It is good that you have spared me the maths and yourself the cramped finger joints. I start to 'glaze over' pretty quickly when the sigmas and backwards sixes start to appear. However, in this case, even you verbalization of the situation leaves me wondering what's wrong -- you say I'm '180 degrees out of phase' but then you seem to re-state my premises almost exactly ;-) To wit:
I have said that you should
use an overlength pipe for starting:
It seems logical to me that once you get a resonant length for cold starts, you should work your way out gradually until you get the longest variation that will still easily catch hold and run.
You stipulate that you may find it advantageous
to use an overlength pipe for starting:
It is easier to start an over-long duct from "cold iron".
Isn't that the same thing?
And again, my saying
So, getting the longest pipe that will sustain on a cold start should come into even better resonance as it heats up, and be able to keep going.
seems to me remarkably like your assertion that
... as the temperature within a duct rises, resonances are only enhanced ...
It does not seem to me that we basically disagree on this point at all.
The assertion of mine which is truly suspect is
This could result in dropping out of resonance after warmup if you start short.
What I mean by this is that, if your engine is some design which relies on a phase relationship between the combustion tube [i.e. chamber and tailpipe] and another resonant element [say, a long tubular intake duct], this phase relationship could be right for a cold engine, but not right when the combustion tube heats up, if you start with the shortest possible overall length. If such a 'phase balanced' design is correctly proportioned cold, there could be a significant phase shift, because the average gas temperature of the air-cooled intake will not rise in keeping with the combustion tube, i.e. its natural frequency might hardly rise at all. [The specific engine design will determine how true this really is in practice.]
Yes, the phrase "dropping out of resonance" was silly and misleading. It is the phase relationship between resonant elements of different temps and dimensions that I was trying to get at. It seems to me that you want to arrive at the perfect phase relationship as you gain temperature, not have the perfect relationship when you start cold and have it degrade as you get running. An extension for starting is a useful "cheat" to get the relationship right for the cold start and then restore that relationship again by removing it after warmup starts to throw it out of whack. This, I believe, is exactly what Rossco has observed.
Another positive effect of an extension for starting might be that it gives you a more massive 'piston' to work against for better compression during the early 'constant volume expansion' phase of combustion.