Taken from one of my old posts on swamped barrels:
I'll jump in with a guess about the reason for a swamped barrels shape.
IMO there was a desire to get weight out of the barrel and because of the high pressures around the powder charge, the breech was not a good place to remove barrel material.
The middle area, being reinforced by material on both sides could be made a lot thinner.
The muzzle was the weak place for two reasons. First, it only has material on one side (the area towards the breech.)
The second reason takes a little explaining:
The old barrels were not only iron (much weaker than any steel) but they were forge welded.
Forge welding, in which the barrel material is heated to the yellow-white hot plastic like condition and then fluxed (to keep the joint clean) and then pounded together to form the weld produces a weld of questionable quality to put it mildly. Simply put, the weld may or may not have bonded so it was weak.
If the weld was weak, the gun barrel should have as much thickness as it can have in the area. This thickness not only increases the likelyhood of having some good weld in the joint but it makes the barrel much stronger just because of the increased material thickness.
That is my theory of the reason for the swamped barrels shape. I'll gladly consider any other ideas (like I like the idea of raising the sight higher).