Explain how a longer tube generates a tighter pattern.
I can't answer that because I don't know how or why, but I do have a theory. All I can go by factually is experience and shooting hundreds of shotguns.
But since I know how to use Google, here's a quote from a custom gunsmith that makes custom shotgun barrels.
"Speaking from a long experience working with shotgun barrels, certain circumstances prove that barrel length has an effect on pattern density and dispersal.
Turkey hunting and turkey match shooting where maximum density patterns are desired will, when all else is equivalent, tend to group more tightly with a longer barrel such as 30" vs. 20".
The 20" barrel is much more maneuverable for hunting and is a typically better choice for turkey hunting as an example.
I work at making those short barrels pattern very tightly for hunters so they can get performance more like a long barrel in a short barrel. That has more to do with the quality and precision of the work and if done to a longer barrel, would likely outperform the shorter barrel. See how many short barrels you see at the patterning contests; some have had to place barrel length limits on shooters to keep the longest barrels out and make the contest practical, the same way that many racing styles have engine size limits to keep the field more level. The proof that precision and talent mean a lot can be seen in the winning performance of certain racing teams, since all must play within the same limitations- but a few are always a bit faster with the same size components.
Shorter barrels may need a slightly tighter amount of choke to get equivalent results of a longer barrel due to the shot spread effect from the muzzle pressure that naturally drops as length increases. I believe it was the Churchill XXV that they said used a bit more choke to pattern the same as the 30" barreled versions.
It is true that most times the pattern, velocity, and hitting power will be little changed by modest barrel length differences, but can be radically different due to quality of bore contours, finish, choke alignment, and the details that make a fine barrel different from an economy barrel. However, shotguns being a fickle servant, we may sometimes find that what was considered likely a plow horse wins the triple crown in pattern density at long range. Thus is the nature of the beast- that no small sample comparison will ever show much to be able to conclude with certainty about most analyzed details. Some details will require much more exposure than anecdotal levels to reduce an uncertainty factor."
So, like I said, factors like bore quality, lengthened forcing cones, backboring, etc., are more important than barrel length when it comes to patterning (hence the example I gave of my 14" Rem870), barrel length can be a factor.