The easy question (longwave propagation) first.  As always, there is plenty of info to be found with Google.

www.google.com/search?q=longwave+propagation

Back to your first question (50' antenna).  The problem with short antennas and long wavelengths (for transmitting) is that the feedpoint impedance becomes very low, the currents extreme, the resistive losses significant, and the efficiency extremely low.

The practical length of vertical antennas becomes a significant issue even at the low end of HF, let alone longwave.  What you're asking for is equivalent to trying to use a 2 foot (60 cm) tall antenna on the 80m (3.5MHz) amateur band.

If you transmit low power, combined with poor efficiency, then your emitted signal strength will be extremely low.  If you transmit high power, then you have to build a huge antenna matching circuit with very heavy wiring (due to the currents, and to reduce loss).

One related hobby is called "lowfer".  Is that what you're up to ?  I think that they're restricted to 160-190 kHz.

Google again...

www.google.com/search?q=lowfer
www.google.com/search?q=lowfer+transmit+antenna

Good luck.

re: Ve1BLL
 Yes that's right 50 feet of antenna or .007 wavelength gives about 17 millioms of radiation resistance, or about 30 db below an isotropic source. Checked into all of the internet references I could find, information on longwave radio is sketchy and sometimes contradictory, also went digging through the U of Manitoba Engineering library and found one book on the subject.
   I am on the lowfer email reflector and have been seeing a few of them using the dsp program Spectran. Was hopping to find an engineering student who did a project in LF radio, or some one that could send me an IEEE article on the subject.

       73 de Ve4XC

I believe that LF and VLF provide fairly good propagation.  For example, the USN uses VLF to communicate with submarines worldwide.  Also, I have a cheap radio controlled clock (in Ottawa) that seems to pick-up the 60kHz signal from Colorado quite easily.  Of course, these guys don't restrict their transmitting antennas to 50 feet.

I've read about people using synchronized systems (DSP) on VLF where they resend the same signal over and over again, and integrate it at the receiving end.  Obviously, if the systems are well synchronized (GPS?), there is essentially no limit to the processing gain.

I'd like to see someone develop a slow communication system using this technology and VLF where, using a pair of little boxes and some wire antennas, one could send a 2-digit number around the world within a day or so.  For example, 17 = 'Happy Birthday', or 89 = 'Coconut crop failed, send money ASAP, usual amount'.  

I think right now the processing gain limit is about 20 DB with currently available software.  As for around the world stuff, UK hams have been seen on the east coast on 73 Khz and Zl6OH has been seen many times in W, K and VE on 137 khz. If you are an active ham you might be interested in this, I am within 350 miles of 4 lowfer stations each running about 2-3 milliwats erp, 1 watt output into an antenna less than 50 feet. I can hear the closest station 287 miles using DSP, the far station is hit and miss on visual cw DSP, but I have yet to see or hear any identifiable signals on skywave, implying that on 185 khz at 2 mw erp a 350 mile groundwave signal is many db stronger than any similar skywave signals.
    The purpose of all of this is to see if an average amateur station, 100 watts into an antenna next to the garage can be provide effective communications on LF. Since it seems likely that Canada and the US support the idea of a Longwave ham band.

The military use of vlf for undersea communications is tied to large vertical radiators (eg 1300ft) with a huge top hat and multi megawatt transmitters.

You might have some fun experimenting with driving a 50 ft vertical though as it's theoretical power gain is only about 1.5dB less than a quarter wave dipole.  Trouble is that it's feedpoint impedance R +jX has R = 17 milliohms as VERNJAC said, while jX is many thousands of ohms Xc.  The only practical way to feed it is to place it on top of a resonant circuit and tap off 50 ohms near the bottom of the coil.

Sounds simple and it is till you give it much power when you'd better have some real good insulation in that capacitor/coil as voltages reach thousands of volts.  Also at low frequencies corona on the antenna top is an issue and a large surface area top hat or ball or whatever is needed or there'll be a purple glow soaking up your power at quite low power input levels.

Good luck.

Don't the military antennas use really long buried radials?  I also thought I remembered reading that the signals were found to have propogated through denser materials better than higher frequencies.  That would explain why VLF is used for communications with submarines.  

The miles long buried radials used on VLF transmitting antennas are part of extracting every bit of efficiency possible by lowering the ground impedance.  

I believe that horizontally polarized signals do not penetrate the earth anywhere near as far as vertically polarized signals (which are used for submarine communications) and to get the best propogation mode and range with vertical polarization a VLF frequency is best.

Submarine VLF receiving antennas are commonly a pair of vertically polarised loops on ferrite cores, crossed at 90 deg to give a 4-leaf clover radiation pattern.  They cannot transmit on VLF.

Suggestion: Visit
http://www.lwca.org/library/articles/k0lr/antennas/loferant.htm
http://www.amrad.org/projects/lf/actant/index.htm
etc. for more info