"Barrel harmonics" is a term used to describe the fact that waves are generated by the act of burning propellant at one end of the barrel to shove a bullet through the tube and out the muzzle. We tend to think of the barrel as a rigid tube but it actually vibrates exactly like a guitar neck vibrates when you pluck a string. If you hang a weight off the end of a guitar neck it alters the wave centers, or nodes, of the neck and it changes the sound. If you hang a weight off the end of the barrel is alters the location of the nodes along the barrel as well. Barrels are supported in places that constructively interfere with the vibration of the neck or don't interfere at all. If you alter the location of those nodes, the support locations may then fall at places that interfere DESTRUCTIVELY with the waves and harmonics (octave of the main or fundamental wave and octaves of that octave).
Three nodes:
An interesting example might be the Lee Enfield N.1 Mk.III.
This rifle has wood over its entire length and a couple of supports within that wood that are located so as to protect the vibrations of the barrel. When tinkerers modify these rifles to "sporterize" them by removing wood to drop their weight and relocating the supports, they can sometimes find that the point of aim has been drastically altered.
Bob