Lorentz Transformation and Waves
The Lorentz Transformation as a Wave FunctionWe will demonstrate that translationdependent transformations, of which the Lorentz Transformation is a special case, arise naturally out of wave systems. As a result, the Lorentz Transformation might be considered a natural consequence of the wave characteristics of matter. Does this concept form the link between quantum theory's waveparticle duality and relativity's LorentzTransformation? You decide. A newer version of this page now exists at my blog. 
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Part 1 Imagine an entity composed of standing waves in one dimension. This system can be characterized by equations in various ways:
The perceptive will already see the implications in the second equation, but we will forgo the discussion until after a few simulation snap shots.

Part 1: Concieved 1983 Posted 2001 Formatted 2010
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To sum:
Lorentz & Standing WavesIn the Lorentz Transformation accelerating an object will contract its ruler, and dilate its clock. In standing waves changing the coefficients to contract the ruler and dilate the clock means changing the same variables that will produce acceleration. Using a little algebra, the wave coefficients will relate to the velocity in the Lorentz Transformation:
With out the use of higher level multidimensional mathematics and special considerations from physical requirements, we can not determine for sure if this result is significant. Evidence that the result is physically significant is above: standing wave systems transform intrinsically when accelerated. Evidence that the result is mathematically trivial: whenever a system is transformed, all waves within that system must transform accordingly. Imagine a violin on a space ship, it should sound correct to the astronauts, regardless of the acceleration of the ship. Physically Significant or Mathematically Trivial? Newtonian Transformation  ie. translation without transformation: Standing waves may produce other transformation including the Newtonian Transformation of classical physics


Part 2: In 1983 I recognized (above) that the Lorentz Translation could result from a wave equation. I was unable to determine the significance of the result. One reason may have been that I assumed that a standing wave system required both the forward and reverse parts to have the same wavelength. Considerations and Special Cases
This pattern is interesting, because it gives us a standing wave overlapping a moving wave. This may be a good general description when we see ripples between rocks in a flowing stream. But this form probably does not apply to the Lorentz Transformation. 2: We could approach the problem from an energy and momentum perspective.
It's hard to see whether the interference pattern can be made to conform to a moving "standing wave." But we can see some interesting considerations in the equation. The total energy of the system should be E1+E2. The total momentum of the system should be P1  P2. From the perspective of matter as a standing wave, we would expect momentum to correlate to two de Broglie wavelengths: = 1  2 = h/P. 3: We could look at the general form and ask what special conditions must the general form meet to conform to the measured characteristics of relativity and quantum theory.
This is the challenge. Did Ivanov take this form far enough to show that it could fit the Lorentz Transformation? If not, what wave characteristics will make it do so? 
Part 2: Found online September 2012 

