How does the number of waves $N$ affect the trade-off between load capacity and deflection range?

The number of waves $N$ has a quartic relationship with the spring rate $k \propto N^4$. Increasing $N$ significantly increases the stiffness, allowing for higher load capacity within a very short axial space. However, as $N$ increases, the maximum allowable deflection $f_{max}$ decreases because the shorter arc length between peaks increases the bending stress for a given displacement. Engineers must balance $N$ to achieve the required $P$ at $H_w$ without exceeding the stress limit $\sigma < 0.8 S_y$. For high-deflection applications, a lower $N$ with a thicker material $t$ is often preferred.