How does the 'number of waves' $n$ influence the spring's stability and its susceptibility to buckling under high axial loads?

The stability of a wave spring is inversely proportional to the wave period. A higher number of waves $n$ increases the radial stiffness and reduces the likelihood of lateral buckling. However, as $n$ increases, the spring rate $k$ increases by a power of 4 ($n^4$), making the spring much stiffer. For multi-turn springs, the ratio of free height $H_f$ to mean diameter $D_m$ is critical; if $H_f/D_m > 1.5$, internal or external guidance (a rod or a bore) is mandatory. The buckling limit is reached when the axial load induces a tangential stress that exceeds the critical buckling load $P_{cr} = _x000c_rac{_x0008_eta imes E imes I}{L^2}$, modified for the periodic geometry of the waves.