About Barrel Harmonics. ( or Sine waves) When you fire any gun there is tremendous amounts of energy release, and it want to go in all directions. Some of it pushes the PRB out the barrel. some of it is recoil on your shoulder. The rest pushes against the sides of the barrel, making it want to vibrate, just the same as it you were to hold a string tight, and then pluck it. We would not have string instruments like violins and guitars without these vibrations.
In a Octagon barrel, the 8 sides form over lapping triangles, which are the strongest geometric structure we have. They do suppress much of the sine waves. When you make a barrel half octagon, and half round, in a half stock gun, the octagon portion keeps any flaws in the barrel mortise from influencing the barrel vibrations, and throwing the shot away from the POA. The fact that the half round portion begins where the forestock typically ends, allows the rounded portion of the barrel, which vibrates freely, to be " Free Floated " just as round barrels are free floated in modern gunstocks. ( and for the same reason). As long as the round exterior of he barrel is concentric with the bore, the vibrations are free to go in any direction, and you get good accuracy. If, however, the exterior shape is not concentric with the bore, you get a barrel that is heavier on one side than on the other, and your ball will go to different POI depending on the velocity created by different loads.
Even with a concentric barrel, there is a sweet spot for every rifle or smoothbore barrel, where it will throw all the shots to the same, small, POI. That is why we spend time adjusting powder charges and components to find the most accurate load in each gun.
Oh, in a full Octagon barrel, the harmonics are suppressed to a large extent for the entire length of the barrel. That is why " Long Rifles " have always had a reputation for being more accurate than other guns.
I bought a full octagon Winchester 94 .32 Winchester Special rifle, just to see if I would get better groups out of a factory gun than I was getting from my much later made Winchester .30-30 with its skinny full round barrel. The old gun can shot rings around the newer one! And, you can load the magazine of the octagon barrel completely with shells, and not see the typical vertical stringing of shots on a target that you get shooting a round barrel .30-30.
Oh, I got the idea that this might be how harmonics work even in modern guns, when a good friend bought a limited production Marlin 39 with a full octagon barrel that was much longer than their normal production. It was the most accurate .22 he owned, and the most accurate lever action rifle he owned. It just did not heat up and move, like happens with his other .22s and other caliber rifles. We decided that the massive extra amount of steel around the bore had to be dispursing the heat better in that gun, so that it was not affecting the harmonics as the barrel warmed. But, then, we thought about that hypothesis, and decided that the structure of the barrel ( Octagon) must have something to do with the continuing small groups, even when the gun was " HOT ". That is when we began experimenting with our ML rifles, and comparing them to round barreled smoothbores, and any half-octagon, half-round barrels we had or could find. We put the octagon barrels under the full pressure of a fully loaded magazine, just as we did with the round barrels, and the octagon was was the better shooter every time.
Something to think about.