Foundations — Normal shock waves — Rankine-Hugoniot relations (all 5) — derivations

Before you can read a single Rankine–Hugoniot line, you must own every symbol in it. This page builds each one from nothing, in the order they lean on each other, and shows the picture behind it.

1. The flow and its two states — subscripts $1$ and $2$

• The gas flows left→right through a tube of unit area (we imagine $1{\text{m}}^2$ of cross-section so we never have to write area — it cancels everywhere).
• State 1 = the "before" snapshot; state 2 = the "after" snapshot.
• The shock itself is so thin (a few molecule-widths) that we treat it as a flat mathematical surface with no thickness.

Why the topic needs it: every relation is a ratio like $p_2/p_1$ — a comparison of "after" to "before". Without the two labels there is nothing to compare.

2. Density $\rho$ — how tightly packed the gas is

• The left box (state 1) has a few dots — low density. The right box (state 2) has the same dots squeezed closer — higher density. The shock compresses the gas, so ${\rho }_2>{\rho }_1$.
• Why the topic needs it: "the flow gets compressed" is "$\rho$ rises". The ratio ${\rho }_2/{\rho }_1$ is one of the five headline results.

3. Velocity $u$ — how fast the gas moves

• In the picture of Section 1, $u_1$ is the long arrow (fast) and $u_2$ is the short arrow (slow). Crossing the shock, the flow slows down: $u_2<u_1$.
• Why the topic needs it: kinetic energy $\frac{1}{2}{u}^2$ and momentum $\rho u^2$ both live on $u$, and $u_1/u_2$ turns out to equal ${\rho }_2/{\rho }_1$ (Section 8).

4. Pressure $p$ and temperature $T$

• Across the shock both jump: $p_2>p_1$ and $T_2>T_1$. The re-entry heating story is $T_2\gg T_1$.
• Why the topic needs it: $p$ carries the momentum push; $T$ carries the thermal energy. Two of the five results are $p_2/p_1$ and $T_2/T_1$.

5. The ideal-gas link $p=\rho RT$

• Why the topic needs it: it converts between the three state variables. It is the bridge used to get Relation 5 (temperature) from Relations 3 and 4.

(For the deeper meaning of $p_0$, $T_0$ see Stagnation properties.)

6. Speed of sound $a$ and Mach number $M$

Why is $M$, and not raw $u$, the star of the whole chapter? Because whether a shock can even exist depends on comparing the flow to sound. A pressure signal (a "warning") travels at $a$. If $u>a$ ($M_1>1$), the gas outruns its own warnings — downstream fluid can't "hear" the obstacle, so nature builds a wall: the shock. That is why all five relations are written as functions of $M_1$ alone.

• The figure shows a source: at $M<1$ the ripples spread everywhere (warning gets ahead); at $M>1$ they pile into a cone/wall (no warning upstream).

(More on $a$ and $M$: Speed of sound and Mach number.)

7. The heat-capacity ratio $\gamma$ and enthalpy $h$

• Why the topic needs it: the energy conservation law reads $h_1+\frac{1}{2}{u}_1^2=h_2+\frac{1}{2}{u}_2^2$. Both the $\gamma$ and the $h$-form appear in every derivation step.

8. The three conservation laws — the whole engine

Read in plain words:

• Mass: whatever mass enters the slab per second must leave — molecules don't appear or vanish. So $\rho u$ (mass through unit area per second) is the same on both sides.
• Momentum: the momentum of the flow changes only because of the pressure pushes on the two faces. So $p+\rho u^2$ (pressure push + momentum flux) matches across.
• Energy: no heat added, no work done outside ⇒ total enthalpy $h+\frac{1}{2}{u}^2$ is conserved.

Why the topic needs it: everything — all five Rankine–Hugoniot relations — is pure algebra on these three lines plus $p=\rho RT$. Nothing else enters.

9. Entropy $s$ — the referee that picks the real shock

• Why the topic needs it: conservation laws alone can't tell you which direction the jump goes. Entropy is the tiebreaker, and it also explains why total pressure $p_0$ falls (see Stagnation properties).

10. Notation you'll meet — quick glossary

11. How the foundations feed the topic

The three laws (mass, momentum, energy) + the ideal-gas link produce the five ratios; the Mach number is the single variable they all depend on; entropy selects the physically real branch.

Equipment checklist

Cover the right side and test yourself — you are ready when every line is instant.

Next: with every symbol earned, go build the five ratios in Normal shock waves — Rankine-Hugoniot relations (all 5) — derivations. Related roads: Isentropic flow relations, Oblique shock waves, Rayleigh & Fanno flow.