Combustion instability — low-frequency (chugging), high-frequency (screaming): Visual walkthrough

This is the visual companion to the parent topic.

Step 1 — The swing: what "in phase" even means

WHAT. Picture a child on a swing. The swing's position $x (t)$ goes back and forth — we call that a smooth oscillation. You, the pusher, apply a force $F (t)$ . The question of the whole page is: when should you push so the swing grows?

WHY this first. Every symbol later ( $p^{'}$ , $q^{'}$ , phase $ϕ$ ) is just a dressed-up version of this swing. If you own the swing, you own the rocket.

PICTURE. In the figure, the blue curve is the swing's velocity $v (t)$ (how fast it moves, positive = moving forward). The green arrows are your pushes.

Figure — Combustion instability — low-frequency (chugging), high-frequency (screaming)

Push when velocity is already forward (green arrow points the same way as motion) → you add energy, the swing grows. This is being in phase.
Push against the motion (velocity backward) → you steal energy, the swing dies.

The single most important idea on this page: growth is a timing question, not a strength question.

Step 2 — From swing to gas: naming $p^{'}$ and $q^{'}$

WHAT. A combustion chamber holds hot gas at a mean pressure $\overset{p}{ˉ}$ (the steady background). On top of that steady value, the pressure wobbles a little. We call that wobble $p^{'}$ (read: "p-prime").

$p (t) = steady mean \overset{p}{ˉ} + tiny wobble p^{'} (t)$

The prime $^{'}$ always means "the fluctuating part, the ripple on top of the average."

WHY split it. The steady $\overset{p}{ˉ}$ does nothing interesting — it just sits there. All the drama (growth or decay) lives in the ripple $p^{'}$ . So we track only $p^{'}$ .

Now the pushing. In a rocket you don't push with your hands — you push with heat. Burning fuel dumps heat into the gas. The rate at which heat is dumped also wobbles; its ripple is $q^{'}$ (read: "q-prime"), the fluctuating heat-release rate per unit volume.

PICTURE. Blue = pressure ripple $p^{'}$ . Yellow = heat-release ripple $q^{'}$ . They are the swing's velocity and your push, re-labelled for a gas.

Step 3 — Why heat is a push: one line of gas physics

WHAT. We must justify that "adding heat" really acts like "pushing the swing." Trap a gas parcel at fixed volume and pour in a little heat $δ Q$ . Its pressure rises: $δ p = \frac{γ - 1}{V} δ Q .$

Term by term, right where each lives:

$δ Q$ — the pinch of heat you added (Joules).
$V$ — the volume of the parcel; a bigger parcel warms less per Joule, so $V$ divides.
$γ$ (gamma) — the specific-heat ratio, a pure number ( $\approx 1.2$ for hot rocket gas) measuring how "springy" the gas is. $(γ - 1)$ is exactly the fraction of added heat that turns into a pressure kick.

WHY it matters. This proves the causal arrow: more heat now → more pressure now. Heat is literally a pressure push. So the swing rule transfers directly.

PICTURE. A sealed cylinder; a red flame dumps heat; the pressure gauge needle jumps up.

Step 4 — The rate equation: how fast acoustic energy grows

WHAT. Let $E$ be the energy stored in the pressure oscillation (the "loudness" of the swing). Combining Step 3 with the acoustic energy balance for a perfect gas gives the master rate:

$\frac{d E}{d t} = \frac{γ - 1}{γ p ˉ} p^{'} q^{'} .$

Reading it symbol by symbol, in place:

$\frac{d E}{d t}$ — the rate of change of oscillation energy. Positive = getting louder. The symbol $\frac{d E}{d t}$ answers the question "is the swing gaining or losing energy right now?"
$\frac{γ - 1}{γ p ˉ}$ — a positive constant (from Step 3's springiness, per unit mean pressure). Being positive, it never changes the sign of the answer — it just scales it. So we can ignore it when we only care about grow-vs-decay.
$p^{'} q^{'}$ — THE product. This is the whole story.

WHY a product, and why this tool. We need a single number that is large-and-positive only when $p^{'}$ and $q^{'}$ are both pointing the same way at the same instant, and negative when they oppose. Multiplication does exactly that:

$p^{'}$	$q^{'}$	product $p^{'} q^{'}$	meaning
+ high	+ high	+	heat lands on a peak → grows
+ high	− low	−	heat lands as pressure peaks — but heat is falling → damps
− low	− low	+	(heat removed while pressure low also reinforces — the swing rule is symmetric)
− low	+ high	−	mistimed → damps

Neither $p^{'}$ alone nor $q^{'}$ alone can tell you this — only their product encodes alignment. That is precisely why the product tool, and no simpler one, appears here.

PICTURE. Blue $p^{'}$ , yellow $q^{'}$ , and their red product $p^{'} q^{'}$ drawn underneath, shaded green where positive (energy in) and red where negative (energy out).

Step 5 — Why the loop $\oint$ : bookkeeping over a whole cycle

WHAT. One instant is not enough. During a single wobble, $p^{'} q^{'}$ is sometimes $+$ and sometimes $-$ . What matters is the net over one full cycle. We add up $p^{'} q^{'}$ across the whole loop:

$\oint p^{'} q^{'} d t .$

Reading the new symbols:

$\oint$ — the loop integral. The little circle on the integral sign means "sum over exactly one complete cycle, returning to the start." We use it (not a plain $\int$ ) because an oscillation is a closed loop in time; we want the running total after the swing comes back around.
$d t$ — "for each tiny slice of time $d t$ "; the integral is the grown-up word for add up all the slices. The green area minus the red area from Step 4, summed over one period.

WHY. A momentary $+$ can be cancelled by a later $-$ . The cycle total is the honest referee:

$\oint p^{'} q^{'} d t > 0 \Rightarrow oscillation GROWS (instability)$

This is Rayleigh's Criterion, now fully earned. See Rayleigh Criterion.

PICTURE. The green (positive) and red (negative) areas of $p^{'} q^{'}$ over one period, with a running tally showing the net area is positive → the swing amplitude ratchets up each cycle.

Step 6 — The phase dial: one knob controls everything

WHAT. Slide the timing between push and swing. Let the pressure be $p^{'} = cos ω t$ and the heat be delayed by a phase angle $ϕ$ : $q^{'} = cos (ω t - ϕ)$ .

$ω$ (omega) — the angular frequency, how fast the oscillation spins through its cycle.
$ϕ$ (phi) — the phase lag: how far behind the pressure the heat arrives, measured as an angle where a full cycle $= 36 0^{\circ}$ .

Do the loop integral (verified below) and everything collapses to a single clean fact: $\oint p^{'} q^{'} d t \propto cos ϕ .$

WHY $cos ϕ$ is perfect. Cosine is the natural "alignment meter": it is $+ 1$ when two waves line up ( $ϕ = 0$ ), $0$ when they are a quarter-cycle apart ( $ϕ = 9 0^{\circ}$ ), and $- 1$ when opposed ( $ϕ = 18 0^{\circ}$ ). The growth/decay verdict is just the sign of $cos ϕ$ .

ALL CASES — no gaps:

phase lag $ϕ$	$cos ϕ$	verdict
$ϕ = 0^{\circ}$ (perfectly in phase)	$+ 1$	maximum growth
$0^{\circ} < ϕ < 9 0^{\circ}$	$+$	grows
$ϕ = 9 0^{\circ}$ (quarter cycle)	$0$	neutral — the exact knife-edge
$9 0^{\circ} < ϕ < 27 0^{\circ}$	$-$	damps (safe)
$ϕ = 18 0^{\circ}$ (dead opposite)	$- 1$	maximum damping
$ϕ = 27 0^{\circ}$	$0$	neutral again
$27 0^{\circ} < ϕ < 36 0^{\circ}$	$+$	grows (loops back to start)

PICTURE. A dial from $0^{\circ}$ to $36 0^{\circ}$ ; green growth wedge, red damping wedge, with the $9 0^{\circ}$ and $27 0^{\circ}$ knife-edges marked. The whole engineering battle is: shove $ϕ$ into the red.

Step 7 — What sets $ϕ$ : the two clocks (chug vs scream)

WHAT. The phase lag $ϕ$ comes from a time delay $τ$ between a pressure blip and the heat it eventually causes. In angle terms, $ϕ = ω τ$ . Different physical delays give different frequencies — same Rayleigh rule, two personalities.

WHY two answers. There are two reservoirs that can store a disturbance and hand it back:

Slow clock — CHUGGING. The delay is the propellant transit + ignition lag $τ$ (milliseconds). A quarter-cycle alignment needs $ω τ \approx π /2$ , giving $f_{c h ug} \sim \frac{1}{4 τ} .$ With $τ \sim$ ms → tens-to-hundreds of Hz. This feedback runs through the injector pressure drop and choked nozzle filling/emptying; its cure is a stiff injector, and its size scale sits inside the characteristic velocity.
Fast clock — SCREAMING. The delay is the acoustic transit time across the chamber (microseconds), set by the cavity's natural modes. With a hot-gas sound speed $a \sim 1000$ m/s and radius $\sim 0.1$ m → kilohertz. Cured by baffles and Helmholtz cavities that add loss (this is the world of Thermoacoustics).

PICTURE. Two loops sharing the same Rayleigh core: the slow feed-line loop (blue, ms) and the fast acoustic loop (red, μs), both feeding the same $p^{'} q^{'}$ integrator.

The one-picture summary

WHAT. Everything on one canvas: swing ⇒ ripples $p^{'}, q^{'}$ ⇒ heat-is-a-push ⇒ product $p^{'} q^{'}$ ⇒ loop integral ⇒ phase dial ⇒ two clocks. The verdict flows left to right.

Recall Feynman retelling — say it back in plain words

A rocket chamber is a swing, and heat is the hand that pushes it. The swing is the pressure ripple $p^{'}$ ; the push is the heat ripple $q^{'}$ . Because pouring heat into trapped gas makes pressure, a push landing on an already-high peak stacks a taller peak — that's a growing scream. To score a whole cycle honestly, we multiply push times swing at every instant ( $p^{'} q^{'}$ , positive when they agree), then add it around the loop ( $\oint$ ). Positive total means the engine gets louder every cycle — instability. The single dial that decides it is the timing lag $ϕ = ω τ$ : if heat arrives within a quarter-cycle of the pressure peak it grows; if it lags into the far side it damps. Engineers don't fight the loudness directly — they twist that dial into the safe red zone. When the storing clock is the slow feed-line lag (milliseconds) you hear a lazy chug; when it's the fast acoustic ring across the chamber (microseconds) you hear a lethal kilohertz scream. One criterion, two clocks.

Recall Quick self-check

Why the product $p^{'} q^{'}$ and not $p^{'} + q^{'}$ ? ::: The product is positive only when both point the same way — it measures alignment (timing). A sum would just measure size, not whether the push lands on a peak. Why the loop $\oint$ and not a plain moment? ::: An oscillation is a closed cycle; a momentary gain can be undone later, so only the net over one full period tells you grow-vs-decay. What single knob controls growth in Step 6? ::: The phase lag $ϕ$ ; the verdict is the sign of $cos ϕ$ — grows for $∣ ϕ ∣ < 9 0^{\circ}$ , damps for $9 0^{\circ} < ϕ < 27 0^{\circ}$ . Chug vs scream difference in one word? ::: The clock — millisecond feed lag $τ$ (chug) vs microsecond acoustic transit (scream); same Rayleigh rule.