# Phi-heterodyning

As the fundamental standing wave frequency stabilizes, additional standing waves called ‘harmonics’ or ‘wave partials’ form in harmonious coexistence creating the following pattern.

Now, in the book I propose the Fibonacci series as a natural damping function in harmonic resonance. This can be proven to be a nominal solution for the second-order equation known as the ‘characteristic wave damping equation.’ When the first two pair of numbers are plugged into this equation and it is solved like a quadratic equation, the golden ratio becomes the eigenvector. This means that a nominal or natural amount of damping is present in a standing wave from the very beginning, induced by reflection inside some kind of container, and thus present in all harmonic partials. The standing wave is slightly warped by this as predicted by Einstein’s geodesics in General Relativity.

I call this “phi-damping” and it occurs strongest in two phi spaced “Landau damping wells” either side of every PI period node.

Zooming into one of the damping wells, we can see how harmonics share energy in the same medium.

This concept was introduced by Russian physicist Lev Landau in 1932. Known as Landau-Zener theory, it explains how energy is transferred between two waves across a damping well. This damping well represents a low pressure zone something like a miniature hurricane or tornado where energy differential between the two waves torques across. I visualize this as something like a whip being snapped.

The damping well region can be approximated by a Fibonacci (or Lucas) spiral {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...}.

For Fibonacci proportions 13:8 and above in the harmonic series, wave formation is suppressed (proven by acoustic experiments and common use). However, harmonics less than 13:8 can and do form. This is expressed in the following image.

This is called Phi-heterodyning and is explained in the paper by Bovenkamp and Giandinoto entitled “Incorporation of the Golden Ratio Phi into the Schrodinger Wave Function using the Phi Recursive Heterodyning Set.” In this, Giandinoto shows how to generate any wave, and thus the Fourier series and Wave function, entirely from golden sections.

The implication is that all partials in any resonating standing wave are damped and attenuated to some degree by the golden ratio. Interfering partials below 12 (and their even multiples) thus follow a Gaussian derivative distribution of energy as described in my book. Partial interference above 12, starting with 13:8, will experience a dramatically increasing degree of damping to suppress wave formation entirely. The remaining harmonics then have space to coexist together with the fundamental resonant frequency, sharing energy in a stationary position in perfect harmony.

The last thing to say about this process is the Interference Function presented in the book can be used to generate a sine wave. Since this function is based on squared integers divided by a Fibonacci estimating function based on Phi and square root of 5, the following offers a simple proof that sine waves and thus all waves may be derived from the Interference Function.