The phenomena of the singing tube and burner and the Rijke tube are
both cases in which the heat source is oscillating at the frequency
of the tone. These are examples of oscillators in which positive
feedback reinforces the oscillation of the source. The feedback is
in the form of a periodic reflection from the upper end of the tube.
The intensity of the oscillation can be tuned by adjusting the phase
of the feedback. In the case of the Rijke tube the optimum placement
of the screen is 1/4 of the length from the bottom of the tube, This
makes the reflection path 1-1/2 times the length of the tube, or at
the speed of sound, 3/4 of a cycle, giving a phase shift of 90
degrees. Getting the sign right involves knowing what the correct
phase shift is on reflection, and I can't quite get my head around
that one this afternoon. Thirty years ago Brian Pippard and I tried
to figure out a good explanation of how a Rijke tube works, but we
were not very successful. I think I've got it now, however. The
screen acts as a thermal reservoir and the surface of the wires
oscillates in temperature as a function of the air velocity which
flows past it. The flow is a combination of a steady convection and
an oscillating term. corresponding to the sound.
I think it likely that the burner version has a very similar
explanation for the phase shift, and that the burner flame output
also varies periodically, though the mechanism corresponding to the
heat capacity and thermal conductance combination of the Rijke tube
wire screen doesn't immediately strike me. As far as mode locking
goes, yes, I suppose you could use that term, but it does not
enhance conceptual grasp in my opinion.