# 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.