Here is my 2c. If the barrel is longer its node should be wider but each node will also be farther apart. If shorter the node will be narrower but closer together. Brian testing at AB showed that it was a factor of barrel weight and not stiffness that made for an accurate barrel.
Think of it like a tuning fork. Long vs short of them.
To be clear, I agree that a longer barrel of a given contour will have a wider node than a short barrel of the same contour. What I don't agree with is the part where weight , not stiffness, makes accuracy.
Perhaps I'm reading more or less into that statement than is intended.
My findings have been that a less stiff barrel yields more muzzle displacement and a lower frequency, hence slowing things down and giving a wider node. I've cut barrels down to a smaller od at a point several inches behind the muzzle to create sort of a "hinge point." The result was a wider tune window and more muzzle displacement. Essentially, making a heavy barrel act like a lighter one in regard to how it responds with a tuner. The idea was/is to get a barrel to respond the same with a lighter tuner as it does a heavier one. A heavier tuner gives a wider tune window than a lighter one, to a point. Yes, a tuner can be too heavy and create accuracy problems instead of benefit it, but from what I've found, that point is north of about 12 ounces on a typical short range br contour barrel and length. Its pretty forgiving in this regard and not much difference between a lv and hv contour.
Keep in mind, stiffness is affected much faster with length than with diameter.
I also agree that nodes are further apart on a less stiff barrel. I don't think of this as a bad thing, as with increased muzzle deflection of a less stiff barrel, or one where more deflection is induced with a tuner, the barrel does a good job of what I call "talking to you." By this I mean that tune issues are more apparent, telling you that the gun is clearly out of tune vs a very stiff barrel that may well shoot within the condition yet be somewhat out of tune, making tuning decisions harder to make because you just can't always clearly see a small amount of out of tune with such small muzzle deflection.
I was lucky enough to have spoken with Harold Vaughn about this very subject. After studying his computer generated results I asked him whether he had found that heavy stiff barrels were more accurate. His answer was that weight produced more accuracy bun not necessarily stiffness.
One thing that I noticed from his material was that the longer the barrel time the more complicated that vibration patterns seemed to become, and to my thought, the less chance that those patterns would be the same, which may partially explain why short range benchrest shooters do not simply take the weight of their tuners as additional barrel length. Using the tuner on a shorter barrel slows the swing as if is was longer without the disadvantage of a longer barrel time. I think that when shooters found that they were getting more accuracy with shorter barrels they assumed that it was because they are stiffer, but as most of us are aware correlation is not causation.
I think the complications with barrel time are likely due to the forced deformation of the barrel coinciding with the barrel's natural frequency. The longer in bore time allows more time to be influenced by those frequencies. They're always there, from the time the trigger begins to move away from the sear, forward but with more amplitude as ignition and combustion occur.
It's a good thing that tuners are way easier to use than they are to fully understand, as there are so many variables that I don't believe its precisely calculable, all that's going on.
Whether using a tuner or not, accuracy is about timing bullet exit with optimal muzzle position. We often refer to this spot as a node. Longer barrels have more muzzle deflection and vibrate at a lower frequency than a short barrel of the same diameter, hence slowing everything down and having a larger arc. This all equates to a wider "node."
Adding a mass at the end of either has the same effect, even more so.
I think you know most of this Boyd, but it's good conversation.
You're idea of hinging the barrel does have a similar effect, as well.
