# Wave Model Assumptions

Despite
the apparent logic of the previous mathematical discussions, i.e. *
Mathematics Behind the Wave Equations**
*and *Relative and Relativity
Transforms*, which may appear supportive of a general WSM model,
there are still many questions as to how the IN-OUT waves actually physically
interact at the wave-centre. For it would seem that the mathematical
model described implicitly assumes an order in terms of both spherical
symmetry and phase arrival of IN waves from all the other wave-centres
in the universe, which is difficult to comprehend given the apparent
chaotic distribution of particles on the quantum scale. Therefore, this
discussion of some of the issues perceived is injected into the review,
such that mathematical logic is not allowed to take automatic precedence
over physical reality. As such, we will start with Wolff’s idea that
the IN waves undergoes a process of spherical rotation that leads to
the OUT waves being inverted, as described in the following paraphrased
quote:

*
The Physical Origin of Electron
Spin (1997) Rotation of the inward wave at the centre to become
an outward wave is an absolute requirement to form a particle structure.
Rotation in space has conditions. Any mechanism that rotates must not
destroy the continuity of the space. …Spherical rotation is an astonishing
property of 3D space. It permits an object structured of space to rotate
about any axis without rupturing the coordinates of space. After two
turns, space regains its original configuration. …. as the inward waves
converge to the centre, rotate with a phase shift to become the outward
wave, and continually repeat the cycle. The required phase shift is
a 180 ^{o} rotation that changes inward wave amplitudes to become
those of the outward wave. *

Based on [1], we might see how a 180^{o} phase shift between the
IN and OUT waves might be represented by an inversion of the amplitude
[A] between the IN and OUT waves, such that the additive process of
wave superposition reduces to a negation of the two amplitudes.

[1]

While limited to a 1-dimensional description, the form of [1] represents
two waves propagating in opposite directions, i.e. ± k_{0}x,
with the required [180^{o}] phase shift effectively causing
an inversion of the wave amplitude at, or near, the wave-centre. Although
the form of the individual wave equations has already been shown to
correspond to waves propagating with constant velocity, see *
Wave
Propagation *for details, we possibly need to question exactly how
these waves are thought to propagate in space [x] and time [t] within
the WSM model.

*
Beyond the Point Particle (1995)
The equation above shows that an electron is comprised of two spherical
scalar waves travelling in space with velocity [c]; one inward and the
other outward. The two superimposed waves form a standing wave, termed
a space resonance (SR). The centre of the wave structure is the nominal
location of the electron. These space resonances are perpetual spherical
oscillators. Each resonance extends throughout space and interacts with
other resonances so that the natural laws result from the properties
of the waves and the medium they travel in, i.e. space or the ether.
*

However, while the quote above appears to support the idea that space
is the propagation medium for matter waves, Wolff‘s description of ‘*space
density’* might seem somewhat ambiguous if based on the following
paraphrased quote:

*
Origin of the Natural Laws in a
Binary Universe Space Density Principle (SDP) defines the wave medium
– space, where the properties of waves depend on properties of the medium.
Thus, space as a medium, is the wellspring of everything. At each point
in space, waves from all particles in the universe combine their intensities
to ‘form’ the wave medium of space. In other words, at every point in
space, the frequency [f] or the mass [m] of a particle depends on the
sum of squares of all wave amplitudes [A] from the [N] particles inside
the ‘Hubble universe of radius [R= c/H], where [H] is the Hubble constant.
*

Based on the description above, there is a suggestion that
it is the combined wave intensity that ‘*forms*’ the medium of
space, while a physical propagation medium is normally described as
existing independently to any wave in propagation. Of course, pursuing
this line of logic also suggests that the WSM model might essentially
be a mechanical wave model, where the media of space becomes an absolute
reference frame, even though this may be far from obvious, if everything
in the universe is made of waves. However, if space is an elastic medium,
then the sum total of wave amplitudes, at any point, might reflect the
potential energy as measured from some equilibrium state of the medium.
In this respect, the square of the wave amplitude [A^{2}], a
scalar quantity, would correspond to the potential energy stored within
the wave medium.

*But in what form do the IN-OUT waves propagate?
*

Based on the first animation left, we might perceive the IN waves
propagating towards a wave-centre as a series of concentric 3D shells,
where the wave amplitude [A] is represented by the light-dark shading,
which based on [1] would actually correspond to [A/x], where [x] corresponds
to the radial distance to the wave centre. However, the second
animation in the centre might provide a better visualization,
albeit reduced to 2-dimensions, of an IN wave propagating towards the
wave-centre as a *‘distortion’* of the space fabric, where this
distortion would correspond to the wave amplitude, i.e. potential energy.
Finally, the third animation right might suggest that the light-dark
shading of the first animations are analogous to a longitudinal wave,
where each region corresponds to the compression or rarefaction of space.
If so, then these compressions and rarefactions might be interpreted
as changes in the density of space, possibly aligning to Wolff’s description
of ‘*space density’*. At this point, it is highlighted that the description above
is based on a highly speculative assumption, i.e. the fabric of space
can be modelled as an continuous elastic medium. While this assumption
may be compatible with the macroscopic description of space-time within
general relativity, it possibly runs into problems within modern
*quantum field theory*
and, for this reason, Wolff’s frequent use of the term ‘*quantum waves’*
may be erroneous. However, the debate as to whether the *‘fabric of
space’* is discrete or continuous possibly needs to be contextualized
in terms of the scale of wave propagation, e.g. electron wavelength.
If this is the case, the WSM model appears to operate on a scale in
the order of 10^{-12} to 10^{-18 }metres based on the
approximate estimate range of the electron radius. In contrast, if we
equate the quantum scale to the
*Planck scale*, the units of dimensions
fall by some 17 orders of magnitude to 10^{-35} metres.

*So, Is there some further requirement for a physical description
underpinning any wave model? *

In LaFreniere's work, he makes reference to the idea that each point
in space can be modelled as a simple harmonic oscillator, i.e. based
on *Hooke’s law*. However, in order for the waves to propagate, any offset
of a point in space from some equilibrium state must affect its neighbours,
such that space may operate as a continuous medium at the scale of the
WSM model, while possibly still being a discrete or granular structure
at a smaller scale, e.g. quantum and/or Planck scale. As far as is known,
Wolff’s work does not speculate on this issue, but the reader interested
in such ideas may wish to review LaFreniere’s webpage entitled ‘*The Aether*’ .

*So what physics supports spherical rotation?
*

Wolff’s idea of spherical rotation appears to be predicated on a
mathematical description defined within the *
SU2 group*, which
goes beyond the scope of this discussion’, although a visualisation
of the concept may be referenced through the work of *
Robert Gray
**. *It is highlighted that while there appears to be an open
issue regarding the superposition of the IN-OUT waves in the region
in which spherical rotation is thought to take place, i.e. close to
the wave-centre, the focus of the following discussion will be orientated
towards an analogous idea of wave reflections. Normally, the idea of
two waves propagating through a common point in space can be described
in terms of the amplitudes being added to form a superposition wave
without any direct affect on the two underlying waves, e.g.

However, we might recognise that a physical object, such as a ball,
will bounce off, or be reflected back, from a rigid wall, or high-density
structure; where the ‘*reflection*’ would then be quantified in
terms of momentum and energy conservation plus any assumption about
the ‘*elasticity*’ of the collision. Clearly, even without acceptance
of the WSM model, it is recognised that waves also carry energy and
momentum, which in turn can be subject to reflection given certain boundary
conditions. Within the current WSM model, we might characterise the
wave-centre as a small volume of space with a very high ‘*density’
*surrounded by space of a much lower density, such that the IN wave
reaches a boundary condition at, or near the wave-centre.

*So how might we quantify the boundary condition
associated with a wave-centre? *

At this point, the wave models of Wolff and LaFreniere appear to diverge. Wolff considers the wave-centre to be totally reflective in as much that the entirety of the IN wave is rotated back towards the IN wave, while LaFreniere’s model appears to be more closely orientated towards wave superposition in that IN waves simply passes through the wave-centre. However, the LaFreniere model is also ambiguous in the sense that the wave equations used to create the simulation do not appear to described the actual causal mechanism physically in operation within the wave-centre.

*Note: At this point, it is unclear
whether the IN-OUT wave models of Wolff or LaFreniere can really describe
what might be perceived as a near chaotic phase relationship of IN waves
arriving at any given wave-centre. In this context, the idea of a wave-centre
being a self-contained ‘space resonance’ that only require IN waves,
of any phase, to maintain the energy lost by OUT waves may be
another possibility, which the LaFreniere model appears to support. *

Despite the note above, it is possible that the wave-centre may act
as a boundary to IN waves. However, it is also assumed that the potential
energy associated with the IN wave, at the wave-centre boundary, has
to adhere to the law of energy conservation, such that the potential
energy associated with the IN wave must equal that of the OUT wave.
If so, it is unclear whether the assertion of the Wolff model that the
OUT wave undergoes a 180^{o} phase shift or amplitude inversion
can be maintained when considered in terms of the following animation.

The animation starts just before the red IN wave reaches the wave-centre boundary, after which the blue OUT wave is reflected and adds in superposition with following IN waves. Simply as a point of reference, the red IN wave is continued as the faint red dotted wave beyond the boundary so that an easier comparison can be made with the actual blue OUT wave reflected, i.e. it is considered to be a mirror image.

*So can the blue OUT wave be amplitude
inverted at this boundary in this case? *

The argument being forwarded is based on the conservation of energy
at the boundary. If we assume that the WSM model is essentially a mechanical
wave model, where each point in space can be modelled as a *
simple
harmonic oscillator*, then the amplitude [A] of the IN wave
at the boundary is proportional to the potential energy of the wave
at that point. Without the boundary, this amplitude-energy would
continue to propagate to the right, as indicated by the dotted red
wave, although the wave boundary now requires this amplitude-energy
to be reflected back to the left. If we also assume that this wave
energy is propagating as a longitudinal wave, i.e. as a series of
compressions and rarefactions of the space density, then any wave
amplitude above the central black line might be considered in terms
of a compression, while any wave amplitude below this line would be
associated with a rarefaction of the space fabric. If this is
the case, then any inversion of the wave amplitude at the point of
'hard' reflection would not maintain energy conservation and
therefore negate the compression-rarefaction state of the space medium
at that point.

*What other implications might follow a change
in the density of space? *

While the assumption may be proved false, the discussion is currently assuming that the WSM model is basically a mechanical longitudinal wave model in terms of its general principles of propagation. The formula for the velocity [v] of a longitudinal wave in a mechanical propagation medium takes the following form:

[2]

For the purpose of this discussion, the quantity [ρ] corresponds to the density of the medium, while the quantity [E] will simply be defined in general terms as the modulus of elasticity of the medium. However, the scope of this modulus depends on whether the propagation medium is likened to a gas, fluid or solid. Easily compressible media, e.g. gases, have a low value of [E], while almost incompressible fluids, e.g. water, have a very high value for [E]. For general assessment, we might simply assume that space is incompressible like a fluid and therefore has a very high value of [E]. However, if [E] is a constant for the medium of space, no matter what its actual value, there is a suggestion that the velocity of wave propagation through space may be an inverse function of its density, such that the waves in question might propagate more slower in the region close to the high-density wave centre.

What else may require further consideration?

Built into the wave equation shown in [1] is the assumption that
the amplitude [A] of both the IN and OUT waves is driven by the function
[A_{0}/x]. We might show that the relationship [A_{0}/x]
is a natural consequence of the inverse square law for radiated energy
in 3-dimensions, i.e. OUT waves, where the energy is distributed over
the surface of a sphere of radius [R]:

[3]

In the case of mechanical waves, the square of the wave amplitude [A^{2}]
is proportional to the energy [E], at some given radius [R], such
that the amplitude [A] must be an inverse function of radius [R]:

[4]

However, in the Wolff model, the amplitude of the OUT
wave has to be directly
related to the IN wave, which undergoes spherical rotation to be its
inverse at the wave-centre [x=0]. As such, we really need to explain,
and justify, the assumption that that the IN wave is both spherically
symmetrical and driven by the function [A_{0}/x]. If we assume
that all the particles in the universe, whose OUT waves form the IN
waves of a specific wave-centre and then apply Huygens principle to
these waves, it is conceptually possible for the sum of these waves
to conform to the function [A_{0}/x], providing that all of
these waves arrived in phase, as required by symmetry of the animation
shown.

*What is the probability of all waves being
in-phase? *

As has been stated, the Wolff and LaFreniere wave models differ on
the issue of spherical rotation, such that LaFreniere argued that once
the wave-centre had been created, as a space resonance, it only needs
IN waves of any phase to restore the energy lost by OUT waves. Of course,
this idea comes with its own set of problems, as it is unclear how the
natural frequency of the wave-centre could be maintained as either a
‘*forced vibration’* or ‘*resonant standing wave’*, if the
IN waves have an arbitrary phase relationship to each other. As such,
we either have to resolve exactly how spherical rotation might work
or consider how out-of-phase IN waves compensate for the energy lost
by the OUT wave. While Wolff’s work does not appear to address such
fundamental issues, LaFreniere’s webpage ‘*Theory of Evolution*’ does outlined some ideas under the heading: ‘*Electron Creation’*,
e.g.

“The first step after the aether creation must have been the electron creation. One can show that according to Huygens’ Principle, many plane waves travelling inside the aether could add themselves and produce an electron. Without any existing electrons, electron creation is not likely to happen, one chance out of billions of billions, because the wavelength and the phase must match during a very short period of time. However, with billions of existing electrons, whose wavelength is almost the same, this situation becomes much more probable. There is still one chance out of billion, but because time is almost infinite in such a universe, this situation is not just probable, it is a certainty.”

In many respects, this discussion, and the ones to follow, should
be seen in terms of a ‘*duty of inquiry’*, which is attempting
to look at the ‘*for and against’* arguments for a
*wave structure* underpinning
all matter particles. At one level, the basic idea of a wave structure
model appears compelling, as the idea of particles with mass becomes
increasingly difficult to support in the sub-atomic domain. If so, the
idea of mass being replaced by energy seems a natural step in which
the concept of a sub-atomic particle is essentially replaced by an energy-density
confined within some given volume of space. This said, if energy is
a scalar quantity, then any
*energy model* requires some form of wave
structure as a transport mechanism to move in space-time. Of course,
the ‘*weight of authority*’ will argue that the standard model
continues to accommodate the idea of particles, primarily for semantic
convenience, while the actual details are subsumed into
*quantum field
theory* without recourse or need for waves to propagate through the ether
of space. However, while this is, and will remain, the mainstream consensus
of science for the foreseeable future, many still have a nagging doubt
that much of the current quantum model is still unverified and questionable,
although it is unclear whether the WSM model is able to provide a workable
alternative, which is able to stand up to scrutiny. On this note, the
*following discussion* will continue the review of the mathematics of
the wave equations based on a number of different simulations.