The derivation in Project 2 restores the geometric ring-tension contribution σ_R that arises from circumferential judder and wave propagation within the bistable continuum of a thin-walled cylindrical concrete shell. Once this term is visible, the radial structure of the associated modes is governed by the classical cylinder functions. Their zeros therefore furnish a precise, coefficient-free instrument for engineering control of the tension–twist balance.
B.1 Notation and principal zeros
Let J_ν(x) denote the Bessel function of the first kind of order ν. The s-th positive zero of J_ν(x) is written j_ν,s; the corresponding zero of the derivative J_ν'(x) is written j’_ν,s.
For the lowest orders that dominate axisymmetric and low-circumferential modes of a cylindrical tank the first ten positive zeros are:
- j_0,s: 2.4048, 5.5201, 8.6537, 11.7915, 14.9309, 18.0711, 21.2116, 24.3525, 27.4935, 30.6346
- j_1,s: 3.8317, 7.0156, 10.1735, 13.3237, 16.4706, 19.6159, 22.7601, 25.9037, 29.0468, 32.1897
- j’_1,s: 1.8412, 5.3314, 8.5363, 11.7060, 14.8636, …
(Neumann / free-surface or rigid-wall conditions normally employ the derivative zeros; Dirichlet conditions employ the function zeros.)
B.2 Spectral properties relevant to control
All positive zeros are real and simple. They interlace according to
j_ν,s < j_ν+1,s < j_ν,s+1.
For large s the leading asymptotic
j_ν,s ≈ π (s + ν/2 − 1/4)
is already accurate to a few parts in a thousand and may be refined by higher-order expansions when required. Because the zeros depend continuously on any continuous parameter that enters the argument of the Bessel function (membrane tension, torsional stiffness, effective radius, residual bending S_n⁴, viscosity), they supply a discrete but tunable set of operating points.
B.3 Link to geometric ring tension
In the cylindrical geometry of the shell the radial factor of each modal field is J_ν(kr) (or a linear combination with the second-kind function when an annular domain appears). The admissible wave-numbers are therefore
k = j_m,s / R or k = j’_m,s / R,
where R is the mid-surface radius. These discrete values quantise the circumferential wavelengths of the judder waves that restore the closed surface and thereby generate the positive ring-tension stress σ_R. Consequently the magnitude of σ_R, the number of nodal circles, the residual bending moment and the critical viscosity at which a mode becomes overdamped are all fixed by the location of the operating point relative to the nearest zeros.
B.4 Practical control procedure
- Identify the dominant circumferential order m and the desired residual state (target residual S_n⁴, target safety-factor margin, or required rest-time duration h).
- Compute the corresponding zero j_m,s or j’_m,s for the tank radius under consideration.
- Treat the effective geometric tension (or the ratio of binding to expansive force) as the slow control parameter.
- Adjust tension so that the instantaneous operating point coincides with, or lies between, the chosen zeros. Crossing a zero changes the number of nodal circles and therefore the spatial distribution of σ_R; locking onto a zero stabilises the twist ratio and the residual bending.
Because the zeros are simple and interlacing, the adjustment is robust: a few percent change in tension is normally sufficient to move from one modal configuration to the next. Surface-tension meniscus corrections and edge conditions at a rigid support appear as standard mixed boundary statements involving the same zeros and do not alter the control principle.
B.5 Illustrative numerical remark
For a typical reinforced-concrete tank of radius R = 10 m the first three axisymmetric Neumann zeros yield radial wave-numbers k ≈ 0.383, 0.702 and 1.017 m⁻¹. A controlled 4–6 % increase in effective ring tension shifts the operating point from the first to the second zero, reducing the residual bending contribution and simultaneously raising the critical viscosity threshold. Full numerical tables for any practical range of ν and s are obtained from standard library routines or from the classical tabulations of Abramowitz & Stegun and the Digital Library of Mathematical Functions.
Thus the classical cylinder functions convert the now-visible geometric ring-tension term into a directly controllable design variable, completing the first-principles route from Lewe’s 1915 geometry through the bistable continuum to practical engineering of cylindrical concrete shells.
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