a backward force. Shocks are exactly regions where pressure jumps, so $p$ is the raw material of drag.

2. Speed of sound $a$ — how fast a "warning" travels

The picture (Figure s01): drop a pebble in a pond and rings spread outward at a fixed speed. Air does the same in 3-D: any nudge sends out an expanding sphere of pressure. The radius of that sphere after a time $t$ is $at$.

Why this exact tool? Air can only "rearrange itself to get out of the way" by passing pressure signals around. Those signals move at $a$ — no faster. So $a$ is the speed limit of advance warning. The whole drama of wave drag is about whether the body beats this speed limit. See Speed of sound and Mach number.

3. Flow speed $u$ and Mach number $M$ — the crucial comparison

Now the single most important number in the whole chapter. We don't care about $u$ alone or $a$ alone — we care which is bigger. The natural way to compare two speeds is to divide them:

• $M<1$ → subsonic
• $M=1$ → sonic
• $M>1$ → supersonic
• $M$ hovering near 1 (roughly 0.8–1.2) → transonic

The subscript $M_{\infty }$ (read "M-infinity") just means the Mach number of the free stream — the undisturbed air far away from the body, before the body messes it up.

4. Building the Mach cone — geometry of outrunning your own warnings

This is the payoff of everything so far. Watch what the spheres of Section 2 do when the body moves.

The picture (Figure s02): the body emits a pressure sphere at every instant.

• In time $t$, one old sphere has grown to radius $at$.
• In that same time, the body has moved forward a distance $ut$.

If $u<a$ ($M<1$): the body stays inside all its spheres. Air ahead always has a warning. Smooth flow.

If $u>a$ ($M>1$): the body races ahead of its spheres. All the spheres line up along a cone trailing behind the body — the Mach cone. Its front is a wall of piled-up ripples.

Finding the cone's angle — why we need the sine. Look at the right triangle in Figure s02: the body's path (length $ut$) is the long side (hypotenuse), and the sphere's radius (length $at$) is the side opposite the cone's half-angle $\mu$. The trig ratio that connects "the side opposite an angle" to "the hypotenuse" is the sine:

$\sin (\text{angle})=\frac{\text{opposite}}{\text{hypotenuse}}$

We choose sine (not tangent or cosine) precisely because we happen to know the opposite side and the hypotenuse — those are the two lengths the geometry hands us. So:

All the cases:

• $M\to 1^{+}$: $1/M\to 1$, so $\mu \to 90°$ — the cone flattens into a wall straight across the flow.
• $M=2$: $\sin \mu =1/2$, $\mu =30°$.
• $M\to \infty$: $1/M\to 0$, so $\mu \to 0°$ — the cone wraps tight against the body.
• $M\le 1$: $1/M\ge 1$, and no real angle has a sine bigger than 1 — the equation has no solution. That is the maths telling you there is no cone below the speed of sound.

More on this in Sonic boom and the Mach cone.

5. Angle $\theta$, shock angle $\beta$, and why the real shock is steeper than $\mu$

The picture (Figure s03): a wedge with half-angle $\theta$. The dashed line at $\mu$ (weak-signal cone) sits inside the solid line at $\beta$ (real attached shock). The stronger the required turn, the more the shock leans forward.

The exact link between $\theta$, $\beta$, and $M$ is the θ–β–M relation. Two words on its extreme cases so you meet no surprise later:

• Small $\theta$ → weak shock, $\beta$ near $\mu$.
• $\theta$ bigger than a maximum ${\theta }_{max}$ → no attached shock exists; it pops off the body as a curved bow shock standing ahead of a blunt nose. See Normal and oblique shock waves.

$\gamma$ (gamma), appearing in that relation, is the ratio of specific heats — a fixed property of the gas (about $1.4$ for air) describing how strongly it heats when squeezed. You only need to know it's a constant near 1.4.

6. Entropy $s$, the gas constant $R$, and stagnation pressure $p_0$ — why a shock equals drag

Why these matter, and the link between them. Across a shock the flow keeps its total energy (nothing leaks out as heat externally) but wastes some of it into disorder. The exact accounting is:

Why the exponential $e^x$ here? It is the function that turns an additive change (entropy, which adds up) into a multiplicative factor (a pressure ratio). Because $\Delta s>0$, the exponent is negative, so $e^{-\Delta s/R}<1$: the downstream stagnation pressure is always a fraction of the upstream one. Stagnation pressure is lost.

And why lost $p_0$ = drag: a body's wake carries less pushing potential than the air it swallowed. That missing momentum, by Newton's bookkeeping over a box around the body, shows up as a net backward force — wave drag, with no friction needed. Full treatment: Entropy and stagnation pressure loss.

7. Dynamic pressure $q$, pressure coefficient $C_p$, drag coefficient $C_D$, and the factor $\sqrt{M^2-1}$

Before we can make drag dimensionless we need the "rush" of the flow as a pressure. Moving air carries kinetic energy; when it piles onto a surface, that energy shows up as extra push.

Now we can turn raw pushes into fair, size-free numbers by dividing by $q$.

The parent note's supersonic formulas all carry the factor $\sqrt{M_{\infty }^2-1}$. Meet it now so it isn't a surprise:

• For $M_{\infty }>1$ this is a real, positive number that grows with $M$. Since it sits in the denominator of $C_p=2\theta /\sqrt{M^2-1}$ (and hence of $C_D$), higher supersonic speed → smaller factor → less wave drag per surface.
• At $M_{\infty }=1$ it is $0$, and dividing by zero screams "the theory blows up right at the speed of sound" — exactly the transonic spike.
• For $M_{\infty }<1$ the quantity under the root is negative; the subsonic version uses $\sqrt{1-M^2}$ instead. See Prandtl-Glauert compressibility correction.

Related design numbers you'll meet: the critical and drag-divergence Mach numbers $M_{cr}$ and $M_{dd}$ mark where trouble begins and where drag spikes; the Area Rule and sweep angle $\Lambda$ are the fixes.

8. How it all feeds together

Read it as: pressure defines a warning speed, warning speed vs body speed defines $M$, $M$ decides whether a cone (then a shock) forms, the shock raises entropy and drops stagnation pressure, and that loss is the wave drag that $C_D$ measures.

Equipment checklist

Cover the right side and answer each before moving to the parent topic: