What is the waveform for a Rubens' tube driven slightly off-resonance?

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.

• 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. – dmckee --- ex-moderator kitten Apr 14 '17 at 15:18
• @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. – Emilio Pisanty Apr 14 '17 at 15:23