How do we calculate the linear expansion of a wave spring's outside diameter during compression?

When a wave spring is compressed, the waves flatten, causing a slight increase in the outside diameter ($OD$). The expansion can be approximated by the formula $\Delta OD = 0.02 \cdot \frac{f \cdot (OD + ID)}{N^2}$, where $f$ is the deflection and $N$ is the number of waves. For springs operating in a bore, this expansion is critical; if the clearance between the $OD$ and the bore is insufficient, the spring will bind, causing a sudden non-linear increase in spring rate and potential fatigue failure. Engineers should specify a bore diameter that accounts for both the manufacturing tolerance and this functional expansion.