Standing Wave Compression

As detailed further in another discussion, the Michelson-Morley experiment (1871/1875) to detect the Ether as a wave propagation medium only appeared to produce negative results. The subsequent development of special relativity (1905), anchored in the Lorentz transforms, were then seen as confirmation that the Ether did not exist, at least, as a wave propagation media. However, the establishment of this premise pre-dated any significant research into the concept of standing waves and certainly long before the idea of standing wave compression had been first forwarded by Yuri Ivanov in 1981. Given that the mathematics underpinning this concept are possibly key to the WSM model, we need to present this rationale before building further on its assumptions. If the following description of compressed standing waves has validity, the definition of 1 metre, as given earlier, would have to be revised as being 3301527.46 times the standing wave length [L] of the 86Кr emission. The implication of this revised statement are defined by equation [16] below, where the length [L] of the standing wave would be a function of the velocity [β=v/c]. However, ultimately this proposal also requires that all inter-atomic distance found in the material structure of our universe must have an underlying standing wave structure, such that the change in the measure of distance goes undetected.

The animations above were produced using a Freebasic program, see 'Simulations' for details, in which the source (left) emits a sinusoidal wave towards the reflector (right). The top animation has set [v/c=0], while the bottom animation has set [v/c=0.5c] in order to show a comparative effect of standing wave compression between the source and the reflector. It is highlighted that the only parameter changed in these two animations is the value of [v/c].

Note: Ivanov based his initial work on field experiments with sound wave propagating through the media of air [c=1], which were then affected by the wind velocity [w]. However, in a wider context, we might recognise the the equivalence of the reference france of the source and reflector moving with velocity [v] from left-to-right, while the wind would blow from right-to-left.

For the moment, we shall initially assume that these waves are analogous to acoustical sound waves, which propagate through the medium of air with a given velocity [c]. In this context, the wave propagation velocity [c] does not change, irrespective of the velocity [v] of the source or reflector, although we might recognise that this relative motion might give rise to a Doppler effect.  If we assume that the velocity [v] is left to right, then the net velocity [v’] of these wave will be perceived to be a function of distance [L] between source and reflector and the time [t] taken:

[1]     

However, the effect summarised in [1] is only assumed to underpin the classical Doppler effect, where the net velocity [v’] is allowed to exceed [c], but which is a violation of special relativity. Of course, we might recognise that any determination of [c, v, v’] will depend on the observer's measurement of space and time, i.e. [L,t], which returns to an earlier historical point made about the use of interferometry, also pioneered by Michelson-Morley, in establishing the standardisation of the ‘metre’ as a unit of length. So, within the context of the basic WSM model, let us table the following question:

If everything is made of standing waves, how absolute is any relative measure of distance?

We shall not attempt to answer this question at this stage, but simply return to the animation above, where the bottom trace shows the aggregation of the middle standing wave with time. Now we appear to see the relative random form of the isolated standing wave stabilise to form  the apparent static standing wave form, which is compressed as a function of the relative velocity [v].

OK, but how might this description be extended beyond the classical Doppler effect?

So, having presented an initial framework, we might now consider the mathematics underpinning Ivanov’s idea, starting with some basic wave amplitude equations:

[2]     

Here we see a fairly standard form of a 1-dimensional travelling wave equations, where the waves are propagating in opposite directions as defined by [±kx]. Within this form, and the previous introductory description, it will be assumed that the waves always propagate with velocity [c] in the medium with a given frequency [f]. However, if the relative velocity [v’] can change, the wavelength [λ=v’/f] must also change, but let us be a little more specific about the geometry on which we shall proceed:

The diagram above is representing a wave propagating out from a source [O] with velocity [c]. However, the diagram also assumes this source is moving with a relative velocity [v] and at some given point arrives at [N], where we wish to evaluate the perceived velocity of the wavefront originating at [O], but now  having reached [A] and [B]. Therefore, an observer at [N] sees the distance to the wavefront at [A] to be [AN] and the wavefront [B] to be [BN]. Clearly, from the geometry, the observer at [N] will perceive the wavefront distances [NA] and [NB] to be different and therefore the propagation velocities [c₁, c₂] to differ in equal time [t]. So, using the diagram, we might establish some basic geometric relationships on which to proceed:

[3]     

So, based on [3], we might construct expressions for [c₁, c₂], where [β=v/c]

[4]     

For general calculation, used in the diagram above, we might simplify the expressions in [4] for the parallel and perpendicular cases associated with the [x,y] axes:

[5]     

Having established some of the implications of the geometry, we might now continue with the mathematical evaluation of the wave equations in [2]:

[6]     

Based on [λ=c/f], we have now transposed [2] into a form being described by the geometry above, i.e. two waves perceived to be propagating in opposite directions with velocities [c₁,c₂].  However, the form of [6] can also be changed using the following trigonometric identity:

[7]     

We can add and subtract the fractional expressions using the following rules:

[8]     

On the basis of [8], the final expression in [7] becomes:

[9]     

The resulting waveform in [9] can be also be described in terms of the trigonometric ‘sum-to-product’ identity of the form:

[10]   

If so, then we might equate the values of [t’, k’] in [10] to the expressions in [9], although it will be more useful to transpose [k’] to the physical wavelength [λ_SW] of the standing wave via the relationship [k=2π/λ]:

[11]   

However, we now have the requirement to normalise the expressions in [11] to the ‘real’ propagation velocity [c] and relative velocity [v] using the expression derived in [4]. Looking at [11], we might see that there are 3 distinct combinations of [c₁, c₂]. but we shall start with [c₁*c₂],

[12]   

Now [c₂+c₁]:

[13]   

Finally  [c₂-c₁]:

[14]   

We can now substitute [12,13,14] into [11] starting with [t’]

[15]   

Now applying the appropriate substitution to [λ_SW]

[16]   

Again, we might rationalise the results in [15] and [16] to the diagram shown above by only considering the 1-dimensional x-axis, where [θ=0]:

[17]   

Note: there may appear to be a discrepancy by a factor of (2) in the final form of [16], when compared against different sources of Ivanov’s work. In later versions of this derivation, the substitution in [11] is based on [λ’=k’/π] rather than [λ’=k’/2π]. At face value, it would seem that this discrepancy is only due to a mixing of the general understanding of the term 'wavelength [λ]' and 'standing wave length [L]'. The former is one complete cycle of a wave, while the latter is the distance between nodes. It is this mixing of terms that causes the factor (2) change below.

[18]   

As a consequence, the final form of [16] becomes:

[19]   

However, if you set [θ=0], then [19] reduces to [20] and setting [β=0] suggests that the factor of (2) does not align to the requirement that [λ=c/f]. In the case of a standing wave, the symbol [λ'] appears to reflect the length [L] between nodes, which is half of one full wavelength cycle.

[20]   

So if we make reference to the animation above, where [β=0.7], the nature of the compressed standing wave can possibly be seen in a different way. In the top trace, we clearly see the different wavelengths of the two travelling waves propagating in different direction calculated based on the following expression

[21]    

These wavelengths are still visible in the middle standing wave trace and based on the form of [9] and [10], it might be argued that the higher frequency wave is acting as a carrier wave, where its amplitude is modulated by the lower frequency wave. In this context, there is no obvious compression of the standing wave in isolation. However, if we aggregate the middle standing wave over a larger period of time, we do see a compressed standing wave of sorts.

Note: Some further evaluation of this effects is discussed in terms of a 'relative Doppler effect' and the conceptual operation of a relativistic 'light clock'.

At this point, we might attempt to provide some comparative measure of the wave-length distances involved in the WSM model of an electron. In this specific case, the fundamental frequency [f₀] of an electron might be estimated based on the following assumptions linked to the experimental determination of electron mass.

[22]   

Note: The frequency shown above can also be translated into Compton's wavelength for the electron based on [λ=c/f] giving a figure of 2.43*10^-12 metres. On the sub-atomic scale, this wavelength appears to be quite large when considered in terms of the assumptions underpinning the determination of the Bohr radius of an hydrogen atom, which may be difficult to reconcile with the following wave model of the electron.

Given the note of caution above, we will not question the assumptions underpinning [22] too closely at this stage. However, it might be recognised that any practical measurements of the electron standing waves, assuming they exist, would presumably only detect the aggregated amplitude as a function of time, as suggested by the bottom (grey) trace in the animation below.

Note: whether the model shown above actually works is still in some doubt,
when extended to relativistic velocities that cause different Doppler effects
 and time-dilation considerations to be taken into account.
This is still work-in-progress.

In the animation above, the standing waves shown in red and grey are compressed as a function of the velocity [β], although for the reason outlined above, the animation is restricted to [β=0]. However, as a speculative assumption, the time-aggregated standing wave may have a tangibility in the sense that it reflects the electron energy density in space-time. While the discussion of this simulation is running ahead of the review in progress, it is only intended to provide an initial visualisation of a 'speculative' IN-OUT wave model of an electron.

Note: The model implied by the animation is based on the idea that any IN wave is subject to 'spherical rotation' in order to become an OUT wave. However, this wave model appears to be contradicted by Gabriel laFreniere - see wave structure for more preliminary details. Again, it is highlighted that there are still many issues not really understood within the generalisation of the WSM model.

As a one-dimensional cross-section, the wave-centre represented by the red dot receives IN waves from the forward and backward directions from which OUT waves are returned, as per the upper traces. However, a 3-D spherical symmetric solution of the wave equation requires the amplitude of the waves to decay with distance by a factor [1/x]. However, as one final note of caution, it is also possible that this speculation will amount to nothing unless all these wave equations can also explain the failure of the Michelson-Morley experiment to detect the Ether. However, the next section of discussions will continue to consider some further implications of a physical 'reality' built on waves.