Legend
- $c$ = velocity of propogation = speed of light (299,792,458 meters/second)
- $f$ = frequency
- $\lambda $ = wavelength
Formulas
The basic formula for calculating wavelength is:
\begin{equation}
\lambda = \frac{c}{f}
\end{equation}
To make the math simpler, frequency ($f$) is expressed in megahertz (MHz) and the velocity of propogation in free space ($c$) for frequencies above 30 MHz is expressed as and rounded to 300 megameters (Mm). This will return a wavelength ($\lambda$) in meters. So for 1 wavelength above 30 MHz:
\begin{equation}
\lambda_{m} = \frac{300}{f_{MHz}}
\end{equation}
However, when $f < 30_{MHz}$, the velocity of propogation ($c$) is expressed as and rounded to 286 Mm because
"[e]lectrical wave propagation in wire is about 95% to 97% the speed
of light. Since wavelength is most commonly used for building antennas
which involve conducting the wave from air into the wire and vice
versa, the calculation is adjusted assuming the slower propagation in
an unshielded conductor.
"However, this 3% to 5% discrepancy is small enough at frequencies
above 30 MHz that it is usually ignored for simplicity, and 300 Mm is
used instead" (Adam Davis, KD8OAS).
When $f < 30_{MHz}$ the discrepancy becomes more significant and the adjusted value, approximately 95% of 300 Mm, is used instead, which is approximately 286Mm (which would actually be $0.95\overline{3}$). This results in the following formula for 1 wavelength below 30 MHz:
\begin{equation}
\lambda_{m} = \frac{286}{f_{MHz}}
\end{equation}
To convert this into feet, multiply $c$ by 3.28084, which results in the following formula for receiving an answer in feet when $f > 30_{MHz}$:
\begin{equation}
\lambda_{ft} = \frac{(3.28084)300}{f_{MHz}} = \frac{984.252}{f_{MHz}}
\end{equation}
This is rounded down to $984/f$ for the sake of simplicity. However, recall that when $f < 30_{MHz}$, the velocity of propogation ($c$) is expressed as and rounded to 286 Mm. Applying this formula results in the following for converting this into feet below 30 MHz:
\begin{equation}
\lambda_{ft} = \frac{(3.28084)286}{f_{MHz}} = \frac{938.32024}{f_{MHz}}
\end{equation}
This is also rounded down to $938/f$ for the sake of simplicity.
Calculating for half and quarter waves is just a manner of of dividing $c/2$ or $c/4$, respectively. So we end up with the following calculation for calculating the length of half wave antennas in feet when $f > 30_{MHz}$:
\begin{equation}
\lambda_{ft} = \frac{(3.28084)(300/2)}{f_{MHz}} = \frac{492.126}{f_{MHz}}
\end{equation}
When calculating the length of half wave antennas in feet where $f < 30_{MHz}$, we have the following formula:
\begin{equation}
\lambda_{ft} = \frac{(3.28084)(286/2)}{f_{MHz}} = \frac{469.16012}{f_{MHz}}
\end{equation}
But this is generally expressed as $468/f$, not as 469. Why is this? First of all, remember that the velocity factor is approximately 95-97% of the speed of light, so adjusting this value results in slightly different results. Also, whether we use the adjusted value of $c$ when $f < 30_{MHz}$ (286 Mm) or apply the velocity factor directly to $c$ will slightly alter our result. So for instance, the following calculation will get us closer to $468/f$:
\begin{equation}
\lambda_{ft} = \frac{(3.28084)((300/2)(0.95))}{f_{MHz}} = \frac{467.5197}{f_{MHz}}
\end{equation}
This would round up easily to $468/f$ when $f < 30_{MHz}$, and it is slightly more accurate.
This shows why there are different equations and when each should be used.
