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:
However, there is possibly some value in reformatting  to initially convert the variables [k,ω] back into the basic units of space [x] and time [t]:
In the form of , we see that the 2nd 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  or  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  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:
However, the dot product of the [∇] operator with itself will create a scalar, which in mathematics is called the ‘Laplacian Operator’ :
Given that there may be an inference that  is only describing a physical wave amplitude [A], we might replaced [A] with the symbol [ψ] to denote some, as yet, unspecified scalar quantity.
As shown,  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  into spherical coordinates:
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  by setting the terms with [θ] and [φ] to zero as follows:
So, now we are in a position to present a revised form of :
It is worth noting that while  may now appear quite abstract, it is still expressing the same relationship shown in . 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  is updated again and makes the assumption that the velocity of propagation [v] equals the speed of light [c]:
As such,  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:
The equation above corresponds to  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]:
In mathematical terms, each equation in  is a possible solution of the partial differential equation, originated in , and shown in a spherical symmetry form in . The second expression on the right of each equation in  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  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=A2.
- 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.
Huygen’s principle, it is possible that the sum of all
these OUT-to-IN waves within the geometry of a spherical surface [4πr2]
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  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 , 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:
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 , 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
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.
While the form of  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 , e.g. ω1, k3, 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 [A1=A3; kIN=kOUT; ωIN=ωOUT]. It is also highlighted that the equation in  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 , which appears to conform to that provided by Milo Wollf and detailed in Robert Gray’s paper:
However, looking at , 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  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  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 . 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:
The derivation proceeds from  by using the following trigonometric identity to map  into  through a series of substitutions:
The following set of substitutions are shown as a series of steps for general reference:
Again, with reference to Robert Gray’s paper, [17e] above appears to align with his equation , which is then manipulated into the form of  below by only considering the ‘real’ part of [18e] such that:
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:
If we now substitute the form of  back into , 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.
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  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:
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.