Two By Four And Flat Top, Please

A convolution plot of the square root of the gradient (\(\because f\propto\sqrt{k}\)),

\((2{ mc^{ 2 } }ln(cosh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x)))^{'}=c{ \sqrt { { 2m } }  }G.tanh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x)\)

with a Gaussian distribution of mean, \(\mu=2\) and spread, \(\sigma=0.25\).

root of the gradient, and resulting convolution plot of these two functions,

The distribution of oscillating frequencies tilt towards higher frequencies.  When we increase the domain of the gradient plot to include more oscillations of higher amplitude,

The distribution of frequencies flat out at the top, as

\(\lim\limits_{x\to\,large}{c{ \sqrt { { 2m } }  }G.tanh(\cfrac { G }{ \sqrt { 2{ mc^{ 2 } } }  } x)}=c{ \sqrt { { 2m } }  }G\)

Particles in oscillations and circular motion inside the \(\psi\) cloud of another particle will have these characteristic frequency distributions.