The "vibrations" are, as Impalacustom suggests, complicated. The transverse whipping and the compressive shock wave discussed from the beginning are independent. Varmint Al's site has good FEA's of all the major modes of vibration except the compression wave, though he does mention in one of his pages that it exists. Nonetheless, his testing showed muzzle deflection imposed by high pressure gas deformation and recoil forces were dominant over the natural harmonic vibration modes. They occur after the "string is plucked" by firing and the bullet is gone. See the description and animated 3D FEA's here
. Still, even the initial deflection comes out very close the 3rd harmonic timing when you run the numbers.
I've long suspected that "hummer" barrels may be those barrels for which the best tuned load has the transverse deflection and compression wave influences in sync. It may explain why the Houston Warehouse experiments found a particular barrel length tended to favor precision POI (21.75", IIRC, but I'd have to re-read the text to be sure my memory hadn't moved the number) regardless of chambering. I also suspect this may be why an adjustable tuner can make many kinds of ammo shoot very well, but still only make a few loads shoot bugholes. Those would, I suspect, be the ones were the vertical muzzle deflection, in particular, and the compression wave nulls would happen to coincide at the moment the bullet exits.
Two complications with Long's theory and calculator are barrel coupling to the receiver and the fact that the speed of sound in steel has a lot of dependency on grain orientation and alloy. He says that typically his calculator gets within 2% of sweet spot loads, so you still have a little tuning to do after you've loaded to the theoretically optimal barrel time. That degree of error could, indeed, be due to differences in the steel. In an email correspondence some years ago, Chris also told me that he had one barrel on a benchrest rifle he'd built that was the only one he'd tried for which the 2% error didn't work out, He believed that was because he had turned ultra high tolerance threads on the barrel that could not even be threaded hand tight. It took an action wrench just to get the barrel screwed down to the shoulder. He believed that in that case he had succeeded in coupling the receiver to the barrel so well that it became contiguous with the barrel as far as the reflecting compression wave was concerned, and its length had to be added to that of the barrel.