Why does a wave spring exhibit a 'Non-Linear' spring rate at the beginning and end of its travel?

While the theoretical spring rate $k$ is linear, real-world wave springs exhibit 'bottoming' and 'top-off' effects. At the start of deflection, the waves may not all engage simultaneously due to manufacturing tolerances (the 'first-wave effect'), resulting in a lower initial rate. Near solid height, the waves begin to touch each other or the 'shim' ends begin to compress, causing the rate to increase exponentially. This is known as 'load-jumping'. Designers should only rely on the linear portion of the curve, typically between $20\%$ and $80\%$ of the available deflection.