The Musical Fire of Rubens' Tubes

In 1905 the German physicist Heinrich Rubens devised a spectacular device to demonstrate the acoustic standing waves that develop in air columns. A long metal tube is drilled with a row of small holes along its top and filled with flammable gas. One end is capped, the other is connected to a speaker or nozzle. As the gas escapes and is ignited, a horizontal line of flames dances above the tube. When a pure tone resonates within, the flames grow taller at pressure antinodes and shrink at nodes, rendering the otherwise invisible wave pattern vividly visible. This calculator estimates the spacing of those peaks and the overall behavior of the wave for any chosen tube length and driving frequency.

The foundation of the calculation is the wavelength $\mathrm{\backslash lambda}$ of the sound within the tube. If the excitation frequency is $f$ and the speed of sound is $v$, then $\mathrm{\backslash lambda}=v/f$. In a tube with both ends effectively closed by the gas membrane and speaker, pressure antinodes form at the boundaries while velocity nodes occur there as well. The fundamental mode therefore fits half a wavelength into the tube length $L$, satisfying $L=\backslash lambda/2$. Higher modes accommodate additional half waves. Knowing the wavelength allows us to compute the spacing between consecutive nodes: it is simply $\backslash lambda/2$. The number of peaks (antinodes) that will appear is approximately $\left(\backslash lfloor\; 2L/\backslash lambda\; \backslash rfloor\; +\; 1\right)$, where the floor function counts how many half waves fit inside plus one for the boundary.

Because real tubes are not perfectly closed, the flame pattern tends to blur near the ends and the effective length differs slightly from the physical length. Our calculator presents idealized values that still provide an excellent starting point for experimenters. For a driving frequency of 440 Hz (concert A) in a one-meter tube, for example, the wavelength is approximately 0.78 m. Only a single pressure antinode appears in the center, matching the dramatic single arch often seen in demonstrations. Doubling the frequency halves the wavelength and yields three antinodes with a node spacing of about 0.39 m.

From Nodes to Flame Heights

To convey more than just counts, the tool samples the standing wave along the tube length at evenly spaced positions. At a point $x$ measured from the speaker end, the pressure variation of a standing wave can be written as $P(x)\; =P$₀ \sin\left(\tfrac{2\pi x}{\lambda}\right). The flame height is roughly proportional to the local pressure because higher pressure pushes more gas through the holes. The calculator normalizes the amplitudes between zero and one and lists them in a table so builders can anticipate the relative size of each flame. This preview aids in choosing hole spacing and tube length for artistic displays.

Historical and Educational Context

Rubens' tube is beloved in classrooms and science museums because it converts abstract wave equations into a mesmerizing kinetic sculpture. Observers can immediately see that certain frequencies produce crisp stationary patterns while others lead to chaotic flickering. By sweeping a tone generator across frequencies, one watches nodes drift and merge, revealing the continuity of resonant modes. The device also highlights the relationship between wavelength and frequency: higher notes create denser flame patterns, a phenomenon intimately tied to the fundamental relation $f=v/\mathrm{\backslash lambda}$. Beyond pure physics education, artists have employed arrays of Rubens' tubes as audio displays and pyrotechnic instruments.

The typical tube ranges from one to four meters in length with hole spacings of 1–2 cm drilled using a jig. Propane or natural gas provides fuel while a small speaker at one end injects sound. Safety is paramount: the tube must be leak-tested, the gas regulated, and a nonflammable environment maintained. Because the flames consume oxygen, demonstrations are best conducted outdoors or with proper ventilation. Though the calculator does not model gas flow or combustion dynamics, the standing wave pattern is largely independent of those details so long as the burn rate is uniform.

Using the Calculator

Enter the physical length of the tube, the frequency of the tone you wish to play, and the speed of sound (which varies slightly with temperature). The calculator returns the wavelength, the fundamental frequency that would place a single antinode in the center, the spacing between nodes, and how many peaks you should observe. It also produces a table listing sample positions and relative flame heights. Adjust the number of sample points to control how fine-grained the table is. Because the math assumes ideal boundary conditions, the actual observed pattern may shift by a few centimeters. Experimenters often fine-tune by cutting the tube slightly long and then trimming while watching the flames.

The following table illustrates sample output for a 1.2 m tube driven at 500 Hz with sound speed 343 m/s. The table lists position along the tube and the normalized flame height predicted at that point.

Position (m)	Relative Flame Height
0.00	0.00
0.30	0.82
0.60	0.97
0.90	0.82
1.20	0.00

In practice the flames near the endpoints will be shorter than ideal due to pressure leakage, yet the central two peaks should appear as predicted. If the driving frequency is changed to 750 Hz, the calculator shows five peaks, demonstrating how quickly the pattern evolves with tone. These values help performers choreograph musical pieces that synchronize with changing flame landscapes, turning the Rubens' tube into a fiery equalizer.

Extending the Model

Advanced users may wish to account for temperature gradients along the tube, the damping effect of gas flow, or nonuniform hole spacing. The speed of sound itself follows $v=\backslash sqrt\{\backslash gamma\; R\; T\; /\; M\}$, where $\mathrm{\backslash gamma}$ is the adiabatic index, $R$ the gas constant, $T$ the absolute temperature, and $M$ the molar mass of the gas mixture. Warmer gas leads to larger wavelengths for a given frequency. Some builders deliberately heat the tube to shift resonances or to create dynamic visual effects. Others construct two-dimensional arrays using multiple parallel tubes to display crude spectral visualizations of music, combining engineering with art.

Even with these complexities, the essential physics remains the interplay between frequency, wavelength, and tube length. By experimenting with the inputs to this calculator, one can quickly predict how a design change will influence the fiery dance. Whether you are a teacher preparing a classroom demonstration, a maker crafting a sound-reactive sculpture, or a performer seeking a pyrotechnic accompaniment, the Rubens' tube offers an unforgettable fusion of sound and flame.