Acoustic modes in combustion chamber — cause of instability

WHY do acoustic modes even exist?

Deriving the mode frequencies from scratch

WHAT we assume: small pressure perturbations $p'$ on top of a mean pressure, gas at rest on average, uniform speed of sound $c$.

Step 1 — Governing equation. Linearizing mass + momentum + the ideal-gas relation gives the wave equation: $\frac{\partial^2 p'}{\partial t^2} = c^2\,\nabla^2 p'$ Why this step? Any small disturbance in a compressible gas obeys the wave equation — pressure changes accelerate gas, gas motion compresses gas, and the loop propagates at $c$.

Step 2 — Separate space and time. Look for standing solutions: $p'(x,t) = P(x)\,\cos(\omega t)$ Why? A resonance oscillates at ONE frequency everywhere; only its amplitude varies with position. Substituting: $-\omega^2 P = c^2 P'' \;\Rightarrow\; P'' + k^2 P = 0,\quad k=\frac{\omega}{c}$

Step 3 — Apply boundary conditions (1-D longitudinal, length $L$). At a rigid wall the gas velocity must be zero, and since velocity $\propto \partial p'/\partial x$, we need a pressure antinode ($P'=0$) there. For a chamber closed at both ends (a good first model, since the injector face and the throat both partly reflect): $P(x)=A\cos\!\Big(\frac{n\pi x}{L}\Big),\qquad k_n=\frac{n\pi}{L}$ Why the cosine with $P'=0$ at both ends? Cosine has zero slope at $x=0$ and at $x=L$ only when $kL=n\pi$. That's the "fits neatly" condition.

Step 4 — Frequencies. Since $\omega = ck$:

For a cylindrical chamber you also get transverse modes solved with Bessel functions: $f_{mn}^{\text{trans}}=\frac{\alpha_{mn}\,c}{2\pi R}$ where $\alpha_{mn}$ are Bessel-derivative roots and $R$ is the chamber radius. These are usually classified as:

• Longitudinal (L): waves along the axis — set by $L$.
• Tangential (T): waves spinning around the axis — most destructive.
• Radial (R): waves in/out from the axis.

WHY do these modes cause INSTABILITY?

HOW instability actually builds — the derivation of "in phase = growth": Model the acoustic energy $E$ gained per cycle as the work done by heat release. Using $p'\propto\cos\omega t$ and $q'\propto\cos(\omega t-\phi)$: $\Delta E \propto \int_0^{2\pi/\omega} \cos(\omega t)\cos(\omega t-\phi)\,dt \;\propto\; \cos\phi$ Why? The time-average of a product of two cosines is $\tfrac12\cos\phi$.

• $\phi=0$ (heat exactly in phase): $\cos\phi=+1$ → maximum drive → unstable.
• $\phi=90^\circ$: $\cos\phi=0$ → neutral.
• $\phi=180^\circ$ (out of phase): $\cos\phi=-1$ → damping → stable.

So the phase between heat release and pressure — not the magnitude — decides the sign.

Worked examples

Common mistakes (Steel-manned)

Flashcards

Connections

• Speed of sound in gases — sets every mode frequency via $c=\sqrt{\gamma R T/M}$.
• Standing waves and resonance — same math as organ pipes/strings.
• Rayleigh Criterion — the phase condition governing thermoacoustics.
• Nozzle flow and acoustic damping — main energy loss stabilizing modes.
• Injector design and baffles — engineering fixes to shift the heat-release phase.
• Rocket Propulsion — combustion chamber overview — parent note.

Concept Map

Hinglish (regional understanding)