# Mathematics Behind the Wave Equations

*Milo Wolff’s* description of the IN-OUT wave process suggests that the
IN wave becomes the OUT wave after undergoing a
*spherical rotation* within
the wave-centre, which it is assumed must also account for a phase difference
of 180° between the IN-OUT waves. If so, the superposition wave, i.e.
IN+OUT, may be modelled as 2 cosines waves that meet at a point in space,
i.e. r=0, t=0, but which then undergo a phase shift [φ] with respect
to each other in time and space. However, it is stated at the outset,
this whole discussion of the WSM model represents a boot-strapping of
my own personal understanding. As such, the following discussion of
the wave equations is a first attempt to understand some of the issues
and will therefore probably contain mistakes and incorrect assumptions,
which will need to be revised over time, as and when corrections are
received. So, to begin, the previous discussions of both *
mechanical waves*
and *quantum waves* have shown how the basic equation for a propagating
wave can be presented in the following 1-dimensional form:

[1]

However, there is possibly some value in reformatting [1] to initially convert the variables [k,ω] back into the basic units of space [x] and time [t]:

[2]

In the form of [2], we see that the 2^{nd} order rate of
change of [A] with respect to [x] and [t] will remain constant, assuming
a constant propagation velocity [v], which suggests that [A] can be
described by a trigonometric function of some sort, i.e. a wave. However,
the form of [1] or [2] is not really suitable for the formulation of
a scalar 3-dimensional spherical symmetrical wave, as required by the
WSM model. While it is highlighted that the basic principles underpinning
[1] do not necessarily change within the WSM model, the notation to
be used may require explanation and some further examination of its
assumptions. As such, we might start by simply introducing the [∇]
operator, which is just a shorthand way of writing the rate of change
occurring in three dimensions, i.e. x, y, z, where [**i, j, k**]
are unit vectors:

[3]

However, the dot product of the [∇] operator with itself will
create a scalar, which in mathematics is called the ‘*Laplacian
Operator’* :

[4]

Given that there may be an inference that [1] is only describing a physical wave amplitude [A], we might replaced [A] with the symbol [ψ] to denote some, as yet, unspecified scalar quantity.

[5]

As shown, [5] is in the form of Cartesian coordinates, which is
not necessarily the most convenient coordinate system for expressing
spherical symmetry, such that we might wish to convert [5] into
*spherical
coordinates*:

[6]

However, if we adopt the WSM assumption that the spherical wave function [ψ] is not only a scalar, but spherically symmetric, then we can simplify [6] by setting the terms with [θ] and [φ] to zero as follows:

[7]

So, now we are in a position to present a revised form of [1]:

[8]

It is worth noting that while [8] may now appear quite abstract,
it is still expressing the same relationship shown in [1]. However,
in order to cross-reference a much more detailed derivation of the wave
equation by Robert Gray entitled ‘*Basic Calculations Of The WSM Theory’*,
the form of [8] is updated again and makes the assumption that the velocity
of propagation [v] equals the speed of light [c]:

[9]

As such, [9] now aligns to the starting point of the paper cited
above, which can be referenced for some of the more detailed steps,
which then allows this discussion to comment on more general issues.
However, if we expand the form of the Laplacian operator, it might be
realised that there is a further expansion required based on the ‘*product
rule’* of differentiation:

[10]

The equation above corresponds to [5] in Robert Gray’s paper and
has the same form as shown in the appendix of *Milo Wolff’s book*.
However, Robert Gray’s paper then continues the derivation using the
stated method of ‘*separation of variables’* and multiple substitutions
until the following stage is reached at his equation numbers [24..27]:

[11]

In mathematical terms, each equation in [11] is a possible solution of the partial differential equation, originated in [1], and shown in a spherical symmetry form in [10]. The second expression on the right of each equation in [11] simply tries to highlight that the basic wave expression [ωt±kx] still exists in each, when substituting [kc=ω].

*But how might we initially interpret these equations?*

At this stage, the following bullets simply represents some initial thoughts about how the mathematical model implied by the wave equations in [11] might be aligned to some physical model:

- The first thing that might be highlighted is that the ‘
*amplitude’*of this scalar wave reduces as an inverse function of radius, i.e. [1/r].

- In terms of the OUT wave, we might consider an earlier discussion
*of 3D waves*, where the square of the wave amplitude corresponds to the potential energy being transported by the wave at a given point in space-time, i.e. E=A^{2}. - If so, the energy of this spherical symmetric wave would conform
to the normal inverse square law as the energy becomes distributed over
the surface of a sphere. This interpretation might also align to the
requirement that the wave model must conform to known observations of
the particle model associated with gravitation and charge.

- However, it is not necessarily obvious as to why the [1/r] expression
would apply to the IN wave, which is described as the sum of the OUT
waves from all other wave-centres in the visible universe. Given this
interpretation, an individual OUT wave would continue to decrease by
[A/r] when approaching another wave-centre and re-labelled as IN waves.
However, using
*Huygen’s principle*, it is possible that the sum of all these OUT-to-IN waves within the geometry of a spherical surface [4πr^{2}] may explain the effective [1/r] increase as the net IN wave approaches another wave-centre, although it would appear to require an extraordinary degree of phase synchronisation given that these other wave-centres would all be at different radii. - As a general assumption, each solution in [11] might interpret
the sign of the [kr] component in terms of the direction of propagation
in space, while the sign of the [ωt] component may be interpreted
as a [±180°] phase rotation or '
*spin*', which might then explain the existence of particles and antiparticles. - Within the WSM model, it is assumed that a resultant standing wave
or space resonance is formed by the superposition of the IN and OUT
waves, where the OUT wave is the IN wave having undergone a 720° spherical
rotation, which it is assumed also results in a 180° phase shift between
the IN and OUT wave within a region close to the wave-centre. However,
the details of spherical rotation within the WSM model needs to be the
subject of a more detailed review - see
*wave structure*.

In order to consider the general solution in [11], we might initially
ignore the sign of [ω, k] and assume that the IN and OUT wave have
the same propagation velocity [c] and frequency for a given wave-particle
type, e.g. an electron. As such, we might then transpose the exponential
form back into a trigonometric function, as *
previously discussed*:

[12]

So, based on the mathematical derivation of a 3D spherically symmetric solution of the wave equation, the form of both the IN and OUT waves would appear to be described by a cosine function. The following simulation is provided as an initial visualisation of an IN-OUT superposition of two cosine waves, conforming to [12], travelling towards and away from a wave-centre, synchronised at t=0, but with a half-wavelength phase shift between the IN and OUT wave. In the context of the simulation below, we are actually observing 2 radial IN (blue) waves, each approaching a wave-centre from opposite directions, which are then assumed to undergo spherical rotation within the wave-centre, denoted by the red dots, to become the OUT (green) waves. Given that the amplitudes, i.e. y-axis, of these waves only represents some, as yet, undefined scalar quantity, the simulation is really a 1D slice through a wave-centre, as represented by the red dot, which extends [r] into space and oscillates in time [t].

Note: The animation above
is thought to conform to the *Wolff wave
structure*,

inclusive of spherical rotation. However,
this model will be questioned

in a later discussion of the
*LaFreniere wave structure*
model,

which appears to reject the idea of spherical rotation.

The superposition of 2 waves is usually described as the result of the interaction between 2, or more, waves travelling through the same medium at the same time. Generally, the waves are thought to pass through each other without being disturbed, although they result in a net displacement, i.e. the superposition wave, which is the sum of the individual wave displacements and may also be described as either a standing or resonance wave. However, in this case, the suggestion is that the IN wave undergoes spherical rotation at the wave-centre, which presumably represents the maximum energy density potential, and then becomes the OUT wave travelling back in the opposite direction. In order to create the IN-OUT superposition wave shown above, the following equation was used.

[13]

While the form of [13] is based on equation [11a+11c], it now requires
a 180° phase shift that is assumed to result from the spherical rotation.
It is also highlighted that the numbering in [11], e.g. ω_{1},
k_{3}, has been dropped based on the WSM model assumption that
when there is no force acting on the particle, i.e. the wave-centre
is not in motion, then [A_{1}=A_{3}; k_{IN}=k_{OUT};**
**ω_{IN}=ω_{OUT}]. It is also highlighted
that the equation in [13] only accounts for the waveforms on the left
or right of the wave-centre, where both sets of [r] values are considered
positive and decrease towards zero for each IN wave and increase for
each OUT wave, i.e. with respect to the wave-centre [r=0]. Conceptually,
each frame of the simulation is drawn for all values of [r] for a given
increment in time [t]. Mathematically, the ‘*amplitude [A/r]*’
of the IN and OUT cosine wave functions would increase towards infinity
as [r] approaches zero; although the simulation suggests that the amplitude
of the superposition wave always remains finite, which may require some
additional interpretation, both mathematically and physically. However,
for now, we shall continue with the mathematical derivation rooted in
[11], which appears to conform to that provided by Milo Wollf and
detailed in Robert Gray’s paper:

[14]

However, looking at [14], it is unclear why this equation appears to describe the superposition of the IN and OUT wave as a subtractive function rather than additive. So while it is highlighted that reversal of the sign in [14] negates the need for the [180°] phase shift included in the simulation, it is still unclear why this is a valid physical description of the superposition of these 2 waves. However, the form of [14] may have something to do with the following function known as the sine cardinal or sinc function:

Note: We might also wish
to question the physical implications

as to why we need the two
infinities associated with [A/r] to be

cancelled with the
mathematical abstraction of the Sinc function.

Comparison of the sinc function, as shown above, with the superposition IN-OUT waves shown in the previous simulation appear to show a striking similarity. However, there does not appear to be any obvious way to create the Sinc function by simply adding 2 cosine functions, as carried out in [13]. This said, the equation below basically aligns itself to equation (28) as detailed in Robert Gray’s paper, which appears to then lead us to a basic Sinc function:

[15]

The derivation proceeds from [15] by using the following trigonometric identity to map [15] into [17] through a series of substitutions:

[16]

The following set of substitutions are shown as a series of steps
for *general reference*:

[17]

Again, with reference to Robert Gray’s paper, [17e] above appears
to align with his equation [43], which is then manipulated into the
form of [18] below by only considering the *‘real’* part of [18e]
such that:

[18]

So the equations and animation appears to reflect a possible IN-OUT standing wave formed as a superposition of two waves travelling in opposite directions. We can expand the superposition form sin(kr)sin(ωt) using the following product identity:

[19]

If we now substitute the form of [19] back into [18], it returns
us to the idea that the standing wave is form by the ‘*subtraction’*
of two underlying cosine waveforms propagating in space [r] and time
[t], but without any obvious explanation as to why this is valid for
a superposition or additive process.

[20]

As far as it is understood, this mathematical model is used by Milo Wolff to support his version of the WSM model. However, at this stage, it is unclear whether this mathematical construct really provides an adequate physical description of the wave-particle model. For example, the following issues still need to be resolved:

- The method of superposition appears to require the subtraction
of the 2 travelling waves rather than an addition.
- The 180° phase shift between the IN and OUT wave is not fully understood,
although it may result from a process described as spherical rotation
rather than the waves simply passing through each other.

- The desired superposition wave appears to be highly dependent on
the aggregation of IN waves from all other wave-centres in the visible
universe. While it is possible that Huygen’s principle can be used to
create the 3D spherical IN wave, it is unclear how spherical symmetry
would be maintained given a random distribution of wave-centres in
the surrounding space.
- There is an additional complexity, alluded but not shown in the animation related to the relative velocity [β=v/c]. In practice, the velocity [β] has to be explained in terms of some sort of Doppler effect within the reference frame selected. Again, this complexity is deferred to a later discussion.

As such, there are still some, possibly many, open questions as to whether this mathematical
model is physically representative of the WSM model. First, it is sensible
to recognise that the animation is only a 1D simulation, which uses
the wave equations derived for 3D space. Therefore, the *‘amplitude’*
of the IN wave, i.e. y-axis, is only representative of some scalar value
at a point in 3D space, which according to [13] corresponds to a travelling
wave, where the amplitude increases towards infinity as [r] approaches
zero, due to the expression [A/r]. However, given that this value is
physically described as a sum of a finite number of OUT-waves sourced
by all the other wave-centres in the visible universe, i.e. another
finite number, then the amplitude value of the IN wave must always remain
finite despite the infinite implication of the mathematical model. Equally,
the mathematical model does not really appear to explain why or how
the OUT wave is a continuation of the IN wave after having undergone
a 720° spherical rotation:

*Note:
Spherical Rotation*
*As the IN wave arrives at the wave-centre, it rotates continuously
into the OUT wave. However, this rotation cannot ‘twist’ space without
limit, therefore the associated scalar amplitude of the wave must continuously
change in the process of rotating from an IN to OUT wave. Within the
wave centre, the amplitude of the IN wave must become the amplitude
of the OUT wave, but now subject to a half cycle phase change. This
process is described as spherical rotation, which is a specific property
of 3D space only, where space can continuously rotate around a point
and return to an initial state after a 720 °, not 360
°, rotation. *

Other than the note above, we shall defer the details of spherical
rotation to later. However, such issues might make us begin to question
the scope of this wave equation to accurately describe what is actually
taking place within the WSM model. It is also highlighted that the
main proponents of the wave model, i.e. *Milo Wolff* and
*Gabriel Lafrniere*,
appear to disagree on the mechanism of how the ‘*space resonance*’ is
maintained over time. Milo Wolff appears to suggest that the IN-OUT wave
process continues indefinitely, while Gabriel LaFreniere suggests that
the wave-centre resonance is self-sustaining even though it radiates
energy, because it also receives energy from the rest of the visible
universe. However, the details of this difference will also be deferred
to a later discussion - see *wave
structure*.