The tube that sings when heated low down

In 1859 Pieter Rijke, a professor at Leiden, put a piece of wire gauze into a glass tube about a metre long, held the tube upright, heated the gauze red-hot with a flame, and took the flame away. The tube sounded a loud, low note, its lowest resonance, for as long as the gauze stayed hot — a few seconds — and it sounded only when the gauze was in the lower half of the tube, and loudest a quarter of the way up. With the gauze in the upper half, or the tube horizontal, or the top closed, there was no sound.

Rijke’s tube is the cleanest case of a phenomenon that turned out to be everywhere heat is released inside a resonator. Bryan Higgins had heard a hydrogen flame sing inside a glass tube in 1777; the howl of a badly adjusted gas burner, the rumble of an industrial furnace, the oscillations that destroyed early liquid-fuel rocket engines and that still trouble the low-emission combustors of gas turbines are the same. All of them obey a rule that Lord Rayleigh stated in The Theory of Sound in 1878: heat given to the air at the moment of greatest compression, or taken from it at the moment of greatest rarefaction, encourages the vibration.

A sound that needs a push #

A resonator left to itself rings down. The width that is a lifetime found that every resonance loses energy at a definite rate, and an organ pipe or an open tube loses its acoustic energy through its open ends, where sound radiates away, and to friction at its walls. For it to sing steadily, something must feed it energy at the same rate, and the feeding must be in step with the oscillation — the frequency that gets an answer found that a periodic push transfers energy only if it has a component in phase with the velocity of what it pushes.

Heat can push air. A small packet of air heated suddenly expands, and that expansion is a push on the surrounding air, a source of sound. The question is when the heat arrives. Heat added while a packet of air is being compressed — while the sound’s pressure there is high — makes it expand against the compression and adds energy to the wave, as a child standing up at the bottom of a swing adds energy to the swing; heat added while the air is expanding, at low pressure, adds expansion when the wave would rather have the air contract, and takes energy out. That is Rayleigh’s criterion, and it reduces the singing tube to two questions: where in the tube is the pressure swinging, and when does the gauze give up its heat?

Pressure where the air is still #

In the tube’s lowest mode, the ends are open to the room and the pressure there cannot swing; the air at the ends moves in and out freely. In the middle the opposite holds: the air is pushed from both sides at once and stays still while its pressure swings most.

A gauze can only add heat in proportion to how much the air moves through it: a hot wire’s heat loss depends on the speed of the air flowing past it. At the ends the air moves most but the pressure does not swing, so nothing is driven. In the middle the pressure swings most but the air does not move through the gauze, so its heat transfer does not oscillate. The driving is the product of the two, and it is largest a quarter of the way along, where both are substantial.

That product changes sign at the middle. In the lower half, when the air is moving upward through the gauze, the pressure above it is rising; in the upper half, upward motion goes with falling pressure. So the same gauze, releasing heat in step with the air’s motion, drives the sound in one half of the tube and damps it in the other — but which half drives depends on which way the air is flowing steadily through the gauze, because the gauze’s heat transfer responds to the total speed of the air past it, steady draught plus oscillation. In an upright tube the hot gauze draws a steady draught upwards, the lower half is upstream of it, and the lower half drives. Lay the tube on its side and the draught stops, the heat transfer responds to the oscillation without a steady flow to ride on, and its response is at twice the sound’s frequency, which drives nothing; the tube falls silent.

Heat that arrives late #

There is still something missing. In the standing wave, the pressure and the air’s velocity are a quarter cycle apart in time: the pressure at a point is greatest when the air there has stopped and is about to reverse. A gauze whose heat release followed the air’s velocity instantly would release its heat a quarter cycle away from the pressure maximum, half of it while the pressure was high and half while it was low, and do nothing on average.

The gauze does not follow instantly. Its heat has to cross the thin layer of air clinging to each wire before it reaches the moving air, and that takes time: the heat release lags behind the air’s velocity by a fraction of a period. A lag shifts part of the heat release towards the moment of high pressure, and the tube begins to be driven.

The growth rate goes as the sine of the lag. A real gauze, with its heat crossing the boundary layer round its wires in a tenth of a period or so, sits well inside the driving range. A gauze that responded with a lag of more than half a period would damp the sound wherever it was put, and the place where the tube sings would move to the upper half.

The tube as an engine #

Rayleigh’s criterion is a statement about a heat engine, and it helps to see the tube that way. Follow a small packet of air near the gauze through one cycle of the sound. As the wave compresses it, the packet’s pressure rises; with the lag, it receives heat from the gauze while its pressure is high; it then expands as the pressure falls, and gives up less heat, or none, while its pressure is low. In a diagram of pressure against volume the packet traces a loop, and the area of the loop is work done on the surrounding air — work that goes into the sound. It is the same loop every heat engine runs, with heat taken in at high pressure and rejected at low, and the ceiling on every engine, set before it was designed applies to it: the work cannot exceed what the temperature difference between the gauze and the air allows.

The Rijke tube is a poor engine. The packet’s temperature swing is tiny, the heat transferred in phase with the pressure is a small fraction of the heat the gauze gives up, and most of the gauze’s heat simply warms the draught. The acoustic power of a singing tube is a fraction of a watt from a gauze giving up hundreds. But the engine has no moving parts, no valves and no timing mechanism: the sound itself sets the timing, by moving the air through the gauze, and the gauze’s lag supplies the phase. Engineered thermoacoustic engines improve the efficiency by making the heat exchange happen along a stack of plates whose spacing is chosen so that the lag is close to the ideal quarter period everywhere along it.

Where it sings, and how loudly #

Whether the tube actually sings depends on the driving exceeding the tube’s own losses, and that fixes a range of positions.

The figure uses a model that has become standard for the Rijke tube — the tube’s lowest acoustic mode, damped by its losses, driven by a gauze whose heat transfer follows the air’s speed past it by King’s law for a hot wire, with a lag. With a weak gauze nothing happens anywhere. With a hotter one the tube sings over a range of positions centred on a quarter of its length, the range widening as the gauze gets hotter. Rijke found the loudest sound at a quarter of the length; he could not have known that this is where $\sin(2\pi x/L)$ peaks, but the experiment found it.

Once the sound grows, something must stop it growing, and the same gauze does. King’s law makes the heat transfer depend on the square root of the air’s speed, so it grows less than in proportion as the oscillation grows. When the air’s to-and-fro speed at the gauze becomes comparable to the steady draught — when the flow through the gauze reverses for part of each cycle — the heat release cannot follow the oscillation any more, and the driving levels off.

The result is a limit cycle: an oscillation whose amplitude is set not by how it was started but by the balance between the driving and the losses at large amplitude. The clock that is pulled into step followed such self-sustained oscillators — clocks, hearts, lasers — and the singing tube is one more: a steady source of heat and a resonator together make a steady note, as a steady wind and a reed make one in a clarinet.

Loud at once, and slow to stop #

The way the note starts has one more surprise. A smoothly growing driving should start the sound quietly, its amplitude growing gradually from zero as the driving passes the threshold. In the model, as in experiments on Rijke tubes, it does not.

Raise the gauze’s heating gradually and the tube stays silent until the linear threshold is passed, then jumps to a loud note. Lower it and the note persists well below the threshold before it collapses. Between the two thresholds, silence and song are both stable, and which one the tube is in depends on its history — a subcritical onset, of the kind the calm that is the ghost of a cycle followed in other oscillators. The nonlinearity of the heat transfer is responsible: at moderate amplitude, King’s law makes the driving relatively stronger than at small amplitude, so a large oscillation can sustain itself where a small one cannot start. For a combustor, this is the dangerous property. A machine running just below its linear threshold can be tipped into a violent oscillation by a single large disturbance — a gust, a pressure pulse from a valve — and then stays there.

Glassblowers’ tubes and singing flames #

Rijke’s tube has relatives that sing for related reasons. Glassblowers knew long before Rijke that a glass tube with a hot bulb blown at one end, its other end open, could sing on its own; Carl Sondhauss studied the effect in 1850. In a Sondhauss tube the heat is at the closed end, and the oscillation is driven by the temperature gradient along the tube’s wall rather than by a draught through a gauze: air moving towards the hot end is heated as it is compressed there and cooled as it moves back towards the cold end and expands. It is the ancestor of the thermoacoustic engines built since the 1980s.

Higgins’s singing flame, a small hydrogen flame at the end of a narrow pipe pushed up inside a wider glass tube, sings at the tube’s resonance when the flame is at the right height, and stops when it is moved. There the flame’s heat release responds to the oscillating flow of gas up its feed pipe, with a lag set by the time the gas takes to burn. Each of these is Rayleigh’s criterion with a different way of making heat follow the sound at the right moment. A resonator’s note itself is the ordinary one — the note a bottle sings whatever its shape found how a cavity’s shape sets its pitch — and what the heat supplies is not the note but the energy that keeps it going, as the regular pushes of the swing that is pumped, not pushed keep a swing going, but here from a source that is not pulsed at all.

Rockets, turbines and refrigerators #

What a gauze does in Rijke’s tube, a flame does in a combustion chamber, more violently. A flame’s heat release responds to pressure and velocity fluctuations through the mixing of fuel and air and the time the mixture takes to burn, and when that response has the right lag at the right place in a chamber’s acoustic modes, the chamber sings. In the development of the F-1 engines of the Saturn V rocket in the early 1960s, combustion instability destroyed engines on the test stand, and it took some two thousand full-scale firings — including detonating small bombs inside running engines to test whether an oscillation, once started, would die away — to find an injector design with baffles that broke up the transverse modes. Modern gas turbines burn lean, premixed fuel to cut nitrogen oxide emissions, and lean flames are especially sensitive; their designers spend much of their effort predicting and suppressing thermoacoustic oscillations, with acoustic dampers, careful placement of fuel injection, and sometimes active control that measures the pressure and adjusts the fuel in antiphase.

The same coupling can be run on purpose. A thermoacoustic engine uses a temperature difference across a stack of plates inside a resonator to sustain a sound with no moving parts, and the sound can then pump heat — a fridge with no work going into it followed such devices — or drive an alternator. They are, at heart, Rijke tubes engineered for efficiency: Rayleigh’s criterion arranged to hold everywhere along the stack, with the phase between heat transfer and pressure controlled by the spacing of the plates rather than left to the lag of a gauze.

What the figures leave out #

The figures use a model with one acoustic mode, a gauze treated as a point, a fixed time lag, and King’s law for the heat transfer. A real Rijke tube has higher modes, which a gauze can also drive; the steady draught warms the air above the gauze, which changes the sound speed in the upper half and moves the mode shapes slightly; and the gauze’s lag depends on the draught’s speed, which itself depends on the gauze’s temperature, which falls as the gauze radiates and is cooled by the oscillation it drives. The heat-transfer strength K bundles the gauze’s temperature, size and the draught into one number. The amplitudes are in the model’s own units, and the thresholds and hysteresis are the model’s; experiments find the same qualitative behaviour with numbers that depend on the details. The domain is a long, narrow tube driven in its lowest mode by a gauze much smaller than a wavelength.

Still open: predicting it in a real engine #

The Rijke tube is understood well enough to be a textbook example and a test bed. Real combustors are not. A flame’s response to the acoustic field — how its heat release changes when the pressure and velocity at its base oscillate — depends on the turbulence, the fuel’s mixing, the flame’s shape and the chamber’s geometry, and is usually measured rather than predicted. Combining measured or simulated flame responses with acoustic models of whole engines, to predict before a machine is built whether it will sing, is an active field, and so is the question of how noise in a turbulent flame interacts with a subcritical onset — whether random fluctuations can tip a stable combustor into oscillation, and how much warning, in the statistics of its noise, a combustor gives before it does. Early-warning signals of this kind, found in the noise’s spectrum and its autocorrelation, are now being tested on real turbines. Noise can do more than warn: in a system with two stable states, it can carry the system from one to the other, as the noise that helps a weak signal through found for a particle hopping between two wells, and in the bistable range of a subcritical combustor it can switch a quiet flame into a roaring one and back, so that the machine spends part of its time singing for no reason an operator can see.

Heat released in a resonator drives its sound when it arrives while the pressure is high, Rayleigh’s criterion; in an open tube’s lowest mode that makes the driving ∝ $\sin(2\pi x/L)\sin(\omega\tau)$ — positive in the lower half for a heat lag under half a period, largest at a quarter of the way up — so a hot gauze at L/4 grows a whisper into a loud note limited by its own saturating heat transfer, a gauze at 3L/4 damps it, and the onset is subcritical: in the model, the note starts past K = 0.63 and stops only below 0.55. A tube sings when its heat is released in the right place and a little late.