The velocity of light, V, is roughly 186,282mi./s. Multiplying V by 5,280ft./mi. yields 
984,000,000* ft./s. Megahertz is a million events per second. One wavelength is also one hertz,
which is also equal to one period of events per second. Therefore, 984,000,000 divided by
frequency equals one wavelength (1?). 

One wavelength = 984 ÷ F(MHz). For the ½? dimension, in feet, 492 is the value to be
divided by the frequency, and 492 ÷ F(MHz) = ½?, in feet.

Example: 984 ÷ 155MHz = 6.35ft. = 1?. ½? = 3.175ft.
3.175ft. × 12 in./ft. = 38.1in. ¼? = 19.05in.

All of the above is relative to the speed of light. If we are going to use wires or 
conductors of any type as the antenna, then things take on a different characteristic. 
Anything other than free space will slow down energy, and the ratio of free-space travel to 
conductor travel is the velocity factor, usually noted as vp.

So, is the free-space formula in error? Not really, it just needs some modification. In
reality, the frequency that provided this zero reactive point was plugged into the formula to
find a "K" factor that would modify the free-space formula, 492 ÷ F(MHz), to bring us to the 
zero point. That "K" factor is 0.95 for wire antennas up to about 30MHz. Beyond 30MHz, the 
diameter of the wire also affects the overall length for a pure resistive impedance. The 
result is the classic 468 ÷ F(MHz) seen in most text books for a ½? antenna.

Note: the "K" Factor (0.95) which alters the free space formula of 492 (491.8) only applies to 
the reasonable and accurate determination of the PHYSICAL LENGTH of a 1/2 wave antenna for a 
given frequency, but DOES NOT APPLY to antennas longer than a 1/2 wl in length.

ref: the arrl antenna book, 15th edition, chapter 2, page 4, end of the section 
(not surprisingly) titled, "end effect." 

the same goes for determining the correct length for sections of feedline operating as a
1:1 transformer, as in the case of the electrical tuned half wave line, or the electrical 
tuned quarter wave line, operating as a 4:1 transformer / matching section. in each of these
cases we need to correct for the ratio of free space travel to conductor travel by factoring
in the velocity factor / vp of the feedline being used to construct the transformer.

for the quarter wave section: 246 ÷ F(MHz) X vp = electrical length in feet.
for the half wave section: 492 ÷ F(MHz) X vp = electrical length in feet.

generally speaking, all patchcords should be kept to =/< than 1/20 of a wavelength.

* rounded
? wavelength