How is the theoretical spring rate $k$ calculated for a multi-turn Crest-to-Crest wave spring, and how do the number of waves $N$ and turns $Z$ affect the result?

The spring rate $k$ for a multi-turn Crest-to-Crest wave spring is derived using a modified beam deflection formula for curved segments. The fundamental equation is $k = \frac{E b t^3 N^4}{4 D_m^3 Z}$, where $E$ is the Young's Modulus, $b$ is the radial wall width, $t$ is the material thickness, $N$ is the number of waves per turn, and $Z$ is the number of active turns. It is critical to note that the rate is inversely proportional to the number of turns $Z$, meaning doubling the turns halves the rate, while the rate increases with the fourth power of the number of waves $N$. Engineers must also apply a correction factor $K$ for curvature when the ratio of mean diameter to radial wall is low, typically $D_m/b < 8$.