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This video has a nice description of resonance inside a Rubens' tube, and it has some pretty nice visuals of the tube being driven at the precise resonance,

as well as just above resonance:

I rather like that last one, which shows that even off-resonance you can still see fringes in the flames (with the contrast degrading the more you go away from the resonance), essentially because the reflectance of the two ends is not perfect, and the resulting decay of the pulse kills the destructive interference because it limits the time available for the reflecting pulse to get out of phase with the driver.

I am curious to see this worked out mathematically, which shouldn't be too challenging - just a scalar wave driven at a close-to-resonance frequency $\omega$ at one end, with non-perfect reflectance $r$ on both ends. It should be a good exercise, so instead of confining it to some notes I'd throw away anyway, I thought I'd open the floor here.

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  • $\begingroup$ Ooooooh. That's quite a nice example of a Rubens' tube, there. They are pretty easy to slap together but not at all easy to get just right. $\endgroup$ – dmckee --- ex-moderator kitten Apr 14 '17 at 15:18
  • $\begingroup$ @dmckee Mould claims that he bought that one, so it's presumably commercially available, though I can't see whether he states where he got it. $\endgroup$ – Emilio Pisanty Apr 14 '17 at 15:23
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The source of sound for modern day Ruben's tubes generally a speaker attached at one end of the tube.

Even if driven with a pure harmonic, the speaker does not necessarily reproduce the tone; there is some harmonic distortion that tends to widen the spectrum of the tone. So if you drive the frequency slightly off of resonance as predicted by tube length, etc., you still have some power being driven by the speaker at the resonant frequency. But since its not full power, the amplitude will slightly decrease until you drive it well away from the frequency.

I build my own tube, here and besides running with an oscillator to observe standing waves, I drive it with music like Frankenstein by the Edgar Winter Group and Jerry Lee Lewis's Great Balls of Fire. A fine show.

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    $\begingroup$ The question asks for an explicit waveform, regardless of the mechanism. You have an alternative theory to the suggestion in the OP and that's all well and good but you still need to back it up with actual hard math. $\endgroup$ – Emilio Pisanty Apr 5 '18 at 23:27
  • $\begingroup$ @EmilioPisanty Well it explains why no answers in about a year. Indeed hard math. Beyond my capability. Sorry. If you are more well versed, recommend you look into harmonic distortion. The difficulty here though may not be the math but rather identifying the physics, the elements causing the nonlinearity. Maybe the speaker. Maybe something in the gas. Maybe the imperfect reflections. No math will answer these hypotheses. Only carefully designed experiments to rule one or the other out. $\endgroup$ – docscience Apr 9 '18 at 20:06
  • $\begingroup$ @EmilioPisanty I learned something very interesting this year. That sound waves can still propagate standing waves in a curved channel. Experiments showed not only harmonic distortion, but also a shift in the harmonics from where tube length predicted them to be. Research afterwards led to a paper that published theory supporting my observations. Have you done any deep digging on published works regarding the Rubens tube? $\endgroup$ – docscience Apr 9 '18 at 20:12

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