How to find the wave period from kinetic energy?

Imagine a swinging pendulum, it's kinetic energy goes from zero at the edge to maximum in the middle, to zero at the other edge, maximum in the middle on the way back etc...

Since kinetic energy is a scalar, without direction, you'll get the same maximum value twice during one full swing.

So you'll need to double the time period of your kinetic energy graph to get the time period of the wave.