# Why is the speed of light rounded to 286 Mm in calculations involving frequencies below 30MHz?

I understand that in order to make the maths simpler, frequency ($f$) is expressed in megahertz (MHz) and the velocity of propagation in free space ($c$) for frequencies above 30 MHz is expressed as and rounded to 300 megameters/second (Mm/s). The actual speed of light is 299,792,458 meters/second, so expressing and rounding this to 300 Mm makes sense.

I'm confused why it is rounded to 286 Mm when $f < 30\ \mathrm{MHz}$. Please explain. An excellent answer will show the maths.

• @KevinReid actually that is how I originally had all of mine but it was recommended to me that I subscript them. – Dan Oct 29 '13 at 18:33
• That's surprising. Could the recommender perhaps provide a citation for that style? – Kevin Reid AG6YO Oct 29 '13 at 19:57
• @KevinReid this was the recommendation – Dan Oct 29 '13 at 21:18
• Ah, I see. That recommendation was referring to writing $f_{\mathrm{MHz}}$ instead of $f\ \mathrm{MHz}$, because the latter means $f$ multiplied by MHz which is wrong because variables for quantities are generally assumed to already have their proper units/dimensions, and the subscript is just a hint about what units the numeric value should be in. That's different from a constant, where you really do want to say $30$ multiplied by $\mathrm{MHz}$, or $30\ \mathrm{MHz}$. – Kevin Reid AG6YO Oct 29 '13 at 22:15

## 1 Answer

Electrical 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 is used instead. For frequencies below 30 MHz it becomes more significant and the adjusted value, approximately 95% of 300 Mm, is used instead - about 286 Mm.

\begin{equation} \lambda_{\mathrm{m}} = \frac{(300\ \mathrm{Mm})(0.95\overline{3})}{f_{\mathrm{MHz}}} = \frac{286\ \mathrm{Mm}}{f_{\mathrm{MHz}}} \end{equation}

• Is this called the 'velocity factor'? – Dan Oct 24 '13 at 20:17
• Yes, this is referred to as Velocity Factor and in Amateur Radio it can be used to discuss speed of propagation in any medium a radio wave might travel, including conductors, "air", etc. – Adam Davis Oct 24 '13 at 20:23
• – Dan Oct 24 '13 at 20:52