Adapted from notes by David Bell:
For the sawtooth to be made functional, the remainders of the generation numbers where the line burns out (modulo 192) must either be constant or else oscillate only over a few "safe" values.
The sawtooth can be easily modified by shifting the glider reflectors forwards or backwards along the path and by adjusting when the beehive is turned into the backward glider. But simply bouncing the gliders back using the glider reflectors might not be good enough; it might be necessary to hold onto them until the right generation numbers before releasing them.
Another sawtooth mechanism: a pair of gliders can arrive from the end of the line to cleanly ignite it while sending a glider back to the source direction. Perhaps the timings involved in this alternative mechanism would be easier:
Update: 16 August 2005
Notes by Dave Greene:
I tried replacing the reflectors at the stationary end of the pattern with a p8 universal regulator, and ended up with an almost-sawtooth that works nicely for 342,000 generations -- and then blows up. Apparently my math is still wrong somewhere...
My one useful result was that a line-igniting glider can be delayed by any multiple of 176 generations, and the burning fuse will still arrive at the active site at the same phase of the p192 lineship. [After 192 generations the fuse is longer by 16 cells, so it takes 16 ticks longer to burn. So to keep the burning fuse from arriving late at the active site, you have to subtract 16 ticks from 192].
-- So I attached a p1056 gun (6 x 176) to the universal regulator, and found a configuration that survives for quite a few cycles... apparently by sheer luck, since there's something wrong with the underlying theory. The problem seems to be that the ever-increasing return time of the glider also extends the length of the fuse by a variable amount -- and the fuse burns four times as fast as a glider travels, so it's easy for the feedback effect to result in several possible arrival times. The burning-out reaction is versatile enough to handle any amount of lateness up to 80 ticks or so after the phase used for the first couple of cycles -- but eventually a phase always seems to come along that is off by more than that.
I also tried p880 and p2112 drive guns [I thought I had accounted for the variable fuse lengths with p2112, which is LCM(176,192) -- but no such luck.] It seems possible that delaying one of these drive guns by some number of generations would get the pattern into a stable cycle -- I just haven't figured out how to predict this in advance yet, and brute-force searching is fairly tedious in this case. Anyway, it wouldn't be quite as interesting to get the right answer by accident!
I tried writing equations to predict the length of later cycles, given the glider travel time and fuse burning time and the phase of the burning fuse's arrival at the active site... but so far I've always ended up with wrong answers after a cycle or two. Haven't given up yet, but would be happy if someone else wanted to figure it out!