Foundations — Under-expanded nozzle — Prandtl-Meyer expansion, efficiency loss

Before we can read a single line of the parent note, we must earn every letter it uses. Nothing below assumes you have seen this notation before. We build in order: each idea leans only on the one above it.

1. Pressure — the "push per patch" $p$

Think of blowing up a balloon. The air inside pushes outward on every bit of rubber equally. That outward push, measured per unit of area, is the pressure.

We will meet three different pressures, and the whole topic is really a story about which one is bigger:

The subscript $\infty$ (infinity) just means "far away, undisturbed" — the ordinary air pressure of the surroundings.

Look at the figure: the same gas has three pressures depending on where you measure. The topic's central question is a simple comparison — is $p_e$ bigger than, equal to, or smaller than $p_{\infty }$?

We need $p$ because "under-expanded" is defined by a pressure comparison. See 3.3.14-Over-expanded-nozzle-shock-diamonds for the opposite case.

2. Area $A$ and the nozzle shape

A rocket nozzle is a tube that first narrows to a tight throat, then widens out — the classic bell shape. As the gas passes through and the area changes, the gas speeds up. That link between area and speed is the 3.2.7-Isentropic-flow-area-Mach-relation.

We need $A_e$ because thrust has a pressure-times-area term: $A_e(p_e-p_{\infty })$. A push per area, multiplied by the area it acts on, gives a total force.

3. Speed of the gas $v$ and mass flow $\dot{m}$

The dot notation is worth pausing on: $\dot{m}$ is not "m times something" — the dot means rate of change in time. So $\dot{m}$ = "mass, per second."

4. Force and Thrust $F$

By Newton's third law, throwing gas backward pushes the rocket forward. Thrust has two contributions, and the topic's key thrust equation just adds them:

Notice: when $p_e>p_{\infty }$ (under-expanded) that second term is positive — it adds thrust. Yet the topic says we still lose efficiency. The resolution of that puzzle is what the whole parent note is about, and we need the 2.5.12-Thrust-coefficient-definition to measure it.

5. Angles — how we describe "turning"

Look at the red arrow in the figure. It started axial (straight back). After the gas expands outside the nozzle, it points outward by angle ${\theta }_{\text{turn}}$. Only the part still pointing straight back helps push the rocket — and that part is shorter than the whole arrow.

What is $\cos \theta$, from zero?

Why cosine and not something else? Because "how much of a tilted arrow points forward" is exactly the definition of cosine — the shadow the arrow casts onto the forward axis. When $\theta =0$ (no turning), $\cos 0=1$: the whole arrow is useful. When $\theta$ grows, $\cos \theta$ shrinks below 1, and the missing fraction $(1-\cos \theta )$ is the thrust thrown sideways and lost.

6. Mach number $M$ — speed measured in "sound-lengths"

Why measure speed this way instead of in metres per second? Because the behaviour of a gas flips completely at $M=1$:

• $M<1$ (subsonic): disturbances (pressure news) can travel upstream — the flow "knows what's ahead."
• $M>1$ (supersonic): nothing can outrun the flow — pressure news cannot travel upstream.

That single fact is why the expansion cannot be smooth and uniform: because the exhaust is supersonic ($M_e>1$), the pressure drop at the exit lip cannot warn the gas in advance. The adjustment must happen through waves fanning out from the lip. That is the Prandtl-Meyer fan.

7. Mach waves and the Mach angle $\mu$

Picture a speedboat: the faster it goes, the flatter and more swept-back its wake lines. Same here — faster gas (bigger $M$) gives a smaller, more swept angle $\mu$.

What is $\arcsin$ (and $\arctan$)?

The parent note is full of $\arctan$ and $\arcsin$. These are "un-doing" functions — they answer the reverse question.

We need them because the Prandtl-Meyer formula computes an angle from a speed ratio — that is precisely the "un-do" direction.

8. Isentropic flow and $\gamma$

This is the key assumption that lets us track the gas: because $p_0$ is preserved, knowing $p_e/p_{\infty }$ tells us the new Mach number.

You do not derive $\gamma$; you look it up for your gas. It appears throughout the Prandtl-Meyer function and the pressure relations.

9. Prandtl-Meyer function $\nu (M)$ — the "turning budget"

Now every piece is earned. The parent's central tool:

Prerequisite map

Every box above is a symbol we built on this page; every arrow is "you need this before that." Follow the arrows into the topic node and you can read the parent note line by line.

Equipment checklist

Cover the right side and test yourself. If you can answer all, you are ready for the parent note.

See also 3.3.13-Optimal-expansion-ratio, 3.3.16-Altitude-compensation-nozzles, and the oblique-shock counterpart 4.1.3-Oblique-shock-theory.