I don't disagree with the transverse wave (fishing rod type bending). What I was looking for was the horizontal compression waves through steel, and how quickly that will disperse. The transverse waves will vibrate with the "natural frequency" of the firearm, while the compression wave will move with the speed of the transmission medium (steel).
That "fishing rod" action is a result of the the exact same force that makes a fishing rod whip, the inertia of the rod resisting change at a different rate because of the elasticity of the rod. This is caused by the internal recoil and gravity. The barrels have this transverse wave in the vertical plane, as there is no forces acting in the horizontal plane. In theory if you could get a rifle to recoil directly to the rear, you could minimize this barrel whip (such as comparing the barrel whip on an M16 with that of an AK).
The "horizontal compression" wave is moving through the barrel, from chamber to muzzle. As the wave moves the effect is seen at 90 degrees to the direction of wave travel, which is supposedly measurable as a diameter increase and decrease.
The frequency of sound doesn't change the speed of sound. The frequency of the transverse wave (fishing rod whip) will be somewhere along the realm of a few hundred Hertz (long tube of steel means a relatively low harmonic frequency), but the horizontal compression wave supposedly travels much faster (the wave that causes the increase/decrease of muzzle diameter).
In air, we have "free space loss" so that things further away get quieter, or dimmer. In the barrel the number of cycles of the muzzle increase/decrease is determined by the speed of the wave and the length of the barrel (multiply the number of cycles and the difference in magnitude of the first to final and you can calculate the signal loss over distance, unfortunately numbers that I do not have for a compression wave in steel). If we go for the low estimate of around 12,000 fps for this wave propagation, then there will be 2 or 3 cycles minimum before the bullets leaves the bore, and depending on barrel length possibly more.
Personally I'm not totally sold on the compression wave idea as the main evidence cited was muzzle velocity SD and group dispersion. It makes sense, but do do other explanations.