A round barrel is almost always going to give you barrel vibration or barrel " whip ", as you call it. All steel barrels exhibit harmonic waves when subjected to high pressure. Remember, that pressure that is pushing your ball or shot down the barrel is also trying to escape by pushing out on the barrel.
Octagon barrels then to have more metal to them than round barrels do. Octagon barrels consist of equal triangles which dampen barrel vibrations much better than any round barrel can do. In modern rifles, you will see very heavy round barrels use in target guns, or, to cut weight, you will see fluted barrels, that began life as a larger diameter heavy round barrel, and then had metal removed from the sides, by milling groove of equal depth in the barrel- sort of the opposite of cutting rifling on the inside of the barre. Those long, straight flutes created triangles, too, and that adds stiffness to the barrel, and dampens vibrations.
There will always be barrel harmonics, or waves when a gun is fired. However, its the thin, round barrels, often found on shotguns, and some smoothbore single barrel guns, that will vibrate the worst. The thinner the barrel is on a modern rifle, the more the barrel vibrates when fired. Those thin, whippy barrels that are put on some rifles help reduce the weight being carried, which is their main selling point, but the vibrate terribly, take some effort to zero in for a load, and then over heat with only a few shots. For the hunter, who may fire only one or two shots during an entire hunt, those qualities are not a problem. For someone who may have to shoot many more shots, and needs a gun with a barrel that will maintain its zero even when it gets hot, a heavier barrel will alomst always shoot better.
All these considerations are trade-offs. Nothing comes for free.
