Foundations — Detached bow shock

Before you can read the parent note Detached bow shock, every letter it uses must mean something you can picture. Below, each symbol is built from nothing, anchored to a picture, and justified — why does the topic even need it? Read top to bottom; each rung leans on the one below it.

1. Speed, sound, and the Mach number $M$

Think of air as a crowd of people. If you nudge one person, the "nudge" passes along person-to-person at a fixed pace — that pace is $a$. It is how fast news of a disturbance can travel.

The parent writes $M_1$ for the Mach number of the flow upstream (in front of) the shock. The little 1 means "state 1 = before the shock"; 2 means "state 2 = after the shock". Likewise $a_1$ is the sound speed before the shock and $a_2$ the (higher, because hotter) sound speed after it.

Figure s01 — outrunning your own sound. The yellow dot is the body flying to the right at $M=2$. Each blue circle is a sound ripple it emitted earlier, spreading outward at speed $a$. Because the body moves twice as fast as those ripples grow, the circles pile up behind it and their common tangent (the pink lines) forms a sharp wedge — the Mach wave. Look at how the pink lines only exist because the body is outrunning the blue circles: at $M<1$ the body would sit inside every circle and no wedge could form.

See Mach angle and Mach waves for the wave-cone picture behind supersonic flight.

2. A vector and its components: $u_n$ and $u_t$

When the air arrow meets a slanted wall (the shock), we split the arrow into two perpendicular pieces:

• $u_n$ — the piece normal (perpendicular) to the shock. "Normal" here means at a right angle to, not "ordinary".
• $u_t$ — the piece tangential (parallel) to the shock, sliding along it.

Figure s02 — split the arrow. The yellow line is the shock wall. The white arrow is the incoming velocity $\vec{u}$. We break it into the pink piece $u_n$ (straight through the wall) and the blue piece $u_t$ (sliding along the wall). Watch that the blue piece lies flat along the yellow wall while the pink piece stabs across it — only the pink piece gets slowed by the shock; the blue one is unchanged on the far side.

3. Trig tools: $\sin$, $\cos$, $\tan$, $\cot$

The parent's key formula is full of these. Each answers a specific question on a right triangle. Picture a right triangle with an angle at one corner: the side across from it is opposite, the side next to it (not the long slanted one) is adjacent, and the long slanted side is the hypotenuse. We build these before the angles of §4 because those angles are measured with exactly these tools.

4. Angles: $\theta$, $\beta$, and $\mu$

Three angles run through the whole topic. Keep them straight. Each is measured with the trig tools of §3.

Note the range of $\beta$: it lives between the weakest shock ($\beta =\mu$) and the strongest ($\beta =90°$, a head-on normal shock).

Figure s03 — the three angles. The dashed white line is the incoming flow direction. The yellow line is the shock, tilted by $\beta$ (yellow arc) from that flow. The pink arrow is the flow after the shock, bent by $\theta$ (pink arc). Compare the two arcs: $\beta$ (yellow) is always bigger than $\theta$ (pink) — the shock wall tilts more than the flow bends. The blue note reminds you $\mu$ is the smallest $\beta$ the wall could ever take.

See Oblique shock waves for how $\theta$ and $\beta$ dance together, and Maximum deflection angle and weak/strong shock solutions for the weak/strong split.

5. Fluid state symbols: $\rho$, $\gamma$

The density ratio ${\rho }_2/{\rho }_1$ is the physical heart of the shock: mass is conserved, so if the flow slows and crowds up, that ratio tells you by exactly how much.

6. Reading subscripts and the "$n$" tag

Put the labels together so the parent's dense notation reads like English:

Both $M_{n1}$ and $M_{n2}$ are normal Mach numbers — the perpendicular velocity component divided by the local sound speed. Precisely: $M_{n1}=u_{n1}/a_1$ (using the cool upstream sound speed $a_1$ from §1) and $M_{n2}=u_{n2}/a_2$ (using the hotter downstream sound speed $a_2$). They let us reuse straight-on normal-shock math on the perpendicular slice of an oblique shock.

7. Stand-off distance $\Delta$ and nose radius $R_n$

How it all feeds the topic

The map below is read bottom to top, like building a tower. Sound speed lets us define the Mach number. Trig lets us measure the three angles and combine them with the velocity components into the θ–β–M relation. That relation has a peak — ${\theta }_{max}$ — and whether the body's demanded turn clears that peak decides whether the shock stays attached or detaches into a bow shock. The stand-off scaling then tells us how far ahead the detached shock sits.

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