Well, there is no simple answer to that question. I've been working on this, but due to political issues, have not posted my findings here yet.
The first idea is based on Stan Meyer's work. If you look at a molecule of water, it is a bar magnet with the two hydrogen atoms at a positive charge on one side, and a negatively charged oxygen atom at the other. The bond between the three atoms being maintained by the orbit of shared electrons. Now imagine pulses of electromagnetic energy timed so that the orbit of these bonding electrons is stretched at the apex of their perspective orbits. Stretching of this orbit enough would break the bond between the atoms and release monatomic hydrogen and oxygen. This is how Stan claimed his device worked, but has not been proven mathematically or otherwise.
After examining his system and associated patents, his system appears to run on a different process. Pure water is a really good insulator. To that end, a large electrostatic field and be developed before the dielectric strength is exceeded, and electricity begins to flow. The strong electrostatic field causes constant polarized stress on the water molecule. This stress causes the water to either pull apart, or spin on an axis through the center of the oxygen atom and on of the hydrogen atoms, causing the other hydrogen atom to become fly loose. To develop the field, a tank LC circuit is constructed consisting of the cell as the capacitor and a bifilar inductor. The hard part is following the proper frequency as it changes when the cell starts producing gas.
Another idea is magnetic field resonance. When electricity is passed through a conductor, a magnetic field is created as the current increases in strength down the conductor. When the pulse reaches the end of the conductor, a small amount is reflected. When the reflected wave reaches the first end, we pulse the electricity back on, adding to the strength of the magnetic pulse with each cycle. How exactly this is supposed to break apart water eludes me, but the theory to develop the field is sound.
I know there are more theories to use resonance than this, but this is all I have studied in detail.
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