Someone might have put together a theoretical model of what barrel vibration might look like.
But this is a very complicated problem with way too many variables to reduce to a calculus equation.
Shooting that barrel and letting your target tell you what it likes will solve the problem
That has been done and, with respect to complexity, yes and no. Yes, if one chases all possible variables like variation in the material's (barrel, in this case) homogeneity from a metallurgical standpoint. It becomes a question of significance (will it matter) and practicality (can it be reasonably measured).
Given a medium (again, a barrel in this case), one need not worry about the variation of energy introduced to the system with respect to the timing of a 'node', since the frequency of the vibrations does not change. A guitar string produces the same note (frequency) regardless of how firmly it is plucked. Variation in energy results in variation in amplitude, the distance between the high and low extremes of the sinusoid. As ladder tests confirm, vertical spread is not a linear function of velocity. The amplitude of barrel vibration is also in play.
'Nodes', as we use the term, are located in places where the velocity of the vibrating system is slowing down, momentarily reaches 0, and reverses direction. Visualize the guitar string again - the velocity of the string's movement, regardless of how firmly it is plucked, is momentarily 0 at the very top and very bottom of its travel and it slows materially relative to its maximum velocity during the transition on either side. In this sense, it is indeed differential calculus. The first derivative of the frequency/amplitude function yields the slope of a tangent line at a given point. Nothing more than the rate of change. That rate of change is 0 at the extremes of amplitude, the highest and lowest points relative to the x axis of the sinusoid, and the rate of change is progressively smaller as these points are approached from either direction. The rate of change is at its maximum where the frequency/amplitude function crosses the x axis. In practical terms, 'nodes' are located at the top and bottom of a barrel's frequency of vibration and the barrel's velocity of vibration is highest when the barrel is 'straight', or right in the middle of its amplitude of vibration.
This is
generally how OBT tables are produced. I'm sure not many are sitting on the edges of their seats waiting for further mention of differential calculus ...