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Pressure along the tube: p(x,t) now (blue), FDTD envelope (orange), closed-form amplitude (green dashed)
Loudness at the pressure antinode versus frequency; dashed lines are the resonances
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How to use
Pick a preset or press Tune to nearest resonance, then watch the flame row: at a resonance the flames are tall and short in alternating groups; N and A markers show the pressure nodes and antinodes. Try: (1) mode 1 and mode 3, count the nodes and compare with the formula; (2) drag the frequency slider off the resonance and see the flames flatten while the response graph shows the speaker is no longer in step with the tube; (3) switch the far end between closed and open and see that the resonances move; (4) change the gas to methane and the temperature to 40 C and note that every resonance shifts with the speed of sound; (5) lower the gas supply pressure to 100 Pa and turn up the loudness until the flames blow out at the pressure troughs. Slow motion sets how many wave periods you see per second. Drag to orbit, wheel to zoom. Space pauses, R resets.
The workshop, tube with flanges and brass burner nozzles, loudspeaker with its cone and surround, gas cylinder with regulator and gauge, signal generator, amplifier, rotating-mirror drum and extinguisher were modelled and baked in Blender; Three.js r160 animates the cone, the 56 flames, the gas motion and the pressure colouring from the same finite-difference tube model as the 2D version.
The science
The demonstration. A long metal tube is filled with a flammable gas (classically propane) and drilled with a row of small holes along the top. A loudspeaker at one end drives a sound wave in the gas. A small flame burns at every hole. At certain frequencies the flame row turns into a picture of the sound wave: the flames are tall where the pressure swings hard and almost unchanged where it does not. Rubens built it in 1905.
One-dimensional acoustics. For wavelengths much longer than the tube diameter (here the bore radius is a = 5 cm, so only plane waves propagate below about 1.4 kHz in propane, where the first transverse mode appears) the gas obeys the damped wave equations for pressure p(x,t) and gas velocity u(x,t):
rho du/dt = - dp/dx - R u, dp/dt = - rho c² du/dx, R = 2 rho c alpha
c = sqrt(gamma R_gas T / M) is the speed of sound in the gas (about 250 m/s in propane, 343 m/s in air, 446 m/s in methane), rho = p M / (R_gas T) its density, and alpha (1/m) the amplitude decay rate that stands in for wall friction, heat loss and the gas escaping through the holes.
Boundary conditions. The speaker cone moves with velocity u(0,t) = u0 sin(2 pi f t). A rigid cap gives u = 0 at the far end (a pressure antinode); an open mouth gives p = 0 a little beyond the tube end, at the effective length L + 0.61 a (end correction for an unflanged pipe).
closed tube: f_n = n c / (2 L), n = 1, 2, 3 ... open mouth: f_n = (2n - 1) c / (4 (L + 0.61 a))
Exactly at f_n the speaker and the reflected wave reinforce each other: a standing wave p(x,t) = P(x) sin(...) with fixed nodes (P = 0, spaced by half a wavelength lambda/2 = c / 2f) and antinodes (|P| maximal). Losses limit the height of the resonance peak; the steady-state solution with a complex wave number k = (w/c) sqrt(1 - 2 j c alpha / w) is evaluated in closed form and drawn as the green dashed curve.
Why the flames move. Gas leaves a hole at a speed given by Bernoulli's law, v = sqrt(2 (P_g + p') / rho_gas), where P_g is the supply (gauge) pressure and p' the acoustic pressure. A laminar flame is roughly proportional to the gas flow, so its height is h = h_ref sqrt((P_g + p') / P_ref). Where p' swings between +A and -A the flame height pulses between h_ref sqrt(1 + A/P_g) and h_ref sqrt(1 - A/P_g); at a node p' stays zero and the flame sits at its still height. If A exceeds P_g the pressure inside the tube briefly drops below the atmosphere, the flow reverses and the flame goes out at the troughs (the "low gas pressure" preset).
Slow motion and the mirror. A 100 Hz wave cannot be seen with the eye (the original apparatus used a rotating mirror). Here simulated time is slowed so that the chosen number of wave periods pass each second. The waterfall panel stacks successive instants of the flame row: stripes that alternate are antinodes, steady bands are nodes.
Numerical method. The wave equations are solved on a staggered grid (pressure at cell centres, velocity on faces, leapfrog time stepping, CFL 0.95, cells of about 5 mm). The speaker drives the first face, the far end is a rigid face or a pressure-free face, and a short pre-run brings the tube to its steady state whenever a parameter changes.
Independent checks in this page (panel at the bottom): (1) the first three resonances found by exciting a lossless tube with a pressure pulse and Fourier-analysing the ring-down are compared with the textbook formulas; (2) the steady-state pressure amplitude along the tube at the nearest mode is compared with the closed-form complex solution (peak and whole profile); (3) the spacing of the pressure nodes is compared with c / 2f; (4) the discrete energy of a lossless tube is checked to stay constant.
Honest simplifications. Plane waves only (no transverse modes, no viscous boundary layer beyond one lump coefficient alpha); the holes are not individual acoustic elements and do not radiate; the flame height law is a simple square-root model with no flame dynamics, buoyancy or flicker instability (the small flicker you see is cosmetic noise); the tube is isothermal (the flames do not heat the gas); the cone is a perfect velocity source. Displacements of the gas dots are exaggerated so they can be seen. Real Rubens tubes need loud sound, about 120 to 140 dB near the antinodes (the readout shows the value for your settings). Flames are drawn, never lit: do not repeat this with real gas without proper supervision and safety equipment.