What is the impact of radial wall thickness on the buckling stability of tall Crest-to-Crest springs?

Tall Crest-to-Crest wave springs (where $L_{free}/D_m > 1.5$) are susceptible to buckling or 'snaking' when compressed. The radial wall thickness $b$ contributes to the lateral stiffness. To prevent buckling, the spring should be guided either by a shaft or in a bore. The clearance should be minimal but sufficient to allow for the diameter expansion during compression. If no guide is possible, the engineer must increase the radial width $b$ to improve the moment of inertia $I = \frac{b t^3}{12}$ relative to the axial direction, effectively increasing the critical buckling load $P_{cr}$.