A Swamped barrel is an octagon barrel which tapers in a special way.
Starting at the breech, it is large and it tapers down as you look further forward. About 12-15 inches from the muzzle it becomes a straight octagon for a few inches.
About 5-10 inches from the muzzle it starts to get larger and continues getting bigger until it reaches the muzzle.
A Colerain barrel for example starts out at the breech measuring 1.062, 11 inches forward it has been reduced to .862, at 25 inches from the breech it measures .772, at 33 inches from the breech it measures .772, at 38 inches (at the muzzle)it measures .957.
I believe the name came from the similarity to a drained swamp where one bank is higher than the other with the low area between them.
I have heard a number of theories about the reason for this shape. The one I like best is that this is a good way to get a lot of weight out of the barrel and still leave the breech strong to withstand the powder burning.
Some of the barrels were forged with this taper built in (rather than filed into it later) and this would have been a practical way of using less material in an age where iron wasn't as available as it is now.
As for why the shape starts getting larger as it approachs the muzzle, some say it's to keep the barrel from being "muzzle light".
I think it had more to do with the quality of the welds produced with the hammer/anvel method of welding which the barrel makers used at the time.
IMO, by leaving more material at the muzzle, they increased the chanch of having a good weld joint (measured from the bore outward to the outside of the barrel). More weld joint length increased the likelyhood that some "good weld" was present.
Barrels of this design were very popular in the 1700s, but in the 1800s many builders changed to the straight octagon for their new guns.
