Level wave speed limit?
Posted: January 26th, 2023, 9:16 pm
Are there any known bounds on what level wave speed limits are possible? By "level" I mean something like:
Contrast with something like:
which I would not describe as level.
In terms of speed orthogonal to the level I find it suspicious that nothing between c/2 and c is known. By reading slanted waves the "wrong" way I am pretty sure it's going to be possible to get speeds between c/2 and c (or even over c!) which is why I limit my question to level waves. Also c itself (reduced) is definitely achievable even in level waves, e.g. 2c/2:
I hacked up a copy of LLSSS to do some weird wrapping searches and barring a bug I believe it has shown there are no level waves of unreduced speeds 2c/3, 3c/4, 3c/5, 4c/5, 4c/6, 5c/6, 4c/7, 5c/7, 6c/7, 5c/8, 6c/8, 7c/8, 6c/9, 7c/9, 8c/9, 7c/10, 8c/10, or 9c/10 (all periods up to 10 except for 5c/9 and 6c/10). It tended to run slower and take more memory closer to c/2 (which is why I did not complete 5c/9 or 6c/10).
I would conjecture there are no level waves between c/2 and c and in fact more strongly that there are no "level back half planes" there either, by which I mean any full plane pattern traveling at a speed between c/2 and c which has a level front half plane of all zero must be all zero everywhere.
Relatedly it is known that there are no back half planes of a specific spatial slope which travel faster than c/2, but the argument fundamentally relies on the spatial slope of the half plane division and any sort of argument like this would some how have to "allow c through".
Also relatedly it is known that for any traveling plane at a speed < c with a level front half of parallel zebra stripes must be all zebra stripes. This is tantalizingly close to what we want to show and how the "< c" part shakes out is maybe interesting, but I see no way to repurpose the argument here.
EDIT: As a perhaps related question is it possible for a slanted wave to have its two orthogonal speeds both be > c/2? I would think not but am not immediately sure how to prove any such thing. The usual argument about c/2 upper bound for finite patterns seems very tied to its specific spatial slope.
EDIT2: I had forgotten about this horrible thing I had found earlier when digging in 2c/3. I would describe it as a "slanted back half plane" for speeds 2c/3 and 4c/7 (both faster than c/2):
It means if my implied conjecture about slanted waves having to have at least one side <= c/2 is true it's true in a very weird way (that depends on the front-to-back finite depth).
Code: Select all
x = 63, y = 4, rule = B3/S23
3o2b3o2b3o2b3o2b3o2b3o2b3o2b3o2b3o2b3o2b3o2b3o2b3o$3obo3bob3obo3bob3ob
o3bob3obo3bob3obo3bob3obo3bob3o$obo7bobo7bobo7bobo7bobo7bobo7bobo$4b2o
b2o5b2ob2o5b2ob2o5b2ob2o5b2ob2o5b2ob2o!
Code: Select all
x = 54, y = 30, rule = B3/S23
bo$b2o$2b4o$2o2bo2bo$3bo3b2o$4bo3b4o$6b2o2bo2bo$9bo3b2o$10bo3b4o$12b2o
2bo2bo$15bo3b2o$16bo3b4o$18b2o2bo2bo$21bo3b2o$22bo3b4o$24b2o2bo2bo$27b
o3b2o$28bo3b4o$30b2o2bo2bo$33bo3b2o$34bo3b4o$36b2o2bo2bo$39bo3b2o$40bo
3b4o$42b2o2bo2bo$45bo3b2o$46bo3b4o$48b2o2bo$51bo$52bo!
In terms of speed orthogonal to the level I find it suspicious that nothing between c/2 and c is known. By reading slanted waves the "wrong" way I am pretty sure it's going to be possible to get speeds between c/2 and c (or even over c!) which is why I limit my question to level waves. Also c itself (reduced) is definitely achievable even in level waves, e.g. 2c/2:
Code: Select all
x = 62, y = 2, rule = B3/S23
62o$2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o2b2o!
I would conjecture there are no level waves between c/2 and c and in fact more strongly that there are no "level back half planes" there either, by which I mean any full plane pattern traveling at a speed between c/2 and c which has a level front half plane of all zero must be all zero everywhere.
Relatedly it is known that there are no back half planes of a specific spatial slope which travel faster than c/2, but the argument fundamentally relies on the spatial slope of the half plane division and any sort of argument like this would some how have to "allow c through".
Also relatedly it is known that for any traveling plane at a speed < c with a level front half of parallel zebra stripes must be all zebra stripes. This is tantalizingly close to what we want to show and how the "< c" part shakes out is maybe interesting, but I see no way to repurpose the argument here.
EDIT: As a perhaps related question is it possible for a slanted wave to have its two orthogonal speeds both be > c/2? I would think not but am not immediately sure how to prove any such thing. The usual argument about c/2 upper bound for finite patterns seems very tied to its specific spatial slope.
EDIT2: I had forgotten about this horrible thing I had found earlier when digging in 2c/3. I would describe it as a "slanted back half plane" for speeds 2c/3 and 4c/7 (both faster than c/2):
Code: Select all
x = 60, y = 78, rule = B3/S23
o$bo$b2o$bo$2ob2o$3b2o$2o3bo$4ob2o$5bobo$7b2o$4bo2bo$5ob2ob2o$9b2o$5ob
2o3bo$10ob2o$11bobo$13b2o$10bo2bo$11ob2ob2o$15b2o$11ob2o3bo$16ob2o$17b
obo$19b2o$16bo2bo$17ob2ob2o$21b2o$17ob2o3bo$22ob2o$23bobo$25b2o$22bo2b
o$23ob2ob2o$27b2o$23ob2o3bo$28ob2o$29bobo$31b2o$28bo2bo$29ob2ob2o$33b
2o$29ob2o3bo$34ob2o$35bobo$37b2o$34bo2bo$35ob2ob2o$39b2o$35ob2o3bo$40o
b2o$41bobo$43b2o$40bo2bo$41ob2ob2o$45b2o$41ob2o3bo$46ob2o$47bobo$49b2o
$46bo2bo$47ob2ob2o$51b2o$47ob2o3bo$52ob2o$53bobo$55b2o$52bo2bo$53ob2ob
2o$57b2o$53ob2o3bo$58obo$59bo2$58bo$59o2$59o$60o!