I have heard about this natural resonant frequency. I think people are trying to use it to send and receive wireless power.
I am guessing that all you want is a pure, low frequency sinusoidal output to drive some sort of low frequency antenna. No modulation required. The link you posted to seems to think that one single frequency is desirable. Modulation causes different frequencies to occur.
I have never built one but here is what I would do. I would build three stages.
7.83 Hz pure sine generator ===> Low harmonic distortion amplifier ===> Low frequency antenna
A chip like this would do for the sine generator. XR2206CP. The data sheet provides a "Circuit for Sine Wave Generation with Minimum Harmonic Distortion". Available on eBay.
http://cgi.ebay.com.au/2-XR-2206-XR2206CP-XR2206-Monolithic-Function-Generator-/280581098961?pt=LH_DefaultDomain_0&hash=item4153efcdd1The output of the chip will not be able to supply very much power so you will need an amplifier or buffer to drive your antenna check out the class A or class AB amplifiers on Wikipedia.
http://en.wikipedia.org/wiki/Electronic_amplifierThe output of the amplifier will connect to one end of the coil, the other will connect to a capacitor, the larger the better. An electrolytic will have to be used, I am talking big, like 20 000 uF. They are available online. Get a couple and put them in parallel. The other end of the capacitor can be connected to ground or your supply voltage.
Now an antenna that transmits 7.83 Hz is an unusual thing, and using a standard type like a dipole/monopole is out of the question, it will be too large. Seeing as the antenna only needs to have a effect close to you I would use a flat coil of copper wire about .5m in diameter with enough turns on it to give about 20 ohms of resistance. Use a thin gauge of copper wire for this like 0.25mm I think about 200grams (60m) will give you about 20 ohms. If you are serious about the pocket nature of the antenna then make the coil smaller but beware of heat generated.
You can then use the amplifier to drive the antenna. I am guessing a voltage of about 40V pk-pk will be as far as you can go before the coil starts to heat up too much because of the I*I*R losses in the copper. 40V pk-pk means 20V pk-pk across 20 ohms (your dc blocking capacitor voltage will settle to 20V)means 1 A pk-pk which means 20W pk-pk of power dissipated in your coil. This is just a guess so start off low and keep going till it is too hot. There will be other effects reducing current, like the impedance of your capacitors and amplifier.
The coils inductance will not feature much at 7.83Hz, so this DC analysis should suffice.
Finally if the idea of this project is surround yourself in 7.83Hz signal why not run a few loops of wire around your whole damn property,say 1cm below the ground, just a bit of a soil covering to hide it. Or around the perimeter of a room you frequent often. and drive this loop with the antenna. Heat dissipation will be better with the spread out coil so you could increase the current in your copper wire.
Right disclaimer time. Now that I have told you how I would do it. I have no idea what such a device may do to you, your neighbors, etc. You should do some serious research into any possible ill affects before going ahead with any construction. Also ask the people who may be affected if they mind. Just common courtesy.
I am guessing the aim is a general feeling of wellness. Protection from electromagnetic "smog". Is this correct?