# Examples

## Assumptions

• The wave center is the fundamental particle, which is possibly the neutrino. Longitudinal in-waves are reflected to become out-waves.  The amplitude of these waves decrease with the square of distance, with each wavelength, or shell (n).
• Particles are created from a combination of wave centers. A number of wave centers (K) form the core of the particle, resulting in a standing wave formation from the combination of in-waves and out-waves.
• Wave centers prefer to reside at the node of a standing wave, minimizing amplitude. They will move to minimize amplitude if not at the node.
• With sufficient energy, wave centers may be pushed together in arrangement to create a new particle (i.e. neutrino oscillation), but will decay (break apart) if the structure does not lend itself to a geometric shape where each wave center resides at the node in a wave.
• When wave centers are spaced in the nodes, at even wavelengths in the core, waves are constructive. A particle’s amplitude is the sum of its individual wave center amplitudes in the particle core.
• If two wave centers are pi-shifted from each other on the wave (1/2 wavelength) it will result in destructive waves. This is an anti-particle.  For example, if the neutrino is the fundamental wave center, then the anti-neutrino is a wave center pi-shifted from the neutrino.
• Particle radius is proportional to the total wave amplitude, and is the edge of where standing waves convert to traveling, longitudinal waves.
• Particle energy is the energy of standing waves within the particle’s radius.

## Example #1 – Tau Electron (calculated rest energy)

The Longitudinal Energy Equation was used to calculate the rest energy of all particles, and then shown for the lepton particles (the neutrino and electron particle families) in the calculations table for each value of K. Each particle is assumed to consist of wave centers, which is given the variable K to describe the unique wave center count for each type of particle.

Each particle was mapped from values of K=1 to K=118. Then, the best fit was determined for a particle matching the rest energy. For the tau electron, it is K=50. Interestingly, all leptons were found at magic numbers also seen in atomic elements.

Tau Electron (K=50)

The only variable in the Longitudinal Energy Equation is K. The remaining are wave constants. Inserting the value 50 for K into the equation results in a value of 2.8137E-10 joules, or when expressed in electron-volts, it is 1.756 GeV (compared to a measured result of 1.777 GeV). This number was added to the leptons table in the calculations table.

## Example #2 – Tau Electron (for linear solution)

The chart in the particle calculations page shows a linear relationship between the particle rest energy and the particle number (K). This section describes the example of how these values are obtained. Although joules could have been used to linearize particle energies, eV was chosen for the chart. Therefore, particle energies were first converted to eV. Then, they were divided by the fourth power of K to linearize. This is the equation:

For the tau electron, a rest energy of 1.756 GeV was calculated from above (2.8137E-10 joules). The value of K that was used in 50. These are inserted into the equation above.

Tau Electron (K=50)

The value of 280.96 was plotted on the chart for modified particle rest energy, where the particle number (K) is 50. This was repeated for each of the subatomic particles. A linear solution was found from the neutrino to the Higgs boson.

Further examples are provided for the calculation of the neutrino rest energy and the electron rest energy on their respective pages.