We have a very interesting explanation of how DE Broglie derived his wave equation here:

http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/De_Broglie_WavelengthHis starting point was to assume E whilst represented by mc^2 could also equal hv which is Planck's constant times velocity. This term operates for light only so examining equation 5 we see that he ended up with essentially h/mv. If we take angular momentum as a stand in for energy and because we have 2 fields in motion we should be able to derive an equation at the Planck scale that equals De Broglie's. This is dependent upon the radius of a particle.

It has to be understood that we are no longer describing the wavelength here. If we are working in natural units the angular momentum will range between 0 and 1. The velocity is forward momentum and so describes the stretch of the wave with time. Hench this formula describes the rate of change in the wavelength.

Now if we go back to [tex]\frac{Pl}{GPm} \frac{\partial ^2 E}{\partial t ^2}[/tex] we also have a mass component. If we have Va as our angular momentum we have h/Va*v as the velocity of a wavelength segment which can then be related to the time over which it evolves. As [tex]\frac{\partial ^2 E}{\partial t ^2}[/tex] describes our wave and [tex]\frac{Pl}{GPm}[/tex] relates this to mass and gravity we should be able to combine these. Remember that the mass was the only true variable in [tex]\frac{Pl}{GPm}[/tex] and in the De Broglie equation it is mass we have modified by replacing it with angular momentum.

Since we are only modifying mass all our calculations are done at a set radials distance as mass dereases. This is because Pl never changes. If we take the earth as an example we need to correlate a change between the surface and twice the radius to our fixed radius. We need to see how the mass should decrease to give us the right gravitational field strength. This factor is important to determine. Because we hold the radius fixed we can then model the wave evolution over a spherical surface as described above. This is not meant to reflect the reality of particles but is a device to determine the relationships.

In our left hand term the mass factor relates to the radial distance the wave travels away from the source. This has to be a density change. It is not only related to the density of the source mass but also the gravitational field density at any radial distance. Our factor will be p with p= 1/r1^2 where r1 is the distance away from the source hence we now have [tex]\frac{Pl}{pGPm}[/tex] as our first term. The use of radial distance now connects the permittivity/permeability term directly to the wave evolution via the vector r1. The starting point will have r0 (our static radius) equal to r1. As r1 moves away the density is reduced by our term p. This will produce a gradual blue shift and can be related to time dilation via an implicit time on the z-axis. This does NOT describe the evolution of a light wave. Other factors come into play for light.

From now on the model uses linear algebra.