> In other words, let's say that to compensate for shorter antenna length, I
> swap out my 1W transmitter with a 2W one. Would that even work? And if it
> would, what happens to the extra power -- does it get converted to heat
> inside the antenna, or does poor SWR cause the energy to be lost inside the
> output stage?
>
> If I sound like an idiot (which I suspect I do), I would appreciate it if
> you could point me to a reference that would help explain this.
>

The idiot is probably the guy who thinks he can explain things about RF.  So
here I am. I'm going to consider your question the way it was asked.  What
everyone else wrote is also good information.

Your premise is that going from 1/2 wavelength to 1/4 wavelength eliminates
half the antenna, and therefore half the signal. This isn't really the
case.  A 1/4-wave antenna by itself would not work properly; there needs to
be a ground plane under it.  This ground plane takes the place of the other
1/4 wavelength's worth of antenna you just tried to cut off.  The ground is
often described as giving the antenna "something to work against"; but the
reality is that whatever you construct still has to resonate at the
frequency you're using.

What may happen instead is that the output match changes.  Dipole antennas
(that's the half-wave) run about 72 ohms, while the 1/4-wave is generally
considered to be 50 (although one source claims a "perfect" one will be 37
ohms!).  The resulting loss in power isn't really mysterious.  If the
impedance is higher than design, the output amplifier just doesn't supply as
much current in the first place.  If you only get a watt out of the antenna,
it's because the output stage is only generating a watt now.  In the other
case, the lower impedance loads the output stage down so that it can't keep
the voltage up, and I x V will still be lower.  The solution to all this is
to change the output circuit to operate at the right impedance, whether that
means changing components, or adding something to transform the impedance
somewhere between the two ends.

The other consideration (and I'll defer to the others here) is that the
radiation pattern changes.  A dipole's peak direction is perpendicular to to
the antenna line.  The "ground place" antenna has its pattern "squashed
upwards" so the peak direction is at an angle above the horizon.  That has a
real effect on the receivers.
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