The Rijke Tube: Turning Heat Into a Screaming Tone

A pipe that sings when you heat it

The apparatus is almost insultingly simple. Take an open-ended vertical pipe — a classic demonstration uses a brass or glass tube roughly 0.3 to 1 m long and a few centimetres across — and fix a disc of wire gauze (iron or copper) across it in the lower half. Heat the gauze to red heat, and within a second or two the whole tube erupts into a loud, sustained, nearly pure musical note that can reach 90–100 dB. Pieter Leonard Rijke, professor of physics at Leiden, first reported the effect in 1859.

In Rijke's original version the gauze is warmed by a Bunsen flame held under the tube; when the flame is slid away the note continues to sing for perhaps ten seconds, fading as the gauze cools. Heat the gauze electrically instead and the tone rings indefinitely. Two quick tests expose the underlying physics before any equations are written. First, tilt the tube to horizontal and the sound stops instantly. Second, move the gauze into the upper half and the tube stays stubbornly silent no matter how hot the wire. Any correct explanation must account for both facts.

Two ingredients: a resonator and a heat source in a draft

A Rijke tube is really two systems that happen to share the same air. The first is an acoustic resonator. An open–open pipe supports standing waves exactly like an organ flue pipe: the fundamental fits half a wavelength in the tube, so λ = 2L and the frequency is f = c / 2L. With the speed of sound c ≈ 343 m/s and L ≈ 0.8 m that is roughly 210 Hz; because the resident air is hot, the effective sound speed is higher and the real pitch sits a little above the cold prediction, typically 150–250 Hz. In this mode the two open ends are pressure nodes and velocity antinodes, while the centre is a pressure antinode and a velocity node.

The second system is the heat source. The glowing gauze warms the air touching it; that air becomes buoyant and rises, and cool air is drawn in from below to replace it. The result is a steady upward mean flow — a convection draft — through the pipe, driven by buoyancy exactly as a chimney is. So we have a localized heat source sitting inside a mean flow, embedded in a resonant air column. The scream comes from the coupling between the unsteady heat release and the sound field, and that coupling has a strict phase rule.

Rayleigh's criterion: heat delivered at the right moment

The governing principle was stated by Lord Rayleigh in a short 1878 note in Nature, "The Explanation of Certain Acoustical Phenomena." In his words: if heat be given to the air at the moment of greatest condensation, or taken from it at the moment of greatest rarefaction, the vibration is encouraged. Put plainly — add heat when the gas is most compressed, and the oscillation grows; add it when the gas is most rarefied, and the oscillation is killed.

The reason is thermodynamic. Each parcel of gas in the standing wave traces a small cycle in the pressure–volume plane. If you inject heat while the parcel is at high pressure, it expands and pushes harder on its neighbours; if you inject the same heat at low pressure, it does less work. Heating in phase with the pressure therefore leaves a net positive amount of p dV work per cycle — each parcel behaves like a tiny heat engine feeding the sound wave. Formally this is captured by the Rayleigh index, the cycle- and volume-integral of the acoustic pressure fluctuation p′ times the heat-release fluctuation q′:

• R ∝ ∮ p′(t) q′(t) dt (weighted by (γ−1)/γp̄, with γ the heat-capacity ratio).

If R is positive and large enough to exceed the tube's damping, the acoustic energy grows and the tube sings. If R is negative, or positive but too small, the wave decays. Everything about where to put the gauze reduces to steering the sign and size of that integral.

Why the lower half drives and the upper half damps

The crux is a time lag. Heat transfer from the hot gauze to the passing air rises with the flow speed across the wires — the classic hot-wire relation is King's law, Q ∝ A + B√U, the same physics used in hot-wire anemometry. So when the air rushes past the gauze faster, more heat is dumped into it. But the transfer is not instantaneous: the thermal boundary layer around each wire takes a fraction of the acoustic period to respond, delaying the heat actually delivered to the gas.

Now follow the phases. In the fundamental standing wave the acoustic velocity leads the pressure by a quarter cycle. In the lower half of the tube, the phase of inflow toward the central pressure antinode is directed upward, adding to the mean convection draft; the total speed past the gauze therefore peaks a quarter cycle before peak pressure. The thermal lag then delays the delivered heat by roughly another quarter cycle, so it lands right at the moment of maximum compression — precisely Rayleigh's condition, R > 0, and the tone grows. In the upper half the compression-phase acoustic velocity points downward, opposing the draft; the phase relationship flips so the extra heat arrives during rarefaction, giving R < 0 and active damping.

This is why the mean draft is essential: it breaks the up–down symmetry of the tube. Lay the pipe horizontal and the buoyant flow vanishes; the two halves become mirror images whose contributions cancel, so no position sings — matching the tilt test. Place the gauze at the exact centre and it sits on a velocity node, where there is no oscillating flow to modulate, so the coupling collapses. The drive is strongest where the acoustic velocity is large and biased by the draft, which is why the optimum is near the lower quarter, x ≈ L/4.

The governing equations and the growth-to-limit-cycle route

Quantitatively the tube obeys the linearized one-dimensional equations of thermoacoustics: momentum, ρ₀ ∂u′/∂t = −∂p′/∂x, and an energy balance with a heat-release source, ∂p′/∂t + γp₀ ∂u′/∂x = (γ−1) q′(x,t). Eliminating u′ yields the ordinary acoustic wave equation but now forced by (γ−1)∂q′/∂t, a term concentrated at the gauze. Solving the resonator with this localized source gives a complex frequency whose imaginary part is the growth rate σ: it is positive (the mode grows) when the Rayleigh drive at the gauze outweighs the combined losses — viscous and thermal dissipation in the boundary layers, heat conduction, and acoustic radiation leaking out of the open ends. That balance sets a threshold heating power: below it the tube is silent, above it the tone erupts.

Left unchecked, exponential growth would give infinite loudness. Instead the amplitude saturates into a limit cycle once the acoustic velocity becomes comparable to the mean flow: the √U heat-transfer law flattens, and at large amplitude the flow at the gauze can momentarily reverse, both of which cap the drive. The Rijke tube is therefore a textbook self-excited oscillator of the same mathematical family as the van der Pol oscillator — unstable at small amplitude, self-limiting at large amplitude, settling onto a steady tone.

From lab curiosity to engines, fridges, and rocket disasters

The Rijke tube belongs to a family of thermoacoustic devices. Its closed-tube cousin is the Sondhauss tube (Karl Sondhauss, 1850), a bulb-and-neck that glassblowers noticed would sing when the hot glass bulb cooled — here there is no mean flow, and the sound is driven by oscillatory heat exchange at the hot closed end. At the opposite temperature extreme, Taconis oscillations appear when a tube is dipped into liquid helium: the huge temperature gradient sets a gas column singing all by itself.

Engineered deliberately, the same p′–q′ coupling becomes useful hardware. Thermoacoustic engines convert a temperature difference into acoustic power with no moving parts; building on Peter Ceperley's 1979 traveling-wave insight, Greg Swift and colleagues at Los Alamos developed practical machines, and the Backhaus & Swift (1999) traveling-wave heat engine reached about 30% efficiency, a healthy fraction of the Carnot limit. Run the cycle backwards and you get a thermoacoustic refrigerator: loudspeaker-driven acoustic power pumps heat up a temperature gradient, offering rugged, environmentally benign cooling — explored for cryocoolers and even spaceflight.

The dark side is the same physics uninvited. When combustion is the heat source, the Rayleigh criterion governs combustion instability: if the flame's heat release happens to peak in phase with a chamber acoustic mode, pressure oscillations grow explosively. The F-1 engine of the Saturn V suffered violent instability; Rocketdyne engineers ran on the order of 2,000 test firings and added injector baffles and acoustic-damping cavities to tame it, because such oscillations can grow by tens of percent and destroy a chamber in milliseconds. The same "screech" and "rumble" constrain modern low-NOx gas-turbine combustors and jet afterburners. A humming pipe on a lecture bench and a rocket engine shaking itself apart are, at heart, the same equation.

How the gauze position sets the Rayleigh index and decides whether the tube sings.

Gauze position	Acoustic velocity there	Heat release vs. pressure	Outcome
Lower quarter (≈ L/4)	Large; reinforces the upward draft	In phase (Rayleigh index ≫ 0)	Loudest, fastest-growing tone
Just below centre	Moderate	Still in phase (index > 0)	Weaker but still sings
Exact centre (L/2)	≈ Zero (velocity node)	Negligible coupling	Silent — nothing to drive
Upper half	Opposes the draft	Out of phase (index < 0)	Damped — any tone dies out
Horizontal / no draft	No mean flow, symmetry restored	Cancels over the cycle	Silent at every position